Hyperordered Structures in Materials: Disorder in Order and Order within Disorder (The Materials Research Society Series) 9819952344, 9789819952342

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The Materials Research Society Series

Koichi Hayashi   Editor

Hyperordered Structures in Materials Disorder in Order and Order within Disorder

The Materials Research Society Series

The Materials Research Society Series covers the multidisciplinary field of materials research and technology, publishing across chemistry, physics, biology, and engineering. The Series focuses on premium textbooks, professional books, monographs, references, and other works that serve the broad materials science and engineering community worldwide. Connecting the principles of structure, properties, processing, and performance and employing tools of characterization, computation, and fabrication the Series addresses established, novel, and emerging topics.

Koichi Hayashi Editor

Hyperordered Structures in Materials Disorder in Order and Order within Disorder

Editor Koichi Hayashi Department of Physical Science and Engineering Nagoya Institute of Technology Nagoya, Aichi, Japan

ISSN 2730-7360 ISSN 2730-7379 (electronic) The Materials Research Society Series ISBN 978-981-99-5234-2 ISBN 978-981-99-5235-9 (eBook) https://doi.org/10.1007/978-981-99-5235-9 © Materials Research Society, under exclusive license to Springer Nature Singapore Pte Ltd. 2024 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 Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Solid matter can roughly be categorized into crystals, which have regular atomic arrangements over long distances and amorphous, which have random atomic arrangements. These materials possess various functional properties which we believe are governed by deviations from the perfect regularity or randomness of their atomic arrangements. Therefore, we decided to focus on these deviations, discuss its correlation with material functions, and summarize it in this technical book. From the crystal side, the deviations from perfect crystallinity are represented by dopants. Dopants are aperiodic structures within regular matrix structures, and in semiconductors, play an important role in creating carriers. It can be said that most materials contain dopants. Although dopants are considered point defects, this book is written with more emphasis on defect complexes, which will play a leading role in materials design in the future. We call this structural feature “Disorder in Order.” On the other hand, there are no amorphous materials that have perfect randomness. A topological analysis of the atomic arrangement in glass has revealed that crystal-like topology can be extracted from the glass network structure, and that this topology correlates with the functional properties of the glass. We call this structural feature “Order within Disorder.” These imperfections, each from perfect regularity and perfect randomness, are integrated and called “Hyperordered Structure” in this book. These ideas and the structural features of “Hyperordered Structure” are described in detail in Part I, Hyperordered Structures: Disorder in Order/Order within Disorder. Hyperordered structures exist in a wide range of material groups, such as dielectrics, functional glasses, zeolites, superconductors, and biological materials, and can be regarded as a treasure trove of material functionality. This book is organized from analytical, theoretical, and material perspectives to provide a comprehensive understanding of hyperordered structures. In Part II, Characterization of Hyperordered Structures, methods that can reveal hyperordered structures are introduced. In particular, five chapters introduce methods using quantum beams. Part III, Computational Approaches to Hyperordered Structures, introduces methods for unveiling electronic states and structural features to explore the function of hyperordered structures. Part IV, Physicochemical Properties of Hyperordered Materials, v

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introduces hyperordered structures hidden in actual materials with examples of a wide range from protein and inorganic matter. We hope that this book, which discusses materials science from a new perspective, will intrigue readers and bring new ideas to researchers. Nagoya, Japan

Koichi Hayashi

Contents

Part I

Hyperordered Structures: Disorder in Order/Order within Disorder

1

From Point Defects to Defect Complexes . . . . . . . . . . . . . . . . . . . . . . . . Koichi Hayashi

2

Topological Order and Hyperorder in Oxide Glasses and Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shinji Kohara

Part II

3

17

Characterization of Hyperordered Structures

3

Atomic-Resolution Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tomohiro Matsushita, Koji Kimura, and Kenji Ohoyama

33

4

X-Ray and Neutron Pair Distribution Function Analysis . . . . . . . . . . Yohei Onodera, Tomoko Sato, and Shinji Kohara

93

5

Angstrom-Beam Electron Diffraction Technique for Amorphous Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Akihiko Hirata

6

Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Ayano Chiba and Shinya Hosokawa

7

Property Measurements of Molten Oxides at High Temperature Using Containerless Methods . . . . . . . . . . . . . . . . . . . . . . 159 Takehiko Ishikawa, Paul-François Paradis, and Atsunobu Masuno

Part III Computational Approaches to Hyperordered Structures 8

Density Functional Theory Calculations for Materials with Complex Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Ayako Nakata and Yoshitada Morikawa

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Contents

Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Yu Takano, Takahiro Ohkubo, and Satoshi Watanabe

10 Reverse Monte Carlo Modeling of Non-crystalline and Crystalline Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Naoto Kitamura and Shinji Kohara 11 Structural-Order Analysis Based on Applied Mathematics . . . . . . . . 265 Motoki Shiga and Ippei Obayashi 12 Machine Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Motoki Shiga and Satoshi Watanabe Part IV Physicochemical Properties of Hyperordered Materials 13 Dielectric Materials with Hyperordered Structures . . . . . . . . . . . . . . . 313 Hiroki Taniguchi 14 Hyperordered Structures in Microporous Frameworks in Zeolites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Masanori Takemoto, Kenta Iyoki, and Toru Wakihara 15 Glasses with Hyperordered Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 353 Atsunobu Masuno and Madoka Ono 16 Biological Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 Hideaki Tanaka 17 Battery and Fuel Cell Materials with Hyperordered Structures . . . . 395 Naoto Kitamura 18 Superconductors with Hyperordered Structures . . . . . . . . . . . . . . . . . 411 Yoshihiro Kubozono and Jun Akimitsu 19 Ordered and Disordered Metal Oxide for Biomass Conversion . . . . 433 Daniele Padovan and Kiyotaka Nakajima Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453

Abbreviations

2D 3DG AAL ABED ADL AFM AIMD ALD AMBER APD AT ATAT ATP BCC BCZT BLC BO BTE BVS C4mim CC CCTO CE CEF CHA CHARMM Chl CI CN CORs CV

Two-dimensional 3-Deoxy-glucosone Aero-Acoustic Levitation Angstrom-beam electron diffraciton Aerodynamic levitation Atomic force microscopy Ab initio molecular dynamics Anharmonic lattice dynamics Assisted model building with energy refinment Avalanche photodiode PaseATP synthase Alloy Theoretic Automated Toolkit Adenosine triphosphate Body-centered cubic (Ba,Ca)(Zr,Ti)O3 Barrier layer capacitor Bridging oxygen Boltzmann transport equation Bond valence sum 1-butyl-3-methylimidazolium Coupled-cluster CaCu3 Ti4 O12 Cluster expansion Cyclic electron flow Concentric hemispherical analyzer Chemistry at Harvard macromolecular mechanics Chlorophyll Configuration interaction Coordination number Chemically ordered regions Cross-validation ix

x

CVD Cyt b6f DAC DF DFT DI DMF DnC DOS DRIFT DRP DTA DW EDX/EDS EML EPDDs ESL EXAFS FAD FCC Fd FDCA FFP FFT FNR FSDP FTIR FWHM GAP GFL GGA GIC GL GP GROMACS GVL hcp HDNNP HF HIP HMF HOMO HREM HR-TEM HSMC

Abbreviations

Chemical vapor deposition Cytochrome b6f Diamond anvil cell Density Functional Density functional theory Distortion index Dimethylfuran Drop and Catch Density of states Diffuse reflectance infrared Fourier transform Dense random packing Differential Thermal Analysis Drop weight Energy dispersive X-ray spectroscopy Electromagnetic levitation Electron-pinned defect-dipoles Electrostatic levitation Extended X-ray absorption fine structure Flavin adenine dinucleotide Face-centered cubic Ferredoxin Furan-2,5-dicarboxylic acid Forward focusing peak Fast Fourier Transform Ferredoxin–NADP+ reductase First sharp diffraction peak Fourier Transform Infrared Full Width at Half Maximum Gaussian approximation potential Gas Film Levitation Generalized gradient approximation Graphite intercalation compound Ginzburg Landau Gaussian process Groningen machine for chemical simulation γ-valerolactone Hexagonal close-packed High-dimensional neural network potential Hartree–Fock Hot isostatic pressure 5-hydroxymethylfurfural Highest occupied molecular orbital High-resolution electron microscopy High-resolution transmittance electron microscopy Hard sphere Monte Carlo

Abbreviations

IMFP INS IPEH IR ISS-ELF IUVS IXS IZA-SC IZO JAXAJ JG KMC KS LASSO LC-ωPBE LDA LET LHCI LHCII LIB LPSO LUMO LVP MAS MBP MC MCPB MD MO MP MRB MRO MTK++ MyPresto NADP+, NADPH NAMD NBO ND NH NH3-SCR NITO-x% NMR NN NPG

xi

Inelastic mean free path Inelastic neutron scattering Inverse photoelectron holography Infrared absorption spectroscopy Electrostatic levitation furnace onboard the ISS Inelastic ultra violet scattering Inelastic X-ray scattering The Structure Commission of the International Zeolite Association In2 O3 -ZnO Japan Aerospace Exploration Agency Johari-Goldstein Kinetic Monte Carlo Kohn-Sham Least absolute shrinkage and selection operator Long-range corrected omega-Perdew–Burke–Ernzerhof Local density approximation Linear electron transfer Light-harvesting complex I Light-harvesting complex II Lithium ion battery Long-period stacking ordered Lowest unoccupied molecular orbital Large volume press Magic-angle spinning Maximum bubble pressure Monte Carlo Metal center parameter builder Molecular dynamics Molecular orbital Møller-Plesset Magnesium rechargeable battery Medium-range order Modeling toolkit plus plus Medically yielding protein engineering simulator Nicotinamide adenine dinucleotide phosphate Nanoscale molecular dynamics Non-bridging oxygen Neutron diffraction Neutron holography Selective catalytic reduction of NOx with NH3 x% Nb+In co-doped TiO2 Nuclear magnetic resonance Nearest neighbor Neopentylglycol

xii

NRIXS NRU NSE NV O(N) oct OEC OSDAs PALS PC PCA PCS PD PD PDB PDF PEF PEH PET PH PID PLA PMN PNRs PP PQ PQH2 PSD PSI PSII PVSVC PXRD PZT QENS RAA RESP RFA RMC RMSD RMSE RMSF SADs SAED SD SDD

Abbreviations

Nuclear resonant inelastic X-ray scattering National Research Universal Neutron spin echo Nitrogen-vacancy Order-N Octahedral Oxygen-evolving complexes Organic structure-directing agents Positron annihilation lifetime spectroscopy Plastocyanin Principal component analysis Photon correlation spectroscopy Pendant drop Persistence diagram Protein Data Bank Pair distribution function Polyethylene furanoate Photoelectron holography Polyethylene terephthalate Persistent homology Proportional-Integral-Differential Polylactic acid Pb(Mg1/3 Nb2/3 )O3 Polar nano regions Principal Peak Plastquinone Plastoquinol Position sensitive detector Photosystem I Photosystem II PV siplit vacancy complex Powder X-ray diffraction Lead ziroconate titanate Quasi-elastic neutron scattering Reduced atomic arrangement Restrained electrostatic potential Retarding field analyzer Reverse Monte Carlo Root-mean-square distance Root-mean-square error Root-mean-square fluctuations Structure-directing agents Selected area electron diffraction Sessile drop Silicon drift detector

Abbreviations

SEM SF Sn-Beta SOAP SOFC SPEA SRO STEM SX TDI TEM tet Ti/SBA-15 TM ToF TOF UV UV-Vis XAFS XFEL XFH XPCS XPS XRD

xiii

Scanning electron microscopy Symmetry function Sn-containing Beta zeolite Smooth overlap of atomic position Solid oxide fuel cell Scattering pattern extraction algorithm Short-range order Scanning transmission electron microscopy Soft X-ray Time-domain interferometry Transmission electron microscopy Tetrahedral TiO2 -grafted mesoporous silica SBA-15 Trans membrane Time-of-flight Turnover frequency Ultra-Violet Ultraviolet-visible X-ray absorption fine structure X-ray free electron laser X-ray fluorescence holography X-ray photon correlation spectroscopy X-ray photoelectron spectrum X-ray diffraction

Part I

Hyperordered Structures: Disorder in Order/Order within Disorder

Chapter 1

From Point Defects to Defect Complexes Koichi Hayashi

Abstract Most of the materials have dopants to create functionalities. However, due to the high demands on material functionality in cutting-edge devices, the conventional concept of point defects is no longer sufficient. Therefore, attempts to break through the limitations of point defects by doping different elements to create defect complexes have begun to progress. This section begins with examples of point defects and their role in a wide range of materials, including semiconductors, superconductors, catalysts, scintillators, metals, and glasses. Then, the limitations of the point defects and their solutions using defect complexes are presented with examples of semiconductor and glass. In the latter part of this section, examples of defect complexes are presented, divided into dopant-vacancy pairs and complex defects with sub-nanometer scales. These confer novel functionalities to materials. For example, nitrogen vacancy (NV ) centers in diamond have been considered as a promising for quantum bits, and Mn4 CaO5 clusters in photosystem II protein play an important role in photosynthesis. Finally, I introduce characterization techniques and theoretical methods to correctly understand the structures and properties of the defect complexes. Keywords Defect complex · Dopant–vacancy pair · Atomic resolution holography · Large scale DFT

1.1 Doping and Functionality Most materials contain dopants that confer some functionalities. Si, which is widely used in the world, is one of the good examples. Pure Si is an insulator. However, by doping different elements such as boron or phosphorus to Si, it exhibits functions as a p- or n-type semiconductor, respectively. In this case, the functionality originates K. Hayashi (B) Department of Physical Science and Engineering, Nagoya Institute of Technology, Nagoya 466-8555, Japan e-mail: [email protected] © Materials Research Society, under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. Hayashi (ed.), Hyperordered Structures in Materials, The Materials Research Society Series, https://doi.org/10.1007/978-981-99-5235-9_1

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from carriers, such as holes or electrons. pn junctions, which are interfaces between pand n-type semiconductors, are basic elements of semiconductor electronic devices, such as diodes, transistors, and integrated circuits. Similarly, the injection of carriers is necessary in many superconducting materials. To realize superconductivity, some dopants are added to host materials in many cases. For example, in the typical cuprate superconductor La2-x Srx CuO4 , Sr is responsible for injecting holes at the LaO plane, which is the carrier doping layer [1]. The carriers generated at the LaO plane are transferred to the superconducting layer CuO2 plane. On the other hand, the electrondoped copper oxide superconductor Nd2-x Cex CuO4 was developed later [2]. Perovskite materials exhibit a variety of functions, such as ferroelectricity or superconductivity. As a catalyst, perovskite oxides have long been known to exhibit gas-phase oxidation reactions. Therefore, they have been expected to have use in removing NOx and CO in automobile exhaust gas [3]. Moreover, their use in photocatalysis has also been studied [4]. Dopants have been used to generate carriers, and in 2000 a water-splitting reaction of more than 50% was demonstrated in La-doped NaTaO3 . Many other such systems have been found, and the search for novel substances is underway. In particular, for Ca, Sr, and La-doped in KTaO3, the dopant sites have been determined by X-ray absorption fine structure (XAFS) and X-ray fluorescence holography (XFH). As a result, it was found that they characteristically occupy multiple sites, including substitutions in both A and B sites [5, 6]. It was reported that the electron-hole recombination can be controlled by dual-site doping [7]. Scintillators, which are implemented in radiation detectors, often require dopants because they need luminescent centers for visible light. For example, in a NaI singlecrystal scintillator, doped Tl+ becomes a luminescent center [8]. Here, Na and I atoms interact with X-rays or γ-rays, generating secondary electrons. The generated secondary electrons spread in a matrix, and Tl+ defects catch them and are subsequently excited. Tl+ luminescent centers emit light owing to the 6sp → 6s2 transition. Since the decay time of scintillation is as short as of a series of silica crystals and densified silica glasses. RT/7.7 GPa glass shows a distinct boson peak at E of ~5 meV (more details about the boson peak are given in Chap. 6) and the peak shifts toward a large value with increasing density. Crystal phases show similar behavior, but both α-cristobalite and α-quartz show a sharp low-energy peak, whereas coesite shows very broad peaks that stretch toward higher energies. We consider that this behavior may be related to topological disorders in coesite. Moreover, the variation of ring size is a key feature in understanding the dynamics manifested by the boson peak in silica glass.

2.3 Cavity Distribution Cavity distribution is a very important topological feature of oxide glasses and liquids. The details of the calculation of cavity distributions are described in Refs. 5 and 6. The cavity volume ratio of silica glass is approximately 32% [10, 11] calculated using the pyMolDyn code [16]. Onodera et al. reported the squeezing of a cavity associated with the densification by hot compression determined using the MD-RMC models of densified silica glasses. Figure 2.5 shows the visualization of cavities (green) together with the histograms of cavity size distributions. It is found that a large cavity is transformed into small cavities by densification [12].

2.4 Persistent Homology Analysis Persistent homology analysis is a relatively new topological analysis method. The details of the analysis are given in Chap. 11. This method was developed to capture the shape of rings and cavities on the basis of a persistent diagram (PD); hence, it is very powerful to combine it with conventional ring size distribution and cavity distribution analyses. Figure 2.2e−h show Si-centric PDs, which provide information on ring shape, of α-cristobalite (d = 2.327 g cm−3 ), α-quartz (d = 2.655 g cm−3 ), coesite (d = 2.905 g cm−3 ), and silica glass (d = 2.2 g cm−3 ) obtained by MD−RMC modeling [11]. The death values in intense death profiles indicated by arrows become small, suggesting that the rings are distorted with the density change from α-cristobalite to coesite (see Fig. 2.2e−g). On the other hand, the Si-centric PD of silica glass shows a distinct profile along with the death axis, suggesting that silica glass includes the

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homology of a series of silica crystal polymorphs. Thus, the combination of ring size distribution analysis and persistent homology analysis provides us with crucial topological information that was hither to available.

2.5 Hyperordered Oxide Glasses and Liquids Recently, extremely ordered (hyperordered) glasses and liquids have been discovered. We introduce hyperordered silica glass, alumina (Al2 O3 ) glass, and erbia (Er2 O3 ) liquid to discuss the relationships among diffraction peaks, topological order, and hyperorder.

2.5.1 Densified Silica Glass Figure 2.6a shows the in situ neutron structure S(Q) of SiO2 glass under high pressure [18]. The details of the high-pressure technique are described in Chap. 4. The first sharp diffraction peak (FSDP) and the principal peak (PP) are observed in the ambient pressure data (black curve) at Q ~1.5 Å–1 and ~3 Å–1 , respectively [4]. The formation of FSDP is the result of atomic ordering along with cavities by corner-sharing SiO4 tetrahedra. The origin of the second PP seems to be some type of orientational

2 Topological Order and Hyperorder in Oxide Glasses and Liquids

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correlation among oxygen atoms that occupy the corner of the tetrahedra, suggesting that PP reflects the packing of oxygen atoms [19]. More details are described in Chap. 4. Upon application of pressure at room temperature, the FSDP shifts to high Q and diminishes. In contrast, the PP becomes very sharp (see Fig. 2.6a) associated with the cavity volume reduction, as illustrated in Fig. 2.5. This extraordinarily sharp PP is a signature of hyperordered orientational correlations formed by oxygen atoms under high pressure. Figure 2.6b shows the X-ray total structure factors S(Q) of densified silica glasses obtained by hot compressions (discussed in this chapter). These data are not in situ diffraction data, but the FSDP is the sharpest in the sample recovered at 1200 °C/

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7.7 GPa. This behavior is very different from in situ neutron diffraction data shown in Fig. 2.6a. We demonstrate that 1200 °C/7.7 GPa glass is the most ordered silica glass in the world, the so-called hyperordered silica glass. This behavior is also captured by the Si-centric PDs obtained by the MD−RMC modeling. Figure 2.7a−c show the Si-centric PDs of densified silica glasses recovered at RT/7.7 GPa (d = 2.24 g cm−3 ), (400 °C/7.7 GPa (d = 2.54 g cm−3 ), and 1200 °C/7.7 GPa) (d = 2.72 g cm−3 ) together with three-dimensional representations of the PDs shown in Fig. 2.7e−g, respectively. The death values of the vertical profile along with the death axis highlighted by rectangles in Fig. 2.2a−c become small after densification, which is similar to the change in a series of silica crystal polymorphs shown in Fig. 2.2e−g. Moreover, the three-dimensional representation shows a marked sharpening in the distribution of multiplicities with increased density. Therefore, we can conclude that the three-dimensional representation of Si-centric PD is a good indicator of FSDP of densified silica glasses.

2 Topological Order and Hyperorder in Oxide Glasses and Liquids

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2.5.2 Intermediate Alumina Glass Silica is a good glass former as mentioned above, whereas alumina is not a glass former and is classified as an intermediate according to Sun [20] (see Chap. 15) for more details). Indeed, it is impossible to form an aluminum glass by the conventional melt−quench technique. However, Hashimoto et al. have recently reported that the amorphous alumina synthesized by the anodization of aluminum metal exhibits glass transition [10]. Figure 2.8 shows neutron (a) and X-ray (b) total structure factors S(Q) of silica and alumina glasses. Silica glass shows a distinct FSDP because of its lower glass-forming ability. In contrast, alumina glass shows an extraordinarily sharp PP in the neutron S(Q) data similar to in situ high pressure data of silica glass shown in Fig. 2.6a, which suggests that the packing density of oxygen atoms is very high in alumina glass. We suggest that the alumina glass is a hyperordered glass. Figure 2.9 illustrates the atomic arrangements of alumina glass in stick representation (a) and with cavity visualization (b). We can see the lattice (crystal)-like atomic arrangement formed by the edge-sharing of AlOn polyhedra highlighted by black dotted lines. In addition, we can also recognize many more sparse regions formed by the tetrahedral corner-sharing motif in Fig. 9a. The cavity volume ratio of alumina glass is 4.5%, which is comparable to those of densified silica glasses recovered at 1200 °C/7.7 GPa. The average Al−O coordination number is 4.7, which is much higher than 4 in

2 Topological Order and Hyperorder in Oxide Glasses and Liquids

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silica glass and the formation of AlO4 , AlO5 , and AlO6 is confirmed. This variation of Al−O coordination is the reason for the formation of edge-sharing Al−O polyhedra, which can disturb the evolution of intermediate-range ordering detected as an FSDP. (a)

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Fig. 2.8 Neutron a and X-ray b total structure factors S(Q) of silica and alumina glasses. Reproduced from [10]. CC BY 4.0

(a)

(b)

Al O

Fig. 2.9 Atomic arrangements of alumina glass in stick representation a and with cavity visualization (highlighted in green) b. Pink and red circles represent Si and O atoms, respectively. Reproduced from [10]. CC BY 4.0

26

S. Kohara

2.5.3 Levitated Erbia Liquid Erbia (Er2 O3 ) is a nonglass-forming material and its melting point is extremely high (T m = 2430 °C). To perform diffraction measurements of such a high-temperature liquid, we have developed several containerless techniques, which allow us to hold a liquid droplet without any container. The variation of the levitation technique is described in Chap. 7. The details of the combination of levitation techniques and diffraction measurements proposed by Price are reviewed in ref 21. The levitation technique has recently been in the investigation of structures of single-component oxide liquids, Al2 O3 [22], Y2 O3 [23], Ho2 O3 [23], La2 O3 [23], ZrO2 [23], UO2 [24], TiO2 [25], B2 O3 [26], Lu2 O3 [27], and Yb2 O3 [27]. Here, we employed the aerodynamics levitation technique for X-ray diffraction measurement of liquid erubia, with which a sample is levitated using an inert gas from a conical nozzle [21]. Figure 2.10 shows X-ray structure factors S(Q) of erbia liquid (2650 °C) [28] (a) and zirconia (ZrO2 ) liquid (2800 °C) [29] (b) together with those obtained using several simulation techniques. We cannot observe any FSDP in both data sets because they are nonglass-forming materials. However, they exhibit a PP at Q ~2 Å−1 . The FWHMs of PP for zirconia and erbia liquids are 0.7669 and 0.4299, respectively. In the case of zirconia liquid, the RMC−density functional (DF)/MD model of 501 particles (magenta curve) reproduces experimental data. However, we need the RMC−MD model of 5000 particles (red curve) to reproduce the extraordinarily sharp PP for erbia liquid. As a benchmark, we reduced the number of particles in the standard RMC approach and confirmed that 500 particles (blue) are insufficient to reproduce such an extraordinarily sharp PP. We consider that erbia liquid is an unusual hyperordered oxide liquid. (a)

(b) 2

FWHM 0.7669

Experimental data 3

FWHM 0.4299

1

S(Q)

S(Q)

2

5000 particles RMC model 3000 particles 1000 particles 500 particles 250 particles

1

3 2 1

0

Experimental data

0

0 -1

501 particles

-1 1.0 0

5

2.0 10

Q (Å 1)

3.0 15

0

5

10

15

Q (Å 1)

Fig. 2.10 X-ray structure factors S(Q) of erbia liquid (2650 °C) [28] a and zirconia (ZrO2 ) liquid (2800 °C) [29] b together with those obtained by RMC−MD, RMC, and RMC−DF/MD modeling. Reproduced from [28]. CC BY 4.0

2 Topological Order and Hyperorder in Oxide Glasses and Liquids

27

Fig. 2.11 Atomic arrangement of erbia liquid in stick representation. Reproduced from [28]. CC BY 4.0

The cation−oxygen coordination numbers are approximately 6 for both liquids, which is extremely larger than 3.9 for SiO2 (2000 °C) liquid and 4.4 for Al2 O3 liquid (2127 °C). Moreover, the oxygen−cation coordination number is 3.0 for zirconia liquid and 4.1 for erbia liquid, suggesting that a large fraction of an OEr4 tetracluster, which cannot be observed in other liquids, is observed. Figure 2.11 illustrates the atomic arrangement of erbia liquid in stick representation. We can see extraordinarily densely packed atomic arrangement, which is found in alumina glass, but we can hardly see the sparse region, which is observed in alumina glass. This extraordinarily densely packed atomic arrangement highlighted by dotted lines is formed by the edge-sharing ErOn polyhedra associated with the formation of OEr4 tetraclusters and is the origin of extremely low glass-forming ability and the extraordinarily sharp PP.

2.6 Concluding Remarks In this chapter, we introduced the concept of topological order and disorder proposed by Gupta [9]. He proposed on the basis of ring statistics. We applied this concept to silica polymorphs and also compared their data with inelastic neutron scattering data

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to understand the origin of topological disorder, which can be important information to understand the boson peak. We have also introduced the analytical methods for cavity distributions and persistent homology because they are very powerful in combination with conventional ring size distribution analyses. We have also reviewed several unusual structures of oxide glasses and liquids in terms of “hyperorder.” Their structures are very different from those of conventional oxide glasses and liquids. The characteristic features of these glasses and liquids are the variations of polyhedra in terms of coordination number, polyhedral connections (corner, edge, and face), ring size, and ring shape in terms of homology. These features should as a whole be called “topological disorder,” which is related to the formation of hyperorder in oxide glasses and liquids. Acknowledgements This work was supported by a JSPS Grant-in-Aid for Transformative Research Areas (A) “Hyper-Ordered Structures Science”: Grants No. 20H05878 and No. 20H05881.

References 1. Kohara S, Salmon PS (2016) Adv Phys: X 1:640 2. Kohara S, Akola J (2021) In World scientific series in nanoscience and nanotechnology advanced characterization of nanostructured materials, Ed by Sinha SK, Sanyal MK, Loong CK (World Scientific Co. Pte. Ltd., Singapore), pp 247−305 3. Ohara K, Onodera Y, Murakami M, Kohara S (2021) J Phys: Condens Matter 33:383001 4. Kohara S (2022) J Ceram Soc Jpn 130:531 5. Benmore CJ (2023) Comprehensive inorganic chemistry III (Third Edition)” ed by Reedijk J, Poeppelmeier KR (Elsevier, Amsterdam), pp 384−424 6. McGreevy RL, Pusztai L (1988) Molec Simul 1:359 7. McGreevy RL (2001) J Phys: Condens Matter 13:R877 8. Kohara S, Pusztai L (2022) Atomistic simulations of glasses: fundamentals and applications, ed by Du J, Cormack AN (Wiley-American Ceramic Society, Hoboken), pp 60−88 9. Gupta PK (1993) J Am Ceram Soc 76:1088 10. Hashimoto H, Onodera Y, Tahara S, Kohara S, Yazawa K, Segawa H, Murakami M, Ohara K (2022) Sci Rep 12:516 11. Onodera Y, Kohara S, Tahara S, Masuno A, Inoue H, Shiga M, Hirata A, Tsuchiya K, Hiraoka Y, Obayashi I, Ohara K, Mizuno A, Sakata O (2019) J Ceram Soc Jpn 127:853 12. Onodera Y, Kohara S, Salmon PS, Hirata A, Nishiyama N, Kitani S, Zeidler A, Shiga M, Masuno A, Inoue H, Tahara S, Polidori A, Fischer HE, Mori T, Kojima S, Kawaji H, Kolesnikov AI, Stone MB, Tucker MG, McDonnell MT, Hannon AC, Hiraoka Y, Obayashi I, Nakamura T, Akola J, Fujii Y, Ohara K, Taniguchi T, Sakata O (2020) NPG Asia Mater. 12:85 13. Sokolov AP, Kisliuk A, Soltwisch M, Quitmann D (1992) Phys Rev Lett 69:1540 14. Sugai S, Onodera A (1996) Phys Rev Lett 77:4210 15. Inamura Y, Arai M, Nakamura M, Otomo T, Kitamura N, Bennington SM, Hannon AC, Buchenau U, Non-Cryst J (2001) Solids 293–295:389–393 16. Heimbach I, Rhiem F, Beule F, Knodt D, Heinen J, Jones RO (2017) J Comput Chem 38:389 17. Wille TF, Rycroft CH, Kazi M, Meza JC, Haranczyk M (2012) Microporous Mesoporous Mater 149:134

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18. Zeidler A, Wezka K, Rowlands RF, Whittaker DAJ, Salmon PS, Polidori A, Drewitt JWE, Klotz S, Fischer HE, Wilding MC, Bull CL, Tucker MG, Wilson M (2014) Phys Rev Lett 113:135501 19. Salmon PS, Zeidler A (2019) J Stat Mech Theory E 2019:114006 20. Sun K-H (1947) J Am Ceram Soc 30:277 21. Price DL (2010) In High-temperature levitated materials (Cambridge University Press, Cambridge), pp 2−19 22. Skinner LB, Barnes AC, Salmon PS, Hennet L, Fischer HE, Benmore CJ, Kohara S, Weber JKR, Bytchkov A, Wilding MC, Parise JB, Farmer TO, Pozdnyakova I, Tumber SK, Ohara K (2013) Phys Rev B 87:024201 23. Skinner LB, Benmore CJ, Weber JKR, Du J, Neuefeind J, Tumber SK, Parise JB (2014) Phys Rev Lett 112:157801 24. Skinner LB, Benmore CJ, Weber JKR, Williamson MA, Tamalonis A, Hebden A, Wiencek T, Alderman OLG, Guthrie M, Leibowitz L, Parise JB (2014) Science 346:984987 25. Alderman OLG, Skinner LB, Benmore CJ, Tamalonis A, Weber JKR (2014) Phys Rev B 90:094204 26. Alderman OLG, Ferlat G, Baroni A, Salanne M, Micoulaut M, Benmore CJ, Lin A, Tamalonis A, Weber JKR (2015) J Phys: Condens Matter 27:455104 27. Pavlik A III, Ushakov SV, Navrotsky A, Benmore CJ, Weber JKR (2017) J Nucl Mater 495:385 28. Koyama C, Kohara S, Onodera Y, Småbråten DR, Selbach SM, Akola J, Ishikawa T, Masuno A, Mizuno A, Okada JT, Watanabe Y, Nakata Y, Ohara K, Tamaru H, Oda H, Obayashi I, Hiraoka Y, Sakata O (2020) NPG Asia Mater. 12:43 29. Kohara S, Akola J, Patrikeev L, Ropo M, Ohara K, Itou M, Fujiwara A, Yahiro J, Okada JT, Ishikawa T, Mizuno A, Masuno A, Watanabe Y, Usuki T (2014) Nat Commun 5:5892

Part II

Characterization of Hyperordered Structures

Chapter 3

Atomic-Resolution Holography Tomohiro Matsushita, Koji Kimura, and Kenji Ohoyama

Abstract This chapter describes details of atomic-resolution holography, which gives a three-dimensional atomic image around a target atomic site. It can visualize the atomic structures of an impurity, a dopant, an adsorbate on a crystal, and a thin-film interface as well as the positional fluctuation of a target atom. These atomic structures cannot be measured by conventional measurement methods, such as X-ray diffraction (XRD), X-ray absorption fine structure (XAFS), and electron microscopy. There are several types of atomic-resolution holography: X-ray fluorescence holography, neutron holography, photoelectron holography, and inverse photoelectron holography. The principle and apparatus of these techniques are introduced together with their features. Some excellent examples of hyperordered structures embedded in crystals are presented, such as the Co suboxidic structure in TiO.2 , the Eu dopant structure in CaF.2 , the P dopant structure in the diamond, the defect structure between an insulating layer and a diamond semiconductor, and so forth. Section 3.1 gives a brief introduction to atomic-resolution holography. Section 3.2 describes its principle in detail, especially, the recording processes of holograms and the reconstruction algorithms. Individual holographic techniques are introduced in Sects. 3.4, 3.5, 3.6, and 3.7. Finally, the features of these techniques are summarized in Sect. 3.8. Keywords Atomic-resolution holography · X-ray fluorescence holography · Neutron holography · Photoelectron holography · Inverse photoelectron holography · Three-dimensional atomic image

T. Matsushita (B) Graduate School of Science and Technology, Nara Institute of Science and Technology, Ikoma, Nara 630-0192, Japan e-mail: [email protected] K. Kimura Department of Physical Science and Engineering, Nagoya Institute of Technology, Nagoya 466-8555, Japan K. Ohoyama Graduate School of Science and Engineering, Ibaraki University, Hitachi 316-8511, Japan © Materials Research Society, under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. Hayashi (ed.), Hyperordered Structures in Materials, The Materials Research Society Series, https://doi.org/10.1007/978-981-99-5235-9_3

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3.1 Introduction Holography is a method for generating a three-dimensional (3D) image from an interference pattern of waves recorded on a physical medium. Hungarian physicist Gabor invented holography in 1948, for which he received the Nobel Prize in Physics in 1971 [1]. Although his study was intended to improve the resolution of an electron microscope by using the interference of electron waves, his proposed principle of holography is applicable to any type of wave. In particular, optical holography is used worldwide in applications such as the anti-counterfeiting of banknotes. The principle of holography has been applied to the 3D reconstruction of atomic-scale images; this is called atomic-resolution holography. In 1986, Szöke presented the basic principle of atomic-resolution holography to record atomic arrangements [2]. In this study, it was proposed that a 3D atomic arrangement can be reconstructed from a holographic oscillation contained in electron waves or electro-magnetic waves emitted from atoms (e.g., photoelectrons and fluorescent X-rays). In 1988, Barton presented a Fourier transform method for reconstructing an atomic image from an atomic-resolution hologram [3, 4]. Figure 3.1 shows a schematic image of the atomic-resolution holography (as noted below, this is the normal-mode measurement method). When a material is irradiated with X-rays, photoelectrons (or fluorescent X-rays) are emitted from the atoms in the material. These particles have wave properties. These waves are scattered by the surrounding atoms, resulting in scattered waves. Naturally, the scattered and direct waves interfere with each other, producing an interference pattern in the emission angular distribution. This interference pattern is an atomic-resolution hologram, which contains information on the 3D arrangement of atoms around the emitter of the particle. By analyzing this hologram, a 3D atomic arrangement around the emitter can be obtained. This method is especially powerful for elucidating local structures around a dopant in materials. Recent advances in experimental and analysis methods for atomic-resolution holography have revealed that doped elements can induce not only a simple point defect but also novel structural features such as dopant clusters, defect complexes, and strongly off-centered atomic arrangements. The following sections describe how these hyperordered structures in crystals (i.e., disorder in order) can be characterized by atomic-resolution holography.

3 Atomic-Resolution Holography

35

d) Angular distribution = Hologram

a) Excitation b) direct wave

c) scattered wave

Emitter atom

Reconstruction calculation

3D atomic image around the emitter atom

Fig. 3.1 Schematic diagram of atomic-resolution holography. a Irradiation with excitation particles (X-rays, etc.). b Emission of secondary particles. c Scattering of waves of secondary particles, producing scattered waves. d Interference pattern between the direct and scattered waves in the emission angular distribution. This interference pattern is a hologram

3.2 Principle of Atomic-Resolution Holography 3.2.1 Recording Process The recording process of atomic-resolution holography can be of two modes: normal mode and inverse mode. Figure 3.2 shows the principle of the recording process. The normal mode utilizes the scattering and interference of spherical waves emitted from a specific element. The inverse mode accesses the scattering and interference of

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(a) Normal mode

(b) Inverse mode

Hologram L

θ,ϕ

Hologram

L

θ,ϕ

F F

E

S

E

a

S a

Fig. 3.2 Recording process of atomic-resolution holography: a normal mode and b inverse mode

incident plane waves. Considering the time-reversal symmetry of the laws of physics, these two methods result in the same hologram being obtained.

3.2.2 Normal Mode Figure 3.2a shows the recording process for a normal-mode hologram. A beam (photons or electrons; labeled as L) excites the secondary particles (electrons or photons) emitted from the atom (labeled as E). Here, the atom that emits the secondary particle is the emitter. The secondary particle has the property of a wave; therefore, a spherical wave spreads around the emitter. The wave is scattered by surrounding atoms to form the scattered wave. One scatterer atom is labeled as S. The scattered and unscattered waves interfere with each other, and an interference pattern is formed by the angular distribution of the emitted secondary electron (labeled as F). Because the scattered and unscattered waves can be regarded as the object waves and reference wave, respectively, the interference pattern can be regarded as the hologram. This hologram contains information on the 3D atomic structure around the emitter. Figure 3.3 shows a schematic view of the formula for interference in the normal mode. The emitted wave (.s-wave: isotropic spherical wave) is described as ϕ(k, r) ≡

.

exp(ik|r|) , |r|

(3.1)

where .k and . r are the wave number and position vector, respectively. There are many scatterer atoms, and their index is written as .h. The wave reaches a scatterer atom

3 Atomic-Resolution Holography

37

Fig. 3.3 Schematic view of formula for interference in normal mode

located at . a h . The wave function at the scatterer is given by ϕ(k, a h ) =

.

exp(ik|a h |) . |a h |

(3.2)

Then, the scattered wave is formed as exp(ik|r − a h |) exp(ik|a h |) · f (θah k ) , |a h | |r − a h |

ψ(k, r, a h ) =

.

(3.3)

where . f (θ ) is the atomic scattering factor (also called the atomic form factor, or atomic structure factor) and .θah k is the angle between vectors . a h and . k. As described below, this scattering factor varies depending on the measurement method. For example, in the case of fluorescent X-rays, the atomic scattering factor has the following relationship with the atomic scattering factor of X-ray diffraction: .

f (θah k ) = − f xray (θah k ).

(3.4)

The interference intensity is expressed by | |2 | | E | | . I (k, r) = |ϕ(k, r) + ψ(k, r, a h )| . | |

(3.5)

h

This interference pattern is observed at a distant detector. (The atomic scale is nm, and the detector is nearly 1 m away, which is distant in terms of the atomic scale.) Therefore, | |2 | | E | | . I (k) = |ϕ(k) + ψ(k, a h )| . (3.6) | | h

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Here, the following relation is used: ϕ(k) = lim r ϕ(k, r).

.

(3.7)

r →∞

When .r approaches infinity, the vector . k becomes parallel to the vector . r; consequently, .(k, r) → k. Then, the hologram function is defined by subtracting the reference wave intensity: | |2 | | E | | .χ (k) = |ϕ(k) + ψ(k, a h )| − |ϕ(k)|2 . | |

(3.8)

h

χ (k) =

E

.

h

| exp(i(kah − k · a h )) . 2R f (θah k ) ah |

(3.9)

This equation contains each atomic position vector, and the hologram is the simple sum of each atom’s hologram. Therefore, a 3D atomic image can be reconstructed from the hologram without requiring additional information such as a model of the initial atomic arrangement. This normal mode is used in photoelectron holography (PEH) and normal-mode X-ray fluorescence holography (XFH), which are described later.

3.2.3 Inverse Mode Figure 3.4 shows a schematic view of the formula for the inverse mode. The incident plane wave is given by .ϕ(k, r) = exp(i k · r). (3.10)

.

. .

Fig. 3.4 Schematic view of formula for interference in inverse mode

3 Atomic-Resolution Holography

39

Here, the origin is the atom that emits the secondary particle (X-ray, .γ -ray, or electron). This plane wave is scattered by a scatterer atom located at . a h , where .h is the index for the atoms. The wave function at the scatterer is given by ϕ(k, a h ) = exp(i k · a h ).

.

(3.11)

Then, the scattered wave is formed as ψ(k, r, a h ) = exp(i k · a h ) f (θah k )

.

exp(ik|r − a h |) . |r − a h |

(3.12)

Here, many atoms are present. The interference intensity is expressed by | |2 | | E | | . I (k, r) = |ϕ(k, r) + ψ(k, r, a h )| . | |

(3.13)

h

This interference pattern is observed at the emitter atom (. r = 0). Therefore, | |2 | | E | | . I (k) = |ϕ(k) + ψ(k, a h )| . | |

(3.14)

h

Then, the hologram function is defined by subtracting the reference wave intensity: | |2 | | E | | .χ (k) = |ϕ(k) + ψ(k, a h )| − |ϕ(k)|2 . | |

(3.15)

h

χ (k) =

E

.

h

| exp(i(kah + k · a h )) . 2R f (θah k ) ah |

(3.16)

The obtained equation is similar to that for the normal mode. When . k/ = −k is adopted, the two equations are identical.

3.2.4 Effect of Thermal Vibration and Fluctuation The positions of atoms fluctuate due to thermal vibration and other factors. If the atoms are vibrating isotropically and the standard deviation of their vibration amplitude of the .hth atom is defined as .σh , the hologram is given by Matsushita et al. [5]

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| | exp[i(kah − k · a h )] exp[−σh2 |k|2 (1 − cos θah k )] . 2R f (θah k ) |ah | h (3.17) From this equation, the larger the angle .θah k , the smaller the term .exp[−σh2 |k|2 (1 − cos θah k )] becomes. This means that the amplitude of backscattering is suppressed by the fluctuation of the atomic positions. χ (k) =

E

.

3.3 Overview of Atomic Image Reconstruction Atomic image reconstruction methods are important in practical uses. Barton [3, 4] proposed a Fourier transform method because the hologram has the term.exp(i(kah + k · a h )). The amount of information is preserved by the Fourier transformation, and a 3D volume hologram is required. When using a single-energy hologram, the twinimage formation, where a false image is generated at the point-symmetrical position, is a serious problem as described later. The normal mode of XFH suffers from this problem because it is difficult to obtain a multi-energy hologram. In contrast, when a multi-energy hologram is available, Barton’s method is effective. Inverse XFH and neutron holography (NH) satisfy this condition. Here, Barton’s method and its application to XFH are described. In addition, an advanced reconstruction method called the scattering pattern extraction algorithm (SPEA) [6, 7] is briefly presented, which is applicable to single-energy X-ray fluorescence holograms to obtain reasonable atomic images. This advanced method was originally developed for the atomic image reconstruction in PEH to deal with complicated scattering processes of electrons (i.e., the complicated form of . f (θah k )), and a detailed explanation will be given in Sect. 3.6.

3.3.1 Barton’s Algorithm Atomic images, .U (r), can be obtained from the Fourier-transform-based formula [3] { .U (r) = w(θkr )χ (k)R[exp(i(kr − k · r)]d k, (3.18) k=|k|

where .w(θkr ) is a weight function. This weight function is related to the scattering factor and therefore depends on the measurement method. For example, in XFH, it is .w(θkr ) = −1. However, the atomic image reconstructed from Eq. (3.18) suffers from the twin-image problem, shown in Fig. 3.5. Suppose a dimer model contains Fe as an emitter atom at the origin and Cu as a scattering atom at .(x, y, z) = (3, 0, 0)Å (Fig. 3.5a). The calculated hologram at a photon energy of 6.4 keV is shown in Fig. 3.5a. Figure 3.5b shows the reconstructed image obtained by substituting this

3 Atomic-Resolution Holography

41

holographic oscillation into .χ (k) in Eq. (3.18). We can observe a twin image at (x, y) = (−3, 0) Å, as indicated by the arrow, in addition to the real atomic image observed at .(x, y) = (3, 0)Å. As shown in Fig. 3.5c, the oscillation of the hologram signal is almost symmetric about .θ , resulting in the twin image at the centrosymmetric position of the real Cu atom, as shown in Fig. 3.5b. Such symmetric oscillation appears when .kah ∼ π N (N = 1, 2, ...). Because twin images are inherent to holography, it is impossible to avoid this problem by enhancing the statistics or increasing the measurement angular range. One of the most effective methods to suppress twin images is multi-wavelength measurements. As shown in Fig. 3.6a, hologram oscillations are dependent on the energy (wavelength), which is included in . k in Eqs. (3.9) and (3.16). In Fig. 3.6a, the oscillations are in-phase for various energies in the forward-scattering region at around .θ = 0◦ and not in-phase in the back-scattering region at around .θ = 180◦ . Therefore, by superimposing the images with different energies by using [4]

.

{ U (r) =

{ dk

.

k

w(θkr )χ (k)R {exp[i(kr − k · r)]} d k,

(3.19)

|k|=k

twin images can be effectively suppressed. Figure 3.6b shows the image reconstructed by multi-wavelength holograms from 7.5 to 11.0 keV in steps of 0.5 keV using Eq. (3.19). We can confirm that the twin image almost completely disappears. Note that Eq. (3.19) includes the integral with respect to .k = |k| = 2π/λ, where .λ is the wavelength of fluorescent or incident X-rays for the normal or inverse mode, respectively. Because the wavelength of fluorescent X-rays cannot be tuned, Eq. (3.19) is not applicable to the holograms recorded by the normal-mode XFH. Therefore, almost all XFH experiments have been performed using the inverse mode [8]. Notably, the twin-image problem can be ignored for PEH because the holographic oscillations of photoelectrons attenuate much more strongly with increasing .θ , and thus, the photoelectron holograms are far from symmetric with respect to .θ . This is why the normal mode works well for PEH. In contrast, the attenuation of the holographic oscillation with increasing .θ is even weaker for NH than that for XFH, because the scattering length .bh of neutrons is constant for .θ . Therefore, twin images in NH have a greater influence on the reconstructed image than in XFH. As described in Sect. 3.5, this difficulty has been successfully overcome by using the time-offlight (ToF) method, which enables us to record numerous holograms with different wavelengths at the same time.

3.3.2 Fitting-Based Atomic Image Reconstruction By using the atomic distribution function, .g(r), the atomic-resolution hologram, χ (k), can be written as

.

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T. Matsushita et al. (a)

(b)

4

6.4 keV

( )

2

Fe

Cu

Fe

0 -2

Twin Image

-4

3

-4

-2

0 2 ( )

4 1.0

0.1

(c) 6.4 keV

peak

peak

peak

peak peak peak 180° dip 150

= 0°

dip

100 50 (deg.)

0 dip dip

Fig. 3.5 Origin of twin images. a Calculated hologram of a dimer model of Fe and Cu for .hν = 6.4 keV. b Atomic image reconstructed from the hologram shown in (a). c Line profile of the hologram along the dashed line indicated in the hologram to the right

(a)

(b)

10.0 keV 9.0 keV 8.0 keV

4 ( )

2

6.4 keV

Fe

0 -2

150

100 50 (deg.)

0

-4 -4 0.1

-2

0 2 ( )

4

1.0

Fig. 3.6 Effect of multi-wavelength holograms on suppressing twin images. a Calculated hologram oscillations of Fe-Cu dimer model (Fig. 3.5a) with different incident energies. b Atomic images using eight energies (7.5 to 11.0 keV in steps of 0.5 keV) obtained from Eq. (3.19)

3 Atomic-Resolution Holography

43

{ χ (k) = 2

g(r) f (θak )R[exp[i(kr − k · r)]]d r.

.

(3.20)

As will be described in Sect. 3.6 (see Eq.e (3.48)), the atomic distribution function g(r) is conveniently expressed by .g(r) = h δ(r − a h )/r . With this definition, Eq. (3.20) is rewritten as E .χ (k) = 2 g(r h ) f (θak )R[exp[i(kr − k · r)]]d r. (3.21)

.

h

e (exp) e By minimizing the evaluation function. E = i |χi − χi(sim) |2 + λ j g j (see Eq. (3.51) in Sect. 3.6), the atomic image can be obtained. The second term is the socalled penalty term in L1 regularization, thus, SPEA with this evaluation function is called SPEA-L1 [10, 11]. The iterative calculation .

g (n+1) (r) = g (n) − α

∂ E (n) ∂r

(3.22)

is used in the minimization process. Here, the parameter .α is optimized using the gradient method. Figure 3.7 shows a comparison of the atomic images around Zn in ZnTe obtained using Barton’s algorithm and SPEA-L1 [10]. The holograms obtained from the XFH experiment were used for the reconstruction. Figure 3.7a shows the hologram obtained at the incident X-ray energy of 12 keV. As schematically shown in Fig. 3.7b, the Te-plane around the Zn atom was reconstructed as shown in Fig. 3.7c–f. Here, atomic images were reconstructed using Barton’s algorithm and SPEA-L1 by using nine holograms from 11 to 15 keV in steps of 0.5 keV (Fig. 3.7c and d) and a single hologram at 12 keV (Fig. 3.7e and f). Using nine holograms, both Barton’s method and SPEA-L1 well reconstruct the Te atomic images, although the reconstructed image obtained using Barton’s algorithm contains sidelobes of the Fourier transform, as indicated by circles in Fig. 3.7c. By contrast, the quality of the reconstructed image from the single-energy hologram obtained by SPEA-L1 (Fig. 3.7f) is markedly different from that obtained by Barton’s method (Fig. 3.7e). In other words, distinct Te atomic images can be observed in Fig. 3.7f, whereas clear atomic images are hardly observed in Fig. 3.7e. In recent years, SPEA-L1 has been successfully used in some studies [12–14], especially in valence-specific XFH studies [13, 14]. This result demonstrates the potential of XFH to obtain reliable atomic images even with a single-energy hologram, and thus, it provides a good opportunity to revisit normal-mode XFH. As described in the next section, normal-mode XFH has some advantages over inverse mode XFH, such as a much shorter measurement time with a 2D detector and the ability to measure small samples with focused incident X-rays. In this context, experimental and analytical techniques based on normal-mode XFH are being investigated further.

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Fig. 3.7 Comparison of Barton’s method and SPEA-L1. a X-ray fluorescence hologram of ZnTe obtained using the incident X-ray energy of 12 keV. The data were taken from Ref. [9]. b Structure of ZnTe. c Cross section of the image reconstructed by Barton’s method. Nine holograms from 11 to 15 keV were used. The distance of the xy-plane from the emitter is 1.5 Å. d Image reconstructed from nine holograms using SPEA-L1. e Image reconstructed from a single hologram obtained at 12 keV using Barton’s method. f Image reconstructed from a single hologram using SPEA-L1. Adapted from Ref. [10] with reprint permission

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3.4 X-ray Fluorescence Holography (XFH) X-ray fluorescence holography was first demonstrated by Tegze and Faigel in 1996 [15]. Because the hologram amplitude is only approximately some 0.1% of the fluorescent X-ray intensity, XFH experiments using laboratory X-ray sources are very time-consuming. However, with the advent of third-generation synchrotron radiation facilities together with significant developments in analytical methods to obtain improved 3D atomic images [16–25] and experimental techniques to efficiently collect fluorescent X-rays [26–30], XFH has been practically applied to various structural [8, 31], functional [8, 12, 31–38], and biological materials [31, 39, 40]. Here, the apparatus of XFH and several hyperordered structures in materials found by XFH will be described. Furthermore, recent developments in experimental and analytical techniques will be presented.

3.4.1 Hologram Oscillations in Fluorescent X-Rays Suppose a material containing a dopant is irradiated with X-rays. If the energy of the X-rays is higher than the absorption edge of the dopant, the dopant emits fluorescent X-rays. Then, part of the fluorescent X-rays are scattered by the surrounding atoms. The interference between the unscattered and scattered waves slightly modulates the intensity of the fluorescent X-rays, which corresponds to the holographic oscillation. The unscattered and scattered fluorescent X-rays correspond to “Direct wave” and “Scattered wave” in Fig. 3.3, respectively, which illustrates the formation process of the interference pattern in the normal mode. Then, the intensity of the fluorescent X-rays . I for the normal mode can be written as [8, 32, 41] .

I (k) = 1 − 2R

E | re f xray (θak )

| exp[i(kah − k · a h )]

ah h | |2 |E r f (θ ) | e xray ak | | +| exp[i(kah − k · a h )]| , | | a h h

(3.23)

where . k is the wave vector of the fluorescent X-rays, .re ≈ 2.8 × 10−5 Å is the classical electron radius,. a h is the position of the.hth atom,. f is the atomic scattering factor, and .θak is the angle between . k and . a h . The second term represents the holographic oscillation, .χ (k): χ (k) = −2R

.

E | re f xray (θak ) h

ah

| exp[i(kah − k · a h )] .

(3.24)

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Here, the amplitude of the holographic oscillation, .χ (k), is only approximately some 0.1% of the intensity of the fluorescent X-rays. Note that Eq. (3.24) can be obtained by substituting . f (θak ) in Eq. (3.9) into the negative value of the atomic scattering factor, .− f xray (θak ). The negative sign originates from the phase shift of X-rays by .π through Thomson scattering [42]. Equation (3.24) can be converted into the holographic oscillation for the inverse mode by defining . k as the wave vector of incident X-rays and replacing . k with .−k, that is, | E | re f xray (θak ) .χ (k) = −2R exp[i(kah + k · a h )] . (3.25) ah h

3.4.2 Apparatus Nowadays, XFH experiments are actively being carried out in synchrotron facilities such as SPring-8 and KEK-PF in Japan [32]. Because the holographic oscillation is only approximately some 0.1% of the intensity of the fluorescent X-rays, the highintensity X-rays available at a synchrotron facility are necessary. Furthermore, the synchrotron X-ray source can offer energy tunability to record multi-wavelength holograms by inverse mode XFH. Figure 3.8 illustrates the experimental setup for inverse mode XFH. A sample on a two-axis (.θ, φ) goniometer is irradiated with X-rays. The fluorescent X-rays from a target element in the sample are selectively extracted by adjusting the position of the analyzer crystal. Owing to the round shape of the analyzer crystal, the fluorescent X-rays are focused on the detector, thereby enhancing the fluorescent signals

Sample

Incident X-rays θ

ϕ

Analyzer crystal Pb shield Detector

Fluorescent X-rays

Fig. 3.8 Apparatus for inverse mode XFH

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Fluorescent X-rays sample 45° pin-hole mirror

s ray

t X

n ide

2D detector SDD

Inc

Camera Fig. 3.9 Apparatus for normal-mode XFH

significantly. An avalanche photodiode or silicon drift detector (SDD) is typically used as a detector. The fluorescent intensities are recorded as a function of .θ and .φ in Fig. 3.8, and thus, 2D data can be obtained, which is referred to as a hologram. Figure 3.9 schematically shows the experimental setup for normal-mode XFH [43, 44]. In normal-mode XFH, fluorescent X-rays from the fixed sample are recorded as a function of the emission direction. Because the sample is fixed during the measurement, holograms from small samples can be readily measured using a focused X-ray beam, which is available in synchrotron facilities. The typical size of the focused X-rays is .∼1 × 1 .µm.2 , and thus, a sample with a size of .∼10 × 10 .µm.2 or less can be a target of normal-mode XFH. A telescopic camera and a 45.◦ pinhole mirror are installed to observe the sample from the direction of the incident X-rays, thereby enabling the sample position to be precisely adjusted with an accuracy of about 1 .µm. Such high accuracy is required to measure small samples such as protein crystals [44]. In a recent setup, a 2D detector was used to efficiently measure fluorescent X-rays emitted in a certain range of solid angles. In addition, the energy spectra are observed using an SDD.

3.4.3 Applications Recently, various hyperordered structures have been found in materials such as thermoelectrics, lightweight structural materials, and superconductors [31]. Here, some examples of hyperordered structures revealed by XFH are presented.

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Co-doped TiO.2 : Dilute Magnetic Semiconductor As already described in Sect. 1.1, Co-doped TiO.2 is a promising ferromagnetic semiconductor with a high Curie temperature of about 600 K. The suboxidic structure shown in Fig. 1.1.3 embedded in this material is a typical hyperordered structure. Figure 3.10a and b present atomic images around Co in a Co-doped TiO.2 thin film [45]. It is confirmed that the positions of atomic images around Co in paramagnetic Ti.0.99 Co.0.01 O.2 (Fig. 3.10a) agree well with those predicted from the rutile-type TiO.2 structure (Fig. 3.10c). This observation indicates that the Co atoms in Ti.0.99 Co.0.01 O.2 are located at the substitutional sites of rutile-type TiO.2 . By contrast, the atomic images around Co in ferromagnetic Ti.0.95 Co.0.05 O.2 (Fig. 3.10b) cannot be explained by the simple substitutional model. Figure 3.10d shows the corresponding atomic configuration around Co, where fewer oxygen atoms are present around Co than in the simple substitutional model shown in Fig. 3.10c; in other words, suboxidic coordination is formed around Co in ferromagnetic Ti.0.95 Co.0.05 O.2 . Although this suboxidic structure is energetically unstable, a first-principles calculation showed that the dimerization of the Co atom stabilizes this suboxidic state, as shown in Fig. 1.3 in Chap. 1. This study on Co-doped TiO.2 demonstrates the capability of XFH to reveal not only simple point defects but also more complex local structures, namely, hyperordered structures.

Zn, Y-doped Mg Alloy: Lightweight Structural Material Mg–Zn-Y alloys [46] are promising candidates for next-generation lightweight structural materials because of their excellent mechanical properties, such as a very high yield strength of more than 600 MPa and reasonable ductility. Scanning transmission electron microscopy (STEM) revealed the Zn and Y atoms from a Zn.6 Y.8 cluster in the .hcp Mg matrix structure as shown in Fig. 1.1.4 [47] in Chap. 1. As already mentioned in Chap. 1, this Zn.6 Y.8 cluster can be regarded as a hyperordered structure. While STEM can visualize the atomic arrangement in a selected area of a sample, XFH gives a statistically averaged structure around the doped elements in the whole sample, which is useful for understanding the effect of doping on the macroscopic properties of materials. For this purpose, XFH was applied to Mg.75 Zn.10 Y.15 alloy [48]. An in-plane atomic image around Zn in Mg.75 Zn.10 Y.15 is shown in Fig. 3.11a. Here, the atomic images obtained from the experimental and calculated holograms are displayed using Barton’s algorithm. The calculated atomic image was derived from the structural model proposed in the STEM study [47]. The upper part of Fig. 3.11a illustrates the three different in-plane atomic configurations of Zn. Here, Zn atoms form a regular triangle along the .ab plane, and we labeled the vertices of this triangle as A, B, and C (Fig. 3.11a and b). The atomic image obtained by XFH corresponds to the superposition of the local structures around these three Zn atoms, as shown in Fig. 3.11b.

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Fig. 3.10 Reconstructed real-space images around Co in a Ti.0.09 Co.0.01 O.2 and b Ti.0.95 Co.0.05 O.2 thin films. c and d Structure models obtained from the images in (a) and (b), respectively. Adapted from Ref. [45] with reprint permission

In both the experimental and calculated results in Fig. 3.11, sixfold atomic images can be clearly observed at the positions marked by the solid circles, corresponding to the positions of the surrounding Zn atoms inside the Zn.6 Y.8 cluster. However, atomic images cannot be confirmed at the positions marked by the dashed circles in the experimental results, which is in contrast to the calculation. These positions correspond to the surrounding Zn atoms included in the neighboring clusters. Therefore, weak intercluster and strong intracluster correlations are indicated, as schematically shown in Fig. 3.11c. According to the calculation of the atomic images, where the intercluster positional fluctuations were expressed as a Gaussian distribution, the magnitude of fluctuation (half width at half maximum of the Gaussian) was evaluated to be at least 0.33 Å. A study of the deformation behavior of Mg-Zn-Y alloys [49] demonstrated that in-plane ordering is closely related to the ductility. Namely, the ductility of this alloy deteriorates as the in-plane ordering of the Zn.6 Y.8 cluster develops. Thus, the intercluster positional fluctuations revealed in the present XFH study can contribute to the enhancement of ductility. Recently, Hagihara et al. [50, 51] reported that the addition of a small amount of Zn and Y (. v2 > v3 . The neutron wavelength is inversely proportional to its velocity or proportional to ToF as .λ(Å) = 3956. × ToF(s)/L1 (m), where L1 is the distance from the neutron source to the sample position. Based on this relation, the wavelength dependence of the intensity of neutrons or .γ -rays from the NH samples can be obtained by measuring the ToF without any mechanical motions. By using white neutrons with the ToF method, one can obtain a large number of holograms with different wavelengths through one measurement: 130 holograms in the range of 0.3–6.6 Å with the typical operating conditions at BL10. This is important to reduce the twin images in reconstructed atomic images from the holograms; the effect of multi-wavelength measurement is shown in Fig. 3.6 using Barton’s algorithm. The total number of holograms of NH can be changed easily by changing the binning of the wavelength or ToF.

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Fig. 3.19 Principle of the ToF method in J-PARC

Atomic images .U (r) can be visualized from the obtained holograms, .χ (k), by using Barton’s method as in XFH explained in Sect. 3.3.1: { U (r) =

.

{ eikr dk

χ (k)e−ikr dk

(3.27)

|k|=k

The second integral in Eq. (3.27) is performed on the surface of a hologram with a fixed radius .k, and the first integral is over an amplitude of .k. For a practical analysis of the experimental data, the first integral over.k replaces to the sum about the discrete .k selected from the 130 holograms obtained by the ToF method. In comparison with XFH experiments in Synchrotron radiation facilities, in which 5–10 wavelengths can be obtained for one XFH experiment, holograms with up to 130 different wavelengths are an excellent advantage of NH to improve the reliability of atomic images despite the weaker beam flux of neutrons than those of photons in synchrotron radiation facilities. In 2017, Hayashi et al. succeeded in improving the accuracy of atomic images drastically by using white neutrons in MLF and in visualizing local atomic structures around Eu in a typical scintillation crystal 2 at% Eu-doped CaF.2 [67]. From the detailed analysis of the visualized atomic images, some characteristic fluctuation of Ca by Eu doping was found [67]. This means that inverse mode NH with white neutrons can be used for practical investigations in materials science. Thus far, many results have already been obtained in J-PARC, such as semiconductor 0.26 at% Bdoped Si [58], the cage-type rare earth compound 2 at% Sm-doped RB.6 (R: Yb, La) [70], thermoelectric materials, 0.75 at% B-doped Mg.2 Si [71], 0.25 at% B-doped Mg.2 Sn [72], and power semiconductor 0.06 at% B-doped SiC [73]. Some results are introduced in a later section.

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3.5.3 Apparatus Figure 3.20 shows the main part of the instruments for NH installed at BL10 of MLF in J-PARC. The instruments are simple; they consist of a .φ-.ω dual-axes goniometer, .γ -ray detectors, and position-sensitive neutron detectors (PSD) for the neutron Laue camera method (not shown in Fig. 3.20). A single crystalline sample is set at the tip of the horizontal aluminum rod on the .φ goniometer set on the .ω goniometer; the .ω (corresponding to .θ in Fig. 3.8) and .φ axes are vertical and horizontal, respectively, so that the sample rotates about the two intersecting axes. The .ω axis for neutron experiments corresponds to the .θ axis for XFH in Figs. 3.3 and 3.8. The .γ -ray intensity is plotted on a sphere in the k-space with a radius of .k = |k.| = 2.π /.λ as a function of .φ-.ω, which is referred as a NH hologram. The angular range of the .ω axis for NH is 0.◦ .≤ ω ≤ 170.◦ , which is larger than the typical range of 0.◦ .≤ θ ≤ 70.◦ for XFH because of the difference in the penetration depth of neutrons and X-rays. The larger .ω range is an advantage of NH for observations of low-symmetry materials, such as materials without inversion symmetry. The directions of the crystal axes can be determined accurately by the neutron Laue camera method using PSD with an accuracy of approximately 0.3.◦ at the beginning of each experiment, which avoids ambiguity in analysis of holograms. Currently, the maximum neutron beam size of BL10 is 25.× 25 mm.2 at the sample position. It is larger than the size of the samples, because the samples have to be fully in the beam during the .φ-.ω rotations; this is a different requirement from those in XFH and PEH. The element selectivity of inverse mode NH is directly indicated by the energy resolution of the.γ -ray spectra. Two types of.γ -ray detectors with different resolutions are used for NH on BL10 at present: a Bi.4 Ge.3 O.12 (BGO) detector for high efficiency and low energy resolution and poor element selectivity (./E/E = 30% at .∼400 keV), and a CeBr.3 detector for higher energy resolution (4% at .∼400 keV). To enhance the counting efficiency in experiments using a CeBr.3 -type detector, a multidetector system is used as shown Fig. 3.20, because prompt .γ -rays emit isotropically. Though

Fig. 3.20 Main part of instruments for NH installed at BL10 of MLF in J-PARC, which consists of a dual-axis goniometer, .γ -ray detectors and Pb shields

Sample

ϕ

γ-ray

γ-ray detectors

ω

Incident Neutrons

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Ge detectors have excellent energy resolution (0.3% at .∼400 keV) and provide ideal element selectivity, they are not effective for NH at J-PARC because of insufficient counting speed.

3.5.4 Emitter Elements for NH For XFH, elements heavier than K can be used as emitters; therefore, a wide variety of materials can be used as targets of XFH. In contrast, only a few emitter elements are available for inverse mode NH. Basically, elements that generate a stronger prompt .γ -ray are suitable for use as the emitters in NH. Figure 3.21 shows the intensity of the prompt .γ -ray from each element. The horizontal and vertical axes indicate the neutron absorption cross section, .σa , and prompt .γ -ray intensity, respectively. Figure 3.21 shows the ease/difficulty of NH experiments by using these elements as emitters. However, the elements in the upper right have a larger .σa , implying that when their concentrations are excessively high, only the surface regions will be observed. Thus far, only elements in the dashed circles in Fig. 3.21 are observable as dopants: -15

10

10

B

Strong -16

B

10 Highest Analytical Sensitivity (cps /atom)

Cd

Gd Sm

Hg -17

Ta W Mn

-18

10

Weak

V Cr

-19

10

Na

Al

Pd

Ge

Ce Y Zn P H

-20

10

Au

Nd

In Sc

Tm Ag

Yb Co Ti

Eu

Dy

Er

Ho

10

Tb

Ir

Cl

Cu Fe

Sn Si

-21

10

-1

10

0

10

1

10

2

10

3

10

4

10

5

10

Absoption Crosssection (barn)

Fig. 3.21 Intensity of .γ -ray from each element [74]. Horizontal axis indicates neutron absorption cross section [75]. Dashed circle indicates elements that have already been used as emitters

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10

B/natural B, Sm, Eu, and Cd with a very small concentration of 1 at% or less. At the same time, the visualization of atomic images of Cu and Si, whose .γ -ray intensity is 4–5 orders weaker than that of.10 B, has been successful as standard samples, implying that for practical experiments, the optimization of the emitter concentration under the balance of the .γ -ray intensity and absorption effects is important. Materials that can serve as targets for NH remain limited; therefore, complementary use with other atomic-resolution holography techniques (e.g., XFH, PEF, IPEH) is indispensable and effective for investigations of local structures/hyperordered structures.

.

3.5.5 Applications Scintillation Crystal Eu-doped CaF.2 This material is important in the history of atomic-resolution holography because it was the first one in which accurate atomic images around a dopant (Eu) were successfully visualized by NH using white neutrons [67]. Subsequently, studies have actively investigated local structures/hyperordered structures of materials containing light elements. Eu-doped CaF.2 is a typical scintillation crystal that is used for radiation measurements because of its high stability and humidity resistance [76]. Because divalent and trivalent ions are stable for Eu, doped Eu ions are naturally expected to be located at Ca.+2 positions in the form of Eu.+2 . However, X-ray absorption fine structure (XAFS) measurements confirmed that Eu ions are trivalent in Eu-doped CaF.2 [67]. Thus, the position of Eu.+3 in the CaF.2 lattice and the origin of electric neutrality should be confirmed. Figure 3.22 shows the multi-wavelength hologram obtained by superimposing holograms for 34 different wavelengths along the radial direction because the radius of a hologram in.k-space is defined as.|k| = 2.π /.λ. From the 3D holograms in Fig. 3.22, atomic images around doped Eu were successfully visualized as shown in Fig. 3.23a.

Fig. 3.22 Holograms of prompt.γ -rays from Eu in 2% Eu-doped CaF.2 [67]. a 3D volume hologram, and b hologram at .k = 4.05Å.−1 . Reproduced with permission from Ref. [67]

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Fig. 3.23 3D atomic images reconstructed a from the obtained holograms and b from simulated holograms using a simple undistorted model of CaF.2 [67]. Reproduced with permission from Ref. [67]

First, the experimentally obtained atomic image shown in Fig. 3.23a is basically consistent with that obtained from simulated holograms by using a model in which Eu is located at the Ca position in the undistorted CaF.2 lattice shown in Fig. 3.23b. This indicates that the doped Eu.3+ is definitely located at the Ca.2+ position as expected, despite the difference in valency. Thus, the origin of the electric neutrality remains unclear. Previous studies claimed that the modification of a local structure is stabilized by the capture of excess fluoride ions that might be located at a neighboring interstitial position. Careful analysis of the atomic images shown in Fig. 3.23a revealed excess F around doped Eu. Figure 3.24 shows the (100) Ca plane including doped Eu (green). The red image in the rectangle in which the maximum scale expanded by a factor of 4 can be found at 2.3 Å from doped Eu. The distance between Eu-excess F is consistent with that obtained by XAFS. Although Eu.3+ -interstitial F.− coupling has been proposed in previous studies, Fig. 3.24 shows the first evidence obtained directly through experiments. From a comparison between the experimental and simulated atomic images, another important finding is that the nearest Ca (dashed circles) in Fig. 3.23a is not obvious and is split into two parts, implying that the positions of Ca fluctuate/distribute around the ideal position of Ca owing to Eu doping, while the nearest neighbor (NN) F is not affected. Figure 3.24b shows the cross sections of the 3D experimental images in the Ca (100) plane. The Ca images split with distance of 1.2 Å. In contrast, the NN F around doped Eu is clearly visualized in Fig. 3.23a, and it indicates that Eu doping does not affect the NN F site despite the shorter distance from Eu than that from the split nearest Ca.

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b

Fig. 3.24 2D images at typical atomic (100) planes. a Ca plane at z = 0.0 Å. Eu (green) was located at the origin. b Ca plane at z = 2.70 Å from Eu [67]. Reproduced with permission from Ref. [67]

Cage-Type Rare Earth Compound: Sm-doped RB.6 (R: Yb, La) Rare earth hexaborides (RB.6 , where R is a rare earth) have attracted interest because of their unique electric, magnetic, and physical properties, including the competition between their magnetic and multipolar orderings, Kondo effects, and valence ¯ in fluctuations. Figure 3.25 shows the crystal structure of RB.6 (space group: Pm.3m), which a rare earth atom (yellow) is surrounded by the cage of 24 B. Inelastic X-ray scattering revealed anomalous phonon softening in GdB.6 , TbB.6 , and DyB.6 [77–79]. Such softening was also observed by Raman scattering, where the energy and intensity of the R motions can be scaled by the free space in the B-cage [80, 81]. Thus, the free space for the R motion in the B-cage in Fig. 3.25 must be an important factor for the unusual properties of this system. In contrast, the dispersion relation of YbB.6 along the [100] direction does not show particular softening. The difference in phonon behavior must be due to the divalency of Yb in YbB.6 , whereas R in RB.6 (R = Gd, Tb, Dy) is trivalent because of the difference in ionic radius.

Fig. 3.25 Crystal structure of RB.6 (R: rare earth)(Space ¯ group: Pm.3m)

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Fig. 3.26 Hologram of Sm dope Yb.11 B.6 at Sm at .λ = 2.37Å [70]. Reproduced with permission from Ref. [70]

Fig. 3.27 Atomic images (red) of Yb (a) and B (b) in the (100) plane of 2% Sm dope Yb.11 B.6 [70]. Circles and dashed square are expected positions of Yb and unit cell. Reproduced with permission from Ref. [70]

Figure 3.26 shows a hologram of 2% Sm-doped Yb.11 B.6 at .λ = 2.37 Å [70]; it is consistent with the simulated one, implying that the hologram patterns were observed successfully. Note that because natural B has a strong neutron absorption cross section, .σa , the sample was prepared using .11 B as it has a negligible .σa . Figure 3.27 shows atomic images (red) of (a) Yb and (b) B in the (100) planes of 2% Sm-doped Yb.11 B.6 [70]. Circles and dashed squares indicate the expected positions of Yb/B and unit cell, respectively. Yb and B are clearly observed around Sm up to approximately 15 Å from the central Sm. From the comparison with atomic images of the undoped Si, the fluctuation of Sm in the B-cage could be estimated; the isotropic mean-square displacement .σ = 0.25(4) Å for YbB.6 . Figure 3.28 shows the distance dependence of the intensity of atomic images of (a) Yb and La and (b) B. The horizontal axes indicate the distance from Sm. The dashed

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Fig. 3.28 Distance dependence of intensity of a Yb and b B. Horizontal axis indicates the distance from Sm. Dashed lines indicate the simulation using the undistorted RB.6 lattice. Reproduced with permission from Ref. [70]

lines indicate the simulation using the undistorted RB.6 lattice. The intensity of Yb and La (Fig. 3.28a) is consistent with the simulation, indicating that the fluctuation of Yb/La is negligible. For B (Fig. 3.28a), the intensity of the NN B shows an obvious reduction of 28%, and those of the second NN and further B are consistent with the simulation (dashed line). Because the intensity is reduced if the atoms are fluctuating, this indicates that the NN B is fluctuating owing to Sm doping, and the second NN and further B show no obvious fluctuations. By quantitative analysis based on comparisons with simulations of the effects of fluctuations, the isotropic mean-square displacement of B is estimated as .σ = 0.28 Å, which cannot be interpreted just by the thermal vibration in undoped RB.6 as determined by the conventional diffraction technique. Thus, the fluctuation of B must be caused by Sm doping. The doping effect on Yb/La and the second NN and further B is suppressed because of strong valence bonding in the B-cage. This means that only the NN B cage has the distortion caused by Sm doping, implying that Sm doping realizes disorder in ordered structure (lattice) for hyperordered structures.

3.6 Photoelectron Holography (PEH) When a sample is irradiated by photons, electrons in the core level are excited, and photoelectrons or Auger electrons are emitted from the atoms in the sample. Photoelectron spectroscopy is used to analyze the kinetic energy of photoelectrons. Because the energy level of a core electron depends on the element and valence, the chemical composition of a sample can be determined. PEH is a method based on core-level photoelectron spectroscopy, from which information on the atomic arrangement around a photoelectron emitter can be obtained.

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The recording process of PEH was mentioned in the previous section. When a monochromatic excitation light is incident on a sample surface, photoelectrons are excited. A spherical wave of photoelectrons centered on the emitter atom is generated. The spherical wave is scattered by the surrounding atoms to produce a scattered wave. The spherical wave and the scattered wave interfere with each other. If the initial spherical wave and the scattered wave are considered to be the reference wave and the object wave, respectively, this interference pattern can be regarded as a hologram. This interference pattern is the photoelectron hologram. In some cases, it is also called photoelectron diffraction, which is referred to as a photoelectron hologram hereafter. This hologram reflects the 3D atomic structure around the atom that emits the photoelectrons. Through computations, a 3D atomic arrangement can be reconstructed without a structural model. In 1990, Harp performed PEH and attempted to reconstruct a 3D atomic image. A photoelectron hologram is easy to observe because the hologram amplitude is more than 10% of the photoelectron intensity [82]. However, owing to the complexity of the electron scattering process, a good atomic image could not be obtained despite the efforts of many researchers until a new method was developed. In 2004, Matsushita demonstrated that an atomic image could be obtained by using a fitting-based calculation method [6, 83] that differs from the Fourier-transform-based method. Subsequently, this method has been improved, and sparse-modeling-based theories have been proposed in recent years. PEH has the excellent feature of being able to reproduce the local structure around the atom that emits the photoelectrons. The binding energy of the core level is element-specific. Furthermore, chemical shifts can be used to distinguish atomic sites with different bonding patterns and chemical environments. Owing to the atomic site selectivity of photoemission spectroscopy, impurities in crystals, interfaces, surface adsorbates, and thin films can be observed if the local structure is oriented even without any long-range order. The energy of excited photoelectrons is attenuated by inelastic scattering in materials. The inelastic mean free path (IMFP) is short. Therefore, information can be obtained from the surface to a depth of about the length of the IMFP. For example, photoelectrons excited with soft X-rays have a kinetic energy of a few hundred eV. In this case, the measurement can be limited to a depth of approximately 1 nm from the surface. When hard X-rays are used, the kinetic energy of the photoelectrons is several keV, which results in a longer IMFP and allows bulk-sensitive measurements. The authors have developed an atomic image reconstruction theory that incorporates the scattering process of photoelectrons [6, 7, 83–87]. This method has a feature that 3D atomic images can be reconstructed from a single-energy photoelectron hologram without using the multiple-energy (wavelength) method.

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3.6.1 Apparatus Some apparatus can be used to measure photoelectron holograms. There are two types of excitation light sources: X-ray tubes in a laboratory and synchrotron radiation at a synchrotron radiation facility. In the case of X-ray tubes, emissions from magnesium (K.α 1253.61 eV) and aluminum (K.α 1486.6 eV) are used. In a synchrotron radiation facility, micrometer-sized, bright, monochromatic X-rays with variable energy are available; this enables the measurement of small samples. To measure a photoelectron hologram, the angular distribution of the photoelectrons emitted from a sample should be detected. Therefore, an electron analyzer with good angular resolution is required. Two types of electron analyzers can be used to measure the two-dimensional (2D) angular distribution. One is a commercial electron energy analyzer, namely, a concentric hemispherical analyzer (CHA), which measures the 2D angular distribution by rotating both the polar and azimuthal angles of the sample as shown in Fig. 3.29a. This energy analyzer provides good energy resolution; however, it requires a long measurement time because of the scanning over the angles. The another is the 2D display-type spherical mirror analyzer (DIANA) [87], whose structure is shown in Fig. 3.29b. Photoelectrons emitted from the sample are bent by the electric field such that they converge on the aperture; after passing through the aperture, the 2D angular distribution of photoelectrons is projected onto the screen. This analyzer can measure 2D angular distributions instantaneously. Recently, a high-resolution retarding field analyzer (RFA) has been developed [88, 89], as shown in Fig. 3.29c. By applying a negative voltage to the spherical grid in the RFA, photoelectrons with a higher kinetic energy are projected onto the screen. The RFA is a photoelectron high-pass filter. Lock-in measurement by varying the negative voltage makes it a bandpass filter. By developing the spherical electrode, both high energy resolution and high angular resolution were achieved. The RFA has a simple structure and is expected to be widely used in future PEH measurements.

3.6.2 Analysis Method To analyze photoelectron holograms, it is important to understand the recording process. Energy is conserved before and after photoelectron excitation (Fig. 3.30). The binding level of the initial state is denoted as . E B . Usually, the origin of . E B is taken at the Fermi energy (. E F ). Similarly, the energy . E f of the final state is simply expressed as follows when the origin is taken from the Fermi surface: .

E f = hν − E B ,

(3.28)

where .hν is the photon energy. When a photoelectron is excited into vacuum, the vacuum level becomes the origin. The difference between the vacuum level (. E V ) and Fermi energy (. E F ), that is, the work function .φ, is expressed as

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(a) CHA Screen

(b) DIANA

(c) RFA Sample Rotations

SX Screen electron orbits

Screen ~40mm

SX Sample

Fig. 3.29 Apparatus for PEH. a Concentric hemispherical analyzer (CHA). The angular distribution of photoelectrons excited by soft X-rays (SX) is measured by scanning the sample angles. b Displaytype spherical mirror analyzer (DIANA). Emitted photoelectrons are focused on a pinhole and projected on the screen. c Retarding field analyzer (RFA). The photoelectron angular distribution is directly projected on the screen

.

E kvac = E f − φ = hν − E B − φ.

(3.29)

The energy from the Fermi level to the bottom of the valence band is called the internal potential .V0 . Therefore, the kinetic energy of electrons in a solid can be written as follows, which is used for the formation of the hologram: .

E kholo = hν − E B + V0 .

(3.30)

The emitted photoelectron or Auger electron can be defined as ϕ L (k, r, θ, φ) = k

E

.

lm

A Llm i l+1 h l(1) (kr )Ylm (θ, φ),

(3.31)

3 Atomic-Resolution Holography

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Fig. 3.30 Energy conservation in PEH

Photoelectron Ef Ekvac

Ekholo

Energy

EV ϕ

EF

Valence band V0

hν

Surface

EB

Vacuum

Core level Solid

Depth

√ where .k is the wavenumber given by .k = 2m E k /h, . A Llm is a coefficient, and . L is an index for multiple excited states. .h l(1) (kr ) is a Hankel function of the first kind. .Ylm (θ, φ) are spherical harmonics. The coefficient . A Llm depends on the transition process. For example, when the emitted wave function is an .s-wave, the coefficient is set to . As00 = 1, and the wave function is given as 1 eikr . ϕ (k, r, θ, π ) = √ 4π r

. s

(3.32)

Here, the angles .θ and .φ can be regarded as the motion vector: ϕ L (k, r, θ, φ) = ϕ L (k, r) = ϕ L (k, r ).

.

(3.33)

Then, the emitted wave is scattered by the surrounding atoms. The scattered wave function caused by the scatterer atom located at the position vector . a is defined as .ψ L (k, r, a). The whole wave function is given by ϕ L (k, r) =

E

.

ψ L (k, r, a h ).

(3.34)

h

Here, .h is the index of the scatterer atom. The wave function at a distant location is given as −ikr .ψ L (k, a) ≡ lim r e ψ L (k, r, a) (3.35) r →∞

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ϕ L (k) ≡ lim r e−ikr ϕ L (k, r).

.

r →∞

(3.36)

The observed intensity is given as | |2 | E || E | . I (k) = ψ L (k, a h )| . |ϕ L (k) + | | L

(3.37)

h

The hologram function .χ (k) is defined by removing the reference wave intensity I (K ). Therefore, EE .χ (k) = 2R[ϕ L∗ (k)ψ L (k, a h )]. (3.38)

. 0

h

L

Photoelectron Emitter According to the perturbation theory, the photoelectron wave function is given by .

A Llm = −2πi

E

,

(3.39)

lm

where .φ L (k) is the initial state of the core-level electron and . L is an index used to distinguish the initial states. For example, in the case of . p-initial states, . L = px , p y , pz should be used. .olm (k, r) is a basis function of the excited photoelectron given by ) ( r 0.01 mm.

.>1 mm.

Yes

No

2

NH

IPEH

Neutron/.γ -ray

Electron/Fluo. X-ray √

√

.

.

Bulk .∼1 cm √

Surface 1–10 nm .×

0.5–1 cm.3 No

.>1 mm.

.

2**

Yes

normal-mode XFH, .10 × 10 .µm.2 or less can be a target. Further, such a size is possible for inverse mode XFH, if a technique to keep the incident X-ray from irradiating a specific area during sample rotation can be developed ** 2 . Ideally, 1 .µm. or less is possible. A technique needs to be developed to keep the electron beam from irradiating a specific area * For

.

Table 3.2 Features and requirements of PEH, XFH, NH, and IPEH Features

Requirements

PEH

– Chemical-state-specific structural analysis is possible

– Surface cleaning is necessary

XFH

– High reliability of atomic images

– Lightest measurable element is potassium

– Sample environment is flexible NH

– High reliability of atomic images

– Observable emitters are limited

– Multi-wavelength holograms can be efficiently collected IPEH

– Laboratory experiment is possible

– Surface cleaning is necessary – Measurement time is long

Acknowledgements The authors gratefully acknowledge the Ministry of Education, Culture, Sports, Science and Technology, MEXT, of Japan for the financial supports through JSPS Grantsin-Aid for Transformative Research Areas (A) “Hyper-Ordered Structures Sciences” via Grant Nos. 20H05878, 20H05881, 20H05884, and 21H05547, Innovative Areas “Hydrogenomics” via Grant Nos. 19H05045 and 21H00013,"Materials Science on Mille-Feuille Structure" via Grant Nos. 20H05878 and 20H05881, “3D Active-Site Science” via Grant Nos. 26105001, 26105006, 26105013, 26105014, and 26105010, “Materials Science on Synchronized LPSO Structure” via Grant No. 23109001, Scientific Research (A) through Grant Nos. 19H00655 and 20H00303, Scientific Research (B) through Grant Nos. 21H01027 and 22H01774, and Young Scientists via Grant No. 20K15024. This work was also supported by the Nippon Sheet Glass Foundation for Materials Science and Engineering and Japan Association for Chemical Innovation. The synchrotron experiments were carried out with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) and Photon Factory Program Advisory Committee. The neutron experiments were performed under the user program of the Materials and Life Science Experimental Facility of J-PARC.

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Chapter 4

X-Ray and Neutron Pair Distribution Function Analysis Yohei Onodera, Tomoko Sato, and Shinji Kohara

Abstract Pair distribution function (PDF) obtained by X-ray diffraction (XRD) and neutron diffraction (ND) measurements enables us to probe the structure of disordered materials, which has no long-range order and periodicity. This chapter provides an introduction to PDF analysis using XRD and ND techniques. A brief outline of the theory of diffraction for disordered materials is given with a focus on the use of various real-space functions. The structures of single-component disordered materials are introduced to understand the origins of characteristic diffraction peaks, i.e., the first sharp diffraction peak (FSDP) and the principal peak (PP), observed in broad halo patterns. Furthermore, the instrumentation of synchrotron X-rays and neutrons for PDF analysis with associated results for the structural studies of disordered materials under high temperature and high pressure are reviewed. Keywords X-ray diffraction · Neutron diffraction · Pair distribution function · Glass · Structure

Y. Onodera Institute for Integrated Radiation and Nuclear Science, Kyoto University, Osaka 590-0494, Japan e-mail: [email protected] T. Sato Department of Earth and Planetary Systems Science, Hiroshima University, Hiroshima 739-8526, Japan e-mail: [email protected] S. Kohara (B) Research Center for Advanced Measurement and Characterization, National Institute for Materials Science, Ibaraki 305-0047, Japan e-mail: [email protected] © Materials Research Society, under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. Hayashi (ed.), Hyperordered Structures in Materials, The Materials Research Society Series, https://doi.org/10.1007/978-981-99-5235-9_4

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4.1 Introduction The complementary use of different quantum beam diffraction (X-ray diffraction (XRD) and neutron diffraction (ND)) measurements, such as synchrotron X-ray, neutron, and electron diffraction measurements, is quite useful in investigating the atomistic structure of functional materials. In the case of crystalline materials, their crystal structure can be determined on the basis of crystallography, which focuses on the symmetry and periodicity of the atomic arrangement in crystalline materials, with the aid of diffraction measurements. On the other hand, the structure of disordered materials, e.g., liquid, glassy, and amorphous solids, has no distinct translational symmetry and periodicity, and therefore cannot be analyzed on the basis of crystallography. Therefore, the measurement of pair distribution function (PDF) using different quantum beam diffraction techniques is a canonical approach to probing atomic arrangements in disordered materials. The PDF expresses the probability of finding atomic pairs separated by a certain distance, which provides us with the real-space information on the structure of disordered materials. Furthermore, PDF analysis has recently been used to examine the local structural disorder in crystalline materials. In this chapter, a brief introduction to PDF analysis using XRD and ND is given. First, the basic theory of diffraction experiments is described. Next, the structures of single-component oxide glasses, such as silica (SiO2 ) glass are reviewed to understand the origins of diffraction peaks in broad halo patterns. Moreover, the structure and diffraction data of oxide liquids and glasses under high temperatures and high pressures are introduced.

4.2 Diffraction Theory In PDF analysis, diffraction patterns can be considered as a function of Q, which gives the magnitude of the scattering vector given by Q=

4π sin θ , λ

(4.1)

where 2θ is the scattering angle and λ is the incident wavelength of X-rays or neutrons. The scattering intensity of materials containing n chemical species is normalized to give the total structure factor, S(Q) [1, 2], S(Q) = 1 +

n ∑ n ∑ [ ] 1 cα cβ wα∗ (Q)wβ (Q) Sαβ (Q) − 1 , 2 |⟨W (Q)⟩| α=1 β=1

(4.2)

where cα is the atomic fraction of chemical species α and wα (Q) is either a Qindependent coherent scattering length in ND or Q-dependent atomic scattering (form) factor with dispersion terms in XRD. S αβ (Q) is a partial structure factor

4 X-Ray and Neutron Pair Distribution Function Analysis

95

and ⟨W (Q)⟩ =

∑

cα wα (Q).

(4.3)

α

Since X-rays are scattered by electron clouds in atoms, the X-ray atomic form factor depends on the atomic number. On the other hand, neutrons are scattered by their interactions with atomic nuclei, and the neutron coherent scattering length varies independent of the atomic number. In other words, X-rays are sensitive to heavy elements whereas neutrons are sensitive to light elements. Therefore, the complementary use of XRD and ND is robust for revealing the atomic arrangements in materials containing several chemical species. The corresponding structural information in real space is contained in the PDF g(r) obtained by a Fourier transform of S(Q), Q max ∫

1 g(r ) = 1 + 2π 2 rρ

Q(S(Q) − 1)sin(Qr )M(Q)d Q,

(4.4)

Q min

where r is the interatomic distance, ρ is the atomic number density in Å–3 , and M(Q) is the Lorch [3] modification function, M(Q) = sin(πQ/Qmax )/(πQ/Qmax ) for Q < Qmax , and M(Q) = 0 for Q > Qmax where Qmax is the maximum value of Q. To obtain structural information with a sufficient real-space resolution, it is indispensable to obtain S(Q) up to the high-Q region because a high real-space resolution is achieved by a Fourier transform of S(Q) with a large Qmax [4]. The structure of a material containing n chemical species is given by n(n + 1)/2 of these partial PDFs. The reduced PDF G(r) is derived from a Fourier transform of S(Q) as follows. 2 G(r ) = π

Q max ∫

Q(S(Q) − 1)sin(Qr )M(Q)d Q.

(4.5)

Q min

The PDF g(r) is derived from the following equation. g(r ) =

G(r ) + 1. 4πrρ

(4.6)

The total correlation function T (r) and the radial distribution function RDF(r) are obtained from the following equations: T (r ) = G(r ) + 4πrρ = 4πrρg(r ),

(4.7)

R D F(r ) = r G(r ) + 4πr 2 ρ = 4πr 2 ρg(r ) = r T (r ).

(4.8)

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The coordination number of type β atoms around a type α atom, N αβ , which is the number of atoms of type β between r 1 and r 2 (r 1 < r 2 ) from an atom of type α, is calculated from the partial radial distribution functions RDF αβ (r) as ∫r2 Nαβ =

∫r2 4πr 2 cβ ρg αβ (r )dr.

R D F αβ (r )dr = r1

(4.9)

r1

Therefore, it is possible to obtain the coordination number when individual peaks, which arise from one partial radial distribution function, are assigned and extracted from RDF(r). T (r) is commonly used to analyze the atomic arrangement in oxide glasses. The reason for choosing T (r), which scales as r, rather than other functions such as g(r), which scales as a constant, and RDF(r), which scales as r 2 , is that it is broadened by thermal vibration [5]. Thus, T (r) is more suitable for peak fitting than g(r) [6]. By using the real-space functions mentioned above, we can obtain interatomic distances and coordination numbers of materials.

4.3 PDF Diffractometers at Advanced Quantum Beam Facilities The advanced instrumentations at neutron and synchrotron facilities provide diffraction data in a wide Q range, which is achieved by using short-wavelength (highenergy) neutrons and X-rays, thereby providing diffraction data in a wide Q range [4]. High-energy X-ray PDF diffractometers are available at several synchrotron facilities, e.g., BL04B2 and BL08W at SPring-8, Japan [7], I15-1 at Diamond Light Source, UK [8], and 11-ID-C at the Advanced Photon Source, USA [9]. In the case of neuron diffractometers, D4 at the Institut Laue–Langevin, France [10], NOMAD at Spallation Neutron Source, USA [11], GEM and NIMROD at ISIS Neutron Source, UK [12], and NOVA at J-PARC, Japan [13] are available for PDF analysis of disordered materials.

4.4 GeO2 Crystal and Glass Figure 4.1a shows the X-ray S(Q) of a GeO2 crystal and glass [14] obtained by synchrotron XRD measurements. Germanium dioxide (GeO2 ) is a prototypical glassforming material and shares the same local structural motif as its crystalline counterpart. The germanium–oxygen coordination number is 4, suggesting that a tetrahedral corner-sharing network is formed. Although the S(Q) of the GeO2 crystal shows Bragg peaks that reflect its long-range periodicity, the S(Q) of the GeO2 glass shows a broad halo pattern owing to the lack of periodicity. The T (r) are shown in

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Fig. 4.1 a X-ray total structure factors S(Q) and b total correlation functions T (r) of GeO2 crystal and glass

Fig. 4.1b. In T (r) data of both the GeO2 crystal and glass, Ge–O correlation peak is clearly observed at 1.74 Å since the GeO4 tetrahedron is the shared structural motif in crystalline and glassy GeO2 . On the other hand, the Ge–Ge correlation peak, which corresponds to the distance between centers of corner-sharing GeO4 tetrahedra, is observed at ~3.2 Å and is obviously diminished in the T (r) for GeO2 glass, indicating that the structure of GeO2 glass is disordered in a length scale larger than the first cation–cation correlation length. Moreover, it is difficult to assign a peak beyond 4 Å in T (r) to an atomic pair correlation for GeO2 glass. To investigate the length scale beyond the nearest-neighbor correlation length, “intermediate-range order,” structure modeling is an effective tool (see Chap. 10 for more information).

4.5 SiO2 Glass Silicon dioxide (SiO2 ) is classified into a glass-forming material along with GeO2 , and it has a network structure in which the interconnections of SiO4 tetrahedral motifs form a network by the corner sharing of oxygen atoms. Figure 4.2a shows the X-ray [7] and neutron [15] structure factors S(Q) for SiO2 glass. The first sharp diffraction peak (FSDP) [16, 17] appears at Q ~ 1.5 Å–1 in both X-ray and neutron S(Q)s, whereas the principal peak (PP) appears at Q ~ 3 Å–1 in only the neutron S(Q) because the PP is correlated with the packing of oxygen atoms in oxide glasses [18]. By using Eq. (4.2), we calculate the weighting factors in the X-ray and neutron S(Q) for SiO2 glass as follows: X-ray S(Q) = 0.218SSi−Si (Q) + 0.498SSi−O (Q) + 0.284SO−O (Q),

(4.10)

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Neutron S(Q) = 0.069SSi−Si (Q) + 0.388SSi−O (Q) + 0.543SO−O (Q),

(4.11)

where the Q-dependent atomic form factors are approximated using atomic numbers. As can be seen in the weighting factors calculated above, the weighting factor of the O–O correlation is large in the neutron S(Q) compared with that in the X-ray S(Q). Indeed, the X-ray- and neutron-weighted partial structure factors S αβ (Q) obtained by a combination of classical molecular dynamics (MD) simulation and reverse Monte Carlo (RMC) [19, 20] modeling indicate that the FSDP is observed owing to the positive contributions of S Si–Si (Q), S Si–O (Q), and S O–O (Q) in both the X-ray and neutron S(Q)s, whereas the positive Si–Si and O–O PPs are canceled out by the negative Si–O PP of the X-ray S(Q) [19, 20]. Therefore, PP is visible in only ND data for oxide glasses, because the oxygen–oxygen correlation is dominant owing to its large weighting factor for ND. The FSDP was first discussed by Wright and Leadbetter in 1976 [21], although it seems that the term “FSDP” was first used by Phillips in 1981 [22]. It is common knowledge that the FSDP of SiO2 glass is associated with the continuous random network model proposed by Zachariasen [23], and the origin of the FSDP of SiO2 glass has long been discussed [22, 24]. Onodera et al. have reported in 2019 that the FSDP in SiO2 glass originates from the arrangement of tetrahedral SiO4 motifs with the periodicity of 4 Å given by 2π/QFSDP , where QFSDP is the position of the FSDP, with a coherence length of 10 Å given by 2π/ΔQFSDP (ΔQFSDP is the full width at half-maximum of the FSDP) [19]. The T (r) of SiO2 glass is shown in Fig. 4.2b. In T (r) data of both the X-ray and neutron, the Si–O correlation peak is clearly observed at 1.62 Å and the Si–O coordination number is 4.0 ± 0.1. The O–O and Si–Si correlation peaks are observed at 2.63 and 3.08 Å, respectively. It is notable that a prominent O–O correlation peak is observed in neutron T (r), whereas a clearer Si–Si correlation peak is observed in X-ray T (r). This is because of the difference between the weighting factors in XRD and ND data as shown in Eqs. (4.10) and (4.11), suggesting that the complementary use of X-ray and neutron enables us to analyze the glass structure more precisely.

4.6 Other Single-Component Disordered Materials Figure 4.3 shows total structure factors S(Q) of glassy (g)-Cu50 Zr50 [19], amorphous (a)-Si [25], g-ZnCl2 [26], g-GeSe2 [27], g-GeS2 [28], g-GeO2 [29], g-SiO2 [15], liquid (l)-CCl4 [30], and l-P [31]. Note that Q is scaled by the nearest-neighbor distance d appearing in the T (r) to eliminate the atomic size differences. The average metal–metal coordination number in g-Cu50 Zr50 is approximately 12 and the local structural unit of g-Cu50 Zr50 can be regarded as an icosahedron. The structural unit of other materials is a regular tetrahedron, although a-Si and all g-AX2 materials have a network structure, whereas the two molecular liquids in which CCl4 and P4 tetrahedra are isolated do not form a network structure. In Fig. 4.3, three peaks, Q1 (FSDP), Q2 (PP), and Q3 , can be observed in the S(Q) for all materials except for a-Si

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Fig. 4.2 a X-ray and neutron total structure factors S(Q) and b total correlation functions T (r) of SiO2 glass

and g-Cu50 Zr50 , whereas the split of PP is observed in the S(Q) for molecular liquids. Both g-ZnCl2 [26] and g-GeSe2 [27] have a small fraction of edge-sharing tetrahedra as well as corner-sharing ones. However, the three (Q1 , Q2 , Q3 )-peak structure seems to arise from mostly corner-sharing tetrahedra because the ratio of edge-sharing to corner-sharing tatrahedra is very small [26, 27]. The FSDP of g-GeO2 observed at a larger Qd value stems from the structure with the higher packing of atoms in gGeO2 [32, 33]. Although the FSDP is observed in oxide glasses with a corner-sharing polyhedral network, it also appears in the S(Q) of l-CCl4 [30], l-P [31] (Fig. 4.3), and l-KPb [34–36] and other molecular liquids [37]. Thus, it is suggested that the FSDP is not a signature of a network formation. A prominent FSDP has appeared in the diffraction patterns of other network-forming materials, such as g-B2 O3 [38] and g-As2 O3 [39], but not in a-Si and a-Se [40]. Therefore, it can be concluded that the FSDP is a signature of a sparse distribution of planes in polyhedra, because the FSDPs of g-SiO2 [41, 42] and l-P [31] diminish with the reduction of cavity volumes associated with the increase in pressure. The origin of the PP is obvious in oxide glasses because the PP is correlated with the packing of oxygen atoms (at the corner of polyhedral units) [18]. For instance, the PP of the neutron S(Q) of g-SiO2 becomes sharper under high pressure [42]. On the other hand, the PP of the neutron S(Q) of l-CCl4 shown in Fig. 4.3 is split, which indicates the presence of intermolecular orientational correlations of CCl4 tetrahedral motifs [37, 43, 44]. The orientational correlations appear in l-P and presumably in g-As2 O3 [39] because their S(Q) shows a split PP. Therefore, it seems that the PP reflects inter-polyhedral correlations observed in a small-length scale compared with FSDP in disordered materials. It is notable that the transition from a low-density

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Fig. 4.3 Total structure factors S(Q) of g-Cu50 Zr50 [19], a-Si [25], g-ZnCl2 [26], g-GeSe2 [27], g-GeS2 [28], g-GeO2 [29], g-SiO2 [15], l-CCl4 [30], and l-P [31]. Scaling to the magnitude of scattering vector Q is applied by multiplying Q by d (first interatomic distance observed in total correlation functions). In the case of l-P, d is calculated on the basis of the side length of a P4 tetrahedron

molecular liquid to a high-density network in l-P under extreme conditions leads to the diminishment of FSDP. On the other hand, a remarkable FSDP appears in the X-ray S(Q) of the low-density molecular l-P (Fig. 4.3). This result implies that the FSDP also appears in the arrangement of tetrahedra without the central atom, because the S(Q) of molecular l-P shows the remarkable FSDP even though a P4 tetrahedron does not contain a central atom as shown in Fig. 4.3. Only Q3 is found in X-ray S(Q) of g-Cu50 Zr50 , which has a dense random packing (DRP) structure [45–47]. Therefore, the local structure of g-Cu50 Zr50 is markedly different from a tetrahedral structure owing to the absence of a strong chemical bond, indicating that the PP is a signature of the formation of a chemical bond because g-Cu50 Zr50 has no strong chemical bond owing to its DRP structure. Kohara et al. have compared the atomic structure of liquid and solidified (crystalline and amorphous) Si with those of SiO2 [48]. Although the short-range structural unit is a regular tetrahedron in a-Si and g-SiO2 , the chemical contrast (Si is fourfold and O is twofold) in the SiO4 unit in g-SiO2 is not found in the SiSi4 unit because the number of all atoms is fourfold in a-Si. Figure 4.4a shows the X-ray S(Q) of

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a-Si [25] together with that of l-Si (1770 K) [19]. In the X-ray S(Q) of a-Si, prominent Q2 (PP) and Q3 are observed at Qr A–X ~ 5 and 8.5, respectively, although no Q1 (FSDP) is observed. However, the S(Q) of l-Si is markedly different from that of a-Si, as shown in Fig. 4.4a. This is because of the large difference in the Si–Si coordination number N Si–Si , 3.9 in amorphous solid [25] versus 5.7 in liquid [19], associated with the remarkable increase in density from 2.30 g/cm3 in amorphous solid to 2.57 g/cm3 in liquid. Figure 4.4b shows the X-ray [7] and neutron [15] S(Q) of g-SiO2 together with the X-ray S(Q) of l-SiO2 (2019 K) [49]. Note that the X-ray S(Q) of l-SiO2 is comparable to that of g-SiO2 . A sharp FSDP is observed in both the S(Q)s of l/g-SiO2 , indicating that strong Si–O covalent bonds are maintained even in l-SiO2 . These behaviors are consistent with the small differences in the Si–O coordination number N Si–O (4.0 in glass and 3.9 in liquid) and density (2.20 g/cm3 in glass and 2.10 g/cm3 in liquid) between glass and liquid. Furthermore, Kohara et al. revealed the difference between a-Si and g-SiO2 in terms of the network topology revealed by quantum beam diffraction measurements, structure modeling based on diffraction data, and a series of topological analyses. They showed that the narrower ring size distribution and the smaller cavity volume ratio in a-Si than in g-SiO2 is a signature of an extremely poor amorphous-forming ability of a-Si [48]. Moreover, they concluded that the chemical contrast in the corner-sharing tetrahedral network in AX2 (A, fourfold cation; X, twofold anion) is crucial for good amorphous-forming ability [48].

4.7 Al2 O3 Glass (Single-Component Intermediate Oxide Glass) The basic concept for the formation of glass is the tetrahedral motif with cornersharing oxygen atoms proposed by Zachariasen in 1932. In Zachariasen’s rule, the coordination number of oxygen atoms (n) around a cation (A) is limited to 3–4 and AOn polyhedra form a corner-sharing network [23]. Fifteen years after Zachariasen’s proposal, Sun reported the classification of single-component oxides into glass formers, glass modifiers, and intermediates [50] (The details of the classification proposed by Sun are discussed in Chap. 15). Alumina (Al2 O3 ) is classified into an intermediate. Al2 O3 acts as a glass former and a glass modifier in binary oxide glasses, although Al2 O3 cannot sorely form glass via a melt-quenching approach. Hashimoto et al. reported that electrochemically synthesized amorphous Al2 O3 shows a glass transition, demonstrating that Al2 O3 is a glass (g-Al2 O3 ) [51]. The density of gAl2 O3 measured using a gas pycnometer is 3.05 g/cm3 , which is smaller than those for crystalline Al2 O3 (α-Al2 O3 , 4.00 g/cm3 ; γ-Al2 O3 , 3.59 g/cm3 ) and slightly larger than that for l-Al2 O3 (2.92 g/cm3 ) [52]. Figure 4.5 shows the X-ray and neutron total structure factors S(Q) of g-Al2 O3 , g-SiO2 [7, 15], and l-Al2 O3 (2400 K) [52]. The FSDP, which originates from a sparse distribution of planes in polyhedra, is clearly observed at Q = 1.52 Å–1 in the S(Q) of g-SiO2 , whereas the FSDP observed at

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Fig. 4.4 a X-ray total structure factors S(Q) of a-Si [25] and l-Si (1770 K) [19]. b X-ray total structure factors S(Q) of g-SiO2 [7] and l-SiO2 (2019 K) [49], together with neutron S(Q) [15] of g-SiO2 . Scaling to the magnitude of scattering vector Q is applied by multiplying Q by r A–X (first interatomic distance observed in total correlation functions). Taken from Ref. [48]

Q ~ 2 Å–1 is not sharp in the S(Q) of g-Al2 O3 , suggesting that the absence of the corner-sharing polyhedral motif with a large cavity volume, which is a signature in typical glass formers. The neutron S(Q) of g-Al2 O3 shows an extraordinarily sharp PP, whereas the X-ray S(Q) of g-Al2 O3 shows no PP owing to the small O–O weighting factor. Since PP is an indicator of the packing of oxygen atoms, the extraordinarily sharp PP in the neutron S(Q) indicates that a structure with densely packed oxygen atoms is formed in g-Al2 O3 . The neutron S(Q) of l-Al2 O3 is broader than that of g-Al2 O3 especially in the PP, whereas the X-ray S(Q) of l-Al2 O3 and g-Al2 O3 are more identical, indicating that oxygen-related structure is different between l-Al2 O3 and g-Al2 O3 . Figure 4.6 shows the X-ray and neutron total correlation functions T (r) of g-Al2 O3 [51], l-Al2 O3 [52], and g-SiO2 [7, 15]. The first peak of the T (r) of g-Al2 O3 observed at 1.81 Å is assigned to the Al–O correlations. The second peak observed at 2.8 Å in the neutron T (r) and that observed at 3.2 Å in X-ray T (r) are assigned to O–O and Al–Al correlations, respectively. The Al–O atomic distance in g-Al2 O3 is longer than the Si–O atomic distance in g-SiO2 . In addition, the Al–O correlation peak shows an asymmetric shape with a tail of ~2.4 Å. These results suggest that distorted AlOn polyhedra are formed with an Al–O coordination number N Al–O greater than 4. Indeed, the N Al–O is found to be 4.6 ± 0.2, which is identical to the results of

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Fig. 4.5 a X-ray and b neutron total structure factors S(Q) of g-Al2 O3 (2400 K) [51], l-Al2 O3 [52], and g-SiO2 [7, 15]

NMR measurements (N Al–O = 4.73) and higher than 4.4 in l-Al2 O3 (2400 K) [52]. Such a high cation–oxygen coordination number, which is frequently observed in high-temperature nonglass-forming oxide liquids [53, 54], can hardly be observed in the typical glass formers and glass-forming liquids. Hashimoto et al. also performed MD-RMC modeling to reveal the structure of g-Al2 O3 . The MD-RMC model suggests that OAl3 triclusters are formed by the edgesharing AlOn polyhedra (n = 4–6). In addition, they reported that the edge-sharing (a)

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AlOn polyhedra forms a lattice-like structure with an O–O distance of ~2.3–2.7 Å. This distance is nearly identical to the periodicity of ~2.3 Å calculated from the peak position of the PP observed in the neutron S(Q) of g-Al2 O3 . Therefore, they concluded that the formation of large amounts of AlO5 and AlO6 polyhedra, which is outside of Zachariasen’s conventional glass-forming concept, might be the origin of the extraordinarily sharp PP in the neutron S(Q) of g-Al2 O3 . The electrochemically synthesized g-Al2 O3 , which is the first successfully synthesized single-component intermediate glass, has many features, such as the densely oxygen-packed structure with a large fraction of edge-sharing polyhedra, that is completely outside of the conventional glass-forming concept [51]. Controlling the formation of a unique structure will provide glass with novel characteristics, e.g., hardness, crack resistance, and permittivity. Therefore, the fabrication of intermediate oxide glasses via an electrochemical approach is a useful tool for creating new glass materials.

4.8 Diffraction Measurements Under High Temperature Understanding the structure of high-temperature liquids provides us with useful information for understanding the nature of glass transition. However, the clarification of atomic arrangements in high-temperature liquid is a challenging scientific task, because chemical reactions of liquids with sample containers are very difficult to avoid. Moreover, the Bragg peaks from a crystalline container disturb the measurement of high-quality diffraction data from liquids. To overcome these problems, several containerless levitation techniques have been developed [55]. The details of various levitation techniques are described in Chap. 7. Levitation techniques also enable the access of deep undercooled liquids and enhance glass formation because heterogeneous nucleation is avoided (glass preparations using a levitation technique are discussed in Chap. 15). The structure of a typical nonglass-forming liquid, l-Al2 O3 , has been investigated by XRD, ND, and MD simulations [52, 55–61]. Furthermore, structures of UO2 [62] and the UO2 –ZrO2 system [63], the common nuclear fuel component of a reactor, have been studied. The structures of ZrO2 [53, 64, 65], HfO2 [65], and Lanthanide oxides [54, 64] have been studied to clarify the physicochemical properties of high-temperature liquids. In this chapter, the dedicated high-energy X-ray diffractometer for diffraction measurements on levitated liquids is described. In addition, representative structural studies of high-temperature oxide liquids by a combination of levitation techniques and diffraction measurements are introduced. The dedicated PDF diffractometer for liquid, glassy, and amorphous materials is developed at the high-energy XRD beamline BL04B2 of SPring-8. The details of the diffractometer are described in Refs. [7, 66]. The diffractometer has four CdTe detectors (low-scattering-angle regions) and three Ge detectors (high-scatteringangle regions, see Fig. 4.7a). The experimental setup for the XRD measurements on levitated liquids at the BL04B2 beamline with an aerodynamic levitator [4, 67] is shown in Fig. 4.7b. A sample of 2 mm diameter is levitated by dry air and heated

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Fig. 4.7 a Dedicated high-energy X-ray PDF diffractometer installed at the BL04B2 of SPring-8. b Aerodynamic levitator [4] installed on the PDF diffractometer. Reproduced from Ref. [66] (CC BY 4.0)

by a CO2 laser. The temperature of the levitated sample is measured by a pyrometer. Note that the use of a levitation technique eliminates the Bragg peaks originating from a crystalline container, realizing the measurements of high-quality diffraction data at high temperatures. In addition to the basic concept of glass-forming materials proposed by Zachariasen [23] and Sun [50], Angell [68] proposed the concept of “fragility” in glassforming liquids. He interpreted the behavior of strong and fragile liquids on the basis of topological differences. SiO2 , GeO2 , and B2 O3 are classified into typical strong liquids. They have a covalently bonded cation–oxygen network, and their viscosities obey the Arrhenius law. In contrast, chalcogenide and iron phosphates are regarded as typical fragile liquids. Their networks are almost ionic, and their viscosities markedly deviate from the Arrhenius behavior. Determining the structure of oxide liquids under high temperatures is crucial for clarifying the fragility of liquids. Er2 O3 , a representative nonglass former, has an exceedingly high melting temperature (T m ) of 2686 K. Koyama et al. reported the results of high-energy XRD and density measurements on l-Er2 O3 [54]. The density measurements on l-Er2 O3 were carried out using an electrostatic levitation furnace at the International Space Station [69] because the measurement of density data for liquid at extremely high temperatures on the ground is impossible (the details of the density measurements are introduced in Chap. 2). The X-ray total structure factors S(Q) of l-Er2 O3 (2923 K) [54], l-SiO2 (2373 K) [70], l-Al2 O3 (2400 K) [52], and l-ZrO2 (3073 K) [53] are compared in Fig. 4.8a. Note that scaling to the magnitude of the scattering vector Q

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is applied by multiplying Q by r A–X . A prominent FSDP is found only in the S(Q) of l-SiO2 at Qr A–X = 2.6, because l-SiO2 is a typical glass-forming liquid. On the other hand, a PP appears in the S(Q) of both l-Er2 O3 and l-ZrO2 at Qr A–X ~ 4.5. On the other hand, the S(Q) of l-Al2 O3 shows a very small peak between the positions of FSDP and PP, indicating that l-Al2 O3 possesses an intermediate structure [54] between l-SiO2 and l-Er2 O3 /l-ZrO2 . Because the PP is an indicator of oxygen packing in ND data due to the large O–O weighting factor for neutrons (see Chap. 2), no PP is observed in the X-ray S(Q) of l-SiO2 and l-Al2 O3 . On the other hand, the origin of the PP in the X-ray S(Q) of l-Er2 O3 and l-ZrO2 is attributed to the packing of heavy elements, since X-rays are sensitive to them. Figure 4.8b shows the X-ray total correlation functions T (r) of l-Er2 O3 [54], l-SiO2 [70], l-Al2 O3 [52], and lZrO2 [53]. The first correlation peak, observed at 2.2 Å in the T (r) of l-Er2 O3 , is assigned to the Er–O correlation, and a tail of the first peak to ~3 Å suggests that the ErOn polyhedral unit is distorted. The second peak observed at 3.7 Å is assigned mostly to the Er–Er correlation, which shows the distance between centers of ErOn polyhedra. The contribution of O–O correlation is almost inappreciable because of its small weighting factor for X-rays. Both the Er–O distance of 2.2 Å and the Zr– O distance of 2.1 Å are longer than those of Si–O (~1.63 Å) at 2373 K and Al–O (~1.78 Å) at 2400 K owing to substantial differences between the radii of the cations. The longer cation–oxygen atomic distance in l-Er2 O3 and l-ZrO2 indicates that the oxygen coordination number around a cation is greater than 4. This is because the Er–O distance (2.2 Å) and Zr–O distance (2.1 Å) are close to the sum of the ionic radii (sixfold zirconium, 0.72 Å; sixfold erbium, 0.89 Å; oxygen:1.35 Å) [71]. Therefore, the structures of l-Er2 O3 and l-ZrO2 comprise the interconnected polyhedral units with high cation–oxygen coordination numbers and are very different from those of l-SiO2 and l-Al2 O3 . This behavior of coordination numbers is in line with the absence of the FSDP, which appears owing to a sparse distribution of planes in polyhedra in typical glass-forming oxide glasses in XRD data (Fig. 4.8a). Indeed, there is no such structural ordering manifested by FSDPs in l-Er2 O3 and l-ZrO2 owing to their very densely packed structure. Koyama et al. also performed a combination of RMC-MD simulations and revealed that the structure of l-Er2 O3 consists of linearly arranged distorted OEr4 tetraclusters, giving rise to a long periodicity, which is the origin of the extraordinarily sharp PP in the X-ray S(Q). Moreover, persistent homology [72] analysis shows that the homology of l-Er2 O3 is similar to that of the crystalline phase [73]. Additional density functional (DF)-MD simulations suggest that the viscosity of this liquid is very low indicating that l-Er2 O3 is an extremely fragile liquid [54].

4.9 X-Ray Diffraction Measurements Under High Pressure Diffraction patterns from non-crystalline materials such as glasses and liquids are broad because their structural periodicity does not continue over long distances. Experimentalists generally need diffraction data with high S/N ratio in order to obtain reliable structural information on glasses and liquids. In situ XRD measurements of

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Fig. 4.8 a X-ray total structure factors S(Q) and b total correlation functions T (r) of l-SiO2 (2373 K) [70], l-Al2 O3 (2400 K) [52], l-ZrO2 (3073 K) [53], and l-Er2 O3 (2923 K) [54]. Scaling to the magnitude of scattering vector Q is applied by multiplying Q by r A–X (first interatomic distance observed in total correlation functions). Reproduced from Ref. [54] (CC BY 4.0)

glass under high pressure are different from those at ambient pressure as follows: the amount of a sample is limited, the effect of scattering from other materials (e.g., anvils, gaskets, and pressure media) is inevitable and significant, and the aperture angle, or 2θ, is limited because the measurements must be conducted using pressuregenerating apparatuses. Most of high-pressure in situ XRD measurements of glasses and liquids are performed using a diamond anvil cell (DAC) apparatus or a large volume press (LVP) in combination with synchrotron X-rays. The methods of XRD measurements with these two types of apparatuses are presented below. A DAC is a very simple apparatus and mainly consists of two jewel-like-shaped diamonds (e.g. brilliant cut, drukker cut), called diamond anvils (Fig. 4.9). Their tips are cut off and flattened, and they face each other to compress a sample uniaxially. Samples are generally held in a hole drilled in a plate (generally metallic) called a gasket. This apparatus is small, easy to handle, and really compatible with XRD measurements using synchrotron radiation because the volume of samples becomes quite small (10–100 μm size). A DAC can generate pressures up to 400 GPa or higher at room temperature. Temperature conditions of several thousand kelvins can be realized simultaneously by focusing a near-infrared or infrared laser beam and irradiating it through the anvil to the sample. For example, XRD measurements of silicate liquids at 60 GPa/3000 K have been reported [74]. Usually, XRD measurements with a DAC are conducted using monochromatic X-rays, which are injected from a direction parallel to the compression axis, and scattered X-rays are recorded by a two-dimensional detector installed downstream (Fig. 4.9a). In this case, the use of single-crystal diamond anvils minimizes diffraction from the diamonds unless diffraction conditions are not fulfilled, which is very convenient, especially in experiments on powder crystalline materials. However, the

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Fig. 4.9 Diamond anvil cell (DAC) apparatus. a Diamond anvils and gaskets. b Various forms of DAC apparatus for loading

thickness of the sample along the X-ray path is typically less than 100 μm, whereas that of two diamond anvils together are typically 3–4 mm; thus, the Compton scattering from diamond anvils is much stronger than that from the sample. In addition, when using high-energy X-rays (e.g. >30 keV), it is impossible to avoid all diffraction conditions of diamond anvils, and it is often difficult to remove these diffraction spots, especially for weak oscillations from non-crystalline materials in the high-Q region. The signal from diamond anvils becomes a serious noise mainly in the highQ region, especially for non-crystalline materials composed of light elements such as silicates. Therefore, it is crucial to remove these noises appropriately to obtain reasonable data. In addition, to obtain an accurate pair distribution function g(r), it is desirable to obtain a total structure factor S(Q) up to a high-Q region, preferably up to about Q = 15 Å–1 . In order to maintain the strength of the base seats supporting the diamonds (e.g., tungsten carbide is commonly used in room-temperature experiments), an aperture for diffraction is usually set to about 60° (i.e., 2θ = 30°). In this case, the 30 keV X-rays, which are commonly used in DAC experiments, will result in as low as Q = 7.9 Å–1 at 2θ = 30°. To remove Compton scattering from diamonds, the most commonly used method is to measure scattering profiles from an empty DAC without a sample before and/or after the experiments and subtract them from the sample profiles as background. Even then, accurate subtraction is not always easy. A perforated diamond anvil, which is an anvil curved along the X-ray path, can also work for reducing diamond Compton scattering [75]. For insufficient aperture angle problems, some attempts to overcome them have been reported by using monochromatic X-rays with high energy (e.g., 100 keV) [76] and by developing a high-aperture-type DAC with machining anvils and base seats to support loads effectively [77]. Measurements up to high-Q region can be also achieved by reducing and subtracting the Compton scattering from diamonds with the energy dispersive method [78], which is often used in experiments with large presses as described below. However, when using either method, the intensity of coherent scatterings rapidly decreases at high Q, and accurate high-pressure in situ measurement of S(Q) at above Q = 10 Å–1 is very difficult, especially at pressures higher than 100 GPa because the sample becomes very small. Furthermore, it is

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difficult to obtain detailed structural information such as interatomic distances and coordination numbers from g(r) alone owing to the overlapping of peaks from many elements. Recently, however, attempts have been made to combine measurements in a limited Q region with first principles and/or classical molecular dynamics (MD) calculations to investigate the details of structural changes [41, 42, 79, 80]. Large volume press (LVP) is an apparatus to compress a sample cell assembly consisting of a sample, pressure-transmitting medium, sample capsule, and heaters to achieve high pressure and high temperature using multiple anvils with a hydraulic press. The anvils are mainly made of tungsten carbide. The sample size is on millimeter order. LVP has advantages in large-sample synthesis, experiments, and precise and stable temperature and pressure control. There are various types of LVP apparatus depending on the pressurization method. Mainly two types of LVP are used for XRD experiments because it is necessary to secure the path of the incident and scattered X-rays. One is the DIA type, which is a cubic-anvil apparatus that compresses a cubic sample cell with six anvils, and the other is the Kawaitype multi-anvil apparatus, which pressurizes an octahedral sample cell assembly with eight second-stage anvils with one corner of the cube cut off using the DIA type as the first-stage anvil (Fig. 4.10). The upper limit of the generated pressure is determined by the strength of the anvil. At high temperatures, the generated pressure becomes lower than that at room temperature because the pressure-transmitting medium softens. For example, pressures of 27 GPa/3000 K [81] and 44 GPa/2000 K [82] with tungsten carbide anvils and over 100 GPa [83] with sintered diamonds have been reported. The Paris–Edinburgh (PE)-type LVP, which was developed to increase the sample volume for high-pressure neutron scattering experiments has also been used for XRD of liquids [84] and glasses [85] (Fig. 4.11). In situ ND measurements of glass under high pressure have also been reported [42, 80, 86]. The sample is uniaxially compressed by two conical anvils using a compact hydraulic press. These conical anvils have a cup in the center to increase the sample size up to a few mm. The usual upper limit of pressure attainable with a PE press is less than 10 GPa. Measurements up to above 100 GPa in sub-mm sizes have also been realized by inserting diamond anvils inside the PE press [85]. In XRD measurements using a Kawai-type apparatus, white X-rays are collimated to about 10–100 μm by a slit usually composed of tungsten carbide or another slit and irradiated to the sample through the gap between the first- and second-stage anvils. Scattered X-rays passing through the anvil gap on the other sides are again collimated by a narrow slit and detected at multiple diffraction angles with a Ge solidstate detector held on a goniometer rotating vertically/horizontally. This method is called the energy-dispersive XRD and is most commonly used for LVP experiments (e.g., Ref. [88]). This technique has the advantage that both incident and scattered X-rays are collimated by slits, thereby eliminating scattered X-rays from regions other than the sample. The lower limit of X-ray energy is about 30 keV owing to absorption by the sample cell and the upper limit is about 60–80 keV, depending on the X-ray energy distribution at the beamline. Due to geometrical constraints, goniometers often have a range of motion up to 2θ = 20–25°, so ideally, structural information up to Q ~ 20 Å–1 would be expected. However, the intensity of coherent

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Fig. 4.10 Example of LVP apparatus. a picture of inner parts of a Kawai-type multianvil apparatus, provided by Kawazoe [87]. b Picture of a multianvil press SPPED-Mk. II at SPring-8 BL04B2, Japan Synchrotron Radiation Research Institute (JASRI)

Fig. 4.11 PE type LVP. a WC anvils and gaskets. b Picture of a PE press used at the sector 16-BMB beamline, High Pressure Collaborative Access Team (HPCAT) at the Advanced Photon Source, Argonne National Laboratory [84], provided by T. Yu

scattering decreases rapidly at high Q and measurements are not actually easy even for Q > 15 Å–1 . Since a PE press is a uniaxial compression-type apparatus, it can realize observations at much wider angles than a multianvil apparatus in the direction perpendicular to the compression axis. For example, precise S(Q) up to Q = 22 Å–1 at 5.3 GPa and 1873 K for liquid NaAlSi3 O8 by energy dispersive XRD in the range of 2θ = 3–37° has been reported [84].

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4.10 Diffraction Measurements of Silica and Silicate Glasses Under High Pressure Structural measurements by XRD and ND of glasses and liquids under high pressures are of particular interest from the perspectives of condensed matter physics, materials science, and Earth and planetary science. In recent years, the idea of “polyamorphism”, which means that there are phase transitions in non-crystalline materials as well as crystalline materials, has been widely accepted. It is considered that liquids and glasses have phases that are thermodynamically stable or metastable but with low potential energy and separated by energy barriers. High-pressure synthesized glasses sometimes exhibit desirable properties that cannot be obtained at ambient pressure, and are of interest from the viewpoint of synthesizing new glass materials. From the viewpoint of Earth and planetary science, determination of physical and chemical properties of silicate liquids (magmas) or metallic liquids (mainly ironbased), which now exist or have existed in the interior of the Earth and other planets, is necessary for clarifying what is happening or has happened in the planets. In this section, some of the works on high-pressure in situ diffraction experiments of silica and silicate glass will be presented. Silica (SiO2 ) glass is the most typical glass with a three-dimensional network structure with SiO4 tetrahedra as the basic unit at ambient pressure. It is definitely one of the most well-studied glasses in the field of high-pressure science. Pressure induces significant changes in the short- and intermediate-range structure of SiO2 glass: the permanent densification at around 10 GPa, the coordination number change from SiO4 tetrahedra to SiO6 octahedra at 20–40 GPa, and the change to over sixfoldcoordinated structures occurring at above 100 GPa are noteworthy. The first highpressure in situ XRD measurement of SiO2 glass was reported in 1992 [89]. At that time, it was difficult to measure S(Q) in a sufficient Q region to obtain an accurate g(r). Technical developments in high-pressure and synchrotron techniques have led to XRD at high temperatures [90], XRD up to 100 GPa [91], ND up to 18 GPa [42], and XRD above 100 GPa [41, 92, 93], revealing details of pressure-induced structural changes in SiO2 glass. X-ray S(Q) of SiO2 glass measured up to 100 GPa are shown in Fig. 4.12. The obvious peak at around 1.6–2.5 Å–1 is the FSDP. As mentioned in Sect. 4.5, the FSDP is observed owing to the positive contributions of the Si–Si, Si–O, and O–O partial structure factors, and it is considered to reflect the arrangement of voids and/ or cages in a SiO4 tetrahedral network, i.e., the intermediate-range structure. The PP at around 3 Å–1 is observed only at a high pressure. This peak is not observed by XRD but clearly observed by ND at ambient pressure, and is considered to reflect the packing of oxygen atoms. With increasing pressure, the PP becomes observable in XRD at about 10 GPa and prominent at above 20 GPa. This behavior is understood in terms of the Si–Si partial structure factor obtained by MD simulation, in which the Si–Si PP increases with increasing pressure [41]. Permanent densification (or densification) [95] is a phenomenon that when glass is subjected to a pressure of about 10 GPa and then recovered, the glass becomes

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Fig. 4.12 X-ray total structure factors S(Q) of SiO2 glass under high pressures [91]. The patterns at 0.1 MPa are from [94]

denser than before compression. This phenomenon has been reported for silicate, aluminosilicate, and borate glasses of various compositions, as well as SiO2 glass. It is also reported for bulk metallic glasses and chalcogenide glasses. In the case of SiO2 glass, density increases of up to 20–25% have been reported [96, 97]. Densification is attributed to the contraction of voids in the intermediate-range network structure by compression. The simultaneous application of temperature significantly promotes densification, resulting in higher densities at lower pressures [89, 98]. FSDP shifts toward higher Q by the densification in oxide glasses with various compositions. Although PP is sharp and distinct under high pressures, it is not observed in the recovered densified glasses, and the profiles of S(Q) at above 8 Å–1 are very similar before and after compression [99, 100]. This finding also supports the hypothesis that densification is caused by the contraction of the intermediate-range structure with little change in the short-range structure. A change in the short-range structure occurs at above 20 GPa; the coordination number is 5 at 27 GPa and the pattern at this pressure can be explained fairly well by

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mixing the patterns at 20 and 35 GPa. On the other hand, the patterns from 35 to 100 GPa are very similar, although the overall pattern gradually shifts toward higher Q owing to compression (Fig. 4.12). g(r) shows that the Si–O distance increases rapidly from 20 to 35 GPa despite compression, and then decreases at above 35 GPa. The coordination number of oxygen to silicon increases from 4 to 6 from 20 to 35 GPa and remains nearly sixfold-coordinated from 35 GPa to least 100 GPa or higher [91], therefore SiO2 glass may be considered to behave as a “sixfold-coordinated phase” in this pressure range. At ambient pressure, it is considered that almost 100% of the Si is fourfold-coordinated. On the other hand, MD calculations have suggested that only about 75% of Si species are sixfold-coordinated and five- and seven-coordinated species also exist in a “sixfold-coordinated phase” [41]. At higher pressures, the coordination number begins to increase above 100–140 GPa [41, 92, 93]; at 200 GPa, it is suggested by a combination of XRD measurements and MD calculations that the average coordination number becomes 7 with a similar short-range structure to pyrite-type silica crystal [41]. Silicate glasses also have a network structure based on SiO4 or AlO4 tetrahedra at ambient pressure, but their network is modified or disconnected by networkmodifying cations (e.g., Na, Mg, and Ca). The structure of silicate liquids under high pressure is particularly important from the geophysical viewpoint, but the melting point of silicates is very high (usually >2000 K), and experiments at high temperatures are often much more difficult than those at room temperature. The high-pressure structure of glasses is expected to resemble those of liquids with the same composition, especially in the case of “strong” glass-forming liquids [49]. High-pressure in situ XRD measurements have been carried out for geophysically important compositions such as MgSiO3 [101], Mg2 SiO4 [102], jadeite (NaAlSi2 O6 ) [103], basalt (aluminosilicate containing cations such as Mg, Ca, and Fe) [86], as well as aluminum-rich compositions such as anorthite (CaAl2 Si2 O8 ) and silica-free CaAl2 O4 [79]. ND measurements have also been reported for MgSiO3 and CaSiO3 [80] and basalt [86] glasses. Owing to the technical difficulties in obtaining high-Q data, the peaks in g(r) become broader, which makes it almost impossible to assign various atomic pairs corresponding to each peak. In most cases, diffraction data are not sufficient for discussing the detailed structure. This difficulty can be partly compensated with the help of MD calculations, but the results of calculations and experiments do not always agree well. Future improvements in measurements are strongly expected. Note that the description below is not necessarily based on the results of experiments but includes many predictions based on MD calculations. FSDP is observed for all compositions and located at around 2 Å–1 at ambient pressure, which is higher than that in SiO2 glass, suggesting that the network-modifying cations cleave the network and reduce voids. PP is not observed in the XRD patterns at ambient pressure for all compositions but begins to be observed at pressures above 5–10 GPa, and becomes more prominent with increasing pressure. An irreversible FSDP shift (densification) has been reported for MgSiO3 , CaSiO3 , jadeite, and basalt glasses, whereas the XRD pattern is considered to be reversible for Mg2 SiO4 . This may be due to the fact that the network structure of Mg2 SiO4 at ambient pressure is dominated by MgOx , not by SiO4 [104]. The FSDP of jadeite glass is not

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so prominent at ambient pressure but becomes sharper with increasing pressure and subsequent temperature increase. The recovered FSDP is sharp and clear. Since crystalline jadeite is thermodynamically stable only at high pressures, the drastic change in the FSDP suggests that liquid (and glass) structures at ambient pressure and high pressure may be very different. The coordination number of Si is suggested to remain four below 20 GPa as in SiO2 glass because the Si–O bond length shows almost no change for MgSiO3 , CaSiO3 , Mg2 SiO4 , anorthite, and basalt. However, there is a study suggesting that the Si–O bond length significantly increases and the coordination number begins to increase already at 10 GPa [101]. NMR measurements for recovered samples indicate that Al consumes non-bridging oxygen at lower pressures than Si to increase its coordination number [105]. The coordination number of Al in both CaAl2 Si2 O8 and CaAl2 O4 starts to increase from 0 GPa and sixfold-coordinated species become dominant at 20 GPa. The approximate average coordination numbers of network-modifying cations for glasses at ambient pressure are reported as follows: 4.5 for MgSiO3 [106], 5 for Mg2 SiO4 with a mixture of 4, 5, and 6 [104], 6 for CaSiO3 [80], and 6.5 for anorthite [79]. The coordination number of Mg and Ca in crystalline phases with the same compositions is 6. Therefore all compositions except CaSiO3 seem to have different coordination states in crystals and glasses. The coordination number will increase with pressure to 6 for MgSiO3 , 7.5 for CaSiO3 , and 9 for anorthite at about 20 GPa. High-pressure in situ diffraction measurements of silicate glass have been limited, and it is not fully understood how structural changes occur under pressure; however, such information will be of great interest for the development of new materials.

4.11 Permanently Densified SiO2 Glass Recovered After Hot Compression As described in the previous section, the FSDP in diffraction data for SiO2 glass shifts toward higher Q and diminishes under high pressure at room temperature (cold compression) as shown in Fig. 4.12. Although this trend observed under cold compression has been understood as a general trend for the behavior of FSDP under high pressure, Onodera et al. reported the unusual behavior of the FSDP in XRD data for SiO2 glass recovered after hot compression. They recovered densified SiO2 glasses after hot compression at a pressure of 7.7 GPa and temperatures up to 1473 K, and probed the glass structure by a combination of diffraction measurements and structure modeling [100]. Figure 4.13a shows X-ray S(Q) for hot-compressed SiO2 glasses. The evolution of FSDP at 7.7 GPa is observed at a temperature higher than 673 K; thus, the sharpness of the FSDP is decreased with increasing a temperature up to 673 K. The density of the hot-compressed SiO2 glasses also changes in behavior at a temperature higher than 673 K. These results indicate the transformation from a low- to high-density amorphous phase in SiO2 glass. Onodera et al. prepared a glass

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with the same density as the hot-compressed glass (1473 K/7.7 GPa) by cold compression at RT/20 GPa. Figure 4.13b shows X-ray S(Q) for two densified glasses with the same high density (hot-compressed glass, 2.72 g/cm3 ; cold-compressed glass, 2.71 g/cm3 ). The position of the FSDP is almost the same corresponding to the same density, whereas the sharpness of the FSDP shows a significant difference between the two glasses. The Si–O coordination number obtained from the corresponding T (r) functions is four in each densified glass, indicating that the structure of the densified glasses comprises a network of corner-sharing tetrahedral SiO4 motifs. The coherence length describing the intermediate-range ordering based on these motifs, given by 2π/ΔQFSDP , reaches 12.7 Å for 1473 K/7.7 GPa. On the other hand, the coherence length is 7.9 Å for RT/20 GPa. Therefore, although the densities of 1473 K/7.7 GPa and RT/20 GPa glasses are comparable, their structures are markedly different, with a coherent length for 1473 K/7.7 GPa that is ~61% longer than that for RT/20 GPa. In addition, hot-compressed glasses were stable for at least 1.5 years at ambient conditions, whereas cold-compressed glass showed a reduction in density of 2.8% after 1.5 years, suggesting that permanently densified SiO2 glasses can be synthesized by hot compression. As mentioned above, the glass structure can be controlled by controlling the processing conditions such as temperature and pressure. The densified SiO2 glass has attracted much attention as a candidate core material for optical fibers capable of reducing loss (see Chap. 15). The knowledge of the glass structure obtained by diffraction measurements may therefore be helpful for the design of new glassy materials.

Fig. 4.13 a X-ray total structure factors S(Q) of SiO2 glasses recovered after hot compression. b X-ray total structure factors S(Q) of hot- and cold-compressed SiO2 glasses

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4.12 Summary This chapter described a brief introduction to the pair distribution function (PDF) analysis by X-ray and neutron diffraction measurements of disordered materials. Although solving the structure of disordered materials such as glasses, liquids, and amorphous materials is difficult owing to the lack of long-range periodicity, the PDF provides real-space structural information, such as interatomic distance and coordination number. Moreover, the complementary use of X-rays and neutrons enables us to analyze disordered structures more precisely by utilizing the contrast between X-ray form factors and neutron coherent scattering length. Attempts to understand the origin of diffraction peaks observed in diffraction patterns of typical disordered materials are made. The first sharp diffraction peak (FSDP) appears as the result of a sparse distribution of planes in polyhedra. The principal peak (PP) is a signature of the formation of chemical bond and reflects inter-polyhedral correlations on a short-length scale compared with FSDP. The extraordinarily sharp PP observed in neutron diffraction data for Al2 O3 glass fabricated by the electro-chemical anodization of Al metal indicates the formation of a densely oxygen-packed structure with a small cavity volume that is completely outside of Zachariasen’s glass-forming concept. The dedicated diffractometer for accurate diffraction measurement on levitated liquid under high temperatures is available at SPring-8. The structure of a representative nonglass-forming liquid, Er2 O3 , was successfully discovered by applying an aerodynamic levitation technique and high-energy X-rays. Experimental methods of X-ray diffraction measurements for disordered materials under high pressures are introduced together with the results of diffraction measurements of silica and silicate glasses under high pressures. In particular, the unusual behavior of the FSDP was found in the X-ray diffraction data for permanently densified SiO2 glass synthesized by hot compression at a pressure of 7.7 GPa and a temperature of 1473 K. The advent of synchrotron and neutron facilities led to the accurate measurement of diffraction of disordered materials from ambient to extreme (high pressure and high temperature) conditions. Acknowledgements This work was supported by a JSPS Grant-in-Aid for Transformative Research Areas (A) “Hyper-Ordered Structures Science”: Grants No. 20H05878, No. 20H05879 and No. 20H05881. The authors greatly appreciate Prof. Hidenori Goto, Prof. Takayoshi Yokoya, Mr. Mitsuki Ikeda, Dr. Ritsuko Eguchi, and Ms. Risa Kawamoto for their valuable suggestions and assistance in completing this section.

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Chapter 5

Angstrom-Beam Electron Diffraction Technique for Amorphous Materials Akihiko Hirata

Abstract This chapter describes a direct measurement method called angstrombeam electron diffraction that detects the local atomic arrangements of amorphous materials using sub-nanometre-sized electron probes. The purpose of this method is to provide additional structural information to global diffraction experiments, such as X-ray or neutron diffraction methods, by taking electron diffraction from local regions. The chapter first provides a brief background on this method, including diffraction from small volumes, as well as the effects of specimen thickness. Subsequently, it summarizes a few applications of this method to local atomic arrangements of structural and functional amorphous materials, such as metallic glasses, oxide electrode materials, and phase-change Materials. This method reveals the distortion of icosahedral atomic clusters in metallic glasses, nanoscale inhomogeneity of silicon monoxide, and structural similarity between amorphous and crystal chalcogenide structures in these applications. Lastly, this chapter presents related analysis techniques, including angstrom-beam electron diffraction mapping, local reverse Monte Carlo simulation, and virtual angstrom-beam electron diffraction method. These techniques seek a deeper understanding of the relationship between diffraction patterns and local atomic arrangements. These facts demonstrate that this method could be a powerful analytical tool to reveal the hyperordered structures between order and disorder. Keywords Amorphous · Glass · Electron diffraction · Short-range order · Medium-range order

A. Hirata (B) Department of Materials Science, Waseda University, Shinjuku, Tokyo 169-8555, Japan e-mail: [email protected] © Materials Research Society, under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. Hayashi (ed.), Hyperordered Structures in Materials, The Materials Research Society Series, https://doi.org/10.1007/978-981-99-5235-9_5

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5.1 Background Many challenges remain in the structural analysis of amorphous materials without translational and point symmetry. The main challenge is that it is difficult to define unit cell because of the lack of translational symmetry. Particularly, a global structure cannot be simply represented by a single local structure. Amorphous structures are generally measured by X-ray or neutron diffraction experiments and analyzed through a pair distribution function (PDF) derived from the measured intensity [1–3]. Accurate intensity measurements up to a high scattering angle provide more detailed structural information in the PDF, as recently performed using a synchrotron X-ray scattering experiment [4, 5]. The diffraction intensity is macroscopically isotropic in three-dimensional reciprocal space. Therefore, it becomes a one-dimensional profile. Furthermore, the resultant PDF profile is one-dimensional and may contain structural information regarding a variety of local structures. Structural modeling through molecular dynamics or reverse Monte Carlo simulation implies the presence of a variety of local structures that may have different coordination numbers or geometries. To verify this, we require different experimental techniques with higher spatial resolution, down to the sub-nanometre scale in real space. Transmission electron microscopy (TEM) is a technique with atomic-scale spatial resolution. TEM provided the first evidence of dislocations in crystals in the 1950s. In the 1970s and the 1980s, high-resolution electron microscopy (HREM) was used to observe local structures in amorphous structures [6–9]. Contrary to the lattice-shaped images in crystals, a maze-like contrast reflecting the disordered nature of atomic arrangements can be observed in amorphous materials. According to Krivanek et al., maze-like images contain both structural information and noise, and local ordered regions larger than 1.5 nm can be observed under their experimental conditions [6]. By combining HREM and computational simulations, Hirotsu et al. revealed the importance of optimum defocus conditions for imaging the local order, particularly in metallic glasses [9–14]. In the 2000s, the development of spherical aberration correctors drastically improved the image quality and enabled depth-resolution imaging [15–17]. Furthermore, fluctuation electron microscopy has been proposed as another analytical technique [18–23]. While imaging techniques have been developed as mentioned above, electron diffraction techniques with a focused electron probe have become usable owing to the development of field emission guns. The field emission guns enable us to obtain electron diffraction patterns from nanometre- or sub-nanometre-scale regions of amorphous materials using TEM/scanning transmission electron microscopy (STEM). This method is called micro- or nano-beam electron diffraction and is widely used to analyse local structures. For example, in the 1970 and 1980s, Cowley’s group used this method to conduct numerous local structure analyses for crystalline materials [24, 25]. However, only a few studies on the application of amorphous materials were reported in the early stages of the analysis [26]. In addition to nanoscale crystalline order in the marginal glass formers, the non-crystalline order in metallic glasses was observed by nanobeam electron diffraction in the late 2000s [27–29].

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5.2 Fundamentals of Angstrom-Beam Electron Diffraction 5.2.1 Global and Local Diffraction of Amorphous Materials The diffraction intensity of the local area of the amorphous materials is quite different from that of the whole area. The former exhibits an anisotropic discrete distribution, whereas the latter exhibits an isotropic spherical distribution. When the beam size is less than several nanometres, spherical halo rings become aggregates of discrete spots, each of which originates from the local structural order in amorphous materials. At this scale, a sufficient number of local structures were not included in the volume to form continuous diffraction intensities. Individual local structures can be seen, especially if the volume through which the beam passes becomes small enough. Independent of the scale, the diffraction intensity from monatomic amorphous materials can be written as I ( Q) =

N  m=1

f m2 ( Q) +

N 

N 

m=1

n=1 (n = m)

f m ( Q) f n ( Q)ei Q•r mn ,

(5.1)

where Q is the scattering vector (| Q| = 4π sinθ/λ), f m is the atomic scattering factor of the mth atom, and r mn is the positional vector between the mth and nth atoms. The first term indicates scattering from the mth atom, whereas the second term indicates the correlation between the mth and nth atoms. If the volume contains sufficient local structures to form continuous intensities in a three-dimensional reciprocal space, the diffraction intensity can be transformed into a one-dimensional representation in polar coordinates as follows:  I (Q) = N f 2 (Q) + N f 2 (Q)

∞ 0

4πr 2 (ρ(r ) − ρ0 )

sin Qr dr, Qr

(5.2)

where ρ0 is the average atom number density, and ρ(r ) is the radial number density function. Equation (5.2), without directional information, can be applied to the interpretation of general X-ray or neutron diffraction intensities obtained from a large volume. However, Eq. (5.2) is no longer valid when the volume is reduced to several nanometres. In this case, we must use Eq. (5.1), which includes the directional information. This is because the diffraction pattern is no longer continuous but instead consists of discrete diffraction spots, as mentioned above. Furthermore, this also applies to the angstrom-beam electron diffraction method [30–39] described in this chapter.

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Figure 5.1 shows a schematic illustration of the angstrom-beam electron diffraction technique, together with the diffraction patterns obtained from Zr–Ni metallic glass with several beam diameters [30]. The diffraction patterns obtained from the local regions were recorded when the focused electron probe was irradiated on a thin area of amorphous material (Fig. 5.1a). Because the electron probe size is variable, diffraction patterns can be acquired from areas of various sizes in amorphous materials, as shown in Fig. 5.1b. The diffraction pattern from the wide area with a diameter of 100 nm (bottom right) shows full halo rings, indicating an isotropic nature of the diffraction intensity. The transmitted beam that is not diffracted by the substance is indicated by the central spot. The diffraction intensity becomes discrete and eventually reduces to only a few diffraction spots as the probe diameter decreases. The diffraction pattern using a probe size of 0.36 nm (upper left) seems similar to that of a typical crystal, where the four strong diffraction spots form a four-fold symmetric pattern. This sort of diffraction pattern, however, can be derived from the local atomic configurations of amorphous materials, containing no crystals, as demonstrated subsequently.

Fig. 5.1 a Schematic illustration of the angstrom-beam electron diffraction technique. b Electron diffraction patterns with different beam sizes obtained from a Zr66.7 Ni33.3 metallic glass. The beam diameters are shown in the patterns. (Reproduced with permission from [30].)

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5.2.2 Diffraction from a Single Atomic Cluster This section discusses the diffraction intensity of a tiny atomic cluster known as the short-range order in metallic glasses. Figure 5.2a, b, c show the simulated diffraction patterns from a single atom, paired atoms, and an atomic cluster found in Zr– Pt metallic glass, respectively [31]. As shown in Fig. 5.2a, the diffraction pattern obtained from a single atom exhibits a featureless intensity distribution, in which the intensity gradually decreases from the center to the outside. In this case, there is no second term that is a correlation term for multiple atoms in Eq. (5.1). Therefore, the intensity was consistent with the square of the atomic scattering factor. Meanwhile, the paired atoms in Fig. 5.2b generated intensity modulation along the vector R12 , connecting one atom with another. This is because the correlation term in Eq. (5.1) appears and modulates intensity. The diffraction intensity of the atomic cluster consisting of 13 atoms, as shown in Fig. 5.2c, shows a distorted hexagonal pattern with six strong diffraction spots. Due to the correlation among the 13 atoms, the intensity modulation in this case is more complicated than in the case of paired atoms and occurs in multiple directions. This pattern resembles that of typical crystals, but the atomic cluster has noncrystalline features and is quite different from typical crystalline clusters. Usually, this kind of atomic cluster can be extracted from structural models of metallic glasses constructed using molecular dynamics or reverse Monte Carlo simulations. As in the case of crystals, the diffraction pattern from a single atomic cluster, shown in Fig. 5.2c, strongly depends on its orientation. Therefore, the spatial distribution of the diffraction intensity can be examined by rotating the cluster about one axis, as

Fig. 5.2 Simulation of electron diffraction patterns for a single Au atom, b pair of Au atoms, and c atomic cluster (icosahedron) in a Zr-Pt metallic glass model. The corresponding atomic configurations are also depicted on the lower side. (Reproduced from [38]. CC-BY 4.0)

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Fig. 5.3 a Sequential changes in simulated electron diffraction patterns obtained by rotating the atomic cluster as shown in b. b Atomic cluster (icosahedron) of the Zr-Pt metallic glass used for the diffraction simulation, together with the rotation axis. c Change in the diffraction spot intensity indicated by an arrow in a. (Reproduced from [38]. CC-BY 4.0)

shown in Fig. 5.3b. Figure 5.3a shows a continuous variation in the diffraction patterns obtained from the single atomic cluster in Fig. 5.3b [38]. The intensity indicated by the arrow in Fig. 5.3a gradually decreases with rotation and almost vanishes with a 16-degree rotation, as shown in Fig. 5.3c. This implies that the diffraction intensity from a cluster exhibits a discrete spotty distribution in three-dimensional reciprocal space. As a result, the atomic clusters in the glass specimens do not always generate strong diffraction intensities because the clusters are oriented randomly in real space. This allows us to selectively observe the oriented on-axis cluster using this technique.

5.2.3 Effect of Specimen Thickness To acquire diffraction patterns from small volumes, it is necessary to limit the area to which the electron beam is irradiated. In the plane normal to the incident direction of the electron beam, the area can be limited by focusing the electron beam down to

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the sub-nanometre scale. However, the resolution along the incident direction (depth resolution) was not as good as that in the plane normal to the incident direction. It is impossible to resolve the structural information along the beam incident direction, particularly for ABED using a small convergence angle. Therefore, it is necessary to prepare the TEM sample as thin as possible to reduce the volume through which the electron passes. Because the sample thickness is a significant factor, it will be discussed below. In this section, we estimated the atom numbers in the column through which the electron passes, as shown in Fig. 5.4 [38]. Furthermore, the average diameter of the atoms and packing density are 0.28 nm and 0.7, which are typical for metallic glasses. When the thickness is less than 5 nm and the beam diameter is less than 5 Å, the atom number can be reduced to less than 50. Under these conditions, it was feasible to observe a single atomic cluster, as shown in Fig. 5.2c. However, when the beam diameter was larger than 10 Å, a rapid increase in the number of atoms led to the overlap of multiple clusters. This implies that reducing the beam size to approximately 5 Å is essential for observing local structures at the cluster level. As a result, this technique is called angstrom-beam electron diffraction and is different from conventional nano-beam electron diffraction. As mentioned previously, the diffraction intensity of a single cluster strongly depends on its orientation. As a result, the symmetric pattern, which is like a crystal zone axis pattern, is referred to as an “on-axis pattern,” while the others are referred to as “off-axis patterns.” As shown in Fig. 5.5, atomic clusters generating on- or off-axis patterns were prepared and artificially stacked along the electron incident direction [31]. The electron diffraction simulation was then performed for five models Fig. 5.4 Plots of the number of atoms in the column where the electron beam goes through to the beam diameters. The sample thicknesses used in the calculations are shown in the graph. (Reproduced from [38]. CC-BY 4.0)

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Fig. 5.5 Effects of overlapping atomic clusters on electron diffraction patterns. (Reproduced with permission from [31].)

with an on-axis cluster stacked 0~4 off-axes. Even when the four off-axis clusters overlapped with the on-axis clusters, the resulting diffraction pattern did not change significantly. In practice, an on-axis cluster is connected to another on-axis cluster, which enhances the intensity of the symmetric on-axis pattern. We further simulated more realistic situations with thicknesses ranging from 2.5 to 30 nm using a large-scale amorphous model made with a molecular dynamics simulation. For the diffraction simulations, we carried out a multislice simulation that allowed us to consider the plural scattering effect. Figure 5.6 shows the simulated electron diffraction patterns for the 2.5-, 5-, 10-, and 30 nm thickness models [38]. The thinner models are included in the thicker models. The model with a 2.5 nm thickness generates an on-axis diffraction pattern with paired diffraction spots, similar to the patterns in Fig. 5.5. The pattern of the 2.5 nm model can be maintained almost entirely up to 5 nm and approximately up to 10 nm, despite the significant increase in thickness. This implies that the off-axis regions do not significantly contribute to the strong diffraction intensity. However, the 30 nm thickness changed the pattern significantly, and the paired diffraction spots became asymmetric. In addition, we observed partial diffraction rings instead of separated diffraction spots. This type of pattern in the experiment shows that the region is too thick and inappropriate for angstrom-beam electron diffraction observations.

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Fig. 5.6 Effects of sample thicknesses on electron diffraction patterns. The diffraction patterns are calculated for a molecular dynamics model of the Zr-Pt metallic glass with thicknesses of a 2.5, b 5.0, c 10.0, and d 30.0 nm. e Side view of the structure model for the pattern d. (Reproduced from [38]. CC-BY 4.0)

5.2.4 Measuring Equipment The angstrom-beam electron diffraction experiment is usually carried out in the STEM mode, where the focused electron beam can be easily aligned using the Ronchigram method. As mentioned previously, a field emission gun is required to focus the beam at the sub-nanometre scale. In our experiments, the custom-made small condenser lens apertures are 3.5 and 5.0 µm in diameter and are used for making a semi-parallel electron beam. The 3.5 and 5.0 µm apertures lead to beam diameters of 0.8 and 0.36 nm with convergence semi-angles of 1.0 mrad and 3.3 mrad, respectively. Note that the convergence semi-angle is generally a trade-off with beam diameter. Therefore, the size of the aperture should be optimized by considering the characteristics of amorphous structures. The resultant diffraction patterns were acquired using a charge-coupled device (CCD) camera.

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5.3 Applications of Angstrom-Beam Electron Diffraction 5.3.1 Metallic Glasses The angstrom-beam electron diffraction technique was initially applied to observations of metallic glasses. The purpose is to directly observe short-range order, which is usually composed of 10~16 nearest neighbor atoms together with a central atom [40–42]. We often call these atomic clusters or coordination polyhedra. In order to see atomic clusters, the focused electron probe is prepared at the cluster level (subnanometre). A full width at half maximum of the probe is about 0.36 nm using the 5 µm condenser lens aperture. The samples for the observation were prepared by low-voltage ion milling with a cooled stage. The sample thicknesses were estimated to be less than 5 nm using electron energy loss spectroscopy. Figure 5.7 shows on-axis electron diffraction patterns typically observed in a Zr66.7 Ni33.3 metallic glass, together with simulated patterns [30]. The simulated patterns are obtained by the diffraction simulation for atomic clusters formed in the glass model made with an ab-initio molecular dynamics simulation. Since the diffraction intensity from an atomic cluster strongly depends on its orientation, as discussed above, it is necessary to examine the diffraction intensity thoroughly by rotating the atomic cluster. From some specific directions, we can see on-axis patterns where two or more paired spots are formed, just like zone-axis patterns in crystals. The diffraction patterns of the simulation are well consistent with those of the experiment, as seen in Fig. 5.7. Among various atomic clusters, an icosahedron consisting of 13 atoms has received considerable attention in the field of metallic glasses. Note that icosahedral symmetry is incompatible with the translational symmetry of the crystals. The icosahedron was initially considered a local atomic arrangement stabilized in a liquid or a supercooled liquid of metals, as suggested by Frank [43]. The icosahedral atomic clusters were later proposed as short-range order structures for metal–metal-type metallic glasses by many researchers [44, 45]. After the discovery of quasicrystals, icosahedral atomic clusters have become increasingly significant. Moreover, quasicrystal formation from glassy states has been reported for several metallic glass systems. Therefore, we observed the local structures of a Zr-Pt metallic glass, where nanoscale quasicrystals form directly from the glassy matrix [31]. Many angstrombeam diffraction patterns acquired from thin regions of Zr-Pt were examined, but none of them showed distinct icosahedral symmetry like quasicrystals. Instead, the diffraction patterns composed of three paired spots are frequently observed as shown in Fig. 5.8. Although the patterns appear to be those of a face-centered-cubic crystal, they are derived from heavily distorted icosahedra, which are partially close to the crystal. Unlike in crystals, the intensity dumping at a large scattering angle is a noticeable feature of distorted icosahedra.

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Fig. 5.7 a-cExperimental angstrom-beam electron diffraction patterns obtained from a Zr66.7 Ni33.3 metallic glass. a’-c’ Simulated diffraction patterns obtained from atomic clusters shown in a”-c”. The atomic clusters are extracted from a molecular dynamics model. The Voronoi indices for each atomic cluster are also shown. (Reproduced with permission from [30].)

5.3.2 Silicon Monoxide Amorphous silicon monoxide (SiO) has received much attention for its application in automotive lithium-ion batteries. Amorphous SiO was synthesized over a hundred years ago, and there has been much debate about its heterogeneity [46]. The main issue is whether amorphous SiO has a unique atomistic structure or phase-separated nanostructure composed of Si and SiO2 . We confirmed that the structural factor of amorphous SiO is not coincident with that of the simple sum of amorphous Si and SiO2 using synchrotron X-ray diffraction [33]. To understand this difference, we conducted an angstrom-beam electron diffraction experiment using a specially designed 3.5 µm condenser aperture. The samples for the observation were prepared

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a

b

c

a’

b’

d

e

Fig. 5.8 a, a’ Typical angstrom-beam electron diffraction pattern of a Zr80 Pt20 metallic glass. b, b’ Simulated diffraction pattern from a distorted icosahedron shown in c. d Simulated diffraction pattern from a face-centered-cubic cluster ([110] direction). e Partial similarity of a distorted icosahedron to a face-centered-cubic cluster. (Reproduced with permission from [31].)

by the conventional crushing method. The sample thicknesses were estimated at 2~3 nm by atomic force microscopy. As shown in Fig. 5.9, this technique enabled us to resolve the local structures of Si-like, SiO2 -like, and their interface nanoscale regions. The results revealed that amorphous SiO is not homogeneous, and the interface structures contribute to the formation of the characteristic structure factor, which differs from the simple sum of Si and SiO2 . The interface regions are composed of silicon suboxides. The critical point is that a homogeneous model can also explain the X-ray structure factor, which is global structural information. As a result, the combined method of electron and X-ray diffraction is quite effective in revealing inhomogeneous structures, such as SiO.

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Fig. 5.9 a-c Angstrom-beam electron diffraction patterns obtained from a bright, b interface, and c dark areas in annular dark-filed STEM images. a’-c’ Simulated diffraction patterns obtained from a’ Si region, b’ interface region, and c’ SiO2 region of the inhomogeneous structure model satisfying the X-ray structure factor. The corresponding atomic configurations are depicted in a”-c”. Dark and bright circles denote Si and O atoms, respectively. d The circumferentially averaged intensity profiles made by collecting several diffraction patterns of bright, interface, and dark areas. (Reproduced from [33]. CC-BY 4.0)

5.3.3 Phase-Change Materials Phase-Change Materials, such as Ge2 Sb2 Te5 , have already been applied to the recording layer of digital versatile discs. The phase change between the amorphous and crystal phases with different optical properties is utilized for recording or erasing digital information. The structural origin of the rapid phase change (~20 ns) is the main issue, and the structural similarity between them has been widely discussed based on experimental and computational studies. In this situation, we struggled to observe the local atomic arrangements of amorphous Ge2 Sb2 Te5 using angstrombeam electron diffraction [36]. The sputter-deposited thin film with a 10 nm thickness was utilized for the observation. Figure 5.10 shows a local structure model constructed using a local reverse Monte Carlo calculation (described later) based on the experimental diffraction pattern. The local structure model created by distorting a rocksalt crystal is well fitted by two types of electron diffraction patterns (fourfold and sixfold) and an X-ray structure factor as well. The octahedron in a rock salt crystal becomes asymmetric in the amorphous structure, leading to a decrease in the coordination number, which should be related to the bonding nature.

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Fig. 5.10 a Schematic illustration of the local reverse Monte Carlo simulation for amorphous Ge– Sb–Te Phase-Change Materials. b, b’ Simulated diffraction pattern and atomic configuration of the initial structure (NaCl). c, c’ Simulated diffraction pattern and atomic configuration of the final structure that satisfies the experimental pattern shown in d. e Distorted octahedron found in c’. f, f’ Simulated diffraction pattern and atomic configuration of the final model shown in c’ from another direction. g X-ray structure factors for the initial model (NaCl) and final model. For comparison, the total structure factor for amorphous Ge2 Sb2 Te5 obtained by synchrotron X-ray scattering [47] is also shown. (Reproduced with permission from [36].)

5.4 Associated Techniques 5.4.1 Angstrom-Beam Electron Diffraction Mapping We obtained angstrom-beam electron diffraction patterns with positional information in real space using the scanning function of STEM to understand the overall features of the local structural order. This technique allows us to discuss the correlation length, spatial distribution, and detailed medium-range order of glassy materials based on diffraction maps. Figure 5.11 shows an example of a diffraction map consisting of over 1,000 diffraction patterns for bulk metallic glass (Pd–Cu–Ni–P) [38]. In the mapping data, each pixel of the real-space image contains a two-dimensional diffraction pattern. Therefore, diffraction maps (dark-field images) can be reconstructed using any position in the diffraction patterns. From the maps obtained, the

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Fig. 5.11 a Schematic illustration of angstrom-beam electron diffraction mapping. b Diffraction map reconstructed by the paired regions on the first halo ring of the integrated pattern. c Diffraction patterns for each pixel in the region surrounded by white lines in b. (Reproduced from [38]. CC-BY 4.0)

brighter regions with a size of 1.0~1.5 nm can be interpreted as the presence of medium-range order structures composed of several atomic clusters.

5.4.2 Local Reverse Monte Carlo Modeling As mentioned previously, it is technically feasible to construct local structure models to satisfy the angstrom-beam electron diffraction experiment through a procedure similar to reverse Monte Carlo simulation [48–50]. As shown in Fig. 5.10a, an initial atomic configuration, which can be a random or crystalline structure, is prepared in a three-dimensional confined space. The number density was carefully set to satisfy the overall density. The atoms were then moved in random directions to satisfy the experimental diffraction pattern. The movement of the atom is rejected if it attempts to move across the boundary. Moreover, constraints can be imposed on the interatomic distances to prevent the atoms from approaching each other. Finally, we obtained the local structure model based only on the local diffraction pattern without using any interatomic potentials. Although this method is still under development, it can at least provide candidate local atomic arrangements that satisfy the two-dimensional local diffraction pattern and the overall density.

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5.4.3 Virtual Angstrom-Beam Electron Diffraction It is possible to replicate the angstrom-beam electron diffraction mapping experiment virtually on a computer [39]. To simulate this procedure, we prepared thin films cut from relatively large structure models made with molecular dynamics or reverse Monte Carlo simulations. The thickness of the films was less than 5 nm, which was comparable to that of the real specimens. We scanned the electron probe with a convergence angle and beam size similar to those used in the experiment on the thin films and calculated the diffraction patterns for each position. Figure 5.12 shows an example of the correspondence between the glass structure model and a series of simulated electron diffraction patterns. An extracted medium-range order consisting of four atomic clusters is also depicted. This analytical technique can be utilized not only to reproduce angstrom-beam diffraction experiments, but also to extract medium-range order structures from structure models constructed by molecular dynamics or reverse Monte Carlo simulation.

5.5 Summary In this chapter, we have shown that the angstrom-beam electron diffraction technique is helpful for determining the local atomic configurations of amorphous materials by combining it with other experimental and computational techniques. The advantage of angstrom-beam electron diffraction over high-resolution TEM or STEM imaging is that the local diffraction patterns can be directly compared with global X-ray or neutron diffraction with a high degree of accuracy. Therefore, the local information obtained from angstrom-beam electron diffraction is useful for confirming the structure model constructed based on global structural information, such as X-ray or neutron diffraction. Furthermore, local diffraction experiments would become much more significant if the materials contained hyperordered structures that are related to spatial inhomogeneity or structural defects that determine the overall physical properties.

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Fig. 5.12 a Example of the structure model for the virtual angstrom-beam electron diffraction analysis. The beam positions are indicated by square marks. b Simulated diffraction patterns obtained from the beam positions in a. c Aggregate of atomic clusters found in the model of a. (Reproduced from [39]. CC-BY 4.0)

References 1. Warren BE (1990) X-Ray Diffraction. Dover Publications 2. Guinier A (1994) X-Ray Diffraction. In Crystals, imperfect crystals, and amorphous bodies. Dover publications 3. Cusack NE (1987) The physics of structurally disordered matter: an introduction. CRC Press 4. Kohara S, Suzuya K (2003) Nucl Instrum Methods Phys Res, B 199:23 5. Kohara S, Salmon PS (2016) Adv Phys X 1:640 6. Krivanek OL, Gaskell PH, Howie A (1976) Nature 262:454 7. Gaskell PH, Smith DJ (1980) J Microsc 119:63 8. Ichinose H, Ishida Y (1983) Trans Jpn Inst Met 24:405 9. Hirotsu Y, Akada R (1984) Jpn J Appl Phys 23:L479 10. Anazawa K, Hirotsu Y, Inoue Y (1994) Acta Metall Mater 42:1997 11. Matsushita M, Hirotsu Y, Ohkubo T, Oikawa T, Makino A (1996) Mater Sci Eng A 217–218:392 12. Hirotsu Y, Ohkubo T, Matsushita M (1998) Microsc Res Tech 40:284 13. Ohkubo T, Kai H, Makino A, Hirotsu Y (2001) Mater Sci Eng A 312:274 14. Ohkubo T, Hirotsu Y (2003) Phys Rev B 67:094201

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15. 16. 17. 18. 19. 20. 21. 22. 23.

Hirotsu Y, Nieh TG, Hirata A, Ohkubo T, Tanaka N (2006) Phys Rev B 73:012205 Hirata A, Hirotsu Y, Nieh TG, Ohkubo T, Tanaka N (2007) Ultramicroscopy 107:116 Yamasaki J, Mori M, Hirata A, Hirotsu Y, Tanaka N (2015) Ultramicroscopy 151:224 Treacy MMJ, Gibson JM (1996) Acta Cryst A 52:212 Treacy MMJ, Gibson JM, Fan L, Paterson DJ, McNulty I (2005) Rep Prog Phys 68:2899 Gibson JM, Treacy MMJ, Sun T, Zaluzec NJ (2010) Phys Rev Lett 105:125504 Voyles PM, Gerbi JE, Treacy MMJ, Gibson JM, Abelson JR (2001) Phys Rev Lett 86:5514 Voyles PM, Abelson JR (2003) Sol Energy Mater Sol Cells 78:85 Stratton WG, Hamann J, Perepezko JH, Voyles PM, Mao X, Khare SV (2005) Appl Phys Lett 86:141910 Cowley JM, Spence JCH (1979) Ultramicroscopy 3:433 Cowley JM (1985) Ultramicroscopy 18:11 Cowley JM (2002) Ultramicroscopy 90:197 Hirata A, Morino T, Hirotsu Y, Itoh K, Fukunaga T (2007) Mater Trans 48:1299 Hirata A, Hirotsu Y, Amiya K, Nishiyama N, Inoue A (2008) Intermetallics 16:491 Hirata A, Hirotsu Y, Amiya K, Inoue A (2008) Phys Rev B 78:144205 Hirata A, Guan PF, Fujita T, Hirotsu Y, Inoue A, Yavari AR, Sakurai T, Chen MW (2011) Nat Mater 10:28 Hirata A, Kang LJ, Fujita T, Klumov B, Matsue K, Kotani M, Yavari AR, Chen MW (2013) Science 341:376 Hirata A, Chen MW, J Non-Cryst Solids (2014) 383:52 Hirata A, Kohara S, Asada T, Arao M, Yogi C, Imai H, Tan YW, Fujita T, Chen MW (2016) Nat Commun 7:11591 Zhu F, Hirata A, Liu P, Song S, Tian Y, Han J, Fujita T, Chen MW (2017) Phys Rev Lett 119:215501 Nishino H, Fujita T, Cuong NT, Tominaka S, Miyauchi M, Iimura S, Hirata A, Umezawa N, Okada S, Nishibori E, Fujino A, Fujimori T, Ito S, Nakamura J, Hosono H, Kondo T (2017) J Am Chem Soc 139:13761 Hirata A, Ichitsubo T, Guan PF, Fujita T, Chen MW (2018) Phys Rev Lett 120:205502 Han J, Hirata A, Du J, Ito Y, Fujita T, Kohara S, Ina T, Chen MW (2018) Nano Energy 49:354 Hirata A (2021) Microscopy 70:171 Hirata A (2022) Quantum Beam Sci 6:28 Miracle DB (2004) Nat Mater 3:697 Sheng HW, Luo WK, Alamgir FM, Bai JM, Ma E (2006) Nature 439:419 Ma E (2015) Nat Mater 14:547 Frank FC (1952) Proc Royal Soc A 215:43 Soklaski R, Nussinov Z, Markow Z, Kelton KF, Yang L (2013) Phys Rev B 87:184203 Ding J, Cheng YQ, Ma E (2014) Acta Mater 69:343 Hohl A, Wieder T, van Aken PA, Weirich TE, Denninger G, Vidal M, Oswald S, Deneke C, Mayer J, Fuess H, J Non-Cryst Solids (2003) 320:255 Kohara S, Kato K, Kimura S, Tanaka H, Usuki T, Suzuya K, Tanaka H, Moritomo Y, Matsunaga T, Yamada N, Tanaka Y, Suematsu H, Takata M (2006) Appl Phys Lett 89:201910 Mcgreevy RL, Pusztai L (1988) Mol Simul 1:359 Keen DA, McGreevy RL (1990) Nature 344:423 McGreevy RL (2001) J Phys: Condens Mat 13:R877

24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.

36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50.

Chapter 6

Dynamics Ayano Chiba and Shinya Hosokawa

Abstract In this chapter, we review the wavenumber and energy ranges covered by various dynamics measurement methods and then outline inelastic neutron scattering (INS) and inelastic X-ray scattering (IXS) in particular. We explain their principles and methods and pick up recent applications and developments. As an example of recent INS research, a study of a solid refrigerant is first introduced, since Ålevel dynamics contributes to thermal properties. For another example, we look at recent advances on the boson peak, which is an excitation that is universally seen in structurally disordered systems. As an example of recent IXS studies, we pick up the measurement of phonon dispersion relations of Mg alloys, which are expected to be the next-generation structural material. In the last part of this chapter, we introduce recently developed measurement techniques of IXS in which Mössbauer nuclei are excited by synchrotron radiation and used for dynamic measurements. Keywords Inelastic neutron scattering · Inelastic X-ray scattering · Relaxation · Phonon · Magnon

6.1 Introduction Hyperordered structures are difficult to understand by ordinary diffraction experiments alone. By combining dynamic measurements with static measurements such as diffraction, it is possible to understand the structures both in space and time axes. Since dynamic measurements often do not require atomic periodicity, they will have great potential in investigating typical hyperordered structures in functional

A. Chiba (B) Department of Physics, Keio University, Yokohama 223-8522, Japan e-mail: [email protected] S. Hosokawa Institute of Industrial Nanomaterials, Kumamoto University, Kumamoto 860-8555, Japan e-mail: [email protected] © Materials Research Society, under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. Hayashi (ed.), Hyperordered Structures in Materials, The Materials Research Society Series, https://doi.org/10.1007/978-981-99-5235-9_6

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materials, such as formations of clusters, the co-doping of impurity elements, or element-vacancy correlations. In Sect. 6.2, we will look at several dynamical methods for investigating different space and time scales. Some traditional macro-, meso-, and microscopic methods are introduced by applying different external fields to target materials, such as optic, electric, magnetic, and acoustic fields. In Sect. 6.3, we explain how inelastic scattering experiments can provide the dynamic structure of materials. In Sects. 6.4–6.7, we describe inelastic scattering measurements using neutron and synchrotron radiation quantum beams, describing their advantages and disadvantages, giving examples of experimental results, and introducing new measurement methods that have recently been developed. Finally, overall features of dynamical methods for investigating hyperordered structures of materials are summarized in Sect. 6.8.

6.2 Space and Time Scales of Dynamical Methods Movements of atoms have various scales in space length (or wavenumber) and time (or frequency). Figure 6.1 shows the wavenumber (space)–energy (time) scales accessible by various measurement techniques, frequently employed in studies such as Ref. [1]. The corresponding correlation length .r and the atomic/molecular motion time .t are also indicated at the top- and right-hand side of the figure. To investigate the dynamics of materials, it is important to choose an appropriate method for the length and time scales of interest, as well as to know the advantages and disadvantages of experimental methods. In this section, we focus on the measurements of atomicscale microscopic dynamics using quantum beams and present typical examples of hyperordered structure measurement. Thus far, optical methods have frequently been used for investigating the dynamics of materials, particularly localized vibration modes. Some examples are Raman scattering, infrared (IR) absorption spectroscopy, and Brillouin scattering, which are shown in Fig. 6.1, depending on the excitation energies of interest. Since the energy of light is low for visible and infrared light, the observable . Q ranges are very limited, as shown in the figure. Photon correlation spectroscopy (PCS) can reach very low energies. The recently developed X-ray photon correlation spectroscopy (XPCS) technique widely expands the observable . Q range up to several 10 nm.−1 with a very low excitation energy. Inelastic ultraviolet scattering (IUVS) is also a relatively new dynamics measurement technique that emerged in the 2000s and has a wavenumber range lower than that of neutrons and an energy range by which one can measure the speed of sound in some materials. Neutron spin echo (NSE) is a well-established technique that covers the lowest energy region in neutron-based dynamics measurements. Since this book mainly covers structures on space scales of around Å order, here we focus on methods that enable the simultaneous measurement of the time axis and the space axis of Å order, namely, we focus on inelastic neutron scattering (INS, Sect. 6.4) and inelastic X-ray scattering (IXS, Sect. 6.5). We would also like to briefly

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IXS

Raman IR IUVS Brillouin scattering

INS NSE TDI

PCS

XPCS

Fig. 6.1 Wide ranges of wavenumber . Q–energy . E space and areas accessible by several experimental techniques. See texts for details. The corresponding correlation time .t and length .r are also indicated at the right and top axes, respectively Fig. 6.2 Schematic diagram of inelastic scattering

kf, Ef

Q ki, E i

2θ sample

introduce two recently developed methods that use the Mössbauer effect excited by synchrotron radiation: nuclear resonant inelastic X-ray scattering (NRIXS, Sect. 6.6), by which one can selectively measure the dynamics of atoms that have Mössbauer nuclei, and time-domain interferometry (TDI, Sect. 6.7), by which one can measure the region around 10 nm.−1 , 10 neV, which was not covered previously.

6.3 Principles of Inelastic Scattering Figure 6.2 shows a typical inelastic scattering experiment. Suppose an incident neutron or X-ray beam with wave vector.ki and energy. E i . A scattered beam changes both the wave vector and energy to .k f and . E f , respectively. Then, the scattering intensity . S(Q, E) can be described as a function of wave vector transfer, .Q = k f − ki , and energy transfer, . E = E f − E i . . S(Q, E) is referred to as the dynamic structure factor.

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S(Q, E) can be separated into two factors as .

S(Q, E) = Sinc (Q, E) + Scoh (Q, E),

where . Sinc (Q, E) and . Scoh (Q, E) represent incoherent and coherent dynamic structure factors, respectively, denoting the self- and pair parts of this function. To discuss the physical meaning of the . S(Q, E) function, we should start with the Van Hove space-time correlation function [2], .

G(r, t) =

1 , N

where n(r, t) =

N Σ

.

δ(r − r j (t))

j=1

is the number density as functions of .r and .t, and .< > represents the average over the system. The .G(r, t) function is also separated into the self- and pair correlation functions (.G s (r, t) and .G p (r, t), respectively) as .

G(r, t) = G s (r, t) + G p (r, t).

Here, .G s (r, t) is the probability of finding a particle at position .r and time .t if the same particle was at position .0 and time 0. On the other hand, .G p (r, t) is that if any particle was at position .0 and time 0. The intermediate scattering function . I (Q, t) introduced by Van Hove [2] can be written as

.

I (Q, t) = =

N N 1 ΣΣ

N i=1 j=1 N N 1 ΣΣ . N i=1 j=1

Note that the self-part of . I (Q, t), . Is (Q, t), means .i = j, i.e., I (Q, t) =

. s

N 1 Σ , N j=1

and the pair part . Ip (Q, t) is the remainder. This function is also connected with a space-Fourier transform of .G(r, t), i.e.,

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∫ .

I (Q, t) =

G(r, t) exp(iQ · r)dr,

(6.1)

V

and the . S(Q, E) function is a time-Fourier transform of . I (Q, t), i.e., 1 . S(Q, E) = 2π

∫∞ I (Q, t) exp(−i Et/ℏ)dt. −∞

As indicated by these equations, the information directly obtained from scattering experiments is the Fourier transform of real space and time. For static structures, the fact that the scattering amplitude is given by the three-dimensional Fourier transform of the real-space potential (first-order Born approximation) is explained as a consequence of simple scattering theory in Ref. [3]. Note that in the above equation, . Sinc (Q, E) and . Scoh (Q, E) can be calculated from . Is (Q, t) and . Ip (Q, t), respectively. Therefore, .G(r, t) can, in principle, be evaluated from the space–time double Fourier transforms of the experimentally obtained . S(Q, E) function. Because of the limited space and time ranges in the experimental data, however, it is not easy to obtain .G(r, t) without special analytical elaborations. The spatial range of primary interest in the structure of hyperordered materials is on the order of Å. Since neutrons have mass whereas X-rays have no mass, the energy . E of wavelength .λ = 1 Å, for example, can be estimated as .

(Neutron) E = P 2 /2m n ∼ 81.8 meV, (X-ray) E = hc/λ ∼ 12.4 keV,

where . P and .m n are the momentum and the mass of the neutron, respectively, .h is Planck’s constant, and .c is the speed of light. Compared with these is the energy we aim to measure, e.g., phonons, on the meV order. To measure such dynamics, scattered waves in which meV-order excitations are added to or subtracted from the incident energy must be resolved from the incident wave. Therefore, INS historically had an advantage in studying phonons, because IXS requires a precise energy resolution of at least .ΔE/E i ∼ 10−7 for the incident energy . E i . In other words, a very sharp monochromatic technique is required. Nowadays, the achievable resolutions of INS and IXS are on the order of 1 .µeV and 1 meV, respectively, and thus dynamics are actively measured by both techniques.

6.4 Inelastic Neutron Scattering (INS) Historically, neutron scattering had been the only available technique for measuring collective dynamics, including phonons and slow dynamics, such as diffusion, because it requires a resolution of at least the meV order. For example, a Nobel Prize

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winner, B. N. Brockhouse, measured INS on liquid water to investigate its diffusive motion in 1959 [4]. Although it is now possible to measure dynamics with X-rays, neutron scattering has the following four advantages: First, high-energy-resolution measurement is possible. An energy resolution up to three orders of magnitude higher than that of X-rays can be achieved. For example, for quasi-elastic scattering instruments, a near-backscattering spectrometer DNA at J-PARC, Japan, has a standard resolution of 2–20 .µeV [5], whereas the best IXS resolutions are barely beginning to fall below meV. Second, magnetic excitations can be observed in . Q−E space [6]. To study magnetic excitations in a wide . Q−E space, INS is almost the only technique available. Even with the availability of highly brilliant synchrotron radiation X-rays, there is still difficulty in completing the study of magnetic static structures by RIXS alone, and, moreover, since inelastic scattering is roughly more than three orders of magnitude weaker than elastic scattering in intensity, INS is likely to continue to be the main method for observing magnetic excitations. Third, coherent and incoherent-scattering cross sections are adjustable by isotopes. By isotopic substitution, one can experimentally pick up dynamics of interest. To investigate self-diffusion, for example, it is advantageous to select nuclides with large incoherent-scattering cross sections. A list of scattering cross sections for each element and isotope, as well as abundance or half-life in nature for each nuclide, can be found in Refs. [7, 8]. With the right combination of nuclides, the diffusion of a particular element in a multi-elemental material can be measured. This technique is often used to study the diffusion of hydrogen in hydrogen storage materials. Fourth, neutron scattering is also suitable for measuring the dynamics of light elements such as hydrogen, because the incoherent neutron-scattering cross section of hydrogen is as large as 80 barn. Elements with small atomic numbers are difficult to see using X-rays, but the order in which they are visible by neutrons varies from nuclides to nuclides, so isotope substitution can be used to adjust the visibility of certain elements. The disadvantages of neutron scattering include the following: First, samples generally need to be mm to cm in size. The reasons for this are the small scattering probability due to fm-sized nuclei, the limited intensity of the incident neutron beam, and the difficulty in focusing neutrons. Second, some elements with high neutron absorption coefficients, such as .3 He, .6 Li, B, Gd, Cd, Au, and Hg, are difficult to measure. In practice, modern spallation neutron sources provide intense beams, so even in the presence of some of these elements static structures are becoming measurable, but the dynamics would still be difficult to observe. Third, since some elements have large incoherent-scattering lengths compared with coherent ones, such as H, .7 Li, and V, only self-diffusive motions can be observed for these elements, and collective information such as phonon dispersions would be difficult to measure. Fourth, the velocity of neutrons at wavelengths of about Å is slow, about several thousand m/s (about 2200 m/s for 25 meV), which limits the . Q–. E region that can be measured. Let us look at this point in more detail below.

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(a)

(b)

Ei

Q3

k2

E

k3

Q2 k1

Q1

2θ

ki

Fig. 6.3 a Schematic diagram of INS, and b typical . Q–. E space of INS with . E i = 50 meV

We briefly explain the. Q–. E constraint on the INS technique. A schematic diagram of inelastic scattering with . E i = constant is shown in Fig. 6.3. . E i s on the order of 10 meV is often used, and the magnitude of .k f changes from that of .ki , as shown in Fig. 6.3a. Thus, the magnitude of .Q changes with not only the usual .2θ but also . E, as expressed by .

Q 2 = ki2 + k 2f − 2ki k f cos 2θ ] √ 2m [ = 2 2E i + E − 2 E i (E i + E) cos 2θ . ℏ

(6.2)

Figure 6.3b shows a typical . Q–. E space of INS with . E i = 50 meV at several 2.θ values. As seen in the figure, . Q–. E relations at constant 2.θ are not straight lines but curved. Note that INS can be performed within the range of these curves. The chain line indicates the dispersion relation of longitudinal acoustic (LA) sound excitations at a low . Q with a velocity of 6,000 m/s; the detailed information on this excitation cannot be evaluated by using this INS spectrometer. For information on theoretical aspects of neutron scattering, see Ref. [9]. For information on quasi-elastic scattering in particular, see Ref. [10]. Let us look at AMATERAS, Fig. 6.4, as an example of an INS (and also quasielastic neutron scattering, QENS) instrument whose main research targets are phonon and magnon dispersion relations, as well as diffusion, in the energy region below 3– 80 meV, depending on. E i . It can achieve a resolution of.∼10.µeV order if the incident energy is on the order of 1 meV [11, 12]. The instrument is characterized not only by its high resolution, but also by its high neutron intensity. To achieve high intensity,

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Fig. 6.4 Schematic of cold-neutron disk chopper spectrometer AMATERAS [12]

a coupled H.2 moderator is used for a source. Also, to achieve high resolution, in addition to a disk chopper for beam monochromatization near the sample, another chopper for pulse shaping is placed near the beam source. As a result, high-intensity peaks are cut out with high resolution. AMATERAS has recently been used to microscopically understand a potential new cooling technology. In general, heat properties in solid materials are due to the dynamics of electrons and atoms, so it is plausible that dynamics measurements play an active role in thermal materials research. The background of this research is that nearly 30% of all electricity is used for cooling these days, and solid refrigerants are attracting attention to reduce environmental impact. It has been known that “orientation-disordered crystals”, such as neopentylglycol (NPG), have a pressureinduced calorific effect 10 times greater than that of conventional solid refrigerants, but the microscopic mechanism had been unknown. Using AMATERAS with an incident energy of . E i = 2.64 meV, Li et al. found that the free rotation motion of NPG molecules begins in the temperature range of 300–320 K, as shown in Fig. 6.5 [13, 14]. The system changes from a monoclinic to a face-centered cubic lattice with increasing temperature in this temperature range, as has been known from X-ray diffraction. However, since it is the dynamics of the atoms that absorb/desorb the heat, ordinary diffraction alone may not be able to reveal its nature as a refrigerant. Combining these quasi-elastic neutron scattering results with molecular dynamics simulations, it was found that the entropy change caused by the phase transition from a state in which only the rotational relaxation of the methyl group is observed (up to 300 K) to a state in which the entire molecule can rotate (above 320 K) is responsible for the solid-state caloric and “barocaloric” effects. This study is an example of how the high energy resolution of neutron scattering can be used to understand thermophysical properties that are directly related to atomic dynamics. For structurally disordered systems, some progress achieved using INS instruments, including AMATERAS, has recently been reported in the study of the “boson

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Fig. 6.5 Observed dynamics and its schematic picture of neopentylglycol (NPG) molecules [13, 14]. a Temperature dependence of . S(Q, E) with incident energy of . E i = 2.64 meV taken at AMATERAS, which shows free orientational motion begins at the temperature range 300–320 K. b Schematic figure of “orientation-disordered crystal”, in which the center of each molecule has a regular crystalline arrangement, but each molecule can rotate freely. c Crystal lattice of NPG in the low-temperature phase. The brown, red, and pink spheres represent carbon, oxygen, and hydrogen atoms, respectively. Reproduced with permission from Ref. [13]

peak”. In general, glass materials are known to have low-energy harmonic vibrations called boson peaks [15]. Many studies have been conducted to investigate the origin of boson peaks [16], where the material most studied is SiO.2 glass. For a recent example, SiO.2 glass with a new structure in the sense of a high first-sharp diffraction peak has been prepared, and changes in the boson peaks have been reported [17]. In 2021, a “magnetic boson peak” was reported in a spin glass state. To answer the fundamental question of whether there are magnetic excitations inherent in spin glass, Kofu et al. measured magnetic excitations in a magnetic ionic liquid C4mimFeCl.4 at a low temperature of 0.3 K in a glassy solid state as well as in a crystalline state [18]. Here, C4mimFeCl.4 contains iron as an anion (Fig. 6.6) and takes the liquid state at room temperature. It can take a crystalline or glass solid state, depending on the cooling rate. The sample is deuterated to avoid the strong incoherent scattering from H atoms. As a result of INS, they found a “magnetic boson peak”, as shown in Fig.

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Fig. 6.6 Molecular structure of magnetic ionic liquid C4mimFeCl.4 and its magnetic excitations [18]. a Molecular structure. The arrow indicates that the.FeCl4 − anion has a spin of 5/2. b Magnetic boson peak in the glass state and spin-wave excitations in the crystalline state. Reproduced from Ref. [18]. CC-BY 4.0

6.6, that is considered to be caused by the lack of magnon propagation, reflecting the lack of periodicity. The boson peak observed here is due to magnons and not vibrations. In 2022, a boson peak seems to have been observed in a material that could be classified as a crystal in a sense [19]. The crystal structure of Ba.1−x Sr.x Al.2 O.4 can be regarded as a network of AlO.4 oxygen tetrahedra penetrating into the Ba sublattice formed by the periodic placement of Ba atoms. Ishii et al. found a “sublattice glassy state” in which the periodicity of Ba is maintained and the AlO.4 network becomes glassy. Notably, Ba.1−x Sr.x Al.2 O.4 , although clearly a crystalline solid, exhibits thermal properties commonly found in amorphous solids, such as low thermal conductivity comparable to that of SiO.2 glass. The “sublattice glassy state” can be created by a simple method of uniformly mixing and heating raw materials. In this study, the boson peak played the role of showing the amorphous nature of the sample, where AMATERAS was used for the measurement.

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6.5 Inelastic X-Ray Scattering (IXS) In the 1990s, IXS was put into practical use for investigating the phonon dynamics of materials, owing to the remarkable development of synchrotron radiation facilities and X-ray spectroscopic techniques [20]. The most characteristic feature of this technique is that the incident X-ray energy is on the order of 10 keV, while the phonon excitation energy of interest is on the order of meV. Thus, a resolving power of 10.7 is necessary, which can be achieved by a sharp monochromatic technique using a large synchrotron radiation facility and sophisticated techniques in X-ray optics. Several high-resolution IXS spectrometers are in operation all over the world, i.e., BL35XU [21] and BL43XU [22] at SPring-8, Japan, ID28 [23] at ESRF, France, and sector 3 [24] and 30 [25] at APS, USA. The minimum energy resolution of IXS is 0.84 meV FWHM [26] at present. IXS measurements have four advantages over INS. First, samples as small as 0.1 mm or less can be measured. In the case of INS, on the other hand, the sample size is typically 1 mm–1 cm. Second, because the incident X-ray energy is very large (about 10 keV), the . E and . Q are separated, in contrast to Equation 6.2 for neutrons, and the accessible . Q–. E range is simple, in contrast to that shown in Fig. 6.3 for INS, making it possible to measure phonon excitations of light elements, such as liquid Li, with sound speeds of about 4500 m/s [27]. Third, there is no need to deuterate the sample. This advantage is relevant only if the sample contains hydrogen atoms. For neutrons, the incoherent-scattering cross section of hydrogen atoms is 80 barn [7, 8], which is more than 10 times larger than that of other elements and causes high background. This can be critical because the intensity of inelastic scattering is low compared with that of elastic scattering. Thus deuterium substitution is usually recommended. However, not only can deuterated substitution be difficult and expensive, but there is also concern that it may alter physical properties such as the dispersion relation of phonons. With X-rays, there is no need for deuterium substitution. Fourth, since synchrotron radiation X-rays are usually horizontally polarized, longitudinal and transverse phonons can be separately observed by selecting appropriate crystal angles with respect to the incident X-rays, as explained later. The disadvantage of IXS to INS is that light elements are difficult to see because the magnitude of the atomic scattering factors is ordered by atomic number; for atomic scattering factors for X-rays, see Ref. [28, 29]. The energy resolution power of about 10.7 is achieved by a sharp monochromatic technique with a backscattering geometry of Si(9 9 9), Si(11 11 11), or Si(13 13 13). With this technique, energy resolutions of about 3.0 [27], 1.5 [1], and 1.0 meV [26] can be achieved at incident X-ray energies of 17.793, 21.747, and 23.724 keV, respectively. Figure 6.7 shows a schematic view [30] of a typical IXS spectrometer at BL35XU of SPring-8 [31]. The X-rays generated from an undulator insertion device at a large storage ring are roughly monochromatized by a liquid-N.2 -cooled Si(111) double crystal and raised up to obtain sufficient working space again by two Si(111) crys-

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Detecters Sample

10 m Synchrotron ring

Undulator

Backscattering Hutch

Analyser Hutch

Curved Si analyzers

Mirror

Backscattering monochromator

Incident X-ray intensity monitor

Optics Hutch

Liq. N2 cooled monochromator

Fig. 6.7 Schematic view of a typical IXS spectrometer. Reproduced with permission from Ref. [30]

tals. The X-ray beam is highly resolved by using a Si(11 11 11) backscattering monochromator with a scattering angle of about 89.98.◦ and focused onto the sample with a bent mirror. The incident beam size is about .0.1 × 0.1 mm.2 . The scattered X-rays travel about 10 m, again are backscattered and focused by 12 Si(11 11 11) curved energy analyzer crystals, travel again about 10 m, and are finally detected near the sample position by 12 CdZnTe detectors. The energy scan is conducted by changing the temperature of the backscattering monochromator crystal by on the order of 1 K, while the temperatures of the analyzer crystals are kept unchanged within .±1 mK during the experiments. The energy is calibrated by using known phonon energies of a diamond crystal, and the temperature/energy ratio of the monochromator is obtained to be about 18 mK/meV. The energy resolution of the IXS spectrometer was obtained as the elastic peak width of a Plexiglas® sample until very recently, but that of TEMPAX® glass has been found to be more correct. In addition, using the horizontal linear polarization of the incident X-ray beam, longitudinal and transverse phonons can be separately observed by adjusting the crystal angles with respect to the incident beam. The above descriptions are for general IXS setups, and much smaller sample sizes of .µm order (about .4 × 4 .µm.2 ) where a diamond anvil cell (DAC) is applicable by using a Kirkpatrick–Baez mirror [32], as well as high temperatures up to 3000 K by using a laser heating system, expand investigation targets to those in the geoscience field within the earth. Here, we introduce recent IXS measurements on typical functional materials from the view point of hyperordered structures science. Mg is a light element, and because this metal is soft, chemically active, and flammable, it was believed that Mg cannot be used as a structural material like Al, although this element can easily be retrieved from the sea. In 2001, however, these disadvantages of Mg were found to be rectified by adding small amounts of Zn and rare-earth metals [33]. Because of the excellent properties together with the ease of recycling, the Mg alloys are expected to be next-generation structural materials for bodies of subway trains or even aircraft, for example.

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151 [0001]

(a)

[1100]

[1120] h h c SF c h h qx h h c SF c h h

(b)

qz A

K

Γ

2π/c M

qy

4π/3a

Fig. 6.8 a Schematic of Zn.6 Y.8 clusters in long-period stacking ordered (LPSO) phase located at stacking faults. b First Brillouin zone of .hcp structure of pure Mg. Reproduced with permission from Ref. [35]

The origin of the remarkable properties was clarified by a scanning transmission electron microscopy (STEM) technique with an atomic-resolution angle annular dark-field (HAAF) function. The formation of clusters of Zn.6 Y.8 fragments of . f cc structure embedded in a long-period stacking ordered (LPSO) phase was found to play an important role (see Fig. 6.8a) [34]. To clarify the impurity effects in the microscopic elastic properties of the revolutionary Mg alloys, IXS experiments were carried out on a Mg.85 Zn.6 Y.9 crystal at room temperature at BL35XU of SPring-8 [35, 36]. Figure 6.9 shows logarithmic plots of the IXS spectra of an Mg.85 Zn.6 Y.9 single crystal for the (a) longitudinal and (b) transverse modes along the .Γ-A direction. The first Brillouin zone of a .hcp structure of pure Mg is given in Fig. 6.8b, and the .Γ-A direction is along the .c-axis. In (a) and (b), several excitation modes are observed in the Brillouin zone. Among them, two types are seen, one showing clear dispersion relations with . Q, as indicated by arrows and the other having no dispersion, i.e., localized modes, given by dashed lines. Figure 6.9c shows the dispersion curves along the .Γ-A direction, in which circles and triangles represent the longitudinal and transverse modes, and closed and open marks denote clear peaks and broad shoulders, respectively. The dispersive excitations are mostly located on those of pure Mg given by solid curves. Additionally, dispersion-less excitations are newly observed at about 4, 10, and 17 meV, as shown by the dashed lines. Exactly the same dispersion relations are observed in the.Γ-K and .Γ-M directions, perpendicular to the .c-axis [35]. An ab initio molecular dynamics simulation was carried out to obtain the vibrational density of states (DOS) for this system. The 4, 10, and 17 meV modes are interpreted as the twisting and stretching modes and between the cluster and host Mg atoms. As another example of phonon dispersion measurements of heavy-element-doped inorganic materials, Kimura et al. reported on Ta-doped Fe.2 VAl Heusler-type thermoelectric materials [37]. Another study that has yielded important results at BL35XU is the measurement of the dispersion relation of rubrene (C.42 H.28 ), a well-studied high-mobility organic semiconductor [38]. This result is important because phonons are considered to play a major role in the physical properties of molecular solids [39]. The mechanism

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of electrical conduction in high-mobility organic semiconductors differs from that of the general band conduction in inorganic semiconductors such as Si, and it is currently believed that the behaviors of phonons inevitably explain it. Nevertheless, phonons in organic molecular crystals generally have more than 100 branches, and even with neutron scattering, there have been few measurements of dispersion relations in organic semiconductors [40]. Since phonons are considered to be strongly involved in the electrical conduction of high-mobility organic semiconductors [41], this research is important to the study of materials such as those of organic electroluminescence displays. Moreover, it is known to be difficult to reproduce weak forces such as intermolecular forces in first-principles calculations, and the measurement of dispersion relations will be useful to verify such forces. Note that it is becoming possible to measure phonon dispersions of nondeuterated single crystals with a size of 1 mm.3 or smaller because one must deuterate the samples to reduce the background level when the measuring method is INS, and also the sample size must be larger for INS. The successful measurement of the dispersion relation of rubrenes is thus a significant development.

6.6 Nuclear Resonant Inelastic X-Ray Scattering (NRIXS) NRIXS is a method by which we can selectively measure . S(Q, E) of atoms that have Mössbauer nuclei that can be excited by synchrotron radiation. Figure 6.10a shows a schematic view of a typical NRIXS setup [42]. The X-rays generated from an undulator insertion device are roughly monochromatized by a Si or diamond double crystal and highly resolved by several types of high-resolution monochromator, such

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as dispersive channel-cut or nested configuration monochromators, depending on the outgoing X-ray energy and resolution of the target element [43]. The intensity of the incident beam is usually monitored with an ionization chamber. The wellresolved X-rays are irradiated onto the sample including an appropriate isotope, and the emitted X-rays are detected with an avalanche photodiode (APD) detector with a time resolution of 1 ns order. Figure 6.10b shows a typical time response of the signal, which is composed of a prompt peak scattered by electrons in the sample and a long-lasting delayed NRIXS signal [44]. Thus, pulsed synchrotron X-rays with a pulse interval of about 100 ns are essential for observing the phonon DOS of the sample. Similar to the nonresonant IXS experiments, NRIXS measurements under high temperatures and pressures are possible by using DAC with laser heating [45]. Atomic dynamics of impurity atoms in a simple metal are one of the key issues concerning hyperordered structure of materials. Seto et al. [44] measured NRIXS spectra of the .57 Fe isotope in highly diluted (0.017 and 0.1 at.%) Fe in Al and Cu metals, respectively, to investigate the local vibrational DOSs of Fe impurities. The measurements were carried out at room temperature at BL09XU of SPring-8. The storage ring was operated in a special timing mode having a bunch interval of 228 ns. A bandwidth of 3.2 meV full-width at half-maximum for the incident X-ray beam was obtained by using a nested high-resolution monochromator consisting of asymmetric Si(511) and asymmetric (975) channel-cut crystals. The energy of the radiation was scanned around the nuclear resonant energy of .57 Fe of 14.413 keV. Figure 6.11 shows the NRIXS spectra of .57 Fe in (a) Al and (b) Cu metals. In both spectra, peaks are observed at the excitation energy of about 14 meV. Another peak is seen in (b) at about 30 meV, while the spectrum in (a) shows only a small tail in this energy range. A theoretical calculation was performed for investigating these special features in the NRIXS spectra, and it was found that the 14 meV peaks originate from the resonance with the vibrational modes of the host Al atoms, while the 30 meV peak is interpreted as a localized mode of the impurity Fe atoms.

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Fig. 6.11 NRIXS spectra of diluted .57 Fe in a Al and b Cu metals. Reproduced with permission from Ref. [44]

6.7 Time-Domain Interferometry (TDI) The Mössbauer effect is an extremely sharp resonance effect. For example, the gamma rays elastically emitted from .57 Fe excited nuclei, as also described in Sect. 6.6, provide very good monochromatic light with an energy width of .Γ0 = 4.7 neV, which could be an ideal incident light for dynamics measurements with neV resolution. The arrangement shown in Fig. 6.12 is used to measure . I (Q, t) for general materials that do not contain Mössbauer nuclides. In general interferometry, light is interfered through different paths in space by splitting the light with a half-mirror and stacking them again. In this method, however, as shown in Fig. 6.12c, the light is interfered through different paths in time, which is called time-domain interferometry (TDI) [46]. The most basic TDI concept can be understood as follows [47]. Assume that there is no sample in the TDI in Fig. 6.12 and a scattering angle .2θ of zero [48]. The excitation of .57 Fe nuclei by synchrotron radiation produces strongly forward-directed Mössbauer gamma rays at Fe-1 in Fig. 6.12. If the energy of the gamma rays emitted from another .57 Fe (Fe-2 in Fig. 6.12) is shifted by the Doppler effect, for example, the interference between the gamma rays from Fe-1 and Fe-2 will produce a beat that can be observed. An avalanche photodiode (APD) is used for the detector, as described in Sect. 6.6. If the sample to be measured is placed between Fe-1 and Fe-2, and the detector is placed in the direction of.2θ after the momentum transfer with the sample, the light passing through path-1 and path-2 in Fig. 6.12 will be scattered by the sample at different times. Therefore, the intermediate scattering function . I (Q, t) (Eq. 6.1) of the sample can be measured as the amplitude of the interference pattern. However, it was shown that the correct . I (Q, t) cannot be obtained by simply using TDI with a single gammaray wavelength because of the finite energy width of the incident light. In practice, therefore, “multiline” TDI was performed to obtain the correct . I (Q, t) by applying a magnetic field to Fe-1 and Fe-2 to exploit the hyperfine structure of the nuclear

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Fig. 6.12 Schematic of time-domain interferometry (TDI). The figure is drawn with reference to Ref. [48]

energy levels and using monochromatic gamma rays of several different energies [49]. The momentum (space)–energy (time) regions covered by TDI are shown in Fig. 6.1, from which one can see that it covers areas that could not be measured previously. Here, we introduce the following discovery due to the ability of measurement in spatial and temporal domains that were previously unmeasurable. Figure 6.13 shows the inverse temperature dependence of the relaxation time. of supercooled o-terphenyl measured by TDI [50]. Here, o-terphenyl is a typical glassforming material that has been intensively studied. In the supercooled liquid state of glass-forming materials, a subnanometer-scale relaxation process called Johari– Goldstein (JG) relaxation can be observed at temperatures below the melting point . Tm as shown in Fig. 6.13a, but the space scale of the branching of JG relaxation from the .α relaxation, i.e., diffusion, had been unknown because of the lack of a measurement technique that covers the space and time range simultaneously. Figure 6.13b shows that there is no branching of JG relaxation in the main peak region of . S(Q). This means that the branching of JG relaxation does not occur at the scale of intermolecular size. However, a crossover from the .1/T -dependence of the JG relaxation from the slope of .α relaxation is observed in the larger . Q region, i.e., on a more localized space scale. This is the first experimental result of observing the branching of the JG relaxation and clarifying its origin from a microscopic viewpoint in molecular glasses. TDI uses meV-resolution synchrotron radiation pulses to excite .57 Fe nuclei and is currently available at BL35XU in SPring-8, Japan.

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Fig. 6.13 Inverse temperature dependence of molecular motion time related to glass transition. a General conceptual picture of the glass transition of liquids. Sup-cooled Liq. denotes supercooled liquid. b TDI measurement results of the relaxation time . for .o-terphenyl. The inset shows the . S(Q) and . Q regions of TDI measurements. Reproduced with permission from Ref. [50]

6.8 Summary and Perspective In this article, we review the remarkable experimental developments in the methods for observing the dynamics of materials using quantum beams of X-rays and neutrons from recently improved sources, e.g., SPring-8 and J-PARC, respectively. Functional materials having the so-called hyperordered structure usually include a hidden atomic configuration that cannot be easily clarified by experimental studies for static and averaged atomic structures. It is well known that traditional dynamic methods of Raman scattering and IR spectroscopy yield useful and additional information about local atomic arrangements of molecules and covalent materials, indicating that dynamic experiments reveal hidden information of materials. Nowadays, we can evaluate many dynamic properties by using improved quantum beams, as introduced in this article. Therefore, it is, at present, very important for scientists to know a wide range of both the properties of a researched material caused by an expected hyperordered structure and the advantages/disadvantages of the experimental methods for investigating the dynamics of materials. We expect that readers will gain new insights for investigating the science of materials by considering the dynamic properties when handling functional materials. Acknowledgements The authors greatly appreciate Prof. Maiko Kofu (J-PARC Center, JAEA) and Prof. Makina Saito (Tohoku Univ.) for their valuable suggestions.

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35. Hosokawa S, Kimura K, Stellhorn JR, Yoshida K, Hagihara K, Izuno H, Yamasaki M, Kawamura Y, Mine Y, Takashima K, Uchiyama H, Tsutsui S, Koura A, Shimojo F (2018) Acta Mater 146:273 36. Hosokawa S, Kimura K, Yamasaki M, Kawamura Y, Yoshida K, Inui M, Tsutsui S, Baron AQR, Kawakita Y, Itoh S (2017) J Alloys Compd 695:426 37. Kimura K, Yamamoto K, Hayashi K, Tsutsui S, Happo N, Yamazoe S, Miyazaki H, Nakagami S, Stellhorn JR, Hosokawa S, Matsushita T, Tajiri H, Ang AKR, Nishino Y (2020) Phys Rev B 101:024302 38. Takada K, Yoshimi K, Tsutsui S, Kimura K, Hayashi K, Hamada I, Yanagisawa S, Kasuya N, Watanabe S, Takeya J, Wakabayashi Y (2022) Phys Rev B 105:205205 39. Fratini S, Nikolka M, Salleo A, Schweicher G, Sirringhaus H (2020) Nat Mater 19:491 40. For example, see Natkaniec I, Bokhenkov EL, Dorner B, Kalus J, Mackenzie GA, Pawley GS, Schmelzers U, Sheka EF (1980) J Phys C: Solid St Phys 13:4265 41. Coropceanu V, Cornil J, da Silva Filho DA, Olivier Y, Silbey R, Brédas J-L (2007) Chem Rev 107:926 42. Seto M, Yoda Y, Kikuta S, Zhang XW, Ando M (1995) Phys Rev Lett 74:3828 43. Alp EE, Sturhahn W, Toellner TS, Zhao J, Hu M, Brown DE (2002) 144/145:3 44. Seto M, Kobayashi Y, Kitao S, Haruki R, Mitsui T, Yoda Y, Nasu S, Kikuta S (2000) Phys Rev B 61:11420 45. Lin J-F, Sturhahn W, Zhao J, Shen G, Mao H-K, Hemley RJ (2005) In: Chen J, Wang Y, Duffy TS, Shen G, Dobrzhinetskaya LP (eds) Advances in high-pressure techniques for geophysical applications. Elsevier, Amsterdam, p 397 46. Baron AQR, Franz H, Meyer A, Rüffer R, Chumakov AI, Burkel E, Petry W (1997) Phys Rev Lett 79:2823 47. Smirnov GV, van Bürck U, Arthur J, Popov SL, Baron AQR, Chumakov AI, Ruby SL, Potzel W, Brown GS (1996) Phys Rev Lett 77:183 48. Saito M, Yamaguchi T, Nagao M (2022) Butsuri 77:690 (in Japanese) 49. Saito M, Masuda R, Yoda Y, Seto M (2017) Sci Rep 7:12558 50. Saito M, Kitao S, Kobayashi Y, Kurokuzu M, Yoda Y, Seto M (2012) Phys Rev Lett 109:115705

Chapter 7

Property Measurements of Molten Oxides at High Temperature Using Containerless Methods Takehiko Ishikawa, Paul-François Paradis, and Atsunobu Masuno

Abstract To understand the hyperordered structures found in new glasses fabricated with containerless methods, it is of paramount importance to know not only their atomic structures but also their macroscopic thermophysical properties. However, due to their high melting temperatures and the risk of contamination from the crucibles, molten oxides whose melting temperatures are above 2000 °C can hardly be processed using conventional methods. This explains that the published data on thermophysical properties are very scarce. In this chapter, four containerless methods capable of measuring several thermophysical properties such as density, surface tension, and viscosity are introduced. Three of them use a gas flow to levitate samples against gravity, while the electrostatic levitation furnace onboard the International Space Station utilizes the Coulomb force to spatially position samples in microgravity. Features of each method are summarized in this chapter, including their advantages and disadvantages. Finally, the measured data of refractory oxides whose melting temperatures are above 2000 °C are summarized. Keywords Levitation · High temperature melts · Density · Viscosity

T. Ishikawa (B) Japan Aerospace Exploration Agency, Tsukuba 305-8505, Japan e-mail: [email protected] P.-F. Paradis INO, Quebec G1P 4S4, Canada e-mail: [email protected] A. Masuno Kyoto University, Kyoto 615-8520, Japan e-mail: [email protected] © Materials Research Society, under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. Hayashi (ed.), Hyperordered Structures in Materials, The Materials Research Society Series, https://doi.org/10.1007/978-981-99-5235-9_7

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7.1 Introduction Importance of levitation techniques is increasing in the research area of oxide glasses since new glasses fabricated with containerless methods exhibit structures that are significantly different from those of conventional glasses. They do not contain the corner-sharing tetrahedral networks which are commonly found in the conventional glasses [1]. Moreover, the new glasses show superior mechanical [2–6], optical [7– 20], and magneto-optical properties [21, 22]. They are fabricated from the deeply undercooled melts with levitation techniques. To clarify the vitrification processes, the atomic structures of their liquidus phases as well as glass phases at high temperature are measured by combining the levitators with synchrotron X-ray and neutron diffraction [23–50], X-ray absorption fine structure (XAFS) [51–54], Raman scattering [55], and nuclear magnetic resonance (NMR) [56–60]. Detailed methods and procedures to obtain atomic structures from the measurements are described elsewhere in this book. Macroscopic thermophysical properties of oxide melts are also important to understand the vitrification mechanism and phase selection processes from deeply undercooled melts. They are also inevitable to conduct MD simulations to obtain atomic structures and explore “hyperordered structures” in disordered atomic structure. For example, density data are necessary to calculate the total pair distribution function G(r) from the measured total structure factor S(Q) by X-ray or neutron diffraction experiments. As highlighted by Angell [47], a fundamental knowledge of both strong and fragile liquids (respectively high and low-glass-forming ability) can be inferred from the temperature dependence of viscosity. The heat capacities can be estimated thorough statistical/quantum mechanics once the binding energies between atoms are determined. Such thermophysical properties as density, surface tension, viscosity, and heat capacity are usually measured using conventional methods with containers. However, to obtain accurate data without contamination from the crucibles, levitation methods are required for refractory oxide melts, in which melting temperatures are above 2000 °C. In this chapter, methods to measure thermophysical properties of molten oxides with containerless processing (levitation) are presented and the data found in the literature summarized.

7.2 Methods for Containerless Processing with Oxide Samples Four levitation methods used for oxide melts, namely the aerodynamic, the aeroacoustic, the gas film, and the electrostatic techniques are briefly described in this section. The electromagnetic levitation method (EML) used for metal/alloys [61–64]

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is not applicable for oxide samples due to their poor electrical conductivities. Similarly, the acoustic levitation method [65, 66] is not suitable for high-temperature melt because it is very difficult to maintain a stable standing wave at high temperatures.

7.2.1 Aerodynamic Levitation Aerodynamic levitation (ADL) is one of the simplest methods among the levitation methods. A sample can be stably levitated in a conical nozzle where a gas flow creates a stable point (pressure minimum) [67–69]. High power lasers are used to heat and melt the levitated samples. The typical sample size is around 1–3 mm in diameter and the molten sample is nearly spherical. By controlling the gas flow rate, steady sample levitation can be achieved. Because of its simplicity, ADLs are widely used to produce spherical samples. Moreover, they are used for research on metastable phase formation from undercooled melts and fabrication of novel glasses which cannot be achieved with conventional methods using containers [2–22]. Originally, the nozzle used was so deep that heating, temperature measurement, and observation of the sample was possible only from the top. In the late 1990s, shallow nozzles were developed for the sake of synchrotron x-ray measurements on aerodynamically levitated liquid, which enabled the sample observation from the side [70]. Even if shallow nozzles were implemented, a full view of the sample is still not possible since the south pole of the sample is obstructed by the levitator. Furthermore, the lower part of the specimen is cooled by the flowing gas which introduces a large gradient in temperature. This issue can be alleviated by directing a second beam through the throat of the levitator. Care shall be taken to ensure that the levitator is leveled and that the laser beams hit the sample head-on. Not doing this would induce a rotation of the sample [71]. A typical setup for thermophysical property measurements with an ADL is shown in Fig. 7.1 [72]. To reduce the temperature gradient, a heating laser beam is divided to heat the aerodynamically levitated sample from the top and the bottom. A surface oscillation excitation system [73] induces an oscillatory pressure fluctuation on the gas flow and excites drop oscillation on the liquid sample. Detailed sample images are taken from a high-speed camera observing from the side with a proper backlight. Such experimental arrangement makes possible the determination of such thermophysical properties as density, surface tension, and viscosity of oxides in their liquid phases [73]. Coupled with X-ray/neutron diffraction facilities [47, 50, 70] and NMR [56–60], aerodynamic levitators were utilized to probe the atomic structure and dynamics of liquids at elevated temperatures. Furthermore, splitable nozzles [71] were successfully utilized in splat-quenching research [74, 75], drop calorimetry [76], and surface tension determination with innovative techniques (namely, droplet impingement [77] and bouncing [78]).

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Fig. 7.1 Schematic of an aerodynamic levitation system for measuring thermophysical properties of molten oxides. Reproduced with permission from Ref. [72]

7.2.2 Aero-Acoustic Levitation The aero-acoustic levitation (AAL) method first levitates a sample at an unstable location in the gas flow and stabilizes it through acoustic forces with surrounding transducers [79–81] (Fig. 7.2). A feedback control, based on the specimen position coordinates, is utilized to ensure adequate acoustic forces from the transducers allowing levitation of spheroids with diameters varying from about 1 to 3 mm. Upon melting, the strong acoustic pressure will make the sample to adopt an oblate-spheroid shape according to mass and surface tension. Full visibility of the sample is available. AALs combined with high-speed cameras were used for studies on metastable phase formation from undercooled oxide melts [82–94]. Developments on thermophysical property measurements have been conducted by Ushakov et al. since the 2010s. The authors were able to determine the density of a non-symmetric specimen through an image analysis [95]. This type of levitator was not used successfully so far for surface tension and viscosity measurements.

7.2.3 Gas Film Levitation The gas film levitation method (GFL), mainly developed in France, floats a molten sample on a thin gas film formed between the sample and a pressurized porous diffuser (Fig. 7.3) [96]. Large samples (up to 10 g of copper, silver, and gold) can be levitated by this method. The molten sample is usually flat shaped, like that of sessile drop (SD) because of gravity. The bottom of the sample is hidden by a diffuser. Combined

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Fig. 7.2 Top: schematic diagram of an aero-acoustic levitation apparatus. 1, levitated sample; 2, gas flow tube and heater; 3, translation stage; 4, flow control system; 5, acoustic transduced (threeaxis); 6, diode laser sample illuminator; 7, sample position detector (three-axis); 8, video camera; 9, vacuum chuck; 10, laser beam heating. Bottom: photograph of the aero-acoustic levitation apparatus showing a molten aluminum oxide sample at a temperature of 2700 K levitated in argon. Reproduced with permission from Ref. [81]

by such heating systems as a furnace or an electromagnetic heating, samples can be melted. Since the distance between the levitated sample and the lower diffuser is small ( 0 at T = 0. The minimum energy necessary to excite one electron from the superconducting state is Δ(0) at T = 0, that is, the energy required to destroy the superconducting state is 2Δ(0) because the superconducting state is produced by two electrons (Cooper pair). In the BCS theory, the attractive interaction is expressed ( )as −V0 , where V0 is −1 a positive value. At N (0)V0 0.5. The value of c in the former compound is smaller than that in the latter compound. The T c value is lower in the former compound than in the latter one. Thus, the T c changes depending on the location of the metal atom, and is clearly correlated with the value of c (or FeSe layer distance).

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The cesium (Cs) atom in (NH3 )y Csx FeSe only occupies the 2a site because of the large space, meaning that the maximum x for the Cs atom is 0.5 [22, 23]. The N atom occupies the 4c site in (NH3 )y Csx FeSe [23]. Thus, the larger atom can only occupy the 2a site, while smaller atoms may occupy both the 4c and 2b sites. Moreover, dense intercalation of the atom with a smaller ionic radius at both the 4c and 2b sites produces a larger c to result in a higher T c . In addition, the lattice constant c in (NH3 )y Nax FeSe is completely separated into two values depending on x [22, 24], because the Na atoms can occupy either 2a or 4c/2b owing to the moderate ionic radius, as described above. The application of pressure to (NH3 )y Csx FeSe results in a monotonic decrease of T c in the low-pressure range, whereas the T c rapidly increases at pressures >15 GPa. A T c value of 49 K was achieved at 21 GPa [25]. This behavior is called “pressure-driven superconductivity” or “pressure-induced superconductivity.” The origin of the emergence of the high-pressure superconducting phase was explored, and the rapid enhancement of the DOS at the Fermi level through a reconstruction of the Fermi surface was found in (NH3 )y Lix FeSe by measuring the Hall effect [26]. The T c value was clearly correlated with c, resulting from the chemical and physical pressure effects [24]. The coexistence of multiple superconducting phases and the coexistence of magnetic order and superconductivity in (Li1−x Fex )OHFeSe were reported [27–29]. Two superconducting phases at T c (SC1) = 40 ± 2 K and T c (SC2) = 35 ± 2 K, as well as the ferromagnetic order (magnetic transition temperature T M = 12 K) were also found in (Li1−x Fex )OHFeSe [29]. The crystal structure of (Li1−x Fex )OHFeSe demonstrated the existence of both superconducting FeSe and insulating (Li1−x Fex )OH layers [27–29]. The space group of the crystal lattice was P4/nmm (No. 129). The emergence of a pressure-driven superconducting phase was also confirmed for (Li1−x Fex )OHFe1−y Se [30] and the crystal structure is shown in Fig. 18.5. The second superconducting phase (SC-II) under pressure emerged at approximately 5 GPa, in which the normal state was assigned to the Fermi liquid state. The T c increased to 50 K at 12.5 GPa. Without any structural transition, the SC-II exhibited an enhanced carrier density, which occurred owing to the pressure-induced Fermi surface reconstruction [30]. Materials with similar crystal structures were reported for (Na1−x OH)Fe1−y Se (Cmma (No. 69)) [31]. The c values of (Li1−x Fex )OHFeSe and (Na1−x OH)Fe1−y Se are 9.2714(4) Å and 9.1629(13) Å, respectively [29, 31]. The former material shows superconductivity, but the latter does not show. These materials were prepared using a low-temperature hydrothermal ion-exchange method. Figure 18.5 shows that an insulating block layer ((Li1−x Fex )OH) is located between the superconducting FeSe layers; therefore, the material is recognized as a “hyperordered structure.” More extended metal-doped FeSe structures were prepared by using organic solvents such as amines [32–34]. Metal doping using ethylenediamine (EDA), trimethylenediamine, tetramethylene diamine, and hexamethylenediamine successfully synthesized metal-doped FeSe, which exhibits a maximum T c value of approximately 45 K for (EDA)y Lix FeSe with a c value of 22.393(8) Å (or FeSe layer distance of 11.197(4) Å) [34]. The materials with larger FeSe layer distances exhibit lower

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Fig. 18.5 Crystal structure of (Li1−x Fex )OHFe1−y Se drawn with the atomic coordinates reported in Ref. [29]

Fig. 18.6 Plot of T c against c in metal doped FeSe (Mx FeSe) prepared with liquid ammonia or amines. The plot is schematically drawn based on the graphs (Figs. 18.5 and 18.6) shown in Ref. [24]. The sample corresponding to each plot (different color) can be found in Figs. 18.5 and 18.6 of Ref. [24]; red circle, green triangle and blue diamond refer to the T c -c plots obtained from pressure dependence of T c in (NH3 )y Csx FeSe [25], low T c phase of (NH3 )y Nax FeSe and high-T c phase of (NH3 )y Nax FeSe [24], respectively

T c values, as seen from the T c -c plot reported by Terao et al. [24]; the plot is drawn in Fig. 18.6. Namely, an exceedingly large FeSe distance is not advantageous for superconductivity in metal-doped FeSe [24, 32–34]. The above-mentioned T c -c plot shows that 17.5 Å is the critical point to achieve a maximum T c value, whereas the critical point for providing a discontinuous jump for T c , which is related to the emergence of SC-II by applying pressure, is 14 Å. Thus, the insertion of a hyperordered layer into the FeSe lattice leads to the following three features (see Fig. 18.6):

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(1) The enhancement of T c through an increase in the distance between the FeSe layers (c ≤ 17.5Å). (2) A slow decrease in T c because of the extreme separation of the FeSe layers (c ≥ 17.5 Å). (3) A discontinuous jump in the T c through the Fermi surface reconstruction (Lifshitz transition) emerges at a certain FeSe layer distance (c = 14 Å). The presence of a hyperordered structure in the FeSe lattice contributes to the precise tuning of T c and provides a good platform to pursue novel physics or electronic mechanisms such as the Lifshitz transition, Fermi surface nesting, and spin fluctuation.

18.3 Bi-S2 Compounds: Layer-By-Layer Structure BiS2 -related compounds, such as LnO1−x Fx BiS2 and Bi4 O4 S3 , exhibit a layer-bylayer structure [35, 36]. In LnO1−x Fx BiS2 , the superconducting BiS2 and insulating LnO layers are alternatively stacked along the c-axis [35]. The insulating LnO layer can be called the “block layer” because the superconducting BiS2 layers are separated by the LnO layer. Figure 18.7 shows the crystal structure of LnO1−x Fx BiS2 and its space group is P4/nmm (No. 129). The insulating LnO layer separates the superconducting BiS2 layers. The T c value of Bi4 O4 S3 , which was the first BiS2 compound to be discovered, is 6 K. LnOBiS2 is a band insulator, and the electron doping through the partial replacement of oxygen (O) by fluorine (F) leads to superconductivity with T c values of 2–10 K [35]. The characteristics of superconductivity in LnO1−x Fx BiS2 can be controlled by varying the x value. In LaO0.5 F0.5 BiS2 , the averaged valence of bismuth Fig. 18.7 Crystal structure of LaO0.5 F0.5 BiS2 drawn with the atomic coordinates reported in Ref. [35]

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(Bi) changes to 2.5 from 3 in LaOBiS2 , that is, Bi2.5+ , which was determined using soft X-ray photoemission spectroscopy, being suggestive of a mixed valence state between Bi3+ and Bi2+ [37]. It is important to stress that the coexistence of magnetic order due to the magnetic moments of atoms in the block layer and superconductivity is observed in LnO1−x Fx BiS2 . The superconductivity in CeO0.3 F0.7 BiS2 (T c = 4.5 K) coexists with ferromagnetic order (ferromagnetic transition temperature (T m ) = 6.54(8) K) [38], demonstrating the emergence of multiple quantum properties produced by the hyperordered structure. The T c of LaO0.5 F0.5 BiS2 rapidly increases from 3.3 K at 0 GPa to 10.1 K at 1.1 GPa [39], and a structural phase transition from tetragonal (P4/nmm (No.129)) at 0 GPa to monoclinic (P21/m (No. 11)) at 0.8 GPa occurs, indicating that the pressure-driven enhancement of T c is triggered by a structural phase transition [40]. The electrical resistivity in the normal state also reduces when pressure is applied [39]. The change from an under-doped state in the tetragonal structure to an optimaldoped state in the monoclinic structure was suggested to be the origin for the rapid enhancement of T c . The superconducting gap function may be an s-wave [41], and electron–phonon coupling (electron–phonon coupling constant (λ ≈ 0.85) calculated for LaO0.5 F0.5 BiS2 can reasonably explain the T c value of approximately 10 K [42]. Therefore, LaO0.5 F0.5 BiS2 may be a conventional BCS-type superconductor. The elements currently capable of replacing the Ln atoms in LnO1−x Fx BiS2 while still exhibiting superconductivity are confined to La, Ce, Pr, Nd, and Yb [43, 44]. As previously described, the key for the superconductivity of LnO1−x Fx BiS2 is electron doping into LnOBiS2 . Superconducting property was observed by electrostatic electron doping into thin crystals of LaOBiS2 [45], with a T c of 3 K being by applying a gate voltage (6.0–8.0 V) for the ionic liquid gate dielectric 1-butyl-3methylimidazolium hexafluorophosphate (bmim[PF6]). Thus, electrostatic electron doping leads to superconductivity in LaOBiS2 single crystals.

18.4 Cluster-Based Superconductors This section introduces superconducting materials consisting of clusters. The fullerene superconductor is one of the most famous superconductors. A fullerene molecule is one of carbon clusters. In fact, among the fullerenes, only C60 exhibits superconductivity through a combination with metal atoms. C60 is a C cluster consisting of 60 C atoms. Mx C60 (M: metal atoms) can be regarded as a hyperordered structure in which C60 is a building block. K3 C60 (T c of 18 K) was the first fullerene superconductor to be discovered [46]. Here, three electrons transferring from the metal atoms to C60 are critical for the emergence of superconductivity, and only face-centered cubic (fcc) crystals exhibit superconductivity. The T c increases monotonously with an increasing unit cell volume (V ), which can be explained using the simple BCS theory and a simple band picture [47]. A smaller transfer integral between the C60 molecules by extending V results in a larger DOS at the Fermi level to enhance T c . RbCs2 C60 exhibits the highest T c (33 K) at ambient pressure [48].

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Although the preparation of Cs3 C60 has been attempted, it could not be obtained using a simple heating technique. Cs3 C60 , which exhibits a T c value of 40 K under pressure, was successfully prepared by using a liquid NH3 technique [49]. Subsequently, Cs3 C60 , which has two crystal structures, namely, the A15 and fcc structures, has been comprehensively studied. A superconducting transition at 38 K was confirmed for A15-type Cs3 C60 (body-centered tetragonal (Pm 3 m (No. 221)) at 0.93 GPa [50], whereas a superconducting transition at 35 K was found in fcc Cs3 C60 (Fm 3 m (No. 225)) at 0.73 GPa [51]. The former superconducting state emerged from the antiferromagnetic insulating state by the application of pressure [52], and a recent study demonstrated that the emergence of superconductivity originated from the combination of Mott localization and the dynamic Jahn–Teller effect [53]. It is considered that the superconducting pairing for C60 superconductors is due to electron-phonon coupling λ, i.e., larger λ provides higher T c . The next target material is Ag6 O8 MX, where M and X are cations and anions, respectively. This material possesses a crystal lattice consisting of AgO8 within the cage structure [54]. The AgO8 cage is a building block for the material, that is, the cage is recognized as a cluster such as C60 . Thus, the structure of Ag6 O8 MX can be regarded as a “hyperordered structure.” The crystals exhibit fcc structures with the space group of Fm 3 m (No. 225) at room temperature [55]. Figure 18.8 shows that the cation (M) is inserted into the spatial site with a cubic structure which is produced by the connection of cages, and the anion (X) is encapsulated in the cage. This material exhibits two features of a hyperordered structure, namely: (1) The intercalation of two ions into the spatial sites formed in the lattice and cluster cage. (2) The cluster cage is the building block for the lattice. Ag6 O8 Ag(NO3 ) crystals exhibit superconductivity with a T c of 1.04 K at ambient pressure [54]. The temperature dependence of R, specific-heat and magnetic susceptibility exhibits anomalies at 185 and 90 K [56], which is due to the structural phase transitions. The NO3 anion rotates freely at room temperature, and it rotates in the (001) plane below 185 K [57]. This superconductor is a type-II superconductor, and

Fig. 18.8 Crystal structure of Ag6 O8 Ag(NO3 ) drawn with the atomic coordinates reported in Ref. [55]. a Part of unit cell and b the cage structure

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BCS-type conventional superconductivity is expected. The other superconductors discovered so far are Ag6 O8 Ag(BF4 ) with a T c of 0.15 K [54] and Ag6 O8 Ag(HF2 ) with a T c of 0.8–1.5 K [58]. Another interesting cage structure R5 T6 Sn18 (R: Sc, Y and Lu, T: Rh and Ir) has been discovered by Remeika et al. [59]. The superconducting properties changed depending on the element R incorporated in the cage; the values of T c are 5 K for Sc, 3 K for Y and 4 K for Lu. All the information is listed in Table 18.1. The most interesting characteristic of this series is that Y5 T6 Sn18 (T: Rh and Ir) has anisotropic gap symmetry, and others (R: Sc and Lu, T: Rh and Ir) have isotropic, of which reasons are not yet clear [60]. Detailed information of superconducting properties of R5 T6 Sn18 series is reviewed [61]. Hydrogen (H)-based superconductors are a recent topical material. The first Hbased superconductor to be discovered was H2 S, in which superconductivity emerges at 203 K by the application of an extremely high pressure of 155 GPa [62]. The characterization of composite was successfully achieved by X-ray diffraction, demonstrating that the exact chemical composition was H3 S. H2 S decomposed to H and S under pressure to produce H3 S. Subsequently, superconductivity with a T c value Table 18.1 Superconducting parameters obtained from experiments in R5 T6 Sn18 Space group

Sc5 Rh6 Sn18 Sc5 Ir6 Sn18 Y5 Rh6 Sn18

Y5 Ir6 Sn18

Lu5 Rh6 Sn18 Lu5 Ir6 Sn18

I41 /acd

Fm3m

I41 /acd

Gap Isotropic asymmetry a (nm)

1.3601

I41 /acd

I41 /acd

Isotropic

Anisotropic Anisotropic Isotropic

Isotropic

1.3595

1.3792

1.3602

1.3735

1.3671

I41 /acd

c (nm)

2.7198

2.7180

2.7498

–

2.7330

3.7315

T c (K)

5.0

1.0

3.0

2.1

4.0

3.0

μ0 H c (0) (mT)

99.1(1)

12.2(1)

50.1(4)

30.4(1)

75.0(2)

47.6(1)

μ0 H c1 (0) (mT)

5.64(1)

–

2.49(2)

1.46(1)

4.38(1)

1.87(2)

μ0 H c2 (0) (T)

7.24(5)

–

4.21(6)

2.916()

5.58(5)

3.90(10)

λ(0) (nm)

34.2(1)

–

51.4(2)

67.2(1)

38.8(1)

59.2(2)

ξ (0) (nm)

6.74(1)

–

8.84(4)

15.9(1)

7.68(2)

13.0(1)

κ(0)

51.7(1)

–

59.1(4)

62.1(1)

50.5(1)

66.8(1)

γ (mJ/mol K2 )

51.0(4)

27.3(4)

37.8(4)

32.1(2)

49.1(4)

31.9(1)

Θ D (K)

276(2)

246(2)

185(1)

212(2)

158(1)

193

Δ(0) (meV)

0.990(3)

0.140(1)

–

–

0.719(2)

0.544(3)

ΔC e /γn T c

2.55(36)

1.39

1.96

1.68

2.02

1.83

2Δ(0)/ kB T c

4.26(3)

3.26(6)

–

–

4.12(2)

4.07(5)

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Fig. 18.9 Crystal structure of LaH10 drawn with the atomic coordinates reported in Ref. [66]. The LaH32 cage is depicted

≥260 K was discovered in LaH10 at 180–200 GPa [63]. In particular, LaH10 is recognized as “cluster compound,” because the La atom is surrounded by 32 H atoms as a building block for the fcc lattice, as shown in Fig. 18.9, that is, the LaH32 cluster forms the fcc crystal lattice. The crystal structure is similar to that of C60 superconductors. The space group of the crystal lattice of H3 S is R 3 m (No. 169) above approximately 150 GPa and Im 3 m (No. 229) below approximately 150 GPa [64]. The Im 3 m crystal lattice exhibits a superconductivity of 200 K. In contrast, LaH10 assumes an fcc crystal structure (Fm 3 m (No. 225)) [65, 66]. The superconductivity of H3 S and LaH10 with T c > 200 K appears to be reliable. The pairing mechanism is suggested to be electron–phonon coupling, and the high θD and large λ may produce a high T c . Additionally, the unambiguous isotope effect was discovered in D3 S [62], suggesting a BCS-type electron-phonon mechanism.

18.5 Superconductivity in Quasicrystals The electrical resistivity, magnetization, and specific-heat measurements of Al-ZnMg quasicrystals exhibit bulk superconductivity with a very low T c value of 0.05 K [67]. Quasicrystals are different from crystals and amorphous. The quasicrystal has no strict translational symmetry to define the crystal but does exhibit long-range order (hyperordered structure). Superconductivity is defined by the periodic arrangement of atoms, that is, a crystal, but this material does not exhibit translational periodicity. Shetchman et al. defined this material as a quasicrystal because of the observation of a five-fold symmetry which does not occur in normal crystals [68]. The Al-Zn-Mg quasicrystals exhibit zero-resistance, and the specific-heat measurement suggests bulk BCS-type superconductivity. These results imply that quasicrystals can be accompanied by an electronic structure with a long-range order. The low T c of the quasicrystals was explained by the low DOS at the Fermi level [67]. The superconductivity of some quasicrystals has been reported, where the icosahedral quasicrystal phases of i-Al-Cu-Li and i-Al-Mg-Zn exhibit superconductivity with T c values of 1.5 and 0.4 K, respectively [69, 70]; ‘i’ means “icosahedral.”

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Additionally, the superconductivity in rapidly-quenched ribbons of Ti-Zr-Ni alloys mainly containing an icosahedral quasicrystal phase exhibit T c values of 1.5–1.94 K [71]. Thus, quasicrystals exhibit superconductivity with low T c values (