Dynamics of Systems on the Nanoscale (Lecture Notes in Nanoscale Science and Technology, 34) 303099290X, 9783030992903

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Lecture Notes in Nanoscale Science and Technology 34

Ilia A. Solov’yov Alexey V. Verkhovtsev Andrei V. Korol Andrey V. Solov’yov   Editors

Dynamics of Systems on the Nanoscale

Lecture Notes in Nanoscale Science and Technology Volume 34

Series Editors Zhiming M. Wang, Chengdu, China Greg Salamo, Fayetteville, AR, USA Stefano Bellucci, Frascati RM, Italy

Lecture Notes in Nanoscale Science and Technology (LNNST) aims to report latest developments in nanoscale science and technology research and teaching – quickly, informally and at a high level. Through publication, LNNST commits to serve the open communication of scientific and technological advances in the creation and use of objects at the nanometer scale, crossing the boundaries of physics, materials science, biology, chemistry, and engineering. Certainly, while historically the mysteries in each of the sciences have been very different, they have all required a relentless step-by-step pursuit to uncover the answer to a challenging scientific question, but recently many of the answers have brought questions that lie at the boundaries between the life sciences and the physical sciences and between what is fundamental and what is application. This is no accident since recent research in the physical and life sciences have each independently cut a path to the edge of their disciplines. As both paths intersect one may ask if transport of material in a cell is biology or is it physics? This intersection of curiosity makes us realize that nanoscience and technology crosses many if not all disciplines. It is this market that the proposed series of lecture notes targets.

More information about this series at https://link.springer.com/bookseries/7544

Ilia A. Solov’yov · Alexey V. Verkhovtsev · Andrei V. Korol · Andrey V. Solov’yov Editors

Dynamics of Systems on the Nanoscale

Editors Ilia A. Solov’yov Carl von Ossietzky University Oldenburg Oldenburg, Germany

Alexey V. Verkhovtsev MBN Research Center Frankfurt am Main, Germany

Andrei V. Korol MBN Research Center Frankfurt am Main, Germany

Andrey V. Solov’yov MBN Research Center Frankfurt am Main, Germany

ISSN 2195-2159 ISSN 2195-2167 (electronic) Lecture Notes in Nanoscale Science and Technology ISBN 978-3-030-99290-3 ISBN 978-3-030-99291-0 (eBook) https://doi.org/10.1007/978-3-030-99291-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Foreword

Atomic Physics, in recent years, has undergone a very fundamental transformation. Since its very origins, it was a ‘pure’ subject, in the sense that the atom was studied in isolation, far away from intruding matter. Researchers generally attempted to observe the narrowest possible spectral lines, the longest possible Rydberg series and sought to measure ‘natural’ linewidths as free as possible from the extraneous influence of collisions. The main exception to this approach was afforded by plasma spectroscopy, where atomic and ionic properties could provide valuable information necessary in the quest for fusion or the study of stellar atmospheres, and by the investigation of atoms in externally applied electric and magnetic fields, which provided the opportunity to refine atomic physics still further by resolving degeneracies and exploring quantum physics in its transition to the semi-classical limit. Throughout all this period, molecular behaviour was treated in a real sense as a separate subject, complementary but different, somehow associated with chemistry rather than physics, although, as Feynman argued, nobody has ever found any real difference between these two subjects. The emphasis, in this respect, has changed radically. Today, the main interest in atomic physics is the study of the atom, no longer as ‘separate’ but rather as an entity integrated into its intimate environment. The question today is really how much light does atomic physics shed on larger systems when these exhibit specific kinds of symmetry which ‘preserve’ atomic properties; how can one track the transition from atomic to solid-state behaviour as the atoms are piled up one by one to form clusters and bridge the gap between a purely atomic system and condensed matter; how do atoms behave when trapped inside hollow spherical molecules, when subjected to high pressures, etc. and how the powerful methods of atomic theory can be used to describe such ‘hybrid’ situations or adapted to extend their applicability into a more extensive ‘intermediate’ range. These new developments are driven by two significant advances. On the one hand, experimental physics has made enormous strides by using novel techniques such as laser and synchrotron radiation spectroscopy, trapping and cooling, time-of-flight methods, deposition on surfaces, etc. New systems such as size-selected clusters of various types, fullerenes and metallofullerenes, nanostructures, nanotubes, graphene v

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layers, etc. have been discovered. On the other hand, the theoretical methods born in atomic physics, initially more limited in scope, have suddenly been transformed into extremely powerful and general approaches, with applications far beyond what had originally been conceived for them, thanks to the extraordinary progress in numerical methods of computer-assisted science. In the second of these revolutions, the Russian school of physicists (particularly the one centred in St. Petersburg) has played a key role. Its innovative contributions trace a long and distinguished history right back to the Founding Father Lev Landau, whose influence on theoretical and mathematical physics not only in Russia but all over the world has remained as strong as ever throughout the changes. Indeed, the contributions of the Russian school have been of crucial importance. The present volume allows us to follow the many and varied facets of the transformation of atomic theory to adapt it to modern times. They are exemplified by a number of different themes pursued by the researchers surrounding Prof. Andrey Solov’yov, one of the leading teams establishing and developing these contemporary theoretical and mathematical physics methods. The book overviews many research areas relevant to selected critical technological applications involving Dynamics of Systems on the Nanoscale (DySoN). In particular, it focuses on recent advances made possible through multiscale theoretical and computational methods. To a large degree, our current understanding of Dynamics of Systems on the Nanoscale was made possible because of the seminal research of Prof. Andrey Solov’yov, his research team and close collaborators. Solov’yov was one of the first to note that although mesoscopic, nano- and biomolecular systems differ in their nature and origin, many fundamental problems are common to all of them. In the quest to find an answer to many fundamental questions addressing foundations of structure and dynamics on the nanoscale, Solov’yov introduced a new interdisciplinary field that lies at the intersection of physics, chemistry and biology, a field now entitled Meso-Bio-Nano (MBN) Science. The cuttingedge research in this area is carried out in the MBN Research Center gGmbH—a specialised research centre founded by Solov’yov in Frankfurt am Main (Germany). The mission of the centre is to deliver a significant breakthrough in the quantitative understanding of challenging interdisciplinary problems in the field of MesoBio-Nano Science through the implementation of modern theoretical and computational multiscale modelling methodologies and high-performance computing and exploiting this knowledge in numerous different applications, novel and emerging technologies. This book serves as an ideal introduction to dive into the research of Dynamics of Systems on the Nanoscale as it gives a comprehensive overview of examples of structure formation and dynamics of animate and inanimate matter on the nano- and the mesoscales. Most of the examples discussed in the book are reviews on standalone research focuses that belong to the broader area of Dynamics of Systems on the Nanoscale, tied to Solov’yov’s seminal research in interdisciplinary Meso-Bio-Nano Science and honouring his 60th anniversary that took place in 2020. Several topics covered in the book are directly linked to the research areas that have been actively explored at the MBN Research Center gGmbH. In many cases,

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the fundamental ideas, which served as a starting point for research, have been generated by Solov’yov. He was the key driving force in the further theoretical and computational elaboration of the topics, promoting and delivering the ideas aiming to broaden the community involved. Due to his efforts, the studies carried out at MBN Research Center have received support from a number of national and international funding agencies. As an example, we can mention the research towards possibilities and perspectives for designing and practical realisation of novel intensive gamma-ray Crystal-based Light Sources (CLS) operating at photon energies from 102 keV and above that can be constructed through the exposure of oriented crystals—linear, bent and periodically bent, to beams of ultra-relativistic positrons and electrons. CLSs can generate radiation in the photon energy range where the technologies based on the motion of the particles in the fields of permanent magnets become inefficient or incapable. To accomplish this task, there has been created a broad collaboration of research groups with different but complementary expertise, such as material science, nanotechnology, particle beam and accelerator physics, radiation physics, X-ray diffraction imaging, acoustics, solid-state physics, structure determination, advanced computational modelling, high-performance computing as well as industries specialising in the manufacturing of crystalline structures and design and construction of complete accelerator systems. In the past decade and a half, the activity of many research groups (theoretical and experimental) in this field has been boosted by the efforts and research carried out by Solov’yov and his colleagues. The studies of various phenomena relevant to the CLS field of research have been supported by several nationally funded projects (Germany, Denmark, Italy, South Africa). A unique experience has been gained within several European collaborative projects focused on the interdisciplinary and highly international R&D activities towards advancing the technologies for the manufacturing of periodically bent crystals, its characterisation, experimental investigation of the relevant channelling phenomena as well as further development of the theoretical methods and numerical algorithms. The book features many more examples of scientific avenues where Solov’yov made the pioneering contribution. The most prominent ones include, but are not limited to, the multiscale approach for the physics of ion-beam cancer therapy, studies of atomic cluster collisions, investigation of structure and dynamics of bioand macromolecules, physics of interface processes and development of novel theoretical approaches such as the irradiation-driven molecular dynamics and stochastic dynamics. In summary, although they treat many different aspects, the chapters in this book form a homogeneous whole that bears a common theoretical signature. Indeed, this is the paramount quality of the work. Implicitly, it demonstrates both the power and the generality of the clutch of novel techniques deployed here.

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Emeritus Professor of Physics at Imperial College London, Honorary Professor at the East China University in Shanghai, Permanent Guest Researcher at the WIPM, Laboratory of the Chinese Academy of Sciences, President of the European Academy of Sciences Arts and Letters London, United Kingdom

Jean-Patrick Connerade

Preface

Nowadays, understanding of Dynamics of Systems on the Nanoscale (DySoN) forms the core of the multidisciplinary research area, represented by many challenging interdisciplinary problems at the interface of physics, chemistry, biology and material science. Recent advances in this research area have often been achieved through multiscale computational modelling and high-performance computing. This book provides a comprehensive introduction to many exemplar research and emerging technological fields within the DySoN research area and presents important recent achievements. Finally, it concludes and gives an outlook for future developments. The book is based on the topics that have been developed intensively over the last two decades. They all concern the highly interdisciplinary scientific problems in which structure formation and dynamics of animate and inanimate matter on the nanometre scale play a central role. Many examples of complex many-body systems of micro- and nanometre-scale size exhibit unique features, properties and functions. These systems may have very different nature and origin, e.g. atomic and molecular clusters, nanostructures, ensembles of nanoparticles, nanomaterials, biomolecules, biomolecular and mesoscopic systems. A detailed understanding of the structure and dynamics of these systems on the nanometre scale is a fundamental and challenging task, the solution of which is required by nano- and biotechnologies, development of new materials with unique properties, plasma and pharmaceutical industries, and medicine. Although mesoscopic, nano- and biomolecular systems differ in their nature and origin, many fundamental problems are common to all of them: What are the underlying principles of self-organisation and self-assembly of matter on the micro- and nanoscale? Are these principles classical or quantum? How does function emerge at the nano- and the mesoscale in systems with different origins? What criteria govern the stability of these systems? How do their properties change as a function of size and composition? How does their environment alter their properties? Seeking answers to these questions is at the core of a new interdisciplinary field that lies at the intersection of physics, chemistry and biology, a field now entitled Meso-Bio-Nano (MBN) Science, which is presented in the book. Special attention in the book is devoted to investigations of the structure, properties and dynamics of complex MBN systems ix

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employing photonic, electronic, heavy particle and atomic collisions. This includes problems of fusion and fission, fragmentation, surfaces and interfaces, reactivity, nanoscale phase and morphological transitions, irradiation-driven transformations of complex molecular systems, collective electron excitations, radiation damage and biodamage, channelling phenomena and many more. The material is arranged as follows. In the introductory chapter, the main ideas, methodologies, problems and exemplar case studies discussed in the book are introduced. It explains the structure of the book and describes in brief the content of its chapters. It emphasises a number of research fields relevant to selected important technological applications involving Dynamics of Systems on the Nanoscale (DySoN) and recent advances achieved there by means of the multiscale theoretical and computational methods. Chapter 2 emphasises the links and the origin of the methodologies utilised in the DySoN research with those well known and widely used in theoretical atomic and molecular physics. Chapter 3 introduces the interdisciplinary MBN Science and the powerful software packages MBN Explorer and MBN Studio that are widely used for multiscale computational modelling of complex molecular structure and dynamics of very different complex MBN systems. Chapter 4 is devoted to the computational techniques for studying biomacromolecules. Several case studies are presented. Particular attention is devoted to the discussion of the interconnection of molecular dynamics approach and statistical mechanics, as well as additional possibilities for multiscale modelling of protein folding based on utilisation of the two methodologies. Illustrative examples of computational research are given for DNA, proteins and polypeptides. Chapter 5 describes quantum effects in biological systems such as photosynthesis, magnetoreception, etc., representing the emerging research area of quantum biology. These effects typically involve both dynamics of electronic and ionic subsystems in which quantum phenomena play a crucial role. Often theoretical and computational analysis of such effects requires sufficiently high precision in treating electronic structure and transport properties. Due to the high complexity of the systems, their computer simulations are typically performed employing high-performance computing. Hybrid quantum/classical mechanics methods for such computer simulations are introduced. Many case studies of biophysical systems and quantum processes therein are presented. Chapter 6 is devoted to the discussion of phase transitions occurring in nanosystems, e.g. melting, solidification, martensitic transitions, multifragmentation, etc. Phase transitions and their conditions are among the most characteristic properties of materials. Computational studies of these important phenomena are often performed by means of molecular dynamics. Such simulations are used to verify and validate the force fields used in simulations of thermomechanical properties of various materials relevant to their experimental research and technological applications. Chapter 7 presents multiscale modelling of a broad range of processes occurring in the course of deposition of various materials on surfaces, as well as during their

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mechanical modifications. Such processes typically involve relatively large molecular systems and require a lot of computer power. This imposes certain limits on the size of the simulated systems utilising molecular dynamics. However, the interconnection of molecular dynamics and Monte Carlo methods enables to overcome these deficiencies and perform simulations through a stochastic dynamics approach beyond the limits of the classical molecular dynamics. Illustrative computational case studies and the validation of simulation results through comparison with experiment are given. Chapters 8–10 are devoted to the description of multiscale theoretical and computational methods that have been developed and successfully applied to atomistic modelling of the critical physical, chemical and biological processes in the research fields represented by important novel and emerging technologies, such as (i) controlled nanofabrication, (ii) ion-beam cancer therapy and (iii) novel intensive gamma-ray light sources, which have been chosen in this book as the exemplar technological case studies. Frankfurt am Main, Germany November 2021

Ilia A. Solov’yov Alexey V. Verkhovtsev Andrei V. Korol Andrey V. Solov’yov

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Andrey V. Solov’yov

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Atomic and Molecular Physics Methods for Nanosystems . . . . . . . . . Alexey V. Verkhovtsev and Andrey V. Solov’yov

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Multiscale Computational Modelling of MesoBioNano Systems . . . . Gennady B. Sushko, Ilia A. Solov’yov, and Andrey V. Solov’yov

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Structure and Dynamics of Bio- and Macromolecules . . . . . . . . . . . . 137 Alexey V. Verkhovtsev, Ilia A. Solov’yov, and Andrey V. Solov’yov

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Quantum Effects in Biological Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Anders Frederiksen, Thomas Teusch, and Ilia A. Solov’yov

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Dynamics and Phase Transitions in Nanosystems . . . . . . . . . . . . . . . . . 249 Alexey V. Verkhovtsev and Andrey V. Solov’yov

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Multiscale Modeling of Surface Deposition Processes . . . . . . . . . . . . . 307 Ilia A. Solov’yov and Andrey V. Solov’yov

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Multiscale Modeling of Irradiation-Driven Chemistry Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 Gennady Sushko, Alexey V. Verkhovtsev, Ilia A. Solov’yov, and Andrey V. Solov’yov

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Multiscale Approach for the Physics of Ion Beam Cancer Therapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 Eugene Surdutovich, Alexey V. Verkhovtsev, and Andrey V. Solov’yov

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10 Novel Light Sources Beyond FELs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 Andrei V. Korol and Andrey V. Solov’yov 11 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 Andrey V. Solov’yov

Chapter 1

Introduction Andrey V. Solov’yov

Abstract This chapter is introductory for the book. It outlines the main ideas, introduces the main methodologies, problems and exemplar case studies discussed in the book. It explains the structure of the book and describes in brief the content of its chapters. It emphasizes a number of research fields relevant to selected important technological applications involving Dynamics of Systems on the Nanoscale (DySoN) and recent advances achieved there by means of the multiscale theoretical and computational methods.

1.1 Dynamics of Systems on the Nanoscale This book provides an overview of recent advances in theoretical and computational physics aiming at the quantitative understanding of Dynamics of Systems on the Nanoscale (DySoN). This multidisciplinary research area is represented by a large number of challenging interdisciplinary problems at the interface of physics, chemistry, biology, and materials science. The advances in this field of modern research often have been achieved by means of the multiscale computational modeling methods and high performance computing. This book provides a comprehensive introduction to a number of exemplar research and emerging technological fields within the DySoN research area and presents important recent achievements therein. Finally, it draws conclusions and gives outlook for the future developments. The book is based on the topics that have been developed intensively over the last two decades. They all concern the highly interdisciplinary scientific problems in which structure formation and dynamics of animate and inanimate matter on the nanometer scale plays a central role. There are many examples of complex manybody systems of micro- and nanometer scale size exhibiting unique features, properties, and functions. These systems may have very different nature and origins, e.g., atomic and molecular clusters, nanostructures, ensembles of nanoparticles, nanomaterials, biomolecules, biomolecular, and mesoscopic systems. A detailed understandA. V. Solov’yov (B) MBN Research Center gGmbH, Altenhöferallee 3, 60438 Frankfurt am Main, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 I. A. Solov’yov et al. (eds.), Dynamics of Systems on the Nanoscale, Lecture Notes in Nanoscale Science and Technology 34, https://doi.org/10.1007/978-3-030-99291-0_1

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ing of structure and dynamics of these systems on the nanometer scale is a difficult and fundamental task, the solution of which is required by nano- and bio-technologies, development of new materials with unique properties, plasma and pharmaceutical industries, and medicine. Although mesoscopic, nano- and biomolecular systems differ in their nature and origin, a number of fundamental problems are common to all of them: What are the underlying principles of self-organization and self-assembly of matter on the microand nanoscale? Are these principles classical or quantum? How does function emerge at the nano- and the meso-scale in systems with different origins? What criteria govern the stability of these systems? How do their properties change as a function of size and composition? How are their properties altered by their environment? Seeking answers to these questions is at the core of a new interdisciplinary field that lies at the intersection of physics, chemistry, and biology, a field now entitled Meso-Bio-Nano (MBN) Science [1] which is presented in the book. Special attention in the book is devoted to investigations of the structure, properties, and dynamics of complex MBN systems by means of photonic, electronic, heavy particle, and atomic collisions. This includes problems of fusion and fission, fragmentation, surfaces and interfaces, reactivity, nanoscale phase and morphological transitions, irradiation-driven transformations of complex molecular systems, collective electron excitations, radiation damage and biodamage, channeling phenomena, and many more. Emphasis in the book is placed on overview of the theoretical and computational physics advances in the DySoN research area and the related state-of-the-art experimental and technological progress. Particular attention in the book is devoted to the utilization of advanced computational techniques and high performance computing in studies of dynamics of systems on the nanoscale. Over the years advances in the DySoN research area have been reported at many related international conferences. Among them the two conference series: International Conference “Dynamics of Systems on the Nanoscale” (DySoN)1 and International Symposium “Atomic Cluster Collisions” (ISACC)2 traditionally represent the DySoN research area in the most consistent and comprehensive way. The book is organized as a collection of reviews on the selected topics outlined below. Each book chapter represents a research field on its own. However, methodologically and scientifically there are many interconnections between all the chapters in this book. Chapter 2 emphasizes the links and the origin of the methodologies utilized in the DySoN research with those well known and widely used in theoretical atomic and molecular physics. In particular, Chap. 2 discusses the application of well-established atomic and molecular physics methods for describing the structure and properties of atomic clusters as well as pristine and endohedral carbon fullerenes. Emphasis is made on the description of electronic properties of such nanosystems and physical

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http://www.dyson-conference.org/. http://www.isacc-portal.org/.

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phenomena that manifest themselves in photon, electron, and ion collisions with the nanosystems and involve dynamics of electrons. Chapter 3 introduces the interdisciplinary Meso-Bio-Nano (MBN) Science and the powerful software packages MBN Explorer and MBN Studio that are widely used for multiscale computational modeling of complex molecular structure and dynamics of very different complex MBN systems. The chapter presents the unique features of MBN Explorer enabling efficient simulations of structure and dynamics of a large variety of molecular systems with the sizes ranging from the atomic up to the mesoscopic scales. It introduces the application areas of MBN Explorer and describes MBN Studio—a special multi-task toolkit for MBN Explorer, which enables construction of computational projects, simple start of simulations with MBN Explorer, as well as visualization and analysis of the results obtained. Chapter 4 is devoted to the computational techniques used for the investigation of biomacromolecules. Several case studies are presented. Particular attention is devoted to the discussion of interconnection of molecular dynamics (MD) approach and statistical mechanics, as well as additional possibilities for multiscale modeling of protein folding based on utilization of the two methodologies. Such a multiscale modeling approach permits simulations of a large variety of biomacromolecular systems (both in vacuum and in ubiquitous environments) and their transformations at different thermal and biologically relevant conditions and various external stresses. Illustrative examples of computational research are given for DNA, proteins, and polypeptides. Chapter 5 describes quantum effects in biological systems such as photosynthesis and magnetoreception, representing the emerging research area of quantum biology. These effects typically involve both dynamics of electronic and ionic subsystems in which quantum phenomena play a crucial role. Often theoretical and computational analysis of such effects requires sufficiently high precision in the treatment of electronic structure and transport properties. Due to the high complexity of the systems their computer simulations are typically performed by means of high performance computing. Hybrid Quantum/Classical mechanics methods for such computer simulations are introduced. The chapter discusses case studies of quantum processes present in real biological systems, such as photoabsorption, electron transfer, protoncoupled electron transfer, and spin chemistry. Chapter 6 is devoted to the discussion of phase and structural transitions that occur in nanosystems, such as melting, solidification, martensitic transitions, and spontelectric phenomena. Phase transitions and their conditions are among the most characteristic properties of materials. Computational studies of these important phenomena are often performed by means of MD. Such simulations are used for the verification and validation of the force fields used in simulations of thermomechanical properties of various materials relevant to their experimental research and technological applications. Chapter 7 presents multiscale modeling of a broad range of processes occurring in a course of deposition of various materials on surfaces, as well as during their mechanical modifications. Such processes typically involve relatively large molecular systems and require a lot of computer power. This imposes certain limits on

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the size of the simulated systems by means of MD. However, interconnection of MD and Monte Carlo methods enables to overcome these deficiencies and perform simulations by means of stochastic dynamics approach beyond the limits of the classical MD. The chapter presents the basic theoretical concepts underlying stochastic dynamics implementation in MBN Explorer and provides several illustrative computational case studies accompanied by characteristic experimental results to validate the computational approaches. Chapters 8–10 are devoted to the description of multiscale theoretical and computational methods that have been developed and successfully applied to atomistic modeling of the key physical, chemical, and biological processes in the research fields represented by important novel and emerging technologies, such as (i) controlled nanofabrication, (ii) ion-beam cancer therapy, and (iii) novel intensive gamma-ray light sources, which have been chosen in this book as the exemplar technological case studies. These case studies and related research fields are briefly introduced in the following sections of this introductory chapter.

1.2 Multiscale Modeling of Complex Molecular Structure Formation and Evolution The aggregation of atoms and small molecules into clusters, nanoparticles and macromolecules, clustering (or coalescence) of nanoparticles and biomolecules into nanostructures, nanostructured materials, biomolecular complexes, and hybrid systems possessing different morphologies are the processes by which a wide range of MBN systems can be created [2]. Some of these systems have been synthesized only recently and have become a subject of intensive investigations due to their unique structural, optical, magnetic, thermomechanical, or thermo-electrical properties and can be utilized in a variety of important applications [2]. Clustering, self-organization, and structure formation are general phenomena manifesting themselves on very different levels and scales of matter organization or self-organization. They appear in many different areas of research: astrophysics, physics, chemistry, biology, materials science, nanoscience, neuroscience, and even in technology (clustering in the wireless, computer, or windmill networks, etc). Apart from the fundamental value of this knowledge it is also highly relevant to the key problems of modern technology. One of such problems is controllable fabrication of nanostructures with nanoscale resolution that remains a considerable scientific and technological challenge [3]. The fabrication of smaller and smaller structures has been the goal of the electronics industry for more than three decades, since the smaller the structures the stronger the operational power of the manufactured nanodevices. To date Moore’s law has held within the semiconductor industry allowing smaller and smaller devices to be fabricated increasing operational power within a fixed size device. However, as the size of the structures falls below 30 nm traditional manufacturing methods (e.g., plasma etching, plasma-enhanced chemical vapor deposition) are struggling to

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meet Moore’s law. Hence there is the urgent need to develop new nanofabrication methods, among which the Focused Electron Beam Induced Deposition (FEBID) is one of the most promising allowing controlled creation of nanostructures with nanometer resolution [4]. Such methods [5, 6] exploit irradiation of nanosystems with collimated electron beams. These can be used to create specific structural motifs of metal nanoparticles for catalytic and nanoelectrochemistry applications [7, 8]; to fabricate metal nanostructures for sensors, nanoantennas and magnetic devices, surface coatings, and thin films with tailored properties to be used in electronic devices and other applications. The bottleneck in this technological development concerns the fact that often the fundamental physicochemical phenomena that govern multistep (e.g., irradiation and replenishment stages) process of fabrication, self-organization, composition, and growth of nanosystems coupled to radiation are not understood well enough [4]. In spite of the significant experimental progress that has been achieved and the large amount of data that have been accumulated, there is still a lack of the underlying irradiation-driven chemistry (IDC) that determines the unique electronic, magnetic, mechanical, catalytic, etc. properties of the irradiated nanosystems. Computational multiscale modeling provides a new methodology for understanding IDC and, consequently, advancing controllable fabrication of nanostructures. A rigorous quantummechanical description of the IDC processes is possible only for relatively small molecular systems containing, at most, a few hundred atoms. Classical MD could be considered as an alternative modeling framework for much larger systems. However, standard classical MD is unable to simulate such IDC processes as it typically does not account for the coupling of the system to the incident radiation nor does it describe the induced quantum transformations. These deficiencies have recently been overcome in a new methodology, Irradiation Driven Molecular Dynamics (IDMD) [9] allowing, for the first time, high accuracy simulation of IDC in complex molecular systems, for further details see Chaps. 3 and 8 of this book. Within this framework, the IDC transformations are treated as random, fast, and local processes which can be incorporated locally into classical MD force fields in a random manner according to the probabilities of the quantum processes that may occur in the system. The IDMD approach has been implemented in the MBN Explorer software package [1, 10] capable of operating with a large library of interatomic force fields and their combinations. The developed computational framework provides a broad range of possibilities for multiscale modeling of the IDC processes, see Chap. 3, that underpin emerging technologies ranging from controllable fabrication of nanostructures, see Chap. 8, to radiotherapy, see Chap. 9. Chapter 8 is devoted to the multiscale computational modeling of the FEBID processes, i.e., the self-organizing molecular medium experiencing chemical transformations in the presence of irradiation (by electrons, ions or photons) that is being validated in experiments on the formation, growth and modification of various nanostructures. This analysis provides atomistic-level insights into structural and dynamical properties of metal, metal-carrying, and organic nanosystems exposed to radiation and their underlying interatomic interactions and chemistry.

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The validated IDMD approach [9] implemented in MBN Explorer together with modern computational facilities constitute a unique powerful tool for the breakthrough computational research in this field as well as in radiotherapy, radiochemistry, astrochemistry, functionalization of nanosystems, catalytic properties, and many more. The computational multiscale modeling performed by means of MBN Explorer is based on (i) the ab initio methods to evaluate parameters of irradiationinduced quantum transformations of system constituent molecules, (ii) classical MD to characterize clusters, nanoparticles, nanostructures, precursor molecules and study their interaction with the substrate, (iii) the reactive CHARMM force field allowing to simulate molecular fragmentation and changes in molecular topology, (iv) the IDMD methodology describing random interactions of deposited precursor molecules with a primary electron beam and with secondary electrons emitted from the substrate as well as possible chemical transformations of precursors caused by interactions with primary and secondary electrons, and (v) stochastic dynamics (including kinetic Monte Carlo algorithms) enabling to build up dynamical models of the system behavior and evolution on the temporal and spatial scales far beyond those arising from the computational limits of the classical MD. The stochastic dynamics [11–13] is designed to model the time-evolution of a many-particle systems stepwise in time. Instead of solving dynamical equations of motion stochastic dynamics approach assumes that with a certain probability, at each step of the evolution, the system undergoes a structural transformation. The new configuration of the system is then used as the starting point for the next evolution step. The transformation of the system is governed typically by several kinetic rates. Their choice is defined by the model considered. This requires some additional justification and analysis which can be usually performed by means of MD. Being applied to the description of complex molecular systems, this methodology due to its probabilistic nature allows to study dynamical processes on the time scales significantly exceeding the characteristic time scales of conventional MD simulations. The stochastic dynamics method is ideal in the situations when certain minor details of dynamical processes become inessential, and the major transition of the system to new states can be described by only a few kinetic rates being determined through the corresponding physical parameters. The stochastic dynamics methodology is very general and by itself can be explored in enormous number of different dynamical systems [1]. It allows to develop models of dynamical systems beyond the limits of MD. Thus, the morphological transitions in nanofractals, nanowires, and many other MBN systems involve dynamics of enormous number of atoms and occur on the time scales which are well beyond the nowadays limits for classical MD simulations [14]. However, the experimentally observed morphological transitions [14] can be successfully studied through the coarse-grained multiscale approach, in which the dynamics of the whole system is reduced to the random walk dynamics of constituent nanoclusters. In this case, the key dynamical processes in the system, occurring on a long-time scale, can be parameterized through only a few kinetic rates. Such systems, processes therein and their modeling are discussed in detail in Chaps. 3, 6 and 7.

1 Introduction

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1.3 Multiscale Modeling of MBN Systems at the Life Science Interface Nowadays radiotherapy is used in around 50% of cancer treatments and relies on the deposition of energy into tumor tissue. Although it is generally effective, some of the deposited energy can adversely affect healthy tissue outside the tumor volume, especially in the case of photon radiation (gamma and X-rays). Improved radiotherapy outcomes have been achieved with the ion-beam cancer therapy discussed in detail in Chap. 9 due to the characteristic energy deposition curve which culminates in a localized, high radiation dose (in form of a Bragg peak) [15]. The development of radiotherapy as a tool for cancer treatment has led to the accumulation of large amounts of data on radiation damage induced in biological systems as a function of supplied dose. Such data are currently included in several empirical Monte Carlo models [15], which are used to establish the protocols underpinning current clinical therapy practice, e.g., Phillips Pinnacle,3 Varian Eclipse,4 Brainlab,5 Raystation.6 These Monte Carlo-based approaches aim to provide a rapid evaluation of the deposited dose distribution in a tumor allowing treatment plans to be prepared, according to clinical protocols, that aim to maximize the dose delivery for tumor remediation while minimizing the total dose to the patient to avoid harmful side effects. At present the dose distributions (and thence biological damage) are based upon a macro- and microscale dosimetry approach, which however is sub-optimal. In the next generation of radiotherapy methods, for example in ion-beam therapy, the dose is delivered in a narrow spatial region known as the Bragg peak, therefore it is necessary to prepare new protocols that are based on nanodosimetry which is itself based on a better understanding of discrete collisions rather than macroscopic energy deposition. Recent research has shown that there are substantial qualitative and quantitative differences between the effects of ions, electrons, and photons on tissue that must be understood at the molecular level if we are to optimize both current and new radiotherapies [15]. Indeed it has become obvious that a fundamental understanding of radiation damage (RADAM) processes cannot be gained from studying any particular mechanism or process taking place at a particular time and space scale but instead, we need to understand the complex cascade of processes triggered by the propagation of particles through biological targets to determine the final radiobiological effect. These developments have boosted in recent years the scientific interest in obtaining deeper understanding of radiation damage with ions [15]. It was realized that a number of fundamental scientific questions related to the assessment of biological damage on the molecular level have not yet been resolved. Therefore, this field 3

www.healthcare.philips.com/main/products/ros/products/pinnacle3. www.varian.com/en-gb/oncology/products/software/treatment-planning/eclipse. 5 www.brainlab.com/en/radiosurgery-products/iplan-rt-treatment-planning-software/. 6 https://www.raysearchlabs.com/. 4

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has attracted much attention from the scientific community all over the world, and especially in Europe. Among these studies the multiscale approach to the physics of radiative damage was an important breakthrough [15]. It is systematically presented and thoroughly discussed in Chap. 9. The unique feature of the multiscale approach to the physics of ion-beam cancer therapy is in the integration of key processes occurring on different temporal, spatial and energy scales within a single inclusive scenario incorporating: (i) propagation of charged particles in biological media, (ii) irradiation-induced fast quantum processes within biomolecular environments, (iii) characterization of time and spatial evolution of track structures and localized energy deposition, (iv) slower nanoscale post-irradiation relaxation, chemical and thermalization processes occurring in the irradiated biological media, (v) evaluation of biodamage accounting for specific radiobiological effects allowing prediction of biological outcomes of irradiation, (vi) multiscale treatment models combining nanoscale descriptions of radiation driven molecular modifications with macroscale dose delivery protocols, and (vii) development of the new algorithms defining novel radiotherapy protocols for clinical targeting of tumors. A quantitative description of radiation damage requires inclusion of both short time (sub)femtosecond/ (sub)nanoscopic scale quantum aspects and longtime/ mesoscopic scale environment effects. Dynamical descriptions of molecular systems on the atomic/subnano- and nano-/mesoscopic scales are currently performed with disconnected theoretical and simulation tools. Examples of current tools that can be used within the multiscale approach to provide a transformative models of radiation damage include: density functional theory (DFT) and time-dependent DFT (TDDFT); the R-matrix method; many-body theory; quantum MD codes such as Wien2k, deMon, Gaussian, ORCA, OCTOPUS; radiation transport MC codes GEANT4 (originally developed for nuclear physics), EGS4, FLUKA, MCNP, Penelope; track structure event-by-event MC codes—KURBUC, PARTRAC, OREC, SHERBROOKE, LEPTS; and MBN Explorer, AMBER, GROMACS, NAMD for multiscale simulations of complex molecular structure and dynamics. These well-established computer codes cannot be reviewed in any detail in a short introductory section, but let us emphasize that in the near future there is no serious hope to explore both atomic/subnano- and nano-/mesoscopic scale ranges by simply extending one computational approach to the other domain. However, the aforementioned multiscale approach for biodamage offers the methodology to establish the necessary interlinks between the key phenomena within the multiscale scenario from the physical processes induced by irradiation up to the biological level. Its development and the current status are overviewed in Chap. 9 of this book.

1 Introduction

9

1.4 Multiscale Modeling of Novel Materials and Material Interfaces for Applications in Novel and Emerging Technologies Nanostructured materials are materials with characteristic size of structural elements on the order or less than several hundreds of nanometers at least in one dimension. Examples of nanomaterials include nanocrystalline materials, nanofibers, nanotubes, and nanoparticle reinforced nanocomposites. Atomistic modeling is based on atoms as elementary units in the models, thus providing the atomic-level resolution in the computational studies of materials structure and properties. The main atomistic methods in material research are (i) molecular dynamics technique that yields “atomic movies” of the dynamic material behavior through the integration of the equations of motion of atoms and molecules, (ii) metropolis Monte Carlo method that enables evaluation of the equilibrium properties through the ensemble averaging over a sequence of random atomic configurations generated according to the desired statistical-mechanics distribution, and (iii) kinetic Monte Carlo method that provides a computationally efficient way to study systems where the structural evolution is defined by a finite number of thermally-activated elementary processes. Examples of exploitation of these algorithms to the simulation of various properties of nanostructured materials is given in Chaps. 6 and 7 of this book. Rapid advances in the synthesis of nanostructured materials combined with reports of their enhanced or unique properties have created, over the last decades, a new active area of materials research. Due to the nanoscopic size of the structural elements in nanomaterials, the interfacial regions, which represent an insignificant volume fraction in traditional materials with coarse microstructures, start to play the dominant role in defining the physical and mechanical properties of nanostructured materials. This implies that the behavior of nanomaterials cannot be understood and predicted by simply applying scaling arguments from the structure—property relationships developed for conventional polycrystalline, multiphase, and composite materials. New models and constitutive relations are therefore needed for an adequate description of the behavior and properties of nanomaterials [1]. Computational modeling is playing a prominent role in the development of the theoretical understanding of the connections between the atomic-level structure and the effective (macroscopic) properties of nanomaterials. Atomistic modeling has been at the forefront of computational investigation of nanomaterials and has revealed a wealth of information on structure and properties of individual structural elements (various nanolayers, nanoparticles, nanofibers, nanowires, and nanotubes) as well as the characteristics of the interfacial regions and modification of the material properties at the nanoscale. Due to the limitations on the time- and length-scales, inherent to atomistic models, it is often difficult to perform simulations for systems that include a number of structural elements that are sufficiently large to provide a reliable description of the macroscopic properties of the nanostructured materials. An emerging key component of the computer modeling of nanomaterials is, therefore, the development of novel

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mesoscopic simulation techniques capable of describing the collective behavior of large groups of the elements of the nanostructures and providing the missing link between the atomistic and continuum (macroscopic) descriptions. MBN Explorer is one of the few software packages that integrates these diverse scales into one program and, therefore, permits multiscale simulations of nanostructured materials. The capabilities and limitations of the atomistic and mesoscopic computational models used in investigations of the behavior and properties of some selected nanomaterials by means of MBN Explorer are discussed below in Chaps. 6 and 7. A material made from two or more constituent materials with physical or chemical properties significantly different from the properties of the constituent materials is called a composite material, or simply a composite. Typically the individual components remain separate and can be identified within the composite structure. However, the new material as a whole acquires specific and often desired properties. It might be stronger, lighter or a lower cost, etc. The well-known composites include concrete, reinforced plastics, composite ceramic, metal composites, metal matrices, etc. The composites are used in many application areas, such as engineering, building, and construction. Different materials exhibit different behavior under the same thermal and mechanical loading. The behavior can be attributed to different properties, or the manifestation of different effects, which can be characterized by certain specific quantities. These characteristics can be grouped into the four major categories. The first category is related to the behavior of materials driven by thermal loading without relation to their mechanical properties. This category includes such characteristics as melting points, specific heat capacities, thermal conductivities, etc. The second category is attributed to the behavior of materials under mechanical loading without fracture. The quantities of this category include elastic moduli, Poisson’s ratios, yield points, viscosity, etc. The third category embraces the thermomechanical characteristics of materials, such as density, thermal expansion coefficients, and energetic characteristics of crystals. The fourth category is related to the fracture behavior of materials and operates with such characteristics as hardness and density of dislocations. The dependencies of all these characteristics of the material properties upon temperature are usually referred to as thermomechanical properties of materials. Examples of simulations and analysis of the thermomechanical properties for selected materials are given below in Chap. 6. Development of novel materials is often driven by their specific technological applications which impose certain criteria on the required properties of materials. Chapter 10 provides an example of such interconnection describing the stateof-the-art development of the emerging technology for intensive gamma-ray light sources [16]. This technology among other requirements relies on the manufacturing of high quality crystalline structures with desired properties. The development of coherent radiation sources for wavelengths λ well below 1 Angstrom (i.e., in the hard X-ray and gamma-ray regimes) is a challenging goal of modern physics [16]. Sub-angstrom wavelength powerful radiation sources will have many applications in the basic sciences, technology, and medicine. In particular,

1 Introduction

11

they may have a revolutionary impact on nuclear and solid-state physics, as well as in the life sciences. Present laser systems are capable of emitting radiation from the infrared to ultraviolet range of the spectrum and several Free-Electron-Laser (FEL) sources are operating (DESY-FLASH, FERMI, LCLS, SACLA) or are planned (XFEL, SwissFEL) for X-ray wavelengths down to λ ∼ 1 Å. However, no laser system has yet been commissioned below 1 Å due to the limitations of permanent magnet and accelerator technologies. Multiscale computational modeling provides a breakthrough in virtual design of novel gamma-ray Light Sources (LSs) operating in the energy range from 102 keV up to GeV that can be constructed through exposure of oriented crystals (Linear Crystals—LC, Bent Crystals—BC, Periodically BC—PBC, quasi PBC—qPBC) to the beams of ultra-relativistic particles (electrons, positrons, etc). This interdisciplinary approach combines theory, multiscale computational modeling, and design of the crystals with desired properties as well as the processes accompanying the crystal irradiation by the beams (propagation, radiation, thermal, damage). This means that the processes are simulated on the atomistic level from the atomistic scale up to the bulk limit. On this basis the subsequent characterization of the emitted radiation, the crystal sustainability and quality after the irradiation is performed. This information guides practical realization of the novel LSs with desired characteristics. This field of research and technology is described in Chap. 10. The oriented Crystal-based LSs (CLSs) [16] can be of the different types including crystalline synchrotron radiation emitters, channeling radiation emitters, crystalline bremsstrahlung radiation emitters, Crystalline Undulators (CU with both Small Amplitude Small Period-SASP and Large Amplitude Long Period-LALP), quasiperiodic Crystalline Undulators (qCU with both SASP and LALP), stacks of CUs and qCUs. These devices can be utilized for generation of intensive radiation with wavelengths orders of magnitudes less than 1 Angstrom, i.e., in the range where the traditional radiation technologies based on the photon emission by charged particles in strong fields of permanent magnets become inefficient, or incapable. The limitations of the permanent magnet-based LSs can be overcome in CLSs by exploiting strong crystalline fields. The field strength in a crystal is 1010 V/cm which is equivalent to a magnetic field of 3000 T while state-of-the-art superconducting magnets are capable to reach only tens of Tesla. The crystal orientation against the beam enhances the strength of the particles interaction with the crystal. This opens possibilities for the guided motion of particles through crystals of different geometry and the enhancement of the radiation. Charged particles propagating in the channeling regime through oriented crystals can emit intensive radiation known as the channeling radiation (ChR). In non-oriented crystals, the main source of radiation is bremsstrahlung (BrS) (incoherent and coherent) [17]. All these radiation mechanisms and their different combinations can be exploited for the generation of Gamma-radiation of different intensities and at different spectral ranges by ultra-relativistic particles. Important is that intensity of the emitted radiation and the brilliance of the corresponding CLSs can be made higher than those constructed on the basis of the permanent magnets [16]. For example,

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the CU-based CLS brilliance in the energy range from 102 keV up to tens of MeV is comparable to that of conventional third-generation LSs operating at much lower energies, see Chap. 10. Moreover, CU may provide a possibility of generating stimulated emission in the same way as in a FEL, thus providing an exciting possibility of developing a CU-based Laser in the photon energy range 102 –103 keV (hard Xand gamma-ray range). More possibilities with construction of the gamma-ray sources with photon energies up to GeV range arise with increase of the particle energy in the new generation of accelerators (up to TeV range for electrons and positrons). Increasing the characteristic energy of emitted radiation involves new physical phenomena, such as radiation damping, and crystal sustainability, which should be carefully accounted for in the virtual design of the novel CLSs. This imposes the ultimate requirements to the theory (atomistic level of description, multiscale modeling approaches), computational methodologies (efficient algorithms and methods), and computational techniques (high performance computing). All these exciting challenges and possibilities are discussed in Chap. 10.

References 1. Solov’yov, I.A., Korol, A.V., Solov’yov, A.V.: Multiscale Modeling of Complex Molecular Structure and Dynamics with MBN Explorer. Springer International Publishing, Cham (2017) 2. Connerade, J., Solov’yov, A.V., Greiner, W.: The science of clusters: an emerging field. Europhys. News 33, 200 (2002) 3. Plant, S., Cao, L., Palmer, R.: Atomic structure control of size-selected gold nanoclusters during formation. J. Am. Chem. Soc. 136, 7559–7562 (2014) 4. Huth, M., Porrati, F., Schwalb, C., Winhold, M., Sachser, R., Dukic, M., Adams, J., Fantner, G.: Focused electron beam induced deposition: a perspective. Beilstein J. Nanotechnol. 3, 597–619 (2012) 5. Utke, I., Moshkalev, S., Russel P. (eds.): Nanofabrication Using Focused Ion and Electron Beams. Oxford University Press (2012) 6. Cui, Z.: Nanofabrication. Principles, Capabilities and Limits. Springer International Publishing, Cham (2017) 7. Xu, W., Kong, J.S., Yeh, Y.T.E., Chen, P.: Single-molecule nanocatalysis reveals heterogeneous reaction pathways and catalytic dynamics. Nat. Mater. 7, 992–996 (2008) 8. Murray, R.W.: Nanoelectrochemistry: metal nanoparticles, nanoelectrodes, and nanopores. Chem. Rev. 108, 2688–2720 (2008) 9. Sushko, G.B., Solov’yov, I.A., Solov’yov, A.V.: Molecular dynamics for irradiation driven chemistry: application to the FEBID process. Eur. Phys. J. D 70, 217 (2016) 10. Solov’yov, I.A., Yakubovich, A.V., Nikolaev, P.V., Volkovets, I., Solov’yov, A.V.: MesoBioNano explorer - a universal program for multiscale computer simulations of complex molecular structure and dynamics. J. Comput. Chem. 33, 2412–2439 (2012) 11. Sushko, G.B., Friis, I., Solov’yov, I.A., Solov’yov, A.V.: Stochastic dynamics algorithm for MBN Explorer. http://www.mbnresearch.com/implemented-algorithms (2021) 12. Dick, V.V., Solov’yov, I.A., Solov’yov, A.V.: Fragmentation pathways of nanofractal structures on surfaces. Phys. Rev. B 84, 115408 (2011) 13. Panshenskov, M.A., Solov’yov, I.A., Solov’yov, A.V.: Efficient 3D kinetic Monte Carlo method for modeling of molecular structure and dynamics. J. Comput. Chem. 35, 1317–1329 (2014)

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14. Solov’yov, I.A., Solov’yov, A.V., Kébaili, N., Masson, A., Bréchignac, C.: Thermally induced morphological transition of silver fractals. Phys. Status Solidi B 251, 609–622 (2013) 15. Solov’yov, A.V. (ed.): Nanoscale Insights into Ion-Beam Cancer Therapy. Springer International Publishing, Cham (2017) 16. Korol, A.V., Solov’yov, A.V., Greiner, W.: Channeling and Radiation in Periodically Bent Crystals, 2nd edn. Springer, Berlin (2014) 17. Korol, A.V., Solov’yov, A.V.: Polarization Bremsstrahlung. Springer, Berlin (2014)

Chapter 2

Atomic and Molecular Physics Methods for Nanosystems Alexey V. Verkhovtsev and Andrey V. Solov’yov

Abstract This chapter discusses the application of well-established atomic and molecular physics methods for describing the structure and properties of nanosystems, particularly atomic clusters as well as pristine and endohedral carbon fullerenes. Emphasis is made on the description of electronic properties of such nanosystems and physical phenomena that manifest themselves in photon, electron, and ion collisions with the nanosystems and involve dynamics of electrons. It is demonstrated that the diffraction and interference phenomena play an important role during the interaction of nanosystems with photons and electrons. The essential role of the multipole surface and volume plasmon excitations in the formation of electron emission spectra of metal clusters and fullerenes is elucidated. The relaxation mechanisms of resonant electronic excitations in metal clusters and fullerenes are discussed in the context of evaluation of widths of the corresponding resonant processes. Finally, the fundamental physical phenomena associated with the confinement of atoms and small molecules inside carbon fullerenes representing a nanometer-sized cavity are discussed.

2.1 Introduction Many theoretical and computational methods discussed in this book originate from the atomic and molecular physics. Methods based on quantum mechanics, such as the Hartree–Fock method [1] and density-functional theory (DFT) [2, 3] have been utilized for many decades to calculate the electronic properties of many-electron atoms and molecules. Since the 1980s these methods have also been applied to bigger systems, such as atomic and molecular clusters. Due to the increased size and complexity of the system, the electronic properties of nanosystems were often A. V. Verkhovtsev (B) · A. V. Solov’yov MBN Research Center gGmbH, Altenhöferallee 3, 60438 Frankfurt am Main, Germany e-mail: [email protected] A. V. Solov’yov e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 I. A. Solov’yov et al. (eds.), Dynamics of Systems on the Nanoscale, Lecture Notes in Nanoscale Science and Technology 34, https://doi.org/10.1007/978-3-030-99291-0_2

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studied within simplified model approaches that are based on fundamental physics principles. In this chapter we discuss the transition from atomic systems to nanosystems, simultaneously presenting the methodology for studying the electronic properties of many-atom systems. Most ab initio electronic structure calculations based on the Hartree–Fock method or DFT assume the electronic configuration is in an instantaneous ground state for each nuclear configuration. Besides, ab initio methods can be utilized for investigating electron dynamics induced by the interaction of radiation with matter. Such studies are commonly based on time-dependent DFT [4]. Another important aspect discussed in this chapter is the separation of slow ionic and fast electronic degrees of freedom of a molecular system according to the Born– Oppenheimer approximation [5]. This fundamental result gave birth to the molecular dynamics (MD) technique which is widely utilized nowadays in many different research areas from atomic cluster physics to materials science and biophysics. Numerous examples of the application of classical MD to these research areas are presented in other chapters of this book. A significant part of this chapter is devoted to discussing one of the most prominent phenomena related to the electronic subsystem—collective electronic excitations, also known as plasmons. In this chapter we focus on the collective phenomena related to electron dynamics in nanosystems, particularly metal clusters and fullerenes, and do not consider quantum phenomena in biological systems. Phenomena related to electron dynamics in biological systems are discussed in Chap. 5 of this book.

2.2 Born–Oppenheimer Approximation Consider an arbitrary molecular system containing N atoms. In the non-relativistic case, the Hamiltonian Hˆ of the system is given by Hˆ =

N   α=1

+

−

   Ne  2 2 2 2 − ∇α + ∇i 2Mα 2m e i=1

Ne N     Z α Z β e2 e2 Z α e2 + − |Rα − Rβ | i< j |ri − r j | α=1 i=1 |ri − Rα | α R2 . In the general case, the volume of the system can be expressed as V =

 4π   4π  3 R2 − R13 = R23 1 − ξ 3 , 3 3

(2.69)

where R1 and R2 are the inner and the outer radii of the system, respectively, and ξ = R1 /R2 ≤ 1 is the ratio of the two radii. The spherical-shell model defined by Eqs. (2.68) and (2.69) is applicable for any spherically symmetric system with an arbitrary value of the ratio ξ . The limiting case of ξ → 0 describes a metallic cluster/nanoparticle with the electron density distribution over the full sphere of radius R: ρ0 (r ) =

N (R − r ) . V

(2.70)

In this limit, R1 → 0 and R ≡ R2 so that the cluster volume is V = 4π R 3 /3. The limiting case of ξ = 1 or R1 → R2 ≡ R represents a system with the electron density distribution over an infinitely thin sphere R. This model was applied previously to investigate the photoionization [63] and the electron scattering [80, 82, 83] processes of carbon fullerenes. The total electron density of the system at the point r and time t is introduced as ρ(r, t) = ρ0 (r) + δρ(r, t), where ρ0 (r) denotes the stationary distribution of the negative charge at the point r , and δρ(r, t) is the density variation caused by the interaction with an external electric field. The collective motion of the electron density is described using the Euler equation and the equation of continuity [88]. The Euler equation couples the acceleration, dv(r, t)/dt, of the electron density with the total electric field E acting on the system at the point (r, t): dv(r, t) = E(r, t) . dt

(2.71)

The electric field E includes both the external field acting on the system and the polarization contribution due to the variation of electron density δρ(r, t):  E(r, t) = −∇φ(r, t) − ∇

δρ(r , t)  dr , |r − r |

(2.72)

where φ(r, t) is the scalar potential of the external field. Introducing Eq. (2.72) in (2.71) and evaluating the full time derivative of the vector v, one obtains

2 Atomic and Molecular Physics Methods for Nanosystems

    ∂v(r, t)  δρ(r , t)  + v(r, t) · ∇ v(r, t) = − ∇φ(r, t) − ∇ dr . ∂t |r − r |

37

(2.73)

The potential of the external field is assumed to satisfy the wave equation and has the monochromatic dependence on t: φ(r, t) = eiωt φ(r) ,

(2.74)

where φ(r) satisfies the equation φ(r) + q 2 φ(r) = 0

(2.75)

with q being the wave vector. The motion of electron density in the system obeys the equation of continuity, which reads   ∂ρ(r, t) + ∇ · ρ(r, t) v(r, t) = 0 . (2.76) ∂t Equations (2.73) and (2.76), being solved simultaneously, determine the variation of electron density δρ(r, t) as well as its velocity v(r, t). The second term on the lefthand side of Eq. (2.73) can be neglected, which means physically that the external field causes only a small spatial inhomogeneity in the electron density distribution within the system. The solutions of Eqs. (2.73) and (2.76) are sought in the following form: &

δρ(r, t) = δρ(r) eiωt . v(r, t) = v(r) eiωt

(2.77)

Substituting these expressions into Eqs. (2.73) and (2.76) and performing some algebraic transformations with the simultaneous use of Eq. (2.75) and |r − r |−1 = −4π δ(r − r ), one derives a set of the following linear equations:    δρ(r ) i  ∇φ(r) + ∇ dr , v(r) = ω |r − r | 

  δρ(r ) ω − 4πρ0 (r) δρ(r) + ∇ρ0 (r) · ∇ dr |r − r | = q 2 ρ0 (r) φ(r) − ∇ρ0 (r) · ∇φ(r) .

(2.78)

2

(2.79)

In the case of the spherically symmetric density distribution, ρ0 (r) = ρ0 (r ), one can exclude angular variables from Eqs. (2.78) and (2.79) by expanding functions φ(r), δρ(r), and |r − r |−1 into spherical harmonics and then integrating over spherical angles of the vector r.

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Fig. 2.1 Left panel: Exposure of a spherical shell of a finite width R = R2 − R1 to an external electric field causes variation of the surface charge densities, σ (1,2) , and that of the volume charge density, δ. Right panel: Representation of the symmetric (a) and the antisymmetric (b) modes of the surface plasmon. Reproduced from Ref. [90] with permission from Springer Nature

Performing some algebraic transformations [90], one obtains a general equation for the variation of electron density in an arbitrary spherically symmetric system: 

∞  ρ0 (r ) gl (r, r  ) δρl (r  ) dr  ω − 4πρ0 (r ) δρl (r ) + 4π l2 2

0

= q 2 ρ0 (r )φl (r ) − ρ0 (r )φl (r ) ,

(2.80)

the scalar potential, ω is the frequency of the external field, and the where φl (r ) is √ notation l = 2l + 1 is introduced. The function gl (r, r  ) is defined as gl (r, r  ) = l

 r l−1 r

(r  − r ) − (l + 1)

  l+2 r (r − r  ) , r

where (x) is the Heaviside step function. Consider a system with a homogeneous charge distribution over a spherical shell of a finite width R = R2 − R1 (see the left panel of Fig. 2.1). This means that the equilibrium electron density distribution, ρ0 (r ), is constant within the interval R1 ≤ r ≤ R2 and equals to zero if otherwise: ρ0 (r ) = ρ0 (r − R1 )(R2 − r ) ,

(2.81)

where ρ0 is defined by Eq. (2.68). The derivative of the function ρ0 (r ) is given by   ρ0 (r ) = ρ0 δ(r − R1 ) − δ(r − R2 ) ,

(2.82)

where δ(x) is the delta function. The solution of Eq. (2.80) for such a system is sought in the following form: δρl (r ) = δl (r )(r − R1 )(R2 − r ) + σl(1) δ(r − R1 ) + σl(2) δ(r − R2 ) , (2.83)

2 Atomic and Molecular Physics Methods for Nanosystems

39

where δl (r ) describes the volume density variation arising inside the finite-width spherical shell, and σl(1,2) are variations of the surface charge densities at the inner and the outer surfaces of the shell, respectively (see Fig. 2.1). The volume density variation causes the formation of the volume plasmon, while the variations of the surface densities lead to the formation of the surface plasmon, which has two normal modes, the symmetric and antisymmetric ones. It has been argued previously [88, 90, 91] that only the surface plasmon can occur in the system interacting with a uniform external electric field, as it happens in the photoionization process. Non-uniformity of the external field causes the formation of the volume plasmon [91] which appears due to compression of the electron density inside the volume of the shell. These two scenarios are described in greater detail below.

2.6.1 Interaction with a Uniform External Field The case of a uniform external field describes the interaction of a system with an electromagnetic field. It is assumed that the wavelength of electromagnetic radiation is much larger than the typical size of the system, i.e., the condition ω R  1 is fulfilled. This condition implies the validity of the dipole approximation. In this limit, the wave vector q = 0 and Eq. (2.80) for the multipole variation of the electron density in a spherically symmetric system turns into the following one: 

∞  ρ0 (r ) gl (r, r  )δρl (r  )dr  = −ρ0 (r )φl (r ) . (2.84) ω − 4πρ0 (r ) δρl (r ) + 4π l2 2

0

Substituting (2.81)–(2.83) into Eq. (2.84) and taking into account that in the dipole approximation the field intensity φl does not depend on the spatial coordinate, φl (R1 ) = φl (R2 ) ≡ φl , one derives the following equation: $ % w − 1 δl (r )(r − R1 )(R2 − r )   l + 1 (1) l (2) l−1 δ(r − R1 ) σ + σ ξ + wσl(1) + Il(1) − l2 l l2 l   l + 1 (1) l+2 l (2) δ(r − R2 ) + wσl(2) + Il(2) + σ ξ − σ l2 l l2 l φ φ = − l δ(r − R1 ) + l δ(r − R2 ) , 4π 4π

(2.85)

where w = ω2 /ω2p with ω p being the volume plasmon frequency associated with the density ρ0 ,

(2.86)

40

A. V. Verkhovtsev and A. V. Solov’yov

ωp =

'

( 4πρ0 =

3N , − R13

(2.87)

R23

and Il(1)

l = 2 R1l−1 l

R2

δl (x) dx , x l−1

R1

Il(2)

l +1 1 = l2 R2l+2

R2 x l+2 δl (x)dx .

(2.88)

R1

Matching the terms of different types on the right- and the left-hand side of Eq. (2.85), one obtains expressions which define the variations of the volume and the surface charge densities. The solution corresponding to the volume density variation reads as: $ % (2.89) w − 1 δl (r )(r − R1 )(R2 − r ) = 0 . This means that no volume plasmon can arise in the system due to interaction with a uniform external field. The volume plasmon manifests itself only when the system interacts with a non-uniform external field, for instance, in collisions with charged particles. Thus, the total variation of electron density is described only by the surface density contributions: δρl (r ) = σl(1) δ(r − R1 ) + σl(2) δ(r − R2 ) ,

(2.90)

which can be defined from the following system of coupled equations:  ⎧ φ l +1 l ⎪ ⎪ w − σl(1) + 2 ξ l−1 σl(2) = − l ⎨ 2 4π l l   .  φ l l ⎪ (1) (2) l l+2 ⎪ σ ξ σ + w − = ⎩ 2 l 4π l l2 l

(2.91)

The determinant of the system (2.91) is  = (w − w1l )(w − w2l ), where w1l and w2l are the roots of the secular equation  = 0:  1 ' 1 1− 1 + 4l(l + 1)ξ 2l+1 2 2l + 1 . 1 ' 1 1+ w2l = 1 + 4l(l + 1)ξ 2l+1 2 2l + 1

w1l =

(2.92)

Variation of the surface charge densities, σl(1,2) , results in the formation of two coupled modes of surface plasmon oscillations [73, 90, 92, 93]. Frequencies of the symmetric, ωl(s) , and the antisymmetric, ωl(a) , surface plasmons of multipolarity l are given by the following expression:

2 Atomic and Molecular Physics Methods for Nanosystems (s/a) ωl

 = 1∓

41

1 ' 1 + 4l(l + 1)ξ 2l+1 2l + 1

1/2

ωp √ , 2

(2.93)

where “−” and “+” stand for symmetric and antisymmetric modes, respectively. In the symmetric mode, the charge densities of the two surfaces oscillate in phase, while in the antisymmetric mode they are out of phase (see Fig. 2.1). Since only dipole excitations may arise in the system due to interaction with the external electromagnetic field, the case of interest is l = 1. The expression for the resonant frequencies of the surface plasmons is then reduced to: ω(s/a) = where p = The values

'



N (s/a) (3 ∓ p) 2(R23 − R13 )

1/2 ,

(2.94)

1 + 8ξ 3 with ξ = R1 /R2 being the ratio of the inner to the outer radius. N (s) = N

p+1 , 2p

N (a) = N

p−1 2p

(2.95)

are the number of delocalized electrons which are involved in each plasmon mode. They obey the sum rule N (s) + N (a) = N , where N is the total number of delocalized electrons participating in the collective excitation. The cross section of photoionization by a single photon is given by the general expression: 4π ω Im α(ω) , (2.96) σγ (ω) = c where α(ω) is the dynamic polarizability. This quantity is related to the external electric field E(ω) and the induced dipole moment, which is defined as: √ ∞ 4π d(ω) = r 3 δρ1 (r ) dr . 3

(2.97)

0

The quantity δρ1 (r ) defines the electron density variation (2.90) created by the dipole mode (l = 1) of surface plasmon oscillations. The dipole polarizability α(ω) can be expressed in the following form [90]:  α(ω) ∝ N

1 ω2 − ω(s) 2

+

1 ω2 − ω(a) 2

 .

(2.98)

This expression clearly shows that the photoionization cross section is defined by the two surface plasmons with the frequencies ω(s) and ω(a) . The final expression for the photoionization cross section within the PRA is obtained by accounting for damping of the plasmon oscillations due to the decay of the collective mode to the incoherent sum of single-electron excitations (see Sect. 2.7.4).

42

A. V. Verkhovtsev and A. V. Solov’yov

This is done by introducing the finite widths,  (s) and  (a) , of the plasmon resonances and making the following substitutions in the right-hand side of Eq. (2.98): 1 ω2

−

ω(s,a) 2

−→

1 ω2

−

ω(s,a) 2

+ iω (s,a)

.

(2.99)

Taking the imaginary parts of the polarizability produces: Im

1 ω2 − ω

(s,a) 2

+ iω (s,a)

ω (s,a) −→ $ . 2 %2 2 ω2 − ω(s,a) + ω2  (s,a)

(2.100)

Thereby, the final formula for the photoionization cross section within the PRA is written in the following form [90, 91]: 4π ω2 σ (ω) = c



N (a)  (a) N (s)  (s) + $ % $ %2 2 ω2 − ω(s) 2 + ω2  (s) 2 ω2 − ω(a) 2 + ω2  (a) 2

 . (2.101)

2.6.2 Interaction with a Non-uniform External Field In the case of the interaction with a non-uniform electric field, e.g., in collisions with charged particles, the multipole variation of the electron density (2.83) is defined as the solution of a general equation (2.80). Carrying out the transformations, similar to those described above for the case of a uniform external field, one derives the following equation: $ % w − 1 δl (r )(r − R1 )(R2 − r )   l + 1 (1) l (2) l−1 δ(r − R1 ) σ + σ ξ + wσl(1) + Il(1) − l2 l l2 l   l + 1 (1) l+2 l (2) δ(r − R2 ) + wσl(2) + Il(2) + σ ξ − σ l2 l l2 l  φl (r ) 1   (r − R1 )(R2 − r ) − φl (R1 )δ(r − R1 ) − φl (R2 )δ(r − R2 ) . = q2 4π 4π (2.102) Matching the terms of different types on the left- and the right-hand side of Eq. (2.102), one obtains three equations: one for the volume density variation and the other two for the variation of the surface charge densities. The solution of Eq. (2.102) corresponding to the volume density variation reads as:

2 Atomic and Molecular Physics Methods for Nanosystems

δl (r ) =

q 2 φl (r ) , w − 1 4π

43

(2.103)

and the density variation due to the surface plasmon modes is σl (r ) = σl(1) δ(r − R1 ) + σl(2) δ(r − R2 ) .

(2.104)

The quantities σl(1) and σl(2) satisfy the following system of coupled equations:  ⎧ l l + 1 (1) ⎪ ⎪ w − σl + 2 ξ l−1 σl(2) = F1 ⎨ l2  l   , l l+2 (1) l ⎪ (2) ⎪ ⎩ 2 ξ σl + w − 2 σl = F2 l l where F1,2 = ∓

φl (R1,2 ) − Il(1,2) 4π

(2.105)

(2.106)

and the functions Il(1,2) were defined above in (2.88). Solutions of the system (2.105) are given by the following expression: σl(1) σl(2)

 1 F1 w − =  1 F2 w − = 

 % F1 l $ l−1 F1 + ξ F2 ≡ 2 l   , % F2 l + 1$ F2 + ξ l+2 F1 ≡ 2  l

(2.107)

where  is the determinant of the system. Finally, using Eqs. (2.103) and (2.107) in Eq. (2.83), one obtains the expression which defines the multipole variation of electron density in a spherically symmetric hollow system under the action of the multipole component φl (r ) of the external field: q 2 φl (r ) (r − R1 )(R2 − r ) w − 1 4π F1 F2 δ(r − R1 ) + δ(r − R2 ) . +  

δρl (r ) =

(2.108)

The first term leads to the formation of the volume plasmon, while the two other terms are responsible for the formation of two coupled modes of the surface plasmon.

44

A. V. Verkhovtsev and A. V. Solov’yov

2.7 Illustrative Case Studies 2.7.1 Application of the Jellium Model for Metal Clusters Essential progress in experimental study of photoabsorption processes of clusters has been achieved by using new sources for free neutral [94] and charged [95] size-selected metal clusters. These measurements were followed by new theoretical calculations of the photoionization cross section [96–98] and of the photoelectron angular distribution for neutral and charged metal clusters [99, 100]. From the theoretical point of view, the valence electron-shell structure and optical properties of various atomic clusters were successfully described by means of the jellium model (see Sect. 2.5). This section highlights some achievements of the applications of jellium model for studying the photodetachment process from metal cluster anions [100]. The first measurements of the angle-resolved photoelectron spectra of negatively charged sodium clusters were reported in Refs. [101, 102]. The experiments were performed in a broad range of cluster sizes, 3 ≤ Z ≤ 147, and allowed one to probe the angular momenta of single-electron orbitals. It was also demonstrated that simple models based on single-electron treatment of the photoionization process fail to describe the angular anisotropy of photoelectrons. The first calculations performed within the framework of the jellium model for the ionic core and the random phase approximation with exchange (RPAE) for the valence electrons demonstrated a crucial role of many-electron correlations in describing the correct behavior of the photoelectron angular distribution [100]. The essential contribution of many-electron correlations in the photoionization process is not a surprise for neutral metal clusters. The correlations become even more pronounced for negative ions because of weaker binding of the valence electrons. As a result, the role of the continuous spectrum of excitations becomes even more important in the description of photoionization of negative ions, which is very interesting from the theoretical point of view. − In Ref. [100] two magic sodium clusters, namely Na− 19 and Na57 , have been studied. These clusters can be treated as spherically symmetric objects due to the completely − filled electronic shells. The ground-state electronic configurations of Na− 19 and Na57 clusters are defined as 1s 2 2 p 6 3d 10 2s 2 for Na− 19 ,

1s 2 2 p 6 3d 10 2s 2 4 f 14 3 p 6 5g 18 for Na− 57 .

The wave functions of a photoelectron were obtained from the HF equations as the solution corresponding to the energy ε = ω − I p and satisfying certain asymptotic behavior (see, e.g., [27]), where I p is the ionization potential and ω is the photon energy. The wave functions of the excited states were calculated either in the field of the “frozen” core with the created vacancy or in the field of the rearranged residual electron structure of the cluster. The obtained HF radial wave functions Pnl (r ) and

2 Atomic and Molecular Physics Methods for Nanosystems

45

Pεl (r ) are used further to calculate the dipole-photon amplitudes as well as the Coulomb matrix elements to account for many-electron correlations. The partial photodetachment cross section σnl (ω) of the nl shell is given by the expression: % 4π 2 α Nnl $ |dl+1 |2 + |dl−1 |2 , (2.109) σnl (ω) = 3(2l + 1)ω where α is the fine-structure constant, ω is the photon energy, and Nnl is the number of electrons in the nl shell. The reduced HF dipole matrix elements dl±1 (in the length form) are defined as follows: '  ≡ εl ± 1| d | nl = (−1) l> Pnl (r )Pεl±1 (r ) r dr , ∞

dl±1

l

(2.110)

0

where l> = l + 1 for the l → l + 1 transition and l> = l for the l → l − 1 transition. The total photodetachment cross section is obtained by the sum over all partial cross sections. Many-electron correlations were taken into account by employing the RPAE scheme (see Sect. 2.4.3), which describes the dynamic collective response of an electron system to an external electromagnetic field [27]. Within the RPAE, the dipole-photon amplitudes εl ± 1| D(ω)| nl are obtained by solving the following integral equation:

εl ± 1| D(ω)| nl = εl ± 1| d | nl ⎞ ⎛  ⎟ ν2 | D(ω)| ν1  ν1 , εl ± 1| U | ν2 , nl ⎜ ⎟ +⎜ − . (2.111) ⎠ ⎝ ω − ε2 + ε1 + ıδ ν2 >F ν1 ≤F

ν1 >F ν2 ≤F

Here the indices ν1 and ν2 denote the quantum numbers {n(ε), l} of the virtual electron–hole states, F is the Fermi energy, and the matrix element . . . | U | . . .  stands for the sum of the direct and exchange Coulomb matrix elements [27]. The angular distribution of photoelectrons detached from the nl shell by an unpolarized photon is determined by the differential cross section of the electron emission into the solid angle d = sin θ dθ dφ:   σnl (ω) 1 dσnl (ω) = 1 − βnl (ε)P2 (cos θ ) , d 4π 2

(2.112)

where P2 (cos θ ) is the Legendre polynomial, βnl (ε) is the angular distribution anisotropy parameter, and ε is the photoelectron energy. The main feature of the total photoabsorption cross section in metal clusters is the giant plasmon resonance at about 2–3 eV which appears due to the collective response of the systems [38]. For neutral and positively charged metal clusters

46

A. V. Verkhovtsev and A. V. Solov’yov

Fig. 2.2 Panel A: The RPAE partial (dashed, dash-dotted, and dotted lines) and total (solid line) photodetachment cross sections for the Na− 19 anion. Panel B: The total photodetachment cross section for Na− calculated within the HF (dashed line) or RPAE (solid line) framework. Redrawn 57 from data presented in Refs. [100, 103]

plasmon resonance lies in the discrete spectrum of electronic excitations. The simple HF scheme produces the set of single-electron excitations below the ionization potential and the corresponding distribution of the oscillator strengths. When the correlations are accounted for, the most of the oscillator strength becomes concentrated at the energy close to 2–3 eV, thus describing properly the plasmon excitation. The distinguishing feature of negatively charged clusters is that they either do not have at all or have just one or two discrete dipole excitations. Therefore, the main part of the oscillator strength of partial transitions is distributed in the continuous spectrum of photodetachment cross section. This feature manifests itself both within the HF approximation and the RPAE. Thus, the plasmon resonance lies in continuum, and accounting for many-electron correlations leads to a significant change in the resonance position, its maximum value, and the shape of the resonance curve [98]. The typical photodetachment cross section behavior for metal cluster anions, − namely Na− 19 and Na57 , is shown in Fig. 2.2 and reveals the powerful maxima due to the collective electron excitation. For the Na− 19 anion (Fig. 2.2A), six dipole transitions were taken into account in the RPAE calculations. The dipole transitions from the outer 3d and 2s shells give the main contribution to the total cross section in the vicinity of the plasmon resonance which is located at about ω = 2.4 eV. However, it should be noted that it is not enough to account only for the transitions from the outer shells in order to obtain the final profile of the plasmon excitation. Instead, one should include the interactions between all valence electrons. For larger cluster anions the number of transitions, which must be accounted for, increases significantly. The total photodetachment cross sections of Na− 57 calculated within the HF approximation and the RPAE are shown in Fig. 2.2B. The main contribution to the total cross section comes from the outer 5g shell. However, similar to the case of the Na− 19 anion, to form the powerful resonance in the total cross section it is essential to take into account the contributions of several shells, namely of the 2s,

2 Atomic and Molecular Physics Methods for Nanosystems

47

Fig. 2.3 The angular anisotropy parameter β(ε) for the 3d shell of the Na− 19 anion as a function of photoelectron energy. Comparison between the HF (dashed line), RPAE (solid line) and the experiment [102] (symbols). Experimental data marked as A, B, C, and D correspond to the photoionization from the sublevels of the 3d orbital split by the crystalline field. Redrawn from data presented in Refs. [100, 103]

3d and 3 p shells. For Na− 57 the position of the plasmon resonance is about 2.6–2.7 eV, and, thus, it does not change significantly with the size of the cluster and lies very close to the classical Mie value. It should be noted that the total photodetachment cross section is not a very sensitive indicator of the applicability of the jellium model to metal clusters. The photoelectron angular distribution represents a much better test for the conventional jellium model since the behavior of the anisotropy parameter β(ε) can show whether the shell structure is real and the orbital momentum l is a good quantum number in the cluster system. The comparison between the HF, the RPAE and the experimental data [102] for the outer 3d shell in the Na− 19 anion is presented in Fig. 2.3. The experimental points marked as A, B, C and D correspond to the photoionization from the sublevels of the 3d orbital which is split by the crystalline field of the cluster in the range 1.75–2.15 eV. It is interesting to note that the general behavior of photoelectron angular distribution is approximately the same for all sublevels. This indicates that all electrons in the sublevels can be characterized by the same orbital quantum number l. The 3d binding energy, equal to 1.92 eV within the HF approximation, is very close to the center of the multiplet. The calculations within the HF framework (dashed line) fail to explain the dependence of the angular anisotropy parameter on the photoelectron energy. On the contrary, the RPAE results (solid line) are in good agreement with the experimental data. However, the HF and RPAE schemes based on the jellium

48

A. V. Verkhovtsev and A. V. Solov’yov

Fig. 2.4 The angular anisotropy parameter for the 4 f (panel A) and 5g (panel B) shells of Na− 57 anion as a function of the photoelectron energy. The results of HF (dashed line) and RPAE (solid − line) calculations are compared with the experiment data for Na55 (symbols) taken from Ref. [102]. Different experimental points correspond to the photoionization from the sub-levels of the orbital split by the crystalline field of the cluster. Redrawn from data presented in Refs. [100, 103]

model ignore the splitting and, thus, provide the average dependence of the angular anisotropy parameter. As for larger clusters, the calculations of the angular anisotropy parameter β(ε) were performed for several outer shells of the Na− 57 anion. The experimental measurements of the photoelectron spectra were performed not for the closed-shell Na− 57 16 anion, but for Na− 55 , which has the unfilled 5g shell [102]. Note that despite different cluster anions, considered theoretically and experimentally, as well as the differences in ionization potentials, the general behavior of the angular anisotropy parameter for inner shells (for instance, for 4 f shell, see Fig. 2.4A) is reproduced quite well by the performed many-body calculations. However, the calculated dependence β(ε) for Na− 57 for the outer 5g shell agrees only qualitatively with the experimental data for Na− 55 (Fig. 2.4B). Several reasons can be indicated which may explain the quantitative deviation. First, the polarization potential was neglected in the calculations performed in Ref. [100] which is rather large for neutral metal clusters and, thus, modifies the dynamics of the outgoing photoelectron. Second, the electrons in the open 5g 16 shell in Na− 55 are much stronger influenced by the crystalline field than the electrons in the filled 5g 18 shell of Na− 57 . The comparison of the existing experimental and theoretical data indicates that further investigation of the process is needed. The theoretical consideration of manyelectron system was performed in Ref. [100] using the simple jellium model for the ionic core. A more detailed analysis of the problem should go beyond this model and account for the realistic geometrical structure of the core. However, the agreement between the experimental data on the angular distribution and the results of the current

2 Atomic and Molecular Physics Methods for Nanosystems

49

theory indicate that the definite shell structure of valence electrons is reproduced quite well even neglecting the interaction with the real crystalline field.

2.7.2 Correction to the Jellium Model for Fullerenes Despite a widespread use of the jellium model for studying the ground-state electronic properties of fullerenes [62–65], it was concluded [104, 105] that these properties cannot be described properly by the standard jellium model which produces, in particular, unreliable values for the total energy [104]. To avoid this, adding of structureless pseudopotential corrections to the jellium potential Ucore (r ), Eq. (2.64), was suggested [104]. As a rule, a phenomenological square-well (SW) pseudopotential has been commonly used in the calculations [65, 74, 104, 106]: & Ucore (r ) →

Ucore (r ) + USW , R1 ≤ r ≤ R2 , Ucore (r ) , otherwise

(2.113)

where USW is an adjustable parameter to correct the electronic structure of the fullerene. It was claimed that accounting for such a pseudopotential increases the accuracy of the jellium-based description [106] and, for instance, allows one to reproduce the experimental value of the first ionization potential of C60 [65]. Nonetheless, the applicability of the jellium model for fullerenes and the choice of parameters of the used SW pseudopotential have not been clearly justified from a physical viewpoint. Besides, this simple approach cannot describe properly the valence electron density distribution (see the solid cyan line in Fig. 2.5) that is crucial for the calculation of dynamical polarizability and the photoionization cross section. In contrast to more precise quantum chemistry methods, the jellium model does not take into account chemical features of the fullerene, such as hybridization of atomic orbitals in the formation of chemical bonding. However, the jellium model can be improved by means of a more sophisticated pseudopotential which would enable describing chemical properties of the real system. In Ref. [67] a structured pseudopotential correction U , originated from the comparison of an accurate ab initio calculation (by means of DFT) with the jelliumbased one, was proposed. The correction was defined as a difference between the total electrostatic potential of the system obtained from the ab initio calculation and the one obtained within the jellium model: jel

QC (r ) − Utot (r ) . U (r ) = Utot

(2.114)

QC (r ) the total electrostatic potential of the system (repreTo derive the potential Utot sented as a sum of the nuclear and electronic parts) was calculated:

50

A. V. Verkhovtsev and A. V. Solov’yov

Fig. 2.5 Radial electron density of C60 obtained from the ab initio calculation (solid black curve) and calculated by means of the jellium model: the standard one (dashed cyan curve), with the additional square well (SW) pseudopotential (solid cyan curve) and with the additional pseudopotential U (dash-dotted orange curve). Redrawn from data presented in Ref. [67]

Utot (r) = Un (r) + Uel (r) = −

 A

ZA + |r − R A |



ρ(r ) dr . |r − r |

(2.115)

Then, the exact potential and the electron density (accounting for the real icosahedral symmetry of the fullerene C60 ) were averaged over the directions of the position vector r: QC (r ) = U n (r ) + U el (r ) , Utot  1 U i (r ) = Ui (r)d (i = tot, n, el) , 4π  1 ρ(r ) = ρ(r)d . 4π jel

(2.116)

QC The total potentials Utot (r ) and Utot (r ) of C60 as well as their difference U (r ) are shown in Fig. 2.6. The radial density of the delocalized electrons in C60 fullerene, obtained within the two approaches, is presented in Fig. 2.5. The figure shows that the standard jellium model without any corrections (dashed cyan curve) fails to represent the results of the ab initio calculation (black curve). The additional SW pseudopotential does not modify the density distribution significantly (solid cyan curve). The introduction of the pseudopotential U allows one to improve significantly the electron density distribution (dash-dotted orange curve in Fig. 2.5).

2 Atomic and Molecular Physics Methods for Nanosystems

51

Fig. 2.6 Panel A: Total electrostatic potential of C60 obtained from the ab initio quantum chemistry calculation (solid curve) and within the jellium model (dashed curve). Panel B: The difference U between the total electrostatic potential of C60 calculated by the ab initio methods and that one calculated within the jellium model (solid curve). The square well pseudopotential USW is also shown for the comparison (dashed curve). Redrawn from data presented in Ref. [67]

As opposed to the square well pseudopotential which affects equally all electrons of the system, U is an alternating-sign pseudopotential (see Fig. 2.6B); therefore it is attractive in the vicinity of the fullerene ionic core and repulsive at larger distances from the fullerene surface. That means that such potential affects differently the σ and π -electrons of C60 which are located on the fullerene’s surface and perpendicularly to it, respectively. Therefore, such potential enables to account, at least partly, for the sp 2 -hybridization of carbon atomic orbitals, and predict the shape of the electron density in a more realistic way. By means of the presented pseudopotential, a relatively simple jellium model acquires more physical sense and parameters of the model obtain a clear physical justification. The photoionization process of C60 has been studied in a way similar to that described in Sect. 2.7.1 for the metal cluster anions. The partial photoionization cross section σnl (ω) of the nl-shell of a fullerene is defined by Eq. (2.109). Using the single-electron wavefunctions (here the LDA approach has been employed) the one-particle transition amplitudes dl±1 can be calculated. The collective electron

52

A. V. Verkhovtsev and A. V. Solov’yov

Fig. 2.7 Photoionization cross section of C60 obtained by means of LDA calculations within the jellium model with structured correction U (solid line) and measured experimentally by Hertel et al. [78], Kafle et al. [107] and Reinköster et al. [108] (symbols). In the calculations many-electron correlations were accounted for within the RPA approach. Redrawn from data presented in Ref. [103]

excitations in fullerenes were taken into account by solving Eq. (2.111). Describing the single-particle interaction within the LDA approach, one should then exclude the exchange interaction in the matrix elements . . . | U | . . . , replacing the latter by the ordinary direct Coulomb matrix element in Eq. (2.111). Thus, one obtains the reduced amplitudes Dl±1 (ω), which include the many-electron correlations within the random phase approximation (RPA). As demonstrated in Fig. 2.5 the correction U improves the description of the ground-state density distribution within the jellium model and gives a more realistic electron density as compared to the conventional model. The total photoionization cross section of C60 was calculated in Ref. [103] within the LDA and RPA approaches using the corrected jellium potential. Results of the calculations and the comparison with experimental data [78, 107, 108] are presented in Fig. 2.7. The calculated cross section describes qualitatively the main features of the spectrum despite some quantitative discrepancies. It should be noted that the cross section calculated using the corrected jellium model gives a better agreement with experimental data as compared to previous calculations performed within the conventional jellium model (see, e.g., Refs. [63, 64]).

2 Atomic and Molecular Physics Methods for Nanosystems

53

2.7.3 Plasmon Excitations in Photo- and Electron Impact Ionization of Atomic Clusters and Fullerenes Photoionization of Fullerenes Photoionization of fullerenes and other nanoscale systems represents a complex phenomenon and involves a number of features which can be studied by means of various theoretical methods. Being by its nature a quantum phenomenon, the photoionization process can be described within the ab initio framework based on the time-dependent density-functional theory (TDDFT) [4], see Sect. 2.4.4. However, it is well established that photoionization of nanoscale carbon systems, fullerenes in particular, as well as various metallic clusters and nanoparticles, takes place through plasmons—collective excitations of delocalized valence electrons which are induced by an external electric field. The plasmon excitations correspond to oscillations of the electron density with respect to the positively charged ions and are described in the classical physics terms [13, 88] (see Sect. 2.6). When a fullerene is ionized either by a photon or by a charged projectile, various types of collective excitations, which are characterized by prominent resonant-like structures in the ionization spectra, are formed in the system. The most prominent structure, positioned in the excitation energy range from 20 to 30 eV, is formed due to collective oscillations of both σ and π delocalized electrons of a system, while a smaller narrow peak in the low-energy region of the spectrum (below 10 eV) is attributed to the collective excitation of only π -electrons. The σ - and π -electrons occupy, respectively, σ - and π -orbitals of a fullerene, which are formed due to the sp2 -hybridization of carbon atomic orbitals. The resonance peaks in the ionization spectra are described by some characteristic widths, , which have a quantum origin and appear due to the decay of the collective excitation modes into the incoherent sum of single-electron excitations (see Sect. 2.7.4 below). In most cases, the excitation spectra calculated within the ab initio framework can be obtained in a broad range of excitation energies range only for small molecules or clusters consisting of a few atoms. For larger system, such as, for instance, fullerenes, a vast majority of contemporary software packages for ab initio based calculations can describe accurately only a limited number of low-lying excited states located below or just above the ionization threshold. A detailed structure of the spectrum at higher excitation energies, where the plasmon excitations dominate the spectrum, could be hardly revealed due to significant computational costs. An alternative approach for the description of electron excitations in many-electron systems is based on the jellium model, see Sect. 2.5. An effective tool for evaluation of the contribution of plasmon excitations to the ionization spectra is based on the plasmon resonance approximation (PRA), see Sect. 2.6. The advantage of this approach is that it provides a clear physical explanation of the resonant-like structures in the photoionization [88, 89] and inelastic scattering cross sections [80–82, 86, 87] on the basis of excitation of plasmons by the photon or electron impact.

54

A. V. Verkhovtsev and A. V. Solov’yov

In Ref. [89] the contributions of various classical and quantum physics phenomena appearing in the photoionization process of C60 fullerene were elucidated. By comparing the ab initio TDDFT results with those based on the PRA, the wellresolved features of the photoabsorption spectrum of C60 were mapped to different types of single-particle and collective electron excitations having the different physical nature. It was demonstrated that the peculiarities arising in the spectrum atop the dominating plasmon excitations have the quantum origin. A series of individual peaks in the continuous part of the excitation spectrum have been assigned to the particular single-electron transitions and caused by ionization of inner molecular orbitals of the fullerene. The photoabsorption spectrum of C60 was calculated by means of the TDDFT approach outlined in Sect. 2.4.4. TDDFT calculations were performed in the linear response regime within the dipole approximation, see Eq. (2.46). The dynamical polarizability tensor αi j (ω), which describes the linear response of the dipole to the external electric field, was calculated according to Eq. (2.58), and the photoabsorption cross section σ (ω) was related to the imaginary part of αi j (ω) through Eq. (2.59). Within the PRA the dynamical polarizability α(ω) has a resonance behavior in the region of frequencies where collective electron modes in a fullerene can be excited, see Eq. (2.98). In Ref. [89] both π - and (σ + π )-plasmons, which involve only π or both σ + π delocalized electrons of the system, respectively, were studied. Thus, the photoionization cross section, σpl (ω) ∝ Im α(ω), is defined as a sum of the two plasmons, σpl (ω) = σ π (ω) + σ σ +π (ω), and the contribution of each plasmon is governed by the symmetric and antisymmetric modes according to Eq. (2.101). Figure 2.8 shows the photoabsorption spectrum of C60 calculated within the ab initio and classical approaches in the photon energy region up to 100 eV. The thin solid (black) line represents the results of TDDFT calculations within the LDA approach, and the thick solid (green) one represents the contribution from the plasmon excitations. The main resonant structure presented in Fig. 2.8 is formed due to collective oscillations of both σ - and π -electrons of the system, while a prominent peak in the low-energy region of the spectrum (shown in the inset) is attributed to the collective excitation of only π -electrons. The dashed (red) and dash-dotted (blue) lines show, respectively, contributions from the symmetric and antisymmetric modes of the plasmons to the cross section. The resonance frequencies, ωs and ωa , for the two modes of the (σ + π )- and π -plasmons as well as the corresponding widths, s and a , are summarized in Table 2.1. The width sσ +π = 11.4 eV of the symmetric (σ + π )-plasmon mode corresponds to the experimental values obtained from the photoionization and energy loss experiments on neutral C60 [78, 83]. For the antisymmetric mode, the value aσ +π = 33.2 eV was used, which corresponds to the widths of the second plasmon resonance obtained in the study of photoionization of q+ C60 (q = 1 − 3) ions [109]. The PRA describes quite well the main features of the spectrum, such as height, width, and position of the plasmon resonance peaks. The spectrum calculated within the TDDFT approach reveals a more detailed structure which is formed atop the plasmon resonances and represents a series of individual peaks. The oscillator strengths, calculated by means of TDDFT and within the plasmon resonance approximation in

2 Atomic and Molecular Physics Methods for Nanosystems

55

Fig. 2.8 The photoabsorption cross section of C60 calculated within the TDDFT method (thin black line) and the plasmon resonance approximation (thick green line) [89]. The curves, obtained within the classical approach, describe the dominating plasmon resonance, which is formed due to collective oscillations of (σ + π ) delocalized electrons of the system, and a narrow low-energy peak below 10 eV (shown in the inset) which is attributed to the collective excitation of only π -electrons. Contributions of the symmetric and antisymmetric modes of the plasmons are shown by the dashed (red) and dash-dotted (blue) lines, respectively. Reproduced from Ref. [89] with permission from American Physical Society Table 2.1 Peak positions and the widths of the two modes of the (σ + π )- and π -plasmons used in the calculations reported in Ref. [89]. All values are given in eV ωs s ωa a (σ + π )-plasmon 20.3 π -plasmon 5.8

11.4 1.2

33.5 7.9

33.2 3.5

the photon energy range up to 100 eV, are equal to 224 and 195, respectively. Analysis of the plasmon contribution to the cross section shows that about 9 π -electrons are involved in the low-energy collective excitation below 10 eV. This value corresponds to the experimentally evaluated sum rule of the oscillator strength up to the ionization threshold of C60 , I p ≈ 7.6 eV, which gives the value of 7.8 [110]. In Fig. 2.9, the theoretical curves are compared to the results of experimental measurements of photoabsorption of C60 [107] (open squares). The oscillator strength, calculated by means of TDDFT, is very close to the experimentally measured value of 230.5 [107]. It should be noted that the detailed structure of the spectrum, which is described within the TDDFT approach, is not seen in the experimental curve due to a high operational temperature of 500–700 ◦ C [110]. In the experiments, the linewidths of single-electron excitations are broadened in the vicinity of the main plasmon resonance due to the coupling of electron excitations with the vibrational modes of

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A. V. Verkhovtsev and A. V. Solov’yov

Fig. 2.9 The photoabsorption cross section of C60 calculated within the TDDFT method (thin black line) and the plasmon resonance approximation (thick green line) [89]. The curve, obtained within the classical approach, describes both the (σ + π )- and π -plasmons. Theoretical curves are compared to the experimental data [107]. Reproduced from Ref. [89] with permission from American Physical Society

the ionic background [60]. As demonstrated in Fig. 2.9, the PRA gives an adequate description of the experimental results. Next, the origin of individual peaks formed in the photoionization spectrum of C60 atop the plasmon resonances has been elucidated. These peaks can be assigned to discrete transitions between particular molecular orbitals (MOs). The C60 fullerene belongs to the icosahedral (Ih ) symmetry group, therefore its MOs can be classified according to the Ih irreducible representations. The icosahedral symmetry allows the maximum orbital degeneracy equal to five. Thus, the MOs can be singly (ag , au ), triply (t1g , t1u ), (t2g , t2u ), fourfold (gg , gu ) and fivefold (h g , h u ) degenerated. The subscripts “g” and “u” denote, respectively, symmetric (“gerade”) and antisymmetric (“ungerade”) MOs with respect to the center of inversion of the molecule. Due to the quasispherical structure of the molecule, the MOs can be expanded in terms of spherical harmonics in the angular momentum l [111] (see Fig. 2.10). Thus, the innermost ag , t1u and h g MOs in the Ih symmetry represent, respectively, the s, p, and d orbitals, which correspond to l = 0, 1, and 2. The orbitals which correspond to higher angular momenta are constructed as a combination of several MOs. The correspondence between the MOs of C60 and the spherically symmetric orbitals with a given value of angular momentum l is given in Table 2.2. In the spherical representation of C60 , the delocalized electrons are considered as moving in a spherically symmetric central field. Therefore, the ground-state electronic configuration is described by the unique set of quantum numbers {n, l} where n and l are the principal and orbital quantum numbers, respectively [62]:

2 Atomic and Molecular Physics Methods for Nanosystems

57

Fig. 2.10 Left panel: The ground-state electronic structure of C60 obtained within the ab initio framework accounting for the real Ih symmetry of the molecule. Each line corresponds to one molecular orbital which accommodates (or may accommodate) two electrons. Black and red lines (in the range −6 · · · − 25 eV) represent the MOs, which are occupied by 240 delocalized valence electrons of C60 . Blue lines (above −5 eV) represent virtual bound states. Right panel: Electronic structure of the corresponding spherically symmetric nl-orbitals. The horizontal lines indicate the occupation numbers for each orbital and correspond to the summarized number of electrons which occupy the corresponding MOs. Reproduced from Ref. [89] with permission from American Physical Society Table 2.2 Molecular orbitals occupied by delocalized electrons of C60 (left column) and the corresponding spherically symmetric orbitals which are obtained by the expansion of real MOs in terms of spherical harmonics in the angular momentum l (right column) [89] ag s (l = 0) t1u p (l = 1) hg d (l = 2) gu + t2u f (l = 3) h g + gg g (l = 4) h u + t1u + t2u h (l = 5) ag + t1g + gg + h g i (l = 6) h u + t1u + t2u + gu k (l = 7) h g + gg + t2g + h g l (l = 8) gu + h u + gu + t1u + t2u m (l = 9)

1s 2 2 p 6 3d 10 4 f 14 5g 18 6h 22 7i 26 8k 30 9l 34 10m 18 2s 2 3 p 6 4d 10 5 f 14 6g 18 7h 10 . The superscripts indicate the occupation numbers for each spherically symmetric orbital and correspond to the summarized number of electrons which occupy the corresponding MOs (see the left and right panels of Fig. 2.10). One may consider the icosahedral symmetry of C60 as a perturbation of the spherical one, so the correspondence between the real MOs and the spherically symmetric nl-orbitals can be explained in terms of splitting of the latter ones due to reduction of the symmetry.

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A. V. Verkhovtsev and A. V. Solov’yov

Fig. 2.11 Upper panel: Excitation energies of the optically allowed discrete transitions to the virtual bound states which correspond in the spherical representation of C60 to the unoccupied or partially occupied 7h, 8i, and 10m orbitals (see the text for details). Lower panel: Ionization thresholds of the highest occupied molecular orbital, h u (partially occupied 7h orbital), as well as of the following innermost valence orbitals: h g and gg (5g), gu and t2u (4 f ), h g (3d), t1u (2 p), and ag (1s). In the LDA calculations, the ionization thresholds for the HOMO (h u ), and the innermost valence (ag ) molecular orbitals are 6.65 eV and 24.8 eV, respectively. Reproduced from Ref. [89] with permission from American Physical Society

A number of virtual bound states of C60 (t1u , t1g , t2u , h g , ag , h u , gu ) were calculated which can be assigned in the spherical representation to the unoccupied or partially occupied 7h, 8i, and 10m orbitals (see the blue dashed lines in Fig. 2.10). The optically allowed discrete transitions should result in the change of the MO’s symmetry (g ↔ u) or satisfy the l → l ± 1 selection rule within the spherical representation. In Ref. [89] all possible optically allowed discrete transitions were listed and the corresponding transition energies were calculated. The results of this analysis are summarized in the upper panel of Fig. 2.11. The peaks in the TDDFT spectrum can be assigned to 5g, 6g → 7h; 7i → 7h; 6h, 7h → 8i; 8k → 8i, and 9l → 10m

2 Atomic and Molecular Physics Methods for Nanosystems

59

transitions.2 The discrete transitions are shown in the upper panel of Fig. 2.11 by thin solid and dash-dotted vertical lines. The six lowest optically allowed π − π ∗ excitations (h u → t1g , h g → t1u , h u → h g , gg → t2u , h g → t2u , and h u → gg ) [111] from 3 to 6 eV correspond to 7h → 8i and 6g → 7h transitions (see solid red and dash-dotted green lines), which are involved in the formation of the π -plasmon. The 9l → 10m, 7i → 7h, and 8k → 8i transitions (violet, aquamarine, and blue lines, respectively) result in the formation of individual peaks in the region from 9 to 14 eV, which are formed atop the (σ + π )-plasmon excitation. Features in the energy range from 14 to 18 eV are assigned to the single-electron 6h → 8i and 5g → 7h transitions from the lower-lying 5g and 6h orbitals. Thus, accounting for the optically allowed discrete transitions it is possible to reveal a detailed structure of the photoionization spectrum of C60 up to 18 eV. However, one can extend the analysis and characterize a number of subsequent peaks. In the lower panel of Fig. 2.11, vertical lines represent the ionization thresholds of several particular orbitals. The highest-occupied molecular orbital (h u ) of C60 corresponds in the spherical representation to the partially filled 7h orbital. In the calculations performed within the LDA approach [89], the h u ionization threshold is 6.65 eV (solid blue line) which is slightly lower than the experimentally measured ionization potential of C60 , approximately equal to 7.6 eV [78]. Since there are no discrete optical transitions with the energy above 20 eV, a series of peaks, arising between 20 and 25 eV, can be assigned to the ionization of the innermost fullerene MOs (the corresponding nl-orbitals are given in parentheses), namely h g and gg (5g), gu and t2u (4 f ), h g (3d), t1u (2 p), and ag (1s). The calculated ionization thresholds of these orbitals are shown in the lower panel of Fig. 2.11. Within the LDA approach, the threshold of the innermost molecular orbital (ag ) equals to 24.8 eV (solid cyan line). The information obtained within the ab initio framework allows one to reveal the origin of the individual peaks in the photoionization spectrum of C60 for the photon energies up to 25 eV. The nature of several subsequent peaks located at about and above 30 eV cannot be explored by the ab initio approach and should be investigated by means of the model one. One may suppose that these peaks should be caused by excitation of particular molecular orbitals to the continuum.

Inelastic Scattering of Electrons from Atomic Clusters and Fullerenes Investigation of the photoionization process allows one to analyze only the dipole plasmon excitation mode (the angular momentum l = 1) [38, 90]. Electron collective modes with higher angular momenta can be studied in the fast electron–cluster collisions if the scattering angle of the electron is large enough [80–83]. Dipole plasmon resonances of the same physical nature as in the case of the photoabsorption or photoionization dominate the electron energy loss spectrum (EELS) if the scattering angle of the electron, and thus its transferred momentum, are sufficiently small. With 2

To simplify the analysis the spherical representation of orbitals is used here.

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A. V. Verkhovtsev and A. V. Solov’yov

increasing the scattering angle, plasmon excitations with higher angular momenta become more probable. The actual number of multipoles coming into play depends on the cluster size. In the process of inelastic scattering the projectile electron undergoes the transition from the initial electron state (ε1 , p1 ) to the final state (ε2 , p2 ) which is accompanied by the ionization (or excitation) of a target from the initial state i with the energy εi to the final state f with ε f . The matrix element, M, which defines the amplitude of the inelastic scattering is given by

,

 1

M = f, 2

1, i

a |r − ra |   (−)∗ 1 = ψ2 (r)ψ ∗f ({ra }) ψi ({ra })ψ1(+) (r){dra }dr , (2.117) |r − r | a a +

where {ra } = r1 . . . r N are the position vectors of the delocalized electrons in the target, r is the position vector of the projectile, ψ1(+) (r) and ψ2(−) (r) stand for the initial- and the final state wave functions of the projectile, respectively. Superscripts (+) and (−) indicate that asymptotic behavior of the wave functions is “plane wave + outgoing spherical wave” and “plane wave + incoming wave”, respectively. The matrix element can be written as follows:

, + 



4π dq 

−iq·r



iq·ra (2.118) 1 f 2 e e i , M= 2 3

q (2π ) a

where q = p1 − p2 is the transferred momentum. If the velocity of a projectile is high and significantly exceeds the characteristic velocities of delocalized electrons in the target, the first Born approximation is applicable [80]. Within this approximation the initial and the final states of the incident electron can be described by plane waves: ψ1(+) (r) = eip1 ·r ,

ψ2(−) (r) = eip2 ·r .

(2.119)

Then, the amplitude of the process reduces to 4π M= 2 q

, +



iq·ra e i f

a

.

(2.120)

q=p1 −p2

The magnitude q 2 is related to the scattering angle as q 2 ≈ p12 θ 2 under the assumption that the energy loss ω = ε1 − ε2 is small, ω  ε1 (which implies p1 ≈ p2 ) and the scattering angle is small, θ  1 rad. Performing the multipole expansion of the exponential factors in Eq. (2.120) (see, e.g., Ref. [112]), one obtains:

2 Atomic and Molecular Physics Methods for Nanosystems

M = 4π



l

i

∗ Ylm (q)

lm

, +



f φl (ra )Ylm (ra ) i ,

a

where the notation φl (r ) = 4π

jl (qr ) q2

61

(2.121)

(2.122)

is introduced and jl is a spherical Bessel function of the order l. Let us consider a general expression for the cross section of the scattering process: dσ =

 dp2 2π |M|2 δ(ω f i − ω) dρ f . p1 (2π )3 pol pol f

Here ω f i = ε f − εi , the sign

(2.123)

i

-

denotes the summation over the projection of the final state f orbital momentum, whereas poli denotes the averaging over the projections of the initial state orbital momentum, and dρ f is the density of final states of the target. Substituting the scattering amplitude (2.121) into Eq. (2.123), one derives the doubly differential cross section:  1 p2  d2 σ = dε2 dp2 π p1 lm

pol f

+

f



, 2





Vlm (ra ) i δ(ω f i − ω) dρ f ,

a



(2.124)

where Vlm (r) = φl (r )Ylm (r)

(2.125)

is the multipolar potential of the fast projectile, dp2 denotes the differentiation over the solid angle of the scattered electron and sign dρ f means the summation over the final states (which includes the summation over the discrete spectrum and the integration over the continuous spectrum). According to Kubo linear response theory [32, 80], the integral on the right-hand side of Eq. (2.124) can be related to the variation of electron density caused by an external electric field, and the following substitution can be performed:

, 2 



1

∗ Im Vlm Vlm (ra ) i δ(ω f i − ω)dρ f → (r)δρl (ω, q; r)dr .

π a (2.126) Here, δρl (ω, q; r) is the density variation due to the exposure of the system to the multipolar potential Vlm (r). In a general case, this variation depends on the transferred energy ω, transferred momentum q and the position vector r. 

+

f

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A. V. Verkhovtsev and A. V. Solov’yov

Using (2.126) in (2.124), the doubly differential cross section acquires the form

1 p2  d2 σ = 2 Im Il (ω, q) , dε2 dp2 π p1 l where Il (ω, q) =



∗ (r)δρl (ω, q; r)dr . Vlm

(2.127)

(2.128)

m

As described in Sect. 2.6 atomic clusters and fullerenes can be considered in the general case as spherically symmetric systems where the charge is distributed homogeneously between two concentric spheres, see Fig. 2.1. For such a geometry, the multipole variation of electron density is given by the sum of the volume density variation arising inside the shell and the variations of the surface charge densities at the inner and the outer surfaces of the shell, respectively, see Eqs. (2.83) and (2.108). The volume density variation causes the formation of the volume plasmon, while the variations of the surface densities correspond to the symmetric and antisymmetric modes of the surface plasmon (see Fig. 2.1). Presenting the multipole variation of the electron density, δρl (ω, q; r), as a sum of three contributions (see Eq. (2.83)), using the explicit expression for the multipolar potential Vlm (r ) (see Eqs. (2.122) and (2.125)), and performing some algebraic transformations, one comes to the formula for the differential inelastic scattering cross section with no damping of plasmon oscillations [90]:   2 2 S1l (q) ω2l S2l (q) d2 σ 2 R2 p2  ω2p Vl (q) ω1l . = Im + + 2 2 dε2 dp2 π q 4 p1 ω2 − ω2p ω2 − ω1l ω2 − ω2l l

(2.129)

In this expression, ω p is the volume plasmon frequency, Eq. (2.87), which is independent of l. ω1l and ω2l are the frequencies of the symmetric and antisymmetric surface plasmon modes of multipolarity l, defined by Eq. (2.93). Functions Vl (q), S1l (q) and S2l (q) are the diffraction factors depending on the transferred momentum q. They determine the relative significance of the multipole plasmon modes in various ranges of the projectile’s scattering angles. The dominant contribution of different multipole modes results in a significant angular dependence for the differential electron energy loss spectrum [87]. Explicit expressions for these functions are given in Ref. [90]. Then, the differential inelastic scattering cross section which accounts for three plasmons and with damping included reads as: d2 σ (v) d2 σ (s1 ) d2 σ (s2 ) d2 σ = + + , dε2 dp2 dε2 dp2 dε2 dp2 dε2 dp2

(2.130)

2 Atomic and Molecular Physics Methods for Nanosystems

63

where ωp2 l(v) Vl (q) 2R2 p2  d2 σ (v) = ω $ %2 $ %2 dε2 dp2 πq 4 p1 ω2 − ωp2 + ω2 l(v) l 2 ω1l 1l(s) S1l (q) d2 σ (s1 ) 2R2 p2  = ω $ % $ %2 2 2 dε2 dp2 πq 4 p1 ω2 − ω1l + ω2 1l(s) l 2 ω2l 2l(s) S2l (q) d2 σ (s2 ) 2R2 p2  = ω $ %2 $ %2 . dε2 dp2 πq 4 p1 ω2 − ω2 + ω2  (s) l 2l

(2.131)

2l

The cross section d2 σ/dε2 dp2 can also be written in terms of the energy loss ω = ε1 − ε2 ≡ ε of the incident projectile of energy ε1 . Integration of d2 σ/dε dp2 over the solid angle leads to the single differential cross section: dσ = dε



d2 σ 2π dp2 = dε dp2 p1 p2

qmax q dq qmin

d2 σ . dε dp2

(2.132)

It was shown [80] that the excitations with large angular momenta l have a singleparticle rather than a collective nature. It follows from the fact that with the increase of l the wavelength of the plasmon excitation becomes smaller than the characteristic wavelength of the delocalized electrons in the fullerene. In the case of C60 , the estimates show [80] that the excitations with l > 3 are formed by single-electron transitions rather than by the collective excitations. Hence, only terms corresponding to the dipole (l = 1), quadrupole (l = 2) and octupole (l = 3) plasmon excitations should be accounted for in the sum over l in Eqs. (2.130) and (2.131). The theory presented relies on the number of multipole terms to be accounted for (v) (lmax = 3 for C60 ) and the widths of the plasmon resonances,  (s) jl and l , which are not just the fitting parameters of the model but the real physical quantities. A precise calculation of the widths can be performed by analyzing the decay of the collective excitation mode into the incoherent sum of single-electron excitations. This process should be considered within the quantum-mechanical framework [85] and cannot be treated within the classical physics framework, as the PRA does. In Ref. [85], such an analysis was made to obtain the values of the surface and volume plasmon widths for a Na40 cluster. This analysis is discussed below in Sect. 2.7.4. Nevertheless, the widths of the plasmon excitations in the fullerene can be estimated using the relation similar to the Landau damping of plasmon oscillations. Such an estimate results in  (s) jl ∼ lv F /R, where v F is the velocity of the fullerene electrons on the Fermi surface. A similar estimate was successfully applied to the investigation of collective excitations in metal clusters [85] (see also Sect. 2.7.4). The formation of the volume plasmon in the electron impact ionization of metal clusters and carbon fullerenes was revealed in Refs. [85–87]. The model accounting for the contribution of different plasmon modes was successfully utilized to describe

64

A. V. Verkhovtsev and A. V. Solov’yov

the experimentally observed variation of the electron energy loss spectra of C60 in collision with fast electrons [86, 87]. Figure 2.12 shows the the differential cross section d2 σ/dε dp2 as a function of the transferred energy ε calculated for the electron-Na40 collision [82]. Panel A shows that at the scattering angle θ = 1◦ the dipole plasmon excitation (l = 1) dominates in the electron energy loss spectrum while the quadrupole excitation (l = 2) provides a relatively small contribution. The contributions of the monopole (l = 0) and all higher multipole excitations (not shown in Fig. 2.12A) are almost negligible at this scattering angle. Figure 2.12B demonstrates that at θ = 6◦ the quadrupole excitation becomes the leading excitation in the electron energy loss spectrum, shifting the maximum of the spectrum towards higher energies and also changing the profile of the resonance. The dipole and the octupole (l = 3) excitations also provide considerable contributions in a wide range of transferred energies broadening the spectrum. The dominance of quadrupole excitation is not as large as for the dipole excitation at θ = 1◦ . At θ = 9◦ (see Fig. 2.12C) the picture becomes more complex. In this case the octupole excitations provide the dominating contribution to the spectrum in the vicinity of the maximum of the energy loss spectrum at ε ≈ 4 eV. Besides this region, the dipole, quadrupole and even excitations with angular momentum l = 4 give comparable contributions to the energy loss spectrum and form a rather broad structure. The monopole excitation and the excitations with angular momentum l = 5 and higher (not shown in Fig. 2.12C) are almost negligible. With increasing scattering angle, excitations with l = 4 become more important. However, the corresponding spectrum does not possess a resonance behavior because it is mainly formed by single-electron transitions. Comparison of the results derived from the RPAE calculations (see solid lines in Fig. 2.12) with those obtained in the PRA (dashed lines) shows that, despite the simplicity, the plasmon resonance treatment is in quite good agreement with the consistent many-body quantum calculation. The main discrepancy between the two approaches arises from the single-particle transitions omitted in the PRA but taken into account in the RPAE calculation. These transitions bring some structure to the final energy loss spectra manifesting themselves over the smooth resonance behavior which is reproduced by the PRA (see also the case of photoionization of C60 described above in Sect. 2.7.3). At larger scattering angles plasmons with larger angular momenta can be excited. However, excitations with large enough angular momenta occur due to single-particle transitions rather than due to collective excitations. Therefore, the agreement between the classical PRA and the quantum-based RPAE is better at small angles. Figure 2.13 represents the electron energy loss spectra of C60 calculated within the PRA (see Eqs. (2.130) and (2.131)). The calculations were performed for the scattering angles θ = 3◦ , 5◦ , 7◦ and 9◦ . The calculated curves are compared to the results of experimental measurements of the inelastic scattering of fast (1 keV) electrons on C60 [86, 87]. For the sake of convenience, both the experimental and the theoretical curves are normalized to unity at the point of maximum. The values of the plasmon frequencies and the corresponding widths for all three collective

2 Atomic and Molecular Physics Methods for Nanosystems

65

Fig. 2.12 The differential cross section d2 σ/dε dp2 as a function of the transferred energy ε calculated for the electron–Na40 collision [82]. The impact electron energy ε = 50 eV. The electron scattering angle is θ = 1◦ (panel A), θ = 6◦ (panel B) and θ = 9◦ (panel C). Solid lines represent the results of the RPAE calculation with the Hartree–Fock jellium model basis wave functions. Thick solid (black) line is the total EELS. Thin solid (gray) lines marked with the angular momentum number represent various multipole contributions to the EELS. Dashed line shows the EELS calculated within the plasmon resonance approximation. Redrawn from data presented in Ref. [82]

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A. V. Verkhovtsev and A. V. Solov’yov

Fig. 2.13 Comparison of the electron energy loss spectra, calculated within the plasmon resonance approximation, with the experimental spectra measured for the incident energy range 1002–1050 eV and for the scattering angles θ = 3◦ . . . 9◦ . The symmetric and antisymmetric modes of the surface plasmon are shown by dashed red and the dash-dotted blue lines, respectively; the volume plasmon contribution is shown by the double-dotted purple line. The total cross section is shown by the thick green line. Open squares represent the experimental data [86, 87]. For the sake of convenience, both the experimental and the theoretical curves are normalized to 1 at the point of maximum. The energy scale is the same for all panels. Reproduced from Ref. [113] Table 2.3 Peak positions and the widths of the two surface plasmon modes and of the volume plasmon used in the calculations reported in Refs. [86, 87]. All values are given in eV l=0 l=1 l=2 l=3 ω1l

0

19.0

25.5

30.5

ω2l

0 37.1

11.4 33.2

15.3 31.0

18.3 29.5

(s) 2l ωp

37.1 37.1

33.2

31.0

29.5

l

26.0–48.3

(s) 1l

(v)

excitations are summarized in Table 2.3. The widths of the volume plasmon were varied to obtain a better agreement with the experimental data. The dashed red and dash-dotted blue curves in Fig. 2.13 represent the symmetric and antisymmetric modes of the surface plasmon respectively; the double-dotted

2 Atomic and Molecular Physics Methods for Nanosystems

67

purple line shows the contribution of the volume plasmon. The sum of the three collective excitations is shown by the thick green line. Open squares represent the experimental data [86, 87]. At the small scattering angle, θ = 3◦ , the symmetric mode of the surface plasmon dominates the cross section. A similar behavior is observed in the photoionization process (see Fig. 2.8). In fact, in the case of the uniform external field (q → 0), there is no volume plasmon excitation in the system (see Sect. 2.6) and the symmetric plasmon mode exceeds significantly the antisymmetric mode. Non-uniformity of the external field causes the formation of the volume plasmon whose contribution to the cross section is insignificant when the scattering angle is small. With increasing the scattering angle (θ = 5◦ and 7◦ ), the symmetric mode of the surface plasmon becomes less relevant and the antisymmetric mode more prominent. At the larger angle (θ = 9◦ ), the symmetric surface plasmon almost does not contribute to the cross section while the volume plasmon becomes dominant. The contribution of the antisymmetric surface and the volume plasmons can explain the origin of the two peaks in the energy loss range from 20 to 30 eV at the scattering angle θ = 7◦ . The peak position of each plasmon resonance (2.131) and the resulting cross section d2 σ/dε2 dp2 is influenced by the manifestation of the diffraction effects. In Ref. [83], it was shown that plasmon modes with different angular momenta provide dominating contributions to the differential cross section at different scattering angles, which leads to the significant angular dependence of the energy loss spectrum. This phenomenon was described in terms of the electron diffraction at the fullerene edge [83]. As it is seen from Eq. (2.131), the resonance peak of each plasmon is defined not only by the plasmon frequencies ω p , ω1l and ω2l , but also by the multipolar diffraction factors Vl (q), S1l (q) and S2l (q) which depend on the transferred momentum q. In the limiting case of an infinitely thin layer, this dependence is described by the spherical Bessel functions jl2 (q R) which oscillate with q and, thus, give suppression and enhancement of the partial plasmon modes at certain angles [83]. The incident energy of the projectile does not influence on this behavior and defines only the absolute value of the cross section.

2.7.4 Electron Energy Loss Spectra of Metal Clusters: Contribution of the Surface and Volume Plasmon Excitations Damping of the plasmon oscillations is related to the decay of the collective electron excitations to the single-particle ones similar to the mechanism of Landau damping in infinite electron gas. Frequencies of the surface plasmon excitations in neutral metal clusters lie in the vicinity of the ionization threshold. For instance, in small sodium clusters, they are below the ionization potential, and single-particle excitations in the vicinity of the surface plasmon resonance have therefore the discrete spectrum. In this case, the width of a surface plasmon excitation caused by the Landau damping should

68

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be treated as the width of the distribution of the oscillator strengths in the vicinity of the resonance. The problem of the formation of the surface plasmon resonance widths in clusters was studied in Refs. [85, 114–116]. The resonance frequencies of volume plasmon excitations in metal clusters are typically located above the ionization threshold. This means that the volume plasmon excitations are quasi-stable and have the real channel of the Landau damping leading to the ionization of the cluster [85]. Thus, the process of inelastic scattering in the region of transferred energies above the ionization threshold can be described as follows. The projectile particle induces the oscillations of the electron density in the cluster; in turn, they cause oscillations of the electric field which result in the ionization of the cluster. A similar scenario takes place with damping of the surface plasmon resonances in fullerenes [117], which also decay via the autoionization channel. The differential cross section of the electron inelastic scattering on metal clusters obtained in the PRA with accounting for both surface and volume excitations [85] reads as: ωl2 ω l(s) 4Rp2  d2 σ 2 2 = (2l + 1) j (q R) $ %2 l dε2 dp2 πq 4 p1 l (ω2 − ω2 )2 + ω2  (s) l

(2.133)

l

ωp2 ω l(v) 2R 3 p2  (2l + 1) + $ %2 πq 2 p1 l (ω2 − ωp 2 )2 + ω2 l(v)   2 jl+1 (q R) jl (q R) , × jl2 (q R) − jl+1 (q R) jl−1 (q R) − qR ' where ωp = 3Ne /R 3 is the volume plasmon resonance frequency and ωl = √ l/(2l + 1)ωp is the frequency of a surface plasmon excitation with the angular momentum l, l(v) and l(s) are the corresponding widths. The cross section (2.133) is similar to the expression obtained in Ref. [118] for electron scattering on small metal particles by means of classical electrodynamics. According to Ref. [85], the width of the surface plasmon resonance in the PRA is equal to: l(s) =

4π ωl 



(s)



2

ψμ ϕl (r) ψν δ(ωl − εμ + εν ) , (2l + 1)R ν,μ

(2.134)

 -  where ϕl(s) (r) = m (r/R)l θ (R − r ) + (R/r )(l+1) θ (r − R) Ylm (n). Note that the same expression was obtained in earlier studies [114, 115] using other methods. Evaluation of the expression (2.134) for sufficiently large clusters leads to the wellknown result for the Landau damping of the surface plasmon oscillations, l(s) = 3l vF /R, where vF is the velocity of the cluster electrons on the Fermi surface [115]. In the PRA one can also determine the autoionization width of the volume plasmon resonance [85], which is equal to:

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l(v) =

- 



(v)



2

ψμ ϕl (r) ψν δ(ωp − εμ + εν )dμ ν

8π ωp q 2 R 3 jl2 (q R) − jl+1 (q R) jl−1 (q R) − 2

2 j (q R) jl (q R) q R l+1

69

,

(2.135)

 -  where ϕl(v) (r) = m jl (qr ) − jl (q R)(r/R)l θ (R − r )Ylm (n). The summation is performed over the occupied single-electron states ν and the integration is performed over the electronic states μ of the continuous spectrum. The projectile particle excites simultaneously numerous modes of the volume plasmon. The sum of the potentials of all the modes gives the resulting potential ϕl(v) (r). It is essential that all normal modes of the volume plasmon have the same resonance frequency ωp , but the excitation probability for these modes depends on the kinematics of collision. This leads to the dependence of the volume plasmon potential ϕl(v) (r) upon the transferred momentum. The oscillations of the volume plasmon potential result in the ionization of the cluster, which probability and the volume plasmon resonance width depend on transferred momentum q. However, the numerical analysis [85] showed that the dependence of l(v) on q is rather weak in the region of q  1, where collective electron oscillations mainly take place. Therefore, the volume plasmon resonance width with the given l can be approximated by the limiting value following from Eq. (2.135) at q = 0: 

 2 π 2 ωp 



(v)

ψμ ϕl |q=0 (r) ψν δ(ωp − εμ + εν )dμ . R ν (2.136) Figure 2.14 shows the dependence of the autoionization width l(v) on the transferred momentum q for the volume plasmon modes, which provide a significant contribution to the EELS. The width of the dipole, quadrupole and octupole volume plasmon resonances has been calculated according to Eq. (2.135). The transferred momentum q plays the role of the wave vector for the volume plasmon excitations. All three plasmon modes have a similar dependence of l(v) upon q. The width grows slowly in the region of small q and it decreases rapidly at larger q. In the latter region, the probability of volume plasmon excitation by the incoming electron is correspondingly reduced. Note that the wavelength of a collective electron oscillation should be larger than the inter-electronic distance in the cluster, i.e., plasmon wave vector should be smaller than the Fermi momentum of cluster electrons. In the region q < 0.5, where the latter condition is fulfilled, the dependence of l(v) upon q is rather weak, and the resonance width can be approximated by the following values: 1(v)  0.5ωp , 2(v)  0.3ωp , and 3(v)  0.23ωp . Contrary to surface plasmons, the autoionization width of a volume plasmon decreases with the growth of the angular momentum. Figure 2.15 shows the EELS of a sodium Na40 cluster in collision with a 50-eV electron at the scattering angle θ = 9◦ . The figure illustrates the region of transferred energy above the ionization potential, ω > 3.3 eV, where volume plasmon modes become significant. Solid curves illustrate the spectrum calculated using the random phase approximation with exchange (RPAE) method [38, 81]—an ab initio method l(v) = (2l + 5)

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(v)

Fig. 2.14 Autoionization width l of the dipole (l = 1), quadrupole (l = 2), and the octupole (l = 3) volume plasmon excitations as a function of transferred momentum q. Redrawn from data presented in Ref. [85]

Fig. 2.15 Differential cross section d2 σ/dε2 dp2 as a function of the transferred energy ω calculated for the collision of a 50 eV electron with a Na40 cluster for the scattering angle θ = 9◦ [85]. Solid black and gray lines represent the RPAE results (see the text for further details). Contributions of the surface and the volume plasmons calculated in the PRA (2.133) are shown by dashed and dash-dotted lines, respectively. A solid green line represents the sum of these two contributions. Redrawn from data presented in Ref. [85]

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which takes into account many-electron correlations in a many-particle system. In Fig. 2.15, the thick curve corresponds to the total EELS calculated with RPAE, while thin curves show various partial contributions corresponding to different angular momenta (l = 0 to 4). These curves are marked by numbers. The partial contribution to the EELS with l < 3 has the broad maximum in the vicinity of ω  5.1 eV. Comparison of the EELS calculated within the ab initio RPAE approach and the PRA confirms the idea that the peculiarity in the EELS in the vicinity of ω ∼ 5 eV is connected with the volume plasmon excitation (see the dash-dotted curve). Figure 2.15 demonstrates that collective excitations provide dominating contribution to the total EELS determining its pattern.

2.7.5 Relaxation of Electronic Excitations in Metal Clusters As described above, the plasmon resonances in metal clusters may lie below the ionization thresholds, i.e., in the region of the discrete spectrum of electron excitations [50]. This fact rises an interesting physical problem about the eigenwidths of these electronic excitations which possess large oscillator strengths and form the plasmon resonances. Knowledge of these widths is necessary for the complete description of the electron energy loss spectra, electron attachment, polarization bremsstrahlung, and photoabsorption cross sections in the vicinity of the plasmon resonances and the description of their dependence on the cluster temperature. The dependence of the plasmon resonance photoabsorption patterns of metal clusters on temperature has been studied experimentally in Ref. [119]. In metal clusters, the origination of the electron excitation widths is mainly connected with the dynamics of the ionic cluster core [59, 60, 120–124]. Let us focus on the influence of the dynamics of ions on the motion of delocalized electrons in metal clusters and discuss it on the basis of the dynamic jellium model suggested in Ref. [59] and developed further in Refs. [58, 60]. This model generalizes the static jellium model [30, 44, 45] which treats the ionic background of an atomic cluster as frozen by taking into account vibrations of the ionic background near the equilibrium point. The dynamic jellium model treats simultaneously the vibration modes of the ionic jellium background, the quantized electron motion, and the interaction between the electronic and the ionic subsystems. In Ref. [59], the dynamic jellium model was applied for a consistent description of the physical phenomena arising from the oscillatory dynamics of ions. An important example of the effect, originating from the interaction of the ionic vibrations with delocalized electrons, is the broadening of electron excitation lines. The interest in the problem of the electron excitation linewidths formation in metal clusters was stimulated by numerous experimental data on photoabsorption spectra, most of which were addressed to the region of dipole plasmon resonances [35, 48, 75]. The dynamic jellium model [59] allows one to calculate widths of the electron excitations in metal clusters caused by the dynamics of ions and their temperature

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dependence accounting for the two mechanisms of the electron excitation line broadening, namely, an adiabatic and a non-adiabatic (or dynamic) ones. The adiabatic mechanism is connected with the averaging of the electron excitation spectrum over the temperature fluctuations of the ionic background in a cluster. This phenomenon has also been studied in a number of papers [59, 60, 120–124]. The adiabatic linewidth is equal to ( =

   4 ln 2

ˆ cth

Vnn . m 2kB T

(2.137)

Here m and  are the mass and frequency corresponding to the generalized oscillatory

considered, T is the cluster temperature, kB is the Bolzmann constant, and

mode

ˆ

Vnn is the matrix element of the electron-phonon coupling, calculated for surface and volume cluster vibration modes in Ref. [60]. The mechanism of non-adiabatic electron excitation line broadening has been considered for the first time in Refs. [59, 60]. This mechanism originates from the real multiphonon transitions between the excited electron energy levels. Therefore, the dynamic linewidths characterize the real lifetimes of the electronic excitations in a cluster. According to Ref. [59], the probability of a multiphonon transition from an excited cluster state with electronic and phononic quantum numbers n and N to all possible states (n  , N  ) is equal to

2(ϕ  −ϕ )

Hn2 n 2π 2

e n n .

|A| =  =  v(q0 )(Vn n − Vnn )

(2.138)

Here Hn n is the half-distance between the electron energy levels εn (q) and εn (q) in the tangent point, v(q0 ) =

.    l 2 − 2S(2N − l − 1) + S 2 /2S

(2.139)

is the ion velocity in the tangent point, which is expressed via the number of emitted phonons l = N  − N , where N  and N are the phonon numbers, and the parameter

2

S = Vˆnn − Vˆn n /2m3 ; ϕn , ϕn are the phases of ionic motion, arising from the distance between the turning points and the tangent point, being equal to   ' ' Z n + Z n2 − 2N − 1 Z n Z n2 − 2N − 1 2N + 1 + ln ϕn = , 2 4 2N + 1

(2.140)

√ where Z n = (l − S)/ 2S.√The expression for ϕn is the same, but the parameter Z n is equal to Z n = (l + S)/ 2S.

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The adiabatic broadening mechanism explains the temperature dependence of the photoabsorption spectra in the vicinity of the plasmon resonance via the coupling of the dipole excitations in a cluster with the quadrupole deformation of the cluster surface. The photoabsorption spectra were calculated within the framework of deformed jellium model using either the plasmon pole approximation [120, 121] or the local density approximation [122–126]. In Ref. [124], the octupole deformation of the cluster surface was taken into account. It was demonstrated that the octupole deformation increases the Landau damping as a result of breaking the selection rule, leading to a mixture of the dipole and the quadrupole electronic excitations. Via this mechanism, the octupole deformations of the cluster surface provide the dominating contribution to the thermal broadening of electron excitation lines in small metal clusters. In Refs. [59, 60] both the adiabatic and non-adiabatic linewidths of electron excitations in the vicinity of the plasmon resonance caused by coupling of electrons with various ionic vibration modes have been calculated. The behavior of the adiabatic and non-adiabatic linewidths as a function of temperature is shown in Fig. 2.16. The non-adiabatic linewidths characterize the real lifetimes of cluster electron excitations. Naturally, the non-adiabatic widths turn out to be much smaller than the adiabatic ones due to the slow motion of ions in the cluster. However, the adiabatic linewidths do not completely mask the non-adiabatic ones because the two types of widths manifest themselves differently. The adiabatic broadening determines the pattern of the photoabsorption spectrum in the linear regime. The non-adiabatic linewidths are important for the processes, in which the real lifetime of electron excitations and the electron-ion energy transfer are essential. The information about the non-adiabatic electron-phonon interactions in clusters is necessary for the description of electron inelastic scattering on clusters [80–82, 85], including the processes of electron attachment [29, 127, 128], the non-linear photoabsorption and bremsstrahlung [129–132], the problem of cluster stability and fission. The non-adiabatic linewidths determined by the probability of multiphonon transitions are also essential for the treatment of the relaxation of electronic excitations in clusters and the energy transfer from the excited electrons to ions, which occurs after the impact- or photoexcitation of the cluster. The role of the volume and the surface vibrations of the ionic cluster core in the formation of the electron excitation linewidths was investigated in Refs. [59, 60]. It was demonstrated that the volume and surface vibrations provide comparable contributions to the adiabatic linewidths, but the surface vibrations are much more essential for the non-adiabatic multiphonon transitions than the volume ones.

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Fig. 2.16 Panel A: temperature dependence of the adiabatic linewidth , calculated according to Eq. (2.137) for the dipole electron excitation with the energy ωn = 3.013 eV in the Na40 cluster. Dashed curves labeled as 1, 2, and 3 show the adiabatic width corresponding to the electron coupling with the three first volume vibration modes, respectively. Dashed-dotted curve shows the adiabatic width arising from the electron coupling with surface vibrations of the cluster. Solid curve shows the total adiabatic linewidth. Panel B: temperature dependence of the non-adiabatic width calculated according to Eq. (2.138) for the dipole excitation with the energy ωn = 3.013 eV in the Na40 cluster. Redrawn from data presented in Refs. [59, 60]

2.7.6 Diffraction of Fast Electrons on Atomic Clusters and Fullerenes The phenomenon of elastic scattering of fast electrons on metal clusters and fullerenes appears because the ionic density distribution in a cluster is typically characterized by a rigid border. The presence of a surface in a cluster results in the specific oscillatory behavior of the electron elastic scattering cross sections, which can be interpreted in terms of electron diffraction by the cluster surface [80, 83]. The detailed theoretical treatment of the diffraction phenomena arising in electron scattering on

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metal clusters and fullerenes was given in Refs. [80–82]. Experimentally, diffraction in electron elastic scattering cross sections on gas-phase fullerenes was observed for the first time in Ref. [83]. Let us explain the physical nature of the diffraction phenomena arising in elastic electron–cluster scattering. For the sake of simplicity, we consider atomic clusters as spherically symmetric systems with a uniform electron density distribution; this model is well applicable, e.g., to highly symmetric “magic”-number metal clusters or carbon fullerenes. The cross section of elastic scattering of a fast electron on a cluster in the first Born approximation (see, e.g., Ref. [6]) reads as 4 dσ = 4 F(q)2 . dp2 q

(2.141)

Here, F(q) is the form factor of the cluster, q = |p2 − p1 | is the momentum transfer, with p1 , p2 being the momenta of the electron in the initial and the final state, respectively, and dp2 denotes the differentiation over the solid angle of the scattered electron. The magnitude of q 2 is related to the scattering angle θ = p 1 p2  1 rad via: (2.142) q 2 = p12 + p22 − 2 p1 p2 cos θ = 2 p12 (1 − cos θ ) ≈ p12 θ 2 . The form factor of the target, F(q), can be expressed as a product of the form factor of the atomic concentration, n(q), and the form factor of a single atom, FA (q): F(q) = FA (q)



eiqr j = FA (q) n(q),

(2.143)

j

where the summation is performed over all coordinates, r j , of all atoms in the cluster. The applicability of this approximation has been examined in Ref. [83] for metal clusters and fullerenes. The form factor of the atomic concentration, n(q), depends on the geometry of the cluster. In the case of the metal cluster, assuming a homogeneous distribution of atoms in the volume of the cluster of the radius R, one derives   j1 (q R) sin (q R) cos (q R) = 3N , (2.144) − n(q) = 3N (q R)3 (q R)2 qR where j1 (q R) is the spherical Bessel function of the first order and N is the number of atoms in the cluster [80]. The simplest approximation for the description of a fullerene is to assume that carbon atoms are uniformly distributed on the surface of a sphere of the radius R. In this case, one derives n(q) = N

sin(q R) . qR

(2.145)

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Fig. 2.17 Experimental (symbols) and theoretical (solid curve) angular dependencies of the differential elastic scattering cross section in collision of a 809 eV electron with the C60 molecule [83]. Symbols correspond to the two independent sets of measurements. Dashed curve is the cross section for the mixture containing 60% of C60 and 40% of equivalent isolated carbon atoms. Redrawn from data presented in Ref. [83]

This form factor oscillates with the period q = 2π/R  1. These oscillations form the diffraction pattern of the differential cross section (2.141) which possesses a series of diffraction maxima and minima whose positions are mainly determined by the radius of the target. Figure 2.17 presents the dependence of the cross section dσ/dp2 on the scattering angle θ for elastic collision of a 809-eV electron with the C60 fullerene [83]. The figure shows that the cross section possesses a series of diffraction maxima and minima. Experimental data points obtained in Ref. [83] in the two sets of measurements are illustrated by open and closed circles. The cross section dependence obtained theoretically is shown by a solid curve. Experimental data have been normalized to the theoretical cross section at the second diffraction maximum (θ = 5◦ ). Figure 2.17 shows quite a good agreement between the experimental and the theoretical results in a position of the first and the second maxima. The entire pattern of the differential cross section obtained theoretically is very similar to that measured in an experiment. In the vicinity of the diffraction maxima at θ < 10◦ the cross section greatly exceeds the elastic scattering cross section on the equivalent number of isolated atoms because of the coherent interaction of the projectile electron with the fullerene sphere. In the region θ > 10◦ , where q > 1, the projectile electron scatters on individual carbon atoms of the fullerene rather than on the entire molecule. Therefore diffraction features of the cross section in the region θ < 10◦ are much more pronounced than in the region θ > 10◦ . In the region θ < 10◦ , where q < 1, the theoretical cross section has zeros while the experimental one does not. The presence of zeros at q ≈ π k/R < 1 where k

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is an integer, in the theoretical curve is the consequence of the coherent scattering of electrons on the fullerene sphere.3 However, in the experiment, zeros in the cross section can disappear because of the presence of carbon atoms or some other impurities in the gas cell. Figure 2.17 also shows the differential cross section for the mixture containing 60% of C60 and 40% of isolated carbon atoms (the dashed curve). The differential electron elastic scattering cross section on single carbon atoms does not have diffraction oscillations and thus it forms the smooth background removing zeroes in the angular dependence of the cross section.

2.7.7 Polarization Effects in Low-Energy Electron–Cluster Collisions The previous section has been devoted mainly to the collisions of fast electrons with metal clusters and fullerenes. In the case of low-energy electron (LEE)–cluster collisions, i.e., when the velocity of the projectile is lower or comparable with characteristic velocities of the delocalized cluster electrons, polarization effects come into play [38]. In Ref. [80] it was demonstrated, on the basis of the Born theory of electron–cluster collisions, that electron collisions with metal clusters in the region of collision energies below 3–5 eV should be treated as slow, while for fullerenes the region of collisions energies extends up to 30 eV. In LEE–cluster collisions the role of the cluster polarization and exchange– correlation effects increases dramatically. The polarization potential of electron– cluster interaction sometimes changes completely the qualitative picture of the collision. For instance, this takes place when considering LEE elastic scattering on metal clusters. In this case, the resonant structures can appear in the energy dependence of the electron elastic scattering cross section due to the presence of the bound or quasi-bound states in the system [133, 134]. During the past decades, considerable attention has been devoted, both experimentally and theoretically, to the problem of electron attachment to metal clusters and fullerenes. The electron attachment process is one of the mechanisms which leads to the negative cluster ion formation in gases and plasmas and thus it attracts the interest of numerous researchers. Low energy electron–fullerene scattering was studied in Refs. [135–138]. For metal clusters, the electron attachment problem has been the subject of the intensive experimental [139–143] and theoretical [29, 127–129, 144] investigations. Below, this problem is discussed in more detail. The very simple picture of attachment is described in many textbooks (see, e.g., Ref. [145]). Let us assume that there exists a Langevin attractive potential of the form In the region θ > 10◦ , where q > 1, the process of elastic scattering on the fullerene shell with the subsequent excitation of surface multipole plasmons becomes dominating. This process is described by the formulas of the second Born approximation which was used to correct the calculated cross section at large values of transferred momentum.

3

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V (r ) = −

α 2r 4

(2.146)

outside the cluster radius. The constant α is the static polarizability of the cluster. One can then show that there is an orbiting cross section,  σ =

2π 2 α ε

1/2 ,

(2.147)

which sets an upper limit bound to the attachment cross section (the so-called Langevin limit). Here, ε is the kinetic energy of the projectile electron. This simple treatment, if valid, would explain the behavior of the cross section in the vicinity of the threshold. It is known that metal clusters possess a high polarizability (see, e.g., Ref. [48]); hence, large capture cross sections are anticipated. However, simple attempts to account for attachment by using the static polarizability α are not in accordance with observations [140]. The great weakness of the Langevin model is the treatment of α as an approximate constant. In fact, it possesses a complicated energy dependence due to the dynamical polarizability of the metallic cluster. The possibility of resonances in the capture cross section was considered theoretically in Refs. [127, 129]. It was demonstrated that low-energy electrons can excite a collective plasmon resonance within the metal cluster in the electron attachment process as a result of a strong dipole deformation of the charge density of the cluster. Later this idea was commented on in the context of the measurements performed in Ref. [139], although no clear evidence of the resonant behavior was found. The total inelastic scattering cross sections measured in Ref. [139] included attachment as only one of several possible contributing channels. The resonant electron attachment mechanism was called in Refs. [127, 129] a “polarizational capture” in analogy with the similar mechanism known in the theory of bremsstrahlung (see, e.g., Ref. [146] and references therein). An important consequence of the polarization mechanism is that the low-energy electron falls into the target and the probability of this process is enhanced. Since the process as a whole is resonant, the enhancement is greatest for energies rather close to the plasmon resonance in the dynamic polarizability of the cluster. In the attachment process the electron loses its excess energy. Emission of the photon via the polarizational mechanism is one of the possible channels of the energy loss [127–132, 147]. The energy of the electron can also be transferred to the excitations of the ionic background of the cluster [60], which may lead to increase of its vibrations and final fragmentation. In spite of the significant physical difference between various channels of the electron energy loss, they have one important common feature: the energy is transferred to the system via the plasmon excitation. Therefore, calculating the total electron attachment cross section including all possible channels of the electron energy loss in the system, one obtains [29, 128] a qualitatively similar dependence of the cross section as it was obtained initially for the radiative channel of electron energy loss [127].

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In Ref. [127] the attachment cross section has been calculated within the jellium model in a scheme which holds best if the kinetic energy of the electrons is somewhat higher than the energy of the resonance. Also, it was assumed that the attached ion is created in the ground state. As a useful step in simplifying the calculation, a Kramers–Kronig transformation procedure was introduced to compute the polarizability from the absorption coefficient, thereby circumventing the need for full ab initio calculations. Within this approximate scheme, it was found [127, 129] that the resonant attachment cross section dominates over the non-resonant one by a factor of about 103 –104 near resonance, and is therefore a very significant pathway for electrons of low enough energy. In Refs. [29, 128], the earlier theoretical work on attachment was extended by including the following improvements: (i) all possible channels of the electron attachment were included and the total cross section of the process was calculated rather than analyzing a particular single channel; no assumption that the system can only return to its ground state had been made; (ii) theoretical approximation was used to treat electron energies not only in the resonance region, but also throughout the range of interest; (iii) an RPAE calculation of the dynamical polarizability was performed along with the corresponding electron attachment cross sections on the basis of the consistent many-body theory with the use of the Hartree–Fock jellium model wave function; (iv) calculations were performed for both neutral and charged cluster targets; (v) the polarization effect on the incoming particle as well as collective excitations of different multipolarity in the target electron system were taken into account; (vi) Dyson’s equation was used to reduce the problem of the interaction of an extra electron with a many-electron target system to a quasi-single-particle problem in a similar way as it was done for negative atomic ions calculations [28]. An example of such a calculation is shown in Fig. 2.18A. This plot represents the total and partial electron capture cross sections calculated for neutral potassium K8 cluster. The inset demonstrates the photoabsorption spectrum of K8 . In Ref. [29] this calculation was performed in various approximations outlined above. It was found that the resonance pattern in the electron capture cross section for the K8 cluster turns out to be similar in various approaches, although for some other sodium and potassium clusters it is more sensitive to the approximations made [29]. The plasmon resonance in the electron capture cross section is shifted on the value of energy of the attached electron as compared to the photoabsorption case shown in the inset. Experimental evidence for the resonance enhancement of the cross sections of electron attachment process has been obtained in Ref. [141]. The experimental points from the cited paper are shown in Fig. 2.18B. Comparison of the two panels of Fig. 2.18 indicates the reasonable agreement between the predictions of theory and the experimental results. However, more precise measurements would be desirable to resolve the more detailed structures in the electron attachment cross sections. The plasmon resonance enhanced mechanism of electron attachment considered above is typical for metal clusters rather than for carbon fullerenes. Although fullerenes have many similarities in the properties with metal clusters and also possess the plasmon resonances, the energies of these resonances are much higher

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Fig. 2.18 Panel A: Total and partial electron capture cross sections in the vicinity of the plasmon resonance calculated for a neutral potassium K8 cluster [29]. The inset shows the photoabsorption spectrum of K8 . Panel B: Experimental evidence for the resonance enhancement of the electron attachment cross section [141]. Redrawn from data presented in Ref. [38]

(∼7 and ∼20 eV) and thus cannot be reached at low kinetic energies of the projectile electron.

2.7.8 Dynamical Screening Effects in Endohedral Fullerenes Another area of vivid interest in the atomic cluster community during the last several decades has been related to endohedral fullerenes—systems made of an atom or a small molecule encapsulated inside a fullerene cage (see, e.g., reviews [148–150] and references therein). Great scientific interest has been driven by numerous possible applications [149] of atoms encapsulated inside spherical or near-spherical fullerenes and the fundamental aspects of an atom confined inside a nanometer-sized cavity [151]. The confinement of atoms inside fullerene cages has been addressed by numerous theoretical approaches (see, e.g., Refs. [93, 106, 152–156]), while the corresponding experimental studies have been relatively scarce [157–161], mainly due to the technical difficulties in producing such targets in sizable quantities and with high purity. The most definitive experimental measurements to directly access the physics of

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confined atoms is the observation of their response to irradiation by photons. A number of recent experimental studies [157, 160] focused on the photoionization of endohedral fullerene ions have considered the mutual influence of the fullerene cage and the encapsulated species on the photoionization process. The most recent review of these studies can be found in Ref. [162]. The fundamental interest in studying endohedral fullerenes is, on one hand, the opportunity to probe spectra of a novel type in which atomic excitations interact with cavity resonances [163], and, on the other hand, the fact that the confined atom is screened by a fullerene shell, whose properties resemble those of a hollow metal sphere [152]. In essence, the fullerene shell behaves like a “Faraday cage” of very small dimensions, inside which, in the static limit, there is no possibility for external fields to penetrate, and thus the whole enclosed volume is at the same potential. Numerous theoretical studies (e.g., [106, 154]) to name a few) have predicted the existence of so-called “confinement resonances” in the photoionization spectra of endohedral fullerenes. Experimentally, such resonances were detected for the first time in the photoionization spectra of Xe@C60 [157, 158]. Confinement resonances appear as a result of interference between a direct wave of the photoelectron escaping the atom and the waves due to scattering from the atoms of the fullerene cage. Depending on the photoelectron momentum the interference can be constructive or destructive. Thus, the spectrum of the encaged atom acquires additional oscillations as compared to the free atom. It was noted [106] that the cage-induced oscillations have the same nature as the extended x-ray absorption fine-structure (EXAFS) for solid-state systems. The role of dynamical screening in the photoionization or photoabsorption processes of an atom confined within a fullerene shell was elucidated in Ref. [152]. When an endohedral fullerene is exposed to an external electromagnetic field, the fullerene shell dynamically screens the confined atom. The atom experiences a field that is enhanced or suppressed depending on the frequency of the light, ω. The result is that, for the same external field, the photoabsorption rate of the confined atom differs from that of the free atom. A dynamical screening factor F ≡ F (ω) can be defined to relate the photoabsorption cross sections of these two atoms: F =

σconf . σfree

(2.148)

The screening is strongly dependent on the frequency of the light (hence the term dynamical screening) and leads to an enhancement of the amplitude of the electromagnetic wave within a specific range of frequencies. This range is determined by the characteristics of the fullerene plasmon (or, more generally, by the plasmon of the confining system). The problem of dynamical screening of an atom confined at the center of a fullerene cage of finite thickness (see Fig. 2.19A) was studied theoretically in Ref. [93]. In a later study [164] the method was generalized for an arbitrary position of the confined atom inside the fullerene (see Fig. 2.19B). Similar to other case studies presented in this chapter, a fullerene has been treated as a spherical shell of

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Fig. 2.19 Panel A: The atom, indicated by the solid circle, is confined at the center of a fullerene of average radius R and thickness R. The inner and the outer radii of the fullerene are given by R1 and R2 respectively. The system is exposed to an external field Eext . Panel B: The confined atom is located at distance ρ from the center of the fullerene. The field at the position of the atom is given by Etot (ρ, ω). Reproduced from Ref. [93] with permission from IOP Publishing and Ref. [164] with permission from American Physical Society

thickness R with a dielectric function  ≡ (ω). Following the Drude model, the dielectric function  is expressed as: (ω) = 1 −

ωp2 ω2

,

(2.149)

where the plasma frequency ωp is described by the total number of delocalized electrons Ne (4 valence electrons from each carbon atom) and the volume of the spherical shell. The point-like atom located at r = 0 is characterized by the polarizability αa ≡ αa (ω). This system is exposed to a monochromatic electromagnetic wave. Using the dipole approximation and neglecting the magnetic part, this is treated as an external uniform electric field Eext (t) = E0 eiωt . An object’s photoabsorption rate is proportional to its photoabsorption cross section. Therefore the dynamical screening factor F (ω) can be calculated by comparing the absorption rate of the confined atom to that of the free atom. In the dipole limit, an object exposed to an electromagnetic wave will absorb its energy with a rate Q given by [165]:  ω (2.150) Q = Im E∗ (r) dD(r). 2 Here the integration is carried out over the volume of the considered system. E(r) is the electric field intensity at the position r and dD(r) is the dipole moment of the integration element at that position. For a point-like atom located at r Eq. (2.150) reads as $ % Q a = ω Im E∗ (r)d(r) /2 = ω E02 Im (αa (ω)) /2, (2.151) where d(r) is the dipole moment of the atom. Thus, the dynamical screening factor becomes

2 Atomic and Molecular Physics Methods for Nanosystems

F (ω) =

83

$ ∗ % Im Etot (r) dtot (r) E 02 Im(αa (ω))

,

(2.152)

where Etot (r) and dtot (r) are the total electric field and total dipole moment at r, the position of the encapsulated atom. As demonstrated in Ref. [93], the dynamical screening factor F (ω) of an atom confined at the center of the fullerene of the finite width (see Fig. 2.19A) is given by:

2

w

,

F (ω) = 1 − α (ω)w˜

(2.153)

a

where 9  2 (2 + )( − 1) + (1 − )(2 + ) ξ 3 w˜ = 3  R1 w=

(2.154) (2.155)

with  = (2 + )(2 + 1) − 2 (1 − )2 ξ 3

(2.156)

and ξ = R1 /R2 ≤ 1 being the ratio of the inner to outer radii of the fullerene shell. The components of the screening factor w and w˜ are transformed into   1 1 2(N1 − N2 ) − R3 ω2 − ω12 + i1 ω ω2 − ω22 + i2 ω 2  2 N2 N1 w˜ = − 3 3 + R1 R2 ω2 − ω12 + i1 ω ω2 − ω22 + i2 ω

w = 1+

(2.157) (2.158)

where ω12 =

  ωp2  ωp2  3 − p , ω22 = 3+ p , 6 6

p=

'

1 + 8ξ 3 .

(2.159)

The frequencies ω1 and ω2 are the eigenfrequencies of the symmetric and antisymmetric surface plasmon modes of the fullerene, respectively. N1 and N2 are the number of delocalized electrons involved in each mode: N1 = N

p+1 2p

N2 = N

p−1 . 2p

(2.160)

1 and 2 are the widths of the two surface plasmon modes. These describe the decay rate from the collective excitation to the incoherent sum of single-electron excitations. The case study described below corresponds to the C60 fullerene with the mean radius R = 3.5 Å and a thickness R = 1.5 Å.

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Fig. 2.20 The dynamical screening factor for the Ar atom located at the center of the C60 cage. The dynamical screening factor was calculated for two cases: (i) neglecting the atomic feedback, labeled as “no α(ω)”, and (ii) accounting for this feedback, labeled as “with α(ω)”, represented by the solid and dashed green lines, respectively. These are compared with TDLDA calculations (thin solid blue line) from Ref. [153]. The ionization potential of the endohedral atom is indicated by the vertical dotted line. The thick solid (yellow) and dashed (red) lines represent the dynamical screening factor calculated within the modified model, which includes the contributions of the σ and π plasmons. Reproduced from Ref. [164] with permission from American Physical Society

Figure 2.20 shows the dynamical screening factor calculated for the Ar@C60 system where the Ar atom located at the center of the C60 cage (see dashed green line) [164]. The dynamical screening factor has been calculated in the photon energy region above the ionization threshold of the argon atom, I = 16 eV. For photon energies ω < I the calculation of the screening factor is meaningless according to Eq. (2.148), since the atomic photoionization cross section σfree is equal to zero. The curve is compared with the limiting case where the interaction between the atom and fullerene is neglected (thin solid green line). There is a noticeable difference between the two cases, which can be attributed to the large dipole polarizability of argon in the energy range considered. The results of model calculations are compared to the results of TDLDA calculations from Ref. [153] (thin blue line). Figure 2.20 indicates good overall agreement between the model results and the TDLDA calculation. The inclusion of the interaction between the polarized atom and fullerene into the dynamical screening factor improves the correspondence. Even better agreement with the results of TDLDA calculations has been achieved by accounting for the σ and π plasmons (which involve σ and π delocalized electrons of the fullerene, respectively), each split into a symmetric and an antisymmetric mode. The modified dynamical screening factors are shown in Fig. 2.20 by thick solid (yellow) and dashed (red) curves. The solid line is the dynamical screening factor for the limiting case when the interaction between the atom and fullerene is neglected,

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Fig. 2.21 The dynamical screening factor for Ar@C60 as a function of photon energy ω and the radial distance of the atom from the fullerene’s center, ρ, calculated for different values of the angle θ between ρ and E0 . Reproduced from Ref. [164] with permission from American Physical Society

and the dashed line accounts for the interaction between the polarized atom and the polarized fullerene. It is evident that the modification of the screening factor due to the accounting for σ and the π plasmons leads to a better correspondence with the TDLDA calculations [153]. In reality, the endohedral atom is not fixed at the center of the fullerene. Thermal vibrations, van der Waals interaction with the fullerene, and other effects, such as electron transfer, can cause the atom to be preferentially displaced from the center. Figure 2.21 illustrates the dynamically screening factor for Ar@C60 where the Ar atom is arbitrarily positioned inside the fullerene. Four panels of Fig. 2.21 show the screening factor as a function of photon energy ω and radial distance from the center, ρ, see Fig. 2.19B. Each panel corresponds to the indicated value of the angle θ between ρ and E0 . Figure 2.21 indicates that the dynamical screening factor depends rather weakly on θ . The radial distance from the center, ρ, has a much larger influence on the dynamical screening factor, particularly at large distances. As the atom is moved away from the center of the fullerene, the screening factor (in the vicinity of the dipole-symmetric plasmon mode) increases. This increase becomes more dramatic as the atom approaches the fullerene shell. The role of the off-center position of an encapsulated atom in the photoionization spectra of endohedral fullerenes (specifically, the presence of confinement resonances

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in the spectra) was elucidated theoretically in Ref. [155]. The results of experiments [157] which contradicted the earlier theoretical predictions on the existence of confinement resonances in particular endohedral systems were explained. The position of the endohedral atom can be characterized by the displacement vector a from the center of the fullerene cage. The equilibrium position of an endohedral atom and the amplitude of thermal vibrations depend on the atom type, the cage size and the temperature. Due to thermal motion the atom is not fixed at any particular position inside the fullerene. For meaningful observable results it is thus necessary to carry out the averaging over the values a = |a|. As demonstrated in Ref. [155], the structure of confinement resonances in the photoionization cross section of an endohedral atom is very sensitive to the mean displacement a of the atom from the cage center, and in many cases averaging over the values a destroys the confinement resonance structure. Qualitatively, this can be explained as follows. The confinement resonances appear due to the interference of the two waves of the photoelectron, the direct and the scattered one, both originating from the encaged atom. The atom at the center can be treated as a point-like source. For a = 0 the source acquires a size D ≈ 2 a, where a is the mean distance from the center. The finite size of the source influences the interference pattern. When D exceeds the half-wavelength, the angular averaging can destroy the oscillatory structure. On this basis the criterion was formulated which allows one to estimate, for a particular endohedral system, the interval of photon energies where the resonances can appear. It was shown [155] that the resonance structure will be suppressed for the electron momenta p > pmin ≈ π/D. The corresponding range of photon energies is 2 /2 + I0 , where I0 is the ionization potential of the atomic shell. ω > pmin

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107. Kafle, B.P., Katayanagi, H., Prodhan, M., Yagi, H., Huang, C., Mitsuke, K.: Absolute total photoionization cross section of C60 in the range of 25–120 eV: revisited. J. Phys. Soc. Jpn. 77, 014302 (2008) 108. Reinköster, A., Korica, S., Prümper, G., Viefhaus, J., Godehusen, K., Schwarzkopf, O., Mast, M., Becker, U.: The photoionization and fragmentation of C60 in the energy range 26–130 eV. J. Phys. B: At. Mol. Opt. Phys. 68, 2135–2144 (2004) 109. Scully, S.W.J., Emmons, E.D., Gharaibeh, M.F., Phaneuf, R.A., Kilcoyne, A.L.D., Schlachter, A.S., Schippers, S., Müller, A., Chakraborty, H.S., Madjet, M.E., Rost, J.M.: Photoexcitation of a volume plasmon in C60 ions. Phys. Rev. Lett. 94, 065503 (2005) 110. Berkowitz, J.: Sum rules and the photoabsorption cross sections of C60 . J. Chem. Phys. 111, 1446–1453 (1999) 111. Saito, S., Oshiyama, A.: Cohesive mechanism and energy bands of solid C60 . Phys. Rev. Lett. 66, 2637–2640 (1991) 112. Varshalovich, D.A., Moskalev, A.N., Khersonskii, V.K.: Quantum Theory of Angular Momentum. World Scientific Publishing, Singapore (1988) 113. Verkhovtsev, A.V., Korol, A.V., Solov’yov, A.V.: Plasmon excitations in photo- and electron impact ionization of fullerenes. J. Phys.: Conf. Ser. 438, 012011 (2013) 114. Lushnikov, A.A., Simonov, A.J.: Surface plasmons in small metal particles. Z. Phys. 270, 17–24 (1974) 115. Yannouleas, C., Broglia, R.A.: Landau damping and wall dissipation in large metal clusters. Ann. Phys. 217, 105–141 (1992) 116. Yannouleas, C.: Microscopic description of the surface dipole plasmon in large Nan clusters (950 ≤ n ≤ 12050). Phys. Rev. B 58, 6748–6751 (1998) 117. Bertsch, G.F., Bulgac, A., Tomanek, D., Wang, Y.: Collective plasmon excitations in C60 clusters. Phys. Rev. Lett. 67, 2690–2693 (1992) 118. Lushnikov, A.A., Simonov, A.J.: Excitation of surface plasmons in metal particles by fast electrons and X rays. Z. Phys. B 21, 357–362 (1975) 119. Ellert, C., Schmidt, M., Schmitt, M., Reiners, T., Haberland, H.: Temperature dependence of the optical response of small, open shell sodium clusters. Phys. Rev. Lett. 75, 1731–1734 (1995) 120. Pacheco, J.M., Broglia, R.A.: Effect of surface fluctuations in the line shape of plasma resonances in small metal clusters. Phys. Rev. Lett. 62, 1400–1402 (1989) 121. Bertsch, G.F., Tomanek, D.: Thermal line broadening in small metal clusters. Phys. Rev. B 40, 2749–2751 (1989) 122. Penzar, Z., Ekardt, W., Rubio, A.: Temperature effects on the optical absorption of jellium clusters. Phys. Rev. B 42, 5040–5045 (1990) 123. Montag, B., Reinhard, P.G., Meyer, J.: The structure-averaged jellium model for metal clusters. Z. Phys. D 32, 125–136 (1994) 124. Montag, B., Reinhard, P.G.: Width of the plasmon resonance in metal clusters. Phys. Rev. B 51, 14686–14692 (1995) 125. Wang, Y., Lewenkopf, C., Tomanek, D., Bertsch, G., Saito, S.: Collective electronic excitations and their damping in small alkali clusters. Chem. Phys. Lett. 205, 521–528 (1993) 126. Pacheco, J.M., Schöne, W.D.: Shape phase transitions in the absorption spectra of atomic clusters. Phys. Rev. Lett. 79, 4986–4989 (1997) 127. Connerade, J.P., Solov’yov, A.V.: Radiative electron capture by metallic clusters. J. Phys. B: At. Mol. Opt. Phys. 29, 365–375 (1996) 128. Ipatov, A., Connerade, J.P., Gerchikov, L.G., Solov’yov, A.V.: Electron attachment to metallic clusters. J. Phys. B: At. Mol. Opt. Phys. 31, L27–L34 (1998) 129. Connerade, J.P., Solov’yov, A.V.: Giant resonances in photon emission spectra of metal clusters. J. Phys. B: At. Mol. Opt. Phys. 29, 3529–3547 (1996) 130. Gerchikov, L.G., Solov’yov, A.V.: Photon emission in electron-cluster collision in the vicinity of plasmon resonance. Z. Phys. D 42, 279–287 (1997) 131. Korol, A.V., Solov’yov, A.V.: Polarizational bremsstrahlung of electrons in collisions with atoms and clusters. J. Phys. B: At. Mol. Opt. Phys. 30, 1105–1150 (1997)

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132. Gerchikov, L.G., Ipatov, A.N., Solov’yov, A.V.: Many-body treatment of the photon emission process in electron-clusters collisions. J. Phys. B: At. Mol. Opt. Phys. 31, 2331–2341 (1998) 133. Ipatov, A.N., Ivanov, V.K., Agap’ev, B.D., Eckardt, W.: Exchange and polarization effects in elastic electron scattering by metallic clusters. J. Phys. B: At. Mol. Opt. Phys. 31, 925–934 (1998) 134. Descourt, P., Farine, M., Guet, C.: Many-body approach of electron elastic scattering on sodium clusters. J. Phys. B: At. Mol. Opt. Phys. 33, 4565–4574 (2000) 135. Lezius, M., Scheier, P., Märk, T.D.: Free electron attachment to C60 and C70 . Chem. Phys. Lett. 203, 232–236 (1993) 136. Huang, J., Carman, H.S., Jr., Compton, R.N.: Low-energy electron attachment to C60 . J. Phys. Chem. 99, 1719–1726 (1995) 137. Elhamidi, O., Pommier, J., Abouaf, R.: Low-energy electron attachment to fullerenes C60 and C70 in the gas phase. J. Phys. B: At. Mol. Opt. Phys. 30, 4633–4642 (1997) 138. Ptasinska, S., Echt, O., Denifl, S., Stano, M., Sulzer, P., Zappa, F., Stamatovic, A., Scheier, P., Märk, T.D.: Electron attachment to higher fullerenes and to Sc3 N@C80 . J. Phys. Chem. A 110, 8451–8456 (2006) 139. Kasperovich, V., Tikhonov, G., Wong, K., Brockhaus, P., Kresin, V.: Polarization forces in collisions between low-energy electrons and sodium clusters. Phys. Rev. A 60, 3071–3075 (1999) 140. Kresin, V., Guet, C.: Long-range polarization interactions of metal clusters. Philos. Mag. B 79, 1401–1411 (1999) 141. Sentürk, S., Connerade, J.P., Burgess, D.D., Mason, N.J.: Enhanced electron capture by metallic clusters. J. Phys. B: At. Mol. Opt. Phys. 33, 2763–2774 (2000) 142. Rabinovitch, R., Xia, C., Kresin, V.V.: Evaporative attachment of slow electrons to alkali-metal nanoclusters. Phys. Rev. A 77, 063202 (2008) 143. Rabinovitch, R., Hansen, K., Kresin, V.V.: Slow electron attachment as a probe of cluster evaporation processes. J. Phys. Chem. A 115, 6961–6972 (2011) 144. Hervieux, P.A., Madjet, M.E., Benali, H.: Capture of low-energy electrons by simple closedshell metal clusters. Phys. Rev. A 65, 023202 (2002) 145. Massey, H.S.W.: Atomic and Molecular Collisions. Taylor and Francis, London (1979) 146. Korol, A.V., Solov’yov, A.V.: Polarization Bremsstrahlung. Springer Series on Atomic, Optical, and Plasma Physics, vol. 80. Springer, Berlin, Heidelberg (2014) 147. Amusia, M.Y., Korol, A.V.: On the continuous spectrum electromagnetic radiation in electronfullerene collisions. Phys. Lett. A 186, 230–234 (1994) 148. Popov, A.A., Yang, S., Dunsch, L.: Endohedral fullerenes. Chem. Rev. 113, 5989–6113 (2013) 149. Yang, S., Wang, C.R. (eds.): Endohedral Fullerenes: From Fundamentals to Applications. World Scientific, Singapore (2014) 150. Jalife, S., Arcudia, J., Pan, S., Merino, G.: Noble gas endohedral fullerenes. Chem. Sci. 11, 6642–6652 (2020) 151. Connerade, J.P.: Confining and compressing the atom. Eur. Phys. J. D 74, 211 (2020) 152. Connerade, J.-P., Solov’yov, A.V.: Dynamical screening of a confined atom by a fullerene. J. Phys. B: At. Mol. Opt. Phys. 38, 807–813 (2005) 153. Madjet, M.E., Chakraborty, H.S., Manson, S.T.: Giant enhancement in low energy photoemission of Ar confined in C60 . Phys. Rev. Lett. 99, 243003 (2007) 154. Dolmatov, V.K.: Photoionization of atoms encaged in spherical fullerenes. In: Advances in Quantum Theory, vol. 58, pp. 13–68. Academic, New York (2009) 155. Korol, A.V., Solov’yov, A.V.: Confinement resonances in the photoionization of endohedral atoms: myth or reality? J. Phys. B: At. Mol. Opt. Phys. 43, 201004 (2010) 156. Korol, A.V., Solov’yov, A.V.: Vacancy decay in endohedral atoms: the role of an atom’s non-central position. J. Phys. B: At. Mol. Opt. Phys. 44, 085001 (2011) 157. Kilcoyne, A.L.D., Aguilar, A., Müller, A., Schippers, S., Cisneros, C., Alna’Washi, G., Aryal, N.B., Baral, K.K., Esteves, D.A., Thomas, C.M., Phaneuf, R.A.: Confinement resonances in photoionization of Xe@C+ 60 . Phys. Rev. Lett. 105, 213001 (2010)

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158. Phaneuf, R.A., Kilcoyne, A.L.D., Aryal, N.B., Baral, K.K., Esteves-Macaluso, D.A., Thomas, C.M., Hellhund, J., Lomsadze, R., Gorczyca, T.W., Ballance, C.P., Manson, S.T., Hasoglu, M.F., Schippers, S., Müller, A.: Probing confinement resonances by photoionizing Xe inside a C+ 60 molecular cage. Phys. Rev. A 88, 053402 (2013) 159. Stefanou, M., Chandler, H.J., Mignolet, B., Williams, E., Nanoh, S.A., Thompson, J.O.F., Remacle, F., Schaub, R., Campbell, E.E.B.: Angle-resolved photoelectron spectroscopy and scanning tunnelling spectroscopy studies of the endohedral fullerene Li@C60 . Nanoscale 11, 2668–2678 (2019) 160. Müller, A., Martins, M., Kilcoyne, A.L.D., Phaneuf, R.A., Hellhund, J., Borovik, A., Jr., Holste, K., Bari, S., Buhr, T., Klumpp, S., Perry-Sassmannshausen, A., Reinwardt, S., Ricz, S., Schubert, K., Schippers, S.: Photoionization and photofragmentation of singly charged positive and negative Sc3 N@C80 endohedral fullerene ions. Phys. Rev. A 99, 063401 (2019) 161. Obaid, R., Xiong, H., Augustin, S., Schnorr, K., Ablikim, U., Battistoni, A., Wolf, T.J.A., Bilodeau, R.C., Osipov, T., Gokhberg, K., Rolles, D., LaForge, A.C., Berrah, N.: Intermolecular coulombic decay in endohedral fullerene at the 4d → 4 f resonance. Phys. Rev. Lett. 124, 113002 (2020) 162. Müller, A., Kilcoyne, A.L.D., Schippers, S., Phaneuf, R.A.: Experimental studies on photoabsorption by endohedral fullerene ions with a focus on Xe@C+ 60 confinement resonances. Phys. Scr. 96, 064004 (2021) 163. Connerade, J.P., Dolmatov, V.K., Manson, S.T.: Controlled strong non-dipole effects in photoionization of confined atoms. J. Phys. B: At. Mol. Opt. Phys. 33, L275–L282 (2000) 164. Lo, S., Korol, A.V., Solov’yov, A.V.: Dynamical screening of an endohedral atom. Phys. Rev. A 79, 063201 (2009) 165. Landau, L.D., Lifshitz, E.M.: Course of Theoretical Physics, vol. 8. Electrodynamics of Continuous Media. Butterworth-Heinemann (1984)

Chapter 3

Multiscale Computational Modelling of MesoBioNano Systems Gennady B. Sushko, Ilia A. Solov’yov, and Andrey V. Solov’yov

Abstract This chapter provides an introduction to the MesoBioNano (MBN) Science—a novel field of interdisciplinary research interlinking research areas in Physics, Chemistry, Biology, Materials Science and related industries. It gives a short overview of the major computational approaches exploited in the field with the emphasis on the multiscale computational modelling by means of MBN Explorer—a multi-purpose software package for advanced multiscale simulations of complex molecular structure and dynamics. The chapter presents the unique features of MBN Explorer enabling efficient simulations of structure and dynamics of a large variety of very different complex molecular systems with the sizes ranging from the atomic up to the mesoscopic scales. It introduces the application areas of MBN Explorer and describes MBN Studio—a special multi-task toolkit for MBN Explorer, which enables construction of computational projects, simple start of simulations with MBN Explorer, as well as visualization and analysis of the results obtained.

3.1 Introduction The MBN Science is the interdisciplinary field of research studying structureformation and dynamics of animate and inanimate matter on the nano- and the mesoscales. Any form of condensed matter, including biological, consists of numerous components linked by different interactions. Important efforts in deepening the molecular-level understanding of different forms of condensed matter and their G. B. Sushko · A. V. Solov’yov (B) MBN Research Center gGmbH, Altenhöferallee 3, 60438 Frankfurt am Main, Germany e-mail: [email protected] G. B. Sushko e-mail: [email protected] I. A. Solov’yov Carl von Ossietzky Universität Oldenburg, Carl-von-Ossietzky-Str. 9-11, 26129 Oldenburg, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 I. A. Solov’yov et al. (eds.), Dynamics of Systems on the Nanoscale, Lecture Notes in Nanoscale Science and Technology 34, https://doi.org/10.1007/978-3-030-99291-0_3

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dynamical behaviour concern the origin, nature and evolution of various complex molecular systems, as well as the emergence of new features, properties, processes and functions involving the systems with increasing their size and complexity. MBN Science bundles up a number of traditional topics in theoretical physics and chemistry at the interface with Life Sciences and Materials Research under a common theme. Any form of condensed matter, including biological, consists of many different components linked by numerous, different interactions. Important efforts in deepening the molecular-level understanding of different forms of condensed matter and their dynamical behaviour concern the origin, nature and evolution of various complex molecular systems, as well as the emergence of new features, properties, processes and functions involving the systems with increasing their size and complexity. On the meso- and nanoscales, the physics and chemistry of biological and biomolecular systems, nanosystems and materials typically deal with such behaviour. Many examples of emergence of qualitatively new features can be quoted, e.g. the development of new collective properties when going from small molecules to large clusters or the cluster aggregation on surfaces leading to the appearance of fractal-shaped morphologies. The fractal morphologies, being emerged in dynamical systems on the nanoscale, remain characteristic for many systems, including biological ones, at practically all larger scales, and are present in practically all living systems. This chapter provides an introduction to MBN Science, which is based on theoretical and computational physics. This field of theoretical research emerged only recently together with development of powerful computers and advanced computational techniques. Namely, the computational aspect of MBN Science provides the methodology for revealing novel features of structure and dynamics of nanoscopic and mesoscopic molecular systems. It also supports the high level of interdisciplinarity of the research, because similar computational methodologies can be easily adopted to molecular systems of very different nature and origin. The range of open challenging scientific problems in MBN Science is very broad and it grows rapidly facilitating also the development of relevant theoretical and computational methods. In spite of a huge diversity of MBN systems, the methodologies for their theoretical description and computer simulations are often similar. This observation hints towards the development of universal computational techniques applicable for modelling all kinds of MBN systems. In most of the application areas, simulations need to operate over a wide range of scales, ranging from the molecular and the nanoscale to the micro- and sometimes even to macro-dimensions in order to describe at a sufficient level of detailed different multiscale physical and chemical phenomena. The development of multiscale modelling tools goes in parallel with the development of new theoretical methods and widening modern methods of high-performance computing. In recent years, the multiscale modelling has become one of the most topical research fields. In order to fully exploit its potential, one often needs to be familiar with a wide range of interdisciplinary topics in

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• Physics: providing the fundamental theories for the matter foundation, the delivery of radiation and its interactions with MBN targets, or explaining the fundamentals of variety of processes occurring during deposition of materials on surfaces and the formation of nanostructures and novel materials. • Chemistry: describing the chemical processes induced at specific physical conditions and providing tools for tailoring nanoscale species to specific functions. • Materials Science: searching for advanced materials with the unique properties or functionalization of the materials on the nanoscale. • Life Sciences: elucidating effects on the molecular and cellular levels and integrating this knowledge into clinical practices. • Software Engineering & High-Performance Computing: providing the basis for advanced computational/virtual modelling of a large variety of systems and phenomena on scales ranging from atomic to macroscopic. This chapter presents an introduction to MBN Science by means of the advanced computational approach enabled by the powerful and universal software packages MBN Explorer and MBN Studio. It contains the introductory information about the software packages and the areas of their application.

3.2 Multiscale Structure and Dynamics of MBN Systems The Born–Oppenheimer theorem [1] states that any molecular system consists of the two weakly bound subsystems—electronic and ionic. The ground state electronic structure of a molecular system defines interatomic interactions, atomic valences, types of interatomic bonds and, finally, the molecular structure. The Born– Oppenheimer theory demonstrates that the motion of ions in a molecular system occurs quasiclassically and slowly with respect to the characteristic times of the electronic motion. This observation is of fundamental importance. It means that the electronic subsystem becomes a source of the electric field acting on ions in addition to their interionic Coulomb interaction and that the ions motion can be described in terms of classical trajectories. The characteristic time scales of the ionic motion stretch from femtoseconds upwards, while the essential electronic dynamics and quantum processes occur on the sub- or femtosecond time scales. Such situation opens remarkable possibilities for the description of molecular systems by means of classical molecular dynamics (MD), which is widely discussed in this book. The limits of classical MD can be extended by the application of various well-justified model approaches establishing its links with quantum processes occurring at certain conditions in molecular systems on the time scales shorter than femtosecond. The limits can also be extended by linking the classical MD to purely probabilistic Monte Carlo (MC) descriptions, which are highly appropriate for the analysis of structure and dynamics of large molecular systems at the long enough time scales. Such multiscale descriptions, which often play the key role in overcoming the complexity of molecular systems, are discussed in this book as well. Figure 3.1 illustrates the

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Fig. 3.1 Multiscale descriptions of MBN systems involve Quantum Models (QM), quantum interactions and effects, elaborated force fields applied for NVE and NVT ensembles of particles (atoms and molecules), simplified force fields and coarse-grained approaches, stochastic dynamics and Finite Element Methods (FEM). Each of the models operates at a given characteristic time and spatial scales

situation and introduces several popular theoretical and computational approaches, which are widely used in multiscale descriptions of MBN systems and will be discussed in this book. There are many examples of complex nano- and microscale molecular systems that exhibit unique features, properties and functions. In spite of a huge diversity of the systems, the methodologies for their description and computer simulations are very similar. Within one chapter, or even one book, it is impossible to describe all this diversity. Therefore, here let us just mention a few popular examples: • • • • • • •

Free atomic and molecular clusters, nanoparticles; Supported clusters and nanoparticles; Nanocarbon systems (fullerenes, nanotubes, graphene, nanowires, etc.); Endohedral atoms and molecules; Bio-macromolecules: peptides, polypeptides, proteins, DNA; Hybrid MBN systems; Solid, liquid, gaseous, plasma systems and their interfaces.

Often the mentioned systems become the key components of the larger scale biological, physical or chemical complex systems. Such molecular building blocks intrude their structural characteristics into the systems on the micro-scale and thus determine also the global, macroscopical features and properties of the corresponding materials. On the nano- and mesoscales the typical examples of such behaviour are related to atomic and molecular clusters, various nano-objects (fullerenes, endohedrals, functionalized nanoparticles, nanotubes, etc.) and bio-macromolecules (proteins and DNA). A detailed understanding of the structure and dynamics of such molecular systems on the nano- and micrometer scales is an important fundamental

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Fig. 3.2 Illustration of selected MBN systems. Reproduced from Ref. [2] with permission from John Wiley & Sons Inc

task, the solution of which is needed in numerous nano- and biotechnology, material science and medical applications. Figure 3.2 illustrates selected molecular systems that are simulated and visualized by means of the theoretical and computational methodologies described below. The research in the field of MBN Science is often focused on the theoretical characterization of structural and dynamical properties of complex MBN systems. Nowadays, experimental, technological and computational capacities of the methods and approaches traditionally associated with atomic and molecular physics can be applied to much more complex MBN systems. A variety of ab initio theoretical methods, model approaches and computational techniques are used in the research. The universality of theoretical research instruments creates close and fruitful interdisciplinary interconnection of the nano- and biomolecular research with the strong mutual feedback. The selection of concrete case studies is often driven by specific properties or characteristic features of the MBN systems and their relevance to certain important applications. This book introduces methodologies relevant to this type of study and gives examples of their use. Many essential properties and applications of MBN systems are related to their dynamics. The dynamics of MBN systems and their related properties are thoroughly discussed in this book. These topics include: • MD in different statistical mechanics ensembles, e.g. canonical and microcanonical; • Nanoscale phase transitions (PT): folding, melting, solidification, sublimation, martensite–austenite, multi-fragmentation, etc.; • Dynamical processes with biomacromolecules: DNA unzipping, polypeptide and protein folding, random walk dynamics of unfolded proteins, quantum properties inside biomacromolecules;

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Molecular systems in ubiquitous environment and external fields; Collision processes involving clusters and biomolecules; Photo processes and optical properties; Collective electron excitations; Fusion, fission, association, dissociation and fragmentation processes; Particle propagation through a medium; Diffusion processes; Surface aggregation and growth processes; Tribological processes.

Each of the above-listed molecular systems and research topics will be introduced in the following chapters in the context of computational studies with MBN Explorer—a universal software package. Again most of these topics have a universal character and concern various molecular systems making the whole research area highly interdisciplinary. Here, for the sake of introduction, one of such highly interdisciplinary topics is briefly highlighted. It concerns nanoscale PTs in finite MBN systems, which manifest themselves in different ways, e.g. folding, melting, solidification, sublimation, multi-fragmentation, etc., see e.g. [3]. The important goal of numerous current investigations is to gain the detailed knowledge about the nanoscale mechanisms leading to global conformational changes of single biomacromolecules, nanoparticles or other MBN systems. Each of the mentioned processes can be interpreted as a first-order PT, because they all are characterized by rapid growth of the systems free energy at certain temperature. As a result, the heat capacity of the system as a function of temperature acquires a sharp maximum at the PT temperature, see Fig. 3.3. The description of PTs was tackled by different theoretical and experimental approaches and methods. Thus, in recent years the problem of protein folding was studied with the an interdisciplinary approach combining MD with statistical mechanics, computational chemical physics, and quantum mechanics aiming to provide a comprehensive description of PTs and cooperative dynamics in peptides, proteins and other MBN systems [4, 5]. Understanding such structural transformations reveals a tremendous amount of useful information about the properties of the molecular systems, including important details about their functioning and regulating. For structural transitions in complex MBN systems, neither an analytical solution nor a brute force numerical computation is feasible. Even for the most advanced computers, the MD simulations typically are limited to a microsecond timescale. Furthermore, such runs often provide inadequate statistics for proper sampling. Alternative approximations are clearly required. Statistical mechanics provides a mature framework for dealing with PTs on larger temporal and spatial scales. It defines the partition function, which allows one to construct a parameter-free description of the observable properties of a system [4, 5]. This approach provides the solid theoretical methodology to study complex MBN systems beyond the computational limitations. This new theoretical method generalizes attempts to understand protein folding process on the basis of direct MD simulations and provides the quantitative description

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Fig. 3.3 An illustration of nanoscale phase transitions in different MBN systems. See the text for explanation. Reproduced from Ref. [3]

of thermodynamic characteristics of complex MBN systems that are not accessible with direct MD simulations. Such multiscale statistical mechanics models for the description of the PTs in MBN systems were justified through the comparison of the results of the statistical mechanics and physical kinetics approach with those obtained by direct MD simulations for a certain range of system sizes accessible for both the methods. Also, the approach was validated by comparison of its outcomes with experiment. In many disciplines and areas of research the key complex physical, chemical and biological processes often involve different time and spatial scales. One has then to deal with problems, mechanisms, processes occurring in such systems on different scales, dimensions, spaces, etc., although joined into one coherent scenario. Understanding such multiscale scenarios often gives the key to major breakthroughs. Nano- and mesoscales define the research area in which the quantum world of single atoms, small atomic molecules and clusters meet the bulk scale of classical physics. The mesoscale is a significant step up in the system size and complexity compared to the nanoscale. Already, on the nanoscale, many new physical and chemical properties of the system emerge and this process continues on the mesoscale. Thus, superconductivity emerges only in the mesoscopically large systems. Thermomechanical properties of matter, magnetism and other collective phenomena cannot be understood at the atomic level. Although they emerge already on the nanoscale, their evolution towards the bulk limit continues and some new features arise on the mesoscale. It is more difficult to attribute a certain scale to the emergence of the

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phenomenon of life, although it is obvious that the building blocks of the living systems as well as their interactions and the principles of their dynamics, behaviour and functioning are determined on the nano- and mesoscales. Functioning of MBN systems and devices often involves different scales too. Indeed, batteries and other energy-related devices such as solar cells, fuel cells and super-capacitors, that are seen as representative mesoscale challenges, rely on a battery’s ability to store electricity due to the nanoscale structure of individual components such as the anode, cathode and electrolyte, but the device’s real-world performance depends on how all the components work together at the mesoscale. In the case of the ion-beam cancer therapy, ions propagate in the biological targets macroscopically large distances, although the irradiation-induced transformations in the irradiated biological systems and living organisms leading to the therapeutic effects happen in the nanoscopic volumes in the vicinity of the ion tracks. The latter example is a nice illustration of the multiscale nature of MBN systems, processes therein and the interconnection of this research with the important medical applications. It will be discussed in Chap. 9 of this book in detail.

3.3 MBN E XPLORER Main Features 3.3.1 MBN E XPLORER MesoBioNano Explorer (MBN E XPLORER) [2, 6–8] is a software package for the advanced multiscale simulations of structure and dynamics of complex molecular MesoBioNano (MBN) systems. It has many unique features and a wide range of applications in Physics, Chemistry, Biology, Materials Science, Industry and Medicine, see Fig. 3.4. It is suitable for classical non-relativistic and relativistic MD, Euler dynamics, reactive and irradiation-driven molecular dynamics (RMD and IDMD) simulations, as well as for stochastic dynamics or Monte Carlo (MC) simulations of various randomly moving MBN systems or processes. These algorithms are applicable to a large range of molecular systems of different kinds, such as nano- and biological systems, nanostructured materials, composite/hybrid materials, gases, plasmas, liquids, solids and their interfaces, with the sizes ranging from atomic to mesoscopic. MBN Explorer can be utilized in numerous industrial applications. The concrete examples include functionalized surface coatings and nanostructured materials, stronger and lighter materials for aircraft and cars (providing high-performance in extreme conditions), superhard nanostructured materials for mechanical engineering, nanostructured implants for medical applications, superplasticizers for the cement industry (needed for the production of concrete with higher compressive strength), highly efficient batteries and catalysers, drugs, etc. In most of these applications, it is necessary to identify and/or design specific properties of the system determined by its molecular structure on the nanoscale and to ensure their transfer to the macroscopic scale in order to make them functional and usable. Such a transition implies a

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Fig. 3.4 Illustration of different application areas of MBN Explorer

multiscale modelling supported in MBN Explorer through a combination of MD and MC simulations. The ultimate goal behind the development of MBN Explorer [2] is to expand the understanding of structure and dynamics of complex molecular systems, mechanisms of their stability, self-organization and growth, multiscale phenomena involved, as well as the ways of their manipulation and control, aiming at a broad spectrum of application of this knowledge in nanotechnology, microelectronics, materials science and medicine. The first release of MBN Explorer has been the heritage of more than a decade of development. The code has been thoughtfully tested and proved to be efficient and reliable in calculations. The structure of MBN Explorer, its main features and capabilities are described in detail in the reference article [2] published by the Journal of Computational Chemistry (JCC). This paper reported about the important milestone in the development of MBN Explorer and the possibilities of its broad utilization in various application areas. Therefore, authors of all published works utilizing MBN Explorer are requested to include the primary citation [2]. The illustrations highlighting MBN Explorer and its 3D kinetic Monte Carlo module (see Fig. 3.5) were picked up for the cover pages of two JCC issues.

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Fig. 3.5 Images selected for cover pages of two volumes of Journal of Computational Chemistry highlighting MBN Explorer, in which the reference article about MBN Explorer [2] (left) and its 3D kinetic Monte Carlo module [9] (right) were published

Since its first release, the code was under continuous development conducted by the world-class scientists and IT developers affiliated with the MBN Research Center gGmbH, see http://www.mbnresearch.com/. The second and the third releases of MBN Explorer announced by the MBN Research Center gGmbH in 2015 and 2017 extended the library of implemented algorithms,1 improved performance of the earlier implemented ones and fixed all the bugs identified in the initial version of the code. The forth release of MBN Explorer took place in 2019. It was an improved version of the third release in which most of the key algorithms of the package were optimized and benchmarked against their realization in other codes. A number of known bugs were fixed and the overall performance of the code was significantly enhanced. The fifth release of MBN Explorer is announced by the MBN Research Center gGmbH in 2022. The general modular structure of MBN Explorer in its current version is presented in Fig. 3.6. MBN Explorer enables calculations of energies of a large variety of MBN systems and optimization of their structures. The software package supports different types of MD for MBN systems. These include classical nonrelativistic, irradiation driven, reactive, Euler and relativistic MD. The aforementioned types of simulations operate with a large library of interatomic potentials implemented in the software thus allowing to model many very different molecular systems. It is also possible to simulate dynamics of MBN systems in the presence of external fields—electric, magnetic, gravitational and electromagnetic waves. Also MBN Explorer supports Mote Carlo (MC) simulations of the random walk (stochastic) dynamics of numerous and very different MBN systems. These 1

http://www.mbnresearch.com/implemented-algorithms.

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Fig. 3.6 The general modular structure of MBN Explorer 5.0

simulations are based on the kinetic Monte Carlo type of algorithms with parameters that can be obtained by means of MD simulations. Such methodology becomes very useful and efficient for the computational modelling of numerous multiscale phenomena that occur in different MBN systems on the temporal and spatial scales inaccessible to conventional classical MD. Figure 3.4 highlights a variety of application areas and examples of molecular systems that can be simulated using MBN Explorer. In the course of development, many specific algorithms have been implemented in MBN Explorer. They open unique possibilities in computational modelling of a large variety of MBN systems and processes therein. Therefore, authors of all published works utilizing the special algorithms are requested to include additionally the following citations. • Relativistic integrator and atomistic relativistic molecular dynamics [10]; • Stochastic dynamics and kinetic Monte Carlo approach [9, 11]; • Reactive CHARMM force field, reactive molecular dynamics or molecular mechanics with dynamical topology [12]; • Irradiation-driven molecular dynamics [13–15]; • Relativistic molecular dynamics with accounting for incoherent quantum processes [16]; • Relativistic molecular dynamics with accounting for radiation damping force [17];

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• Relativistic molecular dynamics with accounting for particle beam emittance [18]; • Molecular dynamics of MBN systems in external fields [19]; • Corrected Embedded Atom Model (EAM) potentials [20, 21]. The detailed description of MBN Explorer can be found in the books dedicated to its different aspects. Thus, all the technical details about the software package and how to apply it to simulations of different MBN systems and their properties are described in the Users’ Guide [7] and in the book of Tutorials [8]. The systematic description of numerous case studies performed by means of MBN Explorer is gathered in the dedicated monograph devoted to multiscale modelling [6]. This chapter provides an introduction to the broad area of the MBN Explorer related computational research, but also serves as a prelude to the particular topics discussed in this book. Therefore, different parts of this chapter are linked to the topic discussed in the follow-up chapters of this book in greater detail.

3.3.2 Universality MBN Explorer is designed for studying a broad range of physical, chemical and biological systems, and materials by computing their energies, optimizing molecular structures, as well as through different types of MD and stochastic dynamics simulations. The latter sometimes is also called random walk or kinetic Monte Carlo dynamics. Universality is an important feature of MBN Explorer, which allows modelling of a large number of very different molecular systems with the sizes ranging from atomic to mesoscopic and multiscale processes therein. Figure 3.6 illustrated the main modules of MBN Explorer software package enabling a universal approach to the modelling of MBN systems.

3.3.3 Library of Interatomic Potentials, Force Fields and External Fields MBN Explorer operates with a large library of interatomic potentials. A distinctive feature of the program is the possibility to combine various interatomic potentials from the library of potentials implemented in MBN Explorer and utilize them in different types of MD, optimization or energy calculations, see Fig. 3.6. These include pairwise and many-body potentials, molecular mechanics force fields (e.g. CHARMM), which are widely accepted for studying bio- and nanosystems, as well as the unique ones, e.g. reactive CHARMM and other reactive force fields. Parameters for the interatomic potentials are usually obtained from calculations based on the density functional theory (DFT) or from the experiment. MBN Explorer also supports simulations of MBN systems dynamics in the presence of external fields— electric, magnetic, gravitational and electromagnetic waves.

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The file format of molecular mechanics force field used by MBN Explorer is the same as the one utilized in the programs CHARMM [22], XPLOR2 and NAMD [23]. This compatibility allows using MBN Explorer for calculations of a broad range of biological molecules. The results of MBN Explorer calculations can be visualized and analysed by means of MBN Studio introduced in Sect. 3.4. They are also compatible with other standard visualization programs such as VMD [24] and Chemcraft.3

3.3.4 Unique Algorithms Apart from the standard algorithms, MBN Explorer is equipped with unique algorithmic implementations that enhance significantly the computational modelling capacities in various research and technological areas. As seen from the list of implemented special algorithms outlined above MBN Explorer supports reactive MD and irradiation-driven MD, advanced kinetic MC modelling, simulations of MBN systems in various external fields, atomistic simulations of ultrarelativistic charged particles’ propagation and channelling through oriented crystals with accounting for ionization energy losses and radiation damping, photon emission by ultrarelativistic charged particles and many more. The complete list of algorithms implemented in MBN Explorer can be found on the webpage: http://www.mbnresearch.com/ implemented-algorithms. Despite the universality, the computational efficiency of MBN Explorer is comparable or even higher than the computational efficiency of other software packages with much more limited scopes and modelling capacities, making MBN Explorer a favourable alternative with respect to such codes. MBN Explorer supports OpenMP and MPI parallelization. The important feature of MBN Explorer is the extensibility of the code. It is achieved through the object-oriented programming with C++. The modular design of the code allows integration of new algorithms and techniques for MD simulations.

3.4 MBN S TUDIO Main Features 3.4.1 MBN S TUDIO In order to facilitate the practical work with MBN Explorer, a special multitask software toolkit, called MBN S TUDIO, has been developed [25]. It simplifies the modelling of MBN systems, setting up and starting calculations with MBN Explorer, monitoring their progress and examining the calculation results. 2 3

http://cns-online.org/v1.3/. http://www.chemcraftprog.com.

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Fig. 3.7 The main features of MBN Studio. Reproduced from Ref. [25] with permission from Elsevier

MBN Studio can be utilized for any type of calculations that are supported by MBN Explorer and introduced in Sect. 3.3. The main features of MBN Studio are presented in Fig. 3.7. MBN Studio enables the Project set-up (standard as well as application-specific). Application-specific projects are usually designed for particular tasks that involve some additional parameters specific to the applications. Such projects are typically designed for utilization in specific application areas, e.g. related to novel or emerging technologies. Often application-specific projects also involve special algorithms. A special modelling plug-in allows one to construct and prepare application-specific projects for simulation quickly and efficiently. MBN Studio has an advanced MBN System modeler, a built-in tool for MBN system design. By means of this plug-in, one can easily assemble molecular systems of different geometries and compositions for their further simulations with MBN Explorer. MBN Studio supports various standard Input/Output data formats and links to numerous online Databases and Libraries with coordinates and geometries for atomic clusters, nanoparticles, biomolecules, crystals and other molecular systems that can be utilized in simulations with MBN Explorer. MBN Studio is equipped with the Output data handling, Visualization and Analytic tools that allow calculation and analysis of specific characteristics determined

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by the output of MD simulations. Examples include calculations of diffusion coefficients, heat capacities, melting temperatures for solids, atomic radial distribution functions and many others. MBN Studio also enables Video rendering of all kinds of dynamics of MBN systems simulated with MBN Explorer. MBN Explorer is distributed together with a large library of illustrative examples, presenting all types of algorithms and tasks which can be simulated by means of the code. One can use these examples as templates for the construction of new projects. MBN Studio makes this work simple and often self-explanatory doable for researchers with the general knowledge in natural sciences, such as physics, chemistry, biology or materials science. The authors of all published works that utilize MBN Studio are requested to include the primary citation [25]. The follow-up subsections provide further details about the main features of MBN Studio presented in Fig. 3.7. The full description of the MBN Studio functionality is given in Ref. [8].

3.4.2 Project Set-Up MBN Studio enables the standard project set-up, as well as setting up the application-specific projects. The standard projects typically are based on the general algorithms implemented in MBN Explorer enabling single-point energy calculations, structure optimization and a range of MD and stochastic dynamic tasks, see Figs. 3.6 and 3.8. Application-specific projects are usually designed for particular tasks that involve some additional parameters and algorithms specific to the applications. For instance, in simulations of charged particles propagation through crystalline media in the channelling regime, one needs to provide crystal orientation parameters for straight, bent or periodically bent crystals. Relativistic atomistic MD simulations of particle propagation and radiation in such systems require specific algorithms as well as the knowledge of some specific parameters, e.g. particle beam emittance. The corresponding plug-in built into MBN Studio capable of set up such kinds of atomistic simulations with MBN Explorer is a unique instrument for atomistic computational modelling of the key processes occurring in the crystal-based intensive gamma-ray light sources. Construction of such light sources is the actual task for the related emerging technology.

3.4.3 Output Data Handling MBN Studio is equipped with convenient tools for output data handling. It has a number of built-in plug-ins for graphical representation of the simulated data, for

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Fig. 3.8 MBN Studio has a form of convenient interface, the main panel of which is illustrated

Fig. 3.9 MBN Studio is equipped with the output data handling tools

instance, for plotting dependencies of simulated characteristics (e.g. potential, kinetic or total energy) on time or on simulation step numbers, thermal characterization of simulated systems, etc. Amount and type of the output data, as well as their representation, are defined by the user (Fig. 3.9).

3.4.4 Visualization Tools MBN Studio is equipped with the tools for visualization of the input and output structures of simulated MBN systems, as well as for visualization of the whole simulated processes. The visualization tools operate with any kind of MBN systems and dynamics that can be simulated by means of MBN Explorer, see Fig. 3.6. The visualization of structure and dynamics of MBN systems assists their modelling, virtual manipulation and design. The MBN Studio interface provides a variety of graphical options for visualization (Fig. 3.10).

3.4.5 Analytic Tools MBN Studio is equipped with tools for the output data analysis. It has a number of built-in algorithms for the calculation of specific characteristics that can be obtained from the simulated data. Examples of analysis that can be performed include

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Fig. 3.10 MBN Studio is equipped with the tools for visualization of the input and output structures of simulated MBN systems, as well as for visualization of the whole simulated processes

Fig. 3.11 MBN Studio is equipped with the output data analysis tools

calculation of diffusion coefficients of atoms and molecular species in different media, heat capacities, melting temperatures for solids, radial distribution functions, etc (Fig. 3.11).

3.4.6 Standard Input/Output Formats MBN Studio supports many standard input/output data formats. The conventional formats are utilized for defining geometries of MBN systems, interatomic interactions involved, outputs of MD and stochastic dynamics simulations, etc. The formats supported by MBN Studio enable simple links and utilization of the input and output data with other packages dealing with similar tasks and systems, as well as to various relevant databases (Fig. 3.12).

Fig. 3.12 MBN Studio supports standard input and output formats

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3.4.7 Links to Databases and Libraries MBN Studio supports links to various online databases and libraries which contain coordinates and geometries for atomic clusters, nanoparticles, organic and inorganic molecules, biomolecules, crystals and other molecular systems, as well as parameters of interaction force fields and potentials. This information is necessary for setting up computational projects with MBN Explorer (Fig. 3.13).

3.4.8 Video Rendering MBN Studio supports video rendering of simulation results for MD and stochastic dynamics of MBN systems. This applies to numerous processes involving MBN systems, such as collisions, protein folding, DNA unzipping, diffusion, etc. This tool operates with any output of MBN Explorer suitable for movie rendering. Users can define the rendering parameters (number of frames, format, duration, etc.) as well as the video format (Fig. 3.14).

Fig. 3.13 MBN Studio supports links to online databases and libraries

Fig. 3.14 MBN Studio enables video rendering of dynamics of simulated MBN systems

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3.4.9 System Modeler MBN Studio is equipped with an advanced MBN system modeler enabling the user to construct a large variety of molecular systems quickly and efficiently. By means of this plug-in, one can easily construct new complex molecular systems of different geometry on the basis of geometries known for various molecular objects, such as organic and inorganic molecules, biomolecules, carbon nanotubes of different chirality, fullerenes, nanoparticles and crystalline samples of different shapes (e.g. spherical, ellipsoidal, cubic, conical) with various atomic compositions, etc. The constructed systems can be used for further simulations with MBN Explorer (Fig. 3.15).

3.5 Areas of Application of MBN E XPLORER and MBN S TUDIO There are many different research and technology areas for applications of MBN Explorer and MBN Studio. Below, these areas are briefly introduced and the key topics and representative case studies in each of the areas are outlined. Selected examples are shown in figures (Figs. 3.4, 3.16, 3.17, 3.18, 3.19, 3.20, 3.21, 3.22 and 3.23).

3.5.1 Crystals, Liquids, Gases and Plasmas MBN Explorer enables simulations of • Crystalline structures, • Liquids and soft matter, • Gaseous systems and plasmas, • Physical and chemical phenomena with solids, liquids, gases and plasmas, • Perform multiscale modelling of aforementioned systems.

Fig. 3.15 MBN Studio is equipped with an advanced MBN system modeler

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Fig. 3.16 Liquid water at the atomistic level

There are many examples of simulations which are gathered in the library of MBN Explorer tests and examples [6–8]. These exemplar projects include simulations of metallic (e.g. Ti, Ni, NiTi, etc.), diamond and silicon crystals, oxides (e.g. MgO, SiO2 ), thin films, surface coatings, liquids (e.g. water, soft matter) and their interfaces with metals and biocompatible materials, as well as their various properties and processes with their involvement. MBN Explorer also provides tools for multiscale modelling of various MBN systems. These tools allow one to model the kinetic behaviour of such systems far beyond the time and spatial limits of the conventional MD simulations [9, 26].

3.5.2 Atomic and Molecular Clusters, Nanoparticles MBN Explorer enables simulations of • Atomic clusters, • Molecular clusters, •Finite nanosystems: fullerenes, endohedral atoms, coated and functionalized nanoparticles, etc., • Deposited and embedded clusters and nanoparticles, • Dynamics of cluster and nanosystems. The range of materials that can be probed in the aforementioned systems with MBN Explorer is very broad. It includes metal, noble gas and semiconductor materials, fullerenes and all other allotropic forms of nanocarbon, composite and functionalized NPs, nanoalloys, etc. The sizes of these molecular systems could be varied from a few atoms up to millions of atoms. Possible simulations include the tasks on the structure analysis and optimization, various thermal effects, mechanical properties, nanoscale phase transitions, diffusion and a broad range of other dynamical and collision processes involving clusters and NPs [27] (Fig. 3.17).

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Fig. 3.17 SiO2 nanoparticle

3.5.3 Biomolecular Systems MBN Explorer enables simulations of • Structure of biomolecules and macromolecules • Biomolecular complexes, • Bio-nano systems, • Structural transitions, biomolecular processes, • Dynamics of DNA, RNA and proteins, • Perform multiscale modelling of aforementioned systems. A large variety of biomolecular systems and hybrid bio-nano systems with various interfaces [28] that can be simulated by means of MBN Explorer make the code a useful instrument for computational research in this research area. Transformations of these systems at different thermal and biologically relevant conditions and various external stresses can be explored. Numerous possible case studies include proteins, DNA, lipid bilayers, interaction of these systems with NPs, external environments and many more. MBN Explorer allows one to simulate structure and dynamics of proteins, DNA, RNA and other biomolecules in ubiquitous environments. Protein folding [29], antigen–antibody bonding [30], DNA unzipping [31], radiation damage phenomena [32] and many other processes involving biomolecules can be explored computationally (Fig. 3.18).

Fig. 3.18 DNA photolyase enzyme repairing a photolesion

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3.5.4 Collisions and Reactions MBN Explorer enables simulations of • Collision processes, fission, fusion fragmentation processes involving clusters, nanoparticles, biomolecules, nanostructures and materials, • Molecular association, dissociation, reactions, • Collision- and irradiation-induced chemistry, • Particles propagation through media, • Collision-induced medium effects. With MBN Explorer, one can investigate a large variety of collision and dynamical processes, reactions involving numerous and very different complex molecular systems aforementioned above. These processes often play an essential role in the related modern technologies, see Sect. 3.5.8 below. This explains why the simulation and quantification of these processes become increasingly important tasks in connection with further developments of the related technologies (Fig. 3.19).

3.5.5 Nanostructured Materials MBN Explorer enables simulations of • Metallic, organic and inorganic nanomaterials, • Crystalline superlattices, • Nanotubes, nanowires, nanofractals, nanofilms, graphene, functional nanoparticles, etc., • Nanoscale phase and structural transitions, • Self-assembly and growth, • Irradiation-driven nanofabrication. Nanoscale molecular objects, such as atomic clusters, NPs, proteins, DNA fragments, etc., provide a possibility to construct new types of materials, the so-called nanostructured materials, thin films, surface coatings with structure and functionality determined by the properties of the molecular building blocks from which they are

Fig. 3.19 Ion induced shock wave damaging a DNA molecule

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Fig. 3.20 Fragment of a composite C60 -based nanowire

Fig. 3.21 Composite antimony fractal on graphite surface

constructed. MBN Explorer allows one to simulate a wide spectrum of the aforementioned nanostructured materials and to study their properties [6]. The concrete examples include nanostructured metals (e.g. Ni or Ti [33]), crystalline superlattices of metal NPs linked by different organic or biological molecules, nanocarbon (nanosilicon)-based nanostructured materials (e.g. trimethylbenzene-C60 nanowires [34]), nanofractals [26] and many more (Fig. 3.20).

3.5.6 Composite Materials and Material Interfaces MBN Explorer enables simulations of • Alloys and composites, • Surface structure and material interfaces, • Tribological properties: nanoindentation, scratching, lubrication, etc., • Deposition, diffusion, aggregation and surface pattern formation, morphological transitions, • Functional surface coatings. MBN Explorer has the necessary tools (appropriate force fields and algorithms) to simulate many novel composite materials consisting of components of different nature, ordered or disordered. Examples of such hybrid systems include nanoalloys, material interfaces, NPs placed into the biological environments (e.g. attached to DNA, protein, or cell membrane), metal clusters deposited on graphite, silicon or other materials and many more. With MBN Explorer, one can simulate and investigate a variety of complex multiscale dynamical processes, such as diffusion, aggregation of atoms, molecules, clusters, on surfaces, surface pattern formation, morphological transitions, etc. (Fig. 3.21).

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Fig. 3.22 Nanoindentation of a titanium crystal

3.5.7 Thermomechanical Properties of Materials MBN Explorer enables simulations of • Thermomechanical properties of materials, • Elastic and plastic deformations, • Defects and dislocations, • Phase and structural transitions, • Irradiation-induced damages, • Materials at extreme conditions. MBN Explorer can be utilized for simulations and investigation of the mechanical properties and thermal effects of a broad variety of the aforementioned materials. This includes analysis of elastic and plastic deformations [36] (e.g., Young’s modulus, Poisson’s ratio, hardness), dynamics of dislocations, phase and structural transitions [37], thermo-mechanical and irradiation-induced damages [38], and many more. For most of the materials, processes and phenomena, the dependence of their various characteristics on thermal conditions is an important effect which can be simulated with MBN Explorer. This knowledge can be utilized in various applications. One can also investigate tribological processes (nanoindentation [35], scratching, lubrication, etc.) involving various surfaces my means of MD simulations (Fig. 3.22).

3.5.8 Emerging Technologies MBN Explorer and MBN Studio can be utilized for computational modelling in • Nanotechnologies, • Biotechnologies, • Material technologies, • Radiation technologies, • Plasma technologies, • Space technologies, • Computational technologies,

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Fig. 3.23 Axial channelling of an ultra-relativistic electron along crystal axis

• Medicine. MBN Explorer and MBN Studio are the powerful tools for computational modelling in numerous different areas of challenging research arising in connection with the development of the aforementioned technologies [10]. There are several such areas, in which simulations performed by means of MBN Explorer and MBN Studio contributed immensely to their development. For instance, one of the areas concerns the construction of novel light sources based on charged particles channelling in crystalline undulators. Another example deals with simulations of the nanoscopic molecular processes playing the key role in the ion-beam cancer therapy [39, 40]. MBN Explorer combined with the visualization interface of MBN Studio in many cases can substitute expensive laboratory experiments by computational modelling making the software play a role of a kind of “computational nano- and microscope” (Fig. 3.23).

3.6 Computational Methods for Studying Structure and Dynamics of MBN Systems Let us now introduce the key methodologies that are used for simulations of structure and dynamics of MBN systems on different scales. Most of these methodologies are indicated in the boxes in Fig. 3.6 showing the modular structure of MBN Explorer. The single-point energy calculation and the structure optimization algorithms mostly concern computational tasks devoted to the structure analysis of MBN systems. MD algorithms and methods are used to study different kinds of dynamical transformations in MBN systems. MBN Explorer supports Newtonian, Euler and Langevin dynamics, reactive and irradiation driven MD, relativistic dynamics. MD algorithms are more diverse as they describe motion of MBN systems under different physical conditions and thus obeying different equations of motion or even principles. For example, Newton’s equations describe the classical molecular dynamics of atoms, while Euler’s equations are used to describe the motion of rigid bodies (e.g. molecules, clusters, nanoparticles). In the latter case, one has to account also for the spatial rotation of rigid bodies via accounting for their three additional rotational degrees of freedom. MBN Explorer also enables stochastic dynamics simulations

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of MBN systems. In this case, particle dynamics within the system is governed by the probabilities of changes or processes that may occur in the system on each simulation step. The aforementioned methodologies are applicable for studying dynamical processes and phenomena in many different MBN systems. Simulations with MBN Explorer can be designed in a way that different types of calculations can inter-rely on each other and use the output from one calculation on a next step or a scale thus enabling multiscale modelling of MBN systems and processes therein. Below in this section, the following major computational tasks implemented in MBN Explorer are introduced and briefly discussed • • • • • • • • • •

single-point energy calculation; structure optimization; Newtonian molecular dynamics; Euler dynamics; Langevin molecular dynamics; reactive molecular dynamics; irradiation-driven molecular dynamics; relativistic dynamics; stochastic dynamics; multiscale modelling.

3.6.1 Single-Point Energy Calculation Single-point energy calculation allows determining the potential energy of a molecular system at a given geometry. The geometry of the system is defined by a set of fixed coordinates of all the atoms composing the system. The potential energy of the system is equal to the sum of overall interatomic interactions in the system. An interatomic interaction between atoms is defined through parametrization of the corresponding interaction energies and may have very different analytic forms depending on the nature of interaction, the type of atoms involved and the system type. The library of interatomic interactions implemented in MBN Explorer is rather large. It covers all the major types of interatomic interactions including pairwise (Lennard–Jones, Morse, Dzugutov, Girifalco, Power, Exponential, quasi-Sutton-Chen, and Coulomb), many-body potentials (Sutton-Chen, Gupta, Brenner, Tersoff, Finnis–Sinclair), and potentials for molecular mechanics in the CHARMM format. The detailed description of the entire library of interatomic interactions supported by MBN Explorer one can find in the books [6, 7]. By choosing relevant interatomic potentials for all interacting atoms one can calculate the system potential energy at a given molecular configuration. This important characteristic can be derived for a broad variety of MBN systems. Numerous examples of the potential energy analysis for different MBN systems can be found in [6] and the following up chapters of this book.

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3.6.2 Structure Optimization Structure optimization (also called energy minimization) algorithms permit to find configurations of a molecular system corresponding to local minima and sometimes even to the global minimum on the multidimensional potential energy surface of the system. The two main algorithms of structure optimization implemented in MBN Explorer are (i) the velocity quenching [7] (VQ) and the conjugate gradient [41] algorithms. The general common idea of the structure optimization algorithms is to employ a mathematical procedure to move atoms from their non-equilibrium, nonoptimized initial positions to the optimized ones at which the net forces (the gradients of potential energy) acting on the atoms become negligible. The main idea of the velocity quenching algorithm is to absorb the most efficiently all the kinetic energy from the system including translational and rotational kinetic energies. This is achieved in the following way. In a course of molecular dynamics simulation, kinetic energies of all the particles in the system are monitored. When the kinetic energy of a particle within the system becomes maximal, the absolute value of its velocity is set equal to zero. After a certain number of steps, this brings the system to equilibrium corresponding to a local energy minimum on the potential energy surface of the system. If found value corresponds to the lowest value among all other local energy minima, then it corresponds to the global energy minimum. Conjugate gradient optimization technique implemented decomposes the Ndimensional minimization task into a series of linear minimization processes with an ultimate goal to establish a local minimum on the multidimensional potential energy surface. For each particle of the system, the algorithm determines a displacement in order to minimize interatomic forces acting on it. The energy minimization procedure consists of a sequence of steps. During these steps, algorithm changes depending on both state of the algorithm and the current state of the entire system. Further details on this procedure can be found in [7, 41] Both algorithms are general and applicable to any type of interatomic interactions supported by the software. Structure optimization is a typical and necessary step that has to be completed prior to any dynamical studies, as those often need to be carried out from an equilibrium configuration of a molecular system. Numerous examples of the velocity quenching and conjugate gradient algorithms for optimization of different MBN systems can be found in [6] and the following up chapters of this book.

3.6.3 Newtonian Molecular Dynamics The Newton’s equations of motion for atoms in the system is described as follows: m i ai = m i

d2 ri = Fi , dt 2

i = 1, . . . , Nat .

(3.1)

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Here, m i is the mass of an atom, ai is its acceleration and Fi is the force that acts on the atom and Nat is the total number of atoms in the system which are not constrained within rigid molecules. The forces Fi are determined by the interatomic potentials and external force fields. These forces may be pairwise or many-body in nature. To solve Eq. (3.1) numerically one needs to provide initial conditions, i.e. define initial positions and velocities of particles at time instance t = 0. The detailed description of the algorithms for creation of different MBN systems, defining interatomic interactions therein and setting up initial conditions, as well as examples of simulations can be found in the books [6–8].

3.6.4 Euler Dynamics of Rigid Bodies Euler dynamics describes the dynamics of a system containing rigid bodies. In application to MBN systems, the Euler dynamics describes dynamics of a number of interacting MBN systems with frozen internal degrees of freedom. This means that typically it deals with the motion of rigid molecules, clusters, nanoparticles, etc. (or their parts), in which distances between any two atoms within the rigid fragment are fixed. In a fixed reference frame (called the lab-frame), the location and orientation of the molecule is described through (i) motion of the center of mass of the molecule with respect to the origin of the lab-frame; (ii) orientation of the molecule with respect to the lab-frame. Therefore, the position and the orientation of a rigid molecule can be defined with three spatial and three angular variables. Thus, the equations describing the motion of a rigid molecule are the Newton equations, Eq. (3.1), for the motion of the center of mass and the Euler equations for its rotation. The rotational motion is governed by the following equations [42]: dLα = Tα , dt

α = 1, . . . , Nrm .

(3.2)

Here, Lα is the total angular momentum of a rigid molecule with respect to the origin of the lab-frame, Tα is its total torque and Nrm is the total number of rigid molecules in the system. Equation (3.2) describes the evolution of the angular momentum of a rigid molecule with time under the action of external torques. This equation is only valid in an inertial frame. However, the frame in which the coordinate axes are aligned along the principal axes of rotation of the molecule and Lα possesses its simplest form is non-inertial. Thus, it is helpful to define two Cartesian coordinate systems: the first one with coordinates (x, y, z), is the fixed lab-frame, while the second molecular-frame, with coordinates (x  , y  , z  ), co-rotates with the molecule so that the x  -, y  - and z  -axes are always pointing along the principal axes of rotation, see Fig. 3.24. Since the molecular-frame co-rotates with the molecule, its instantaneous angular velocity coincides with that of the molecule. Hence,

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Fig. 3.24 Two coordinate frames used to describe the motion of a rigid molecular object: (x, y, z) represents the fixed laboratory frame, while the non-inertial coordinate frame (x  , y  , and z  ) is the object frame, in which its tensor of inertia is diagonal

dLα dLα = + ωα × Lα . dt dt

(3.3)

Here, ωα and Lα are the angular velocity and the angular momentum of a rigid molecule in the molecular-frame. Introducing the Cartesian components of the vec    tors Tα ≡ Tαx  , Tα y , Tαz , ωα ≡ ωαx  , ωα y  , ωαz  , and Lα ≡ Iαx  x  ωαx  , Iα y y ωα y ,  Iαz z ωαz , where Iαx  x  , Iα y y and Iαz z are the principal moments of inertia of the molecule, one writes the components of Eq. (3.2) as   Tαx  = Iαx  x  ω˙ αx  − Iα y y − Iαz z ωα y ωαz ,

(3.4)

where (x  , y  , z  ) undergo cyclic permutations. Here, we have used the fact that moments of inertia of a rigid molecule are constant in time in the co-rotating molecular-frame.

3.6.5 Langevin Molecular Dynamics According to the equipartition theorem [43], every degree of freedom, f , in a system being at the equilibrium at temperature T possesses the same kinetic energy equal to K f = kB T /2, where k is the Boltzmann constant. Therefore, the effective temperature T of the system is given by the ensemble average of its kinetic energy:  g   N +N  Nrm at rm  2  1 2 Kf = m i vi + ωi · Iˆi · ωi . (3.5) T = gkB f =1 3(Nat + 2Nrm )kB i=1 i=1

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Here, g = 3Nat + 6Nrm is the number of degrees of freedom, Nat is the number of atoms having three degrees of freedom and Nrm is the number of rigid molecules with six degrees of freedom. It is often desirable that a simulation is conducted in a way that the temperature is kept constant. This requires some algorithm to maintain the system average kinetic energy constant for a system at the thermal equilibrium. MBN Explorer offers several options for temperature control. The common technique of velocity scaling (the Berendsen thermostat) is suitable for use during the equilibration period of a simulation, while the Langevin dynamics describes the evolution of a molecular system which experiences random collisions with “virtual” particles that mimic the environment (the Langevin thermostat). A simple velocity scaling thermostat is that of Berendsen et al. which is referred to as the Berendsen thermostat in literature [44]. At periodic intervals linear and angular velocities of all particles in the system are multiplied by a constant factor  λ=

t 1+ τT



T0 −1 . T

(3.6)

Here, T0 is the thermostat temperature, T is the temperature of the system and t is the integration time step, The parameter τT , called the “rise time” of the thermostat, characterizes the strength of system’s coupling to a virtual heat bath. Larger values of τT correspond to weaker coupling; in other words, the larger τT is, the longer it takes for a system to achieve a given T0 . By repeatedly correcting particle velocities according to Eq. (3.6) during the simulation, the average kinetic energy is made to approach a constant value. In the Langevin dynamics, atoms in the system are considered to be embedded into a “sea” of fictional particles. In this case, the dynamics of atoms in the system is described by the Langevin equations of motion which include additional terms accounting for the friction force and for the noise: d2 ri 1 m i 2 = Fi − m i vi + dt τd

 2kB T0 m i Ri (t), τd

i = 1 . . . Nat .

(3.7)

Here, Fi is the physical force acting on the atom, kB T denotes the thermal energy in the system, τd is the characteristic viscous damping time, and Ri (t) represents a delta-correlated stationary Gaussian process with zero-mean, satisfying Ri (t) = 0,

Ri (t)Ri (t  ) = δ(t − t  ),

(3.8)

where ... denotes time-averaging. The Langevin equation of motion, Eq. (3.7), gives a physically correct description of a many-particle system interacting with a heat bath, maintained at a constant temperature T0 . The viscous damping time parameter, τd , describes the characteristic time of energy exchange between particles and the heat bath. This parameter should be chosen carefully. If τd is small, the Brownian dynamics tends to dominate over the

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Newtonian dynamics of the system, potentially leading to physically wrong results. In the opposite limit, when τd is chosen to be large, the system requires a long period to attain the desired temperature. It is straightforward to couple the rigid molecule equations of motion to a thermostat. The thermostat is coupled to both the translational and rotational degrees of freedom and so both the translational and angular velocities can be treated similarly. The Langevin dynamics of the translational degrees of freedom of a rigid molecule obeys Eq. (3.7). Interaction of rigid molecules with the environment introduces two additional terms in the Euler Eq. (3.4) which modify the torque acting on the molecule:  2kB T0 Iˆα  1 ˆ  Rα (t), i = 1 . . . Nrm . (3.9) Tα = T0α − Iα ωα + τd τd Here T0α is the torque acting on the rigid molecule according to Eq. (3.2), ωα is the angular velocity of a rigid molecule, Iˆα is the diagonalized tensor of moments of inertia and Rα (t) represents a delta-correlated stationary Gaussian process with zero-mean, satisfying conditions (3.8). Berendsen and Langevin thermostats provide for the basic functionality to control the temperature of the system. For systems composed of particles with three degrees of freedom, these thermostats are rather standard and are also available in most computational packages [22, 23, 45–47]. The thermostats implemented in MBN Explorer allow generalization for systems comprising rigid molecules (having six degrees of freedom). Note, that it is important to choose thermostat appropriately depending on the physical problem to be solved. The Langevin Eq. (3.7) puts constraints on the random forces and the friction applied to the particles in the system such that the random force and the friction terms become related, thereby satisfying the fluctuation–dissipation theorem and guaranteeing the NVT statistics. Figure 3.25a shows distribution of temperature fluctuations computed for a C60 molecule subject to the Langevin thermal bath; the data points were sampled from a 1 ns long MD simulation at 300 K. To compute the distribution shown in Fig. 3.25a, the following procedure was employed: during the simulation, the immediate value of temperature of C60 molecule was stored on every step of the simulation. Next, the recorded temperatures were sorted in bins of 10 K width and plotted in a form of a distribution function. The computed distribution can be compared with the fundamental distribution of temperature fluctuations [43], having the form  (T )2 1 . (3.10) exp − p(T ) = √ 2σ 2 2πσ Here, T = T − T0 is the deviation of the system temperature T from that of the thermostat, T0 , and σ 2 = 2T02 /g defines the width of the distribution through the number of degrees of freedom in the system, g. For a finite system consisting of particles with only three degrees of freedom, g = 3N , where N is the number of

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Fig. 3.25 Distribution of temperature fluctuations computed for a C60 molecule (shown in the inset) subject to a Langevin thermostat (a) and Berendsen thermostat (b) during a 1 ns long MD simulation at 300 K. The parameter τd of the Langevin thermostat, see Eq. (3.7), and the parameter τT of the Berendsen thermostat, see Eq. (3.6), were set equal to 100 fs. The solid lines show the profile of the NVT canonical temperature fluctuations distribution computed using Eq. (3.10). Reproduced from Ref. [2] with permission from John Wiley & Sons Inc

particles in the system. Figure 3.25a compares the results of the numerical simulations (the histogram) carried out with the Langevin thermostat with the fundamental distribution Eq. (3.10) where the parameters T0 = 300 K, N = 60 are used (the solid curve). It is seen that the Langevin thermostat correctly describes the NVT canonical ensemble. Contrary to the Langevin thermostat, the Berendsen thermostat fails to reproduce the distribution function (3.10), as it is illustrated by Fig. 3.25b where the histogram represents the numerical simulations of temperature fluctuations in C60 equilibrated using the Berendsen thermostat. Since the canonical distribution can not be reproduced, the Berendsen thermostat is not commonly used for long-run MD simulations. However, an advantage of the Berendsen thermostat is that it provides a faster equilibration of a molecular system, which is typically necessary to carry out prior to any production MD simulation. The present version of MBN Explorer provides two basic thermostats which are sufficient to mimic the effects of the thermal bath in the majority of computational tasks. For the sake of completeness, let us nevertheless mention that there are also other ways to implement thermostats, see e.g. Lowe–Andersen [48] or NoséHoover thermostat [49, 50]. These implementations might carry some additional useful features. For example, the Lowe–Andersen thermostat is appropriate for problems related to diffusive effects in molecular systems and environments. In this case, it is essential to conserve the momentum transfer in the system, that is not the case in the stochastic Langevin thermostat.

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3.6.6 Reactive and Irradiation-Driven Molecular Dynamics Reactive Molecular Dynamics (RMD) and Irradiation-Driven Molecular Dynamics are new types of atomistic MD [13] based on the utilization of reactive CHARMM (rCHARMM) force field (FF) [12] or any other appropriate force field represented in the CHARMM format [12]. RMD, IDMD and rCHARMM represent the unique algorithms implemented in MBN Explorer which enable simulations of chemical transformations and irradiation-driven chemical transformations that may occur in many different MBN systems. The main difference between RMD and IDMD is that IDMD accounts for both irradiation-induced chemical transformations in the system and pure chemical transformations, while RMD accounts only for pure chemical transformations between atoms and molecules (i.e. chemical reactions) that may occur in the course of MD simulation. Therefore, below mostly the more general IDMD case study is discussed, considering RMD as just the limiting case of IDMD corresponding to the absence of irradiation. The IDMD methodology is applicable to any MBN system exposed to radiation. Within the framework of IDMD, various quantum collision processes (e.g. ionization, electronic excitation, bond dissociation via electron attachment or charge transfer) are treated as random, fast and local transformations incorporated into the classical MD framework in a stochastic manner with the probabilities elaborated on the basis of quantum mechanics. This can be achieved because the aforementioned quantum processes occur on the sub- to femtosecond time scales (i.e. during the periods comparable or smaller than a typical single time step of MD simulations) and involve typically a relatively small number of atoms. The probability of each quantum process is equal to the product of the process cross section and the flux density of incident particles [1]. The cross sections of collision processes can be obtained either from (i) ab initio calculations performed by means of various dedicated codes, or (ii) analytical estimates and models, or (iii) experiments, or (iv) found in atomic and molecular databases. The flux densities of incident particles are usually specific to the concrete problem and the system considered. The properties of atoms or molecules (energy, momentum, charge, valence, interaction potentials with other atoms in the system, etc.) involved in such quantum transformations become changed according to their final quantum states in the corresponding quantum processes. In a quantum process, the energy and momentum transferred to the system through irradiation are absorbed by the involved electronic and ionic degrees of freedom resulting in the creation of chemically reactive sites (atoms, molecules, molecular sites). The follow-up dynamics of the reactive sites is described by the classical MD at a certain thermodynamic state of the system until the system experiences further irradiation-driven quantum transformation or a chemical reaction. The chemically reactive sites are involved in the chemical reactions leading to the change of their molecular and reactive properties, and after some period of time to establishing the chemical equilibrium in the system.

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The IDMD methodology accounts for the major dissociative transformations of irradiated molecular systems and possible paths of their further reactive transformations [13] which can be simulated by means of MD with reactive force fields [12]. The necessary input parameters for such simulations can be elaborated on the basis of the quantum chemistry methods. IDMD simulations are sensitive to various statistical mechanics factors, like the number density of the reactive species, their mobility, diffusion, temperature and pressure of the medium, etc. All these factors are accounted for through either Newtonian or Langevin framework of MD. IDMD simulations allow accounting for the dynamics of secondary electrons and the mechanisms of energy and momentum transfer from the excited electronic subsystem to the system’s vibrational degrees of freedom, i.e. to its heat. For small molecular systems being in the gas phase, the ejected electrons can often be uncoupled from the system and excluded from the analysis of the system’s post-irradiation dynamics on the large time scales. For the extended molecular and condensed phase systems, the interaction of secondary electrons with the system can be treated within various electron transport theories, such as diffusion [39, 51] or Monte Carlo (MC) approach [52], and be considered as additional irradiation field imposed on the molecular system [13, 15]. Such an analysis provides the spatial distribution of the energy transferred to the medium through irradiation. Finally, immobilized electrons and electronic excitations transfer the deposited energy to the system’s heat via the electron–phonon coupling, which lasts typically up to the picosecond time scale [53]. The IDMD approach accounts for the main pathways of this relaxation process determining its duration, the temporal and spatial dependence of the amount of energy transferred into the system’s heat. As such, IDMD allows the computational analysis of physicochemical processes occurring in the systems coupled to radiation on time and spatial scales far beyond the limits of ab initio quantum mechanics based computational schemes (e.g. TDDFT, nonadiabatic MD, Ehrenfest dynamics, etc.). In spite of the model assumptions, the IDMD approach is still based on the atomistic description as any other form of traditional MD. IDMD relies on several input parameters such as the bond dissociation energies, molecular fragmentation cross sections, amount of energy transferred to the system upon irradiation and its spatial distribution, energy relaxation rate. These characteristics originating from smaller spatial and temporal scales can be obtained by accurate quantum mechanical calculations by means of the aforementioned computational schemes. If such calculations become too expensive, the required parameters can still be obtained from experimental data or by means of analytical models/methods. Due to the limited number of parameters that enter IDMD and the reactive molecular force fields, in comparison with a huge number of various systems and output characteristics accessible for simulations and analysis, the IDMD turns out to be the very useful methodology allowing the computational analysis of the system properties in the spatial and temporal domains inaccessible by other computational means. As such, it opens unique possibilities for modelling irradiation-driven modifications and chemistry of complex molecular systems beyond the capabilities of either pure quantum or pure classical MD. Moreover, the IDMD approach enables to link outputs of numerous MC codes (e.g. Geant4 [54, 55]) simulating radiation and particle

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transport in different media with the inputs of IDMD and thus to achieve the multiscale description of irradiation driven molecular dynamics, chemistry and structure formation in many different MBN systems. These important capabilities of MBN Explorer have been demonstrated in the recent work on the FEBID case study [15]. A similar methodology can be used for simulations of numerous molecular systems placed into radiation fields of different modalities, geometries and temporal profiles. The IDMD theoretical and computational framework provides a broad range of possibilities for multiscale modelling of the irradiation-driven chemical processes that underpin emerging technologies ranging from controllable fabrication of nanostructures with nanometer resolution (see e.g. Refs. [56, 57] for FEBID and Refs. [58, 59] for UVL) to radiotherapy cancer treatment (see, e.g. [39, 40, 60, 61]), both discussed further in this book. The IDMD algorithm has been validated through a number of case studies of collision and radiation processes including atomistic simulations of the FEBID process and related IDC [13, 15], collision-induced multifragmentation of fullerenes [62], electron impact-induced fragmentation of W(CO)6 [63], thermal splitting of water [12], radiation chemistry of water in the vicinity of ion tracks [64], DNA damage of various complexity induced by ions [65, 66] and other [6]. Some of these case studies list are discussed further in the follow-up chapters of this book.

3.6.7 Relativistic Molecular Dynamics The relativistic dynamics implemented in MBN Explorer permits atomistic simulations of relativistic particles’ motion in different media. This methodology enables relativistic molecular dynamics simulations of propagation of different (positively and negatively charged, light and heavy) particles through various media such as crystals, hetero-crystalline structures (including superlattices), bent and periodically bent crystals, amorphous solids, liquids, nanotubes, fullerites, biological environment and many more. This has been achieved through an appropriate choice of the interaction potential between projectile particle and target medium, and the utilization of the dynamic boundary conditions [10]. The equations of relativistic particle motion in a medium read as ⎧ ⎨

1 mγ ⎩ r˙ = v v˙ =

 F·v F−v 2 , c

(3.11)

−1/2  where γ = ε/mc2 = 1 − v 2 /c2 stands for the relativistic Lorentz factor with ε and m being the projectile energy and mass, respectively. The force F = −∇U (r) acting on the projectile is due to its interaction with the surrounding atoms. For further details, see Ref. [67].

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Applied to the propagation in a medium, Eq. (3.11) describes the classical motion of a particle in the electrostatic field of the medium atoms. They do not account for random events of inelastic scattering of a projectile from individual atoms leading to the atom’s excitation or ionization. The impact of such events on the projectile motion is twofold. First, they result in a gradual decrease in the projectile energy due to ionization losses. Second, they lead to a chaotic change in the direction of the projectile motion. Rigorous treatment of the inelastic collision evens can only be achieved by means of quantum mechanics. However, taking into account that such events are random, fast and local they can be incorporated into the classical mechanics framework according to their probabilities [68]; this approach is implemented in MBN Explorer. In the exemplar case study by these means, the channelling effect and related phenomena (channelling radiation, crystalline undulator effect, volume capture and reflection, multiple scattering, etc.), which take place when a charged particle enters a crystal at small angles with respect to a crystallographic direction, can be modelled at the atomistic level of detail [67]. The particle becomes confined and forced to move through the crystal preferably along the crystallographic direction, experiencing collective action of electrostatic field of the lattice ions. Since the field is repulsive for positively charged particles, they are steered into the interatomic region, while negatively charged projectiles move in close vicinity of ion strings or planes. The unique possibilities for multiscale computational modelling of the aforementioned physical processes and related radiation phenomena provided by MBN Explorer facilitate the design and practical realization of novel gamma-ray crystal-based light sources [67, 69].

3.6.8 Stochastic Dynamics Stochastic dynamics is designed to model the time evolution of many-particle systems stepwise in time. Instead of solving dynamical equations of motion, the stochastic dynamics approach assumes that the system undergoes a structural transformation at each step of evolution with a certain probability. The new configuration of the system is then used as the starting point for the next evolution step. The transformation of the system is governed by several kinetic rates which are chosen according to the model considered. Due to its probabilistic nature, this methodology permits studying dynamical processes involving complex molecular systems on the time scales significantly exceeding the characteristic time scales of conventional MD simulations. The stochastic dynamics suits well the situations when certain details of dynamical processes become inessential and the major transitions of a system to new states can be described by a certain number of kinetic rates that can be attributed to the main physical processes driving the system transformation. The stochastic dynamics module implemented in MBN Explorer [2, 9, 11] enables to simulate the following stochastic processes in a system of particles composed of groups of particles of different types:

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• • • • • •

free diffusion in a 2D or 3D system; peripheral diffusion of a particle bound to a group of particles; detachment (evaporation) of a particle from a group of bound particles; particle annihilation, i.e. removal of a particle from the system ; particle creation, i.e. addition of a particle to the system; particle type change, i.e. the process in which a particle of one type is transformed into a particle of another type; • fission of one particle into two different particles; • fusion of two particles into one particle; • substitution reaction involving two particles, which leads to the creation of two new particles. Each of the aforementioned processes can be characterized in terms of kinetic rates. One can design simulations with different boundary conditions and choose specific geometries for particle sources and absorbers. The well-known Kinetic Monte Carlo (KMC) method [9] corresponds to a special case of the stochastic dynamics. It allows simulations of diffusion-driven processes (Brownian-like motion) in different systems with the Monte Carlo (MC) algorithm [9, 70–72]. The diffusion coefficients for different diffusion regimes entering the KMC models can be obtained from all-atom MD simulations by analysing particle trajectories. This gives an excellent example of multiscale modelling based on descriptions of quantum interatomic interactions (sub-nanoscale), molecular dynamics (nanoscale) and KMC (mesoscale and higher).

3.6.9 Multiscale Computational Modelling Finalizing this overview of the main algorithms implemented in MBN Explorer let us emphasize a broad range of methodologies for multiscale computational modelling of MBN systems which can be exploited by means of these algorithms. The multiscale computational modelling with MBN Explorer deals with the following theoretical and computational methods and their interlinks: • Ab initio quantum methods for electronic structure calculations (e.g. Hartree–Fock theory, many-body theory, density functional theory) can be used for the validation of interatomic interaction potential functions and force fields for classical singlepoint energy calculations and all types of classical molecular dynamics operating on the time and spatial scales inaccessible for simulations based on the ab initio quantum methods; • Reactive molecular dynamics and irradiation-driven molecular dynamics enables to embed random, fast and local quantum transformation occurring in molecular systems due to chemical reactions or irradiation-induced quantum processes into the classical molecular dynamics framework. This provides possibilities for simulations of chemical and irradiation-driven transformations of MBN systems on

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the temporal and spatial scales inaccessible for simulations based on the ab initio quantum methods, quantum collision, reaction and scattering theories. Coupling of electron transport theories based on either the Monte Carlo method, or the diffusion equation approach, or any other relevant theoretical scheme with the irradiation-driven molecular dynamics approach through accounting for the electron–phonon coupling open possibilities to explore novel features in dynamics of MBN systems arising on the pico- and nanosecond time scales; Coarse graining of interatomic interactions on the basis of the pure Euler molecular dynamics or a combination of the Euler molecular dynamics with conventional molecular dynamics enables to exclude a considerable number of inessential interatomic interactions and degrees of freedom from simulations and thus proportionally increase sizes of simulated systems; Coarse graining of molecular systems by introducing their building block structure and describing building blocks (e.g. amino acids in proteins, nuclear bases in DNA or RNA molecules, etc.) as objects with certain effective mass, geometry and interactions. This allows to exclude a considerable number of inessential interatomic interactions and degrees of freedom from simulations and thus proportionally increase sizes of simulated systems and/or simulations length; In the case of relativistic dynamics of charged particles propagating through different media, the implemented algorithms enable simulations of particle motion on the macroscopically large distances with the atomistic accuracy. These algorithms enable to obtain the necessary atomistic insides into macroscopically large systems and processes occurring therein, including essential quantum processes and processes in strong atomic fields. On this basis, one can computationally explore the operation of novel intensive sources of high-energy monochromatic gammaray based on irradiation of oriented crystals of different geometry (linear, bent and periodically bent) by beams of ultrarelativistic electrons and positrons; By means of stochastic dynamics algorithms MBN Explorer allows one to perform simulations of MBN systems dynamics on the temporal and spatial scales significantly exceeding the limits for the conventional atomistic MD simulations. Such multiscale dynamics approach is ideal for systems in which details of their atomistic dynamics become excessive and the overall behaviour of a system can be reproduced through kinetic rates for the dominating modes of motion and probabilities of the key processes occurring in the system. This important feature of MBN Explorer expands significantly its application area and goes beyond the limits of the MD codes that unable to support the multiscale modelling; By increasing the size of MBN systems, one can simulate very many different types of dynamics and related phenomena that are well known from the infinite condensed matter theories (hydrodynamics, thermodynamics, acoustics, thermal conductivity, material sciences, etc.). Through establishing interlinks between these theories and molecular dynamics simulations, one derives a very useful tool for atomistic multiscale analysis of various phenomena (including their emergence) in finite MBN systems on various temporal and spatial scales.

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This is not the exclusive list of algorithms and methods suitable for multiscale computational modelling of MBN systems, but rather illustrative. Other methodologies can be suggested or developed. However, the outlined approaches clearly demonstrate how broad and versatile is the research area of multiscale computational modelling. There are many open exciting and challenging problems in it and MBN Explorer provides a solid platform for their further exploration. Acknowledgements The authors are grateful to the Deutsche Forschungsgemeinschaft for partial financial support of this work (Projects no. 415716638 and 413220201).

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39. Surdutovich, E., Solov’yov, A.V.: Multiscale approach to the physics of radiation damage with ions. Eur. Phys. J. D 68, 353 (2014) 40. Solov’yov, A.V. (ed.): Nanoscale Insights into Ion-Beam Cancer Therapy. Springer International Publishing, Cham (2017) 41. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.: Numerical Recipies. The Art of Scientific Computing. University Press, Cambridge (1988) 42. Landau, L.D., Lifshitz, E.M.: Course of Theoretical Physics, vol. 1. Mechanics. Elsevier, Oxford (2003) 43. Landau, L.D., Lifshitz, E.M.: Course of Theoretical Physics, vol. 5. Statistical Physics, Part I. Butterworth-Heinemann, Oxford (1980) 44. Berendsen, H.J.C., Postma, J.P.M., van Gunsteren, W.F., DiNola, A., Haak, J.R.: Molecular dynamics with coupling to an external bath. J. Chem. Phys. 81, 3684–3690 (1984) 45. Case, D.A., Cheatham III, T.E., Darden, T., Gohlke, H., Luo, R., Merz Jr, K.M., Onufriev, A., Simmerling, C., Wang, B., Woods, R.J.: The Amber biomolecular simulation programs. J. Comput. Chem. 26, 1668–1688 (2005) 46. Gale, J.: Gulp - a computer program for the symmetry adapted simulation of solids. JCS Faraday Trans. 93, 629 (1997) 47. Refson, K.: Moldy: a portable molecular dynamics simulation program for serial and parallel computers. Comput. Phys. Commun. 126, 309–328 (2000) 48. Koopman, E.A., Lowe, C.P.: Advantages of a Lowe-Andersen thermostat in molecular dynamics simulations. J. Comput. Chem. 124, 204103 (2006) 49. Nose, S.: A unified formulation of the constant temperature molecular-dynamics methods. J. Comput. Chem. 81, 511–519 (1984) 50. Hoover, G.W.: Canonical dynamics: equilibrium phase-space distributions. Phys. Rev. A 31, 1695–1697 (1985) 51. Surdutovich, E., Solov’yov, A.V.: Transport of secondary electrons and reactive species in ion tracks. Eur. Phys. J. D 69, 193 (2015) 52. Dapor, M.: Transport of Energetic Electrons in Solids, 3rd edn. Springer International Publishing, Cham (2020) 53. Gerchikov, L., Ipatov, A., Solov’yov, A., Greiner, W.: Non-adiabatic electron-ion coupling in dynamical jellium model for metal clusters. J. Phys. B: At. Mol. Opt. Phys. 33, 4905 (2020) 54. Agostinelli, S., Allison, J., Amako, K., Apostolakis, J., Araujo, H., Arce, P., Asai, M., Axen, D., Banerjee, S., Barrand, G., Behner, F., Bellagamba, L., Boudreau, J., Broglia, L., Brunengo, A., Burkhardt, H., Chauvie, S., Chuma, J., Chytracek, R., Cooperman, G.: GEANT4 - a simulation toolkit. Nucl. Instrum. Methods A 506, 250–303 (2003) 55. Allison, J., et al.: Recent developments in GEANT4. Nucl. Instrum. Methods A 835, 186–225 (2016) 56. Huth, M., Porrati, F., Schwalb, C., Winhold, M., Sachser, R., Dukic, M., Adams, J., Fantner, G.: Focused electron beam induced deposition: a perspective. Beilstein J. Nanotechnol. 3, 597–619 (2012) 57. Utke, I., Hoffmann, P., Melngailis, J.: Gas-assisted focused electron beam and ion beam processing and fabrication. J. Vac. Sci. Technol. B 26, 1197–1276 (2008) 58. Wu, B., Kumar, A.: Extreme ultraviolet lithography: a review. J. Vac. Sci. Technol. B 25, 1743–1761 (2007) 59. Hawryluk, A.M., Seppala, L.G.: Soft x-ray projection lithography using an x-ray reduction camera. J. Vac. Sci. Technol. B 6, 2162–2166 (1988) 60. Schardt, D., Elsässer, T., Schulz-Ertner, D.: Heavy-ion tumor therapy: physical and radiobiological benefits. Rev. Mod. Phys. 82, 383–425 (2010) 61. Surdutovich, E., Solov’yov, A.: Multiscale modeling for cancer radiotherapies. Cancer Nanotechnol. 10, 6 (2019) 62. Verkhovtsev, A., Korol, A.V., Solov’yov, A.V.: Classical molecular dynamics simulations of fusion and fragmentation in fullerene-fullerene collisions. Eur. Phys. J. D 71, 212 (2017) 63. de Vera, P., Verkhovtsev, A., Sushko, G., Solov’yov, A.V.: Reactive molecular dynamics simulations of organometallic compound W(CO)6 fragmentation. Eur. Phys. J. D 73, 215 (2019)

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64. de Vera, P., Surdutovich, E., Mason, N.J., Currell, F.J., Solov’yov, A.V.: Simulation of the ion-induced shock waves effects on the transport of chemically reactive species in ion tracks. Eur. Phys. J. D 72, 147 (2018) 65. Friis, I., Verkhovtsev, A., Solov’yov, I.A., Solov’yov, A.V.: Modeling the effect of ion-induced shock waves and DNA breakage with the reactive CHARMM force field. J. Comput. Chem. 41, 2429–2439 (2020) 66. Friis, I., Verkhovtsev, A.V., Solov’yov, I.A., Solov’yov, A.V.: Lethal DNA damage caused by ion-induced shock waves in cells. Phys. Rev. E 104, 054408 (2021) 67. Korol, A.V., Solov’yov, A.V., Greiner, W.: Channeling and Radiation in Periodically Bent Crystals, 2nd edn. Springer, Berlin (2014) 68. Korol, A.V., Solov’yov, A.V., Greiner, W.: The influence of the dechannelling process on the photon emission by an ultra-relativistic positron channelling in a periodically bent crystal. J. Phys. G: Nucl. Part. Phys. 27, 95–125 (2001) 69. Korol, A.V., Solov’yov, A.V.: Crystal-based intensive gamma-ray light sources. Eur. Phys. J. D 74, 201 (2020) 70. Solov’yov, I., Solov’yov, A.V., Kebaili, N., Masson, A., Brechignac, C.: Thermally induced morphological transition of silver fractals. Phys. Status Solidi B 609, 251 (2014) 71. Dick, V.V., Solov’yov, I.A., Solov’yov, A.V.: Fragmentation pathways of nanofractal structures on surfaces. Phys. Rev. B 84, 115408 (2011) 72. Moskovkin, P., Panshenskov, M.A., Lucas, S., Solov’yov, A.V.: Simulation of nanowire fragmentation by means of kinetic Monte Carlo approach: 2D case. Phys. Status Solidi B 251, 1456–1462 (2014)

Chapter 4

Structure and Dynamics of Bio- and Macromolecules Alexey V. Verkhovtsev, Ilia A. Solov’yov, and Andrey V. Solov’yov

Abstract This chapter overviews the computational techniques and theoretical models used for exploring structure and dynamics of complex biomolecular systems at the atomistic level of detail. Particular focus is put on the application of statistical mechanics methods combined with the classical molecular mechanics approach to the description of phase transitions in polypetides and proteins. The molecular mechanics approach permits simulating a large variety of biomacromolecular systems (both in vacuum and in ubiquitous environments) and their transformations at different thermal and biologically relevant conditions as well as at various external stresses. This chapter presents several illustrative examples of such computational research related to protein folding, unbinding of protein-ligand complex, and DNA unzipping.

4.1 Introduction Biomolecular systems are highly dynamical in nature, contrary to what may be implied by the static illustrations of proteins, nucleic acids, and other biomolecular structures printed in textbooks. Life occurs above absolute zero, and the biomolecular components in and around a cell—proteins, nucleic acids, lipids, carbohydrates—are continuously sampling, via intramolecular interactions, the myriad conformational states that are thermally accessible at physiological temperatures. Simultaneously, a given biomolecule also samples (and is sampled by) a rapidly fluctuating local environment comprised of other biopolymers, small molecules, water, ions, etc. that A. V. Verkhovtsev (B) · A. V. Solov’yov MBN Research Center gGmbH, Altenhöferallee 3, 60438 Frankfurt am Main, Germany e-mail: [email protected] A. V. Solov’yov e-mail: [email protected] I. A. Solov’yov Department of Physics, Carl von Ossietzky Universität Oldenburg, Carl-von-Ossietzky-Str. 9-11, 26129 Oldenburg, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 I. A. Solov’yov et al. (eds.), Dynamics of Systems on the Nanoscale, Lecture Notes in Nanoscale Science and Technology 34, https://doi.org/10.1007/978-3-030-99291-0_4

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diffuse over the nanometer distances, leading to intermolecular interactions and the formation of supramolecular assemblies. These intra- and intermolecular contacts are governed by the same physical principles (forces, energetics) that characterize individual molecules and interatomic interactions, thereby enabling a unified picture of the physical basis of molecular interactions from a small set of fundamental principles. From just a few physical laws, and several plausible assumptions describing covalent and non-covalent (non-bonded) interactions and their relative magnitudes, much can be learnt about molecular interactions and dynamics as the means by which proteins fold into thermodynamically stable “native” structures (that are structures with the lowest free energy under a given set of conditions), bind other proteins or small molecules to trigger various cellular responses, act as allosteric enzymes, participate in metabolic pathways and regulatory circuits, and so on—in short, all of cellular biochemistry. Computational approaches are well suited for studying molecular interactions, from the intramolecular conformational sampling of individual proteins (such as membrane receptors or ion channels) to the diffusion dynamics and intermolecular collisions that occur in the early stages of formation of cellular-scale assemblies. In this chapter, we discuss the dynamics of biomolecular systems simulated within the classical molecular mechanics approach, achieved through the molecular mechanics force fields. Several examples of the simulations of such systems and the analysis of simulation outcomes are presented. This chapter gives also an overview of an interdisciplinary approach combining the methods of classical molecular dynamics, statistical mechanics, computational chemical physics, and quantum mechanics aiming to provide a comprehensive description of phase transitions and cooperative dynamics in polypeptides, proteins, and other biomacromolecules. Understanding such structural transformations reveals a tremendous amount of useful information about the properties of these systems, including how they function and how they are regulated.

4.2 Methodologies for Describing Structure and Dynamics of Biomolecular Systems Biochemical processes occur on different scales of length and time [1] ranging from a few angstroms, the size of the active site of proteins where the ultrafast triggering steps usually take place, up to the level of the cells and organs, where their macroscopic effects are detectable by the naked eye. Intermediate steps are the structural rearrangement of biomolecules (approximately nanometer and 10–100 ns scales), their aggregation/separation and folding/unfolding (10 nm to micrometer and greater than microsecond) and internal cell diffusion and dynamics (micrometers to millimeters and milliseconds to hours). This inherent hierarchical organization is responsible for the complexity of living matter: a single process involves a multiscale cascade of events whose description requires the combination of different methodologies in

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multiscale approaches [2–4]. At any resolution, the quality of a model depends on the accuracy with which the two following issues are addressed: the description of the interactions and the sampling of the configurations of the system. In this respect, there are a few concepts that iteratively occur. First concept regards the potential energy surface (PES). The method used to evaluate the PES strictly depends on the resolution level. For small molecules (up to several tens of atoms), both the nuclear and electronic degrees of freedom can and must be explicitly treated in order to describe the electronic structure of the molecule. The concept of the potential energy surface is related to the Born–Oppenheimer approximation assuming that the much faster electrons adiabatically adjust their motion to that of the atomic nuclei (see Chap. 2). Thus, at any time, the Schrödinger equation for the electron system is solved in the external field generated by the atomic nuclei considered as frozen. As a result, one obtains a nuclear-configuration-dependent set of energy eigenvalues E i ({Ri }) that define the PES of the ground and excited states. In turn, the PESs are effective electronic structure-dependent potential energy functions that determine the dynamics of the nuclei. Being computationally demanding, the methods of quantum mechanics cannot be applied straightforwardly to describe the dynamical behavior of large molecular systems such as proteins. However, one can distinguish the principal coordinates in the molecules that correspond to the quantum nature of the covalent chemical bonds in the system. These coordinates are usually the distances between atoms, the angles between two neighboring chemical bonds, and the dihedral angles that correspond to the twisting along chemical bonds. The dynamics of the system in the coordinates of bond lengths, angles between bonds and dihedral angles can be described classically at moderate temperatures (ω  2kB T , i.e., at temperatures at which one can omit the quantum corrections to the vibrations). Such a description implies the construction of a classical Hamiltonian of the system that describes the interactions between the atoms. The classical Hamiltonian for the description of the dynamics of atoms in a molecule is usually constructed on the basis of so-called Molecular Mechanics (MM) potential, which is discussed in Sect. 4.2.1. The parameters of the MM potential that describe the “stiffness” of chemical bonds, angles between bonds, etc. are usually obtained on the basis of quantum mechanical calculations of the fragments of a large molecule. The important goal of numerous current investigations is to generate new, detailed knowledge about the nanoscale mechanisms leading to global conformational changes of single biomacromolecules and, in particular, mechanisms of polypeptide and protein folding and general structural transitions in proteins and biomacromolecular complexes. Building a comprehensive understanding of phasetransition-like phenomena in finite systems is a challenging problem, with a variety of applications in biophysics and nanosystems, including protein folding and misfolding. For structural transitions in complex biomacromolecules, neither an analytical solution nor a numerical computation is feasible. Even for the most advanced computers, molecular dynamics (MD) simulations are nowadays challenging for timescales longer than microseconds. Furthermore, such simulations often provide inadequate statistics for a proper sampling. As a

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result, alternative approximations are clearly required. Statistical mechanics provides a solid framework for dealing with such processes. It defines the partition function that is the sum over all possible conformational states of the system with the corresponding statistical weights. Knowing the partition function of the system one can describe all its thermodynamic characteristics, e.g., evaluate its energy and heat capacity at different temperatures. This methodology is briefly described in Sect. 4.2.2. Establishing fundamental connections between the statistical mechanics methods for calculating partition functions with the modern computational techniques for MD is a promising research direction. As illustrated further in this chapter via several case studies, the combined statistical mechanics and MD methods provide a useful tool for the quantitative description of phase transitions and cooperative changes in large biomacromolecules.

4.2.1 Molecular Mechanics Force Field Molecular mechanics (MM) potential is a special form of the potential energy of the system widely used to describe the structure and properties (see also Chap. 5) of macromolecular systems, such as polypeptides, proteins, lipids, and DNA [5–8]. The principal difference between the MM potential and many other interatomic potentials (see Chap. 3) is that the MM potential requires also the specification of the system’s molecular topology, a set of rules that impose constraints on the system and permit maintaining its natural shape as well as its mechanical and thermodynamical properties. The basic idea behind the MM potential is to account for all physically important interactions in a molecular system, i.e., the covalent interactions and the long-range non-bonded interactions, using a simple parametric form of the potential. The total energy of the system interacting via the MM potential can be written as Utot = Ucov + UvdW + UC ,

(4.1)

where the terms Ucov , UvdW and UC describe the energy of the covalent, van der Waals, and electrostatic (Coulomb) interactions, respectively. The van der Waals interaction between two neutral atoms or molecules is usually modeled in terms of the Lennard–Jones (LJ) potential:  ULJ = 

r0 ri j

12



r0 −2 ri j

6  ,

(4.2)

where  is the depth of the potential energy well, r0 is the equilibrium distance, and ri j is the distance between atoms. The term Ucov in Eq. (4.1) parametrizes the energy associated with the covalent interactions. It is only applicable to the systems with a predefined molecular topology

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since the latter defines the rules which determine the chemical bonds in the system. Thus, the energy of the covalent interactions is a sum over all such interactions. Real chemical bonds have a complex quantum mechanical nature. Therefore, the potential Ucov provides only a relatively simple parametrization describing the covalent interactions through a number of empirical parameters and fitting functions. The parameters of the potential are derived either from experimental measurements of crystallographic structures, infrared spectra or on the basis of quantum mechanical calculations carried out for relatively small systems [5, 9, 10]. The potential energy Ucov is constructed as follows: Ucov =

Nb 

Na Nub

2  2  2  (a)  (ub) (ub) rik(ub) − rik,0 r θ ki(b) − r + k − θ + kik ij i j,0 i jk i jk,0 j i jk

α=1 i, j∈α

+

Nd  α=1 i, j,k,l∈α

α=1 i, j,k∈α

α=1 i,k∈α

Nid    2 ki(d) ki(id) jkl 1 + cos(n i jkl χi jkl − δi jkl ) + jkl Si jkl − Si jkl,0 .

(4.3)

α=1 i, j,k,l∈α

(a) (ub) (d) (id) (ub) Here ki(b) j , ki jk , kik , ki jkl , ki jkl , ri j,0 , θi jk,0 , rik,0 , Si jkl,0 , n i jkl , δi jkl are the parameters of the potential. The collection of parameters for different molecular systems is often referred to as a force field [5, 10]. Different force fields have been developed during the years and include, for example, the well-established CHARMM [5] and AMBER [10] force fields. The quantities ri j , θi jk , χi jkl , Si jkl are four independent variables (coordinates) which are illustrated by Fig. 4.1. The first term on the right-hand side of Eq. (4.3) describes the potential energy arising due to stretching of the bonds between pairs of atoms in the system (see Fig. 4.1a). Here, ki(b) j is the stiffness parameter, ri j,0 is the equilibrium distance, and ri j is the actual distance between atoms i and j, which form the bond with index α. The summation is carried out over all topologically defined bonds of the total number Nb . The second term represents the potential energy arising due to the change of angles between every topologically defined triple of atoms in the system. Here, θi jk stands for the angle between ri j and r jk (see Fig. 4.1), θi jk,0 is the equilibrium value of this angle, and ki(a) jk is the stiffness parameter. The summation is performed over different triples of the total number Na . The third term, commonly referred to as the Urey-Bradley term, is also related to the “angular” interactions in the system. It corresponds to a fictional interaction between the first, i, and the third, k, atoms in every topologically defined triple (ub) is the stiffness parameter for this of atoms in harmonic approximation. The kik (ub) interaction, rik,0 is the equilibrium distance between the atoms, and Nub is the total number of such interactions. The fourth term in Eq. (4.3) describes the torsion energy. It is characterized through a dihedral angle χi jkl formed by four atoms, i, j, k, and l, connected via chemical bonds (see Fig. 4.1). The “multiplicity” n i jkl is typically set to 1, 2, or 3, while ki(d) jkl is

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Fig. 4.1 a Internal coordinates for molecular mechanics interactions: ri j governs bond stretching; θi jk represents the angle term; χi jkl gives the dihedral angle; the small out-of-plane angle Si jkl is governed by the so-called “improper” dihedral angle. b Dependencies of the potential energy on coordinates used in the molecular mechanics potential, Eq. (4.3), describing the bonded, angle, dihedral angle, and improper dihedral angle interactions. Reproduced from Ref. [11] with permission from Springer Nature

the dihedral spring constant. The potential energy corresponding to torsion degrees of freedom is usually assumed to be periodic because several stable conformations of the molecule with respect to these degrees of freedom are possible [5, 10]. The last term in Eq. (4.3) describes the so-called improper dihedral angles, which are used in the molecular topology to maintain planarity. As such, the harmonic form with a large spring constant ki(id) jkl and equilibrium value Si jkl,0 (typically equal to zero) is used to restrain configuration of an atom and three atoms bonded to it. Similar to the case of dihedral angles, Si jkl is the angle between the plane containing the first

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three atoms with indices (i jk) and the plane containing the last three atoms with the indices ( jkl) which define this dihedral angle. The form of the potential energy functions, which are used to describe the components of Ucov in the standard CHARMM force field, is demonstrated in Fig. 4.1b. To model molecular interactions in a wider range of biological processes, the standard molecular mechanics potentials should be improved. The CHARMM force field employs harmonic approximation for describing the interatomic interactions, thereby limiting its applicability to small deformations of the molecular system. In case of larger perturbations, the potential should decrease to zero as the valence bonds rupture. The rupture of valence bonds causes the involved angular and dihedral interactions to vanish as well. In Ref. [11], the so-called reactive CHARMM (rCHARMM) force field was presented that permit classical MD simulations of the rupture and formation of covalent bonds by means of the MBN Explorer software package [12]. This approach goes beyond the harmonic approximation, thus describing the physics of molecular dissociation more accurately, and permits the construction of dynamic molecular topology, which instructs MBN Explorer how the existing covalent bonds can break and new covalent bonds can be formed. These features make MBN Explorer rather unique, for example, for simulating irradiation- and collision-induced biodamage by means of classical MD. Further details on rCHARMM and its application for modeling radiation-driven and chemical transformations in complex molecular and biomolecular systems are given in Chap. 8.

4.2.2 Statistical Mechanics Model for Studying Phase Transitions in Polypeptide Chains Phase transitions (PT) in finite complex molecular systems, i.e., the transition from a stable three-dimensional (3D) molecular structure to a random coil state or vice versa (also known as the folding process), have a long-standing history of investigation. The PT of this nature occur or can be expected in many different complex molecular systems and in nano-objects, such as polypeptides, proteins, polymers, DNA, fullerenes, nanotubes. They can be understood as the first-order PTs, which are characterized by a rapid growth of the system’s free energy at a certain temperature. As a result, the heat capacity of the system as a function of temperature acquires a sharp maximum at the PT temperature. In Ref. [13] a theoretical method based on the statistical mechanics was developed for treating the α-helix↔random coil transition in polypeptide chains. The model describes essential thermodynamical properties of the system such as heat capacity, the phase-transition temperature, and others from the analysis of the polypeptide PES calculated as a function of two dihedral angles responsible for the polypeptide twisting. The suggested theory is general and with some modification can be applied for the description of phase transitions in other complex molecular systems.

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Section 4.4 describes the application of this model for studying folding of proteins. The key principles of this theory are outlined below. Consider a polypeptide consisting of n amino acids. The polypeptide can be found in one of its numerous isomeric states that have different energies. A group of isomeric states with similar characteristic physical properties is called a phase state of the polypeptide. Thus, a regular bounded α-helix state corresponds to one phase state of the polypeptide, while all possible unbounded random conformations can be denoted as the random coil phase state. By definition, the phase transition (PT) is a transformation of the polypeptide from the regular state to a random coil conformation. To study thermodynamic properties of the system, one needs to investigate its PES with respect to all degrees of freedom. A Hamiltonian function of a polypeptide chain is constructed as a sum of the potential, kinetic, and vibrational energy terms. For a polypeptide chain in a particular conformational state j consisting of n amino acids and N atoms, one obtains Hj =

−6

3N  pi2 P2 1 ( j) 2 ( j) ( j) I1 1 + I2 22 + I3 23 + + U ({x}) , + 2M 2 2m i i=1 ( j)

(4.4)

where P, M, I1,2,3 , 1,2,3 are the momentum of the whole polypeptide, its mass, its three main momenta of inertia, and its rotational frequencies. pi , xi , and m i are the momentum, the coordinate, and the generalized mass describing the motion of the system along the ith degree of freedom. U ({x}) is the potential energy of the system, being the function of all atomic coordinates in the system. All degrees of freedom in a polypeptide can be grouped into two classes: “stiff” (“hard”) and “soft” degrees of freedom. The degrees of freedom corresponding to the variation of bond lengths, angles, and improper dihedral angles (see Fig. 4.1) are called “stiff”, while degrees of freedom corresponding to the angles ϕi and ψi (see Fig. 4.2) are classified as “soft” degrees of freedom. The “stiff” degrees of freedom can be treated within the harmonic approximation because the energies needed for a noticeable change of the system structure with respect to these degrees of freedom are

Fig. 4.2 Dihedral angles ϕ and ψ used for characterization of the secondary structure of a polypeptide chain. Reproduced from Ref. [13] with permission from Springer Nature

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about several eV which is significantly larger than the characteristic thermal energy of the system at room temperature, kB T = 0.026 eV. The Hamiltonian of the polypeptide can be rewritten in terms of the “soft” and “stiff” degrees of freedom. Transforming the set of Cartesian coordinates {x} to a set of generalized coordinates {q}, corresponding to the “soft” and “stiff” degrees of freedom, one obtains Hj =

P2 1 ( j) 2 ( j) ( j) I1 1 + I2 22 + I3 23 + 2M 2 +

ls ls  

gi j

pis p sj 2

i=1 j=l

+

l s +l h

l s +l h

gi j

i=ls +1 j=ls +1

+

l ls s +l h 

gi j pis p hj +

i=1 j=ls +1

pih p hj 2

+ U ({q s }, {q h }) ,

(4.5)

where q s and q h are the generalized coordinates corresponding to the “soft” and “stiff” degrees of freedom, and p s and p h are the corresponding generalized momenta. ls and lh are the number of the “soft” and “stiff” (“hard”) degrees of freedom in the system, satisfying the relation 3N − 6 = ls + lh . U ({q s }, {q h }) in Eq. (4.5) is the potential energy of the system as a function of the “soft” and “stiff” degrees of freedom. 1/gi j has a meaning of the generalized mass, while gi j is defined as follows: gi j =

3N −6  λ=1

1 ∂qi ∂q j . m λ ∂ xλ ∂ xλ

(4.6)

Here xλ and m λ are the generalized coordinate in the Cartesian space and the generalized mass of the system, corresponding to the degree of freedom with index λ. qi and q j denote the “soft” or the “stiff” generalized coordinates in the transformed space. The motion of the system with respect to its “soft” and “stiff” degrees of freedom occurs on the different timescales as was discussed in [15]. The typical oscillation frequency corresponding to the “soft” degrees of freedom is on the order of 100 cm−1 , while for the “stiff” degrees of freedom it is more than 1000 cm−1 [15]. Thus, the motion of the system with respect to the “soft” degrees of freedom is uncoupled from the motion of the system with respect to the “stiff” degrees of freedom. Therefore, the fifth term in Eq. (4.5) which describes the kinetic energy of the “stiff” motions in the polypeptide can be diagonalized. The corresponding set of coordinates {q˜ s } describes the normal vibration modes in the “stiff” subsystem:

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Hj =

1 ( j) 2 P2 ( j) ( j) + I1 1 + I2 22 + I3 23 + 2M 2

 2  2  l h  p˜ ih μih ωi2 q˜ih + 2 2μih i=1 +

ls ls  

gi j

i=1 j=1

pis p sj 2

+ U ({χ }) + U ({ϕ, ψ}) .

(4.7)

Here ωi and μih are the frequencies of the ith “stiff” normal vibrational mode and the corresponding generalized mass. Note that the fourth term in Eq. (4.5) vanishes if the “soft” and the “stiff” degrees of freedom are uncoupled. The last two terms in Eq. (4.7) describe the potential energy of the system with respect to the “soft” degrees of freedom. For every amino acid, there are at least two “soft” degrees of freedom, corresponding to the angles ϕi and ψi (see Fig. 4.2). Some additional “soft” degrees of freedom involve the rotation of the side radicals in amino acids. A typical example is the angle χi , which describes the twisting of the side-chain radical β along the Ciα − Ci bond (see Fig. 4.2). The angle χi is defined as the dihedral angle  β β between the planes formed by the atoms (Ci − Ciα − Ci ) and by the bonds Ciα − Ci β β and Ci − Hi1 . Note that the notations χ , ϕ, and ψ are used for the simplicity and for the further explanation of the theory. The set of these dihedral angles builds up the set of “soft” degrees of freedom of the polypeptide: {q s } ≡ {χ , ϕ, ψ}. If the system is considered in the vicinity of its equilibrium state, then the motion of the polypeptide with respect to the “soft” degrees of freedom can be considered as the motion of the system of coupled non-linear oscillators. In the vicinity of the system’s equilibrium state, the generalized mass can be written as follows:   ls   s  ∂ 1/gi j  1 1  s + qk − qks0 + · · · , (4.8) =  s gi j ∂qk  s s gi j {qi } k=1

0

qk =qk

0

where qks0 denotes the value of the kth “soft” degree of freedom at the equilibrium position. The second and higher terms in Eq. (4.8) describe the dependence of the generalized mass on coordinates and can be neglected if the system is in the vicinity of its equilibrium. All the information about the non-linearity of the oscillations is contained in the potential energy functions U ({χ }) and U ({ϕ, ψ}) in Eq. (4.7). All thermodynamic properties of a system are determined by its partition function, which can be expressed via the system’s Hamiltonian in the following form [16]:  Z=

  H d , exp − kB T

(4.9)

where H is the Hamiltonian of the system, kB and T are the Boltzmann constant and the temperature, respectively, and d is an element of the phase space. Substituting (4.7) into (4.9), one obtains an expression for the partition function of a polypeptide

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in a particular conformational state j. Thus, the partition function of the system can be factored as follows: Z=

1 Z1 · Z2 · Z3 · Z4 · Z5, (2π )3N

(4.10)

where

 M21 1 M22 P2 − Z 1 = exp + + − ( j) ( j) kB T 2M 2I1 2I2  M23 + ( j) d3 P · d3 Q · d3 M · d3  = 2I3  ( j) ( j) = 64π 5 V j M 3/2 I1 I2 I3 ( j) (kB T )3





lh 1  exp − kB T i=1

Z2 =



p˜ ih

2

2μih

 2  μih ωi2 q˜ih + 2

(2π kB T )lh , dlh p˜ h · dlh q˜ h = lh i=1 ωi  Z3 = =



(4.11)

(4.12)

 ls  s 2 p˜ i 1  exp − dls p˜ s = kB T i=1 2μis ls   μis , (2π kB T ) ls

(4.13)

i=1

 Z4 =

 Z5 =

  U ({χ}) ˜ exp − dlχ χ˜ s , kB T

(4.14)

 ˜ U ({ϕ, ˜ ψ}) exp − dlϕ ϕ˜ s · dlψ ψ˜ s . kB T

Z 1 , Eq. (4.11), describes the contribution to the partition function originating from the motion of the polypeptide as a rigid body. Here V j is the specific volume of the polypeptide in conformational state j and M is the angular momenta of the polypeptide. Z 2 , Eq. (4.12), accounts for the “stiff” degrees of freedom in the polypeptide.

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Z 3 , Eq. (4.13) describes the contribution of the kinetic energy of the “soft” degrees of freedom to the partition function. Z 4 , Eq. (4.14), and Z 5 , Eq. (4.15) describe the contribution of the potential energy of the “soft” degrees of freedom to the partition function. Integrating over the phase space in Eqs. (4.11)–(4.15) is performed over generalized coordinates and momentum space. The final expression for the partition function of a polypeptide reads as (the complete derivation can be found in Ref. [13])

 3N −3−ls /2

Z = AB(T ) (kB T )

Z un

+β

Z bn−1 Z u

+

n−2 

 (n

− i)Z bi Z un−i

i=4

+

(n−3)/2  i=2

βi

n−i−3  k=i

⎤ (k − 1)!(n − k − 3)! Z k+3i Z un−k−3i ⎦ . i!(i − 1)!(k − i)!(n − k − i − 3)! b

(4.15)

The first and the second terms in the square brackets describe the partition function of the polypeptide in the random coil and the α-helix phases, respectively. The third term accounts for the situation of the phases coexistence. The lowest limit of summation in this term corresponds to the shortest α-helix that has only i = 4 amino acids. The last term in the square brackets accounts for the polypeptide conformations in which a number of amino acids in the α-helix conformation are separated by amino acids in the random coil conformation. The outer sum over i in this term runs over the separated helical fragments of the polypeptide, while the inner sum goes over individual amino acids in the corresponding fragment. The conformations with two or more helical fragments are energetically unfavorable (see the discussion in Ref. [13] and in Sect. 4.3). Therefore, the last term can be omitted when constructing the partition function. The factor A in Eq. (4.15) is determined by the specific volume, momenta of inertia, and frequencies of the normal vibration modes of the polypeptide in different conformations, ls is the total number of the “soft” degrees of freedom in the system, and the function B(T ) describes the rotation of the side radicals in the polypeptide [13]. The quantity β in Eq. (4.15) stands for the helix initiation factor: β = exp (−3E HB /kB T ) ,

(4.16)

where E HB is the energy of a single hydrogen bond. The notations Z b and Z u stand for the contributions to the partition function arising from a single amino acid in the bounded (“b”) or unbounded (“u”) states, respectively. They can be written as π π Z b,u = −π −π

  b,u (ϕ, ψ) dϕ dψ, exp − kT

(4.17)

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where b (ϕ, ψ) and u (ϕ, ψ) are the potential energies of a single amino acid in the bounded and unbounded conformations calculated as functions of the twisting degrees of freedom ϕ and ψ. These degrees of freedom are defined for each amino acid of the polypeptide (except for the boundary ones) and are described by two dihedral angels ϕi and ψi which are defined by four neighboring atoms in the polypeptide chain, see Fig. 4.2. The angle ϕi is defined as the dihedral angle between the planes   formed by the atoms (Ci−1 − Ni − Ciα ) and (Ni − Ciα − Ci ), while ψi is the dihedral   angle between the (Ni − Ciα − Ci ) and (Ciα − Ci − Ni+1 ) planes. For an unambiguous definition most commonly used [17–20], ϕi and ψi are counted clockwise if one looks at the molecule from its NH2 -terminus. Substituting Eq. (4.17) into Eq. (4.15), one obtains the final expression for the partition function of a polypeptide experiencing an α-helix↔random coil PT. An alternative theoretical approach for the study of α-helix↔random coil PT in polypeptides was introduced by Zimm and Bragg [21]. It is based on the construction of the partition function of a polypeptide involving two parameters s and σ , where s describes the contribution of a bounded amino acid relative to that of an unbounded one, and σ describes the entropy loss caused by the initiation of the α-helix formation. Assuming that the polypeptide has a single helical region, the partition function derived within the Zimm–Bragg theory reads as Q =1+σ

n−4 

(n − k − 3)s k ,

(4.18)

k=1

where n is the number amino acids in the polypeptide. The partition function (4.15) can be rewritten in a similar form:  Z = 1 + βs (T ) 3

n−4 

 (n − k − 3)s (T ) Z un (T ). k

(4.19)

k=1

Here s(T ) = Z b /Z u with Z b,u defined in (4.17). By comparing Eqs. (4.18) and (4.19), one can evaluate the Zimm–Bragg parameter σ as (4.20) σ (T ) = β(T )s 3 (T ), with β(T ) from (4.16).

4.3 Phase and Structural Transitions in Polypeptide Chains The partition function of a polypeptide chain constructed by means of the theoretical method described in Sect. 4.2.2 permits the complete thermodynamic description of the system, which includes calculation of all essential thermodynamic variables

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and characteristics, such as heat capacity, PT temperature, free energy, etc. This section illustrates the applicability of the method for the description of the PT in polyalanine chains of different lengths. The theoretical predictions are successfully compared with the results of several independent experiments and with the results of MD simulations.

4.3.1 Energetics of Alanine Polypeptide To construct the partition function Eq. (4.15), one needs to calculate the PES of a single amino acid in the bounded, b (ϕ, ψ), and unbounded, u (ϕ, ψ), conformations versus the twisting degrees of freedom ϕ and ψ (see Fig. 4.2). The potential energies of alanine in different conformations determine the Z b and Z u contributions defined in Eq. (4.17). The PES of alanine depends both on the conformation of the polypeptide and on the amino acid index in the chain. The PES for different amino acids of the 21-residue alanine polypeptide calculated as a function of twisting dihedral angles ϕ and ψ is shown in Fig. 4.3. These surfaces were calculated with the use of the CHARMM27 force field for a polypeptide in the α-helix conformation. The PESs presented in Fig. 4.3a–e correspond to the variation of the twisting angles in the second, third, fourth, fifth, and tenth amino acids of Ala21 , respectively. Amino acids are numbered starting from the NH2 terminus of the polypeptide. On the PES corresponding to the tenth amino acid in the polypeptide (see Fig. 4.3e), one can identify a prominent minimum at ϕ = −81◦ and ψ = −71◦ . This minimum corresponds to the α-helix conformation of the corresponding amino acid, and energetically, to the most favorable amino acid configuration. In the α-helix conformation, the tenth amino acid is stabilized by two hydrogen bonds (see Fig. 4.4). With the change of the twisting angles ϕ and ψ, these bonds become broken and the energy of the system increases. The tenth alanine can form hydrogen bonds with the neighboring amino acids only in the α-helix conformation, because all other amino acids in the polypeptide are in this particular conformation. This fact is clearly seen from the corresponding PES, Fig. 4.3e, where all local minima have energies significantly higher than the energy of the global minima (the energy difference between the global minimum and a local minimum with the closest energy is E = 0.736 eV, which is found at ϕ = 44◦ and ψ = −124◦ ). The PES depends on the amino acid index in the polypeptide as it is clearly seen from Fig. 4.3. The three boundary amino acids in the polypeptide form a single hydrogen bond with their neighbors (see Fig. 4.4) and, therefore, are less bounded than the amino acids inside the polypeptide. The change in the twisting angles ϕ and ψ in the corresponding amino acids leads to the breaking of these bonds, hence increasing the energy of the system. However, the boundary amino acids are more flexible than those inside the polypeptide chain, and, therefore, their PES is smoother. Figure 4.3 shows that the PESs calculated for the fourth, fifth, and the tenth amino acids are very close and have minor deviations from each other. Therefore, the PESs

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Fig. 4.3 Potential energy surfaces for different amino acids of the Ala21 polypeptide calculated as the function of twisting dihedral angles ϕ and ψ in: a second, b third, c fourth, d fifth, and e tenth alanine molecule. Amino acids are numbered starting from the NH2 terminal of the polypeptide. Energies are given with respect to the lowest energy minimum of the PES in eV. Reproduced from Ref. [14] with permission from Springer Nature

Fig. 4.4 Alanine polypeptide in the α-helix conformation. Dashed lines show the hydrogen bonds in the system. The figure shows that the second alanine forms only one hydrogen bond, while the fifth alanine forms two hydrogen bonds with the neighboring amino acids. Reproduced from Ref. [14] with permission from Springer Nature

for all amino acids in the polypeptide, except the boundary ones, can be considered identical. Each amino acid inside the polypeptide forms two hydrogen bonds. However, since these bonds are shared by two amino acids, there is only effectively one hydrogen bond per amino acid (see Fig. 4.4). Therefore, to determine the potential energy surface of a single amino acid in the bounded and unbounded conformations, the

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potential energy surface calculated for the second amino acid of the alanine polypeptide (see Fig. 4.3a) was used, because only this amino acid forms single hydrogen bond with its neighbors. The PES of the second amino acid Fig. 4.3a has a global minima at ϕ = −81◦ and ψ = −66◦ and corresponds to the bounded conformation of the alanine. Therefore, the part of the PES in the vicinity of this minima corresponds to the PES of the bounded state of the polypeptide, b (ϕ, ψ). The potential energy of the bounded state is determined by the energy of the hydrogen bond, which for an alanine is equal to E HB = 0.142 eV. This value is obtained from the difference between the energy of the global minima and the energy of the plateau at ϕ ∈ (−90◦ .. − 100◦ ) and ψ ∈ (0◦ ..60◦ ) (see Fig. 4.3a). Thus, the part of the PES which has an energy less than E HB corresponds to the bounded state of alanine, while the part with energy greater than E HB corresponds to the unbounded state. Figure 4.5 presents the PESs for alanine in both the bounded (a) and unbounded (b) conformations. The PESs were calculated from the PES for the second amino acid in the polypeptide, which is shown in Fig. 4.5(c).

Internal Energy of Alanine Polypeptide Knowing the PES for all amino acids in the polypeptide, one can construct the partition function of the system using Eq. (4.15). Figure 4.5a, b shows the dependence of b (ϕ, ψ) and u (ϕ, ψ) on the twisting angles. These quantities define the contributions of the bounded and unbounded states of the polypeptide to the partition function of the system, see Eq. (4.17) where the integrals are to be evaluated numerically. Once the partition function is constructed, one can define all essential thermodynamical characteristics of the system. The first-order PT is characterized by an abrupt change of the internal energy of the system with respect to its temperature. In the course of the first-order PT, the system either absorbs or releases a fixed amount of energy while the heat capacity as a function of temperature has a pronounced peak [16, 17, 22, 23]. The manifestation of these peculiarities for alanine polypeptide chains of different lengths is discussed below. Figure 4.6 shows the dependencies of the internal energy on temperature calculated for the Alan polypeptides consisting of n = 21, 30, 50, and 100 amino acids. Dashed lines correspond to the results obtained using the statistical approach, while the symbols show the results of MD simulations. Figure 4.6 demonstrates that the internal energy of Alan rapidly increases in the vicinity of a certain temperature corresponding to the temperature of the first-order PT. The value of the step-like increase of the internal energy is usually referred to as the latent heat of the PT, i.e., the energy absorbed by the system during PT is denoted as Q. Figure 4.6 shows that the latent heat increases with the growth of the polypeptide length. This happens because in the α-helix state, long polypeptides have more hydrogen bonds than short ones and, for the formation of the random coil state, more energy is required.

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Fig. 4.5 Potential energy surfaces for alanine in α-helix a and random coil b conformation. The PES for the second amino acid of the polypeptide is shown in panel c and is used to determine the PESs in the α-helix and random coil conformations. The part of the PES shown in plot c, with energy less than E HB , corresponds to the α-helix conformation (bounded state) while that with energy greater than E HB corresponds to the random coil conformation (unbounded state). Reproduced from Ref. [14] with permission from Springer Nature

The characteristic temperature region of the abrupt change in the internal energy (half-width of the heat capacity peak) characterizes the temperature range of the PT. This quantity is denoted as T . With the increase of the polypeptide length the dependence of the internal energy on temperature becomes steeper and T decreases. Therefore, the PT in longer polypeptides is more pronounced. In the following subsection, the dependence of T on the polypeptide length is discussed in detail. Within the molecular dynamics framework one can evaluate the dependence of the total energy of the system (that is, the sum of the kinetic, potential, and vibrational energies) on temperature. Then the heat capacity can be factorized into two terms

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Fig. 4.6 Dependencies of the internal energy on temperature calculated for Alan polypeptides consisting of n = 21, 30, 50, and 100 amino acids. Dashed lines correspond to the results obtained within the framework of the statistical model. Symbols correspond to the results of MD simulations, which are fitted using Eq. (4.21). The fitting functions are shown with thin solid lines. The fitting parameters are compiled in Table 4.1. Redrawn from data presented in Ref. [14]

which correspond to the internal dynamics of the polypeptide and to the potential energy of the polypeptide conformation. The conformation of the polypeptide influences only the term related to the potential energy and the term corresponding to the internal dynamics is assumed to be independent of the polypeptides conformation. This factorization allows one to distinguish from the total energy the potential energy term corresponding to the structural changes of the polypeptide. The formalism of this factorization is discussed in detail in Ref. [13]. The energy term corresponding to the internal dynamics of the polypeptide neither influence the PT of the system, nor does it grow linearly with temperature. The term corresponding to the potential energy of the polypeptide conformation has a step-like dependence on temperature that occurs at the temperature of the PT. To clearly observe the manifestation of the PT, the linear term was subtracted from the total energy of the system so that only its non-linear part was considered. The slope of the linear term was obtained from the dependencies of the total energy on temperature in the range of 300–450 K, which is far beyond the PT temperature, see Fig. 4.6. Note that the dependence shown in the figure corresponds only to the non-linear potential energy terms.

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Table 4.1 Parameters used in Eq. (4.21) to fit the results of MD simulations n E 0 (eV) E (eV) γ (K) T0 (K) 21

11.38 ± 0.23

4.30 ± 0.10

79.37 ± 7.62

670.0 ± 2.0

30

13.61 ± 0.58

4.70 ± 0.16

37.92 ± 7.31

747.4 ± 3.3

40

16.80 ± 0.39

6.26 ± 0.08

26.59 ± 2.25

785.7 ± 1.8

50

19.94 ± 0.79

8.15 ± 0.21

29.36 ± 5.51

786.6 ± 2.9

100

29.95 ± 0.67

12.58 ± 0.16

10.49 ± 2.00

801.1 ± 1.1

a (eV/K) 0.0471 ± 0.0003 0.0699 ± 0.0008 0.0939 ± 0.0005 0.1178 ± 0.0010 0.2437 ± 0.0009

The heat capacity of the system is defined as the derivative of the total energy on temperature. However, as seen from Fig. 4.6, the MD data is scattered in the vicinity of a certain expectation line. Therefore, the direct differentiation of the energy obtained within this approach would lead to non-physical fluctuations of the heat capacity. To overcome this difficulty, a fitting function for the total energy of the polypeptide is defined as follows:   T − T0 E arctan + aT , (4.21) E(T ) = E 0 + π γ where E 0 , E, T0 , γ , and a are the fitting parameters. The first and the second terms on the right-hand side of Eq. (4.21) are related to the potential energy of the polypeptide conformation, while the last term describes the linear increase of the total energy with temperature. The fitting function (4.21) was also used in other studies [24, 25] for the description of the total energy of polypeptides. The results of fitting are shown in Fig. 4.6 with thin solid lines. The corresponding fitting parameters are compiled in Table 4.1. Figure 4.6 shows that the results obtained using the MD approach are in a reasonable agreement with the results obtained from the statistical mechanics formalism. The fitting parameter E corresponds to the latent heat of the PT while the temperature width of the PT is related to the parameter γ . With the increase of the polypeptides length, the temperature width decreases while the latent heat increases (see the corresponding columns in Table 4.1). These features are correctly reproduced in MD and the statistical mechanics approach. Furthermore, MD simulations demonstrate that with an increase of the polypeptide length, the temperature of the PT shifts toward higher temperatures (see Fig. 4.6). The temperature of the PT is described by the fitting parameter T0 in Table 4.1. Note also that the increase of the PT temperature is reproduced correctly within the framework of the statistical mechanics approach, as seen from Fig. 4.6.

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Table 4.2 Parameters characterizing the heat capacity peak in Fig. 4.7 calculated using the statistical approach. Heat capacity at 300 K, C300 , the transition temperature T0 , the maximal value of the heat capacity C0 , the temperature range T of the PT, and the specific heat Q are shown as a function of polypeptide length, n n C300 T0 C0 T Q (meV/K) (K) (eV/K) (K) (eV) 21 30 40 50 100

2.034 2.840 3.736 4.631 9.111

770 805 815 825 835

0.020 0.037 0.060 0.080 0.308

134 96 86 66 35

2.70 4.15 5.77 7.38 15.47

Nonetheless, the results of MD simulations and the results obtained using the statistical mechanics formalism have several discrepancies. As seen from Fig. 4.6, the latent heat of the PT for Ala100 polypeptide obtained within the framework of the statistical approach is higher than that obtained in MD simulations. This happens because the potential energy of the polypeptide is underestimated within the statistical mechanics approach. Indeed, long polypeptides (consisting of more than 50 amino acids) tend to form short-living hydrogen bonds in the random coil conformation. These hydrogen bonds lower the potential energy of the polypeptide in the random coil conformation. However, the “dynamic” hydrogen bonds are neglected in the present formalism of the partition function construction. Additionally, the discrepancies between the two methods arise due to the limited MD simulation time and to the small number of different temperatures at which the simulations were performed. Indeed, for Ala100 26 simulations were performed, while only 3–5 simulations correspond to the PT temperature region (see Fig. 4.6).

Heat Capacity of Alanine Polypeptide The dependence of the heat capacity on temperature for Alan of different lengths is shown in Fig. 4.7. The results obtained using the statistical approach are shown with the dashed lines, while the results of MD simulations are shown with the solid lines with symbols. Since the classical heat capacity is constant at low temperatures, this constant value has been subtracted for a better analysis of the phase transition (PT) in the system. The constant contribution to the heat capacity, denoted as C300 , is calculated as the heat capacity value at T = 300 K. The C300 values for Alan are compiled in the second column of Table 4.2. As seen from Fig. 4.7, the heat capacity of the system as a function of temperature acquires a sharp maximum at certain temperature corresponding to the PT. The peak is characterized by the transition temperature T0 , the maximal value of the heat capacity C0 , the temperature range T , and the latent heat Q. These parameters were extensively discussed in Ref. [13]. Within the framework of the two-energy

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Fig. 4.7 Dependencies of the heat capacity on temperature calculated for Alan polypeptides consisting of n = 21, 30, 50, and 100 amino acids. Dashed black lines represent the results obtained using the statistical approach, solid red lines with symbols—the results of MD simulations. Dotted blue curves show the heat capacity calculated within the framework of the Zimm–Bragg theory [21]. Values of the heat capacity, C300 , at T = 300 K are given in Table 4.2. Redrawn from data presented in Ref. [14]

level model describing the first-order PT, it was shown that E ∝ n −1 . S 2 (4.22) Here E and S are the energy and the entropy changes between the α-helix and the random coil states of the polypeptide, n is the number of amino acids in the polypeptide. Figure 4.8 shows the dependence of the PT characteristics on the length of the Alan polypeptide. The maximal heat capacity and the temperature range are plotted against n 2 and 1/n, respectively, while the temperature and the latent heat of the PT are shown as functions of n. Squares and triangles represent the PT parameters calculated using the statistical approach and those obtained from the MD simulations, respectively. The results obtained within the framework of the statistical model are in good agreement with the MD simulations. However, since the latter are computationally time demanding it is difficult to simulate PT in large polypeptides. The difficulties T0 ∼

E = const, C0 ∼ S 2 ∝ n 2 , S

Q ∼ E ∝ n, T ∼

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Fig. 4.8 Phase-transition parameters C0 , T , T0 , and Q calculated as a function of polypeptide length (n stands for the number of amino acids). Squares and triangles represent the parameters calculated using the statistical approach and those obtained from the MD simulations, respectively. Reproduced from Ref. [14] with permission from Springer Nature

arise due to the large fluctuations which appear in the system at the PT temperature and due to the large timescale of the PT process. The relative error of the PT temperature obtained on the basis of MD approach is in the order of 3–5%, while the relative error of the heat capacity is about 30% in the vicinity of the phase PT (see Fig. 4.7). The heat capacity peak is asymmetric. At higher temperatures, beyond the peak, the heat capacity forms a plateau (see Fig. 4.7) due to the conformations of the amino acids with larger energies [32]. At T = 1000 K, the differences in the heat capacities are 1.00, 1.60, 2.83, and 5.47 meV/K for Ala21 , Ala30 , Ala50 , and Ala100 , respectively. The magnitude of the plateau increases with the growth of the polypeptide length. This happens because the number of energy levels with high energies rapidly increases for longer polypeptide chains. The dependence of the heat capacity calculated within the framework of the Zimm–Bragg theory is shown in Fig. 4.7 by dotted blue lines for polypeptides of different lengths. Figure 4.7 shows that results obtained on the basis of the Zimm– Bragg theory are in a good agreement with the results of the statistical approach from Ref. [13]. The values of the PT temperature and of the maximal heat capacity in both cases are close. The values of heat capacity obtained within the framework of the Zimm–Bragg model at temperatures beyond the PT window are slightly lower than those calculated within the framework of the statistical model. An important difference between the two theories is due to the accounting for the states of the polypeptide with more than one α-helix fragment. These states are often referred to as multi-helical states of the polypeptide. However, their statistical weight

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in the partition function is suppressed. The suppression arises because of entropy loss in the boundary amino acids of a helical fragment. The boundary amino acids have weaker hydrogen bonds than amino acids in the central part of the α-helix. At the same time, the entropy of such amino acids is smaller than the entropy of amino acids in the coil state. These two factors lead to the decrease of the statistical weight of the multi-helical states. The contribution of the multi-helical states to the partition function broadens the heat capacity peak and decreases its height. The multi-helical states become important in longer polypeptide chains that consist of more than a hundred of amino acids. As seen from Fig. 4.7, the maximal heat capacity obtained within the framework of the Zimm–Bragg model for the Ala100 polypeptide is significantly lower than that obtained within the statistical approach. For Alan consisting of less than 50 amino acids, the multi-helical states of the polypeptide can be neglected as seen from the comparison performed in Fig. 4.7. Omission of the multi-helical states significantly simplifies the construction and evaluation of the partition function.

Helicity of Alanine Polypeptides Helicity is an important characteristic of the polypeptide which can be measured experimentally [26–29]. It describes the fraction of amino acids in the polypeptide that is in the α-helix conformation. With the increase of temperature the fraction of amino acids being in the α-helix conformation decreases due to the PT. Within the framework of statistical approach the helicity of a polypeptide is defined as follows [13]: n (n − i + 1)(i − 1)Z bi−1 Z un−i+1 β i=4 , (4.23) fα = nZ where Z is defined in Eq. (4.15) and Z b , Z u —in Eqs. (4.17). The dependence of f α on temperature T obtained for Alan of different length is shown in Fig. 4.9. It is also possible to evaluate the dependence of helicity on temperature within the framework of the MD approach. In this case, the helicity is defined as the ratio of the number of amino acids in the α-helix conformation to the total number n of amino acids averaged over the MD trajectory. The amino acid is considered to be in the conformation of an α-helix if the twisting angles are within the range of ϕ ∈ [−72◦ , −6◦ ] and ψ ∈ [0◦ , −82◦ ] [13]. The helicity for Ala21 obtained within the framework of MD approach is shown in the inset of Fig. 4.9. From this plot, it is seen that at T ≈ 300 K, which is far below the temperature of the PT, the helicity of the Ala21 polypeptide is 0.82. The fact that at low temperatures the helicity of the polypeptide obtained within the MD approach is smaller than one arises due to the difficulty of defining the α-helix state of an amino acid. Thus, the helicity obtained within the MD approach rolls off at lower temperatures compared to the helicity of the polypeptide of the same length obtained using the statistical mechanics approach.

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Fig. 4.9 Helicity f α versus temperature obtained using the statistical approach (Eq. (4.23)) for Alan polypeptides consisting of n = 21, 30, 40, 50, and 100 amino acids. The inset shows the helicity for n = 21 obtained within the framework of the MD approach. Reproduced from Ref. [14] with permission from Springer Nature

The kink in the helicity curve corresponds to the temperature of the PT. As seen from Fig. 4.9, with an increase of the polypeptide length, the helicity curve becomes steeper in the vicinity of the PT. In the limiting case of an infinitely long polypeptide chain, the helicity behaves as a step function. This is yet another feature of a first-order PT.

4.3.2 Correlation of Different Amino Acids in the Polypeptide

An important question concerns the statistical independence of the amino acids in a polypeptide at different temperatures. Quantitatively, the independence can be characterized in terms of root-mean square deviations (RMSD) of the twisting angles: RMSD(χi ) =

M  j=1



(ϕi − ϕi0 )2 , M

(4.24)

where χ stands for either ψ or ϕ, index i enumerates amino acids, i 0 is the index of an arbitrary chosen reference amino acid in the polypeptide, and M is the number of the MD simulation steps.

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Fig. 4.10 The root-mean-square deviation of angles ϕ and ψ calculated with respect to the central (i 0 = 10) amino acid in Ala21 . The data refer to T = 300 K (top) and 1000 K (bottom). Redrawn from data presented in Ref. [14]

Figure 4.10 presents the RMSDs calculated at T = 300 K (top) and 1000 K (bottom) for Ala21 polypeptide. The deviations were calculated with respect to the twisting angles ϕ10 and ψ10 of the central, i 0 = 10, amino acid. At T = 300 K all amino acids in the polypeptide are in the α-helix conformation, and the RMSD of either angle is less than 16◦ for all amino acids but the boundary ones, for which the deviation is 28◦ for ϕ and 34◦ for ψ. This happens because, while the boundary amino acids are loosely bounded, the central amino acids in the polypeptide are close to the minima, which corresponds to an α-helix conformation. In the α-helix state, all central amino acids are stabilized by two hydrogen bonds, while the boundary amino acids form only one hydrogen bond. At T = 1000 K the polypeptide is, virtually, in the random coil phase and, therefore, becomes more flexible. In this phase, the stabilizing hydrogen bonds are broken, and the RMSDs increase significantly. The RMSDs for the central and boundary amino acids are almost the same, confirming the assumption that short alanine polypeptides do not build hydrogen bonds in the random coil phase. Also to be noted is that in the random coil phase (and in the central part of the α-helix), the RMSDs of ϕ and ψ do not depend on the distance between amino acids in the polypeptide chain. For instance, the deviation between the angles of i 0 = 10th and i = 11th amino acid is almost the same as of the i 0 = 10th and i = 17th ones. Thus, it can be concluded that in a certain phase of the polypeptide (α-helix or random coil), amino acids can be treated as statistically independent.

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Fig. 4.11 Transition energy (upper row) and heat capacity (lower row) versus temperature for the Val30 polypeptide. Symbols show the results obtained from MD simulations. The thin curves in the upper row stand for the interpolating function, in the lower row—for its derivative. Panels a and c show the results obtained within short simulations while panels b and d present the results of the total simulation. Reproduced from Ref. [30] with permission from Springer Nature

4.3.3 Molecular Dynamics Simulations of π-Helix↔Random Coil Phase Transition The α-helix conformation is not the global energy minimum for valine and leucine polypeptides in vacuo, since the π -helix conformation has lower energy, according to the CHARMM27 [5] force field. Therefore, in Ref. [30], the study was carried out on the π -helix↔random coil transition in the polypeptide consisting of 30 amino acids (Val30 and Leu30 ). In the π -helix conformation, the N-H group of an amino acid forms a hydrogen bond with the C=O group of another amino acid being placed five residues away, while in the α-helix conformation this hydrogen bond is formed between amino acids being four residues away from each other. Two graphs in the upper row of Fig. 4.11 show the dependence of the transition energy E t on the temperature calculated for the Val30 polypeptide. The transition

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energy is defined as E t = E tot − 2E k ,

(4.25)

where E tot and E k are, respectively, the total and kinetic energies of a polypeptide [30]. The lower row graphs show the temperature dependence of the heat capacity: CV = ∂E t /∂ T |V=const

(4.26)

obtained from the analysis of the polypeptide energy fluctuations. In all graphs, the symbols indicate the MD results (details of the MD simulations performed one finds in appendix of the cited paper). The solid curves in the upper row stand for the interpolating function, the derivative of which is plotted in the lower row also with solid line. The total simulation time should be chosen to be long enough in order to ensure that the heat capacity does not depend on it. Panels (b) and (d) correspond to the data obtained over the total simulation time τ , whereas the panels (a) and (c) show the results obtained in simulations which were 16 times shorter. The energy fluctuation approach is more general than the method based on differentiation of the internal energy on the temperature. This approach does not depend on the number of data points (simulations at different temperatures) and allows one to determine the absolute values of CV . Indeed, for the Val30 polypeptide, five to nine simulations are sufficient to reproduce both peaks in the heat capacity on temperature dependence, whereas the method based on differentiating of the energy of the system requires at least twice as many data points. The dependence CV (T ) shown in Fig. 4.11d exhibits two well pronounced peaks. Each peak is a result of certain structural transformation [30]. The peak at the higher temperature is due to the π -helix↔random coil transition of the polypeptide which is accompanied by the breaking of hydrogen bonds in the backbone of the polypeptide chain. The smaller peak at the lower temperature can be explained by the dynamics of side-chain radicals. At low temperature, the side-chain radicals of the Val30 polypeptide form the ordered state in which they are aligned along the backbone of the polypeptide. With increase of temperature the ordering of side chains becomes broken and the radicals rotate. The transition from the ordered state to the disordered one can be interpreted as a PT. More details on the structural transformations in the Val30 polypeptide can be found in the cited paper [30]. Note that there is no second peak in the heat capacity dependence of Ala30 and Leu30 polypeptides (see Section “Heat Capacity of Alanine Polypeptide” and Fig. 4.13). This can be explained as follows [30]. In alanine polypeptides, the sidechain radicals are small and thus weakly bound. In leucine polypeptides, these radicals are larger and, therefore, the structural transition is shifted toward higher energies and takes place simultaneously with the π -helix↔random coil of the backbone of the chain. To analyze the dependence of the numerical error of the heat capacity on MD simulation time τ , one can introduce the function

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Fig. 4.12 The MD simulated and the parameterized dependencies of the deviation χ, Eq. (4.27), as a function of simulation time τ . Note the double decimal log scale. Squares show the results of MD simulations, while the dashed line corresponds to the linear fit, Eq. (4.28). Redrawn from data presented in Ref. [30]

χ (τ ) =



(Ci (τ ) − Cref )2 ,

(4.27)

i

where the summation is performed over all data points, Ci (τ ) is the value of the heat capacity obtained from MD simulation of duration τ . In Ref. [30], the reference value of the heat capacity, Cref , corresponded to the longest simulation. Assuming that χ (τ ) obeys the power law, Eq. (4.27) can be parametrized as follows: log10 (χ ) = α + β log10 (τ ),

(4.28)

where α and β are the constants. Figure 4.12 shows the MD simulated (symbols) and the parametrized (dashed line) dependencies of log10 (χ ) on log10 (τ ). It is seen that in its central part the simulated dependence is well approximated by the linear function (4.28). The corresponding coefficients are α = 1.39 ± 0.29 and β = −0.89 ± 0.06. Thus, it can be concluded that χ τ −1 . The deviations from the linear behavior can be attributed to the following facts: (i) at log10 τ < 4.2 the simulation time is too short and thus is insufficient for statistical description of the system; (ii) at log10 τ > 4.6, the deviations arise due to the remaining statistical errors in the reference heat capacity Cref . The analysis similar to the presented above was performed in Ref. [30] for a leucine polypeptide with n = 30. Figure 4.13a shows the dependence of the transition energy on temperature which exhibits a step-like dependence intrinsic for the first-order PT.

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Fig. 4.13 Transition energy (graph a) and heat capacity (graph b) versus temperature calculated for the Leu30 polypeptide. In a, the squares stand for the MD results, and the solid curve shows the corresponding interpolating function. In b, the symbols show the dependence of the heat capacity calculated from the energy fluctuations, the solid line corresponds to the derivative of the energy interpolating function. Reproduced from Ref. [30] with permission from Springer Nature

In the case of a leucine polypeptide in vacuo, this transition corresponds to the π -helix↔random coil transition. Figure 4.13b shows the temperature dependence of the heat capacity calculated from the energy fluctuations (symbols) and from the differentiation of the energy interpolating function (solid line). The use of energy interpolating function allows one to reduce the simulation time needed for the description of the PT. The temperature dependence of the heat capacity obtained with this method can be used to identify the “temperature regions of interest”. In these regions, more systematic analysis of heat capacity should be performed with the use of energy fluctuations method. Figure 4.13 shows that the PT in the Leun polypeptide is more pronounced than in the Alan and the Valn ones. For leucine, the peak value of CV is the largest one, while the temperature range of the PT is approximately the same for the three polypeptides. This happens because the leucine side-chain radical is larger than that in alanine and, therefore, the peak in the heat capacity is more powerful.

4.4 Thermodynamics of Protein Folding In Ref. [31], a statistical mechanics method for the theoretical description of the protein folding process was presented. The process of protein folding was considered as a first-order phase transition in a finite system. The suggested method is based on the theory developed for the helix↔coil transition in polypeptides [13, 14, 18– 20, 30, 32, 33] (see Sects. 4.2.2 and 4.3) and applied for folding↔unfolding phase transition in single-domain proteins. In this section, the statistical model for protein folding is overviewed.

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The approach developed in Ref. [31] starts with the construction of the partition function of a protein in vacuo, which is the further generalization of the formalism developed in [32], accounting for folded, unfolded, and prefolded states of the protein. The model is based on a number of assumptions about the system. Most of the assumptions are necessary for the factorization of the partition function of the system. For the correct description of the protein folding in water environment, it is of primary importance to consider the interactions between the protein and the solvent molecules. The hydrophobic interactions are known to be the most important driving forces of protein folding [34]. The method presented in Ref. [31] permits to construct the partition function of a protein that accounts for the protein interaction with solvent, i.e., accounts for the hydrophobic effect. In Ref. [31], the hydrophobic interactions in the system were treated using the statistical mechanics formalism developed in Ref. [35] for the description of the thermodynamical properties of the solvation process of aliphatic and aromatic hydrocarbons in water. The water molecules only form the protein’s first solvation shell which are considered to be interacting with the protein hydrophobic surface. However, accounting solely for hydrophobic interactions is not sufficient for the proper description of the energetics of all conformational states of the protein and one has to take electrostatic interactions into account. Hence, the electrostatic interactions were treated within a similar framework as described in Ref. [36]. The developed statistical mechanics model of protein folding was applied for two globular proteins, namely, staphylococcal nuclease and metmyoglobin. These proteins have simple two-stage-like folding kinetics and demonstrate two folding↔unfolding transitions, referred to as heat and cold denaturation [37, 38]. The comparison of the results of the theoretical model with that of the experimental measurements shows the applicability of the suggested formalism for an accurate description of various thermodynamical characteristics in the system, e.g., heat denaturation, cold denaturation, increase of the reminiscent heat capacity of the unfolded protein, etc.

4.4.1 Partition Function of a Protein The most relevant degrees of freedom in the protein folding process are the twisting degrees of freedom along its backbone chain as discussed in Sects. 4.2.2 and 4.3. These degrees of freedom are defined for each amino acid of the protein except for the boundary ones and are described by two dihedral angles ϕi and ψi (see Fig. 4.2). According to Ref. [31], the partition function of a protein Z p (without any solvent) can be written in the following way: Z p = A(kB T )

3N −3− l2s

ξ  a   j=1 i=1

π −π

 ( j) i (ϕi , ψi ) exp − dϕi dψi , kB T −π



π

(4.29)

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where the summation over j includes all ξ statistically relevant conformations of the ( j) protein, a is the number of amino acids in the protein and i is the potential energy surface as a function of twisting degrees of freedom ϕi , and ψi of the ith amino acid ( j) in the jth conformational state of the protein. The exact construction of a (ϕi , ψi ) for various conformational states of a particular protein will be discussed below. The angles ϕ and ψ are considered as the only two soft degrees of freedom in each amino acid of the protein, and therefore the total number of soft degrees of freedom of the protein ls = 2a. Partition function in Eq. (4.29) can be further simplified if one assumes (i) that each amino acid in the protein can exist only in two conformations: the native state conformation and the random coil conformation; (ii) the potential energy surfaces for all the amino acids are identical. This assumption is applicable for both the native and the random coil state. It is not very accurate for the description of thermodynamical properties of single amino acids, but is reasonable for the treatment of thermodynamical properties of the entire protein. The judgment of the quality of this assumption could be made on the basis of comparison of the results obtained with its use with experimental data (see below). Amino acids in a protein being in its native state vibrate in a steep harmonic potential. Here it is assumed that the potential energy profile of an amino acid in the native conformation should not be very sensitive to the type of amino acid and thus can be taken as the potential energy surface for an alanine amino acid in the α-helix conformation [14]. Using the same arguments the potential energy profile for an amino acid in unfolded protein state can be approximated by the potential of alanine in the unfolded state of alanine polypeptide (see Ref. [14] for discussion and analysis of alanine’s potential energy surfaces). Indeed, for an unfolded state of a protein, it is reasonable to expect that once neglecting the long-range interactions all the differences in the potential energy surfaces of various amino acids arise from the steric overlap of the amino acid’s side chains. This is clearly seen on alanine’s potential energy surface at values of ϕ > 0◦ presented in Ref. [14]. But the part of the potential energy surface at ϕ > 0◦ gives a minor contribution to the entropy of amino acid at room temperature. This fact allows one to neglect all the differences in potential energy surfaces for different amino acids in an unfolded protein, at least in the zero-order approximation. This assumption should be especially justified for proteins with the rigid helix-rich native structure. The staphylococcal nuclease studied in Ref. [31] has definitely high α-helix content. Another argument which allows to justify the assumption for a wider family of proteins is the rigidity of the protein’s native structure. Below, the assumptions made are validated by comparing the results of the theoretical model with the experimental data for α/β-rich protein metmyoglobin [38]. For the description of the folding ↔ unfolding transition in small globular proteins obeying simple two-state-like folding kinetics, it is assumed that the protein can exist in one of three states: completely folded state, completely unfolded state, and partially folded state where some amino acids from the flexible regions with no prominent secondary structure are in the unfolded state, while other amino acids are in the

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folded conformation. With this assumption the partition function of the protein reads as a  κ! (4.30) Zp = Z0 + Zi , (i − (a − κ))!(a − i)! i=a−κ where Z 0 is the partition function of the protein in completely unfolded state, a is the total number of amino acids in a protein, and κ is the number of amino acids in flexible regions. The factorial term in Eq. (4.30) accounts for the states in which various amino acids from flexible regions independently attain the native conformation. The summation in Eq. (4.30) is performed over all partially folded states of the protein, where a − κ is the minimal possible number of amino acids being in the folded state. The factorial term describes the number of ways to select i − (a − κ) amino acids from the flexible region of the protein consisting of κ amino acids attaining native-like conformation. Finally, the partition function of the protein in vacuo has the following form: Z p = Z˜ p A(kB T )3N −3−a ,

(4.31)

where a  κ!Z bi Z ua−i exp (i E 0 /kB T ) (i − (a − κ))!(a − i)! i=a−κ    π π b (ϕ, ψ) Zb = dϕdψ exp − kB T −π −π    π π u (ϕ, ψ) dϕdψ. Zu = exp − kB T −π −π

Z˜ p = Z ua +

(4.32) (4.33) (4.34)

Here the trivial factor describing the motion of the protein center of mass, which is of no significance for the problem considered, is omitted. b (ϕ, ψ) (“b” stands for bound) is the potential energy surface of an amino acid in the native conformation and u (ϕ, ψ) (“u” stands for unbound) is the potential energy surface of an amino acid in the random coil conformation. The potential energy profile of an amino acid is calculated as a function of its twisting degrees of freedom ϕ and ψ. Let us denote by b0 and u0 the global minima on the potential energy surfaces of an amino acid in folded and in unfolded conformations, respectively. The potential energy of an 0 + u,b (ϕ, ψ). E 0 in Eq. (4.32) is defined as the energy amino acid then reads as u,b difference between the global energy minima of the amino acid potential energy surfaces corresponding to the folded and unfolded conformations, i.e., E 0 = u0 − b0 . The potential energy surfaces for amino acids as functions of angles ϕ and ψ were calculated and thoroughly analyzed in [14]. In nature, proteins perform their function in the aqueous environment. Therefore, the correct theoretical description of the folding↔unfolding transition in water environment should account for solvent effects.

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4.4.2 Partition Function of a Protein in Water Environment The partition function of the infinitely diluted solution of proteins Z can be constructed as follows: ξ  ( j) (4.35) Z˜ p( j) Z W , Z= j=1 ( j)

where Z W is the partition function of all water molecules in the jth conformational ( j) state of a protein and Z˜ p is the partition function of the protein in its jth conformational state, in which the factor describing the contribution of “stiff” degrees of freedom in the system is further omitted. This is done in order to simplify the expressions, because “stiff” degrees of freedom provide a constant contribution to the heat capacity of the system since the heat capacity of the ensemble of harmonic oscillators is constant. For the simplicity of notations, one puts Z˜ p ≡ Z p . There are two types of water molecules in the system: (i) molecules in pure water and (ii) molecules interacting with the protein. It is assumed that only the water molecules being in the vicinity of the protein’s surface are involved in the folding↔unfolding transition, because they are affected by the variation of the hydrophobic surface of a protein. This surface is equal to the protein’s solvent accessible surface area (SASA) of the hydrophobic amino acids. The number of interacting molecules is proportional to SASA and includes only the molecules from the first protein’s solvation shell. This area depends on the conformation of the protein. The main contribution to the energy of the system caused by the variation of the protein’s SASA is associated with the side chains of amino acids because the contribution to the free energy associated with solvation of protein’s backbone is small [22]. Thus, in Ref. [31] the main attention was paid to the accounting for the SASA change arising due to the solvation of side chains. All water molecules are treated as statistically independent, i.e., the energy spectra of the states of a given molecule and its vibrational frequencies do not depend on a particular state of all other water molecules. Thus, the partition function of the whole system Z can be factorized and reads as Z=

ξ 

Z p( j) Z sYc ( j) Z wNt −Yc ( j) ,

(4.36)

j=1

where ξ is the total number of states of a protein, Z s is the partition function of a water molecule affected by the interaction with the protein, and Z w is the partition function of a water molecule in pure water. Yc ( j) is the number of water molecules interacting with the protein in the jth conformational state. Nt is the total number of water molecules in the system. To simplify the expressions, water molecules that do not interact with the protein in any of its conformational states are not considered, i.e., Nt = max j {Yc ( j)}.

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Table 4.3 Parameters of the partition function of water according to Ref. [35] Number of 0 1 2 3 hydrogen bonds Energy level, E i (kcal/mol) Energy level, E is (kcal/mol) Translational frequencies, (T ) νi , cm−1 Librational frequencies, νi(L) , cm−1

4

6.670

4.970

3.870

2.030

0

6.431

4.731

3.631

1.791

−0.564

26

86

61

57

210

197

374

500

750

750

Following the formalism from Ref. [35], the partition function of a water molecule in pure water reads as Zw =

4    ξl fl exp(−El /kB T ) ,

(4.37)

l=0

where the summation is performed over five possible states of a water molecule (the states in which water molecule has 4, 3, 2, 1, or 0 hydrogen bonds with the neighboring molecules). El are the energies of these states and ξl are the combinatorial factors being equal to 1, 4, 6, 4, 1 for l = 0, 1, 2, 3, 4, respectively. They describe the number of choices to form a given number of hydrogen bonds. fl in Eq. (4.37) describes the contribution due to the partition function arising to the translation and libration oscillations of the molecule. In the harmonic approximation fl are equal to  −3  −3 × 1 − exp(−hνl(L) /kB T ) , fl = 1 − exp(−hνl(T ) /kB T )

(4.38)

where νl(T ) and νl(L) are translation and libration motion frequencies of a water molecule in its lth state, respectively. These frequencies are calculated in Ref. [35] and are given in Table 4.3. The contribution of the internal vibrations of water molecules is not included in Eq. (4.37) because the frequencies of these vibrations are practically not influenced by the interactions with surrounding water molecules. The partition function of a water molecule from the protein’s first solvation shell reads as 4    ξl fl exp(−Els /kB T ) , (4.39) Zs = l=0

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where fl are defined in Eq. (4.38) and Els denotes the energy levels of a water molecule interacting with aliphatic hydrocarbons of protein’s amino acids. Values of energies Els are given in Table 4.3. For simplicity, all side-chain radicals of a protein are treated as aliphatic hydrocarbons because most of the protein’s hydrophobic amino acids consist of aliphatic-like hydrocarbons. The developed theoretical model [31] accounts also for the electrostatic interaction of protein’s charged groups with water. The presence of electrostatic field around the protein leads to the reorientation of H2 O molecules in the vicinity of charged groups due to the interaction of dipole moments of the molecules with the electrostatic field. The additional factor arising in the partition function of water molecules reads as  ZE =

1 4π



 α  Ed cos θ , sin θ dθ dϕ exp − kB T

(4.40)

where E is the strength of the electrostatic field, d is the absolute value of the H2 O molecule dipole moment, and α is the ratio of the number of water molecules that interact with the electrostatic field of the protein (N E ) to the number of water molecules interacting with the surface of the amino acids from the inner part of the protein while they are exposed to water when the protein is being unfolded (Nw ), i.e., α = N E /Nw . Note that the effects of electrostatic interaction turn out to be more pronounced in the folded state of the protein. This happens because in the unfolded state of a protein opposite charges of amino acid’s side chains are in average closer in space due to the flexibility of the backbone chain, while in the folded state the positions of the charges are fixed by the rigid structure of a protein. Integrating Eq. (4.40) allows one write the factor Z E for the partition function of a single H2 O molecule in pure water in the following form: ⎛ ZE = ⎝

 kB T sinh Ed

Ed kB T

⎞α ⎠ .

(4.41)

This equation shows how the electrostatic field enters the partition function. In general, E depends on the position in space with respect to the protein. However, this dependence is neglected and the parameter E is treated as an average characteristic electrostatic field created by the protein. Let us denote by Ns the number of water molecules interacting with the protein surface in its folded state, i.e., Nt = Ns + Nw , where Nt is defined in Eq. (4.36). It is assumed that the number of water molecules interacting with the protein (Yc ) is linearly dependent on the number of amino acids being in the unfolded conformation, i.e., Yc = Ns + i Nw /a, where i is the number of the amino acids in the unfolded conformation and a is the total number of amino acids in the protein. Thus, the partition function (4.36) with the accounting for the factor (4.41) reads as

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Fig. 4.14 Structure of a staphylococcal nuclease (PDB ID 1EYD [39]) and b horse heart metmyoglobin (PDB ID 1YMB [40]). Reproduced from Ref. [31] with permission from Springer Nature

Z = Z sNs

ξ  

Nw

Nw

i( j)

Z b Z wa Z Ea exp (i E 0 /kB T )

a−i( j)  Nw × Zu Zs a ,

(4.42)

j=1

where i( j) denotes the number of the amino acids being in the folded conformation when the protein is in the jth conformational state. Accounting for the statistical factors for amino acids being in the folded and unfolded states, similar to how it was done for the vacuum case (see Eq. (4.32)), one derives from Eq. (4.42) the following final expression: Z = (Z s ) Ns ×  × Z ua Z sNw +

a  i=a−κ



κ! exp (i E 0 /kB T ) N /a Z b Z wNw /a Z E w (i − (a − κ))!(a − i)!

i

(4.43)  (Z u Z sNw /a )a−i ,

where the term in the square brackets accounts for all statistically significant conformational states of the protein. The constructed partition function of the system can be used to evaluate all thermodynamic characteristics of the system. In Ref. [31], it was utilized to calculate the temperature dependence of the heat capacity for two globular proteins, metmyoglobin and staphylococcal nuclease. The model predictions were compared with experimental data from Refs. [37, 38]. The structures of metmyoglobin and staphylococcal nuclease proteins are shown in Fig. 4.14. These are relatively small globular proteins consisting of ∼150 amino acids. Under certain experimental conditions (salt concentration and pH) the metmyoglobin and the staphylococcal nuclease experience two folding↔unfolding transitions, which induce two peaks in the dependence of heat capacity on temperature (see the following section). The peaks at lower temperature are due to the cold denaturation of the proteins. The peaks at higher temperatures arise due to the ordinary folding↔unfolding transition. The availability of experimental data for the heat

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capacity profiles of the mentioned proteins, the presence of the cold denaturation, and simple two-stage-like folding kinetics are the reasons for selecting these particular proteins as case studies for the verification of the developed theoretical model.

4.4.3 Heat Capacity of Staphylococcal Nuclease Staphylococcal or micrococcal nuclease (S7 Nuclease) is a relatively nonspecific enzyme that digests single-stranded and double-stranded nucleic acids, but is more active on single-stranded substrates [41]. This protein consists of 149 amino acids. Its structure is shown in Fig. 4.14a. To calculate the SASA of staphylococcal nuclease in the folded state, the 3D structure of the protein was taken from the Protein Data Bank (PDB ID 1EYD). Using CHARMM27 [5] force field and NAMD program [6] the structural optimization of the protein was performed and SASA was calculated with the solvent probe radius 1.4 Å. The value of SASA of the side chains in the folded protein conformation is equal to S f = 6858 Å2 . In order to calculate SASA for an unfolded protein state, the value of all angles ϕ and ψ was put equal to 180◦ , corresponding to a fully stretched conformation. Then, the optimization of the structure with the fixed angles ϕ and ψ was performed. The optimized geometry of the stretched molecule has a minor dependence on the value of dielectric susceptibility of the solvent, and therefore the value of dielectric susceptibility was chosen to be equal to 20, in order to mimic the screening of charges by the solvent. SASA of the side chains in the stretched conformation of the protein is equal to Su = 15813 Å2 . The change of the number of water molecules interacting with the protein due to the unfolding process can be calculated as follows: Nw = (Su − S f )n 2/3 ,

(4.44)

where Su = 15813 Å2 and S f = 6858 Å2 are the SASA of the protein in unfolded and in folded conformations, respectively, and n ≈ 30 Å−3 is the density of the water molecules. To account for the effects caused by the electrostatic interaction of water molecules with the charged groups of the protein, it is necessary to evaluate the strength of the average electrostatic field E in Eq. (4.41). The strength of the average field can be estimated as Ed = kB T , where d is the dipole moment of a water molecule, kB is Boltzmann constant, and T = 300 K is the room temperature. According to this estimate, the energy of characteristic electrostatic interaction of water molecules is equal to the thermal energy per degree of freedom of a molecule [31]. At physiological conditions, staphylococcal nuclease has eight charged residues [42]. The value of α for this protein varies within the interval from 1.29 to 31.27 for λd ∈[10..30] Å, where λd is the Debye screening length of the charge in

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(M)

Table 4.4 Values of E 0 for staphylococcal nuclease (E 0 ) and metmyoglobin (E 0 values of solvent pH (S) (M) pH value E 0 (kcal/mol) E 0 (kcal/mol) 7.0 5.0 4.5 4.10 3.88 3.84 3.70 3.5 3.23

) at different

0.789 0.795 0.803 1.128 0.819 1.150 1.165 1.2 0.890

electrolyte. For numerical analysis in Eq. (4.43), a characteristic value of α equal to 2.5 was used. An important parameter of the model is the energy difference between the two states of the protein normalized per one amino acid, E 0 , introduced in Eq. (4.32). This parameter describes both the energy loss due to the separation of the hydrophobic groups of the protein which attract in the native state of the protein due to van der Waals interaction and the energy gain due to the formation of van der Waals interactions of hydrophobic groups of the protein with H2 O molecules in the protein’s unfolded state. Also, the difference of the electrostatic energy of the system in the folded and unfolded states is accounted for in E 0 . The difference of the electrostatic energy may depend on various characteristics of the system, such as concentration of ions in the solvent and its pH, on the exact location of the charged sites in the native conformation of the protein and on the probability distribution of distances between charged amino acids in the unfolded state. Since the exact calculation of E 0 is rather difficult, the energy difference between the two phases of the protein has been considered as a parameter of the model. E 0 is treated as being dependent on external properties of the system, in particular, on the pH value of the solution. In Ref. [31], the value of E 0 was fitted to reproduce the experimental measurements at different pH values. Another characteristic feature the protein folding↔unfolding transition is its cooperativity. In the model, it is described by the parameter κ in Eq. (4.30). κ describes the number of amino acids in the flexible regions of the protein. The staphylococcal nuclease possesses a prominent two-stage folding kinetics, and therefore only 5–10% of amino acids is in the protein’s flexible regions. Thus, the value of κ for this protein is small. It can be estimated as being equal to 149 · 7% ≈ 10 amino acids. The values of E 0 for staphylococcal nuclease at different values of pH are given in Table 4.4. For the analysis of the variation of the thermodynamic properties of the system during the folding process, one can omit all the contributions to the free energy of

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Fig. 4.15 Dependencies of the heat capacity on temperature for staphylococcal nuclease at different values of pH. Solid lines show results of the calculation, while symbols present experimental data [37]. Reproduced from Ref. [31] with permission from Springer Nature

the system that do not alter significantly in the temperature range between −50 ◦ C and 150 ◦ C. Therefore, from the expression for the total free energy of the system F, one can subtract all slowly varying contributions F0 as follows:  δ F = F − F0 = −(kB T ln Z − kB T ln Z 0 ) = −kB T ln

Z Z0

 .

(4.45)

It follows from Eq. (4.45) that the subtraction of F0 corresponds to the division of the total partition function Z by the partition function of the subsystem (Z 0 ) with slowly varying thermodynamical properties. Therefore, in order to simplify the expressions, one can divide the partition function in Eq. (4.43) by the partition function of fully unfolded conformation of a protein (Z ua Z sNw ) and by the partition function of Ns free water molecules (Z wNs ). Thus, Eq. (4.43) can be rewritten as follows:  Z=

Zs Zw

 Ns  a  1+ i=a−κ

κ! exp (i E 0 /kB T ) (i − (a − κ))!(a − i)!



Zb Zu

i 

Zw ZE Zs

i Nw /a 

.

(4.46) Equation (4.46) is the final equation that is used for calculation of the partition function of the protein. The exact expressions used for evaluation of the dependencies of heat capacity on temperature are presented in Ref. [31]. The dependence of heat capacity on temperature calculated for staphylococcal nuclease at different pH values is presented in Fig. 4.15 by solid lines. The results of experimental measurements from Ref. [37] are presented by symbols. From Fig. 4.15, it is seen that staphylococcal nuclease experiences two folding↔transitions in the

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range of pH between 3.78 and 7.0. At the pH value 3.23 no peaks in the heat capacity are present. It means that the protein exists in the unfolded state over the whole range of experimentally accessible temperatures. Comparison of the theoretical results with experimental data shows that the theoretical model reproduces experimental behavior better for the solvents with higher pH. The heat capacity peak arising at higher temperatures due to the standard folding↔unfolding transition is reproduced very well for pH values being in the region 4.5–7.0. The deviations at low temperatures can be attributed to the inaccuracy of the statistical mechanics model of water in the vicinity of the freezing point. The accuracy of the statistical mechanics model for low pH values around 3.88 is also quite reasonable. The deviation of theoretical curves from experimental ones likely arise due to the alteration of the solvent properties at high concentration of protons or due to the change of partial charge of amino acids at pH values being far from the physiological conditions. Despite some difference between the predictions of the developed model and the experimental results arising at certain temperatures and values of pH, the overall performance of the model can be considered as extremely good for such a complex process as structural folding transition of a large biological molecule.

4.4.4 Heat Capacity of Metmyoglobin Metmyoglobin is an oxidized form of a protein myoglobin. This is a monomeric protein containing a single five-coordinate heme whose function is to reversibly form a dioxygen adduct [43]. Metmyolobin consists of 153 amino acids and its structure is shown in Fig. 4.14b. In order to calculate SASA of side chains of metmyoglobin, exactly the same procedure as for staphylococcal nuclease was performed (see discussion in the previous subsection). SASA in the folded and unfolded states of the protein has been calculated and is equal to 6847 Å2 and 16926 Å2 , respectively. Thus, there are 984 H2 O molecules interacting with protein’s hydrophobic surface in its unfolded state. The electrostatic interaction of water molecules with metmyoglobin was accounted for in the same way as for staphylococcal nuclease. The parameter α in Eq. (4.41) was chosen to be the same as for staphylococcal nuclease, i.e., equal to 2.5. On this basis, it was derived that 10950 H2 O molecules involve in the interaction with the electrostatic field of metmyoglobin in its folded state. The strength of the field was chosen the same as for staphylococcal nuclease. The parameter κ for metmyoglobin in Eq. (4.30), describing the cooperativity of the folding↔unfolding transition, differs significantly from that for staphylococcal nuclease. The transition in metmyoglobin is less cooperative than the transition in staphylococcal nuclease because metmyoglobin has intermediate partially folded states [44]. Thus, while the rigid native-like core of the protein is formed, a significant fraction of amino acids in the flexible regions of the protein can exist in the unfolded

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Fig. 4.16 Dependencies of the heat capacity on temperature for horse heart metmyoglobin at different values of pH. Solid lines show the results of the calculation. Symbols present the experimental data from Ref. [38]. Reproduced from Ref. [31] with permission from Springer Nature

state. It is assumed that ∼1/3 of metmyoglobin’s amino acids are in the flexible region, i.e., the parameter κ in Eq. (4.30) is equal to 50. The values of E 0 in Eq. (4.32) differ from that for staphylococcal nuclease and are compiled in Table 4.4. Solid lines in Fig. 4.16 show the dependence of the metmyoglobin’s heat capacity on temperature calculated using the developed theoretical model. The experimental data from Ref. [38] are shown by symbols. Metmyoglobin experiences two folding↔unfolding transitions at the pH values exceeding 3.5 which can be called as cold and heat denaturations of the protein. The dependence of the heat capacity on temperature therefore has two characteristic peaks, as seen in Fig. 4.16. Figure 4.16 shows that at pH lower than 3.84 metmyoglobin exists only in the unfolded state. The comparison of predictions of the developed theoretical model with the experimental data on heat capacity shows that the theoretical model is well applicable for metmyoglobin case as well. The good agreement of the theoretical and experimental heat capacity profiles over the whole range of temperatures and pH values shows that the model treats correctly the thermodynamics of the protein folding process. The developed theory includes a number of parameters, namely, the energy difference between two phases E 0 , strength of the electrostatic field E, number of interacting H2 O molecules α, the parameter describing the cooperativity of the phase transition κ, as well as other parameters introduced in Ref. [35] to treat the partition function of water. Three parameters, E, E 0 , and κ, are dependent on the properties of a particular protein and on the pH of the solvent. The values of these parameters were adjusted to reproduce the experimental data. All other parameters of the model

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describing the structure of energy levels of water molecules, their vibrational and librational frequencies, etc. are considered as fixed, being universal for all proteins. Despite the model features of the developed approach, it should be stressed that the complex behavior and the peculiarities in dependencies of the heat capacity on temperature are all well reproduced by the developed model with only a few parameters. Therefore, the developed model can be used for the prediction of new features of phase transitions in various biomolecular systems. Indeed, from Figs. 4.15 and 4.16, one can extract a lot of useful information on the heat capacity profiles: the concave bending of the heat capacity profile for a completely unfolded protein, the temperature of the cold and heat denaturation, the absolute values of the heat capacity at the phase-transition temperature, and the broadening of heat capacity peaks. Another peculiarity which is well reproduced by the developed statistical mechanics model is the decrease of the heat capacity of the folded state of the protein in comparison with that for unfolded state and asymmetry of the heat capacity peaks.

4.5 Unbinding of a Protein–Ligand Complex Biological processes are driven by interactions between the molecular components of cellular machinery, commonly between proteins and their target molecules (generically termed ligands). Most of these processes portray a cascade of protein-ligand association/dissociation events, and thus knowledge and control of their energetics and kinetics is of key importance in molecular biology, proteomics, and therapeutic research, to name a few. Protein-ligand dissociation is, in essence, a fragmentation of complex multiatomic aggregates. Many-body aggregates are very ubiquous in Nature, and have been the object of extensive experimental and theoretical studies in a wide range of research fields from nuclear fission to atomic clusters fragmentation to dissociation of insulin from its receptor on the cell membrane, etc. Despite a vast amount of data has been accumulated, there is a need for an efficient and physically sound theoretical approach that could possibly rationalize it and make insightful predictions. In Ref. [45], the unbinding process of a protein-ligand complex of major biological interest was investigated computationally at the atomistic level by means of molecular mechanics method. The unbinding of a complex biomolecular system was described in terms of a reduced set of relevant generalized coordinates while restricting most of its conformational internal degrees of freedom. The complex problem was reduced to a low-dimensional scanning (based on the energy minimization technique) along a selected distance between the protein and the ligand. A remarkable protein-ligand system is the antibody–antigen system involved in a fundamental recognition process during the body immune response. This response is triggered by foreigner molecules—the antigens (AG). One key mechanism whereby the immune system recognizes and targets them for destruction is by releasing antibodies (AB) [46], very large proteins featuring a basic scaffold: each consists of two identical “light” (L) and “heavy” (H) chains of amino acids Y-shaped folded as shown

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Fig. 4.17 Overall ribbon representation of a complete AB structure. The two pairs of H-chains are depicted in red and blue, and the corresponding L-chains in yellow and gray. The dashed ellipse highlights one of the AG-binding fragments (the so-called Fab), in an all-atom representation; the trapezoidal region puts in evidence the Fab variable domains (with added hydrogens), and the dashed arc illustrates the chains’ cleavage sections for these variable domains to be detached. A simplified scheme of AB-AG binding is presented in the inset. Reproduced from Ref. [45] with permission from Springer Nature

in Fig. 4.17. The two tips of the Y branches display a distinctive variable region, i.e., the specific AB “lock” for which the target AG has the “key” (see the schematic inset in Fig. 4.17). This “key” can be a small protein fragment or a low molecular weight compound named hapten. Upon exposure to a particular AG, a set of ABs is refined to target it, via a mutation process [47, 48], mainly occurring in the referred variable region. Along a maturation series, the increase in affinity strongly correlates with an increase in the corresponding AB-AG dissociation times τ [46, 49–51]. Usually, τ is expressed in terms of the rate of spontaneous dissociation, koff = 1/τ . Not surprisingly, much effort has been devoted to the determination of those koff values, with some of the most innovative experiments involving force probe micromanipulation techniques like atomic force microscopy (AFM) to measure ABAG-binding forces [46, 52–55]. Some further insight into the molecular structure, interactions, and unbinding pathways underlying such single molecule experiments has been gained from computer simulations using “force probe” molecular dynamics (FPMD) [56]. However, the question arises of to what extent the measured unbinding force in the mechanically speeded up process of pulling out the ligand relates to the thermodynamic or kinetic parameters describing the spontaneous dissociation.

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The later arises in the minute timescale [50] in contrast to the timescales of AFM (millisecond) and FPMD (nanosecond). There is also the matter of across which pathway is unbinding being forced. In the absence of a pulling force, one regains the spontaneous (natural) mode of AB-AG dissociation, a thermally activated barrier-crossing along a preferential path in a multidimensional energy landscape. The contributing activated states (which determine koff ) may well be described in terms of a few collective coordinates, in close analogy to other studied fragmentation processes [33, 57, 58]. Within this context, it would be reasonable to constrain the many other degrees of freedom that only contribute to the negligible fine structure of the energy landscape. It is a rational approach to probe the unbinding of a complex system like the AB-AG one, in order to determine the corresponding energetic barrier and derive koff . For that, the ABAG system seems particularly appropriate: experimental reports suggest that AB evolution results in a rigidified “lock-and-key” mature structure [59], a result that is corroborated by the structural comparison of the X-ray resolved conformations of the same mature antibody in the bound and unbound forms, which exhibit a RMSD (root-mean-square deviation) of 0.38 Å for the α-trace [60]. As a case study, fluorescein (Flu)—a synthetic hapten—extensively used in kinetic measurements of off-rates (koff ) [61] and a valuable reference system in immunology was considered in Ref. [45]. Anti-fluorescein AB-AG complexes are also clear-cut models in the sense that Flu is a small inert and rigid ligand (see Fig. 4.18) and the offrates of a number of anti-Flu complexes have been found to display an Arrhenius-like behavior [49]. The study has been carried out for the anti-fluorescein IgG monoclonal antibody 4-4-20 (mAb4-4-20) [62], for which two crystallographic structures of its Fab fragments (see Fig. 4.17) have been reported [63, 64]. The two variable domains of a Fab fragment (labeled V L and V H ) constitute the so-called Fv-fragment (highlighted in Fig. 4.17), which is the minimal antigen-binding fragment. In fact, many engineered ABs feature only the V L and V H domains [65]. This practice further endorses the idea of a system with a restricted number of binding-determinant degrees of freedom. It also makes it realistic (and computationally less demanding) to consider just the mAb4-4-20 variable domains: V L with 112 amino acids and V H with 118 amino acids. The all-atom representation of Fv-fragment of the mAb4-4-20-Flu complex structure is shown in Fig. 4.19. The interatomic interactions were described using the widely used CHARMM force field for proteins [5]. Its potential energy function has been defined above in Sect. 4.2.1. The bonding and Lennard-Jones parameters for fluorescein (which are not available in the commonly used CHARMM parameterizations) were determined in Ref. [45] on the basis of DFT calculations using the B3LYP functional with the 6-31G(d) basis set. Partial atomic charges were fitted to reproduce the molecular electrostatic potential (MEP). The corresponding CHARMM atom types and partial charges are indicated in Fig. 4.18.

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Fig. 4.18 Structural formula and assigned atom labels for Fluorescein {2-(6- hydroxy-3-oxo-(3H)xanthen-9yl) benzoic acid}. The CHARMM force field atom types and the partial charges (units of e) are listed in the table. The dashed line puts in evidence the two aromatic (ring) fragments labeled and grouped in the table. Reproduced from Ref. [45] with permission from Springer Nature

4.5.1 Accounting for Implicit Solvent Solvation, stability, and dissociation of proteins in water (the physiological solvent) are largely governed by electrostatic interactions [66]. Introducing explicit water molecules in a computational simulation increases significantly the calculation time. Moreover, when the calculations involve any energy minimization-based technique like calculating minimum energy reaction paths, the explicit water molecules will arrange in a single conformation matrix, exerting forces on the solute that are very different from the solvent mean force. Alternatively, a continuum treatment of the solvent as a uniform dielectric may provide an accurate enough description of such interactions, a most physically correct implicit solvent model arising from solving the so-called Poisson–Boltzmann (PB) equation [67, 68]. The protein is treated as a low-dielectric cavity bounded by the molecular surface and containing partial atomic charges—typically taken from the

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Fig. 4.19 All-atom representation of the Fv-fragment of the mAb4-4-20-Flu complex structure. The ribbon representation highlights the backbone of the two chains, the H-chain in blue, and the L-chain in gray. The Flu molecule is depicted in ball-and-stick mode. Reproduced from Ref. [45] with permission from Springer Nature

classical molecular mechanics force field. The solvent is implicitly introduced by assuming a high-dielectric surrounding of the protein. Since under physiological conditions macromolecules are dissolved in dilute saline solutions, a term for the average charge density due to the mobile ions of the dissolved electrolyte is also included. This continuum treatment relies on the reasonable assumption that it is possible to replace the ionic potential of mean force with the mean electrostatic potential, and it neglects non-Coulomb (e.g., van der Waals) interactions and ion correlations. The actual PB equation reads as ∇ · [ε(r)∇ϕ(r)] = −4πρ(r) − 4π

N 

e qi n i (r)λ(r) ,

(4.47)

i=1

with ϕ(∞) = 0 and

n i (r) = n i0 exp(e qi ϕ(r)/kB T ) .

(4.48)

Equation (4.47) relates the electrostatic potential ϕ to the protein’s charge density ρ, the dielectric properties of both the protein and solvent (ε, position-dependent (r)), and the charge density due to the mobile ions given by the summation term; qi is the charge of ion type i, n i (r) its local concentration, e the elementary charge,

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and λ(r) a parameter that describes the ions’ accessibility at position r. In Ref. [45], the boundary potentials (at the lattice edge) were approximated by the sum of the Debye–Hückel potentials of all the charges, meaning ϕ=

N  i=1

e qi

exp(−ri /λ D ) , εwater ri

(4.49)

where λ D is the Debye length. As for the Boltzmann distribution (4.48), n i0 is the ion’s concentration in bulk solution, kB the Boltzmann constant, and T the absolute temperature. Any point within one ionic radius from the macromolecular surface (and inside it) is inaccessible, i.e., λ(r) = 0; the remaining region outside has λ(r) = 1.

4.5.2 Reaction Coordinate for System’s Unbinding In Ref. [45], the reaction coordinate describing the system’s unbinding was defined as the distance between the groups of a specific amino acid, Arg39 L (Arg39 L refers to Arginine number 39 in the L-chain) and Flu. As described in Ref. [45], Arg39 L has its +1-ionized group (centered on atom Cζ ) directly involved in hydrogen bonding to the hydroxyl group of Flu (atoms O1–H12 in Fig. 4.18). Thus, it is logical to consider the distance between the specific groups of Arg39 L and Flu engaged in that driving hydrogen bonding as a most likely unbinding coordinate. On this basis, the distance between the Cζ atom of Arg39 L and the Flu’s hydroxyl oxygen (O1) was then set as the appropriate coordinate for scanning. The scanning started from the distance in the reference conformation and progressed in increments of 0.25 Å until a ∼40 Å distance. At this distance and for the set cutoff, the interaction energy between the hapten and the AB becomes zero. For each scanning step, the system was energy-minimized to an energy gradient tolerance ≤ 4×10−4 eV·Å−1 . A 12 Å cutoff on long-range interactions was used. During minimization, the hapten was free to move, while the AB was kept frozen for all but the side-chain atoms of a few key amino acids gating the passage of the hapten. The unconstrained side chains belong to His31 L , Asn33 L , Arg52 H , Tyr56 H , Glu59 H , Tyr102 H , and Tyr103 H . Minimizations were performed for ε = 1, and solvent effects were introduced as corrections a posteriori. Calculation of koff . In compliance with the experimentally reported Arrhenius-like behavior [49] and within the context of the reaction-rate theory [69], the off-rate constant along the scanned pathway was computed using the expression koff = ω exp (−E/kB T ) ,

(4.50)

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Fig. 4.20 Energy profiles (in vacuo) for different distance scanning runs, corresponding to different constraining schemes on the protein atoms. Each curve corresponds to a different number of gating amino acid side chains that were allowed to move during each scanning, namely, seven side chains (c1), six (c2), five (c3), and four (c4) (see the text for details). Reproduced from Ref. [45] with permission from Springer Nature

where E is the activation energy (i.e., the barrier height) and ω the pre-exponential factor, which was estimated using the harmonic approximation, i.e., ! 1 ω= 2π

k . μ

(4.51)

Here μ is the reduced mass of the system and k the harmonic force constant obtained from parabolic fit of the data (the bounding region of the well in the energy profiles). For systems similar to the one described in this section (with reduced masses in the 200–500 range and binding pocket’s length within 3–7 Å), an estimate for the frequency ω falls in the 1011 –1012 s−1 range.

4.5.3 Energetic and Structural Analysis The energy profiles resulting from the scanning runs with and without solvent correction are plotted in Figs. 4.20 and 4.21. Figure 4.20 displays the in vacuo results for four different scanning runs, corresponding to different constraining schemes on the protein atoms. The scheme referring to the seven unconstrained side chains (His31 L , Asn33 L , Arg52 H , Tyr56 H , Glu59 H , Tyr102 H , and Tyr103 H ) has been labeled “c1” in Fig. 4.20. To better assess on the influence of those seven amino acids on the escaping profile, they were subject

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Fig. 4.21 Comparison of the distance scanning energy profiles in vacuo and with implicit solvent corrections (with and without dissolved electrolyte), for the “c2” and “c3” constraining schemes. Reproduced from Ref. [45] with permission from Springer Nature

to successive constraining procedures, exemplified in Fig. 4.20 for three representative cases, labeled “c2”, “c3”, and “c4”, that correspond to six, five, and four unconstrained side chains (out of the initial seven). The constraining limit is the set of amino acids Asn33 L , Tyr56 H , Tyr102 H , and Tyr103 H corresponding to the “c4” curve. Within this limit, no general significant differences on the energy profile arise from the explored different schemes. These four amino acids always experience significant conformational changes upon the hapten’s passage, in comparison to the remaining moving ones which just slightly adjust positioning. The plane defined by the side-chain oxygens of the four amino acids in question can be taken as the outmost limit of the protein’s pocket, and it is

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Table 4.5 Kinetic and equilibrium parameters obtained from calculations [45] based on the computational scanning. Available experimental values are also presented for comparison: (a) and (c) determined in solution (Refs. [70] and [71], respectively), and (b) at a surface by SPR [51] Parameter

Simulations In vacuo

koff

(s−1 )

c2

c3

c2

c3

3.4 × 10−6

6.8 × 10−6

4.1 × 10−3

5.4 × 10−3

Equilibrium 3.50

Experimental

T (K)

1.9 × 10−3 (a)

291

6.8 × 10−3 (b)

298

4.3 × 10−3 2.5 × 10−2 (c)

298

3.65

291

Solvent corrected

3.55

3.60

3.65

distance (Å)

intersected at a ∼15 Å scanning distance. Below this separation distance, the total energy plots in Fig. 4.20 depict the expected profile for an activated process. For the different curves, the height and shape of the energetic barrier at ∼7 Å is essentially the same: 1.029, 1.027, 1.060, and 1.026 eV, respectively, for 7, 6, 5, and 4 moving side chains. Beyond the 15 Å distance, the in vacuo profiles depict an asymptotic increase to a final plateau above the referred energetic barrier, making unbinding unfeasible. Predictably, the inclusion of solvent effects rectifies the asymptotic behavior depicted in the in vacuo profiles, as exemplified in Fig. 4.21 for two scanning runs. At larger separation distances, the energy profile has been significantly flattened, and it is also for the larger distances that the effect of the dissolved electrolyte becomes perceptible. In solution, the electrostatic interactions between the protein and the escaping hapten are effectively screened allowing for unbinding to happen. Of relevance is also the decrease in the height of the energetic barrier at ∼7 Å: with implicit solvent effects, this barrier value is 0.863 and 0.871 eV, respectively, for the “c2” and “c3” schemes. koff determination. Table 4.5 presents the calculated values of koff , with and without solvent correction, resulting from parabolic fit to the profiles, considering the energy barrier at ∼7 Å. Experimentally available koff values are also presented for comparison. It is evident that, even for an extensively studied system like mAb4-4-20–fluorescein, experimental koff values may differ by an order of magnitude, depending on setup conditions and techniques [51, 70, 71]. As for the estimated values, while the in vacuo results are off-range, the solvent-corrected ones are comparable to the experimental results. The equilibrium distance between the antibody and the hapten (the well minimum) also compares better to the experimental value in the case of the solvent-corrected simulations. As it follows from the data presented in Table 4.5 the different constraining schemes have little influence on the order of magnitude of the koff values. The results reported in Ref. [45] and reviewed above open a practical and physically sound procedure to compute energy profiles along the selected reaction coor-

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dinate(s). For the exemplar case study of the biologically relevant antigen–antibody complex, it was possible to find a distance-dependent escaping channel in the multidimensional potential energy landscape, thus reducing the unbinding to a lowdimensional problem: the system seems to be efficiently bound by this one distance coordinate. The effect of the solvent was also accounted for. Despite the fact that it was introduced as a correction a posteriori, it permits to ascertain that this is one effect that needs to be included, because it has a significant influence on the overall energetic profile and subsequent parameters derived from it. With solvent effects, the derived off-rates are in reasonable agreement with the experimentally determined ones, a result that can be regarded as an indicator of validity of the developed approach.

4.6 DNA Unzipping The DNA double helix unzipping is a compulsory stage of the biological processes, such as the DNA transcription and replication, in which the genetic information is transferred. In these processes, the double-stranded (ds) DNA is separated into two single strands (ss), making it possible to read and to reproduce the genetic information recorded in the nucleic bases sequence [72]. The ds- to the ss-state transition is one of the most studied processes in molecular biophysics (see, e.g., [73]). It is observed in vitro under the temperature increase up to 70–100 ◦ C (DNA melting) or under the decrease of ions concentration in solution [73]. However, the process of DNA melting differs from the double helix unzipping by enzymes. In the course of the former, the DNA regions with rich content of A-T pairs melt at lower temperatures and then the regions with high content of G-C pairs which are more stable. On the contrary, the enzymes open the DNA double helix in the processes of transcription or replication sequentially, i.e., unzip one complementary pair after another. Correct understanding of the DNA unzipping process would shed light on problems in molecular biology and DNA physics. However, the mechanism of strands separation in dsDNA is not quite clear yet [74]. Significant progress in the experimental study of the DNA unzipping process was achieved by means of the single molecule manipulation techniques [75–78]. In the cited experiments, the unzipping was performed via mechanical separation of the DNA strands driven by an external force applied to a paramagnetic bead connected chemically to one of the strands of the double helix. The experiments show that the DNA unzipping proceeds as a threshold process, i.e., it begins at a certain critical value of the applied force and has a cooperative character. Despite the sufficiently detailed study of DNA unzipping in experiments with single molecules, the process of the DNA stands separation remains not entirely understood. This is due to the fact that the utilized experimental methods deal only with the macroscopic characteristics of the unzipping, such as the distance between the unzipped strands, the level of the critical force, and the velocity of strand pulling. However, full understanding of the unzipping mechanics and convincing interpretation of the experimental results can be achieved if one is able to describe the micro-mechanics of the base pairs opening in the double helix.

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Fig. 4.22 DNA duplex constructed from Drew–Dickerson dodecamer with hairpin AAG (DD-h). a nucleotide content in DD-h; b additional external forces, applied to the opposite phosphates groups of different strands, drive the directed unzipping process of dsDNA. Reproduced from Ref. [84] with permission from IOP Publishing

So far, the theoretical description of DNA unzipping was devoted to the analysis of the dependence the critical value of applied force on the base pair [76], the origin of irregularities in the unzipping process [79], the influence of the double helix torque on DNA unzipping [80], and the temperature dependence of unzipping [81]. The mechanisms of cooperative effects in DNA strands separation were analyzed in [82]. The cited study revealed the important role of the pathway of transition of the base pairs in dsDNA from closed to open states in the course of the unzipping process and the formation of metastable states of preopen base pairs in water environment. The first all-atom MD simulations of DNA duplex unzipping was performed in Ref. [83], but only the macroscopic characteristics for the double helix separation were obtained. MD simulations of the directed unzipping process of the DNA duplex in water and counterions solution were carried out in Ref. [84]. The object chosen for the simulations was the DNA duplex consisting of the Drew–Dickerson (DD) dodecamer: d(CGCGAATTCGCG)2 , with the hairpin (AAG) at one of the double helix ends [85]. This system includes 27 nucleic bases that form 12 complementary base pairs. The structure of the duplex, its nucleic content, and sequence are shown in Fig. 4.22. The simulations were performed with accounting for an additional external force driving the directed unzipping process. The DD duplex with hairpin (DD-h) was unzipped on the pair-by-pair basis by means of the steered MD which was successfully used in earlier studies of biomacromolecular mechanics, e.g., [86]. Below in this section we overview the results of Ref. [84] and consider the DNA unzipping process as a case study of MD simulations which can be achieved by applying the MBN Explorer software package [12] for modeling transformations in complex biological systems. The MD simulations [84] were performed using the CHARMM27 force field [5] (see also Sect. 4.2.1) to describe the interactions between atoms. This is a common

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Table 4.6 Structural parameters of the Drew–Dickerson dodecamer: experimental values obtained via NMR and X-ray methods are taken from PDB, the MD results are from Ref. [84] Parameter NMR PDB data X-ray PDB data MD Twist (◦ ) Rise (Å) Roll (Å) Stretch (Å) Stagger (Å) Shear (Å) Opening (◦ ) Buckle (◦ ) Propeller (◦ )

33.9 ± 0.6 3.35 ± 0.03 3.8 ± 0.9 −0.18 ± 0.03 −0.11 ± 0.05 −0.00 ± 0.04 1.7 ± 0.4 −0.1 ± 0.5 −11.8 ± 1.3

35.6 ± 0.6 3.35 ± 0.03 −0.3 ± 0.9 −0.15 ± 0.02 0.16 ± 0.03 −0.01 ± 0.03 1.1 ± 0.5 1.2 ± 0.9 −12.0 ± 1.2

34.0 ± 1.0 3.30 ± 0.05 5.2 ± 1.0 −0.09 ± 0.01 −0.11 ± 0.07 −0.02 ± 0.05 0.6 ± 0.7 1.6 ± 2.3 −8.5 ± 2.0

empirical force field for treating DNA, proteins, and lipids [5, 87, 88]. The set of parameters used in the simulations can be found in Refs. [6, 14, 89, 90]. The duplex was solvated in the water box with dimensions 72 × 102 × 100 Å3 . The water was described using the TIP3P model [5]. Sodium and chloride ions with concentration of 100 mM were added to the water. The system was equilibrated in the NPT canonical ensemble for 500 ps at 310 K and pressure 1 atm. The total number of atoms (including water and hydrogen molecules) in the system exceeded 80,000. The simulations were performed with a time step of 1 fs, the uniform dielectric constant was set to 1, and the Coulomb forces cutoff was implemented using the switching function which goes from 1 to 0 at the distances between 10 and 12 Å. The steered MD utilizes an additional constraining harmonic potential applied to the phosphates of different strands located between the first and second base pairs of the DNA duplex. Introducing the dependence on time t, one writes the potential as follows: κ U (r, t) = (r − r0 − νt)2 . 2 In the simulations, the stiffness k was set to 500 kcal/mol/Å2 to make the distance r between the phosphates much larger than its thermal fluctuation δr ∼ (kB T /k)  0.1 Å. The initial distance r0 was set to 20 Å. The unzipping of DNA with nearly constant speed ν was performed by the stepwise increase of the νt value by 0.1 Å at the end of each 500 fs interval. In the course of simulations, the equilibrium structure of the DNA duplex in the water solution with counterions was relaxed during 2 ns by running the MD simulations. The parameters of the DD structure were averaged over the last 0.3 ns of the simulation trajectory by means of the software package “Curves+” [91]. In Table 4.6, the structural parameters obtained are compared with the experimental data available from Protein Data Bank (PDB).

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The data listed in Table 4.6 indicate good overall agreement between the MD simulations and experiments. The parameters of the mutual base pairs disposition in the DNA duplex (Twist, Rise) are in good agreement with the PDB data for DD conformation both in solution (NMR) and in crystal (X-ray). The average roll parameter obtained from MD calculations is close to NMR result. The MD results of the translational parameters of the bases intrapair (Stretch, Stagger, Shear) are also in agreement with experiment. There is deviation in the averaged values of the rotational parameters Buckle and Propeller from the experimental data. This discrepancy is due to significant variation of the parameters in the two last pairs in DD-h caused by the presence of hairpin at the end of studied DNA duplex. Using the steered MD approach with the force applied, the separation of the DNA duplex strands was performed for the structure of DD-h being initially in the equilibrium state with all the pairs closed [84]. The unzipping process stops when fully open form of DD-h is achieved. The simulations were repeated three times and the resulting trajectories (runs) of the DD-h unzipping were recorded.

4.6.1 Macroscopic Parameters of the Duplex Strands Separation The simulations performed allowed the authors of Ref. [84] to carry out analysis of the macroscopic parameters of the double helix such as the stretching distance between the phosphates in the separated strands, the distance between the atoms bonded by central H-bonds in the complementary pairs, and the rotation angle of the base pairs in the course of the unzipping process. The illustrative results of the steered MD simulations for the first six pairs are presented in Fig. 4.23. The simulations show the transformation of the double helix into two single chains of nucleotides. At the beginning of simulations, the distance between the terminating phosphates is 20 Å. During the strand separation process this distance gradually increases for each pair of separated phosphates, see Fig. 4.23a. The increase is due to the unwrapping of the DNA strands (the distance between the phosphates of one strand along the DNA backbone is ca. 7Å) under the action of the applied force. Similar changes can be observed for the stretching of base pairs. Figure 4.23b shows the growth of the N–H distance in the central H-bond of the pairs in duplex with time. As it is seen, the distance grows not so smoothly as it happens for the phosphates. This is a result of involvement of additional degrees of freedom of the bases in the DNA strand. It can be noted that the phosphates to which the force is applied are located in different strands between the first and second bases. Therefore, the stretching trajectories for the first pair of phosphates are close to straight lines, see Fig. 4.23(a) and the lengths for H-bonds in the first and second base pairs are close to each other, see Fig. 4.23(b).

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Fig. 4.23 a Distances between the phosphates, b distances between hydrogen atoms in the central H-bond of the base pairs, c the variation of the base pairs rotation angle under unzipping of the first six base pairs in DD-h. The common horizontal axis reflects the simulation time. Reproduced from Ref. [84] with permission from IOP Publishing

Due the action of the applied force, each base pair of the duplex rotates around the double helix axis to the position of the unzipped pair (being collinear with the direction of the applied force). Figure 4.23c illustrates the evolution of the rotation angle for the pairs of phosphates in DD duplex from its initial value. In the course of simulation, the angles between the line of the force applied to the first pair of phosphates in DD-h and all the subsequent pairs of phosphates were monitored during the process of the duplex unzipping. As seen, at the initial stage of the unzipping process, the angles between the lines of the phosphates in neighboring pairs have the characteristic difference of ca. 34◦ being in agreement with the results of the simulations of the DD-h structure, see the DD structural parameters in Table 4.6. This analysis shows that prior to the start of the unzipping process, the DNA duplex rotates as a rigid body by the helix twist angle to bring the pair to the correct initial position. This result confirms the long-standing prediction [92] on the double helix rotation in the process of DNA transcription. The dependencies presented in Fig. 4.23c also suggest that during the unzipping process each pair experiences two types of motion. First, there is rotation together

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with all other closed pairs in the duplex. The second motion is the relatively slow rotation of the unzipping pair during the stage when it becomes a part of the unzipping fork. These two rotations are revealed in Fig. 4.23 (c) as the change in the slope of the dependencies at the instants at which the corresponding inter-phosphate distances starts to grow (Fig. 4.23 (a)) and the corresponding base pairs begin to unzip. These instants are marked with arrows in Fig. 4.23 (c). The decrease of the rotation speed in the separating bases is caused by the transition of each base pair from the closed state in the double helix into the unzipped single strand state which is not stabilized by hydrogen bonds. This transition takes place on the characteristic scale of several angstroms.

4.6.2 Internal Parameters of Base Pairs Unzipping For better understanding of the mechanism of the strands separation in dsDNA, one can analyze the motion of nucleic bases in the complementary base pairs in the course of unzipping. This can be done by analyzing the evolution of translational and rotational parameters of the nucleic bases within the complementary pairs. In Ref. [84], the variation of these parameters along the unzipping trajectories was calculated for the third and fourth (G-C), and the fifth and sixth (A-T) pairs in DD-h and the analysis of the appropriate tendencies and features was presented. To illustrate the results obtained, Fig. 4.24 shows the dependencies of the structural parameters (as indicated) on simulation time for the fourth G-C pair. In each plot (top and bottom, see the caption), the three rows correspond to the data collected during three runs of simulations. The figure shows that when the unzipping process begins the translational parameters, Stretch or Shear, start to grow. The amplitudes of fluctuations of the translational parameters (Stagger, in particular) increase steadily as the inter-phosphate distance starts growing. This can be due to diminishing of the stacking interactions with neighboring bases along the chain. Sharp variation of the rotational parameters at the beginning of the unzipping process is clearly seen for the Opening, and in several cases for the Buckle and Propeller parameters. The increase of the distance between the strands leads to the formation of some sort of “shelves” in the dependencies for the Stretch and Shear parameters. As seen, the shelves arise after 225 ps (the first run), 215 ps (the second run), and 240 ps (the third) in the Stretch dependencies, and somewhat later (the second and third runs) in the Shear one. The process of shelves formation in the unzipping trajectories is accompanied by the change of fluctuation amplitudes for corresponding coordinates. These increase significantly on the shelves, which is another clear indication of the formation of a new state of the pair. The results of the simulations show that stretching and shearing motions of the bases are very essential for the unzipping process. They have larger amplitudes as compared to those of other types of motion. The analysis of the structural parameters of base pair unzipping for the simulated trajectories in all runs indicates that the

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Fig. 4.24 Variation of translational and rotational parameters during the unzipping of the fourth G-C base pair of DD-h. In each plot (the top plot shows Stretch, Stagger, and Shear, the bottom one—Opening, Buckle, and Propeller), three rows correspond to the three runs of the steered MD simulations. Reproduced from Ref. [84] with permission from IOP Publishing

translational motion of the bases is the dominant motif in the process. The variation of rotation parameters accompanies the base pair unzipping and facilitates the formation of certain local conformational states of the base pairs at some parts of the unzipping trajectories. Analysis of a number of trajectories carried out in Ref. [84] allowed the authors to formulate the following two characteristic scenarios for the unzipping process. The first one starts with the base pair unzipping via the Stretch growth, followed by the shelf formations (at about 2 Å) after which the Shear value also starts increasing. The

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Fig. 4.25 Base stretching coordinate (lines) and number of water molecules (symbols) vs. simulation time for the fourth (left) and sixth (right) base pairs (bp) of DD-h. Three rows correspond to the three runs of the simulations. Reproduced from Ref. [84] with permission from IOP Publishing

second scenario is characterized by the growth of the Stretch and Shear parameters, and the formation of the shelves at about 5 Å on the Shear coordinate dependence. The first scenario occurs for G-C pairs in the most cases and less frequently for the A-T pairs. The second scenario takes place for the A-T pairs, and to some degree it varies for different trajectories. Such variations reflect the fact that the opened bases create new interactions between themselves and with water molecules in solution.

4.6.3 The Role of Water Molecules in Base Pair Unzipping Formation of shelves in the unzipping coordinate dependence on time and the enhancement of the fluctuations of the translational parameters characterize the transition of an unzipped base pair to another conformational state. At some distance between the bases within the pair, when the H-bonds stretching reaches a certain value, a new type of bonding between the bases can arise in which the bases interaction occurs via water molecules. Thus, the water molecules can form bridges between the bases in the unzipped pair and stabilize the new stretched state. A special study was performed in [84] aimed at understanding the role of water molecules in the complementary base pairs (bp) opening. A part of water molecules that forms the bridges between the atoms of different bases in the pair undergoing the unzipping process were marked in the course of the simulations. To form the H-bond, the following condition was assumed to be met. The bond was considered as created between the oxygen (or nitrogen) and the hydrogen atoms if the distance between the atoms does not exceed 3.7 Å and, simultaneously, the angle between the directions of the O–H bond and the H–N (or H–O) bonds is less than 30◦ . The role of water molecules in the base pair unzipping was studied for the fourth (G-C) and the sixth (A-T) pairs of DD-h. The results of calculations for the three runs of simulations are presented in Fig. 4.25. In all graphs, the vertical axis scales the base stretching distance coordinate (the corresponding dependencies are plotted

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with lines) and the number of water molecules that link the bases in the unzipping pair (symbols). The results presented demonstrate that the fourth G-C bp (Fig. 4.25left) in each run gets unzipped following a similar path. Each unzipping trajectory exhibits a metastable state [82] of the bp linked by one or two water molecules. That state exists for about 30 ps of simulation time and correlates with the formation of the shelves discussed above. In the process of the A-T bp unzipping, the water molecules are involved differently in different runs. In first run, the water molecules behave similarly to the case of the G-C bp although only single water molecules links are formed; in the second run, the unzipping proceeds without noticeable involvement of water molecules; in the third run, water molecules become involved in the stagger type of motion of the bases, small stretching and sharp change of the propeller coordinates [84]. Sharp changes of the shear coordinate (on the level of 5 Å) in the second and third runs destroy possible water bridges in base pairs. The differences in the A-T and G-C bp unzipping arise due to the different number of the H-bonds in these base pairs and as a consequence the different mobility.

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82. Volkov, S.N., Solov’yov, A.V.: The mechanism of DNA mechanical unzipping. Eur. Phys. J. D 54, 657–666 (2009) 83. Santosh, M., Maiti, P.: Force induced DNA melting. J. Phys.: Condens. Matter 21, 034113 (2009) 84. Volkov, S.N., Paramonova, E.V., Yakubovich, A.V., Solov’yov, A.V.: Micromechanics of base pair unzipping in the DNA duplex. J. Phys.: Condens. Matter 24, 035104 (2012) 85. Drew, H.R., Wing, R.M., Takano, T., Broka, C., Tanaka, S., Itakura, K., Dickerson, R.E.: Structure of a B-DNA dodecamer: conformation and dynamics. Proc. Natl. Acad. Sci. U.S.A. 78, 2179–2183 (1981) 86. Sotomayor, M., Schulten, K.: Single-molecular experiments in vitro and in silico. Science 316, 1144–1148 (2007) 87. Sotomayor, M., Corey, D.P., Schulten, K.: In search of the hair-cell gating spring: elastic properties of ankyrin and cadherin repeats. Science 13, 669–682 (2005) 88. Gullingsrud, J., Schulten, K.: Lipid bilayer pressure profiles and mechanosensitive channel gating. Biophys. J. 86, 3496–3509 (2004) 89. Rapaport, D.C.: The Art of Molecular Dynamics Simulation. Cambridge University Press (2004) 90. Frenkel, D., Smit, B.J.: Understanding Molecular Simulation, 2nd edn. Academic Press, San Diego, San Francisco, New York, Boston, London, Sydney, Tokyo (2002) 91. Lavery, R., Moakher, M., Maddocks, J.H., Petkeviciute, D., Zakrzewska, K.: Conformational analysis of nucleic acids revisited: curves+. Nucl. Acids Res. 37, 5917–5929 (2009) 92. Levinthal, C., Crane, H.R.: On the unwinding of DNA. Proc. Natl. Acad. Sci. U.S.A. 42, 436–438 (1956)

Chapter 5

Quantum Effects in Biological Systems Anders Frederiksen, Thomas Teusch, and Ilia A. Solov’yov

Abstract This chapter overviews and discusses an emerging area of quantum biology. The chapter provides examples of multiscale biophysical phenomena in which quantum mechanics (QM) plays a crucial role. Among others, the chapter discusses case studies of quantum processes present in real biological systems, such as photoabsorption, electron transfer, proton-coupled electron transfer and spin chemistry. For all the case studies, both a quick rundown of the theory behind each example is provided accompanied with real-life applications. Finally, ideas and possibilities, where the knowledge from quantum effects in biological systems may stimulate advances in novel devices for diverse applications on the nanoscale, are provided.

5.1 Possible Quantum Effects in Biological Systems Naturally, quantum mechanics has been used successfully to describe the physical properties of various phenomena at the scale of atoms and subatomic particles. It is striking that an increasing number of experimental evidence has accumulated in the last decades suggesting that quantum mechanics may also play a decisive role in wet and noisy biological environment [1–4]. This chapter deals with several of such phenomena and discusses how quantum mechanics may become crucial for certain biological functions. The chapter provides an introduction to the multiscale concepts that can be applied to deliver a detailed description of such effects. Let us start by introducing several general problems that exist in the field of quantum biology.

A. Frederiksen · T. Teusch · I. A. Solov’yov (B) Department of Physics, Carl von Ossietzky Universität Oldenburg, Carl-von-Ossietzky-Str. 9-11, 26129 Oldenburg, Germany e-mail: [email protected] A. Frederiksen e-mail: [email protected] T. Teusch e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 I. A. Solov’yov et al. (eds.), Dynamics of Systems on the Nanoscale, Lecture Notes in Nanoscale Science and Technology 34, https://doi.org/10.1007/978-3-030-99291-0_5

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5.1.1 Photosynthesis Solar energy harvested by photosynthetic organisms is the primary source of energy in the biosphere. The Earth receives solar energy at a rate of 120,000 TW, vastly exceeding the current worldwide energy consumption rate of 15 TW [5]. Solar energy, harvested by biological, artificial, and bio-hybrid systems, constitutes a renewable source for mankind’s future energy needs. In photosynthetic purple bacteria, light harvesting is performed by membrane domains called chromatophores (Fig. 5.1), comprising supra-molecular assemblies of hundreds of cooperating protein subunits that convert short-lived electronic excitations resulting from photon absorption to stable chemical energy. Chromatophores constitute the simplest known photosynthetic system and, therefore, are ideal for fundamental investigations [6, 7]. Experiments on photosynthesis generated renewed interest in quantum coherence [10–14]; quantum coherent beating in chromatophores suggests that coherence is possibly not as sensitive to environmental noise as previously expected [15]. State-of-the-art research seeks to quantify the impact of biological environment on the quantum coherence within the chromatophores and to deliver understanding of the unprecedented efficiency of inter-complex excitation transfer which is presently not achievable in any artificial system created by mankind [8, 9]. Modeling of the excitation transfer in biological systems relies on multiscale methods of quantum mechanics to account for the quantum processes, accompanied with the classical methods used to describe the motion of the biological environment [16].

Fig. 5.1 Photosynthetic molecular machinery chromatophores. A Spherical chromatophore from Rb. sphaeroides consisting of LH2s (green), LH1-RC cores (red, blue), bc1 (purple) and ATP synthase (orange) [8]. The chromatophore, including water molecules and lipids (not shown), contains approximately 136 million atoms. B Lamellar chromatophore from Rsp. photometricum consisting of LH2s (green) and LH1-RC cores (blue) [9]. The system contains, including water (not shown) and lipids, approximately 20 million atoms. Reproduced from [8, 9] with permission from Elsevier

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5.1.2 Charge Transfers in Biological Systems Electron transfer reactions have a vital importance in biological systems, being, for example, responsible for such acts as activation of sensory proteins [1], DNA UVdamage repair [17], energy harvesting [8, 9, 18, 19], magnetic field sensing [20–24] and many others. Three of these exemplary functions are illustrated in Fig. 5.2: the electron transfer reaction activates enzyme photolyase which then repairs a UVdamaged DNA, Fig. 5.2A; a charge-transfer processes through the cytochrome bc1 complex leads to formation of an electrostatic gradient through a membrane [3, 25, 26], Fig. 5.2B; a light-triggered electron transfer induces activation of a photoreceptor protein cryptochrome. Even though the role of electron transfer reactions has been established in various biological systems (see citations above), it is difficult to observe such reactions experimentally under controlled conditions. In particular, experimental studies alone cannot describe electron transfers on the level of atomistic details, which, however, is often necessary for completing the interpretation of the underlying biophysical mechanisms. Alternatively, computational models of electron transfer processes provide reasonably robust approaches [4] to characterize electron transfer reactions. It has been revealed that for a quantitative description of the electron transfer processes in a biological system, it is necessary to consider the entire system and not just the electron donor and acceptor sites that are directly involved in the electron transfer process. This has been recently demonstrated for several different exemplary systems [22, 23, 27–31]; however, it remains largely unknown what interactions between the moving electron and the rest of the protein constitute the driving force for the electron transfer reaction. The cytochrome bc1 complex (see Fig. 5.2B) is an important example of a biological system whose function relies heavily on electron and proton transfers [26]. The bc1 complex is a catalytic transmembrane protein that, through a series of

Fig. 5.2 Examples of electron transfer processes in biological systems. A Electron transfer initiating DNA UV-lesion repair by enzyme photolyase [17]. B Electron transfer triggering a cascade of charge-transfer reactions in the cytochrome bc1 complex that lead to a formation of an electrostatic gradient through the plasma membrane [3, 25, 26]. C Activation of cryptochrome protein initiated by blue light excitation of the flavin cofactor leading to a formation of a radical pair [20–24]. Figure reproduced from [22]

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proton and electron transfer reactions, oxidizes quinol (QH2 ) cofactors at the socalled Qo active site and reduces quinone (Q) cofactors at the Qi active site, in an overall process referred to as the Q-cycle [25]. The initial step of the Q-cycle corresponds to the binding of a QH2 molecule to the Qo -site followed by two electron transfer reactions, taking place in a bifurcated manner, towards different prosthetic groups of the bc1 complex subunits. During each QH2 oxidation, the bifurcated electron transfer occurs alongside two proton transfers from the QH2 to the positive side of the membrane, in order to create a transmembrane potential that is necessary for ATP synthesis [3, 22, 27, 28, 32–34]. The bc1 complex is present both in photosynthetic units like chromatophore and in respiratory chains [3, 25, 26]. It is striking that despite its importance the details of the mechanism of the proton and electron transfers at the Qo -site of the bc1 complex still remains elusive. However, it is believed that the bifurcation of the electrons is accompanied by the proton transfers, i.e. that electrons and protons are transferred simultaneously from QH2 to the bc1 complex in a coupled fashion [3, 26]. Such coherent dynamics has fundamental importance for the understanding of protein function, but is also providing a very illustrative example in quantum biology, where interplay of quantum and classical scales plays a decisive role.

5.1.3 Magnetoreception Magnetic sensing is a type of sensory perception that has long captivated the human imagination, although it seems inaccessible to humans. Over the past 50 years, scientific studies have shown that a wide variety of living organisms have the ability to perceive magnetic fields and can use information from the Earth’s magnetic field in orientation behaviour [1, 35–41]. Perhaps the most well-studied example of animal magnetoreception is the case of migratory birds (e.g. European robins (Erithacus rubecula), Australian silvereyes (Zosterops l. lateralis) and garden warblers (Sylvia borin)), who use the Earth’s magnetic field, as well as a variety of other environmental cues, to find their way during migration [41]. The core for the magnetic compass sense is located in the eyes of a bird, and furthermore it is light-dependent, i.e. a bird can only sense the magnetic field if certain wavelengths of light are available [42–44]. Specifically, many studies have shown that birds can only orient if blue light is present [45–47]. The avian compass is also an inclination-only compass, meaning that it can sense changes in the inclination of magnetic field lines but is not sensitive to the polarity of the field lines [48, 49]. Under normal conditions, birds are sensitive to only a narrow band of magnetic field strengths around the geomagnetic field strength, but can orient at higher or lower magnetic field strengths given accommodation time [44]. Despite decades of study, the physical basis of the avian magnetic sense remains elusive [1, 41, 50, 51]. The most supported mechanism to explain the magnetic compass in birds is the so-called radical pair-based model [1, 50]. The underlying idea of this model is that the avian compass may operate through a chemical reaction in the eyes of a bird, involving the

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Fig. 5.3 Schematic illustration of the avian radical pair-based compass. Magnetoreceptive molecules in the birds eyes host a non-equilibrium pair of radicals (R1 , R2 ) and endow the bird with capabilities to sense the Earth’s magnetic field. The radical pairs participate in spin-dependent chemical reactions that are sensitive to the angle  between this z-axis and the direction of the geomagnetic field B, which in turn could be related to the direction of bird motion, denoted by v. Figure reproduced from [52]

production of a non-equilibrium radical pair. A radical pair, most generally, is a pair of molecules, each having an unpaired electron (Fig. 5.3). An assumption of the radical pair mechanism (RPM) is that a pair of nonequilibrium radicals is created in a pure singlet quantum state and the pair starts to relax towards its equilibrium state in which the two electron spins are completely uncorrelated, i.e. when an external geomagnetic field can no longer influence the spin dynamics. A sensitive compass, therefore, requires to maintain a non-equilibrium state of the radical pair for as long as possible, which in practice means for at least a microsecond [53]. Normally, weak magnetic interactions, like the ones discussed here, have negligible influence on the outcome of chemical reactions because they pale in comparison to the random fluctuations in energy experienced by all molecules at physiological temperatures. Proof-of-principle experiments have demonstrated the sensitivity of a model radical pair system to the direction of an Earth-strength magnetic field (ca. 50 µT) via magnetic interactions that are a million times smaller than the thermal energy, kB T [1]. The sensitivity of the radical pair is high due to the fact that the radical pair operates sensibly to magnetic fields in a regime that is far from thermal equilibrium. In migratory birds this non-equilibrium radical pair is expected to reside in a protein called cryptochrome [23, 45, 50, 54, 55]. Cryptochrome 4 from European robin is especially interesting in this respect due to its peculiarities in the molecular structure [54]. Cryptochrome binds at its active site to flavin adenine dinucleotide (FAD) that absorbs blue light and triggers cryptochromes’ biological activation (see Fig. 5.2C). After the light absorption by FAD, an initial radical pair is quickly formed by

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electron transfer between FAD and a nearby tryptophan residue (Trp) resulting in a [FAD•− TrpH•+ ] radical pair inside cryptochrome. To be magnetic field sensitive, the radical pairs in avian cryptochromes are theorized to be further stabilized via sequential electron hopping along a chain of four tryptophans (see Fig. 5.2C) [4, 54]. The RPM is a genuine mechanism solidly backed by theory and experiment, with hundreds of laboratory studies reporting effects of external magnetic fields on radical pairs [1, 56–58]. Several behavioural observations support the RPM hypothesis with one finding that birds require light to use their compass sense [59–62]. Another important result is that birds have an inclination compass: instead of distinguishing north from south, they detect the difference in the directions of the magnetic pole and magnetic equator [1, 49, 63, 64]. This accords precisely with the RPM in which the yield of the reaction product is unaffected by an exact inversion of the magnetic field vector.

5.1.4 Artificial Systems Inspired by Nature Quantum effects in biological systems have inspired many scientists to focus on a magnitude of artificial systems that were inspired by Nature. Such inspirations were specifically targeted to address the important applications, such as energy storage and production. Water oxidation on tungsten trioxide is an example from the field of surface chemistry. This example is not only inspired by the natural processes but is closely linked methodologically to the biological phenomena mentioned above. It is remarkable that many processes in Nature occur with an unprecedented efficiency although they are hosted in highly disordered and flexible environment. To make a better link between quantum processes that occur in real life and those that are expected in laboratory, here an example of a chemical system is discussed that relies on intramolecular electron transfers and has distinct similarity in operation to photosynthetic and magnetoreceptive biological systems. In times of increasing CO2 concentrations in the atmosphere, alternative, green energy sources receive increased attention [65]. A steadily growing area of research is concerned with energy production inspired by Nature [66]. The photocatalytic generation of hydrogen is one promising approach in this context, since hydrogen is considered to be an efficient energy storage medium which is used in fuel cells. In the photocatalytic water splitting process, water is cleaved in hydrogen and oxygen in the presence of a photocatalyst with the help of sunlight. In 1972, Fujishima and Honda demonstrated that hydrogen can be produced with TiO2 as a catalyst [67], but the economic efficient production of green hydrogen is still a major challenge today. Since then, many different transition metal-based materials have been studied to increase the efficiency of hydrogen production [66]. One example of a promising material is tungsten trioxide (WO3 ), which enables the possibility of the visible light to be used for hydrogen production purposes. The basic principle of water splitting on a semiconductor surface essentially includes four steps [66, 68–77]. First, a photon is absorbed by the semiconductor.

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Fig. 5.4 Mechanism of excitation of tungsten trioxide by sunlight. The position of the redox potentials for H+ /H2 and O2 /H2 O is shown relatively to the valence band and conduction band position of WO3

The energy of this photon must be greater than the bandgap to create an electron–hole pair. Ideally, the bandgap ranges between about 1.6 and 2.9 eV, as this allows sunlight to be used for the excitation. This process is shown schematically in Fig. 5.4 on the left. In the second step, the electron and hole are separated by exciting the electron into the conduction band of the photocatalyst. The third step involves the transport of electron and hole towards the interfaces, where the partial reactions (water oxidation and hydrogen reduction) take place in the last step. These processes are shown in Fig. 5.4 (right). It is remarkable that the photocatalytic water splitting example has many analogies with the quantum biological processes discussed above. As in photosynthesis and magnetoreception, visible light is important to trigger the process, which leads to charge dynamics; this is also the case in the biological examples. It is therefore beneficial to consider water splitting reaction jointly with the other examples as the problems can naturally benefit from each other.

5.2 Walking the Thin Line of Complexity: Complexity Versus Feasibility Having mentioned a few relevant problems in quantum biology and related areas it is now instrumental to discuss the toolbox necessary for investigating these problems computationally. Below we summarize the concepts that are instrumental to address the problems mentioned above.

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5.2.1 Highly Accurate Ab initio Methods Density Functional Theory One of the basic tools for the description of quantum properties of biological systems is the density functional theory (DFT). It is remarkable that this is exactly the same theory as one would expect to apply to study nanoscale systems (see Chap. 2 of this book). DFT relies on the Hohenberg–Kohn theorem, which states that the ground state electronic energy of a complex system, E[ρ], can be determined by the density of electrons, ρ(r ) [78]. The most important equation of DFT is the Kohn–Sham equation, in which the kinetic energy of the Hamiltonian is calculated from the electron density by approximating the electron density to be a set of orbitals and through the unknown quantity, the exchange–correlation energy E XC [ρ] [79]:    Nn ZI ρ(r2 ) 1 2  − ∇i − + dr2 + VXC ψi (r1 ) = εi ψi (r1 ). (5.1) 2 r r12 I =1 I Here ∇i2 denotes the Nabla operator, Z I represents the charge of the I ’th nucleus, r I is the distance between the I ’th nucleus and an electron at position r1 in space. r12 is the distance between two electrons at positions r1 and r2 and εi is the energy of an orbital i. Equation (5.1) is written in atomic units, where the reduced Planck constant , elementary charge e, Bohr radius a0 and electron mass m e are set to 1. The electron density ρ can be described by the orbitals ψi used in the Kohn–Sham equation ρ(r ) =

n 

|ψi (r )|2 ,

(5.2)

i

where the summation is performed over all orbitals in the system. The VXC in Eq. (5.1) is the exchange–correlation potential defined as VXC [ρ] =

δ E XC [ρ] . δρ

(5.3)

A general assumption of DFT is that it describes systems in the electronical ground states. This assumption is not particularly useful for the calculation of electronic transitions, which involve excited states. However, DFT can be extended to include timedependent (external) electric potentials thereby permitting calculation of excited states, among other properties. The starting point for such time-dependent calculations is the time-dependent Schrödinger equation, i

∂  = H(r, t), ∂t

(5.4)

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with  being the wave function of the system. The Hamiltonian can be written as H(r, t) = H 0 (r) + V ext (r, t).

(5.5)

Here H 0 (r) is the time-independent Hamiltonian and V ext (r, t) is a time-dependent external potential. Equation (5.4) can be expressed through time-dependent molecular orbitals φi (r, t) as i

∂ φi (r, t) = (F + V ext (t)) φi (r, t), ∂t

(5.6)

which can further be reformulated as [80]: i

  ∂ φi (r, t) = T + V e f f (t) φi (r, t). ∂t

(5.7)

Here the effective potential V e f f (r, t) is given by V e f f (r, t) = V ne (r, t) + J(r, t) + V xc (r, t, 0 ) + V ext (r, t),

(5.8)

where V ne (r, t) denotes the operator of the time-dependent potential energy of the electron–nuclei attraction. The equation assumes that the set of orbitals describing the system is given by a Slater Determinant composed of Kohn–Sham orbitals, i.e. ρ(r, t) =

N 

|φi (r, t)|2 ,

(5.9)

i=1

and the Coulomb potential in Eq. (5.8) can be described as  ρ(r  , t) J(r, t) = dr  . |r − r  |

(5.10)

Equation (5.8) can be solved self-consistently by guessing an electron density ρ and an exchange–correlation energy E XC .

Wave Function Theory It is well known that various DFT functionals have problems in the description of charge-transfer processes [81], which are rather important in the case of quantum effects in biological systems. In order to be able to describe such processes correctly, even along a complete potential energy surface (PES), the complete active space self-consistent field (CASSCF) method [82] can be used. In contrast to the DFT methods, CASSCF is based on wave functions and not on the electron density and is particularly suitable for calculating systems with a multi-reference character in a physically meaningful way. Possible examples include molecular systems with

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degeneracies, bond breaking and modeling excited states. Furthermore, the CASSCF method is both variational and size-consistent [83]. Within the CASSCF method, the wave function is constructed as a linear combination of all possible determinants k [84]:  CkI | Sk , (5.11) |IS  = k

where CkI represents the expansion coefficients. The set of excited determinants | Sk  consists of Configuration State Functions (CSF) in an electronic state I. The CSFs are constructed from molecular orbitals ϕi , which in turn are expanded in basis functions:  cji φj , (5.12) ϕi = j

where cji represent the molecular orbital coefficients. The | Sk  are uniquely defined by their spin S and occupation. The energy of the system is then calculated according to the Rayleigh quotient [84] as E (c, C) =

IS |Hel |IS  , IS |IS 

(5.13)

with Hel being the electronic Hamiltonian. The energy is optimized so that its gradients vanish ∂ E (c, C) ∂ E (c, C) = = 0. ∂cji ∂CkI

(5.14)

From a technical point of view, a complete active space (CAS) is defined in which a full configuration interaction (Full-CI) calculation is performed. The selection of the orbital space allows a reduction of the problem to the significant orbitals. Since a Full-CI, due to the factorial scaling, quickly reaches its limits, a conventional CASSCF calculation without approximations is currently limited to about 14 orbitals for systems comprising just under 100 atoms [84]. The notation of such a CAS is done by specifying m electrons in n orbitals, from which a CAS(m,n) is formed, which is the so-called active space. It is obvious that the severe size limitations do not permit to apply the CAS method to any realistic biological system; this approach can nevertheless be very useful in a hybrid or multiscale description, where several methods of various complexities are used and are coupled with each other. It is possible that the orbitals are not only optimized for a certain state, but represent an averaging over many states. Such situations correspond to the so-called state-averaging (SA) and offer an efficient solution to calculate several states simultaneously with one set of orbitals.

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It should be noted at this point that a CASSCF calculation is quite complex to carry out. It is not only mathematically a difficult optimization problem, but also technical problems occur due to undesired orbital rotations and the so-called root flipping. A disadvantage of the CASSCF method is that although it provides a qualitatively good wave function, the total energies obtained are typically overestimated [85]. This is due to the fact that the dynamic correlation is only taken into account within the active space, but is neglected outside. To overcome this problem, various correlation methods are available. However, methods like multi-reference CI (MRCI) [86] and multi-reference coupled cluster (MRCC) [87] are computationally even more expensive [84], permitting nowadays calculations of systems with about 20 atoms, so that only multi-reference perturbation theory methods (MRPT) [88] can be used sensibly in the case of biological systems.

5.2.2 From the Real World to Model Systems Molecular Dynamics A key part of a biological system as well as its artificial human-made analogues is the environment [4, 16, 22, 89]. The environment is largely responsible for driving the function of biological machinery and, therefore, should be accounted for. The simplest way to do so is through the method of classical molecular dynamics. Classical molecular dynamics (MD) simulations are controlled by Newton equations and therefore a biological system consisting of N atoms can be described by solving a set of differential equations Fi = −

∂Utotal (r1 , r2 , . . . , rN ) = m i r¨i ∂ri

i = 1, 2, . . . , N .

(5.15)

With m i and ri being the mass and position of the i’th atom, respectively. Utotal is the total potential energy depending on all atomic positions thus coupling the positions of all atoms, and Fi is the force acting on the i’th atom. In order to solve Eq. (5.15), one needs to know the initial positions and velocities of all the atoms in the system. In the case of the simulation of a biological system, the initial coordinates are typically defined from the coordinate specification of the system, while the velocities are randomly assigned, following the Boltzmann–Maxwell distribution at a given temperature. Equation (5.15) can be rewritten for the NVT and NPT statistical ensembles, and transformed to its stochastic counterpart, known as the Langevin equation [90, 93]  dvi 2λkB T = −λvi + Fi (ri ) − Ri (t). (5.16) mi dt mi

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Here vi is the velocity, m i is the mass of a given atom, Fi (ri ) is the force acting on that atom, Ri (t) is a Gaussian random function, and λ is the damping coefficient. For the right-hand side of Eq. (5.16), the first term describes the dissipative motion and the last term describes random thermal fluctuations.

Force Field Functions Classical MD simulations of complex biological assemblies can, nowadays, be routinely carried out utilizing programs like NAMD [90, 91], Gromacs [92] and MBN Explorer [93]. The potential energy of the system in all these programs is defined through a so-called force field, which is often factorized as Utotal = Ubond + Uangle + Udi hedral + UvdW + UCoulomb .

(5.17)

Here the first three functions are potentials created by stretching, bending and twisting specific degrees of freedom in the system; these terms can be written as [90, 93]: Ubond =



kibond (ri − r0i )2 ,

(5.18)

bonds i

Uangle =



angles j

Udi hedral =

angle

kj





di hedral l

(θ j − θ0 j )2 ,

(5.19)

kldi hedral [1 + cos(n l φl − γl )], n = 0, n = 0. kldi hedral (0l − γl )2 ,

(5.20)

The degrees of freedom involved in these equations are illustrated in Fig. 5.5. The coefficients k are the spring constants that depend on the atom types in a given energy term. The nonbonded interactions in Eq. (5.17) are represented through the van der Waals and Coulomb interactions, which are usually factorized as

UvdW =

 i

UCoulomb =

j>i



εi j

σi j ri j

  qi q j . 4π 0 ri j i j>i

12



σi j −2 ri j

6  ,

(5.21) (5.22)

Here σi j is the equilibrium distance between two particles, i.e. the distance where the van der Waals potential has a minimum equal to εi j . ri j is the distance between atoms i and j, 0 is the vacuum permittivity and qi is the partial charge of an atom i. MD simulations are considerably cheaper than their quantum mechanical counterparts and may serve to obtain the different statistical states of molecular systems, as well

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Fig. 5.5 Bonded interactions in the classical molecular mechanics force field. Here r governs bond stretching; θ represents the bond angle term; φ gives the dihedral angle; the out-of-plane angle α may be controlled by an “improper” dihedral ϕ, see Eqs. (5.18)–(5.20)

as to describe the of overall protein dynamics. The more delicate information coming from electron excitations are not really accessible by the classical MD methods and are thus left to the more costly quantum mechanical computations.

5.2.3 Taking the Best from Both Worlds Electrostatic Embedding To reduce the complexity, but yet still be able to describe possible quantum effects in a complex biological environment, it is possible to use the so-called embedding schemes. The electrostatic embedding scheme consists of the three methodological layers shown in Fig. 5.6 and is described in detail in an earlier publication [94]. Here we just mention its basic concepts. The innermost region, depicted by the green hemisphere in Fig. 5.6, is the cluster model. It represents an area in the system that is treated with an accurate quantum chemical method. The model should be chosen as large as necessary and as small as possible in order to keep the computational effort low. This is by no means trivial, as many physical properties, such as the adsorption geometries and energies, have to be taken into account, which requires extensive convergence studies. The second region, visualized in red in Fig. 5.6, surrounds the cluster model and consists of so-called effective core potentials (ECP). The effective core potentials prevent the electron density from flowing out of the cluster model into the outer layer. Technically, the ECPs consist of point charges and a surrounding core potential. The last region is the point charge field, which is shown in blue in Fig. 5.6. The point charges are used to account for the long-range Coulomb interactions within the system. This region is characterized by the fact that although it has by far the largest

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Fig. 5.6 Schematic representation of an electrostatically embedded cluster model (left). The green region represents the cluster model, the red region the effective core potentials (ECP) and the blue region the point charges. A solvated protein is shown on the right as an analogy of a biological system with a QM region indicated in green, the protein scaffold in red and the remaining solvent molecules in blue

dimension of all the layers in terms of size, the calculation is not time-consuming. Petersen et al. have shown that the point charge field is essential in order to obtain correct excitation energies [95]. The authors also provide a recipe on how a cluster model can be constructed. Cluster models provide a powerful approach to describe electronic processes in small molecules in both ground and electronically excited states [94, 96, 97]. For example, Mitschker was able to show that this method can be used to describe photochemistry of water on a rutile-TiO2 surface [98–100]. In a recent study, Kick et al. [101] show that even small, unoptimized cluster models without polarization between the layers are sufficient to correctly reproduce the binding energy and electronic structure of the rutile-TiO2 (110) surface and OH, OOH and H2 O as adsorbates. In the case of biological systems, cluster models are, for example, successfully applied for studies of electron transfer in plant cryptochrome [24, 102], see Fig. 5.6 (right).

Hybrid QM/MM Methods Realization of the importance of both QC and MD simulations in application to complex biomolecules was achieved by Warshel and Levitt, who came up with a method to combine the two methods that later became known as the QM/MM method [103]. This is especially useful when a part of a system is of interest, but the surroundings are influencing the part in question. An example could be a part of a protein in an aqueous solution, if we take the rest of the protein or the water out of the simulation a crucial part might be missing and doing MD simulation can leave out certain effects

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only observed in QC simulations. This can be solved by combining the QC and MD methods, giving the potential energy of the system to consist of a sum of potentials for the different methods: VQ M/M M = VM D + VQC + VCoupling .

(5.23)

Here VM D is the potential from the MD simulation described in Eq. (5.17), VQC is the total potential energy from the QC simulations, and VCoupling is the potential gained from the coupling between the QC and the MD parts given by [103]

VCoupling =

 qi q j i, j

ri j

+

 i, j

⎡ i j ⎣

ri0j ri j



12 −2

ri0j ri j

6 ⎤ E W ⎦ + Vind + Vind ,

(5.24)

where the index i is used for atoms inside the quantum region and j are used for E is the atoms outside the quantum region (i.e. they are described classically); Vind W is the potential energy, defined by the charges and dipoles in the protein and Vind potential created by the dipoles induced by the surrounding water molecules. In the classical region, a point-induced dipole μ is assigned to each atom as

μj = α j Ej ,

(5.25)

where α j is the atomic polarizability of an atom j and Ej is the electric field given by    ri  r j j Ej = qi 3 + −∇ μ j  3 . (5.26) ri j r j j i j  = j Here qi is a charge in the quantum region and ri j is the distance between an atom i in the quantum region and an atom j outside the quantum region. When running a QM/MM calculation the self-consistent method here requires determining the correct dipole moments μj and charges qi of the water molecules and the protein. Here the induced dipole moment from Eq. (5.25) can be used to determine the charges in Eq. (5.26) from which the dipole moment for iteration n + 1 can be found and used in an iterative manner up until a convergence criterion is reached.

5.3 Electron Transfers in Biological Systems One of the most intriguing processes in quantum biology is the transfer of charge from one part of a biological system to another. The transfer of a single charge can introduce a major change in certain biological systems [3, 22, 23, 26, 30, 33, 104–106]. Some illustrative exemplary case studies are discussed below.

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5.3.1 Electron Transfers in Cryptochromes Time-dependent (TD) DFT was used to describe electronic transitions in exemplary biological systems [22, 23]. In particular, the electron transfer in Arabidopsis thaliana cryptochrome 1 (AtCry1) [23, 107] and European robin cryptochrome 4 (ErCry4) was explored both experimentally and computationally [4, 54]. Electron dynamics in cryptochrome leads to different radical pair formations in which one radical constitutes the FAD•− cofactor and the other radical is either a tryptophan or tyrosine residue [4, 23, 31], see also Fig. 5.2. In the case of AtCry1, it was established that the ASP396 residue (D396(H)) was involved in a structural transformation once the flavin part of the FAD moiety gains a negative charge and the Trp400 residue becomes a positive radical [23, 24, 102]. For AtCry1, the different radical pair formations arise through a sequential electron transfer involving a chain of three tryptophans, Trp400, Trp377 and Trp324, where each is associated with a certain radical pair state of AtCry1 and denoted as RP-A, RP-B and RP-C, respectively. Figure 5.7 summarizes the results of electron transfer dynamics in AtCry1 and shows the population of the radical pair states RP-A, RP-B and RP-C over an interval of 1 ns. The population of a radical pair is a key quantity to define which tryptophan is forming a radical with FAD•− in AtCry1 at a given time instance. Per definition, radical pair population varies between 0 and 1, where 0 is characteristic for a neutral tryptophan, while 1 denotes the cation radical state. Initially, AtCry1 is assumed to be in the RP-A state, as the RP-A state is formed in less than a picosecond after flavin photoexcitation [108]. In the course of the simulation, the FAD cofactor remains

Fig. 5.7 Formation and decay of three possible radical pair states in Arabidopsis thaliana cryptochrome 1 (AtCry1). The initially occupied RP-A state (blue) decays rapidly giving rise to the RP-B state (red) followed by the formation of the RP-C state (green). The radical pair population is obtained as an ensemble average of 32 simulations of 1 ns length which were calculated from snapshots taken from an MD simulation of AtCry1 resting state at 500 ps intervals to represent the ensemble of AtCry1 structural variety. Error bars denote the standard deviation and indicate the variance of the time evolution of the population observed for the underlying individual simulations. The average populations (light blue, orange, light green lines) represent the fitting curves from a two-step kinetic model, which has been fitted to the simulation data. Reproduced from [23] with permission from American Chemical Society

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negatively charged while the electron is free to move between Trp400, Trp377 and Trp324. As it follows from Fig. 5.7, the radical pair RP-A (blue) decays quickly, within 20 ps, giving rise to the formation of the RP-B state (red) and finally the RP-C state (green). The radical pair RP-B is only built up to about 40% when it starts decaying, giving rise to the population of the RP-C state. After 150 ps, the RP-C population remains stable at about 80% for the rest of the simulation. The observed behaviour could be understood as a two-step electron transfer process with the transfer Trp377 →Trp400(H)•+ followed by a second transfer Trp324→Trp377(H)•+ yielding RP-C as the ultimate radical pair. k If I

k If I I

kbI I

kbI I I

Trp400  Trp377  Trp324,

(5.27)

where k I I and k I I I denote the rate constants for electron transfer steps II (Trp377 →Trp400(H)•+ ) and III (Trp324→Trp377(H)•+ ), and the subscripts f and b indicate the forward and backward transfers, respectively. The two-step kinetic process above including back transfer could be described with coupled rate equations [23]: d [A] = −k If I [A] + kbI I [B] dt d [B] = −k If I I [B] + k If I [A] + kbI I I [C] dt d [C] = −kbI I I [C] + k If I I [B]. dt

(5.28) (5.29) (5.30)

Here, the square brackets [...] denote the normalized concentration of the corresponding radical pair or, in other words, the radical pair population. Equations (5.28)–(5.30) were solved numerically by varying the four rate constants k If I , kbI I , k If I I and kbI I I until the deviation of the average radical pair population from the simulation data points in Fig. 5.7 was minimized. The final numerical fit of the radical pair population is shown in Fig. 5.7 with light colour lines while the obtained rate constants are summarized in Table 5.1. The upper panel of Fig. 5.8 shows that after the formation of the radical pair, i.e. after the creation of FAD•− in the active site of AtCry1, the COOH group of the D396(H) residue turns spontaneously towards the flavin, and the COOH group

Table 5.1 Rate constants of the electron transfer steps in Arabidopsis thaliana cryptochrome. The error bars denote the deviation of the numerical fit from the simulation data e− transfer step Donor Acceptor k f [ns−1 ] kb [ns−1 ] II III

Trp377  Trp400 Trp324  Trp377

190 ± 18 70 ± 18

85 ± 18 10 ± 18

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Fig. 5.8 Protonation of flavin by aspartic acid 396 (D396). Shown is the spontaneous rotation of the COOH group of D396(H) towards flavin subsequent to the radical pair formation in AtCry1 which ultimately leads to flavin protonation. Relative orientations of D396 and flavin obtained through MD simulation of A the resting state of AtCry1 and B AtCry1 in the radical pair state [FAD•− +Trp400(H)•+ ]. A hydrogen bond (orange) is formed between D396HD2 and flavinO4 . Protonated flavin (C) is obtained from a QM/MM calculation, which yields the energy profile (D) for flavin nitrogen, N5, protonation. The distance between the hydrogen of the COOH group and the flavin nitrogen, N5, atom was considered as the reaction coordinate. Reproduced from [23] with permission from American Chemical Society

approaches the nitrogen, N5 and oxygen, O4, atoms of the flavin group, see Fig. 5.8B. Afterwards, a hydrogen bond of 1.8 Å length is formed between the O4 atom of the flavin and the COOH group of D396. The free energy profile for the protonation reaction of the flavin by D396 was then obtained from a QM/MM simulation where the hydrogen of the COOH group of D396 was pulled towards the N5 of the flavin group [23, 24, 102]. The distance between the hydrogen of D396 and nitrogen, N5, atom of the flavin group was considered as the reaction coordinate and was sampled from 2.6 down to 0.9 Å to describe the proton transfer process completely. From the energy profile, a rate constant of the proton transfer was obtained, and proved to be close to the experimental values found earlier [109]. Another study [4] focused on electron transfer in the ErCry4 protein. In that case, the activation process of the protein undergoes similar electron transfer processes and relies on four tryptophan residues (TrpA , TrpB , TrpC and TrpD ) with the residue numbers 395, 372, 318 and 369, respectively. The free energy of the radical pair states in ErCry4 was obtained following a combined QM and MD description [4]. The formalism relied on statistical mechanics and is well described elsewhere

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[4, 103, 110]. The free energy of a given redox radical pair state of ErCry4 was defined as G i (E i ) = −kB T ln [ p(E i )] + G i(0) ,

(5.31)

where G i is the free energy of a redox state i; E i is energy gap between two energy states describing two radical pair configurations in ErCry4, which is defined as the total energy difference between two redox states; kB T is the Boltzmann constant and temperature in Kelvin; p(E i ) is a probability distribution of the energy gap and G i(0) is a constant. The computed free energy profiles for the radical pair states in ErCry4 are shown in Fig. 5.9D and allow determining the reorganization energies λ and driving forces G for all electron transfer processes of interest [4, 110–113]. The obtained values of reorganization energies and driving forces can be used to estimate the electron transfer rate constants as [4, 111–113] log10 k = 13 − 0.6(R − 3.6) − 3.1

(G + λ)2 , λ

(5.32)

where R is the edge-to-edge distance between two Trp residues (see Fig. 5.9A–C), measured in Å, the reorganization energy λ is measured in eV and G is the driving force defined by the free energy difference between two redox states. In Eq. (5.32), it is measured in eV. The electron transfer process in ErCry4 [1, 4] is believed to be the core for explaining the putative magnetic sensitivity of the protein as compared to a significantly lower magnetic field effect in cryptochromes from other species [4]. The electron transfer rate constants and the G values are summarized in Table 5.2. The results reveal that the first two reactions, RPA −→ RPB and RPB −→ RPC , are exergonic and significantly faster than their corresponding back reactions. The reaction RPC −→ RPD is barely exergonic but the electron transfer constants in that case as well as for the reverse RPD −→ RPC are close, (15 ± 4) ns−1 , indicating that the RPC and RPD states in ErCry4 could exist in dynamic equilibrium. This result leads to a “composite” radical pair whose spin dynamics and reaction rates are weighted averages of the properties of RPC and RPD . The corresponding weights are given by the fractional populations of the two states [114]. The electron transfer rate constants for the two different systems AtCry1 and ErCry4, summarized in Tables 5.1 and 5.2, can directly be compared. The reaction Trp377 −→ Trp400 in AtCry1 has a rate constant of (190 ± 18) ns−1 and the back reaction Trp400 −→ Trp377 is equal to (85 ± 18) ns−1 . The forward reaction is therefore slightly slower than the reaction constants for forward reaction RPA −→ RPB in ErCry4 with (740 ± 180) ns−1 , but the backward reaction is faster than RPB −→ RPA with (0.013 ± 0.05) ns−1 . It should be noted that the forward reaction electron transfer rate constants are within the same order of magnitude, but differ for the backward reaction by a factor of 103 . Both forward reactions outpace any other electron transfers within their respective system. The forward reaction Trp324 −→ Trp377 in

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Fig. 5.9 A–C Probability density distributions for the pairwise edge-to-edge distances between Trp A , Trp B , TrpC and Trp D in ErCry4 calculated from the MD trajectories in the protein for five redox states. D Free energies of the different redox states in ErCry4, obtained following Eq. (5.31). The key reorganization energies, λ X Y and the driving forces GXY , where X is the initial state and Y is the product state, are indicated and were used to estimate the corresponding electron transfer rate constants for ErCry4, see Table 5.2. Figure reproduced from [4]

AtCry1 equals to a forward reaction constant of (70 ± 18) ns−1 and the backward reaction Trp377 −→ Trp324 equals to (10 ± 18) ns−1 . In ErCry4, the reaction RPB −→ RPC has an electron transfer rate constant of (50 ± 18) ns−1 and the backward reaction RPC −→ RPB equals to (0.9 ± 0.41) × 10−3 ns−1 . Therefore, both forward reactions are within the same order of magnitude with the Trp324 −→ Trp377 reaction being slightly faster. On the other hand, the backward reaction Trp324 −→ Trp377 in AtCry1 is a factor of 104 faster than the backward reaction RPB −→ RPC in ErCry4. The reaction RPC  RPD in ErCry4 has a forward electron transfer rate constant of (13 ± 4) ns−1 and a backward reaction constant of (15 ± 4) ns−1 and ranges in the

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Table 5.2 Summary of the electron transfer rate constants k, the values of the reorganization energy λ, the driving force G and the electron donor–acceptor distance R for the different electron transfers in ErCry4 [4], see Fig. 5.9. The electron transfer rate constant was determined using Eq. (5.32) Energy/kcal mol−1 Edge-to-edge Electron transfer rate distance/Å constant/ns−1 λ G R k RPA −→ RPB RPB −→ RPA RPB −→ RPC RPC −→ RPB RPC −→ RPD RPD −→ RPC

17.7 ± 0.2 28.7 ± 0.7 30.8 ± 0.3 29.7 ± 0.4 22.8 ± 0.3 19.5 ± 0.2

−5.814 ± 0.006 5.814 ± 0.006 −8.146 ± 0.004 8.146 ± 0.004 −0.717 ± 0.004 0.717 ± 0.004

3.7 ± 0.2 4.1 ± 0.3 3.7 ± 0.3 4.5 ± 0.4 3.6 ± 0.2 3.6 ± 0.2

740 ± 180 0.013 ± 0.05 50 ± 18 (0.9 ± 0.41) × 10−3 13 ± 4 15 ± 4

Fig. 5.10 Reaction cycle for water splitting on surfaces [115] (left). The star symbolizes an adsorbed species. Characteristic illustration of a water molecule adsorption on WO3 is shown on the right. Reproduced from [116] with permission from American Chemical Society

same order of magnitude than the RPB −→ RPC in ErCry4 and the reactions Trp377 −→ Trp400 and Trp324  Trp377 in AtCry1.

Electron Transfers in Artificial Systems After we have discussed a few examples of electron transfers in biological systems, we would like to provide an illustration of a completely different system that operates on similar rules. An illustrative example that relies on molecular electron transfers is the charge transfer between water deposited on a WO3 surface. In this case, the example deals with a deposited single molecule on a surface, but permits to apply the high-level methods of quantum mechanics to study charge-transfer and bonddissociation processes. Consider the reaction cycle of the water splitting mechanism shown in Fig. 5.10 (left). An illustrative molecular rendering of the system is shown in Fig. 5.10 (right).

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Table 5.3 Activation and recombination barriers computed for the key steps of the water splitting reaction deposited atop of WO3 [116]. Results were obtained using the DFT approach employing the PBE0 functional [118] Reaction Activation energy Recombination energy/eV H2 O∗ −→ HO–H∗ OH∗ −→ O–H∗ O–H2 O∗ −→ HOO–H∗ OOH∗ −→ O2 –H∗

0.30 eV/22 kJ/mol 0.12 eV/7 kJ/mol 0.25 eV/20 kJ/mol 0.10 eV/4 kJ/mol

0.41 eV/32 kJ/mol 0.36 eV/24 kJ/mol 0.63 eV/60 kJ/mol 0.16 eV/9 kJ/mol

Fig. 5.11 Reaction pathway for the water splitting reaction on WO3 in terms of BSSE-corrected adsorption energy and Gibbs free energy for one monolayer. Dotted lines refer to the diffusion of H atoms on the surface. Reproduced from [116] with permission from American Chemical Society

The adsorption energy of all molecular species in Fig. 5.10 (left) can be calculated as [94] follows: E ads = E WO3 −Ads − E WO3 − E Ads ,

(5.33)

where E WO3 −Ads represents the DFT energy of the adsorbate on the WO3 surface and E WO3 and E Ads the self-energies of the respective subsystems. The energy can further be corrected using the a posteriori counterpoise-correction scheme proposed by Boys and Bernadi [117]. In order to investigate the reaction cycle, not only the reaction products, but also all the intermediates and transition states were considered [116]. The activation and recombination barriers of the reaction are compiled in Table 5.3 and the overall results are summarized in the reaction path diagram in Fig. 5.11. The energies shown in Table 5.3 and Fig. 5.11 were obtained using a pure quantum DFT method. Figure 5.11 shows that all recombination barriers for the studied reactions are higher than the activation energies along the reaction path. The initial water dis-

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sociation has the highest activation barrier of 0.30 eV, whereas the dissociation of O–H2 O∗ has the highest recombination barrier of 0.63 eV. The Gibbs free energy shows a similar behaviour as the adsorption energy: although there are minor differences for the relative energetic behaviour, the overall trend of the reaction turns out to be close. Since the first reaction step shows the highest activation barrier, without inclusion of any diffusion of hydrogen atoms, it can be assumed that the water dissociation is the rate-determining step. From an experimental point of view, it is necessary to perturb the system to initiate the reaction, for example, to apply voltage, temperature or radiation. Here, one possibility is to utilize the Computational Hydrogen Electrode (CHE) Nørskov model [119]. According to that model, the following quantities are included in the contribution for the Gibbs free energy of reaction G 0 : G 0 = E + Z P E + E T − T S.

(5.34)

Here E is the DFT energy, Z P E is the zero-point energy, E T is the thermal contribution to the vibrational energy and T S is temperature×entropy of the adsorbate-WO3 system. An electrical voltage can be added to the Gibbs free energy of the reaction as follows: G = G 0 − eU,

(5.35)

where e is the elementary charge and U is the external voltage. The potential of the last reaction step in Fig. 5.10 (left) 2H2 O −→ 2H2 + O2 was shifted to the experimentally known value (293 K results in 4.92 eV), since O2 can only be insufficiently described within DFT [120]. Eventually, the overpotential is calculated as [121] E = max(G reaction steps ) − 1.23 eV.

(5.36)

The following equation holds at standard conditions ( p = 101.3 hPa and T = 298.15 K) and can be applied to calculate the energetic difference between the product states of the studied reaction: E(H+ ) + E(e− ) =

1 E(H2 ). 2

(5.37)

Under standard conditions the first two reaction steps, the formation of OH and O, show the highest reaction energy with 2.02 and 2.30 eV, respectively, see Fig. 5.12. In contrast to the first steps, the subsequent formation of O2 has a much smaller energetic barrier and the formation of OOH is actually exergonic. Figure 5.12 shows in red the reaction process for an applied voltage of 1.23 V, since an ideal catalyst for all reaction steps would require this potential [122]. For U = 2.16 eV, all reaction steps are exergonic (green curve). The overpotential of these reactions is 1.07 eV. In the calculations, only the educts and products were included in the CHE model, without considering transition states [116]. Therefore, microkinetic studies using the

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Fig. 5.12 Dependence of the Gibbs free energy for water splitting on WO3 reaction on the reaction coordinate studied at different values of externally applied voltages. Shown are the exemplary results obtained at standard conditions. Reproduced from [116] with permission from American Chemical Society

CATMAP software [123] were carried out [116]. The reaction cycle in Fig. 5.10 (left) has been taken as a reference under standard conditions (293 K, 1 bar). The Gibbs free energies from Fig. 5.11 were adopted as adsorption energies. The microkinetic studies showed that the formation of H2 and O2 is virtually non-existent. The strong adsorption energy of atomic hydrogen of −2.51 eV is the reason why virtually no H2 is formed. The water splitting process on surfaces can be simulated in the ground state with little effort, but the inclusion of charge-transfer states increases the complexity dramatically. The ground state and the charge-transfer (CT) state of water on WO3 (001) using a cluster model is discussed here [124]. To model the interaction of a photogenerated hole in the surface with the H2 O molecule and the subsequent electron transfer of water to the surface, the CASSCF method can be used since the charge-transfer reaction is a multi-reference problem. The calculations were performed using the Orca program package version 4.2 [125]. State-averaged (SA)-CASSCF was used to calculate the potential energy surface. The studied model consists of a cluster, which is electrostatically embedded into an environment consisting of 280959 point charges. The corresponding charges obtained from periodic bulk calculations are qW = +3 and qO = −1, respectively [116]. The coordinates of the atoms were adopted from 0.25 ML H2 O adsorption on WO3 (001) [116]. The appertaining cluster model as well as the nine different degrees of freedom of a water molecule with respect to its centre of mass are depicted in Fig. 5.13 (left). The electronic Schrödinger equation for a water molecule and a selection of atoms of the WO3 surface around it (cluster model) was solved pointwise, considering the three major Jacobi coordinates ϑ (rotation), Y (movement on the surface) and Z (desorption coordinate). First, the energetic ground state of the cluster model was determined, by considering different electronic configurations of the cluster model with the spin-polarized PBE0 functional [118]. It turns out that the cluster model exhibits two unpaired

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Fig. 5.13 Depiction of a water molecule atop the WO3 surface. The shown system is studied quantum mechanically using CASSCF and a cluster model. All nine different Jacobi coordinates are indicated (left). Relevant coordinates for this example are highlighted in light blue. Tungsten atoms are depicted in grey, oxygen atoms are depicted in red, the water molecule is depicted in dark and light blue and ECPs are depicted in green. For more visibility, the point charge field is not shown. The geometry of the water molecule in the ground state is shown on the left and for the charge-transfer state on the right. Reproduced from [124]

electrons in the ground state, resulting in a triplet state. The energetic difference between this electronic configuration and the configuration with no unpaired electron is minimal being equal to 0.007 eV, but it is important to determine the correct ground state for subsequent CAS calculations in order to prevent unwanted orbital rotations and artificial root flipping. In order to obtain both a good wave function for the initial guess of the system and to define the correct orbitals for the CAS calculations, natural orbitals were determined for the H2 O–WO3 (001) system at a non-interacting distance of 18.5 Å. The active space within the CAS calculations was then constructed from these orbitals, see Fig. 5.14. The active space consists of five molecular orbitals (MOs) of the water molecule with six electrons and two MOs of the cluster with two electrons resulting in a CAS(8,7). Using the CASSCF ansatz, it was then possible to calculate the chargetransfer state as the adsorption of H2 O+ on WO− 3 (001) along the entire 3D potential energy surface. A total of 9259 points were calculated on the GS-PES as well as on the CT-PES. Figure 5.15 shows three 2D surfaces for the combinations (ϑ, Z), (ϑ, Y) and (Y, Z) for the ground and charge-transfer states. The most prominent characteristic is the deep minimum at ϑ = 54.54◦ , Z = 2.28 Å and Y = 0.42 Å with an adsorption energy of −1.63 eV, which is visible in Fig. 5.15 (top). In Fig. 5.15, bottom the potential energy surfaces for the CT are presented. The minimum of the CT-PES is at very small Y-values. The oxidation process is hence energetically stabilized when close to the characteristic onefold coordination oxygen atoms at the surface. The energetically most favourable position of the molecule is reached at ϑ = 168◦ , Z = 2.65 Å and Y = −3.18 Å. The minimum of the CT-PES is much wider localized than in the GS. The diagonal excitation energy

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Fig. 5.14 Depiction of the active space used for the CAS(8,7)-3D-PESs calculation of a H2 O molecule deposited on WO3 surface. Reproduced from [124]

Fig. 5.15 2D cuts of the 3D-PES for ground state (top) and charge-transfer state (bottom) of a H2 O molecule deposited on a WO3 surface. Shown are the 2D cuts in (ϑ, Z), (ϑ, Y) and (Y, Z) coordinate spaces. Reproduced from [124]

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equals 3.78 eV from the ground state minimum. All excitation energies were calculated using CASSCF and serve as an example of how pure quantum chemistry methods can be used to describe charge-transfer reactions.

5.4 Proton-Coupled Electron Transfers Electron transfers in biological systems are sometimes coupled with a proton transfer meaning that both particles move in tandem [3, 126, 127]. The biological gain for such coupled transfers is still not completely understood, though in this section we seek to discuss several recent advances in the subject.

5.4.1 Understanding Energy Conversion in Photosynthesis—The Case of bc1 Complex Proton Transfer Studies of electron transfers in the photosynthetic cytochrome bc1 complex hypothesized that the primary reaction was a concerted proton-coupled electron transfer (PCET) reaction because of the apparent absence of intermediate states associated with single proton or electron transfer reactions [3]. More recent studies have focused on the kinetics of the primary bc1 complex PCET reaction with a vibronically nonadiabatic PCET theory in conjunction with all-atom molecular dynamics simulations and electronic structure calculations [3]. The studies implicated a concerted PCET mechanism with significant hydrogen tunneling and nonadiabatic effects in the bc1 complex. The series of charge-transfer reactions performed by the QH2 , denoted the Q-cycle [3, 22, 27, 28, 32–34], is initiated upon transmembrane diffusion and binding of a quinol (QH2 ) cofactor to the Qo active site. The QH2 is located at the interface between the cytochrome b and the iron–sulphur protein (ISP) subunits of the bc1 complex, as depicted in Fig. 5.16 (upper panel). A double quinol oxidation takes place, alongside two proton transfer reactions, in which electrons are transferred to prosthetic groups that are covalently bound to the bc1 complex. Two heme b groups, one heme c group and one Fe2 S2 cluster bound to the ISP are involved in the electron transfer reactions, while amino acid groups and water channels assist the proton transfers [3, 22, 27, 28, 32–34]. Classical MD simulations have previously been performed on a computational model consisting of 0.5 million atoms, including the X-ray crystal structure of the bc1 complex of Rhodobacter capsulatus [128], embedded in a bilayer lipid membrane the bc1 complex with QH2 and Q substrate molecules bound to the active sites [25, 26]. Recent MD simulations and electronic structure calculations, the kinetics of the primary PCET reaction that takes place in the bc1 complex were characterized. The

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Fig. 5.16 The bc1 complex and its reaction cycle. During the Q-cycle, two substrate quinol molecules (QH2 ) are oxidized to quinone (Q) at the Qo -site, while one Q is reduced to QH2 at the Qi -site of the bc1 complex. In this process, two protons are absorbed from the negative side of the membrane, and four are released to the positive side, hence maintaining the transmembrane electrochemical gradient. An oxygen molecule may occasionally bind in a pocket near the Qo -site [28], which could lead to superoxide production. Reproduced from [27]

PCET rate constant and associated hydrogen/deuterium (H/D) kinetic isotope effect (KIE) were calculated using a vibronically nonadiabatic PCET theory designed to describe such reactions [3, 126, 127, 129]. The principal equation for the PCET rate constant calculation reads as

k

PCET

=

 μ

where

  2   |Vel |2  2   π G ◦ + ν − μ + λ  Sμν  exp − Pμ ,  λkB T 4λkB T ν (5.38)    ∞    Sμν 2 =  Sμν (RDA )2 P (RDA ) d RDA .

(5.39)

0

Here Sμν is the overlap integral between the reactant and product proton vibrational wave functions labeled as μ and ν, respectively. Vel the electronic coupling between the diabatic electronic reactant. μ and ν are the energies of the reactant and product proton vibrational states relative to their respective ground states, the four lowest energy levels are shown in Fig. 5.17. λ is the total reorganization energy, which is the measure of structural reorganization of the environment upon the transition and also includes the reorganization of the reaction complex itself. G ◦ is the reaction free energy for the pair of ground reactant and product vibronic states including the

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Fig. 5.17 Proton vibrational wavefunctions and energy levels in the diabatic reactant and product potentials of the Qo -site of the bc1 complex. The four lowest energy levels and the corresponding proton vibrational wavefunctions for the reactant (blue) and product (red) proton potential. The dependence of the overlap integrals Sμν (see Eq. (5.39)) on the proton donor–acceptor distance is modeled by displacing the reactant and product diabatic potentials along the proton coordinate in opposite directions, as indicated by orange arrows. Reproduced from [3] with permission from American Chemical Society

proton zero-point energies. RDA is the proton donor–acceptor distance. The weight factor Pμ in Eq. (5.38) is given by   μ exp − k T .  B Pμ = (5.40)  i exp − kB T i Here, kB is the Boltzmann constant and T is the temperature of the system. The theory treats the electrons and transferring proton quantum mechanically and incorporates the effects of environmental reorganization and motions of the proton donor–acceptor mode. The agreement of the computed rate constants and KIE values, see Fig. 5.18, with experimental data on related systems provides validation for this PCET theory and support for a concerted PCET mechanism. The calculations also imply the significance of hydrogen tunnelling and nonadiabatic effects in this biochemically essential process. Previous experimental studies of systems undergoing quinol reduction reactions [130, 131] have reported KIE values comparable to the values in the bc1 complex theoretical model. Even though the experimental system studied is quite different from the model used in computational studies, the high values of the KIE seem to prevail, indicating that this may be a common characteristic of quinol oxidation reactions. The qualitative agreement of the calculated values of the H/D KIE with experimental measurements also indicates that the PCET reaction studied here could

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Fig. 5.18 A PCET rate constant of the primary charge-transfer reaction at the Qo -site of the bc1 complex with hydrogen (solid line) and deuterium (dashed line) calculated for a range of temperatures from 270 to 312 K using the formalism described in earlier studies [3]. B KIE, defined as the ratio of the PCET rate constant for hydrogen to the rate constant for deuterium, calculated for the same temperature range. Reproduced from [3] with permission from American Chemical Society

be the rate-determining step of the bc1 complex reaction mechanism. This possibility is consistent with previous experimental studies indicating that the primary quinol oxidation is indeed the rate-limiting step [3]. Moreover, earlier investigations of the KIE of the quinol oxidation reaction in the cytochrome b6 f complex observed KIE values lower than those computed in the newer computational studies [3, 132, 133]. In spite of the structural differences between b6 f complex and bc1 complex, the earlier investigations also concluded that the KIE implicates PCET as the rate-limiting step of the overall reaction. Other qualitative conclusions made in earlier papers agree with the analysis of more recent studies, namely, that the proton and electron transfer reactions are coupled and the Fe atoms of the iron–sulphur cluster are antiferromagnetically coupled. The calculations behind the PCET transfer invoke many approximations in both the derivation of the PCET rate constant expression and the calculation of the input quantities. For example, the quantum mechanical description of the active site included only a limited number of atoms. More residues could be included within the active site, and hybrid quantum mechanical/molecular mechanical (QM/MM) methods could be used to include the effects of the environment in

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those calculations. In addition, the electronic coupling calculation could also be performed at a higher level of theory using multi-reference wave function methods. Despite these approximations, the most recent analysis [3, 26] represents a practical implementation of vibronically nonadiabatic PCET theory for a complex biological system and provides insights into this essential bioenergetic process. The resulting KIE values shown in Fig. 5.18 are qualitatively consistent with experimental data on related biomimetic systems in the temperature region explored [3]. The agreement between the calculations and the experimental data provides validation for the theoretical model and the strategies for computing the input quantities. Such agreement also implies a concerted PCET mechanism in the charge-transfer reaction of the bc1 complex and suggests that this PCET step may be rate limiting. The relatively large KIE values indicate the importance of hydrogen tunneling and nonadiabatic effects for this system. These insights into this biochemically essential system have broader implications for other related bioenergetic systems. Further studies of secondary proton and electron transfer reactions after semiquinone formation would provide more information about the complex Q-cycle.

5.5 Spin Chemistry Spin chemistry is another illustrative example that may be instrumental to various biophysical phenomena. It is especially relevant for avian magnetoreception as the radical pair dynamics relies heavily on spin chemistry. Here we explore how spin dynamics may be involved in this biological phenomenon and associated applications.

5.5.1 Unveiling the Avian Compass A non-equilibrium radical pair inside a cryptochrome protein is generated in an entangled quantum state, such as the singlet state [1, 114, 134]. Entangled states are characterized by a large coherence between the two involved electrons: one electron cannot be described without describing the other electron at the same time, i.e. the two electrons are not independent of each other, but must be considered part of the same physical entity. Such coherence between particles is a quantum mechanical phenomenon without any macroscopic analogue [1, 114, 134]. Coherent states are highly non-equilibrium states, and as such they will tend to relax towards an equilibrium state where the two electrons can again be considered as two completely independent particles, i.e. a state without any coherence [1, 134–137]. The physical process that makes the system—the radical pair quantum state—tends towards thermal equilibrium is spin relaxation, thus it is an effect of the thermal motions in the protein [135–137]. Since spin relaxation is a process that removes coherence from the radical pair, it is also often referred to as

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decoherence which is the more generally used term when considering interactions between arbitrary quantum systems and their interactions with their environment (so-called open quantum systems) [135–137]. The effect of thermal motion is that certain interactions become time dependent, for example, the hyperfine interactions between the spin of the unpaired electron in a radical and the magnetic moments of nearby magnetic atomic nuclei could change over time as the atoms move around [1, 114, 134]. Solving equations such as the Liouville–von Neumann equation with interactions possessing a complex time dependence becomes too computationally expensive when the effect of a realistic amount of magnetic nuclei is to be accounted for [114, 134]. Thus, it is common to use the so-called Redfield theory to describe the effects of fluctuating interactions, as it removes the explicit time dependence from equations [135, 136]. Spin systems can be affected by external magnetic fields and processes involving spin dynamics may, therefore, in some cases depend on both the relative orientation of the spin system and the strength of such external magnetic fields. When spin system processes can lead to a variety of different products, a common quantity of interest is the quantum yield, which describes the probability of a spin system to end up in a specific product state. For the generic radical pair in Fig. 5.19A, the quantum yields P1 and P2 predict the relative yields of the two chemical species that would be obtained in an experiment; it is these quantum yields that may depend on the external magnetic field vector. The quantum yields of spin chemical processes are defined as follows:  ∞ Tr (Pi ρ(B, t)) dt . (5.41)

i = ki 0

Here ki is the rate constant for the process i, Pi is the projection operator onto the quantum state |i that may participate in the process (or the identity operator for spin-independent processes). The time evolution of the density operator ρ is given by describing the spin system ensemble, ρ(t), which is governed by the Liouville–von Neumann equation [134]: i dρ = − [H, ρ] + K(ρ) , dt 

(5.42)

where the density operator and hence the quantum yield also depend on the orientation and field strength of the external magnetic field, B. The index i refers to a specific process, i.e. one would define quantum yields P1 (B) and P2 (B) in order to represent the processes shown in Fig. 5.19A where the projection operators P P1 and P P2 are the singlet and triplet projection operators, respectively. MolSpin [134] contains a variety of methods for solving Eq. (5.41), relying on different assumptions to speed up the calculations. The most general method is the numerical integration using a leapfrog algorithm [138], while a faster method based on a Laplace transformation may be used when there are no time-dependent interactions. The fastest method, specifically designed for calculations on radical pairs, furthermore assumes

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Fig. 5.19 Magnetic field effects in a radical pair process. A A generic reaction scheme where the radical pair is produced in the singlet state and interconversion between singlet and triplet states is possible. Different reactions may happen from the singlet and triplet states, leading to chemically distinct reaction products, and external magnetic fields may influence the singlet-triplet interconversion, thereby affecting the relative amount of reaction products, P1 and P2 . B A calculation of the singlet quantum yield, P1 , for the radical pair [FAD•− ...W•+ ] in AtCry1 cryptochrome, shown in C, including 14 magnetic nuclei. C The orientation of the external magnetic field is varied in the x z-plane of the molecular reference frame as defined on the isoalloxazine moiety of FAD•− . The 14 magnetic nuclei included in the calculation are labeled. Reproduced from [134] with permission from AIP Publishing

that no spin-dependent reactions are present as well as the assumption that no coupling exists between the two unpaired electrons of the radical pair such that each radical can be treated individually. All of these calculation methods rely on basic procedures such as diagonalization of the Hamiltonian or solving a linear system of equations. Radical pair processes such as those defined in Fig. 5.19A are thought to be responsible for endowing migratory song birds with their magnetic compass sense [1, 50, 52, 64, 134–136], where one could imagine that one of the reaction products, say P1 from the singlet spin state, would produce the compass signal that would through some still unknown means be perceived by the bird. It would, therefore, be relevant for magnetoreception to evaluate the singlet quantum yield as a function of orientation relative to the geomagnetic field, in order to gauge the orientational dependence of the radical pair processes. The putative magnetosensor radical pair of the avian magnetic compass is an [FAD•− ...W•+ ] radical pair hosted by the flavoprotein cryptochrome [1, 23, 24, 50, 52, 64, 102, 134–136] and is illustrated in Fig. 5.19C. A calculation of the singlet yield was performed earlier [139], and, more recently, the calculation was repeated [134]; the result is illustrated in Fig. 5.19B. The two independent calculations show good correlation with a dip in the singlet quantum yield at the same magnetic field orientation, namely, θ = 90◦ . This change in singlet quantum yield depending on the orientation of the magnetic field is an important result in the story of bird navigation as the orientational change of the cryptochrome protein with respect to the magnetic field can be related to the head motion of a bird [48, 52, 134]. The main challenge posed by calculations in spin chemistry is the number of nuclear spins included. It is possible to include more nuclear spins in such a calculation, though the calculation time may become unpleasantly long for a complete quantum mechanical description.

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5.5.2 Radiofrequency Effects on Radical Pair Dynamics Weak radiofrequency (RF) magnetic fields in the MHz range have been shown to influence the concentrations of reactive oxygen species (ROS) in living cells [140– 142]. Remarkably, the energy that could possibly be deposited by such radiation is orders of magnitude smaller than the energy of molecular thermal motion. A plausible explanation to the observed effect relies on the interaction of RF magnetic fields with transient radicals within the cells, affecting the ROS formation rates through the radical pair mechanism [140, 143]. Prediction of the RF magnetic field effects in biomolecular systems is, however, not straightforward, as it relies on multiple interlinked scales ranging from electrons to the whole cell. This gap in our understanding of RF field effects on biological systems is, however, important and needs special attention because wireless charging has already been commercialized in various sectors such as portable consumer electronics [144] and manufacturing facilities [145]. Since radicals could possibly exhibit a sufficiently strong interaction with the weak RF magnetic fields [140], the search can be somewhat limited by focusing on molecular intracellular processes involving radicals. Radicals inside a cell may be created in pairs in a coherent state far from thermal equilibrium, and the relaxation pathway towards thermal equilibrium can be altered by weak external magnetic fields [1, 140]. Due to the high reactivity of radicals, various reaction pathways with radical involvement will normally be available, and external magnetic fields would thus modulate the corresponding reaction probabilities [1, 140]. The external magnetic RF fields would, therefore, lead to a difference in the relative amounts of intracellular reaction products that in turn would affect cellular functioning. The effect is expected to be dependent on the strength of the external magnetic fields, as well as their polarization and oscillation frequency. For example, in previous studies, a radical pair with flavin adenine dinucleotide •− . . .O•− (FAD) and superoxide O•− 2 , [FAD 2 ] was suggested to be responsible for an observed effect of RF magnetic fields in human umbilical vein endothelial cells [141, 142], being involved in avian magnetoreception [1, 23, 24, 50, 52, 64, 102, 134–136], or in the cytochrome bc1 complex [3, 22, 27, 28, 32–34]. may be generated as a side reacA recent study indicated that superoxide O•− 2 tion in the cytochrome bc1 complex [27, 28, 32–34], and it is not unlikely that the O•− 2 production rate as well as its chance to escape the reaction sites within the protein complex might be affected by RF magnetic fields. The singlet and triplet states of the radical pair are the so-called eigenstates of the Zeeman interaction with a homogeneous magnetic field along the z-axis, and eigenstates of the Hamiltonian do not mix over time. It is, therefore, necessary to include the hyperfine interactions of at least one magnetic nucleus in order to enable any possibility for singlet–triplet mixing, since singlet and triplet states will no longer be eigenstates of the total Hamiltonian in that case. In practice this means that radical pairs without any magnetic nuclei, and therefore no hyperfine interactions, such as a pair of superoxide radicals, would not have any singlet–triplet mixing (unless it is introduced by other means) and, therefore, no RF magnetic field effects would be possible. A single hyperfine

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coupling on at least one of the two radicals is enough to enable singlet-triplet mixing. The reason that hyperfine interactions are necessary for state mixing is due to the conservation of angular momentum: the singlet and triplet states have different angular momenta by definition, and a conversion between singlet and triplet states can, therefore, only happen if an external angular momentum, such as a nuclear spin, is changed simultaneously, in order to conserve the total angular momentum. RF magnetic fields can also be described in terms of photons, and since photons carry angular momentum, interactions with photons can cause transitions between spin states as well. In the model radical pair system investigated, each radical has a single magnetic nucleus, and the effect of changing the internal magnetic interactions within the radical pair, i.e. the hyperfine interaction strength of one of these nuclei, is explored [140]. The obtained change in the singlet product probability can be understood by rewriting the hyperfine interaction for the first radical as H(1) H F = gμ B a1 S1 · I1 =

 gμ B a1  2 R1 − S21 − I21 , 2

(5.43)

where S21 and I21 are the total angular momentum operators of the electron and nucleus on the first radical, respectively, both having an eigenvalue of 43 2 . R21 = (S1 + I1 )2 is the total spin angular momentum of the first radical consisting of an unpaired electron and a magnetic nucleus; the allowed eigenvalues of R21 are 02 and 22 . Thus, the possible energy states of the first radical are split by the hyperfine interaction in Eq. (5.43), such that one state has the energy E 1 = − 43 gμ B a1 , corresponding to the eigenvalue 02 of R2 , and three states have the energy E 2 = 14 gμ B a1 , corresponding to eigenvalue 22 of R2 . The transition frequency between these energy states is therefore given as ν=

gμ B MHz E2 − E1 + ν = a1 + ν ≈ 28 a1 + ν, 2π  2π  mT

(5.44)

where ν is a contribution from the static external magnetic field, which is needed because only the hyperfine interactions were included in E 1 and E 2 while the Zeeman interaction also impacts on the energy difference between the possible states in the radical; ν is approximately 0.7 MHz at B0 = 50 µT. Note that the form of Eq. (5.44) only holds when the static magnetic field is weak compared to the hyperfine interactions, such that it only contributes the small perturbation ν; a more rigorous treatment of the impact of the static external magnetic field on the transition frequencies is much more involved. The correspondence between transition frequency and the isotropic hyperfine coupling manifests itself in Fig. 5.20: for a1 = 0.25 mT one observes a large change in the singlet yield at 0.25 · 28 MHz + 0.7 MHz = 7.7 MHz, for a1 = 0.75 mT at 21.7 MHz, and for a1 = 1.50 mT at 42.7 MHz, as indicated by the dashed lines in Fig. 5.20. For all values of a1 , the probability of forming the singlet product additionally has a large change at 28.7 MHz. This additional feature is caused by the second radical, which has a magnetic nucleus with a fixed isotropic hyperfine coupling of 1 mT. The simple correspondence between hyperfine interactions and the

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Fig. 5.20 Hyperfine interactions impact the singlet product probability change in the studied model reaction. The strength of the isotropic hyperfine coupling constant a1 is modified here. Pronounced changes in the singlet probability are always seen in the low-frequency limit, and for ω ≈ a1 · 28 MHz mT + 0.7 MHz. Reproduced from [140]

frequency dependence of the singlet yield change becomes more complicated when multiple magnetic nuclei reside on the same radical, due to second-order interactions where nuclei may interact with each other through the hyperfine interactions with the unpaired electron. It should nevertheless be clear from Fig. 5.20 that the hyperfine interactions in a radical pair are crucial in determining whether an RF magnetic field might potentially influence a radical pair reaction, and therefore indicate which of the internal molecular parameters are involved in the process.

5.6 Photobiology Photoactivation of proteins is one of the most important processes for many biological processes. Among the most known is photosynthesis, though this process only works for light of certain wave length. Here we will focus on an illustrative example of photobiology that can be explored for light of different wavelengths.

5.6.1 Photoabsorption of the Cryptochrome Protein Family Experiments with migratory birds have shown that the wavelength of the ambient light is crucial for the magnetic compass sense: the birds could utilize the magnetic compass when exposed to blue or green light but not when only red light was available [146]. This is in partial agreement with the absorption spectrum of FAD, which only absorbs in the blue and UV regions of the spectrum, but as FAD is bound within cryptochrome, its absorption spectrum could differ significantly from the spectrum of the FAD cofactor in cryptochrome in isolation, as is already seen, for example,

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in the experimentally obtained spectrum of Arabidopsis thaliana cryptochrome 1, where an absorption shoulder at 470 nm appears in the bound FAD (see Fig. 5.22) [147]. It would therefore be interesting to know whether a significant red shift of the absorption spectrum of FAD could be caused by its interactions with a specific cryptochrome matrix, such as cryptochrome 4 from the European robin, increasing the range of light conditions where the magnetic compass would be operational. The absorption spectra of FAD within six different cryptochromes were earlier calculated [29], namely, cryptochromes from Drosophila melanogaster (DmCry), Arabidopsis thaliana (AtCry1), Mus musculus (MmCry) and Xenopus laevis (XlCry), as well as cryptochromes 1 and 4 from the European robin, Erithacus rubecula (ErCry1a and ErCry4). Magnetic field effects have been reported for all of these species [4]. An important aspect in the studies of the absorption spectra of cryptochromes is the accurate and systematic consideration of the protein environment on the absorption properties of the flavin chromophore. Calculations of the absorption spectra are computationally expensive protocol which can be summarized in the following four steps: (i) obtaining the equilibrated structure of cryptochrome, (ii) performing a quantum mechanics/molecular mechanics (QM/MM) geometry optimization of the core quantum region of the protein shown in Fig. 5.21, (iii) calculating the polarizable embedding (PE) potential to describe the surroundings of the core quantum region and (iv) performing the spectrum calculation. The six computed absorption spectra of FAD embedded in different cryptochromes are shown in Fig. 5.21 with the account for the thermal motion of the proteins. The averaging was achieved through the extended molecular dynamics (MD) simulations [29] and corresponds to statistically independent configurations of the proteins. All of the six studied cryptochromes show very similar absorption

Fig. 5.21 FAD within cryptochrome with the quantum (QM) region highlighted. Cryptochrome matrix is shown in the background, and the flavin part of FAD is highlighted. This highlighted part consists of 30 atoms and is used as the core quantum region in the calculations, whereas the rest of FAD and the protein, along with water and ions not shown here, are considered to be the environment and are represented by a polarizable embedding potential in the spectrum calculations. The N5 nitrogen atom is labeled. Reproduced from [29] with permission from American Chemical Society

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properties for the flavin part of the FAD cofactor. One may note that the spectrum from ErCry4 is slightly blue-shifted relative to AtCry1 and DmCry for the two peaks at wavelengths longer than 350 nm, but the shift is δL d, where δu is the atomic displacement, δL is the Lindemann parameter typically equal to 0.10 − 0.15, and d is the interatomic distance [69]. The analysis revealed that interatomic interactions at distances, exceeding the equilibrium distance by a characteristic vibration amplitude defined by the

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Fig. 6.11 Potential energy isolines (converted into temperature) for 6 nm-radius titanium, gold, and magnesium NPs whose constituent atoms interact via the original, Eq. (6.18) (dashed lines) or the modified, Eq. (6.17) (solid lines) EAM-type Gupta potentials. Purple, cyan, green, yellow, and orange curves correspond to 400, 800, 1200, 1600, and 2000 K, respectively. The red curves denote the energy difference (converted to kelvin) corresponding to the predicted bulk melting temperatures (see Table 6.4). Reproduced from Ref. [78] with permission from IOP Publishing

Lindemann criterion, significantly affect the accuracy of simulations. To elaborate on this issue, the potential energy surfaces (PES) were analyzed for 6 nm-radius Mg, Ti, and Au NPs with the optimized structure. Positions of all atoms but one were fixed. The movable atom was displaced from its equilibrium position and the interaction energy was calculated. Then, the energy of the perturbed system was subtracted from the energy of the fully optimized system. The resulting PES for the titanium and gold NPs are presented in Fig. 6.11. Each panel shows several isolines corresponding to a given energy difference between the optimized and the perturbed systems. For the sake of clarity, this quantity has been converted into temperature. Due to the additional linear term, the modified many-body potential (solid curves) makes the resulting potential steeper at large interatomic distances, as compared to the original potential (dashed curves). For instance, for the titanium nanoparticle the displacement of an atom for about 0.3 Å (i.e., by approximately a tenth of the closest interatomic distance, dTi = 2.95 Å), results in the energy difference of about 0.17 eV that corresponds to 2000 K. Thus, interatomic interactions at distances, exceeding the equilibrium one by a characteristic vibration amplitude δu, are overestimated by conventional many-body potentials and should be corrected in order to reproduce the quantitatively correct value of the melting point. A more accurate description of the interatomic interaction in the region beyond the equilibrium distance allows one to handle the problem of the accurate description of thermomechanical properties of metal materials. Generalized modification of EAM-type potentials. In Ref. [86] the above-described methodology was generalized and a new modification of an EAM-type potential was proposed. The new modification keeps features of the linear correction, Eq. (6.17), i.e., maintains its behavior in the vicinity of atomic equilibrium points

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Fig. 6.12 Potential energy Umod , Eq. (6.20), as a function of interatomic distance r . Solid lines show Umod (r ) for different values of the parameters, which were derived using the procedure described in the text. Dotted curves show the piecewise linear approximation to Umod (r ). Dashed gray line depicts the potential Ulin , Eq. (6.17). Redrawn from data presented in Ref. [86]

and enhances the repulsive interactions with an increase of atomic displacements. The modification has been constructed in such a way that it contains a parameter describing the characteristic range of the potential, thus eliminating the dependence of the potential on the choice of the cutoff distance. These conditions are fulfilled by multiplying Ulin by a sigmoid function which is equal to unity at small interatomic distances and asymptotically approaches zero beyond a given distance. The modified EAM-type Gupta potential then reads as U = UGup + Umod

N 1  B˜ ri j + C˜ ≡ UGup + , 2 i, j=1 1 + eλ(ri j −rs )

(6.20)

where UGup is given by Eq. (6.18). The parameters B˜ and C˜ have the same meaning as B and C in Eq. (6.17): B˜ defines the additional force acting on the nearest atoms and C˜ adjusts the depth of the potential well in the vicinity of the equilibrium point where U = 0. The parameter λ describes the slope of Umod at large interatomic distances, while rs defines the sigmoid’s midpoint and hence the range of this potential. Figure 6.12 shows the potential Umod for a pair of atoms as a function of interatomic distance r . Due to its sigmoid-type shape, Umod (r ) asymptotically approaches zero and its range serves as a natural cutoff distance for this interaction. For each pair of atoms the potential Ulin in Eq. (6.17) grows monotonically with interatomic distance up to the cutoff rc , and all atoms located within the sphere of radius rc experience a constant force exerted by a given atom. On the contrary, Umod has a maximum at interatomic distances of about 5–8 Å depending on the choice of λ and rs (see Fig. 6.12). Thus, the force exerted by an atom due to Umod enhances

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Table 6.5 Parameters of the potential Umod , Eq. (6.20), used to analyze the melting temperature and equilibrium properties of silver, gold and titanium [86] B˜ (eV/Å) C˜ (eV) λ (Å−1 ) rs (Å) Ag Au Ti

0.009 0.026 0.052

−0.048 −0.145 −0.269

5.93 4.68 2.77

7.10 7.36 6.68

interaction with several nearest atomic shells while the interaction with more distant atoms weakens. The strength of this interaction is governed by steepness of the potential beyond the maximum, i.e., by the parameter λ. Therefore, the force acting on the nearest neighbors due to Umod should exceed (by the absolute value) the force Flin as its effect is compensated by the weaker interaction with more distant atoms. Thus, for each pair of atoms interacting via Umod (r ) the initial slope of the potential should be steeper than the slope of Ulin (r ), i.e., B˜ > B. In Ref. [86] parameters of the new modification were derived analytically by approximating the sigmoid-type function Umod (r ) with a piecewise linear approximation U¯ mod (r ). Then, parameters of this function were expressed through the parameters B and C of the linear correction. As a last step of this procedure, U¯ mod (r ) was fitted with Umod (r ) to derive λ and rs . Further technical details of this procedure are given in Ref. [86]. The modified potential Umod , Eq. (6.20), was validated in [86] by analyzing melting temperature and near-equilibrium properties of silver, gold and titanium nanosystems. The parameters of Umod used for this analysis are summarized in Table 6.5. Benchmarking the modified potential. Tables 6.6 and 6.7 summarize the results on structural and energetic properties of silver, gold, and titanium nanocrystals obtained with the sigmoid-type modification Umod (6.20). These results are compared to those obtained by means of the original EAM-type Gupta potential (6.18) and the linear correction Ulin (6.17). The calculated bulk cohesive energies are summarized in Table 6.6. Both the linear correction and the sigmoid-type modification almost do not change the values predicted by the original Gupta potential, and all these values are in good agreement with experimental data [85] with the relative discrepancy of less than 0.5%. Table 6.7 presents equilibrium lattice constants for silver, gold and titanium calculated with UGup , UGup + Ulin and UGup + Umod . The force created by the linear correction causes a uniform strain on the crystals, which become uniformly compressed. For silver and gold this effect is rather small (the relative change in the lattice parameters is less than 1%) while the relative shortening of titanium crystals is about 2.5%. This can also be attributed to the very steep linear correction (i.e., the large force) that should be used to reproduce the experimental bulk melting temperature of Ti. Note also that geometry optimization of a Ti crystal using the original Gupta potential yields the structure which is elongated along the [0001] axis as compared to the experimental value (the calculated lattice parameter c = 4.75 Å vs. the exper-

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Table 6.6 Bulk cohesive energy (in eV per atom) calculated with the original Gupta potential, Eq. (6.18), as well as with the Gupta potential corrected by Ulin , Eq. (6.17), and by the sigmoid-type modification Umod , Eq. (6.20), proposed in Ref. [86] UGup UGup + Ulin UGup + Umod exp. [85] Ag Au Ti

2.96 3.78 4.87

2.96 3.77 4.87

2.97 3.78 4.83

2.96 3.78 4.85

Table 6.7 Equilibrium lattice constants (in Å) calculated with the original Gupta potential (UGup ), as well as with the Gupta potential corrected by Ulin and the new modification Umod [86]. Two lattice parameters, a and c, are listed for titanium UGup UGup + Ulin UGup + Umod exp. [85] Ag Au Ti (a) Ti (c)

4.07 4.06 2.91 4.75

4.05 4.03 2.83 4.63

4.07 4.09 2.89 4.77

4.09 4.08 2.95 4.68

imental value of 4.68 Å). Geometry optimization by means of the linear correction results in a uniform compression of the crystal, which brings the parameter c in a better agreement with the experimental value. The sigmoid-type modification Umod has a small impact on the equilibrium lattice parameters, which almost coincide with those predicted by the original Gupta potential and agree reasonably well with the experimental results. Contrary to the linear correction, Umod does not induce strong compression of the Ti crystal and its lattice parameters obtained by means of Umod are similar to those calculated with UGup . As discussed above, this is due to the functional form of Umod wherein the plays a role at small interatomic distances (which span positive contribution of Umod plays a role over a few nearest atomic layers) while the negative contribution of Umod at larger values of r . Figure 6.13 shows the melting temperatures of finite-size Ag, Au and Ti nanoparticles as functions of their inverse diameter D. For all the metals, the bulk melting temperature predicted by the original Gupta potential is significantly lower than the experimental values. The most illustrative example is titanium (see the lower panel of Fig. 6.13) whose melting temperature calculated with UGup is approximately 1380 K. It is more than 500 K lower than the experimental value of 1941 K (marked by a star symbol) which yields the relative discrepancy of about 30%. A similar feature has been observed for gold and silver—the absolute discrepancy is smaller for these metals (about 330 and 100 K, respectively) while the relative discrepancy for gold is as large as 25%. These results justify further the necessity of correcting the EAMtype potential to bring the calculated bulk melting temperatures in closer agreement with the experimental values. The modification Umod produces a similar effect as the linear correction—it leads to an increase in nanoparticles’ melting temperatures and,

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Fig. 6.13 Melting temperature of Ag, Au, and Ti nanoparticles of diameter D calculated by means of the original EAM-type Gupta potential (Eq. (6.18)), its linear correction Ulin (Eq. (6.17)) and the new modification Umod (Eq. (6.20)) proposed in Ref. [86]. Lines represent the extrapolation of the calculated numbers to the bulk limit. Experimental values of bulk melting temperature are shown by stars. Redrawn from data presented in Ref. [86]

as a result, to an increase of the bulk melting temperatures. The new modification improves the calculated bulk melting temperature for the three metals considered. Good agreement with the experimental values has been obtained for titanium and silver (the relative discrepancies from the experimental values are 0.8 and 1.5%, respectively) while a somewhat larger discrepancy of about 6% has been observed for gold. This is linked to the observation that the sigmoid-type modification increases the slope of the Tm (1/D) dependence for silver and titanium nanoparticles but it almost does not change the slope for gold nanoparticles. The utilized parameters of Umod for gold have been chosen in Ref. [86] such that all the quantities considered agree better with experimental data as compared to the original Gupta potential. A better agreement might be achieved by performing a more detailed analysis of the multidimensional parameter surface of Umod . A finer tuning of the parameters should bring the calculated Tmbulk for gold to a better agreement with experimental data.

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6.2.4 Melting Phase Transition in Nanoalloys Much interest has emerged recently in studying alloys on the nanoscale [65, 87]. Alongside composition and temperature, the state space is defined also by the system size [88, 89]. The additional complexity of nanoalloys facilitates steering their physicochemical properties, such as reactivity or phase transition points. Due to the associated martensitic phase transition (discussed in detail in Sect. 6.3), the near-equiatomic nickel-titanium (NiTi) alloy stands out among bimetallic alloys as the most prominent instance of a shape-memory (SM) material [90]. Besides this solid–solid martensitic phase transition, the melting phase transition and phase segregation are relevant physical phenomena in the fabrication and processing of NiTi [91]. Conventional NiTi is already of great importance as an adaptive material in biomedical devices due to its good corrosion resistance and low stiffness. However, its nanostructured variant consisting of crystallites below 100 nm can exhibit yet enhanced thermomechanical properties [90]. Insights into the structural and thermodynamic properties of nanoalloys can be obtained by means of advanced computer simulations. However, despite the extensive amount of research carried out so far on NiTi alloys, the study of thermodynamic properties of NiTi and other Ti-based (nano-)alloys has been very limited [13]. Pasturel et al. have attempted to deduce the composition–temperature phase diagram of bulk NiTi using DFT calculations with periodic boundary conditions [92]. Atomistic simulations of melting in Ni-based nanoalloys have been reported in Refs. [88, 93, 94]. In Ref. [95] different regions of the composition–temperature-size phase diagram of NiTi were investigated by means of the linear correction to the EAM-type potential [78], described above in Sect. 6.2.3. The modified potential has been utilized to evaluate the melting point of several spherical NiTi nanoalloys of different sizes. These results were then used to evaluate the bulk melting temperature to compare with experimentally determined phase diagrams for bulk NiTi materials. This analysis provided atom-level insights into the structural and thermal properties of NiTi nanoalloys. It was demonstrated that accounting for distant atomic interactions, which are neglected in many other interatomic potentials, is crucial for the accurate assessment of NiTi melting, as it is largely attributed to the inclusion of the interaction between second-nearest neighbor and more distant Ti atoms [95]. Interatomic potential for NiTi systems. Interatomic interactions involving Ni and Ti atoms have been modeled in Ref. [95] by means of the EAM-type Finnis–Sinclair potential: N 

rep  Ui + Uiattr , (6.21) Uorig = i=1 rep

where Ui and Uiattr are the repulsive pairwise and the attractive many-body contributions, respectively:

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Table 6.8 Original parametrization [84] of the EAM-type Finnis–Sinclair potential, Eqs. (6.21)– (6.23), alongside the parameters B and C of the linear correction, Eq. (6.17) [78] interaction D (nm) A (eV) p ξ (eV) q B C (eV) (eV/nm) Ti-Ti Ni-Ni Ni-Ti

0.2950 0.2490 0.2607

0.153 0.104 0.300

rep Ui

9.253 11.198 7.900

=

 j=i

Uiattr

1.879 1.591 2.480

2.513 2.413 3.002

0.114 0.000 0.000

   ri j Aαβ exp − pαβ −1 , Dαβ

      ri j 2 = − ξαβ exp −2qαβ −1 . Dαβ j=i

−0.0595 0.0000 0.0000

(6.22)

(6.23)

In these expressions summation is performed over pairs of atoms i and j, and the interatomic interaction between distinct chemical elements is tagged by α and β. The interatomic interactions in NiTi nanoalloys have been described by employing the parameterization from Ref. [84], which is given in Table 6.8. In the cited study, the values have been selected to match bulk ground-state properties, such as cohesive energy, vacancy formation energy as well as lattice and elastic constants determined using LDA calculations of crystal supercells under periodic boundary conditions. Fitting has involved pure Ni, pure Ti, and independently NiTi. Furthermore, fitting has embraced the cutoff radius, where all the interatomic interactions are truncated, as a free variable. Since no interpolation in the vicinity of the resulting short cutoff, rc = 4.2 Å, has been discussed in Ref. [84], this truncation leads to a severe discontinuity in the interatomic potential energy. The original EAM-type potential, Eqs. (6.21)–(6.23), was augmented in Ref. [95] by an additional repulsive term [78], N 

rep  U= Ui + Uiattr + Ulin ,

(6.24)

i=1

where

 1  Bαβ ri j + Cαβ . = 2 j=1 N

Ulin

(6.25)

The resulting parameterization for Ti is given in Table 6.8. As discussed in Sect. 6.2.3 the correction Ulin represents a minor change to the potential energy but has nevertheless significant impact on collective system properties, like the melting temperature. Figure 6.14 illustrates the minor role of this modification. The figure shows the potential energy profile of a titanium dimer,

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Fig. 6.14 Potential energy of the monatomic and heteroatomic interactions in NiTi for original EAM-type potential, Eqs. (6.21)–(6.23), and the potential augmented with the linear correction, Eq. (6.24). Thick black line shows the effective potential Ueff (r ), Eq. (6.27), that describes the segregation of nickel on the nanoparticle surface (see text for details). Two insets show illustrative cross sections through the nanoparticle below (top left) and above (right) the melting point; the higher the local Ni concentration the darker the atom. Reproduced from Ref. [95] with permission from American Chemical Society

calculated with the original (the dashed curve) and the modified (the solid blue curve) potentials. The long cutoff, rc = 7 Å, has been chosen to diminish the discontinuity which emerges in the original potential. Thermal and structural properties of NiTi nanoalloys were studied by means of MD simulations using MBN Explorer. Standard velocity-Verlet integration with a typical step size t = 2 fs has been employed. In the canonical MD calculations, a Langevin thermostat has been used with a typical damping constant τ = 40 fs.

Assessing the Ni-Ni Interaction Both the modified [78] and original [84] EAM-type potentials in Eqs. (6.22)–(6.24) rest upon the additive cohesion scheme which models a binary system by considering three distinct interactions: the two monatomic (Ti-Ti, Ni-Ni) and the heteroatomic (Ni-Ti) one. The modification of the EAM-type potentials for Ti-Ti interactions has been discussed in Ref. [78], while the Ni-Ni interaction has been assessed in Ref. [95]. For that, icosahedral particles comprising from 147 up to about 18,000 Ni atoms have been subject to a sequence of canonical MD runs at subsequent thermostat temperatures Ti+1 = Ti + T . Several different interatomic potentials have been employed; the EAM-type potential with original parametrization [84] and

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Fig. 6.15 Calculated melting temperatures of pure Ni N (N = 147 − 18, 000) nanoparticles (symbols) as a function of N −1/3 . The linear least-squares fit to MD results are indicated by straight lines. Redrawn from data presented in Ref. [95]

the short cutoff, rc = 4.2 Å, is referred to as a short-cutoff Gupta potential. For the sake of comparison, additional simulations have been performed with the widely used Sutton–Chen [39] and the so-called quantum Sutton–Chen [96] potential; the latter represents a different parametrization of the original [39] potential. With growing system size, the bulk crystal lattice becomes the most favorable geometry in finite-size NPs. Therefore, MD simulations have been carried out for a spherical Ni nanocrystal (particle radius R = 2 nm, N = 3043 atoms) with cubic lattice (lattice constant a = 3.524 Å), where the long-cutoff Gupta potential (rc = 7 Å) was employed. Both factors, namely the more favorable geometry as well as the long-cutoff radius, enforce cohesion and are thus expected to increase the melting point. The duration of the individual simulation runs t = 20 ps and the resulting heating rate T /t = 0.1 K/ps have been chosen low enough to not influence the calculated melting point. Melting is a first-order phase transition which exhibits a peak in the heat capacity CV . In order to obtain the specific heat from MD simulations, the caloric curves E(T ) were spline-smoothed and then differentiated with respect to T . The global maximum of the specific heat is hence taken as the thermodynamic indication of melting. Plotting the melting temperature Tmelt of the finite systems as a function of the inverse particle radius R (or inverse cubic root of N ) according to Eq. (6.19) bulk ≡ Tmelt (N → ∞). The allows one to evaluate the bulk transition temperature Tmelt results are depicted in Fig. 6.15. bulk = 1759 K, which is in good agreement The short-cutoff potential yields Tmelt with the experimental bulk melting temperature of 1728 K [97]. It was found that the long cutoff and the crystalline geometry have an insignificant effect on the simulated

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Fig. 6.16 Calculated melting temperatures of equiatomic NiTi nanoparticles (symbols) as a function of inverse nanoparticle diameter. The linear fit to MD results toward the bulk limit is indicated by a straight line. Redrawn from data presented in Ref. [95]

melting point of Ni. Neither the original Sutton–Chen potential nor its quantum variant yields reasonable melting points (see Fig. 6.15). Hence, it was concluded that both the short-cutoff and the long-cutoff Gupta potentials are, already without modification, suitable for the atomistic simulation of the solid–liquid phase transition in monatomic Ni systems.

Melting of Equiatomic NiTi Alloys The modified EAM-type potential, Eqs. (6.24)–(6.25), has been applied to systematically explore the melting of equiatomic NiTi nanoalloys. For that purpose, austenitic nanocrystals comprising from 1061 up to about 67,000 atoms have been subject to a sequence of canonical MD runs. The maximal heating rate T /t = 0.125 K/ps has been chosen low enough to not influence the calculated melting point. Again, the global maximum of CV was taken as the thermodynamic indication of melting, and the bulk melting temperature was determined according to Eq. (6.19). The Ni-Ti interaction has been kept unchanged. The simulation results are shown in Fig. 6.16. bulk = The modified potential yields an extrapolated bulk melting temperature Tmelt 1564 K, which is close to the experimental bulk value [97] of 1583 K. Figure 6.16 illustrates that both the linear correction to the potential and the increased cutoff have a pronounced effect on the simulated melting point. The impact of the cutoff distance rc for the simulation of melting was analyzed in a representative crystalline nanoparticle with the radius R = 2 nm; the original

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Fig. 6.17 Melting temperature of the spherical equiatomic NiTi nanoparticle with the radius of 2 nm as a function of cutoff distance as well as the corresponding radial distribution function. The dotted line is given to guide the eye. Reproduced from Ref. [95] with permission from American Chemical Society

Gupta potential has been employed for this analysis. The connection of the melting point Tmelt (R) with the atomic structure of the crystal was explored by calculating the RDF g(r ), Eq. (6.8). The distribution of atoms of the same chemical element (Ti-Ti, Ni-Ni) and the distribution of atoms of different elements (Ni-Ti) have been analyzed at the temperature of 600 K. The results are presented in Fig. 6.17. The melting point exhibits a minimum in the vicinity of the short cutoff (rc = 4.2 Å) and converges at the larger distances. The minimum coincides with the inclusion of distinct monatomic interactions. Since the Ni-Ni interaction is significantly weaker than the Ti-Ti interaction (see Fig. 6.14) and since the inclusion of Ni-Ni interactions beyond the short cutoff shows little effect (see the comparison of shortcutoff and long-cutoff potentials in Fig. 6.15), the sensitivity of the melting point in the vicinity of minimum has been largely attributed to the inclusion of the interaction between second-nearest neighbor Ti atoms.

Melting of Near-Equiatomic NiTi Alloys Apart from the equiatomic case, different chemical compositions have also been investigated in Ref. [95]. The NiTi phase diagram is intricate in the vicinity of equiatomic compositions below the melting point [97]. Near-equiatomic NPs have been created by starting from the known austenite crystal structure of equiatomic NiTi. First, atoms inside the equiatomic nanoparticle of radius R have been replaced randomly until the desired chemical composition was reached. The nanoparticle was

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A

B

Fig. 6.18 Panel A: RMSD at the core of equiatomic NiTi nanocrystals of different diameters. The jump in the RMSD indicates locally the phase transition. Panel B: Calculated bulk melting temperatures as a function of composition (symbols) versus the experimental phase diagram [97] (colored background). Reproduced from Ref. [95] with permission from American Chemical Society

then subject to an equilibration comprising 1000 macrosteps. At each macrostep, (i) a random pair of atoms swapped their positions, (ii) the resulting trial configuration was relaxed via a short microcanonical MD run, and (iii) the trial configuration was accepted based on its internal energy and the Metropolis criterion [98]. This criterion defines the acceptance probability  that one should accept a trial move from a configuration with energy E i to another configuration having energy E i+1 :    E i − E i+1 , (i → i + 1) = min 1, exp kT

(6.26)

where k represents Boltzmann’s constant. Temperature T has been set equal to 600 K. For the simulation of melting, the equilibrated NPs have been subject to a sequence of canonical MD runs with the heating rate of 0.125 K/ps. In this analysis, the maximum of CV was taken as the thermodynamic indication of melting, but also the atomic RMSD at the core (center) of the NP was monitored as a local structural indication. Since melting proceeds via surface nucleation, the core transforms not until the end of the phase transition. As an illustration, Fig. 6.18A shows the RMSD at the core of equiatomic nanocrystals. The RMSD was averaged over several atoms in the core region and over several MD time steps. Plotting the NP melting temperature as a function of the inverse NP radius allows to evaluate the bulk melting point according to Eq. (6.19). Results of the analysis performed employing the modified EAM-type potential, Eqs. (6.24)–(6.25), are shown in Fig. 6.18B by symbols. The calculated melting temperatures derived from the specific heat (black circles in Fig. 6.18B) are positioned in the solid–liquid coexistence portion of the experimental phase diagram [97]. This result is reasonable because the specific heat peak is a global indication lying between the onset and end of the melting transition. On the Ni rich side of the phase diagram, the calculated melting temperatures which

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Fig. 6.19 Local Ni concentrations as a function of the distance from the nanoparticle center (below and above the melting temperature of 1444 K). In the left panel, a linear fit is indicated by a straight line so as to highlight the concentration gradient. In the right panel, the nanoparticle was discretized into different layers and the mean concentration of Ni was evaluated in each layer (see text for details). Reproduced from Ref. [95] with permission from American Chemical Society

are derived from the RMSD (blue triangles) are in good overall agreement with the experimental liquidus. However, on the Ni poor side, the calculated melting temperatures derived from the RMSD are below the experimental liquidus. It is possible that especially for the Ni poor compositions, the aforementioned MC based equilibration method yields configurations which do not corresponds to nearly optimal configurations. Less favorable geometries should lead to decreased melting points.

Surface Segregation Surface segregation is known to be an obstructive effect in the fabrication and thermal processing of NiTi [91]. Since MD simulations lead to detailed physical trajectories, surface segregation can be studied using the MD data discussed above. The local concentration of chemical elements inside the NiTi nanoparticle can serve as an indication of phase separation. Such an analysis has been performed in Ref. [95]: each atom in an equiatomic NP (5 nm diameter, overall Ni concentration 50%) has been assigned a local Ni concentration during the simulation of melting. The local concentration is given by the fraction of Ni atoms among the neighbor atoms in a sphere of radius 0.5 nm around the atom of interest. This temperature-dependent local concentration was then averaged over several subsequent MD time steps so as to improve statistics. For example, the nanoparticle’s melting temperature Tmelt ≈ 1444 K can be extracted from Fig. 6.18A based on the jump in the RMSD. In the left panel of Fig. 6.19, the local atomic Ni concentrations are plotted as a function of the distance from the particle’s center. This data is given below (at T = 1350 K) and above (at T = 1500 K) the melting point.

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Above the melting point in the liquid phase the local Ni concentration at the center of the particle decreases on average, whereas the local Ni concentration at the surface increases (solid orange line in the left panel of Fig. 6.19). This effect has been observed for nanoparticles of different chemical composition. The right panel of Fig. 6.19 shows an alternative representation of the results where the trend for the nickel atoms to segregate to the surface is confirmed. The equiatomic NP is discretized into different layers (ranging from its core to the surface), and the mean Ni concentration is evaluated in each layer. The layers have 0.2 nm thickness except for the core layer, which subsumes all the atoms with radial distance smaller than 1.2 nm. The segregation of nickel toward the nanoparticle surface might also be relevant to the modeling of toxicity of nanostructured nickel-titanium alloys. In Ref. [99] a criterion for segregation in liquid binary alloys based on the relative strength of interatomic potential energies was provided. According to that criterion segregation occurs if the effective potential Ueff (r ) = UNi−Ti (r ) −

UNi−Ni (r ) + UTi−Ti (r ) 2

(6.27)

is positive at distance r close to the nearest-neighbor distances D. This happens because at elevated temperatures, regions of the interatomic potential where monatomic interactions by trend are stronger than the heteroatomic interaction become accessible to the vibrating atoms. In these regions the atoms favor the monatomic local environment, leading to phase separation. The potential energy for the monatomic and heteroatomic interactions in NiTi is shown in Fig. 6.14, which illustrates that the aforementioned criterion is indeed true for NiTi. It is important to stress again that the modification of Ti–Ti interaction appears as a slight correction of the pairwise potential energy but has major implications for collective properties, such as the melting temperature. The insets of Fig. 6.14 show an illustrative cross section through the nanoparticle below and above the melting point. High local Ni concentrations are colored in black, while low local Ni densities are colored white. The plotted cross sections show that below the melting point (top-left inset), dark and light spots are distributed evenly, while above the melting point (bottom-right inset) the core appears lighter than the surface.

6.3 Martensitic Phase Transition in Solids Shape-memory (SM) alloys can, after initial inelastic deformation, reconstruct their pristine lattice structure upon heating. The near-equiatomic NiTi alloy is the most prominent instance of SM alloys with applications in sensors, antennas, medical wires, and stents [100]. Its nanostructured equivalents are polycrystalline structures with grain sizes typically under 100 nm. They exhibit remarkably altered characteristics, such as improved thermomechanical properties [90]. The underlying phenomenon of the SM effect is a reversible structural phase transition: At first,

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cooling (or a deformative stress) triggers the martensitic transition from a solid higher symmetry phase, dubbed austenite, to another solid but lower symmetry phase, dubbed martensite. Upon heating above the austenitic transition temperature TA (or, if T > TA , unloading the stress) the initial structure is recovered by the martensite– austenite transition. However, no plausible atomistic theory of the SM effect in NiTi exists, especially for nanoscale systems. The low-temperature crystal structure remains controversial as well as the intermediate phases along the transition path. A cubic lattice structure (designates as B2) is widely believed to be the high-temperature austenite phase in equiatomic NiTi. Extensive ab initio calculations [12] favor an orthorhombic structure, named B33, to be the lowest energy martensite phase whereas B19 , frequently observed in experiments (see, e.g., [101]), is supposed to be metastable and stabilized by residual stresses. This poses the questions on whether the experimentally observed bulk structures are stable in nanosystems and whether the SM effect can be observed at the nanoscale [102]. MD simulations of spherical equiatomic NiTi nanocrystals under free boundary conditions were performed in Ref. [103] aiming to investigate the B19 → B2 transition. The many-body Gupta-type potential was used but with a modified cutoff behavior that yields a specific inclination angle for the monoclinic martensite lattice. A single heating rate of 1 K/ps was considered for nanoparticles of up to N ≈ 2 × 105 atoms. A local order parameter based on nearest-neighbor distances was constructed to estimate the size-dependent phase transition temperature TA . The values of TA calculated for the nanocrystals of different sizes were then extrapolated to the bulk limit according to Pawlow’s law, Eq. (6.19). In Ref. [13] the thermally induced martensite–austenite phase transition in free equiatomic NiTi nanocrystals comprising up to N ≈ 4 × 104 atoms was investigated by means of MD simulations using the MBN Explorer software package. Following Ref. [103], the Gupta-type potential was used to describe the interatomic interactions employing the parameters listed in Table 6.8. Structure optimization was accomplished in Ref. [13] by generating spherical equiatomic NiTi particles with varying lattice parameters a, b, c and the angle α between them. The constructed systems were subsequently subject to local energy minimization. The resulting optimized monoclinic martensite lattice was in agreement with other available values (see Table 2 in the cited paper and references therein). By varying the cutoff radius rc it was found that martensite is more stable than a cubic lattice only in the range from rc = 4.08 to 4.23 Å. Similar to the melting, the martensitic phase transition is the first-order phase transition which exhibits a peak in the heat capacity CV , Eq. (6.1). The anharmonicity of the potential functions is a prerequisite for simulating these phase transitions, but the classical schemes lack the capability of reproducing quantum features of the heat capacity expected in experiments, for example, due to the quantization of energy levels. To complement the simulation results, a quantum model for the crystalline nanoparticles, namely, the harmonic Einstein solid (see, e.g., Ref. [104]), was considered in Ref. [13]. Within this model it is assumed that atoms behave as independent harmonic oscillators of the same frequency ω0 . Within this model the

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heat capacity (per atom) is given by  C V = 3kB x

2

ex (e x − 1)2

 x=ω0 /kB T

,

(6.28)

where kB and  are Boltzmann and Planck constants, respectively. The model parameter ω0 can be obtained from the autocorrelation function R(τ ) [105] which is defined and transformed as follows: R(τ ) =

v j (t) · v j (t + τ ) ≈ 1 + ω02 τ 2 /2 . v2j (t)

(6.29)

Here v j (t) is the velocity of a jth atom, . . .  denotes averaging with respect to time t, and τ stands for the time lag. Within the model, the dependence v j (t) is harmonic, therefore, assuming τ to be smaller than the period of oscillations, 2π/ω0 , one derives the indicated approximate equality. Thus, evaluating the autocorrelation function for the simulated MD trajectories and analyzing its slope with respect to τ 2 in the region of small time lags, one can extract the Einstein frequency ω0 . In Ref. [13], MD simulations were performed for equiatomic spherical NiTi nanocrystals of various radii R. The simulations were carried out using the canonical ensemble without boundary constraints; the temperature T was controlled by a Langevin thermostat with damping constant 40 fs. To determine the temperatures of the martensite–austenite (“m-a”) transition and of the melting (“m”), a R = 3 nm NiTi nanoparticle was heated from T = 2 K to 2000 K with heating rate of 1.25 K/ps corresponding to the simulation time of 1.6 ns. Figure 6.20A shows the caloric curve E(T ). The figure indicates that the linear dependence E ∝ T is interrupted by step-like increments at T ≈ 200 K and

A

B

Fig. 6.20 Panel A: the caloric curve for 3 nm equiatomic NiTi nanocrystal. Panel B: the corresponding dependence of heat capacity C V (measured in units of kB ) on the thermostat temperature. See also explanations in the text. Redrawn from data presented in Ref [13]

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T ≈ 1400 K corresponding to the latent heat of the martensite–austenite E m−a and melting E m phase transitions, respectively. The linear segments are highlighted by fits (dashed lines) reflecting the distinct phases of NiTi (martensite, austenite, liquid). The simulated value E m−a ≈ 20 meV. Experimental values for bulk NiTi include 13 meV [106] and 16 meV [107]. Two sudden changes in the caloric curve correspond to two maxima in the dependence of heat capacity CV on temperature, see Fig. 6.20B. The simulation statistical error (not indicated in the figure1 ) due to finite number of independent MD runs is on the level of 20%. Within this level of uncertainty the simulated dependence coincides with the calculated one (the dashed line) between T = 300 and 1200 K, where the NP experiences no structural changes and stays in the austenite phase. The calculated dependence was obtained from Eq. (6.28) with the parameter ω0 = 22.5 meV calculated as explained in the text below Eq. (6.29). For high temperatures the harmonic approximation asymptotically follows the empirical Dulong– Petit law, CV ≈ 3kB [41]. Major deviations from this background dependence are the martensite–austenite transition at TA = 243 K and the melting phase transition at Tm = 1385 K. The latter value is smaller than the experimentally measured bulk melting temperature Tmbulk = 1583 K [97], which is in agreement with Pawlow’s law (6.19). It is clearly seen from Fig. 6.20 that the shape of the martensite–austenite transition peak differs remarkably from the spike-like melting peak. In particular, it possesses a second less pronounced shoulder. This feature was observed earlier in the simulation of small atomic clusters [108] where the shoulders are associated with premelting (see also Sect. 6.2.2) and other structural rearrangements occurring below the onset of the actual transition process.

6.4 Spontaneously Electrical Solids Another example of a solid–solid phase transition is the existence of the spontelectric state of matter [109]. This state of matter is exemplified by the presence of static, spontaneous electric fields extending throughout thin films of dipolar solids. The key characteristic of the spontelectric state is that molecular material, when laid down as a thin film on a low-temperature surface, spontaneously develops an electric field; hence the name “spontelectric”. This field may exceed 108 V m−1 and is created through molecular dipole orientation in the solid, giving rise to polarization charge, and consequently a measurable potential, on the surface of the film [109]. The spontelectric state was discovered using a low energy electron beam technique, using the ASTRID storage ring at the Aarhus University [110]. Molecular modeling of the spontelectric effect is challenging because of the subtle nature of the respective long-range and many-body forces [109]. The first atomistic simulation of the spontelectric effect has been performed recently in Ref. [111]. The 1

The error data can be found in the original Fig. 2 in Ref. [13].

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Fig. 6.21 Illustrative time evolution of the system’s dipole alignment. The last 200 ps of each stable MD trajectory have been used to calculate the thermodynamic average alignment, see Fig. 6.22. Reproduced from Ref. [111] with permission from Springer Nature

atomistic model based on results of DFT calculations has been employed to study the spontelectric effect in the molecular film made of methyl formate (MF). Since the spontelectric effect is thought to be a bulk phenomenon, the molecular model developed in Ref. [111] was applied to a simulation box with periodic boundary conditions. The box contained N = 216 molecules, randomly oriented at the beginning of the simulation. This corresponds on average to 6 molecular layers in each spatial direction which roughly represents a minimal extent for observing spontelectricity. Initially the populated simulation box matches a pre-specified density ρ = 0.987 g/cm3 (being the density of liquid MF). The simulation box is allowed to adapt its side length in the course of the simulation. The box size spans 2 times the cutoff radius of the van der Waals and electrostatic interactions, but at the same time it is small enough to allow collection of some statistics from re-running simulations. The system has been simulated using periodic boundary conditions for 500 ps at a given temperature T . 200 independent runs have been conducted at each temperature between T = 40 and T = 90 K. Intramolecular parameters have been defined through the CGenFF force field, whereas intermolecular parameters were determined by the custom global estimation method described in Ref. [111]. Figure 6.21 shows time evolution of the system’s dipole alignment for several exemplar trajectories. The initial random orientation of MF molecules spontaneously relaxes to a certain alignment level. Here, dipole alignment is defined as the average net electrical dipole moment of the system considering all molecules. It is normalized relative to the dipole moment of the MF monomer:

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Fig. 6.22 Simulated average alignment, Eq. (6.30), in bulk MF as a function of temperature. The experimentally observed dependence [112] with the anomalous increase toward higher temperatures is shown by orange circles. The gray squares are error bars indicating one standard deviation; they are based on 30 independent trajectories for a given value of T . Redrawn from data presented in Ref. [111]

μ |μmonomer |

with μ =

N 1  µi . N i

(6.30)

In respective experiments, the z-axis, oriented perpendicular to both the substrate and the film surface, is the natural principal direction of alignment. Because no substrate and no film surface are present in the bulk simulation, the system as a whole will not exhibit a preferential alignment in the z-direction. Hence, in the simulations the vector μ can point in any direction, not necessarily along the zaxis. The comparison of simulation results with the experimental data is shown in Fig. 6.22. The simulated average alignment is the thermodynamic average deduced from the 30 independent MD runs. Even though rather big uncertainty exists in the calculated curve, it reproduces the anomalous polarization for MF with an increase toward higher temperatures. Overall, the alignment shows higher values than the experimental curve. This fact might be due to the difference in boundary conditions between the simulated bulk system and a thin film deposited in experiments. This deviation might also be due to lower or higher packing of the dipoles in the periodic boundary bulk simulation. Furthermore, the parametrization of the force field based on quantum calculation of MF dimers may not adequately reflect the situation in the solid film with an increased number of nearest neighbors, e.g., in terms of the molecule’s polarizability. There is great scientific interest regarding microscopic details of the spontelectric state, especially concerning properties which are currently not directly observable

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Fig. 6.23 Statistical distribution h(α, T ) of orientation angles relative to the principle alignment direction μ. Predictions in bulk MF by the MD-based molecular model developed in Ref. [111]. The thermodynamic average shape originates from a superposition of multimodal distributions. The histogram at 90 K (higher overall alignment in Fig. 6.22) is more skewed than the histogram at 70 K (lower overall alignment in Fig. 6.22). Redrawn from data presented in Ref. [111]

in experiment. Such a quantity is the temperature-dependent distribution of dipole orientations, h(α, T ), measured relative to the preferred alignment direction. The average angle directly related to the average orientation via  α = arccos

μ |μmonomer |

 .

(6.31)

can be extracted from experiments solely. The molecular model developed in Ref. [111] has been deployed to monitor and predict the distribution of relative angles in the course of the simulation. Results of this analysis are shown in Fig. 6.23. The simulations predict a broad and slightly skewed distribution of orientation angles suggesting that the appreciable electric surface potentials of MF nanofilms observed in the experiment originate from a subtle alignment tendency on the microscopic level. The developed model can hereby assist in better interpreting findings related to experiments on the spontelectric effect.

6.5 Tribology Different materials exhibit different behavior under the same thermal and mechanical loading. The behavior can be attributed to different properties or the manifestation

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of different effects, which can be characterized by certain specific quantities. These characteristics can be grouped into four major categories. 1. The first category is related to the behavior of materials driven by thermal loading without relation to their mechanical properties. This category includes such characteristics as melting points, specific heat capacities, and thermal conductivities. 2. The second category is attributed to the behavior of materials under mechanical loading without fracture. The quantities of this category include elastic moduli, Poisson’s ratios, viscosity, etc. 3. The third category embraces the thermomechanical characteristics of materials, such as density, thermal expansion coefficients, and energetic characteristics of crystals. 4. The fourth category is related to the fracture behavior of materials and operates with such characteristics as hardness and density of dislocations. The dependencies of all the material properties’ characteristics upon temperature are usually referred to as thermomechanical properties of materials. The field of materials science and, in particular, the research area related to the thermomechanical properties of materials is a broad field of research being closely related to numerous industrial applications [90, 113–115]. Tribology is another related domain of research and engineering focused on investigating interacting surfaces in relative motion. This includes the study and application of the principles of friction, lubrication, and wear. The tribological interactions of a solid surface due to its interfacing with other materials and the environment may result in loss of material from the surface. The process leading to loss of material is known as “wear”. Major types of wear include abrasion, friction (adhesion and cohesion), erosion, and corrosion. Wear can be minimized by modifying the surface properties of the solids by one or more “surface engineering” processes or using lubricants (for frictional or adhesive wear). All these processes have great industrial importance. The properties of materials can be experimentally examined by means of different tests. The most common ones are indentation, scratching, and wear tests. These tests might include micro- and nano-probing. The latter can be conducted by means of atomic force microscopy or other nanoindentation and nano-scratching techniques. Nanoindentation represents various hardness tests that are applied to small volumes of a material in order to study its mechanical properties [116]. The most common use of nanoindentation is to measure hardness and elastic modulus [117]. In the course of the nanoindentation process a specified force (on the order of millinewtons) is applied to the indenter. Simultaneously, the indenter displacement (on the order of hundreds of nanometers) is monitored in order to obtain the “applied force—indenter displacement” curve as an output (see Fig. 6.24). The derived curve is also commonly referred to as the “load-displacement” (P − h) curve. The main advantage of the nanoindentation process is the possibility to test very small volumes of the material and its applicability to the characterization of thin films. The principle of the nanoindentation experiment is similar to the “classical”

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Fig. 6.24 Schematic representation of the nanoindentation process (left) [24, 118] and a typical load–displacement (P − h) curve (right) obtained during nanoindentation of a single-crystal Pt(100) [117]. The left panel is reproduced from Ref. [118]. The right panel is redrawn from data presented in Ref. [117] with permission from Elsevier

hardness measurement of materials, where a rigid indenter with a defined geometry is pushed into the sample material causing the elastic–plastic deformation of a small volume of the sample. The indenter is usually made of a diamond crystal, free of impurities or inclusions, in order to avoid spurious effects of the indenter tip deformation. Diamond is selected for nanoindentation due to its exceptional hardness, thermal conductivity, and chemical inertness, which surpass any other known material. Various geometries are commercially available for the indenter shape such as three-sided and four-sided pyramids, wedges, cones, cylinders, and spheres. The indenter tip can be made flat, sharp, or rounded to a cylindrical or spherical shape. The typical size of the tip is of the order of several tens of nanometers for the sharpest tips and a few hundreds of nanometers for the rounded tips. The residual imprint on the sample surface after nanoindentation is on the order of a micrometer even with large forces applied. Generally, the following three stages of the indentation cycle are distinguished: (i) the loading part where the applied force increases until a peak value, (ii) the holding part where the peak load is maintained for a prescribed amount of time, and (iii) the unloading part where the applied force decreases gradually to zero, see the right panel of Fig. 6.24. A nanoindentation experiment lasts generally only for some tens of seconds in order to remain in the high precision domain of the measuring equipment. The indenter velocity varies between tens of nm/s and μm/s. Over the last decades the computational modeling of materials properties has been pursued by different methodologies of the computational condensed matter physics and materials science. Modeling of the mechanical, thermal, and transport properties of materials was in focus in these studies. The most popular theoretical approaches that were utilized for the computational modeling of thermomechanical properties of materials are the Finite Element Method (FEM), the Monte Carlo (MC)

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approach, and different forms of Molecular Dynamics (MD), including its quantum and classical versions. Typically FEM and MC methodologies are applied to model the materials’ properties on the larger spatial and temporal scales. In contrast, the MD approach, particularly its all-atom version, is used to simulate materials’ properties on the nanometer scale and processes lasting up to the microsecond time scale. The combination of the aforementioned methodologies forms a basis for the multiscale modeling of thermomechanical properties of materials, which is one of the hot topics in this field of research. For instance, through the multiscale approach the mechanical models from continuum-based elasticity theory can be examined by the comparison with the outcomes of MD simulations and the limits of validity of the continuum-based elasticity theory can be established. In this way the pertinent concepts from continuum theories such as the stress analysis by means of the stress tensor, elastic constants, and elastic moduli can be re-examined at the atomic level. Despite the numerous studies conducted in this research area for different materials and their properties, many questions remain open. The fact is seen already from the permanent searching in many laboratories worldwide for novel materials, as well as for the materials with improved properties. Essential is that this work is pursued both experimentally and theoretically. This section provides several illustrative examples of computational modeling of the nanoindentation process by means of the MBN Explorer [19] software package. Different aspects of the nanoindentation process can be studied, for instance, elastic and plastic deformations of pure metals and nanoalloys can be simulated for different types of indenters, at different temperatures, etc. Apart from the nanoindentation process, many other tribological processes (e.g., wear resistance or scratching) can also be simulated.

6.5.1 Thermomechanical Properties of NiTi Alloys For many decades, biocompatible metal materials have been used for medical implants in trauma surgery, orthopaedic and dental medicine due to good formability, high strength, and resistance to fracture. The important disadvantage of metals is their tendency to corrode in physiological conditions, therefore a large number of metals and alloys are unsuitable for implantation being too reactive in the human body. The list of metals currently used in implantable devices is limited to the following three systems: stainless steels, cobalt–chromium-based alloys, and titanium and its alloys [90, 119]. Another important criterion for successful incorporation of these materials into clinical practice is the ability to provide sufficient mechanical strength, especially under the cyclic loading conditions, to ensure the durability of medical devices made thereof [119]. The advantages and disadvantages of the mentioned systems for implant fabrication are described in Ref. [90]. According to the cited paper titanium is superior to other surgical metals from the point of view of corrosion resistance. Another favorable property of titanium is low elastic modulus which is two times less than that

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of stainless steel and Co-Cr. This results in less stress shielding and associated bone resorption around Ti orthopedic and dental implants. Furthermore, titanium is more light-weight than other surgical metals and produces fewer artifacts on computer tomography and magnetic resonance imaging (see, for example, review [120]). However, the static and fatigue strengths of pure titanium are too low to use it for commercial production of the implants. To this end, a special group of Ti alloys, namely, bimetallic NiTi compound containing 54–60% of Ni (wt.), has gained popularity in biomedical applications. This compound exhibits unique properties of shape-memory and super elasticity [12] being, at the same time, highly biocompatible due to the high titanium content. In addition, elastic modulus of NiTi can be made very low, twice lower than that of pure titanium. This makes NiTi an excellent candidate for implant material capable of mimicking the mechanical behavior of bones [121, 122]. A fundamental understanding of the deformation mechanisms and construction of reliable computational models for the numerical testing of titanium and its compounds are essential for further developments of these materials for technological applications. This step should be based on the study of the full atomistic dynamics in the materials of interest. MD simulations of dynamic processes with large systems allow one to overcome the problem of computational demand by calling for classical interatomic potentials to enable the treatment of appropriate spatial and time scales. Thermomechanical properties of Ti and NiTi materials were investigated by means of all-atom MD simulations of the nanoindentation process using the MBN Explorer package [24, 118, 123]. The load–displacement curves for amorphous, crystalline, and nanostructured Ti and NiTi were obtained for different geometries of indenter, and the results were used further to calculate the hardness and Young’s modulus of the materials. In what follows, the main results obtained in the cited papers are discussed in greater detail.

6.5.2 MD Simulation of the Nanoindentation Process The indentation process was modeled for three different shapes of the indenter: (i) a square indenter with a cross section of 1.8 nm × 1.8 nm, (ii) a conical indenter with both the base radius and the height equal to 7.4 nm (the cone angle is equal to 90◦ ), and (iii) a spherical indenter of the radius of 7 nm. The indenters were modeled as absolutely rigid bodies constructed from carbon atoms in the fcc lattice with the lattice constant of 1 Å. Such tight packing was chosen to avoid possible penetration of titanium atoms into the indenter. The deformation of the indenters and the thermal oscillations of their atoms were neglected. The hardness of a diamond is much higher than that of titanium, therefore the rigid body model for the indenter is fully applicable for titanium and NiTi samples. The Ni-C and Ti-C interactions were described using the repulsive power potential with the cutoff radius of 3 Å. Prior to the simulation of nanoindentation, the systems were equilibrated. In each case study, the indenter tip was positioned 3 Å from the surface of the sample,

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Fig. 6.25 Visualization of nanoindentation of the crystalline [24] (top) and nanostructured [123] (bottom) titanium samples with a square indenter. Left panels provide general view of the nanoindentation process, middle and right panels illustrate the structural deformations at intermediate (middle) and the maximum, h ≈ 30 Å (right) penetration depths. Reproduced from Ref. [24] with permission from Elsevier and from Ref. [123] with permission from American Chemical Society

and the constant temperature simulations were performed at 30 K for 60 ps. After equilibration, the atomic positions of two bottom layers of each sample were fixed to avoid translational motion of the sample. During the simulations, the indenters were moving 3 nm downwards with a constant speed of 40 m/s, and then retracted upwards with a speed of 10 m/s. These values correspond to a typical range of velocities, which are feasible for studying using modern MD simulations of nanoindentation [37]. Figures 6.25 and 6.26 illustrate the indentation process of the crystalline and nanostructured Ti samples, and of the crystalline and amorphous NiTi alloys with a square indenter [24, 123]. A general view of the samples in the course of nanoindentation is shown in the left panels of the figures, while the middle and right panels show structural deformations occurring at intermediate and the largest penetration distances, respectively. Figure 6.27 presents the load–displacement curves (i.e., dependence of the force exerted on the indenter on the indentation depth) obtained for indenters of different geometries (top, middle, and bottom rows) applied to the titanium (left column) and NiTi (right column) materials. The black curves in the left-column graphs represent results of the indentation of an ideal titanium crystal, while the red curves correspond to the nanostructured material. The right-column graphs allow one to compare the dependencies obtained for the crystalline (black) and the amorphous (red) NiTi. For the square indenter, the initial stage of indentation for all materials is characterized by a linear dependence of the force F on the indentation depth h. This

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Fig. 6.26 Same as in Fig. 6.25 but for crystalline (top) and amorphous (bottom) nickel-titanium samples. Titanium and nickel atoms are shown by red and blue colors, respectively. Reproduced from Ref. [123] with permission from American Chemical Society

means that the initial stage of the process is performed in the elastic regime, and the dependence F(h) can be approximated in this region by a linear function, F ∝ h. After the initial stage, the load–displacement curve reaches a maximum value and then fluctuates around a constant value.2 The latter feature is related to the geometry of the square indenter: due to the constant cross section further penetration into the sample does not affect the resulting force. In general, the jitter-like behavior of all presented curves can be explained in terms of the relaxation of stresses in the samples. In turn, the relaxation can be associated with the process of dislocation migration from the deformed site [124]. This process is illustrated by Fig. 6.28 where surfaces comprising most stressed atoms are shown by blue color. Initially, the stresses atoms form a relatively symmetric connected group in the vicinity of indenter tip, see Fig. 6.28a. Further indentation leads to the formation of arc-like structures directed outwards the indentation site, Fig. 6.28b. These structures represent the dislocations that are formed in the vicinity of the indenter and propagate outward from the indentation region. In Fig. 6.28c, two dislocation regions are seen that are separated from the central stressed part and one more dislocation that is almost separated on the right from the indenter tip.

2

For amorphous NiTi, the dependence does not reveal a prominent maximum but fluctuates around the value of 0.04 µN/nm, indicating that it is much easier to deform the amorphous sample as compared to the ideal crystalline one.

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Fig. 6.27 Dependence of the force applied to the square (upper row), conical (middle row), and spherical (lower row) indenters on the indentation depth. Left column presents the dependencies obtained for crystalline (black curves) [24] and nanostructured (red cures) [123] titanium, right column—for crystalline (black) [118, 124] and amorphous [123] NiTi. Reproduced from Ref. [123] with permission from American Chemical Society

For all samples, the load–displacement curves for the conical and spherical indenters demonstrate gradual increase of the force with the penetration depth, so that no elastic regime has been observed. For the conical indenter, the simulated curves can be fitted with the quadratic function, F ∝ h 2 [125]. For the spherical indenter, in the region h  1 nm, the curves can be well fitted by the dependence F ∝ h 3/2 [125]. In both cases, the indentation of titanium and NiTi structures enters the plastic regime starting from the very first contact of the indenter tip with the sample. The presented load–displacement curves for Ti show that the forces, which act on the indenters interacting with the nanostructured sample, are 1.5–2 times smaller than those exerted by the ideal crystal, see the left-column plots in Fig. 6.27. Indeed,

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Fig. 6.28 Stressed fragments in the Ni-Ti crystalline structure formed in the course of nanoindentation. Panels a, b and c correspond to the indentation depths h = 16, 24 and 32 Å, respectively. Two separated stress regions shown in panel c correspond to two dislocations that are moving away from the indentation site. Reproduced from Ref. [124] with permission from Elsevier

since the ideal crystalline structure has neither defects nor irregularities, the work done by the indenter to deform the ideal sample is larger than that in the case of the nanostructured crystal. In the latter case, the force value depends strongly on the size of nanocrystals and their packing. For NiTi structures the force acting of the amorphous sample is 2–3 times smaller than its value in the crystalline one. As discussed in the following section, this difference reveals itself in calculating mechanical properties of the amorphous sample.

6.5.3 Quantification of Mechanical Properties of Ti and NiTi Samples Direct comparison of the parameters of the indented systems, such as the force acting on the indenter, obtained experimentally and numerically is hardly possible due to difference in the spatial scales probed in experiments and in MD simulations [24]. A typical system size, which is feasible to investigate within the MD framework, is several orders of magnitude smaller than the size of experimentally studied samples. By means of the MD approach, it is possible to simulate penetration of an indenter at depths h of about several nanometers, while experiments deal with the h values ranging from hundreds of nanometers up to few micrometers. Nevertheless, the results of MD simulations can be used for estimation of elastic parameters of the sample which then can be compared with the experimental values. To provide estimations for mechanical properties of the titanium and nickeltitanium samples, the MD simulations of the unloading stage of the indentation procedure were performed in Ref. [123]. In these simulations the indenter was moved in the upward direction with a constant speed of 10 m/s. The initial part of the unloading stage is shown in Fig. 6.29. The results of these simulations can be used to calculate hardness H and Young’s modulus E of the investigated materials.

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Fig. 6.29 Load–displacement curves calculated for the case of the square (top left), conical (top right) and spherical (bottom panel) indenters. Black and red curves represent results of indentation of the nanostructured titanium and amorphous nickel-titanium samples, respectively. The unloading stage was performed starting from the indentation depth of approximately 1.5 nm. Dashed green and dash-dotted blue lines represent the analytic fit by a linear function to the initial part of the unloading curves for titanium and nickel-titanium samples, respectively. Reproduced from Ref. [123] with permission from American Chemical Society Table 6.9 Hardness H (in GPa) of titanium and nickel-titanium samples calculated for the indenters of different shapes [123] Sample/indenter Square Conical Spherical Ti crystalline Ti nanostructured Ni-Ti crystalline Ni-Ti amorphous

16 14 39 16

12 12 24 9

5.6 4.3 15 6

Table 6.9 presents the values of hardness, defined as the ratio of the force acting on the indenter to its contact area with the material, for the Ti and NiTi materials. It is seen that for the same material the values of H vary with the shape of the indenter. This can be related to the fact that for the considered indenters with small contact area with (the square indenter and the conical one at small penetration depths) the channel of stress release associated with dislocation migration from the deformation site is suppressed. In the case of the large spherical indenter, a signif-

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Table 6.10 Young’s modulus E (in GPa) of titanium and nickel-titanium samples calculated at the beginning of the unloading stage for the indenters of different shapes [123] Sample/indenter Square Conical Spherical Ti crystalline Ti nanostructured Ni-Ti crystalline Ni-Ti amorphous

67 69 110 78

111 49 120 56

82 74 148 130

icantly higher dislocation activity has been observed. For the titanium samples, the H values calculated for the spherical indenter are close to the experimentally measured values: H = 3.32 ± 0.59 GPa and H = 4.68 ± 0.93 GPa for commercially available microstructured Ti and titanium crystals, respectively [126]. With respect to the investigated Ni-Ti structures, hardness of the ideal crystal is about 2.5 times higher than that of the amorphous sample. The results of MD simulations can be also used for calculating Young’s modulus (elastic modulus) E, which is defined as the ratio of stress to strain along a given axis, and is a quantitative characteristics of stiffness of a material. This quantity can be related to the so-called reduced Young’s modulus, E r , of the material and the elastic modulus of the indenter, E i , as follows: 1 1 − νi2 1 − ν2 + = , Er E Ei

(6.32)

where ν and νi are Poisson’s ratios of the material and the indenter, respectively (see, e.g., [127]). Since the indenters are modeled as rigid, infinitely stiff bodies, the last term on the right-hand side of the equation can be omitted. The value of the reduced Young’s modulus is proportional to the derivative of the initial part of the uploading curve [125]: √ 2 dF = √ Er A , dh π

(6.33)

where A is the contact area. In Fig. 6.29, the slopes dF/dh are indicated by dashed (green) and dash-dotted (blue) lines. Substituting relation (6.32) in (6.33) and using the values ν = 0.32 and ν = 0.35 for Ti and equiatomic NiTi alloy respectively, one calculates the Young’s modulus of the studied samples, see Table 6.10 [123]. Values of E for the ideal crystalline and nanostructured titanium samples are generally lower than the experimental values measured for the microstructured (E = 122 ± 6 GPa) and nanostructured (E = 132 ± 8 GPa) titanium, respectively [126]. The discrepancies can be attributed to different types of samples studied numerically and experimentally as well as to the different shapes of the utilized indenters. In the experiment [126], the indentation was performed by means of the so-called Berkovich

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indenter tip, which has the shape of a three-sided pyramid with a very flat profile and a total included angle of 142.3◦ . A more accurate reproduction of experimental data can be achieved by carrying out more elaborated simulations of the unloading stage. The deviations of the calculated values of hardness and Young’s modulus from the experimental values are most prominent for the square indenter, which was the smallest one among the indenters considered in Ref. [123]. The origin of such discrepancies can be explained by the fact that the linear size of this indenter is comparable to the lattice constants of the crystalline sample or to the size of a grain in the nanostructured sample. Thus, the finite-size effects should play the most prominent role in the case of small indenters [24]. As follows from Table 6.10, the E values calculated for the ideal crystalline fcc B2 NiTi phase for different indenters are on the level of 110 − 150 GPa, which generally corresponds to the experimentally reported values, E = 100 − 160 GPa, obtained from the indentation by a large (2.8 mm in diameter) spherical indenter [128]. For amorphous NiTi sample, the obtained results vary in the wide range of 55 − 130 GPa, although these values also generally correspond to the experimental results for the monoclinic B19 phase of the shape-memory NiTi alloy (E = 68 ± 5 GPa) [128] and for the nanostructured NiTi alloy (E = 55 ± 8 GPa) [126]. Acknowledgements The authors gratefully acknowledge the possibility to perform computer simulations at Goethe-HLR cluster of the Frankfurt Center for Scientific Computing.

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111. Kexel, C., Solov’yov, A.V.: Predicting dipole orientations in spontelectric methyl formate. Eur. Phys. J. D 75, 89 (2021) 112. Plekan, O., Cassidy, A., Balog, R., Jones, N.C., Field, D.: Spontaneous electric fields in films of cis-methyl formate. Phys. Chem. Chem. Phys. 14, 9972–9976 (2012) 113. Hussain, C.M. (ed.): Handbook of Functionalized Nanomaterials for Industrial Applications. Elsevier (2020) 114. You, Z. (ed.): Materials Science and Industrial Applications. Trans Tech Publications Ltd., Switzerland (2019) 115. Verlinden, B., Driver, J., Samajdar, I., Doherty, R.D.: Thermo-Mechanical Processing of Metallic Materials. Elsevier (2007) 116. Fischer-Cripps, A.C.: Nanoindentation, 3rd edn. Springer Science+Business Media, LLC, New York (2011) 117. Schuh, C.A.: Nanoindentation studies of materials. Mater. Today 9, 32–40 (2006) 118. Sushko, G.B., Verkhovtsev, A.V., Yakubovich, A.V., Solov’yov, A.V.: Molecular dynamics simulations of nanoindentation of nickel-titanium crystal. J. Phys.: Conf. Ser. 438, 012021 (2013) 119. Kramer, K.H.: Implants for surgery – a survey on metallic materials. In: Materials for Medical Engineering, vol. 2, pp. 9–29. Wiley-VCH, Weinheim (2005) 120. Geetha, M., Singh, A.K., Asokamani, R., Gogia, A.K.: Ti based biomaterials, the ultimate choice for orthopaedic implants - a review. Prog. Mater. Sci. 54, 397–425 (2009) 121. Prymak, O., Bogdanski, D., Koller, M., Esenwein, S., Muhr, G., Beckmann, F., Donath, T., Assad, M., Epple, M.: Morphological characterization and in vitro biocompatibility of a porous nickel-titanium alloy. Biomaterials 26, 5801–5807 (2005) 122. Brailovski, V., Prokoshkin, S., Gauthier, M., Inaekyan, K., Dubinskiy, S., Petrzhik, M., Filonov, M.: Bulk and porous metastable beta Ti-Nb-Zr(Ta) alloys for biomedical applications. Mater. Sci. Eng. C 31, 643–657 (2001) 123. Sushko, G.B., Verkhovtsev, A.V., Solov’yov, A.V.: Validation of classical force fields for the description of thermo-mechanical properties of transition metal materials. J. Phys. Chem. A 118, 8426–8436 (2014) 124. Verkhovtsev, A.V., Sushko, G.B., Yakubovich, A.V., Solov’yov, A.V.: Benchmarking of classical force fields by ab initio calculations of atomic clusters: Ti and Ni-Ti case. Comput. Theor. Chem. 1021, 101–108 (2013) 125. Oliver, W.C., Pharr, G.M.: An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J. Mater. Res. 7, 1564–1583 (1992) 126. Levashov, E.A., Petrzhik, M.I., Shtansky, D.V., Kiryukhantsev-Korneev, P.V., Sheveyko, A.N., Valiev, R.Z., Gunderov, D.V., Prokoshkin, S.D., Korotitskiy, A.V., Smolin, A.Y.: Nanostructured titanium alloys and multicomponent bioactive films: mechanical behavior at indentation. Mater. Sci. Eng. A 570, 51–62 (2013) 127. Landau, L.D., Lifshitz, E.M.: Course of Theoretical Physics, vol. 7. Theory of Elasticity. Elsevier, Amsterdam (2006) 128. Rajagopalan, S., Little, A.L., Bourke, M.A.M., Vaidyanathan, R.: Elastic modulus of shapememory NiTi from in situ neutron diffraction during macroscopic loading, instrumented indentation, and extensometry. Appl. Phys. Lett. 86, 081901 (2005)

Chapter 7

Multiscale Modeling of Surface Deposition Processes Ilia A. Solov’yov and Andrey V. Solov’yov

Abstract This chapter is devoted to the discussion of multiscale modeling of processes occurring during the deposition of nanoparticles on surfaces. The modeling relies on the method of stochastic dynamics. Stochastic dynamics describes processes in complex systems where the dynamics is represented through a number of kinetic processes occurring with certain probabilities. The chapter discusses the concept of stochastic dynamics and illustrates its implementation in a popular program MBN Explorer. In MBN Explorer, stochastic dynamics relies on the Monte Carlo approach and describes physical, chemical, and biological processes on multiple temporal and spatial scales. The chapter presents the basic theoretical concepts underlying stochastic dynamics implementation and provides several computational case studies accompanied with characteristic experimental results to validate the computational approaches.

7.1 Introduction to Stochastic Dynamics Scanning probe microscopy and high-resolution surface-sensitive diffraction are undoubtedly the tools of choice for probing adlayer and thin-film morphology formation in structures of varying levels of complexity. The studies of the last decades revealed that far-from-equilibrium structures formed during deposition of atoms [1], nanoparticles [2], and biomolecules [3] on various substrates, often possess unique properties [4–9], e.g., unusual morphology or chemical reactivity. It is, however, remarkable that despite extensive research efforts, the mechanisms of self-organization in many of these systems are not entirely understood [7–15]. This understanding is critical for the experimental control of the self-organization process I. A. Solov’yov (B) Department of Physics, Carl von Ossietzky Universität Oldenburg, Carl-von-Ossietzky-Str. 9-11, 26129 Oldenburg, Germany e-mail: [email protected] A. V. Solov’yov MBN Research Center, Altenhöferallee 3, 60438 Frankfurt, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 I. A. Solov’yov et al. (eds.), Dynamics of Systems on the Nanoscale, Lecture Notes in Nanoscale Science and Technology 34, https://doi.org/10.1007/978-3-030-99291-0_7

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and could be obtained through a combination of experimental measurements supported by model calculations and simulations [8, 9, 12–16]. In this respect, several methods that permit simulations of bio-nanosystems were developed [8, 12, 17–21]. Atomistic processes that govern macroscopic phenomena in materials often have significant activation barriers and, therefore, occur at time scales beyond the reach of conventional atomistic molecular dynamics (MD) simulation techniques. Although recent developments of MD-based accelerated dynamics [22–25] have successfully extended the simulation time scales to microseconds, it remains computationally inefficient to employ atomistic MD for a large class of important problems such as, e.g., diffusion, nucleation, growth, crystallization, defect evolution, and chemical reactions. Stochastic dynamics is often the tool of choice for studying dynamic of processes occurring on long time scales, e.g., milliseconds to hours [4, 12–15, 26– 32]. Instead of propagating individual atoms in time, as done in MD, the stochastic dynamics models the evolution of a coarse-grained molecular system in a probabilistic way. Stochastic dynamics can be used to simulate processes of epitaxial thin film growth [4, 33], crystal growth [34], defect diffusion in metals and semiconductors [35], nanoparticle diffusion and were employed for several biophysical applications [36, 37]. These existing methods deliver a good agreement with experimental observations, but it should be noted that depending on the studied problem, the methodological realization of the algorithms underlying stochastic dynamics is usually different [4, 30–32]. Modeling of the stochastic dynamics of surface deposition processes often relies on the so-called kinetic Monte Carlo (KMC) approach [8, 12–15]. Here stochastic processes drive the temporal and spatial evolution of a system described through constituent sub-components deposited on a surface. The crucial steps in performing stochastic dynamics are defining the sub-components of a system and identifying the underlying stochastic processes and the associated probabilities. Stochastic description of a complex system is often simplified if it is assumed to consist of certain particles representing its sub-components. The probabilities for processes modeled within the framework of stochastic dynamics may be obtained from accurate all-atom MD simulations, quantum chemistry (QC) calculations, or taken directly from experiments. Naturally, in such a description, multiple degrees of freedom in a complex system are substituted through several effective processes, governed by predefined rate constants. The idea here is to drastically reduce the complexity of a system but capture all essential features of a complex dynamical process. For example, diffusion coefficients that characterize the kinetics of diffusing molecules can be converted into a stochastic probability for random translation of a particle into an adjacent position; diffusion coefficients can be routinely obtained from all-atom MD simulations [38–40]. Binding and activation energies could also be obtained directly from MD-simulations [7, 41–44] and converted to a stochastic probability of particle detachment that can be used to coalescence and fragmentation processes in a complex system. This chapter discusses the theoretical foundations of stochastic dynamics and introduces its implementation in the MBN Explorer software suite [8]. The chapter

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serves as an overview of multiscale modeling techniques applied to study the deposition of particles on surfaces and investigate the self-assembly processes. Basics of algorithms for such simulations are introduced, and the physical meaning of the underlying parameters is explained.

7.1.1 Adsorption and Desorption Processes An important goal of nanotechnology is the development of controllable, reproducible and industrially transposable, nanostructured materials. In this context, controlling the final architecture of such materials by tuneable parameters is a fundamental problem. The traditional technique of thin-film growth by deposition of atoms [45, 46], small atomic clusters [45, 47] and molecules [46, 48, 49] on surfaces gives a possibility to construct materials with pre-defined properties. Recent experiments show that patterns with different morphology can be formed in the clusters deposition process on a surface [45, 47, 50]. Among other possible shapes, droplet-like and fractal islands have been observed in various systems [45, 47, 50]. It was shown that the island morphology depends on various factors, such as temperature [50, 51], particles size [52], particles deposition rate [51, 53, 54], substrate roughness [54–56], concentration of impurities in the system [47, 52, 57] and interparticle interaction energies [47, 51]. Among the many possible morphologies that can manifest on a surface in the course of particle deposition process, the fractal shape is one of the most non-trivial ones. The investigation of the dendritic structures (fractals) has attracted considerable attention of many scientists in the last decades due to their abundance in Nature [50, 52, 57–64]. The formation of such systems provides a natural framework for studying disordered structures on the surface because fractals are generally observed in far from equilibrium growth regime. During the last years, the fractal shape has been recorded for a variety of systems. Thus, fractals consisting of Ag [52, 57, 62], Co [61], Au [65], Fe-N [63] clusters and C60 molecules [65, 66] have been fabricated on different surfaces with the use of the cluster deposition technique [46, 48]. In this chapter, we focus specifically on the example of fractal formation and stability on surfaces and discuss details of multiscale modeling approaches used to study these systems. Using MBN Explorer [8], it is possible to model the growth process by deposing particles on a surface. To compare with the experimental measurements [52, 57, 62], one should use model parameters in the simulations consistent with the experiment. Thus, for example, a diameter of a particle of 2.5 nm corresponds to the size of an Ag500 cluster used in experimental investigations [52, 57, 62]. In that experiment, the deposition flux decreased linearly from Fstar t = 7.2 × 1013 particle/cm2 s to Fend = 1.1 × 1011 particle/cm2 s. By choosing the flux value and defining a surface, it is possible to computationally model the deposition process as illustrated in Fig. 7.1A. A deposited silver cluster experiences Brownian-like motion until it collides with another cluster or a group of clusters in which case the cluster could be attached

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Fig. 7.1 Key elementary kinetic processes of nanocluster dynamics on a surface. A nanoparticle is deposited on a surface with a characteristic deposition flux F (A). A deposited nanocluster diffuses over the surface (B). For the sake of a concrete example Ag500 is illustrated from an earlier investigation [15]. The cluster center of mass trajectory obtained from MD simulations is shown. It indicates that the deposited silver cluster experiences the random, Brownian-like, motion that can be parameterized by the corresponding kinetic rate. The long time-scale motion of an ensemble of the deposited clusters can be parameterized through three different kinetic rates d ,  pd and de (C), corresponding to: the diffusion of a freely deposited cluster over a surface—d , the diffusion rate of a cluster along the periphery of an island on surface— pd , and the detachment rate of a cluster from an island—de . Peripheral diffusion depends on the number of broken bonds (m) and the number of maintained neighboring bonds (n). The particle detachment rate depends on the number of broken bonds (l). In the depicted example m = 1, n = 1, l = 1. Figure is redrawn from data presented in [9]

and experience peripheral diffusion, or be detached. The diffusion coefficients for these different diffusion regimes of a cluster can be deduced from all-atom MD simulations by analyzing trajectory of the cluster center of mass, as illustrated in Fig. 7.1B. The mechanism of cluster diffusion over a surface is an interesting nontrivial problem as the clusters can experience different types of motion, leading to different diffusion mechanisms [67–71], for example the single jump mechanism [68], dislocation mechanism [69], concentrated rotation and translation mechanism [70], or dimer shearing mechanism [71]. Figure 7.1B shows an example of a typical cluster center of mass trajectory obtained with MBN Explorer [8] for the Ag500 cluster moving during 10 ns at

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T = 800 K. In this case, the interaction of silver atoms within the cluster was modeled through the Sutton–Chen potential [72], while the interaction of the cluster with graphene was described using the Morse potential [8], adapted from earlier studies [73, 74]. The atomistic MD simulation provides the description of the dynamics of each atom in the cluster. However, such a description can not be applied directly to model pattern formation on a surface because the characteristic temporal and spatial scales of the whole multi-cluster system are typically far beyond the limits of nowadays all-atom MD simulations. This deficiency of MD approach can be overcome if the random MD motion of constituent clusters in the system is substituted with the stochastic description, in which the kinetic rates are determined from the all-atom MD simulations performed for single constituent clusters as illustrated in Fig. 7.1B.

7.1.2 Modeling Surface Diffusion Processes: Kinetic Parameters To simulate surface diffusion processes, the following procedure could be adopted [8, 9, 15]. At every step of the simulation, new particles are deposited on the surface according to a given deposition rate. Computationally, the surface is modeled as a regular grid whose cells could accommodate one particle at maximum. The deposited particles occupy some of the free cells on the grid. Simultaneously, the already deposited particles diffuse on the surface, with the rate [8, 12–15]   Ea , d = ν1 exp − kT

(7.1)

where E a is the activation energy, ν1 is the attempt escape rate, T is the system’s temperature, and k is the Boltzmann constant. The process of particle diffusion on a surface is schematically illustrated in Fig. 7.1C. An important quantity in the stochastic dynamics is the time step, t, which defines the characteristic time for particle diffusion on a surface as t = 1/ d .

(7.2)

The time step t is related to the diffusion coefficient D for a particle on a surface. The diffusion coefficient for a particle on a surface is the proportionality coefficient in the diffusion equation [75, 76]. Solving the diffusion equation in two dimensions allows one to write the probability to find a particle at time t on a distance [r , r + dr ] from its initial location as   r2 1 exp − r dr. (7.3) ω(r, t)dr = 2Dt 4Dt

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The probability function defined in Eq. (7.3) allows one to calculate the mean square displacement of a particle as [75, 76]  (r1 − r0 )2 =

∞

ω(r, t)r 2 dr = 4D(t1 − t0 ),

(7.4)

0

where r0 and r1 are the distances to a particle from the initial position at two successive time instances t0 and t1 . Equation (7.4) allows to express the diffusion coefficient as D=

r 2  , 2zt

(7.5)

where r 2  is the mean-square displacement of a particle per time t, and z is defined by the dimensionality of space [75, 76]. In the case of particle diffusion on a surface z = 2 (see Eq. (7.4)). On the other hand, the mean-square displacement depends on the diffusion rate and on the particle hopping length, which can be considered equal to the particle diameter d0 : (7.6) r 2  = d d02 t. Here t has a meaning of a single simulation step defined in Eq. (7.2). Substituting Eq. (7.6) into Eq. (7.5), one obtains D=

d d02 . 2z

(7.7)

Equation (7.7) allows to estimate  (and therefore t) once the diffusion coefficient is known: d2 (7.8) t = 0 . 2z D For example, the diffusion coefficient of an Ag500 cluster on graphite at room temperature was measured as 2 · 10−7 cm2 /s [57]. Substituting this value into Eq. (7.8), one obtains t = 78 ns. Substituting Eq. (7.1) into Eq. (7.7), one relates the diffusion coefficient to the activation energy and temperature: D=

    Ea Ea d02 ν1 exp − = D0 exp − . 2z kT kT

(7.9)

From Eq. (7.9), it follows that the diffusion coefficient decreases as the activation energy grows. This results in an exponential growth of the time step t with E a , since t ∼ 1/D (see Eq. (7.8)). Equation (7.8) introduces the optimal time step for the computations because it defines the characteristic time at which a freely deposited particle is displaced on d0 , i.e. in the neighboring lattice cell (see Fig. 7.1C).

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Particles that are in contact with other particles could also experience diffusion processes. In that case, one can distinguish between peripheral diffusion and particle detachment. Peripheral diffusion and particle detachment are the two processes that occur on a surface. Both processes are schematically illustrated in Fig. 7.1C. The diffusion and the detachment rates depend on the activation energy and the particleparticle interaction. In Arrhenius approximation, the diffusion rate of a particle along the periphery of an island reads as:  n Ea m Eb − − ,  pd (m, n) = ν2 exp − kT kT kT 

(7.10)

where m is the number of bonds that are broken due to the particle motion, E b > 0 is the binding energy of two particles, n is the number of maintained neighboring bonds between two particles and  ≤ E b is the diffusion energy barrier [62, 77], ν2 is the attempt escape rate for peripheral diffusion. Equation (7.10) describes the probability of a particle to overcome a potential energy barrier, which for a particle diffusing along the island periphery is parameterized by the energies E b , , and E a . Note that the parameter E a , which enters Eq. (7.10), depends on the simulation time step t. Therefore, only the parameters E b and  define the potential energy barrier for particle diffusion along the island periphery, while E a characterizes the overall time scale of the process. The detachment rate of a particle from the island is given by   μ Ea l Eb − − , de (l) = ν3 exp − kT kT kT

(7.11)

where l is the number of bonds broken after particle detachment from the island, μ is the chemical potential of particle detachment [47, 62, 77, 78], ν3 is the attempt escape rate of a particle from its equilibrium position. Equation (7.11) can be understood within the framework of the classical nucleation theory [78], which studies the liquid↔gas transition in droplets. It is derived in the Arrhenius approximation, similarly to Eq. (7.10). For the further description, we assume: ν1  ν2  ν3 = 0 .

(7.12)

This approximation occurs in the situation when the characteristic attempt escape rates of a particle leading to its diffusion or detachment are close. From Eqs. (7.10)– (7.11) one derives that the probability of different kinetic processes in the system depends on the values of E a , E b , , μ, which are called the kinetic parameters. For convenience, the kinetic parameters are often defined in units of kT (1 kT = 0.026 eV) at room temperature (300 K) [9, 12–15]. The interaction energy between the deposited particles and the substrate is responsible for the particle mobility on a surface, as follows from Eq. (7.1). The interaction Ag energy of Ag500 (E a ), C60 (E aC60 ), and Sb2300 , (E aSb ) clusters with graphite surface at Ag room temperature has been estimated as E a = 6.6 kT [15], E aC60 = 6.9 kT [79] and

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E a(Sb) = 27.1 kT [53]. The significant spread of the values indicates the essential role of interatomic interactions in defining the activation energy. The value of E a defines the time scale of the surface pattern formation and fragmentation processes. In this chapter, we will focus the discussion on the case study of silver cluster deposition and diffusion, and will therefore assume E a fixed. Another important quantity characterizing the particle diffusion on a surface is its attempt escape rate 0 (see Eqs. (7.10)–(7.12)), which can be estimated as [9, 12, 15]   2Dz Ea . (7.13) 0 = 2 exp kT d0 For a silver nanoparticle with d0 = 2.5 nm deposited on graphite the diffusion coefficient at room temperature D  2 · 10−7 cm2 s−1 [57], resulting in 0 = 0.94 · 1010 s−1 . The interaction energy of two particles, E b , depends on the atomic composition of the particles and on the presence of impurities in the system [52, 57, 62]. It was shown that the presence of oxygen impurities in a silver cluster deposited on graphite leads to the decrease of E b and consequently to the degradation of surface pattern stability. We note, that in experiment [52, 57] silver cluster fractals are formed and may decay on comparable time scales. This is only possible if E b is of the same order of magnitude as E a . The diffusion barrier energy  depends on the atomic composition of the cluster and usually amounts 0.05–0.2 of the bonding energy of two clusters [80]. The change in the chemical potential μ arises due to the energy difference caused by the change in the number of particles in the system. The chemical potential characterizes the ability of particles to diffuse from regions of high chemical potential to those of low chemical potential and is defined as the partial derivative [81]  μ=

∂U ∂N

 ,

(7.14)

V,S

where U and S are the total energy and the entropy of the system, V is its volume and N is the number of particles in the system. The variation of the chemical potential arising due to a structural transformation in the system can be calculated from the known values of the chemical potential of individual components of the system before and after the transformation. For example, for the evaporation of a silver nanoparticle from an island with N particles on graphite surface Ag(islandN ) + C(graphite) → Ag(islandN−1 ) + C(graphite) + Ag(particle), (7.15) the corresponding change of the chemical potential can be calculated as a difference between the chemical potential of the products and the educts. With μAg(islandN ) ≈ μAg(islandN−1 ) one obtains (7.16) μ = μAg(particle) .

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The chemical potential can be measured experimentally [82] and is tabulated for many substances [83, 84]. It depends on the phase state of the system: for the gas of (gas) (liquid) silver atoms μAg = 2.55 eV, while for the silver in the liquid phase μAg = 0.8 eV [83]. These values and Eq. (7.16) allow one to suggest that the change of the chemical potential in the silver fractal fragmentation process, at room temperature lies within the range 30–100 kT [13–15].

7.2 Pattern Formation, Evolution, and Fragmentation Processes at Interfaces 7.2.1 Pattern Formation: Case Study of Fractal Growth To illustrate stochastic dynamics in application to the process of particle surface deposition and modeling of pattern formation, we now consider a case study of silver cluster fractal growth and stability by varying the kinetic parameters E b , , μ. Using the computational approach outlined above, we have obtained several fractal structures that are similar to the silver cluster fractals on graphite surface reported to be observed in experiment [52, 57, 62]. An example of a fractal grown computationally using the stochastic dynamics approach is shown in Fig. 7.2B. We have also chosen that fractal structure for the further investigation of the post-growth relaxation processes, discussed below. The diameter of the fractal is ∼635 nm, which is close to the diameter of the experimentally grown structures [52, 57, 62]. For the sake of illustration in Fig. 7.2A, we show the experimentally grown silver cluster fractal prior thermal annealing, which triggers the fractal fragmentation [52, 57, 62]. An important characteristic of the fractal is the fractal dimension d f . The Hausdorff fractal dimension is generally defined as [85, 86]:

Fig. 7.2 Structure of silver cluster fractal experimentally grown by clusters deposition technique on graphite surface [62] (A) in comparison with a smaller fractal obtained computationally using the stochastic dynamics approach (B). Both structures are shown in one scale that is indicates. Figure is redrawn from data presented in [15]

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log[N (l)] . l→0 log[1/l]

d f = lim

(7.17)

Here N (l) is the number of self-similar structures of linear size l needed to cover the whole structure. In practice, the fractal dimension is usually calculated by the box-counting method or using the mass-formula for fractal dimension [87]:  N =ρ

2R d0

d f ,

(7.18)

where N is the number of particles in the fractal, R is the radius of the minimal circumscribed circle of the fractal structure, d0 is the diameter of a particle and ρ is the ratio of the covered surface to the entire surface area (packing √ density). In the case of simulations performed on the hexagonal grid ρ = π/2 3. Equation (7.18) can be used to calculate the fractal dimension of the structure shown in Fig. 7.2B. With R = 319 nm, d0 = 2.5 nm and N = 5182, one obtains d th f = 1.6. This value is in a good agreement with experiment for silver cluster fractals grown on the graphite ex p surface (Fig. 7.2A), which gives d f = 1.7 ± 0.1 [52]. As illustrated in Fig. 7.2, the topology of the fractal simulated using stochastic dynamics is close to the fractal topology seen in experiment. In both cases the fractals shown in Fig. 7.2 have several main branches, growing from the center of the fractal. The branch width of the fractal simulated by the means of stochastic dynamics is ∼6.5 nm, while the typical experimental width of the branch is 10–20 nm [52]. The difference arises because the particles in the simulation were deposited on a surface at a somewhat higher rate than in experiment [15].

7.2.2 Pattern Evolution on the Surface Let us now consider fractal post-growth relaxation. According to the estimates above, for Ag500 one-time step in stochastic dynamics corresponds to t = 78 ns, which allows one to calculate the simulation time as t = Nstep t,

(7.19)

where Nstep is the number of simulation steps. The rate of fractal decay depends on the interparticle interaction, and it defines the morphology of the fragments that are formed during the process. Snapshots of the structures arising at different stages of the fragmentation process simulated at different parameters of interparticle interactions are shown in Fig. 7.3. This example shows how different can be the fragmentation paths and the fragments morphology. Figure 7.3 shows that for E b = 1 kT,  = 0.2 kT one observes an entire defragmentation of a fractal, which is the fastest fragmentation path. In this case, the interaction energy between the particles is relatively weak and the probability to evaporate

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Fig. 7.3 Evolution of fractal structure on a substrate with periodic boundary conditions. The initial fractal structure shown in the middle undergoes fragmentation in different final states depending on the interparticle interactions in the system. Numbers above the corresponding images indicate the kinetic parameter values of E b and  used in the simulations (in units of kT ), μ = 2 kT in all cases. The simulation time t is given for each path of the fragmentation. Reproduced from [15] with permission from American Physical Society

a particle from the fractal is much higher than the probability of newly deposited particles to nucleate. This fragmentation scenario can be realized in experiment if the temperature of the system is rapidly elevated after the fractal was created. Figure 7.3 shows that for E b ≥ 2 kT the fractal melts in a number of compact droplets. Depending on the energies of interparticle interactions, the shape of the droplets becomes different. Thus, for E b = 2 kT,  = 0.4 kT three large, almost spherical, droplets of a similar size are formed. In this case, the binding energy E b between the particles is rather small, allowing relatively easy detachment of particles, but at the same time, it is large enough to make the characteristic particle detachment time comparable with the characteristic particle nucleation time, thereby preventing the system from entire defragmentation, observed at E b = 1 kT. Thus, the fragmentation path at E b = 2 kT goes via the rearrangement of the entire system and the formation of large stable droplets.

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A further increase of the interparticle interaction energy leads to the change in the fractal fragmentation pattern. As seen in Fig. 7.3 at E b = 4 − 6 kT the fractal fragments into several compact droplets. The analysis of morphology of the created patterns leads us to the following main conclusions: (i) the growth of E b leads to the increase of the number of droplets on a surface (see E b = 4 kT and E b = 6 kT) and to the decrease of their average size. This happens because the detachment of particles from the fractal becomes energetically an unfavorable process, and the fractal fragments mainly due to the peripheral diffusion of particles, initiated at the peripheral defect sites. (ii) The increase of the peripheral diffusion barrier energy  suppresses the diffusion of particles, resulting in a slower evolution and fragmentation of the fractal shape. It is remarkable that at E b = 6 kT and  = 1.2 kT one observes the formation of elongated islands on a surface, which follow the direction of the fractal branches. A further increase of the interparticle binding energy with the simultaneous lowering the barrier energy for the particle peripheral diffusion favors the formation of elongated islands on a surface. Figure 7.3 illustrates this for E b = 12 kT and  = 1 kT. In this case, the timescale for the particles to detach from the fractal is significantly larger than that for the peripheral particle diffusion. A simultaneous increase of the interparticle binding energy and the barrier energy for the particle peripheral diffusion leads to the growth of the fractal lifetime. Figure 7.3 shows that for E b = 24 kT and  = 12 kT the fractal has no noticeable changes in its morphology after 4 s of simulation. In the case when the interparticle energies are large, the fractal fragmentation is expected to occur on a larger time scale and can be simulated numerically if the value of the simulation time step is increased. The important characteristic of the fractal fragmentation is the number of fragments at a given time. The smallest fragment is a single particle. The time evolution of the number of fragments calculated for different sets of kinetic parameters is shown in Fig. 7.4a, d. Curve 1 in Fig. 7.4a shows the time evolution of the number of fractal fragments at E b = 1 kT. The number of fragments in this case rapidly approaches the asymptotic value, approximately equal to the half of the total number of particles in the fractal. This means that the system dominantly consists of dimers. With increasing E b the number of fragments at the equilibrium decreases, as seen in Fig. 7.3. It is interesting to note that at E b = 2 kT there are three dominating large islands (see Fig. 7.3). The total number of fragments at the end of the simulation in this case is equal to 100, being much smaller than the total number of particles in the system. This feature arises in the situation when a large number of single particles detach from the large droplets but later stick back. In this case, the number of single particles fluctuates rapidly resulting in the large fluctuations of N f r (t) dependence shown in Fig. 7.4a by curve 2. Figure 7.4d shows that there is no dramatic change in N f r (t) dependence with the growth of  at a constant value E b . This analysis shows also that the growth of  preventing particles’ peripheral diffusion hinders the fast transformation of droplets into compact islands which eventually results in the increase of the number of fragments on a surface.

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Fig. 7.4 Time evolution of the number of fragments N f r , Rmax  introduced in Eq. (7.20) and of the S/P ratio introduced in Eq. (7.28) calculated for the fractal structure shown in Fig. 7.2B. The fractal fragmentation has been analyzed at μ = 2 kT for different values of the binding energy E b and the barrier energy . Plots a–c show the results of calculation obtained for  = 0.2E b and the different values of the binding energies between two particles. Lines 1–6 correspond to E b = (1, 2, 3, 4, 5, 6) kT, respectively. Plots d–f represent the results obtained at E b = 4 kT for different values  = (0, 0.4, 0.8, 1.0, 3.2, 4) kT. The direction of growth of  is shown in these plots. Reproduced from [15] with permission from American Physical Society

As seen in Fig. 7.3, in the course of fractal fragmentation the mobile particles can coalescence into islands, i.e. groups of particles bound together. The size and the number of islands on the substrate depend on the binding energy E b and the barrier energy . The important characteristic of the fragmentation pattern on a surface is the average maximal radius of the created islands, which reads as Nfr 1  (i) R , Rmax  = N f r i=1 max

(7.20)

(i) where N f r is the total number of islands on a surface, Rmax is the maximal radius of the ith island. The dependencies of Rmax (t) calculated at different values of E b and  are shown in Fig. 7.4b, e. These figures show that in average Rmax  approaches the equilibrium value at the chosen values of kinetic parameters except for E b = 3 kT,  = 0.6 kT when the large fluctuations of Rmax  develop and grow with time. This happens because at E b = 3 kT the rate of single-particle detachment turns out to be so that only several particles are able to overcome the detachment energy barrier at one simulation step. The escaped particles freely diffusing over the surface after a short period of time return to the same or some other island. Although the number of fluctuating fragments on the surface in this case is relatively small (see Fig. 7.4a), the fluctuations of Rmax  become considerable because small

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islands can be spontaneously created but most of them disappear just after several simulation time steps. Thus, for example, for t1 = 3.484 s Rmax 1 = 21.5 nm, while for t2 = 3.485 s Rmax 2 = 33.4 nm. The change of the maximal radius Rmax  is thus 11.9 nm within 1 ms time interval. This happens because for the time frame t1 , there are N S(1) = 6 single particles and N L(1) = 10 fragments of a larger size with approximately equal diameter on the surface. For the time frame t2 , the number of large fragments is N L(2) , still equal to 10, while there are no single particles on the surface (i.e. N S(2) = 0). With R L(1) = R L(2) = R L being the characteristic radius of the large island, R S(1) = R S(2) = R S the radius of a single particle, and N L(1) = N L(2) = N L , one derives N S N L Rmax  = (7.21) (R L − R S ) , N1 N2 where N S = N S(1) − N S(2) is the change of the number of single particles, N1 = N L + N S(1) is the total number of particles at instance t1 , and N2 = N L + N S(2) is the total number of particles at instance t2 . Substituting values for Ns , N L , N1 and N2 in Eq. (7.21) for the special case considered one obtains Rmax  =

15 (R L − R S ) . 42

(7.22)

Substituting R L  = 32 nm and R S = 1.25 nm in Eq. (7.22), one derives Rmax  = 11 nm. Equation (7.22) shows that Rmax  increases with R L which grows with time until it reaches the equilibrium value. Equation (7.21) can also be rewritten as Rmax  =

N12

N S N L (R L − R S ) , (1 − N S /N1 )

(7.23)

which shows that for N S  N1 the fluctuation of the average radius Rmax  can be several times larger than the value of the average radius. Note that although the largest islands are observed for the kinetic parameter E b = 2 kT (see Fig. 7.3), the largest average maximal radius is expected for E b = 3 kT as depicted in Fig. 7.5. This happens because the number of single particles on the surface for E b = 3 kT is about 10, while for E b = 2 kT it is exceeding 100. Figure 7.4e shows some dependence of Rmax  on . The growth of  leads to the decrease of Rmax , which is a natural result of a lower peripheral mobility of particles. Figure 7.6 shows the distributions of island sizes in the system after 4 s of simulation. In order to improve the statistics, the distributions shown in Fig. 7.6 have been averaged over a time interval τ = 0.78 s as follows N f r (t) =

1 τ



τ/2

−τ/2

N f r (t − x)d x.

(7.24)

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Fig. 7.5 Dependence of N f r (squares, left scale) and Rmax  (dots, right scale) on the binding energy E b calculated for the barrier energy  = 0.2E b after 4 s simulation, corresponding to the dependencies shown in Fig. 7.4a, b. Reproduced from [15] with permission from American Physical Society

Fig. 7.6 Distributions of island sizes formed on the substrate after 4 s of simulation. The distributions were calculated at the fixed values of E b = 4 kT, and μ = 2 kT for different values of  as indicated. Reproduced from [15] with permission from American Physical Society

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Fig. 7.7 Size distributions of islands calculated at different stages of the fractal fragmentation (see Fig. 7.4) for E b = 2 kT,  = 0.4 kT and μ = 2 kT. The corresponding simulation time is given in the insets to the plots. Reproduced from [15] with permission from American Physical Society

The histograms in Fig. 7.6 have been calculated with different barrier energies. The maxima in the distributions show the most abundant island sizes. Figure 7.6 shows that the sizes of the islands created in the fractal post-growth fragmentation process depend strongly on the binding energy E b and the barrier energy . At some values of E b and  one can identify two maxima in the island size distributions. Especially clear this feature manifests itself at  = 0 kT,  = 0.4 kT and  = 3.2 kT. The presence of two maxima in the island size distributions tells that there are two groups of islands on the surface having different preferential island size. Let us also analyze the time evolution of the distributions shown in Fig. 7.6. Figure 7.7 shows distributions of the island sizes calculated at different fragmentation stages for a fixed set of the kinetic parameters. Figure 7.7 illustrates the evolution of the island size distribution simulated at E b = 2 kT and  = 0.4 kT. After fast fragmentation of the fractal into a subset of noncompact islands which occurs on the time scale greater than 0.04 s, the distribution of islands sizes has a Gaussian-like shape with the maximum centered at 25 nm. In the course of the fractal fragmentation process the magnitude and the position of the maximum of the distribution change because the morphology of the system changes due to the evaporation of single particles from the islands and the nucleation of single particles. Figure 7.7 illustrates that small islands nucleate into larger droplets resulting in a shift of the maximum of the distribution towards larger island sizes. Interesting that the fragmentation/nucleation dynamics leads in this case study to the formation of two maxima which correspond to the presence in the system of the droplets of different radii. Figure 7.7 shows the evolution of the fractal fragmentation process. The initial fragmentation of the fractal is very rapid. It involves the rearrangement of single

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particles in the fractal which form the defects at the fractal periphery. The evolution of the shape of the large droplets slows down with the growth of their size due to the decrease of the droplets mobility (see Fig. 7.7b–c). At the stage when only a few large-size droplets remain their dynamics is governed to large extend by the interchange of peripheral particles from these droplets (see Fig. 7.7d–f). The large droplets diffuse slowly over a surface and may eventually merge. The characteristic time scale for diffusion of an entire large droplet is significantly larger than the characteristic diffusion time of single constituent particle, and therefore practically can not be resolved within the simulation time limit. However, note that this motion can also be simulated with a larger time step. The appropriate value of the time step can be estimated using Eq. (7.8). Another useful quantity for the characterization of surface structures is the ratio between the area and the perimeter of the structure (S/P ratio) [52]. This ratio characterizes the island topology. Thus, the S/P ratio for a linear chain of N spherical particles is equal to d0 S = , (7.25) P 4 where d0 is the diameter of a particle. Note that the S/P ratio for a linear chain is always a constant. The S/P ratio for a compact droplet of the radius, Rd , is equal to S Rd = . P 2

(7.26)

It can be easily expressed via the number of particles N in the droplet: d0 √ S = N. (7.27) P 4 √ In this case, the S/P ratio increases as N with the growth of the system size. The S/P ratio for a fractal consisting of N particles should be larger than in Eq. (7.25) and smaller than in Eq. (7.27). Let us now analyze the time evolution of the average S/P ratio of the system during the fractal fragmentation. The S/P ratio for a system of N islands is defined as Nfr 1  Si , S/P = N f r i=1 Pi

(7.28)

where Si and Pi are the area and the perimeter of ith island, and N f r is the number of islands in the system. The S/P ratio is a useful characteristic for the structure morphology, often used in experiment [52]. The dependence of the S/P ratio on time calculated for different sets of the kinetic parameters is shown in Fig. 7.4c, f. Curve 1 in Fig. 7.4c shows time evolution of the S/P ratio during the fractal relaxation in the case of the relatively small binding energy between the particles being equal to 1 kT. The S/P ratio in this

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case rapidly decreases until it reaches the minimum value 0.78 nm, i.e. the S/P ratio that is slightly smaller than the value for a dimer of particles with d0 = 2.5 nm. Figure 7.4 shows that the S/P dependencies to large extend follow the dependencies calculated for Rmax . The performed analysis provides a lot of useful information on the dynamical evolution of the system during fragmentation. However, its direct comparison with experimental measurements is rather difficult because the calculated distributions vary with time, but the experimental measurements are usually performed for stationary (or quasi-stationary) systems. Nevertheless, the comparison with experiment is possible if the average life-time Tl of the studied configuration is greater than the characteristic measurement time Tm : Tl  Tm .

(7.29)

Here, Tl is defined as the characteristic time period at which an observable characteristic, e.g., the number of fragments in the system, changes within the statistical uncertainty, and Tm is the minimal time period required to perform an experimental measurement. An important characteristic of the system’s stability is the total number of fragments N f r in the system. At the equilibrium, N f r fluctuates around the average constant value. Note that N f r may have similar behavior in a so-called kinetically trapped state, or a quasi-equilibrium state that is separated from the equilibrium state by an energy barrier. The energy barrier between the kinetically trapped state and the equilibrium state may be significantly larger than the thermal vibration energy, therefore the trapped system may spend a noticeable lifetime in the kinetically trapped state. This lifetime can be sufficient for experimental measurements and for holding Eq. (7.29). This means that the quasi-equilibrium value of N f r may come out different for varied initial distributions of particles on a surface, demonstrating that diverse evolution paths may lead the system to different final quasi-equilibrium states. Below we analyze two examples supporting this hypothesis. Figure 7.8 depicts the time evolution of the number of fragments/nucleation islands, N f r , calculated (line 1) for the fractal having the initial shape as plotted in Fig. 7.2A, and (line 2) during the nucleation process of randomly distributed particles. The total number of constituent particles in both cases is equal to 5182. The size of the substrate used in the simulation is identical in both cases, equal to 650 × 750 nm2 . Figure 7.8 shows that the inter-particle interaction influences significantly the system dynamics. Thus, in the case of the weak bonding between particles (i.e. E b = 1 kT,  = 0.2 kT), see Fig. 7.8a line 1, the fractal fragments into ∼2320 islands, i.e. most of the particles in the system are bound in a form of dimers. Remarkably, that at these kinetic parameters particles randomly distributed over a surface nucleate to approximately the same quasi-equilibrium value N f r (line 2 in Fig. 7.8a). The insets in Fig. 7.8a illustrate the distribution of particles at the instant t = 4 s in the case of nucleation and at t = 2.8 s for the fragmentation. Figure 7.8a shows that the system can evolve from the very different initial states to the same final state.

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Fig. 7.8 Time evolution of the number of fragments/nucleation islands on a surface, N f r , during the fractal fragmentation process (line 1) and during the nucleation process of randomly distributed particles (line 2). Plots a and b have been calculated at different values of the kinetic parameters: a E b = 1 kT,  = 0.2 kT, μ = 2 kT; b E b = 4 kT,  = 0.4 kT, μ = 2 kT. The insets show the morphology of the system at the end of the simulation. Reproduced from [15] with permission from American Physical Society

The fragment number evolution with time depend on the inter-particle interaction as seen from Fig. 7.8b, obtained at larger E b , E b = 4 kT,  = 0.4 kT. The quasiequilibrium value of N f r in this case depends on the initial distribution of particles on a surface. The inset to Fig. 7.8b shows that both systems have evolved in a group of droplets, whereby the size of the droplets created from the initial fractal distribution of particles is larger than the size of the droplets created via the nucleation. Figure 7.8 shows that for the chosen kinetic parameters, the number of fragments in the system becomes constant or changes slowly with time at sufficiently large t value. The resulting static or quasi-static distributions of particles can be compared with experimental observations. In the cases when the initial distribution of particles on a surface influences the final morphology of the system means the system occupies one of the kinetically trapped state. Although the quasi-equilibrium kinetically trapped states do not have the lowest free energy, they may live for sufficiently long time to perform experimental measurements of the system characteristics. Figure 7.9 shows the island size distributions and the corresponding S/P ratio distributions calculated for the fractal fragmentation on the 650 × 750 nm2 with periodic boundary conditions. The distributions plotted in Fig. 7.9a–b have been obtained with the kinetic parameters E b = 3 kT,  = 0.6 kT, μ = 10 kT at t = 4 s, i.e. well after the fractal fragmentation when the system evolves in the almost stationary equilibrium or quasi-equilibrium state. In this case diffusion of particles along the fractal periphery is the dominating process. The increased rate of particle peripheral diffusion leads to the faster island rearrangement, and the formation of islands of different sizes, as seen in Fig. 7.9b. The insets in Fig. 7.9a–b show the results of experimental measurements obtained for silver fractal fragmentation via

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Fig. 7.9 S/P ratio distributions calculated after the fractal fragmentation with different sets of the kinetic parameters and the corresponding distributions of island sizes. Distributions a and b are calculated with E b = 3 kT,  = 0.6 E b , μ = 10 kT; c and d with E b = 4 kT,  = 0.4 E b , μ = 2 kT. Insets show the results of experimental measurements for silver fractal fragments created via annealing (a) and (b), and by adding of oxide impurities to silver clusters c and d [52]. Reproduced from [15] with permission from American Physical Society

annealing at 600 K. The experimentally measured distribution of the silver cluster island sizes is rather broad, with the most probable radius of silver islands ∼25 nm. A close value of 23 nm follows from the theoretical analysis. The discrepancy may arise due to the thinner branches of the fractal used in the simulations as compared to the ones analyzed in experiment. Figure 7.9c–d shows the island size distribution and the corresponding S/P ratios distributions calculated with E b = 4 kT,  = 0.4 kT, μ = 2 kT. The results of numerical calculation are compared with the experimental data shown in the insets to Fig. 7.9c–d on silver fractals grown with the oxidized silver nanoparticles [52]. In the experiment the most abundant radius of the silver cluster islands is 18 nm, being in good agreement with the results of our calculations as seen from Fig. 7.9d. Note that the width and the position of the maximum in the calculated distributions shown in Fig. 7.9 are rather close to the experimentally observed ones while the absolute value of the experimental and theoretical distributions differ quite significantly. This happens because in the discussed case study we analyze the dynamics

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of a single fractal, while the experimental measurements deal with many fractals on a surface.

7.3 Thermally Induced Morphological Transition of Silver Fractals We have just discussed that the nanofractal morphologies are not stable but evolve with time and are subject to the conditions at which fragmentation occurs [52]. The understanding of the post-growth relaxation processes allows one to govern the selforganization processes of particles on a surface for the purpose of obtaining patterns with predictable properties. Possible instability arising in fractals’s branches is similar to the Rayleigh instability in liquid cylindrical shapes [52, 88, 89]. Studies related to this general and fundamental phenomenon address various disciplines (e.g., hydrodynamics, solidstate physics, and materials science) and have been applied to systems characterized by different length scales, ranging from the macroscopic to the nanoscale [52, 88, 89]. The fundamental questions addressed by these studies concern issues of the interplay and balance governing the bulk and surface contributions to the energetics of materials, their morphological stability, and the relaxation dynamics in nonequilibrium complex systems. This knowledge has important technological applications concerning the formation and control of materials with tailored properties, such as nanowires [90–92], thin films [93, 94], and other surface-supported structures. In this section, we discuss experimental and theoretical study of a particular structural and dynamical transformation of non-equilibrium fractal shapes, which we call a morphological transition. The concrete case study again deals with silver fractals created on graphite surfaces, however, in this case, we consider deposition and dynamics of Ag800 nanoparticles instead of Ag500 to be consistent with experiment [9, 52].

7.3.1 Experimental Observation and Characterization of Morphological Transition Let us first give a short summary of the experimental observations of the morphological transition as reported in [9]. Gas-phase neutral silver fractals can be produced by a gas aggregation technique [62], where silver nanoparticles are deposited at low impact energy (0.05 eV/atom), on a cleaved graphite surface maintained in vacuum. Different cluster deposition fluxes F can be reached typically in a range between 1010 to 1011 clusters/cm2 /s at different coverage as explained in an earlier publication [55]. Since the per-atom kinetic energy of the incident clusters is very low compared with their binding energy (1.2 eV per atom), the clusters migrate on the surface as

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Fig. 7.10 Time-temperature (T, t) diagram. The diagram illustrates the choice of temperature and temporal conditions at which experiments have been performed. Red curve corresponds to the measurement regime at a fixed temperature T1 with different times of heating. Blue curve corresponds to the measurement regime performed with a fixed period of time t = t1 − t0 for different temperatures T . In all the cases, the sample is heated up to a given temperature during the initial period of time t0 . Reproduced from [9] with permission from John Wiley & Sons

a whole and grow into islands through diffusion limited aggregation (DLA) process into islands exhibiting fractal morphologies. After deposition, the sample is heated in vacuum by Joule effect, where the heat is produced by an electrical current flowing through the samples holder. The measured temperature T of the holder equates the temperature of the sample. The sample is heated from the room temperature T0 up to a temperature T during initial period of time t0 , as shown in Fig. 7.10. Then the sample is kept at this fixed temperature for a given period of time t and cooled down. The periods of heating and cooling down are typically shorter than the explored periods of annealing. During the interval t0 , no significant changes in the fractal morphology occur, although they are chosen to be sufficient for the fractal equilibration to the temperature of annealing. After cooling down the sample is transferred in air and imaged in a scanning electron microscope (SEM). The advantages of this annealing method allow one to precisely monitor the heating process and observe the structural transformations that take place during annealing, by interrupting the thermal treatment at any desired point, leading to samples with controlled and reproducible nanostructures. Several temperatures and annealing times were investigated. Thus, each measurement is characterized by the parameters (T, t). The range of studied temperatures was chosen such that evaporation of individual silver atoms from clusters could be

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Fig. 7.11 Morphological evolution of silver nanofractal islands during annealing. Characteristic SEM images of silver nanofractal islands (a) and their morphological evolution during annealing at T1 = 555 K for t = 5 min (b), 15 min (c), 30 min (d), 60 min (e) and 120 min (f). Reproduced from [9] with permission from John Wiley & Sons

neglected, i.e., we have considered temperatures below ∼700 K. For analysis, the morphology of the studied samples was compared to the morphology of the sample produced under the same deposition conditions and kept at room temperature T0 . The latter fractal morphology was used as a reference as it corresponds to the initial morphology prior to the annealing process. For the reference sample kept at T0 = 300 K, the initial width of fractal branches ξ0 = 12 nm does not change during the whole duration of the experiment lasting typically from minutes up to several hours. For the annealed samples, the initial branch width does not change much during the initial annealing periods t0 , being ∼1 min, see Fig. 7.10. The consideration of the branch width evolution with respect to the initial branch width for every (T, t) enables the reduction of the results’ dispersion, since ξ0 can vary in different measurements due to flow dispersion. Figure 7.11 shows SEM images corresponding to different times of the annealing process. The performed experimental analysis shows that at the given temperature each fractal shape experiences fragmentation into a number of islands located randomly in the vicinity of the original core of the fractal. On the other hand, the experimental results suggest that after a certain time interval, sufficient for fractal fragmentation, the observed fragmentation pattern strongly depends on the temperature at which the evolution process takes place. Notably, the time of such transformation becomes shorter with an increase in the temperature. The dynamical evolution of fractal shapes can be described through various characteristics. For instance, these could be the fractal dimension, or the normalized width of the fractal branches. These characteristics experience a variation with respect to time and temperature at which the fractal fragmentation process is observed. We call

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Fig. 7.12 SEM images of nanofractals at several annealing temperatures. Characteristic SEM images of nanofractals obtained by a low flux silver cluster deposition of 4 nm diameter just after deposition (a) and after t = 2 h annealing under ultra-high vacuum at several temperatures T : 490 K (b), 555 K (c) and 685 K (d). Reproduced from [9] with permission from John Wiley & Sons

the dynamical evolution of nanofractal shapes, leading to their fragmentation, as the morphological transition. Figure 7.12 shows images of fractal morphology obtained just after deposition (reference sample) and after 2 hours annealing under vacuum at several temperatures: 490, 555, 685 K. The figure illustrates that smooth evolution of the fragmented fractal shapes changes at temperatures ∼700 K at which, due to the higher mobility of deposited nanoparticles, the small islands disappear and the bigger ones emerge. After annealing at T  700 K, the recrystallization occurs during sample downcooling, as seen in Fig. 7.12d. Below, we do not discuss such transformations and focus mainly on the characterization of the morphological transition of fractal shapes occurring at lower temperatures T  700 K.

7.3.2 Theoretical Description of Morphological Transition As seen from the experimental data, the nanofractal morphological transition involves dynamics of enormous number of atoms and occurs on the time scales which are well beyond the nowadays limits for classical MD simulations. However, the fractal dynamics can be studied through the multiscale approach, in which the dynamics of the whole system is reduced to the random dynamics of constituent nanoclusters. The fractal structure shown in Fig. 7.2B was now used to investigate the thermally induced morphological transition [9]. It should, however, be noted that for a better comparison with experiment, this fractal was assumed to be formed from Ag800 nanoclusters instead of the Ag500 discussed above. The change of the constituent particles does affect the overall size of the fractal, as well as the kinetic parameter values. The fractal morphology can be characterized by various quantities. The average diameter of the fractal is equal to 900 nm, being close to that of the initial experimentally grown fractal structures [52, 57, 62] shown in Figs. 7.11a and 7.12a. Let us, however, note that the average branch width of the simulated fractal is equal to

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ξ0th = 4.16 nm. This number is smaller than the corresponding experimental value ex p ξ0 = 12 nm, being although sufficient for the simulation of the fractal morphological transition and the elucidation of its general features at the less computational costs. Let us now estimate and compare the characteristic times of the peripheral diffusion τ pd = 1/  pd and the particle detachment τde = 1/ de from an island. From Eqs. (7.10) and (7.11), one derives       Eb E b + μ Eb τde 2 = e · exp ≈ 7.39 · exp

1 ∼ exp τ pd kT kT kT

(7.30)

Here, we have used the estimate [l − (m + n/5)]E b /kT ∼ E b /kT , where l ∼ 2, m ∼ 1 and n ∼ 1 are the characteristic average numbers of l, m and n in the course of particle dynamics. The change of the chemical potential μ is assumed fixed, equal to 2 kT [9]. This estimate shows that the characteristic time of particle detachment from an island exceeds significantly the characteristic time of the particle peripheral diffusion, which can be written as follows  τ pd (T, E a , E b ) = t (T, E a ) · exp

 λE b , kT

(7.31)

where λ is a dimensionless model constant of the order of unity. Let us now consider the important instant τ0 (T, E a , E b ) of the fractal evolution in the course of the morphological transition at which the first breaks in fractal branches arise. The estimate performed in Eq. (7.30) shows that these breaks are caused mainly by the particle peripheral diffusion. Therefore, it is natural to assume that such rearrangements of the fractal morphology occur after a certain number of periods of time τ pd (T, E a , E b ), i.e. τ0 (T, E a , E b ) = A · τ pd (T, E a , E b ) ≡

  A E a + λE b , · exp 0 kT

(7.32)

where T is the temperature at which the morphological transition occurs and A is a dimensionless model constant, which may depend on the fractal morphology, the width of the fractal branches, the number of constituent particles, but not on the variables which characterize the system dynamics, such as the activation energies, particle attempt escape rates, temperature of the system etc. 0 is the attempt escape rate defined in Eqs. (7.12) and (7.13). Now let us assume that at a given temperature T0 the time τ0 (T0 , E a , E b ) is known either from experiment or from simulations. Then, one can rewrite Eq. (7.32) in the form convenient for the evaluation of the corresponding energy E b from the known E a and τ0 (T0 , E a , E b )   Ea kT0 τ0 (T0 , E a , E b )0 − Eb = · ln . λ A λ

(7.33)

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Now, assuming that the time instant τ0 (T1 , E a , E b ) is also known at temperature T1 , one can express the ratio 0 /A through the known values as 1/(T1 −T0 )  0 τ0 (T0 , E a , E b )T0 . = A τ0 (T1 , E a , E b )T1

(7.34)

The evolution of a fractal shape can be characterized by the variation of the fractal dimension. As seen from Fig. 7.11 the fractal dimension of the initial pattern changes in the course of the morphological transition. The evolution of the fractal shapes begins with the value of d f = 1.7, which grows up approaching its limit d f → 2, being the characteristic value for droplet-like shapes. The morphological transition corresponds to the period of evolution accompanied by the significant and systematic variation of fractal shape, its fragmentation into many smaller islands, and their subsequent evolution towards the spherical-like droplet shapes. This period of the system evolution is characterized by relatively rapid variation of the fractal dimension at the initial stages and its little variation at the later stages, when it approaches the asymptotic value. It is worth noting that the evolution of the system does not stop even at this moment and is being continued. At this latest stage of the evolutionary process, the detachment and particle exchange between sphericallike droplets in the system play the most important role. This stage of the evolution scenario is similar to the process known as the Ostwald ripening [95, 96]. The fractal shapes and their morphological evolution can also be characterized by other morphological parameters different from the fractal dimension. Thus, the width of fractal branches turns out to be a rather convenient morphological parameter for the characterization of the fractal pattern evolution on a surface [52]. This parameter can be introduced through the ratio between the area S and the perimeter P of a structure on a surface. The average S/P ratio for a pattern consisting of N islands is defined as N 2  Si , (7.35) ξ  ≡ 2S/P = N i=1 Pi where Si and Pi are the area and the perimeter of the ith island. The morphological parameter ξ  defined in Eq. (7.35) characterizes the width of fractal branches. Being applied to spherical-like islands arising at the latest stages of the fractal evolution, the parameter ξ  gives an estimate of the radius of such islands, which has tendency to grow up with time. Therefore, it is very convenient for the characterization of both initial and the final stages of the fractal morphological transition. The parameter ξ  has been used for the characterization of both the experimentally observed morphological transition of the nanofractal shapes and the simulated evolution of nanofractal structures. It is also possible to estimate the instant τ1 (T, E a , E b ) at which the fractal morphological transition ends and the ripening regime in the evolution of the system morphology starts to prevail. This instant should be proportional to the characteristic time of particle detachment from an island, which can be estimated similarly to

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Eq. (7.31), as follows:  λ1 E b + μ , τde (T, E a , E b ) = t (T, E a ) · exp kT 

(7.36)

where λ1 is a dimensionless model constant, describing an average number of neighbors in the detachment process. This number should be within the range between 1 and 2. From Eq. (7.36), one estimates τ1 (T, E a , E b ) as:   (λ1 − λ)E b + μ B . τ1 (T, E a , E b ) = B · τde (T, E a , E b ) ≡ τ0 (T, E a , E b ) · · exp A kT

(7.37) From Eq. (7.37) with λ1 >λ, B>A and μ = 2 kT, one derives τ1 (T, E a , E b ) τ0 (T, E a , E b ). This means that there is a sufficiently long time interval of the evolution during which the fractal morphological transition occurs. Let us now apply the outlined formalism for the analysis of the experimental data presented in the previous section. For the easier reading, we have compiled the values of all essential discussed parameters in Table 7.1. They are divided into two major categories: (i) parameters and characteristics relevant to (or obtained from) experiment and (ii) parameters and characteristics relevant to (or obtained from) simulations, if they are different from category (i). The experiments reported in the previous section deal with Ag800 clusters. Being soft-landed on the graphite surface such clusters become d0 = 4 nm in diameter. The diffusion coefficient of a single such cluster can be derived through the analysis of mobility of the cluster on a surface, as illustrated in Fig. 7.2B. Alternatively, the diffusion coefficient can be experimentally measured through the scanning electron microscopy technique. Thus, the diffusivity parameters for Ag clusters on Ag(100) [97], Re clusters on Re(0001) [98], Rh clusters on Rh(100) [99], Ir clusters on Ir(111) [100–102], and metal atoms on W(211) [103] were established. The value of the diffusion coefficient for the Ag500 cluster on graphite at room temperature was estimated as D Ag500  2 · 10−7 cm2 /s [57]. Rescaling this value to Ag800 [104], one derives D Ag800  1.25 · 10−7 cm2 /s. For other metal clusters, such measurements have been also performed. For example, the diffusion coefficients for Au250 , Sb2300 on graphite at 300 K are equal to 4 · 10−6 cm2 /s, 1.7 · 10−8 cm2 /s correspondingly [105]. Knowing D and d0 , one can determine the time step t in the stochastic dynamics for the Ag800 cluster. Thus from Eq. (7.8), one derives t = 3.2 · 10−7 for T = 300 K. The temperature dependence of the diffusion coefficient and the rate d is determined by the energy E a as seen from Eq. (7.1). From the measurements of silver clusters diffusion over graphite surfaces, the energy E a has been evaluated as E a = 0.12 eV. Note that this value may depend on the graphite crystalline plane over which the cluster diffusion takes place and its quality. We do not analyze these dependencies and choose the parameter E a as a characteristic value for the silver cluster diffusion over graphite. The values of E a and d at room temperature for a cluster of diameter

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Table 7.1 Parameters and characteristics relevant to both the experimental and computational case studies of the morphological transition of silver fractals Parameters and characteristics relevant to (or obtained from) experiment Initial fractal geometry ex p Fractal dimension d f

Figs. 7.11a and 7.12a 1.7 ± 0.1

Radius d0 of Ag800 on graphite 4 nm Chemical potential μ 2 kT ex p Initial average width of fractal branches ξ0 = 12 nm Diffusion coefficient D Ag800 at T = 300 K 1.25 · 10−7 cm2 /s Time step t in stochastic dynamics at T = 300 K 3.2 · 10−7 s Energy E a 0.12 eV Constant 0 in Eqs. (7.32) and (7.12) 3.242 · 108 s−1 Pre-exponential constant D0 in Eq. (7.9) 1.297 · 10−5 cm2 /s Experimental value of ratio 0 /Aex p 4.605 · 105 s−1 s−1 Dimensionless constant Aex p 703.954 Dimensionless constant λ from Eq. (7.32) 1.15 Dimensionless constant λ1 from Eq. (7.37) 1.4 Dimensionless constant B ex p 1000 Energy E b 0.58 eV Energy  E b /5 Parameters and characteristics relevant to (or obtained from) simulations Initial fractal geometry Fig. 7.2B Number of particles in the fractal 5182 Fractal dimension d th 1.76 f Initial average width of the fractal branches Dimensionless constant A in Eq. (7.32) Dimensionless constant B in Eq. (7.37) Total number of steps in the simulations Nstep

ξ0th = 4.16 nm 5 7 235,920,000

d0 determine the pre-exponential constants 0 = ν1 = 3.242 · 108 s−1 in Eq. (7.32) and D0 = 1.297 · 10−5 cm2 /s in Eq. (7.9). According to Eqs. (7.1), (7.7), and (7.8), the time step t grows exponentially with the temperature decrease. Therefore, the time duration of the evolutionary process covered by the stochastic dynamics simulations at a given number of steps depends on the temperature at which the simulation is performed. The simulations of the fractal fragmentation dynamics have been carried out at different temperatures but at the given total number of time steps Nstep = 235,920,000. The total number of steps t defines the maximum time of the fractal evolution achieved in the simulation, see Eq. (7.19). Let us consider the numbers relevant to the experiments described in the previous section. In this case, the values of temperatures and times are as follows: T0 =

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300 K, T1 = 555 K, τ0 (T0 , E a , E b ) = 365 · 24 · 60 · 60 = 3.1536 · 107 s (one year), τ0 (T1 , E a , E b ) = 30 s. With these values, one derives from Eq. (7.34) 0 /Aex p = 4.605 · 105 s−1 . From Eq. (7.34) and the value of 0 obtained above, one derives the dimensionless constant Aex p = 703.954. Equation (7.33) can be used for the evaluation of the energy E b from the experimental measurements of the parameters of the morphological transition. The value of E b and the kinetic parameter λ ∼ 1 can be determined from best fit of τ0 (T, E a , E b ) given by Eq. (7.32) to the corresponding experimentally observed instants of appearance of the first breaks in the fractal branches. At T = 555 K they appear after t ∼ 1 min as seen from the experimental data presented in Fig. 7.11a, where some breaks are clearly visible. This analysis suggests the values E b = 0.58 eV and λex p = 1.15. The values E a , E b and the ratio 0 /A determine τ0 (T, E a , E b ) at arbitrary temperature T . This dependence is shown in Fig. 7.13a by red line. Blue line in Fig. 7.13a corresponds to τ1 (T, E a , E b ) defined by Eq. (7.37). It represents the instants in the (t; T ) diagram at the vicinity of which the morphological transition of fractals comes

Fig. 7.13 Morphological transition as seen in experiment. In panel a, red and blue lines feature τ0 (T, E a , E b ) and τ1 (T, E a , E b ) defined by Eqs. (7.32) and (7.37) correspondingly. These lines were rendered with the values of E a and E b presented in Table 7.1. They define the borders of the region in which the morphological transition occurs. Structure insets show the experimentally measured patterns characteristic for the regions of stability and ripening. Stars indicate the points (T, t) for which the experimental patterns presented in Figs. 7.11 and 7.12 have been captured. Green and magenta arrows indicate temperature and time intervals within which the evolution of the fractal structures with temperature and time has been recorded respectively. Panels b and c show the variation of the normalized average area-to-perimeter ratio ξ  with respect to temperature and time. They correspond to the green and magenta arrows indicated in panel a. In panel b, the normalized constant ξ0 = 12 nm corresponds the fractal branch width at T0 = 300 K. In panel c, the normalized constant ξ1 ≈ ξ0 = 12 nm corresponds the area-to-perimeter ratio at (T, t1 ) = (555 K, 2 min). The dashed line in panel a limits from above the area which could be covered by the simulations. Reproduced from [9] with permission from John Wiley & Sons

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to an end and turns into the droplet ripening regime. The parameters B ex p = 103 and λ1 = 1.4 entering Eq. (7.37) have been chosen such as to achieve the best fit of τ1 (T, E a , E b ) to the corresponding experimentally observed instants of the morphological transition. Structure insets in Fig. 7.13a show the experimentally measured patterns characteristic for the regions of stability and ripening, the dashed line limits from above the area which could be covered by the stochastic dynamics simulation with the chosen number of steps Nstep . Stars in Fig. 7.13a indicate the points (T, t) for which the experimental patterns presented in Figs. 7.11 and 7.12 have been captured. Green and magenta arrows indicate temperature and time intervals within which the evolution of the fractal structures with temperature and time has been recorded respectively. Figure 7.13b, c shows the measured variation of the normalized average areato-perimeter ratio ξ  with respect to temperature and time. They correspond to the green and magenta arrows indicated in Fig. 7.13a. In Fig. 7.13b, the normalized constant ξ0 = 12 nm corresponds the fractal branch width at T0 = 300 K. The time instant at which this value was measured is not important, as the fractal stays stable at room temperature for about a year. Now let us present the results of stochastic dynamics simulations of thermally induced fractal morphological transition and their analysis performed with the use of the above described theoretical model. The parameters 0 , D0 , E a , E b , μ, , λ1 , and λ2 that have been used in the simulation are the same as discussed above in connection with the experimental observation of the morphological transition and summarized in the upper part of Table 7.1. Parameters that are specific for the simulation are summarized in the lower part of Table 7.1. Some of these parameters concern the geometry of the fractal chosen for simulations, e.g., the total number of particles in the fractal, the fractal dimension and the initial average width of the fractal branches ξ0th . Table 7.1 also includes the empirical constants A and B obtained from fitting of τ0 (T, E a , E b ) and τ1 (T, E a , E b ) defined by Eqs. (7.32) and (7.37) to the corresponding instants of the simulated morphological transition, and the total number of steps in the simulation. Although the fractal dimension of the simulated fractal is close to the experimental value the initial average width of the fractal branches ξ0th = 4.16 nm is significantly smaller than that in experiment. The smaller width of fractal branches leads to their faster fragmentation. The faster fragmentation of the fractal branches results in the smaller values of the empirical constants A and B as compared to those relevant to experiment. The faster fragmentation of the fractal allowed us to simulate the morphological transition in the fractal up to the stages when ripening regime starts. Also, we extended our simulations towards the region of higher temperatures, where all the processes in the system (morphological transition, ripening, etc) proceed faster. Figure 7.14 illustrates the time evolution of the silver fractal structure shown in Fig. 7.2B at different temperatures within the range from 300 up to 1000 K. The chosen temperature range covers the temperatures relevant to the experiment extending it to the region of higher temperatures, where the morphological transition goes faster. The yellow region in Fig. 7.14 corresponds to the time and temperature ranges within which the morphological transition occurs. The regions of stability

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Fig. 7.14 Simulated evolution of fractal structure. The yellow region corresponds to the time and temperature ranges within which the morphological transition occurs. The regions of stability and ripening are shown by green and red, respectively. The red and blue lines feature τ0 (T, E a , E b ) and τ1 (T, E a , E b ) defined by Eqs. (7.32) and (7.37) correspondingly. These lines were rendered with the values of the parameters presented in Table 7.1. The evolution of the fractal morphology is illustrated at T = 1000 K and for t = 7.5 · 10−3 s. Red dots indicate the points (T, t) to which the presented patterns correspond. The dashed line limits the area covered by the performed simulation. Reproduced from [9] with permission from John Wiley & Sons

and ripening are colored green and red respectively. The red and blue lines feature τ0 (T, E a , E b ) and τ1 (T, E a , E b ), defined by Eqs. (7.32) and (7.37) correspondingly, were rendered with the values of parameters presented in Table 7.1. Figure 7.14 illustrates the evolution of the fractal morphology by snapshots of the fractal morphology taken at the fixed temperature T = 1000 K in different moments of time and at the fixed instant of evolution t = 7.5 · 10−3 s at different temperatures. Red dots indicate the points (T, t) to which the presented snapshots correspond. The dashed line limits the area covered by the performed simulation. The period of the fractal degradation depends on the system temperature. At T = 1000 K, the fractal structure melts completely during t ∼ 10−2 s, while the fractal structure exhibits very slow perturbations at T = 300 K evolving towards fractal fragmentation and subsequent ripening during the time periods lasting for many days. Such behavior explains why this simulation of these processes becomes increasingly expensive at lower temperatures. Indeed, in spite of the fact that the time step in the simulation increases at lower temperatures, this rise is exponentially less significant as compared to the growth of the characteristic times τ0 (T, E a , E b ) and τ1 (T, E a , E b ) of the morphological transition as easy to figure out from the

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Fig. 7.15 Time and temperature evolution of the average area-to-perimeter ratio ξ . Time evolution of ξ  calculated for the fractal shown in Fig. 7.2B fragmenting on a surface at different temperatures (a). Parameters at which the simulation is performed are given in Table 7.1, the reference parameters for the plot are ξ0 = ξ1 = 4.16 nm and t1 = 10−5 s. Normalized ξ /ξ0 calculated at the time instance t = 0.22 s (b). Time evolution of the calculated normalized ratio ξ /ξ1 at a fixed temperature T = 1000 K (c). Reproduced from [9] with permission from John Wiley & Sons

comparison of the corresponding equations. It is also seen from the comparison of the behavior of the dashed line, indicating the limiting values of time at each given T in the performed simulation, with the behavior of τ0 (T, E a , E b ) and τ1 (T, E a , E b ) curves plotted in Fig. 7.14. The evolution of the fractal morphologies shown in Fig. 7.14 clearly indicates the morphological transition in the region of times t > τ0 (T, E a , E b ) within the temperatures range covered by the simulation. In this region, the fractal structure and its geometrical characteristics begin to experience significant morphological change leading to the multifragmentation of the fractal into a large number of islands and the their further evolution at t > τ1 (T, E a , E b ) towards the spherical shapes (ripening process). The experimental results presented in Fig. 7.11 as well as the simulation results shown in Fig. 7.14 indicate that the width of the fractal branches characterized by the parameter ξ , introduced in Eq. (7.35), grows with time at any given temperature. Below we simulate this behavior and compare it with the experimental observations. The dependence of the average area-to-perimeter ratio ξ  for a fragmenting fractal as a function of time t and temperature T calculated within the interval of temperatures 300 K ≤ T ≤ 1000 K is shown in Fig. 7.15 as a 2D surface. Figure 7.15 shows that the ratio ξ  growing with time at each given temperature experiences a rapid rise at a certain interval of times τ0 (T, E a , E b )  t  τ1 (T, E a , E b ). Figure 7.15 features the dependence of the simulated ξ /ξ0 ratio with time and temperature. In order to compare the topology of the simulated surface with exper-

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iment, we analyze ξ /ξ0 at fixed time (t = 0.22 s) and temperature (T = 1000 K) instances. The simulated behavior of ξ /ξ0 as a function of temperature and time agree reasonably well with the experimentally observed dependencies presented in Fig. 7.13, however, certain discrepancies should be noted. In particular, (i) the variation of ξ /ξ0 in the simulation is somewhat smaller than observed experimentally and (ii) the time scale of the simulated morphological transition is shorter than in experiment. These differences arise due to the smaller fractal branch width in simulations as compared to that in the experiment, as seen from Table 7.1. Figure 7.15b shows the dependence of the simulated ξ /ξ0 as a function of temperature at t = 0.22 s. The shape of the curve mimics the experimentally observed dependence presented in Fig. 7.13b. The interval of temperatures in which the morphological transition takes place is practically the same in both cases. At higher temperatures, both curves saturate at a plateau indicating the end of the morphological transition. Thus, the value of ξ /ξ0 at the plateau is the characteristic quantity for the morphological transition. Comparison of these quantities derived from the experiment and the simulation shows that the experimental value of ξ /ξ0 is 2.3 times larger than the simulated one as is expectable for a fractal with 2.86 times wider branches, see Table 7.1. The further rise of the curve in Fig. 7.13b at temperatures above 700 K indicates the start of the droplet ripening regime. Figure 7.15c shows the dependence of the simulated ξ /ξ0 as a function of time at T = 1000 K. This temperature is higher than T = 555 K at which the corresponding time evolution of fractals has been measured. However, due to the fact that at high temperatures, the morphological transition occurs relatively fast, we were able to simulate it up to the later evolutionary stages at which ripening process starts, as illustrated in Fig. 7.14. Therefore, the simulated time range in Fig. 7.15c covers the later stages of the fractal evolutionary scenario as compared to that measured in the experiment. Indeed, experimental points presented in Fig. 7.13a, c correspond to a relatively narrow time region within the morphological transition interval at the chosen temperature. Therefore, the experimental data shown in Fig. 7.13c are related to only a part of the curve Fig. 7.14 above the red line.

7.4 Surface-Assisted Chemical Transformation and Catalytic Reactions In this chapter, we have extensively reviewed the computational possibilities for modeling of surface deposition processes. It is, however, also important to stress that stochastic dynamics can be used to model surface-assisted chemical transformation and catalytic reactions. One possibility here is to consider fusion of particles in a system. This process is especially important for modeling synthesis reactions, where two particles are transformed into a new particle type upon a contact. In the terms of stochastic dynamics, such a process corresponds to two particles’ interaction and

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producing a new particle with a certain probability. It is possible to connect this probability with the parameters of the system, following the theory of chemical reactions and the principle of detailed balance [106]. The rate of a synthesis reaction involving two compounds with types i and j can be calculated as sr = ksr [i][ j],

(7.38)

where [i] and [ j] are the concentrations of each reactant and ksr is the so-called reaction rate coefficient or rate constant [106–108], although it is not really a constant, because it includes all the parameters that affect reaction rate, except for time and concentration. The rate constant can be derived for specific processes. In the case of chemical reactions in the gas phase, it can be obtained from the collision theory [108] and reads as   Eb , (7.39) ksr = Zρ exp − kT where E b is the activation energy of the reaction, k is the Boltzmann factor, T is the temperature, Zρ is the pre-exponential factor which has the meaning of the total number of collisions that collide with the right orientation. The collision frequency Z that has units of collisions per unit time can be calculated as [108]  Z = Ni N j σi j

8 kT . π μi j

(7.40)

Here, Ni and N j are the total numbers of particles of type i and j in the simulation box, the square root represents the mean velocity of molecules obtained from the Maxwell–Boltzmann distribution for thermalized gases, σi j is the averaged sum of the collision cross-sections of molecules of type i and j. The collision cross-section represents the collision region presented by one molecule to another, μi j in Eq. (7.40) is the reduced mass of two colliding particles. Once the rate of a synthesis reaction for a specific particle type pair is known, it is then possible to calculate the probability of this reaction to occur over one step of stochastic dynamics simulation as Psr = sr t. For the complete description of possible reactions happening in the course of particle deposition process, it is also important to include particle substitution and replacement reactions. These reactions occur when several particles interact and lead to products that in the most general case involve several new products. The rates of the substitution reactions can also be obtained for specific examples, for example in the case of chemical reactions, it can be derived through the collision theory [108]. This rate would then determine the probability of a substitution reaction over one simulation step. Substitution reactions within the framework of stochastic dynamics can be used to model single and double substitution reactions. Moreover, it can be used to model combustion reactions.

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7.5 Conclusion and Outlook This chapter overviews the state of the art computational approaches for multiscale modeling of surface deposition processes. It has a special emphasis on several case studies that were obtained using the program MBN Explorer [8], which permits simulations of multiscale physical, chemical, and biological processes in the 2D and 3D spaces. The chapter illustrates that surface deposition processes can be modeled nowadays on time scales comparable with those achieved in experiment through stochastic dynamics. An illustrative case study of silver nanoparticle self-organization on the surface was discussed. The simulations revealed that silver nanoparticles form islands that possess certain fractal properties. Although the chapter is focused extensively on one particular example, the described stochastic simulations can be used in versatile simulations. For example, it is possible to study system formation at nonequilibrium regime, such as thin films’ growth performed with a method of physical vapor deposition. Another interesting process to examine is the temperature-induced structure evolution, and specifically nanowire fragmentation, fractal branching, and defect formation in the time scale of a few minutes to hours. Finally, the method can be used to study various diffusion-driven processes occurring in both organic and inorganic environments, for instance, diffusion of nanoparticles in a biological cell, [109] as an important application in drug delivery studies. Despite universality, the discussed method of stochastic dynamics can also be developed further, to address even a broader spectrum of problems. Thus, for example, one can extend the description to account for chemical reactions and for structural changes of particles in the system. Another enhancement would be to introduce particle geometry reconfiguration and spatial rotation, which would allow for more realistic simulations of biomacromolecules. In this respect, the universality of the MBN Explorer package is crucial to permit further development of stochastic dynamics in the future.

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Chapter 8

Multiscale Modeling of Irradiation-Driven Chemistry Processes Gennady Sushko, Alexey V. Verkhovtsev, Ilia A. Solov’yov, and Andrey V. Solov’yov Abstract This chapter gives an overview of Irradiation-Driven Molecular Dynamics (IDMD)—the novel computational technique enabling atomistic simulations of the irradiation-driven transformations of complex Meso-Bio-Nano (MBN) systems exposed to various radiation modalities. Within the IDMD framework, various quantum processes occurring in irradiated systems are treated as random, fast, and local transformations incorporated into the classical MD framework in a stochastic manner with the probabilities elaborated on the basis of quantum mechanics. Major transformations of irradiated molecular systems (such as topological changes, redistribution of atomic partial charges, alteration of interatomic interactions) and possible paths of their further reactive transformations can be simulated by means of MD with reactive force fields, particularly with the reactive CHARMM (rCHARMM) force field implemented in the MBN Explorer software package. This chapter provides several exemplary case studies illustrating the utilization of IDMD. Particular examples include irradiation-induced chemical transformations (including fragmentation) of organic and biomolecular systems and controlled fabrication of nanostructures using the Focused Electron Beam-Induced Deposition (FEBID) technique.

8.1 Introduction There are numerous examples where chemical transformations of complex molecular systems are driven by irradiation. Often such modifications carry important outcomes to the functional properties of the irradiated molecular systems. Examples G. Sushko · A. V. Verkhovtsev · A. V. Solov’yov (B) MBN Research Center gGmbH, Altenhöferallee 3, 60438 Frankfurt am Main, Germany e-mail: [email protected] A. V. Verkhovtsev e-mail: [email protected] I. A. Solov’yov Department of Physics, Carl von Ossietzky Universität Oldenburg, Carl-von-Ossietzky-Str. 9-11, 26129 Oldenburg, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 I. A. Solov’yov et al. (eds.), Dynamics of Systems on the Nanoscale, Lecture Notes in Nanoscale Science and Technology 34, https://doi.org/10.1007/978-3-030-99291-0_8

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include the inactivation of living cells due to the irradiation-induced complex DNA strand breaks [1–3]; the formation of cosmic ices and dust in the interstellar medium and planetary atmospheres due to the interplay of the molecular surface adsorption and surface irradiation [4]; the formation of biologically relevant molecules under extreme conditions involving irradiation [5], and many more. Irradiation-driven chemistry (IDC) is nowadays utilized in modern nanotechnology, such as focused electron beam deposition (FEBID) [6–8] and ultraviolet lithography (UVL) [9, 10]. These technologies belong to the next generation of nanofabrication techniques, allowing the controlled creation of complex three-dimensional nanostructures with nanometer resolution that is attractive in basic and applied research. Fabrication of increasingly smaller structures has been the goal of the electronics industry for more than three decades and remains one of this industry’s biggest challenges. Furthermore, IDC is a key element in nuclear waste decomposition technologies [11] and radiation therapies of cancer [1, 3, 12]. IDC studies transformations of molecular systems induced by their irradiation with photon, neutron, or charged particle beams. IDC is also relevant for molecular systems exposed to external fields, mechanical stress, or plasma environment. A rigorous quantum mechanical description of the irradiation-driven molecular processes, e.g. within time-dependent density functional theory (TDDFT), is feasible but only for relatively small molecular systems containing, at most, a few hundred atoms [13–16]. This strong limitation makes TDDFT of limited use for the description of the IDC of complex molecular systems. Classical molecular dynamics (MD) could be considered an alternative theoretical framework for modeling complex molecular systems. For instance, by employing the classical molecular mechanics approach, it is feasible to study the structure and dynamics of molecular systems containing millions of atoms [17, 18] and evolving on time scales up to hundreds of nanoseconds [19–21]. In the molecular mechanics approach, the molecular system is treated classically, i.e., the atoms of the system interact with each other through a parametric phenomenological potential that relies on the network of chemical bonds in the system. This network defines the so-called molecular topology, that is, a set of rules that impose constraints on the system and permit maintaining its natural shape as well as its mechanical and thermodynamical properties. The molecular mechanics method has been widely used throughout the past decades and has been implemented, for instance, in the well-established computational packages CHARMM [22], AMBER [23], GROMACS [24], NAMD [25], and MBN Explorer [26]. Despite the manifold advantages, standard classical MD cannot simulate irradiation-driven processes as it typically does not account for coupling of the system to incident radiation, nor does it describe quantum transformations in the molecular system induced by the irradiation. These deficiencies have been overcome recently by introducing Irradiation-Driven Molecular Dynamics (IDMD) [27], a new methodology allowing atomistic simulation of IDC in complex molecular systems. The IDMD approach has been implemented in the MBN Explorer software package [26] capable of operating with a large library of classical potentials, manybody force fields, and their combinations. The reactive CHARMM (rCHARMM)

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force field [28, 29] implemented in MBN Explorer enables the description of bond rupture events and the formation of new bonds by chemically active atoms in the system, monitoring all the changes of the system’s topology that occur during its transformations. Chemically active atoms carry information about their partial charges, interactions with other atoms in the system, valences, and multiplicities of the bonds that can be formed with other reactive atoms in the system. Being an extension of the commonly used standard CHARMM force field [30–32], rCHARMM is directly applicable to organic and biomolecular systems. Its combination with other force fields [33] enables simulations of an even broader variety of molecular systems experiencing chemical transformations while monitoring their molecular composition and topology changes [27, 29, 33–37]. This chapter provides a brief overview of the IDMD methodology exploiting the rCHARMM force field and complements it with several illustrative examples of multiscale modeling of irradiation-driven transformations involving complex molecular systems.

8.2 Key Principles of Irradiation-Driven Molecular Dynamics IDMD methodology has been designed for the atomistic simulations of the IDC processes, and it is applicable to any molecular system exposed to radiation [27, 37– 39]. The general principles of IDMD have been described in Chap. 3. In this section, the most essential details of IDMD relevant for the atomistic simulations described further in this chapter are briefly reiterated. The IDMD methodology accounts for the major dissociative transformations of irradiated molecular systems and possible paths of their further reactive transformations [27] which can be simulated by means of MD with reactive force fields [28]. The necessary input parameters for such simulations can be elaborated on the basis of the quantum chemistry methods. IDMD simulations allow to account for the dynamics of secondary electrons and the mechanisms of energy transfer from the excited electronic subsystem to the system’s vibrational degrees of freedom, i.e., to its heat. For small molecular systems being in the gas phase, the ejected electrons can often be uncoupled from the system and excluded from the analysis of the system’s post-irradiation dynamics. For the extended molecular and condensed phase systems, the interaction of secondary electrons with the system can be treated within various electron transport theories, such as diffusion [2, 40] or Monte Carlo (MC) approach [41], and be considered as additional irradiation field imposed on the molecular system [37]. Such an analysis provides the spatial distribution of the energy transferred to the medium through irradiation. IDMD relies on several input parameters such as the bond dissociation energies, molecular fragmentation cross sections, amount of energy transferred to the system

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upon irradiation, energy relaxation rate, and spatial region wherein the energy is relaxed. These characteristics originating from smaller spatial and temporal scales can be obtained by accurate quantum mechanical calculations, e.g., by means of DFT and TDDFT, nonadiabatic MD, etc. IDMD approach enables to link outputs of numerous MC codes (e.g., Geant4 [42, 43]) simulating radiation and particle transport in different media with the inputs of IDMD and thus to achieve the multiscale description of irradiation-driven molecular dynamics, chemistry, and structure formation in many different MBN systems [37]. A similar methodology can be used for simulations of numerous molecular systems placed into radiation fields of different modalities, geometries, and temporal profiles. The IDMD framework provides a broad range of possibilities for multiscale modeling of the IDC processes that underpin emerging technologies ranging from controllable fabrication of nanostructures with nanometer resolution (see e.g. Refs. [8, 44] for FEBID and Refs. [9, 10] for UVL) to radiotherapy cancer treatment (see, e.g. [1–3]), both discussed further in this chapter. The IDMD algorithm has been validated through a number of case studies of collision and radiation processes including atomistic simulations of the FEBID process and related IDC [27, 37], collision-induced multifragmentation of fullerenes [33], electron impact-induced fragmentation of W(CO)6 [36], thermal splitting of water [28], radiation chemistry of water in the vicinity of ion tracks [34], DNA damage of various complexity induced by ions [29], and other [38]. Several case studies from this list are discussed in greater detail further below.

8.3 Generalization of Standard Molecular Force Fields for Reactive Molecular Dynamics MBN Explorer allows simulations of dissociation and formation of covalent bonds through the reactive CHARMM (rCHARMM) force field [28], which is an extension of the standard CHARMM force field [30–32]. This extension requires specification of two additional parameters that define the dissociation energy of a covalent bond and the cutoff radius for bond breaking or formation. By specifying the additional parameters for the bonded interactions, MBN Explorer considers all molecular mechanics interactions, i.e., bonded, angular, dihedral, using an alternative parametrization. If the distance between a given pair of atoms becomes greater than the specified cutoff radius, this particular bonded interaction is removed from the system’s topology and not considered in future calculations. The standard CHARMM force field [30] employs harmonic approximation for describing the interatomic interactions, thereby limiting its applicability to small deformations of the molecular system. In case of larger perturbations, the potential should decrease to zero as the valence bonds rupture. In order to permit rupture of covalent bonds in a molecular mechanics force field, MBN Explorer uses a

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modified interaction potential describing the interaction of atoms connected by chemical bonds. The standard CHARMM force field describing covalent bonds is defined as (8.1) U b (ri j ) = kibj (ri j − r0 )2 , where kibj is the force constant of the bond stretching, ri j is the distance between atoms i and j, and the parameter r0 is the covalent bond length. This parametrization describes well the bond stretching regime in the case of small deviations from r0 but gives an erroneous result for the larger distortions. For a satisfactory description of the covalent bond rupture, it is reasonable to use the Morse potential. This potential requires one additional parameter if compared to the aforementioned harmonic potential; this parameter accounts for energy of the bond dissociation. For a pair of atoms, the Morse potential reads as   U b (ri j ) = Di j e−2βi j (ri j −r0 ) − 2e−βi j (ri j −r0 ) ,

(8.2)

1/2  where Di j is the dissociation energy of the covalent bond and βi j = kibj /Di j determines the steepness of the potential. Figure 8.1 illustrates the Morse potential for the CN7–CN8B bond that is one of the covalent bonds in the DNA backbone; for small deviations from r0 the Morse potential and the harmonic approximation are close to each other.

Fig. 8.1 The pairwise carbon–carbon (type CN7–CN8B) interaction potential in harmonic approximation (8.1) and modeled with the Morse potential (8.2)

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The rupture of covalent bonds in the course of simulation automatically employs an improved potential for the valence angles. In the CHARMM force field, the potential associated with the change of a valence angle between bonds with indices i j and jk reads as (8.3) U a (θi jk ) = kiθjk (θi jk − θ0 )2 , where kiθjk and θ0 are parameters of the potential and θi jk is the actual value of the angle formed by the three atoms. This potential grows rapidly with increasing the angle and it may lead to non-physical results when modeling the covalent bond rupture. In order to avoid such cases, the harmonic potential, Eq. (8.3), is substituted in the modified force field with an alternative parametrization, which reads as   U a (θi jk ) = 2kiθjk 1 − cos(θi jk − θ0 ) .

(8.4)

At small variations of the valence angle, this parametrization is identical to the harmonic approximation (8.3) used in the standard CHARMM force field. For larger values of the angle, the new parametrization (8.4) defines an energy threshold which becomes important for accurate modeling of bond breakage. The rupture of a covalent bond is accompanied by ruptures of the angular interactions associated with this bond. The effect of bond breakage on the angular potential can be described through a smoothed step function σ (ri j ) defined as σ (ri j ) =

 1 1 − tanh(βi j (ri j − ri∗j )) , 2

(8.5)

with ri∗j = (RivdW + r0 )/2. This function introduces a correction to the angular interj action potential, assuming that the distance between two atoms involved in an angular interaction increases from the equilibrium value r0 up to the van der Waals contact value R vdW . Since an angular interaction depends on two bonds connecting the atoms with indices i j and jk, the potential energy describing the valence angular interaction that is subject to rupture is parameterized as U˜ a (θi jk ) = σ (ri j ) σ (r jk ) U a (θi jk ) .

(8.6)

As seen from Eq. (8.6), the angular potential decreases with the increase of the bond length between any of the two pairs of atoms i j or jk. As an illustration, Fig. 8.2A shows the CN8B–ON2–P angular potential which arises, for instance, when modeling DNA nucleotides. The presented angular potential is calculated using Eq. (8.6) assuming the breakage of the bond between the oxygen and the phosphorous atoms. For the sake of illustration, the CN8B–ON2 bond length in this case is taken equal to its equilibrium value r0 . Dihedral interactions arise in the conventional molecular mechanics potential due to the change of the dihedral angles between every four topologically defined atoms. Let us consider a quadruple of atoms with indices i, j, k, and l, bound through an interaction which is governed by a change of the dihedral angle. In this case, the

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(B)

Fig. 8.2 A The CN8B–ON2–P angular potential calculated using Eq. (8.6) with account for the ON2–P bond rupture. B The CN4–P–ON2–CN7 dihedral potential calculated using Eq. (8.8) with account for the ON2–P bond rupture

dihedral angle stands for the angle between the plane formed by the atoms i, j, and k, and the plane formed by the atoms j, k and l. In the harmonic approximation, the dihedral energy contribution reads as:    Uidjkl = kidjkl 1 + cos n i jkl χi jkl − δi jkl ,

(8.7)

where kidjkl , n i jkl , and δi jkl are parameters of the potential, and χi jkl is the angle between the planes formed by atoms i, j, k and j, k, l. The dihedral interactions also become disturbed upon covalent bond rupture; therefore, Eq. (8.7) should be modified to properly account for this effect. The rupture of a dihedral interaction between a quadruple of atoms i, j, k, and l should take into account three bonds that contribute to this interaction. Thus, the potential energy describing the dihedral interaction with account for the bond rupture reads as U˜ idjkl = σ (ri j ) σ (r jk ) σ (rkl ) Uidjkl ,

(8.8)

where Uidjkl is the potential in Eq. (8.7) describing the dihedral interaction within the framework of the standard CHARMM force field. The functions σ (ri j ), σ (r jk ),and σ (rkl ) are defined by Eq. (8.5); they are used to limit the dihedral interaction upon increasing the corresponding bond length. Figure 8.2B shows a profile of a CN4– P–ON2–CN7 dihedral interaction potential with accounting for the rupture of the central ON2–P bond. This dihedral interaction is also important for modeling bond breakages in DNA nucleotides. Establishing force field parameters through quantum mechanical calculations. The rCHARMM parameters (particularly dissociation energy and cutoff distance for bond breaking and formation) are commonly determined from density functional theory (DFT) calculations. In Ref. [36], rCHARMM was employed for modeling irradiation-driven fragmentation of W(CO)6 molecule, one of the widely used

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(C)

+ Fig. 8.3 Panel A Potential energy curves for W–C and C–O bonds in the W(CO)+ 6 and WC(CO)5 molecules. Symbols show the results of DFT calculations performed using the B3LYP functional and a LanL2DZ/6-31+G(d,p) basis set. Lines show the fit to these data with the Morse potential, + Eq. (8.2). Panels B and C show, respectively, the optimized structures of W(CO)+ 6 and WC(CO)5 . In the latter case, three different types of W–C bonds (labeled as (1), (2) and (3)) appear after one oxygen atom has been removed. Reproduced from Ref. [36] with permission from Springer Nature

precursors for FEBID. DFT calculations were conducted to determine the rCHARMM for neutral and singly charged W(CO)6 molecules and their fragments; the results obtained were benchmarked against experimental data [45]. The DFT calculations were performed employing the B3LYP exchange–correlation functional and a mixed LanL2DZ/6-31+G(d,p) basis set, wherein the former set described the W atom and the latter was applied to C and O atoms. As the simulations conducted in Ref. [36] aimed to reproduce appearance energies from electron-impact ionization experiments, singly charged parent molecule and molecular fragments were considered. Geometry of each molecule was optimized first and a potential energy surface scan was then performed for different W–C and C–O bonds to calculate equilibrium bond lengths, dissociation energies, and force constants. Atomic partial charges were obtained through the natural bond orbital analysis [46]. To benchmark the methodology, equilibrium bond lengths in a neutral W(CO)6 molecule, for which there is plenty of reference data, were analyzed first. The calculated values, r0W−C = 2.07 Å, r0C−O = 1.15 Å, are in good agreement with experimental data [47] and with the results of earlier DFT calculations [48]. The chosen exchange-correlation functional and the basis set were then used for DFT calculations of a singly charged molecule. The calculated dissociation energies of W–C bond in + the parent cation W(CO)+ 6 and in different fragments such as W(CO)n (n = 1 − 5) + and WC(CO)n (n = 0 − 5) are in good agreement with the electron-beam mass spectrometry results of Ref. [45] with the relative discrepancy of a few kilocalories per mole. Figure 8.3A shows potential energy curves for W–C and C–O bonds in the parent + W(CO)+ 6 molecule as well as in WC(CO)5 formed upon removal of an oxygen atom from one of the ligands. Optimized structures of these molecules are shown in Fig. 8.3B and C, respectively. Symbols illustrate the results of DFT calculations while lines show a fit to this data with the Morse potential, Eq. (8.2). The W–C bond

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Table 8.1 Parameters of the rCHARMM force field used in the simulations of W(CO)+ 6 fragmentation [36]. In the case of WC(CO)+ n , W–C bonds labeled as (1) and (2) refer to the notations of Fig. 8.3C W(CO)+ WC(CO)+ n n Bond type W–C C–O W–C (1) W–C (2) C–O r0 (Å) D (kcal/mol) k b (kcal/mol Å−2 )

2.11 40.7 119.9

1.14 212.8 1493.9

1.83 143.3 369.4

2.31 22.6 67.7

1.14 212.8 1493.9

in the parent molecule is significantly weaker (D = 40.7 kcal/mol) than the C–O bond (D = 212.8 kcal/mol), which is a common feature of metal carbonyls [49]. In the case of WC(CO)+ 5 , three different types of W–C bonds, labeled as (1), (2), and (3), can be distinguished, see Fig. 8.3C. When an oxygen atom is removed, the remaining carbon atom of the ligand becomes stronger bound to the tungsten atom (bond (1)); the dissociation energy of this W–C bond increases and varies from 119.9 to 143.3 kcal/mol depending on the WC(CO)+ n fragment considered. The opposite CO group becomes weakly bound to the metal atom (W–C bond (2)) and the equilibrium distance between W and C increases from 2.11 Å to 2.31 Å. Four W–C bonds in the perpendicular plane (bond (3)) remain unaffected and their dissociation energy does not change with respect to that in the parent molecule. Table 8.1 summarizes the rCHARMM parameters used in the calculations [36]. Since the dissociation energy of W–C bond in different W(CO)+ n (n = 1 − 5) fragments does not vary significantly [45], the value for the formation of W(CO)+ 5, D = 40.7 kcal/mol, has been used in the simulations. Several DFT potential energy scans were also performed to evaluate the equilibrium angles. It was found that the equilibrium values for the C–W–C and W–C–O angles are 90 and 180 ◦ C, respectively. In Ref. [29], rCHARMM was employed for modeling the DNA damage due to the nanoscale shock wave induced by a passing ion. It is widely established that one of the key events of radiation-induced DNA damage concerns the formation of single- and double-strand breaks (SSBs and DSBs) as well as more complex damages. Therefore, the rCHARMM force field was used to describe covalent interactions in the DNA backbone while covalent interactions in other parts of the DNA molecule were modeled using the standard CHARMM force field. Dissociation energy and cutoff distance parameters of rCHARMM for the specific bonds were determined from DFT calculations of a guanosine monophosphate (GMP, C10 H14 N5 O8 P) molecule, whose structure is shown in Fig. 8.4. The calculations were performed using Gaussian 09 software [46] employing the B3LYP exchange–correlation functional [50, 51] and the 6-31G(d,p) basis set for wavefunction expansion. Geometry of a neutral GMP molecule was optimized first and then saved in the Z-matrix format to enable a fixed scan over different covalent bonds. For each bond, a 50-step scan was performed in steps of 0.1 Å, starting from

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Fig. 8.4 Optimized geometry of guanosine monophosphate (GMP) molecule that is a structural unit of the DNA [29]. Bonds between the labeled atoms have been considered reactive in the simulations conducted in Ref. [29]. The black dashes split the molecule into three parts: the phosphate group, sugar part, and the nucleobase Table 8.2 Dissociation energy D, equilibrium distance r0 , and the cut-off distance rcutoff for breakage/formation of covalent bonds in the sugar–phosphate backbone of guanosine monophosphate, calculated at the B3LYP/6-31G(d,p) level of theory [29]. Cutoff distance is defined for each bond as the distances at which the bond energy is equal to 0.1D Bond type D (kcal/mol) r0 (Å) rcutoff (Å) C3 –O C5 –O C4 –C5 P–O

160.22 160.22 146.71 146.07

1.415 1.415 1.518 1.609

3.60 3.60 3.90 3.75

the interatomic distance of about 0.6 Å. It was found that the covalent bonds of the DNA backbone, along with the glycosidic bond between the sugar ring and the guanine nucleobase, have the lowest dissociation energies and thus likely have a higher probability to break. The results for C3 –O, C5 –O, C4 –C5 and P–O bonds of the DNA backbone (see the labeled atoms in Fig. 8.4) are listed in Table 8.2. A rupture of the C4 –C3 bond does not cause a break in the DNA strand, as the bond is a part of a sugar ring, therefore, it has not been parametrized by the reactive CHARMM force field in this case. Spring constants for the considered bonds have been taken from the standard CHARMM force field for nucleic acids [31].

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8.4 Irradiation-Induced Chemical Transformations of MBN Systems 8.4.1 Fragmentation of Organometallic FEBID Precursors Molecular fragmentation as a result of energy deposition involves several stages that take place on different time scales. First, the incident radiation (e.g., a photon, electron, or ion) interacts with the molecule and transfers energy to it by means of different mechanisms, e.g., electronic excitation, ionization, or (in the case of electron irradiation) dissociative electron attachment (DEA). These are fast processes that happen on the sub-femtosecond scale and leave the molecule in an excited electronic state. In the case of ionization, some fraction of deposited energy is spent in overcoming the ionization threshold, another fraction is carried away by the ejected electron, while the remaining part is stored in the target in the form of electronic excitations. The latter can involve different molecular orbitals, being of either bonding or antibonding nature. An excitation involving an antibonding molecular orbital evolves quickly through cleavage of a particular bond on the femtosecond timescale. The fragmented parent molecule may still keep some amount of the deposited energy which can lead to the sequential fragmentation of other bonds on the picosecond or even longer timescales. The excited electronic state may also involve a bonding molecular orbital or may not be localized on a particular bond. In this case, the excess energy can be redistributed over a larger part or even the entire volume of the system and be transferred later into its vibrational degrees of freedom. Relaxation of the deposited energy due to electron–phonon coupling mechanism [52] leads to an increase in the amplitude of thermal vibrations which, in turn, leads to evaporation of loosely bound CO ligands. As it was shown in the case of small metal clusters [52], the electron–phonon coupling is a slow process (as compared to a characteristic time of electron–molecule interaction) that happens on a picosecond time scale (see Chap. 2). The subsequent evaporation process may last up to microseconds. In Ref. [36] radiation-induced fragmentation of a W(CO)6 molecule was studied by means of reactive MD simulations employing MBN Explorer. Focus was made on the events happening from the cleavage of a particular bond to the redistribution of excess energy over the internal degrees of freedom of the molecule, i.e., on the time scales spanning from pico- to microseconds. Although quantum mechanical calculations can simulate the process of energy deposition, excitation of specific electronic orbitals, and the initial stage of the fragmentation process, the span of the entire fragmentation process goes far beyond the limits of quantum MD. Thus, classical MD remains the only computational technique allowing exploration of the process within the required time frame at the atomistic level of detail. The following approach within the framework of classical MD was proposed [36] to describe the aforementioned fragmentation stages, i.e., fast cleavage of an individual bond (referred hereafter as stage I) and slow energy redistribution over all the molecular degrees of freedom (stage II from now on). Both processes result in

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(A)

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Fig. 8.5 Schematic illustration of the two stages of fragmentation of W(CO)6 . A Stage I: the deposited energy is transferred into the kinetic energy of atoms in a specific bond. The resulting velocities of the atoms are shown by arrows. B Stage II: the deposited energy is redistributed over the entire molecule and shared between the kinetic energy of all the atoms. Reproduced from Ref. [36] with permission from Springer Nature

an increase in internal energy of the molecule after the energy deposition, which is treated as an initial increase of the kinetic energy of atoms. Within the approach used to model stage I, the energy has been deposited locally into a specific covalent bond of the target and converted into kinetic energy of the two atoms forming the bond (see Fig. 8.5A). Velocities of these atoms have been defined to obey the total energy and momentum conservation laws:  v1 =

2μE dep u, m1

 v2 = −

2μE dep u. m2

(8.9)

Here, E dep is the amount of deposited energy remaining in the system after ionization (i.e., excess energy over the first ionization potential), m 1 , m 2 and μ = m 1 m 2 /(m 1 + m 2 ) are, respectively, masses and the reduced mass of the atoms forming the bond, and u is a unit vector defining the direction of the relative velocity of these atoms upon bond cleavage. The orientation of u may be determined by the field resulting from the local molecular configuration around the bond. In the model described in Ref. [36], it has been chosen randomly. Stage II is governed by the thermal mechanism of fragmentation where the energy is distributed over all degrees of freedom of the target. In this case, equilibrium eq velocities of atoms corresponding to a given temperature, vi , are scaled1 by a factor α depending on the amount of energy deposited (see Fig. 8.5B). The kinetic energy of N atoms is then given by 1

Note that only the absolute values of the velocities are scaled, while their directions are unaltered; thus, the total momentum is conserved.

8 Multiscale Modeling of Irradiation-Driven Chemistry Processes N 1 i

2

eq

m i (α vi )2 =

3N kB T + E dep . 2

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(8.10)

The first term on the right-hand side of Eq. (8.10) corresponds to the initial kinetic energy of the atoms at equilibrium (e.g., T = 300 K in the simulations), with kB being the Boltzmann’s constant. The second term on the right-hand side is the excess energy deposited in the molecule during the collision. Each of the above-described mechanisms leads to the formation of a particular group of experimentally observed fragments, while the whole experimental picture is reproduced well when both stages of the fragmentation process are considered. Notations ‘stage I’ and ‘stage II’ reflect the different time scales for the processes which precede the bond cleavage, i.e., bond cleavage on the femtosecond timescale after the electronic excitation to an antibonding molecular orbital, and fragmentation induced by the relaxation of the deposited energy due to electron–phonon coupling, which happens on the picosecond timescale. These stages can happen sequentially in the same excited molecule so that the initial cleavage of a specific bond (stage I) is followed by redistribution of the remaining energy over the molecular fragment (stage II). However, the energy deposited initially into the molecule may be directly redistributed among all degrees of freedom on the picosecond timescale and lead to the fragmentation pattern described by stage II. The MD simulations of W(CO)+ 6 fragmentation were performed using the MBN Explorer software package [26]. First, the structure of the molecule was optimized using the parameters obtained from the DFT calculations, see Sect. 8.3 above. Then the molecule was equilibrated at T = 300 K for 100 ns. The equilibration simulation was performed using the Langevin thermostat with damping time of 2 ps. Atomic coordinates and velocities were recorded every 100 ps. The equilibrated trajectory was sampled to generate random initial geometries and velocity distributions for the simulation of fragmentation. About 1000 constant-energy simulations, each of 1 µs duration was conducted at different values of E dep ranging from 0 to 475 kcal/mol. The upper limit is several times larger than the energy needed to break one W–C bond (see Fig. 8.3A) which enables simulation of evaporation of several CO ligands. Fragments produced after 1 µs of simulation were analyzed, and the corresponding appearance energies were evaluated from this analysis and compared to experimental data. In the following, the fragmentation pathways of a W(CO)+ 6 molecule upon electron impact ionization are discussed using the energy deposition model described above. Simulated appearance energies are compared to the most recent set of experimental data obtained from mass spectrometry for positively charged fragments produced by electron beams of energy ≤ 140 eV [45], as well as to the previously reported experimental values [53–57]. Figure 8.6 shows the number of CO fragments, n, produced upon dissociation of the parent W(CO)+ 6 molecule for a given amount of excess energy E dep . Open blue squares show the results of simulations of stage I (see Fig. 8.5A). In this case, a given amount of energy was deposited into a W–C bond resulting in a prompt release of

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Fig. 8.6 Appearance energies for n CO ligands produced upon fragmentation of a W(CO)+ 6 molecule. Results of the reactive MD simulations are shown by symbols (see the text for details). Experimental appearance energies from Ref. [45] are shown by black dashed lines and from Refs. [53–57] by shaded areas. Dashed red line shows the largest number of CO released at each E dep . Reproduced from Ref. [36] with permission from Springer Nature

one CO group. However, no further fragmentation has been observed even at high values of E dep . Due to a large difference in masses of a carbon and a tungsten atoms, more than 90% of deposited energy is transferred into kinetic energy of the C atom and carried away by the released CO group. Therefore, independently of the value of E dep given to the molecule through cleavage of a W–C bond, only a small amount of energy is transferred to the remaining W(CO)+ 5 fragment, which is not sufficient to observe further fragmentation events. A single atom or a small fragment produced after cleavage of a specific bond can hit the remaining large fragment upon its escape, redepositing some amount of energy into the large fragment and thus triggering further fragmentation at stage II (see Fig. 8.5B). Filled blue squares describe the situation when the CO group was released due to the cleavage of a W–C bond collided with the remaining W(CO)+ 5 molecule, which led to the loss of another CO group. However, not more than two CO ligands have escaped the parent molecule in this case. The results of simulations describing stage II are shown in Fig. 8.6 by filled red circles. Distribution of deposited energy over all degrees of freedom of the target leads to evaporation of multiple CO fragments. These results are in good agreement

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with the appearance energies reported in a recent experimental study by Wnorowski et al. [45] (black dashed lines) as well as with appearance energies obtained in earlier experiments [53–57] (shaded areas). Note that the first ionization potential was subtracted from the experimental appearance energies to convert them into the excess deposited energy. Due to the statistical nature of the fragmentation process, emission of a given number of CO groups was observed at different values of E dep . The lowest values at which five or less CO molecules were recorded after 1 µs agree nicely with the experimental results. However, complete fragmentation (i.e., loss of six CO molecules) has been observed only in a few trajectories at E dep = 375 and 400 kcal/mol, which are significantly higher than the experimental appearance energy for W+ . This indicates that the complete fragmentation takes place on a larger time scale and longer simulations are needed to observe the complete fragmentation at smaller energies. The results of simulations describing stage II have been used to evaluate branching ratios for the production of different fragments for a given amount of excess energy E dep , see Fig. 8.7. This analysis provides information on how many carbonyl groups will be most likely evaporated at a given E dep after 1 µs. Figure 8.7 shows that emission of one, two, and three CO groups (n = 1, 2, 3) takes place in rather wellseparated energy “windows”. Fragments corresponding to n = 1 and 2 were recorded in these energy ranges with the maximal probability corresponding to the branching ratio of 1. For the larger numbers of emitted CO, the maximal branching ratios drop down to 0.8 suggesting an increased probability of observing different fragments. Note also that the characteristic energy ranges for the emission of n CO groups increase with n. This analysis allows for the evaluation of a typical amount of energy that should be deposited into the W(CO)+ 6 molecule to observe a specific fragment. The experimental fragmentation mass-spectra [45, 53–57] revealed the formation + of not only W(CO)+ 6−n (n = 0 − 6) but also WC(CO)5−n (n = 2 − 5) molecules, which, however, have not been observed in the simulations of stage II even after 1 µs of simulation. This is due to the very low probability for observing a statistical cleavage of a C–O bond, owing to the much lower strength of the W–C bond. It was therefore assumed that a C–O bond can break after a localized energy deposition into it. As has been shown in Fig. 8.3 and Table 8.1, loss of an oxygen atom from a CO group makes the opposite ligand weaker bound to the W atom (bond (2)), while the W–C bond corresponding to the cleaved C–O (bond (1)) becomes much stronger. Therefore, localized energy deposition into one C–O bond leads to a prompt release of an O atom, together with the recoil of the C atom to the parent molecule, which ends up being vibrationally excited and subsequently releases a CO fragment. This scenario explains also the production of smaller fragments due to the subsequent loss of several CO groups from WC(CO)+ 5 which were observed experimentally and in the simulations at E dep > 180 kcal/mol, as shown in Fig. 8.8. It should be noted that, in this case, the reactive force field is rather sensitive to the parameters used, particularly to the dissociation energies, and variation of these parameters can impact the results of simulations. Black squares in Fig. 8.8 depict the results of simulations using the dissociation energies calculated by DFT (see Sect. 8.3), specifically DC−O = 212.8 kcal/mol, while the appearance energy for

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Fig. 8.7 Branching ratios for the formation of n CO ligands upon fragmentation of a W(CO)+ 6 molecule during stage II (see Fig. 8.5B) as a function of the excess energy E dep . At a given E dep , the sum of branching ratios for all the fragments is equal to 1. Results of the reactive MD simulations are shown by symbols. Experimental appearance energies from Ref. [45] are shown by dashed lines. Reproduced from Ref. [36] with permission from Springer Nature 2 WC(CO)+ 4 is well reproduced, the rest are shifted towards larger energies. However, the C–O bond dissociation energy of about 180 kcal/mol can be estimated from the analysis of appearance energies reported by Wnorowski et al. [45]. It should be noted that the C–O bond in metal carbonyls is rather complex and can be characterized as a mixture of a triple and a double bond; DC−O varies from 126 kcal/mol for a double bond to 257 kcal/mol for a triple bond [58, 59]. Thus, values within this range are meaningful. Simulation results using DC−O = 180 kcal/mol are shown in Fig. 8.8 by open red circles. The calculated appearance energies are shifted to lower energies with respect to the results employing the dissociation energy obtained from DFT, being closer to the experimental results. Filled blue triangles show the results of simulations where the collision of the escaping O atom with the parent molecule has been taken into account. In this case, + the first four appearance energies (WC(CO)+ 4 to WC(CO) ) are well reproduced, 2

Although this fragment was not observed experimentally, its appearance energy (gray dotted line in Fig. 8.8) can be estimated from the data reported in Ref. [45]. The calculated first appearance energy is smaller than the expected experimental value. This is due to the thermal energy of the molecule which also contributes to fragmentation together with the deposited energy.

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Fig. 8.8 Comparison between simulated (symbols) and experimental appearance energies for the different fragments of the WC(CO)+ 5 molecule. Open black and red symbols correspond to simulations in which the oxygen atom escapes the system without further collision (see the text for details). Filled blue and green symbols describe events where the O atom collides with the parent molecule. Experimental appearance energies from Ref. [45] are shown by dashed lines and from Refs. [53–57] by shaded areas. Reproduced from Ref. [36] with permission from Springer Nature

but the last one (WC)+ is not observed. This happens because the deposition of large amounts of E dep larger than 390 kcal/mol leads to the cleavage of not only C–O bond but also the W–C bond labeled as (1) in Fig. 8.3C. One may expect that this bond becomes stronger as the coordination number of W decreases (i.e., when smaller fragments are formed) what happens at large values of E dep . Thus, DW−C could be larger than 143 kcal/mol, the value obtained from the DFT calculations for the WC(CO)+ 5 molecule. A set of additional simulations has been performed using DW−C = 180 kcal/mol, that is, similar to the dissociation energy of the C–O bond; these results are shown by filled green diamonds in Fig. 8.8. Under these conditions, the experimental appearance energies for W(CO)6 fragments are well reproduced. Although the reactive force field is rather sensitive to the parameters used, an appropriate choice of the parameters has led to a quantitative agreement with experiments. This shows that reactive classical MD simulations are appropriate to simulate the fragmentation patterns of W(CO)+ 6 molecule. The method, being general, can be applied to other organometallic precursors for FEBID as well as to many other inorganic, organic, and biological molecules.

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8.4.2 Fragmentation of the DNA Molecule by the Ion-Induced Nanoscale Shock Wave The reactive MD and IDMD methodologies can be utilized to evaluate radiobiological damage created by heavy ions propagating in different media, including biological. An energetic charged particle propagating through the medium loses its energy in inelastic collisions with the medium constituents. The transport of produced secondary particles, as well as the radiation damage induced by them, are the objects of experimental, numerical, and computational studies [3, 60]. In particular, the physics and chemistry of radiation damage caused by irradiation with protons and heavier ions have recently become a subject of intense interest because of the use of ion beams in cancer radiotherapy [1–3, 12]. Recent review papers [2, 61] and the book [3] presented an overview of the main ideas of the MultiScale Approach to the physics of radiation damage with ions (MSA). This approach has enabled to develop the knowledge about biodamage at the nanoscale and molecular levels and to find the relation between the characteristics of incident particles and the resultant biological damage. An overview of the MSA is given Chap. 9. Radiation damage due to ionizing radiation is initiated by the ions incident on tissue. The initial kinetic energy of the ions ranges from a few to hundreds of MeV per nucleon. In the process of propagation through tissue, they lose energy due to ionization, excitation, nuclear fragmentation, etc. Most of the energy loss of the ion is transferred to tissue. For an energetic ion propagating in a medium, the dependence of the energy deposited into the medium on the penetration distance is characterized by a sharp maximum, the Bragg peak, in the region close to the end of ion’s trajectory. It is commonly understood that the secondary electrons and free radicals produced in the processes of ionization and excitation of the medium with ions are largely responsible for the vast portion of the biodamage. In the Bragg peak region, the secondary electrons lose most of their energy within 1–2 nm of the ion’s path [62]. After that, the electrons continue propagating, elastically scattering with the molecules of the medium until they get bound or solvated electrons are formed [2, 3]. Such lowenergy electrons remain important agents for biodamage since they can attach to biomolecules like DNA causing dissociation [63]. The energy lost by secondary electrons in the processes of ionization and excitation of the medium is transferred to its heating (i.e., vibrational excitation of molecules) due to the electron–phonon interaction. As a result, the medium within a 1–2-nm region (for ions not heavier than iron) surrounding the ion’s path is heated up rapidly [62, 64]. The pressure inside this narrow region increases by several orders of magnitude (e.g., by a factor of 103 for a carbon ion at the Bragg peak [64]) compared to the pressure in the medium outside that region. This pressure builds up by about 10−14 − 10−13 s and it is a source of a cylindrical shock wave [65] which propagates through the medium for about 10−13 − 10−11 s. Its relevance to the biodamage is as

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follows. If the shock wave is strong enough,3 it may inflict damage directly by breaking covalent bonds in a DNA molecule [29, 62, 66–70]. Besides, the radial collective motion of the medium induced by the shock wave is instrumental in propagating the highly reactive molecular species, such as hydroxyl radicals and solvated electrons, to large radial distances (up to tens of nanometers) thus increasing the area of an ion’s impact [34, 40]. Further details on this phenomenon are given in Chap. 9. The direct thermomechanical damage of the DNA molecule as a result of interaction with the ion-induced shock wave has been explored using MBN Explorer [29, 62, 66, 67, 70]. In the earlier investigations [62, 66, 67], the DNA damage by ion-induced shock waves was studied by means of classical MD simulations using non-reactive molecular mechanics force fields. In those simulations, the potential energy stored in a particular DNA bond was monitored in time as the bond length varied around its equilibrium distance [62, 67]. When the potential energy of the bond exceeded a given threshold value, the bond was considered broken. In the pioneering study [62], the MD simulations were focused on the interaction of the cylindrical shock wave originating from ion’s path with a fragment of a DNA molecule situated on the surface of a nucleosome, see Fig. 8.9A. Nucleosomes, histone–protein octamers wrapped about with a DNA double helix, are the primary structural units of chromatin, which is a principal component of the cell nucleus in eukaryotic cells. The simulations [62] were done for four values of LET, namely, 900, 1730, 4745, and 7195 eV/nm, corresponding to the Bragg peak values for carbon, neon, argon, and iron ions, respectively. Carbon ions are clinically used for cancer treatment, whereas heavier ions up to iron are present in galactic cosmic rays, being potentially damaging for humans during space missions [71, 72]. The simulations were performed using the standard CHARMM force field [30] which implies the harmonic approximation for describing the interaction potentials for covalent bonds and thus does not allow to observe bond-breaking events directly. Therefore, in order to study whether the covalent bonds in the DNA backbone can be broken during the shock wave action, the energy temporarily deposited to these bonds was calculated. The analysis of MD simulations performed for four values of LET (900, 1730, 4745, and 7195 eV/nm) gives the distributions of the bond energy records. These records can be represented by a histogram that assigns to every interval of energy (ε, ε + δε); the number of records corresponding to the bond energies from this interval. For each value of LET, the bond energy distribution was constructed. These distributions (normalized to the total number of records Nr for each value of LET) are shown in Fig. 8.9B, where ln (1/Nr d N /dε) is plotted versus the corresponding energy interval. Next, the number of energy records of selected covalent bonds of the DNA backbone exceeding a given threshold was counted. This was done by direct counting of bond energy records, for which E > E 0 , where E 0 is a variable threshold; e.g., the records counted for E 0 = 2.5 eV are shown to the right of the dashed vertical line in Fig. 8.9B. 3

The strength of the shock wave depends on the distance from the ion’s path and the value of linear energy transfer (LET), that is the amount of energy that a projectile ion transfers to the medium traversed per unit distance.

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Fig. 8.9 A The cylindrical shock wave front in water (on the right; ion’s path is the axis of this cylinder, perpendicular to the figure plane) interacts with a nucleosome (on the left) with a segment of a DNA molecule on the surface. The yellow dot indicates the place where damage occurs. The medium is very dense following the wave front and is rarefied in the wake. B The dependence of the logarithm of the normalized number of the covalent bond energy records for the selected DNA backbone region per 0.01 eV energy interval on the bond energy for four values of LET: 900, 1730, 4745, and 7195 eV/nm, corresponding to the Bragg peak values for carbon, neon, argon, and iron ions, respectively. Straight lines correspond to the fits of these distributions. The figures are reproduced from Ref. [62]

A more quantitative description of the ion-induced shock wave phenomenon has become possible [70] by means of reactive MD simulations that permitted explicit simulation of covalent bond rupture and formation [28]. A recent study [29] presented a detailed computational protocol for modeling the shock wave-induced DNA damage by means of the rCHARMM force field [28]. The target DNA molecule studied in [29, 70] contained 30 complementary DNA base pairs. The molecule was placed in a water box extending 17 nm from the DNA in the x- and y-directions and 8 nm in the z-direction. The total system size was 1,010,994 atoms. It is widely established that one of the key events of radiation-induced DNA damage concerns the formation of single- and double-strand breaks (SSBs and DSBs) of the sugar–phosphate backbone. Therefore, the rCHARMM force field was used to describe interatomic interactions in the C3 –O, C4 –C5 , C5 –O and P–O bonds in the DNA backbone, which connect the sugar ring of one nucleotide and the phosphate group of an adjacent nucleotide, see Fig. 8.4 and Table 8.2 above. Bond dissociation energies and cutoff distances for bond breakage/formation were determined from density functional theory (DFT) calculations [29]. Covalent interactions in other parts of the DNA molecule were modeled using the standard CHARMM force field. In the MD simulations, the energy lost by the propagating ion is deposited into the kinetic energy of water molecules located inside a “hot” cylinder of 1 nm radius around the ion’s path. The equilibrium velocities of all atoms inside the “hot” cylinder are increased by a factor α such that the kinetic energy of these atoms reads as [62, 66, 73]:

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Fig. 8.10 A Illustration of the propagation of the ion-induced shock wave in the vicinity of the DNA segment containing 30 base pairs [29]. Water molecules inside the “hot” cylinder surrounding the ion track (shown by the yellow arrow) are highlighted. B The density of water in the radial direction from the ion’s path is shown at different instances, ranging from 0 to 12 ps after irradiation

N 1 i

2

m i (αvi )2 =

3 N kB T + Se l . 2

(8.11)

Here, Se is the LET of the projectile ion, l is the length of the simulation box in the z-direction (parallel to the ion’s path), and N is the total number of atoms within the “hot” cylinder. The first term on the right-hand side of Eq. (8.11) is the kinetic energy of the 1-nm radius cylinder at the equilibrium temperature, T = 300 K, whereas the second term describes the energy loss by the ion as it propagates through the medium. The shock wave propagates in the molecular system radially away from the ion track, see Fig. 8.10A. The range of the shock wave propagation in the aqueous environment can be determined by monitoring the radial density of the water molecules in time. The ion track was in this case placed directly through the geometrical center of the DNA strand. The results of this analysis are shown in Fig. 8.10B. The wave front moves toward the edge of the simulation box as time passes, and the wave profile becomes lower and broader, showing that the shock wave relaxes as time passes. This indicates that the impact of the shock wave weakens over time. Figure 8.10B shows that the maximal density of the wave does not change significantly in the range of 2–6 nm from the ion track, thus a DNA strand placed in this range is expected to receive the strongest impact from the shock wave. The impact of a shock wave on DNA can be characterized through the probability of strand breaks formation, which can be used to quantify the amount of biodamage induced by the shock wave mechanism [2]. Figure 8.11 shows the number of strand breaks in the DNA segment, caused by a shock wave induced by an argon ion with LET of 2890 eV/nm, a function of simulation time. The number of breaks rises quickly within the first 7 ps of the simulation, where the shock wave front hits

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Fig. 8.11 The number of strand breaks in the 30 base pairs-long DNA segment induced by the thermomechanical stress by the argon ion with a LET of 2890 eV/nm as a function of simulation time [29]. Two insets in the top show the breakage of the DNA strands at simulation time instances of 4 ps and 15 ps

both DNA strands of the target molecule. After this moment, the DNA damage rate becomes lower lasting until approximately 15 ps, hereafter, the number of breaks remains steady. Even if the number of breaks generally rises with time, some fluctuations in this quantity can be seen locally. This effect might be attributed to the broken bonds which can be rejoined if the atoms involved get close to each other after the initial bond breakage. This analysis shows that multiple or even complex stand breaks might be induced by the generated shock waves. The methodology reviewed here can be applied in further computational studies considering irradiation of DNA with different ions and different orientations between the ion’s path and the DNA molecule. Such analysis is important for understanding the radiation damage with ions on a quantitative level, focusing on particular physical, chemical, and biological effects that bring about lethal damage to cells exposed to ion beams [2, 74, 75]. The outcomes of such analysis are described in detail in Chap. 9.

8.5 Multiscale Modeling of the Focused Electron-Beam-Induced Deposition (FEBID) Process The controllable fabrication of nanostructures with nanoscale resolution remains a considerable scientific and technological challenge [76]. To address such a challenge, novel techniques have been developed [7] which exploit irradiation of nanosystems

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with collimated electron and ion beams. One of such techniques, Focused Electron Beam-Induced Deposition (FEBID) [6–8], is based on the irradiation of precursor molecules [77] by high-energy electrons while they are being deposited on a substrate. Electron-induced decomposition of the molecules releases its metallic component which forms a deposit on the surface with a size similar to that of the incident electron beam (typically, a few nanometers) [44]. As a result, FEBID enables reliable directwrite fabrication of complex, free-standing 3D structures [44, 78]. To date, FEBID has mainly relied on precursor molecules developed for chemical vapor deposition (CVD)—a process mainly governed by thermal decomposition, while dissociation mechanisms in FEBID are predominantly electron-induced reactions. While primary electron (PE) energies during FEBID are typically between 1 keV and 30 keV, chemical dissociation is most efficient for low-energy (up to several tens of eV) secondary electrons (SE) created in large numbers when a highenergy PE beam impinges on a substrate. Secondary electrons are emitted from the substrate and the deposit, making the electron-induced chemistry that governs FEBID substantially complicated. As a result, as the intended nanostructure resolution falls below 10 nm FEBID struggles to fabricate structures with the desired size, shape, and chemical composition [44], which mainly originates from the lack of molecularlevel understanding of irradiation-driven chemistry (IDC) underlying nanostructure formation and growth. FEBID operates through successive cycles of precursor molecules replenishment on a substrate and irradiation by a tightly focused electron beam, which induces the release of metal-free ligands and the growth of metal-enriched nanodeposits. This process involves a complex interplay of phenomena, taking place on different temporal and spatial scales: (i) deposition, diffusion, and desorption of precursor molecules on the substrate; (ii) transport of the primary, secondary, and backscattered electrons; (iii) electron-induced dissociation of the deposited molecules; (iv) the follow-up chemistry; and (v) relaxation of energy deposited into electronic and vibrational degrees of freedom, and resulting thermomechanical effects. Each of these phenomena requires dedicated computational and theoretical approaches. Further advances in FEBID-based nanofabrication require a deeper understanding of the relationship between deposition parameters and physical characteristics of fabricated nanostructures (size, shape, purity, crystallinity, etc.). Advanced experiments combined with molecular-level computational modeling can provide the required insights into the fundamental mechanisms of electron-induced precursor fragmentation and the corresponding mechanism of nanostructure formation and growth using FEBID. Until recently, most computer simulations of FEBID and the nanostructure growth have been performed using a Monte Carlo (MC) approach and diffusion–reaction theory [7, 79, 80], which allow simulations of the average characteristics of the process concerning local growth rates and the nanostructure composition. However, these approaches do not provide any molecular-level details regarding structure (crystalline, amorphous, mixed) and the IDC involved. At the atomic level, quantum chemistry methods have been utilized to analyze the adsorption energies and optimized structures of different precursor molecules deposited on surfaces [81, 82].

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Fig. 8.12 Snapshot of the MD simulation [27] of adsorption of W(CO)6 precursor molecules atop the SiO2 surface at the early stage of irradiation by an electron beam (a transparent green cylinder). The interaction of deposited precursor molecules with the beam leads to the fragmentation of precursors and to the formation of tungsten clusters, shown in blue

Nevertheless, ab initio methods are applicable to relatively small molecular systems with a typical size of up to a few hundred atoms. This makes ab initio approaches of limited use to describe the irradiation-induced chemical transformations occurring during the FEBID process. A breakthrough into the atomistic description of FEBID was achieved recently by means of the IDMD approach [27], described above in Sect. 8.2. This approach overcomes the limitations of previously used computational methods and describes FEBID-based nanostructures at the atomistic level by accounting for quantum and chemical transformation of surface-adsorbed molecular systems under focused electron beam irradiation [27, 37, 38]. Major transformations of irradiated molecular systems are simulated by means of MD with the rCHARMM force field [28, 29] using the MBN Explorer [26] and MBN Studio [83] software packages. In the pioneering study [27], IDMD was successfully applied for the simulation of FEBID of W(CO)6 precursors on a SiO2 surface and enabled to predict the morphology, molecular composition and growth rate of tungsten-based nanostructures emerging on the surface during the FEBID process. A snapshot of the MD simulation of the first irradiation phase is shown in Fig. 8.12. The irradiation by an electron beam of the cylindrical shape has been considered. Only those precursors molecules that move inside the cylinder are exposed to radiation that may induce their dissociation. The dissociation rate of the precursor molecules was evaluated from the experimental data [79]. The follow-up study [37] introduced a novel multiscale computational methodology that couples MC simulations for radiation transport with IDMD for simulating the IDC processes with atomistic resolution. The developed multiscale modeling

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approach has enabled simulation of radiation transport and effects in complex systems where all the FEBID-related processes (deposition, irradiation, replenishment) are accounted for. The spatial and energy distributions of secondary and backscattered electrons emitted from a SiO2 substrate were used to simulate electron-induced formation and growth of metal nanostructures obtained after deposition of W(CO)6 precursors on SiO2 . Within the IDMD framework, the space-dependent rate for bond cleavage in molecules on the substrate surface is given by: P(x, y) = σfrag (E 0 ) JPE (x, y, E 0 ) σfrag (E i ) JSE/BSE (x, y, E i ) , +

(8.12)

i

where E 0 is the initial energy of the electron beam, E i < E 0 a discrete set of values for the electron energies lower than E 0 ; JPE/SE/BSE (x, y, E i ) are space- and energydependent fluxes of primary (PE), secondary (SE), and backscattereted (BSE) electrons per unit area and unit time, and σfrag (E i ) is the energy-dependent molecular fragmentation cross section. The PE beam flux at the irradiated circular spot of radius R is I0 , (8.13) J0 = e S0 where I0 corresponds to the PE beam current, S0 = π R 2 to its area and e is the elementary charge. The electron distributions were simulated using the MC radiation transport code SEED (Secondary Electron Energy Deposition) [41, 84]. Molecular fragmentation and further chemical reactions were simulated by means of MBN Explorer [26], while MBN Studio [83] was employed to construct the molecular system, perform the precursor molecule replenishment phases, and analyze the IDMD simulation results. Figure 8.13 illustrates the space-dependent fragmentation rates induced by uniform 1 keV (panel A) and 30 keV (panel B) beams of unit PE flux J0 = 1 nm−2 fs−1 within a circular area of radius R = 5 nm. Although the number of BSE/SE electrons for 30 keV is small, their large cross section (in relation to PE) produces a significant fragmentation probability, but less than that due to PE at the beam area. However, for 1 keV, the fragmentation probability due to BSE/SE (∼80–90% exclusively due to SE) is very large and significantly extends beyond the PE beam area. These results clearly demonstrate the very different scenarios to be expected for beams of different energies and which will importantly influence the deposit properties, as well as the prominent role of low-energy SE on molecular fragmentation. Each irradiation phase lasts for a time known as dwell time, whose typical duration in experiment (≥μs) is still computationally demanding for MD. To address this challenge, the irradiation phase was simulated for 10 ns and simulated PE fluxes J0 (and hence PE beam currents I0 ) were then scaled to match the same number of PE per unit area and per dwell time as in experiments [27]. As for replenishment,

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Fig. 8.13 Electron-induced fragmentation rates for W(CO)6 precursor molecules irradiated with PE beams of E 0 = 1 keV (A) and E 0 = 30 keV (B). The green transparent surface depicts the PE beam area. Reproduced from Ref. [37]

its characteristic times are also typically very long (∼ms). In simulations, the CO molecules desorbed to the gas phase are simply removed during the replenishment stages and new W(CO)6 molecules are deposited. As described above, the tabulated space-resolved fragmentation probability per primary electron is used in the IDMD simulation of the irradiation phase to link the bond dissociation rate to the electron flux. As the realistic experimental time scale for τd is challenging for MD, the simulated PE fluxes J0 (and hence PE beam currents I0 ) are rescaled to match the same number of PE per unit area and per dwell time as in experiments. The correspondence of simulated results to experimental ones is established through the correspondence of the electron fluence per dwell time per unit area in simulations and experiments [27]. Such an approach is valid in the case when different fragmentation events occur independently and do not induce a collective effect within the system. In this case, the irradiation conditions for the deposited precursor molecules are the same in simulations and in experiments. This correspondence condition gives Iexp = Isim

2 Rexp Sexp = Isim 2 λSsim λRsim

(8.14)

exp

λ=

τd , τdsim

(8.15)

where Sexp and Ssim are the electron beam cross sections used in experiments and simulations, respectively; Rexp and Rsim are the corresponding beam spot radii. The rCHARMM force field used to model the structure and dynamics of irradiated W(CO)6 molecules atop the hydroxylated SiO2 surface, requires specification of several parameters (the equilibrium bond lengths, bonds stiffness, and dissociation energies). Additionally, one needs to define the dissociative chemistry of precursors

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Fig. 8.14 Time evolution of the size of the largest W-enriched island (A), the number of W atoms (B), the number of W(CO)6 (C) and CO (D) molecules in the system during the irradiation periods. The irradiation periods are marked by the successive numbers. The duration of each period is 10 ns. In simulations, the electron beam radius R is equal to 5 nm and the beam current I0 = 4 µA. Redrawn from data presented in Ref. [27]

including the definition of the molecular fragments and atomic valences. In the model considered, only the dissociation and formation of the W–C and W–W bonds were permitted, while the C–O bonds were treated within the harmonic approximation, Eq. (8.1), preventing those bonds from breakage. The FEBID process was modeled [27] with the rescaled computationally accessible parameters (the irradiation time and the beam current). These parameters may differ from experimental values, but they have to be chosen so that they correspond to a given (in experiment) number of electrons Ne targeting the system (the electron fluence) and thus producing the irradiation-induced effects on the same scale as in the experiment. Following this idea, the irradiation time in IDMD simulations is typically decreased as compared to the corresponding experimental values. Figure 8.14A shows the time dependence of the size of the largest W-enriched island emerged in the simulations. Panels B, C, and D show this evolution for the numbers of W atoms, W(CO)6 , and CO molecules in the system during the irradiation periods, respectively. The irradiation periods are marked in the plots by the successive numbers. The duration of each period is 10 ns. The replenishment periods are excluded from the plots. However, the drops in the number of CO molecules and the number of W(CO)6 , corresponding to the changes in the system that occur during the replenishment periods, are well seen. Both the size and the number of W atoms in the islands grow due to the attachment of new atoms to the islands in the course of the FEBID process. Figure 8.14A and B indicate the coalescence of smaller islands

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Fig. 8.15 Evolution of the number of atoms in the largest simulated islands for PE beams of energies 1, 10, and 30 keV, for different currents as indicated. Redrawn from data presented in Ref. [37]

into a larger single nanostructure (ripening process) during the initial stage of the FEBID process. The irregular spikes on the curves arise at the instants when separate islands merge together. The W–(CO) bonds in precursors dissociate during the irradiation periods leading to the appearance of the CO molecules. Most of these are created in the vicinity of the surface and later are evaporated into the vacuum chamber. The evaporation process continues during the replenishment periods. Figure 8.14D shows that during each irradiation period the number of CO molecules grows nonlinearly. To account for the evaporation process of CO during the replenishment periods, the CO molecules are removed from the simulation box after each irradiation phase. After that the new W(CO)6 molecules are deposited on the surface according to the chosen deposition rate and the duration of the deposition process. This results in the abrupt decrease in the CO molecules and increase in W(CO)6 numbers before the start of each new cycle irradiation. Figure 8.15 shows the results of IDMD simulations conducted in Ref. [37]. The number of atoms (either W, C or O) in the largest island is shown for three simulation conditions close to the experimentally reported conditions [85]: 30 keV at I0 = 0.3 nA, 10 keV at I0 = 2.3 nA, and 1 keV at I0 = 3.7 nA. Smaller clusters tend to merge with time giving rise to larger structures and, eventually, to the largest island

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(C) Fig. 8.16 Compositions and morphologies of the deposits created by FEBID. A Dependence of the deposit metal content on the beam energy E 0 and current Iexp from experiments (open symbols) [85] and simulations (full symbols) [37]. Numbers next to symbols represent the beam energy in keV for each case. Panels B and C show the top views of the deposits produced by 10 [email protected] nA and 1 [email protected] nA beams, respectively. The green area marks the PE beam spot while blue, white, and red spheres represent, respectively, W, C, and O atoms; the SiO2 substrate is represented by a yellow surface. Reproduced from Ref. [37]

displayed in the figure. The jumps in the island size observed with some frequency are due to the merging of independent clusters that grow on the substrate. Experimental measurements performed to date have been limited to particular values of energy and current due to the characteristics of the electron source [85]. In contrast, the IDMD simulation method permits the exploration of a much broader range of electron beam parameters. Full symbols in Fig. 8.16A depict the simulated metal contents of the deposits as a function of experimentally equivalent current Iexp . Error bars show the standard deviations obtained from three independent simulations for each case. Experimental results [85] are shown by open symbols. Numbers next to symbols represent the primary beam energies in keV. It is clearly seen that the results from simulations are within the range of experimental uncertainties, which indicates the predictive capabilities of the simulations. This analysis provides a detailed “map” of the attainable metal content in the deposits as a function of the beam parameters, which is a valuable outcome for the optimization of FEBID with W(CO)6 on SiO2 . Dashed lines in Fig. 8.16A correspond to the limiting values of PE beam energy and current studied in Ref. [37]. These results clearly show that, within the analyzed energy domain, a decrease in the beam energy and an increase in the current promote the faster growth of the deposit, as well as the augment in its metal content. Simulation results provide the grounds for clearly understanding such trends: an increment in the current means a larger number of PE per unit time, while a reduction in the energy produces an increase in the SE yield. These lead to both the greater size of the deposit and its larger metal content due to the increased probability of bond cleavage (see Fig. 8.13).

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Figure 8.16B and C show top views of the simulated deposits for 1 keV at 3.7 nA and 10 keV at 2.3 nA, after 5 and 7 irradiation cycles, respectively (the number of atoms in the largest island is similar in these cases, ∼12000). The green circular line marks the area covered by the PE beam (having a radius of 5 nm). These figures show that different energy–current regimes lead to distinct deposit microstructures and edge broadenings. While the more energetic 10 keV energy beam produces a deposit almost entirely localized within the PE beam area, the 1 keV beam produces a more sparse deposit (at least during the early stage of the FEBID process) that significantly extends beyond the PE beam area, producing an undesired edge broadening of the structure. As described above, investigation of the physicochemical phenomena that govern the formation and growth of nanostructures coupled to radiation is a complex multi-parameter problem. Indeed, a vast number of parameters (e.g., different precursor molecules, substrate types, irradiation and replenishment regimes, additional molecular species that may facilitate precursor decomposition, etc.) can be varied with the aim of improving the purity of grown deposits and increasing the deposition rate. It is therefore essential to develop a comprehensive computational protocol for atomistic simulations of the FEBID process. In the recent study [86], a detailed computational methodology for modeling the formation and growth of metal-containing nanostructures during FEBID by means of IDMD has been formulated. Different computational aspects of the methodology as well as the key input parameters describing the precursor molecules, the substrate, and the irradiation conditions have been systematically described. A step-bystep simulation and parameter determination workflow represented a comprehensive computational protocol for simulating and characterizing a broad range of nanostructures created by means of FEBID. The formulated computation protocol has been applied to simulatie the FEBID of Pt(PF3 )4 —a widely studied precursor molecule [87–92]—on a SiO2 surface. As such, this work extends the above-described IDMD-based studies [27, 37] of the FEBID of W(CO)6 towards another precursor molecule which has been commonly used to fabricate platinum-containing nanostructures. In contrast to the earlier studies [27, 37], the study performed in Ref. [86] has focused on the case of low precursor surface coverage (below one monolayer) which is of interest for the FEBID-based fabrication of free-standing 3D nanostructures [93, 94]. Particular focus has been made on the atomistic characterization of the initial stage of the FEBID process, including nucleation of Pt atoms, formation of small metal clusters on the surface followed by their aggregation, and, eventually, the formation of dendritic platinum nanostructures. IDMD simulations have been performed for 23 cycles of the FEBID process with a total simulation time of 230 ns. The accumulated fluence of PE is ∼7.5 × 1018 cm−2 . This corresponds to the equivalent total experimental irradiation time of 20 ms with the following beam parameters taken from Ref. [88]: electron current Iexp = 2.8 nA and the estimated beam spot radius Rexp = 40 nm. Although the experimental irradiation was performed with a continuous beam, dwell time of a single irradiation cycle exp exp was set to τd = 1 ms. This corresponds to the PE flux J0 ≈ 3480 nm−2 ms−1 .

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Following the earlier studies of FEBID using IDMD [27, 37], the dwell time value τd = 10 ns is used in the simulations. The rescaled electron current in the simulations is Isim ≈ 4.4 µA. A snapshot of the system at the end of the 20th FEBID cycle is presented in Fig. 8.17. Panels A–D correspond to different amounts of energy transferred to the medium via the Pt–P bonds fragmentation. The energy parameter E dep governing the bond fragmentation is equal to 300 kcal/mol and 205 kcal/mol for the cases presented in panels A & C and B & D, correspondingly. As shown in Fig. 8.17A, B, three spatial regions can be distinguished where different structures are formed on the surface depending on the spatial distributions of the fragmentation probability and deposited precursor molecules. Inside the beam spot area with the diameter of 10 nm (indicated by the blue circle in Fig. 8.17A and B), high probability of Pt–P bond fragmentation leads to dissociation of Pt(PF3 )4 molecules and formation of metal clusters. The clusters grow, merge, and interconnect during the irradiation process, forming a network of thread-like metallic nanostructures. The transition region of 1 nm radius outside the beam spot area contains smaller metal clusters with a larger number of PF3 ligands attached due to lower fragmentation probability in this spatial region. The presence of the ligands prevents dense packing and aggregation of isolated metal clusters. As a result, the height of the deposited structures in this spatial region is higher than that within the beam spot area. The region beyond the transition region (near the simulation box boundaries) contains mostly intact or less fragmented precursor molecules. The largest cluster formed at the end of the simulation is selected to study its evolution during the FEBID process. The cluster’s evolution is tracked back to the initial nucleation stage using coordinates of the center of mass of the metalcore. The cluster structure and the number of Pt atoms in the cluster at the end of each FEBID cycle are shown in Fig. 8.18. The platinum-containing nanostructure grows via coalescence of the neighboring metal clusters of different sizes. The coalescence takes place via an interplay of the following mechanisms: an addition of a fragmented precursor molecule with a single Pt atom (see the evolution of the cluster structure at FEBID cycles 3–4 and 8–9 in Fig. 8.18), merging of two clusters of the comparable size (see the cluster structure at cycles 4–5, 9-10, and 10–11), as well as a combination of both. Metal clusters containing up to about 30 Pt atoms preserve a spherical shape and tend to rearrange after an elongation caused by the merging of clusters of comparable size (see cycles 1–9). The sequential coalescence of larger clusters containing several tens of Pt atoms results in the formation of randomly oriented branched structures (see cycles 10–20). The coalescence of all the isolated Pt-containing clusters into a bigger structure continues after the 20 simulated cycles. It is expected that the continuation of the irradiation and replenishment processes will lead to further growth and interconnection of the branched clusters into a single metal network. Next, the process of initial cluster growth and coalescence has been studied for an ensemble of the deposited clusters. Figure 8.19 shows the size distribution of grown clusters as a function of the number of Pt atoms (panel A) and the difference between the distributions for two consecutive FEBID cycles (panel B) for the first

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Fig. 8.17 A snapshot of the multiscale IDMD simulation of the FEBID process of Pt(PF3 )4 at the end of the 20th cycle (after 200 ns of the simulation). Panels A and C show the top view and the side view of a 6 nm thick slice through the beam center of the grown structure for the regime of the energy transferred to the medium during fragmentation E dep = 300 kcal/mol with 223 precursor molecules added on average per FEBID cycle. Panels B and D show the corresponding views for the the regime of E dep = 205 kcal/mol with 68 precursor molecules added on average per FEBID cycle. The scale bar is the same for all figures. The blue cylinder indicates the PE beam spot. Reproduced from Ref. [86]

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Fig. 8.18 Evolution of the largest cluster in the course of simulation at the end of each FEBID simulation cycle (shown in bold). The number of Pt atoms is also indicated for each structure. Reproduced from Ref. [86]

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Fig. 8.19 A Size distribution of Pt clusters during the first 7 FEBID cycles. B Difference of size distributions for Pt clusters between two consecutive FEBID cycles for the first 7 cycles. Reproduced from Ref. [86]

seven FEBID cycles. In the course of these cycles, small metal clusters start to nucleate and reach the size of about 20–30 atoms. As demonstrated in Fig. 8.18, clusters of substantially larger size are formed during the several follow-up FEBID cycles by merging the clusters of similar size, which have been formed over the first seven cycles. The distributions shown in Fig. 8.19 are obtained at the end of the irradiation stage at each FEBID cycle. Positive values in Fig. 8.19B indicate an increased number of clusters of a given size in comparison with the previous cycle; negative values indicate a decreased number of the clusters of such size. As more irradiation cycles are performed and more Pt atoms are accumulated on the surface due to replenished precursors, the Pt-containing structures start to merge and consistently increase in size. The size distributions shown in Fig. 8.19A are peaked at the number of platinum atoms N = 2 for all the FEBID cycles considered. This feature is attributed to the constant addition of precursor molecules during the replenishment stage at each FEBID cycle. During the following cycles, Pt-containing structures are formed mainly via coalescence of larger clusters containing about 20– 30 Pt atoms. When the clusters reach a certain size, they become less mobile and behave as centers of attraction for new molecules.

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Fig. 8.20 A Atomic content of the deposited nanostructure is analyzed by splitting the nanostructure into two layers of 0.7 nm thickness each, corresponding to the height of a Pt(PF3 )4 monolayer. B Relative Pt content in the beam spot area for the first two layers of 0.7 nm thickness as a function of electron fluence. Panels C and D show the evolution of the radial distribution of relative Pt content for the first and second layers, respectively. Reproduced from Ref. [86]

The height and the metal content of nanostructures are the main FEBID characteristics, measured experimentally. The experimental results [87] indicate that the growth rate and atomic content of the deposited material strongly depend on the electron flux and the amount of precursor molecules. As shown in Fig. 8.17, the metal clusters in the simulations grow nearly isotropically in the (x y)-plane parallel to the substrate surface, but the cluster size and morphology depend on the radial distance from the beam center. Thus, the height and the relative Pt content of the deposited nanostructures are evaluated in concentric bins with a width of 1 nm around the beam spot axis. The nanostructure growth during the FEBID process is characterized by the atomic content of the deposited material. The relative atomic content is calculated considering all atoms in the beam spot area layer by layer. The thickness of each layer is set to 0.7 nm corresponding to the height of Pt(PF3 )4 monolayer deposited on SiO2 (see Fig. 8.20A). The relative Pt content is calculated by dividing the number of Pt atoms by the total number of atoms in the considered volume. The evolution of the average Pt content in the first two layers with the number of FEBID cycles is presented in Fig. 8.20B–D. Panel B shows the dependence of the relative Pt content as a function of electron fluence. Figure 8.20C and D show the evolution of the radial distribution of Pt atoms for the first two layers.

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Fig. 8.21 Evolution of the height of the grown Pt-containing structure at different FEBID simulation cycles for E dep = 300 kcal/mol. A Maximum height of deposited Pt atoms within concentric bins of 1 nm thickness from the electron beam axis at the end of each simulated FEBID cycle. B Maximum height of Pt structures within the distance of 3 nm from the electron beam axis as a function of electron fluence fitted by a linear function. Reproduced from Ref. [86]

Figure 8.20B reveals that the relative Pt content in the first layer increases linearly during the first 11 FEBID cycles and then starts to grow slower, presumably coming to saturation. This indicates that the formation of the first layer has not been completed within the 20 FEBID cycles. The first layer continues to undergo structural transformations that are expected to last until the metal clusters merge into a single structure. Afterward, the evolution of the structure will concern mainly the second and further layers. The Pt distribution within the first layer is mainly homogeneous within the first 12 simulation cycles, while a more dense region proximal to the beam axis appears after the 13th cycle. This region corresponds to the location of the largest cluster in the simulation. Platinum atoms start to fill the second layer at the fifth cycle at the edge of the beam spot. The thickness of the deposited material is calculated by the maximum z coordinate of Pt atoms within the concentric bins. Figure 8.21A shows the evolution of the dependence of the nanostructure height as a function of radius from the PE beam axis in the course of the FEBID simulation. One can distinguish the two regions, one—in the center of the beam spot and another one closer to its edge (see also Fig. 8.20A). The larger height of the structures at distances 4–6 nm from the beam center arises due to the presence of attached PF3 ligands, which do not allow dense packing of the Pt clusters. The height of the deposited material in the center of the beam spot (at radial distances below 3 nm) as a function of electron fluence is shown in Fig. 8.21B.

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8.6 Conclusion This chapter has reviewed the novel computational multiscale modeling approach based on reactive molecular dynamics [28] and Irradiation-Driven Molecular Dynamics (IDMD) [27], the novel computational technique enabling atomistic simulations of the irradiation-driven transformations of complex Meso-Bio-Nano (MBN) systems exposed to various radiation modalities. IDMD relies on several input parameters with a clear physical meaning, such as the bond dissociation energies, molecular fragmentation cross sections, amount of energy transferred to the system upon irradiation, energy relaxation rate, and spatial region wherein the energy is relaxed. Major transformations of irradiated molecular systems (such as topological changes, redistribution of atomic partial charges, alteration of interatomic interactions) and possible paths of their further reactive transformations can be simulated by means of MD with reactive force fields, particularly with the rCHARMM force field [28] implemented in the MBN Explorer software package. The utilization of IDMD and reactive MD using the rCHARMM force field has been illustrated through several exemplary case studies related to irradiation-induced fragmentation of molecular and biomolecular systems, and to controlled fabrication of nanostructures using the Focused Electron Beam-Induced Deposition (FEBID). The results presented in this chapter demonstrate that IDMD provides a powerful computational tool to model the irradiation-induced transformations of complex MBN systems at the atomistic level of detail. The IDMD simulations of the FEBID process demonstrate great predictive power, yielding the morphology of the simulated metal nanostructures, their composition, and growth characteristics in good agreement with available experimental data. The presented multiscale methodology opens a broad range of possibilities for modeling irradiation-driven modifications and chemistry of complex molecular systems. This methodology provides a wide range of possibilities for atomistic-level study of FEBID and many other processes in which the irradiation of molecular systems and irradiation-driven chemistry play the key role. Acknowledgements The authors are grateful to Deutsche Forschungsgemeinschaft for the partial financial support of this work (Project no. 415716638). The authors gratefully acknowledge the possibility to perform computer simulations at the Goethe-HLR cluster of the Frankfurt Center for Scientific Computing.

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Chapter 9

Multiscale Approach for the Physics of Ion Beam Cancer Therapy Eugene Surdutovich, Alexey V. Verkhovtsev, and Andrey V. Solov’yov

Abstract This chapter provides a comprehensive review of the MultiScale Approach (MSA) to the physics of radiation damage with ions and its application in ion beam cancer therapy. The assessment of radiation damage is based on a series of effects that take place on a variety of scales in time, space, and energy starting from ion entering tissue and ending with analysis of the irreparable DNA damage events that lead to cells inactivation. The MSA allows one to predict survival probabilities for cells irradiated with ions. These probabilities depend on types of cells, type of ions, the fluence of the beam, the depth in the tissue, the initial energy of ions, the concentration of oxygen in tissue, and other external conditions. According to MSA, the scenario of radiation damage with ions includes the ion-induced shock waves. Predictions of the calculated quantities are compared with different sets of experiments including involving different cell lines. The chapter shows applications of MSA to the analysis of dependence of cell survival probability on the depth in the case of spread-out Bragg peak and the dependence of relative biological effectiveness on the linear energy transfer, and the overkill effect in particular. The application of the MSA in medical treatment planning is also discussed.

9.1 Introduction The physics and chemistry of radiation damage caused by irradiation with protons and heavier ions have become a subject of intense interest in recent decades because of the use of ion beams in cancer therapy [1–5]. Ion-beam cancer therapy (IBCT) was

E. Surdutovich Oakland University, Rochester, MI, USA e-mail: [email protected] A. V. Verkhovtsev · A. V. Solov’yov (B) MBN Research Center gGmbH, Altenhöferallee 3, 60438 Frankfurt am Main, Germany e-mail: [email protected] A. V. Verkhovtsev e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 I. A. Solov’yov et al. (eds.), Dynamics of Systems on the Nanoscale, Lecture Notes in Nanoscale Science and Technology 34, https://doi.org/10.1007/978-3-030-99291-0_9

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first realized in the 1950s s as proton-beam therapy after being suggested by Robert R. Wilson in 1946 because of the favourable shape of the depth-dose distribution due to the fundamental difference in the energy deposition profile between charged projectiles and photons. For ions, this shape is characterized by the Bragg peak, which is a sharp maximum in the dose deposition close to the end of their trajectories. This peak corresponds to the maximum in the linear energy transfer (LET), and we will refer to this peak as the Bragg peak. Due to this key feature, IBCT allows a delivery of high doses into tumours, maximising cancer cell destruction, and simultaneously minimising the radiation damage to surrounding healthy tissue. The depth-dose curve can be described by three features: the peak value of LET, the proximal plateau value of LET, and the length (in depth) of a tail distal to the peak. Usually, projectile ions are stripped off electrons. The LET is proportional to the square of charge of the projectile, therefore, ions heavier than protons have a taller Bragg peak. This makes the use of heavier ions more desirable. However, the corresponding increase of LET in the plateau region and the increasing size of the tail hinder the usage of heavier ions. As a result, carbon ions, besides protons, are the most clinically used modality [3, 4]. Despite its high cost, proton-beam therapy is widely spread around the world with about 100 operational centres.1 More proton centres are under construction. Although heavy ion therapy was adopted in the 1990s, there are only 12 clinical centres (in Austria, China, Germany, Italy, and Japan) where carbon ions are used [6]. Physically, the Bragg peak appearance is explained by the increase of crosssections of inelastic interactions of projectiles with the molecules of the medium as the speed of the projectile decreases; the cross-sections reach their maximum values just before they sharply drop to zero. As a result, the deposition of destructive energy to the tissue per unit length of the ion’s path is maximized within 1 mm of the end of ion’s trajectory. The location of the Bragg peak depends on the initial energy of ions. Typical depths for carbon ions (in liquid water representing tissue) range from about 2.5 to 28 cm as the initial energy ranges from 100 to 430 MeV/u [4, 7–11]. Hence, a deeply-seated tumour can be scanned with a well-focused pencil beam of ions with minimal lateral scattering. Over the past 30 years, technological and clinical advances of IBCT have developed more rapidly than the understanding of radiation damage with ions. Although empirical approaches have produced exciting results for almost 200 hundred thousand patients thus far, many questions concerning the mechanisms involved in radiation damage with ions remain open and the fundamental quantitative scientific knowledge of the involved physical, chemical, and biological effects is, to a significant extent, missing. Indeed, the series of works that elucidated the importance of low-energy (below ionization threshold) electrons appeared in ca. 2000, while the treatment of patients at GSI2 started in 1997. The dominant molecular mechanism of a double strand break (DSB), the most important DNA lesion [12, 13], still remains unknown. Even the significance of the relation of DNA damage (including DSBs) compared 1 2

As of April 2021 [6]. Gesellschaft für Schwerionenforschung, Darmstadt, Germany.

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to the damage of other cellular components to the cell inactivation or sterilization is not entirely clear. This list can be continued. Besides IBCT, the mechanisms of biodamage due to irradiation with heavy ions have attracted attention in regards to radioprotection from galactic cosmic rays, especially during potential long-term space missions [4]. Over many decades of using radiation with photons, vast data relating the radiation damage to deposited dose were accumulated. These data are currently used to describe the biological damage due to ions [4]. Nonetheless, there are substantial qualitative and quantitative differences between the effects of ions and photons on tissue. The first difference is in the localization of the dose distribution for ions distinguished from the mostly uniform dose distribution for photons. This feature reveals itself longitudinally (along the ion’s path) as the Bragg peak. Radially (with respect to the ion’s path), it shows up as the sharply decreasing (within several tens of nm) radial dose distribution, while the average distance between adjacent ions in clinically used beams is several hundreds of nm. The second difference is a consequence of the first. Secondary particles such as electrons, free radicals, etc., produced as a result of the interaction (ionization and excitation) of ions with the medium, emerge at the location of the Bragg peak in much larger number densities than those produced by photons, and their distribution is also non-uniform. These secondary particles are largely responsible for biological damage, and in order to assess the damage, it is important to distinguish the biological effects of the locally deposited dose and the local number density of secondary particles. In other words, the (radial) dose is not the only characteristic that determines the biological damage. For instance, clustered damage, deemed to be more lethal than isolated damage, can be caused by several low-energy electrons, which are not associated with a large dose deposition. This qualitatively and quantitatively changes the effect of the radiation [1, 4, 14]. There are also differences in the chemical interactions related to a different balance between free electrons, free radicals, and other agents for ions versus photons. These differences, for example, affect the resistivity of cells to radiation and thus are important for the assessment of radiation damage. Finally, the Bragg peak leads to thermomechanical effects, which stem from large gradients in the radial dose deposition. One of the most important questions in the foundation of science devoted to radiation damage with ions is the question about molecular mechanisms leading to DNA damage, or more generally, biodamage. While “whether the biodamage leads to cell sterilization?” is a biological question, the question about mechanisms of biomolecular damage belongs to the realms of physics and chemistry. The role of low-energy (sub-15 eV) electrons has been especially emphasized in Refs. [15– 18]. A number of quantum effects, such as dissociative electron attachment (DEA), formation of electronic and phononic polarons, are discussed in the context of the interaction of these electrons with biomolecules. DEA is deemed to be the leading mechanism for DNA single strand breaks (SSBs) at low energies, while a number of ideas, including the action of Auger electrons, in relation to the mechanism of double strand breaks (DSBs) has been suggested [17, 19]. The Auger effect along

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Fig. 9.1 Features, processes, and disciplines, associated with radiation therapy shown in a space– time diagram, which shows approximate temporal and spatial scales of the phenomena. The history from ionization/excitation to biological effects on the cellular level is shown in the main figure, and features of ion propagation are shown in the inset. Reproduced from Ref. [30] with permission from Springer Nature

with intermolecular Coulombic decay (ICD) is discussed not only in relation to the mechanism of DSBs but also as important channels for the production of secondary electrons, especially in the presence of nanoparticles as sensitizers [20, 21]. Still more understanding is needed for the interaction of electrons of higher energies. This chapter is devoted to the overview of the main ideas of the multiscale approach to the physics of radiation damage that has been designed with the goal of developing knowledge about biodamage on the nanoscale and molecular level and finding the relations between the characteristics of incident particles and the resultant biological damage [1, 22]. This approach is unique in distinguishing essential phenomena relevant to radiation damage at a given time, space, or energy scale and assessing the resultant damage based on these effects.

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The MultiScale Approach (MSA) was formulated and then elaborated upon, as different aspects of the scenario were added in a series of works [10, 11, 14, 22–35]. A number of review papers were written on different stages of the development [30, 36, 37]. The name of the approach emphasizes the fact that important interactions involved in the scenario happen on a variety of temporal, spatial, and energy scales. Temporal and spatial scales are schematically shown in Fig. 9.1. Because of the nature of the subject, most of other methods devoted to assessment of biological damage are multiscale as well [38–43]. What singles this approach is its inclusiveness with respect to effects that happen on all scales relevant to the radiation damage. From the very beginning, the approach was formulated as phenomenon-based and was aimed at elucidating the physical, chemical, and biological effects that are important or dominating on each scale in time, space, and energy. The practical goal of the MSA is the calculation of survival probabilities for cells irradiated with ions. These survival probabilities are directly related to the relative biological effectiveness (RBE) [3, 4, 44, 45], one of the key integral characteristics of the effect of ions compared to that of photons. The RBE is defined as a ratio of doses delivered by photons and those by given projectiles leading to the same biological effect, such as inactivating a given percentage of cells in an irradiated region. This is why the calculation of survival probabilities is so important. The path to the calculation of survival probabilities has been marked out in Ref. [30]. Then, in Ref. [32], the calculations were successfully compared with a series of experiments and the linear-quadratic model coefficients were calculated on the basis of the MSA. Further progress with the analysis of RBE and the overkill effect has been reported in Ref. [35]. The oxygen enhancement ratio (OER), which compares the biological action of given projectiles to that at different aerobic or hypoxic conditions of irradiated targets, is calculated as a byproduct since the survival probabilities are calculated at different conditions [32]. This chapter is organized in the following way. In Sect. 9.2, the scenario of radiation damage with ions is described. Section 9.3 is devoted to the ion’s transport in the medium. The Bragg peak details are discussed as well as the secondary electron production. Section 9.4 is devoted to the transport of secondary electrons. Radial dose equal to pressure is also derived there. Section 9.5 is devoted to thermomechanical effects, which play an important role in the scenario of radiation damage with ions. Section 9.6 is a quintessential part where the distributions and fluences of reactive species are calculated for different values of LET. Then, the calculation of cell survival curves is done on this basis. Section 9.7 presents two examples of applications of MSA. It is followed by conclusions and outlook.

9.2 Multiscale Scenario of Radiation Damage Radiation damage due to ionizing radiation is initiated by ions incident on tissue. Initially, they have energy ranging from a few to hundreds of MeV. In the process of propagation through tissue, they lose their energy in the processes of ionization, excitation, nuclear fragmentation, etc. Most of the energy loss of the ion is transferred

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to tissue.3 Naturally, radiation damage is associated with this transferred energy, and the dose (i.e., deposited energy density) is a common indicator for the assessment of the damage [1, 4, 44]. The profile of the LET along the ion’s path is characterized with a plateau followed by a sharp Bragg peak. The position of this peak depends on the initial energy of the ion and marks the location of the maximum radiation damage. In the process of radiation therapy, a tumour is being “scanned” with the Bragg peak both laterally and longitudinally. The active longitudinal scanning is achieved by changing of initial energy of projectiles, while passive is achieved by using scatterers of variable thickness in front of the target.4 in order to deposit a large dose to the target and spare healthy tissues surrounding it. However, the deposition of large doses in the vicinity of the Bragg peak does not explain how the radiation damage occurs, since projectiles themselves only interact with a few biomolecules along their trajectory and this direct damage is only a small fraction of the overall damage. It is commonly understood that the secondary electrons and free radicals produced in the processes of ionization and excitation of the medium with ions are largely responsible for the vast portion of the biodamage. Secondary electrons are produced during a rather short time of 10−18 –10−17 s following the ion’s passage. The energy spectrum of these electrons has been extensively discussed in the literature [10, 11, 46–48], and the main result (relevant for this discussion) is that most secondary electrons have energy below 50 eV (more than 80% for an ion energy5 of 0.3 MeV/u) and only a few (less than 10% for 0.3 MeV/uions) have energy higher than 100 eV. Moreover, this is true for a very large range of ion energy. This has several important consequences. First, the ranges of propagation of these electrons in tissue are rather small, around 10 nm [49]. Second, the angular distribution of their velocities as they are ejected from their original host, and as they scatter further, is largely uniform [41]; this allows one to consider their transport using a random walk approach [14, 19, 22, 50, 51]. The next time scale 10−16 –10−15 s corresponds to the propagation of secondary electrons in tissue. These electrons (which start with about 45–50 eV energy) are called ballistic. In liquid water, the mean free paths of elastically scattered and ionizing 50-eV electrons are about 0.43 and 3.5 nm, respectively [41]. This means that they ionize a molecule after about seven elastic collisions, while the probability of second ionization is small [10]. Thus, the secondary electrons are losing most of their energy within first 20 collisions and this happens within 1–1.5 nm of the ion’s path [28]. After that they continue propagating, elastically scattering with the molecules of the medium until they get bound or solvated electrons are formed. It is important to notice that these low energy electrons remain important agents for biodamage since they can attach to biomolecules like DNA causing dissociation [18, 52]. The solvated electrons may play an important role in the damage scenario as well [13, 53, 54]. 3

The only part that is not transferred is emitted as radiation. This part, in the case of ions interacting with tissue, is deemed to be insignificant. 4 The longitudinal scanning produces the so-called spread-out Bragg peak (SOBP). 5 This value corresponds to the kinetic energy of ions near the Bragg peak.

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Additionally, the energy lost by electrons during the previous stage in the processes of ionization, excitation, and electron–phonon interaction is transferred to the medium. As a result of this relaxation, the medium within about a 1–1.5-nm cylinder (for ions not heavier than iron) around the ion’s path becomes very hot [25, 28]. This cylinder is referred to as the hot cylinder. The pressure inside this cylinder increases by a factor of about 103 compared to the pressure in the medium outside the cylinder. This pressure builds up by about 10−14 –10−13 s and it is a source of a cylindrical shock wave [27]. This shock wave propagates through the medium for about 10−13 – 10−11 s. Its relevance to the biodamage is as follows. If the shock wave is strong enough (the strength depends on the distance from the ion’s path and the LET), it may inflict damage directly by breaking covalent bonds in a DNA molecule [28]. Besides, the radial collective motion that takes place during this time is instrumental in propagating the highly reactive species such as hydroxyl radicals, just formed solvated electrons, etc. to a larger radial distance (up to tens of nm) thus increasing the area of an ion’s impact. The assessment of the primary damage to DNA molecules and other parts of cells due to the above effects is done within the MSA. This damage happens within 10−5 s from the ion’s passage and consists of various lesions on DNA and other biomolecules. Some of these lesions may be repaired by the living system, but some may not and the latter may lead to cell sterilization. The scenario described above is illustrated in Fig. 9.2.

9.3 Propagation of Ions in Tissue and Primary Ionization of the Medium 9.3.1 The Main Characteristics of Ion’s Propagation in the Medium The scenario starts with the traverse of an ion through tissue. Ions enter the medium with a sub-relativistic energy (for therapy, the carbon ion energy ranges through 100–420 MeV/u and the proton energy can be up to 250 MeV, while the ions of galactic cosmic rays are much more energetic). Then, the ions lose energy propagating in the tissue. This process is described by the stopping power, S, of the medium, equal to −d E/d x, where E is the kinetic energy of the ion and x is the longitudinal coordinate. For projectiles such as protons or heavier ions, there is not much difference between the location of the energy loss by projectiles and that absorbed by the medium longitudinally, i.e., along the ion’s path.6 Therefore, the linear energy transfer (LET), i.e., the energy absorbed by the medium per unit length of the projectiles’s trajectory becomes similar to the stopping power. Hence, the terms “LET” 6

This is so because the energy is mostly transferred to electrons and other secondary particles, whose longitudinal ranges are many times smaller than the characteristic scale of x.

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Fig. 9.2 The scenario of biological damage with ions. Ion propagation ends with a Bragg peak, shown in the top right corner. A segment of the track at the Bragg peak is shown in more detail. Secondary electrons and radicals propagate away from the ion’s path damaging biomolecules (central circle). They transfer the energy to the medium within the hot cylinder. This results in the rapid temperature and pressure increase inside this cylinder. The shock wave (shown in the expanding cylinder) due to this pressure increase may damage biomolecules by stress (left circle), but it also effectively propagates reactive species, such as radicals and solvated electrons to larger distances (right circle). A living cell responds to all shown DNA damage by creating foci (visible in the stained cells), in which enzymes attempt to repair the induced lesions. If these efforts are unsuccessful, the cell dies; an apoptotic cell is shown in the lower right corner. Reproduced from Ref. [30] with permission from Springer Nature

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and “stopping power” are used synonymously. The energy loss occurs due to ionization of the medium, nuclear fragmentation in collisions with nuclei, excitations of the medium, etc. The LET profile for ions is characterized by a plateau followed by the sharp Bragg peak, where the LET reaches its maximum. The tail is caused by the energy loss of the lighter products of nuclear fragmentation, such as protons, neutrons, α-particles, etc. The behaviour of the LET is explained by features of inelastic cross-sections of the projectile in the medium. The Bragg peak in the stopping power of massive charged particles is described by the Bethe-Bloch formula [56–58]:   2 mV 2 4π n e z 2 e4 dE 2 ln = −β , − dx mV 2 I (1 − β 2 )

(9.1)

where m and e are the mass and charge of electron, V is the velocity of the projectile, β = V /c (c is the speed of light in vacuum), z|e| is the charge of projectile, n e is the number density of electrons in the target, and I  is the mean excitation energy of its molecules. This formula provides the dependence of the stopping power on the energy of the ion and practically depends on a single parameter, the mean excitation energy. This parameter for liquid water is chosen empirically somewhere between 70 and 80 eV [9, 59]. The use of such a parameter is sufficient for the calculations of the position of the Bragg peak and its shape, and Eq. (9.1) is used in many Monte Carlo (MC) simulations [9] for that purpose. This parameter, however, hides all physical processes such as ionization and excitation of the medium, even though these same processes are important for the understanding of the scenario of radiation damage. In Refs. [10, 11, 22], a different approach has been used and the singly-differentiated (with respect to the secondary electron energy) ionization cross-sections of water molecules in the medium have been employed as a physical input. This uncovers the physics integrated in the empirical parameter and allows not only describing the features of the Bragg peak but also obtaining the energy spectrum of secondary electrons, which are very much involved in subsequent radiation damage.

9.3.2 Singly-Differentiated Cross-Sections of Ionization The total ionization cross-section, σt , differentiated with respect to secondary electron kinetic energy, W , i.e., singly-differentiated cross-section (SDCS) is the main quantity in our analysis. Besides the kinetic energy of secondary electrons and the properties of water molecules, the SDCS depends on the velocity V of the projectile and its charge, z|e|.

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Calculation of the SDCS Using a Parametric Semiempirical Approach In Refs. [10, 11, 60], the semi-empirical Rudd’s expression [61] for the calculation of SDCS has been used. This analytic expression, containing a number of parameters, is a combination of the experimental data and calculations within the plane-wave Born approximation and other theoretical models [61]. Since this model was developed for non-relativistic protons, it had to be modified to include heavier ions at relativistic velocities. The original SDCS is given in the following form [61]:  4πa 2 Ni dσt 0 = z2 dW I i i ×



I0 Ii

2 (9.2)

F1 (vi ) + F2 (vi )ωi  , (1 + ωi ) 1 + exp(α(ωi − ωimax )/vi ) 3

where the sum is taken over the electron shells of the water molecule, a0 = 0.0529 nm is the Bohr radius, I0 = 13.6 eV, Ni is the shell occupancy, Ii is the ionization potential of the shell, ωi = W/Ii is the dimensionless normalized kinetic energy of the ejected electron, vi is the dimensionless normalized projectile velocity given by vi =

mV 2 . 2Ii

(9.3)

When V  c, V = 2E (where M is the mass of a projectile), and, hence vi = M

m E . When V approaches c, the definition of vi , given by (9.3), holds, however, the M Ii projectile’s velocity V is given by βc, where β 2 = 1 − 1/γ 2 = 1 − (Mc2 /(Mc2 + E))2 , and γ is the Lorentz factor of the projectile. Functions F1 and F2 in (9.2) are given by F1 (v) = A1

ln(1 + v2 ) C 1 v D1 + , 2 2 B1 /v + v 1 + E 1 v D1 +4

(9.4)

and F2 (v) = C2 v D2

A2 v2 + B2 . C2 v D2 +4 + A2 v2 + B2

(9.5)

The fitting parameters A1 ... E 1 , A2 ... D2 , and α depend on the medium. In Ref. [61], they are given for water vapour. The comparison of positions of Bragg peaks for different initial carbon ion energies with those measured in experiments provided sufficient material for refitting of these parameters for liquid water medium [11]. These parameters are listed in Table 9.1. The cut-off energy ωmax is given by

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Table 9.1 Fitting parameters and ionization energies for three outer and two inner shells (symbols 1a1 , 2a1 , 1b1 , 3a1 , and 1b2 represent corresponding molecular orbitals), of water molecules in a liquid water environment [11] Shells Ionization A1 B1 C1 D1 E1 A2 B2 C2 D2 α energies (eV) Outer: 1b1 , 3a1 , 1b2 Inner: 2a1 , 1a1

10.79, 13.39, 16.05 32.3, 539.0

1.02

82

0.5

−0.78 0.38

1.07

14.5

0.61

0.04

0.64

1.25

0.5

1.0

1.0

1.1

1.3

1.0

0.0

0.66

3.0

ωimax = 4vi2 − 2vi −

I0 , 4Ii

(9.6)

where the first term on the right-hand side represents the free-electron limit, the second term represents a correction due to electron binding, and the third term gives the correct dependence of the SDCS for vi  1 [61]. For vi  1, Eq. (9.2) should asymptotically approach the relativistic Bethe-Bloch formula (9.1). This is accomplished when F1 , given by (9.4), is replaced by the following expression, 2

F1 (v) = A1

1+v 2 ln( 1−β 2) − β

B1

/v2

+

v2

+

C 1 v D1 . 1 + E 1 v D1 +4

(9.7)

Indeed, the asymptotic behaviour of (9.7) at v  1 is given by     v2 A1 2 ln − β , v2 1 − β2  which, after being substituted to Eq. (9.2) and the understanding that ddEx ∼ i (W + dσt Ii ) dW dW , leads to Eq. (9.1). The correction of Eq. (9.7) reveals itself as an increase of the cross-section at high energies. An alternative method has been used in Ref. [29], where the dielectric formalism based on the experimental measurements of the energy-loss function (ELF) of the target medium, Im (−1/(E, q)), where (E, q) is the complex dielectric function, and q and E are the momentum and energy transferred in the electronic excitation, respectively [62, 63]. This formalism allows obtaining the SDCS not only for liquid water but for a real biological medium containing sugars amino acids, etc. If the ELF is experimentally known, many body interactions and target physical state effects are naturally included in these calculations. This method is reproduced in Ref. [36].

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9.3.3 The Position of the Bragg Peak The stopping cross-section, defined as σst =

 i

∞

(W + Ii )

0

dσt,i dW , dW

(9.8)

where the sum is taken over all electrons of the target, gives the average energy lost by a projectile in a single collision, which can be further translated into energy loss within an ion’s trajectory segment, d x: S(E) = −

dE = nσst (E) , dx

(9.9)

where n is the number density of molecules of the medium. This quantity is known as the stopping power [44, 59]. As was discussed above in Sect. 9.3.1, for ions this quantity is similar to the linear energy transfer (LET). The LET found from Eq. (9.9) is a function of the kinetic energy of the ion rather than the ion’s position along the path in the medium. The dependence of LET (and other quantities) on this position, however, is more suitable for cancer therapy applications. Integrating inverse LET, given by (9.9), yields7

E0

x(E) = E

d E , |d E  /d x|

(9.10)

where E 0 is the initial energy of the projectile. We obtain the correspondence between the position of the ion along the path and its energy. This allows one to obtain all quantities of interest in terms of x rather than E. The depth dependence of the average LET (stopping power S) as a function of x is shown in Fig. 9.3. The calculations of the LET include the effects that were discussed above, such as SDCS calculated using semi-empirical parametrization (9.2), modified for relativistic energies (9.7) with the use of the effective charge described below in Sect. 9.3.4. The effect of energy straggling due to multiple ion scattering, described in the Sect. 9.3.5 is also taken into account. This effect explains why the height of the Bragg peak decreases with the increasing initial energy of ions and thus increasing depths of the corresponding Bragg peaks. The contribution of non-ionization processes, such as excitation of neutral molecules, is also included in these calculations. In order to accomplish this, the excitation cross-sections for proton projectiles [65] were scaled using the ratio of the effective charges for carbon and proton at a given energy E. In Fig. 9.3, our calculated LET is compared with the experimental results [7]. As can be seen from the figure, the experimental dots at the Bragg peak are systematically lower than the calculated curve, the difference being due to in the nuclear fragmentation component, which has not been included in the analytical calculations. 7

This is known as the continuous slowing down approximation (CSDA) range [64].

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Fig. 9.3 The dependence of the LET on depth with the Bragg peak, plateau, and tail for carbon ions in liquid water. The calculations (solid line) are done for ions with the initial energy of 330 MeV/u and with use of Eqs. (9.8) and (9.9). Experimental results [7] for the same energy are shown with dots. The dashed line depicts the LET dependence without the effect of energy straggling. In the inset, two almost coinciding curves show the agreement between the analytical calculations and MC simulations [9] for 420 MeV/u carbon ion projectiles with straggling being included. Reproduced from Ref. [30] with permission from Springer Nature

It is feasible to include it, as has been done in Ref. [66] for protons, if the appropriate fragmentation cross-sections are known. As confirmed by MCHIT MC simulations [9], nuclear fragmentation reactions become important for heavy-nuclei beams and deeply-located tumours. For example, both experimental data [7] and MCHIT calculations [9] indicate that more than 40% of primary 200 MeV/u 12 C6+ nuclei undergo fragmentation before they reach the Bragg peak position, and this fraction exceeds 70% for a 400 MeV/u 12 C6+ beam. As a result of nuclear reactions the beam is attenuated. New projectiles such as protons, neutrons, and α-particles are formed. Since these particles are lighter than the incident ions, after fragmentation they carry a larger portion of the energy and their penetration depths are larger than that of the original ions [7]. This results in a tail after the Bragg peak also seen in Fig. 9.3.

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9.3.4 Charge Transfer Effect The incident ions are usually stripped of all electrons, but as they slow down they pick electrons off and their charge reduces. The dependence of the charge of ions on their velocity has been suggested by Barkas [67], where the following empirical formula for the effective charge, z eff , is introduced, z eff = z(1 − exp(−125βz −2/3 )) ,

(9.11)

where z is the charge of the stripped ion. This formula is a result of studies of energy loss of ions in emulsions. More detailed descriptions of charge transfer effects have become available recently [68]. These studies allow one to not only estimate the effective charge of the ion but also find its fluctuations. These fluctuations are important since LET increases proportionally as z 2 and if LET becomes large enough, qualitative differences related to thermomechanical effects may become substantial (see Sect. 9.5 below). Regardless of the method of the calculation of the effective charge, in order to find the stopping power and estimate the secondary electron spectra (in the first approximation) z in Eq. (9.2) should be replaced by an effective charge z eff , which decreases with decreasing energy making the ionization cross-section effectively smaller. In Refs. [10, 11] the parameterization (9.11) was used. The effective charge given by this expression slowly changes at high projectile velocity but rapidly decreases in the vicinity of the Bragg peak. As a result, charge transfer significantly affects the height of the Bragg peak, and only slightly shifts its position towards the projectile’s entrance. This happens because the stopping cross-section as a function of velocity 2 . has a sharp peak as velocity decreases. At the same time, σst is proportional to z eff If the latter decreases with decreasing V , the Bragg peak shifts towards the direction of the beam’s entrance to the tissue. For instance, with the account for charge transfer, for carbon ions, the Bragg peak occurs at E = 0.3 MeV/u rather than at E = 0.1 MeV/u.

9.3.5 The Effect of Ion Scattering Tracks of ions emerging from clinically used accelerators do not interfere, i.e., the effects of a single ion do not spread far enough to reach the area affected by adjacent ions. Therefore, it is usually sufficient to study a single ion interacting with tissue and then combine these effects relating the action of the beam with the dose. Even though the Bragg peak is a feature of every ion’s LET, each peak cannot be observed separately. Since each of the projectiles in the beam experiences its own multiple scattering sequence, peaks for different ions occur at a slightly different spatial location and only the Bragg peak, averaged over the whole beam, is observed

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experimentally. Therefore, in order to compare the shape of the Bragg peak with experiments, the whole ion beam should be considered. In Ref. [10], the Bragg peak for an ion beam was obtained via introduction of the energy-loss straggling due to ion scattering. The energy straggling, described by a semi-analytical model [69], is given by 

  

x0 dE 1 dE  (x  − x)2 dx , (x) = (x ) exp − √ dx 2λ2str λstr 2π 0 d x

(9.12)

where x0 is a maximum penetration depth of the projectile and λstr = 0.8 mm is the longitudinal-straggling standard deviation computed by Hollmark et al. [70] for a carbon ion of that range of energy. The Bragg peak shown in Fig. 9.3 was calculated using Eq. (9.12).

9.3.6 Energy Spectra of Secondary Electrons The most important effect that takes place during the propagation of the ion in tissue is the ionization of the medium. This is how, when, and where the secondary electrons, the key player in the scenario of radiation damage, are produced. The information, required for the understanding of phenomena related to secondary electrons, is the number of electrons produced per unit length of the ion’s trajectory and their energy distribution. This section is devoted to the analysis of the electron energy distributions obtained from ionization cross-sections discussed above. The emission of electrons in collisions of protons with atoms and molecules has been under theoretical and experimental investigation for decades [48, 61, 71, 72]. The quantity of interest is the probability to produce Ne secondary electrons with kinetic energy W , in the interval dW , emitted from a segment x of the trajectory of a single ion at the depth x corresponding to the kinetic energy of the ion, E. This quantity is proportional to the singly-differentiated cross-ionization section (SDCS),8 discussed in Sect. 9.3.2. d Ne (W, E) dσt = n x dW dW

(9.13)

where n is the number density of molecules of the medium (for water at standard conditions n ≈ 3.3 × 1022 cm−3 ). Equation (9.13) relates the energy spectrum of secondary electrons to the SDCS regardless of the method, by which the latter are obtained. One important characteristics that can be obtained from the SDCS is the average energy of the secondary electrons, W , which is given by

8

The SDCS are integrated over full solid angle of electron emission.

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W (E) =

1 σt

∞

W 0

dσt dW . dW

(9.14)

The dependence of W  indicates that the energy of secondary electrons is somewhere below 50 eV for the whole range of the ion’s energy and it levels out as the energy of projectiles increases. There are several consequences from this. First, since the dependence of W  on the ion’s energy E on a relevant range of projectile energies (0.3–400 MeV/u) is weak for the large range of the ion’s energy, the number of produced secondary electrons is largely proportional to the value of LET, more precisely to the electronic component of the LET, Se , that excludes nuclear stopping. Indeed, if the ion is destroyed in a nuclear collision, ionization due to its debris should be discussed instead; if it survives then its ionizing capabilities do not change too much, unless it slows down considerably; then, its stopping power may change correspondingly. Second, the expression for W  is independent of the charge of the projectile, e.g., the difference between, say, protons and iron ions is in their values of Se , i.e., in the number of secondary electrons, but not in their relative energy spectra. Therefore, the difference between the effects of these different ions will be in the number of secondary electrons produced by these ions per unit length of path. Third, most of the secondary electrons are capable of ionizing just one or two water molecules; thus, there is no significant avalanche ionization effect [10]. This can be explained by a simple estimate. Since the average energy of secondary electrons in the vicinity of the Bragg peak is about 40 eV (somewhat below this value), the maximum average energy that can be transferred to the next generation secondary electron is just (40 eV − Ii )/2, which is about 15 eV for the outermost electrons, an energy barely enough to cause further ionization. Finally, what is of crucial importance for the consideration of the next scale of electron propagation is that at sub-50 eV energies, the electrons’ cross-sections are nearly isotropic [41, 73] and it is possible to use the random walk approximation in order to describe their transport [14, 22, 50, 51]. This transport is described in the next section.

9.4 Transport of Secondary Electrons in Ion Tracks The next stage of the scenario is related to secondary electrons ejected from the molecules of the medium as a result of ionization. As has been discussed above, most of these electrons have energies below 50 eV. They are called ballistic electrons until their energy becomes sufficiently small and coupling with phonons, recombination, and other quantum processes start dominating their transport. While the electrons are ballistic, their interactions with molecules can be described as a sequence of elastic and inelastic collisions. Many works, by and large using MC simulations, describe the transport of ballistic electrons. They are known as track structure codes [41]. Some of them describe chemical reactions in the medium including the production of radicals and their propagation. However, regardless of how sophisticated these codes are, they do not contain the whole physical picture as will be shown below.

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In this section, a rather simple analytical approach is applied to the description of the propagation of ballistic electrons and its results are compared to MC simulations. It is also demonstrated how to make sense of radiation damage based on these calculations. The main mechanism of radiation damage by ballistic electrons is inelastic collisions with targets. A target in this discussion is a biomolecule, such as DNA. Therefore, the probability of biodamage is a combination of the number of electrons (or other secondary particles) colliding with a given segment of a biomolecule and the probability of a certain inelastic process on impact. The first part is described by the fluence of electrons or other particles on the target. Fluence is the integral of the flux of particles (the number of particles hitting a part of the target’s surface per unit time) over the entire time after the ion’s passage and over the surface of the target. In general, the fluence depends on the distance of the target from the ion’s path and its geometrical orientation. It will be shown that this part can be calculated analytically with accuracy, sufficient for understanding the scenario of radiation damage. The second part, i.e., the probability of a certain inelastic process on impact, is more difficult to assess mainly because of the diverse variety of possible processes. However, there are plenty of data that allow one to make reasonable quantitative estimates for this probability. We start with the analysis of transport of secondary electrons. We also consider the production and transport of reactive species such as solvated electrons and radicals. Then the fluence for relevant configurations is calculated. It will be shown that important characteristics of the track structure such as radial dose can also be calculated as a byproduct of transport analysis. The random walk approach [31, 74] used for these problems allows one to make simple analytical calculations. The main requirement for the use of this approach is that the elastic and inelastic scattering of secondary electrons is isotropic. The anisotropy in the angular dependence of the cross-sections for sub-50-eV electrons appears to be insignificant [41]. As was noted above, in the Bragg peak region, more than 80% of secondary electrons satisfy this condition and only for less than 10% of δ-electrons with energies higher than 100 eV is this condition violated significantly. This section is largely devoted to the transport of sub-50-eV electrons. Moreover, unless specifically stated to the contrary, these secondary electrons are produced by carbon ions in the vicinity of the Bragg peak in liquid water. At this part of the ion’s trajectory, while a 0.3-MeV/u carbon ion passes 1 µm along the path, a typical radius within which the secondary electrons propagate is about 2–3 nm [31]. This allows one to assume that the electron diffusion is cylindrically symmetric with respect to the ion’s path. The electronic component of the LET, Se , remains nearly constant along this 1 µm of ion’s path described by the coordinate ζ . Therefore, the number of ejected secondary electrons per unit length ddζNe is independent of ζ . A typical elastic mean free path of sub-50-eV electrons ranges between 0.1 and 0.45 nm [41, 73]. Since the scale along the Bragg peak is measured in tens of μm, while the radial scale is only tens of nm, therefore one can assume ζ to be ranging from −∞ to +∞. As an ion passes through tissue, it ionizes molecules and ejected secondary electrons (first generation) start with the average energy of ∼45-eV (the average energy

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of secondary electrons formed in the vicinity of the Bragg peak [60]). These electrons lose most of this energy within 1–1.5 nm of the ion’s path, ionizing more molecules (including biomolecules). The second generation of slower electrons is thus formed. The transport of these two generations can be represented by coupled diffusion equations [31]: n 1 (r, t) ∂n 1 (r, t) = D1 ∇ 2 n 1 (r, t) − , ∂t τ1 ∂n 2 (r, t) n 1 (r, t) n 2 (r, t) = D2 ∇ 2 n 2 (r, t) + 2 − . ∂t τ1 τ2

(9.15)

Here, index “1” marks secondary electrons of the first generation; D1 is their diffusion coefficient, τ1 is their lifetime (before they convert to the second generation); accordingly, D1 = vl/6 = 0.265 nm2 fs−1 and τ1 = lion /v = 0.64 fs (all mean free path data are taken from Refs. [73, 75]. They are capable of ionizing another water molecule. In this process, these electrons lose ionization energy to the medium and share the rest of their energy with electrons of the second generation. We assume the remaining energy of first-generation electrons and the energy acquired by the second-generation electrons to be equal, and we refer to both of these kind of electrons as second-generation electrons. The factor of two that appears in front of the second term on the r.h.s. of the second equation of (9.15) formalizes this assumption. Index “2” in this equation corresponds to the second-generation electrons; their energy is about 15 eV and, correspondingly, D2 = 0.057 nm2 fs−1 and τ2 = 15.3 fs. The solutions of the above equations are [31],   ρ2 t d Ne 1 , exp − − dζ 4π D1 t 4D1 t τ1

1 1 d Ne t n 2 (t, ρ) = 2π τ1 dζ 0 D1 t  + D2 (t − t  )   t − t ρ2 t dt  . − × exp − − 4(D1 t  + D2 (t − t  )) τ2 τ1 n 1 (t, ρ) =

(9.16)

From the solutions (9.16), it follows [31] that a few fs after the ion’s passage, all secondary electrons lose energy and the number density of secondary electrons is by and large given by n 2 (ρ, t). With time the distribution becomes a little broader, but the main effect is the exponential decrease with time. As a result of these decrease, the so-called pre-solvated electrons are formed. A pre-solvated stage of electrons is a transition stage between low-energy ballistic electrons and a relatively stable compound of electrons with water molecules known as solvated electrons. This transition takes about 1 ps. It is also possible to assess the energy deposition density (dose), ε(ρ, t), in the medium by ions and secondary electrons. In order to do this, we assume that the average energy, w, ¯ is deposited to the medium with each ionization. Moreover, since the first-generation electrons start with about 45 eV and w¯ ≈ 15 eV, the

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second-generation electrons energy is also approximately (45 − 15)/2 = 15 eV. Then, the rate of energy deposition is proportional to the rate of inelastic events,   d Ne (2) n 1 (ρ, t) n 2 (ρ, t) ∂ε(ρ, t) . = w¯ δ (ρ)δ(t) + + ∂t dζ τ1 τ2

(9.17)

The first energy deposition occurs right at the ion’s path where the molecules are ionized by the ion; this corresponds to the first term on the right-hand side of (9.17), where δ’s are the corresponding δ-functions. The second deposition (second term on the r.h.s.) is the ionization by secondary electrons at the end of their ionization mean free paths. Finally, the third deposition is due to remaining energy loss due to excitation of molecules by the electrons of second generation. After that, the electrons enter the pre-solvated stage. The time integration of (9.17) gives the dependence of the radial dose on time. The radial dose distributions at times 5, 10, 20, and 50 fs are shown in Fig. 9.4. At small radii, the distribution is due to primary ionization and it does not change with time. At larger radii, the dose slowly increases because of energy loss by the second generation of electrons. The shown results are obtained with w¯ = 16.5 eV, which corresponds to the normalization

∞ 0

0

tr

∂ε(ρ, t) dt2πρdρ = Se , ∂t

(9.18)

where Se = 890 eV/nm is the LET of a single ion at the Bragg peak for carbon ions [10] and tr is the time by which most of electrons stop being ballistic (this time is taken to be 50 fs). In Fig. 9.4, the results of integration of (9.17) are compared with the radial dose of Ref. [76]. While the shapes of these distributions are alike, the absolute values are different by the factor of about four. Part of this disagreement (factor of about 2 [10]) can be explained by the fact that in [76], the radial dose presented for 2-MeV/u, i.e., proximal of the Bragg peak for carbon ions. The remaining factor can be due to the energy straggling present in Ref. [76], but absent for a single ion data of (9.17). Also, while the solid line represents the radial dose at 50 fs, in Ref. [76] there is no information about the time. This is typical for Monte Carlo simulations, but since they are compared with experimental data the time corresponding to dots is likely to be on a ps scale. Curiously, the solid curve in Fig. 9.4 gives the upper estimate of pressure near the ion’s path. Indeed, the energy U (ρ) inside a cylinder of radius ρ and length L, coaxial with the ion’s path, is given by

ρ

U (ρ) = 0

where ε(ρ) is the radial dose given by

ε(ρ  )2πρ  dρ  L ,

(9.19)

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Fig. 9.4 The radial dose (in MGy) distributions as functions of the distance from the ion’s path at times 50, 20, 10, and 5 fs are shown with solid and dashed lines with a diminishing dash size, correspondingly. The solid line also represents the initial radial distribution of hydroxyl radicals (the scale in on the right). The corresponding labels are shown on the right side of the frame. Solid squares mark the radial dose data for 2-MeV/u [76] multiplied by the factor of four. Empty circles represent the radial dose by a single generation of secondary electrons calculated in Ref. [50]. Reproduced from Ref. [31] with permission from Springer Nature

ε(ρ) = 0

tr

∂ε(ρ, t) dt. ∂t

(9.20)

Then, the force, normal to the surface of this cylinder is given by F(ρ) = −

∂U (ρ) = ε(ρ)2πρ L , ∂ρ

(9.21)

and the pressure P(ρ) is P(ρ) =

F(ρ) = ε(ρ). 2πρ L

(9.22)

Thus, the radial dose is equal to the pressure in the medium. This notion as well as the results shown in Fig. 9.4 justify the idea of the ion-induced shock waves that will be discussed in the next section. The pressure given by the initial (at 50

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fs) radial dose is used as the initial condition for the pressure [28, 30, 77]. The time ≤50 fs within which the radial dose distribution evolved is sufficient for the temperature and pressure distributions to form around the ion’s path following the process of energy relaxation, in which the energy stored in electronic excitations is transferred into vibrational excitations and then to translational degrees of freedom. The characteristic time for the decay of an electronic excitation in Na clusters is estimated to be about 0.4 ps [78], and it is likely to be several times longer for the liquid water, which is consistent with the analysis performed in Ref. [25]. The units MGy used in Fig. 9.4 correspond to GPa. At one point, ρ = 2 nm, this pressure can be compared to that assessed in Ref. [25]. Figure 9.4 gives the value of 1.8 GPa, which is by a factor of 1.8 larger than the estimate of Ref. [25].

9.5 Thermomechanical Effects Energy relaxation in the medium has been studied in Ref. [25], where the inelastic thermal spike model was applied to liquid water irradiated with carbon ions. This model has been developed to explain track formation in solids irradiated with heavy ions and it studies the energy deposition to the medium by swift heavy ions through secondary electrons [79–89]. In this model, the electron-phonon coupling (strength of the energy transfer from electrons to lattice atoms) is an intrinsic property of the irradiated material. The application of the inelastic thermal spike model to liquid water predicted that the temperature increases by 700–1200 K inside the hot cylinder by 10−13 s after the ion’s traverse [25]. However, within this model, only coupled (between electrons and atoms of the medium) thermal conductivity equations are solved, while the further dynamics of the medium is missing. This dynamics is the consequence of a rapid pressure increase inside the hot cylinder around the ion’s path up to 104 atm, while the pressure outside of it is about atmospheric. A substantial pressure gradient in a liquid medium prompts a rapid expansion. This effect referred to as ion-induced shock waves has been analyzed in different aspects in Refs. [27, 28, 90, 91], thus the ion-induced shock waves were predicted in the framework of MSA. Later, it turned out that a possibility of a shock wave produced in ion tracks in liquid water was discussed back in the 1970s [92]. However, these predictions have not been studied and their role in the ion-induced biodamage was overlooked. In general, the interest in the phenomena of shock or blast waves, featuring propagating surfaces of discontinuity, has an old history dating back to the nineteenth century [93, 94]. However, it was during the twentieth century, and particularly in the 1940s, when crucial contributions were made by renowned physicists such as Zel’dovich, Bethe, Sedov, von Neumann and Taylor [95–99]. As a result of these works, hydrodynamic solutions were derived to analytically describe spherical shock waves, as comprehensively summarized in Refs. [100, 101]. This research was mainly focused on macroscopic phenomena related to detonation. In Ref. [27], these solutions were re-derived for the cylindrical case and applied to ion-induced shock waves. Similarly, thermo-mechanical phenomena induced by ion beams also happen

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in solid targets, where their effects are easier to detect and which present important technological applications [102, 103]. Experimentally, the ion-induced shock waves have not yet been observed directly; however, there have been a number indirect observations that suggest that their effect is significant for any assessment of biodamage due to ions. Therefore, their effect is included and discussed below.

9.5.1 Hydrodynamic Expansion on the Nanometer Scale The problem of the expansion of the medium driven by the high pressure inside the hot cylinder is in the realm of hydrodynamics and it has been thoroughly analyzed in Ref. [27]. It has been shown that the expansion is cylindrically symmetric. If the ratio of pressures inside and outside the hot cylinder is high enough, as happens for large values of LET, the cylindrical expansion of the medium is described as a cylindrical shock wave, driven by a “strong explosion” [101]. For an ideal gas, this condition holds until about t = 1 ns, but in liquid water the shock wave relaxes much sooner. In Ref. [90], the molecular dynamics (MD) simulations of liquid water expansion showed that the shock wave weakens by about 0.5 ps after the ion’s passage. The solution of the hydrodynamic problem describing the strong explosion regime of the shock wave, as well as its mechanical features and limitations, are very well described in Refs. [100, 101, 104]. In Ref. [27], the solution for the cylindrical case has been reproduced and analyzed in order to apply it for the nanometre-scale dynamics of the DNA surroundings. In this section, only the results pertinent to the further discussion of biodamage are presented. A self-similar flow of water and heat transfer depends on a single variable, ξ . This variable is a dimensionless combination of the radial distance, ρ, from the axis, i.e., the ion’s path, the time t after the ion’s passage, the energy dissipated per unit length along the axis, which is equal to the LET per ion, Se , and the density of undisturbed water,  = 1 g/cm3 . This combination is given by ξ=

ρ √

β t



 Se

1/4 ,

(9.23)

where β is a dimensionless parameter equal to 0.86 for γ = C P /C V = 1.222 [27] (where C P and C V are molar heat capacity coefficients at constant pressure and volume, respectively). The radius R and the speed u of the wave front are given by √



R = ρ/ξ = β t and

R β dR = = √ u= dt 2t 2 t

Se  

1/4

Se 

(9.24) 1/4 ,

(9.25)

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Fig. 9.5 Radial distance travelled by the front of the shock wave induced by the five studied ions in the Bragg peak region. The results are obtained using Eq. (9.24). Reproduced from Ref. [105] with permission from American Physical Society

respectively. The position of the wave front calculated according to Eq. (9.24) for different projectile ions in the Bragg peak region is illustrated in Fig. 9.5. It is also worthwhile to combine Eqs. (9.25) and (9.24) and obtain the expression of the speed of the front in terms of its radius R, β2 u= 2R



Se 

1/2 .

(9.26)

Using Eq. (9.26), pressure P at the wave front can be obtained as P=

2 1 β 4 Se u 2 = . γ +1 γ + 1 2 R2

(9.27)

Then, one can solve the hydrodynamic equations in order to obtain the expressions for speed, pressure, and density in the wake of the shock wave, i.e., behind the wave front [27]. The are two important intriguing lines of questions related to the ion-induced shock waves [25, 27]. (i) What can a shock wave do to biomolecules such as DNA located in the region of its propagation through the medium? Can it cause biodamage, e.g., a strand break, by mechanical force? The forces acting on DNA segments were predicted to be as large as 2 nN, which is more than enough to break a covalent bond; however, these forces are only acting for a short time and it remained unclear whether this is sufficient to cause severe damage to DNA molecules. A series of papers were devoted to answering these questions [28, 91, 105]. The answers to these questions are positive, but the extent of significance of this effect depends on the value of LET. The direct mechanical bond breaking is negligible for protons and α-particles, and it plays a minor role for heavier ions such as carbon and oxygen. However, this effect becomes dominant for heavy ions like iron [105]. The direct DNA damage

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due to the ion-induced shock waves is discussed below in sections “Quantification of the Number of Bond Breaks in the DNA Double Twist” and “Force Exerted by the Shock Wave on the DNA”. (ii) The other line of questions is: how significant can the transport due to the collective flow of this expansion be compared to the diffusion of secondary particles? How much does the transport of reactive species (such as hydroxyl radicals) by the shock wave change the initial conditions for the chemical phase of radiation damage? These questions were studied in Refs. [28, 34, 77, 105]. It turns out that this effect is important even for light ions such as protons. This effect is reviewed in Sect. 9.5.2.

Quantification of the Number of Bond Breaks in the DNA Double Twist In the recent study [105], reactive MD simulations of the shock wave induced damage of 30 base pairs long DNA molecule have been conducted by means of the MBN Explorer software package [106] employing the reactive CHARMM (rCHARMM) force field [107]. The simulation geometry is illustrated in Fig. 9.6A. The simulations [105] revealed that the projectile ion propagating in close proximity to the geometrical centre of the molecule produces significant damage within the central segment containing 20 DNA base pairs. The DNA damage produced in segments of such size may lead to complex irreparable lesions in a cell [30, 32, 109]. The number of bond breaks in the DNA double twist was counted after each completed MD simulation and analyzed as a function of the distance dgeo from the ion’s path to the principal axis of inertia of the DNA molecule (see Fig. 9.6A). The results of the performed analysis are shown in Fig. 9.7. Figure 9.7 shows that the number of bond breaks produced in the DNA double twist by the ion-induced shock wave increases with the LET of a projectile ion. Simulation results obtained for the scaled bond dissociation energies De /6 (Fig. 9.7A) reveal that up to two DNA backbone bonds break due to the shock wave induced by the carbon ion whereas up to 50 bonds may be broken due to the iron ion impact. For every combination of the bond dissociation energy and ion’s LET, the average number of bond breaks fluctuates around certain values NSW (De , Se ) within a certain distance range from the ion’s path; the values NSW (De , Se ) are summarized in Table 9.2. As the ion passes at larger distances from the principal axis of inertia of the DNA molecule the average number of bond breaks within the DNA double twist gradually decreases. Figure 9.7 shows that the shock wave induced thermomechanical stress of the DNA mostly occurs at 1 nm from the ion’s path for ions lighter than iron.

Force Exerted by the Shock Wave on the DNA The characteristic range of the shock wave induced thermomechanical damage, RSW , can be evaluated by analyzing the pressure on the shock wave front, Eq. (9.27), and the corresponding force exerted by the shock wave on covalent bonds in the DNA backbone by means of the following model, described in Ref. [105]. For the sake

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A

B Fig. 9.6 Geometry of the DNA molecule and the studied parameters. Panel A shows an ion (C, O, Si, Ar, and Fe) propagating in close proximity to the DNA molecule consisting of 30 complementary base pairs. The ion track is oriented along the z-axis; the x-axis is oriented along one of the principal axis of inertia of the chosen DNA molecule, and the y-axis is along the line defining the shortest distance between the ion track and the selected principal axis of inertia. The collision parameter dgeo is defined as the displacement of the ion’ path along the y-axis with respect to the principal axis of inertia. The specific collision parameters dA and dB are defined as the shortest distances from the ion’s path to DNA strand A and strand B, respectively. Panel B illustrates C3 –O, C4 –C5 , C5 – O and P–O bonds in the DNA sugar-phosphate backbone and the corresponding potential energy curves obtained by means of density-functional theory (DFT) calculations [108]. Bond dissociation energy, De , defined as the depth of the associated potential energy well of the covalent bond has been considered in the simulations as a variable parameter; the values of De determined from the DFT calculations have been scaled by a factor of 2/3, 1/2, 1/3 and 1/6. Reproduced from Ref. [105] with permission from American Physical Society

of simplicity, a molecular bond oriented parallel to the direction of the shock-wave propagation was considered in Ref. [105]. The force stretching the bond is given by F = (πa02 )

∂P l, ∂r

(9.28)

where πa02 is the transverse area of the molecular bond exposed to the pressure created by the shock wave front, ∂∂rP is the pressure gradient, and l is a characteristic interatomic distance in the medium on which the pressure gradient is evaluated. The

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A

B

C

Fig. 9.7 Average number of bond breaks in the DNA double twist calculated as a function of the collision parameter dgeo for the five studied ions. dgeo is the distance from the ion track to the principal axis of inertia of the DNA segment as shown in Fig. 9.6A. Panels A, B, and C show the results of simulations employing the bond dissociation energies De derived from the DFT calculations (see Fig. 9.6B) and the values of De scaled by the factors of 1/2 and 1/6. Two independent MD simulations were performed for each ion and each collision geometry; error bars indicate the corresponding standard deviation. Reproduced from Ref. [105] with permission from American Physical Society Table 9.2 Characteristic number of bond breaks, NSW , occurring in the DNA double twist within a certain distance range r0 from the ion’s path to the principal axis of inertia of the DNA segment [105]. Different columns correspond to the results of simulations where the default bond dissociation energies De [108] as well as the scaled bond dissociation energies De /2 and De /6 were used De /6 De /2 De NSW r0 (nm) NSW r0 (nm) NSW r0 (nm) Carbon Oxygen Silicon Argon Iron

1.1 ± 0.5 17.8 ± 1.4 28.3 ± 4.9 32.9 ± 4.5 34.0 ± 4.9

1.2 0.6 0.9 0.9 0.9

0.05 ± 0.05 0.7 ± 0.4 3.5 ± 1.2 8.8 ± 1.7 24.2 ± 1.7

1.2 0.6 0.9 0.9 0.9

0 0 0 0.8 ± 0.5 5.4 ± 1.5

1.2 1.2 1.2 0.9 1.2

calculations described below have been performed using the value a0 = 0.15 nm corresponding to the van der Waals radius for atoms forming the DNA backbone [110] and l ≈ 0.15 nm, which is a characteristic length of covalent bonds in the DNA backbone. Let us consider potential energy of a DNA backbone bond being under the pressure created by the shock wave front:   U (r ) = De e−2κ(r −r0 ) − 2e−κ(r −r0 ) −

r

F · dr .

(9.29)

r0

The first term on the right-hand side of Eq. (9.29) is the bond potential energy described by the Morse potential, De is the bond dissociation energy, r0 is the equilibrium bond length, and κ defines the steepness of the potential energy curve. These

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parameters for the C3 –O, C4 –C5 , C5 –O and P–O bonds in the DNA backbone (see Fig. 9.6B) are determined from the potential energy curves obtained by means of DFT [108]. The second term on the right-hand side of Eq. (9.29) describes the work against the force F, caused by the pressure gradient on the distance from r0 to r . The force F should not depend on the interatomic distance r as it originates from the medium and is defined by its properties at given thermodynamic conditions and at any given location of the bond in space. Considering the geometry when the bond is oriented along F one derives   U (r ) = De e−2κ(r −r0 ) − 2e−κ(r −r0 ) − F × (r − r0 ) .

(9.30)

Stretching the DNA backbone bond by the force F results in lowering the energy barrier for bond rupture. The energy barrier height reads as

E = U (r0 + r2 ) − U (r0 + r1 ) ,

(9.31)

where r1 = r1 − r0 is the shift of the potential energy minimum with respect to r0 , and r2 = r0 + r2 is the position of the potential energy maximum. The threshold value of the external force at which the bond rupture becomes possible depends on the amount of energy accessible for atoms forming the bond at a given temperature T and on the amount of energy deposited into the medium by the projectile ion. The condition for the bond rupture due to the shock wave induced thermomechanical stress of the DNA can be formulated as follows [105]: 2

μ ( v)2 kB T + ≥ E . 2 2

(9.32)

The first term on the left-hand side of Eq. (9.32) is the average energy available for one degree of freedom in a thermodynamic system being at the equilibrium at T = 300 K. The factor 2 arises since both kinetic and potential energies of the bond are equal to 21 kB T according to the equipartition theorem. The second term on the left-hand side is the kinetic energy of the relative interatomic motion caused by the shock wave; μ = m i m j /(m i + m j ) is the reduced mass of a pair of atoms i and j. The analysis described below has been performed for the C3 –O, C4 –C5 , C5 –O and P–O bonds in the DNA backbone which are shown in Fig. 9.6B. The energy barrier E can be evaluated by equating the derivative of U (r ), Eq. (9.30), over r to zero. Performing some algebraic transformations described in detail in Ref. [105], one derives the following condition for bond rupture: η

l P(R)



∂P ∂r

 α≥

√ 2α 1 − 2α + α ln  2 , √ 1 + 1 − 2α

where η=

μκ 1 γ + 1 πa02 ρ

(9.33)

(9.34)

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and α=

F κ De

(9.35)

are the dimensionless parameters. The left-hand side of Eq. (9.33) is a function of distance from the ion track, r , and the position of the shock wave front, R, at a given time moment t. The parametric inequality (9.33) is fulfilled in the region α ≥ α, ¯ where the threshold value α¯ is determined by equating the left- and right-hand sides of Eq. (9.33). The pressure gradient ∂∂rP in the vicinity of the shock wave front can be related to ∂∂ PR , that is the derivative of the pressure on the shock wave front P(R), Eq. (9.27), with respect to the shock wave front radius R. As demonstrated for the carbon ion at the Bragg peak [27] and derived in Ref. [105] for the heavier ions at the Bragg peak, the following relation applies:    ∂ P  ∂ P    (9.36) = ν ∂R  , ∂r  r =R

where the proportionality factor ν ≈ 5.95 is independent on ion’s LET and time. Differentiating Eq. (9.27) over R and combining Eqs. (9.28) and (9.35), the condition defining the solution of Eq. (9.33) can be expressed as: π νβ 4 a02 l Se ≥ α¯ . γ + 1 κ De R 3

(9.37)

From this expression, one derives the threshold distance from the ion track, RSW , below which the bonds in the DNA backbone can be broken by the shock wave imposed thermomechanical stress: RSW = b Se1/3 ,

(9.38)

where the pre-factor b reads as  b=

π νβ 4 a02 l γ + 1 κ De α¯

1/3 .

(9.39)

The distance RSW depends on the parameters of a specific covalent bond (De and κ) and on the ion’s LET. Figure 9.8 shows the dependence of RSW on Se for the five studied ions in the Bragg peak region [105]. Symbols show the RSW values for cleavage of the C3 –O, C4 –C5 , C5 –O and P–O bonds in the DNA backbone, calculated according to Eq. (9.38) and (9.39). The coefficient b varies in the range (0.072 − 0.081) nm4/3 eV−1/3 for 1/3 the four bonds considered. The RSW = b Se dependence with the average value b = 0.077 nm4/3 eV−1/3 is shown in Fig. 9.8 by the dashed line. The calculated values for RSW for the C3 –O, C4 –C5 , C5 –O and P–O bonds and their average value for different LET values are listed in Table 9.3. The table lists also the corresponding

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Fig. 9.8 The threshold distance RSW for cleavage of the C3 –O, C4 –C5 , C5 –O and P–O bonds by the shock wave induced thermomechanical stress by the five studied ions at the Bragg peak region. The RSW values are calculated according to Eq. (9.38) (symbols). The coefficient b, Eq. (9.39), varies in the range (0.072 − 0.081) nm4/3 eV−1/3 for the four bonds considered. Dashed line shows 1/3 the RSW = b Se dependence with the average value b = 0.077 nm4/3 eV−1/3 . Reproduced from Ref. [105] with permission from American Physical Society Table 9.3 The critical distance from the ion track, RSW , below which covalent bonds in the DNA backbone can be broken by the shock wave induced thermomechanical stress in the vicinity of the Bragg peak. The RSW values are given in nanometers. The bottom line lists the corresponding threshold values of the external force F, Eq. (9.28) C3 –O C4 –C5 C5 –O P–O Average Carbon Oxygen Silicon Argon Iron F (nN)

0.68 0.77 0.94 1.03 1.16 6.7

0.73 0.83 1.01 1.11 1.26 5.2

0.71 0.81 0.99 1.08 1.23 5.7

0.76 0.86 1.05 1.15 1.30 4.8

0.72 0.82 1.00 1.09 1.24 5.6

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threshold values of the external force F at which the bond rupture becomes possible. The threshold values of F are evaluated using Eq. (9.28) with the pressure gradient ∂P being related to the derivative of the pressure at the shock wave front, ∂∂ PR , via ∂r Eq. (9.36). The results presented in Fig. 9.8 and Table 9.3 indicate that the threshold distance from the ion’s path for the bond rupture by the pressure gradient on the shock wave front varies from 0.7 nm for a carbon ion at the Bragg peak to 1.3 nm for an iron ion at the Bragg peak. The estimated RSW values are consistent with the results of MD simulations shown in Fig. 9.7. Note, however, that no strand breaks have been observed in the simulations for carbon and oxygen projectile ions for the default bond dissociation energies De (Fig. 9.7C), which can be attributed to a small number of simulated trajectories and low number of the events. Accounting for different possible orientations of the molecular bonds in the DNA backbone should also increase their average stability. Simulations performed with the scaled bond dissociation energies De /2 and De /6 (Fig. 9.7B and Fig. 9.7A, respectively) indicate the formation of bond breaks in the DNA backbone by the carbon- and oxygen-ion induced shock wave within the range of distances from the ion track, which are consistent with the RSW values determined by Eq. (9.38) and listed in Table 9.3.

9.5.2 Transport of Reactive Species by the Radial Collective Flow The analysis done in Ref. [27] suggests that a considerable collective radial flow emerges from the hot cylinder region of medium. The maximum mass flux density  is the matter carried by the cylindrical shock wave is given by  f u, where  f = γγ +1 −1 density on the wave front. This expression is proportional to√ u and its substitution from Eq. (9.26) yields that the mass flux is proportional to the Se . This √ flux density is inversely proportional to radius ρ and is linear with respect to the Se . It sharply drops to zero in the wake of the wave along with the density. A sharp rarefaction of the volume in the wake of the wave follows from the results of Ref. [27]. This is the effect of cavitation on a nanometer scale and due to this effect, the water molecules of the hot cylinder along with all reactive species formed in this cylinder are pushed out by the radial flow. Such a mechanism of propagation of reactive species, formed within the hot cylinder, is competitive (and becomes dominant at relatively small values of LET) with the diffusion mechanism, studied in MC simulations done using track structure codes [41]. First estimates of the transport were done in Ref. [28]. The time at which the √ wave front reaches a radius ρ can be derived from Eq. (9.24) as it is equal to (ρ 2 /β 2 ) /Se . This time has to be compared to diffusion times, which can be estimated for different reactive species as ρ 2 /D, where ρ is the distance from the ion’s path and √ D is the corresponding diffusion coefficient. The ratio of these times is equal to ( /Se )D/β 2 . For all relevant species, the diffusion coefficient is less than 10−4 cm2 /s [111].

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√ Therefore, the above ratio is less than 10−3 / Se (keV/nm), which is much less than unity even for protons. For instance, for carbon ion projectiles, the wave front reaches 5 nm from the path in 2.8 ps after the ion’s traverse, while hydroxyl radicals reach the same distance via the diffusion mechanism in about 9 ns, a more than 3000 times longer time. In fact, the lifetime of hydroxyl free radicals is shorter than 5 ns [13, 44, 111], therefore the shock wave transport may be the only means to deliver hydroxyl radicals to distances farther than 3.5 nm of the ion’s path. The next stage of this study was a series of works that were by and large based on MD simulations of shock wave propagation. In Ref. [113], δ-electrons were included in the dynamics following the ion’s traverse. These radial doses were obtained at different values of LET for carbon ions, far away and in the vicinity of the Bragg peak. δ-electrons take energy away from the track and, consequently, from the shock waves. The radial dose (or pressure) distributions look similarly to those in Fig. 9.4 when ions are in the Bragg peak region, but they differ significantly if the ion is far from the Bragg peak. These radial dose distributions were used for initial conditions for MD simulations. Depending on these initial conditions, the shock wave was stronger or weaker, sharper or more dispersed. Expectedly, in the vicinity of the Bragg peak (where δ-electrons are suppressed), the results were similar to the step function (a.k.a. “hot cylinder”) initial conditions. All simulations were performed using the MBN Explorer software package [106]. The following work in this series [34] was done using the rCHARMM force field [107] impelemented in MBN Explorer. With the use of rCHARMM, it was possible to accomplish the MD simulations following the main chemical reaction involving hydroxyl radicals, OH• + OH• −→ H2 O2 .

(9.40)

Again, the transport of radicals in a track was studied at LET away from the Bragg peak and in its region. The following two figures tell the story. Figure 9.9 shows the simulation results for 500 keV protons (with low LET), where shock waves were not present [34]. Figure 9.9A shows the mean square displacement (MSD) of OH radicals, with the transport being purely due to diffusion. From this plot, the hydroxyl diffusion coefficient DOH was obtained by means of the Einstein relation, MSD = 6DOH t. A diffusion coefficient of 0.22 Å2 /ps was obtained from these simulations [34], which very well compares with with the results of another simulations giving 0.3 Å2 /ps [116] and with the value used in the popular simulation packages PARTRAC [117] or GEANT4-DNA [118], 0.28 Å2 /ps. The numbers of OH and H2 O2 molecules during simulations are shown by histograms with error bars in Fig. 9.9B, representing the average and standard deviations of three independent runs. These are compared with results reported using the GEANT4-DNA package [119]: dashed lines depict the original results, in which many reactions among reacting species are possible, while dotted lines are scaled in a way in which only reaction (9.40) is included (see Ref. [34] for more details). Even though it is computationally expensive to reach long times with molecular dynamics simulations, it is clear that its results at low LET (and no shock wave) agree very well with the evolution predicted by Monte

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Fig. 9.9 Diffusion and reactivity of the OH radicals produced by 500-keV protons in liquid water, where a shock wave is not expected to occur due to the very low LET. A Mean square displacement of OH radicals and B number of OH and H2 O2 molecules as a function of simulation time. MBN Explorer results (histograms with error bars) are compared to GEANT4-DNA simulations by Karamitros et al. (dashed and dotted lines) [119]. See the text for details. Reproduced from Ref. [77]

Carlo simulations, where shock waves are not included. These findings presented in Fig. 9.9 validate the molecular dynamics model used in Ref. [34]. Similar quantities were found for the case of a carbon ion in the Bragg peak region, where the shock waves are substantial. The shock waves develop naturally in the molecular dynamics simulations, as a result of the energy deposited along the ion’s track. However, the computer simulations allow one to artificially “switch off” this process in order to distinguish its influence in the evolution of the radiation chemistry scenario. Such an artificial switching off was used in Ref. [34] to assess how the presence of shock wave modifies the radiation chemistry as compared to the traditional picture of transport of reactive species where shock waves are not included. Figure 9.10A compares the mean square displacement of radicals in simulations with the wave included (i.e., in presence of a substantial collective flow) versus that switched off (i.e., with the transport being purely due to diffusion). The effective diffusion coefficient for the case of shock wave induced collective flow is about 80 times larger than the diffusion coefficient obtained for the simulation with the shock wave switched off. The reactivity of the OH radicals in the presence of the shock wave is illustrated in Fig. 9.10B. It depicts the average number of OH and H2 O2 molecules, together with the standard deviations obtained after three independent runs, in the two cases where the shock wave is artificially switched off and naturally allowed to develop, respectively. As is clearly seen, the evolution of the number of molecules is different for the simulations with the shock waves on and off. The transport of radicals by the collective flow does not only propagate the radicals much faster than diffusion, but also prevents their recombination, both by their spreading and by creating harsh conditions for the formation of O–O bonds [34].

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Fig. 9.10 A Mean square displacement of the OH radicals produced around a 200-keV/u carbon ion’s path. B Time evolution of the number of OH radicals and produced H2 O2 molecules. Results of simulations where the shock wave is allowed to develop (transport by the collective flow, solid lines) and where it is artificially “switched off” (transport by diffusion, dashed lines) are shown, in order to demonstrate the shock wave effects on the radiation chemistry. Reproduced from Ref. [77]

These results confirm the evaluations made in Ref. [31] that in absence of ioninduced shock waves, the presence of hydroxyl radicals outside of ion tracks would be strongly suppressed. These results also allow one to infer the radial range of spread of reactive species. Further step has been made in Ref. [105] where the range of shock wave propagation in liquid water and hence the range of shock wave driven propagation of reactive species have been evaluated from MD simulations by means of MBN Explorer. A shock wave propagation was simulated in a pure water box with dimensions of 49.5 nm × 49.5 nm × 8.0 nm. The evolution of radial density of water around the tracks of the five projectile ions (carbon to iron at the Bragg peak) is shown in Fig. 9.11. Water molecules located in the vicinity of the ion’s path are transported away from their initial positions, which results in the formation of a cylindrical cavity around the ion’s path. The radius of the cavity grows with time up to the values of about 6 nm for carbon and oxygen ions, while the density of water increases at larger distances from the ion’s path. Following the mass conservation law, the mass of water molecules transported from the region in the vicinity of the ion track is equal to the mass of excess water molecules at larger distances from the track [105]. The simulation of the propagation of the carbon ion-induced shock wave revealed [105] that at a certain time instance, the shock wave has stopped propagating away from the ion’s path and started to move slowly in the inward direction. This happens when the pressure of the shock wave front drops below a certain value determined by the balance of the pressure at the wave front and the water surface tension pressure [114, 115]. Figure 9.12A shows the radial position of the maximal density of water as a function of simulation time for the case of the projectile carbon ion. The radial displacement of the maximal density from the ion track axis increases rapidly during the first 15 ps of the simulation, then reaches the maximal value and

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Fig. 9.11 The average density of water as a function of radial distance from the ion track; the dynamics of the water medium is caused by propagation of the shock wave induced by iron, argon, silicon, oxygen and carbon ions inside a 49.5 nm × 49.5 nm × 8.0 nm water box. The simulation time (measured in ps) is depicted as a colour scale. The shock wave induced by iron, argon, and silicon ions (top row) has reached the simulation box boundary much faster than the shock wave induced by oxygen and carbon ions (bottom row). Therefore, the simulation time for iron, argon and silicon ions is about 3 times shorter than for the lighter ions. Reproduced from Ref. [105] with permission from American Physical Society

starts to decrease at later time instances. The analysis shown in Fig. 9.12A suggests that the maximal radial displacement of the density corresponds to the time instance t = 16.7 ps. Note that for t > 20 ps the radial displacement of the maximal density stops decreasing but fluctuates around the value of 21 nm. This behaviour is attributed to interference with the outer part of the shock wave front, which reaches the simulation box boundary and gets reflected. The behaviour of the system within the simulation time range t ≤ 20 ps is nevertheless physically meaningful as the shock wave has not yet reached the simulation box boundary within this time interval. According to Eq. (9.24), the front of the carbon ion induced shock wave propagates by the time t = 16.7 ps to the distance R = 11.9 nm from the ion track. This characteristic distance defines the propagation range of free radicals, Rr , which are transported by the shock wave driven collective flow (see also Sect. 9.6.3 below). To evaluate the range of shock wave propagation for heavier ions, one would need to run longer simulations and consider much larger simulation boxes than the one used in Ref. [105]. Alternatively, the range of the shock wave propagation induced by high-LET ions can be estimated from the analysis of the pressure on the shock wave front according to Eq. (9.27). Figure 9.12B shows the pressure induced by the shock wave front generated by the different ions in the Bragg peak region. Coloured lines

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A

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B

Fig. 9.12 A The radial position of the maximal density of water as a function of simulation time for the shock wave induced by a carbon ion in the Bragg peak region. The maximal distance of the shock wave propagation is defined by a quadratic fit function (see the dashed line). The nonphysical region due to the shock wave reflection from the simulation box boundaries is marked with grey colour. B The pressure exerted by the shock wave front generated by different ions in the Bragg peak region as a function of the wave front radius R. The red dot depicts the maximal propagation distance for the shock wave front generated by a carbon ion (the “turning point”), calculated using Eq. (9.24). The corresponding time instance, t = 16.7 ps, has been determined from the MD simulations as shown in panel A. The dashed line shows the surface tension pressure on the surface of a cylindrical wake region with radius R. Reproduced from Ref. [105] with permission from American Physical Society

correspond to the results derived using Eq. (9.27). A red dot depicts the pressure P = 0.115 GPa at the distance R = 11.9 nm from the ion track, that is the maximal distance of the wave front propagation for a carbon ion. The indicated value of R corresponding to the instant t = 16.7 ps has been evaluated using Eq. (9.24). The shock wave propagation in the radial direction away from the ion’s path causes cavitation in its wake, leading to the formation of a rarefied cylindrical region [27, 114]. This effect has been observed in MD simulations described in Fig. 9.11. In the course of the shock wave propagation, the pressure at the shock wave front becomes balanced by the surface tension pressure building up at the border of the wake region. As a result, the growth of the wake region stops and this region shrinks after the instant when the pressure of the wave front becomes equal to the water surface tension pressure on the surface of the wake region. The latter can be estimated as ξ , (9.41) P= R where ξ is the coefficient of surface tension and R is the distance from the ion track. Using the aforementioned values P = 0.115 GPa and R = 11.9 nm for the carbon ion at the Bragg peak, one obtains the surface tension coefficient ξ = 1.37 N/m. Note that the medium in the vicinity of the shock wave front is far from equilibrium,

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Table 9.4 The maximum radii of the shock wave region for the five studied ions at their Bragg peak energies. These characteristic radii define the propagation range of free radicals, Rr , which are transported by the propagating shock wave Carbon Oxygen Silicon Argon Iron Rr (nm)

11.9

17.5

31.6

41.5

60.8

and the density of the medium is significantly higher than the density of water at ambient conditions. The high water density in the vicinity of the shock wave front and the large amount of energy deposited into the medium explain the large value of the corresponding surface tension coefficient. The dependence of the surface tension pressure on the distance from the ion track, calculated using Eq. (9.41), is shown in Fig. 9.12B by a dashed line. Assuming that ξ depends weakly and smoothly on LET (or does not depend at all) at the pressures balance point one can evaluate the radii Rr for different ions. The radii Rr define the propagation ranges of free radicals transported by the shock wave induced by different ions. Equating the pressure on the shock wave front, Eq. (9.27), and the surface tension pressure, Eq. (9.41), one obtains a linear dependence of Rr on LET: β4 Se . (9.42) Rr = 2(γ + 1) ξ The calculated values of Rr for the five studied ions at the Bragg peak region are summarized in Table 9.4. The results indicate that for an iron ion the free radicals are transported by the shock wave to the distance of ∼60 nm. This value exceeds by an order of magnitude typical distances that radicals can diffuse in the medium being at the equilibrium during the time corresponding to the duration of formation of the shock wave wake region with the maximum radius.

9.6 Assessment of Radiation Damage Using MSA 9.6.1 Calculation of Number of Secondary Electrons Incident on a DNA Target As has been shown in Ref. [31], the number densities of the first and second generations of secondary electrons are given by Eqs. (9.16). A target is chosen to be a rectangle of area ξ η, where ξ = 6.8 nm and η = 2.3 nm are the length of two twists and the diameter of a DNA molecule, respectively. Thus electrons or radicals hitting such a target would be hitting two rungs of a DNA molecule masked by this target. The plane of the target is chosen to be parallel to the ion’s path with dimension ξ along and η perpendicular to the path. This can be seen in Fig. 9.13. Then angle inscribes the target in a plane perpendicular to the ion’s path, where φ = 2 arctan η/2 r r is the distance between the target and the path.

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The number of first-generation electrons hitting the described target segment of area r φξ ≈ ξ η parallel to the ion’s path per unit time is,   ∂n 1 (r, t) r2 φ d N1 r 2 ξ t . exp − 1 = −φξr D1 = − ∂r 2π d x 4D1 t 2 4D1 t τ1

(9.43)

Its integral over time,

∞ 0

φ 1 dt = 2π

  d N1 r 2 ξ r2 t dt exp − − d x 4D1 t 2 4D1 t τ1   r φ d N1 r ξ , = K1 √ √ 2π d x D1 τ1 D1 τ1

∞ 0

(9.44)

where K 1 is the Macdonald function (modified Bessel function of the second kind) [120], gives the total number of first generation secondary electrons that hit this area. The second-generation contribution is obtained similarly: ∂n 2 (r, t) ∂r

ξr 2 D2 d N1 t 1 =φ 4π τ1 d x 0 (D1 t  + D2 (t − t  ))2   t − t r2 t dt  , − × exp − − 4(D1 t  + D2 (t − t  )) τ2 τ1

2 (t, r ) = −r φξ D2

and then

∞ 0

1 ξr 2 D2 d N1 ∞ t   2 4π τ1 d x 0 0 (D1 t + D2 (t − t ))   2  t −t r t dt  dt − × exp − − 4(D1 t  + D2 (t − t  )) τ2 τ1

(9.45)

2 dt = φ

(9.46)

gives the number of second-generation secondary electrons that hit the same area. The average number of simple lesions due to a single ion, Ne (r ), can now be obtained as the sum,

∞

∞ 1 dt + e (Se ) 2 dt , (9.47) Ne (r ) = N1 (r ) + N2 (r ) = e (Se ) 0

0

where N1 (r ) and N2 (r ) are the average numbers of simple lesions produces by secondary electrons of the first and second generations, respectively, and e (Se ) is the average probability for an electron to induce a simple lesion on a hit. The probability e depends on kinetic energy of secondary electrons incident on the DNA double twist and hence on ion’s LET. The dependencies of N1 (r ) and N2 (r ) are shown in Fig. 9.14.

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Fig. 9.13 The geometry of the problem in the plane perpendicular to ion’s path. The target cylinder that encloses a DNA twist is shown as a circle. Its diameter is η. The dimension ξ is perpendicular to the plane of the figure

Fig. 9.14 The average numbers of simple lesions due to a single carbon ion with a Bragg-peak energy propagating through a uniform chromatin as functions of radial distance from the ion’s path. The lesions are produced by secondary electrons of the first (solid line) and second (dashed line) generations, N1 (r ) and N2 (r ). These dependencies are calculated using the corresponding number of hits, Eqs. (9.44) and (9.46), multiplied by the probability of the production of a simple lesion per hit, e = 0.03 (used in Ref. [30]). Straight (dotted) lines are the values for reactive species, Nr (r ), calculated using Eq. (9.54) with numbers from Ref. [32]. Reproduced from Ref. [124] with permission from Springer Nature

Equation (9.47) gives the average number of simple DNA lesions due to secondary electrons of the first and second generations as a function of the distance of the target DNA segment from the ion’s path. The next step is adding to this the contribution of reactive species, which is a product of the average ∞number of hits ∞ on the chosen area by reactive species Nr (this value is similar to 0 1 dt and 0 2 dt) and the probability of lesion production per hit, r . The number Nr depends on the value of the LET, since at small values of LET the transport of radicals is defined by diffusion and at high values the collective flow is expected to dominate this process.

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9.6.2 Calculation of the Reactive Species Contribution for Small Values of LET The number of produced reactive species, such as free radicals and solvated electrons, depends on the LET. If the LET is not very high, it is expected that the number of reactive species is proportional to the secondary electron production, d N1 /d x, and, therefore, increases nearly linearly with the value of LET [30]. At sufficiently high values of LET, extra production of radicals is possible due to water radiolysis at locations adjacent to the ion’s path. This effect has not yet been quantified and will be accounted for in future works along with the definition of the domain of the LET, where this effect becomes significant. In this work, a linear dependence between the number of reactive species and LET is assumed and the difference between high and low values of LET is defined only by the mechanism of transport of the reactive species; at low LET, this transport is defined by diffusion. Moreover, this means − + OH → OH− are rare and that chemical reactions such as 2OH → H2 O2 and eaq their frequency can be neglected compared to the diffusion term in the diffusion equation, [31]. Thus, the transport of reactive species in the low-LET case can be calculated, by solving a diffusion equation, ∂n r = Dr ∇ 2 n r , ∂t

(9.48)

where n r is the number density and Dr is the diffusion coefficient for reactive species. The initial conditions for this equation can be taken from Ref. [31], ∂n r (r, t) d N1 (2) n 1 (r, t) n 2 (r, t) = δ (r )δ(t) + + , ∂t dx τ1 τ2

(9.49)

where the first term describes the species formed at sites of original ionizations by the projectile, while other two terms are due to inelastic processes involving secondary electrons of the first and secondary generations, respectively. Ionizations and excitations that lead to the production of reactive species, nr (r, t), through the mechanism of Eq. (9.49) take place by about 50 fs [31]. By that time, the forming reactive species are localized within 3 nm of the ion’s path. These are the initial conditions for the following propagation of reactive species by the diffusion and/or collective flow, that happen on much larger scales, up to 100 ps in time and 50 nm in distance. Therefore, in this paper, a simplified initial condition is used, d N1 (2) ∂n r (r, t) =K δ (r )δ(t) , ∂t dx

(9.50)

where K is the number of reactive species produced due to each secondary electron of the first generation ejected by an ion. The value of K ≈ 6 can be evaluated as follows. The primary ionization produces H2 O+ , which is likely to produce a hydroxyl radical [13]. The same thing happens when the secondary electron of the

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first generation ionizes a water molecule (and thus becomes an electron of the second generation). Then two electrons of the second generation (the ionizing and ejected) can produce about four reactive species, two as a result of further energy loss in inelastic processes and two more if they become solvated electrons. A more accurate number for K can be obtained if the probabilities of the above processes are combined following a comprehensive radiochemical analysis. The solution to Eq. (9.48) with the initial condition (9.50) is given by,   r2 d N1 1 exp − . n r (r, t) = K d x 4π Dr t 4Dr t

(9.51)

The next step is to find the number of reactive species, r , incident on the target at a distance r from the ion’s path per unit time. We proceed similarly to Eqs. (9.43) and (9.44). r = −φξr Dr

  φ d N1 r 2 ξ ∂n r (r, t) r2 = K , exp − ∂r 2π d x 4Dr t 2 4Dr t

(9.52)

and its integral over time is simply,

0

∞

r dt =

d N1 ξ η/2 φξ d N1 K =K arctan . 2π dx dx π r

(9.53)

9.6.3 Calculation of the Reactive Species Contribution for Large Values of LET If the reactive species are formed in large quantities as a result of a high-LETion’s traverse, the collective flow due to the shock wave is the main instrument for the transport of these species away from the ion’s path. Interestingly, ranges of propagation of radicals used to be in the realm of chemistry [13, 44, 111]. However, in the case of high LET, this issue is addressed by physicists; the MD simulation (with a use of MBN Explorer package [106, 107]) showed that the range depends on the value of LET [34, 105], but a more extensive investigation is needed to obtain a more detailed dependence. In Ref. [32], a simple model was used to describe this transport. The value of the average number of lesions at a distance r from the ion’s path, Nr = r Nr , was considered to be a constant within a certain LET-dependent range Rr (Se ) defined by the radius of shock wave propagation: Nr (r, Se ) = Nr,0 (Se ) θ [Rr (Se ) − r ] .

(9.54)

θ (x) on the right-hand side of Eq. (9.54) is the Heaviside function. A linear dependence Rr ∝ Se was explored in the earlier study [114], and a conservative estimate

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Rr ≈ 10 nm was derived for carbon ions in the Bragg peak region [30]. As described in Sect. 9.5.2, the value of Rr for carbon ions was recently evaluated more precisely on the basis of MD simulations [105], and the Rr values for heavier ions were estimated from the Rr ∝ Se dependence from the analysis of the pressure at the shock-wave front (see Eq. (9.42) and Table 9.4). The value Nr,0 (Se ) depends on the number of formed radicals, which, in turn, is proportional to the number of generated secondary electrons and hence proportional to LET. Nr,0 (Se ) depends also on the degree of oxygenation of the medium since the concentration of oxygen dissolved in the medium affects the number of formed radicals and, consequently, the creation of DNA lesions. The value of Nr,0 (Se ) = 0.08 for the environment with the normal concentration of oxygen was derived earlier [32] from the comparison of the experimental results [121] for plasmid DNA, dissolved in pure water and in a scavenger-rich solution, and irradiated with carbon ions at the Bragg peak region. A large number of cell survival experiments performed at hypoxic conditions were reproduced with the twice smaller value of Nr,0 = 0.04 [32, 36, 112]. In principle, more information about Nr is needed. For example, at high LET, more reactive species are expected to be produced through the radiolysis of water in the cores of the ion tracks at times ≥50 fs after the energy transfer from secondary electrons to the medium has taken place. This process can now be studied by MD simulations using the MBN Explorer package [106, 122], which is capable of resolving the corresponding temporal and spatial scales. The comprehensive picture of transport of reactive species includes diffusion (dominant at low values of LET), collective flow (dominant at high values of LET), and chemical reactions. With this understanding, as LET increasing Eq. (9.53) should gradually transform into Eq. (9.54). In addition to these equations, the effective range of reactive species is limited by the criterion of lethality that requires a minimal fluence at each site. More discussion on this topic can be found in Ref. [35].

9.6.4 Evaluation of the Number of Ion-Induced DNA Lesions and Cell Survival Probability This section outlines the procedure for evaluating the number of lesions of the DNA molecule produced upon its irradiation with ions and the corresponding cell survival probabilities. The previously developed formalism described in detail in Refs. [30, 36, 37] has been extended recently [105] towards accounting for the DNA lesions produced by the thermomechanical stress imposed on the DNA molecule by the propagating shock wave. The starting point for this theory is the calculation of N (r )—the total average number of simple lesions, i.e. single-strand breaks (SSBs), produced in a DNA double convolution (a DNA double twist) located at distance r from the ion’s path. According to the MSA analysis [30, 36] this number is equal to

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N (r, Se ) = Ne (r, Se ) + Nr (r, Se ) + NSW (r, Se ) ,

(9.55)

where Ne (r, Se ), Eq. (9.47), is the number of lesions produced by secondary electrons, Nr (r, Se ), Eq. (9.54), is the number of lesions produced by free radicals, and NSW (r, Se ) is the number of DNA lesions produced by the thermomechanical stress imposed on the DNA molecule by the propagating shock wave. The creation of DNA lesions by secondary electrons, free radicals, and the shock wave are statistically independent events taking place at different time scales after the ion passage [30, 36]. Therefore, the total average number of simple lesions in a DNA double twist, N (r, Se ), is a cumulative quantity derived by integrating all the events over time. Ne (r, Se ) and Nr (r, Se ) were worked out earlier within the MSA [30–32, 36, 37], whereas NSW (r, Se ) has been quantified recently by means of the MD simulations [105]. Knowing N (r, Se ) at a given distance r , one can use the Poisson statistics to calculate probabilities of different independent events. The probability to produce k lesions in a DNA double twist placed at a distance r from the ion track is equal to Pk (r, Se ) =

N k (r, Se ) −N (r,Se ) e . k!

(9.56)

A lethal DNA lesion is defined within the MSA framework as one double-strand break (DSB) plus at least two additional single lesions occurring within a DNA double twist [30]. This definition relies on earlier findings [109, 123] that complex DNA damage is irreparable for a cell if the damage occurs in a localized DNA segment, which typically consists of two helical turns containing 20 base pairs. Lesions within the DNA double twist may occur on one DNA strand or be present on both strands. Let us define n as the number of vulnerable covalent bonds in one strand in a DNA double twist. In Ref. [105], this number was set equal to 80 as four vulnerable covalent bonds (C3 –O, C4 –C5 , C5 –O and P–O) in the sugar-phosphate backbone correspond to each nucleotide, see Fig. 9.6B. Therefore, the total number of such bonds in both strands in the DNA double twist is 2n = 160. The total number of events Nν for ν = 0, 1, ..., 2n lesions occurring within the DNA double twist is equal to the number of combinations for ν choices taken out of 2n places: (2n)! ν , ν = 0, 1, ..., 2n . (9.57) ≡ Nν = C2n (2n − ν)! ν! Similarly, the number of events Nν(1) of ν lesions being all located on one strand within the DNA double twist can be calculated as ⎧ ⎪ 1 , ν=0 ⎪ ⎨ n! (9.58) Nν(1) = 2 Cnν ≡ 2 , ν = 1, 2, ..., n ⎪ (n − ν)! ν! ⎪ ⎩0 , ν = n + 1, n + 2, ..., 2n .

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The number of events N0(1) corresponding to the absence of lesions (ν = 0) is naturally equal to one. For ν = 1, 2, ..., n lesions, the factor 2 accounts for the two strands within the DNA double twist. The larger number of lesions (ν = n + 1, n + 2, ...2n) will necessarily occur on both DNA strands, thus the corresponding numbers Nν are equal to zero. One can also calculate the number of events Nν(2) when ν lesions result in at least one DSB within the DNA double twist: ⎧ 0 , ν = 0, 1 ⎪ ⎪ ⎪ ν−1 ⎪ ⎪ ⎨  k ν−k Cn Cn , ν = 2, 3, ..., n (9.59) Nν(2) = ⎪ k=1 ⎪ ⎪ (2n)! ⎪ ⎪ ⎩ , ν = n + 1, n + 2, ..., 2n . (2n − ν)! ν! The numbers Nν , Nν(1) and Nν(2) from Eqs. (9.57), (9.58) and (9.59) obey the obvious relationship (9.60) Nν = Nν(1) + Nν(2) . Knowing Nν(1) and the total number of events for ν lesions, Nν , one derives the probability Pν(1) to create ν SSBs located on one DNA strand within the double twist: Pν(1) =

Nν(1) , Nν

ν = 0, 1, ..., 2n .

(9.61)

Substituting here Nν and Nν(1) from Eqs. (9.57) and (9.58), respectively, one derives

Pν(1)

⎧ ⎪ 1 , ν=0 ⎪ ⎨ (2n − ν)! n! = 2 , ν = 1, 2, ..., n ⎪ (n − ν)! (2n)! ⎪ ⎩0 , ν = n + 1, n + 2, ..., 2n .

(9.62)

Analogously, the probability Pν(2) that ν lesions result in at least one DSB within the DNA double twist reads as Pν(2) =

Nν(2) , Nν

ν = 0, 1, ..., 2n .

(9.63)

Substituting Nν(2) from Eq. (9.59) and using Eqs. (9.60)–(9.62) one derives

Pν(2)

⎧ ⎪ 0 , ν = 0, 1 ⎪ ⎨ (2n − ν)! n! = 1 − Pν(1) ≡ 1 − 2 , ν = 2, 3, ..., n ⎪ (n − ν)! (2n)! ⎪ ⎩1 , ν = n + 1, n + 2, ..., 2n . (9.64)

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Now following the above introduced criterion for a lethal DNA lesion, one can derive the probability of such event as follows: Pl (r, Se ) = λ

νmax 

Pν(1)

ν=3

N ν (r, Se ) −N (r,Se ) N 3 (r, Se ) −N (r,Se ) e e + λ P3(2) ν! 3! +

νmax 

Pν(2)

ν=4

N ν (r, Se ) −N (r,Se ) . e ν! (9.65)

Here λ is the probability that a SSB can be converted to a DSB and νmax = 2n. Accounting for λ relies on the experimental findings [15, 17] that the DSBs caused by low-energy electrons with energies higher than ∼5 eV happen in one hit. In that case the subsequent break in the second DNA strand occurs due to the action of debris generated by the first SSB. Following Refs. [15, 17], λ is set equal to 0.15 within the MSA framework [30]. The first term on the right-hand side of Eq. (9.65) describes the sum of probabilities to have all ν (ν = 3, 4, ..., 2n) lesions on one DNA strand with the subsequent conversion of one SSB into a DSB. The second term is the probability of three lesions with at least one DSB among them and the subsequent conversion of one SSB into a DSB, i.e. creating two DSBs. The third term is the sum of probabilities of ν lesions (ν = 4, 5, ..., 2n) with creation of at least one DSB. After simple algebraic transformations Eq. (9.65) can be rewritten in the form: Pl (r, Se ) = λ

νmax  N ν (r, Se ) ν=3

ν!

e−N (r,Se ) + (1 − λ)

νmax  ν=4

Pν(2)

N ν (r, Se ) −N (r,Se ) e , ν! (9.66)

with Pν(2) defined above in Eq. (9.64). At small LET values when the number of lesions in a DNA double twist N (r, Se )  λ  1, the probability of lethal events Pl (r, Se ) is simplified to Pl (r, Se )  λ

N 3 (r, Se ) 7 N 4 (r, Se ) + . 3! 8 4!

(9.67)

In the region 1  N (r, Se )  νmax , the probability of lethal lesion is given by Pl (r, Se )  1 − 2(1 − λ) e−

N (r,Se ) 2

−

3 (1 − λ) e−N (r,Se ) N 3 (r, Se ) . 24

(9.68)

This means that the probability of lethal lesions Pl (r, Se ) → 1 within the entire region where 1  N (r, Se )  νmax . Knowing Pl (r, Se ) one can calculate the number of lethal events in a cell nucleus traversed by a projectile ion. Equation (9.66) represents the probability to create a lethal lesion in a DNA double twist located at the distance r from the ion track. Integrating Pl (r, Se ) over the area perpendicular to the ion’s trajectory and convoluting

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the result with the number density of DNA double twists in a cell nucleus one derives the average number of lethal lesions per unit length of the ion’s trajectory: d Nl = ns dx

∞

Pl (r, Se ) 2πr dr ≡ n s σl (Se ) .

(9.69)

0

Here n s is the number density of DNA double twists in a cell nucleus, which is equal to the number of DNA base pairs accommodated in a cell nucleus, Nbp , divided by the number of DNA base pairs in one double twist and by the nuclear volume Vn [32], Nbp ns = . (9.70) 20 Vn The function σl (Se ) is the cross-section of producing lethal DNA damage in a cell nucleus, which depends on LET and the concentration of oxygen in the target. The σl (Se ) dependence originates from the dependence of N (r, Se ) on LET; this dependence is discussed further below in this section. The number of lethal events in a cell nucleus at a given dose d produced by Nion ions is equal to [30]: dNl (Se ) z¯ Nion (Se ) , (9.71) Yl (Se ) = dx where z¯ is the average distance traversed by Nion ions through the cell nucleus. The average number of ions hitting the nucleus, Nion , depends on the nucleus area An , the dose and LET: d , (9.72) Nion (Se ) = An Se where  is the mass density of the irradiated medium taken equal to the density of liquid water,  = 1 g/cm3 . The probability of cell survival is given by the probability of zero lethal lesions occurrence [30]. According to the Poisson statistics, it is equal to (9.73) surv = e−Yl (Se ) . Substituting Nion into Yl and taking the logarithm of surv one obtains ln surv = −Yl (Se ) = −

d dNl (Se ) z¯ An , dx Se

(9.74)

 l  z¯ is the average number of lethal events created by a single ion in a cell where dN dx nucleus. A successful comparison of calculated survival curves at a range of LET values for a number of different cell lines has been demonstrated in Refs. [32, 35]. An overview of this analysis is given below in Sect. 9.7.1. The dependence of the cross-section of a DNA lethal lesion σl and the number of lethal lesions in a cell nucleus Yl on LET has been analyzed in Ref. [105]. For

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the low-LET irradiation, the number of lesions in a DNA double twist N (r, Se )  1 and hence Pl,Se (r ) ∼ N 3 (r, Se ) according to Eq. (9.67). At low LET values, e.g. for protons in the Bragg peak region, the number of lesions in a DNA double twist is linearly proportional to LET, N (r, Se ) ∝ Se . This dependence arises because (i) the number of secondary electrons incident on a DNA double twist, and hence Ne (r ), is proportional to LET [30], and (ii) the number of formed free radicals is proportional to the number of secondary electrons [31]. The shock wave induced by protons at the Bragg peak does not transport free radicals and other reactive species to the distances much larger than the secondary electron propagation range Re . Note, however, that the diffusion of free radicals on the picosecond timescale might be affected by the temperature increase in the vicinity of the ion tracks. As follows from the analysis described above, the free radicals propagation range Rr is smaller than Re ∼ 1 − 2 nm in the Se region up to 70–140 keV/µm. Combining Eqs. (9.67) and (9.69) and using the N (r, Se ) ∝ Se dependence one obtains that, in this case, σl depends on LET as σl (Se ) ∝ Se3 .

(9.75)

The number of lethal lesions in a cell nucleus Yl , Eq. (9.71), thus increases with LET as σl (Se ) ∼ Se2 . (9.76) Yl (Se ) ∝ Se The quantity Nr (r, Se ) might grow with the growth of LET due to the increase of the SW radius and correspondingly Rr . The growth of Rr results in lowering the density of free radicals and thus their recombination rate constant. The additional growth of Nr (r, Se ) with Se will result in the faster growth of σl and Yl with increasing Se . Even steeper dependencies of σl and Yl on LET may arise at higher Se values when the number of lesions in a DNA double twist N (r, Se )  1 due to a steeper dependence of Pl (r, Se ) on N (r, Se ). Finally, let us consider the case N (r, Se )  1 when the probability Pl (r, Se ) → 1. This case describes iron and heavier ions at the Bragg peak. In this case, multiple lesions are created by the shock wave induced thermomechanical stress of the DNA double twist within the distance range r < RSW (Se ) from the ion track. The number of lesions produced by the shock wave in the region r < RSW (Se ) is much bigger than the number of lesions produced by secondary electrons and free radicals, i.e. NSW (r, Se )  Ne (r, Se ) and NSW (r, Se )  Nr (r, Se ). As described in detail in section “Quantification of the Number of Bond Breaks in the DNA Double Twist”, the number of lesions NSW (r, Se ) has been evaluated from the MD simulations for the five ions with different LET values at the Bragg peak region. The critical distance RSW is analyzed below in section “Force Exerted by the Shock Wave on the DNA”. These results suggest the following stepwise dependence of N (r, Se ) on distance r from the ion track: N (r, Se ) = NSW (Se ) θ (RSW (Se ) − r ) .

(9.77)

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Since at large LET values N ≈ NSW  1 within the range r < RSW (Se ), the probability Pl (r, Se ) → 1 at r < RSW (Se ). Then for high-LET irradiation, one obtains

σl (Se ) =

∞

0

2 Pl (r, Se ) 2πr dr = π RSW (Se ) .

(9.78)

In this case, the number of lethal events in a cell nucleus at a given dose d produced by Nion ions, Eq. (9.71), transforms into: 2 Yl (Se ) = π RSW (Se ) n s z¯ An

ρd Se

(9.79)

with the probability of cell survival being given by Eq. (9.73). The characteristic range for inducing bond breakage by the shock wave induced thermomechanical stress, RSW , has been discussed above in sections “Quantification of the Number of Bond Breaks in the DNA Double Twist” and “Force Exerted by the Shock Wave on the DNA”. As demonstrated in Ref. [105], RSW depends on 1/3 LET as RSW = b Se , where b is the proportionality factor determined in section “Force Exerted by the Shock Wave on the DNA”. In the case of large LET values (where the condition N ≈ NSW  1 is fulfilled) the number of lethal events in a cell nucleus can be written as (9.80) Yl (Se ) = α Se−1/3 where α = π b2 n s z¯ An ρ d .

(9.81)

This means that the number of lethal events in a cell nucleus at a given dose d decreases slowly with high LET, which corresponds to the experimental observations for iron and heavier ions at the Bragg peak region, see Sect. 9.7.6. Now let us come back to the yield of lethal lesions in a cell nucleus Yl , Eq. (9.71). That expression can be rewritten in several ways: Yl (Se ) =

σ (Se ) d Nl π π z¯ Nion (d) = Ng N g σ (Se )Fion , d= dx 16 Se 16

(9.82)

where Fion is the ion fluence. Now we want to dwell on the universality and versatility of this expression. Its first representation, ddNxl z¯ Nion (d), indicates that the yield is just a product of two quantities, the number of lethal lesions per unit length of ion’s trajectory and the total length of ions path through the cell nucleus, which can be broken into average length of a traverse by the number of ions passing through the nucleus. This number depends on the dose. However, the dose in the case of ions is not an independent parameter, it is regulated by the ions fluence: d=

Se Fion Se z¯ Nion = , V 

(9.83)

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where  is the mass density of the nucleus. Let us analyze a number of effects and limits. First, it is interesting to analyze the limits of Nion ; the minimum (nonzero) value for it is one. Then, both the dose and yield are defined by LET, at that the former is linear with it and the latter is linear if the LET is small, but may be quadratic if the LET is larger. This enhancement is expected as a result of the transport of reactive species to larger distances by ion-induced shock waves. If the LET is too large, the lethal damage may happen already on a fraction of z¯ . This means that the “rest” of the dose is wasted, the relative biological effectiveness is reduced, and the so-called overkill effect is observed. On the other side, when Nion is very large, ion tracks are likely to overlap. This corresponds to the case of large ion fluences, which was discussed in Ref. [124]. This limit may be important in the case of applications of laser-driven proton beams. Second, Eq. (9.83) is only valid when the LET is the same for all ions; when it is not, e.g., in the case of a spread-out Bragg Peak (SOBP), then d=

 Sej z¯ F j , ρ j

(9.84)

where a subscript j indicates a corresponding component of the ion beam. This dependence was exploited in Ref. [33], and it is discussed below in Sect. 9.7.4 because the SOBP is used clinically and in many experiments as well. π e) N g σ (S d. Third, more intriguing effects are seen in the second representation, 16 Se d Nl As was mentioned, at relatively small values of LET, the d x is linear with LET. Then the yield is linear with the dose. However, when LET increases, σ (Se ) = κ Seδ , where κ is a coefficient and δ is some positive power that depends on the range of LET, as explained above. As a result, the yield is proportional to Seδ−1 d. This effect is purely due to the ion-induced shock wave effect due to reactive species propagation as well as due to direct bond-breaking effect at large LET values. Fourth, N g in this representation is the number of base pairs in the whole cell nucleus, which gets in this formula from the expectation that the cell is in the interphase and chromatin is uniformly distributed over the nucleus. In particular, this means that the yield for all human cells would be the same. As this may be true for healthy cells of normal tissue, this may not be true for cancerous cells. More research is needed to clarify this point. Fifth, the oxygen concentration dependence is “hidden” in the value of ddNxl . It affects the reactive species effect through the value of Nr that enters Eq. (9.54). The map of oxygen concentration automatically produces the map of the oxygen enhancement ratio (OER), which is the ratio of doses required to achieve the same biological effect with a given oxygen concentration to that with the maximum oxygen concentration (at normal aerobic conditions). The map of OER is deemed to be an important component of therapy optimization. The analysis of OER performed within the MSA framework is described in Sect. 9.7.2.

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Sixth, if the LET is fixed, Eq. (9.82) suggests that the yield, and therefore the logarithm of survival probability is linear with dose, thus making survival curves in their traditional coordinates straight lines. A comparison of a number of survival curves at a range of LET values supports this observation (see Sect. 9.7.1); however, there are experiments that the so-called shouldered survival curves are observed. At this point, it is worthwhile to remind a reader that a vast research of x-rays survival curves [44] that the straight survival curves indicate a single-hit scenario of radiation damage. This means that a single hit of a target (in our case with an ion) leads to cell inactivation with a given probability. This probability includes the probability of DNA damage repair. In the framework of molecular theories developed from 1950s s to 1990s [44], including the microdosimetric kinetic model (MKM) [125, 126] the shouldered survival curves are the result of either non-linear damage or repair. It is interesting to place the MSA on this map; some first studies in this direction are described in Sect. 9.7.3. Thus, the MSA methodology has been discussed. The main result is given by Eq. (9.82), which gives the expression for the yield of lethal lesions. This expression is obtained as a result of analysis of physical, chemical and biological effects on the corresponding scales. Each of its components can be further refined, but its scientific clarity is sound. For instance, in recent years, the product of LET and dose, i.e., Se d is used for proton therapy optimization [127]. In the Bragg peak region, σ (Se ) = κ Seδ and this optimization parameter is a consequence of Eq. (9.82). While we are leaving the outlook of what has to be done along the MSA in the future to the Conclusion section, we get to some applications of MSA promised above.

9.7 Examples of Application of MSA 9.7.1 Survival Probability for Different Cell Lines Irradiated with Ions The above-presented formalism has been utilized in Refs. [32, 35] to evaluate survival probability for various cell lines irradiated with ions. Figure 9.15a, b shows the survival curves for human adenocarcinomic A549 cells and normal fibroblasts AG1522, irradiated with protons and alpha particles at different values of LET. The calculated curves (lines) are compared to the experimental data (symbols) on survival of the same cells in the same conditions. Different cell lines have different cross-sectional area of their nuclei, and, thus, the average distance z¯ of the ion’s traverse through the nucleus [32]. This results in different slopes of the survival curves calculated for A549 and AG1522 cell lines at comparable values of LET. More comparisons of calculated survival curves for other human cell lines, such as glioblastoma A172 cells and normal skin fibroblasts NB1RGB, with experiments are presented in Fig. 9.15c, d.

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Fig. 9.15 Survival curves for different human cell lines: a adenocarcinomic A549 cells, b normal fibroblasts AG1522, c glioblastoma A172 cells, and d normal skin fibroblasts NB1RGB. The calculated survival probabilities are shown with lines and experimental data from Refs. [128, 129] (A549), Refs. [130–132] (AG1522), Refs. [135, 136] (A172) and Refs. [136, 137] (NB1RGB) are shown by symbols. Reproduced from Ref. [32]

For a more complete picture, we also analyzed the widely studied Chinese hamster V79 cells irradiated with protons and alpha particles (see Fig. 9.16), thus confirming the capability of the MSA to reproduce a large number of experimental results, based on the understanding of fundamental molecular and nanoscale mechanisms of radiation damage. With this understanding, it becomes possible to evaluate the probability of cell survival under different environmental conditions of irradiated targets. This issue is crucial for medical applications because in many clinical cases, especially in the centre of large tumours, one can find regions with reduced oxygen concentration. It is established that the presence of molecular oxygen substantially changes chemical interactions with biological molecules as it affects both the content of reactive species and the possibility of damage fixation. The survival curves calculated for the V79 cells irradiated under aerobic and hypoxic conditions are presented in Fig. 9.16 alongside with the corresponding experimental data [130, 138–140]. Under hypoxic conditions, the experimental studies (closed symbols) were performed at high level of hypoxia, since they were carried out in the atmosphere of nitrogen with no addition of pure oxygen.

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Fig. 9.16 Survival curves for Chinese hamster V79 cell line. The calculated survival probabilities are shown by lines and experimental data from Refs. [130, 138–140] are shown by symbols. Experiments performed under normal and hypoxic conditions are depicted by open and closed symbols, respectively. Reproduced from Ref. [32]

9.7.2 Evaluation of the Oxygen Effect Evaluation of cell survival under different environmental conditions allows one to analyze the oxygen enhancement ratio (OER). The OER is about 3 for low-LET radiation and gradually approaches unity as the LET of the radiation increases. Figure 9.17 shows the OER at the 10% survival level calculated for Chinese hamster CHO and V79 cells irradiated with carbon ions. The calculated curves cover a broad range of LET and are compared to existing experimental results for carbon and heavier ions. The MSA adequately describes the main features of the OER as a function of LET, namely it predicts the decrease of the OER with increasing the LET and its asymptotical value equal to unity at high LET. It also provides good quantitative agreement with experimental data [141] in a broad range of LET. At the LET ranging from approximately 100 to 150 keV/µm, where the RBE for carbon ion beams reaches its maximal value [141], the OER is within the range from 1.5 to 2.0 and nicely agrees with different experimental measurements [142–144]. The probability for lesion production by free radicals is sensitive to environmental conditions of irradiated targets. At the early stages of the radiation-matter interaction, a decrease of the concentration of diluted oxygen in the cell environment can modify the water radiolysis process that results in modification of primary DNA damage yields [146]. On the other hand, it has been discussed that the effect of oxygen can be explained mainly by chemical repair or oxygen fixation of primary DNA damages, which come into play at later stages of the radiation-matter interaction depending on the oxygen concentration [145, 146]. In the case of hypoxic conditions, the damage

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Fig. 9.17 Oxygen enhancement ratio at the 10% survival level for V79 and CHO cells irradiated with carbon ions. Symbols denote the experimental data taken from Refs. [141–144]. Reproduced from Ref. [32]

induced by secondary species may be repaired chemically through the reduction of DNA radicals by endogenous thiols such as glutathione or other sulfur-containing cellular constituents [147], thus decreasing the number of individual and clustered DNA lesions processed by enzymatic repair mechanisms. All these mechanisms suggest that in hypoxic conditions, the average probability for radical-induced lesion production at a given distance from the ion’s path should be smaller than that in the aerobic environment. Experimental survival probabilities of cells irradiated under hypoxic conditions (Fig. 9.16) are nicely described with the probability which is two times smaller than that used to describe aerobic conditions; this corresponds to experimental data on the induction of DSBs and non-DSB clustered DNA lesions in mammalian cells at normal concentration of oxygen and at deep hypoxia [145]. Reduction of the oxygen concentration under hypoxia results in a decrease in the rate of formation of free radicals and, thus, in a decrease in the effectiveness of free radicals to produce DNA damage.

9.7.3 Analysis of Survival of Repair-Efficient Cells The criterion of lethality and Eq. (9.82) produce linear (on a semi-log plot) survival curves for cells irradiated with ions (see Sect. 9.7.1). This model includes the probability of enzymatic repair, embedded into the criterion. The criterion itself can be different for different cell lines, but it will lead to straight lines nevertheless. The “shoulderness through damage” translates into MSA language as tracks overlap. In

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this case, the ddNxl depends on fluence and therefore or dose and Eq. (9.82) becomes nonlinear with dose and predicts a shouldered survival curve [124]. However, this happens at very large values of fluence and dose, far larger than those used clinically. Therefore, it is more likely that shouldered curves in ion therapy may be due to repair process. The solution to this problem was suggested in Ref. [32], and it is as follows. This solution does not change the expression for the yield Yl , Eq. (9.82), except for a constant coefficient. What changes is the logarithm of survival probability (9.74); instead of being linear with the yield, it becomes a quadratic function, − ln surv = Yl − (χ0 − χ1 Yl )Yl = (1 − χ0 )Yl + χ1 Yl2 ,

(9.85)

where χ0 and χ1 are positive constants. The first representation can be phenomenologically interpreted in such a way that the cell lines for which the survival curves are shouldered are more resistive than those for which the survival is linear. At small values of yield the r.h.s. of Eq. (9.85) is linear with respect to Yl with a coefficient 1 − χ0 < 1. However, as the yield increases the resistivity decreases linearly and when (χ0 − χ1 Yl ) turns to zero, the survival becomes “normal”. This is formalized as, (9.86) − ln surv = (1 − χ )Yl = Yl − (χ0 − χ1 Yl ) (χ0 − χ1 Yl ) Yl . The coefficient χ = (χ0 − χ1 Yl ) (χ0 − χ1 Yl )

(9.87)

gradually approaches zero with increasing number of lesions until it becomes equal to zero at a critical value, Y˜l = χ0 /χ1 , which depends, in particular, on dose and LET. Above this critical value, Eq. (9.74) remains valid. Thus, the critical yield Y˜l is the transition point in the survival curve from the linear-quadratic to the linear regime. The examples of application of this model are shown in Fig. 9.18. For Yl < χ0 /χ1 , the survival probability given by Eq. (9.86) can be rewritten as [32], π 2 d 2 d π σ Ng + χ1 . (9.88) − ln surv = (1 − χ0 ) σ Ng 16 Se 16 Se2 At this point, the famous empirical parameters α and β of the linear-quadratic model [44] given by − ln surv = αd + βd 2 ,

(9.89)

can be introduced. Equation (9.88) provides the molecular-level expressions for these Se χ0 : parameters at doses d ≤ 16 π σ N g χ1 α = (1 − χ0 )

1 π , σ Ng 16 Se

β = χ1

π 16

σ Ng

2 1 . Se2

(9.90)

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Fig. 9.18 Survival curves for repair-efficient HeLa and CHO cell lines. The calculated survival probabilities are shown with lines and experimental data from Refs. [148, 149] are shown by symbols. The survival curves are calculated using Eq. (9.86) with χ0 = 0.08 and χ1 = 0.07 for HeLa cells and χ0 = 0.4 and χ1 = 0.045 for CHO cells. Reproduced from Ref. [32]

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At Yl > χ0 /χ1 , i.e., for d > α is given by

16 Se χ0 , π σ N g χ1

α=

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survival curves are linear, and the parameter π σ Ng . 16 Se

(9.91)

9.7.4 Survival Curves Along a Spread-Out Bragg Peak The goal of Ref. [33] was to suggest an algorithm for choosing the energy distribution of ion fluence at the entrance in order to achieve the uniform cell survival distribution throughout the SOBP. In the beginning, it was shown that the uniform dose distribution leads to an increase of cell inactivation along the SOBP towards a sharp maximum at its distal end. In this section, we will just show the algorithm in order to achieve the uniform cell survival at a constant oxygen concentration along the SOBP. Let the maximum initial energy at the entrance be E 0 and let it change by step

E to construct the SOBP; the depth of each pristine Bragg peak can be denoted by x j , where j = 0, 1, 2, ..., J . According to Eqs. (9.82) and (9.84), at a given depth x, the yield is Yl =

 π Ng σ (S j (x))F j = Y0 , 16 j

(9.92)

where Y0 is the target yield throughout the SOBP. The goal is to obtain the distribution of F j . Clearly, F0 = Y0

16 , π N g σ (S0 (x0 ))

(9.93)

the fluence at maximum energy corresponds to the desired yield at the distal end of the Bragg peak. Then, π π N g (σ (S1 (x1 ))F1 + σ (S0 (x1 ))F0 ) = N g σ (S0 (x0 ))F0 , 16 16

(9.94)

which gives F1 =

σ (S0 (x0 )) − σ (S0 (x1 )) F0 , σ (S1 (x1 ))

on the next step we find F2 from

(9.95)

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Fig. 9.19 The solid line shows the profile of dependence of yield of lethal lesions in cells along the SOBP as a function of distance on the distal end of the SOBP. The dashed line shows the profile of the depth-dose curve that produced the above result. Reproduced from Ref. [33] with permission from Springer Nature

π N g (σ (S2 (x2 ))F2 + σ (S1 (x2 ))F1 + σ (S0 (x2 ))F0 ) 16 π = N g σ (S0 (x0 ))F0 , 16

(9.96)

and so on. If the oxygen concentration depends on x, this affects all S j (x) and can be easily included in the algorithm. Figure 9.19 shows the application of the algorithm for a proton SOBP example. This example shows a potential for the use of MSA for treatment planning and optimization purposes. Besides, it shows how important the notion of ion-induced shock wave is. In a uniform beam, the dose is linear with the LET (and the production of reactive species). In order for the observed effect of increase of cell inactivation probability with the depth along the SOBP to take place in the case of the uniform dose throughout the SOBP, there should be some dependence of the cell survival probability on LET beyond the linear dependence. The only physical reason for this dependence can be the dependence of radial range of propagation of reactive species depending on LET, Rr (Se ), and only the shock waves can make such a dependence possible.

9.7.5 The Overkill Effect at Large LET In this section, we want to briefly discuss the limit of large values of LET, so large that Nion is close to one. In this limit, it is important that even though Nion in Eqs. (9.82)

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and (9.83) is an average number of ions traversing the cell nucleus, in reality the number of ions is integer. Therefore, Nion can be redefined as the minimum number of ions required to cause the damage reflected by the survival fraction of 0 and the corresponding yield Y0 . Then (since Fion = Nion /An , where An is a cross-sectional area of a cell nucleus) Eq. (9.82) can be solved for Nion as,  16Y0 An +1, = π N g σ (Se ) 

Nion

(9.97)

where square brackets denote the integer part of their content. The relative biological effectiveness (RBE) is given by the ratio of dose delivered by photons, dγ to that delivered by ions in order to achieve the same survival fraction or yield. Then in virtue of Eq. (9.83), RBE =

dγ V dγ dγ V  .  = = 16Y0 An d Se z¯ Nion + 1 Se z¯ π N g σ (Se )

(9.98)

This equation explains the overkill effect. When LET is small, the integer part in the numerator is large compared to unity. In this limit, RBE is given by RBE =

π N g dγ  σ (Se ) . 16 Y0 Se

(9.99)

Since σ (Se ) ∝ Se in this limit, RBE is independent of LET. Then, with increasing LET, σ (Se ) ∝ Se2 and RBE becomes linear with LET until π16NgYσ0 (SAne ) becomes close to unity. This is the limit of large LET, in which RBE becomes inversely proportional to LET, dγ  V RBE = . (9.100) Se z¯ This dependence is discussed in more detail in Ref. [35], and the dependence of RBE corresponding to Eq. (9.98) is shown in Fig. 9.20. A piecewise dependence at increasing values of LET corresponding to small values of Nion deserves a comment. Nothing is wrong with such a dependence mathematically; physically, the uncertainty in LET leads to a continuous curve traced in Fig. 9.20.

9.7.6 Shock Wave Induced DNA Lethal Damage of Cells Irradiated with High-LET Ions This section describes the results of evaluation of survival probabilities of cells irradiated with high-LET ions within the MSA formalism [105]. This analysis reveals

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Fig. 9.20 RBE at 10% cell survival for human normal tissue cells irradiated with carbon ions. The results are obtained using Eq. (9.98). In the high-LET region, the RBE becomes inversely proportional to LET, and the absolute values of RBE depend on the number of ions that traverse the cell nucleus. The values of Nion corresponding to different segments of the calculated curve are indicated. The dashed line is a guide to the eye connecting median points of the hyperbolas. Symbols depict experimental data from [136, 137, 150, 151]. Reproduced from Ref. [35]

the significant role of the shock wave induced thermomechanical mechanism of DNA damage in the cell inactivation. Figure 9.21 shows the average number of simple lesions per a DNA double twist as a function of radial distance from the ion’s path for irradiation with a carbon ion (Fig. 9.21A) and with an iron ion (Fig. 9.21B) in the vicinity of the corresponding Bragg peaks. The average number of simple lesions created by secondary electrons and free radicals (Ne (r, Se ) and Nr (r, Se )), is calculated according to Eqs. (9.47) and (9.54), respectively. The average number of lesions created by the shock wave thermomechanical stress of the DNA, NSW , is taken from the MD simulations described above in section “Quantification of the Number of Bond Breaks in the DNA Double Twist” (see Table 9.2). The number of breaks corresponds to the bond dissociation energies De obtained from the DFT calculations [108]. As follows from the MD simulations (see Fig. 9.7 and Table 9.2), the thermomechanical stress by the carbon ion induced shock wave does not produce any lesions within the DNA double twist for the bond dissociation energies De . In the case of irradiation with a carbon ion, the lesions are created by secondary electrons, free radicals and other reactive species which are spread over the large distance range by the shock wave, see Fig. 9.21A. This is in agreement with the results of earlier studies [28, 30], which demonstrated that at the values of LET typical for a single carbon ion at the Bragg peak (Se = 830 keV/µm), most of ion-induced DNA damage occurs via the chemical effects involving interactions of DNA molecules with

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A

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B

Fig. 9.21 Average number of simple lesions per DNA double twist due to a single carbon ion (A) and iron ion (B) at their Bragg peak energies, as a function of radial distance from the ion’s path. Ne (r ) and Nr (r ) are the numbers of simple lesions produced by secondary electrons and free radicals, respectively. NSW (r ) is the average number of lesions produced due to direct thermomechanical damage by the ion-induced shock wave. The value NSW = 5.4 at r ≤ RSW = 0.4 nm was obtained from MD simulations, as summarized in Fig. 9.7 and Table 9.2. Reproduced from Ref. [105] with permission from American Physical Society

secondary electrons, free radicals, solvated electrons, etc. In contrast, the number of lesions produced by the thermomechanical stress caused by the iron ion-induced shock wave outweighs the number of lesions produced by the chemical effects at distances r ≤ RSW from the ion’s path, as shown in Fig. 9.21B. The analysis described below has been performed using the effective radius RSW = 0.4 nm for the iron ion at the Bragg peak, which is lower than the values reported in section “Force Exerted by the Shock Wave on the DNA”. One should stress that the model presented in section “Force Exerted by the Shock Wave on the DNA” gives the maximal values of RSW for the five studied ions at the Bragg peak, corresponding to the ideal orientation of the molecular bond parallel to the direction of the shock wave propagation. Accounting for different orientations of the bonds in the DNA backbone with respect to the direction of a shock wave propagation should lead to lowering the RSW values. The value RSW = 0.4 nm has been obtained by averaging the number of lesions produced due to direct thermomechanical damage by the iron ion-induced shock wave, NSW , over the range of distances from the ion track to the principal axis of the DNA molecule, see Fig. 9.7C. On the basis of the non-reactive MD simulations and subsequent estimates for the energy deposited into the DNA backbone bonds, it was concluded earlier [28] that the bond breaking due to the shock wave induced thermomechanical stress becomes dominant for ions heavier than argon, propagating in liquid water. This result has been confirmed in Ref. [105] by means of reactive MD simulations. The calculated probabilities Pl (r, Se ) of producing the lethal DNA damage in a DNA double twist located at distance r from the ion’s path, Eq. (9.66), are shown

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Fig. 9.22 Probability for producing lethal lesions in a DNA double twist as a function of radial distance from the ion’s path for irradiation with a carbon ion (panel A) and with an iron ion (panel B) in the vicinity of the corresponding Bragg peaks. Solid gray, solid black, and dashed red curves show, respectively, the contribution of only secondary electrons, secondary electrons and free radicals, as well as these agents together with the shock wave (SW) induced thermomechanical stress of the DNA. Reproduced from Ref. [105] with permission from American Physical Society

in Fig. 9.22 for carbon and iron ions. In the case of irradiation with iron ions (see Fig. 9.22B), accounting for the shock wave induced thermomechanical stress results in a significant increase of the probability of lethal DNA damage within the characteristic distance r ≤ RSW from the ion track. A conservative estimate for the number of bond breaks produced by the iron ion-induced shock wave thermomechanical stress, corresponding to the largest bond dissociation energy De (see Fig. 9.7 and Table 9.2), reveals that five or more bond breaks within the DNA double twist are created when the iron ion propagates at distances smaller than RSW = 0.4 nm from the principal axis of inertia of the DNA molecule. The indicated number of breaks exceeds the minimal number of lesions needed to produce the lethal DNA damage, and hence the probability Pl (r, Se ) = 1 at r ≤ 0.4 nm from the iron ion track. This means that even a single hit of a cell nucleus by a high-LET ion will be sufficient to inactivate the cell. Figure 9.23 shows survival probabilities for two human fibroblast cell lines irradiated with carbon ions at high values of LET; the probabilities were evaluated within the MSA using Eqs. (9.66)–(9.74). Lines show the survival curves obtained with accounting for the DNA damage produced by the secondary electrons, free radicals, and the shock wave mechanism. For carbon ion irradiation, the shock wave mechanism enhances transport of radicals and thus reduces their fast recombination thereby increasing the damaging effect of projectile ions. However, the direct thermomechanical DNA damage by the shock wave plays a minor role in the case of carbon ion irradiation. One should stress a good agreement of the calculated survival probabilities with experimental data [136, 151]. These calculations were performed

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Fig. 9.23 Survival probability as a function of deposited dose for the normal tissue human fibroblast cell lines, NB1RGB and M/10, irradiated with carbon ions. Survival probabilities calculated within the MSA using Eqs. (9.66)–(9.74) at the indicated values of LET are shown with lines. Experimental data for the NB1RGB [136] and M/10 [151] cells measured at a specific dose are shown by symbols. Reproduced from Ref. [105] with permission from American Physical Society

using the range of shock wave-driven propagation of reactive species, Rr = 11.9 nm, which was determined from the reactive MD simulations (see Table 9.4). The shock wave mechanism plays even bigger role in producing lethal damage to cells by high-LET ions as demonstrated in Fig. 9.22B. Figure 9.24 shows survival probabilities for two normal rodent cells, V79 and CHO, irradiated with high-LET iron ions in the vicinity of the Bragg peak. Solid red lines show the probabilities calculated with accounting for the shock-wave induced thermomechanical damage. These probabilities were calculated within the MSA using the number of lethal lesions Yl , defined by Eq. (9.79), and RSW = 0.4 nm, as discussed above. Dashed black lines show the probabilities calculated with accounting for DNA damage produced only by secondary electrons and free radicals. It is apparent that for the irradiation with high-LET ions, the shock wave induced thermomechanical stress of the DNA has a significant impact on the cell survival probabilities. If this mechanism is not taken into consideration, the calculated survival probabilities deviate by orders of magnitude from the experimental values [152, 153]. Indeed, according to Eqs. (9.69)– (9.74), the number of lethal lesions Yl produced by secondary electrons and free radicals in a cell nucleus grows with an increase of LET. As a consequence, the slope of cell survival curves would monotonically increase with an increase of LET. This behaviour contradicts with experimentally observed phenomenon known as the “overkill” effect, which manifests itself when cells are irradiated with high-LET ions. At higher LET a given dose can be delivered with the smaller number of ions. This increases chances that some cells remain non-targeted, i.e. the cell survival

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Fig. 9.24 Survival probability as a function of deposited dose for normal rodent cells, V79 and CHO, irradiated with iron ions at the indicated values of LET in the vicinity of the Bragg peak. Solid red lines show the probabilities calculated within the MSA framework with accounting for the shock-wave induced thermomechanical damage. Shaded areas illustrate variation of cell survival probabilities due to variation in the cell nucleus area (see text for details). Dashed lines show the cell survival probabilities calculated with accounting for the DNA damage produced only by secondary electrons and free radicals. Symbols denote experimental data for irradiation of the V79 [152] and CHO [153] cells. Reproduced from Ref. [105] with permission from American Physical Society

probability should increase. This leads to a less steep dependence of cell survival probability on the deposited dose. Different approaches have been adopted in existing radiobiological models to account for the “overkill” effect. For instance, empirical saturation corrections due to non-Poisson distribution of lethal lesions in the cell nucleus were introduced in the commonly used LEM and MKM models to describe the radiobiological response to high-LET irradiation [154, 155]. In contrast to other models, the MSA describes quantitatively the “overkill” effect through accounting for the shock wave induced thermomechanical stress of the DNA. As follows from Eq. (9.79), the quantification of the number of lethal lesions produced by the ion-induced shock wave in a cell requires data on nucleus area for a particular cell line. Solid red curves in Fig. 9.24A, B are obtained with the values An (V79) = 88 µm2 and An (CHO) = 127 µm2 taken, respectively, from the experimental studies [148, 156]. As it was reported by Konishi et al. [156], the distribution of nucleus areas for the CHO cells is characterized by a rather broad Gaussian-like profile, and the measured nucleus areas vary from about 80 µm2 up to 160 µm2 with the average value of 127 µm2 . The variation of the calculated cell survival probabilities related to the variation of the nucleus size is illustrated in Fig. 9.24 by the shaded areas. Note also that no data on the experimental uncertainties of the measured cell survival probabilities were provided in the earlier experimental studies [148, 156]. Therefore, the characteristic uncertainties for the cells irradiated at doses up to about 10 Gy have been estimated based on the typical experimental uncertainties arising in

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such measurements with carbon ions (Fig. 9.23). The estimated uncertainties for the iron ion irradiation are shown in Fig. 9.24 by grey colour. One may thus conclude that, within the experimental uncertainties, the calculated survival probabilities for cells irradiated with iron ions are in a very good agreement with the experimental results [152, 153]. This agreement provides a strong experimental evidence for the biodamage effects caused by ion-induced shock waves upon irradiation of biological targets with high-LET ions.

9.8 Conclusions and Outlook We reviewed the major methodological concepts of the MultiScale Approach to the physics of ion beam therapy (MSA) and demonstrated that the whole approach converges to a single formula that calculates the yield of lethal lesions in a cell irradiated with ions. This yield, equal to the logarithm of the inverse probability of survival of the cell, depends on the depth, the composition of tissue in front of the cell, oxygen concentration, and the type of the cell. It was demonstrated that the MSA allows one to calculate the probability of cell survival in a variety of conditions, such as high and low values of LET, large and small values of fluence, aerobic, and hypoxic environments. MSA generically predicts linear survival curves, but can explain shouldered curves in special cases. Thus, it is a truly universal and robust method of assessment of radiation damage with ions. Besides its effectiveness, the method answers many questions about the nature of effects that are taking place on a plethora of scales in time, space, and energy. It is tempting to give the last example to this effect. Reference [157] is a review of proton therapy planning with an emphasis on biological effectiveness. There and in many other works, it is noticed that a quantity LET·d, where d is the local dose, is the quantity that should be maximized at tumour region and minimized at healthy tissue. The review does not discuss the physical origin of such a product. However, if we take Eq. (9.82) with σ (Se ) = κ Seδ , we obtain: Yl =

σ (Se ) π π Ng N g κ Seδ−1 d , d= 16 Se 16

(9.101)

i.e., the yield of lethal damage is proportional to the product of LET to a positive power and dose. Since the review [157] is devoted to proton therapy, according to Ref. [105], δ = 3 for protons in the Bragg peak region. Thus, the MSA explains an optimization criteria in different ranges of LET. Not surprisingly, the physical source of Eq. (9.101) is related to the ion-induced shock waves. This review was not intended to compare the MSA with other approaches leading to calculations of survival curves, such as Microdosimetric-Kinetic Model (MKM) [125, 126] and following Modified MKM [158], Local Effect Model (LEM) [159–161], and track structure simulations [38, 162, 163]. Such compar-

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isons are desirable for many reasons, but it will require efforts on different sides. All of these approaches are based on dosimetry (nanodosimetry or microdosimetry), i.e., one way or another assuming that the dose per se does the damage. Other parameters and assumptions are present as well, depending on the approach. MSA is the only phenomenon-based approach, i.e., the radiation damage is deemed to be a consequence of series of effects. By design MSA has to answer why certain effects (e.g., the decrease in cell survival probability at the distal end of the SOBP with a uniform physical dose) take place. Other methods may “include” effects (like the above mentioned) in updated versions and claim that their approach can be used for therapy optimization. However, the optimization of therapy planning deserves a solid theoretical base rather than a solution that somehow works and hopefully treats patients well. Our claim is that the MSA is uniquely designed in response to this quest; it has outstanding predictive qualities and its reliance on the fundamental science makes it exceptionally valuable for the optimization of treatment planning as was demonstrated in a number of examples in this review. In general, if different methods containing different physics manage to predict comparable cell survival curves, it would be at least interesting to know why. The MSA was designed as an inclusive scientific approach and so far it lives to the expectations. Its additional strength is in its capability of adjustment to changing external conditions, e.g., the presence of radiosensitising nanoparticles [21] (of given composition, size, and density). In such cases, additional effects are just included in the scenario. The ion-induced shock wave phenomenon may change the initial conditions for the chemical phase of radiation damage. This prediction could be compared with the track structure simulations if the shock waves were included effectively in their scenario (e.g., by increasing of diffusion coefficients for reactive species depending on their positions in the track for some time on picosecond scale). To summarize, the MSA methodology provides a comprehensive picture of radiation biodamage with ions by integrating all up to date known effects and phenomena. As such, the MSA enables to describe different irradiation scenarios, including irradiation with low-, medium-, and high-LET ions, spread out Bragg peak, the LET dependence of oxygen enhancement ratio, and the overkill effect. The future directions for the MSA development concern, first, the elaboration of different elements of the approach such as quantum processes induced by low-energy secondary electrons and chemically reactive species, the formation of strand breaks, repair mechanisms, etc. Second, the discovery of ion-induced shock waves predicted and already included in the scenario of radiation damage would be the most significant step towards the recognition of the MSA. Some steps were made towards designing experiments in which the ion-induced shock waves could be observed directly [114, 115]. Third, a more elaborated scenario of transport of reactive species including the collective flow due to the shock waves as a function of LET and its comparison with MC simulations will also be an important development. Fourth, a comprehensive study of survival curves for a large variety of cell lines and conditions is definitely desired. Fifth, experiments with high fluences and disabled DNA repair function could explore the effects of tracks overlap, measure the effective radii of ion tracks, which can help better understanding of transport of reactive species. Sixth, a better understanding

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of SOBP features will certainly improve the optimization of therapy planning and bring it to a more scientific level. Seventh, the radiosensitising effect of nanoparticles should be further explored in contact with experimentalists. Finally, the MSA should be applied on the next, larger, scale to optimize the achievement of tumour control as a function of relevant external and internal conditions.

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Chapter 10

Novel Light Sources Beyond FELs Andrei V. Korol and Andrey V. Solov’yov

Abstract The chapter is devoted to the discussion of possibilities to construct novel powerful light sources (LSs) operating in the sub-angstrom wavelength range (the corresponding energies of radiation from hundreds of keV up to tens GeV region) which is far beyond the limits achievable in modern facilities (synchrotrons, undulators, and free-electron lasers, XFEL). The novel LSs (synchrotron-like, undulatorlike) are based on the channeling phenomenon for ultra-relativistic particles in oriented crystals (linear, bent, and periodically bent). These LSs can emit intensive radiation in gamma-ray region. Additionally, the crystal undulator LS has a potential to generate coherent laser-type radiation with wavelengths orders of magnitudes less than 1 Angstrom. Such LSs will have many applications in the basic sciences and the life sciences. Illustrative theoretical and computational results obtained as well as the overview of the relevant experimental activities and achievements are discussed.

10.1 Introduction The development of light sources (LS) for wavelengths λ well below 1 angstrom (corresponding photon energies E ph  10 keV) is a challenging goal of modern physics. Sub-angstrom wavelength powerful spontaneous and, especially, coherent radiation will have many applications in the basic sciences, technology, and medicine. They may have a revolutionary impact on nuclear and solid-state physics, as well as on the life sciences. At present, several X-ray Free-Electron-Laser (XFEL) sources are operating (European XFEL, FERMI, LCLS, SACLA, PAL-XFEL) or planned (SwissFEL) for X-rays down to λ ∼ 1 Å [1–14]. However, no laser system has yet been commissioned for lower wavelengths due to the limitations of permanent

A. V. Korol · A. V. Solov’yov (B) MBN Research Center gGmbH, Altenhöferallee 3, 60438 Frankfurt am Main, Germany e-mail: [email protected] A. V. Korol e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 I. A. Solov’yov et al. (eds.), Dynamics of Systems on the Nanoscale, Lecture Notes in Nanoscale Science and Technology 34, https://doi.org/10.1007/978-3-030-99291-0_10

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magnet and accelerator technologies. Modern synchrotron facilities, such as APS, SPring-8, PETRA III, ESRF [8, 15, 16], provide radiation of shorter wavelengths but of much less intensity which falls off very rapidly as λ decreases. Therefore, to create a powerful LS in the range well below 1 Å, i.e., in the hard X and gamma-ray band, one has to consider new approaches and technologies. In this book we discuss possibilities and perspectives for designing and practical realization of novel gamma-ray Crystal-based LSs (CLS) operating at photon energies E ph  102 keV and above that can be constructed through exposure of oriented crystals (linear Crystals—LC, Bent Crystals—BC, Periodically Bent Crystals—PBC) to beams of ultra-relativistic charged particles. CLSs include Channeling Radiation (ChR) emitters, crystalline synchrotron radiation emitters, crystalline Bremsstrahlung (BrS) radiation emitters, Crystalline Undulators (CU) and stacks of CUs. This interdisciplinary research field combines theory, computational modeling, beam manipulation, design, manufacture, and experimental verification of high-quality crystalline samples and subsequent characterization of their emitted radiation as novel LSs. In an exemplary case study, we estimate the characteristics (brilliance, intensity) of radiation emitted in CU-LS by positron beams available at present. It is demonstrated that peak brilliance of the CU Radiation (CUR) at E ph = 10−1 –102 MeV is comparable to or even higher than that achievable in conventional synchrotrons but for much lower photon energies. Intensity of radiation from CU-LSs greatly exceeds that available in the laser-Compton scattering LSs and can be made higher than predicted in the Gamma Factory proposal to CERN [17– 19]. The brilliance can be boosted by orders of magnitude through the process of superradiance by a pre-bunched beam. We show that brilliance of superradiant CUR can be comparable with the values achievable at the current XFEL facilities which operate in much lower photon energy range. CLSs can generate radiation in the photon energy range where the technologies based on the charged particles motion in the fields of permanent magnets become inefficient or incapable. The limitations of conventional LS are overcome by exploiting very strong crystalline fields that can be as high ∼1010 V/cm, which is equivalent to a magnetic field of 3000 T Tesla while modern superconducting magnets provide 1–10 Tesla [20]. The orientation of a crystal along the beam enhances significantly the strength of the particles interaction with the crystal due to strongly correlated scattering from lattice atoms. This allows for the guided motion of particles through crystals of different geometry and for the enhancement of radiation. Examples of CLSs are shown in Fig. 10.1 [21]. The synchrotron radiation is emitted by ultra-relativistic projectiles propagating in the channeling regime through a bent crystal, panel (a). A CU, panel (b), contains a periodically bent crystal and a beam of channeling particles which emit CUR following the periodicity of the bending [22–24]. A CU-based LS can generate photons of E ph = 102 keV–101 GeV range (corresponding to λ from 0.1 to 10−6 Å). Under certain conditions, CU can become a source of the coherent light within the range λ = 10−2 –10−1 Å [23, 25, 26]. An LS based on a stack of CUs is shown in panel (c) [27]. Practical realization of CLSs often relies on the channeling effect. The basic phenomenon of channeling is in a large distance which a projectile particle penetrates

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Fig. 10.1 Selected examples of the novel CLSs: a bent crystal, b periodically bent crystal, c a stack of periodically bent crystals. Black circles and lines mark atoms of crystallographic planes, wavy curves show trajectories of the channeling particles, shadowed areas refer to the emitted radiation

moving along a crystallographic plane or axis and experiencing collective action of the electrostatic fields of the lattice atoms [28] (see also reviews [29–31] and references therein). A typical distance covered by a particle before it leaves the channeling mode due to uncorrelated collisions is called the dechanneling length, L d . It depends on the type of a crystal and its orientation, on the type of channeling motion, planar or axial, and on the projectile energy and charge. In the planar regime, positrons channel in between two adjacent planes whereas electrons propagate in the vicinity of a plane thus experiencing more frequent collisions. As a result, L d for electrons is much less than for positrons. To ensure enhancement of the emitted radiation due to the dechanneling effect, the crystal length L must be chosen as L ∼ L d [22–24]. The motion of a projectile and the radiation emission in bent and periodically bent crystals are similar to those in magnet-based synchrotrons and undulators. The main difference is that in the latter the particles and photons move in vacuum whereas in crystals they propagate in medium, thus leading to a number of limitations for the crystal length, bending curvature, and beam energy. However, the crystalline fields are so strong that they steer ultra-relativistic particles more effectively than the most advanced magnets. Strong fields bring bending radius in bent crystals down to the cm range and bending period λu in periodically bent crystals to the hundred or even ten microns range. These values are orders of magnitude smaller than those achievable with magnets [1]. As a result, the radiators can be miniaturized thus lowering dramatically the cost of CLSs as compared to that of conventional LSs. Modern accelerator facilities make available intensive electron and positron beams of high energies, from the sub-GeV up to hundreds of GeV. These energies combined with large bending curvature achievable in crystals provide a possibility to consider novel CLSs of the synchrotron type (continuous spectrum radiation) and

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of the undulator type (monochromatic radiation) of the energy range up to tens of GeV. Manufacture of high-quality bent and periodically bent crystals is at the edge of current technologies. A number of theoretical and experimental studies of the channeling phenomenon in oriented linear crystals have been carried out (see, e.g., a review [34]). A channeling particle emits intensive ChR, which was predicted theoretically [35] and shortly after observed experimentally [36]. Since then there has been extensive theoretical and experimental investigation of ChR. The energy of emitted photons E ph scales with the beam energy as ε3/2 and thus can be varied by changing the latter. For example, by propagating electrons of moderate energies, ε = 10–40 MeV, through a linear crystal it is possible to generate ChR with photon energy E ph = 10–80 keV [37, 38]. This range can be achieved in magnetic undulators but with much higher beam energy. High-quality electron beams of (tunable) energies within the tens of MeV range are available at many facilities. Hence, it has become possible to consider ChR from linear crystals as a new powerful LS in the X-ray range [37]. In the gamma range, ChR can be emitted by higher energy ε  102 MeV beams. However, modern accelerator facilities operate at a fixed value of ε (or, at several fixed values) [39–42]. This narrows the options for tuning the ChR parameters, in particular, the wavelength. Hence, the corresponding CLS lack the tunability option. From this viewpoint, the use of bent and, especially, periodically bent crystals can become an alternative as they provide tunable emission in the hard X- and gamma-ray range. Strong crystalline fields give rise to channeling in a bent crystal. Since its prediction [43] and experimental support [44], the idea to deflect high-energy beams of charged particles by means of bent crystals has attracted a lot of attention [30, 34]. The experiments have been carried with ultra-relativistic protons, ions, positrons, electrons, π − -mesons [32, 45–60]. Steering of highly energetic electrons and positrons in bent crystals with small bending radius R gives rise to intensive synchrotron radiation with E ph  100 MeV [61, 62]. The parameters of radiation can be tuned by varying R within the range 100 –102 cm [63, 63–66]. Even more tunable is a CU-LS. In this system CUR and ChR are emitted in distinctly different photon energy ranges so that CUR is not affected by ChR. The intensity, photon energy, and line-width of CUR can be varied and tuned by changing ε, bending amplitude a and period λu , type of crystal, its length and detector aperture [26]. Since introducing the concept of CU, major theoretical studies have been devoted to the large-amplitude large-period bending λu  a > d [22–24]. In this regime, a projectile follows the shape of periodically bent planes. CUR is emitted at the frequencies ωu well below those of ChR, ωch . By varying a, λu , ε and the crystal length one can tune the CUR peaks positions and intensities. Small-amplitude small-period regime, which implies a  d and λu less than period of channeling oscillations [67– 71]. This scheme allows the emission of photons of the higher energies, ωu > ωch , makes feasible construction of a CLS which radiates in the GeV photon energy range [72].

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Initially, the CU feasibility was justified for positrons [22, 24]. Positrons channel over larger distances passing larger number of bending periods and, thus, increasing the CUR intensity. Experiments carried out so far to detect CUR from positrons have not been successful due insufficient quality of periodic bending, large beam divergence and high level of the background bremsstrahlung radiation [73–78]. The feasibility of CU for electrons was also proven [79, 80] but it was indicated that to obtain better CUR signal high-energy (GeV and above) electron beams are preferable. The CUR signal was detected in the experiments with electron beam of much lower energies at the Mainz Microtron [81, 82]. The radiation excess due to CUR was detected although it was not as intense as expected. In part, this discrepancy can be attributed to insufficient quality of the crystalline lattice although this issue has to be investigated in more detail. Also the beam energy used was low (sub-GeV range) and as a consequence photon energies, as well as the choice of particles (electrons) were not optimal. A CU based on the heavy-projectile channeling is also feasible although in this case the main restrictive factor is photon attenuation in a crystalline medium [26] It has been demonstrated that the most feasible devices are the proton-based CU (for the projectile energies ε  1 TeV) and the muon-based CU (for ε  102 GeV). In both cases the use of light crystals (diamond, silicon) is most promising. The first experimental evidence a proton channeling in periodically bent crystal was reported in Ref. [83]. The experiments were carried out with a 400 GeV proton beam at CERN and the evidence of planar channeling in the CU was firmly stated. Theoretical and experimental studies of the CU and CUR phenomena has ascertained the importance of the high quality of the undulator material needed to achieve strong effects in the emission spectra [84]. Up to now, several methods to create periodically bent crystalline structures have been proposed and/or realized. Figure 10.2 provides schematic illustration of the ranges of a and λu within which the emission of intensive CUR is feasible. Shadowed areas mark the ranges currently achievable by different technologies. Several approaches have been applied to produce static bending. The greenish area marks the area achievable by means of technologies based on surface deformations. These include mechanical scratching [77], laser ablation technique [85], grooving method [83, 86, 87], tensile/compressive strips deposition [86, 88–90], ion implantation [91]. The most recent techniques proposed are based on sandblasting one of the major sides of a crystal to produce an amorphized layer capable of keeping the sample bent [92]. Another technique, which is under consideration for manufacturing periodically bent silicon and germanium crystals, is pulsed laser melting processing that produces localized and high-quality stressing alloys on the crystal surface. This technology is used in semiconductor processing to introduce foreign atoms in crystalline lattices [93]. Currently, by means of the surface deformation methods the periodically bent crystals with large period, λu  102 microns, can be produced. To decrease the period λu one can rely on production of graded composition strained layers in an epitaxially grown Si1−x Gex superlattice [94–97]. Both silicon and germanium crystals have the diamond structure with close lattice constants. Replacement of a fraction of Si atoms with Ge atoms leads to bending crystalline

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Fig. 10.2 Shadowing indicates the ranges accessible by means of modern technologies: superlattices (gray), surface deformations (green), and acoustic waves (blue). Sloping dashed lines indicate the boundaries of the stable channeling for ε = 0.5 and 50 GeV projectiles. For each energy, the periodic bending corresponding to the CU regime (characterized by Large Period, (LP), λu  a) lies to the right from the line. The horizontal line a/d = 1 (d is the interplanar spacing) separates the Large-Amplitude (LA) and Small-Amplitude (SA) bending. The boundaries of the most favorable CU regime, LALP, are marked by thick red lines. SASP area stands for Small-Amplitude Short-Period bending

directions. By means of this method sets of periodically bent crystals have been produced and used in channeling experiments [82]. A similar effect can be achieved by graded doping during synthesis to produce diamond superlattice [98]. Both boron and nitrogen are soluble in diamond, however, higher concentrations of boron can be achieved before extended defects appear [98, 99]. The advantage of a diamond crystal is radiation hardness allowing it to maintain the lattice integrity in the environment of very intensive beams [34]. The gray area in 10.2 marks the ranges of parameters achievable by means of crystal growing. The bluish area indicates the range of parameters achievable by means of another method, realization of which is although still pending, based on propagation of a transverse acoustic wave in a crystal [26]. In a Crystalline Undulator (CU), a projectile’s trajectory follows the profile of periodic bending. This is possible when the electrostatic crystalline field exceeds the centrifugal force acting on the projectile. This condition, which entangles bending amplitude and period, the projectile’s energy and the crystal field strength, implies that the bending parameter C is less than one. The bending parameter is defined as the ratio of the interplanar force, which keeps a projectile in a channel, to the maximum centrifugal force acting on the projectile in a bent channel. Two sloping dashed lines in Fig. 10.2 show the dependences a = a(λu ) corresponding to the extreme value C = 1 for ε = 0.5 and 50 GeV projectiles. For each energy, the CU is feasible in the domain lying to the right from the line. In this domain, periodic bending is characterized by a Large Period (LP), which implies (i) λu  a, and (ii) λu greatly exceeds the period

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of channeling oscillations. The horizontal line a/d = 1 (d stands for the interplanar distance) divides the CU domain into two parts: the Large-Amplitude (LA), a > d, and the Small-Amplitude (SA), a < d, regions. Larger amplitudes are more favorable from the viewpoint of achieving higher intensities of CUR. The red lines delineate the domain where the LALP periodic bending can be considered. The necessary conditions, which must be met in order to treat a CU based on the LALP periodic bending a feasible scheme are as follows [22, 24, 100, 101]: ⎧  C = 4π 2 εa/Umax λ2u < 1 – stable channeling, ⎪ ⎪ ⎪ ⎪ – large-amplitude regime, ⎪ ⎨ d < a  λu – N = L/λu  1 (10.1)  large number of periods, ⎪ ⎪ L ∼ min L (C), L (ω) – account for dechanneling and photon attenuation, ⎪ d a ⎪ ⎪ ⎩ ε/ε  1 – low-energy losses. The formulated conditions are of a general nature since they are applicable to any type of a projectile undergoing channeling in PBCr. Their application to the case of a specific projectile and/or a crystal channel allows one to analyze the feasibility of the CU by establishing the ranges of ε, a, λu , L, N and ω which can be achieved. • A stable channeling of a projectile in a periodically bent channel occurs if the  maximum centrifugal force Fcf is less than the maximal interplanar force Umax ,  i.e., C = Fcf /Umax < 1. Expressing Fcf through the energy ε of the projectile, the period and amplitude of the bending one formulates this condition as it is written in (10.1). • The operation of a CU should be considered in the large-amplitude regime. The limit a/d > 1 accompanied by the condition C  1 is mostly advantageous, since in this case the characteristic frequencies of UR and ChR are well separated: 2 ∼ Cd/a  1. As a result, the channeling motion does not affect the paramωu2 /ωch eters the UR, the intensity of which can be comparable or higher than that of ChR. A strong inequality a  λu ensured elastic deformation of the crystal. • The term “undulator” implies that the number of periods, N , is large. Only then the emitted radiation bears the features of an UR (narrow, well-separated peaks in spectral-angular distribution). This is stressed by the third condition. • A CU essentially differs from a conventional undulator, in which the beams of particles and photons move in vacuum, In CU the both beams propagate in crystalline medium and, thus, are affected by the dechanneling and the photon attenuation. The dechanneling effect stands for a gradual increase in the transverse energy of a channeled particle due to inelastic collisions with the crystal constituents [28]. At some point the particle gains a transverse energy higher than the planar potential barrier and leaves the channel. The average interval for a particle to penetrate into a crystal until it dechannels is called the dechanneling length, L d . In a straight channel this quantity depends on the crystal, on the energy and the type of a projectile. In a periodically bent channel there appears an additional dependence on the parameter C. The intensity of the photon flux, which propagates through a crystal, decreases due to the processes of absorption and scattering. The interval

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within which the intensity decreases by a factor of e is called the attenuation length, L a (ω). This quantity is tabulated for a number of elements and for a wide range of photon frequencies (see, e.g., Ref. [102, 103]). The fourth condition in (10.1) takes into account severe limitation of the allowed values of the length L of a CU due to the dechanneling and the attenuation. • Finally, let us comment on the last condition, which is of most importance for light projectiles, positrons, and electrons. For sufficiently large photon energies (ω  101 . . . 102 keV depending on the type of the crystal atom) the restriction due to the attenuation becomes less severe than due to the dechanneling effect. Then, the value of L d (C) effectively introduces an upper limit on the length of a CU. Since for an ultra-relativistic particle L d ∝ ε (see, e.g., [34]), it seems natural that to increase the effective length one can consider higher energies. However, at this point another limitation manifests itself [100]. The coherence of UR is only possible when the energy loss ε of the particle during its passage through the undulator is small, ε  ε. This statement, together with the fact, that for ultrarelativistic electrons and positrons ε is mainly due to the photon emission, leads to the conclusion that L must be much smaller than the radiation length L r , the distance over which a particle converts its energy into radiation. For a positron-based CU a thorough analysis of the system (10.1) was carried out for the first time in Refs. [22–24, 100, 101, 104]. Later on, the feasibility of the CU utilizing the planar channeling of electrons was demonstrated [79, 80]. Recently, similar analysis was carried out for heavy ultra-relativistic projectiles (muon, proton, and ion) [26]. Another regime of periodic bending, Small-Amplitude Short-Period (SASP), can be realized in the domain a < d and λu < 1 micron (these values of λu are much smaller that channeling periods of projectiles with ε  1 GeV). In the SASP regime, in contrast to the channeling in a CU, channeling particles do not follow the shortperiod bent planes but experience regular jitter-type modulations of their trajectories which lead to the emission of high-energy radiation. As mentioned, dynamic bending can be achieved by propagating a transverse acoustic wave along a particular crystallographic direction [22, 24, 105–109]. This can be achieved, for example, by placing a piezo sample atop the crystal and generating radio frequencies to excite the oscillations. The advantage of this method is its flexibility: the bending amplitude and period can be changed by varying the wave intensity and frequency [22, 24, 100]. Although the applicability of this method has not yet been checked experimentally, we note that a number of experiments have been carried out on the stimulation of ChR by acoustic waves excited in piezoelectric crystals [110]. The range of bending period λu ∼ 1 . . . 103 microns corresponds to the frequencies ν = vs /λu ∼ 1 . . . 103 MHz, which are achievable experimentally (vs = 4.67 × 105 cm/s is the speed of sound) [110–115].

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10.2 Overview of Numerical Approaches to Simulate Channeling Phenomenon Various approximations have been used to simulate the channeling phenomenon in oriented crystals. While most rigorous description relies on can be achieved within the framework of quantum mechanics (see recent paper [116] and references therein) quite often classical description in terms of particles trajectories is highly adequate and accurate. Indeed, the number of quantum states N of the transverse motion of √ a channeling electron and/or positron increases with its energy as N ∼ A γ where A ∼ 1 and γ = ε/mc2 stands for the relativistic Lorentz factor of a projectile of energy ε and mass m [117, 118]. The classical description implies strong inequality N  1, which becomes well fulfilled for projectile energy in the hundred MeV range and above. Simulation of channeling and related phenomena has been implemented in several software packages within frameworks of different theoretical approaches. Below we briefly characterize the most recently developed ones.1 • The computer code Basic Channeling with Mathematica [125] uses continuous potential for analytic solution of the channeling related problems. The code allows for computation of classical trajectories of channeled electrons and positrons in continuous potential as well as for computation of wave functions and energy levels of the particles. Calculation of the spectral distribution of emitted radiation is also supported. • A toolkit for the simulation of coherent interactions between high-energy charged projectiles with complex crystalline structures called DYNECHARM++ has been developed [126]. The code allows for calculation of electrostatic characteristics (charge densities, electrostatic potential and field) in complex atomic structures and to simulate and track a particle’s trajectory. Calculation of the characteristics is based on their expansion in the Fourier series through the ECHARM (Electrical CHARacteristics of Monocrystals) method [127]. Two different approaches to simulate the interaction have been adopted, relying on (i) the full integration of particle trajectories within the continuum potential approximation, and (ii) the definition of cross-sections of coherent processes. Recently, this software package was supplemented with the RADCHARM+ module [128] which allows for the computation of the emission spectrum by direct integration of the quasi-classical formula of Baier and Katkov [129]. • The CRYSTALRAD simulation code, presented in Ref. [130] is an unification of the CRYSTAL simulation code [131] and the RADCHARM++ routine [128]. The former code is designed for trajectory calculations taking into account various coherent effects of the interaction of relativistic and ultra-relativistic charged particles with straight or bent single crystals and different types of scattering. The program contains one- and two-dimensional models that allow for modeling of 1

The list of earlier codes developed to simulate the channeling phenomenon includes, in particular, Refs. [119–124].

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•

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classical trajectories of relativistic and ultra-relativistic charged particles in the field of atomic planes and strings, respectively. The algorithm for simulation of motion of particles in presence of multiple Coulomb scattering is modeled accounting for the suppression of incoherent scattering [132]. In addition to this, nuclear elastic, diffractive, and inelastic scattering are also simulated. In Ref. [133] the algorithm based on the Fourier transform method for planar radiation has been presented and implemented to compute the emission spectra of ultra-relativistic electrons and positrons within the Baier-Katkov quasi-classical formalism. Special attention has been given to treat the radiation emission in the planar channeling regime in bent crystals with account for the contributions of both volume reflection and multiple volume reflection events. The simulation presented took into consideration both the nondipole nature and arbitrary multiplicity of radiation accompanying volume reflection. A large axial contribution to the hard part of the radiative energy loss spectrum as well as the strengthening of planar radiation, with respect to the single volume reflection case, in the soft part of the spectrum have been demonstrated. The codes described in Ref. [134] (see also [26]) allow for simulation of classical trajectories of ultra-relativistic projectiles in straight and periodically bent crystals as well as for computing spectra-angular distribution of the radiated energy within the quasi-classical formalism [129]. The trajectories are calculated by solving three-dimensional equations of motion with account for (i) the continuous interplanar potential; (ii) the centrifugal potential due to the crystal bending; (iii) the radiative damping force; (iv) the stochastic force due to the random scattering of the projectile by lattice electrons and nuclei. Recently presented code [135] allows one to determine the trajectory of particles traversing oriented single crystals and to evaluate the radiation spectra within the quasi-classical approximation. To calculate the electrostatic field of the crystal lattice the code uses thermally averaged Doyle-Turner continuous potential [136]. Beyond this framework, included are multiple Coulomb scattering and energy loss due to radiation emission. It is shown that the use of Graphics Processing Units (GPU) instead of the CPU processors speeds up calculations by several orders of magnitude. In Ref. [118, 137] a Monte Carlo code was described which allows one to simulate the electron and positron channeling. The code did not use the continuous potential concept but utilized the algorithm of binary collisions of the projectile with the crystal constituents. However, as it has been argued [26, 138, 139], the code was based on the peculiar model of the elastic scattering of the projectile from the crystal atoms. Namely, atomic electrons are treated as point-like charges placed at fixed positions around the nucleus. The model implies also that the interaction of an projectile with each atomic constituent, electrons included, is treated as the classical Rutherford scattering from a static, infinitely massive point charge. It was demonstrated in the cited papers that in practical simulations, non-zero statistical weight of hard collisions with spatially fixed electrons overestimates the increase of the

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root-mean square scattering angle with increasing the propagation distance of the channeling particle. As a result, the model over-counts dechanneling-channeling events resulting from the hard collisions.

10.3 Atomistic Modeling of the Related Phenomena Numerical modeling of the channeling and related phenomena beyond the continuous potential framework can be carried out by means of the multi-purpose computer package MBN Explorer [140, 141]. The MBN Explorer package was originally developed as a universal computer program to allow investigation of structure and dynamics of molecular systems of different origin on spatial scales ranging from nanometers and beyond. In order to address the channeling phenomena, an additional module has been incorporated into MBN Explorer to compute the motion for relativistic projectiles along with dynamical simulations of the propagation environments, including the crystalline structures, in the course of the projectile’s motion [139]. The computation accounts for the interaction of projectiles with separate atoms of the environments, whereas a variety of interatomic potentials implemented in MBN Explorer supports rigorous simulations of various media. The software package can be regarded as a powerful numerical tool to reveal the dynamics of relativistic projectiles in crystals, amorphous bodies, as well as in biological environments. Its efficiency and reliability has been benchmarked for the channeling of ultra-relativistic projectiles (within the sub-GeV to tens of GeV energy range) in straight, bent and periodically bent crystals [27, 64, 72, 139, 142–153]. In these papers verification of the code against available experimental data and predictions of other theoretical models was carried out.

10.3.1 Methodology The description of the simulation procedure is sketched below. Within the framework of classical relativistic dynamics propagation of an ultrarelativistic projectile of the charge q = Z e and mass m through a crystalline medium implies integration of the following two coupled equations of motion (EM): ⎧ ⎨

1 mγ ⎩ r˙ = v v˙ =

 F·v F−v 2 , c

(10.2)

The force F is the sum of two terms: F = −q ∂U/∂r + f .

(10.3)

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The first term is due to the electrostatic interaction of the particle with crystal atoms; U = U (r) stands for the electrostatic potential. The second term is the radiative reaction force. In MBN Explorer, the EM are integrated using the forth-order Runge-Kutta scheme with adaptive time step control. At each step, the potential U = U (r) is calculated as the sum of potentials Uat (r) of individual atoms U (r) =



 Uat r − R j

(10.4)

j

where R j is the position vector of the jth atom. The code allows one to evaluate the atomic potential using the approximations due to Molière [154] and Pacios [155]. A rapid decrease of these potentials with increasing the distances from the atoms allows the sum (10.4) to be truncated in practical calculations. Only atoms located inside a sphere of the (specified) cut-off radius ρmax with the center at the instant location of the projectile. The value ρmax is chosen large enough to ensure negligible contribution to the sum from the atoms located at r > ρmax . The search for such atoms is facilitated by using the linked cell algorithm implemented in MBN Explorer [33, 140]. EM (10.2) describe the classical motion of a particle in the crystalline environment. They do not account for random events of inelastic scattering of a projectile from individual atoms leading to the atomic excitation or ionization. The impact of such events on the projectile motion is twofold. First, they result in a gradual decrease in the projectile energy due to the ionization losses. Second, they lead to a chaotic change in the direction of the projectile motion. Rigorous treatment of the inelastic collision evens can only be achieved by means of quantum mechanics. However, taking into account that such events are random, fast, and local they can be incorporated into the classical mechanics framework according to their probabilities. This approach, which is similar to the one developed in connection with the Irradiation-Driven Molecular Dynamics [156], is implemented in MBN Explorer following the scheme described in Ref. [134]. The differential probability (per path ds = cdt) of the relative energy transfer μ = (ε − ε )/ε by an ultra-relativistic projectile due to the ionizing collisions with the quasi-free electrons is defined by the following expression [40, 157]) n e (r) d2 P 2 = 2πr02 Z 2 , dμ ds γ μ2

(10.5)

where n e (r) stands for the local concentration of electrons in the crystal. In the points away from the positions of the nuclei it can be calculated from the Poisson equation ∇ · E = −4π en e (r), where E is the field strength in the point r. In a single collision with a quasi-free electron at rest, the relative energy transfer is sought within the interval [μmin , μmax ], where

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μmax

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⎧ ⎪ ⎪ ⎨

2γ ξ for ξ  1 1 + 2γ ξ + ξ 2 = 1 − γ −1 for a positron (ξ = 1) ⎪ ⎪ ⎩ (1 − γ −1 )/2 for an electron (ξ = 1)

(10.6)

Here I is the (average) ionization potential of the crystal atom and ξ stands for the ratio of m to the electron mass m e . Formally, expression (10.5) is valid for μ  1. However, since a probability of collisions with μ ∼ 1 is negligibly small, it can be applied to the whole interval of μ. At each step s = c t of integration of the EM (10.2) the ionizing collisions are treated as probabilistic events. Once the event occurs and the value μ is determined, one calculates a round scattering angle θ measured with respect to the instant velocity v of the projectile:  cos θ =

1 + γ −1 1 − γ −1



1 − μ − γ −1 μ(ξ − 1)γ −1   − 1 − μ + γ −1 1 − γ −2 (1 − μ)2 − γ −2

(10.7)

The magnitude of the second (the azimuthal) scattering angle φ (also with respect to v) is not restricted by any kinematic relations, and is obtained by random shooting (with a uniform distribution) into the interval [0, 2π ]. The twofold probability d P 2 /dμ ds must satisfy the normalization condition  0

L ion



μmax

μmin

d2 P dμ ds = 1, dμ ds

(10.8)

where L ion is the spatial interval within which the probability of an ionizing collision accompanied by arbitrary energy transfer is equal to one. This quantity is expressed through μmin , μmax and the local electron density: L −1 ion =

2πr02 Z 2 n e (r) 1 , γ μ0

μ0 =

μmax μmin ≈ μmin . μmax − μmin

(10.9)

Hence, probability of an ionizing collision with the energy transfer μ occurring within s can be written as dP =

s W (μ) dμ , L ion

W (μ) =

μ0 . μ2

(10.10)

Here s/L ion defines the probability of the collision (with arbitrary μ) to happen on the scale s, whereas the factor W (μ)dμ represents the normalized probability of the energy transfer between μ and μ + dμ. To simulate the probability of the event to happen one generates a uniform random deviate rs ∈ [0, 1] and matches it to s/L ion . If x ≤ s/L ion then the event occurs and one generates the random deviate μ with the probability distribution μ0 /μ2 . The generated μ value is used in (10.7) to calculate the scattering angle θ . The second

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scattering angle φ is generated via φ = 2π rφ with rφ standing for a uniform random deviate rφ ∈ [0, 1]. The values of μ, θ , and φ are used to modify the velocity and energy of projectile at the start of the next integration step of the EM. The trajectory of a particle entering the initially constructed crystal at the instant t = 0 is calculated by integrating equations (10.2). Initial transverse coordinates, (x0 , y0 ), and velocities, (vx,0 , v y,0 ), are generated randomly accounting for the conditions at the crystal entrance (i.e., the crystal orientation and beam emittance). A particular feature of MBN Explorer is in simulating the crystalline environment “on the fly”, i.e., in the course of propagating the projectile. This is achieved by introducing a dynamic simulation box which moves following the particle (see Refs. [26, 139] for the details). As a first step in simulating the motion along a particular direction, a crystalline lattice is generated inside the rectangular simulation box of the size L x × L y × L z . The z-axis is oriented along the beam direction. To simulate the axial channeling the z-axis is directed along a chosen crystallographic direction klm (here integers k, l, m stand for the Miller). In the case planar channeling, the z-axis is parallel to the (klm)-plane, and the y-axis is perpendicular to the plane. The position vectors of the nodes R(0) j ( j = 1, 2, . . . , N ) within the simulation box are generated in accordance with the type of the Bravais cell of the crystal and using the pre-defined values of the lattice vectors. The simulation box can be cut along specified faces, thus allowing tailoring the generated crystalline sample to achieve the desired form of the sample. Several build-in options, characterized below, allow one to further modify the generated crystalline structure [150]. • The sample can be rotated around a specified axis thus allowing for the construction of the crystalline structure along any desired direction. In particular, this option allows one to choose the direction of the z-axis well away from major crystallographic axes, thus avoiding the axial channeling (when not desired). • The nodes can be displaced in the transverse direction: y → y + R(1 − cos φ) where φ = arcsin(z/R). As a result, a crystal bent with a constant radius R is generated. • Periodic harmonic displacement of the nodes is achieved by means of the transformation r → r + a sin(k · r + ϕ). The vector a and its modulus, a, determine the direction and amplitude of the displacement, the wave-vector k specifies the axis along which the displacement to be propagated, and λu = 2π/k defines the wavelength of the periodic bending. The parameter ϕ allows one to change the phase-shift of periodic bending. In a special case a ⊥ k, these options provide simulation of linearly polarized periodically bent crystalline structure which is an important element of a crystalline undulator. These transformations, are reversible and, therefore, allow for efficient construction of a crystalline structure in an arbitrary spatial area. Also, the simulation box can be cut along specified faces, thus allowing tailoring the generated structure to achieve the desired form of the sample.

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In addition to the aforementioned options, MBN Explorer allows one to model binary structures (for example, Si1−x Gex or diamond-boron superlattices) by introducing random or regular substitution of atoms in the initial structure with the dopant atoms. Once the nodes are defined, the position vectors of the atomic nuclei are generated with account for random displacement from the nodes due to thermal vibrations corresponding to a given temperature T . For each atom the displacement vector  is generated by means of the normal distribution w() =

 1 exp −2 /2u 2T , 2 3/2 (2π u T )

(10.11)

where u T denotes the root-mean-square amplitude of the thermal vibrations. The values of u T for a number of crystals are summarized in [29]. By introducing unrealistically large value of u T (for example, exceeding the lattice constants) it is possible to consider large random displacements. As a result, the amorphous medium can be generated. An important methodological issue concerns formulation of a criterion for distinguishing between channeling and non-channeling regimes of projectiles’ motion. Depending on theoretical approach used to describe interaction of a projectile with a crystalline environment, the criterion can be introduced in different ways. For example, within the continuous potential framework [28] the transverse and longitudinal motions of the projectile are decoupled. As a result, it is straightforward to define the channeling projectiles as those with transverse energies ε⊥ less than the height U of the interplanar (or, inter-axial) potential barrier, see Fig. 10.3 left. Within this framework, the acceptance A is determined at the entrance to the crystal and can be defined as the ratio of the number of particles with ε⊥ < U to the total number of particles.

Fig. 10.3 Left. Continuous interplanar Si(110) potential U (y) for a positron calculated in the Molière approximation at T = 300 K. The coordinate y is measured along the 110 axial direction. The channeling regime corresponds to the transverse energies ε⊥ < U . Right. The positron potential in Si(110) is calculated following Eq. (10.4) as a sum of individual atomic potentials with account for thermal vibrations of the atoms. The red arrow is aligned with the 110 direction

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Within the framework of molecular dynamics, the simulations are based on solving the EM (10.2) accounting, as in reality, for the interaction of a projectile with individual atoms of the crystal. The potential U (r) (10.4) experienced by the projectile varies rapidly in the course of the motion (see Fig. 10.3 right) coupling the transverse and longitudinal degrees of freedom. Therefore, other criteria must be provided to identify the channeling segments in the projectile’s trajectory. For a particular case of planar channeling, one can assume that a projectile is captured in the channeling mode when the sign of the transverse velocity v y changes at least two times inside the same channel [118, 139].

10.3.2 Statistical Analysis of Trajectories Taking into account randomness in sampling the incoming projectiles and in positions of the lattice atoms due to the thermal fluctuations, one concludes that each simulated trajectory corresponds to a unique crystalline environment. Thus, all simulated trajectories are statistically independent and can be analyzed further to quantify the channeling process as well as the emitted radiation. Figure 10.4 shows two simulated trajectories of 10 GeV electrons which enter straight oriented diamond crystal along the (110) crystallographic planes. Dashed horizontal lines in Fig. 10.4 mark the cross section of the (110) crystallographic planes separated by the distance d = 1.26 Å. Thus, the y-axis is aligned with the 110 crystallographic axis. The horizontal z-axis corresponds to the direction of the incoming particles. A projectile enters the crystal at z = 0 and exits at z = L. The crystal is considered infinitely large in the x and y directions. The curves shown in the figure represent the projections of the 3D trajectories on the (yz)-plane. These exemplary trajectories illustrate a variety of features which characterize the motion of a charged projectile in an oriented crystal: the channeling motion, the over-barrier motion, the dechanneling and the rechanneling processes, rare events of hard collisions. Apart from providing the possibility of illustrative comparison, the simulated trajectories allow one to quantify the channeling process in terms of several parameters and functional dependencies which can be generated on the basis of statistical analysis of the trajectories [26, 27, 64, 139, 142, 143, 147, 148]. Randomization of the “entrance conditions” leads to a different chain of scattering events for the different projectiles at the entrance to the bulk. As a result, not all trajectories start with the channeling segments. In Fig. 10.4, trajectory (a) refers to an accepted projectile that changes the direction of transverse motion more than two times while moving in the same channel. In contrast, trajectory (b) corresponds to the non-accepted projectile. To quantify this feature we define acceptance as the ratio A=

Nacc , N0

(10.12)

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Fig. 10.4 Selected simulated trajectories of 10 GeV electrons propagating in straight oriented diamond(110) crystal. The trajectories illustrate the channeling and the over-barrier motion as well as the dechanneling and rechanneling effects. The z-axis of the reference frame is directed along the incoming projectiles, the (x z)-plane is parallel to the (110) crystallographic planes (dashed lines) and the y-axis is perpendicular to the planes. The diamond(110) interplanar distance is d = 1.26 Å. For the accepted trajectory (a) the characteristic lengths of the channeling motion are indicated: the initial channeling segment, z ch,0 , and channeling segments in the bulk, z ch,1 and z ch,2 . The non-accepted trajectory (b) corresponds to z ch,0 = 0. Encircled are the parts of trajectories that do not satisfy the criterion adopted for the definition of the channeling mode: the projectile stays in the same channel but changes the direction of the transverse motion only once

where Nacc stands for the number of accepted particles and N0 is the total number of the incident particles. The non-accepted particles experience unrestricted overbarrier motion at the entrance but can rechannel at some distance z. As defined, acceptance depends on the beam emittance, on the type of the crystal and the channel, and on the bending radius. In a bent crystal, the channeling condition [43] implies that the centrifugal force Fcf = pv/R ≈ ε/R, acting on the particle in the co-moving frame (R stands for the bending radius) is smaller than the maximum interplanar force Fmax . It is convenient to quantify this statement by introducing the dimensionless bending parameter C: C=

Fcf ε Rc . = = Fmax R Fmax R

(10.13)

The case C = 0 (R = ∞) characterizes the straight crystal whereas C = 1 corresponds to the critical (minimum) bending radius Rc = ε/Fmax [43]. Figure 10.5 shows acceptance as a function of the bending parameter C for 855 MeV electrons channeling in several oriented crystals as indicated. The symbols mark the data that were obtained by statistical analysis of the simulated trajectories. For each crystal, the corresponding values of bending radius can be calculated from Eq.

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Fig. 10.5 Acceptance versus bending parameter C (see Eq. (10.13)) for 855 MeV electrons. The presented data, obtained by means of MBN Explorer, are taken from: Ref. [27] for diamond(110), Refs. [64, 147] for Si(110), Ref. [150] for Ge(110), and [145] for W(110)

(10.13) using the following values of Fmax calculated in the Molière approximation and at room temperature: 7.0, 5.7, 10.0, and 42.9 GeV/cm for diamond, silicon, germanium, and tungsten, respectively. An accepted projectile stays in the channeling mode of motion over some interval z ch,0 until an event of the dechanneling (if it happens). The initial channeling segment is explicitly indicated for trajectory (a). For the non-accepted particle this segment is absent, z ch,0 = 0. To quantify the dechanneling effect for the accepted particles, one can introduce the penetration length L p [139] defined as the arithmetic mean of the initial channeling segments z ch,0 calculated with respect to all accepted trajectories:  Nacc Lp =

( j) j=1 z ch,0

Nacc

.

(10.14)

For sufficiently thick crystals the penetration length approaches the so-called dechanneling length L d that characterizes the decrease of the fraction of channeling particles in terms of the exponential decay law, ∝ exp(−z/L d ) [158]. The concept of exponential decay has been widely exploited to estimate the dechanneling-channeling lengths for various ultra-relativistic projectiles in straight and bent crystals [30, 45, 55–57, 65, 159–161]. Random scattering of the projectiles can result in the rechanneling process, i.e., capturing the particles into the channeling mode of motion. In a sufficiently long crystal, the projectiles can experience dechanneling and rechanneling several times, as it is illustrated by both trajectories in Fig. 10.4. These multiple events can be quantified by introducing the total channeling length L ch , which characterizes the channeling process in the whole crystal. This quantity is calculated by averaging

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Fig. 10.6 Channeling fractions f ch0 (z) (dashed curves) and f ch (z) (solid curves) calculated for 855 MeV electrons in straight (C = 0) and bent (C > 0) Ge(110) channels. Redrawn from data presented in Ref. [150]

the sums of all channeling segments z ch0 + z ch1 + z ch2 + · · · , calculated for each trajectory, over all trajectories. To quantify the impact of the rechanneling effect, one can compute the channeling fractions f ch,0 (z) = Nch,0 (z)/Nacc and f ch (z) = Nch (z)/Nacc [139, 147]. Here Nch,0 (z) stands for the number of particles that propagate in the same channel where they were accepted up to the distance z where they dechannel. The quantity Nch is the total number of particles that are in the channeling mode at the distance z. As z increases the fraction ξch0 (z) decreases due to the dechanneling of the accepted particles. In the contrast, the fraction ξch (z) can increase with z when the particles, including those not accepted at the entrance, can be captured in the channeling mode in the course of the rechanneling. These dependencies simulated for 855 MeV electron in straight and bent Ge(110) channels are presented in Fig. 10.6. A striking difference in the behavior of the two fractions as functions of the penetration distance z is mostly pronounced for the straight channel. Away from the entrance point, the fraction f ch0 (z) (dashed curve) follows approximately the exponential decay law (see discussion below). At large distances, the fraction f ch (z) (solid curve), which accounts for the rechanneling process, decreases much slower following the power law, f ch (z) ∝ z −1/2 [118]. As the bending curvature increases, C ∝ 1/R, the rechanneling events become rarer, and the difference between the two fractions decreases. For C  0.1 both curves virtually coincide. We note here that the impact of rechanneling was also highlighted by other authors in connection with the experimental studies with both straight [159] and bent [65] crystals.

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10.3.3 Calculation of Spectral Distribution of Emitted Radiation Spectral distribution of the energy emitted within the cone θ ≤ θ0  1 with respect to the incident beam is computed numerically using the following formula:  N0  1

dE(θ ≤ θ0 ) d3 E n = . dφ θ dθ dω N0 n=1 dω d 2π

0

θ0

(10.15)

0

Here, ω is the radiation frequency,  is the solid angle corresponding to the emission angles θ and φ. The quantity d3 E n /dω d stands for the spectral-angular distribution emitted by a particle that moves along the nth trajectory. The sum is carried out over all simulated trajectories, and thus it takes into account the contribution of the channeling segments of the trajectories as well as of those corresponding to the non-channeling regime. The numerical procedures implemented in MBN Explorer to calculate the distributions d3 E n /dω d [139] are based on the quasi-classical formalism [129]. In the limit ω/ε  1 the quasi-classical formula reduces to that known in classical electrodynamics (see, e.g., [162]). The classical description of the radiative process is adequate to characterize the emission spectra by electrons and positrons of the subGeV and GeV energy range. The quantum corrections lead to strong modifications of the radiation spectra of multi-GeV projectiles channeling in bent and periodically bent crystals [27, 72, 142]. The calculated spectral intensity can be normalized to the Bethe-Heitler value (see, for example, Ref. [163]) and thus can be presented in the form of an enhancement factor over the bremsstrahlung spectrum in the corresponding amorphous medium. Figure 10.7 presents results of an exemplary case study of the emission spectra from 6.7 GeV positrons (left) and electrons (right) channeled in L = 105 µm thick oriented Si(110) crystal. The spectra were computed for the emission cone θ0 = 0.4 mrad [164] taht exceeds the natural emission cone γ −1 by a factor of about five. Solid black and dashed red curves present the results of two sets of calculations. The first set corresponds to the case of zero beam emittance, when the velocities of all projectiles at the crystal entrance are tangent to Si(110) plane, i.e., the incident angle ψ is zero [139]. The second set of trajectories was simulated allowing for the distribution of the incident angle within the interval ψ = [−θL , θL ] with θL = 62 µrad being Lindhard’s critical angle. The calculated enhancement factors are compared with the experimental results presented in Ref. [164] and the results of numerical simulations for positrons from Ref. [165]. Figure 10.7 demonstrates that the simulated curves reproduce rather well the shape of the spectra and, in the case of the positron channeling, the positions of the main and the secondary peaks. With respect to the absolute values both calculated spectra, ψ = 0 and |ψ| ≤ ψL , exhibit some deviations from the experimental results. For positrons, the curve with ψ = 0 perfectly matches the experimental data in vicinity

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Fig. 10.7 Enhancement factor of the channeling radiation over the Bethe-Heitler spectrum for 6.7 GeV positrons (left) and electrons (right) in straight Si(110) crystal. Open circles stand for the experimental data [164]. The calculations performed with MBN Explorer [139, 150] are shown with black solid curves, which present the results obtained for fully collimated beams (zero emittance), and red dashed curves, which correspond to the emittance of 62 µrad as in the experiment. The symbols (closed circles and rectangles) mark a small fraction of the points and are drawn to illustrate typical statistical errors (due to a finite number of the trajectories simulated) in different parts of the spectrum. Green dashed curve, shown on the left figure, corresponds to the results presented in Ref. [165]. The data refer to the emission cone θ0 = 0.4 mrad

of the main peak but underestimates the measured yield of the higher (the second) harmonic. Increase in the incident angle results in some overestimation of the main maximum but improves the agreement above ω = 40 MeV. For electrons, the ψ = 0 curve exceeds the measured values, however, the increase in ψ leads to a very good agreement if one takes into account the statistical errors of the calculated dependence (indicated by symbols with error bars). The aforementioned deviations can be due to several reasons. First, the emission spectra can be sensitive to the choice of the approximation scheme used to describe the atomic potentials when constructing the crystalline field as a superposition of the atomic fields, Eq. (10.4). The results presented in Fig. 10.7 were obtained for the trajectories simulated within the Molière approximation framework. Though this approximation is a well-established and efficient approach, more realistic schemes for the crystalline fields, based, for example, on X-ray scattering factors [136, 166] or on accurate numerical approaches for In Ref. [150] a comparison has been carried out of the experimentally measured spectra with those simulated numerically using the Molière and the Pacios approximations for atomic potentials. It has been shown that for positrons both approximations result in virtually the same dependences. In the case of electrons, the spectra obtained with the Pacios potential

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are 5–10 percent less intensive. Within the statistical errors both results are in a good agreement with the experimental measurements. Another source of the discrepancies can be attributed to some uncertainties in the experimental setup described in [164, 167]. In particular, it was indicated that the incident angles were in the interval [−ψL , ψL ] with the value ψL = 62 µrad for a 6.7 GeV projectile. However, no clear details were provided on the beam emittance which becomes an important factor for comparing theory vs experiment. In the calculations a uniform distribution of the particles within the indicated interval of ψ was used, and this is also a source of the uncertainties. The spectra were also simulated for a larger cut-off angle equal to 2ψL (these curves are not presented in the figure). It resulted in a considerable (≈30%) decrease of the positron spectrum in the vicinity of the first harmonic peak. On the basis of the comparison with the experimental data, it can be concluded that the code produces reliable results and can be further used to simulate the propagation of ultra-relativistic projectiles along with the emitted radiation. In the Paper below we present several case studies of the channeling phenomena and radiation emission from ultra-relativistic projectiles traveling in various crystalline environments, incl. linear, bent, and periodically bent crystals as well as in crystals stacks. In most cases, the parameters used in the simulations, such as crystal orientation and thickness, the bending radii R, periods λu , and amplitudes a, as well as the energies of the projectiles, have been chosen to match those used in past and ongoing experiments. Wherever available we compare results of our simulations with available experimental data and/or those obtained by means of other numerical calculations.

10.4 Light Sources at High Photon Energies 10.4.1 Main Characteristics of Light Sources One of the radiometric units, frequently used to compare different LS in the short wavelength range, is brilliance, B. It is defined in terms of the number of photons Nω of frequency ω within the interval [ω − ω/2, ω + ω/2] emitted in the cone  per unit time interval, unit source area, unit solid angle, and per a bandwidth (BW) ω/ω [168–170]. To calculate this quantity is it necessary to know the beam electric current I , transverse sizes σx,y and angular divergences φx,y as well as the divergence angle φ of the radiation and the “size” σ of the photon  beam. Explicit  2 2 expression for B measured in photons/s/mrad /mm /0.1% BW reads [171] B=

103

I Nω , 2 (ω/ω) (2π ) εx ε y e

where e is the elementary charge. The quantities

(10.16)

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x,y =



481

 2 2 σ 2 + σx,y φ 2 + φx,y

(10.17)

are the √ total emittance of the photon source in the transverse directions with φ = /2π and σ = λ/4π φ being the “apparent” source size calculated in the diffraction limit [172]. To ensure the aforementioned units for brilliance, the quantities σ, σx,y should be considered measured in millimeters, and φ, φx,y —in milliradians. Another quantity, frequently used to characterize a light source, is flux F. It stands for the number of photons per second emitted in the cone    and in a given bandwidth. Measured in the units of photons/s/0.1% BW , the flux is related to Nω as follows [171]: F=

I Nω . 103 (ω/ω) e

(10.18)

The product Nω I /e on the right-hand sides of Eqs. (10.16) and (10.18) represents the number of photons per second (intensity) emitted in the cone  and frequency interval ω. Using the peak value of the current, Imax , on the right-hand sides of Eqs. (10.16) and (10.18) one calculates the peak brilliance, Bpeak , and flux, Fpeak . The number of photons Nω emitted within BW ω is related to the spectral distribution d3 E/dω d of the radiated energy in the forward direction: ω . Nω = d3 E/dω d θ=0  ω

(10.19)

The driving force behind the development of light sources is the optimization of their brilliance (or spectral brightness), which is the figure of merit of many experiments [170].

10.4.2 Modern Light Sources Syncrotron Radiation Light Sources Synchrotron radiation is the electromagnetic radiation emitted by charged particles when the particles’ trajectories are subjected to a magnetic field, which is for example generated in bending magnets in circular accelerators.2 In an ultra-relativistic limit, the radiation is collimated in a thin cone with an opening angle ∼ γ −1 . Synchrotron radiation covers a wide spectral range and can be tuned from the infrared to the X-rays. 2

In literature, one can find another term for this type of radiation,—magnetic bremsstrahlung. This term is more frequently used in application to the astrophysical problems, see Ref. [173].

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The first description of the emission of relativistic particles following a circular orbit was given in the early 1910s [174]. Later on, in the 1940s, the problem attracted a lot of attention as it became clear that the rapid growth of the radiated power with electron energy (proportional to ε4 ) brings a problem in the construction of higher electron energy accelerators. The modern derivation of the basic expressions used in the description of synchrotron radiation was provided by Ivanenko and Pomeranchuk [175] and by Schwinger [176]. The first direct observation of the radiation was reported in 1947 [177] at the 70 MeV synchrotron in the General Electric Laboratories (USA). The radiation was seen as a spot of a brilliant white light by an observer looking into the vacuum tube tangent to the electron’s orbit. Initially, some electron storage rings designed and built for nuclear and subnuclear physics started to be for some fraction of the time, as sources of photons for experiments in atomic, molecular and solid-state physics. These machines are nowadays referred to as “first generation light sources” [170]. The experimental results obtained with the synchrotron radiation stimulated the construction of dedicated rings, designed and optimized to serve exclusively as light sources. Examples of these “second generation” machines are the BESSYI ring in Berlin, the two National Synchrotron Light Source rings in Brookhaven, NY (USA), the SuperACO ring in Orsay, near Paris, and the Photon Factory in Tsulcuba (Japan). One of the most important challenges for synchrotron radiation sources has always been to reduce emittance (that is, the product of the spatial size and the angular spread) of the electron beam, because the brilliance of a source is inversely proportional to its emittance (10.16). The typical emittances of “third-generation” synchrotron radiation sources, which started operation in the 1990s (such as the European Synchrotron Radiation Facility (ESRF) in France, the Advanced Photon Source (APS) in the US and SPring-8 in Japan, BESSY II), were initially several nanometer radian (nm rad), eventually decreasing to a few nm rad after operation conditions were optimized. However, further improvements were not feasible [16]. This situation changed in the last decade with the ambitious proposal of the MAX IV ring in Sweden, which aims to significantly reduce emittance down for a 3 GeV storage ring [178]. The key innovation is to introduce a series of miniature dipole magnets are employed to increase the total number of the magnets along the synchrotron ring and thus reduce the bend angle per dipole magnet. This brings important benefits in reducing emittance, which is proportional to the cube of the bend angle. The MAX IV facility was inaugurated in June 2016. Modifications aimed at decreasing emittance have been implemented (or are planned to be introduced) at many other synchrotron radiation facilities including the ESRF, APS, SPring-8 and PETRA III at DESY. These new sources are often called diffraction-limited synchrotron radiation (DLSR) sources, because their emittances approach the limitations set by the wavelength of light. The high beam quality of these facilities will stimulate increased use of coherence-related imaging technologies. Among them, coherent X-ray diffraction imaging is a method for enabling ultrahighresolution imaging of isolated objects without the use of any imaging optics. Another aspect of DLSR sources is that their reduced emittance decreases the horizontal size of the source from the submillimetre scale to a few tens of micrometers, allowing for

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a nearly circular-shaped beam. In contrast, the horizontally elongated beams seen in existing synchrotron radiation sources can only be turned into a circular shape by the use of a pinhole filter. Although the realization and the utilization of DLSR sources is technically challenging due to the requirement for ultimate stability down to a 10 nrad level, the resulting gain of photon flux for the nanobeam drastically increases by three orders of magnitude. Furthermore, a new optical scheme for harmonic separation, instead of a conventional X-ray monochromator with a narrower bandwidth, enables high-quality undulator radiation at a specific harmonic to be extracted, which will give an additional gain by two orders of magnitude for a broad range of applications [16].

Undulators and Wigglers The intensity of synchrotron radiation can be increased reducing the curvature radius of particle’s trajectory. This is realized in the so-called insertion devices, undulators, and wigglers, which create a periodic permanent magnetic field with a sinusoidal dependence along the electron trajectory.3 The field forces the beam particles to move periodically in the transverse direction with a spatial period λu . As a result, the particle undulates, i.e., moves along periodic, sine-like trajectory. The periodicity of the motion gives rise to the electromagnetic radiation of a specific type, the undulator radiation (UR). Due to the interference effects the UR is emitted only at particular wavelengths, λn = λ1 /n (where n = 1, 2, 3 . . . ). The fundamental wavelength λ1 is given by λ1 = λu (1 + K 2/2 + γ 2 θ 2 )/2nγ 2 where θ denotes the emission angle with respect to the axis. The undulator parameter K is related to the magnetic field period λ0 and the amplitude value B0 of the magnetic flux density as follows: K = 93.4λ0 (m)B0 (T) [169]. In the “undulator” regime K  1, the radiation emitted at each inversion interferes with the one produced in the previous inversions. These interferences are constructive for the resonance wavelength and the radiation is produced in a very intense spectral lines (harmonics) form. The sharpness of the harmonics can be affected by the observation angle, the energy spread, and emittance of the electron beam. In the “wiggler” regime (K 2  10), the radiation of the different harmonics overlaps and the spectrum approaches the incoherent sum of the synchrotron radiation spectra formed in the fields of individual magnets. The wavelength of the emitted radiation can be varied by a modification of the undulator magnetic field (by changing the gap for permanent magnet insertion devices or the power supply current for electromagnetic insertion devices). Undulator radiation can be characterized as quasi-monochromatic and tunable. The particular choice of the undulator characteristics and technology enables to optimize the desired spectral range for a given beamline.

3

Operational principle of a magnetic undulator was proposed by Ginzburg [179] and verified experimentally by Motz and co-workers [180, 181].

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The electric field of the radiation is in the plane of the electron trajectory. For a vertical magnetic field, the electron follows an undulator trajectory in the horizontal plane. Combining magnetic fields in both planes with a possible phasing between them enable to provide various type of polarization from linear vertical, linear horizontal to circular one, or more generally elliptical one. Different technologies that provide the possibility of any type of polarization are discussed in Ref. [8]. Also, in the cited paper one finds extensive list of existing synchrotron radiation facilities around the world.

X-Ray Free Electron Lasers The free-electron laser (FEL) concept was introduced by Madey in 1971 [182]. He has demonstrated that if an additional electromagnetic wave of appropriate wavelength and phase is propagating parallel to an electron in an undulator, the electron is coupled to this field exchanging energy. The energy exchange can result in deceleration of the electron and amplification of the radiation field. In this sense, the electron moving through the undulator operates as an amplifier. Placing the undulator inside an optical resonator with mirrors at both ends leads to the production of coherent light analogous to conventional lasers. Madey calculated the gain factor g that defines the increase in the number of emitted photons at a resonance frequency due to the emission stimulation of the beam particles. In a small-signal regime, the gain factor is proportional to the undulator length, to the volume density of the beam particles and scales as γ −3 with the beam energy. In this operational mode (referred to as the oscillator mode), the laser field is stored in an optical cavity, enabling interaction with the electron beam on many passes. FEL oscillators cover a spectral range from the microwaves to vacuum ultra-violet, where mirrors are available. The first FEL oscillator reported [183] operated in the infrared wavelengths. One of the major advantages of the FEL LSs is the tunability of the wavelength of the emitted radiation. It is achieved by modifying the magnetic field of the undulator in a given spectral range set by the electron beam energy. Operation at short wavelengths requires high beam energies for reaching the resonant wavelength, and thus long undulators and high beam density for ensuring a sufficient gain. As the wavelength of radiation decreases so does the effectiveness of storing the emitted light in an optical cavity due to limited performance of the mirrors. To overcome this difficulty a so-called “high-gain” operational regime has been proposed. It was shown that to achieve sufficient gain for a short wavelength radiation, the FEL could operate as an amplifier in the so-called self-amplified spontaneous emission (SASE) mode, the high-gain amplification of the initial spontaneous radiation is obtained with only one pass through a long undulator (100 m range for 1 angstrom) until saturation is reached [184–186]. Details on the theory and physics behind the high-gain operational regime of FELs one finds in [2, 11, 187–192] and references therein. Recent reviews on the achievements made in constructing and further advancing of the X-ray FEL facilities, both commissioned and planned, that

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generate coherent photon pulses with time duration of a few to 100 fs over a wavelength range extending from about 100 nm to less than 1 Å as well as on a number of new scientific results obtained in atomic and molecular sciences, in areas of physics, chemistry, biology and applied science can be found in Refs. [1, 2, 6, 8–13, 16]. The main difference between spontaneous undulator radiation and FEL radiation is that in an undulator there is constructive interference between the electromagnetic waves emitted by one electron at different points of its trajectory. In addition to this, in the FEL process, the waves of different electrons also interfere constructively. This happens due to a positive feedback process in which electrons self-organize. Through interaction with the initially incoherent radiation emission, the intensity of which is proportional to the electron density, electrons form into micro-bunches separated by the radiation wavelength. The narrow-bandwidth emission is then coherent, scaling as N 2 where N is the number of electrons emitting collectively (N  106 for X-ray FELs [11]). The amplification process results in almost full transverse coherence and as linac-based accelerators for FELs deliver bunches with very high peak current, the output peak brightness can exceed that of storage ring sources by orders of magnitude. Nowadays, four FEL facilities, FERMI@Elletra, Italy [7], FLASH at DESY, Germany [4], LCLS at SLAC, USA [1, 2, 5], and SACLA at SPring-8, Japan [14], provide femtosecond short laser-like photon pulses to user experiments. Their wavelengths range from the EUV and soft X-rays (FERMI, FLASH) to hard Xrays (LCLS, SACLA). The peak brilliance usually exceeds 1030 photons s−1 mrad−2 mm−2 per 0.1% BW, orders of magnitude more than third-generation synchrotronbased light sources can provide.

10.4.3 Alternative Schemes for Short-Wavelengths Light Sources Compton Alternative Schemes for Short-Wavelengths Light Sources Another type of modern LS, which does not utilize magnets, is based on the Compton scattering process [193, 194]. In this process, a low-energy (eV) laser photon backscatters from an ultra-relativistic electron thus acquiring increase in the energy proportional to the squared Lorentz factor γ = ε/mc2 . This method has been used for producing gamma-rays in a broad, 101 keV–101 MeV, energy range [195, 196]. Applying the four-momentum conservation law to a photon-electron collision one finds the following relationship between the energies of the incoming, ω, and the scattered, ω , photons: ω =

1 − β cos θ ω. 1 − β cos θ  + δ(1 − cos φ)

(10.20)

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Here β = v/c, δ = ω/ε, θ and θ  are the angles between the momenta of the incoming and scattered photons and that of the incident electron, and φ stands for the angle between the two photons. For a collision between an ultra-relativistic electron (γ = (1 − β 2 )−1/2  1, β ≈ 1) and a low-energy photon, ω  ε, the energy of the scattered photon is peaked along the direction of the incident electron. The back-scattered photon has the maximum energy in a head-on collision with θ = π , θ  = 0 and φ = π : ω ≈

4γ 2 ω ≈ 4γ 2 ω . 1 + 4γ 2 δ

(10.21)

The latter equation is written in the limit of small recoil (i.e., when γ ω  mc2 ). The energy of a back-scattered photon scales with the incoming electron energy as ε2 , so that 0.1 . . . 102 MeV photons can be obtained scattering 1 eV laser photons from ∼(0.1 . . . 1) GeV electrons. The first experimental demonstrations of gamma-ray production due to the Compton scattering were carried out by several groups over fifty years ago [198–200]. The first Compton gamma-ray source facility for nuclear physics research, was brought to operation in Frascati [194]. This facility produced gamma-ray beams with energies up to 80 MeV and an on-target flux of up to 5 × 105 photons/s. Following the success of the facility at Frascati, several more Compton gamma-ray source facilities for nuclear physics research were brought to operation around the world starting in the 1980s. Reviews on Compton gamma-ray beams and some of the commissioned facilities are available [201–206]. To be mentioned is the High Intensity Gamma-ray Source (HIGS) at Duke University that is the first dedicated Compton gamma-ray facility employing as the photon driver a high-power FEL [207]. The HIGS facility is a high-flux, nearly monochromatic, and highly polarized gamma-ray source within 1–100 MeV photon energy range. A maximum total flux of about 3 × 1010 photons/s at 10 MeV has been achieved at HIGS, which is two or three orders of magnitude more than produced by other existing facilities [206]. During the last decade or so, while a few Compton gamma-ray source facilities (e.g., LEGS and Graal) ceased operation after completing their research missions, other facilities continue to flourish with accelerator and laser system upgrades that improve beam performance and enable new capabilities. In the meantime, a few new facilities are under construction around the world. A list of major operational laser-Compton gamma-ray sources and new development projects one finds in Ref. [206], Sect. 5.

10.4.4 Gamma Factory The Compton scattering also occurs if the scatterer is an atomic (ionic) electron which moves being bound to a nucleus. This phenomenon is behind the Gamma Factory (GF) proposal for CERN [17, 18] that implies using a beam of ultra-relativistic ions in the backscattering process.

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Fig. 10.8 Compton backscattering from free electron (left) versus from partially stripped ion (right). Redrawn from data presented in Ref. [208]

The GF project is aimed at creating, storing, and exploiting relativistic beams of partly stripped atomic ions (PSI) that can be stored in the Super Proton Synchrotron (SPS) or the Large Hadron Collider (LHC) storage rings at very high energies (the corresponding relativistic factors within the range γ = 30 . . . 3000), at high bunch intensities (number of ions per bunch 108 . . . 109 ), and at high bunch repetition rate (up to 20 MHz). The GF scheme is based on a resonant excitation of a PSI with the laser beam tuned to the atomic transitions frequencies, followed by the process of spontaneous emission of photons, see Fig. 10.8. Due to the relativistic Doppler effect, the energy of photons emitted in the direction of the beam is boosted by a factor of up to 4γ 2 as compared to the energy of the laser light. Due to huge excess (a factor up to 109 ) of resonant photon absorption cross-sectional compared to that of photon scattering from a free electron, the intensity of an atomic-beam-driven LS is expected to be several orders of magnitude higher than what is possible with Compton gamma-ray sources driven by an electron beam. The proposed LS could be realized at CERN by using the infrastructure of the existing accelerators. It could push the intensity limits of the presently operating light-sources by at least 7 orders of magnitude, reaching the flux of the order of 1017 photons/s, in the particularly interesting gamma-ray energy domain of 1 ≤ E ph ≤ 400 MeV [18]. This domain is out of reach for the FEL-based light sources. The energy-tuned, quasi-monochromatic gamma beams, together with the gammabeams-driven secondary beams of polarized positrons, polarized muons, neutrons, and radioactive ions would constitute the basic research tools of the proposed GF. To prove experimentally the concepts underlying the Gamma Factory proposal, feasibility tests have been and will continue to be performed at the SPS and at the LHC [206]. Since 2017 the experimental beam tests have started with various PSI beams. In 2018, For the first time, the 208 Pb81+ ions were injected into the LHC [209] aiming at demonstrating that bunches of hydrogen-like lead atoms can be efficiently produced

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and maintained at the LHC top energy with the lifetime and intensity fulfilling the GF requirements. Thus, the pivotal concept of the GF initiative that relativistic atomic beams can be produced, accelerated, and stored in the existing CERN SPS and LHC rings has been experimentally proven. It is planned that the SPS and LHC beam tests will be followed by the GF “proof-of-principle” SPS experiment [19] in which a beam of lithium-like lead ions, 208 Pb79+ , will be collided with the photon laser beam tuned to resonantly excite the 2s → 2 p1/2 atomic transition of the ions. It is expected that this experiment will provide a decisive proof and an experimental evaluation of the achievable intensities of the atomic-beam-based gamma-ray source.

10.4.5 Extremely Brilliant GeV γ -Rays from a Two-Stage Laser-Plasma Accelerator Strong electric fields for acceleration of particles can be produced by separation of electrons and ions in dense plasma. Powerful laser pulses propagating in plasma generate such charge separation through the excitation of wakefields due to the action of non-linear ponderomotive force [210, 211]. Wakes with electric fields orders of magnitude larger than in conventional accelerators are feasible allowing for reducing the size of accelerators. Compact laser-wakefield accelerators (LWFAs) have been developed [212] that offers a radically different approach: the acceleration length in plasmas is about three orders of magnitude smaller as compared to conventional accelerators, providing the ability to drive the acceleration and radiation of highenergy particles on a much smaller scale. Multi-GeV electron beams have been produced using LWFA, and X/γ -ray pulses in the keV to MeV range can be produced via LWFA-based betatron radiation and Compton backscattering (see Ref. [213] and references therein). The resulting radiation sources have typical peak brilliance of 1019 . . . 1023 photons/(s mm 2 mrad 2 0.1% BW), while the photon number per shot is limited to 107 . . . 108 photons due to low level of the laser-to-photon energy conversion efficiency. Still, it remains a great challenge to significantly increase the energy conversion efficiency and to generate collimated γ -rays with high peak brilliance with energies in the MeV to GeV range. Although continuous development in ultrahigh-power laser technology provides possibilities for producing brilliant high-energy gamma-ray LSs, there are unavoidable physical limitations on the peak brilliance of gamma-rays produced by means of various methods based on laser pulses [213]. It has been noted in the cited paper that to produce GeV photons an exceptionally high laser intensity of 1023 . . . 1025 W/cm2 (2–4 orders of magnitude higher than the highest intensities available to date) is required. As soon as the laser intensity is reduced to the levels achievable in current high-power laser systems, the methods mentioned become intrinsically inefficient for gamma-ray emission. To overcome the restrictions, a new scheme, based on a two-stage LWFA driven by a single multi-petawatt laser pulse, has been proposed recently [213]. In the first

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stage, the plasma electrons are self-injected and accelerated in the plasma bubble excited by the laser pulse propagating in an under-dense plasma, resulting in a lowdivergence multi-GeV electron beam with a particle density close to the critical plasma density, 1021 cm−3 . The laser-to-electron energy conversion efficiency is quoted at the level of 40%. In the second stage, the laser pulse propagates into the relatively high-density plasma, resulting in a shrunken plasma bubble as the density increases. Besides the accelerated GeV electrons from the previous stage, additional electrons are injected, which further increases the total charge of the accelerated electron beam with a peak density well above the critical density. The efficiency increases to above 50% for the total accelerated GeV electrons as well. This results in large quasi-static electromagnetic fields around the electron beam, which gives rise to the emission of a collimated beam of γ -rays with photon energies up to the GeV level. Based on the results of numerical simulations carried out in Ref. [213] the authors predict that the photon number, peak brilliance, and power of the γ -rays emitted in the two-stage LWFA light source are several orders of magnitude higher than current LWFA betatron radiation and Compton sources. Numerical simulations, carried out in Ref. [213], demonstrate that more than 1012 γ -ray photons/shot are produced for photons above 1 MeV, and the peak brilliance is above 1026 photons s−1 mm−2 mrad−2 per 0.1% bandwidth at 1 MeV. Dependence of the calculated peak brilliance on the photon energy is shown in Fig. 6B in Ref. [213].

10.5 Crystalline Undulators 10.5.1 Crystalline Undulator: Basic Concepts, Feasibility A Crystalline Undulator (CU) device contains a periodically bent crystal (PBCr) and a beam of ultra-relativistic positrons or electrons undergoing planar channeling. In such a system, there appears, in addition to the channeling radiation (ChR) [35], the undulator radiation due to the periodic motion of the particles which follow the bending of the planes. A light source based on a CU can generate photons in the energy range from tens of keV up to the GeV region [21, 72] (the corresponding wavelengths range starts at 0.1 and goes down to 10−6 Å). The intensity and characteristic frequencies of the CU radiation (CUR) can be varied by changing the beam energy, the parameters of bending, and the type of a crystal. Under certain conditions, a CU can become a source of the hard X- and gamma-ray laser light within the wavelength range 10−2 –10−1 Å [21, 25, 26], which cannot be reached in existing and planned FELs based on magnetic undulators. The mechanism of the photon emission in a CU is illustrated by Fig. 10.9 [26]. The z-axis is aligned with the midplane of two neighboring non-deformed crystallographic planes (not drawn in the figure) spaced by the interplanar distance d. The closed circles denote the nuclei of the planes which are periodically bent with the

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Fig. 10.9 Schematic representation of a CU [23, 24, 26]. Closed circles mark the atoms of crystallographic planes which are periodically bent with the amplitude a and period λu . Thin dotted line illustrates the trajectory of the particle (open circles), which propagates along the centerline (the undulator motion) and, simultaneously, undergoes so-called channeling oscillations. The periodic mode leads to the emission of the undulator-type radiation, and, under certain conditions, may result in the stimulated radiation

amplitude a and period λu . The harmonic (sine or cosine) shape of periodic bending, y(z) = a cos(2π z/λu ), is of a particular interest since it results in a specific pattern of the spectral-angular distribution of the radiation emitted by a beam of ultra-relativistic charged particles (the open circles in the figure) propagating in the crystal following the periodic bending. The operational principle of a CU does not depend on the type of a projectile. Provided certain conditions are met the particles will undergo channeling in PBCh [22, 24]. The trajectory of a particle contains two elements which are illustrated by Fig. 10.9. First, there are oscillations due to the action of the interplanar force,—the   /dε (c is so-called channeling oscillations [28], whose frequency ch = c 2Umax the speed of light) depends on the projectile energy ε and on the parameters of the  and the interplanar channel: the maximal gradient of the interplanar potential Umax distance d. Second, there are oscillations due to the periodicity of the bending, the undulator oscillations, whose frequency is u ≈ 2π c/λu . The spontaneous emission is associated with both of these oscillations. The typical frequency of the ChR is ωch ≈ 2γ 2 ch and [35, 214], where γ = ε/mc2 is the relativistic Lorentz factor of the projectile. The undulator oscillations give rise to photons with frequency ωu ≈ 4γ 2 u /(2 + K 2 ), where K = 2π γ a/λu is the so-called undulator parameter. If u  ch , then the frequencies of ChR and UR are well separated. In this case the characteristics of undulator radiation are practically independent on channeling oscillations [22, 24, 26], and the operational principle of a crystalline undulator is the same as for a conventional one (see, e.g., [169, 179, 180, 215, 216]) in which the monochromaticity of radiation is the result of constructive interference of the photons emitted from similar parts of trajectory. Although the motion of a projectile and the process of photon emission in a CU are very similar to that in an conventional undulator based on the action of periodic magnetic (or, electric) field there is an important distinguishing feature. Namely, the electrostatic fields inside a crystal are so strong that they are able to steer the particles

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much more effectively than even the most advanced superconductive magnets. The field strength is on the level of 1010 V/cm which is equivalent to the magnetic field of approximately 3000 tesla. The present state-of-the-art superconductive magnets produce the magnetic flux density of the order of 100 –101 tesla [20, 217]. Strong crystalline fields allow one to bring the period λu of bending down to the hundred or even ten-micron range, which is 2–5 orders of magnitude smaller than the period of a conventional undulator. As a result, the size of the undulator itself can be reduced by orders of magnitude as illustrated by Fig. 10.2, which matches the magnetic undulator for the X-ray laser XFEL [218] with a CU manufactured in University of Aarhus and used further in channeling experiments [32, 81, 161].

10.5.2 Positron and Electron-Based CUs: Illustrative Material Radiation from Diamond-Based CU by Multi-GeV Electrons and Positrons In Ref. [148] the results have been presented in the statistical analysis of the channeling properties and of the spectral intensities of the radiation formed by 195...855 MeV electrons and positrons in the crystalline undulator with the parameters used in the experiments at the MAMI facility [75, 219]. The CUs were manufactured in Aarhus University (Denmark) using the molecular beam epitaxy technology to produce strained-layer Si1−x Gex superlattices with varying germanium content [96]. Later, similar calculations have been extended to the range of multi-GeV projectile energies [150, 220]. To a great extent, this activity was inspired by the plans to carry out channeling experiments with diamond crystals at the SLAC facility (USA) [221] using high-intensity 4...20 GeV electron and positron beams. The sets of simulations have been performed aiming at providing benchmark data for the emission spectra formed by projectile electrons and positrons in silicon-based and diamond-based crystalline undulators with the parameters similar to those used in the experiments with sub-GeV electron beams [75, 219]. The parameters of the CU used in the simulations were as follows: • Bending period amplitude λu = 40 microns. • Number of periods Nu = 8, hence the crystal thickness L = Nu λu = 320 microns. • Bending period amplitude a = 2 . . . 6 Å. Table 10.1 provides the values of acceptance and penetration length L p obtained via statistical analysis of the trajectories simulated in straight (a = 0) and periodically bent Si(110) crystal. In the latter case, the bending parameter C is defined as the ratio of the maximum values of the centrifugal force, Fcf ≈ ε/Rmin with  . The latter was taken equal to Rmin = a −1 (λu /2π )2 , and the interplanar force Umax 5.7 GeV/c, which corresponds to the (110) interplanar potential calculated within the Molière approximation at T = 300 K. The data shown indicate that most of the

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Table 10.1 Acceptance A and penetration length L p for 10 GeV positrons and electrons in straight (a = 0) and periodically bent (λu = 40 microns) planar channels Si(110). The bending parameter  ) stands for the ratio of the centrifugal force ti the interplanar force C = a(2π/λu )2 (ε/Umax Projectile

a (Å)

C

A (%)

L p (µm)

Positron

0 2 4 6 0 4

0 0.08 0.16 0.24 0.0 0.16

97.1 ± 0.9 89.8 ± 2.1 81.6 ± 2.6 71.9 ± 5.8 65.8 ± 2.3 42.9 ± 3.3

302 ± 4 301 ± 5 287 ± 7 273 ± 15 82 ± 4 52 ± 4

Electron

Fig. 10.10 Enhancement factor of the radiation over the Bethe-Heitler spectrum for 10 GeV positrons (left) and electrons (right) positrons in straight Si(110) and in Si(110)-based CU with different bending amplitudes as indicated. The bending period is set to λu = 40 microns. All curves refer to the emission angle θ0 = 7/γ ≈ 0.36 mrad

positrons travel in the channeling mode through the whole crystal. For electrons, both acceptance and channeling segments length are much lower. These features reveal themselves in the emission spectra of the projectiles. Figure 10.10 shows the enhancement factor (over the Bethe-Heitler background) of the radiation emitted by positrons (left panel) and electrons (right panel) [150]. For both projectiles, the spectra in the straight channel (red curves) are dominated by powerful peaks due to the channeling radiation. The peak is more pronounced for positrons since their channeling oscillations are quasi-harmonic resulting in the emission within comparatively narrow-bandwidth centered at ωch ≈ 70 MeV. Strong anharmonicity of the electron channeling oscillation leads to the noticeable broadening of the peak with the maximum located at ωch ≈ 120 MeV. Periodical bending of the crystal planes gives rise to the CU Radiation (CUR). Since the dechanneling length of positrons greatly exceeds that of electrons, the CUR peaks in the positron spectra are much more pronounced. The energy ω1 of the first harmonic of CUR can be estimated from the relation (see [26], Eq. (6.14)):

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ω1 [MeV ] =

ε2 9.5 , 1 + K 2 /2 λu

(10.22)

where ε is substituted in GeV and λu in microns. The quantity K stands for the total undulator parameter due to both channeling oscillations and those due to the bending periodicity [134] K =



2 K u2 + K ch ,

(10.23)

where K u = 2π γ a/λu and K ch ∝ 2π γ ach /λch with ach ≤ d/2 and λch being the amplitude and period of channeling oscillations. In the case of positron channeling, assuming harmonicity of the oscillations one can derive the following expression for 2 averaged over the allowed values of ach (see Ref. [26], Eq. (6.14)): K ch 2 K ch =

2γ U0 , 3mc2

(10.24)

where U0 is the depth of the interplanar potential well. For ε = 10 GeV in Si(110) 2 ≈ 0.56. (U0 ≈ 22 eV one obtains K ch Using (10.22)–(10.24) one estimates ω1 = 16, 11.7, 8 MeV for a = 2, 4, 6 Å, respectively. These values correlate nicely with the positions of the first peaks of CUR seen in Fig. 10.10 left. In Ref. [220] channeling of 4...20 GeV electron and positron beams in oriented diamond(110) crystal, both straight and periodically bent, was simulated and analyzed. As mentioned, this activity has been carried out to produce theoretical benchmarks for the experimental measurements planned to be carried out at the SLAC facility. From this viewpoint, the use of diamond crystals looked preferential since diamond bears no visible influence from being irradiated by the high-intensity beams. The spectral distributions of radiated energy were computed for two values of the emission cone θ0 : (i) a “narrow” cone θ0 = 1/γ , and (ii) a “wide” cone θ0 = 5/γ , which collects virtually all radiation emitted by ultra-relativistic particles. The results of calculations for 10 GeV projectiles, presented in the form of the enhancement factor over the emission spectra in amorphous medium, are shown in Fig. 10.11. For the sake of comparison the spectra formed by a positron moving in an “ideal undulator” (i.e., along the sine trajectory with the given values of a and λu ) are also shown in the upper figures.

Interplay and Specific Features of Radiation Mechanisms for Electrons in Crystalline Undulators In recent papers [144, 146, 151] an accurate numerical analysis has been performed on the evolution of the channeling properties and the radiation spectra for diamond(110) based CUs. Drastic changes in the radiation spectra with a variation of the bending amplitude a have been observed for different projectile energies and

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Fig. 10.11 Enhancement factor of the radiation over the Bethe-Heitler spectrum for 10 GeV positrons (upper row) and electrons (lower row) positrons in straight diamond (thick black lines) and in diamond-based CU (thick red lines) with amplitude a = 4 Å and period λu = 40 microns. Thin blue solid lines show the emission spectra from ideal undulator with the same a and λu . Left column corresponds to the emission angle θ0 = 1/γ = 51.1 µrad; right column—to θ0 = 5/γ ≈ 256 µrad. All spectra refer to the crystal thickness L = 320 microns

their sensitivity to the projectile’s charge has been noted. Some of the predictions made can be verified in channeling experiments with electrons at the MAMI facility. The calculations were performed for 270–855 MeV electrons and positrons propagating in the 20 microns thick diamond crystal. The bending period was fixed at λu = 5 microns whereas the bending amplitude was varied from a = 0 (straight crystal) up to a = 4.0 Å in accordance with the parameters of crystalline samples used in the experiments at MAMI [222]. Figure 10.12 presents the emission spectra for the positrons and electrons with ε = 855 MeV calculated for the opening angle θ0 = 0.24 mrad, which is smaller than the natural emission angle γ −1 = 0.59 mrad. For both types of projectiles the spectra formed in the straight crystal, graph (a), are dominated by the peaks of ChR, the spectral intensity of which by far exceeds that of the incoherent bremsstrahlung background 2.5 × 10−5 in the amorphous medium. For positrons, nearly perfect harmonic channeling oscillations give rise to the narrow peak at ωChR ≈ 3.6 MeV. Strong anharmonicity of the electron channeling oscillations makes the ChR peaks (marked with the upward arrows) less pronounced and significantly broadened (note the scaling factor ×5 applied to the electron spectra). In periodically bent crystals, Fig. 10.12b–d, the spectra exhibit additional features some of which evolve differently with increase in a.

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Fig. 10.12 Spectral distributions of radiation by 855 MeV electrons (dashed blue curves, multiplied by a factor of 5) and positrons (solid red curves) in straight (a) and periodically bent (b)–(d) diamond (110) crystals. The upward arrows indicate the maxima of ChR for electrons, the downward arrows show the positions of the additional maxima appearing in the bent crystals (see explanations in the text). The error bars shown in graph (a) illustrate the statistical errors due to the finite number of the simulated trajectories. The spectra correspond to the opening angle θ0 = 0.24 mrad. Redrawn from data presented in Ref. [146]

• For both types of projectiles there are CUR peaks in the low-energy part of spectra. The most powerful peaks correspond to the emission in the first harmonic at ωCUR ≈ 1 MeV. To be noted is the non-monotonous dependence of the peak values on bending amplitude a. This feature has been discussed in detail in Refs. [146, 151]. • For positrons, the intensity of ChR becomes strongly suppressed as bending amplitude increases: for a = 1.2 Å the intensity is two times less than in the straight crystal. For larger amplitudes, ChR virtually disappears [144, 146]. This happens because the (mean) amplitude of channeling oscillations is a decreasing function of a. Indeed, as a increases, the centrifugal force, especially in the points of maximum curvature, drives the projectiles oscillating with large amplitudes away from the channel resulting in a strong quenching of the oscillations. A quantitative analysis of this feature one finds in Ref. [146]. • For electrons, the peak value of ChR does not fall off so dramatically. As a increases, the peak (marked with the upward arrow) becomes blue shifted and there appears additional structure (the downward arrow) on the right shoulder of the spectrum. The analysis has shown that both features are due to the emission by dechanneled electrons [151] . In a periodically bent crystal, a dechannel particle can experience (i) the volume reflection (VR) [223, 224], occurring mainly at the points of maximum curvature, and (ii) the over-barrier motion in the regions with small curvature. These types of motion contribute to different parts of the radiation spectrum. The radiation, which accompanies VR, is emitted in same energy domain as the ChR. The over-barrier particles radiate at higher energies and this radiation reveals itself as an additional peak in the spectrum. The radiation emission by over-barrier particles in the field of a periodically bent crystal was

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Fig. 10.13 Left. An exemplary trajectory of a 855 MeV electron in a diamond (110) crystal bent periodically with a = 2.5 Å and λu = 5 µm. Highlighted are the segments corresponding to (i) the channeling regime (dashed green curve), (ii) the over-barrier motion (dashed-dotted blue curves), (iii) to the VR events (solid orange curves). Thin wavy lines mark the boundaries of the electron channels. Right. Solid black curve with open circles shows the enhancement factor of the total radiation emitted by 855 MeV electrons in the diamond (110) crystal bent as described above. Dashed green, dashed-dotted blue, and solid orange curves show the contributions coming from the segments of the channeling and over-barrier motions and due to the VR, respectively. Redrawn from data presented in Ref. [151]

discussed qualitatively in Ref. [225] within the continuous potential framework. More detailed quantitative analysis of the phenomena involved can be provided by means of all-atom molecular dynamics. Below we present a brief overview of the results obtained and conclusions drawn in Ref. [151]. To compare the contributions to the total emission spectrum coming from channeling and non-channeling particles the following procedure has been adopted. Each simulated trajectory has been divided into segments corresponding to different types of motion. Namely, we distinguished the following parts of the trajectory: (i) the channeling motion segments, (ii) segments corresponding to the over-barrier motion across the periodically bent crystallographic planes, (iii) segments corresponding to the motion in the vicinity of points of maximum curvature where a projectile experiences VR. For each type of the motion, the spectrum of emitted radiation has been computed as a sum of emission spectra from different segments. Thus, the interference of radiation emitted from different segments has been lost. The aforementioned procedure is illustrated by Fig. 10.13. Its left panel presents a selected trajectory of a 855 MeV electron propagating in periodically bent crystal with bending amplitude 2.5 Å. Different types of segments are highlighted in different colors and types of the line as indicated in the caption. The emission spectra corresponding to different types of motion (calculated accounting for all simulated trajectories) are shown in the right panel. The dependencies presented allow one to associate the maxima in the total spectrum (black solid curve) with the corresponding type of motion. The radiation emitted from segments of channeling motion (dashed green curve) govern the spectrum in the vicinity of the CUR peak (ωCUR ≈ 1 MeV) and contributes greatly to the ChR at ωch ≈ 6 . . . 12 MeV. Numerical analysis of the

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Fig. 10.14 Enhancement factor of the radiation over the Bethe-Heitler spectrum for 855 MeV electrons in straight (a) and periodically bent (graphs (b)–(e) correspond to a = 1.2, 2.5, 4.0, 5.5 Å) diamond(110) crystals. Solid black curves show the total spectra, dashed green ones correspond to the radiation emitted from the channeling segments only, and dashed-dotted blue curves present the spectra due to all non-channeling parts of the simulated trajectories. Redrawn from data presented in Ref. [151]

simulated trajectories has shown that the curvature of the trajectories segments in the points of VR is close to that of the channeling trajectories. As a result, the radiation from the VR segments is emitted in the same energy interval as ChR so that the peak centered at ≈ 9 MeV is due both to the channeling motion and to the VR events. The over-barrier particles experience quasi-periodic modulation of the trajectory when crossing the periodically bent channels. The (average) period of these modulations is smaller than that of the channeling motion and decreases with the increase of the bending amplitude. For a = 2.5 Å this period is approximately two times less than the (average) period of channeling oscillations. As a result, radiation emitted from the over-barrier segments (dashed-dotted blue curve) is most intensive in the range ωch ≈ 15 . . . 20 MeV. This contribution results in the additional structure in the total spectrum. Figure 10.14 illustrates the evolution of the contributions from the channeling and non-channeling particles to the emission spectrum with bending amplitude. In the figure, each graph presents the total spectrum (solid curve) as well as the contributions of the channeling segments (dashed curve) and the non-channeling segments (both over-barrier and VR, dash-dotted curve). In the straight crystal as well as in the periodically bent one with small bending amplitude (a = 1.2 Å) the emission spectrum above 1 MeV is dominated by the channeling particles which provide main contributions to the ChR peak. As a increases, the role of the non-channeling segments becomes more pronounced whereas the

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channeling particles contribute less. The increase in a leads to (i) increase of the curvature of a particle’s trajectory in the vicinity of the VR points, (ii) decrease in the period of the quasi-periodic modulation of the trajectories of over-barrier particles. As a result, two maxima seen in the graphs (b)–(e) become blue shifted as a increases: the maxima marked with upward arrows are due to the channeling motion and to the VR, those marked with downward arrows are associated with the over-barrier particles. For large bending amplitudes, graphs (d)–(e), these maxima are virtually due to the emission of the non-channeling particles only. The low-energy part of the spectrum formed in periodically bent crystals is dominated by the peak at ≈ 1 MeV. For moderate amplitudes, a ≤ 2.5 Å, when the bending parameter C is small (see first equation in (10.1)), this peak associated with CUR and is due to the motion of the accepted particles which cover a distance of at least one period λu in a periodically bent channel. For larger amplitude, a = 4.0 Å (C = 0.77), the penetration length L p of the accepted particles become less than half a period leading to noticeable broadening of the CUR peak. For even larger amplitudes, there are further modifications of the peak related to the phenomenon different from the channeling. Graph (e) shows the dependences for a = 5.5 Å which corresponds to the bending parameter larger than one, C = 1.15. As a result, only a small fraction of the incident electrons is accepted, and channels over the distance less than λu /2 having very small amplitude of channeling oscillations, ach  d/2. Therefore, these particles virtually do not emit ChR but nevertheless contribute to the CUR part of the spectrum (see the dashed curve). However, this contribution is not a dominant one. The main part of the intensity comes from the non-channeling particles, see the dash-dotted curve. The explanation is as follows [151]. As discussed above, a trajectory of a non-channeling particle consists of short segments corresponding to VR separated by segments z ≈ λu /2 where it moves in the over-barrier mode. In the course of two sequential VR the particle experiences “kicks” in the opposite directions, see the lower trajectory in Fig. 10.13a). Therefore, the whole trajectory becomes modulated periodically with the period 2z ≈ λu . This modulation gives rise to the emission in the same frequency as CUR. These effects, which are due to the interplay of different radiation mechanisms in periodically bent crystals, can be probed experimentally. In this connection one can mention recent successful experiments on detecting the excess of radiation emission due to VR in oriented bent Si(111) crystal by 855 MeV electrons [65] and 12.6 GeV electrons [226].

Channeling and Radiation Emission in Diamond Hetero-Crystals As mentioned in Sect. 10.1, periodic bending can be achieved by graded doping during synthesis to produce diamond superlattice [98]. Both boron and nitrogen are soluble in diamond, however, higher concentrations of boron can be achieved before extended defects appear [99]. The advantage of a diamond crystal is radiation hardness allowing it to maintain the lattice integrity in the environment of very intensive beams [34]. Boron-doped diamond layer cannot be separated from a

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Fig. 10.15 a Sketch of the crystal geometry. The diamond single crystal is cut with its surface perpendicular to the [100] direction. The crystal is tilted by 45◦ to orient the (110) planes along the incident beam, panel (a). The hetero-crystal consists of two segments: a straight (S) L S = 141 microns thick single crystal substrate and a boron-doped L PB = 20 microns thick periodically bent (PB) segment which accommodates four bending periods [228]. Gradient shading shows the boron concentration which results in the PB of the (110) planes. Panels (b) and (c) show two possible orientations of the hetero-crystal with respect to the incident beam. In panel (b) the beam enters the PB segment, in panel (c)—the S segment. These two orientations are called “PB-S crystal” and “S-PB crystal”, respectively. An exemplary trajectory of a positron channeled through the whole PB-S crystal is shown in panel (b). An exemplary trajectory of an electron in the S-PB crystal is presented in panel (c). Note that several channeling and over-barrier parts the electron’s trajectory that are outside the drawing are not shown. Redrawn from data presented in Ref. [229]

straight/unstrained substrate (SC) on which the superlattice is synthesized. Therefore, unlike Si1−x Gex superlattice, a diamond-based superlattice has essentially a hetero-crystal structure, i.e., it consists of two segments, a straight single diamond crystal substrate and a periodically bent (PB) layer [228]. Reference [229] presents the results of computational analysis of channeling and radiation properties in experimentally realized diamond-based CU, Fig. 10.15. Special attention has been paid to the analysis of the new effects which appear due to the presence of the interface between the straight and PB segments in the hetero-crystal. The experiment has been carried out with the 270–855 MeV electron beams [222, 227, 230]. For the sake of comparison, the simulations have been carried out for both electron and positron beams. The positron beam of the quoted energy range is available at the DANE acceleration facility [75, 231]. Panel (a) in the figure shows the geometry of the system. The incident beam can enter the crystal at either PB or straight (S) part, panels (b) and (c), respectively. To distinguish the crystal orientation with respect to the incident beam, in the text below the crystal shown in panel (b) is labeled as the PB-S crystal and the one in panel (c) as the S-PB crystal. To illustrate the particle’s propagation through the crystal, the selected trajectories of a positron (red curve, panel (b)) and an electron (blue curve, panel (c)) are shown. The parameters of the hetero-crystal used in the simulations of channeling along the (110) plane matched those used in the experiment [228]: total thickness in the beam direction is L = 161 microns out of which 141 microns corresponds to the straight segment and 20 microns—to the PB segment; the bending amplitude and period are a = 2.5 Å and λu = 5 microns, respectively.

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Fig. 10.16 Spectra of radiation emitted within the cone θ0 = 0.24 mrad by ε = 855 MeV positrons (left) and electrons (right) in propagating in the oriented PB-S and S-PB hetero-crystals. The intensity of the background incoherent bremsstrahlung estimated within the Bethe-Heitler approximation is 2.5 × 10−5 (not indicated in the figure). Redrawn from data presented in Ref. [229]

Figure 10.16 compares the calculated spectral dependences of the radiation emitted within the cone θ0 = 0.24 mrad by ε = 855 MeV positrons (left panel) and electrons (right panel) channeling in the PB-S and S-PB hetero-crystals. In the straight segment of the crystal a projectile can experience channeling oscillations. In addition to these, in the PB segment a projectile is involved in the undulator motion due to the periodic bending of the channels. Spectral distributions of the radiation bear features of both types of the oscillatory motion. The features are more pronounced in the case of positron channeling. The positron spectra clearly exhibit two main peaks: the one of ChR centered at about ω ≈ 3.6 MeV and the CUR peak at ω ≈ 1.1 MeV. These peaks correspond to the fundamental harmonics of two types of radiation. The second harmonics are seen as a small bumps around ω ≈ 2.2 MeV (CUR) and ω ≈ 7.2 MeV (ChR). It is seen that the intensity of CUR is virtually insensitive to the hetero-crystal orientation (PB-S or S-PB) whereas the intensity of ChR for the S-PB crystal is ca 2 times higher than for the PB-S one. The provide qualitative explanation of these features one can consider the following arguments. The intensity of ChR is proportional to the (average) amplitude of channeling oscillations ach squared and to the (average) length L ch of the channeling segment: 2 L ch . When a projectile enters the S-PB crystal it moves initially in the dE ∝ ach straight part of the system so that the values of ach consistent with the channeling condition are within interval [0, d/2], where d = 1.26 Å is the interplanar distance. If entering the PB segment, the centrifugal force acting on a projectile decreases the allowed amplitude values by the factor equal approximately to (1 − C), where  is the bending parameter. For the bending amplitude and period C = 4π 2 aε/λu Umax  ≈ 8 GeV/cm in diamond(110) one finds indicated above and for the value Umax C ≈ 0.4. Further, let us note that dechanneling length L d of a 855 MeV positron in a straight diamond(110) channel is about 500 microns [26], and in the PB one is it ca (1 − C) less, i.e., equals to ≈300 microns. Therefore, in the case of the SPB orientation, most of the accepted particles move in the channeling mode over

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the straight segment (i.e., the largest part of the hetero-crystal, L S = 141 microns out of total L = 161 microns) with the values of ach distributed within the interval [0, d/2]. At the interface, S-to-PB, the particles oscillating with ach  (1 − C)d/2 dechannel. The remaining positrons channel in the short L PB = 141 microns PB segment with the amplitudes distributed within the narrower interval. For the PB-S orientation, the amplitude of the accepted particles is distributed over the narrower interval (1 − C) × [0, d/2], and most part of these particles channel through the whole crystal. Taking into account these arguments one estimates the ratio of the peak intensities of ChR for the two orientations of the hetero-crystal as follows: L S + (1 − C)2 L PB dE S−PB ≈ 2.5 . ≈ dE PB−S (1 − C)2 (L S + L PB ) The ratio obtained corresponds to the ratio of the peak intensities of ChR seen in Fig. 10.16left. Weak dependence of the CUR intensity on the orientation of the crystal is also clear. In this case, the radiation is mostly determined by the number of particles undergoing undulator motion in the PB segment. Due to the strong inequality L d  L, for either orientation the number of particles accepted at the PB segment entrance can be estimated as (1 − C)N , where N is the total number of incoming particles. Thus, the CUR intensities are (approximately) the same in both cases. Compared to positrons, electrons have significantly shorter dechanneling lengths. As a result, the primary fraction of channeled electrons virtually dies out for crystal thicknesses greater than 50 microns for both PB-S and S-PB crystals. This explains why CUR radiation is seen in the spectral distribution only in the case of PB-S crystal, Fig. 10.16right. More discussion on the differences in the electron spectra as well as comparison of the spectral dependences obtained for different energies of the beam one finds in Ref. [229].

Channeling and Radiation Emission in SASP Periodically Bent Crystals The original concept of a CU assumes the projectiles channel in the crystal following the periodically bent planes or axes. For such motion, the undulator modulation frequencies u are smaller than frequencies ch of the channeling oscillations. This regime implies periodic bending with large-amplitude, a > d, and large-period, λu  a. As a result, the CUR spectral lines appear at the energies below those of ChR [22–24]. Another regime of periodic bending, termed as Small-Amplitude Short-Period (SASP), was suggested recently [67]. This regime implies bending with a  d and λu shorter than the period of channeling oscillations. In contrast to the motion in a CU, the channeling trajectory in a SASP crystal does not follow the short-period bent planes but acquires a short-period jitter-type modulations resulting from the bending. These modulations lead to the radiation emission at the energies exceeding the energies of the channeling peaks [59, 67–69, 71, 72, 143]. Interestingly, a similar

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radiative mechanism has recently been studied with respect to the radiation produced by relativistic particles in interstellar environments with turbulent small-scale fluctuations of the magnetic field [232, 233]. In Ref. [143] results of a thorough study of channeling and radiation by 855 electrons and positrons passing through a SASP silicon crystal has been presented. Comprehensive analysis of the channeling and radiation processes has been carried out on the grounds of numerical simulations. Specific features which appear due to the SASP bending have been highlighted and elucidated within an analytically developed continuous potential approximation (see Appendix A in the cited paper). The parameters of the SASP bending were chosen to match those used in the experiment with 600 and 855 electrons carried out at MAMI [68]. A SASP crystal used in the experiment was produced by using Si1−x Gex graded composition with the Ge content x varied from 0.3 to 1.3% to achieve a periodic bending of (110) planes with the amplitude a = 0.12 ± 0.03 Å and period λu = 0.43 ± 0.004 µm. The number of periods quoted was 10. No further details on the actual characterization of the profile of periodic bending were provided although in a more recent paper [59] it was noted that “…the shape is roughly sinusoidal”. In the simulations [143] a thicker crystalline sample, L = 12 µm, was probed assuming perfect cosine bending with period 400 nm and amplitude varied from a = 0 (straight channel) up to a = 0.9 Å, which is close to the half of the (110) interplanar spacing in silicon crystal (d = 1.92 Å). The calculated emission spectra cover a wide range of the photon energies, from 1 MeV up to 40 MeV. The integration over the emission angle θ was carried out for two particular cones determined by the values θ0 = 0.21 and 4 mrad. For a 855 MeV projectile the natural emission angle is γ −1 ≈ 0.6 mrad. Therefore, the smallest value of θ0 refers to a nearly forward emission, whereas the largest value, being significantly larger than γ −1 , provides the emission cone which collects almost all the radiation emitted. The spectra computed show a variety of features seen in Fig. 10.17. To be noticed are the pronounced peaks of ChR in the spectra for the straight crystal (the black solid-line curves). Nearly perfectly harmonic channeling oscillations in the positron trajectories (the examples of the simulated trajectories can be found in [26, 139, 148]) lead to the undulator-type spectra of radiation with small values of the undulator parameter, K 2  1. The positron spectra in straight Si(110) clearly display the fundamental peaks of ChR at the energy ω ≈ 2.5 MeV, whereas the higher harmonics are strongly suppressed. In particular, for the smaller emission cone the peak intensity in the fundamental harmonic is an order of magnitude larger than that for the second harmonics at ω ≈ 5 MeV, and only a tiny hump of the third harmonics can be recognized at about 7.5 MeV (see the top left plot in the figure). For electrons passing through the straight crystal, the ChR peaks are less intensive and much broader than these for the positrons, as a result of stronger anharmonicity of the channeling oscillations in the trajectories. The radiation spectra produced in the SASP crystals display additional peaks, which emerge from the short-period modulations of the projectile trajectories. These peaks, more pronounced for the smaller emission cone, appear at the energies larger than the energies of the channeling peaks. For both types of the projectiles, the

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Fig. 10.17 Spectral distribution of radiation emitted by 855 MeV positrons (upper row) and electrons (lower row) in a L = 12 μm thick straight (a = 0) and periodically bent (a > 0) oriented silicon (110) crystal. The period of the SASP bending is 400 nm and various bending amplitudes are indicated in the common legend is given in the right bottom graph. The left and right columns refer to the emission cones θ0 = 0.21 and 4 mrad. The intensity of the incoherent bremsstrahlung radiation in amorphous silicon (not shown) is 0.016 × 10−3 and 0.15 × 10−3 for the smaller and larger cones, respectively

fundamental spectral peaks in the radiation emergent from the SASP bending corresponds to the emission energy about 16 MeV significantly above the ChR peaks. For positrons, the peaks of radiation due to the bending are displayed in the spectra for the amplitude values 0.1 . . . 0.9 Å. For smaller values of a, the spectral peaks disappear because the positrons experience mainly “regular” channeling staying away from the crystalline atoms and being therefore less affected by the SASP bent planes (see Ref. [143] below for the details). In contrast, the electrons experience the impact of the SASP bending at lower values of a. As seen in the right upper plot for the fundamental spectral peaks emergent from the bending, the peak for a = 0.1 Å is only two times lower than the maximal peak displayed for a = 0.4 Å. To be noted are the spectral properties for smaller aperture value (upper plots in Fig. 10.17). The electron spectra display peaks at the energies around 32 MeV for the values of a exceeding 0.2 Å. These peaks are clearly the second harmonics of the radiation emergent from the SASP bending. In addition, the peaks of channeling radiation decrease in heights and shift toward the lower emission energies. The positron spectra, in contrast to the electron ones, exhibit less peculiarities and gradually converge to the Bethe-Heitler background with increasing radiation energies. For the larger aperture value, θ0 = 4 mrad, a sizable part of the energy is radiated

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at the angles θ > γ −1 . The harmonics energies decrease with θ approximately as (1 + K 2 /2 + (γ θ )2 )−1 . As a result, the peaks of ChR and those of the radiation due to the SASP bending broaden and shift toward softer radiation energies.

Experiments with SASP Periodically Bent Crystals The impact of the radiation collimation on the intensity of the SASP peaks has been measured in experiments with a 855 MeV electron beam at MAMI [69]. The crystal was produced by adding a fraction x of germanium atoms to a silicon substrate. By alternating successively a linear increase of x from 0.5 to 1.5% with a linear decrease a sawtooth pattern of the SASP bending was achieved with 120 periods each of a λu = 0.44 µm. It was indicated in the paper that “the expected oscillation amplitude” of the (110) planes is a ≈ 0.12 Å. The measurements were performed (i) with collimation to an emission angle 0.24 mrad, and (ii) with no collimation. It was noted that the latter case corresponded to the emission cone 4 mrad  γ −1 considered in [143]. In Fig. 10.18 we compare the experimental data (symbols) with the results of simulations carried out with MBN Explorer (solid lines). Shown are the dependences of the enhancement factor on the photon energy for straight and SASP bent Si(110). Left graph corresponds to the narrow emission cone, θ0 = 0.24 mrad, the right graph presents the data for the wide cone, θ0 = 4 mrad. The simulations were performed for several values of the bending amplitude as indicated in the common legend is shown in the right graph. The same SASP bent Si(110) with 120 undulations with period λu = 0.44 µm was used in the experiment at the SLAC facility with a 16 GeV electron beam [59]. In the experiment, the SASP signal can only be expected to appear when the crystal is

Fig. 10.18 Enhancement of radiation emitted by 855 MeV electrons in straight and SASP bent Si(110) with respect to the amorphous silicon. The data refer to the crystal thickness L = 52.3 µm and bending period λu = 436 nm (total number of periods equals to 120). Left and right graphs refer to the emission cones with opening angle θ0 = 0.24 and 4 mrad, correspondingly. Symbols stand for the experimental data taken from Ref. [69] where the bending amplitude was assumed to be a = 0.12 Å

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Fig. 10.19 Spectral distribution of radiation emitted by 20 GeV electrons and positrons in straight and SASP bent Si(110). The data refer to the crystal thickness L = 52.3 µm, bending period λu = 436 nm and bending amplitude a = 0.12 Å. Left and right graphs refer to the emission cones with opening angle θ0 = 13 and 130 µrad, correspondingly. Common legend is presented in the right graph

properly aligned and should reveal itself in a narrow emission angle and the presence of a peak in the spectrum. Therefore, in Ref. [59] the enhancement was looked for as the crystal was rotated in the beam, passing through the aligned condition, and a narrow radiation cone when scanning the horizontal angular distribution with the SciFi detector, both measurements feasible at high beam intensity. However, it was mentioned in the cited paper that the measurements of the spectrum were not successful due to difficulties with the experimental setup and variations in beam energy that had not been expected. In connection with these experiments, which initially had been planned to be carried out with both electron and positron beams, the channeling simulations were carried out for 15–35 GeV projectiles [153] by means of the MBN Explorer package. The simulations of trajectories were supplemented with computation of the spectra of the emitted radiation for various detector apertures. It was recommended to carry out experiments with electrons and with the smallest aperture possible. In this case the SASP signal in the spectrum was expected to be the highest. Figure 10.19 illustrates theoretical predictions by presenting the spectral distribution of radiation energy emitted by 20 GeV projectiles in the narrow 13 µrad ≈ 1/2γ (left graph) and wide 13 µrad ≈ 5/γ (right graph) cones along the incoming beam direction. The peaks centered around ω = 7 GeV are due to the SASP bending. The peaks at much lower energy (0.2–0.5 GeV) correspond to ChR.

10.5.3 Stack of SASP Periodically Bent Crystals In recent series of experiments at MAMI [68] with 600 and 855 MeV electrons the effect of the radiation enhancement due to the SASP periodic bending has been observed (see discussion in section “Channeling and Radiation Emission in SASP

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Fig. 10.20 Spectral distribution of radiation emitted by 20 GeV electrons and positrons in straight and SASP periodically bent 4 microns thick diamond(110) (left) and silicon(110) (right) oriented crystals. Bending amplitude and period are 0.4 Å and 0.4 µm, respectively. Redrawn from data presented in Refs. [27, 150]

Periodically Bent Crystals”). Another set of experiments with thin SASP diamond crystals was planned within the E-212 collaboration at the SLAC facility (USA) with 10–20 GeV electron beams [71]. As a case study aimed at producing theoretical benchmarks for the SLAC experiments, a series of numerical simulations have been performed of the planar channeling of 10–20 GeV electrons and positrons in straight and SASP periodically bent thin crystals of silicon and diamond [27, 150]. The crystal thickness L was set to 4 microns, the period of bending λu = 0.4 microns and the bending amplitude a = 0.4 Å, which is lower than half of the (110) interplanar distance in both cases. In Fig. 10.20 the results of the simulation of radiation of 20 GeV projectiles are compared for the cases of straight and periodically bent diamond(110) crystals. The beam emittance was taken equal to ψ = 5 µrad. The spectra presented refer to the emission cone θ0 = 150 µrad, which is 5.8 times higher than natural emission angle 1/γ = 25.6 µrad and thus collects virtually all radiation emitted. In both figures the peaks located below 1 GeV corresponds to the channeling radiation. For periodically bent targets, the peaks at ω ≈ 6 GeV and above are due to the SASP bending. Note, that bending of a crystal leads to significant suppression of the channeling peak. This effect can be explained qualitatively in terms of the continuous potential modification in a SASP channel (see Appendix A in Ref. [143]). With increase of bending amplitude the depth of the potential well decreases and the width of the potential well grows resulting in decrease of the number of channeling projectiles and in lowering frequencies of channeling oscillations. Another factor that leads to the suppression of ChR is that for the 20 GeV projectiles the characteristic period of channeling oscillations, deduced from the simulated trajectories, is about λch ≈ 10 microns for both diamond and silicon crystals, so that the crystal is too thin to allow for even a single channeling oscillation. Therefore, the peak of channeling radiation is not that pronounced as in the case of thicker, L > λch . The emission spectra formed in thick (L = 24 microns) and thin (L = 4 microns) crystals are compared in Fig. 10.21 where the latter spectrum is multiplied

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Fig. 10.21 Emission spectra for 20 GeV electrons in SASP bent diamond(110) crystal calculated for two different thicknesses, as indicated. The value L = 24 µm exceeds characteristic channeling oscillations period while L = 4 µm is lower than that. Note the absence of channeling radiation peak around 150 MeV in the latter case. For the sake of comparison, the curve for L = 4 µm is multiplied by six. Redrawn from data presented in Refs. [27, 150]

by a factor of 6 for the sake of convenience. A sharp peak of the channeling radiation at ω ≈ 150 MeV is present for the thick crystal whereas for the thin one it reduces to a small hump, which is due to the synchrotron-type radiation emitted by projectiles moving along the one-ark trajectory. Remarkable feature, seen in the figure, is that the peaks due to the SASP bending in both curves virtually coincide. The effects of suppression of the channeling radiation but maintaining the level of undulator radiation in thin crystals can be used to produce intensive radiation at much higher energies corresponding to the SASP bending. To increase the latter intensity one can increase the crystal thickness L. However, this approach is not optimal from the viewpoint of technological complications (increase in the time of the crystal growth as well as in the costs associated, accumulation of the defects in the crystalline structure, etc.) Alternatively, instead of a single thick crystal, a stack of several aligned thin crystals can be used [27, 150], as illustrated by Fig. 10.22. A projectile passes sequentially several layers of periodically bent crystals, which constitute the stack, and the radiation produced in each element of the stack adds to the total radiation emitted by the projectile. For SASP undulator the thickness of layers can be taken in the interval between the bending period λu and the characteristic channeling period of projectile. Such choice of the parameters leads to absence of full channeling oscillation periods in each channeling segment of trajectory of projectile which results in suppression of channeling radiation. The effect of undulator radiation in the system remains and grows with increase of number of layers. Thickness of each crystal layer can be chosen to be smaller than the period of channeling oscillations of the projectile thus suppressing ChR. The intensity of radiation due to the SASP periodic bending increases with the number the stack layers. To simulate the radiation emission from the stack the following system was modeled [27, 150]. A set of several L = 4 microns thick layers of SASP periodically bent

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Fig. 10.22 Stack of n SASP bent crystal layers each of the thickness L separated by the gaps l. Periodic bending of the crystalline structure is illustrated by thin cosine curves. Blue line illustrates a projectile’s trajectory which consists of the modulated parts (inside the crystals) and of the straight line segments in between the layers

crystals (bending period λu = 0.4 microns) separated with l = 4 microns gaps were generated in the simulation box. The 20 GeV projectiles (positrons) entered the first layer tangent to the (110) crystallographic plane. Due to the multiple scattering in the layer, a projectile leaves is at some non-zero angle with respect to the initial direction and this angle serves as the incident angle at the entrance to the second layer, etc. The multiple scattering leads to a gradual increase of the angular dispersion of the transverse velocity of projectiles and to the decrease of the number of channeling particles with the growth of the layer’s number. As a result, for moderate number of layers the destructive effect of the multiple scattering is not too pronounced, so that the radiation intensity increases being proportional to n. For sufficiently large n values, the intensity reaches its saturation level and the peak intensity becomes independent on n. Figure 10.23 compares the radiation spectra calculated for different number of layers in stack, as indicated. For the smaller emission angle (θ0 = 0.3/γ = 15.3 µrad, left panel) the radiation intensity scales linearly with the number of stack layers until n = 6. For larger values of n the spread of the projectiles’ transverse velocities gets wider so that the intensity of radiation emitted within a narrow cone along the initial beam direction saturates. For the larger emission angle (θ0 = 5/γ = 256 µrad) the nearly linear growth of the intensity continues up to n = 24. Left panel in Fig. 10.24 compares the intensities of radiation emitted by 20 GeV positrons traveling in a single 24 microns thick SASP diamond(110) crystal and in a stack of six thin (L = 4 microns) layers. It is seen, that for the channeling peak in the spectrum is suppressed in the case of the stack of layers, where the undulator peaks are of the same intensity for both targets. Right panel in the figure compares the emission spectra for positrons and electrons of the same energy, 20 GeV, and in the same target (stacks of n = 1, 2 and 24 layers. The curves presented illustrate weak sensitivity of the spectra formed in SASP periodically bent crystals to the sign of a projectile’s charge. Therefore, this regime is favorable for the construction of light sources with the use of intensive electron beams which, at present, are more available than positron beams.

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Fig. 10.23 Radiation spectra for the small θ0 = 0.3/γ (left panel) and large θ0 = 5/γ (right) apertures calculated for different number n of diamond(110) periodically bent crystals in stack (as indicated in the common legend). The data refer to 20 GeV positrons, the bending amplitude and period are 0.4 Å and 400 nm, respectively.

10.6 Brilliance of the CU Radiation In this chapter, we present quantitative estimates for the CUR brilliance using the parameters of high-energy positron beams either available at present or planned to be commissioned in near future, see Table 10.2. The table compiles the data for the following facilities: VEPP4M (Russia), BEPCII (China), DANE (Italy), SuperKEKB (Japan) [40], SLAC (the FACET-II beams, Ref. [39]), SuperB (Italy) [103], and CEPC (China) [42]. Note that the SuperB data are absent in the latest review by Particle Data Group [40] since its construction was canceled [234]. Following Ref. [21], we demonstrate that by means of CU-LS, which operates in the LALP regime, one can achieve much higher photon yield as compared to the values achievable in modern facilities operating in the range E ph  102 keV. The relevant modern facilities are synchrotrons and undulators based on the action of magnetic field.4 Another type of modern LS, which does not utilize magnets, is based on the Compton scattering process [193]. In this process, a low-energy (eV) laser photon backscatters from an ultra-relativistic electron thus acquiring increase in the energy proportional to the squared Lorentz factor γ = ε/mc2 . This method has been used for producing gamma-rays in a broad, 101 keV–101 MeV, energy range [195, 196]. The Compton scattering also occurs if the scatterer is an atomic (ionic) electron which moves being bound to a nucleus. This phenomenon is behind the Gamma 4

Fore the sake of comparison we also match our data to the brilliance available at the XFEL facilities for much lower energy of the emitted radiation.

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Fig. 10.24 Upper graph. Comparison of radiation spectra formed by 20 GeV positrons in a L = 24 microns thick crystal and in a stack of six L = 4 microns thick layers. Dashed curves present the spectra calculated for the smaller emission angle, θ0 = 15.3 µrad, solid curves—for the larger emission angle θ0 = 256 µrad. Refs. [27, 150]. Lower graph. Comparison of radiation spectra from 20 GeV positrons (solid curves) and electrons (dashed curves) emitted in stacks of diamond(110) SASP periodically bent crystals with different number of layers as indicated. The data refer to the emission angle θ0 = 5/γ = 256 µrad.

Factory proposal for CERN [17–19] that implies using a beam of ultra-relativistic ions in the backscattering process. In this case, an ionic electron is resonantly excited by absorbing a laser photon. The subsequent radiative de-excitation produces a gammaphoton. With an account for the dechanneling and the photon attenuation, the number of photons Nωn of the frequency within bandwidth ωn emitted in the forward direction within the cone n by a projectile in a CU is given by the following expression (see Refs. [26, 104] for the details):  2 ωn n+1 (nζ ) (nζ ) − J Neff (Nd ; x, κd ) , (10.25) Nωn = A(C) 4π α nζ J n−1 2 2 ωn where ζ = K 2 /(4 + 2K 2 ), Jν (nζ ) is the Bessel function and K = 2π γ a/λu is the undulator parameter. The frequency ωn = nω1 of the nth harmonic is expressed in terms of the fundamental harmonic given by

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Table 10.2 Parameters of positron (“p”) and electron (“e”) beams: beam energy, ε, bunch length, L b , number of particles per bunch, N , beam size, σx,y , beam divergence φx,y , volume density n = N /(π σx σ y L b ) of particles in the bunch, peak current Imax = eN c/L b . In the cells with no explicit reference to either “e” or “p” the data refer to both modalities Facility

VEPP4M

BEPCII

DANE

SuperKEKB

SuperB

FACET-II CEPC

Ref.

[40]

[40]

[40]

[40]

[103, 234]

[39]

[42]

ε (GeV)

6

1.9–2.3

0.51

10

45.5 8

N

15

3.8

(units 1010 ) L b (cm)

5

1.2

p: 4

p: 6.7

e: 7

e: 4.2

p: 2.1

p: 9.04

p: 6.5

p: 0.375

e: 3.2

e: 6.53

e: 5.1

e: 0.438

1–2

p: 0.6

0.5

p: 0.00076

e: 0.5 σx (μm) σ y (μm)

1000 30

347 4.5

260 4.8

e: 0.00011

p: 10

8

p: 10.1

e: 11

8

e: 5.5

p: 0.048

0.04

e: 0.062 φx (mrad) φ y (mrad) Imax (A)

0.2 0.67 144

n 3.2 (1013 cm−3 )

0.35 0.35 152

65

1 0.54

0.85

p: 7.3

6 0.04

e: 5.9

p: 0.32

p: 0.250

p: 0.178

e: 0.42

e: 0.313

e: 0.073

0.03

p: 0.18

p: 0.125

p: 0.044

e: 0.21

e: 0.150

e: 0.044

p: 50–100

p: 723

p: 624

p: 452 12.1×103

e: 77–154

e: 627

e: 490

e: 75.5×103

p: 54

p:1.0 × 106

p:1.3 × 106

p: 2 × 105

e: 82

e:0.6 × 106

e:1.0 × 106

e: 3.9 × 106

ω1 =

2γ 2 2π c . 1 + K 2 /2 λu

0.04

1.25 × 106

(10.26)

The quantity A stands for the channel acceptance, which is defined as a fraction of the incident particles captured into the channeling mode at the crystal entrance (another term used is surface transmission, see, e.g., Ref. [55]). Apart from the factor A, the difference between (10.25) and the formula for an ideal undulator (see, e.g., [171]) is that the number of undulator periods Nu , which enters the latter, is substituted with the effective number of periods, Neff (Nd , x, κd ) ≡ Neff , which depends on the number of periods within the dechanneling length, Nd = L d /λu , and on the ratios x = L d /L a and κd = L/L d where L d denotes the

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dechanneling length and L a is the photon attenuation length. The effective number of periods is given by [26, 104]: Neff

4Nd = xκd



xe−xκd e−κd 2e−(2+x)κd /2 − + (1 − x)(2 − x) 1 − x 2−x

 1 + κd2

(x − 1)2 + 1 . 4π 2 (10.27)

In the limit L d , L a → ∞, i.e., when the dechanneling and the attenuation are neglected, Neff → Nu = L/λu , as it must be in the case of an ideal undulator. In this case one can, in principle, increase infinitely the number of periods by considering larger values of the undulator length L. This will lead to the increase of the number of photons and the brilliance since these quantities are proportional to Nu . The limitations on the values of L and Nu are mainly of a technological nature. The situation is different for a CU, where the number of channeling particles and the number of photons, which can emerge from the crystal, decrease with the growth of L. It is seen from (10.27), that in the limit L → ∞ the parameters κd and xκd = L/L a also become infinitely large leading to Neff → 0. This result is quite clear, since in this limit L  L a so that all emitted photons are absorbed inside the crystal. Another formal (and physically trivial) fact is that Neff = 0 also for a zerolength undulator L = 0. Vanishing of a positively defined function Neff (Nd , x, κd ) at two extreme boundaries suggests that there is a length L(x) which corresponds to the maximum value of the function. To define the value of L(x) or, what is equivalent, of the quantity κ d (x) = L(x)/L d , one carries out the derivative of f (x, κd ) with respect to κd and equalizes it to zero. The analysis of the resulting equation shows that for each value of x = L d /L a ≥ 0 there is only one root κ d . Hence, the equation defines, in an inexplicit form, a single-valued function κ d (x) = L(x)/L d which ensures the maximum of Neff (x, κd ) for given L a , L d and λu . Note that the crystal length enters Eq. (10.25) only via the ratio κd . It was shown [26, 104] that the quantity L(x) ensures the highest values of the number of photons Nωn and of the brilliance Bn of the CUR. Therefore, L(x) can be called the optimal length that corresponds to a given value of the ratio x = L d /L a . The following multi-step procedure has been adopted to calculate the highest brilliance of CUR. • Fix crystal and crystallographic direction. In the current paper we have focused on the (110) planar channels in diamond and silicon crystals, which are commonly used in channeling experiments. We note that other crystals/channels, available or/and studied experimentally, can also be considered [24, 56, 238]. • Fix parameters of the positron beam: energy ε, sizes σx,y and divergence φx,y , peak beam current Imax . • Scan over photon energy ω. For each ω value: 1. Determine the attenuation length L a (ω) (for the photon energies above 1 keV the data are compiled in Ref. [102]).

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2. Scan over a and λu consistent with the stable channeling condition [22, 24]: C = 4π 2

a ε < 1.  λ2u Umax

(10.28)

 where Fcf ≈ ε/R The bending parameter C is defined as the ratio Fcf /Umax  is is the centrifugal force in a channel bent with curvature radius R and Umax the maximum force due to the interplanar potential. Channeling motion in the bent crystal is possible if C < 1. In a periodically bent crystal, the bending radius in the points of maximum curvature equals to λ2u /4π 2 a which explains the right-hand side of (10.28). 3. Determine dechanneling length L d (C). The data on the dechanneling length can be extracted (when available) from the experiments [57, 227] or obtained by means of highly accurate numerical simulation of the channeling process [26, 33, 139]. For positrons, a very good estimation for L d (C) can be obtained by means of the following formulae [26, 30]:

L d (C) = (1 − C)2 L d (0),

L d (0) =

256 aTF d ε 9π 2 r0 m e c2 

(10.29)

Here L d (0) is the dechanneling length in the straight channel, r0 cm is the classical electron radius, Z and aTF are, respectively, the atomic number and the Thomas-Fermi radius,  = 13.55 + 0.5 ln(ε[GeV]) − 0.9 ln(Z ). 4. Determine the maximum value of Neff and the optimal length L . 5. Determine the channel acceptance. The acceptance A(C) of a bent channel can be estimated as follows [30]: A(C) = (1 − C) A0 .

(10.30)

Here A0 = 1 − 2u T /d (u T is the amplitude of thermal vibrations of the crystal atoms) is the acceptance of the straight channel. 6. Substituting the quantities obtained into Eq. (10.25) and Eq. (10.16) one calculates the highest available peak brilliance Bpeak (ω). As formulated, the items (iii)–(vi) listed above are applicable for a fully collimated positron beam with zero divergence. In reality, the beams have non-zero divergence φ, see Table 10.2, so that only a fraction ξ of the beam particles gets accepted into the critical angle L for channeling. To estimate this fraction we assume the normal distribution of the beam particles with respect to the incident angle and calculate ξ as follows:   L θ2 (10.31) exp − 2 dθ. ξ = (2π φ 2 )−1/2 2φ −L

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Table 10.3 Fraction ξ of the beams particles with incident angle less than Lindhard’s critical angle L (in mrad). For each beam indicated the parameter φ (in mrad) stands for the minimum of two divergences φx and φ y , see Table 10.2 Facility

VEPP4M BEPCII

DANE

SuperKEKB

SuperB

FACET-II CEPC

φ L ξ

0.2 0.08 0.31

0.54 0.28 0.40

0.18 0.1 0.42

0.125 0.08 0.48

0.044 0.063 0.85

0.35 0.14 0.31

0.03 0.03 0.67

The values of ξ calculated for the beams listed in Table 10.2 √ are presented in Table 10.3. For each beam, Lindhard’s critical angle L = 2U0 /ε is estimated using the value U0 = 20 eV (which corresponds, approximately, to the interplanar potential depth in Si(110) and diamond(110)) and the indicated values of the beam divergence is calculated as φ = min[φx , φ y ]. To account for the non-zero divergence one multiplies the value Bpeak (ω), calculated as described above, by the factor ξ . Figure 10.25 illustrates the results of calculations performed for diamond(110)based CU using and for the positron beams specified in Table 10.2. The dependences presented were obtained by maximizing the brilliance of CUR emitted in the fundamental harmonic. It is seen, that within the range of moderate values of the bending amplitude (a/d varies from several units up to several tens, graphs (e); d = 1.26 for diamond(110)) it is possible to construct a CU with a sufficiently large number of effective periods, Neff ≈ 10 . . . 100, graphs (c). These values correspond to the range of undulator periods λu ≈ 101 . . . 102 µm (graphs (d)) which is achievable by different methods of preparation of periodically bent crystalline structures, see Sect. 10.1. It is seen from Fig. 10.25 that out of all calculated quantities the peak brilliance, graph (f), is the most sensitive to the parameters of the positron beam. The variation in the magnitude of Bpeak (ω) is over six orders of magnitude, from 1018 up to 1026 ph/s/mrad2 /mm2 /0.1% BW (compare the DANE and CEPC curves). Let us compare the brilliance of CUR with that available at modern synchrotron facilities. Figure 10.26 left presents the peak brilliance calculated for positron-based diamond(110) and Si(110) CUs and that for several synchrotrons. The CUR curves refer to the optimal parameters of CU, i.e., those which ensure the highest values of Bpeak (ω) of CUR for each positron beam indicated. To be noted is that for the well-collimated intensive beams with small transverse sizes (SuperB, FACET, SuperKEK, CEPC) the peak brilliance of CUR in the photon energy range from 102 keV to 102 MeV (the corresponding wavelengths vary from 10−1 down to 10−4 Å) is comparable to (the case of SuperB, FACET, and SuperKEK beams) or even higher (CEPC beam) than that achievable in conventional LS for much lower photon energies. Figure 10.26 right presents the peak intensities, Nω Imax /e, of the first (solid lines) and third (dashed lines) harmonics of CUR from diamond(110)-based CU with the optimized parameters. Different curves correspond to different positron beams as specified in the caption. Most of the curves presented show orders of magnitude higher intensities in the photon energy range one to tens of MeV than that from the laser-Compton scattering LS (open circles). Within the same photon energy interval

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Fig. 10.25 Parameters of the diamond(110)-based CU, - C, λu , a (measured in the interplanar distances d), Neff , L (measured in the units of L d (C)), that ensure the highest peak brilliance Bpeak (ω), graph (f). Different curves correspond to several currently achievable positron beams as indicated in the legend (see also Table 10.2). Redrawn from data presented in Ref. [21]

the CUR intensity can be comparable with or even higher (see the curves for the SuperB, SuperKEK, and FACET-II beams) than the value predicted in the Gamma Factory proposal (marked with the horizontal dash-dotted line). Figure 10.26 demonstrates also the tunability of a CU-LS. For any positron beam with specified parameters the photon yield can be maximized (more generally, varied) over broad range of photon energies by properly choosing parameters of the CU (bending amplitude and period, crystal, plane).

10.6.1 Results of Atomistic Simulations of the CU Light Sources To verify the predictions made in Ref. [21], accurate numerical simulations have been performed [235] at providing reliable quantitative data on the brilliance of CUR. The simulations have been performed for the FACET [39] and SuperKEKB [40] positron beams using the channeling module [139] of the MBN Explorer computational package [140].

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Fig. 10.26 Left: Comparison of the peak brilliance available at several synchrotron radiation facilities (APS, ESRF, PETRA, SPring8) with that for CUR from diamond(110)- and Si(110)-based CUs for several positron beams listed in Table 10.2. The CUR data refer to the emission in the fundamental harmonic. The data on APS (USA), ESRF (France), PETRA (DESY, Germany), SPring8 (Japan) are from [15, 168]. Right: Peak intensity (number of photons per second, Nω Imax /e) of diamond(110)-based CUs calculated for positron beams at different facilities: 1—DANE, 2— VEPP4M, 3—BEPC-II, 4—SuperB, 5—SuperKEK, 6—FACET-II, 7—CEPC. Solid and dashed lines correspond to the emission in the first and third harmonics, respectively. Open circles indicate the data on the laser-Compton backscattering [195]. The horizontal dash-dotted line marks the intensity 1017 photon/s indicated in the Gamma Factory (GF) proposal for CERN [18]. Redrawn from data presented in Ref. [21] Table 10.4 Parameters of diamond-based CU used in the simulations, [235] Facility Beam energy Bending Bending Number of ε (GeV) amplitude a period λu periods Nu (angstrom) (micron) SuperKEKB FACET

4 10

11 21

35 85

84 83

Crystal length L (mm) 2.84 7.01

The parameters of PB diamond (110) crystals used in the simulations are summarized in Table 10.4. For each beam, the indicated values of bending amplitude and period result in the first harmonic energy of the CU radiation emitted in the forward direction equal to ω1 = 2 MeV. The trajectories of projectiles have been simulated and the resulting emission spectra have been calculated. The calculated spectra together with the data on the beams emittance have been used further to calculate the photon flux and brilliance of the CU radiation. For the sake of comparison, the trajectories of 4 and 10 GeV positrons propagating in straight diamond(110) crystals have also been simulated. To reduce the computational time, the straight crystals were chosen ten times shorter than the corresponding PBCs. Figure 10.27 presents the emission spectra calculated for the SuperKEKB beam using the parameters indicated in Table 10.4. The spectra presented correspond to the natural emission cone θ = 1/γ = 130 µrad, correspondingly. Graphs (a) show the spectral dependences over the broad range of the photon energies. For each positron beam the spectrum (red solid line) consists of two main

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Fig. 10.27 Radiation spectra for the SuperKEKB positron beam propagating in diamond(110) crystal. In all graphs the solid line, labeled as “Simulated CU”, stands for the emission spectra calculated using simulated trajectories in the PBCs whose bending parameters and thicknesses are indicated in Table 10.4. On the panel (a), the dashed line (“reference straight crystal”) marks the emission spectrum simulated in a ten times shorter straight diamond crystals (L = 0.284 mm). Shading indicates the statistical error due to the finite number of the trajectories simulated. On the panel (b), the simulated spectrum of CUR is compared to that formed in the “ideal CU” of the same length (see text for more details). All spectra correspond to the natural emission angle θ = 1/γ equal to 130 µrad

parts: (i) narrow peaks (harmonics) of CUR, which dominate in the lower part of the spectrum (up to ω ≈ 10–15 MeV), and (ii) a wider structure (less accented in the case the SuperKEKB beam) at higher photon energies. The latter feature is due to the channeling radiation. For the sake of comparison, the spectra of ChR formed in a straight crystal (of the thickness ten times smaller than that for the PBCs) are also shown in the panels, see blue dashed lines. We do not discuss here the visible transformation in the profiles of ChR emitted in the PBCs as compared to the straight crystals (see Ref. [235] for the discussion) but focus on the spectral dependence of CUR which is shown in graphs (b) in more detail in narrower ranges of photon energies. For the sake of comparison, we also present the spectra of radiation emitted by a projectile which moves along the cosine trajectory y = a cos(2π z/λu with given values of a, λu and Nu . These spectra, labeled as “ideal CU”, are shown with dash-dotted lines. It is worth noting that shapes of the spectra formed in real CU and in the ideal one are quite similar differing mainly in the peak intensities: for simulated trajectories the radiation intensity is approximately 2 times lower. This is due to the dechanneling effect which leads to the gradual decrease in the number of channeling particles with the penetration distance z. Introducing the calculated peak intensities of CUR together with the values of beam emittance and divergence (see Table 10.2) in Eqs. (10.16) and (10.18) one calculates the photon flux and brilliance of CUR. Table 10.5 provides the peak values of these parameters for th first harmonics of CUR. Figure 10.28 compares the brilliance of CUR with that available at modern synchrotron facilities. The data on the APS (USA), ESRF (France), PETRA (DESY,

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Table 10.5 Simulated values of photon energies, peak fluxes and peak brilliances for diamondbased CU-LS [235] Facility Photon energy Peak flux F1 Peak brilliance B1 ω1 (MeV) (photons/s/0.1%BW) (photons/s/mrad2 /mm2 /0.1%BW) SuperKEKB FACET

1.95 1.94

3.50 × 1018 7.75 × 1019

3.50 × 1024 7.95 × 1024

Fig. 10.28 Comparison of the peak brilliance available at several synchrotron radiation facilities (APS, ESRF, PETRA,SPring8) with that for CUR from diamond(110)-based CUs. Black and blue curves stand for the model estimations [21], symbols refer to the results of numerical simulations [235]

Germany), SPring8 (Japan) facilities are from Refs. [15, 168] Black and blue curves stand for the peak brilliances of CUR (solid and dashed lines represent the first and the third harmonics, correspondingly) estimated within the model approach [21]. Solid and dashed lines represent, correspondingly, the first and the third harmonics. Symbols indicate the values obtained by rigorous numerical simulations [235]. Remarkable result is that in most cases the CUR brilliance obtained in the simulations exceeds the values obtained in the estimations [21]. Thus, it has been confirmed that CUs can be considered as novel bright gamma-rays LSs.

10.7 Emission of Coherent CU Radiation 10.7.1 Introduction The radiation emitted in an undulator is not coherent with respect to the emitters, i.e., the undulating particles of total number Np . Indeed, the intensity of the emit Np ted radiation, proportional to the square of the total electric field Etot = j=1 Ej, where E j stands for the electric field of the electromagnetic wave emitted by the jth particle. In an undulator the positions of the particles (in particular, in the

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longitudinal direction) are not correlated.5 As a result, the phasefactors exp(iψ j ), contained in E j , are not correlated as well. Therefore, the sum over the cross terms  ∗ 2 j=k E j · Ek ∝ exp(i(ψ j − ψk )), which appear in |Etot | , cancels out and the intensity is proportional to the number of emitters:

Iinc

Np

2 E j = Np |E1 |2 ∝ Np N 2 . ∝ |Etot | → 2

(10.32)

j=1

This relation points out the two important features of the incoherent spontaneous UR (the subscript “inc” on the left-hand side of stands for “incoherent”). First, Iinc ∝ |E1 |2 ∝ N 2 , i.e., the radiation is coherent (at the harmonics frequencies) with respect to the number of undulator periods, N . The proportionality to N 2 makes the UR a powerful source of spontaneous electromagnetic radiation. In modern undulators, based on the action of magnetic field, the number of undulator periods is on the level of 103 . . . 104 [168]. The second feature is that the UR is incoherent with respect to the number of the radiating particles, Iinc ∝ Np . Hence, the increase in the beam density will cause a moderate (linear) increase in the radiated energy. Even more powerful and coherent radiation will be emitted if the probability density of the particles in the beam is modulated in the longitudinal direction with the period λ, equal to the wavelength of the emitted radiation. In this case, the electromagnetic waves emitted in the forward direction by different particles have approximately the same phase (more exactly, φ j − φk ≈ nλ where n is an integer) [179]. Therefore, the total amplitude of the emitted radiation is a coherent sum of individ Np E j ∝ Np E1 , so that the intensity Icoh ual electromagnetic waves, i.e., Etot = j=1 becomes proportional to the square of the radiating particles: Icoh ∝ |Etot |2 ∝ Np2 N 2 .

(10.33)

Comparing (10.33) and (10.32) one sees, that Icoh /Iinc ∝ Np . Thus, the increase in the photon flux due to the beam modulation (other terms used are “bunching” [3, 6, 169, 188] or “microbunching” [5]) can reach orders of magnitudes relative to the UR of an unmodulated beam of the same density. In what follows we assume that the beam is fully modulated at the crystal entrance. The description on the methods of preparation of a pre-bunched beam with the parameters needed to amplify CUR one finds in Refs. [25, 26].

5

To be specific, we assume the emission in the forward direction. This is why the longitudinal coordinate, i.e., the one along the undulator axis, plays the key role.

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10.7.2 Beam Demodulation In a CU, a channeling particle, while moving along the channel centerline, undergoes two other types of motion in the transverse directions with respect to the CU axis z. First, there are channeling oscillations along the y direction perpendicular to the crystallographic planes. Second, the particle moves along the planes (the x direction). To be noted is that different particles have different (i) amplitudes ach of the channeling oscillations, and (ii) momenta px in the (x z)-plane due to the distribution in the transverse energy of the beam particles as well as the result of multiple scattering from crystal atoms. Therefore, even if the speed of all particles along their trajectories is the same, the difference in ach or/and in px leads to different values of the velocities with which the particles move along the undulator axis. As a result, the beam loses its modulation while propagating through the crystal. For an unmodulated beam, the CU length L is limited mainly by the dechanneling process. A dechanneled particle does not follow the periodic shape of the channel, and, thus, does not contribute to the CUR spectrum. Hence, it is reasonable to estimate L on the level of several dechanneling lengths L d (see panels (b) in Fig. 10.25). Longer crystals would attenuate rather than produce the radiation. Since the intensity of CUR is proportional to the undulator length squared, the dechanneling length and the attenuation length are the main restricting factors which must be accounted for. For a modulated beam, the intensity is sensitive not only to the shape of the trajectory but also to the relative positions of the particles along the undulator axis. If these positions become random because of the beam demodulation, the coherence of CUR is lost even for the channeled particles. Hence, the demodulation becomes the phenomenon which imposes the most restrictions on the parameters of a CU. In Ref. [236] an important quantity,—the demodulation length, was introduced. It represents the characteristic scale of the penetration depth at which a modulated beam of channeling particles becomes demodulated. Within the framework of the approach developed in the cited papers the demodulation length L dm is related to the dechanneling length L d (C) in a bent channel: L dm =

L d (C) . √ α(ξ ) + ξ /j0,1

(10.34)

Here j0,1 = 2.4048 . . . is the first root of the Bessel function J0 (x). The dimensionless parameter ξ is expressed in terms of the emitted radiation frequency ω, the dechanneling length L d (C) and Lindhard’s critical angle L (C) in the bent channel: ξ = ωL d (C)2L (C)/2c (see [238] for the details). The function α(ξ ) is related to the real and imaginary parts of the first root (with respect to ν) of the equation [238] F(−ν, 1, z)

= √ z=(1+i) j0,1 ξ /2

0,

where F(.) stands for Kummer’s confluent hypergeometric function [237].

(10.35)

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Fig. 10.29 The ratio L dm /L d (C) versus photon energy for diamond(110) (left panel) and silicon(110) (right panel) channels calculated for various values of bending parameter C. Redrawn from data presented in Ref. [21]

Equations (10.34) and (10.35) can be analyzed numerically to derive the dependence of the demodulation length on the radiation energy ω for a particular crystal channel. The result of such analysis is illustrated by Fig. 10.29 where the dependences of the ratio L dm /L d (C) on the photon energy are presented for the (110) planar channels in diamond and silicon and for several values of the bending parameter C as indicated. To be noted, is that for all values of the bending parameter C and over broad energy range of the emitted radiation, the demodulation length is noticeably less than the dechanneling one. To preserve the beam modulation during its channeling in a crystal and, as a result, to maintain the coherence of the radiation the crystal length L must be less than the demodulation length: L  L dm < L d (C) .

(10.36)

It follows from (10.34) that in a periodically bent crystal L dm depends on the crystal type, on the parameters of the channel (its width, strength of the interplanar field), on the bending amplitude and period, on the projectile energy and its type (these are “hidden” in L d (C), C, and ξ ) as well as on the emitted photon energy (enters the parameter ξ ). Therefore, Eq. (10.36) imposes addition restriction on the CU length as compared to the case of the CUR emission by the unmodulated beam. In Ref. [238] it is also shown that the phase velocity of the modulated beam along the CU channel is modified as compared to the unmodulated one. The modification changes the resonance condition which links the parameters of the undulator and the radiated wavelength (energy). The expression for the fundamental harmonic frequency ω ≡ ω1 acquires the following form (compare with Eq. (10.26)): ω=

1+

2π c 2γ 2 , + 2β /2 λu

K 2 /2

(10.37)

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Fig. 10.30 Parameters of the diamond(110)-based CU, - λu , a (measured in units of the interplanar distance d = 1.26 Å), C, the undulator parameter K = 2π γ a/λu , the demodulation length L dm (C) ≡ L dm and the number of periods within L dm , Ndm = L dm /λu that ensure the highest peak brilliance of the radiation emitted by the fully modulated FACET-II positron beam. Redrawn from data presented in Ref. [21]

where the additional term in the denominator is given by  2β

= 4γ

2

2L (C)

β(ξ ) +

1

√ 2 j0,1 ξ

(10.38)

with β(ξ ) being another function related to the real and imaginary parts of the first root of Eq. (10.35) (details can be found in Refs. [26, 238]). The quantity ξ = ωL d (C)2L (C)/2c depends on ω. Therefore, Eq. (10.37) represents a transcendent equation which relates ω to the projectile energy and to the bending amplitude and period. Analysis of the formulae written above shows that for given values of ω and ε all other quantities which characterize the CU and the demodulation process can be expressed in terms of a single independent variable, for example, the bending amplitude a. Then, scanning over the a values it is possible to determine the whole set of the parameters (these include a, λu , C, L dm (C)) which maximize the peak brilliance of the superradiant emission (see Sect. 10.7.3). Figure 10.30 shows the results of calculations of the parameters of the diamondbased CU that maximizes the peak brilliance of the radiation of energy ω emitted by the FACET-II positron beam, see Table 10.2. The dependences presented correspond to the emission in the fundamental harmonic. The crystal thickness was set to the

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demodulation length L = L dm (C), graph (e). The quantity Ndm stands for the number of undulator periods within the demodulation length, Ndm = L dm (C)/λu . Only the data corresponding to Ndm ≥ 10 are shown in the panels. The dependences presented refer to the Large-Amplitude regime of the periodic bending, which implies that the amplitude a exceeds the interplanar distance d. Noticing that the factor 2π/λu can be written in terms of the undulator parameter K = 2π γ a/λu , one writes Eq. (10.37) as a quadratic equation with respect to K . Resolving it one finds that K is a two-valued function of ω, which is reflected in graph (f). As a result, all dependences presented contain two branches related to the smaller (black curves) and larger (blue curves) allowed values of K .

10.7.3 Pre-bunching and Super-Radiance in CU Powerful superradiant emission by ultra-relativistic particles channeled can be achieved if the probability density of the particles in the beam is (uniformly) modulated in the longitudinal direction with the period equal to integer multiple to the wavelength λ of the emitted radiation [192]. To prevent the demodulation of the beam as it propagates through the crystal, the crystal length L must satisfy condition (10.36). In a wide range of photon energies, starting with ω ∼ 102 keV, the demodulation length is noticeably less than the dechanneling length L d . In addition to this, in this energy range the photon attenuation length L a in silicon and diamond greatly exceeds the dechanneling length of positrons with energies up to several tens of GeV [26]. Therefore, on the spatial scale of L dm the dechanneling and the photon attenuation effects can be disregarded. In what follows, we carry out quantitative estimates of the characteristics of the superradiant CU radiation (CUR) emitted by a fully modulated positron beam channeled in periodically bent diamond and silicon (110) oriented crystals in the absence of the dechanneling and the photon attenuation. The beam represents a train of bunches each of the length L b containing N particles. The crystal length (along the beam direction) is set to the demodulation length, L = L dm . The transverse sizes of a crystal are assumed to be larger than those of the beam, i.e., than σx,y . For the sake of clarity, below we consider the emission in the first harmonics of CUR, see Eq. (10.37) Final width ω of the CUR peak introduces a time interval τcoh = 1/ω within which two particles separated in space can emit coherent waves. Hence, one can introduce a coherence length [168] L coh = cτcoh =

λ ω , 2π ω

(10.39)

where λ is the radiation wavelength, and the bandwidth ω/ω ≈ 1/Ndm with Ndm = L dm /λu standing for the number of periods within the demodulation length.

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The number of the particles that emit coherently is calculated as Ncoh =

L coh N. Lb

(10.40)

2 . The number of such sub-bunches Their radiated energy is proportional to Ncoh is L b /L coh , therefore, the energy emitted by the whole bunch contains the factor 2 (L b /L coh )Ncoh = N Ncoh . Another important quantity to be estimated is the solid angle coh within which the waves emitted by the particles of the sub-bunch are coherent. This angle can be chosen as the minimum value from the natural emission cone of the first harmonics  = 2π λu /L dm and the angle ⊥ which ensures transverse coherence of the emission due to the finite sizes σx,y of the bunch. Assuming the elliptic form for the bunch cross section one derives ⊥ ≤ λ2 /4π σx σ y . Therefore, the solid angle coh is found from   coh = min ⊥ ,  . (10.41)

The number of photons Nω emitted by the bunch particles one obtains multiplying the spectral-angular distribution of the energy emitted by a single particle by the factor N Ncoh coh (ω/ω). The result reads Nω = 4π α N Ncoh ζ [J0 (ζ ) − J1 (ζ )]2 Ndm

coh ω ,  ω

(10.42)

where ζ = (K 2 + 2β )/2(2 + K 2 + 2β ) with 2β defined in (10.38). The number of photons emitted by the particles of the unmodulated beam in a CU of the same length and number of periods one calculates from Eq. (10.25) written for n = 1 by setting Neff = Ndm , substituting K 2 → K 2 + 2β and multiplying the right-hand side by N . Comparing the result with Eq. (10.42) one notices that the enhancement factor due to the coherence effect is Ncoh coh /.  Another quantity of interest is the flux Fω of photons. Measured in the units of photons/s/0.1%BW , it is related to Nω as follows: Fω =

1 103 (ω/ω)

Nω , tb

(10.43)

where tb = L b /c = eN /Imax is the time flight of the bunch and Imax stands for the peak current. Figure 10.31 shows peak brilliance of radiation formed in the diamond(110)-based CU as functions of the first harmonic energy. Four graphs correspond to the positron beams (as indicated) the parameters of which are listed in Table 10.2. In each graph, the dashed line refers to the emission of the spontaneous CUR formed in the undulator with optimal parameters, see Fig. 10.25. The thick curves present the peak brilliance

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Fig. 10.31 Peak brilliance of superradiant CUR (thick curves) and spontaneous CUR (thin dashed curves) emitted in periodically bent oriented diamond(110) crystal. The graphs refer to four positron beams (as indicated). Redrawn from data presented in Ref. [21]

of the superradiant CUR maximized by the proper choice of the bending amplitude and period (as described in Sect. 10.7.2). Two branches of this dependence, seen in graphs (a)–(c), are due to the two-valued character of the dependence of undulator parameter K on the radiation frequency ω. For the CEPC beam, graph (d), this peculiarity manifests itself in the frequency domain beyond 40 MeV, therefore it is not seen in the graph.

10.7.4 Brilliance of the CU-Based LSs Quantitative analysis and numerical data on the parameters of a CU which maximize the brilliance of CUR in presence of the demodulation process is presented in Sect. 10.7.3. These data have been used to calculate the peak brilliance of the superradiant CUR. Figure 10.32 illustrates a boost in peak brilliance due to the beam modulation. Thick curves correspond to superradiant CUR calculated for fully modulated positron beams (as indicated) propagating in the channeling mode through diamond(110)based CU. In the photon energy range 10−1 . . . 101 MeV the brilliance of superradiant CUR by orders of magnitudes (up to 8 orders in the case of CEPC) exceeds that of the

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Fig. 10.32 Peak brilliance of superradiant CUR (thick solid curves) and spontaneous CUR (dashed lines) from diamond(110) CUs calculated for the SuperKEKB, SuperB, FACET-II, and CEPC positron beams versus modern synchrotrons, undulators, and XFELs. The data on the latter are taken from Ref. [15]

spontaneous CUR (dash-dotted curves) emitted by the random beams. Remarkable feature is that the superradiant CUR brilliance can not only be much higher that the spontaneous emission from the state-of-the-art magnetic undulator (see the curves for the TESLA undulator) but also be comparable with the values achievable at the XFEL facilities (LCLC (Stanford) and TESLA SASE FEL) which operate in much lower photon energy range.

10.8 Conclusion Construction of novel Crysta-Based LSs is a challenging technological task, which requires a highly interdisciplinary approach combining theoretical analysis and computational modeling, development of technologies for crystalline sample preparation (engaging material science, nanotechnology, acoustics, solid-state physics) together with a detailed experimental program (for CLS characterization, design of incident particle beams, experimental characterization of the emitted radiation). To accomplish this task, one has to assemble a consortium that has the necessary broad range of competences to realize the science-toward-technology breakthrough that will enable the practical realization of the CLS. The practical realization of CLS will include elaboration the key theoretical, experimental, and technological aspects, demonstration of the device functionality, designing the novel technology for construction of CLS that will allow for their mass production in the future and establishing the standards required for adoption of CLS by different user communities.

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Realization of this program implies a broad range of correlated and entangles activities including 1. Fabrication of linear, bent, and periodically bent crystalline structures with lattice quality necessary for delivering pre-defined bending parameters within the ranges indicated in Fig. 10.2; 2. Advanced control of the lattice quality by means of the highest quality nondestructive X-ray diffraction techniques. The same techniques to be applied to detect possible structural modification following particle irradiation; 3. Validation of functionality of the manufactured structures through experiments with high-quality (low-energy spread, low emittance, high particle density, and current) beams of ultra-relativistic electrons and positrons with ε = 10−1 –101 GeV, including an authoritative study of the structure sustainability with respect to beam intensity, as well as explicit experimental characterization of the emission spectra; 4. Advance in computational and numerical methods for multiscale modeling of nanostructured materials with extremely high, reliable levels of prediction (from atomistic to mesoscopic scale), of particle propagation, of irradiation-induced solid-state effects, and for calculation of spectral-angular distribution of emitted radiation and for modeling [140]. Ultimately, this will enable better experimental planning and minimization of experimental costs. The knowledge gained the studies (1)–(4) will provide CLSs prototypes and a roadmap for practical implementation by CLS system manufacturers and accelerator laboratories/users worldwide. Sub-angstrom wavelength ultrahigh brilliance, tunable CLSs will have a broad range of exciting potential cutting-edge applications [213]. These applications include exploring elementary particles, probing nuclear structures and photonuclear physics, and examining quantum processes, which rely heavily on gamma-ray sources in the MeV to GeV range [206]. Gamma-rays induce nuclear reactions by photo-transmutation. For example, in the experiment [239] a long-lived isotope can be converted into a short-lived one by irradiation with a gamma-ray bremsstrahlung pulse. However, the intensity of bremsstrahlung is orders of magnitudes less than CUR. Moreover, to increase the effectiveness of the photo-transmutation process is it desirable to use photons whose energy is in resonance with the transition energies in the irradiated nucleus [195, 197]. By varying the CU parameters one can tune the energy of CUR to values needed to induce the transmutation process in various isotopes. This opens the possibility for a novel technology for disposing of nuclear waste. Photo-transmutation can also be used to produce medical isotopes. Another possible application of the CU-LSs concerns photo-induced nuclear fission where a heavy nucleus is split into two or more fragments due to the irradiation with gammaquanta whose energy is tuned to match the transition energy between the nuclear states. This process can be used in a new type of nuclear reactor—the photonuclear reactor [197]. A non-destructive assay system for radioactive waste management by means of nuclear resonance fluorescence triggered by gamma-rays generated from the Compton scattering of laser photons by relativistic electrons has been discussed

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[240]. This problem can also be attacked by means of the CLSs radiation. Powerful monochromatic radiation within the MeV range can be used as an alternative source for producing beams of MeV protons by focusing a photon pulse on to a solid target [239, 241]. Such protons can induce nuclear reactions in materials producing, in particular, light isotopes which serve as positron emitters to be used in Positron Emission Tomography (PET). The production of PET isotopes using CUR exploiting the (γ ; n) reaction in the region of the giant dipole resonance (typically 20–40 MeV) is an important application of CLS since PET isotopes are used directly for medial PET and for Positron Emission Particle Tracking (PERP) experiments. Irradiation by hard X-ray strongly decreases the effects of natural surface tension of water [242]. The possibility to tune the surface tension by CUR can be exploited to study the many phenomena affected by this parameter in physics, chemistry, and biology such as, for example, the tendency of oil and water to segregate. The last but not least, a micron-sized narrow CLS photon beam may be used in cancer therapy [243] to improve the precision and effectiveness of the therapy for the destruction of tumors by collimated radiation allowing delicate operations to be performed in close vicinity of vital organs. The exemplary case study of a tunable CU-based LS considered in Sect. 10.5 demonstrates that peak brilliance of CUR emitted in the photon energy range 102 keV up to 102 MeV by currently available (or planned to be available in near future) positron beams channeling in periodically bent crystals are comparable to or even higher than that achievable in conventional synchrotrons in the much lower photon energy range. Intensity of CUR greatly exceeds the values provided by LSs based on Compton scattering and can be made higher than the values predicted in the Gamma Factory proposal in CERN. By propagating a pre-bunched beam the brilliance in the energy range 102 keV up to 101 MeV can be boosted by orders of magnitude reaching the values of spontaneous emission from the state-of-the-art magnetic undulators and being comparable with the values achievable at the XFEL facilities which operate in much lower photon energy range. Important is that by tuning the bending amplitude and period one can maximize brilliance for given parameters of a positron beam and/or chosen type of a crystalline medium. Last but not least, it is worth to mention that the size and the cost of CLSs are orders of magnitude less than that of modern LSs based on the permanent magnets. This opens many practical possibilities for the efficient generation of gamma-rays with various intensities and in various ranges of wavelength by means of the CLSs on the existing and newly constructed beam-lines. Though we expect that, as a rule, the highest values of brilliance can be reached in CU-based LSs (or, in those based on stacks of CUs) the analysis similar to the one presented can be carried out for other types of CLSs based on linear and bent crystals. This will allow one to make an optimal choice of the crystalline target and the CLS type to be used in a particular experimental environment or/and to tune the parameters of the emitted radiation matching them to the needs of a particular application. The case study presented has been focused on the positron beams, which have a clear advantage since the dechanneling length of positrons is order of magnitude larger than that of electrons of the same energy. This allows one to use thicker crystals

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in channeling experiments with positrons thus enhancing the photon yield. Nevertheless, experimental studies of CLSs with electron beams are worth to be carried out. Indeed, high-quality electron beams of energies starting from sub-GeV range and onward are more available than their positron counterparts. Therefore, these laboratories provide more options for the design, assembly and practical implementation of a full suite of correlated experimental facilities needed for operational realization and exploitation of the novel CLSs. In this connection we note that in the course of channeling experiments at the Mainz Microtron facility with ε = 190–855 MeV electrons propagating in various CUs, which have been carried out over the last decade within the frameworks of several EU-supported collaborative projects (FP6-PECU, FP7-CUTE, H2020-PEARL), a unique experience has been gained. This experience has ascertained that the fundamental importance of the quality of periodically bent crystals, which, in turn is based on the cutting-edge technologies used to manufacture the crystalline structures, modern techniques for non-destructive characterization of the samples, of the necessity of using advanced computational methods for numerical modeling of a variety of phenomena involved. On the basis of this experience the bottlenecks on the way to practical realization of the CLSs concept have been established. To quantify the scale of the impact within Europe and worldwide which the development of radically novel CLSs might have, we can draw historical parallels with synchrotrons, optical lasers, and XFELs. In each of these technologies, there was a time lag between the formulation of a pioneering idea, its practical realization, and follow-up industrial exploitation. However, each of these inventions has subsequently launched multi-billion dollar industries. The implementation of CLS, operating in the photon energy range up to hundreds of MeV, is expected to lead to a similar advance and CLSs have the potential to become the new synchrotrons and lasers of the mid to late twenty-first century, stimulating many applications in basic sciences, technology, and medicine. The development of CLS will therefore herald a new age in physics, chemistry, and biology.

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139. Sushko, G.B., Bezchastnov, V.G., Solov’yov, I.A., Korol, A.V., Greiner, W., Solov’yov, A.V.: Simulation of ultra-relativistic electrons and positrons channeling in crystals with MBN Explorer. J. Comp. Phys. 252, 404 (2013) 140. Solov’yov, I.A., Yakubovich, A.V., Nikolaev, P.V., Volkovets, I., Solov’yov, A.V.: MesoBioNano explorer - a universal program for multiscale computer simulations of complex molecular structure and dynamics. J. Comp. Chem. 33, 2412–2439 (2012) 141. http://mbnresearch.com/get-mbn-explorer-software 142. Sushko, G.B., Korol, A.V., Solov’yov, A.V.: Nucl. Instrum. Methods B 355, 39 (2015) 143. Korol, A.V., Bezchastnov, V.G., Sushko, G.B., Solov’yov, A.V.: Nucl. Instrum. Meth. B 387, 41–53 (2016) 144. Korol, A.V., Bezchastnov, V.G., Solov’yov, A.V.: Eur. Phys. J. D 71, 174 (2017) 145. Shen, H., Zhao, Q., Zhang, F.S., Sushko, G.B., Korol, A.V., Solov’yov, A.V.: Nucl. Instrum. Meth. B 424, 26 (2018) 146. Pavlov, A.V., Korol, A.V., Ivanov, V.K., Solov’yov, A.V.: J. Phys. B: At. Mol. Opt. Phys. 52, 11LT01 (2019) 147. Sushko, G.B., Bezchastnov, V.G., Korol, A.V., Greiner, W., Solov’yov, A.V., Polozkov, R.G., Ivanov, V.K.: J. Phys.: Conf. Ser. 438, 012019 (2013) 148. Sushko, G.B., Korol, A.V., Greiner, W., Solov’yov, A.V.: Sub-GeV electron and positron channeling. J. Phys.: Conf. Ser. 438, 012018 (2013) 149. Sushko, G.B., Korol, A.V., Solov’yov, A.V.: St. Petersburg Polytechnical Uni. J.: Phys. Math. 1, 332 (2015) 150. Sushko, G.B.: Atomistic Molecular Dynamics Approach for Channeling of Charged Particles in Oriented Crystals (Doctoral dissertation), Goethe-Universität, Frankfurt am Main (2015) 151. Pavlov, A.V., Korol, A.V., Ivanov, V.K., Solov’yov, A.V.: Eur. Phys. J. D 74, 21 (2020) 152. Haurylavets, V.V., Leukovich, A., Sytov, A., Mazzolari, A., Bandiera, L., Korol, A.V., Sushko, G.B., Solovyov, A.V.: MBN Explorer atomistic simulations of electron propagation and radiation of 855 MeV electrons in oriented silicon bent crystal: theory versus experiment. Europ. Phys. J. Plus. 137, 34 (2022). arXiv preprint arXiv:2005.04138 153. Korol, A.V., Sushko, G.B., Solov’yov, A.V.: Eur. Phys. J. D 75, 107 (2021) 154. Molière, G.: Z. f. Naturforsch. A 2, 133–145 (1947) 155. Pacios, L.F.: J. Comp. Chem. 14, 410–421 (1993) 156. Sushko, G.B., Solov’yov, I.A., Solov’yov, A.V.: Europ. Phys. J. D 70, 217 (2016) 157. Rossi, B., Greisen, K.: Rev. Mod. Phys. 13, 241 (1941). Prentice-Hall, Inc., New York 158. Beloshitsky, V.V., Kumakhov, M.A., Muralev, V.A.: Radiat. Eff. 20, 95 (1973) 159. Backe, H., Kunz, P., Lauth, W., Rueda, A.: Nucl. Instrum. Method B 266, 3835–3851 (2008) 160. Bogdanov, O.V., Dabagov, S.N.: J. Phys.: Conf. Ser. 357, 012029 (2012) 161. Backe, H., Lauth, W.: Nucl. Instrum. Meth. B 355, 24–29 (2015) 162. Jackson, J.D.: Classical Electrodynamics. Wiley, Hoboken (1999) 163. Tsai, Y.-S.: Rev. Mod. Phys. 46, 815 (1974) 164. Bak, J., Ellison, J.A., Marsh, B., Meyer, F.E., Pedersen, O., et al.: Nucl. Phys. B. 254, 491–527 (1985) 165. Tikhomirov, V.V.: A benchmark construction of positron crystal undulator. arXiv preprint arXiv:1502.06588 (2015) 166. Chouffani, K., Überall, H.: Phys. Status Sol. (b) 213, 107–151 (1999) 167. Uggerhøj, E.: Rad. Eff. Def. Solids 25, 3–21 (1993) 168. Schmüser, P., Dohlus, M., Rossbach, J.: Ultraviolet and Soft X-Ray Free-Electron Lasers. Springer, Berlin/Heidelberg (2008) 169. Rullhusen, P., Artru, X., Dhez, P.: Novel Radiation Sources Using Relativistic Electrons. World Scientific, Singapore (1998) 170. Altarelli, M., Salam, A.: Europhysicsnews 35, 47–50 (2004) 171. Kim, K.-J.: Characteristics of synchrotron radiation. In: X-ray Data Booklet, pp. 2.1–2.16. Lawrence Berkeley Laboratory, Berkley (2009). http://xdb.lbl.gov/xdb-new.pdf 172. Kim, K.-J.: Nucl. Instrum. Meth. A 246, 71–76 (1986)

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173. Ginzburg, V.L.: Theoretical Physics and Astrophysics. International Series in Natural Philosophy, vol. 99. Pergamon Press, Oxford (1979) 174. Schott, G.A.: Electromagnetic Radiation. Cambridge University Press, Cambridge (1912) 175. Ivanenko, D.D., Pomeranchuk, I.Ya.: On the maximum energy achievable in a betatron. Doklady Acad. Nauk 44, 343 (1944) (in Russian) 176. Schwinger, J.: Phys. Rev. 75, 1912 (1949) 177. Elder, F.R., Gurewitsch, A.M., Langmuir, R.V., Pollock, H.C.: Phys. Rev. 71, 829 (1947) 178. Tavares, P.F., Leemann, S.C., Sjöström, M., Andersson, Å.J.: Synchrotron Rad. 21, 862 (2014) 179. Ginzburg, V.L.: Radiation of microwaves and their absorption in air. Bull. Acad. Sci. USSR, Ser. Phys. 11, 165 (1947) (in Russian) 180. Motz, H.: J. Appl. Phys. 22, 527–534 (1951) 181. Motz, H., Thon, W., Whitehurst, R.N.: J. Appl. Phys. 24, 826–833 (1953) 182. Madey, J.M.J.: J. Appl. Phys. 42, 1906–1913 (1971) 183. Deacon, D.A.G., Elias, L.R., Madey, J.M.J., Ramian, G.J., Schwettman, H.A., Smith, T.I.: Phys. Rev. Lett. 38, 892 (1977) 184. Kondratenko, A.M., Saldin, E.L.: Part. Accel. 10, 207 (1980) 185. Bonifacio, R., Pellegrini, C., Narducci, L.M.: Opt. Commun. 50, 373–378 (1984) 186. Kim, K.J.: Phys. Rev. Lett. 57, 1871 (1986) 187. Bonifacio, R., Casagrande, F., Cerchioni, G., de Salvo Souza, L., Pierini, P., Piovella, N.: Rivista del Nuovo Cimento 13, 1–69 (1990) 188. Luchini, P., Motz, H.: Undulators and Free-Electron Lasers. Oxford University Press, New York (1990) 189. Saldin, E.L., Schneidmiller, E.A., Yurkov, M.V.: The Physics of Free-Electron Lasers. Springer, Berlin/Heidelberg (1999) 190. Huang, Zh., Kim. K-J.: Phys. Rev. ST Accel. Beams 10, 034801 (2007) 191. Pellegrini, C., Marinelli, A., Reiche, S.: Rev. Mod. Phys. 88, 015006 (2016) 192. Gover, A., Friedman, A., Emma, C., Sudar, N., Musumeci, P., Pellegrini, C.: Rev. Mod. Phys. 91, 035003 (2019) 193. Federici, L., Giordano, G., Matone, G., Pasquariello, G., Picozza, P., et al.: Lett. Nuovo Cimento 27, 339 (1980) 194. Federici, L., Giordano, G., Matone, G., Pasquariello, G., Picozza, P.G., et al.: Nuovo Cimento 59B, 247 (1980) 195. ur Rehman, H., Lee, J., Kim, Y.: Ann. Nucl. Energy 105, 150 (2017) 196. Krämer, J.M., Jochmann, A., Budde, M., Bussmann, M., Couperus, J.P., et al.: Sci. Reports 8, 139 (2018) 197. ur Rehman, H., Lee, J., Kim, Y.: Int. J. Energy Res. 42, 236–244 (2018) 198. Kulikov, O.F., Telnov, Y.Y., Filippov, E.I., Yakimenko, M.N.: Phys. Lett. 13, 344 (1964) 199. Bemporad, C., Milburn, R.H., Tanaka, N., Fotino, M.: Phys. Rev. 138, B1546 (1965) 200. Ballam, J., Chadwick, G.B., Gearhart, R., Guiragossian, Z.G.T., Klein, P.R., et al.: Phys. Rev. Lett. 23, 498 (1969) (Erratum: Phys. Rev. Lett. 23, 817 (1969)) 201. D’Angelo, A., Bartalini, O., Bellini, V., Levi Sandri, P., Moricciani, D., Nicoletti, L., Zucchiatti, A.: Nucl. Instrum. Meth. A 455, 1 (2000) 202. Schaerf, C.: Phys. Today 58, 44 (2005) 203. Weller, H.R., Ahmed, M.W., Gao, H., Tornow, W., Wu, Y.K., Gai, M., Miskimen, R.: Prog. Part. Nucl. Phys. 62, 257 (2009) 204. Krafft, G.A., Priebe, G.: Rev. Accelerator Scie. Technol. 3, 147 (2010) 205. Sei, N., Ogawa, H., Jia, Q.K.: Appl. Sci. 10, 1418 (2020) 206. Howell, C.R., Ahmed, M.W., Afanasev, A., Alesini, D., Annand, J.R.M., et al.: International Workshop on Next Generation Gamma-Ray Source. arXiv preprint arXiv:2012.10843 (2020) 207. Wu, Y.K., Vinokurov, N.A., Mikhailov, S., Li, J., Popov, V.: Phys. Rev. Lett. 96, 224801 (2006) 208. Krasny, M.W.: The Gamma Factory proposal for CERN. Photon-2017 Conference, May 22–29, 2017 8CERN, Geneva. https://indico.cern.ch/event/604619/contributions/2474166/ attachments/1463495/2261413/Witold_Krasny_Photon_2017.pdf

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209. Schaumann, M., Alemany-Fernández, R., Bartosik, H., Bohl, Th., Bruce, R. et al: First partially stripped ions in the LHC (208 Pb81+ ). In: Proceedings, 10th International Particle Accelerator Conference (IPAC2019), page MOPRB055 (Melbourne, Australia, May 19–24, 2019) 210. Tajima, T., Dawson, J.: Phys. Rev. Lett. 43, 267 (1979) 211. Pukhov, A., Meyer-ter-Vehn, J.: Appl. Phys. B 74, 355 (2002) 212. Esarey, E., Schroeder, C.B., Leemans, W.P.: Rev. Mod. Phys. 81, 1229 (2009) 213. Zhu, X.-L., Chen, M., Weng, S.-M., Yu, T.-P., Wang, W.-M., He, F., et al.: Sci. Adv. 6, eaaz7240 (2020) 214. Kumakhov, M.A., Komarov, F.F.: Radiation from Charged Particles in Solids AIP. New York (1989) 215. Barbini, R., Ciocci, F., Datolli, G., Gianessi, L.: Rivista del Nuovo Cimento 13, 1–65 (1990) 216. Alferov, D.F., Bashmakov, Yu.A., Cherenkov, P.A.: Sov. Phys. - Uspekhi 32, 200–227 (1989) 217. Schneider-Muntau, H.J., Toth, J., Weijers, H.W.: IEEE Trans. Appl. Supercond. 14, 1245– 1252 (2004) 218. The European X-ray Laser Project XFEL. http://www.xfel.eu/ 219. Backe, H., Lauth, W., Kunz, P., Rueda, A., Esberg, J., Kirsebom, K., Hansen, J.L., Uggerhøj, U.K.I.: Photon Emission of Electrons in a Crystalline Undulator. In: Dabagov, S.B., Palumbo, L., Zichichi, A. (eds.) Proceedings of the 51st Workshop Charged and Neutral Particles Channeling Phenomena Channeling 2008, Erice, Italy, Oct. 2008, pp. 281–290. World Scientific, Singapore/Hackensack (2010) 220. Mao, F., Sushko, G.B., Korol, A.V., Solov’yov, A.V., Cheng, W., Sang, H., Zhang, F.-S.: Radiation by ultra-relativistic positrons and electrons channeling in periodically bent diamond crystals. Unpublished (2015) 221. https://www6.slac.stanford.edu/facilities/facet.aspx 222. Backe, H., Lauth, W.: Channeling experiments with electrons at the mainz microtron. In: 4th International Conference on “Dynamics of Systems on the Nanoscale” (Bad Ems, Germany, Oct. 3–7 2016) Book of Abstracts, p. 58 (2016) 223. Taratin, A.M., Vorobiev, S.A.: Phys. Lett. 119, 425 (1987) 224. Taratin, A.M., Vorobiev, S.A.: Nucl. Instrum. Meth. B 26, 512 (1987) 225. Shul’ga, N.F., Boyko, V.V., Esaulov, A.S.: Phys. Lett. A 372, 2065–2068 (2008) 226. Nielsen, C.F., Uggerhøj, U.I., Holtzapple, R., Markiewicz, T.W., Benson, B.C., Bagli, E., Bandiera, L., Guidi, V., Mazzolari, A., Wienands, U.: Phys. Rev. Acc. Beams 22, 114701 (2019) 227. Backe, H., Lauth, W., Tran Thi, T.N.: J. Instrum. (JINST) 13, C04022 (2018) 228. Boshoff, D., Copeland, M., Haffejee, F., Kilbourn, Q., Mercer, C., Osatov, A., Williamson, C., et al.: The search for diamond crystal undulator radiation. In: 4th International Conference on “Dynamics of Systems on the Nanoscale” (Bad Ems, Germany, Oct. 3–7 2016) Book of Abstracts, p. 38 (2016) 229. Pavlov, A., Korol, A., Ivanov, V., Solov’yov, A.: St. Petersburg Polytechnical Uni. J.: Phys. Math. 14, 190 (2021). (arXiv.org: arXiv:2004.07043) 230. Backe, H., Krambrich, D., Lauth, W., Andersen, K.K., Hansen, J.L., Uggerhøj, U.I.: J. Phys. Conf. Ser. 438, 012017 (2013) 231. The DANE Beam-Test Facility. http://www.lnf.infn.it/acceleratori/btf/ 232. Medvedev, M.V.: Astrophys. J. 540, 704–714 (2000) 233. Kellner, S.R., Aharonian, F.A., Khangulyan, D.: Astrophys. J. 774, 61 (2013) 234. Banks, M.: Italy cancels e1bn SuperB collider. Physics World (2012). https://physicsworld. com/a/italy-cancels-1bn-superb-collider/ 235. Pavlov, A.V., Korol, A.V., Ivanov, V.K., Solov’yov, A.V.: Unpublished (2020) 236. Kostyuk, A., Korol, A.V., Solov’yov, A.V., Greiner, W.J.: Phys. B At. Mol. Opt. Phys. 43, 151001 (2010) 237. Abramowitz, M., Stegun, I.E.: Handbook of Mathematical Functions. Dover, New York (1964) 238. Kostyuk, A., Korol, A.V., Solov’yov, A.V., Greiner, W.: Nucl. Instrum. Method B 269, 1482– 1492 (2011) 239. Ledingham, K.W.D., McKenna, P., Singhal, R.P.: Science 300, 1107 (2003)

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Chapter 11

Conclusions and Outlook Andrey V. Solov’yov

Abstract This chapter concludes the book and provides a brief outlook for the further research and technological challenges in the field of Dynamics of Systems on the Nanoscale (DySoN) and related multiscale computational modelling. The tasks of further development of the multiscale computational methods are closely related to further development of MBN Explorer and MBN Studio. These two aspects are closely bundled and thus they are addressed simultaneously in this chapter. The chapter discusses the improvement of existing and implementation of new algorithms and computational methods, extension of ranges of multiscale methodologies, development of new modules for specific application areas, utilization of GPU, development of databases on the computational outputs within the DySoN research area, availability of the software for cloud computing, computational exploration of new areas of research and technology. Finally, the guidelines are provided on how to acquire the MBN Explorer and MBN Studio software, as well as how to contact their development team.

11.1 Dynamics of Systems on the Nanoscale: Further Horizons The further horizons in revealing the nature of multiscale dynamics of complex molecular systems by means of MBN Explorer and MBN Studio will concern major breakthroughs in the quantitative understanding of challenging interdisciplinary problems at the interface of physics, chemistry, biology, and material science through the implementation and exploitation of the computational multiscale modelling methodologies and high-performance computing. These goals could be realized through the methodologies presented in this book, their further development, broadening their application areas and exploring new case studies. The knowledge gained through these studies will be exploited in the important applications. This will contribute to the general modern trend of emerging the numerical modelling as A. V. Solov’yov (B) MBN Research Center gGmbH, Altenhöferallee 3, 60438 Frankfurt am Main, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 I. A. Solov’yov et al. (eds.), Dynamics of Systems on the Nanoscale, Lecture Notes in Nanoscale Science and Technology 34, https://doi.org/10.1007/978-3-030-99291-0_11

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the dominant field of the twenty-first Century Science and technology as we model more and more the natural world in traditionally intractable cases and design new industrial processes (‘The Virtual Factory’) [1]. MBN Explorer and MBN Studio form a unique computational suite for unravelling structure and dynamics of complex MBN systems of very different nature. It provides a wide range of possibilities for the computational studies of various systems and their properties, making MBN Explorer and MBN Studio a powerful and universal instrument of computational research and technology capable to explore both structure and dynamics of the MBN-matter through its simulation, analysis and visualization. Numerous case studies performed by means of MBN Explorer and MBN Studio demonstrated the software effectiveness in simulations of the known features of MBN systems and in predicting the new ones. A few such exemplar case studies are presented in this book in sufficient detail. The software enables computational studies of the MBN systems on the different scales, thus supporting investigations of multiscale processes and multiscale modelling of complex MBN systems. In the course of further development the current capabilities of MBN Explorer and MBN Studio will be further extended enabling the virtual multiscale design/construction and visualization of any imaginable MBN system as well as simulation and detail analysis of the system structure, properties and dynamical processes therein. In the more distant future the limits of multiscale modelling could be further extended and embrace the physical systems on the scales (smaller and larger) beyond those related to the MBN Research area, as there is no principle limitation for such generalization. Thus, the presented MBN software suite could be a prototype of a bigger scope software, which could be named as Physica. In its most general realization, Physica will be capable to deal with all kind of physical systems and phenomena therein computationally, similarly to how it is realized for mathematical solutions in the well-known software package called Wolfram Mathematica. Finally, let us also emphasize that the future of a significant fraction of the European nanotechnology industry is associated with the creation of an integrated environment for numerical design and modelling. This encompasses a wide range of end-products and applications in nanoelectronics, nanomaterials and their adoption within transportation, avionics, polymer technologies, medicine, etc. In most of these areas simulations need to operate over a wide range of scales, ranging from the molecular and the nanoscale to the micro and sometimes even to macro-dimensions. Such multiscale modelling integrating different physical and chemical phenomena is currently one of the hot topics of theoretical and computational research as it is clear from the material presented in this book. The further development of multiscale modelling tools will be conducted in parallel with the development and widening of modern methods of high-performance computing in order to achieve the major breakthroughs and to guide the realization of related emerging or optimization of existing modern technologies. The implementation and success of the versatile numerical design and modelling will require a close and wide cooperation with industrial players. Multiscale modelling may save crucial time and money in product development processes, and hence play a key role in industrial competitiveness. Functioning of MBN systems

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and devices often involves different scales too. Indeed, batteries and other energyrelated devices, such as solar cells, fuel cells, and super-capacitors, that are seen as representative mesoscale challenges, reply on a battery’s ability to store electricity due to the nanoscale structure of individual components such as the anode, cathode, and electrolyte, but the device’s real-world performance depends on how all the components work together at the mesoscale [2]. In the case of the ion-beam cancer therapy presented in Chap. 9 ions propagate in the biological targets macroscopically large distances, although the irradiation induced transformations in the irradiated biological systems and living organisms leading to the therapeutic effects happen in the nanoscopic volumes in the vicinity of the ion tracks. There are many more topics that are relevant to the DySoN research area. An overview of all of them goes beyond the scope of this book. However, concluding the book let us outline the major directions for the further development of MBN Explorer and MBN Studio as the main instrument of the computational research and multiscale modelling in the DySoN research area.

11.2 Further Development of Multiscale Computational Modelling with MBN E XPLORER and MBN S TUDIO There are many possible extensions and generalizations that could be implemented in the software [3, 4]. This development is, to large extent, driven by the actual needs of its users and by the novelty of its potential applications. The most obvious and general directions along which the software will be further developed are outlined below. These development directions include • Further optimization of implemented algorithms. The algorithms already implemented in MBN Explorer and MBN Studio will be further optimized, compared with their or similar realizations in other codes, and benchmarked. This will maintain the objective evaluation of the software effectiveness. This will concern, in particular, the standard and most often exploited algorithms, like integrators, thermostats, fast Fourier transform, linked cell, etc. • New algorithms and methods. In the process of software optimization and further development, many new algorithms will be implemented. The list of the already identified possible future implementations is rather long and it grows continuously. Here, its discussion is limited to mentioning only a few obvious lines for further advances. The capacities of MBN Studio in constructing and visualizing various MBN systems on the basis of the already available information about atomic, molecular, and crystalline systems and the interatomic interactions therein will be advanced further. This will be achieved through the integration of additional functionalities into the systems modeller tookit and interfacing MBN Studio with various databases collecting information relevant for simulations. The systems constructed by means of the toolkit can be used for initializing simulations of structure and dynamics of

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numerous MBN systems by means of MBN Explorer. The current capacities of the toolkit will be advanced up to the level permitting a simple construction of any imaginable MBN system from the predefined molecular elements and subsystems utilizing a sufficiently rich set of the construction and manipulation rules, like in the famous Lego constructor. There will be further advances in MBN Studio towards possible automatization of multistep simulation procedures, sampling of simulation statistics and their processing, as well as various types of analysis of the simulated data. MBN Explorer has a large library of force fields which will be enriched further. With this, new algorithms will be implemented as well, for instance, the algorithms allowing the force field coarse graining for different simulated systems. Such algorithms diminish the level of details for the simulated systems, but increase their size and the time scale of the simulated processes. A similar strategy will be taken with regard to the external fields in which structure and dynamics of MBN systems could be simulated, analysed, and visualized. Advances in this direction will allow to increase the number of various physical and chemical systems and processes therein that could be simulated by means of MBN Explorer and MBN Studio. • Extension of ranges of multiscale methodologies. The algorithms implemented in MBN Explorer operate at different space-and-time scales. With their interlinks one can build up multiscale models for the description of various MBN systems and processes therein as it was demonstrated in several chapters of this book for a number of case studies. A lot more could be achieved in this research area through the interlinks of different theoretical and computational frameworks, such as Quantum Mechanics, Particle Transport Models, Molecular Dynamics, Coarse Graining, Stochastic Dynamics, Finite Element Method, Continuous Medium Models, and others. The mentioned well-established theoretical and computational frameworks can be applied to the description of the MBN systems at different temporal and spatial scales as illustrated in Fig. 11.1. In the figure, lines indicate schematically the limits of the current version of MBN Explorer and arrows show the directions for further development. Here we do not discuss all possible links between different methodologies and the application areas in which such combined multiscale methods can be applied. Examples of such possibilities have been given in several chapters and it is obvious that many more challenging interdisciplinary and multiscale problems could be explored by the aforementioned theoretical and computational means. The further advances in the multiscale modelling will be realized through interfacing MBN Explorer with various existing codes dealing with DFT, TDDFT, quantum MD, MC, FEM, particle transport, etc. These developments will be supplemented by implementation of the corresponding utilities in MBN Studio. • Development of the software modular structure for specific applications. The further technical development of MBN Explorer, MBN Studio and their application in various technological areas (e.g. bio-, nano, material, plasma technologies) will target the creation of new or further elaboration of the existing modules dedicated to specific application areas of the software. This will further simplify setting

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Fig. 11.1 Temporal and spatial system scales and the corresponding simulation methods

up the specific simulation tasks, increase their variety, speed up the numerical analysis of simulation results and their exploitation. An example of such advances may concern the further development of the multiscale approach for the description of radiation damage effects and its application to ion-beam cancer therapy. This approach introduced briefly in Chap. 9 allows quantitative understanding of the radiotherapy medical treatments on the molecular level. The recent advances in this direction reported in [5–9] form an excellent basis for the construction of a special module integrated with MBN Explorer and MBN Studio supporting such simulations. This module can be utilized for advanced studies of molecular processes behind the ion-beam cancer therapy and for the optimization of the existing treatment planning protocols. Similar modules have already been implemented in MBN Explorer and MBN Studio for other application areas, e.g. for the relativistic MD simulations of particle propagation in crystalline media, channelling processes, radiation emission and other phenomena relevant for the virtual design and practical realization of the novel crystal-based light sources that are briefly discussed in Chap. 10, for further details see [10–12]. The new modules may concern applications of MBN Explorer and MBN Studio in computational modelling of biomolecular systems, nanostructured materials, collision and reaction processes, surface deposition processes, irradiation driven processes, and others. • Computational methods for graphics processing units (GPUs). This book does not discuss the realization of MBN Explorer algorithms with GPUs, in spite of the fact that some of the key algorithms have already been adopted for the OpenCL and CUDA languages suitable for GPUs and tested for simulations of various MD processes, like solid-liquid phase transitions in metal clusters [13].

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Such MD simulations of large MBN systems typically require a lot of CPU time. The computational methods based on GPUs help to advance certain type of MD simulations immensely. In Ref. [13] the advantages of GPUs in comparison with CPUs were demonstrated and benchmarked for MD simulations of the kinetics of melting of Ni clusters. It was reported that the speedup in the MD simulations reaches factors greater than 400. The GPU technology opens new paths towards exploration of a larger number of scientific problems inaccessible earlier with the CPU-based computational technology by means of MD and other suitable algorithms implemented in MBN Explorer. Therefore, the work on the further utilization of GPU technology with MBN Explorer and MBN Studio will be continued. • MBN Explorer Projects Database. The MBN Research Center1 has created and maintains a database to which the computational projects realized by means of MBN Explorer and MBN Studio can be deposited. It contains the exemplar, testing and benchmark projects for all the implemented algorithms, as well as the case study projects that were fulfilled for particular application areas. This database is open for the holders of the MBN Explorer and MBN Studio licences. The licence holders have also a possibility to deposit to the database their newly developed projects for sharing them with the broader MBN Explorer and MBN Studio community. The users of MBN Explorer and MBN Studio have also a possibility to deposit their generated data on the thematic databases in the areas of MBN Research that are being developed and maintained by the MBN Research Center with the aim of data widespread dissemination, their further use, analysis and exploitation. An example of such thematic database is the RADAM portal2 for interfacing a network of RADAM (RAdiation DAMage) Databases collecting data on interactions of ions, electrons, positrons, and photons with biomolecular systems, on radiobiological effects and related phenomena occurring at different time, spatial and energy scales in irradiated targets during and after the irradiation. This database was created by the members of the MBN Research Center in the course of COST Action MP1002 (Nano-IBCT: Nanoscale insights into Ion Beam Cancer Therapy) during 2011–2014 according to the Virtual Atomic and Molecular Data Center (VAMDC) standards. The collected data aim to improve our understanding of radiation damage mechanisms of biological targets on the molecular level. Some of the nodes of the RADAM Portal are integrated into the VAMDC (see www.vamdc.eu and Ref. [14]). The RADAM portal is maintained by the team of the MBN Research Center. • MBN Explorer and MBN Studio on-line. In the future MBN Explorer and Studio will be available for cloud computing. This will provide on-line services for the most popular types of simulations, such as virtual design of materials, surface

1 2

http://www.mbnresearch.com. http://radamdb.mbnresearch.com/.

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coatings, simulation of their thermo-mechanical properties, surface deposition, irradiation driven processes and many more. • Exploration of the new research areas and technologies. The current release of MBN Explorer and MBN Studio covers an enormous area of interdisciplinary research as it is obvious from the content of this book. However, there are still many open areas for interdisciplinary research in the field of MBN Science that requires further exploration. Therefore, this whole field will be developed further both scientifically and computationally, and MBN Explorer and MBN Studio will play an important role in this process. The realization of the plans outlined above will increase the number of application areas, case studies, the universality, efficiency, and accessibility of the software beyond its current limits. Some of the above-mentioned multiscale methodologies have already been successfully implemented in the latest release of MBN Explorer and MBN Studio. This process will be continued in the future. The complete realization of this programme means a long term development aiming at a large number of customers and wide exploitation of this universal and powerful software in numerous areas of its application. If you are interested to monitor this exciting development, explore its advantages in research and technology, or even participate in its further development, you should visit the website of MBN Research Center. MBN Explorer and MBN Studio are being continuously developed by the MBN Research Center in Frankfurt. The software can be acquired from MBN Research Center via its website3 following the guidelines.4,5 MBN Explorer exists in different versions. The standard version is limited to 4 cores and suits for workstations and notebooks. The high-performance computing version can exploit all cores of several supercomputer nodes. Each of these versions is available for any existing operating system (Windows, Linux, MacOS). One can acquire different types of licences for MBN Explorer and MBN Studio depending on duration of exploitation, number of users, commercial or noncommercial exploitation. The MBN Research Center website referenced above contains a lot of useful, relevant and up-to-date information about the software including all the references on the published articles, reviews and books, case studies, tutorials, related conferences, etc. Any enquiries about MBN Explorer and MBN Studio, types of licences, prices as well as the possibilities of cooperation with the team of MBN Research Center should be forwarded to the e-mail address: [email protected].

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http://www.mbnresearch.com/. http://www.mbnresearch.com/get-mbn-explorer-software. 5 http://www.mbnresearch.com/mbn-studio. 4

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References 1. Solov’yov, I.A., Korol, A.V., Solov’yov, A.V.: Multiscale Modeling of Complex Molecular Structure and Dynamics with MBN Explorer. Springer International Publishing, Cham (2017) 2. Berrueta, A., Urtasun, A., Ursúa, A., Sanchis, P.: A comprehensive model for lithium-ion batteries: from the physical principles to an electrical model. Energy 286, 144 (2018) 3. Solov’yov, I.A., Yakubovich, A.V., Nikolaev, P.V., Volkovets, I., Solov’yov, A.V.: MesoBioNano explorer - a universal program for multiscale computer simulations of complex molecular structure and dynamics. J. Comput. Chem. 33, 2412–2439 (2012) 4. Sushko, G., Solov’yov, I., Solov’yov, A.: Modeling MesoBioNano systems with MBN studio made easy. J. Mol. Graph. Model. 88, 247–260 (2019) 5. Solov’yov, A.V. (ed.): Nanoscale Insights into Ion-Beam Cancer Therapy. Springer International Publishing, Cham (2017) 6. Surdutovich, E., Solov’yov, A.V.: Multiscale approach to the physics of radiation damage with ions. Eur. Phys. J. D 68, 353 (2014) 7. Surdutovich, E., Solov’yov, A.: Multiscale modeling for cancer radiotherapies. Cancer Nanotechnol. 10, 6 (2019) 8. de Vera, P., Surdutovich, E., Solov’yov, A.V.: The role of shock waves on the biodamage induced by ion beam radiation. Cancer Nanotechnol. 10, 5 (2019) 9. Verkhovtsev, A., Surdutovich, E., Solov’yov, A.V.: Phenomenon-based evaluation of relative biological effectiveness of ion beams by means of the multiscale approach. Cancer Nanotechnol. 10, 4 (2019) 10. Sushko, G.B., Bezchastnov, V.G., Solov’yov, I.A., Korol, A.V., Greiner, W., Solov’yov, A.V.: Simulation of ultra-relativistic electrons and positrons channeling in crystals with MBN explorer. J. Comput. Phys. 252, 404–418 (2013) 11. Korol, A.V., Solov’yov, A.V., Greiner, W.: Channeling and Radiation in Periodically Bent Crystals, 2nd edn. Springer, Berlin (2014) 12. Korol, A.V., Solov’yov, A.V.: Crystal-based intensive gamma-ray light sources. Eur. Phys. J. D 74, 201 (2020) 13. Yakubovich, A.V., Sushko, G.B., Schramm, S., Solov’yov, A.V.: Kinetics of liquid-solid phase transition in large nickel clusters. Phys. Rev. B 88, 035438 (2013) 14. Dubernet, M., Antony, B., Ba, Y., Babikov, Y., Bartschat, K., et al.: The virtual atomic and molecular data centre (VAMDC) consortium. J. Phys. B: At. Mol. Opt. Phys. 49, 074003 (2016)