Intense Shock Waves on Earth and in Space (Shock Wave and High Pressure Phenomena) 3030748391, 9783030748395

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Shock Wave and High Pressure Phenomena

Vladimir Fortov

Intense Shock Waves on Earth and in Space

Shock Wave and High Pressure Phenomena Founding Editor Robert A. Graham

Honorary Editors Lee Davison, Tijeras, NM, USA Yasuyuki Horie, Los Alamos National Laboratory, Los Alamos, NM, USA Series Editors Frank K. Lu, University of Texas at Arlington, Arlington, TX, USA Naresh Thadhani, Georgia Institute of Technology, Atlanta, GA, USA Akihiro Sasoh, Department of Aerospace Engineering, Nagoya University, Nagoya, Aichi, Japan

Shock Wave and High Pressure Phenomena The Springer book series on Shock Wave and High Pressure Phenomena comprises monographs and multi-author volumes containing either original material or reviews of subjects within the field. All states of matter are covered. Methods and results of theoretical and experimental research and numerical simulations are included, as are applications of these results. The books are intended for graduate-level students, research scientists, mathematicians, and engineers. Subjects of interest include properties of materials at both the continuum and microscopic levels, physics of high rate deformation and flow, chemically reacting flows and detonations, wave propagation and impact phenomena. The following list of subject areas further delineates the purview of the series. In all cases entries in the list are to be interpreted as applying to nonlinear wave propagation and high pressure phenomena. Development of experimental methods is not identified specifically, being regarded as a normal part of research in all areas of interest. Material Properties Equation of state including chemical and phase composition, ionization, etc. Constitutive equations for inelastic deformation Fracture and fragmentation Dielectric and magnetic properties Optical properties and radiation transport Metallurgical effects Spectroscopy Physics of Deformation and Flow Dislocation physics, twinning, and other microscopic deformation mechanisms Shear banding Mesoscale effects in solids Turbulence in fluids Microfracture and cavitation Explosives Detonation of condensed explosives and gases Initiation and growth of reaction Detonation wave structures Explosive materials Wave Propagation and Impact Phenomena in SolidsShock and decompression wave propagation Shock wave structure Penetration mechanics Gasdynamics Chemically Reacting Flows Blast waves Multiphase flow Numerical Simulation and Mathematical Theory Mathematical methods Wave propagation codes Molecular dynamics Applications Material modification and synthesis Military ordnance Geophysics and planetary science Medicine Aerospace and Industrial applications Protective materials and structures Mining

More information about this series at http://www.springer.com/series/1774

Vladimir Fortov

Intense Shock Waves on Earth and in Space

Vladimir Fortov Joint Institute for High Temperatures of the Russian Academy of Sciences Moscow, Russia

ISSN 2197-9529 ISSN 2197-9537 (electronic) Shock Wave and High Pressure Phenomena ISBN 978-3-030-74839-5 ISBN 978-3-030-74840-1 (eBook) https://doi.org/10.1007/978-3-030-74840-1 Translation from the Russian language edition: V.E Fortov, Mownye udarnye volny na Zemle i v kosmose by Vladimir Fortov, © Moscow Fizmatlit 2018. Published by Moscow Fizmatlit. All Rights Reserved. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 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 reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedicated to my daughter Svetlana

Preface

Dear reader, this book is a monograph on physics and mechanics of intense shock waves—a unique physical phenomenon that anyone of us has faced or heard of: lightning discharges, motion of hypersonic aircraft, and natural and man-made explosions of various origin. The scale and diversity of shock wave phenomena can catch even the most daring imagination. The monograph will focus on femtometer-sized (10−15 m) relativistic shock waves in nuclear matter and waves in galactic clusters of gigaparsec length. This range includes many other unconventional manifestations of intense shock waves some of which are considered in this book. Despite such an amazing variety, shock waves have a common ideological foundation—strict nonlinearity of their origin and development. In this monograph, an attempt was made to consider the most diverse manifestations of shock waves from these unified positions. At the same time, the author focused on less familiar cases (see the contents), avoiding the classical examples of shock waves in internal and external ballistics and hypersonic aerodynamics. The author disregarded the issues of explosive action, high-speed penetration, detonation, welding, strengthening and blast cutting, which have already passed into the category of applied, well-developed ones, as well as extensive issues related to shock waves and crater formation. When selecting the topic of this monograph, the author was naturally guided by the subject area where he had to work and which is to a lesser extent covered in the literature available for the general reader. Another driving factor for writing this book was that currently the stream of new data on intense shock waves has greatly increased due to the emergence of new experimental methods of their generation in laboratory conditions and also new intriguing observational astrophysical data obtained by the latest generations of ground-based and space telescopes and automatic spacecraft. This has greatly expanded the number of scientists who are interested in shock waves beyond their classical manifestations. Interest in shock wave physics has always been very high due to the natural desire of people to achieve more and work with record high parameters and also due to a significant number of astrophysical, energetic and defense applications. It was indeed vii

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Fig. 1 First successful experiment with shock waves. Battle of David and Goliath, Old Testament, Chap. 17, Book of Kings

Fig. 2 Explosion

military application that fostered the first successful experiment with shock waves which was conducted more than 3000 years ago—during the battle of David and Goliath (Fig. 1) described in the Old Testament Book of Kings. As a result of a high-speed impact of a stone thrown from David’s sling against the head of Goliath, a shock wave with an amplitude pressure of about 1.5 kbar occurred in the sling (according to the results of later 3D computer simulation). This pressure was more than twice the strength of Goliath’s frontal bone, which determined the outcome of the duel, to the great joy of the army and people of Israel. The scheme of shock wave action that was then successfully found is still a platform for all further experiments in the field of intense shock waves and dynamic high-energy density physics (ref. Figs. 2, 3, 4).

Preface Fig. 3 Scheme of shock wave formation. Courant R., Friedrichs K. Supersonic Flow and Shock Waves. Moscow: Foreign Languages Publishing House, 1950

ix

Super-launch flow

P, T, E

Fig. 4 Shock waves from a shot

Since David’s time, the use of more powerful and sophisticated energy cumulation means—chemical and nuclear explosives, powder, light gas, and electrodynamic guns, charged particle fluxes, laser, and X-ray radiation—has enabled the strikers’ throwing speed to be raised by three to four orders of magnitude, and the pressure in the shock wave, by six to eight orders of magnitude, thereby reaching the megabar– gigabar pressure range and “nuclear” energy densities of matter. In modern relativistic accelerators of heavy ions, this range expanded to sub-light velocities and the occurring amplitude pressures of nuclear shock waves reach ∼1030 bar. Essential progress in understanding shock wave phenomena was achieved in the late nineteenth century as a result of the application of the methods of gas dynamics and thermodynamics in describing shock wave processes. This is when the very

x

Preface

Fig. 5 Supersonic flight

concept of shock wave discontinuity was formed, which resolved many paradoxes related thereto. One of them consisted in the fact that without introducing any assumptions on energy dissipation mechanisms from elementary considerations, shock wave relations were obtained that contain the irreversibility of shock compression processes. Remarkably, the basic concepts of shock compression and the respective relations were obtained from gas dynamic equations in the absence of any experimental material. According to Émile Jouguet: “A shock wave first appeared at the point of theorists’ pen.” Just as academician Ya. B. Zel’dovich, we are astonished at the depth of analysis, intuition, and power of theoretical insight of the great minds of the nineteenth century, first of all the German mathematician Bernhard Riemann, the English physicist William John Rankine, the French professor of mechanics and ballistics of the Lorient artillery school Pierre-Henri Hugoniot. From different points and independently of each other, they created the theory of shock waves that has not lost its significance until now. In the twentieth century, the renaissance in the high-energy density physics of intense shock waves was closely associated with the entry of our civilization into the aviation, atomic, and space eras (ref. Figs. 5, 6, 7). The high compression of nuclear fuel in nuclear charges, generated by intense shock waves, is used to initiate nuclear chain reactions. In thermonuclear charges and microtargets of controlled thermonuclear fusion, high-energy states generated by shock and radiation waves are the main tool for compression, heating of thermonuclear fuel, and initiation of thermonuclear reactions in it. The studies of intense shock waves and high-energy densities that started in the mid-forties within the framework of nuclear defense projects have now received a significant new development with the advent of new pulse generators of highenergy densities. Respective sophisticated and expensive experimental devices made it possible to substantially advance along the scale of energy concentrations available

Preface

xi

Fig. 6 Launch of R-36M rocket. Compression shock waves are seen in combustion products

Fig. 7 Nuclear explosion

for physical experiment and to obtain, in laboratory or quasi-laboratory conditions, the states of the mega- and gigabar pressure ranges unattainable for the traditional technique of physical experiment. The use of new generators for the cumulation of high-energy densities leads to interesting physical phenomena: to the radical restructuring of the energy spectrum and composition of compressed and heated matter, to new collective effects and various instabilities in the interaction of directed energy flows with dense plasma,

xii

Preface

its non-stationary motion in the conditions of significant radiation transfer of energy, and to gravity and nuclear phenomena and a number of other exotic effects that can be now predicted only in the most general form. The physics of intense shock waves and the high-energy concentrations created by them have turned now into an extensive and rapidly developing branch of modern science where the most sophisticated means of generation, diagnostic, and computer simulation techniques on the most powerful supercomputers are applied. It is no coincidence that half of the 30 problems of “the physical minimum as of the beginning of the twenty-first century” proposed by Academician V. L. Ginzburg are, to a greater or lesser degree, dedicated to high-energy density physics and shock wave physics. Traditionally, energy concentrations in matter are referred to as “high” if they exceed 104 –105 J/cm3 , which corresponds to the binding energy of condensed matter, e.g., explosives, H2 or metals) and a pressure level of millions of atmospheres. For comparison, the pressures in the center of the Earth, Jupiter, and the Sun are about 3.6 Mbar, 40–50 Mbar, and 200 Gbar, respectively. The largest parameters of shock waves correspond to “ultra-extreme” states inside an atomic nucleus and conditions occurring in microscopic (femtoscopic, to be more precise) volumes of shock wave compression in case of individual collision of nuclei accelerated in accelerators to the near-light velocities. This is the subject of relativistic nuclear physics. As a rule, matter in the conditions of high-energy densities is in the plasma (ionized) state due to the processes of thermal and/or pressure-induced ionization. In astrophysical objects, such compression and heating are carried out by gravitational forces and nuclear reactions, and in laboratory conditions—by intense shock waves, which are excited by a wide variety of “drivers”—from two-stage gas and explosion guns to lasers and high-current Z-pinches with a power of hundreds of terawatts. At the same time, in astrophysical objects, the lifetime of extreme states varies from milliseconds to billions of years, making it possible to perform their detailed observation and measurement using space probes, orbital, and ground-based telescopes (ref. Figs. 8 and 9). But in terrestrial conditions, we are talking about micro-, femto, and attosecond ranges of shock wavelengths, which requires the use of specific extremely fast means of diagnostics. Modern physics of intense shock waves is closely related to such branches of science as plasma and condensed matter physics, relativistic physics, physics of lasers and charged particle beams, nuclear, atomic, and molecular physics, radiation, gas and magnetic hydrodynamics, astrophysics, mathematical physics, computational mathematics, etc. At the same time, a characteristic feature of this science is extreme complexity and strong nonlinearity of the processes occurring in it, the significance of collective interparticle interaction, and relativism, which makes the investigation of the phenomena in this field an interesting and compulsive work, which attracts a constantly increasing number of researchers. The limiting pressures of laboratory shock-compressed plasma still differ from the maximum astrophysical values by 20–30 orders of magnitude, but this gap is being rapidly bridged. Physical processes in the laboratory and in space often demonstrate an astonishing variety and at the same time striking similarities evidencing at least

Preface

xiii

Fig. 8 Explosion of supernova 1987 (SN 1987A). Photo by Hubble space telescope

Fig. 9 Shock wave in space

the uniformity of physical principles of the behavior of matter in an extremely broad range of densities (approximately 42 orders of magnitude) and temperatures (up to 1013 K) (ref. Fig. 10). All this is true, but, as Voltaire pointed out “...nothing in nature is simpler or more orderly. The sovereign states of Germany or Italy, which one can traverse in a half-hour, compared to the empires of Turkey, Moscow, or China, are only feeble reflections of the prodigious differences that nature has placed in all beings” (Voltaire. Micromega. Romans. V. 1.—Paris, 1887).

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Preface

Big Bang

lg P, Mbar Relativism of nucleons

Red giants (3-4)

Sun (2.2) Exoplanets

Nuclear collisions (15.7)

Neutron starts (11-15.6)

Atomic nucleus (14.5)

White dwarfs (6-9)

SHOCK WAVES

ICF TF

Jupiter (1.5)

Dynamics

Electroma gnetic plasma

Electrons

Statics Earth (1)

Nuclear explosions

Quarkgluon plasma

Relativism Т ~mc2 Nucleons

Magnetic CTF (8.5)

lg T, eV

Fig. 10 Phase diagram of states of matter. Numbers in brackets are density logarithm (g/cm3)

It is important that along with fundamental problems, the number of technical problems related to shock wave physics is constantly rising. Shock waves determine the operation of pulsed thermonuclear reactors with inertial confinement of hot plasma, powerful magnetic explosion and magnetohydrodynamic generators, power units and hypersonic jet engines, plasma-chemical and microwave reactors, and plasmotrons and powerful sources of optical and X-ray radiation. In the energy devices of the future, highly compressed and heated plasma will be used as a working medium just as water vapor in modern thermal power plants. Moreover, the study of shock waves forms the scientific basis for explosion safety, natural and man-made disasters, pulse technologies for the explosive synthesis of super-hard materials (synthetic diamonds, boron nitride), explosive welding, strengthening and cutting of materials, etc. Intense shock waves are becoming today the main tool for generating and studying extreme states of matter. These exotic states occur when matter is exposed to powerful shock, detonation and electrical explosion waves, concentrated laser radiation, electron and ion beams, powerful chemical and nuclear explosions, pulse evaporation of pinch liners and magnetic cumulative generators, hypersonic motion of bodies in dense atmospheres of planets, high-velocity shock, and in many other situations characterized by extremely high pressures and temperatures. High-energy densities determine the behavior of matter in a wide region of the phase diagram, occupying the area from a solid and liquid to a neutral gas, encompassing the phase boundaries of melting and boiling, as well as the metal–dielectric

Preface

xv

Red giant New star inception Supernova

Black hole

Interstellar gas Neutron star

Fig. 11 Matter transformation in space. Wide range of parameters. T ≈ 10–1011 K, ρ = 10−30– 5 1015 g/cm3 , p = 1017–1027 bar

transition region. The problem of the metal–dielectric transition has now received significant development in experiments on multiple (quasi-isentropic) shock wave compression of dielectrics, their metallization and on previously discovered plasma phase transitions in the megabar pressure range, and also on dielectrization of highly compressed simple metals. Along with the pragmatic interest, today we can see a general scientific interest in shock waves and high-energy plasma, since the overwhelming majority of visible matter in the Universe is in this exotic state. According to modern estimates, about 95% of the mass (without taking into account “dark” matter) is the plasma of usual and neutron stars, pulsars, black holes, and giant planets of the Solar System as well as recently discovered hundreds of exoplanets—planets beyond the Solar System (ref. Fig. 11). Before becoming a star, the Universe matter undergoes various physical transformations: from quarks and elementary particles to complex molecules and again to atoms and particles; from relativistic energies to absolute zero and back to the state of high-energy and dense plasma; from huge densities to deep vacuum and back to superhigh densities of the atomic nucleus and quarks. Often, these transformations occur when exposed to shock waves or they generate high-power shock waves themselves. Therefore, our fundamental knowledge relative to the structure, evolution, and history of the Universe directly depends on understanding the behavior of matter in all its transformations up to ultra-high-energy concentrations, which forms not only specific physical models but also the worldview concepts of modern natural science.

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Preface

Fig. 12 Time machine. Big Bang scheme

Consistently increasing the available powers of shock waves and high-energy densities on Earth and in space, we seem to move to the past in a time machine to singular conditions of the Big Bang—to a point when the Universe was born 15 billion years ago (ref. Fig. 12). Today, we can see that intense shock waves and the state of matter in extreme conditions are one of the most pressing and intensively developing fundamental scientific disciplines at the intersection of plasma physics, nonlinear optics, condensed state, nuclear, atomic and molecular physics, and relativistic and magnetic hydrodynamics with a high diversity of physical effects stimulated by shock wave compression and heating and with a constantly expanding set of objects and states. Despite an extremely high diversity of objects, experimental and astrophysical situations, they are all combined by the decisive role of intense shock waves and the nonlinearity of phenomena with high-energy density, generated by them. These circumstances are a constant stable stimulus for intensive theoretical and experimental studies resulting in a large number of new and, which is more important, reliable data on the dynamics and specific features of intense shock waves, thermodynamic, structural, gas dynamic, optical, electro-physical, and transport properties of matter in extreme conditions. This specific data is contained in a large number of original publications of recent times and also in publications of previous years some of which are almost inaccessible to Russian specialists, especially young people.

Preface

xvii

The progress in understanding shock waves, achieved by 1961, is very well generalized in the classical monograph by Ya. B. Zel’dovich and Yu. P. Raizer (Physics of Shock Waves and High Temperature Hydrodynamic Phenomena). 2nd Edition, Moscow: Nauka, 1966; 3rd Edition, Moscow: FIZMATLIT, 2008). The book underwent three editions and has long become the “Bible of Shock Waves” which has been used by many generations of physicists including the author of this text. The author wanted to cover, perhaps, a wider range of mutually related issues concerning the physics of intense shock waves and high-energy densities. Many interesting astrophysical, laser, and nuclear physics problems, as well as technical applications, are presented briefly with references to original papers and monographs. There was no intent to include all information known as of today about intense shock waves. The emphasis is on those issues that are of greater interest for the author and which he directly dealt with during his work. It is clear that, because of the extremely rapid development of the physics of intense shock waves, the material touched upon in this book will be constantly expanded and elaborated and will inevitably become outdated. I will be thankful to the readers who will send their critical comments and wishes. The author tried to introduce the reader to a fairly wide range of various problems of the physics of intense shock waves, and, therefore, many topics are covered here not in full, because the book is intended to help the reader feel free in a vast area of actual information accumulated now, see the perspective, and help young scientists in selecting the areas of further research. I hope that the book will be useful to a wide circle of scientists, postgraduates, and students of natural scientific specialties, providing access to original papers and allowing them to unravel the fascinating problems of modern physics of shock waves and extreme states of matter. While preparing the book, I used the assistance, stimulating discussions, and advice of A. M. Bykov, S. N. Vasilyev, N. A. Inogamov, G. I. Kanel, M. B. Kozintsova, A. V. Konyukhov, A. P. Likhachev, I. V. Lomonosov, S. A. Pikuz, S. V. Razorenov, A. M. Sergeev, V. G. Sultanov, H. Shtoker, and others, to whom I am sincerely grateful. I express my special thanks to Natalya Ivanovna Sokolova for the great and qualified work on preparing this manuscript. Moscow, Russia August 2017

Vladimir Fortov

Contents

1

Classical Hydrodynamics of Shock Waves . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 18

2

Shock Compression Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19 30

3

Shock Waves in High-Pressure Physics . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 66

4

Shock Waves in Condensed-Matter Physics . . . . . . . . . . . . . . . . . . . . . . 71 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5

Laser-Driven Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6

Shock Waves in Nuclear Explosions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

7

Cosmic Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

8

Electromagnetic Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

9

Nuclear Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

10 Shock Waves in Traffic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 11 Fermi-Zeldovich Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 12 Shock Wave Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386

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Chapter 1

Classical Hydrodynamics of Shock Waves

There is a great variety of manifestations of shock wave phenomena in various situations. Hydrodynamic manifestations of shock waves are currently most investigated manifestations of these complex and purely non-linear phenomena. It is the field where so far the most complete understanding of reasons for occurrence, formation, and evolution of the structure, dynamics, thermodynamics of shock wave processes has been achieved. Multiple monographs [1–6] that will be described later are dedicated to the modern hydrodynamic theory of shock waves. For reasons of coherence, let us focus on qualitative results, omitting calculation details and some important specific manifestations and features that we will describe in the following chapters. Equations of relativistic hydrodynamics. Equations include conditions for energy and momentum conservation [1]: ∇β T αβ = 0,

(1.1)

where the energy–momentum tensor T αβ = (e + p)u α u β − g αβ p.

(1.2)

Here, e and p are the internal energy density and pressure determined in their own coordinate system, gαβ = diag (1, − 1, − 1, − 1) is the metric tensor, uα = (, v)T is the four-dimensional velocity vector;  = 1/(1 − v2 )1/2 is the Lorenz factor, v = (v1 , v2 , v3 ) is the velocity vector; these equations use a system of units of measurement where the velocity of light is 1. The system of Eqs. (1.1), (1.2) is supplemented by the baryonic number conservation law:   ∇α nu a = 0,

(1.3)

where n is the baryonic number density.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Fortov, Intense Shock Waves on Earth and in Space, Shock Wave and High Pressure Phenomena, https://doi.org/10.1007/978-3-030-74840-1_1

1

2

1 Classical Hydrodynamics of Shock Waves

Let us note that the energy–momentum tensor (1.2) does not take into account dissipative processes, including viscosity and thermal conductivity. Therefore, we speak about perfect-fluid equations. The Eqs. (1.1)–(1.3) are closed by the equation of state [9, 10]. It should be noted that specific physical values in relativistic hydrodynamics, such as enthalpy, entropy, do not belong to the unit of mass, but to the particle or unit of baryonic charge, i.e. to the value for which the conservation law is formulated (1.3). The Eqs. (1.1)–(1.3) can be written in the Cartesian coordinate system (t, x, y, z) in vector form: ∂U ∂F1 (U) ∂F2 (U) ∂F23 (U) + + + = 0, ∂t ∂x ∂y ∂z

(1.4)

where U = (D, S 1 , S 2 , S 3 , τ )T is the vector of conservative variables whose flow densities are recorded as follows: T  Fi = Dvi , S 1 v1 + pδ 1i , S 2 v1 + pδ 2i , S 3 v1 + pδ 3i , S i − Dvi .

(1.5)

Conservative variables D, S 1 , S 2 , S 3 and τ are related with the baryonic number density, components of the velocity vector and relativistic enthalpy, h = 1 + (ε + p)/n, by the following ratios: D = n, S i = nh 2 , τ = nh 2 − p − n

(1.6)

The system (1.4) is hyperbolic [2]. Own values of the matrix ∂Fn /∂U, where Fn = F1 nx + F2 ny + F3 nz , representing velocities of entropic-vortex and acoustic waves are as follows: λ1 = λ2 = λ3 = vn ,

λ4 =

λ5 =

      vn 1 − c2 − c 1 − v2 1 − v2 c2 − vn2 1 − c2 1 − v2 c 2

      vn 1 − c2 − c 1 − v2 1 − v2 c2 − vn2 1 − c2 1 − v2 c 2

,

(1.7)

,

where vn is the projection of the velocity vector on the direction determined by the vector n = (nx , ny , nz ). Relativistic velocity of sound c being a part of (1.7) is determined with derivatives of the equation of state:  c2 =

∂p ∂e



 = x

∂p ∂e

 = n

  ∂p n . p + e ∂n e

(1.8)

1 Classical Hydrodynamics of Shock Waves

3

The above description of the relativistic hydrodynamics was developed to describe extreme astrophysical objects and phenomena [7, 8] and to study collisions of heavy cores at near-light velocity (ref. Chap 9) where the issues of the generation of quarkgluon plasma and the formation of nuclear shock waves are discussed (for details, see [7, 8] and references contained therein). Limit passage to classical hydrodynamics. The limit passage to classical hydrodynamics is based on fulfilling the following conditions. 1.

Flow is slow as compared to the velocity of light. In the adopted system of units where the velocity of light is 1, this condition is recorded as follows: v1  1.

2.

Enthalpy (as well as internal energy and the square of the velocity of sound) is much lower than the square of the velocity of light, h nr  1.

The second condition can be substantiated as a part of the relativistic kinetic theory, and it is postulated in transition to the classical limit in conservation laws (1.4). Relativistic enthalpy h differs from the classical one by that it includes energy related with the rest mass (mc2 ), i.e. h = 1 + hnr . Taking into account these two conditions, components of the vector of conservative variables U = (D, S 1 , S 2 , S 3 , τ )T are recorded as follows:

v2 D =n →n 1+ 2



  = n + O v2 ,



2   v2 S = nh vi → n(1 + h nr ) 1 + vi = nvi + O v 2 , 2 i

2

(1.9)



2

v2 v2 τ = nh − p − n → n(1 + h nr ) 1 + − p−n 1+ 2 2

 4 v2 +O v =n ε+ 2 2

The last expression uses the link between the specific enthalpy hnr (per particle) with non-relativistic energy density ε = e − n and pressure p: hnr = (ε + p)/n. Multiplying (1.9) by particles rest mass m (average baryonic mass) taking into account that ρ = mn, we have U = (ρ, ρv, ε + 0.5ρv2 )T . Flows of conservative variables (1.5) are expressed through components of vector U. In this manner, equations of relativistic hydrodynamics are converted into Euler equations of classical hydrodynamics. The velocity of sound in the classical limit tends to the value c2 = (∂p/∂ρ), and the Jacobi matrix spectrum ∂Fn /∂U (velocities of entropic-vortex and acoustic waves) (1.7) looks as follows:

4

1 Classical Hydrodynamics of Shock Waves

λ1 = λ2 = λ3 = vn , λ4 = vn −, λ5 = vn + .

(1.10)

When adopting regular—non-relativistic hydrodynamics, we note that the mathematical description of the state of a fluid in motion is done [1] using functions that determine the distribution of fluid velocities u = u(x, y, z, t) and any two thermodynamic values, such as pressure p(x, y, z, t) and density ρ(x, y, z, t). It is commonly known that all thermodynamic values are determined by values of any two of them, using the equation of state of matter [9, 10]; therefore, setting five values (three components of velocity u, pressure p and density ρ) fully determines the state of fluid in motion. Velocity u(x, y, z, t) is the fluid velocity at each given point x, y, z of space at time t, i.e., it belongs to specific space coordinates (Eulerian variables), rather than specific fluid particles moving in that space with time (Lagrange variables). Motion equations are based on Euler equations expressing laws of mass, momentum and energy conservation in a non-viscous, non-heat-conductive and compressible continuous medium [1–5]: ∂ρ + divρu = 0, ∂t ∂u 1 + (u∇)u = − grad p, ∂t ρ   2 

 ∂ ρ ε + u2 u2 + div ρu ε + + pu − ρ Q = 0, ∂t 2

(1.11) (1.12)

(1.13)

where derivatives ∂t∂ are calculated in the given point of space (local derivative), and the substantive derivative calculated in a moving point is as follows: ∂ d = + (u∇). dt ∂t In the Lagrange system of coordinates, the conservation laws (1.11)–(1.13) look as follows [1, 3]: dρ + ρdivu = 0, dt ρ

du = −∇ p, dt

dε dV +p = Q, dt dt

(1.14) (1.15) (1.16)

where ε is the internal energy, Q is the energy release per second in grams of matter from external sources, V = 1/ρ.

1 Classical Hydrodynamics of Shock Waves

5

In case of no energy release Q, the flow is adiabatic. By neglecting external energy release Q, we see that processes of energy dissipation that can occur in flowing fluid due to internal friction (viscosity) and heat exchange between its various areas are not taken into account. Such motions of fluids and gases for which thermal conductivity and viscosity processes are not relevant can be regarded as motions of ideal fluid. In case of adiabatic motion, entropy of each area of fluid remains constant when moving in space. By designating s as entropy belonging to the unit of fluid mass, we can express adiabaticity of motion by the following equation: ds = 0, dt where the full time derivative means the change of entropy of the defined moving area of fluid. This derivative can be recorded as follows [1]: ∂s + ugrads = 0. ∂t This is the general equation expressing motion adiabaticity of ideal fluid. Using (1.11), it can be recorded as the continuity equation for entropy, ∂(ρs) + div(ρsu) = 0 ∂t If entropy is the same at all points of fluid volume at some initial moment of time, it also remains everywhere the same and constant in time in case of further fluid motion. In these cases, the adiabaticity equation can be recorded as s = const. For smallness of dissipative processes, it is necessary [1, 3, 4] that the Reynolds number and Peclet number are high. According to [1, 3], let us use the expression for the molar viscosity coefficient η = νρ =

1  ρc l, 3

where l is the length of free path of molecules in gas, c is the velocity of molecules equal to the velocity of sound in magnitude order, ν is the kinematic viscosity (cm2 /s). By substituting the expression for viscosity into the Reynolds number formula, we obtain Re =

Ud Ud d Uρd = =3  ≈ , η ν cl l

where d is the characteristic dimension, U is the characteristic velocity of motion under consideration.

6

1 Classical Hydrodynamics of Shock Waves

In the area of interest where the motion velocity is of the order of the velocity of sound, the Reynolds number appears to be equal, in magnitude order, to the ratio of the dimensions of the system d to the path length of molecules l. If the motion velocity of fluid becomes comparable with the velocity of sound or exceeds it, effects related with the fluid compressibility are brought to the foreground. In practice, such motions are observed in gases. Therefore, high-velocity hydrodynamics is usually referred to as gas dynamics. The condition of neglecting dissipative forces requires the system dimensions to be significantly higher than the free path of molecules. However, as we will see later, fulfillment of this condition, i.e. large size of the system, does not always provide for smallness of dissipative forces and possibility of considering only adiabatic processes in reality. In case of shock waves, extremely high gradients of considered values occur in the flow, whereas the values of these gradients no longer depend on system dimensions and do not fall down as the system dimensions are increased. In these cases, however high is the Reynolds number, we will have to deal with possible change of entropy. Despite the fact that in these cases, the possibility of entropy growth is basically related with dissipative forces, all observed microscopic properties in the flow, in particular, the numerical value of entropy gain in a shock wave, do not depend on viscosity and thermal conductivity. They are automodeling relative to viscosity and thermal conductivity. Therefore, the laws of change of state in a shock wave can be obtained without considering the structure of its front from some equations of conservation of mass, momentum and energy applied to conditions before and after the wave front passes. In case of high Reynolds numbers, we could expect a significant effect of turbulence. In reality, the studies of combined actions of turbulence and rather high velocities (approximately the velocity of sound) represent a complex task, therefore there are few of them. On the one hand, this is due to complexity of such a boundary area. On the other hand, in most problems of dynamics of shock waves, we deal with relatively small dimensions when turbulence fails to develop even at a high value of Re. No heat exchange between individual areas of fluid (and, certainly, between fluid and surrounding bodies in contact with it) means that motion occurs adiabatically in each of fluid areas. Thus, the motion of ideal fluid should be considered as adiabatic. Equations of hydrodynamics are well studied and have a clear physical sense. Equations of continuity (1.11), (1.14) show that the change in the density of matter, ρ, in this element of volume occurs due to the flow of matter into that volume or outflow from it. Euler Eqs. (1.12), (1.15) express Newton laws for a continuous medium. The latter Eqs. (1.13), (1.16) express the first law of thermodynamics—the law of energy conservation. For the Euler formula, it looks as follows: the change in the full energy of a unit of volume at a given point in space occurs as a result of the outflow (inflow) of energy during the movement of matter, work of pressure forces and energy release from external sources.

1 Classical Hydrodynamics of Shock Waves

7

In the Lagrange form, the energy conservation law (1.16) shows that the change in specific internal energy ε of that particle of matter occurs due to compression work done by the medium above it and also due to energy release from external sources. This is fully equivalent to the Euler form (1.13). Equations of continuity, motion and energy form the system of five equations (the motion equation is vector and equivalent to three-coordinate equations) relative to five unknown functions of coordinates and time: ρ, ux , uy , uz , p. External sources of energy Q are deemed to be defined, and internal energy ε can be expressed through density and pressure, ε = ε(p, ρ), since thermodynamic properties of a substance are assumed to be known, and they are described by [9, 10] equations of state that define physical properties of a specific substance and are external relative to the formal description of gas dynamics. Equations of state thus close equations of gas dynamics. Differential equations of gas dynamics [(1.11)–(1.13) and (1.14)–(1.16)) supplemented by equations of state ε = ε(p, ρ) and defined boundary and initial conditions [5, 6] make it possible to solve a ‘direct’ problem of gas dynamics—the calculation of fields of velocities, densities and pressures of a moving medium. This ‘direct’ problem includes most practically important analytical and numerical gas-dynamics studies [5, 6]. It’s worth noting that all motions realized in nature must not only satisfy equations of hydrodynamics but also be stable. Respective issues of stability relative to shock waves will be considered in Chap. 12. However, there is a different (reverse) formulation of gas dynamics problems (Chap. 3, [7–10]) when fields of gas dynamic flows recorded in experiments are used for experimental studies of thermodynamic properties of substances in extreme conditions—at extremely high pressures and temperatures. This is a basis for a complex scientific area—dynamic physics of high pressures [1–3, 3–5, 5–11] that will be described in Chap. 3. We will see how compression and irreversible heating of matter by intense shock waves result in valuable experimental data and allow for studying physical properties of matter in these exotic conditions. Compressibility effects are implemented at significant motion velocities comparable to or exceeding the velocity of sound. We note again that extremely high Reynolds values are almost always dealt with in gas dynamics [1]. If the characteristic velocity of gas dynamic problem approximately equals the velocity of sound or exceeds it, the Reynolds value Re ∼ Lu/v ∼ Ud/(cl), i.e. it contains a knowingly high ratio of characteristic dimensions d to the free path length l. For very high Re values, viscosity is insignificant for gas motion almost in the entire space and thereafter we will consider (except as specified otherwise) gas as ideal (non-viscous) fluid. The nature of gas motion can be various depending on whether it is subsonic or supersonic, i.e. whether its velocity is lower or higher than the velocity of sound. One of the most essential principal differences of supersonic flow is the existence of shock waves in it whose properties we are considering here. According to [1–5], let us give primary specific features of gas dynamic flows required for further studies and resulting from gas-dynamic Eqs. (1.11)–(1.16). First

8

1 Classical Hydrodynamics of Shock Waves

of all, by linearizing [1–6] these equations of relatively low (sonic) disturbances, we consider ρ, p and velocity u as low. Let us record density and pressure as ρ = ρ 0 + ρ, p = p0 + p. Then one can easily come to the wave equation 2 ∂ 2 ρ 2 ∂ ρ = c , ∂t 2 ∂x2

(1.17)

permitting two groups of solutions:

ρ = ρ(x − ct), p = p(x − ct), u = u(x − ct),

(1.18)

ρ = ρ(x + ct), p = p(x + ct), u = u(x + ct),

(1.19)

where c is the positive root, c = + (∂ p/∂ρ)s . The first group describes the disturbance propagating to the positive axis x, and the second one describes the disturbance propagating in the opposite direction. Indeed, in the first case, for example, the defined value of density corresponds to a specific value of argument x − ct, i.e. the disturbance runs towards positive x with time at

velocity c. In this manner, c = + (∂ p/∂ρ)s − is the sound propagation velocity. Its dependence on state parameters will later define the occurrence of shock waves [1–5, 9] as a non-linear development stage of rather strong disturbances. By generalizing (1.17)–(1.19), we can show that if arbitrarily low disturbances of velocity, pressure or density are created at the initial moment t 0 at any point x 0 of quiet gas, two waves carrying disturbances will run from that point in both directions at the velocity of sound. In a wave propagating towards positive x to the right, low changes of all values are interrelated by the following ratios:

1 u =

1 p c = 1 ρ = f 1 (x − ct). ρ0c ρ0

In a wave propagating to the left,

2 u =

2 p c = − 2 ρ = − f 2 (x + ct). ρc ρ

Arbitrary disturbances u and p occurring at the initial moment can be decomposed into two components: u = 1 u + 2 u, p = 1 p + 2 p are subject to these ratios so that the initial disturbance propagates in different directions as two waves. In a general case of flat isentropic motion of gas, f 1 and f 2 are arbitrary functions. By considering flow in a small vicinity of the selected point x 0 , t 0 , we can neglect changes of undisturbed functions u(x, t), p(x, t) in the first approximation in this vicinity and consider them constant and equal values at point x 0 , t 0 . The above picture of disturbance propagation is extended to this case as well. If disturbances

u(x 0 , t 0 ), p(x 0 , t 0 ) are arbitrary, they are also decomposed into two components,

1 Classical Hydrodynamics of Shock Waves

9

one of which will start propagating to the right at a velocity u0 + c0 and the other to the left at a velocity u0 − c0 . Since u and c vary from point to point, paths of disturbance propagation during a long period of time on the x, t plane, described by equations dx/dt = u + c and dx/dt = u − c will be bent. These lines on the x, t plane, along which low disturbances propagate are called characteristics [1–6]. In case of flat isentropic gas flow, we see that there are two families of characteristics that are described by equations dx dx = u + c, = u − c, dt dt and are called C + -i C − -characteristics, respectively. Two characteristics belonging to C+ - and C− -families can be drawn through each point on the x, t plane. In the area of constant flow where u, p, c, ρ are constant in space and time, the characteristics of both families are straight lines. Characteristics as lines where low disturbances are conveyed have an illustrative physical sense [1–6]. Their role in gas dynamics, however, is much deeper since not only small disturbances are conveyed along these wonderful paths but also specific combinations of gas dynamic events [1–6]. According to [1, 3], let us go over to derivatives along characteristics to obtain as follows:     ∂u 1 ∂p ∂p ∂u + (u + c) + + (u + c) = 0, ∂t ∂x ρc ∂t ∂x     ∂u ∂u 1 ∂p ∂p + (u − c) − + (u − c) = 0. ∂t ∂x ρc ∂t ∂x The first of these equations contains derivatives only along C + -characteristics while the second one only along C − -characteristics. By noting that the adiabaticity equation dS/dt = 0 can be considered as an equation along C 0 -characteristics, let us record the gas dynamic equation in the following form du +

dx 1 dp = 0 alongC+ : = u + c, ρc dt

(1.20)

du −

1 dx dp = 0 alongC− : = u − c, ρc dt

(1.21)

ds = 0 along C0 :

dx = u. dt

In Lagrange coordinate equations, characteristics look as follows: C+ :

ρ ρ da da da = c , C− : = c , C0 : = 0. dt ρ0 dt ρ0 dt

(1.22)

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1 Classical Hydrodynamics of Shock Waves

A particularly important example occurs in case of isentropic flow when differential expressions du + dp/ρc and du − dp/ρc represent full differential values [1–6]: 

 J+ = u +

dp =u+ ρc

 c

dρ J− = u − ρ



dp =u− ρc

 c

 dρ , ρ

(1.23)

Riemann invariants. Using thermodynamic ratios, integral values  that are called  dp/ρc = cdρ/ρ can be expressed through one of thermodynamic variables, for example, the velocity of sound c. In ideal gas with constant heat capacity p = p0 ργ , c2 = γ · const · ρ γ −1 , J± = U ±

2 c. γ −1

(1.24)

Equations (1.23) and (1.24) result in that Riemann invariants are constant along characteristics in case of isentropic flow: dx = u + c; dt dx d J− = 0, J− = const along C− : = u − c; dt

d J+ = 0, J+ = const along C+ :

(1.25)

This provision can be deemed a generalization of ratios true for the propagation of acoustic waves in gas with constant parameters. Invariants J + and J − can be considered as new functions instead of variables u and c as per (1.23), (1.24): u= F+ =

γ −1 J+ + J− ,c = (J+ + J− ), 2 4

γ +1 3−γ 3−γ γ +1 J+ + J− , F− = J+ + J− . 4 4 4 4

Equations (1.23) show that characteristics can convey constant values of one of the invariants. Since J + = const along a specific C + -characteristic, the change in the characteristic slope is determined by the change of only one value – invariant J − . J − is also constant along the C − -characteristic, and the change in the slope in case of transition from one point of the x, t plane to the other one is determined by the change in the invariant J + . The characteristic form of gas dynamic equations makes cause-effect relationships in gas dynamics illustrative. It allows identifying areas of influence and tracing the kinematics of the formation of shock waves [1–6]. Multiple numerical methods are based on the characteristic form of methods using the conditions for the conservation of Riemann invariants along characteristics of the same family [6]. It is necessary that the characteristics of the same family never cross in the plane x, t. Otherwise, an ambiguity would occur at the intersection point

1 Classical Hydrodynamics of Shock Waves

11

since intersecting characteristics at one intersection point would have two different invariants [1–6], and therefore, two values of flow parameters at the same time. Meanwhile, only one value J + and one value J − belong to each point of the x, t plane, which are related with only values of gas velocity and sound velocity at that point. Such intersection leads to the disruption of flow continuity and the appearance of discontinuities in gas dynamic values. As we see below, the intersection of the characteristics of the same family is an ideological basis for introducing the notion of discontinuous solutions—shock waves into gas dynamics [1–6]. Papers [1–6] show that in case of the propagation of small acoustic disturbances (1.17), if the wave travels in a single direction only, one of Riemann invariants is constant in space and time. If the wave travels to the right and

u(x, t) =

p(x, t) = f 1 [x − (u 0 + c0 )t], ρ0 c0

the invariant J − is constant: J− = u −

p + const = const. ρ0 c0

If the wave travels to the left, the invariant J + is constant: It is important that the possibility of the existence of waves traveling in a single direction is not limited by a suggestion of low amplitude, whereas in a general case of a traveling wave, one of the Riemann invariants remains constant. From this condition, we can obtain [1–6] that in such a wave  dp = const, J− = u − ρc and values of u and c(u) propagate in gas at constant velocity u + c(u). This ratio describes the wave traveling to the right: u = f {x − [u + c(u)]t}, c = g{x − [u + c(u)]t}. The form of functions f and g is defined by initial and boundary conditions of the problem. Riemann invariants also show that pH V = VH + p

du dp, dp

(1.26a)

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1 Classical Hydrodynamics of Shock Waves

pH E = EH −

du dp

2 dp,

p

index H designates the state behind the shock-wave front (Hugoniot) and integrals are taken along isentrope s = const. Unlike the traveling wave of low amplitude (1.10), various values of gas velocity and thermodynamic variables are conveyed at different velocities so that initial profiles u(x,0), c(x,0) are distorted with time. This is a manifestation of non-linearity of gas dynamic equations [1, 3] and leads to the occurrence of shock waves. The obtained solution in the form of a traveling wave is called a simple wave. Let us trace the qualitative pattern of the propagation of disturbance derivatives in the form of a traveling simple wave, based on precise equations of gas dynamics rather than linearized (acoustic) Eq. (1.11). Distortion of profiles in a traveling wave of finite amplitude. According to [3], let us take the initial profiles of the velocity u(x, 0) and the velocity of sound c(x, 0) in the form depicted in Fig. 1.1. These functions are interrelated in such a manner that J − (x, 0) = const (we consider a wave traveling to the right). According to formula (1.24), we have c = ((γ − 1) /2) u + c0 , where the constant value of the invariant J − is selected as per the condition that u = 0, c = c0 in undisturbed gas. Since p ∼ c2γ/(γ−1) , ρ ∼ cγ/(γ−1) (for c = c0 , p = p0 , ρ = ρ 0 ), the profiles of pressure and density are similar to the velocity of sound profile in qualitative terms. Being constant at the initial moment, the invariant J − (x, t) is constant at all other moments of time, so that motion represents a simple wave traveling to the right. The characteristics of the C + -family are straight lines in Fig. 1.1, dx (γ + l)u =u+c = + c0 . dt 2 They leave points A0 , B0 and D0 , where u = 0, in parallel to each other: dx/dt = c0 (and parallel to the C + -characteristics leaving from the points of the x-axis and corresponding to the undisturbed area). Profiles u and c at the moment t 1 –u(x, t 1 ), c(x, t 1 ) are given in the middle chart (Fig. 1.1). Since constant values u and c are conveyed along the C + -characteristics, values u and c at points A1 , E 1 etc. are equal to respective values at points A0 , E 0 , etc. By plotting points as shown in Fig. 1.1, let us find profiles u and c at moment t 1 . We see that the apex (D) and pedestal (A) of the wave in contact with the areas of constant flow where u = 0, and c = c0 , have shifted along the x-axis by segments equal to c0 t 1 (propagated along characteristics D0 D1 , A0 A1 on the x, t plane). Heights of maximums and minimums u and c have not changed but relative positions of maximums and minimums have become different: the profiles have distorted. There is no such distortion in the acoustic theory where equations of gas dynamics are linearized: the profiles are shifted as a frozen picture. Distortion of the profiles occurs due to the nonlinearity of gas dynamic equations. The physical reason for

1 Classical Hydrodynamics of Shock Waves

13

Fig. 1.1 Propagation of traveling wave to the right. Plotting that makes it possible to determine the distortion of profiles in the wave. Above are profiles of the velocity u(x, 0) and the velocity of sound at the initial moment. Below are distorted profiles at the moment t1 . In the middle is the scheme of the C+ -characteristics [1, 3]

distortion lies in the fact that wave crests travel relatively faster due to a higher propagation velocity in matter (higher velocity of sound) and by a faster drift forwards together with matter (higher gas velocity). On the contrary, wave troughs travel relatively slower since both velocities in them are lower. It should be noted that a similar effect of wave overlapping is well observed in nature during tidal bore when sea waves come over the shore [5]. Smooth littoral waves are described by non-linear equations of the shallow water theory [1]. Their propagation velocity c can be evaluated from considerations of dimensionality: c∼

gl,

14

1 Classical Hydrodynamics of Shock Waves

Fig. 1.2 Diagram showing the rise in steepness and overlapping of a wave of finite amplitude in nonlinear theory [2]. Velocity profiles are shown at successive moments of time. To align waves at different times, the combination x − c0 t is drawn along the x-axis. Profile d meets the physically unreal state. Indeed, the profile looks as e with discontinuities at the point in time t 3

where g is the gravity acceleration; l is the local sea depth that tends to zero when approaching the shore. Rear parts of the wave propagate faster than front ones whose c is lower and they try to outrun them. This is how overlapping occurs. But in this case, it is not accompanied by a ‘gap’ since water continuity is impaired in case of such overlapping. It is broken into large ‘drops’, foam is formed, and this cannot be described by initial equations of the continuum mechanics. Profiles are distorted more heavily with time as shown in Fig. 1.2. If the analytical solution is formally extended to rather long time periods, wave overlapping will occur, as shown in Fig. 1.2, d. This picture has no physical sense since the solution in it is ambiguous. For example, there are three velocity values u at point x = x  at the same point in time: u = 0, u1 and u2 . The occurrence of such ambiguity is mathematically related with intersecting the characteristics of the same family (C + ).

1 Classical Hydrodynamics of Shock Waves

15

Fig. 1.3 Katsushika Hokusai The Great Wave off Kanagawa from the series Thirty-Six Views of Mount Fuji, 1823–1830

In fact, overlapping does not occur, and when front and rear parts of profiles become very steep, discontinuities are formed–shock waves–as shown in Fig. 1.2, e, and Fig. 1.3. In a continuum, during wave steeping, the gradients of the parameters grow at the front, which requires taking into account dissipative processes such as viscosity and thermal conductivity [1, 3]. The consideration of such processes is complicated by the fact that the discontinuity front thickness is rather narrow—just a few lengths of free path. In case of such great gradients when the mass velocity and other values change substantially at the distance of the path of molecules, gas dynamic equations must not be used even taking into account viscosity [3, 4]. The process must be considered based on the molecular kinetic theory of gases, namely based on the kinetic equation for the velocity distribution function of molecules. This analysis shows that the thickness of a leap in a shock wave of any strength x ∼ l [3, 4]. This seems rather natural [2, 3]. In a very strong shock wave, a cold undisturbed gas flows into the discontinuity with high velocity u0 , which is much higher than the chaotic velocity of the initial thermal motion that approximately equals the velocity of sound. After the discontinuity, gas flows relative to the discontinuity at velocity u1 that is less than thermal velocity c1 . This means that the directional velocity u has greatly turned into a chaotic velocity. In other words, the kinetic energy of directional gas motion turned into thermal energy in case of passing through the discontinuity. This transformation occurs as a result of molecule scattering in case of collision. At least one collision must occur for the molecular velocity to change its direction. Therefore, the leap thickness approximately equals the length of the path of molecules for collision. For shock wave thickness in real gas where molecules have internal

16

1 Classical Hydrodynamics of Shock Waves

degrees of freedom, refer to [3, 4]. In Chap. 7, we will see that in space shock waves the dissipative processes are defined by plasma collective effects, which decreases the front thickness of the plasma shock wave by multiple orders of magnitude. Indeed, a competing process of viscous attenuation (dispersion) works in parallel with the front steepening process. This process may occur before shock wave formation, in which case the shock wave will fail to form. Thus, the solution as a simple wave in this case is true only during a limited time period before discontinuities occur. The solution never loses its effects only when the wave has the nature of an expansion shock wave, i.e. it contains no areas where gas velocity, pressure and density are decreased towards the wave propagation direction. Such areas (AE and FD) represent compression waves in Fig. 1.1. To eliminate the physical senseless situation caused by overlapping of flow parameters, this ambiguity area [1–6] is usually not considered and replaced by a structureless mathematical discontinuity, i.e. a shock wave. In this case, general laws of mass, momentum and energy conservation must be ensured at the front of this shock wave, which have a simple algebraic form for the stationary movement of the wave instead of differential equations [1–9]. We should again note [1–10] that the laws of mass, momentum and energy conservation that are taken as a basis for the equations of dynamics for non-viscous and non-heat-conducting gas do not suggest compulsory continuity of gas dynamic values. The same laws can be applied to the areas where gas dynamic values have a discontinuity. From a mathematical point of view, the discontinuity can be considered as a limit case of extremely high gradients of gas dynamic values when the thickness of the layer where these values are finally changed tends to zero. Since there are no characteristic lengths in the dynamics of non-viscous and non-heat-conducting gas, i.e. provided that we abstain from the molecular structure of matter, the possibility of appearance of any number of thin transition layers resulting into a discontinuity in the limit are most restricted. These discontinuities are shock waves. The conditions of mass, momentum and energy conservation from the RankineHugoniot equation must be met at the front of that discontinuity [1, 3]: ρ1 u 1 = ρ0 u 0 ,

(1.27)

p1 + ρ1 u 2 = p0 + ρ0 u 20 ,

(1.28)

ε1 +

p1 u2 p0 u2 + 1 = ε0 + + 0, ρ1 2 ρ0 2

(1.29)

where u1 is the rate of gas outflow from that discontinuity, indexes 0 and 1 designate states of matter before and after passing through the discontinuity. The Rankine-Hugoniot equation supplemented by the equation of state ε = ε(p, v) defines a curve on the p–V plane (shock adiabat) that is an aggregate of states

1 Classical Hydrodynamics of Shock Waves

17

occurring as a result of shock compression of matter [1–4]. The properties of that curve and qualitative specific features of the discontinuity in matter are defined by the sign of the derivative (∂ 2 v/∂p2 )s (ref. Chap. 2) depending on whether the velocity of sound grows or falls as the wave intensity rises. In Chap. 2, we will consider in more details respective hydrodynamic effects in media with an arbitrary equation of state. It is remarkable that Eqs. (1.27)–(1.29) do not contain any restrictive suggestions on the properties of substances and express only general laws of conservation resulting from the most general considerations on symmetry of our space and time. In particular, the analysis of the Hugoniot adiabat shows that in case of (∂2 V/∂p2) > 0, when a substance goes through a shock wave, it is compressed and its pressure and density rise. Inequation u1 > c1 means that the shock wave moves relative to gas in front of it at the supersonic velocity. Therefore, no disturbances from the shock wave can penetrate this medium not compressed by the wave. In other words, the presence of a shock wave has no effect on the state of the gas ahead of it. The above nature of changes in the parameters of matter in the shock wave also results from a rather general requirement of their evolutionism following from the causality of phenomena [1–6]. Along with other thermodynamic values, entropy is also exposed to the discontinuity in the shock wave. It rises due to the entropy increase law. Therefore, entropy s2 of the gas passed through the shock wave must be more than its initial entropy s1 , s2 > s1 . Below we will see (Chap. 2) that this condition implies significant restrictions on the nature of changes in all values in the shock wave. According to [1], let us underline the following circumstance. The presence of shock waves leads to increased entropy in case of such movements that can be considered in the entire space as the motion of ideal fluid having no viscosity and thermal conductivity. Increased entropy means irreversibility of motion, i.e. energy dissipation. Thus, discontinuities are a mechanism resulting in energy dissipation in case of ideal fluid motion. In this sense, shock waves are a paradox. They are paradox in the way that without introducing any suggestions about dissipative forces (viscosity and thermal conductivity, plasma dissipations), we obtain laws of shock waves from general elementary assumptions, which are based on entropy increase, i.e. the laws that contain irreversibility of processes occurring in the shock wave [3]. It is obvious that the true mechanism of entropy increase in shock waves lies in dissipative processes occurring in those rather thin layers of matter that are actually physical shock waves. It is notable that the magnitude of that dissipation is fully defined only by laws of mass, energy and momentum conservation applied to both sides of these layers: their width is set so as to increase the entropy as required by these conservation laws. Shock waves are of peculiar interest also from another point of view. When the efforts to integrate equations without introducing discontinuities (i.e. shock

18

1 Classical Hydrodynamics of Shock Waves

waves) cause paradoxes such as intersections of characteristics and make these equations impossible to solve, the theory of shock waves resolves paradoxes and helps constructing the mode of motion in any conditions. In this chapter we have considered in more details respective hydrodynamic effects in media with an arbitrary equation of state.

References 1. Landau LD, Lifshits EM (1986) Gidrodinamika (Hydrodynamics). Moscow, Nauka [Landau L. D., Lifshits E. M. Gidrodinamika. - M .: Nauka, 1986 (in Russian)] 2. Anile AM, Russo G (1986) Corrugation stability for plane relativistic shock waves. Phys Fluids 29:2847–2852. https://doi.org/10.1063/1.865484 3. Zel’dovich YB, Raizer YP (2008) Teoriya udarnykh voln i vysokotemperaturnykh gidrodinamicheskikh yavleniy (Theory of Shock Waves and High-Temperature Hydrodynamic Phenomena. 3rd ed., corrected), 3rd edn. FIZMATLIT, Moscow [Zel’dovich Ya.B., Raizer Yu.P. Teoriya udarnykh voln i vysokotemperaturnykh gidrodinamicheskikh yavleniy. Izd. 3-ye, ispr. - M .: FIZMATLIT, 2008 (in Russian)] 4. Zel’dovich YB (1938) Teoriya udarnykh voln i vvedeniye v gazodinamiku (Theory of shock waves and introduction into gas dynamics). Published by USSR Academy of Sciences, Moscow-Leningrad [Zel’dovich Ya.B. Teoriya udarnykh voln i vvedeniye v gazodinamiku. - M. – L .: Izd. AN SSSR, 1938 (in Russian)] 5. Raizer YP (2011) Vvedeniye v gazodinamiku i teoriyu udarnykh voln dlya fizikov (Introduction into gas dynamics and theory of shock waves for physicists). Intellect, Dolgoprudny [Raizer Yu.P. Vvedeniye v gazodinamiku i teoriyu udarnykh voln dlya fizikov. —Dolgoprudnyy: Intellekt, 2011 (in Russian)] 6. Krayko AN (2007) Krayko A.N. Kratkiy kurs teoreticheskoy gazovoy dinamiki (A short course on theoretical gas dynamics). Moscow Institute of Physics and Technology [Krayko A.N. Kratkiy kurs teoreticheskoy gazovoy dinamiki. - M .: MFTI, 2007 (in Russian)] 7. Fortov VE (2013) Fizika vysokikh plotnostey energii (High energy density physics). FIZMATLIT, Moscow [Fortov V.E. Fizika vysokikh plotnostey energii. - M .: FIZMATLIT, 2013 (in Russian)] 8. Fortov VE (2016) Extreme states of matter. High energy density Physics. 2nd edn. Springer International Publishing, Switzerland 9. Fortov VE (2012) Uravneniye sostoyaniya veshchestva. Ot ideal’nogo gaza do kvarkglyuonnoy plazmy (Equation of state of matter. From ideal gas to Quark-Gluon Plasma). FIZMATLIT, Moscow [Fortov V.E. Uravneniye sostoyaniya veshchestva. Ot ideal’nogo gaza do kvarkglyuonnoy plazmy. – M .: FIZMATLIT, 2012 [in Russian]] 10. Fortov VE (2016) Thermodynamics and equations of states for matter. From ideal gas to Quark-Gluon Plasma. World Scientific Publications, Singapore 11. Altshuler LV (1968) Uspekhi fizicheskikh nauk (Progress in physical sciences). 85(2):469 [Altshuler L.V. Uspekhi fizicheskikh nauk. 1965. T. 85, vyp. 2. S. 469 [in Russian]]

Chapter 2

Shock Compression Thermodynamics

In Chap. 1, we recognized that the formation of shock waves and expansion waves is defined by the sign of the path derivative (∂ 2 V/∂p2 )s indicating whether the sound velocity rises or falls down as the matter is heated. In this chapter, we will consider in more details in a general form [1–5] how thermodynamic properties of the medium affect the properties of compression and expansion pulses. We will also consider the dynamic effects occurring when the compression and unloading pulses cross the phase boundaries as well as the thermodynamic limitations on the ability to reach various phase boundaries of media by compression pulses. We do not introduce limitations to the form of the equilibrium diagram in advance having in mind a large diversity of phase transitions and we also use purely thermodynamic ratios for analysis, which are not related with specific models of medium state equation. From now onwards, we will follow paper [6] focusing on specific experimental devices for the dynamic generation of high pressures [1–6]. We will consider three methods of pulse impact on matter: shock and isentropic compression, and adiabatic expansion of matter preliminarily compressed by a shock wave. As we saw in Chap. 1, these methods correspond to the automodeling solutions of gas dynamic equations. Position of isentropies and phase boundaries in p–V plane. Let us consider a mixture of two phases in thermodynamic equilibrium [7–9]. For entropy S c in a two-phase area, we can record Sc = S1 + (Vc − V1 )

S , V

(2.1)

where S = S 1 − S 2 and V = V 1 − V 2 is the change in entropy and specific volume during the transition from the first phase (p1 , V 1 ) to the second phase (p2 , V 2 ). Changes in entropy dS 1 /dT (full derivatives are calculated along the phase boundary) are related with the slope of the phase interface curve dp/dT by ratio

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Fortov, Intense Shock Waves on Earth and in Space, Shock Wave and High Pressure Phenomena, https://doi.org/10.1007/978-3-030-74840-1_2

19

20

2 Shock Compression Thermodynamics





∂ S1 ∂T

 + p

∂ S1 ∂p



d S1 dp = . dT dT T

(2.2)

By differentiating (2.1) upon p taking into account the Clapeyron-Clausius equation dp/dT = S/V and (2.2), we obtain.     ∂ Vc d V1 dp d S dT 2 d = − (Vc − V1 ) In − ∂p dp dp dT dT dp   2    dT ∂ S1 ∂ Sc dT . + + ∂ T p dp ∂ p T dp Hence, it follows for isentropy slopes in the two-phase area (∂V/∂p)s for V c − V 1 and phase boundary dV 1 /dp: 

∂ Vc ∂p



  ∂ S1 dT 2 ∂ V1 = − . ∂p dT dp

s

(2.3)

Using thermodynamic identities [10] for the slope of the isentropy and isotherm in the one-phase area, we will record: 

∂ V1 ∂p



 −

s



∂V ∂p

T

  T ∂ V1 = . c p1 ∂ T p

(2.4)

is always steeper than in the p–V plane  Consequently,   the isentropy   the isotherm   ∂ V 2  ∂ p  ∂p cv > 0 : ∂ V < 0 according to the condition c p = cv − T ∂ T p ∂ V >0 T T of matter stability. Slopes of the isotherm and boundary curve are interrelated by ratio ∂ V1 − ∂p



∂V ∂p



 = T

∂V ∂T



dT . p dp

(2.5)

Subtracting Eq. (2.3) from (2.5) and using the ratio c p1 d S1 = − dT T



∂V ∂T



dp , p dT

we obtain 

∂ Vc ∂p



 −

s

∂ V1 ∂p



 =2 T

∂ V1 ∂T



  c p1 dT 2 dT − . T dp p ∂p

(2.6)

Similarly, using (2.4), (2.5) and the Maxwell ratio (∂S 1 /∂p)T = − (∂V/∂T )p , we can obtain a link between the slopes of the isentropy of the first phase and the boundary curve:

2 Shock Compression Thermodynamics



∂ V1 ∂p



21

  ∂ V1 d S1 T dT d V1 =− . dp ∂ T p dT c p1 dp

− s

(2.7)

Subtracting Eq. (2.6) from (2.4), we obtain a ratio for isentropy break when it goes from the one-phase to the two-phase area: 

∂ V1 ∂p



 −

s

∂ Vc ∂p



 = s

2   c p1 dT T ∂ V1 − > 0. cp ∂T T dp

(2.8)

The signs of ratios (2.4) and (2.8) can also be determined using the Le Chatelier principle (pressure change along isentropy plays the role of external impact impairing the system’s thermodynamic equilibrium). The system of Eqs. (2.3)–(2.8) can be used for the analysis in the p–V plane of processes of isentropic compression and expansion of non-ideal plasma. The system of Eqs. (2.3)–(2.8) shows that for the defined phase boundary in the p–V plane, the possibility of phase transition in isentropic processes is defined by the signs of derivatives 

∂V ∂T

 , p

dT d S1 , . dp dT

Let us consider several examples. Let us suggest that for plasma or vapor (phase 1) (∂V 1 /∂T )p > 0 (a case when (∂V1/∂T )p changes the sign is considered below), most probable is the case of phase transition with decreasing volume and setting an order S > 0 in the system so that the phase curve slope is dT/dp = S/V > 0. However, we cannot consider this case only since experimentally we know transitions with dT/dp < 0 and with temperature maximum on the phase curve [11]. Let us first consider transitions with dS 1 /dT > 0. Irrespective of the sign of dT/dp, it follows that   ∂ Vc ∂ V1 . (2.9) < ∂p s ∂p If the phase equilibrium curve in the p–V plane has a negative slope and is located to the right of the initial state in case of compression and to the left in case of expansion, the in Eq. (2.9) means that it is impossible to move from the one-phase to the two-phase area by means of isentropic pressure change. This becomes possible only if dV 1 /dp > 0. If we adopt that the initial states of isentropies belong to the first phase, let us consider the condition of intersecting the phase boundary by them. For dT/dp > 0, the Eq. (2.7) gives 

∂ V1 ∂p

 < s

d V1 , dp

(2.10)

22

2 Shock Compression Thermodynamics a

b

Phase boundary

с

Liquid

Solid body

Fig. 2.1 The position of isentropies and phase boundaries in the p–V plane dS/dT > 0: two-phase area is dashed; S is the first phase isentropy; dashed curve is its continuation into the two-phase area; S c is the isentropy of the two-phase area; T is the isotherm; a dV 1 /dp < 0, dT/dp < 0; b dV 1 /dp > 0, dT/dp > 0; c melting in unloading wave dT/dp > 0, dS 1 /dT > 0 [6]

úwhich leads to the situation shown in Fig. 2.1a. Boundaries with a positive slope in this case are impossible since they contradict (2.10) taking into account that 

∂p ∂V



 =

s

∂p ∂V

 T

T − cV



∂p ∂T

2 < 0.

(2.11)

V

For dT/dp > 0, as per Eq. (2.7), 

∂ V1 ∂p

 < s

d V1 , dp

which leads to a situation shown in Fig. 2.1b. The figure shows what phase boundaries can be studied in case of isentropic pressure change. Based on the empirical equations of states of solid and liquid phases, the authors in [12] have obtained a solid phase entropy growth along the metal melting curve, dS 1 /dT > 0. Since the transition entropy is almost always constant on the melting curve [11–13], we can assume for a liquid phase along the melting curve that dS 2 /dT > 0. The above analysis (dT/dp > 0) shows (Fig. 2.1c) that if the substance has not melted during shock compression, it can go into a liquid phase in case of isentropic

2 Shock Compression Thermodynamics

23

Fig. 2.2 Position of isentropies and phase boundaries in the p–V plane dS 1 /dT < 0: a dT/dp > 0; b dT/dp < 0

expansion. If the matter behind the front is in a liquid state, it cannot go into a solid phase in case of unloading. These results comply with semi-empirical computations [12]. Let us consider transitions with dS 1 /dT < 0. In this case we assume the condition (∂V 1 /∂T )p > 0 as earlier, and we need to consider only negative slopes of the twophase area boundaries, dV 1 /dp < 0. Indeed, according to (2.3), 

∂V ∂p

 > s

∂ V1 . ∂p

(2.12)

However, taking into account (2.8) (∂V 1 /∂p)s > (∂V c /∂p)s that along with (2.11), it excludes the possibility that ∂V 1 /∂p > 0. Isentropies of the initial phase cross the two-phase area provided that dT/dp > 0. Otherwise, their slope in the p–V plane is more gradual than the phase interface curve, and such phase transition in case of isentropic pressure change cannot be recorded (Fig. 2.2). For the boiling curve of single-component systems, usually dT/dp > 0, and for the right boundary curve (evaporation), dS 1 /dT < 0. Selecting the initial states for isentropic compression on the saturation curve (which corresponds to the maximum density of the gas phase at this temperature), according to (2.12) we obtain that the isentropy of the p–V plane is steeper than the right boundary boiling curve. Hence, we come to an important practical conclusion that the vicinity of a critical point cannot be studied using compression isentropies.

24

2 Shock Compression Thermodynamics

As a rule, for most matters the relations (∂p/∂S)v > 0, (∂ 2 p/∂V 2 )s > 0 are satisfied, and the shock adiabat goes in the p–V plane above the Poisson adiabat exiting from the same point [2, 3]. Therefore, the critical point area cannot be studied in case of shock compression of matter in states lying on the saturation curve. The above inequations show that when a shock wave propagates through matter with normal thermodynamic properties whose initial states lie in the p–V plane on (or above) the saturation curve, liquid condensate cannot fall out behind the shock wave front. If the initial states are in the gas–vapor area, the possibility of partial or full condensation of the medium in the process of shock compression is defined by the degree of vapor supersaturation and shock wave intensity. Breakdown of discontinuity in plasma with arbitrary equation of state. Let us consider pressure breakdown in plasma with an arbitrary equation of state— an arbitrary sign of the derivative ((∂ 2 p/∂V 2 )s (this condition is usually called the Bethe-Weil condition [4]), meaning such experimental setups where shock-wave or adiabatic pressure changes are undertaken [1–5]. Direct surges are studied in details in media with a constant sign (∂ 2 p/∂V 2 )s [2, 10, 14, 15]. In this case, the analysis is built on the basis of differential thermodynamic ratios along the shock adiabat or on the basis of considerations on mechanical stability of the discontinuity: both considerations lead to the same results (ref. for example [1–6, 10, 14, 15] and Chap. 12). In case of the first-type phase transition, the Poisson adiabat can experience such a break at the phase interface boundary that the average value of the derivative in some vicinity of the phase boundary is negative along isentropy (∂ 2 p/∂V 2 )s . Just as this is the case for media subject to the van der Waals equation of state and having high heat capacity, an area where (∂ 2 p/∂V 2 )s < 0 [4, 5, 10] may adjoin the area of two-phase states in the p–V plane. Furthermore, we cannot eliminate a possibility of phase transition of the second type in non-ideal plasma with a breakpoint on the Poisson adiabat. Let us assume [12] that before shock compression the medium has normal thermodynamic properties with (∂ 2 p/∂V 2 )s > 0. Figure 2.3 [6] shows multitude of states (wave adiabat) [16] in the p–V plane occurring in the medium after the breakdown of pressure discontinuity. It also shows density profiles in respective waves. In the pressure area p0 < p < p1 , when the ratio (∂ 2 p/∂V 2 )s > 0 is fulfilled, the solution is built in a regular way [10, 14, 15], so that the sought curve is the shock adiabat (Hugoniot adiabat) Oa (p0 < pa < pb < p2 ). For p1 < p < p2 , shock adiabat Oa is supplemented by a section of isentropy S corresponding to the entropy value at point a, and p2 is the second break point on that isentropy. Before state a, matter is compressed in the straight √ shock wave propagating through an unperturbed medium at velocity Da = V0 ( p1 − p0 )/(V0 − V1 ). Further, matter is compressed to state b (along isentropy section ab) in the centered wave of compression that is separated from the area of progressive advance (at point b) by the surface of low discontinuity. Since the propagation velocity of this wave’s tail and head (defined by slopes of tangential lines to the isentropy at points α and b) differ, pressure profile αb will spread out during movement in space.

2 Shock Compression Thermodynamics

25

Fig. 2.3 Wave adiabat in a medium with variable (∂ 2 p/∂V 2 )s . Density profiles corresponding to various areas of the wave adiabat are shown next to it [6]

In the area p2 < p < p3 , the flow pattern differs from the previous cases in that the boundary between the compression wave and the area of progressive advance is a surface of high discontinuity. The breakdown of an arbitrary discontinuity in this case occurs along the curve HS representing a geometrical locus of the intersection points of the Hugoniot adiabats H going from states on the isentropy S and tangential lines in this isentropy drawn from the same origin points. In this case, matter is at first compressed in the shock wave to the state a, then to the state b along the isentropy in the centered compression wave and then to the final state C in the second shock wave. The propagation rate of the second shock wave through pre-compressed matter is defined by the slope of the Rayleigh line bC, so this wave will go behind the first shock wave moving at the velocity determined by the Rayleigh line Oa; such configuration will be stable. In case of p3 < p, the compression is done in a regular shock wave similar to the first considered case (for p < p1 ). Let us note that the derivate break occurs at point (p3 , V 3 ) along the curves HS and H only if d S = 0 at that point [17]. In this manner, in case of discontinuity breakdown in a medium with an arbitrary equation of state, complex flows are possible, which contain several shock waves divided by continuous compression waves. If the time to steady-state equilibrium behind the discontinuity surface is long, the density profiles shown in Fig. 2.3 can be significantly distorted (dashed line in

26

2 Shock Compression Thermodynamics

Fig. 2.3). The theory of shock waves in media with slow excitation of internal degrees of freedom are given in papers [14, 15]. The most studied case is currently a break at the phase boundary of the wave adiabat so that the isentropy section S in Fig. 2.3 converges to a single break point. In this case, shock compression to pressures p1 < p < p3 causes propagation of two shock waves without an intermediate centered compression wave being formed. The formation of a two-wave structure has now been experimentally recorded in two cases: splitting of a shock wave into elastic and plastic waves occurring due to the final shear modulus of solid bodies (ref. Chap. 4) and a two-wave structure caused by polymorphic transitions with volume decrease (ref. Chap. 4, [14, 17]). To form a two-wave structure, the adiabat slope at the break point must be less than the Rayleigh line slope in modulus at the same point (Bethe-Weil condition [2–5]). In case the shock wave is split, the two-wave structure area can be used to obtain quantitative information on phase transitions [18] to study elastic–plastic and wave phenomena (Chap. 4). A thermodynamic consideration similar to the analysis of formula (2.3) leads to (at V = V 1 ) quadratic equation relative to dp/dT: 

dp dT

2 +

2(∂ V1 /∂ T ) p c p1 dp  = 0. +  (∂ V1 /∂ p)T − (∂ V /∂ p) H C dT T (∂ V1 /∂ p)T − (∂ V1 /∂ p) H C (2.13)

The thermodynamic description of the first (initial) phase is usually known with sufficient accuracy; shock compressibility of the mixture of phases is related with the experimentally determined velocity of the weak secondary wave D2 by the ratio. 

∂ V1 ∂p

 = HC

V12 D22

Thus, the coefficients of Eq. (2.13) are determined, which allows experimentally defining the slope of the phase equilibrium curve at the shock adiabat break point. If we know the position of the shock adiabat in the two-phase area and there is an acceptable thermodynamic description of the initial phase, the phase equilibrium curve is determined by numerical integration of the ratio ⎡ T ⎤  c V1c  pc ∂p dp ( pc − pc )(V1 − Vc ) ⎣ = + cV dT + T d V + V dp ⎦ , dT 2Tc (V1c − Vc ) ∂T V T1

V1

p1

1

where integrals are calculated for initial phase parameters along the phase equilibrium curve: index 1 corresponds to the adiabat break point when it intersects the phase boundary [19]. For the slopes of shock adiabats and isentropies in the one-phase area, formula [18] is true

2 Shock Compression Thermodynamics



dp1 dV

 =

27

 dp1/∂ V s + (γ /2V1 )( p1 − p0 ) 1 − (γ /2V1 )(V0 − V1 )

H

,

(2.14)

where p, V are parameters before shock compression, Grüneisen coefficient  γ =V

∂E ∂p

−1 H

=−

V T



∂p ∂V

 s

∂T ∂p

 . s

Using the formula for thermodynamic values in a two-phase area, we obtain a formula for the state at the phase boundary: 

dp dV

 = HC

(∂ pc /∂ V )s + (γc /2V1 )( p1 − p0 ) , 1 − (γc /2V1 )(V0 − V1 )

(2.15)

where the mixture’s Grüneisen coefficient is as follows:   V ∂ pc dT γc = − . T ∂ V s dp Ratios (2.13, 2.14) show that the shock adiabat break is defined by the properties of the first phase and the slope of the phase equilibrium curve dT/∂p at the break point; for the defined phase equilibrium curve, information about properties of the second phase, as for isentropies, becomes unnecessary. Change in the Grüneisen parameter during transition from one-phase to two-phase area       ∂T ∂p V ∂ pc dT , (2.16) γ − γc = − − T ∂V s ∂p s ∂ V s dp where 

∂T ∂p

 − S1

dT (∂ V1 /∂ p)s − (d V1 /dp) = dp (∂ V1 /∂ T ) p

(2.17)

and the link between (∂V 1 /∂p)s and ∂V 1 /∂p is established by the ratio (2.7). The analysis of given formulas shows that any difference sign  = (dp/dV )HC − (dp1 /dV )H is possible depending on specific conditions. The case  > 0 results in a two-wave structure if additional conditions are present. In some cases, for  < 0, surges may appear on the shock adiabat, which are caused by a phase transition. With a strong break of the shock adiabat ( < 0) when the shock compression curve in the p–V plane goes to the right from the mirror reflection from the Rayleigh line break point (Fig. 2.4),    dp  p − p0   dV  < R = V − V 0

(2.18)

28 Fig. 2.4 Abnormalities of shock compression: H Hugoniot adiabat; R Rayleigh line, (pA − p0 )/(V A − V 1 ) = const; R its mirror reflection; AC instability boundary; AB boundary of spontaneous sound emission by shock discontinuity. The slope AB can be negative

2 Shock Compression Thermodynamics

Two waves

Ambiguity of shock compression

and shock compression becomes abnormal, which is exhibited by decreased mass velocity with pressure rise on the Hugoniot adiabat. In this case, the condition of singularity of solving the Riemann problem on the breakdown of arbitrary discontinuity [2–5, 10, 14, 15] is impaired, so that shock compression becomes ambiguous. The analysis made in [20] shows that in this case, the wave may also break down into two discontinuities. Thus, the conditions of shock compression abnormality become symmetric [19]. The compression is abnormal if the shock adiabat goes beyond the angle formed by the Rayleigh line and its mirror reflection (Fig. 2.4) [6]. When exiting this angle to the left, the wave is bifurcated, and when it exits the angle to the right, the conditions of shock compression become unambiguous. Stability conditions can additionally narrow the area of the medium’s normal compression by the shock wave (ref. Chap. 12). In the case of the isentropic expansion of shock-compressed matter, the unloading isentropy can intersect the area of states with abnormal thermodynamic properties (∂ 2 p/∂V 2 )s < 0 caused by the phase transition. In this event, to build an expansion wave, a surge of expansion must be introduced within some interval of pressures, while a compression surge is impossible [15]. A clear experimental proof of the existence of expansion surges was obtained in [21] where the phenomena of split-off during blasting of charges on the surface of steel plates were studied. Results are given for precision experiments intended to record the structure of expansion waves of the highly critical area of freon, which have proved the prediction of [22] on the occurrence of expansion surges near the critical point. In experiments for adiabatic expansion of shock-compressed plasma in an unloading wave, states close to near-critical ones are realized. In this case, the formation of expansion shock waves in the flying-apart substance is also possible. Explanation of an unusual nature of current drop and abnormal behavior of resistance in experiments with exploding wires is associated in [23] with the formation of expansion shock waves when a volume of metal heated by electric current expand. It appeared that the formation of expansion shock waves is highly improbable in such

2 Shock Compression Thermodynamics

29

Fig. 2.5 Shock wave with plasma temperature drop: inline ab (∂V/∂T )p = 0; H is shock adiabat, S is isentropy

gas, although in the case of high heat capacities (cv ≥ 17R) there is an area with (∂ 2 p/∂V 2 )s < 0, which was previously indicated in [14]. The possibility of the formation of the so-called electro-adiabatic expansion shock waves in the flash chamber of the shock tube is theoretically considered in [24]. The reason for abnormality in thermodynamic functions is caused by strong electric current flowing in plasma, which results in unequal electronic and ionic temperatures. Shock waves with the temperature drop. In the previous consideration, it was assumed that (∂V/∂T )p > 0. A completely similar analysis can be done for (∂V/∂T )p < 0. If (∂V/∂T )p changes the sign in the considered area of parameters, difficulties emerge related to the ambiguity of representing thermodynamic states in the p–V plane, which results in the possibility of the thermodynamically equilibrium systems to intersect isentropies in the p–V plane (Fig. 2.5) without violating the second law of thermodynamics in the Planck-Kelvin formulation [10, 25]. For strongly non-ideal media, the ambiguity of state presentation in the p–V plane is also possible, so we will specifically consider this issue here. Equilibrium states of a single-component system are represented in a space of three variables using a two-dimensional parametric surface (equation of state) and different equilibrium thermodynamic processes are depicted on this surface using respective curves. Isentropies on such a surface cannot intersect without violating the second law of thermodynamics (proved using Pfaffian differential equations for the isentropy [26]). However, in the projection onto the p–V plane, the intersections of isentropies become possible, which is associated with an unsuccessful choice of this pair of variables. Sufficient conditions for selecting unambiguous thermodynamic coordinates are formulated in [26]: such coordinates are pairs (T, p), (T, V ), (S, p), (S, V ). (p, V )- and (T, S)-variables widely used for the analysis of thermodynamic processes, and the proof of theorems in thermodynamics are unambiguous. Consideration of processes in these variables leads, as is known, to a number of paradoxes. Thus, Sommerfeld proposed the Carnot cycle for water at a temperature of about 4 °C; this cycle resulted in extremely useful work using heat from a single source [27].

30

2 Shock Compression Thermodynamics

The thing is that water H2 O at atmospheric pressure has the maximum density for a temperature of about 4 °C, which evidently impairs the ambiguity of the representations of thermodynamic states in the p–V plane. Let us note that thermodynamic states in dynamic experiments are recorded rather unambiguously, since the internal energy E is determined along with p, V. Equilibrium temperature is determined using these variables in the same unambiguous fashion (ref. Chap. 11). In media having a variable sign (∂V/∂T )p , shock waves can form with a drop in temperature [2, 6]. Figure 2.5 schematically shows isentropy and Hugoniot adiabat for the case when (∂V/∂T )p < 0 in the medium ahead of the shock wave. When the states behind the front lie in the area (∂V/∂T )p > 0, possible conditions of shock waves with a temperature drop correspond to the area with p1 . In this case, pressure, density, and entropy in the shock wave increase. The analysis of these conditions for water (empirical equation of state was used) shows [28] that such shock wave is similar in all aspects (except for temperature surge) to a regular shock wave discontinuity. Apparently, the first experimental observation of shock waves with a drop in temperature is represented in [28] where measurements of thermal radiation from liquid nitrogen compressed by primary and reflected shock waves showed a decrease in the plasma temperature. This effect [29, 30] is related to the injection of a significant amount of energy into the internal degrees of freedom in the area of high-temperature dissociation of nitrogen.

References 1. Altshuler LV (1965) Altshuler L.V.Use of shock waves in high-pressure physics. Adv Phys Sci 85(2):197 [Altshuler L.V. Primeneniye udarnykh voln v fizike vysokikh davleniy // UFN. 1965. T. 85, №2. S. 197 (in Russian)] 2. Fortov VE (2012)Equation of state of matter. from ideal gas to Quark-Gluon Plasma. FIZMATLIT, Moscow [Fortov V.E. Uravneniye sostoyaniya veshchestva. Ot ideal’nogo gaza do kvarkglyuonnoy plazmy. - M .: FIZMATLIT, 2012 (in Russian)] 3. Fortov VE (2016) Thermodynamics and equations of states for matter. From ideal gas to Quark-Gluon Plasma. World Scientific, New York, London, Tokio 4. Fortov VE (2013) High energy density physics. FIZMATLIT, Moscow [Fortov V.E. Fizika vysokikh plotnostey energii. - M .: FIZMATLIT, 2013 (in Russian)] 5. Fortov VE (2016) Extreme states of matter. High energy density physics, 2nd edn. Springer, Heidelberg, New York, London 6. Fortov VE, Yakubov IT (1994) Physics of non-ideal plasma. Energoatomizdat, Moscow [Fortov V.E., Yakubov I.T. Fizika neideal’noy plazmy. - M .: Energoatomizdat, 1994 (in Russian)] 7. Fortov VE (1972) Thermal physics of high temperatures. 10(1):168–180 [Fortov V.E. Teplofizika vysokikh temperatur. 1972. T. 10, vyp. 1. S. 168–180 [in Russian]] 8. Horie J (1967) J Phys Chem Solids 28:1569–1574 9. Davison L, Graham R (1979) Phys Rept 55:256–268 10. Landau LD, Lifshits EM (2004) Course of theoretical physics, vol. 6. FIZMATLIT, Mexico [Landau L.D., Lifshits E.M. Kurs teoreticheskoy fiziki. T. 6. Gidrodinamika. Izd. 6. - M .: FIZMATLIT, 2004 (in Russian)] 11. Stishov SM (1968) Adv Phys Sci 96(3):467–496 [Stishov S.M. // UFN. 1968. T. 96, vyp. 3. S. 467–496 (in Russian)]

References

31

12. Urlin VD (1965) J Exp Theor Phys 49(2):485–492 [Urlin V.D. // ZHETF. 1965. T. 49, vyp. 2. S. 485–492 (in Russian)] 13. Kormer SB (968) Adv Phys Sci 94(4):641–689 [Kormer S.B. UFN. 1968. T. 94, vyp. 4. S. 641–689 (in Russian)] 14. Zel’dovich YB, Raizer YP (2008) Theory of shock waves and high-temperature hydrodynamic phenomena, 3rd edn. FIZMATLIT, Mexico [Zel’dovich Ya.B., Raizer Yu.P. Teoriya udarnykh voln i vysokotemperaturnykh gidrodinamicheskikh yavleniy, Izd. 3-ye, ispr. - M .: FIZMATLIT, 2008 (in Russian)] 15. Zel’dovich YB (1938) Theory of shock waves and introduction into gas dynamics. Published by USSR Academy of Sciences, Moscow-Leningrad [Zel’dovich Ya.B. Teoriya udarnykh voln i vvedeniye v gazodinamiku. - M. – L .: Izd. AN SSSR, 1938 (in Russian)] 16. Gelin GA (1958) Report to USSR Academy of Sciences. 119(5):1106–1110; 120(4):730–733 [Gelin G.A. // Dokl. AN SSSR. 1958. T. 119, №5. S. 1106–1110; T. 120, №4. S. 730–733 (in Russian)] 17. Sidorenko AD (1968) Report to USSR Academy of Sciences. 178(4):818–821 [Sidorenko A.D. // Dokl. AN SSSR. 1968. T. 178, №4. S. 818–821 (in Russian)] 18. Zharkov IV, Kalinin VA (1968) Equation of state of solid bodies at high pressures and temperatures. Nauka, Moscow [Zharkov I.V., Kalinin V.A. Uravneniye sostoyaniya tverdykh tel pri vysokikh davleniyakh i temperaturakh. - M .: Nauka, 1968 (in Russian)] 19. Zababakhin EI, Simonenko VA (1967) J Exp Theor Phys 52(5):1317–1321 [Zababakhin E.I., Simonenko V.A. // ZHETF. 1967. T. 52, vyp. 5. S. 1317–1321 (in Russian)] 20. Swan CW, Fowler G (1975) R Phys Fluids 18:28–32 21. Ivanov AG, Novikov SA, Garasover YI (1962) Phys Solid Body 4(1):249–256 [Ivanov A.G., Novikov S.A., Garasover Yu.I. Fizika tverdogo tela. 1962. T. 4, vyp. 1. S. 249–256 (in Russian)] 22. Kutateladze SS, Borisov AA, Nakoryakov VE (1980) Reports to the USSR Academy of Sciences. 252(3):595–597 [Kutateladze S.S., Borisov A.A, Nakoryakov V.E. // DAN SSSR. 1980. T. 252, №3. S. 595–597 (in Russian)] 23. Bennet FD, Kahil GD, Wedemeyer EH (1964) In: Chace WC (ed) Exploding wires. Plenum, New York; 3:65; 1968. 4:1 24. Paxton GW (1967) Phys Fluids 10:2360–2378 25. Thomsen Js (1970) Amer J Phys 38:560–587 26. Kestin J (1961) Amer J Phys 29:329–341 27. Thomsen JS, Hartka IJ (1962) Amer J Phys 30:26–29; Curry SM, Henrey GK (1968) Amer J Phys 36:838–843 28. Thomsen JS (1968) Phys Fluids 11:1338–1360 29. Ross M (1988) In: Shmidt SC, Holmes NC (ed) Shock waves in condensed matter. Amsterdam, North Holland, p 87 30. Nellis WJ Ibid. p 43

Chapter 3

Shock Waves in High-Pressure Physics

The most important and quite efficient field of application of modern shock wave physics in science is high energy density physics [1] where powerful shock waves simultaneously act as a means for both generation and diagnostics of extreme state of matter. The final goal of extensive global experiments in macroscopic high energy density physics consists [1] in the generation of extreme parameters of matter the values of which are maximum possible in modern experiments (Table 3.1). Even now, the exotic states of matter with peak pressures of hundreds to thousands of megabars, temperatures up to 10 billion degrees Celsius, and energy densities of 109 J/cm3 , which are comparable to the energy density of nuclear matter, have become the subject of laboratory investigations [1–6]. According to the ideas developed to date [7–11], to implement a controlled thermonuclear reaction with inertial hot plasma confinement requires an energy of several megajoules to be delivered to a spherical target in 10–9 s to generate at its center a deuterium–tritium plasma with extremely high parameters: T ≈ (1–2) · 108 K, ρ ≈ 200 g/cm3 , p ≈ 150–200 Gbar, which is close to the conditions in the center of the Sun. The respective laser power must ten times exceed the powers of all power stations on the Earth. These conditions necessary to ignite a controlled thermonuclear reaction are extremely unusual for terrestrial conditions. However, they are rather typical of the rest of the Universe where the overwhelming mass of matter is compressed by gravity forces and heated by thermonuclear reactions in the interiors of stars and other astrophysical objects. There is a number of interesting problems in high energy density physics the solution of which determine progress in this field of knowledge and adjacent fields [11]. Ignition of thermonuclear reactions with inertial plasma confinement in controlled terrestrial conditions is the primary pragmatic goal of papers in high energy density physics. The leaders here are lasers [7–19], although electrodynamic methods (Z-pinches [19–24]) and heavy-ion accelerators [25–27] are also being rapidly developed. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Fortov, Intense Shock Waves on Earth and in Space, Shock Wave and High Pressure Phenomena, https://doi.org/10.1007/978-3-030-74840-1_3

33

Compressed gas

Nuclear explosives

60 107

40 0.3 1 2

10−2 104 10 5 0.3 2 × 10−5 10−4 10−6

Nuclear explosives

Neutron-induced heating

Shock waves in solids

Shock waves in gases

Combustion-driven shock tubes

Pneumatic shock tubes

Adiabatic compression

Explosive plasma generator

200

2.5

25 MOe magnetic field

50

50

0.3

4×

60

1 MOe magnetic field

10−2

0.5

10−2 0.3

Chemical explosives

Chemical explosives

Temperature, eV

Energy density, MJ/cm3

Metal plates

Final form of the energy source

Primary energy source

3

107

107

2 × 107

10

250

2 × 10−5

10−2

103

104

5 × 107

150

103

1011 107

2×

1010

30

1

5 × 107 105

5

104

5×

102

Total-energy, MJ

5 × 10−5

Pressure, 105 Pa

Table 3.1 Energy sources and experimental devices used in high energy density physics [1, 2]

3 × 104

10−4

6 × 104

(continued)

108

3 × 104

105

1018

10−5

1015

10−6

1015

1022

10−6

3 × 104

1012

1013

10−7 10−6

5 × 1012

1010

10−6 10−6

1010

Power, W

10−7

Duration, s

34 3 Shock Waves in High-Pressure Physics

–

–

–

Rotary machine

Inductive storage device

Storage battery

Electron beam

Laser

–

Capacitors

– – – – 4

10−8

10−3

10−4

2 × 10−4 10−2

10 10−3 2 0.3 – 106 – 5 × 103

10−3

10−2

10−5

10−3

10−6

104

10−6

5 × 10

Pulsed discharges

Plasma focus

High-pressure arcs

–

Target

–

Target

Experiments in furnaces

Slow wire explosion

0.5

2×

Temperature, eV

1

107

–

108

–

5 × 103

104

10

104

5 × 102

105

–

–

–

–

10−2

Total-energy, MJ

0.l

1

0.5

5 × 10−3

10−3

10−4

10−4

10−4

10−3

10−3

1000

100

100

40

Total-energy, MJ

Pressure, 105 Pa

Pressure, 105 Pa

2

10−7

Energy density, MJ/cm3

Temperature, eV

Energy density, MJ/cm3

2 × 10−2

Fast wire explosion

Final form of the energy source

Shock tubes, electric discharge

Final form of the energy source

Primary energy source

Primary energy source

Table 3.1 (continued)

1014 1013

10−8

5 × 1014 10−8

103 10−10

103

104

1010

109

107

109

1012

5 × 10−12

1012

1012

Power, W

108

Power, W

–

∞

∞

10−5

10−3

10−4

10−6

10−3

10−4

10−4

10−5

Duration, s

104

Duration, s

3 Shock Waves in High-Pressure Physics 35

36

3 Shock Waves in High-Pressure Physics

The operation of such fusion targets is basically close to supernovae explosions, allowing the vast wealth of experimental results and sophisticated computer codes created for the calculation of fusion microtargets and nuclear military charges to be employed in astrophysics. Astrophysicists need reliable experimental data on the properties of plasma in the ultramegabar pressure range to construct and verify models of the structure and evolution of planets and exoplanets. For Jupiter and other planets, it is important to ascertain or disprove the existence of a hard core and estimate the size of the domain occupied by metallic hydrogen, to establish the metallization bound for H2 and H2+ He mixture. Of great significance is the analysis of Jupiter’s energetics with the allowance for the phase separation of mixtures He–H, C–O, etc., as well as the study of the origin and dynamics of its magnetic field. Similar problems are also encountered in studies of giant planets and exoplanets. In this case, quite important are shock-wave experiments, which enable the metallization bounds to be determined and the occurrence of plasma phase transitions to be ascertained. For almost 80 years, there has been an intriguing issue of first-order phase transitions in strongly non-ideal plasma [1, 3–6]—ref. the experimental results for recording such transition in deuterium [28]. Ultrahigh power levels in experiments can bring closer the prerequisites for the observation of relativistic gravitational effects. This list can be easily expanded. It is perhaps limited by our imagination only. It is now hard to say, within the accuracy of dozens of years, which of these fascinating problems of high energy density physics will be solved in existing and designed laboratory facilities and whether they will be solved at all. However, as the academician P.L. Kapitsa advises, “when you go fishing, take a rod with the biggest hook hoping to catch the biggest fish.” Dynamic Methods. Modern intensive progress in advancing the scale of high energy densities is related to the transition to the dynamic methods of investigation [1–6, 29–36], which are based on pulsed energy cumulation in matter under study by means of powerful shock waves. The plasma temperatures and pressures arising in this case significantly exceed the thermal and mechanical strengths of the structural materials of the facilities resulting in limitations on the characteristic lifetime of the matter under study in dynamic experiments, which is determined by the target expansion dynamics and is equal to 10–10 –10–5 s. In the dynamic approach, there are no fundamental limitations on the maximum energy densities and pressures produced in the target. They are limited only by the power of the energy source, i.e., the “driver”. The most common tool for producing high energy densities is high-power shock waves [1, 3–5, 29–31], which emerge due to nonlinear hydrodynamic effects in matter during its motion caused by a pulsed energy release (Chap. 1). The main role here is played by the shock wave, a viscous compression shock, in which the kinetic energy of the incoming flow is converted into the thermal energy of compressed and irreversibly heated matter. The shock-wave technique plays a leading role in high-pressure physics today, making it possible to produce maximum pressures of the megabar and gigabar ranges for many chemical elements and compounds (Fig. 3.1). The current range of peak

3 Shock Waves in High-Pressure Physics

37

TF Radiation waves

M-Diel.

Landau Zel’dovich

I-guns Plasma

Debye

CP

Solid

Liquid Vapor

kbar

kbar

kbar

Fig. 3.1 Thermodynamic trajectories of dynamic matter investigation techniques [1, 2]

dynamic pressures is six orders of magnitude higher than those occurring upon the impact of a bullet and three orders of magnitude higher than those in the center of the Earth, and is close [35–39] to the pressure in the central layers of the Sun and inertial thermonuclear fusion targets [1, 2]. These exotic states of matter emerged during the birth of our Universe, within several seconds after the Big Bang [11, 40, 41]. To a certain degree, we can assume that by successively increasing pressure and temperature in shock wave experiments, we seem to move back along the axis of time approaching the moment of the Universe inception—the Big Bang [1, 3–5]. The critical point (CP) parameters of several metals are given at the bottom. Shock waves not only compress matter, but also heat it to high temperatures (ref. Chap. 1), which is of particular importance for the production of plasma, i.e. the ionized state of matter. Currently, a number of dynamic techniques is being employed to experimentally study strongly non-ideal plasmas [1–6, 29, 30, 33, 35, 36, 49]. The shock compression of initially solid or liquid matter enables states of non-ideal degenerate (Fermi statistics) and classical (Boltzmann statistics) plasmas compressed to peak pressures of ≈4 Gbar and heated to temperatures of ≈107 K [37–39] to be produced behind the shock front; at these parameters, the density of the inertial plasma energy is comparable with the nuclear energy density, and the temperatures are close to the conditions under which the energy and pressure of equilibrium radiation begin to play a noticeable role in the total thermodynamics and dynamics of such ultrahigh-energy states.

38

3 Shock Waves in High-Pressure Physics Casemates

Casemates

X-ray source

X-ray source

Camera Explosive

Recorder

90 m

Fig. 3.2 Cylindrical explosion devices for quasi-adiabatic plasma compression: 1 cylindrical specimen; 2 explosive charge; 3, 4 external and internal metal liners; 5 X-ray radiation source; 6 X-ray recorders

To reduce the irreversible heating effects, it is expedient to compress a material by a sequence of incident and reflected H k shock waves (Fig. 3.1) [1, 3, 28, 32, 43], when compression becomes closer to the “softer” isentropic compression, making it possible to obtain substantially higher compression ratios (10–300 times) and lower temperatures (≈10 times) in comparison with a single-stage shock-wave compression [32]. Multiple shock compression has been used successfully for the experimental study of pressure-induced plasma ionization [22, 28, 32, 42] and matter dielectrization [45] at megabar pressures. Quasi-adiabatic compression S 1 has also been realized in the highly symmetric cylindrical (Fig. 3.2) and spherical explosive compression of hydrogen and inert gases [2, 4, 28, 43]. The highest plasma parameters were obtained using spherical (Figs. 3.3 and 3.4) explosive compression [43, 45]. The experiment was performed using an X-ray complex of three betatrons and a multichannel optoelectronic system for recording the X-ray images of the process of spherical deuterium compression. The experiments in “soft” adiabatic compression of plasma by a megagauss magnetic field are discussed in [48, 49]. In another limiting case, when a high-temperature plasma is required, it is expedient to subject lower (in comparison with solid) density targets, for example, porous metals H m [6, 29–31, 33–35, 76] or aerogels H A [49] (Fig. 3.1) to shockwave compression. This makes it possible to sharply strengthen the irreversibility effects of shock compression and thereby increase the entropy and temperature of the compressed state.

3 Shock Waves in High-Pressure Physics Fig. 3.3 Scheme of experiment on explosive spherical compression of D and He plasma: 1 X-ray radiation sources; 2 protective facility; 3 recorders; 4, 5 collimators (Pb); 6 cones (Al); 7 experimental device

39

Gas

Shells Explosive

Figure 3.5 shows the experimental data on the thermodynamics of high-energy states in the (unconventional for plasma physics) [50] range of solid-state densities and high temperatures obtained by the shock-wave compression of porous nickel samples [49]. Interestingly, these experimental data [49] correspond to the metal-dielectric transition region (Fig. 3.6), where pressure-induced and temperature ionization effects are significant for the description of plasma thermodynamics [1, 3–5]. The shock compression of noble gases and saturated alkali metal vapors by incident H 1 and reflected H 2 shock waves (Fig. 3.1) allows the plasma to be studied in the domain with developed thermal ionization, where the electrons obey the Boltzmann statistics [5]. Explosive shock tubes used for the purpose are given in Fig. 3.7. A characteristic feature of the shock-wave technique is that it permits high pressures and temperatures to be obtained in compressed media, while the low-density domain (including the boiling curve and the vicinity of the critical point) turns out to be inaccessible for them (ref. Chap. 2). The plasma states intermediate between a solid body and a gas are studied using the isentropic expansion technique based on the generation of plasma in the adiabatic expansion S of condensed matter precompressed and irreversibly preheated at the front of a powerful shock wave [1–5, 52, 53]. A shock wave propagating through matter M under study causes its compression and irreversible heating to state a (Fig. 3.8). The shock wave reaching the boundary G of the interface with a softer (in dynamic terms) barrier F causes a Riemann centered wave to be formed (ref. Chap. 1) C + C − where the adiabatic expansion of shock-compressed plasma from state a to state i takes place. This expansion generates a shock wave in the barrier, which propagates at the velocity Di . Recording Di makes it possible to determine the pressure and mass velocity of the barrier from the known shock adiabat hi, which coincide with the respective characteristics of expanding matter due to continuity [8] at the contact boundary G.

Fig. 3.4 a Experimental images of shells with deuterium plasma, b X-ray patterns of the spherical compression of spherical targets [43] (not to scale): a singlecascade [54, 55] in the initial state; b at the time of maximum compression; dashes show the results of the functional processing of the image of a compressed cavity of the single-cascade device; c two-cascade [54–56]: 1 and 2 steel shells of internal and external cascades, respectively

The studied plasma (mD = 9.6 g) is compressed 200 times to a density of 5.6 g/cm3 and a pressure of 55 mln atm (5500 GPa)

The studied plasma (mD = 22 g) is compressed 100 times to a density of 4 g/cm3 and a pressure of 18 mln atm (1800 GPa)

Compressed shell with plasma

Upper cascade

Two-cascade structure

Compressed shell with plasma

Initial state of shell with gas

Initial state of shell with gas

Imposition of images on initial and final points in time

Single-cascade structure

A

40 3 Shock Waves in High-Pressure Physics

3 Shock Waves in High-Pressure Physics

41

kbar

Fig. 3.4 (continued)

cm

g/cm Fig. 3.5 Thermodynamics of a non-ideal nickel plasma [49]. Symbols are the results of the shock compression of porous (m = ρ 0 /ρ 00 ) specimens; α is the degree of ionization

Using barriers of various dynamic rigidity and recording p and u taking place in this case, we can continuously determine the advancement of the expansion isentropy p = ps (u) from states on the Hugoniot adiabat to lower pressures and temperatures. The use of throwing systems of various capacity allows varying the entropy transformation in a shock wave and thereby studying various isentropies overlapping the selected area of the phase diagram. The transition from hydrodynamic variables pu- to thermodynamic ones p–v-E can be carried out when calculating the Riemann integrals (ref. Chap. 1) expressing the conservation laws for this type of self-similar flow. Figure 3.9a, b gives entropy diagrams and curves of adiabatic expansion of shockcompressed copper.

42

3 Shock Waves in High-Pressure Physics

E, kJ/g

Experiment

Thermal ionization

Gas-like plasma

Pressureinduced ionization

ρ, g/cm3 Fig. 3.6 Energy density of shock-compressed nickel plasma [49]

Barrier

Gas

Gas

Solenoid Explosive Solenoid Plasma plug Explosive Detonator Fig. 3.7 Explosive shock tubes to measure low-frequency and Hall conductivity as well as the thermodynamic properties of shock-compressed plasma ([52])

This technique was first used to experimentally study the high-temperature portions of the boiling curves, the near-critical states, and the metal-dielectric transition domains for a large number of metals (for details, ref. [1–5, 33, 53]). As an example of this kind, Fig. 3.10 gives entropy (a) and kinematic (b) diagrams of bismuth expansion [1–5, 33, 53], and Fig. 3.11 gives regions of high-temperature uranium evaporation [1–5, 33, 53]. We see that dynamic techniques in their different combinations permit a broad spectrum of plasma states with a variety of strong interparticle interactions to be realized experimentally and investigated (non-ideality). In this case, it is possible not only to experimentally realize conditions with high energy densities, but also to

3 Shock Waves in High-Pressure Physics a

43 b

Мi

М0

M

Пi

П Г

М0

Мi

Пi Г

Fig. 3.8 a, b Gas dynamic scheme of experiments on adiabatic expansion [1–3]

diagnose sufficiently completely these high-energy states, since shock and adiabatic waves are not only a means of generation, but also a specific tool for diagnosing extreme states of matter with a high energy density [1–6] (Chap. 1). They make it possible to determine the thermodynamic properties of plasma by fixing the mechanical parameters of the motion of shock waves and contact discontinuities and also to determine many other physical characteristics of matter with extreme parameters with the use of modern high-velocity diagnostic techniques (ref. Chaps. 4, 11). Using strong lasers to generate shock waves became today a separate and perspective scientific field that we will consider in Chap. 5. Dynamic diagnostic methods are based on the expansion of the “reverse” problem of gas dynamics (ref. Chaps. 1, 11)—on using the relationship between the thermodynamic properties of the medium under study and the hydrodynamic phenomena observed in the experiment, which occur during the cumulation of high energy densities in matter [1, 3]. In a general form, this relationship is expressed by a system of non-linear (3D in partial coordinates) differential equations of non-stationary gas dynamics the complete solution of which is difficult even for the most powerful modern computers. For this reason, dynamic studies tend to use the self-similar solutions of the type of a plane stationary shock wave and a Riemann centered rarefaction wave (ref. Chap. 1) that express conservation laws in simple algebraic or integral forms. In this case, to apply such simplified relations in the experiment, the conditions of the self-similarity of the respective flow regimes must be provided—the shock wave must be plane and stationary. When a stationary shock wave discontinuity propagates through matter, the laws of conservation of mass, momentum and energy [1–3] (Chap. 1) are fulfilled at its front, which relate the kinematic parameters—the shock-wave velocity D and the mass velocity u of matter behind its front—with the thermodynamic parameters—specific internal energy E, pressure ρ and specific volume v:

44

3 Shock Waves in High-Pressure Physics

GPa

a

Al’tshuler, 1980 Fortov, 1989 Laser 2010

J/(g·K)

kbar

b

km/s Fig. 3.9 Entropy diagram (a) and curves of copper adiabatic expansion (b)



{v/v0 = D−u ; p = p0 + Dv0u ,  1D E − E 0 = 2 ( p + p0 )(v0 − v),

(3.1)

where index 0 indicates the parameters of matter at rest ahead of the shock-wave front. These equations help finding the hydro- and thermodynamic characteristics of matter by recoding any two of the five parameters E, p, V, D, u describing the

3 Shock Waves in High-Pressure Physics

45

GPa

a

J/(g·K)

GPa

b

km/s Fig. 3.10 Entropy (a) and kinematic (b) diagrams of bismuth

shock wave discontinuity. The shock wave front velocity D can be easily and precisely measured by basis methods. The selection of the second measured parameter depends on specific experiment conditions. The analysis of the errors in relationships in (3.1) shows [1–3] that in case of strongly compressible (gas) media, it is reasonable to record the density ρ = v−1 of

46

3 Shock Waves in High-Pressure Physics km/s

ρ, g/cm3

kbar

Electrical explosion of conductors:

ρ, g/cm3

Т, 103 K

Fig. 3.11 High-temperature evaporation of uranium in the near-critical domain. The data were obtained using the adiabatic expansion method [53, 87]

shock-compressed matter. Currently, there is a method of such measurements based on recording the absorption of cesium, argon [1, 5] and air of soft X-ray radiation by plasma. In case of lower system compressibility (condensed media), the acceptable accuracies are ensured by recording mass velocity u. Thus, the states of a degenerate plasma of metals and dense Boltzmann plasma of argon [55] and xenon [1, 3–5] were studied. The experimental methods of studying the compressibility of solid bodies using powerful shock waves and measuring front parameters have been developed by Soviet scientists [2, 6, 57–61] and by American authors [62] (except for the deceleration method, see below). However, the Soviet scientists have studied a much wider range of pressures—up to 4 Mill atm. At a later stage, we will follow paper [1]. Papers [1–3] describe three methods of measuring shock wave parameters. 1.

The spallation method is based on measuring the motion velocity of a free body surface unloaded after the shock wave come to the surface and using the rule of velocity doubling according to which mass velocity u approximately amounts to half the free surface motion velocity u1 . This method has limited applicability since in case of very high pressures, significant deviations from the doubling rule occur, which leads to experimental errors in u determination. The principal scheme of the experiment is as follows. A flat plate of the material under study comes in contact with an explosive charge as shown in Fig. 3.12 (the respective diagram of motion on the plane (x, t) is

3 Shock Waves in High-Pressure Physics Fig. 3.12 Scheme of the experiment carried out using the spallation method [1]

47

Explosive Metal

Sensors Sensors

Fig. 3.13 Diagram (x, t) for the spallation method experiment [1]

SW

Explosive

Metal

shown in Fig. 3.13). When the detonation wave comes from explosive to the boundary with the metal, the decay of discontinuity takes place; a shock wave (line AB) goes across the metal at velocity D, the velocity of the contact boundary between explosive and metal (line AE) amounts to the mass velocity of metal u (reflected wave AC propagates across explosive). After the shock wave comes to the free surface (point B), the decay of discontinuity occurs again. Unloading wave BF runs across the specimen backwards, and the metal boundary acquires double velocity u1 ≈ 2u (line BH). To measure the front velocity D at specific distances inside the specimen as shown in Fig. 3.12, electric contact sensors

48

3 Shock Waves in High-Pressure Physics

Fig. 3.14 Scheme of the experiment carried out using the deceleration method [1]

Fig. 3.15 Diagram (x, t) for the experiment carried out using the deceleration method [1]

2.

were installed, which were closed when wave fronts were passing by and which sent a pulse recorded using a special electrical circuit and oscillograph. Dividing the distance d by time, we could find the average front velocity on the base of the measurement of d (d ≈ 5–8 mm, velocity D ~ 5–10 km/s, time ~ 10–6 s). To measure time, it was required to develop special methods for recording short times. The moments of passing through the specified coordinate points of the boundary of unloaded matter were measured in a similar way with the use of electric contact sensors (Fig. 3.12). In this manner, the iron shock adiabat was determined up to pressures p ~ 1.5 · 106 atm (D ~ 7.5 km/s, u ~ 2.4 km/s). The spallation method is not suitable for studying porous materials since the additional velocity i’ is much lower than the velocity i in case of unloading and the doubling rule is not fulfilled. Deceleration method. To study more powerful shock waves for which the velocity doubling rule introduces a noticeable error, the authors of paper [1] used another method called the deceleration method. In principle, this method is absolutely accurate and suitable for studying any materials, including porous ones. In this method, a plate made of the material under study is accelerated to the velocity w using an explosive charge. The plate (striker Y ) impacts the other resting plate (target M) made of the same material. The scheme of the experiment is shown in Fig. 3.14, and the diagram (x, t)— in Fig. 3.15. At the time of impact, two shock waves are generated, which propagate along both bodies (AB and AC in the diagram (x, t)). Pressures p and mass velocities u on both sides of the contact boundary between the bodies are the same and equal to the same values at the front of both shock waves until the latter reach other boundaries of the specimens). The contact boundary has the

3 Shock Waves in High-Pressure Physics

49

Fig. 3.16 Pressure and velocity profiles after the impact in the deceleration method [1]

same velocity u (line AE). Pressure and velocity profiles after the impact are shown in Fig. 3.16. Due to the identity of the materials, both shock waves are also identical, i.e. the jumps of mass velocities in both waves are the same. For the target, the velocity jump coincides with the motion velocity of compressed matter u since the target was initially at rest. As for the striker, matter moves with the striker flying velocity w in front of the shock wave and with velocity u after the wave, so the velocity jump equals w−u in absolute value. Therefore, w−u = u and u = w/2. Thus, the problem is reduced to measuring the front velocity D in the target and the velocity u after striker w has passed. This problem was experimentally solved in the same manner as in the spallation method—using a system of electric contact sensors. The deceleration method [6] was used to obtain a shock adiabat for iron up to pressures p ~ 5 · 106 atm (D ~ 12 km/s, u ~ 5 km/s, V 0 /V ~ 1.75). Porous iron with the density 1.4 times below the normal level was also studied. The deceleration method can be extended to the case when the target under study and the striker are made of different materials, but in this case matter with the known shock adiabat must be taken as the striker. In some cases, this seems to be much more expedient than making a striker from the material under study, since the same explosive charge can yield a more powerful shock wave in the studied substance by selecting the appropriate material for the striker. If striker and target materials differ, then, despite the equality of pressures in both shock waves, velocity jumps are not similar, so that w−u= u. However, if the shock adiabat of the striker is known, then the dependence of the pressure on the mass velocity jump is also known, i.e. the function p = ƒ(w−u). On the other hand, the pressure p is related by the formula p = Du/V 0

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3 Shock Waves in High-Pressure Physics

with a mass velocity jump in the target, which is, in its turn, equal to the contact boundary velocity u. Measuring, as before, the shock wave velocity in the target and the striker velocity w, we can find the velocity u from the equation f (w − u) = Du/V0 .

3.

For this purpose, a graphical method is convenient which is based on using the pressure–velocity diagram (Chap. 1). These diagrams are widely applied in considering various processes with shock waves where two contacting media are used, since the pressures and velocities at the contact boundary between media are the same. Let us consider a collision of the striker and target using the diagram (p, u), where u is the mass velocity of matter in the laboratory coordinate system— in this case, in the system where the target is initially at rest. If the measured velocity of the shock wave on the target is D, then the geometric place of the target state of matter in the shock wave is the straight line p = Du/V 0 with the known slope D/V 0 . Let us depict the shock adiabat of the striker substance taking into account the dependence of the pressure not on the volume but on the velocity jump which is in this case w − u: p = f(w − u). The crossing point B of both lines defines the state (pressure and mass velocity) in both shock waves. If the striker and target are made of the same material, then, as we already know, the crossing point lies right in the middle between the x-coordinates of points O and A (u = w/ 2). Reflection method. This method uses the laws governing the process of the decay of an arbitrary discontinuity (ref. Chap. 1) occurring in case of shock wave reflection from the boundary of two media [1, 6]. It has an advantage that, as compared to the previous ones, requires no measurement of mass velocities, which is more complicated in experimental terms than the measurement of the shock-wave front velocity. However, this method requires a reference substance with a known equation of state. Let us consider a transition of an intense shock wave from medium A to medium B. A shock wave always goes through matter B, and the reflected wave in A can be either a shock wave if matter B is harder than A, or a rarefaction wave if B is softer than A (this can be easier to imagine if we consider the following limit cases: A—gas, B—solid body and A—solid body, B—gas). The velocity and pressure profiles in both cases are depicted in Fig. 3.17. The respective diagrams (x, t) are also given there. Let us consider this process using the pressure–velocity diagram (in the initial state, both matters A and B are at rest in the laboratory coordinate system). Let us assume that the equation of state of matter A is known. Let us depict a shock adiabat of matter A pA (u) on p, u-diagram (Fig. 3.18) [1] for the first shock wave propagating through the unperturbed material. If we measure the front velocity of the initial shock wave D1 in the experiment, then the state in it

3 Shock Waves in High-Pressure Physics

51

Fig. 3.17 Profiles for pressure p and velocity u: diagram (x, t) for the experiment with “reflection”: a case when the reflected wave is a shock wave: OC shock wave in A; CM shock wave in B; CN reflected shock wave in A; KCK A and B contact line; b case when the reflected wave is a rarefaction wave: OC shock wave in A; CM shock wave in B; CN head of the rarefaction wave; CT tail of the rarefaction wave; KCK A and B contact line [2]

Fig. 3.18 Diagram (p, u) for the experiment with “reflection” [1]

will be depicted by point a(pa , ua )—crossing point between the straight line p = D1 u/V 0A and the shock adiabat pA (u). After this shock wave is reflected from the boundary between media A and B, a new state occurs in matter A. If the reflected wave is a shock wave, then the state lies

52

3 Shock Waves in High-Pressure Physics

Fig. 3.19 Scheme of the experiment with reflection [1]

on the shock adiabat of secondary compression for which the initial state is a(pa , V a , ua ); this shock adiabat is depicted by the curve pH outgoing from point a. If the reflected wave is an adiabatic rarefaction wave, then the new state lies on the expansion isentropy coming from point a downwards (curve pS ). Since the equation of state of matter A is assumed to be known, we can convert both the shock adiabat of secondary compression pH (V, V a , pa ) and the rarefaction isentropy with entropy S a = S(pa , V a ) so that velocity is entered as an argument instead of volume. In the first case, this is done by using the relations at the shock-wave front, while in the second case, by using the relations that are true for the rarefaction wave (ref. Chap. 1). If we measure the velocity of the shock wave D in matter B experimentally, then the geometric place of states in this wave is the straight line p = Du/V 0B . The crossing point b between this line and the curve pH apS is the geometric place of possible states in matter A after shock wave reflection—it defines the pressure and velocity in the shock wave in section B, which are equal to the pressure and velocity of the A and B contact boundary (Fig. 3.17). Diagram (p, u) shows the second case when a rarefaction wave occurs during reflection (Fig. 3.18). In the first case, the straight line p = Du/V 0B passes above the straight line p = D1 u/V 0A and the crossing point b lies above point a on the shock adiabat of secondary compression of matter A that is described by the curve apH . The reflection method consists in the following. A shock wave is created in a plate of material A with the known equation of state either directly from an explosive charge or by impacting with another plate preliminary accelerated by the explosive to a high velocity. This wave enters the specimens of materials B under study that include, among other things, a specimen of material A (the experimental scheme is shown in Fig. 3.19). The front velocities D1 and D are determined by recording the moments of closure of electric contact sensors or optical signals when shock wave discontinuities go to the surfaces of the targets located in the places shown in Fig. 3.19 by arrows. Constructing the shock adiabat pA (u) on the p, u-diagram and drawing the straight line p = D1 u/V 0A , we find point a—the state in the shock wave in A. Then, the shock adiabat of secondary compression is drawn through point a upwards, a simple adiabat is drawn downwards, and the straight line p = Du/V 0B is plotted thereby finding the sought state b(p, u) in the shock wave in the specimen under study. In fact, pressure changes between states a and b are always small. As calculations showed, the curve pH apS can be represented in these conditions with high accuracy as a mirror reflection of the shock adiabat of primary compression at point a. Let us note that the slope of the curve pH apS at point a is determined by the velocity of sound behind the front of the primary shock wave in A. Indeed, dp = ± pc du in the rarefaction wave just as in the weak compression wave, i.e. the slope of the curve

3 Shock Waves in High-Pressure Physics

53

pH apS at point a is |dp/du| = pc = c/V, where c and V are the velocity of sound and the specific volume in matter A compressed by the first shock wave. Methods of the experimental determination of the velocity of sound behind the shock-wave front will be considered below. The reflection method was used in paper [6] to record shock adiabats of a number of metals. This method [1–5] was also used in most papers of foreign scientists. Iron, aluminum or brass was most frequently used as a material for screen A. The states in a centered unloading wave in experiments on determining the curves of isentropic expansion of shock-compressed matter, as we saw above, are described by Riemann integrals (ref. Chap. 1): p H v = vH + p

du dp

2

p H  2 du dp; E = E H − p dp, dp

(3.2)

p

that are calculated along the measured isentropy pS = pS (u). Taking measurements at various initial conditions and intensities of shock waves, we can find the caloric equation of state E = E(p, V ) in the field of p–v-diagram overlapped by Hugoniot and/or Poisson adiabats. In the experiments on dynamic effect on plasma, carried out to date, the change in the intensity of shock waves was done by varying the power of excitation sources—driving gas pressure, types of explosives, throwing devices and targets [2, 3]. Moreover, various methods were used to change the parameters of the initial states: changing initial temperatures and pressures (plasma of inert gases, cesium, liquids [2, 5]), using fine targets to increase irreversibility effects [1, 6]. Powerful lasers [74, 75] and electrodynamic [76, 77] methods of acceleration in Z-pinches (ref. Chap. 5) gave new life to the dynamic methods of studying high pressures. Throwing velocities were increased, and electrical contacts were replaced by superfast optical methods of laser interferometry and optical methods of recording shock wave motions in stepped targets. However, the ideological background of the dynamic methods of studying high pressures is the same as before. Thus, the dynamic methods of diagnostics based on the general conservation laws allow reducing the problem of finding the caloric equation of state E = E(p, V ) to measuring the kinematic parameters of shock wave motion and contact surfaces, i.e. to recording distances and times, which can be done with high accuracy. However, the internal energy is not a thermodynamic potential towards variables p, V, so to construct the closed thermodynamics of the system, the additional dependence of temperature T (p, V ) must be known. In optically transparent and isotropic media (gases), the temperature changes together with the other parameters of shock compression. Condensed media and, first of all, metals are usually non-transparent, so the light radiation of a shock-compressed medium cannot be recorded. As it is discussed in Chap. 12, the thermodynamically complete equation of state can be constructed directly from the results of dynamic measurements, without introducing a priori considerations on the properties and nature of matter under study

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3 Shock Waves in High-Pressure Physics

Shock wave

Fig. 3.20 Time profile of a compression pulse registered by a differential laser interferometer at its outlet to the free surface. Indicated is the experimental information obtained from the analysis of this profile

[1–6], based on the first law of thermodynamics and the dependence E = E(p, V ) (or more convenient function γ (p, V ) = pV/E(p, V )) known from the experiment. In addition to this method of determining the thermodynamic properties of matters compressed and heated by powerful shock waves, more advanced experimental methods are used [2–5]. A special place is taken by differential interferometric methods VISAR, ORVIS [N + l, N + 2] (Chap. 5) that make it possible to measure the time profile of a shock wave when it reaches the free surface of the specimen under study. Figure 3.20 shows that the processing of shock wave shapes yields abundant information on the thermodynamic, kinetic and mechanical properties of the material being studied. The mechanical properties of shock-compressed materials will be considered in Chap. 4. Let us now briefly discuss the experimental technique for generation of powerful shock waves in dense media (Fig. 3.21) and some results. Light-Gas Guns and Chemical High Explosives. Today, the technique of powerful shock waves generated by the collision of metal strikers accelerated to velocities of several kilometers per second (Fig. 3.21) with a target made from the matter in question is the main source of physical information about the matter behavior at pressures of the megabar range. Here, we shall not describe in detail the striker acceleration technique and the means of diagnostics—they are dealt with in comprehensive reviews and monographs [1–6, 29, 30, 32–35, 42]. We note only

3 Shock Waves in High-Pressure Physics

EXPLOSIVE EQUIPMENT, GUNS Plane wave lenses Plate acceleration Cumulation One-stage guns Powder guns Two-stage guns Rail guns

55

Striker Specimen

Sensor Striker

ELECTRICAL EXPLOSION Cylinders Foils

}

+shell

RADIATION Laser radiation Electron beams Ion beams Soft X-ray Nuclear explosions

Sensor Specimen

Fig. 3.21 Scheme of powerful shock wave generation

that in the shock-wave experiments of this kind it is possible to carry out sufficiently complete measurements of compressed and heated matter. The equation of state is determined by the electrocontact and optical recording of the time intervals in the motion of shock-wave discontinuities and contact surfaces. Pyrometric, spectroscopic, protonographic, X-ray diffraction, and adsorption measurements are performed using pulsed X-ray and synchrotron radiation sources; laser interferometric measurements are made; low- and high-frequency Hall conductivities are recorded; and piezo- and magnetoelectric effects are studied. Unique information has been obtained in studying the mechanical properties of shock-compressed media: elastic–plastic properties, spallation, polymorphism, viscosity, fracture and fragmentation (ref. Chap. 4). Many unexpected data have been also obtained when studying chemical reactions and the kinetics of physical–chemical transformations in shock and detonation waves [67]. Figure 3.22 schematically shows the research areas in dynamic physics, chemistry and high-pressure mechanics obtained using powerful shock waves. In the USA, for these purposes, gunpowder and light-gas throwing devices— “guns” (Fig. 3.24)—became the most widespread, while in the USSR preference was given to explosive propellant devices [67]. Powder gases formed in charge combustion in a two-stage light-gas gun (Figs. 3.23, 3.24) accelerate a heavy piston that compresses hydrogen while moving in a chamber 10 m long and 90 mm in inner diameter. After membrane rupture,

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3 Shock Waves in High-Pressure Physics

Fig. 3.22 Primary trends in shock wave studies of high pressures [67]

Shock wave shape Equation of state

Elasticity, velocity of sound

Velocity

Phase transition

Fracture Evaporation ionization

Plasticity moduli Yield strength Braking ability

Chemical kinetics Open porosity Elasticity moduli

Heat conductivity

Time

Fig. 3.23 Diagram of a two-stage ballistic facility [32]: a before experiment; b during experiment; 1 powder gases; 2 heavy piston; 3 high-pressure chamber; 4 hydrogen; 5 striker; 6 accelerating channel; 7 target under study

the expanding hydrogen accelerates a light striker 28 mm in diameter in the accelerated vacuumized channel 9 m long. Thus, a highly symmetric smooth and nonheating acceleration of a striker with a mass of ~ 20 g occurs to speeds of 7–8 km/s, which are somewhat higher than those that can be obtained with plane throwing in explosive devices. An impact of the strikers accelerated in this manner generates plane stationary shock waves in liquid argon and xenon the motion velocities of which are recorded by the system of electric contact sensors having the time resolution of hundreds of picosends. These measurements together with the laws of decay of discontinuity at the screen-plasma interface as well as conservation laws were used to obtain [2–5] the equation of state for highly compressed plasma of hydrogen, argon and xenon at pressures up to 0.13 TPa and maximum temperatures up to 30,000 K (ne ~ 5.7 · 1022 cm−3 ). In these conditions of strong compression (maximum concentration of cores ~ 5.7 · 1022 cm−3 ), a distinct deformation of the plasma electronic energetic spectrum is observed. In this case, a significant number of electrons is in a thermally excited state, and, in general, a highly compressed plasma has semi-conducting properties [32]. To increase the throwing velocity and hence the shock-compressed plasma pressure, highly sophisticated gas-dynamic techniques are used. Thus, the method of

3 Shock Waves in High-Pressure Physics

57

Fig. 3.24 Schematic representation of a light-gas gun, Livermore, USA [32]

“gradient” cumulation (Fig. 3.25) is based on a successive increase in the velocity of strikers in plane alternating layers of heavy and light materials. This method is not related to the phenomenon of geometrical energy focusing and therefore exhibits a higher stability of acceleration and compression in comparison with the spherical one. An explosive three-stage “layer cake” constructed in this way [63, 64] accelerates a 100 μm molybdenum striker to speeds of 5 to 14 km/s. An impact of such striker excites in a target plasma a plane shock wave or a series of reverberating shock waves with an amplitude pressures of the megabar range. The geometrical parameters of these experimental devices are selected in such a way as to eliminate the distorting effect of side and rear unloading waves and to ensure the one-dimensionality and stationarity of gas dynamic flow in the region of recording. It is curious that the kinetic energy of a metal striker moving at 10 km/s is close [61] to the kinetic energy of a proton beam in the cyclotron accelerator of the Fermi laboratory. High kinetic energy in shock wave experiments generates a highly compressed plasma of high temperature in the same way as an individual collision of relativistic ions creates a quark-gluon plasma with huge energy densities (ref. Chap. 9). To increase the parameters of shock compression, several experiments used explosive-driven generators of counterpropagating shock waves (Fig. 3.26), where the material under study was loaded on both sides by a synchronous impact of steel strikers symmetrically accelerated by explosive charges [70, 71]. High-precision spherical explosive-driven generators of powerful shock waves (Fig. 3.27) were made in the USSR [29, 30, 34, 72] to study thermodynamic material

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3 Shock Waves in High-Pressure Physics

4.7 km/s

6.7 km/s 12.8 km/s

Steel 2.5 mm

5 mm

Steel 1 mm

mm

Target

1 mm

Fig. 3.25 Principle of “gradient” cumulation [65] and an explosive three-stage “layer cake” [2, 3, 29, 30]

Metal striker Explosive charge Detonator

Detonation lens

Target Explosive charge Detonator

Detonation lens

Fig. 3.26 Explosive-driven generator of counterpropagating shock waves [70, 71]

properties at pressures ranging up to 10 Mbar. Using the geometrical cumulation effects in the centripetal motion (implosion) of detonation products and hemispherical shells, it was possible to accelerate metal strikers to speeds of about 23 km/s in devices weighing about 100 kg with an energy release of about 300 MJ. As we noted, the use of the experimental technique of powerful shock waves for studying extreme states of matter [1–5] is today the main source of information about the behavior of strongly non-ideal highly compressed plasmas in the region of recordhigh temperatures and megabar-to-gigabar pressures. Being exotic for terrestrial conditions, these ultraextreme states are quite typical of the majority of astrophysical

3 Shock Waves in High-Pressure Physics

“Big assembly”

59

“Small assembly”

15.9 km/s

8.5 km/s

90 + 90 kg

Air gap Striker

Screen

Sensor To oscillographs

Computer modeling result

Fig. 3.27 Explosive-driven generators of spherically converging powerful shock waves [34, 35, 72]

objects and define the structure, evolution, and luminosity of stars, the planets of the Solar system and more than 2,000 recently discovered exoplanets [10, 73]. In addition, promising energy projects to create controlled thermonuclear fusion with inertial plasma confinement [2, 3, 7–12] and the realization of high-temperature states in compressed hydrogen [2–5, 43] are associated with plasmas of ultramegabar range. These projects are a permanent factor that gives an impetus to experimental studies into the properties of non-ideal plasmas of hydrogen, deuterium, and inert gases highly compressed by powerful shock waves, which are excited by light-gas (Figs. 3.21, 3.23, 3.24) and explosion plane or hemispherical devices (ref. [2–5] and Fig. 3.21), high-power lasers [74, 75] (Chap. 5), and electrodynamic accelerators [76, 77]. Significantly higher pressures exceeding the pressures of a single shock-wave compression by almost an order of magnitude are implemented, in case of significant reduction of effects of irreversible shock wave heating, with quasi-isentropic compression of matters by a sequence of incident and reflected shock waves in devices having plane, cylindrical and spherical geometry [2–5, 43]. In experiments using this technique, the density of compressed deuterium of about 10 g/cm3 was achieved in the region of pressures up to 800 GPa [54–56].

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3 Shock Waves in High-Pressure Physics

Subsequently, record-high plasma parameters were obtained in deuterium [43]. At the initial pressure of gaseous deuterium p0 = 267 atm and T 0 = 10.5 °C, the density of the shock-compressed deuterium plasma ρ = 10 g/cm3 was recorded at record-high pressure p = 114 Mbar. In these conditions, plasma is strongly non-ideal (G ≈ 4.5 · 102 ) with degenerate (nλ3e ≈ 8 · 102 ) electronic component + number of electrons ≈ 2.8 × 1023 cm−3 . The kinematic and thermodynamic parameters of liquid nitrogen shock compression were measured behind the plane shock wave front using plane-wave and semispherical generators of shock waves. High compression parameters were achieved during the experiments: the shock-compressed liquid nitrogen density ρ ≈ 3.25 g/cm3 and the temperature T ≈ 56,000 K at pressure p ≈ 265 GPa. Density, pressure, temperature and electrical conductivity of the non-ideal plasma of shock-compressed liquid nitrogen were measured. Almost isochoric behavior of the shock adiabat of nitrogen was recorded within the pressure range p = 100 to 300 GPa. The experimental results are interpreted as the observation of the boundary of the transition of shock-compressed nitrogen from the polymeric phase to the state of dense strongly non-ideal plasma at p ≈ 100 GPa, ρ ≈ 3.4 g/cm3 . The experimental data obtained during cylindrical and spherical shock-wave compression concerning the thermodynamic properties of the non-ideal plasma of deuterium are given in Fig. 3.28. Estimates show that the shock-compressed plasma in these experiments is strongly non-ideal, G > 1, with developed ionization, nc /nD ≈ 1, and strong degeneration, nλ3 ≈ 3 It can be seen (lines in Fig. 3.28) that non-ideal plasma models [2, 3] reasonably describe the data of dynamic experiments. In higher-stability conical explosive-driven generators, cumulation effects in the irregular (“Mach”) convergence of cylindrical shock waves (SW) were used (Fig. 3.29). The combination of the effects of irregular cylindrical and “gradient” cumulation enables a shock wave to be excited in copper with an amplitude of ≈20 Mbar, which is comparable with pressures in the near zone of a nuclear explosion (ref. Chap. 4). Papers [83–85] are of extremely high interest since they use high-symmetry electrodynamic acceleration of metal strikers. The photo and scheme of striker acceleration by a powerful magnetic field from impulse megabar current is shown in Fig. 3.30. An important advantage of this method is the high symmetry of acceleration and the ability to set the time form of specimen loading. Comparison of the capabilities of various methods of high-speed throwing is given in Fig. 3.31. Underground Nuclear Explosions. Plasma energy densities that are record-high for terrestrial conditions were obtained in the near zone of a nuclear explosion (ref. Chap. 6). Some physical settings of such experiments are given in Figs. 3.32 and 3.33 [35, 37–39, 68, 86]. The combination of experimental data on a shock-compressed aluminum plasma is shown in Fig. 3.34, where the highest points correspond to the record-high parameters in terrestrial conditions [2–5]. The density of the internal energy of this plasma is E ≈ 109 J/cm3 , which is close to the energy density of nuclear matter, and the pressure P ≈ 4 Gbar is close to the pressure in the internal layers of the Sun. The plasma under these conditions (ne ≈ 4

3 Shock Waves in High-Pressure Physics

a

61

P, GPa

P, GPa

ρ, g/cm3 ρ, g/cm3 b

P, GPa

P, GPa

ρ, g/cm3 ρ, g/cm3 Fig. 3.28 Quasi-isentropic compression of deuterium plasma in the pressure range of up to 5500 GPa [44]. Experiment: ◯ [78, 79]; ● paper [43];  [54–56]; light star [80]; dark star point from paper [78] recalculated for pressure; lines are approximations. Here: HM1 and HM B1 are shock adiabats in experiments M 1 and MB1 [44]; S M1 and S M B1 are isentropies from states M1 and MB1; H is the state on shock adiabats, F are deuterium parameters in SW focusing state

62

3 Shock Waves in High-Pressure Physics

Detonation products SW

SW

Mach shock wave

Fig. 3.29 Shown on the left are explosive-driven conical generators of “Mach” shock waves [82]. Shown on the right are the results of a two-dimensional hydrodynamic calculation

× 1024 cm3 , T ≈ 8 · 106 K) is non degenerate,nλe ≈ 0.07; moreover, it is twelve times ionized, and the non-ideality parameter is small ( ~ 0.1), which is an experimental illustration of the thesis about the simplification of the physical properties of plasma in the limit of ultrahigh energy densities. It is curious that the studied parameter range is adjacent to the region where the energy and pressure of equilibrium light radiation make an appreciable contribution to the thermodynamics of the system:   4/ 3ρT 4 4σ T 4 ER ; pR = = . ER = c 3 c Thus, the plasma dynamics mode is implemented, which is close to the radiation gas-dynamic one [1, 33]. The pressures obtained by means of nuclear explosions [34, 35, 37–39] (Fig. 3.32) belong to the multimegabar pressure region and are close to the characteristic “phys2 ical” pressure, which may be found from dimensionality considerations, p ≈ ae4 ≈ B

300 (aB = è2 /(me2 ) is the Bohr electron radius); starting with these pressures, the Thomas–Fermi model [4, 5] becomes applicable, which implies a simplified quantum-statistical description of highly compressed matter and the “self-similarity” of its physical properties. This model [4, 5] based on the quasiclassical approximation to the self-consistent field (SCF) method, is a substantial simplification of the many-particle quantum– mechanical problem and therefore has become widely used in the solution of astrophysical and special problems.

Specimen

Cathode

Target

Anode/Striker

Fig. 3.30 Scheme of the electrodynamic acceleration of metal strikers on 2-pinch facility, Sandia laboratory, USA [76, 77, 83, 84]

Anode

3 Shock Waves in High-Pressure Physics 63

64

3 Shock Waves in High-Pressure Physics

High-power lasers

Speed, km/s

Space garbage Two-stage lightgas guns

Explosive guns

Powder guns

Weight, g Fig. 3.31 Possibilities of various methods of high-velocity throwing

Nuclear charge

Rock

Rock

Foam plastic Striker (Fe) Screen (Fe) Target

35 km/s

Fig. 3.32 Schematic representation of experiments on the generation of powerful shock waves in the near zone of a nuclear explosion [2–5]

3 Shock Waves in High-Pressure Physics

Concrete plugging Block А1

Reference layer

65

Polyethylene coat γ-detectors Collimating slits

Cables Lead protection

Magnesium cylinder

Rock

Concrete

Explosive chamber

Nuclear charge Fig. 3.33 Schematic representation of experiments with an underground nuclear explosion using a gamma-active reference layer [37–39]

P, Mbar

Radiation

cm3

J/cm3

SCF

Plasma model

TF limit

TFS

Nuclear exp. Explosive, guns

HFS

Ideal plasma

Fig. 3.34 Shock-wave compression of aluminum to gigabar pressures [37–39]

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42. Fortov V.E, Ternovoy V.Y. Zhernokletov MV et al (2003) Pressure-Induced Ionization of nonideal plasma in Megabar range of dynamic pressures. JETP 124(2):288 [Fortov V. E., Ternovoy V. Ya., Zhernokletov M. V. i dr. Ionizatsiya davleniyem neideal’noy plazmy v megabarnom diapazone dinamicheskikh davleniy // ZHETF. 2003. T. 124, №2. S. 288 (in Russian)] 43. Mochalov MA, Il’kayev RI, Fortov VE et al (2017) JETP Lett 151:592–620 [Mochalov M. A., Il’kayev R. I., Fortov V. E. i dr. ZHETF. 2017. T. 151. S. 592–620 (in Russian)] 44. Maksimov EG, Magnitskaya MV, Fortov VE (2005) Difficult behavior of simple metals at high pressures. Phys Usp 175(8):793 [Maksimov E. G., Magnitskaya M. V., Fortov V. E. Neprostoye povedeniye prostykh metallov pri vysokikh davleniyakh// UFN. 2005. T. 175, № 8. S. 793 (in Russian)] 45. Mochalov MA, Il’kayev RI, Fortov VE et al (2010) Measurement of the compressibility of a Deuterium plasma at a pressure of 1800 GPa. JETP Lett 92(5):336–340 [Mochalov M. A., Il’kayev R. I., Fortov V. E. i dr. Izmereniye szhimayemosti deyteriyevoy plazmy pri davlenii 1800 GPa // Pis’ma ZHETF. 2010. T. 92, № 5. S. 336–340 (in Russian)] 46. Fortov VE, Gryaznov VK, Mintsev VB et al (2001) Thermophysical properties of shock compressed argon and xenon. Contrib Plasma Phys 41(2–3):215–218 47. Fortov V, Mintsev VB, Ternovoi VY et al (2004) Conductivity of nonideal plasma. Contrib Plasma Phys 8(3):447–459 48. Hawke PS, Burgess TJ, Duerre DE et al (1978) Observation of electrical conductivity of isentropically compressed hydrogen at megabar pressures. Phys Rev Lett 41(14):994–997 49. Pavlovski A, Boriskov G. et al (1987) Isentropic solid hydrogen compression by ultrahigh magnetic field pressure in megabar range. In: Fowler C, Caird R, Erickson D (eds) Megagauss technology and pulsed power applications. Plenum Press, London, 255; Gryaznov VK, Zhernokletov MV, Iosilevski IL et al (1998) Shock-wave compression and thermodynamics of highly non-ideal metal plasma. JETP 114(4):1242 [Gryaznov V. K., Zhernokletov M. V., Iosilevski I. L. i dr. Udarno-volnovoye szhatiye sil’noneideal’noy plazmy metallov i yeye termodinamika // ZHETF. 1998. T. 114, №4. S. 1242 (in Russian)] 50. Gryaznov VK, Nikolayev DN, Ternovoy VYa et al (1998) Generation of non-ideal plasma by shock-wave compression of highly porous SiO2 -aerogel. Chem Phy 17(2):33–37 [Gryaznov V. K., Nikolayev D. N., Ternovoy V. Ya. i dr. Generatsiya neideal’noy plazmy putem udarnovolnovogo szhatiya vysokoporistogo SiO2-aerogelya // Khimicheskaya fizika. 1998. T.17, №2. S. 33–37 (in Russian)] 51. Fortov VE (ed) (2000) Encyclopedia of low-temperature plasma. Nauka, Moscow [Entsiklopediya nizkotemperaturnoy plazmy / Pod red. V.Ye. Fortova. — M.: Nauka, 2000 (in Russian)] 52. Mintsev VB, Fortov VE (1982) Explosive shock tubes. High Temp 20(4):745 [Mintsev V. B., Fortov V. E. Vzryvnyye udarnyye truby // TVT. 1982. T.20, №4. S. 745 (in Russian)] 53. Zhernokletov MV (1998) Shock compression and isentropic expansion of natural uranium. High Temp 36(2): 231 [Zhernokletov M. V. Udarnoye szhatiye i izoentropicheskoye rasshireniye prirodnogo urana // TVT. 1998. T.36, №2. S. 231 (in Russian)] 54. Mochalov MA, Il’kayev RI, Fortov VE et al (2010) JETP Lett 92(5):336 [Mochalov M. A., Il’kayev R. I., Fortov V. E. i dr. // Pis’ma ZHETF. 2010. T. 92, № 5. S. 336 (in Russian)] 55. Mochalov MA, Il’kayev RI, Fortov VE et al (142) JETP Lett 696 (2012) [Pis’ma ZHETF. 142. S. 696 (2012) (in Russian)] 56. Mochalov MA, Il’kayev RI, Fortov VE et al (2014) JETP Lett146:169 (2014) [Mochalov M. A., Il’kayev R. I., Fortov V. E. i dr. // Pis’ma ZHETF. T. 146. S. 169 (2014) (in Russian)] 57. Al’tshuler LV, Krupnikov KK, Ledenev VN, Zhuchikhiv VI, Brazhnik M I (1958) JETP 34:874 [Al’tshuler L. V., Krupnikov K. K., Ledenev V. N., Zhuchikhiv V. I., Brazhnik M. I. // ZHETF. 1958. T. 34. S. 874] 58. Al’tshuler LV, Krupnikov KK, Brazhnik MI (1958) JETP V. 34:886 [Al’tshuler L. V., Krupnikov K. K., Brazhnik M. I. // ZHETF. 1958. T. 34. S. 886 (in Russian)] 59. Al’tshuler LV, Kormer SB, Bakanova AA, Trunin RF (1960) JETP 38:790 [Al’tshuler L. V., Kormer S. B., Bakanova A. A., Trunin R. F. // ZHETF. 1960. T. 38. S. 790 (in Russian)]

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60. Al’tshuler LV, Kormer SB, Brazhnik MI, Vladimirov LA, Speranskaya MP, Funtikov A I (1960) JETP 38:1061 [Al’tshuler L. V., Kormer S. B., Brazhnik M. I., Vladimirov L. A., Speranskaya M. P., Funtikov A. I. // ZHETF. 1960. T. 38. S. 1061 (in Russian)] 61. Al’tshuler LV, Kuleshova LV, Pavslovskiy MN JETP 39 [Al’tshuler L. V., Kuleshova L. V., Pavslovskiy M. N. // ZHETF. 1960. T. 39 (in Russian)] 62. Walsh JM, Christian RH (1955) Phys Rev 97:1544 63. Walsh JM, Rise MH, McQueen RG, Yargen FL (1957) Phys Rev 1957 108:196 64. Goranson W, Bankroft D et al (1960) J Appl Phys 31:1253 65. Mallory D (1960) J Appl Phys 31:1253 66. McQueen RG, Marsh S (1960) P J Appl Phys 31:1253 67. Kanel GI, Rasorenov SV, Fortov VE (2004) Shock-wave phenomena and properties of condensed matter Springer, New York 68. Fortov VE, Khrapak AG, Yakubov IT (2004) Physics of non-ideal plasma. FIZMATLIT, Moscow [Fortov V. E., Khrapak A. G., Yakubov I. T. Fizika neideal’noy plazmy. — M.: FIZMATLIT, 2004 (in Russian)] 69. Zababakhin EI, Zababakhin IE (1988) Phenomenon of unbounded cumulation. Nauka, Moscow [Zababakhin E. I., Zababakhin I. E. Yavleniya neogranichennoy kumulyatsii. — M.: Nauka, 1988 (in Russian)] 70. Nabatov SS, Dremin AN, Postnov VI, Yakushev V V (1979) Measurements of sulfur electrical conductivity at super-high dynamic pressures. JETP Lett 29(7): 407 [S. S., Dremin A. N., Postnov V. I., Yakushev V. V. Izmereniye elektroprovodnosti sery pri sverkhvysokikh dinamicheskikh davleniyakh // Pis’ma ZHETF. 1979. T. 29, № 7. S. 407 (in Russian)] 71. Fortov VE, Yakushev VV, Kagan KL et al (1999) Abnormal electrical conductivity of lithium in Quasi-Isentropic compression up to 60 GPa (0.6 Mbar). Transition to molecular phase? JETP Lett 70(9):620 [Fortov V. E., Yakushev V. V., Kagan K. L. i dr. Anomal’naya elektroprovodnost’ litiya pri kvaziizoentropicheskom szhatii do 60 GPa (0,6 Mbar). Perekhod v molekulyarnuyu fazu? // Pis’ma ZHETF. 1999. T. 70, № 9. S. 620 (in Russian)] 72. Al’tshuler LV, Trunin RF, Krupnikov KK, Panov NV (1996) Explosive laboratory devices for studying matter compression in shock waves. Phys Usp 166(5):575 [Al’tshuler L. V., Trunin R. F., Krupnikov K. K., Panov N. V. Vzryvnyye laboratornyye ustroystva dlya issledovaniya szhatiya veshchestv v udarnykh volnakh// UFN. 1996. T. 166, № 5. S. 575 (in Russian)] 73. Fortov VE (2009) Extreme states of matter. FIZMATLIT, Moscow [Fortov V. E. Ekstremal’nyye sostoyaniya veshchestva. — M.: FIZMATLIT, 2009 (in Russian)] 74. Da Silva LB, Celliers P, Collins GW et al (1997) Absolute equation of state measurements on shocked liquid deuterium up to 200 GPa (2 Mbar) Phys Rev Lett 78(3):483–486 75. Collins GW, Da Silva LB, Celliers P et al (1998) Measurements of the equation of state of deuterium at the fluid insulator-metal transitionScience 281(5380):1178–1181 76. Knudson MD, Hanson DL, Bailey JE et al (2004) Principal Hugoniot, reverberating wave, and mechanical reshock measurements of liquid deuterium to 400 GPa using plate impact techniques. Phys Rev B 69:144209 77. Knudson MD, Hanson DL, Bailey JE et al (2003) Use of a wave reverberation technique to infer the density compression of shocked liquid deuterium to 75 GPa. Phys Rev Lett 90:035505 78. Grigoryev FB, Kormer SB, Mikhailova OL et al (1972) JETP Lett 16:286 [Grigoryev F. B., Kormer S. B., Mikhailova O. L. i dr. // Pis’ma ZHETF. 1972. T.16. S. 286 (in Russian)] 79. Grigoryev FB, Kormer SB, Mikhailova OL et al (1999) JETP Lett 199916(9):286 [Grigoryev F. B., Kormer S. B., Mikhailova O. L. i dr. // Pis’ma ZHETF. 1999. T.16, №9. S. 286 (in Russian)] 80. Fortov VE, Ilkaev RI, Arinin VA et al (2007) Phys Rev Lett 99:185001 81. Mochalov MA, Zhernokletov MV, Il’kayev RI et al (2010) Experimental measurement of density, temperature and electrical conductivity of shocked non-Ideal plasma of nitrogen in megabar pressure range. JETP 137(1):77–92 [Mochalov M. A., Zhernokletov M. V., Il’kayev R. I. i dr. Eksperimental’noye izmereniye plotnosti, temperatury i elektroprovodnosti udarnoszhatoy neideal’noy plazmy azota v megabarnom diapazone davleniy // ZHETF. 2010. T. 137, № 1. S. 77–92 (in Russian)]

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82. Bazanov OV, Bespalov V et al (1985) Irregular reflection of conic converging shock waves in plexiglass and copper.High Temp 23(5):976[Bazanov O. V., Bespalov V. i dr. Neregulyarnoye otrazheniye konicheskikh skhodyashchikhsya udarnykh voln v pleksiglase i medi // TVT. 1985. T. 23, № 5. S. 976 (in Russian)] 83. Knudson MD, Hanson DL, Bailey JE et al (2001) Equation of State Measurements in Liquid Deuterium to 70 GPa. Phys Rev Lett 87(22):225501 84. Knudson MD, Derjarlais MP et al (2015) Sciences 348(6249) 85. Knudson MD, Derjarlais MP et al www.sciencemag.org/conters. 86. Trunin RF, Podurets MA, Simakov GV et al (1972) Experimental check of Thomas-Fermi model for metals at high pressures. JETP 62(3):1043–1048 [Trunin R. F., Podurets M. A., Simakov G. V. i dr. Eksperimental’naya proverka modeli Tomasa-Fermi dlya metallov pri vysokikh davleniyakh// ZHETF. 1972. T. 62, № 3. S. 1043–1048 (in Russian)] 87. Fortov VE, Lomonosov IV (1997) Thermodynamics of extreme states of matter. Pure Appl Chem 69(4):893–904

Chapter 4

Shock Waves in Condensed-Matter Physics

Intense shock waves have become nowadays a single tool for studying the physical properties of matters at extremely high pressures of megabar–gigabar range [1]. The development of materials processing technologies, experimental techniques and computer modeling methods for processes of inelastic deformation and fracture at the atomistic level stimulated the growing interest to studies of the strength and plasticity of solid bodies at extremely low duration of loading. In particular, one of the perspective trends of precision material processing is related with the use of femtosecond laser technology. In case of so low durations, resistance to the deformation and fracture of solid bodies exceeds the respective values by orders of magnitude at normal deformation rates and approaches maximum possible (“ideal”) strength values. To optimize the technology, it is required to know that aluminum can have the same hardness as sapphire in case of short-term exposure. The effect of temperature on the flow stress of crystalline bodies becomes unusual, which is in some cases even reverse in sign as compared to normal conditions. In particular, this explains why some metallic materials are prone to develop deformation instability and form stripes of adiabatic shift, while others are not. First-year studies in dynamic high-pressure physics were stimulated by a practical need to obtain equations of state (Chap. 3) for a wide range of matters in a megabar range of pressures. Shock compression methods have been used to make a high number of measurements of the states of metals and most widely spread chemical compounds within a vast area of the phase diagram at pressures reaching several million and billion atmospheres, or even up to four billion atmospheres in unique experiments using atomic explosions [2, 3]. Wide-range semi-empirical equations of state [4] have been developed which describe with high precision the thermodynamic properties of most practically important matters in the field of polymorphic transformations, melting, evaporation, ionization and in the vicinity of a critical point. However, experiments with shock waves are characterized not only by a wide range of achievable pressures and temperatures but also by extremely high rates of their variation. These circumstances open unique opportunities for research in the physics

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Fortov, Intense Shock Waves on Earth and in Space, Shock Wave and High Pressure Phenomena, https://doi.org/10.1007/978-3-030-74840-1_4

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of phase and polymorphic transformations, physics of fracture, strength and plasticity [5–9]. At a later stage, we will use overviews from [5–9]. Usually, extreme states of matter are associated with high pressures and temperatures [1, 10] that are achieved in experiments as a result of matter compression in intense shock waves. Low duration of shock wave exposure is usually considered as a limitation of the method. On the other hand, low duration of mechanical loading makes it possible to implement exotic states of matter far from equilibrium and to study behavior of various materials at extremely high deformation rates. Studies of the temperature-rate dependences of the resistance to deformation and fracture of metals and alloys are intended to study the main regularities of the motion of carriers of plastic deformation of dislocations, to identify determinants and regularities of the formation and development of material damages. This information is required to optimize conditions of mechanical processing of metals and solve problems of high-velocity shock and punching. The results of researches in this scientific area are generalized in a number of overviews and monographs; [2–7, 11] can be mentioned as recent ones. Results are published for multiple measurements of flow stress dependence upon pressure in the conditions of shock compression (ref. [12] along with publications cited therein) and non-shock (quasi-isentropic) compression [13, 14]. New definition has been developed for a stress state of a material at various stages of elastic–plastic shock compression involving pulse X-ray diffraction diagnostics [5, 15, 16]. This chapter is dedicated to most exotic results. To discuss them, it is useful to remind some basics of modern concepts of the mechanisms of high-rate deformation and fracture of solid bodies. In terminology of the dislocation theory, the plastic deformation rate γ is defined by the average movement rate of mobile dislocations vd and their density N m interrelated by the Orowan ratio γ = bNm vd ,

(4.1)

where b is the Burgers vector. The average rate of mobile dislocations is a function of stress, temperature and concentrations of various defects preventing movement of dislocations, including dislocations themselves. Obviously, apart from dislocations, the mechanism of plastic deformations can have a significant contribution from twinning that is especially substantial for crystals with hexagonal close-packed (HCP) and body-centered cubic (BCC) structures. However, in most cases, it makes no radical changes in interpretation of rate and temperature dependences of flow stress at highrate deformation of metals. The limited integral information on the regularities of high-rate deformation obtained in experiments, as a rule, prevent from identifying the contributions of various types of dislocations. For these reasons, we will use the dislocation terminology in some averaged and simplified sense without devoting to details of the high-rate deformation mechanism inaccessible for modern experiments. Modern physical theories of plasticity consider also the so-called disclinations and

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73

rotary modes that are in some cases employed for describing large plastic deformations, however this is an excessive complication for the processes and phenomena discussed in the chapter. It is known that the flow stress of crystalline solid bodies rises as the loading rate increases. For many metals, this dependence abruptly rises as the deformation rate ∼103 –104 s−1 increases, which is interpreted as a consequence of changes in the dislocation movement mechanism [3, 17]. In case of low deformation rates, dislocations overcome Peierls barriers and other obstacles as a result of the combined action of applied stress and thermal fluctuations. Due to this, the temperature rise is accompanied by a drop in the yield strength of materials. For deformation at high rate, higher stresses must be applied. For sufficiently high deformation rate, existing stresses are so high that dislocations become capable of overcoming barriers and obstacles without the additional contribution of thermal fluctuations. In this case, the average rate of dislocations, v, becomes linear or close-to-linear function of the applied shear stress, τ, and is controlled by deceleration forces of various nature as per the ratio [18]: Bv = bτ,

(4.2)

where B is the coefficient of dynamic deceleration that includes contributions created by obstacles, electrons and photons. The interaction of moving dislocations with electrons is deemed substantial at low temperatures only. At normal and elevated temperatures, the coefficient of deceleration of phonons by gas Bp (phonon friction or phonon viscosity) can be represented [8] as a linear function of temperature: Bp =

k B T ω2D , π 2 c2

(4.3)

where k B is the Boltzmann constant value, ωD is the Debye frequency, c is the velocity of sound. A comprehensive discussion of the manifestations of phonon viscosity can be found in [19]. Since phonon viscosity is proportional to temperature, we can expect the linear rise of flow stress at extremely high deformation rates as the temperature rises [19]. For a sufficiently high stress called ideal or ultimate shear strength, the material must lose stability to shear stresses and can deform without any contribution of dislocations. The value τ id of ideal shear strength is proportional to the shear modulus G and is τ id ≈ G/10 − G/2π according to various estimations. Since the shear modulus falls down with temperature, the ideal shear strength must also decrease in case of heating. The ratio of the contributions of thermofluctuation and above-barrier mechanisms of dislocation movement depending on temperature and deformation rate is illustrated in Fig. 4.1. The second, no less important, parameter defining the resistance to plastic deformation is the total dislocation density. Figure 4.2 explains the dependence of the yield strength on the dislocation density. Defect-free crystals are characterized by highest

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Fig. 4.1 To explanation of the mechanisms of temperature–time dependences of yield strength

Ideal strength

Stress

Phonon friction Thermoactivation deformation

Quasi-statics, Hopkinson bars Deformation rate

Ideal strength

Yield strength

Fig. 4.2 Schematic representation of the dependence of the plastic flow stress of a crystalline body on the dislocation density [20]

Strength of metal “whiskers” Strength of technical metals Deformed metals and alloys Pure metals Dislocation density

yield strengths. Similar or close to them high-strength structural states are implemented in metal “whiskers” of micrometric thickness [20]. As dislocations appear, flow stress rapidly falls down to the minimum at critical density N c , and as the density rises further, dislocations start blocking each other which results in the flow stress increase. The density of mobile dislocations rises during plastic deformation. Unlike fracture in normal conditions that occurs by the propagation of one or several cracks, high-rate fracture at ultra-short duration of loading due to the insufficient time for information exchange occurs by simultaneous generation, growth and coalescence of multiple cracks or pores. Since the generation and growth have a specific rate, the magnitude of the fracture stress rises as the tensile rate increases. The values of fracture resistance implemented in this case represent a result of competition between the growth of applied tension stresses and their relaxation due to the generation and growth of discontinuities in the material. There is a maximum possible value of the fracture stress—ideal tensile strength determined by zeroing the derivative of pressure with respect to the specific volume on the matter’s isotherm.

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The value of ideal strength is found by the extrapolation of the equation of state into the area of negative pressures or by first-principle calculations of compressibility. The issue of possible polymorphic transformations in the area of negative pressures and their effect on fracture resistance remains practically open. The studies of the mechanical properties of materials in the sub-microsecond range of duration at deformation rates of > 103 c−1 are conducted in the conditions of shock wave loading of samples under test. The measurements are based on the fact that the structure of waves and the dynamics of wave interactions are defined, in addition to the thermodynamic processes described by the equation of state of matter, by the processes of elastic–plastic deformation and fracture in the material [4, 6, 21, 22]. A high number of such measurements are done globally using the methods of continuum mechanics and molecular dynamics, accompanied by computer modeling of shock wave phenomena [7]. Every two years, the American Physical Society holds large international conferences resulting in more than 400 reports being published. New research results weekly appear in leading international media. It is practically impossible to discuss or even mention all these papers in this chapter. Here, we mainly present the results of own experimental researches of the last decade that are pioneering to a certain degree, fall out from a number of routine measurements in this area, are partially debating and attract the attention of specialists of related sciences. We believe that all presented researches lie in the area of the primary development of this field, since setting the problem of studying the effects of structural factors, temperature and duration is natural and important. In the experiments under discussion, a flat shock wave is created in one way or another in a plate of the investigated material and its structure is measured at the outlet from the sample—usually by recording the free surface velocity as a function of time. The flat sample thickness defining the propagation time of the shock wave can vary between approximately 50–100 µm and 10 mm and more, and the free surface velocity profile is measured with a resolution of 1 ns (10−9 s) when using a modern experimental technique. Lately, it became possible to experiment with the samples of micrometric and sub-micrometric thickness, where the time resolution of measurements reaches a picosecond level (10−12 s). Free surface velocity profiles ufs (t) of the flat samples of magnesium alloy Ma2-1 of various thickness, measured [23] in experiments at room temperature are given in Fig. 4.3. A shock compression pulse was generated in samples by the impact of a plate the thickness of which was several times less than that of the sample. Due to abrupt increase in compressibility during the transition from elastic uniaxial compression to plastic compression, the shock wave loses its stability (ref. Chap. 12) and split into an elastic precursor which propagates at the velocity close to the longitudinal velocity of sound c1 and its trailing plastic shock wave whose velocity is determined by the volumetric compressibility of the material. Compression stress in the elastic precursor equals the Hugoniot elastic limit (HEL). With respect to the measured free surface velocity profile, the Hugoniot elastic limit is determined as σHEL =

ρc1 u HEL , 2

(4.4)

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Surface velocity, km/s

0.443 mm

10.78 mm

Time, ns

Fig. 4.3 Measurement results of free surface velocity profiles for the samples of magnesium alloy Ma2–1 [23]. Numbers near wave profiles indicate sample thicknesses

where uHEL is the free surface velocity behind the front of the elastic precursor, ρ is the material density. However, the term “Hugoniot elastic limit” is generally acceptable, it is not completely correct since the value of σ HEL is not constant. Reference to the comparison of wave profiles presented in Fig. 4.3 shows that despite a rise of parameters on each wave profile behind the front of the elastic precursor, stress on its front decreases as the wave propagates. Such attenuation of the elastic precursor is caused by the relaxation of stresses during plastic deformation directly behind the elastic shock wave. With the ratio of striker and sample thicknesses, used in experiments, the loading conditions near the free rear surface of the sample in Fig. 4.3 correspond to the beginning of the shock wave attenuation under the action of the rarefaction wave that overtakes it, the exit of which to the surface causes the velocity to fall. Only part of the rarefaction wave is recorded on the free surface velocity profile, which is limited by the material dynamic tensile strength. The thing is that the reflection of a compression pulse from the free surface gives rise to tension stresses inside the sample, which initiates its fracture—a spallation. This is accompanied by the relaxation of tension stresses and the formation of a compression wave (spallation pulse) the exit of which to the sample surface causes the second rise of its velocity. A decrement of the surface velocity ufs at its fall from the maximum to the value ahead of the front of the spallation pulse is proportional to the fracture stress value—spallation strength of the material in these loading conditions. A ratio used to determine the spallation strength with respect to the measured value ufs is obtained from the analysis of wave interactions by the characteristics method (ref. Chap. 1); for elastic–plastic bodies, the ratio is somewhat complicated by the need to take into account various wave velocities. Further variations of surface velocity are resulted from the multiple reflections of waves inside the spalled layer

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of the sample between its rear surface and fracture surface. The velocity oscillation period is determined by the spallation thickness and the velocity of sound. The shock wave technique is a powerful tool in studying properties of materials at extremely high deformation rates with well controlled loading conditions. The methodology of that scientific area is based on the link between the matter flow parameters recorded in experiments with the physical and chemical processes occurring in matter. Progress of research into the high-rate deformation, fracture, and physicochemical transformations in shock waves is largely related to the development of modern measuring techniques with high spatial and time resolutions of wave profiles [5, 12]. While the first experiments with shock waves included the successful measurement of wave and mass velocities for stationary shock waves only, the further development of methods for continuous recording of kinematic parameters enabled consideration of the structural details of compression and rarefaction waves and their evolution thereby radically increasing the informative value of dynamic research methods for material properties. The methods for analyzing wave profiles are now also well developed and described [2–7, 11]. To date, a great body of experimental data has been gained about the elastoplastic and strength properties of technical metals and alloys, geological materials, ceramics, glasses, polymers and elastomers, plastic and brittle single-crystals in the microsecond and nanosecond ranges of exposure duration. Considerable progress has been achieved in the development of methods for obtaining information about the kinetics of energy release in detonation and initiating shock waves. The experimental data are used to construct phenomenological rheological models of deformation and fracture, and macrokinetic models of physicochemical transformations, which are required for calculating the processes of explosions, high-velocity impact, and the interaction between high-power radiation pulses and matter. This chapter gives new and, in some cases, unexpected results of studies of polymorphic transformations and strength properties of metals and alloys at submicrosecond durations of shock wave loading. Varying the test temperature significantly expanded the range of studied processes and phenomena. Experiments with structural metallic materials and highly pure metallic single crystals demonstrated a number of specific features of high-rate (> 104 s−1 ) deformation and fracture in shock waves, which are extremely interesting in terms of physics of strength and plasticity. High pressures and temperatures reached in case of shock compression of solid bodies can cause phase transitions and polymorphic transformations. For some crystalline materials, including iron, polymorphic transformations were first found in the conditions of shock compression at high pressures. As for the systematic studies of phase diagrams of solid bodies, we must admit that low exposure time makes the shock wave technique less efficient in these terms than, for instance, the static technique for obtaining high pressures in diamond anvils [13]. On the other hand, it seems interesting and intriguing to study how a sufficiently complicated reconstruction of the crystalline structure can occur over so short periods. It is shown that in case of

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b

Sample

Striker

a

Fig. 4.4 a Scheme of shock wave generation in flat metal samples of armco iron and recording of wave profiles at the exit of compression pulse to the free surface of the sample (ARMCO—American Rolling Mill Corporation). Armco iron is technically pure iron containing an extremely low amount of carbon and other impurities; b evolution of the shock compression pulse in a fluid-like material that has not endured physical and chemical transformations

shock compression, structural rearrangement in solid bodies can occur over 10−7 – 10−9 s and less. We can hope that the answer to this question will be given by the studies of polymorphism of materials in various structural states. This chapter also discusses the possibilities and observation results of exotic overheated solid-body states, premelting and polymorphic transformations in the tensile field. High attention has been paid to the behavior of brittle materials during the past decade. The chapter presents some new data on the behavior of glasses and ceramics at shock compression beyond routine studies. Specific fracture waves have been found at shock compression of glasses, methods have been developed to diagnose the nature (brittle fracture or plastic deformation) of the behavior of highly solid materials under compression. Research Methods. There are two methods to get information on ratio between the plastic deformation rate and flow stress, based on measurements of elastic precursor attenuation [24, 25] and measurements of plastic shock wave width [26, 27]. Highrate fracture is studied by the analysis of spallation phenomena [28, 29]. These research methods are briefly described below followed by the presentation of the most interesting results. To get additional information on the macro-kinetic principles of high-rate deformation and fracture, methods of computer modeling of shock wave experiments are used with various hypothetic models and determinant ratios. This area requires further discussion and is not considered in this book. The organization of shock wave measurements is schematically shown in Fig. 4.4a. Impact loading pulses (Fig. 4.4b) are created in flat samples of studied materials by flat metal strikers accelerated by explosive devices [5, 12], pneumatic barrel units (“gas guns” [6]) or by exposure to highly intensive pulse laser or corpuscular radiation [14, 15]. The ratio between striker and target sample thicknesses and diameters is selected so as to ensure strict unidimensionality of the shock wave process during the entire measurement period. The change in the impact velocity (from hundred meters per second to several kilometers per second) varies the maximum pressure

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of shock compression while the striker thickness varying from 50 µm to 5–10 mm determines the duration of the generated compression pulse. In this manner, the discussed measurements have been carried out within the range of shock compression pressures to several dozens of gigapascals with durations of 10−6 to 10−8 s. The objects of measurements and analysis are the shock compression wave, followed by the rarefaction wave (Fig. 4.4b), and also the wave interactions during the reflection of the compression pulse from the free rear surface of the test sample. The processes of structural transformations, plastic flow and fracture are associated with changes in matter compressibility and, therefore, appear in the structure of compression and rarefaction waves. The experiments involve continuous recording of the velocity profiles ufs of the free sample surface, for which purpose the laser Doppler velocity meters VISAR [16] or ORVIS [17] with subnanosecond time resolution are used. It became possible during the last decade to have such measurements within the test temperature range up to 700 °C [18] and in some cases even higher. Moreover, when manganin sensors [5] were used in the experiments being discussed, shock compression pressure profiles were recorded in the internal cross-sections of samples. Typical examples of measurement results are given in Fig. 4.5 that shows the velocity profiles of the free surface of armco-iron samples loaded by the impact of an aluminum plate at temperatures of 20–600 °C [8], and the pressure profiles measured in the internal cross-sections of samples [19]. The velocity profiles exhibit the moments at which three compression waves sequentially reach the sample surface. Owing to the increase in longitudinal compressibility with the transition from elastic to plastic deformation, the shock wave loses its stability (Chap. 12) and splits into an elastic precursor and the following plastic compression wave. At a pressure of ∼13 GPa, iron endures polymorphic transformation α → ε: (body-centered cubic lattice (BCC) → hexagonal close-packed lattice (HCP)) with a decrease in the specific volume, as a result of which the plastic compression wave in this pressure range is split into two. The pressure behind the front of the first plastic shock wave corresponds to the onset of the transformation, while its attenuation and compression rate in the second plastic wave are determined by the kinetics of the structural transformation. Figure 4.4b shows full pressure profiles in iron, obtained using manganin sensors [19]. These experiments were the first where the rarefaction shock wave was observed, which is formed due to the reverse structural transformation of iron into the low-pressure phase. After the shock wave circulation in the striker, a rarefaction wave is formed, which then propagates through the sample following the shock wave. The arrival of the rarefaction wave at the sample surface results in a decrease in the surface velocity. The reflection of a compression pulse from the free surface gives rise to tension stresses inside the sample. The material fracture (spallation) under the action of tension is accompanied with stress relaxation and gives rise to the appearance of a compression wave which arrives at the surface in the form of the so-called “spallation pulse” to increase its velocity

80

Surface velocity, km/s

a

Polymorphic transformation

Armco iron, 2.46 mm Time, μs

b Armco iron, 10 mm

Compression stress, GPa

Fig. 4.5 Examples of the measurement results of pressure and velocity wave profiles. a Free surface velocity profiles for the flat samples of armco iron 2.46 mm thick after an impact by an aluminum plate 2 mm thick at a velocity of (1.9 ± 0.05) km s −1 . Measurements were carried out using VISAR laser Doppler velocity meter at normal and high temperatures (indicated near the respective curves). The shock compression pressure was equal to 19 GPa. HEL is the Hugoniot elastic limit; b pressure profiles in shock compression pulses generated in armco iron samples by the impact of aluminum plates at a velocity of 1.05 km s−1 and 2.06 km s−1 [19]. Measurements were carried out using manganin pressure sensors

4 Shock Waves in Condensed-Matter Physics

Time, μs

once again. Measurements of the resistance to spallation fracture provide information about the strength properties of a material at submicrosecond load durations. The longitudinal stress at the front of the elastic precursor or the Hugoniot elastic limit is given by σxe = 0, 5u fse ρ0 c1 , where ufse is the free surface velocity surge in the precursor, ρ 0 is the initial material density, and c1 is the longitudinal velocity of sound in the material; the stresses are assumed to be positive. The elastic limit in case of a one-dimensional deformation is related to the yield strength σ T , in the ordinary sense of the word, by the equation [5, 10]   cb2 3 σT = σxe 1 − 2 , 2 c1

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√ where cb = K /ρ is the “bulk” velocity of sound and K is the volumetric elasticity modulus. Additional information on the behavior of a material in case of high-rate deformation is given by the analysis of elastic precursor attenuation as it propagates and by the analysis of the structure of a plastic shock wave. Analysis of spallation effects during the compression pulse reflection from the free surface of a body makes it possible to determine the value of fracture stress (spallation material strength) at submicrosecond load durations with respect to the measured profile of the free surface velocity ufs (t). The spallation strength σ sp is determined by the velocity decrease ufs from the maximum value to the value ahead of the front of the spallation pulse [5, 6]. In the linear (acoustic) approximation, the simplified formula for determining the spallation strength has the following form σsp =

1 ρ0 cb (ufs + δ), 2

where δ is a correction for the velocity profile distortion due to the difference between the velocities of the front of the spallation pulse (c1 ), which propagates through the tensile material, and the velocity of the plastic part of the incident unloading wave in front of it (cb ) [6, 20]. The calculation of the value σ sp to take into account the nonlinearity of material compressibility is given using equation-of-state extrapolations to the negative-pressure domain. In this manner, the experimental technique developed today and used in researches of shock wave physics in condensed media gives various information on structural transformations and strength properties of materials at load durations of 10−8 –10−6 s, deformation rates of 104 –108 s−1 and within a wide range of temperatures. Temperature Effects on Yield Strength of Metals at Shock Compression. The importance of researching into the processes of inelastic deformation of solid bodies in case of shock wave loading is defined as a unique possibility of researches in the field of strength and plasticity physics at the highest and reliably measurable deformation rates and as an ability to satisfy various practical needs not limited by shock actions only. Recording of the compression wave structure gives the necessary information on the dynamic limits of elasticity of solid bodies in the microsecond range of load durations. Figure 4.6 gives the results of processing the velocity profiles of the free surface of armco-iron samples given in Fig. 4.4a. For comparison, the data of quasi-static tests is given. It is known that deformation resistance of crystalline solid bodies rises as the loading rate increases, which is explained by a limited number of “carriers of plastic deformation”—dislocations and their rates. As Fig. 4.6 shows, in case of armco-iron, the submicrosecond yield strength is about 1.5 times higher than the quasi-static yield strength. In both cases, the yield strength falls down when heated, but the yield strength drop occurs slower in the conditions of high-rate compression. It is known that for many metals, the dependence of the flow stress on the deformation rate abruptly rises as the deformation rate ∼103 –104 s−1 increases, which

Fig. 4.6 Change in the elastic precursor amplitude (HEL) and the dynamic yield strength (σ m ) of armco-iron together with temperature according to the results of processing the wave profiles shown in Fig. 4.4. The dash-dotted line shows the data of quasi-static tests

4 Shock Waves in Condensed-Matter Physics

Yield strength, GPa

82

Shock compression

Low-rate deformation

Temperature, T °C

is interpreted as a consequence of changes in the dislocation movement mechanism [21]. In case of low deformation rates, dislocations overcome obstacles as a result of the combined action of applied stress and thermal fluctuations. Due to this, the temperature rise is accompanied by a drop in the yield strength of materials. For deformation at high rate, higher stresses must be applied. At deformation rates above a certain threshold (∼104 s−1 for pure metals), the acting stresses turn out to be rather high to overcome obstacles without an additional contribution of thermal fluctuations. Phonon viscosity becomes the dominating mechanism of dislocation deceleration. Since phonon viscosity is proportional to temperature, we can expect flow stress to rise at extremely high deformation rates as the temperature rises [22]. The velocity profiles of the free surface of single-crystal aluminum samples, measured in the conditions of shock wave loading at various temperatures are given in Fig. 4.7 [23, 24]. The figure shows that as the temperature rises, the elastic precursor amplitude increases exponentially, i.e. the dynamic yield strength of the material rises. The precursor shape also changes: as the temperature increases, a characteristic peak appears on its front, which is most prominent in case sample thicknesses are low. This precursor shape is usually related with the accelerating relaxation of stresses [7]. In experiments with relatively thick samples (Fig. 4.6a), the time for the growth of parameters in a plastic shock wave (from 0.1 to 0.9 of its amplitude) increases from 4–6 ns at the room temperature to 12–16 ns near the melting temperature, which corresponds to the change in the average deformation rate from ∼7 × 106 s−1 to ∼3 × 106 s−1 . The measurement results of the dynamic yield strengths of single crystals are summarized in Fig. 4.7 as compared with similar data for the aluminum alloy AlMr. Wave profiles are processed taking into account dependences of the elasticity coefficients of single-crystal aluminum on temperature. In contrast to what is observed in quasi-static conditions, the results of shock wave measurements demonstrate an increase as the temperature of the aluminum dynamic yield strength rises. The

4 Shock Waves in Condensed-Matter Physics

a

Velocity, km/s

Fig. 4.7 Velocity profiles of the free surface of samples of single-crystal aluminum of various thickness. The test temperature is shown by numbers near the respective wave profiles. a Results of experiments with flat strikers [23]; b generation of short compression pulses by an ion beam [24]

83

Time, μs

b

m Velocity, km/s

m m

Time, ns

dynamic yield strength near the melting temperature is at least 4 times higher than that at the room temperature. Since the shear modulus in this temperature range decreases approximately twice, then, accordingly, the ratio between the yield strength and the shear modulus increases almost by an order of magnitude. For comparison, we shall note that in quasi-static conditions, the ratio between the stress at the start of plastic deformation in the single crystals of aluminum and the shear modulus is kept almost constant within the temperature range of 300–600 K, but then decreases 1.5–2 times when temperature further rises to 900 K [25]. The plastic deformation of crystalline bodies is carried out by movement of dislocations. The equation associating the movement velocity v of dislocations with the active shear stress τ and friction forces looks as follows [25, 26]: Bv = bτ,

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where the friction force is in the left part, and the projection of external load on the slide vector belonging to the unit length of dislocations is in the right part, B is the friction coefficient, b is the Burgers vector. The higher is the friction coefficient, the higher stress is required to ensure this movement velocity of dislocations and, hence, this deformation rate. Without analyzing possible mechanisms of high-rate deformation in details, let us compare the observed effect of temperature on the dynamic yield strength with the contributions of various factors to the deceleration of dislocations. The movement of dislocations is decelerated by various obstacles and by friction forces caused by electrons and phonons [25, 26]. The interaction of a moving dislocation with electrons is low and plays an important role at low temperatures only. The phonon friction coefficient Bp within a high temperature range rises in a linear manner together with temperature [26]: Bp =

k B T ω2D , π 2 c3

where k B is the Boltzmann constant value, ω0 is the Debye frequency, c is the velocity of sound. Molecular-dynamic modeling [29–31] proves that the dependence of the coefficient of dislocation deceleration on temperature is close to linear. The resistance exerted by obstacles is obviously proportional to the concentration of these obstacles in the crystal structure and, therefore, it changes with the latter. In particular, the thermodynamically equilibrium concentration of point defects in a crystal grows exponentially with temperature [32]:  HF , Cd = Aexp − kB T 

where H F is the enthalpy of defect formation. The data in Fig. 4.8 show that the dynamic yield strength of aluminum almost linearly changes with temperature, which is reasonably consistent with the dependence of the phonon friction coefficient on temperature. In this manner, it seems probable that the deceleration of dislocations in the conditions of high-rate deformation in shock waves is mainly related with the thermal oscillations of atoms. On the other hand, molecular dynamic calculations showed [33] that thermal fluctuations reduce the shear stress necessary for the formation of dislocations, which must result in reduced yield strength with heating for an ideal defect-free crystal. Figure 4.8 shows also the results of similar measurements for aluminummagnesium alloy AMg6M [27]. Alloy AMg6M was selected for tests because its yield strength in normal conditions is close to that observed for single-crystal aluminum near the melting temperature. The above data show that the dependence of the dynamic yield strength of the alloy on temperature has a weak minimum and the values of alloy yield strengths and single-crystal aluminum coincide at maximum temperatures.

Fig. 4.8 Precursor stress (elastic limit during uniaxial deformation) and dynamic yield strength of single-crystal aluminum [24] and aluminum alloy AMg6M [27] depending on the test temperature

Yield strength, GPa

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85

Precursor stress

АМg6М

Al-single crystals

Fig. 4.9 Dependence of the dynamic yield strength of highly pure titanium and titanium alloys VT1-0 and Ti6–22-22S at the front of elastic precursors on the test temperature [28]

Yield strength, GPa

Temperature, °C

Titanium VT1-0

Temperature, °C

As we believe, this confirms the determinant contribution of phonon friction to the high-rate deformation resistance at high temperatures. Figure 4.9 compares the dependences of dynamic yield strengths on temperature for titanium and its two alloys, corresponding to average deformation rates in elastic– plastic compression waves of about 3 × 105 –106 s−1 [28]. The measurement results show the abnormal rise of the dynamic yield strength in case of the shock compression of soft highly pure titanium while the behavior of highly strong alloys is similar to what takes place in regular conditions at low deformation rates. In general, this does not contradict the assumption that the primary mechanism of deceleration of dislocations changes at high deformation rates. In a pure metal, the flow stress is low and comparable to phonon friction forces so their increase together with temperature makes a solid contribution to the deceleration of plastic deformation carriers—dislocations. To increase flow stress in alloys,

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multiple obstacles are introduced on purpose, such as inclusions and interface boundaries. The stresses necessary to overcome such large obstacles significantly exceed the phonon friction forces. The comparison of the results of shock wave tests of the Ti-6–22-22S alloy and the values of the yield strength at lower deformation rates [34] shows that all data correspond to the unified logarithmic dependence of the yield strength on the deformation rate within 10−4 to 105 s−1 . The preservation of unified dependences of the alloy flow stress on temperature and deformation rate indicates the efficiency of the thermoactivation mechanism of overcoming obstacles by dislocations at deformation rates at least up to 105 s−1 . The difference in the dependences between high-rate deformation stress and temperature explains why alloys in the conditions of high-rate deformation are more prone to the loss of stability and deformation localization in adiabatic shear stripes than pure metals. The abnormal rise of the dynamic yield strength during heating was also observed in experiments with the shock wave compression of uranium [35] and crystals of potassium chloride [36]. Uranium [35] and heat-resistant super-alloys [37] showed a non-monotonous change having a prominent peak in the area of structural transformations, with the heating of the dynamic yield strength. Approaching Ideal Strength of Condensed Matter. Modern experimental methods for shock waves allow for researching into the properties of materials at negative pressures of up to − 15 to – 20 GPa and more [6, 24, 38–40]. The term of “negative pressure” sounds absurd when applicable to gases where pressure has the same physical nature as temperature and is defined by the kinetic energy of molecules. In condensed media, pressure is one of the manifestations of the interatomic forces and can be no-zero at zero absolute temperature. Unlike gases, solid bodies and fluids have the final specific volume at zero pressure and have both compression and tension stress resistance. This allows for negative pressures in condensed matters. The transition to the area of negative pressures in itself is not accompanied by any qualitative changes in the state and properties of matter. However, states with negative pressures are always metastable with regard to the fluid/gas or solid body/gas mixture. Metastable states are thermodynamically stable relative to slight disturbances. The transition to a stable state is accompanied by the formation of nucleus bubbles or pores, which suggests overcoming of the energy barrier [41]. Another important specific feature of negative pressures is a natural barrier of their magnitude. Matter compression is accompanied by almost unlimited rise of pressure. Tensioning quickly leads to an absolute loss of mechanical stability of condensed state at the point where the pressure derivative with respect to volume turns zero. The surface in the pressure–temperature-specific volume coordinates, where the derivative (dp/dV )T = 0 is referred to as the spinodal, and the respective values of tension stresses (negative pressures) as “ideal strength”. The states of matter corresponding to its spinodal are unachievable: matter disintegrates near the spinodal quicker than local thermodynamic equilibrium can be reached. The states of matter with negative pressures are not as exotic as it might seem. For example, parts of bridges are in tension for long periods. Negative pressures in fluid are created in nature by capillary forces that make water go up in the pores

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and cracks of rocks or reach the tops of tall trees [42]. Current experiments allow for creating tensions of condensed matters, at which negative pressures reach − 20 GPa and more for solid bodies and − 100 MPa and more for fluids. Therefore, the issues of the equation of state, phase transitions and polymorphic transformations in the area of negative pressures become topical. Obviously, the issues of condensed matter discontinuity (fracture under the action of tension stress) are burning. These issues are actively discussed in the literature of recent years. We can talk about the formation of the physics of negative pressures, having its own specific tasks. In this connection, it’s worth noting that prominent achievements in high-pressure physics were obtained at the stage when measurements were possible only within a limited range—up to 4 to 10 GPa [43]. The metastable state with negative pressures is destroyed by the origination and growth of nucleus bubbles or pores that occur due to thermal fluctuations of density (homogeneous nucleation) [41, 44, 45] or thanks to impurities and structural irregularities in matter. Nucleus bubbles or pores of small size are irreversible. They can close or even disappear under the action of surface tension forces. Nuclei with a radius exceeding the critical radius r c are unstable and grow with the release of free energy. The nucleus critical radius decreases rapidly as it moves to the area of negative pressures. As a result, the expression for the threshold pc of the formation of bubbles or pores with a size exceeding the critical limit can be represented as follows [46]:  pc (T ) = −

16π γ 3 3kT ln (J0 V t)

1/2 ,

where γ is the surface tension. It follows therefrom that large values of negative pressures can be obtained only in a small volume V and during a short period of time t. An exhaustive discussion of metastable states with negative pressures and an extensive reference list can be found in papers [47, 48]. Significant negative pressures generated in the samples of solid bodies or fluids when a shock compression pulse is reflected from their surface are achievable for measurements in experiments with shock waves of submicrosecond duration. In this case, it is possible to vary the duration of load and temperature within wide limits and, therefore, it becomes possible to get primary information about the kinematics of phenomena. At present, it becomes possible to take measurements at the level of tension stresses comparable to the ultimate or “ideal” strength of condensed matter, which defines the upper boundary of the possible resistance to fracture. It is important to underline that shock wave measurements are not complicated by the use of any a priori models of the phenomenon and are based only on the fundamental conservation laws and the equation of state of matter. The most natural area of research of high-rate tensioning is apparently the elucidation of dependences of fracture resistance on time or rate of tensioning, temperature and structural state for materials of various classes. Figure 4.10 gives measurement results for the spallation fracture resistance of molybdenum in various structural

88

Ideal strength

Spallation strength, GPa

Fig. 4.10 Measurement results for dependence of the resistance to spallation fracture (spallation strength) of single crystals on the deformation rate for molybdenum of various orientations, deformed single crystals and polycrystal molybdenum [38]

4 Shock Waves in Condensed-Matter Physics

Single crystals Deformed single crystals Polycrystals Quasi-statics

Deformation rate

V –1 ,c V0

states depending on the deformation rate [38]. The deformation rate means the rate of matter expansion in a rarefaction wave determined as u fsr V =− , V0 2cb where ufsr is the measured rate of free surface velocity fall of the test sample in the unloading part of the shock compression pulse. Whereas in reality the tensile rate in case of the interaction of counter running rarefaction waves varies greatly, this presentation of experimental data has an advantage of being used for evaluation of fracture kinetics. It was shown [5, 6] that the initial growth rate of the relative volume of discontinuities in case of spallation with the accuracy up to the constant multiplier ∼2–4 equals the rate of matter expansion in the unloading wave, calculated in such a manner. The fracture resistance values implemented in case of spallation represent a result of competition between the growth of tension stresses during wave interactions and their relaxation due to the generation and growth of discontinuities in the material. Since fracture being a kinetic process of the nucleation, growth and coalescence of discontinuities cannot occur immediately, an increase in the rate of load application allows creating ever higher excess stresses in the material. As a result, fracture resistance within the microsecond range of load durations becomes almost twice as high as the strength in case of low-rate tensioning. As the loading pulse duration falls down, the spallation strength rises and becomes compared with the ideal strength in the microsecond range. In its turn, the bigger is the achieved overstress in the material, the smaller and more numerous fracture centers are activated and give contributions to the process rate increase. Due to the short duration of the load on the sample (which limits the possibility of data exchange and stress concentration in the crack top) and high overstresses in

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it as the tensile rate increases, the fracture mechanism changes: instead of fracture due to the growth of single catastrophic cracks in the submicrosecond range, an evolution of dissipated fractures takes place. The most probable centers of fracture nucleation are structural discontinuities in the body. Single crystals are free from inner defects that could become centers of heterogeneous formation of discontinuities and therefore they demonstrate a higher strength. On the other hand, polycrystalline metals and alloys contain multiple boundaries of grains, particles of the impurity phase, micropores and other material defects which are potential centers for the initiation of discontinuities under tension and therefore reduce the overall fracture resistance. The difference in the values of the spallation strength of single crystals and polycrystals for aluminum [24], copper [49, 50] and molybdenum [38] can be two or three-fold. Homogeneous nucleation of discontinuities or initiation of fractures in the areas of accumulated dislocations and other microdefects in polycrystalline materials are unlikely, simply because loading due to fracture initiation at larger defects prevents from achieving the required level of tension stresses. Figure 4.11 represents the experimental results for copper within the nanosecond range of shock wave loading duration. The samples were highly pure copper in single-crystal and poly-crystal states, a copper single crystal with 0.1% silicone and Cu single crystal with 0.1% silicone subjected to heat treatment at 1030 °C in copper oxide for 24 h. It is known that silicon with copper form a solid substitutional solution the retention of which in Cu2 O powder at high temperature results in the formation of SiO2 dissipated particles 350 nm in size in a single-crystal matrix [51]. The measurement results given in Fig. 4.11 show a clear difference in the values of spallation strength (proportional, as discussed above, to the rate decrement ufs ) and the nature of spallation fracture depending on the structural state of test samples. A single crystal of pure copper has the highest strength and is characterized by a rapid completion of the fracture process. The surface velocity of a spalled plate oscillates due to wave reverberation, but its average value quickly becomes constant, which means that the plate deceleration stops 5 × 10 ns after the spallation starts. Polycrystalline copper has the lowest fracture resistance but the fracture process develops slowly. In this case, the link between the spalled layer and the remaining part of the sample is preserved during a relatively long period, which follows from the observed prolonged deceleration of the spallation plate. A single crystal of the solid solution of Cu + 0.1% Si has a somewhat lower strength than a pure copper single crystal and a viscous prolonged nature of fracture. The formation of fine SiO2 particles is associated with the further reduction of the resistance to spallation fracture and its acceleration. The chart in Fig. 4.11b shows that copper crystals with fine SiO2 inclusions behave abnormally: their spallation strength is reduced as the tensile rate rises. It is obvious that the abnormal behavior of spallation strength in the latter case is related with the initiated fracture on brittle inclusions. The resistance to spallation fracture of relatively thick (2.5 mm) single crystals of the solid solution of Cu + 0.1% Si and single-crystal copper with fine SiO2 inclusions is the same, significantly exceeds the strength of polycrystals and is close to the spallation strength of the single crystals of pure copper. In these experiments, fracture took place 0.3 mm

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a

Surface velocity, m/s

Copper single crystals

Copper polycrystals

Time, ns

Spallation strength, GPa

b

Single crystals

Copper polycrystals

Deformation rate

V –1 ,c V0

Fig. 4.11 Experimental results with copper in various structural states. a Profiles of the free surface velocity of single-crystal samples of copper 0.2 mm thick, copper with 0.1% silicone 0.5 mm thick and copper with inclusions of silicone oxide 0.5 mm thick as well as a sample of high-purity polycrystal copper 1 mm thick. The thickness of aluminum strikers varied from 0.05 to 0.2 mm in proportion to the thickness of the test sample; the impact velocity was (1.2 ± 0.05) km/s and (0.66 ± 0.03) km/s, respectively; b dependence of copper spallation strength in various structural states on deformation rate. The data for single crystals of pure copper are taken from [49, 50], the data for polycrystal copper—from [5, 50] and (upper point) [52], the data for the solid solution of silicone in copper and for copper with inclusions of silicone oxide are the results of the measurements taken by the authors

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Epoxy resin

σ sp σ id

Plexiglass

Water

Deformation rate

, s-1

Fig. 4.12 The degree of realization of the ideal spallation strength σ id for homogeneous materials (single crystals, amorphous polymers and fluids) as a function of the deformation rate

from the sample surface. As the deformation rate increases thrice and the spallation thickness decreases thrice (sample thickness of ∼0.5 mm), the spallation strength of the solid solution rises approaching the strength of single-crystal copper while the strength of copper crystals with SiO2 particles falls down approaching the strength of polycrystalline copper. It is hard to explain this abnormality within any kinetics of fracture. Its most probable reason seems to be the dependence of the concentration and size of particles (inclusions)—centers of fracture—on the distance to the surface. Taking into account the contribution of diffusion processes to the formation of oxide inclusions in case of annealing solid solution samples surrounded by Cu2 O, it is quite natural to expect that the size of the inclusions decreases and the material strength, accordingly, increases as the distance from the surface rises. In this manner, the experimental data unambiguously demonstrate the effect of structural factors on the resistance to high-rate deformation and fracture. It seems useful to formalize the observation results by introducing a concept of material defect spectrum—potential centers of fracture characterized by various levels of stress necessary to activate them. It is interesting to compare the measured values of the spallation strength of homogenous condensed media with the maximum possible values of tensile strength. Figure 4.12 represents dependences of the normalized values σ sp /σ id of the spallation strength σ sp of metal single crystals, amorphous polymers and fluids from the deformation rate. The values of ideal strength σ id were evaluated as pressure in the minimum of shock adiabat of matter, extrapolated to the tensioning area [53]:

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σid =

ρ0 c02 , 4b

where c0 , b are coefficients of linear expression for shock adiabat in the form of U s = c0 + bup (U s is the shock wave front velocity, up is the massive matter velocity behind the front [1, 10]). As the comparison with the results of first-principles calculations shows, the error of such evaluation of σ id is about 20% and it tends to uprate σ id . The values of spallation strength σ sp are taken from [24] for aluminum (σ id = 13.4 GPa), [49, 50] for copper (σ id = 23.3 GPa), [38] for molybdenum (σ id = 55 GPa), [54] for iron (σ id = 31.6 GPa), [55] for plexiglass and epoxy resin (σ id = 1.39 GPa and 1.34 GPa, respectively), [56] for water (σ id = 0.28 GPa). Although the measured values of the spallation strength of these materials differ by more than two orders of magnitude, in the normalized coordinates the data scatter is not so high. The data in Fig. 4.12 suggest that up to 30% of the ideal strength of condensed matter is implemented for nanosecond load durations. The FCC-structured single crystals of plastic copper and aluminum exhibit a somewhat higher degree of realization of the ideal strength than iron and molybdenum which possess a BCC crystal lattice. This is supposedly due to the possibility of higher stress densities in the neighborhood of microdefects for BCC-lattice metals with higher yield strengths. The degree of realization of the ideal spallation strength for amorphous polymers and fluids is at least no smaller than for metals, which is probably explained by their higher structural homogeneity. It should also be noted that in media with low or zero yield strength, stresses cannot concentrate in the vicinity of microscopic defects. The difference in the degree of realization of the ideal strength becomes smaller as the load duration becomes shorter. In recent years, a number of measurement results for fluid strength in the conditions of static and dynamic tensions is published. The highest water densities (up to 140 MPa [57]) are obtained in cavities from several microns to several dozen microns. Such cavities are obtained in quartz during the crystal-growing process. Cavities are partially filled with fluid and contain a vapor bubble. Negative pressures in fluid are created by heating until the vapor bubble disappears and further cooling far below the saturation curve. Among other data, the highest values of strength (39–48 MPa) are obtained by the spallation method [56]. The reasons that make possible the high strength realization in microcavities at high temperatures remain incomprehensible. Possibly, water interaction with cavity walls in the quartz crystal plays some part. Possible contribution of dissolved salts is also discussed. In these terms, shock wave measurements have an advantage that wall effects can be excluded, the composition of the studied fluid can be strictly controlled, and temperature, tensile rate and pressure of previous shock compression can vary within wide limits. Currently, new problems appear, related, in particular, with the future energy sector technologies that suggest the use of laser-induced thermonuclear fusion [3, 4, 11], and problems of cavitation become urgent in the conditions of pulse energy emission. It can be expected that this circumstance will attract the attention of researchers to a more detailed study of the strength properties of fluids, including metal melts.

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Fig. 4.13 Example of calculating the fracture of a stretched crystal using the molecular dynamics method [62]. Layer inside a cubic sample is shown

The fracture mechanism and kinetics at stresses close to ideal strength are studied theoretically using molecular dynamics methods (ref., for example [58–64]). It has been found that the spontaneous formation of fracture is in many cases preceded by the formation of areas with disordered crystalline structure where pores are formed. In [58, 64], this effect was identified as melting; the melting temperature at negative pressures becomes lower than in normal conditions. The fracture nucleation is caused by density fluctuations and has a probabilistic nature as a result of which the average waiting time for the appearance of discontinuity in this volume of the material is defined by both active stress and the volume. The frequency of occurrence of discontinuities rises as the tensile rate and, therefore, overstress in the material increase. In case of polycrystalline metals, the calculations [59, 60] proved the trend to the predominant nucleation of fractures at boundaries of grains, although the difference in fracture stresses for polycrystals and single crystals at lower deformation rates is not so large as compared to that observed in experiments. The regularities of growth and coalescence of pores [63, 64] have been studied, which is important for building kinetic models of fracture [64]. As an example, Fig. 4.13 shows one of the results [62] of fracture modeling using the molecular dynamics method. An increase in the resolution capacity of shock wave measurements and movement to the area of more and more short-term loads, on the one hand, and the sophistication of atomistic models of fracture, on the other hand, draw nearer the possibility of direct comparison of experiment and theory. Progress in computer experiments creates a real basis for the transition from simplified representations of the determining factors and basic mechanisms of fracture such as accumulation of thermofluctuation breaks of links [65] or incubation time of fracture [66] to a more real and prognostic theory of the phenomenon. Currently, the analysis of states of matter when a shock compression pulse is reflected from the surface is done using the extrapolation of data concerning equations of state [4, 11] obtained in case of high compression into the tensile area. A more

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reliable way to objectively evaluate the equation of state of matter at high negative pressures (up to spinodal) are first-principals calculations (ref. for example, [67, 68]). For example, the zero isotherm of aluminum was calculated [67] under the theory of density functionality with generalized gradient corrections, which does not contain any specific features and has a minimum at a pressure of − 11.2 GPa. Extrapolation of the shock adiabat of aluminum gives a close value of pressure at the minimum, which equals − 13.4 GPa. Paper [69] offers a new method to measure the velocity of sound in condensed matters in case of their shock compression, unloading and further tension, based on the analysis of wave reverberation in a flat sample one of the surfaces of which is in contact with a high-impedance material. This method is used to find dependences of the velocity of sound in single-crystal zinc on compression/tension stress within − 2 to 13 GPa. The lower boundary of that range corresponds to ∼25% of the maximum possible value of tension stresses in a zinc crystal of this orientation. A shock wave with a rectangular pressure profile was introduced into a thin sample of zinc towards crystalline axis 001 through a thick intermediate molybdenum screen. A shock wave goes to a free surface of the target and is reflected from it as a simple centered rarefaction wave. The shock-compressed substance is unloaded in that wave. Due to the difference in dynamic impedances on the surface of the striker contact and target, reflection takes place once again. Since the striker has a higher impedance, the wave is reflected from the contact surface keeping the same sign [5], i.e. the reflected wave is a rarefaction wave. Since tension stresses are impossible on the contact surface between the striker and the target, the striker influence on the wave process stops after the pressure in it falls to zero. The remaining part of the rarefaction wave is reflected from the formed free surface as a compression wave. The process of wave interactions becomes easily analyzed and interpreted [69], which results in obtaining the values of sound velocities. The experiments conducted have shown that the velocity of sound within the measurement error of ± 1% in the tension area corresponds to the extrapolation of its dependence on stress in the area of compression. Studies of Polymorphic Transformations in Case of Shock Compression. The issue of the mechanism and kinetics of high-rate transformations is apparently the most interesting fundamental problem of the polymorphism of solid bodies in the conditions of shock compression. In this connection, the behavior of materials is studied in various initial structural states within a wide temperature range. We will consider in details the results of experiments with iron, given in Fig. 4.14a. A decrease in the divergence between waves at elevated temperatures is caused by a decrease in transformation pressure (and, therefore, in the velocity of the first plastic shock wave) and, consequently, a rise of pressure increment in the second plastic wave (and, therefore, its velocity). The values of pressure and compression ratio of iron after the first and second plastic shock waves at various temperatures, calculated upon the results of measurements of free surface velocity profiles are given in Fig. 4.14a. It is seen that the total volume decrement in case of shock compression to the final pressure remains almost

4 Shock Waves in Condensed-Matter Physics

a

Pressure, GPa

Shock adiabat

b

Relative specific volume,

0

γ (FCCL) Temperature, K

Fig. 4.14 Results of studies of polymorphic transformation of iron in shock waves at various temperatures. a Pressure and compression ratio of iron after the first and second plastic shock waves at various test temperatures; b phase diagram of iron. Continuous lines show the data of Johnson et al. [70]; measurement results at static compression are taken from the guide book [71]; points show processing results of wave profiles; α (BCCL)—body-centered cubic lattice, γ (FCCL)—face-centered cubic lattice, ε (HCPL)—hexagonal close-packed lattice

95

Statics α (BCCL)

Shock compression ε (HCPL)

Pressure, GPa

unchanged as the temperature rises. The results of measurement of transformation pressure α → ε at various test temperatures are summed and compared with literature data in Fig. 4.14b. Our measurements show a greater drop of transformation pressure with heating than it follows from the data [70]. Since an almost linear dependence of the transformation pressure on temperature is retained in the studied temperature range, it is likely that the α → ε transformation takes place within the entire studied range, and a transition to the FCCL-structure occurs at temperatures above 590 °C. According to compression wave steepness shown in Fig. 4.14a, the transformation time α → ε in iron decreases as the temperature rises. This is partially related with an increase in the amplitude of the second plastic compression wave as the transformation start pressure falls down. More systematic studies of the effect of temperature on the rate of polymorphic transformation in case of shock compression, conducted for potassium chloride crystals [36] showed, contrary to expectations, some slowingdown of the transformation with heating. Since, along with the slowing-down of the

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Single crystal 1.69 mm

Surface velocity, km/s

Pyrographite 4.22 mm GPa

Natural graphite 4.25 mm

Time, μs

Fig. 4.15 Experimental results for the samples of pressed natural graphite (density of 2.17 g cm−3 ), pyrographite UPV-1 and graphite of monochromator grade (density of 2.25 g cm−3 ). The measurements were done at the boundary between the test sample and “window”—a crystal of lithium fluoride. Loading by the impact of an aluminum plate at a velocity of 3.33 km s−1 . Pressures in the first wave are given at the boundary with the “window” and inside the sample (in brackets)

transformation, a similar slowing-down of the relaxation of shear stresses in an elastic precursor was also observed, the authors [36] made a conclusion of the dislocation mechanism of the structural rearrangement of shock-compressed crystals KC1. In [72], the behavior of polycrystalline iron and single crystals of various orientation was compared. A slight difference in the transformation parameters, observed under shock compression of thin (0.7 mm) samples disappears with the transition to a slower quasi-isentropic compression. In contrast to iron that has a lattice with low anisotropy, the parameters of graphite transformation into diamond are rather sensitive to the structure of the material. Due to practical relevance, considerable efforts were devoted to measuring the parameters of this transition (ref. [73–75] and references in these papers). Nevertheless, this mechanism is not yet fully understood. The results of experiments with the samples of graphite of various structure (pressed natural graphite of grade OSCh-T1, pyrographite UPV-1 and synthetic graphite of monochromator grade) are given in Fig. 4.15. A dominating phase in natural graphite is a hexagonal structure with an interlayer distance d 002 = 0.3354 nm corresponding to the total three-dimensional ordering of the graphite lattice (3D ordering coefficient p3 = l). Pyrographite is characterized by high ordering of basal planes (disorientation angle is about 1°) and low ordering in other directions. A long interlayer distance d 002 = (0.3420 ± 0.0002) nm indicates a low degree of 3D ordering p3 = 0.12. The parameter of 3D ordering of graphite of monochromator grade equals p3 = 0.68 (interlayer distance d 002 = (0.3363 ± 0.0001) nm). In all

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cases, the basal planes of the graphite lattice were oriented, to a fairly high degree, in parallel to the sample surface, i.e. perpendicular to the shock compression direction. The measurement results demonstrate a clear correlation between the transformation pressure and the degree of 3D ordering of the graphite crystalline structure. In this case, the large steepness of the second compression wave in highly oriented graphite indicates the highest rate of its transformation into a diamond-like phase at least at the final stage of the process. In the case of pyrolytic graphite, there are no strong indications of the transformation at a pressure below 30 GPa, however it is known [74, 75] that at higher pressures, a rather fast transition to a dense phase occurs. Probably, an explanation of such a strong effect of structural factors on the kinetics of graphite transformation into diamond must be sought in terms of ratios of the rates of nucleation and growth of the high pressure phase. If the nucleation rate is mainly determined by how far the state of graphite compressed by a shock wave enters the area of its instability, the growth rate is apparently determined to a greater extent by the ordering of the structure of the material surrounding the nucleus. The additional information on the kinetics of transformation is given by the measurements of compression wave evolution [76] and the computer modeling of shock wave phenomena. The results of experiments [77, 78] with highly pure titanium at various temperatures and pressures of shock compression are given in Fig. 4.16. It is known that in case of titanium compression, its structural transformation α → ω occurs [71]. Since α → ω, the transition in titanium is accompanied by a very insignificant decrease in volume—about 1.2%. The velocities of wave propagation in the low pressure phase and in the area of mixed phases differ insignificantly. There is a specific feature in the structure of plastic compression waves, as shown in Fig. 4.16, apparently related with the loss of shock wave stability as a result of transformation (ref. Chap. 12). At the same time, the behavior of titanium differs from the behavior of other materials enduring polymorphic transformation under shock compression. The measurement results show at least 1.5× difference in the dependence of the pressures corresponding to the suggested onset of the transformation on the maximum shock compression pressure. Moreover, we see a different effect of temperature at various shock compression pressures: at low impact velocity, a pressure rise takes place with heating, when a polymorphic transformation occurs, which is compliant with the titanium phase diagram [71], whereas a reverse dependence is observed in experiments with high impact velocity. On the other hand, in case of high impact velocity, the increment time in the second wave is reduced with heating, therefore, the transformation rate in this case increases as the temperature rises. As a result, in case of high temperatures, approaching the equilibrium pressure of transformation must be faster and should cause an apparent change in the temperature dependence sign. Unfortunately, while such explanation is consistent with the data on the attenuation of the first plastic wave in pre-heated samples, such attenuation at room temperature is too slow to be able to explain, among other things, a rise of apparent pressure of the onset of the transformation as the maximum compression pressure rises.

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a

Surface velocity, m/s

Fig. 4.16 Free surface velocity profiles of high-purity titanium samples at room and increased temperatures [77, 78]. a Loading of sample by impact of an aluminum plate 0.4 mm thick at a rate of (640 ± 20) m s−1 ; b experiments with impact of an aluminum plate 1.4 mm thick at a velocity of (1200 ± 50) m s−1

4 Shock Waves in Condensed-Matter Physics

GPa GPa

Time, ns

Surface velocity, m/s

b

GPa GPa

Titanium 3.05 mm

Time, ns

It seems extremely promising to search structural transformations under tension similar to those having place under compression. Apart from a natural need to obtain materials with unusual properties, this search is stimulated by the issues of material strength. Some calculations predict structural transformations in the area of negative pressures, in particular, diamond graphitizing under tension [79] and transition to a clathrate phase in silicone [80]. Recent first-principles calculations of the zero isotherm of iron [81] found its abnormal behavior in the area of negative pressures. The results of calculations given in Fig. 4.17a demonstrate a surge of volume related with the rearrangement of the crystal energy spectrum at a pressure of − 3.4 GPa and a minimum at a pressure of − 13.4 GPa. The extrapolation of a shock adiabat of α-iron [16] gives a minimum at a pressure of − 31.6 GPa, which ∼2.3 times differs from the first-principles calculations. The presence of an area of abnormal compressibility near − 3.4 GPa should result in the formation of shock surges when rarefaction waves propagate in iron, which allows for the experimental check of this abnormality. A series of experiments has

4 Shock Waves in Condensed-Matter Physics

a

Pressure, GPa

Shock adiabat

GPa

GPa

Relative specific volume, V/V0

b

Surface velocity, m/s

Fig. 4.17 Results of studying iron behavior at high negative pressures. a Rated [81] zero isotherm (continuous line) and extrapolated shock adiabat of iron [16] (dashed line). Dashes show the measured values of the fracture stresses (spallation strength) of iron at various durations of shock wave loading. AB is the area of abnormal compressibility where the formation of a rarefaction shock wave was expected; b free surface velocity profiles of high-purity iron plates 0.19 and 0.77 mm thick when impacted by aluminum plates 0.05 mm and 0.4 mm thick at a velocity of 650 m s−1 and 1200 m s−1 , respectively

99

Time, ns

been conducted for the purpose. Figure 4.17b gives the examples of the measured profiles of the free surface velocity of single-crystal samples of iron at 10× difference of shock loading durations that are similar in general to those obtained for other metals and contain no evidence of the formation of rarefaction shock waves (ref. Chap. 12). As the tensile rate rises from ∼105 s−1 to ∼5 × 106 s−1 , the iron fracture resistance grows from 2.9 to 7.6 GPa, and the spallation thickness decreases from ∼400 to 40 µm. Comparison with the data for single crystals of molybdenum and aluminum demonstrates the similarity of the dependences of the spallation strength on the tensile rate and does not reveal any specific features in the area of the assumed abnormality in the compressibility of iron. The range of achieved values of tensile stresses significantly covers the abnormality area at the rated zero isotherm of iron, as shown in Fig. 4.17a. Since electron topological transitions [81] must be always inertia-free, it seems unlikely that the abnormality in the compressibility related with it did not show up due to the low duration of negative pressures. It is possible that the abnormality in the compressibility of

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Fig. 4.18 Changes in the state of aluminum under shock compression and further rarefaction for the conditions of the experiments shown in Fig. 4.17

iron takes place only at low temperatures and disappears with heating. In this case, the instability of the crystalline structure can be one of the factors determining the phenomenon of iron cold shortness. Spallation Strength of Single Crystals and Polycrystals Near Melting. Figure 4.18 shows a part of the phase diagram of aluminum calculated under the equation of state [82] for experimental conditions [23, 24]. The position of the boundaries of the area of coexistence of solid and liquid phases at negative pressures is determined, as in the compression area, by the equality of the chemical potential of phases. In this sense, the boundary of the melting area, including its part located in the area of negative pressures, is equilibrium. Since all states with negative pressures are metastable, melting under tension (if observed) is the transformation of a metastable solid phase into a metastable liquid phase. The onset of melting must be accompanied by an increase in the compressibility and a decrease in the flow stress, which must cause the appearance of abnormalities on the rarefaction wave profile when entering the two-phase area. However, the wave profiles near the melting temperature are completely similar to those measured at lower temperatures. The intersection of the calculated boundary of the melting area under tension is not accompanied by an abrupt drop in the tensile strength of single crystals. The result of measurement of the spallation strength of metals in polycrystalline and single-crystalline states at temperatures up to the melting point are summarized in Fig. 4.19. The thermodynamic evaluations of fracture thresholds [83] associated with the melting onset under tension are also given. The given data show that polycrystalline aluminum and magnesium lose their strength when approaching the melting temperature while single crystals preserve high tensile strength even after crossing the melting phase boundary in the area of negative pressures. It can be assumed that the measured values of the dynamic strength of single crystals at high temperatures characterize the properties of partially melted aluminum.

4 Shock Waves in Condensed-Matter Physics

a

Spallation strength, GPa

Fig. 4.19 Dependence of spallation strength of metals on temperature. The lines show the estimates made in the assumption of fracture at the moment of the onset of melting under tension [83]. a Measurement results for polycrystalline aluminum and magnesium [18]; b results of experiments with the single crystals of zinc [84] and aluminum [23, 24]. Data for aluminum correspond to two different deformation rates: 5 × 105 s−1 and 3 × 106 s−1

101

Aluminum AD1 Magnesium Mg95

Homologous temperature, T/Tm

Spallation strength, GPa

b

Temperature, °C

However, this assumption does not explain the difference in the behavior of singlecrystalline and polycrystalline aluminum. If melting centers occur inside the crystal, the crystal ceases to be homogenous anymore and its strength properties must become the same as in a polycrystalline material. Nevertheless, even at the highest temperatures, single crystals of aluminum have a higher strength than polycrystalline aluminum at room temperature and the same deformation rate. It is far more likely that the material did not melt in the experiments conducted. The measured strength in all cases corresponds to the strength of a solid body. If the expected melting did not take place in the process of high-rate tension at high temperatures, then, consequently, the states of overheated solid body were implemented in experiments with single crystals. The overheating value reached 60–65 °C at the shortest durations of the impact loading. It is believed that the crystal surface where the activation energy is close to zero plays a critical role in melting. Melting of a uniformly heated solid body always starts with its surface. Overheated solid-body states can be created only inside a body,

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provided that its surface has a temperature below the melting point. This condition was implemented in the experiments conducted. After melting centers are nucleated in the overheated solid body, they must grow quickly causing plastic deformation of the surrounding crystalline material and invoking its fracture. On the other hand, “premelting” effects are possible in polycrystalline materials at the grain boundaries where the highest concentration of impurities exists [85]. Molecular-dynamic calculations [86] show that melting in the volume of polycrystals starts at the grain boundaries at temperatures far lower than the melting temperature. In thermodynamics terms, this means the presence of excessive internal energy of the near-surface layers of grains where the crystalline structure is distorted due to the asymmetry of acting forces. We can hope that measurements of the dynamic strength of materials near the melting temperature will allow for quantitative evaluation of the distortion energy of near-surface layers of grains. Issues concerning the maximum possible overheating of the crystalline state and also issues of how the melting curve ends in the area of negative pressures are actively discussed in the literature of recent years. Several evaluations of the maximum possible overheating of a crystal are known. Paper [87] introduces the concept of entropy catastrophe that defines the thermodynamic limit of crystal stability as temperature Tms at which melting of an overheated crystal ceases to be accompanied by entropy growth anymore. Later, Talon [88] modified this criterion by introducing a g new point of stability loss Tm at which the entropy of the overheated crystal becomes equal to the entropy of a glass-like phase (no diffusion fluid). This temperature is somewhat lower than Tms . He also showed that the temperature of the onset of the shear instability of a crystal Tmr is below the temperature Tmv of “isochoric catastrophe” at which the specific volume of a crystal becomes equal to the fluid volume and the temperature of entropy catastrophe Tms . These overheating limits for most solid bodies are within the limits of 1.3T m –2.0T m , where T m is the thermodynamic melting temperature. Based on the analysis of the kinetics of the homogenous initiation of melting nuclei in an overheated crystal, Lu and Li [89] expressed a criterion of stability in terms of the critical temperature Tms ≈ 1.2T m at which mass formation of melt nuclei occurs in the overheated crystal. Luo et al. [90] systematized the data on the kinetic limit of possible overheating of crystals and found that for various elements and simple compounds its value varies within (1.08–1.43) T m with a rather weak dependence on the heating rate. Reliable experimental data on the effects of crystal overheating and premelting in polycrystalline materials are extremely not numerous. Experiments [91] with spherical single crystals of silver 0.1–0.2 mm in diameter, covered with a layer of gold are cited over and over again. Since the melting temperature of gold is higher than that of silver, it was assumed that the gold plating must suppress the nucleation of melt on the surface of a silver crystal. In experiments, silver overheating reached 25 K. The period of an overheated state (about one minute) is limited by mutual diffusion processes. It has been reported that a 2× overheating of aluminum was reached on exposure to a laser pulse of picosecond duration [92] however this was not confirmed by later measurements [93] and analysis [94, 95]. In our experiments with aluminum,

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discussed above, overheating at negative pressures reached 65–70 K, which is 7–8% of the melting temperature. Modern achievements in the method of shock wave experiment for reaching deep negative pressures make conceptual the discussion of an issue about how the metastable melting curve of real matter ends within T → 0. It is suggested that as the melting temperature decreases in the area of negative pressures, the melting curve can reach the zero isotherm of matter [96]. On the other hand, the results of a more detailed analysis based on a single-component model of plasma [97] and molecular dynamic calculations [98] indicate a break of the equilibrium melting line in the low-temperature limit due to reaching the boundary of fluid stability. Shock Compression of Brittle Materials. Fracture Waves. While the mechanisms and determinant factors of fracture of brittle materials under tension are well understood, the processes of quasi-static and especially dynamic inelastic compression seem unclear to a significant extent. Research results summed up in previous reviews [99, 100] show that there are still significant gaps in understanding the mechanisms and determinant factors of compression fracture. For this reason, and due to a number of practical applications, greater attention was paid recently to the behavior of highly solid brittle materials (rocks, ceramics, glasses) in case of shock wave loading. It’s worth noting that there are no absolutely brittle materials. For example, all brittle matters including diamond show a significant plasticity when exposed to high pressure. However, unlike plastic metals, highly solid oxides and intermetallics with covalent-ionic interatomic bonds and low symmetry of crystals are characterized by a high energy of dislocation formation and a small number of planes where glide of dislocations is possible. For this reason, plastic deformation is extremely complicated which enables high concentration of stresses on microfractures and other irregularities. Blockage of the glide inside a body at the grain boundary or at the intersection with another active shear system can result in cracking. According to its definition, fracture means material discontinuity. Cracks, as other discontinuities, can form in a non-porous medium only under the action of tension stresses. It is known however that in case of general compression local stresses near irregularities can become tensile. According to the Griffith’s criterion [101], fracture under compression is initiated when the highest local tension stress reaches its threshold. For a biaxial stress state, the Griffith’s criterion looks as follows (σ1 − σ2 )2 + 8K (σ1 + σ2 ) = 0, where σ 1 , σ 2 are main stresses, K is the constant value of the material assumed to be equal to the standard ultimate tensile strength. Model experiments with glass and polymeric plates [102, 103] have shown that a local shear along an inclined crack surface leads to the formation of a configuration of three cracks, which is called a wing-shaped crack. At the edges of the local shear area, separation cracks are initiated which grow along the curved surface beyond the shear plane deflecting in the compression direction. As the growth direction approaches the compression direction, the process is slowed down until full stop,

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but the crack growth can resume if the compression stress rises. The imposition of additional transverse compression suppresses material cracking. Research of the behavior of brittle materials in case of shock wave loading include measurements of shock compressibility, recording and analysis of wave profiles of compression stress or velocity of matter, measurements of stressed state in a shockcompressed material and spallation strength after being loaded by impact loading pulses of varying intensity. The results of studies of the shock compression of very hard ceramics, minerals and glasses are partially summarized in survey papers [104, 105] and monographs [7]. The most controversial is the issue of the nature of inelastic deformation in a shock wave, namely: whether brittle cracking or plastic flow occurs during the compression of a brittle material in a flat shock wave. The point is that both longitudinal and transverse stress components rise under uniaxial shock compression. In the elastic area, the change in the longitudinal σ x and transverse σ y stresses occurs in a consistent manner: σx = σ y

1−v , v

where ν is the Poisson coefficient. The fracture threshold quickly rises as the transverse compression stress increases, and at a certain value σ y the so-called brittleplastic transition takes place: shear stresses become sufficient for activating plastic deformation mechanisms, and crack extension is suppressed by transverse stresses. Resistance to inelastic deformation during fracture, on the one hand, and plastic flow, on the other hand, have a different physical nature and is described in different ways. For this reason, to calibrate rheological models, it is very important to know the nature of observed inelastic deformation. Examples of recording wave profiles during shock compression of two very hard materials representing aluminum oxide—ceramics and single crystal—are given in Fig. 4.20. The results of experiments with ceramics do not generally contain any specific features that could be identified as indicators of fracture. Aluminum oxide ceramics have a high elastic limit (5–10 GPa) and low spallation strength (below 1 GPa). There is conflicting data on whether the spallation fracture resistance of ceramics is preserved in the conditions when the Hugoniot elastic limit is exceeded in a compression wave. The disappearance of the tensile strength could be considered as a sign of fracture under the previous shock compression, but even in this case, it is still not clear whether the fracture occurs during compression or further unloading. Single-crystalline aluminum oxide has a very high spallation strength (up to 20 GPa [106, 107]) below the elastic limit. Smooth wave profiles without specific features are usually recorded in this area. If the maximum stress in the compression pulse exceeds the elastic limit, the spallation strength drops almost to zero; in this case, irregular oscillations appear on wave profiles, which is obviously caused by the significant heterogeneity of the inelastic deformation process. The analysis of the samples of ceramic oxide of aluminum and sapphire kept after shock wave exposure

4 Shock Waves in Condensed-Matter Physics a 3.46 g/cm3 Surface velocity, km/s

Fig. 4.20 Profiles of shock compression of aluminum oxide samples at various pressures of shock compression. a Results of experiments with ceramic samples 3.46 g cm−3 in density; b experiments with ruby [106] and sapphire in contact with water

105

4.5 mm

2.2 mm

Time, μs

Surface velocity, km/s

b

Sapphire

Expected spallation

Ruby

Time, μs

[108] showed that along with cracking, shock waves have a significant plastic deformation, which is proved by multiple dislocations, shear stripes and duplicates inside fragments. One of possible ways to identify fracture during shock compression could be recording of fracture waves [9]. In 1960–, a hypothesis was introduced concerning the process of material fragmentation in a relatively thin layer propagating along an undestroyed medium at the velocity of sound. This fracture front, as it propagates, inculcates a lot of new cracks in an undestroyed material. This self-sustaining fracture wave in which the potential elastic energy of a stressed brittle body is transformed into the surface and kinetic energy of its fragments is similar to a detonation wave the stable propagation of which occurs due to the release of the chemical energy of matter. A relevant bibliography and a critical analysis of various methods of describing the supposed fracture waves can be found in papers [109, 110]. We should say that the first theoretical works failed to create a non-controversial closed model of the phenomenon and properly assess the propagation rate and other kinematic parameters

Fig. 4.21 Profiles of free surface velocities for K8 glass samples 6.1 mm thick [114]. Loading conditions: 1 impact with an aluminum plate 2 mm thick at a velocity of (1900 ± 50) m s−1 ; 2 impact with an aluminum plate 0.9 mm thick at a velocity of (670 ± 30) m s−1 . A dash line shows the result of numerical modeling of experiment 2

4 Shock Waves in Condensed-Matter Physics

Surface velocity, km/s

106

Calculation

Time, μs

of fracture waves. Based on qualitative assumptions, it was suggested that the fracture wave velocity equals the velocity of sound in the material or even exceeds this value. Late in the 1980s, the authors of this chapter found the formation of fracture waves in glass under unidimensional compression of a flat shock wave [111–113]. These works attracted interest and were further developed for a number of reasons. Glass is a traditional model material to study the consistent patterns of deformation and fracture of brittle media. A fracture wave is an example of a self-sustaining fracture under compression, which is important for understanding the mechanisms of earthquakes, rock bump and other catastrophic phenomena. The developed technique of recording the processes accompanying the shock-wave compression of solid bodies makes it possible, in principle, to obtain exhaustive information on the kinetics of the phenomenon. The velocity profiles of the free surface of glass sample K8 measured [114] in the conditions of shock compression being higher and lower than the Hugoniot elastic limit are given in Fig. 4.21. In both cases, the measured wave profiles repeat the compression pulse shape in the sample. Incomplete unloading in the experiment with a higher intensity of a shock wave is explained by the fact that a thick layer of wax adjoined the rear surface of the striker in this experiment. A stage-wise nature of velocity drop in the rarefaction wave at a lower rate of impact is explained by a difference in dynamic impedances of glass and steel striker. Since spallation phenomena do not occur on the free surface velocity profiles, the dynamic tensile strength of glass exceeds 6.8 GPa in case of shock compression below the elastic limit and remains extremely high when the latter is exceeded. For comparison: the quasi-static tensile strength of glass is about 0.1 GPa. The reason for so great a difference is that in normal conditions glass fracture is initiated on its surface where nuclear microcracks are always present. At the same time, in experiments with shock waves, spallation fracture can be initiated only in the volume of the material without any participation of surface defects.

4 Shock Waves in Condensed-Matter Physics

107

Single crystals and glasses in the initial state have a high homogeneity in the material volume. In both cases, centers of fracture nucleation can be formed only during compression. In this sense, these two homogeneous materials differ in that anisotropic highly solid single crystals have a limited number of planes and directions that can be exposed to regular mechanisms of plasticity, while plasticity of amorphous glasses is fully isotropic. The impossibility of plastic shears in arbitrary directions leads to a concentration of stresses in places of intersection of glide stripes or duplicates, which, in its turn, can lead to cracking during compression or unloading from the shock-compressed state. Isotropic glasses are free from this limitation. The retention of the high spallation strength of glass when the Hugoniot elastic limit is exceeded means that the plasticity of the material is preserved during unloading from the shock compressed state and further tension. Comparison with the results of numerical modeling in Fig. 4.21 shows that the reverberation time of a compression/tension pulse in a glass plate is less than expected for a longitudinal elastic wave. This is explained by the fact that a layer of fractured material ∼1.5 mm thick was formed at the impact surface. Such cracking does not occur under shock compression above the elastic limit: the re-reflected compression pulse comes to the surface in this case a little later than the expected elastic wave. Experiments with glass samples of various thickness loaded by shock-compression pulses of long duration [111–113] showed that the fractured layer is expanded with time. This process can be represented as the propagation of a fracture wave. For fifteen years after the discovery of this phenomenon, the fact of the formation of fracture waves in shock-compressed glasses was confirmed many times and a vast volume of empirical information was collected concerning the kinematics of their propagation and initiation limits. A fracture wave is a network of cracks initiated by stress applied to the glass surface with multiple nucleus microcracks and propagating into the material volume. The propagation velocity of a fracture wave is less than longitudinal and shear velocities of sound, is close to the limiting rate of crack growth (∼1.5 km s−1 for glass) and depends on stress. Fracture waves are formed at compression stresses above some threshold that can be identified as a fracture threshold and below the glass elasticity limit. As the acting stress falls down, the fracture wave stops. In the fracture wave, there is a consistent increase in the stress and density of the material in accordance with the laws of conservation of mass and momentum, and the relaxation of shear stresses occurs. The fracture wave velocity slightly rises as the compression stress increases. After the fraction wave passes, the material is completely or almost completely losing its resistance to tensile stress. Plastic deformations suppress material cracking. Although it is shown that the fracture wave is indeed a wave, as it is understood in the mechanics of a continuous medium, its kinematics differs from the kinematics of elastoplastic waves. A shock wave in an elastic–plastic body loses stability due to the abrupt growth of compressibility when the yield strength is reached. As a result, the shock wave is split into elastic and plastic compression waves (ref. Chap. 12). Stress behind the elastic wave front is determined by the value of the yield strength of the material. Such a wave structure must be formed in a polycrystalline brittle material where fracture is initiated in each grain as soon as the stress reaches the fracture

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Fig. 4.22 Transformation of a shock compression pulse as a result of the formation of fracture waves. The free surface velocity profiles are given for a plate of lime-soda glass 5.9 mm thick and a stack of 8 glass plates, each about 1.21 mm thick, in the same loading conditions [115]

Time, μs

threshold. In both cases, the propagation velocity of the second wave is determined by the volumetric compressibility of the material. The fracture wave propagation rate is defined by the crack growth rate that is not related with volumetric compressibility. On the other hand, ultimate tensile stress behind the fracture wave is defined by the conditions on the impact surface (or contact surface between the screen and glass plate). As a result, since the fracture wave propagation velocity and the final state of the material behind it are fixed, the stress ahead of its front (i.e. the stress in the leading elastic wave) is defined by these conditions and is not necessarily equal to the fracture threshold. The glass surface is a source of cracks and plays an important role in the wave process of fracture. In this sense, the process is similar to diffusion. However, the diffusion rate is not kept constant unlike the observed velocity of fracture waves. For this reason, the self-sustaining propagation of subsonic fraction waves is more like combustion. Probably, the role of surfaces and specificity of fracture wave kinematics can be well illustrated when compression waves propagate through a stack of glass plates. When going through each surface, a compression wave is split into a leading elastic wave and a low-velocity fracture wave following it. As a result, the stress at the front of the leading elastic wave is stepwise decreased on each surface in the stack. This must continue until the stress in the elastic wave falls down to the fracture threshold. The results of measurements [115] of the wave profiles generated by an impact in a thick glass plate and in a stack of thin plates are compared in Fig. 4.22. The superposition of fracture waves in a stack of glass plates forms of a two-wave configuration of compression: an elastic wave is transformed into an elastic–plastic one. The compression time in the second wave almost corresponds to the propagation time of counter cracks through a plate in the stack. The stress behind the front of the leading elastic wave in the stack is 4 GPa. Apparently, this stress is close to the glass fracture threshold in these conditions. The final values of the free surface velocity

4 Shock Waves in Condensed-Matter Physics

a 4∙1.5 m

Surface velocity, km/s

Fig. 4.23 Comparison of the surface velocity profiles of a thick plate and a stack of thin plates for ceramic aluminum oxide (a) and boron carbide (b). Loading by the impact of an aluminum plate 2 mm thick at a velocity of 1.9 km s−1 [119]

109

6 mm

g/cm3

Time, μs

b

Surface velocity, km/s

5 mm 4∙1.5 mm 6.07 mm

g/cm3

Time, μs

are almost the same for a thick glass plate and a stack of thin plates. Thus, experiments with shock compression of a stack of plates represent a simple and illustrative method to detect fracture waves. The formation of fracture waves under shock compression of composite glass samples was directly recorded by high-speed photography in transmitted light [116]. The obtained streak photographies show the initiation of fracture on the internal surface of a composite sample and the propagation of two fracture fronts both in the direction of compression and in the reverse direction. It is not clear whether fracture waves can form in other brittle materials apart from glasses. Some papers reported [117] about the detection of delayed relaxation of deviator stresses in shock-compressed ceramics at low distances (2–4 mm) from the impact surface. By analogy with the similar behavior of glasses, the relaxation of deviator stresses was interpreted as evidence of the formation of a fracture wave. These observation results were disputed in [118]. Experimental results [119] with Al2 O3 and B4 C ceramics are given in Fig. 4.23. In contrast to glasses, experiments

110

4 Shock Waves in Condensed-Matter Physics

with stacks of ceramic plates show no decreased amplitude of the elastic precursor due to the formation of fracture waves. In fact, the mass velocity and, therefore, compressive stresses behind the elastic precursor front in experiments with stacks turned out to be even higher than in experiments with monolithic samples. Probably, the initiation of fracture waves under compression is possible only in homogeneous materials where the concentration of nucleus microcracks is possible in surface layers with preserved homogeneity and practical absence of material defects inside a body. In any case, the fact that fracture waves are not recorded in ceramics does not mean that ceramics are deformed plastically in case of shock compression. In paper [119], it was proposed to diagnose the nature of inelastic deformation during shock compression of brittle materials by varying the magnitude of the transverse compression stress and measuring its effect on the Hugoniot elastic limit. It is known [120] that in the area of brittle fracture, the elastic limit strongly depends on pressure while this dependence almost disappears with the onset of plasticity. In case of plastic deformation, the amplitude of the elastic precursor must comply with the flow criterion, for example, Mises or Tresca criterion [7], according to which the stress at the Hugoniot elastic limit is related with the yield strength σ m by the ratio σHEL = σm

1−v , 1 − 2v

where v is the Poisson coefficient. Relatively low side pressure p causes a slight rise in the precursor amplitude: duct σHEL =

(σm + p)(1 − v) . 1 − 2v

In case of brittle behavior, we can use the Griffiths fracture criterion, which gives σHEL = σT

1−v . (1 − 2v)2

In this case, the imposition of lateral pressure leads to a much larger increase in the elastic precursor amplitude: brit = [σT + (1 − 2v)(3 − 2v) p] σHEL

1−v , (1 − 2v)2

which is approximately two and a half times greater than the lateral pressure effect in plastic behavior. The measured velocity profiles of free and prestressed samples of aluminum oxides and boron carbide under shock-wave loading are given in Fig. 4.24. The controlled lateral pressure p ≈ 0.3 GPa in the samples was created by the method of shrunk fit into steel rings that initially had a lower (by 0.1 mm) internal diameter than the samples.

4 Shock Waves in Condensed-Matter Physics

m/s

a

Surface velocity, m/s

Fig. 4.24 Comparison of the velocity profiles of free (dashed curves) and prestressed (continuous curves) surfaces and plates of ceramic aluminum oxide (a) and boron carbide (b) [119]. The box shows a change in the difference of velocities with time

111

μs Time, μs

b

Surface velocity, km/s

Prestressed sample

Free sample

Time, μs

The experiment results shown in Fig. 4.24 demonstrate a difference in the response of aluminum oxide and boron carbide to the lateral pressure. Thus, measurements show that aluminum oxide behaves as a plastic material under unidimensional compression in a shock wave while brittle fracture by compression takes place in boron carbide. Conclusion. The importance of investigating the processes of inelastic deformation and fracture of solid bodies under shock wave loading is defined both by a unique opportunity of pursuing researches in the field of the physics of strength and plasticity at the highest and reliably measurable deformation rates and by various practical needs not limited by shock actions only. Experiments with shock waves make it possible to obtain information about the most fundamental strength properties of materials in conditions that prevent the effect of the surface on the processes of deformation and fracture. This method allows realizing the states of solid bodies and

112

4 Shock Waves in Condensed-Matter Physics

fluids close to the maximum possible strength and thereby experimentally assessing their strength span. The first researches of the elastic–plastic and strength properties of metals and alloys at elevated temperatures and extremely high rates of shock wave loading revealed interesting effects that could be expected, but were still not predicted by the theory. The measurement results show that in these conditions, the effect of temperature on the yield strength can be opposite to the one that takes place at low and moderate rates of deformation. Temperature dependences of the strength of singlecrystalline and polycrystalline metals near the melting point have a substantially different nature, which is explained by the phenomena of crystalline state overheating and premelting. New opportunities are opening up for studying overheated solid-body states and melting in the area of negative pressures. There is reason to hope that the observed effect of premelting can be used for the quantitative description of the state of matter at grain boundaries. Recent studies confirm the informative value and fruitfulness of shock-wave studies of materials and getting information about fast physical–chemical transformations, principles of deformation and fracture at extremely high loading rates. It has been shown that in case of shock compression, polymorphous transformations in solid bodies can occur for 10−9 –10−7 s and less. Modern shock wave technology provides unique opportunities for determining the limiting velocities and high-rate mechanisms of the transformational change in the crystalline structure. On the other hand, studies of some polymorphic transformations such as the transformation of graphite into diamond are obviously practically important due to the implemented and multiple potential technological applications of shock wave effects. From this point of view, it is important to find out the effect of structural and other factors on the kinetics and completeness of transformation. Since polymorphism under compression is typical of many structural and other practically important materials, its studies are important in terms of the calculation prediction of intensive impulse effects. High attention has been paid to the behavior of brittle materials during the past decade. In particular, fracture waves discovered by Russian researchers are being actively studied. The formation of fracture waves is one of the mechanisms of catastrophic loss of strength of very hard brittle materials and is an example of non-local response of a material to loading. A number of methods for diagnosing the states of shock-compressed brittle materials is substantially expanded by the development of a methodology of testing prestressed samples and testing by diverging shock waves. In the coming decade, we should expect a significant expansion of the use of shock wave technology for solving the problems of materials science, physics of strength and plasticity. Further studies of strength variations at the meso-level and clarification of the mechanism of localized shift stripe formation will contribute to the construction of new high-strength materials and improvement of their processing technology. Elucidation of the details of the brittle fracture mechanism under compression will facilitate progress in the creation and use of extra-solid materials, prediction of earthquake consequences and many other problems of modern mechanics of deformation state.

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56. Bogach AA, Utkin AV (2000) J Appl Mech Tech Phys 41(4):198 [Bogach A.A., Utkin A.V. // PMTF. 2000. T. 41, №4. S. 198 (in Russian)] 57. Zheng Q, Durben DJ, Wolf GH, Angell CA (1991) Science 254:829 58. Lynden-Bell RM (1995) J Phys Comdens Matter 7:4603 59. Belak JJ (1998) Comput Aided Mater Design 5193 60. Rudd RE, Belak JF (2002) Comput Mater Sci 24:148 61. Marian J, Knap J, Ortiz M (2004) Phys Rev Lett 93:165503 62. Kuksin AY, Morozov IV, Norman GE, Stegailov VV, Valuev IA (2008) Mol Simul 31:1005 63. Seppälä ET, Belak J, Rudd RE (2004) Phys Rev B 69:134101 64. Kuksin AY, Yanilkin AV (2007) Rep Russ Acad Sci 413(5) [Kuksin A.Yu., Yanilkin A.V. // Dokl. RAN. 2007. T. 413, №5 (in Russian)] 65. Regel VR, Slutsker AI, Tolmashevskiy EE (1972) Adv Phys Sci 106(2):193 [Regel V.R., Slutsker A.I., Tolmashevskiy E.E. // UFN. 1972. T. 106, №2. S. 193 (in Russian)] 66. Morozov N, Petrov Y (2000) Dynamics of fracture. Springer, Berlin 67. Sin’ko GV, Smirnov NA (2002) Lett J Exp Theor Phys 75:217 [Sin’ko G.V., Smirnov N.A. // Pis’ma v ZHETF. 2002. T. 75. S. 217 (in Russian)] 68. Clatterbuck DM, Chrzan DC, Morris JW (2003) Scripta Mater 49:1007 69. Bezruchko GS, Kanel GI, Razorenov SV (2004) J Thermal Phys High Temp 42(2):262 [Bezruchko G.S., Kanel G.I., Razorenov S.V. // TVT. 2004. T. 42, №2. S. 262 (in Russian)] 70. Johnson PC, Stein BA, Davis RS (1962) J Appl Phys 33:557 71. Tolkov EY (1988) Phase transformations of compounds at high pressure. In: Ponyatovsky EG (ed) Reference book. Metallurgy, Moscow [Tolkov E.Yu. Fazovyye prevrashcheniya soyedineniy pri vysokom davlenii. Spravochnik / Pod red. E. G. Ponyatovskogo. - M.: Metallurgiya, 1988 (in Russian)] 72. Jensen BJ, Rigg PA, Knudson MD, Hixson RS, Gray III GT, Sencer BH, Cherne FJ (2006) In: Furnish MD et al (eds) Shock compression of condensed matter-2005. AIP conference proceedings, vol 845. American Institute of Physics, Melville, New York, p 232 73. Duvall GE, Graham RA (1977) Rev Mod Phys 49:523 74. Gust WH (1980) Phys Rev B 22:4744 75. Erskine DJ, Nellis WJ (1992) J Appl Phys 714882 76. Gogulya MF, Batukhtin DG, Voskoboynikov IM (1987) Lett J Tech Phys 13:786 [Gogulya M.F., Batukhtin D.G., Voskoboynikov I.M. // Pis’ma v ZHTF. 1987. T. 13. S. 786 (in Russian)] 77. Kanel GI, Razorenov SV, Zaretsky EB, Herrmann B, Mayer L (2003) J Phys Solid Body 45:625 [Kanel G.I., Razorenov S.V., Zaretsky E.B., Herrmann B., Mayer L. FTT. 2003. T. 45. S. 625 (in Russian)] 78. Bezruchko GS, Razorenov SV, Kanel GI, Fortov VE (2006) In: Furnish MD et al (eds) Shock compression of condensed matter-2005. AIP conference proceedings, vol 845. American Institute of Physics, Melville, New York, p 192 79. Roundy D, Cohen ML (2001) Phys Rev B 64:212103 80. Wilson M, McMillan PF (2003) Phys Rev Lett 90:135703 81. Sin’ko GV, Smirnov NA (2004) Lett J Exp Theor Phys 79:665 [Sin’ko G.V., Smirnov N.A. // Pis’ma v ZHETF. 2004. T. 79. S. 665 (in Russian)] 82. Asay JR, Hayes DB (1975) J Appl Phys 46:4789 83. Kanel GI (2000) J Therm Phys High Temp 38:512 [Kanel G.I. // TVT. 2000. T. 38. S. 512 (in Russian)] 84. Bogach AA, Kanel GI, Razorenov SV, Utkin AV, Protasova SG, Sursayeva VG (1998) Phys Solid Bodies 40:1849 [Bogach A.A., Kanel G.I., Razorenov S.V., Utkin A.V., Protasova S.G., Sursayeva V.G. // FTT. 1998. T. 40. S. 1849 (in Russian)] 85. Dash JG (1999) Rev Mod Phys 71:1737 86. Besold G, Mouritsen OG (1994) Phys Rev B 50:6573 87. Fecht HJ, Johnson WL (1988) Nature 334:50 88. Tallon JL (1989) Nature 342:658 89. Lu K, Li Y (1998) Phys Rev Lett 80:4474

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90. Luo S-N, Ahrens TJ, Gagin T, Strachan A, Goddard III WA, Swift D (2003) Phys Rev B 68:134206 91. Daeges J, Gleiter H, Perepezko JH (1986) Phys Lett A 119:79 92. Williamson S, Mourou O, Li JCM (1984) Phys Rev Lett 52:2364 93. Herrnan JW, Elsayed-Ali HE (1992) Phys Rev Lett 69:1228 94. Rethfeld B, Sokolowski-Tinten K, von der Linde D, Anisimov SI (2002) Phys Rev B 65:092103 95. Ivanov DS, Zhigilei LV (2003) Phys Rev 868064114 96. Skripov VP, Fayzullin MZ (2003) Phase transformations crystal-fluid-vapor and thermodynamic similarity. FIZMATLIT, Moscow [Skripov V.P., Fayzullin M.Z. Fazovyye perekhody kristall–zhidkost’–par i termodinamicheskoye podobiye. - M.: FIZMATLIT, 2003 (in Russian)] 97. Iosilevskiy IL, Chigvintsev AY (2003) Electron J Res Russ 3:20. https://zhurnal.ape.relarn.ru/ articles/2003/003.pdf [Iosilevskiy I.L., Chigvintsev A.Yu. // Elektronnyy zhurn. «Issledovano v Rossii» 2003. T. 3. S. 20; https://zhurnal.ape.relarn.ru/articles/2003/003.pdf (in Russian)] 98. Kuksin AY, Norman GE, Stegaylov VV (2007) J Therm Phys High Temp 45 [Kuksin A.Yu., Norman G.E., Stegaylov V.V. // TVT. 2007. T. 45 (in Russian)] 99. Kranz RL (1983) Tectonophysics 100:449 100. Wang EZ, Shrive NG (1995) Eng Fract Mech 52:1107 101. Griffith AA (1925) The theory of rupture. In: Biezeno CB, Burgers JM (eds) Proceedings of the 1st international congress for applied mechanics, Delft, The Netherlands, 22–26 Apr 1924, p 55 102. Brace WF, Bombolakis EG (1963) J Geophys Res 68:3709 103. Horii H, Nemat-Nasser S (1985) J Geophys Res 90(B4):3105 104. Grady DE (1998) Mech Maler 29:181 105. Kanel GI, Bless SJ (2002) In: McCauley JW et al (eds) Ceramic armor materials by design. Ceramic transactions, vol 134. American Ceramic Society, Westerville, Ohio, p 197 106. Razorenov SV, Kanel GI, Yalovet TN (1993) Chem Phys 12(2):175 [Razorenov S.V., Kanel G.I., Yalovet T.N. // Khim. fiz. 1993. T. 12, №2. S. 175 (in Russian)] 107. Kanel GI, Razorenov SV, Utkin AV, Baumung K, Karow HU, Licht V (1994) In: Schmidt SC et al (eds) High-pressure science and technology 1993. AIP conference proceedings, vol 309. American Institute of Physics, New York, p 1043 108. Wang Y, Mikkola DE (1992) In: Meyers MA, Murr LE, Staudhammer KP (eds) Shock-wave and high-strain-rate phenomena in materials. M. Dekker, New York, p 1031 109. Grigoryan SS (1977) Gaz USSR Acad Sci Mech Solids (1):173 [Grigoryan S.S. // Izv. AN SSSR Mekh. tverdogo tela. 1977. №1. S. 173 (in Russian)] 110. Slettyan LI (1977) Gaz USSR Acad Sci Mech Solids (1):181 [Slettyan L.I. // Izv. AN SSSR Mekh. tverdogo tela. 1977. №1. S. 181 (in Russian)] 111. Kanel GI, Razorenov SV, Fortov VE, Abazekhov MM (1988) In collection of IV all-union meeting in detonation, vol 2. United Institute of Chemical Physics, USSR Academy of Sciences, Chernogolovka, p 104 [Kanel G.I., Razorenov S.V., Fortov V.E., Abazekhov M.M. // V sb. IV Vsesoyuz. soveshchaniye po detonatsii. T. 2. - Chernogolovka: OIKHF AN SSSR, 1988. S. 104 (in Russian)] 112. Razorenov SV, Kanel GI, Fortov VE, Abasehov MM (1991) High Press Res 6:225 113. Kanel GI, Rasorenov SV, Fortov VE (1992) In: Schmidt SC et al (eds) Shock compression of condensed matter 1991. North-Holland, Amsterdam, p 451 114. Kanel GI, Razorenov SV, Utkin AV, Hongliang H, Fuqian J, Xiaogang J (1998) High Press Res 16:27 115. Kanel GI, Bogatch AA, Razorenov SV, Zhen C (2002) J Appl Phys 92:5045 116. Bourne NK, Rosenberg Z, Millett JCF (2004) In: Furnish MD, Gupta YM, Forbes JW (eds) Shock compression of condensed matter 2003. AIP conference proceedings, vol 706. American Institute of Physics, Melville, New York, p 723 117. Bourne N, Millett J, Pickup I (1997) J Appl Phys 81:6019

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Chapter 5

Laser-Driven Shock Waves

Powerful pulse lasers designed to carry out controlled thermonuclear fusion and simulate a nuclear explosion provide a unique opportunity to generate light pulses with an energy of the mega-joule range and a power of hundreds of terawatts and higher [1, 2]. When these pulses act upon condensed media, record local energy densities are achieved which at present cannot be obtained in laboratory experiments by any other methods available. This circumstance makes powerful laser systems a unique tool to generate powerful shock waves and study matter in a plasma state with extremely high energy densities. The rapid development of laser equipment led to the possibility of generating supershort laser pulses and nano-, pico-, femto-1 and attosecond ranges (Fig. 5.1) and to the transition (ref. Table 5.1, Fig. 5.2) of existing laser complexes and laser complexes being constructed to the petawatt and zetawatt range of powers (Fig. 5.2). This made it possible to obtain a wide spectrum of power densities up to the maximum values as of today, q ≈ 1022 –1023 W/cm2 [1–5] that will undoubtedly increase with time. The impact of such giant light fluxes on targets leads to various new physical effects [3–7] such as multiphoton ionization, self-focusing [8] and filamentation of various nature, generation of giant electrical and magnetic fields, acceleration of electrons and ions to relativistic velocities, nuclear reactions caused by these fast particles, relativistic plasma “enlightenment”, nonlinear modulation and multiple generation of harmonics, ponderomotive effects, formation of “hollow” ions, relativistic effects in hydrodynamics and much more, which is today a subject of intensive studies (ref. reviews [2, 9] and references contained therein). Growth of Laser Radiation Intensity. A consecutive growth of laser radiation power density will be accompanied by qualitatively new phenomena [7, 10] such as the spontaneous production of electron–positron pairs (vacuum “boiling” and loss of its transparency [3, 6, 9–24]), the occurrence of microscopic quantities of relativistic 1 For

comparison: the ratio of 10 femtoseconds to 1 min is the ratio of 1 min to the Universe age—14 billion years. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Fortov, Intense Shock Waves on Earth and in Space, Shock Wave and High Pressure Phenomena, https://doi.org/10.1007/978-3-030-74840-1_5

119

120

5 Laser-Driven Shock Waves

Nuclear power plant

Vehicle engine Electrical lamp

All power sources on the Earth

Petawatt laser

Power, W

Fig. 5.1 Power scales

Table 5.1 Comparison of petawatt laser concepts. Symbols “ + ”, “−”, and “0” are indicative of above-average, below-average, and average characteristics, respectively [30] Amplifying medium

Nd:glass

Ti:sapphire

DKDP

Cr:YAG ceramics

Energy source

Nd:glass

Nd:glass

Nd:glass

Nd:glass

Pump

(+)

2ω

(−)

2 ω Nd

(−)

1 ω Ndb

Pump duration, ns

(+)

(0)

> 10

(0)

1

(−)

> 10

(0)

Nda

Amplifier aperture, cm

40

(0)

8

(−)

40

(0)

> 50

(+)

Minimum duration, fs

250

(−)

25

(+)

25

(+)

25

(+)

Efficiency (1 ω Nd → fs), %

80

(+)

15

(0)

10

(−)

25

(0)

Number of petawatt out of 1 kJ 1ω Nd

3.2(3)c

6(1.5)d

4

10

Power reached, PW

1.36

1.1

1.0

-

a Second

harmonic of a neodymium laser harmonic of a neodymium laser c From the pulse of the first harmonic of a neodymium laser to a femtosecond pulse d The radiation resistance of diffraction gratings and sapphire crystals limits the peak power at levels of 3 PW and 1.5 PW, respectively b First

matter, the generation of shock waves and relativistic flows of plasma, solitons, jets and γ -flashes similar to astrophysical ones, and, in the long run, the implementation of quantum gravity conditions [25–27]. At this new level of intensities, the appearance of other schemes of controlled thermonuclear fusion, nuclear reactions and new ways to obtain short-lived isotopes is possible along with unusual schemes of effective compact accelerators [1, 2]. The progress in increasing the intensity of affecting the target as well as the emerging opportunities for generating powerful shock waves and studying the processes of high energy density physics are shown in Fig. 5.2 [2, 7, 10].

5 Laser-Driven Shock Waves

121 Characteristic energy of electrons Era of electroweak interaction

Nonlinear quantum electrodynamics: E ∙ θ ∙ λc = 2m0c2

1 PeV

Zetawatt laser

Specific power, W/cm2

Quark era Limit laser intensity per cm2 of laser medium

Relativistic optics Vosc ~ c (high ponderomotive pressure)

1 TeV Positron electronic era 1 MeV

Bound electrons

Plasma era

Chirping Mode synchronization

1 eV

Q-factor modulation

Atomic era

Fig. 5.2 Growth of laser radiation intensity with time [3, 10]. The rapid growth of laser radiation intensity in the 1960s resulted in the discovery of many nonlinear effects stimulated with bound electrons (characteristic energies of eV level). The modern quick growth of intensity allows studying processes at relativistic energies (W ≈ mec2 ≈ 0.5 meV) of electrons

Immediately after the invention of lasers, multiple nonlinear optical phenomena were recorded [8, 18–24], such as self-focusing, deformation of intra-atomic and molecular fields by laser radiation, stimulated Raman, Brillouin and Thomson emissions, multiphoton ionization and a number of other nonlinear phenomena belonging to bound electrons. Since the advent of the first laser, one of the most important goals of quantum electronics has been and is still an increase in the peak power of laser radiation [28]. The very notion of a “high peak power” is permanently changing, and today means a power of no less than 1 PW (1015 W). The rapid progress in laser power in the 1960s and 1970s was based on the principles of Q-switching and mode locking, which made it possible to reduce the duration of laser pulses from microseconds to picoseconds over 40 years [3]. Further progress in this direction was limited by the large dimensions and cost of lasers, and the need to operate at the limit of the radiation resistance of optical elements. The present-day “renaissance” in laser physics is due to the invention in 1985 of the method of amplification of chirped (frequency-dispersed) optical pulses (Figs. 5.2

122

5 Laser-Driven Shock Waves Several nanojoules, several dozens of femtoseconds (spectrally limited pulse) Generator of femtosecond pulses

0.01 to 100 J, several dozens of femtoseconds (spectrally limited pulse)

Compressor

Several nanojoules, ca. 1 ne (chirped pulse)

Stretcher

0.01 to 100 J, ca. 1 ns (chirped pulse)

CPA or OPCPA amplifiers

Fig. 5.3 General schematic representation of femtosecond lasers [30]. The chirping principle is used without exception in all lasers with a power of 1 TW and higher [30]. It made it possible to increase the intensity of laser radiation by five–six orders of magnitude and to dramatically lower the cost and dimensions of lasers, which have become “desktop” devices affordable for small university laboratories. Furthermore, these lasers combine well with big facilities for laser-induced controlled thermonuclear fusion (CTF) (“fast” ignition—[1, 2]) and charged-particle accelerators, providing a way for recording nonlinear quantum-electrodynamic effects such as pair production from vacuum [6, 20–23, 31] as well as intense optical radiation for studying photon–photon collisions [32]

and 5.3) [29], which opened up the way for multiterawatt, petawatt, and even exawatt laser systems [7, 28] bringing the maximum power densities on the target to q ≈ 2 × 1022 W/cm2 . In this method [29], an initially short laser pulse is stretched in time, passing through dispersive elements (Fig. 5.3, [30]). The pulse is decomposed into spectral components, each of which travels a somewhat different path depending on its wavelength, stretching in time and space several ten thousands of times due to the separation of its spectral components. Such a stretched pulse has a much lower intensity as compared to the initial one and can be easily strengthened. Spectral clipping (“chirping”) of this time-stretched pulse also occurs, i.e. the frequency continuously varies from the beginning to the end of the pulse. The stretched pulse, having a lower power density, is amplified in the ordinary way by a laser active medium, which now operates under the conditions of a much lower flux power, and enters another nonlinear element for optical compression in the next dispersion system— the compressor, which is the most energetically stressed element. The peculiarity of such a scheme is that the laser medium amplifies the stretched pulse with a lower intensity. While conventional techniques enabled the radiation to be focused by lenses in two mutually perpendicular directions, then the new technique does this simultaneously in three dimensions, sharply increasing the resultant power density on the target.

5 Laser-Driven Shock Waves

123

Synchronously with the development of the chirp technique, a new method for obtaining superhigh-power pulses was elaborated, based on optical parametric chirped pulse amplification (OPCPA) in nonlinear optical crystals [33]. The advantage of parametric amplification is an unprecedentedly high chirped-pulse gain: an energy gain of up to three–four orders of magnitude in one pass through the crystal. Another positive point is that the technology of growing wide-aperture crystals of the KDP family (potassium dihydrogen phosphate) has been thoroughly elaborated, which allows the energy of the generated pulses to be increased by scaling the amplifying stages. Currently, three classes of amplifiers—CPA based on titanium-sapphire crystals, CPA based on neodymium glass and OPCPA based on KDP and DKDP (deuterated potassium dihydrogen phosphate) crystals—are approximately at the same level of power: 1 PW. Existing and designed petawatt lasers are divided into three types according to the amplifying medium: (1) neodymium glass, (2) sapphire and (3) parametric KDP and DKDP crystal amplifiers (ref. Table 5.1 from [30]). In all three laser types, the energy (in the form of population inversion) is stored in neodymium ions in glass. In the first case, this energy is directly converted into the energy of a chirped pulse, which subsequently undergoes compression. In the second and third cases, the stored energy is converted into the energy of a narrowband nanosecond pulse, which is then converted into the second harmonic to serve a pump for chirped-pulse amplifiers. This pump either ensures population inversion in a sapphire crystal or decays parametrically into two chirped pulses in a nonlinear crystal. The peak power is determined by the duration of a compressed pulse and its energy. The maximum energy is achieved in neodymium glass lasers since the energy accumulated in the form of population inversion is directly converted into a chirped pulse. However, the narrow spectral band of laser neodymium glass amplification limits the compressed pulse duration at the level of several hundreds of femtoseconds. As a result, the optical resistance of diffraction gratings limits the advancement into the multipetawatt range. Unlike neodymium glass lasers, sapphire lasers provide broadband amplification, which allows pulses to be compressed down to 10–20 fs. At the same time, the aperture of sapphire crystals does not exceed 10 cm for the existing crystal growth technology. In an attempt to cross the petawatt threshold, this small aperture will limit the chirped-pulse energy due to optical decay and self-focusing. Currently, there are dozens of powerful lasers in the world that are either operating (ref. Table 5.2) or will soon be launched, and the promising technologies developed now give hope to obtain in the future an ultra-high power density of about 1028 W/cm2 [3, 7, 9]. Table 5.2 gives characteristics of such systems indicating planned experiments. The HERCULES laser system [34] can generate light pulses with a power of ≈ 300 TW once in 10 s. This laser shows the highest radiation intensity on a target of ≈ 2 × 1022 W/cm2 , obtained by pulse focusing in a focal spot ∼ 1 µm in size. The German PHELIX neodymium glass laser [35] allows obtaining laser pulses with an energy of up to 1 kJ and a power of > 500 TW. The laser operates in two

124

5 Laser-Driven Shock Waves

Table 5.2 Existing units with peak radiation power > 250 TW Country Laser name Scientific center

Laser type

Peak power

Pulse Minimum Maximum Recurrence Main areas of energy, duration, intensity, rate research J fs W/cm2

DKDP

1 PW

70

70

Several times per day

Physics of laser-induced thermonuclear fusion (LTF), extreme states of matter, laser acceleration of particles

560 TW 24

43

Several times per day

Electron acceleration, biomedical applications, laboratory astrophysics

Several times per day

Laser-induced thermonuclear fusion, high energy density physics

Russia Femta-Luch Russian Federal Nuclear Centre All-Russian Research Institute of Experimental Physics (RFNC VNIIEF)

PEARL DKDP RAS Institute of Applied Physics (IAP RAS) USA NIF Lawrence Livermore National Laboratory (LLNL)

Nd:glass 600 TW 1.8 × 106

3 × 106

Trident Los Alamos National Laboratory (LANL)

Nd:glass 250 TW 14

500

5 × 1020

TITAN Lawrence Livermore National Laboratory (LLNL)

DKDP

1000

1021

Several times per day

Ion acceleration, laboratory astrophysics

NIF ARC Lawrence Livermore National Laboratory (LLNL)

Nd:glass 4 × 1 PW

1000

1022

Several times per day

LTF diagnostics, direct ignition in LTF, astrophysics

250 TW 250

4× 100

Ion acceleration, laboratory astrophysics

(continued)

5 Laser-Driven Shock Waves

125

Table 5.2 (continued) Country Laser name Scientific center

Laser type

Peak power

Pulse Minimum Maximum Recurrence Main areas of energy, duration, intensity, rate research J fs W/cm2

ZPetawatt Sandia National Laboratory (SNL)

DKDP

1 PW

500

500

1021

OMEGA EP Laboratory for Laser Energetics (LLE), Rochester University

DKDP

1 PW

1000

1000

> 1020

50

2 × 1022

HERCULES Ti:Sa Center for Ultrafast Optical Science (CUOS) in the University of Michigan

300 TW 17

Several times per day

CTF diagnostics

New schemes of LTF ignition, high energy density experiments

0.1 Hz

Relativistic laser plasma, particle acceleration, X-ray radiation

TEXAS PETAWATT Texas University in Austin (UT Austin)

Nd:glass 11 PW

186

165

Several times per hour

Particle acceleration, biomedical applications

BELLA Lawrence Berkeley National Laboratory (LBNL)

Ti:Sa

11 PW

43

40

1 Hz

Electron acceleration, betatron emission

Diocles Nebraska University in Lincoln

Ti:Sa

700 TW 20

30

DKDP

1 PW

500

Several times per hour

Physics of LTF, extreme states of matter, laser acceleration of particles, laboratory astrophysics

UK VULCAN PW Rutherford Appleton Laboratory (RAL STFC)

500

1021

(continued)

126

5 Laser-Driven Shock Waves

Table 5.2 (continued) Country Laser name Scientific center

Laser type

Peak power

Pulse Minimum Maximum Recurrence Main areas of energy, duration, intensity, rate research J fs W/cm2

ASTRA GEMINI Rutherford Appleton Laboratory (RAL STFC)

Ti:Sa

2× 0.5 PW

2 × 20 40

ORION Nd:glass 2 × Atomic 0.8 PW Weapon Establishment (AWE)

1022

1/20 Hz

Particle acceleration, coherent sources of X-ray radiation, laboratory astrophysics

2× 500

600

3000

1500

Once per day

Laboratory astrophysics, CTF

CTF, high energy density physics

France PETAL Nd:glass 2 PW Commissariat a l’Energie Atomique Cesta Center SAPHIR Ti:Sa Laboratory of Applied Physics (LOA CNRS)

500 TW 20

40

1 Hz

Ion acceleration, proton therapy

LASERIX Paris Sud University

500 TW 30

60

0.1 Hz

X-ray lasers, particle acceleration Interaction of laser radiation with heavy ion beams

Ti:Sa

Germany PHELIX GSI Helmholtz Centre for Heavy Ion Research

Nd:glass 1 PW

500

500

3–4 times per day

PEnELOPE Helmholtz Center Dresden Rosendorf

Yb:CaF2 1 PW

150

150

0.1 Hz

DRACO Helmholtz Center Dresden Rosendorf

Ti:Sa

60

0.1 Hz

500 TW 30

High harmonics, particle acceleration (continued)

5 Laser-Driven Shock Waves

127

Table 5.2 (continued) Country Laser name Scientific center

Laser type

Peak power

Pulse Minimum Maximum Recurrence Main areas of energy, duration, intensity, rate research J fs W/cm2

Ti:Sa

1 PW

25

25

0.1 Hz

Particle acceleration, betatron radiation

Ti:Sa

1 PW

25

25

0.1 Hz

Particle acceleration, betatron radiation

200 TW 5

25

10 Hz

Interaction with matter within the range from X-ray to IR radiation

4× 0.5 PW

1000

1022

3–4 times per day

LTF, high energy density physics, generation of EM fields

33

1022

0.1 Hz

Particle acceleration, relativistic optics

2 × 15 30

1022

Romania CETAL-PW Center of Advanced Laser Technologies Spain Vega 3 Centro de Láseres Pulsados, University of Salamanca (CLPU) Canada ALLS Ti:Sa Institut National de la Recherche Scientifique (INRS) Japan LFEX Institute of Laser Engineering (ILE), Osaka University

DKDP

4× 500

J-KAREN-P Ti:Sa Kansai Photon Science Institute (KPSI JAEA)

850 TW 29

HERMES RIKEN Spring8

2× 0.5 PW

Ti:Sa

High energy density physics, together with free-electron laser (continued)

128

5 Laser-Driven Shock Waves

Table 5.2 (continued) Country Laser name Scientific center

Laser type

Peak power

Pulse Minimum Maximum Recurrence Main areas of energy, duration, intensity, rate research J fs W/cm2

Korea KLF Ti:Sa Advanced Photonics Research Institute (APRI), GIST

100 TW 3 1 PW 30

30 30

10 Hz 0.1 Hz

Generation of hard X-ray radiation, electron acceleration

Several times per hour

Relativistic laser plasma, particle acceleration, generation of X-ray radiation

China Quangguang Shanghai Institute of Optics and Fine Mechanics (SIOM)

Ti:Sa

2 PW

53

26

SILEX Laser Fusion Research Center (LFRC CAEP)

Ti:Sa

300 TW 10

30

1.1 PW

28

XL-III Ti:Sa Institute of Physics of the Chinese Academy of Sciences

32

Electron acceleration, high harmonics

1 Hz

modes—long (0.7–20 ns, 0.3–1 kJ, 1016 W/cm2 ) and short (0.5–20 ps, 120 kJ, 1020 W/cm2 ). As of 2017, record laser pulse powers of 2.0 PW were obtained at LFEX units in the Osaka Institute of Laser Engineering (ILE) [LFEX] and Quangguang units of the Shanghai Institute of Optics and Fine Mechanics (SIOM). In the latter case, the power was increased in the single amplifier channel with an active medium diameter of 100 mm by applying special antireflective coatings and precise synchronization and profiling of pump pulses [Siom OptExp]. The Lawrence Livermore National Laboratory commissioned an ARC (Advanced Radiographic Capability) picosecond multipetawatt laser combining four beams 1 PW each [ARC-SPIE]. The laser is created to expand diagnostic capabilities in controlled fusion experiments at the NIF facility, namely, to ensure a bright pulse source of ionizing radiation for probing fusion targets during their inertial compression and confinement.

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129

Let us consider a number of other projects intended to create powerful lasers [30]. First of all, let us note the Russian project [36] carried out in the Russian Federal Nuclear Centre (Sarov) with the participation of the Institute of Applied Physics of the Russian Academy of Sciences (IAP RAS). One of the channels of the “Luch” facility is used for pumping (second-harmonic pulse energy of 1 kJ, pulse duration of 2 ns). As a master oscillator for the “Luch” facility channel, a special laser was developed that ensures synchronization of pumping both with a femtosecond laser and pump laser of the previous stages of parametric amplification. The minimum energy of a chirped pulse after the final amplification stage is about 100 J at present. The efficiency of the four-grating compressor is 68%. Upon completion of all works for pulse compression, a power of about 2 PW was reached. In 2017, at the Higher Polytechnic School (France) with the support of CNRS and the French Atomic Energy Commission, the Appolon project was implemented to create a femtosecond laser with a record power of 5 PW [37]. A specific feature of this project is its hybrid architecture combining front-end OPCPA based on DKDP crystals with the primary Ti:Sapphire amplifier. The largest European laser project ELI (Extreme Light Infrastructure) is being successfully implemented [38]. The project is intended to create a network of three scientific centers with a super-powerful (10–30 PW) femtosecond laser constructed in each of them to conduct unique scientific research in the field of high energy density physics. In these pan-European projects, the femtosecond laser architecture (parametric amplification of chirped laser pulses with a center wavelength of 910 nm in a DKDP crystal) is assumed to be optimal for further scaling up. The OPCPA scheme is not the only one discussed scheme intended to create multipetawatt and exawatt lasers in the future. Sapphire crystals with an aperture of 30–40 cm became possible along with the use of several brands of neodymium glasses to enlarge the amplification band. Of special interest is the concept related with laser ceramics—a new optical material combining the advantages of glass and single crystal. The use of laser ceramics has substantially modified high average power laser systems. This area of laser technique is intensively developing and there is a large number of publications on low-power femtosecond ceramic lasers. In powerful lasers, ceramics has never been used in practice so far, but in the future, ceramics-based lasers may well compete with neodymium glass, sapphire and OPCPA lasers. In particular, a concept was proposed [30] to create super-powerful femtosecond lasers based on Cr:YAG ceramics combining both conventional principles (energy source is nanosecond neodymium glass laser pulses, CPA) and new opportunities opened up due to the use of laser ceramics. As shown in Table 5.1, Cr:YAG ceramics simultaneously have three key properties: a wide amplification band that allows amplifying pulses to 20 fs; a large aperture making it possible to amplify chirped pulses to multikilojoule level; high efficiency of converting the narrow-band radiation of neodymium glass lasers. These properties open the way for a unique laser with a peak power of 100 PW and a pump energy of 10 kJ. It’s worth noting that elements made of Cr:YAG ceramics have not been used as active elements, but they are widely applied as passive Q-switches.

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In addition to a large aperture, an extremely important advantage of ceramics is an opportunity to create active media that practically cannot be grown as a single crystal. An example is oxides of rare-earth elements doped with neodymium and ytterbium: Nd: Y2 O3 , Nd: Lu2 O3 , (Nd, Yb): Sc2 O3 , Yb: Y2 O3 , etc. Another option was proposed for constructing a multipetawatt laser: CPA in wide-aperture ceramics (Nd, Yb): Lu2 O3 or (Nd, Yb): Sc2 O3 with lamp pumping similar to neodymium glass lasers the pulse duration of which is much longer. Excitation from neodymium ions is transferred to ytterbium ions that ensure a wide band. Direct pumping of ytterbium is possible only with diode lasers, which makes it difficult to scale up. An even wider band can be obtained by the simultaneous use of several oxide crystals (Sc2 O3 , Y2 O3 , Lu2 O3 , etc.) similar to the use of several brands of neodymium glasses or several garnets with a chromium ion. Thus, new petawatt and multipetawatt projects based on CPA in laser ceramics can appear in the nearest future. Petawatt lasers created throughout the world will soon become a tool for mastering a new realm of knowledge in high energy density physics—physics of extreme light fields. In the future, petawatt lasers may be used as charged particle accelerators in basic research, defense technology, and medical applications. Among the last-named ones, mention should be made of the construction of an isotope factory for positron emission tomography as well as of a compact inexpensive ion source for hadron therapy. These and other potential applications as well as significant progress in the field of petawatt lasers generate interest among commercial companies in mastering the petawatt range, which lends additional impetus to the development of laser technologies. This gives a hope that in the nearest years petawatt lasers (including OPCPA-lasers) will become an available tool for fundamental and applied studies. Transition to shorter (attosecond) durations of laser pulses opens new interesting opportunities in high energy density physics, chemistry, biology and medicine [28]. It seems that the first sources of attosecond pulses were sources based on the generation of high harmonics during the interaction of laser radiation with a gas jet. Due to the nonlinear effects of radiation on the atom, high-frequency harmonics appear already at 1013 W/cm2 during its ionization. The use of a supershort (1–2 optical periods) laser radiation made it possible to obtain pulses of ≈ 100 attoseconds on jets, which is interesting for studying atomic and molecular processes at ultrashort times. It is more efficient to generate supershort radiation during laser interaction with solid-state targets [28]. The Doppler effect was considered in paper [28] as one of the mechanisms to generate such radiation. The next substantial step in studying the mechanisms of generation of high harmonics on the surface of solid-state plasma was paper [39] the authors of which paid attention to the fact that the radiation of electrons in the ultra-relativistic limit has a synchrotron nature and occurs during a very short period of time, so that the generated spectrum cut-off frequency is determined by the cube of the relativistic factor of electrons rather than by its square. This theoretical result was confirmed experimentally at the Vulcan facility in the RAL [40–42].

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One of the most intriguing goals of converting optical radiation into attosecond pulses is to solve the task of getting superhigh intensities to observe vacuum nonlinearity effects. In particular, the results of recent calculations and evaluation [18] show that to observe avalanches of electron–positron pair production in vacuum, an intensity of about 1026 W/cm2 is required, which is three orders higher than the intensity that will be obtained by international projects in the short term [38]. One of the obvious ways to increase the intensity is to decrease the wavelength, which allows decreasing the volume where the energy is concentrated in case of extreme focusing limited by the diffraction limit. In addition to the scheme of focusing attosecond pulses generated on the plasma spherical surface [43], an idea was proposed to focus a counter propagating laser pulse reflected from an electron mirror moving at the relativistic velocity and occurring in case of wakefield breaking [44] or ponderomotive expulsion of electrons from thin films [45]. In practice, this mechanism is highly limited by a relatively low efficiency and poor spatial coherence of the generated radiation. From the point of view of improving the energy transformation efficiency, the mode of high-power attosecond pulse production in case of the oblique radiation of the supercritical plasma surface appears to be highly attractive [46, 47]. An electromagnetic wave incident on the layer causes electrons to be shifted deep into the plasma by means of the ponderomotive force. In contrast to the normal-incidence case, in case of oblique incidence, the presence of a plasma flow in a moving frame of reference gives rise to the appearance of a magnetic field created by bare ions when electrons are shifted deep into the plasma [28]. This, in turn, determines the difference in the ponderomotive action of the wave on electrons at two half-periods of the field. The emergence of internal fields in the plasma and the accelerated motion of some plasma electrons leads to the accumulation of the incident wave energy by the plasma at this stage. Since, in case of linear polarization, the force of light pressure oscillates during the field period, a nano beam formed from shifted electrons breaks loose by the charge separation force at some point in time and travels towards the incident wave to become a source of the attosecond burst. In this case, the energy accumulated at the first stage is radiated for a time of about several tens of attoseconds. This threestage description of the process is called the model of a relativistic electron spring due to its similarity with the scenario of energy accumulation by a mechanical spring [48]. Numerical simulation based on the particle-in-cell method shows that a radiation intensity of 1.8 × 1026 W/cm2 can be achieved in the area of several nanometers in size when using a ten-petawatt laser pulse. Another promising mechanism [28] for the generation of high harmonics is the so-called coherent wakefield radiation [49] that consists in the following. When laser radiation interacts with the solid-state target surface, electron bunches are generated and propagate deep into the plasma [50]. In case of charge concentration gradient, these bunches can [28] excite plasma oscillations the frequency of which is a multiple of the bunch repetition rate and the latter, in turn, is determined by the frequency of a laser pulse. These plasma oscillations occur at a certain charge concentration

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gradient. Therefore, they are capable of emitting electromagnetic waves of the same frequency. In this manner, the generation of harmonics takes place, and their frequency reaches the plasma frequency corresponding to the maximum concentration of electrons in the target as the concentration of electrons increases. The generation of coherent wake radiation was observed in a number of experiments. In an experiment carried out in 2004 [51], harmonics were observed on the reverse side of the irradiated target, when thin carbon and aluminum foils were exposed to laser radiation with an intensity of about 1018 W/cm2 . In case of irradiation with a laser pulse with an intensity of only 2 × 1016 W/cm2 , harmonics up to the 18th were observed. To conclude, we can note that in the nearest future we should expect the emergence of multipetawatt laser sources capable of achieving a radiation intensity of 1023 W/cm2 . Today, several countries in the world simultaneously develop laser facilities with a rated peak power of 10 PW. These are Vulcan-10PW in the UK, ILEApollon in France, and PEARL-10 in Russia. Within the framework of the European ELI megaproject, three superhigh-power laser complexes are being constructed in Czechia, Hungary and Romania, which will be used to study fundamental physics in super-intense fields, problems of generation of attosecond pulses, and photonuclear processes. Anticipated is the generation of monoenergetic electron beams with energies of several gigaelectronvolts, ion beams with an energy at a level of 1 GeV, ultrabright gamma-ray radiation with photon energies of the order of several gigaelectronvolts, ultrashort pulses of subattosecond duration, as well as the attainment of record-high radiation intensities at a level of 1026 W/cm2 with the use of attosecond pulses [28]. Further advancement along the laser intensity scale (Fig. 5.2) is now hard to predict, since it is limited by our knowledge of the structure of matter in the immediate spatiotemporal neighborhood of the Big Bang at ultrahigh energy concentrations. Laser-Driven Shock Waves. Apart from the problem of creating and using highpower lasers, a special problem is the measurement of the parameters of compressed and heated matter. Partially, this is related with small spatial dimensions, strong inhomogeneity and a short lifetime of matter in a highly compressed and heated state. In principle, this difficulty can be overcome by increasing the laser energy. However, it can be easily seen that the required energy is proportional to the cube of characteristic size (or to the cube of inertial confinement time, which is the same). Therefore, the practical possibilities of increasing the scale of experiments are rather limited. Another difficulty in diagnosing compressed and heated matter is due to the geometry of spherical laser experiments in which the compressed inner layers of the target are “screened” by the hot and relatively tenuous expanding plasma of the corona. The diagnostics is substantially simplified in case of transition to planar compression geometry since in this case, to determine the parameters of compressed matter, the laws of conservation of mass, momentum and energy can be used which have the

5 Laser-Driven Shock Waves a

b

Laser radiation

Ablation front

Shock-wave front

Fig. 5.4 Schematic representation of the first experiments with laser impact on the target. a The vapor expands along arrows 4 towards the laser beam shown by arrows 1. If the vapor is transparent for radiation, then absorption takes place in surface layer 2. This layer separates the target and the vapor. Vapor pressure 4 sets target substance 3 in motion; b laser target design in the experiment

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simplest form of a system of algebraic equations in case of a unidimensional quasistationary flow (ref. Chap. 1). The approach based on the application of conservation laws to stationary shock wave discontinuity is the main one in dynamic high-pressure physics [51] as we could see in Chaps. 1 and 3. It was successfully used for studying the thermodynamic and kinetic properties of matters in the megabar pressure range. In this case, chemical explosives, nuclear explosions or light-gas launching devices were used to generate shock waves. The employment of high-power lasers to generate shock waves makes it possible to substantially extend the attainable pressure range. It is clear that a classical approach in laser experiments, based on using plane quasi-stationary shock waves (ref. Chap. 3) is also the most promising for carrying out quantitative measurements. To date, most of the experiments in the laser generation of powerful shock waves have been performed in such a traditional formulation that imposes certain restrictions on target and laser pulse parameters. In this formulation, it is currently possible to generate plane laser-driven shock waves in metals with a pressure behind the front of tens of megabars and even gigabars, which several times exceeds the values available for the conventional technique of condensed explosives and light-gas guns and is comparable only with the pressures achieved in the near zone of underground nuclear explosions (ref. Chap. 6). Immediately after the advent of lasers, their unique properties have been successfully used to generate powerful shock waves and obtain extreme states of matter with their help [11]. The scheme of the first experiments is shown in Fig. 5.4. Prolonged exposure of the target to a laser radiation (Fig. 5.4) produces a cloud of vaporized matter above its surface. In case of moderate laser intensity, this cloud

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Fig. 5.5 Laser crown structure near the target: 1 laser beam; 2 vicinity of the critical density surface where laser energy is absorbed; ρ matter density profile in target 6 and corona 5; profiles 8 and 9 are referred to cases of thick and thin targets, respectively; 4 ablation front dividing cold and dense matter to the right from the front and hot, low-dense plasma to the left in the figure. Arrow 3 is the heat flow from critical surface 2 to ablation front 4. Left arrow u designates plasma outflow from the ablation front. Hot plasma pressure in subsonic “cushion” 7 moves the dense matter of the target towards right arrow u. In this sense, the corona acts as a piston pushing the target substance

is a vapor consisting of neutral molecules or atoms, which is usually transparent for radiation [52]. Energy is absorbed in condensed matter of the near-surface layer on the target. The layer is heated, melts and evaporates. There is a vapor flow moving towards the laser beam [52]. This flow exerts a pressure under the action of which the vapor expands towards the laser beam. At high intensities I, the matter removed from the condensed target surface ionizes to form a plasma cloud (or a plume). This cloud is also called the laser plasma corona. The issues of the corona dynamics have been studied in detail for many years in connection with the problem of laser-induced inertial thermonuclear fusion, see, for example [11, 53, 54]. The plasma is usually opaque to the radiation—absorption takes place inside the corona (with density falling down in the direction from the target) near the critical surface. This surface is so named because the electron density decreases to a critical value at which the plasma frequency ωp falls down to laser  radiation frequency. ω ∼ ω p ∼ 4π e4 n e /m e This density is about ≈ 10–100 times (depending on the hardness of a laser photon) lower than the solid-state density, with which the plasma is delivered to the corona. The heat absorbed on critical surface 2 in Figs. 5.5 and 5.6 is transported by the electron thermal conductivity mechanism to the target surface. Heat transfer by heat conductivity is shown by arrow 3 in Fig. 5.5. The respective profile of a heat wave is shown in Fig. 5.6. Ablation front 4 forms the target boundary. The solid-density substance begins to heat up at this front, its density lowers, and a new portion of the substance is delivered to the plasma flow from the target to the corona. The velocity of this portion of the substance is shown with left arrow u in Fig. 5.5. This part goes to critical surface 2, crosses it and continues expansion in vacuum.

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Fig. 5.6 Profiles of density ρ, pressure p, and temperature T in the case of the action of a long pulse of sufficiently high power. This case is of interest for laser-induced thermonuclear fusion problems; 1 laser radiation beam reaching critical density surface 2; 4 ablation front. This is the leading front of the thermal wave, beyond which the temperature rises sharply. The pressure of the corona sustains the quasi-stationary propagation of the shock wave (SW) in the homogeneous substance of the target

The target surface is referred to as the ablation front. A relatively cold target substance of solid-state density is located to the right of front 4 (Fig. 5.5). The plasma flow is subsonic in gap 7 between the surface with critical density 2 and ablation front 4. Hot plasma pressure ahead of the ablation front is transferred to the substance of solid-state density behind the ablation front. As in the case of vapor, the corona pressure causes the flow in the target shown by the right arrow u in Fig. 5.5. Arrows u indicate the flow direction in the coordinate system related with the initially immobile substance of the target. In case of a thick target, the profile of density ρ extends deep into the target along straight line 8 in Fig. 5.5. The corona pressure sustains the shock wave that propagates into the target depth. This shock wave (SW) is shown in Fig. 5.6. Recording the parameters of this wave in the experiment, we can obtain information about the thermodynamics of shock-compressed matter (ref. Chap. 3). The recording of the shape of the compression pulse as it exits the free surface yields ample information about the mechanical properties of the medium at high pressures, temperatures, and deformation rates (Chap. 4). In case of a thin target, the pressure turns zero on the rear side of the target (reverse to ablation front 4). Therefore, the substance pressure and density fall down in the direction of curve 9 in Fig. 5.5. The pressure gradient in dense substance to the right of the ablation front in Fig. 5.5 accelerates the target substance. This is a basis for the famous principal of the acceleration of spherical shells. A qualitatively occurring flow of substance can be represented as consisted of three sections: a stationary shock wave (1) followed by the Chapman Jouget deflagration wave (2) where light energy is absorbed, with a simple centered fracture wave (3) in the end, being adiabatic or isothermic. In this case, the highest attainable pressure is pmax = I 2/3 ρc1/3 ,

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where I is the laser radiation intensity with frequency ω, ρ c is the critical plasma  density (ω0 = ω p = 4π e2 n e /m e ). Analyzing this relationship, we can see that the pressures attainable with the laser method of generation depend slightly on the chemical composition of the target. Herein lies a significant difference between the method of laser generation and the classical methods in dynamic physics, which use the impact of metal plates or the detonation products of condensed explosives (ref. Chap. 3). It is characteristic that high-frequency radiation has significant advantages in terms of obtaining maximum plasma pressures. However, the main advantage of short-wavelength radiation consists in lowering the effect of nonthermal electrons with an increase in laser frequency. One should bear in mind that the above qualitative estimate was obtained for the interval of laser pulse parameters in which the above-mentioned simplified interaction model is purely schematic [11]. The point is that the absorption of light is essentially nonlinear at radiation intensities of 1013 –1017 W/cm2 . In this case, a substantial fraction of light is reflected from the plasma, and the reflection coefficient also depends on the radiation intensity. As a result, the formula for the absorption coefficient, used to estimate the screening time turns out to be oversimplified. At high laser intensities, a whole series of complications arise, which can be fully taken into account only within the framework of complex numerical simulation. The results of these calculations are approximated by the relation: −β

pmax = α I α λ0 , where α ≈ 0.3–0.7, β ≈ 0.3–2.0, depending on the model and on the range of radiation intensity [11]. In [11], requirements were formulated for laser radiation and the size of targets required to obtain in the substance under study plane and stationary shock wave discontinuities, propagating through a relatively cold substance. This will make it possible to use the dynamic diagnostic method (Chap. 3) based on the application of conservation laws to the flow of a shock-compressed plasma. The limitations emerging in this case will define the highest pressure level attainable with the aid of modern lasers. Taking into account the attenuation and curvature of the shock wave due to the rear and side unloading waves as well as due to the nonhydrodynamic target heating by nonthermal electrons, the authors of paper [11] obtained that the laser energy E las at wavelength λ0 required for generating pressure p can be represented as: E las ∼ p 6 λ11 0 . Therefore, the constructive way of advancing further along the pressure scale of laser-driven shock waves involves the use of shorter wavelengths and layered targets to suppress the role of electron heating. Therefore, in experiments with laser-driven

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shock-waves, experimenters aspire to use the high-order harmonics of the fundamental radiation or laser-to-soft X-ray radiation conversion in “hohlraum” schemes [1, 2]. In accordance with the general ideology of dynamic experiments ([51], Chap. 3), to study the equation of state, we must independently record any two of five parameters characterizing the propagation of a plane stationary shock wave discontinuity: D, u, p, V, E with further calculation of other parameters according to general conservation laws at the shock-wave front. This allows us to find the equation of state of matter under study in the caloric form E = E(p, V ). The phase velocity D of motion of a shock wave can be easily and precisely recorded in dynamic experiments using basic electro-contact and optical methods. In case of spacing of ≈ 20–50 µm, characteristic for laser-driven shock waves and discontinuity velocities of 20–50 km/s, the characteristic time of the experiment is about 1 ns, which makes the use of optical methods preferable in the experiment. In opaque materials, a shock wave can be recorded when it reaches the free surface of stepped targets. In this case, it becomes visible if it approaches a distance of several lengths of light run l r from the free surface. For the optical range in metals lr ≈ 10– 5 cm, at the velocity D ≈ 20 km/s, this will lead to the rise of a light signal for the time τ ∼ lr/D ≈ 5 ps, which defines the necessary time resolution in these experiments of electronic-optical converters. The measurement of the second dynamic parameter is associated with much greater difficulties. In dynamic experiments using explosive equipment and lightgas launching devices, the reflection method (ref. Chap. 3) based on the applications of general gas dynamic principles occurring during the decay of an arbitrary discontinuity is widely used to determine mass velocity u of a shock-compressed substance. Specific measurements are reduced to recording phase velocities of shock waves in the substance under study and in “standard” (substance with a known shock adiabat), which, due to the condition of pressure and velocity discontinuity at the contact boundary, makes it possible to fund the mass velocity of the substance under study. Independent determination of a shock adiabat of the “standard” is performed by the “deceleration” method in which the velocity of the shock wave in the standard along with the flight speed w = 2u of the striker made of the “standard” material are recorded. The “reflection” method is the main one in dynamic high-pressure physics and it was used to find the shock compressibility of a large number of chemical elements and compounds. Guidebooks [55, 56] give characteristics for more than 300 substances studied so far at pressures up to ≈ 5 Mbar. The use of the technique of strong underground explosions and detonation of nuclear charges (ref. Chap. 6) made it possible to obtain shock pressures of tens of megabars for which the problem of standards is quite acute since the use of the “reflection” method here is difficult. In this case, it is possible to measure only the relative compressibility of substances using the standard adiabat extrapolated beyond the area of direct measurements. Elements with a high atomic number are used as standards, for which quantum-statistical calculations are more reliable. It is substantial that far extrapolations (5–300 Mbar) must be used in constructing standard shock adiabats.

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The method of recording u proposed in paper [57] is based on measuring the shift of the contrast interface of heavy and light matter using an X-ray velocity recorder with lateral X ray exposure of a layered target [58]. According to estimates in paper [57], this method may give the measurement accuracy of u of about 6–8%. In next-generation laser experiments, a more complete set of physical parameters is recorded apart from kinematic characteristics. In case of significant compression the matter in shock waves of extreme intensities, the recording of wave and mass velocities is not optimal in terms of accuracy of determining the equation of state (ref. Chap. 3). In this case, it is desirable, along with D, to directly measure the plasma density in the shock wave either by the absorption of X-ray radiation as it was done in papers [57, 59] for non-ideal plasma or by the Stark broadening of spectral lines in the X-ray region of the spectrum [60]. Temperature (pyrometric or spectral Doppler) measurements are quite helpful since, in view of the manifestation of the shell structure at high pressures, it is precisely the temperature dependence which would be expected to exhibit significant nonmonotonicities. Experiments with plane stationary shock waves are the simplest and easily interpreted type of experiments to study matter properties at high pressures and temperatures. However, the use of power lasers opens other interesting opportunities in experimental high energy density physics. Irradiation of thin (1–10 µm) metallic targets by laser radiation with I λ20 > 106 Bt leads to heating of the opposite side of the target by non-thermal electrons [61, 62] and ions. Theoretical models belonging to this poorly studied phenomenon can be corrected by comparing the measured temperature with calculation data [63, 64]. At lower values of I λ20 , this method can be used to study the specific features of electronic heat conductivity in laser plasma [64–66]. It should be noted that non-thermal electrons are an efficient source of fast volumetric exchange of condensed matter to temperatures of several electron-volts. The decay of such high-temperature states was used in paper [67] to generate intense shock waves. In case of single shock compression, the density of compressed matter cannot exceed a certain limiting value. Therefore, in the simplest experiments with a single stationary shock wave, an extremely interesting region of the phase diagram corresponding to super-compressed but relatively cold matter appears to be unattainable. To move to this region, a number of approaches was proposed having a common feature—low entropy growth during compression. In experiments with high-power lasers, a sequence of several pulses optimally selected in time, amplitude and frequency or a single continuous pulse of special shape with growing intensity can be used instead of a single pulse [68, 69]. Instead of programming a laser pulse shape, a layered or shell-type target can be used with a specially selected initial density profile. Some other possibilities are discussed in paper [70]. Let us note here that the problem of measuring the parameters of compressed matter becomes rather complicated in all experiments with quasiisentropic compression. In resembling experiments aimed at solving the problems of laser-induced thermonuclear fusion, the use was made of X-ray photography, spectroscopy of multicharged ions, recording of thermonuclear fusion products and indirect methods based on measuring corona parameters. Certain possibilities are

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offered by the method of recording shock waves at an appreciable distance from the place of laser radiation focusing. The application of the point explosion theory [71] allows evaluating the effective explosion energy [72]. Laser methods can be useful not only in the study of highly compressed matter with a density exceeding the solid state. Using an expansion of a material heated by a laser-driven shock wave or nonthermal electrons [67], a wide range of states can be obtained in an isentropic unloading wave, including the region of the Boltzmann strongly non-ideal plasma, the vicinity of the high-temperature boiling curve, and the metal–dielectric transition region [59, 73]. Below, we will consider some experiments on laser generation of shock waves in solids and discuss the physical effects associated with such method of shock production. Currently, two sets of experiments are used for laser generation of powerful shock waves—direct generation (direct drive—Fig. 5.4) and conversion into soft X-ray (indirect drive). Let us start with the first one (Figs. 5.4, 5.5 and 5.6). The first experiments on the excitation of shock waves in solid hydrogen and plexiglass were carried out with a low-power neodymium laser [74] for the energy E ≈ 12 J and the pulse duration τ = 5 ns. Owing to the small (≈ 40 µm) size of the focal spot, the shock waves with a peak pressure of ≈ 2 Mbar rapidly became spherical and decayed. In paper [75], at the radiation intensity I = 3.5 × 1014 W/cm2 , a shock pressure of ≈ 1.7 Mbar was obtained in polyethylene, and in paper [76], at the radiation intensity I = 2 × 1014 W/cm2 , the amplitude pressure in hydrogen and plexiglass was equal to ≈ 2 and 4 Mbar, respectively. Measurements of the plasma corona ion energy and the target recoil momentum (integral methods) made it possible [77] to estimate the pressure in an aluminum target at I ≈ 1014 W/cm2 . A more powerful Janus laser system [78] based on neodymium glass with a pulse energy of up to 100 J and a pulse duration of 300 ps was used in [79–81] to generate plane shock waves. Radiation intensities I of up to 3 × 1014 W/cm2 were produced in a focal spot 300–700 µm in diameter. The velocities of the shock wave discontinuity front of 13 km/s (corresponding to a pressure of ≈ 2 Mbar) were recorded and the velocity of the plasma corona expansion was measured from the time when the shock wave passed through the stepped aluminum specimen. The recording of radiation intensity buildup with time as the shock wave reaches the free surface ( t ≤ 50 ps) allowed estimating the thickness of shock wave discontinuity (thickness is  0.7 µm). Then, the shock pressures were increased by an order of magnitude. The authors of paper [81] used a small-diameter target to lower, in the authors’ opinion, the effect of surface currents [82, 83], and an agreement was obtained between theory [84] and experiment [81]. Plasma pressures p ≈ 35 Mbar under the action of neodymium laser radiation with λ0 = 1 05 µm were obtained [85] by irradiating a target consisting of a 22 µm thick aluminum layer and a 32 µm thick gold layer by ten overlapping beams of the Shiva laser facility [86]. The peak intensity was equal to 2.9 × 1015 W/cm2 with a pulse duration of 625 ps. The recorded velocity of the shock wave in gold was 17.3 ± 0.3 km/s and was consistent with a two-dimensional hydrodynamic calculation [84], in which the fraction of the absorbed in plasma energy was equal to 30% and

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the laser beam convergence was taken into account [87]. The measurement of the X-ray emission spectrum in this experiment made it possible to estimate the heating of the rear target side by nonthermal electrons. It turned out to be under 500 °C, while the temperature of the shock-compressed plasma was about 5 eV. A version of the “reflection” method was implemented in experiments [88] with a layered target, when a shock wave with the amplitude p ≈ 3 Mbar underwent transition from aluminum to gold (p ≈ 6 Mbar) under low-attenuation conditions. Like in [79], the laser radiation (E ≈ 20–30 J, τ = 300 ps) was nonuniformly distributed over the focal spot, its shape varying from circular (c ∅100 µm) to elliptical (with axes of 200 and 500 µm). In the authors’ opinion, the latter circumstance was the main reason for errors in these experiments (δD = 15%, δp = 30%). Systematic investigations of the shock compressibility of aluminum and copper were carried out using a comparative method on the Janus laser facility [89] in the intensity range I ≈ 5 × 1013 –4 × 1014 W/cm2 (E ≈ 30 J, τ = 300 ps). A thin layer of gold in the target was employed to absorb nonthermal electrons and increase the duration of the shock wave (somewhat lowering, however, the peak pressure). The resultant data refer to a 2–6 Mbar pressure range in aluminum and 4–8 Mbar in copper and are in good agreement with the results of dynamic experiments performed using high-power explosives and light-gas launching facilities [90, 91]. Along with the reflection method, a technique for measuring the velocity of plasma motion [58] by pulsed X-ray radiography is being elaborated in laser experiments. A 17 µm thick aluminum target was irradiated by a laser beam of the Shiva facility with the intensity I = 6 × 1014 W/cm2 (E = 110 J, τ = 600 ps). In this case, one of the beams of this facility irradiated a tantalum target, which gave rise to X-rays with a characteristic energy of 1.9 keV. The motion in the field of X-ray radiation was recorded with a fast-response X-ray camera (an X-ray microscope [58]) having a time resolution of 15 ps and a spatial resolution of 4.5 µm, which permitted the velocity of plasma motion to be measured at 8 × 106 cm/s. Mention should be made of the experiments on the generation of shock waves by short-wavelength laser radiation [80], since for these modes an increase in the amplitude pressures of the shock-compressed plasma is predicted due to an increase in the fraction of absorbed laser energy and suppression of nonthermal electrons. The laser radiation with λ0 = 0.35 µm and an intensity of (1–2) × 1014 W/cm2 (τ = 700 ps) was used to irradiate a 25 µm thick aluminum target in which a shock wave was generated with a pressure of 10–12 Mbar. In this case, the energy absorbed by the plasma amounted to ∼ 95% of the incident flow. Irradiation by long-wavelength (λ0 = 1.06 µm) radiation with I = 3 × 1014 W/cm2 (the absorbed intensity was 1.2 × 1014 W/cm2 ) for the same experimental set-up produced a pressure of ≈ 6 Mbar. In the paper, NiKe laser KzF radiation was focused on the target giving a power density of 1013 –1014 W/cm2 . The “reflection” method (ref. Chap. 3) was used to find the parameters of the equation of state for deuterium plasma under a pressure of 1–6 Mbar. The recorded high plasma compressibility is related [92] with deuterium dissociation processes. Paper [93] is dedicated to measuring the optical properties of diamond within the shock-wave pressure range of 6–10 Mbar. OMEGA laser was used to generate

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(reflection)

(own emission)

Lenses

(pyrometer)

Prisms

Target HIPER-laser

Fig. 5.7 Experimental scheme for GEKKO/HIPER laser facility [94]

shock waves (ref. Table 5.2), and VISAR laser interferometer was used to record the kinematic characteristics of shock wave motions and the reflective properties of compressed diamond. The obtained data made it possible to specify the carbon phase diagram within the megabar pressure range. The scheme of the experiment carried out at the GEKKO/HIPER laser facility is shown in Fig. 5.7 [94]. In these experiments, 9 laser beams were optically mixed on the target ensuring power density of about 1014 W/cm3 for a wavelength of 351 nm and a pulse duration of 2.5 ns. Data were obtained for the shock compressibility of some metals under pressures of tens of megabars. In paper [95], laser-driven shock waves with an intensity of up to 14 Mbar were used to study MgO phase diagram and its metallization applicable to creating exoplanet models. Experiments for the “direct” generation of laser-driven shock waves at the OMEGA facility (200–1130 J, 2 ns, λ ∼ 351 nm) allowed finding the parameters of the equation of state, temperature and reflective ability of the strongly non-ideal plasma of polypropylene within the pressure range of 1–10 Mbar using the reflection method (ref. Chap. 3). The pressure range of 0.45–2.2 Mbar for deuterium that causes interest was studied in [96]. The OMEGA laser (3 kJ, 3.7 ns, 350 nm) and aluminum standards for the reflection method were used (Chap. 3). In [97], interesting experiments on the direct recording of shock-compressed plasma density determined upon X-ray absorption were conducted. This helped determining the parameters of the equation of state at high pressures of shock compression. Laser experiments on the direct generation of shock waves in shock compression of targets preliminarily compressed in diamond anvils were further developed (ref. for example [98]) (Fig. 5.8).

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Diamond anvil 100-300μm

Target Shock

Laser Diamond anvil

Fig. 5.8 Scheme of laser experiment with a specimen pre-compressed in diamond anvils [58]

Preliminary compression of targets in diamond anvils allowed increasing several times the initial density of the specimen before shock compression and thereby significantly expanding the region of the matter’s phase diagram available for shock wave experiment. Taking into account the construction of adequate models of giant planets, experiments were carried out to generate laser-driven shock waves in water specimens preliminary compressed to 10 kbar (1 GPa). Changes in the optical properties and growth of H2O reflection coefficient in the megabar (above 1.3 Mbar) range of pressures were recorded. Effects of changes in helium electronic structure were experimentally studied [99]. A specimen of liquid helium was compressed before lasering (3 kJ, 1 ns, 351 nm) in diamond anvils by 3.3 times in order to increase the shock compression density. Pressures of about 1 Mbar were obtained. In [100], a similar scheme was used for experiments on measuring optical temperature and light reflection of shock-compressed helium at a density of up to 1.5 g/cm3 and T = 60,000 K. A conclusion was made on the semi-conductive nature of electron processes in the strongly non-ideal plasma of helium. A number of other early works for “direct” laser generation of powerful shock waves can be found in papers [11, 51]. A restrictive specific feature of laser-driven shock waves in case of their direct excitation is that it is hard to ensure high homogeneity of target laser exposure. The heterogeneities unavoidable in operation of high-power lasers, when a laser beam hits the target, generate noticeable heterogeneities (irregularities) of the shock-wave front, which prevents from using dynamic diagnostics methods (ref. Chaps. 1 and 3)

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based on the use of conservation laws of mass, momentum and energy in algebraic form. A way out from this situation is using the “indirect drive” scheme [1, 2, 101–108] when laser radiation of low spatial homogeneity irradiates walls of a closed volume (hohlraum) generating “soft” X-ray radiation that, in its turn, ensures exposure of the laser target to soft laser radiation of very high homogeneity. This scheme of conversion into “soft” X-ray appeared to be rather efficient not only for laser-induced thermonuclear fusion (Fig. 5.9a), but also for the high-quality laser generation of laser-driven shock waves [101, 109]. This conversion idea was successfully used for the ablation acceleration of golden foil that excited pressure of ∼ 0.7 Gbar [104] when impacting a stepped target (Fig. 5.9b), which is however lower than the record value of 4 Gbar obtained in the near zone of the nuclear explosion (ref. Chap. 6). Equations of states of shock-compressed polystyrene (40 Mbar) and beryllium (15 Mbar) were then obtained according to a similar experimental scheme of “conversions” (Figs. 5.10 and 5.11, [105]) in the field of the parameters characteristic for nuclear explosion shock waves (ref. Chap. 6). A similar experimental set up was used in [106] to heat a converter where two laser beams of the HELEN facility were used (an energy of 500 J, a wavelength of 0.53 µm). High (∼ 1%) accuracies of recording the kinematic parameters of laser-driven shock waves were implemented in the experiments. Measurements of kinematic shock waves resulted in the parameters of shock compressibility of Cu, Au, Pb and plastics within the megabar range of pressures. Powerful shock waves are the necessary (or unavoidable) tool for the practical implementation of the idea of laser controlled thermonuclear fusion with inertial confinement of hot plasma [1, 2, 101, 102] in spherical geometry. These unique possibilities of powerful laser-driven shock waves became a basis for an individual area of laser thermonuclear “shock-wave ignition” [101, 102, 110– 116]. In this scheme, a single high-energy laser pulse “slowly” compresses a spherical target and its trailing short pulse excites the second, more powerful shock wave in the target, which provides the thermonuclear ignition of the precompressed target [101, 110–116]. Therefore, a high attention is paid now to studying laser-driven shock waves in spherical targets [101, 117]. Paper [103] analyzes the characteristics of the shock waves generated in CTF microtargets when exposed to laser radiation. It was demonstrated that at the stage of centripetal convergence, possible intensities of shock waves can reach several gigabars per implosion radii of ∼ 100 µm. Experiments carried out using the OMEGA laser [107] showed a significant amplification of spherical shock waves when they converge to the target center. As a result, the velocity of the spherical shock wave reached 135 km/s, which corresponded to a pressure of ∼ 25 Gbar. Paper [118] gives results for OMEGA laser experiments to generate the spherical shock waves of high intensity and symmetry. In case of specific density and power of

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Fig. 5.9 a Experimental scheme for laser radiation conversion into “soft” X-ray; b, c scheme of conversion of laser radiation into “soft” X-ray with further generation of shock waves in stepped targets [101, 109]

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Fig. 5.10 Scheme of laser acceleration of the striker [104]

Fig. 5.11 Scheme of experiments on the generation of powerful laser-driven shock waves using laser radiation conversion into “soft” X-ray radiation [105]

3 × 1015 W/cm2 , maximum pressures exceeding 1 Gbar were recorded in the center of the microtarget. Paper [119] presents interesting experiments on the compression of a spherical fusion target by a sequence of three shock waves. It was demonstrated that such scheme has an increased compression stability and gives a yield of thermonuclear neutrons (D–I) of about 9.3 × 1015 . An example of the practical implementation of the shock wave ignition scheme is given in [119]. In experiments with the OMEGA laser facility, the spherical microtarget was compressed by a low-power laser beam and further heating of compressed plasma was done using a highly-intensive beam. A significant, though weakly dependent on the intensity of laser radiation, contribution of non-thermal (∼ 30 keV) electrons during compression was noted.

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Even a brief enumeration of possible and already carried out experiments at very high local energy densities attainable with modern lasers shows that this technique has enormous advantages over other methods of obtaining high pressures and allows acquiring new physical information about extreme states of matter. Naturally, such measurements are at the forefront of modern fast-response recording instruments and the interpretation of these experiments calls for new physical models and complex numerical calculations. High-power lasers are not only a modern physical instrument for producing powerful shock waves and high energy densities, but also a unique tool for implementing and studying ultrafast processes in condensed matter states. The physics and mechanics of interaction processes strongly depend on the duration of a laser pulse. For a duration of ≈ 1 ns and above, the main mechanism is the evaporative one (Fig. 5.4), while the thermomechanical mechanism is basic for the femtosecond range. In both cases, ultrahigh deformation rates are dealt with. Mechanical Properties During Ultrafast Deformations. The shock wave technique is a powerful tool for studying the properties of materials at extremely high deformation rates (ref. Chap. 4). The methodology of that scientific area is based on the link between the matter flow parameters recorded in experiments and the physical and chemical processes occurring in matter. Progress of research into the high-velocity deformation, disruption, and physicochemical transformations in shock waves is largely related to the development of modern techniques for measuring wave profiles with high spatial and temporal resolutions [92–94]. To date, a wealth of experimental knowledge has been gained about the elastoplastic and strength properties of technical metals and alloys, geological materials, ceramics, glasses, polymers and elastomers, plastic and brittle single crystals in the microsecond and nanosecond ranges of exposure duration. Considerable progress has been achieved in the development of methods for obtaining information about the kinetics of energy release in detonating and initiating shock waves. The experimental data are used to construct phenomenological rheological models of deformation and fracture, macrokinetic models of physicochemical transformations, which are required for calculating the processes of explosions, high-velocity impact, and the interaction between high-power radiation pulses and matter. Today, the science of the mechanical properties of matter at record-high deformations is rapidly developing. We have considered these interesting issues in details in Chap. 4, and now let us discuss the problems of thermodynamics at ultrafast laser compression pulses [1, 2]. In the previous section we have considered the case of the target exposure to laser when the flow in its dense matter is caused by the pressure of hot vapors or plasma. In case of ultrashort laser pulses (USLP), the situation is completely different. USLP heats the electron subsystem inside the skin-layer [120] of an initially hard target (ref. Fig. 5.12). Heat from the skin-layer propagates in a supersonic manner along the electron subsystem into the target depth. Supersonic advancement of an electron heat wave continues until temperature relaxation occurs between electron

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Fig. 5.12 USLP effect principally differs from the effect of a long laser pulse. For the short period of time τ L , USLP greatly heats electrons in the skin-layer δ skin ≈ 10–20 nm. Typical temperatures T e ≈ 1–5 eV. These temperatures are typical for the thermomechanical ablation threshold. In this case, electron heat conductivity heats the layer d T ≈ 50–150 nm. It is important that formation of the layer d T occurs over the time t eq ≈ 2–10 ps that is short as compared to the sound time t s = d T /cs ≈ 15–50 ps. Thus, we can say that the layer d T with high pressure is formed in a supersonic manner. All further hydrodynamic phenomena are related with the hydrodynamic decay of the layer d T of high pressure and solid-state density. This is the difference from the case with long pulses when the motion of the matter of the target’s high-dense part is caused by the pressure of the corona

and ion subsystems [117, 121]. During relaxation, electrons transfer heat to ions and their local values of temperature T e and T i gradually approach. Aligning of temperatures T e and T i continues during time t eq of about 2–10 ps (depending on the specific metal). It is very important that the depth of the heated layer d T is significant—“acoustic time” t s = d T /ss necessary for the sound to cover the distance d T , exceeds the time t eq of heated layer formation; here cs is the velocity of sound in condensed target matter. For example, in case of aluminum, the heated depth d T ≈ 100 nm, cs ≈ 5–6 km/s, t s ≈ 20 ps, t eq ≈ 2 ps. Then, at times t > t eq , the electron heat wave (EHW) passes into a usual mode of strongly subsonic propagation. Therefore, for the time of about the sound time t, the heated layer depth d T changes slightly (ref. Fig. 5.13) [117, 122, 123]. Hydrodynamics of Ultrashort Pulses. As described above, rapid (supersonic) heating results in the formation of a heated layer with the thickness d T . In case of isochoric heating, ρ(x, t) ∼ const = ρ 0 , t < t eq  t s , high pressures are related with high temperatures. The heating time t eq is less than the time t s = d T /cs of the hydrodynamic decay of the high-pressure layer. Therefore, the description of the hydrodynamic stage is reduced to the description of the problem of the decay of the high-pressure layer. Two situations are possible. In one of them, the USLP beam goes through a transparent dielectric (for example, glass) and is absorbed in metal at the dielectric-metal boundary.

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Acoustic wave

Fig. 5.13 (x, t) Diagram of the propagation of electron thermal wave and sonic disturbance. Sonic disturbance starts from the point x = 0, t = − τ L . Before the start of pulse action, the target substance occupies the first half-space x > 0, vacuum is on the left. The laser beam comes to the target from the left. A rectangle in the coordinate origin shows the USLP heating area in space (to the skin layer depth δ skin ) and time (pulse duration τ L ). The EHW trajectory consists of a steep (along axis t) supersonic and flat strongly subsonic areas—compare with the sonic wave slope. EHW supersonic propagation continues for the time period t eq . For the time t eq , the sonic wave covers small distance cs t eq as compared to the heating layer thickness d T . For the sound time t s = d T /cs , the acoustic disturbance leaves the heating layer d T and the entire heating layer is involved in hydrodynamic motion

In this case, dielectric is on the left and metal is on the right in Fig. 5.13. A shock wave goes into the dielectric depth (i.e. to the left in Fig. 5.13). Another compression wave goes into the metal. Figure 5.13 considers a case when vacuum or low-pressure gas (for example, atmospheric) is located on the left. A situation with a high-pressure layer at the vacuum-metal boundary is shown in Fig. 5.14. If the formation time t eq of the high-pressure layer d T thick is neglected as compared to the characteristic acoustic scale of time t s = d T /cs , we come to a problem with specified initial data. At the instant of time t = 0, we have a homogeneous density profile ρ(x, t = 0) = ρ 0 , x > 0 and a heterogeneous pressure profile shown in Fig. 5.14. The solution of the one-dimensional problem of the decay of the high-pressure layer inside a homogeneous medium is given by the d’Alembert formulas with two diverging compression waves if the pressure is low as compared with the volumetric modulus K. It is shown in Fig. 5.15. The solution looks as follows p(x, t) =

p0 (x − cs t) p0 (x + cs t) + 2 2

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Fig. 5.14 Reducing the problem of hydrodynamic flow initiated by USLP to the problem with initial data. A problem with initial data is required to explain the principal points related with hydrodynamics. The flow at acoustic times and later is formed as a result of high-pressure layer decay at the vacuum-metal boundary. Vacuum is located on the left, the high-pressure level has effective thickness d T . In the target depth (on the right), the temperature returns to the room temperature and the pressure falls to zero

Fig. 5.15 Decay of the initial state with a high-pressure level in homogeneous medium (upper panel) into two diverging waves of semi-amplitude (second panel from above). The wave is reflected from the boundary x = 0 with vacuum p = 0. It is important that in case of reflection, the compression wave p > 0 transforms into the rarefaction wave p < 0. This is how tensile stresses occur. In case of a rather high amplitude of such tensions, spallation occurs—a layer near the boundary x = 0 breaks off the main part of the target. The thickness of the separated layer is defined by the effective thickness of the compression wave and its shape. In case of a strictly rectangle wave (as shown in the figure), the spallation layer thickness is equal to the half of the rectangle wave thickness

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provided that before decay, the medium was at a standstill. The function p0 (x) sets the initial pressure distribution. In our case of the USLP high-pressure layer created at the vacuum-metal boundary (ref. Fig. 5.14), the d’Alembert solution for the times t > 0, t < t s is a combination of three waves. There is a wave running to the right, a wave running to the left and a reflected rarefaction wave: p(x, t) =

p0 (x − cs t) p0 (cs t − x) p0 (x + cs t) + + . 2 2 2

This combination corresponds to three lower panels in Fig. 5.15. In the linear approximation, we neglect the displacement of the vacuum-metal boundary. The sum of the first and third addends in the solution with three waves gives zero pressure at the boundary with vacuum in case of arbitrary function p0 (x). In our case, the initial pressure distribution is shown on the lower panel of Fig. 5.14. For short times t  t s , the flow near the boundary is determined by the addition of the wave running to the left and the wave reflected from the boundary. In this case, the decay of the half-space with a uniform non-zero pressure takes place at the half-space boundary with vacuum since in case of short times the pressure drop into the target depth can be neglected (ref. Fig. 5.15) and pressure distribution can be deemed homogeneous across the target. A rarefaction wave occurs near the boundary, in which matter expands into vacuum. The respective solution is well studied in hydrodynamics [77, 124, 125]. The velocity of sound in gas turns zero when gas pressure drops to zero. In case of condensed matter, the velocity of sound and density remain finite when pressure drops to zero. These circumstances are associated with the formation of a specific rarefaction wave shown in Fig. 5.16 (ref. [126–128]). Density jump occurs at the condensed phase boundary 3 expanding towards vacuum in Fig. 5.16. Pressure identically equals zero from jump 3 to the slope of rarefaction wave 4 when expanding into the zero-pressure medium. The density is constant in this case. The layer of constant thickness forms a plateau on the rarefaction wave density profile [126, 127]. A specific feature of the rarefaction wave is plateau formation. If pressure in the target is homogeneous everywhere (beyond the layer with motion), the rarefaction wave is self-similar with self-similar variable x/t. This wave is called a centered rarefaction wave (Chap. 1) since its profile is covered by a fan of straight characteristics diverging from the coordinate origin. The coordinate origin is at the vacuum-metal boundary at the zero point in time. A non-trivial area of the fan corresponds to slope 4 in Fig. 5.16. The centered wave is an example of a self-similar simple Riemann wave (Chap. 1). Matter inside the plateau is in an equilibrium state belonging to the binodal (the curve of coexistence between vapor and condensed phase). In this case, there is no tension beyond the binodal into the two-phase area and, therefore, no negative pressures [126, 127] (Chap. 4). If we take into account the inhomogeneity of initial pressure distribution on the lower panel of Fig. 5.14, then negative pressures appear [129]. In our case, the

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Fig. 5.16 Decay of the initial pressure profile at short times when the rarefaction wave has covered a small (as compared to the characteristic size d T ) distance deep into the target. Then one can neglect the inhomogeneity of the initial pressure distribution shown on the lower panel in Fig. 5.14: 1 the initial pressure distribution taken from Fig. 5.14; 2 stepped pressure drop to the expansion medium pressure. In case of vacuum this pressure equals zero. The density decreases abruptly together with pressure. The jump takes place in the linear acoustic approximation. If we go beyond this approximation, jump 2 diffuses into rarefaction wave slope 4. It is important that a rarefaction wave in a condensed medium has the area of flow with constant parameters—a step or plateau. The plateau ends at boundary 3 of condensed matter. The reasons for area 5 to appear with tensile stresses p < 0 are discussed in the text

pressure drops into the target volume. Therefore, at the initial stage, boundary 3 in Fig. 5.16 moves at the velocity corresponding to a higher initial pressure near the boundary. Due to pressure drop into the target depth, this velocity must fall down as time goes by. This results in plateau matter tension and an area with negative pressures 5 is formed (ref. Fig. 5.16). Pressure deviation from zero value in area 5 is caused by the deviation of the initial pressure distribution from the maximum value which at point in time t = 0 was at the boundary (ref. Fig. 5.14). Let us abandon the discussion of the growth of negative pressure, the attainment of ultimate strength and the description of the spallation phenomenon (ref. Chap. 4). Let us recall the comparison of USLP with long pulses and the role of evaporation. Role of Evaporation. Let us estimate the amount of evaporated matter in energy contributions of about the threshold of thermomechanical ablation. As we said in connection with Figs. 5.15 and 5.16, the reflection of the compression wave leads to the formation of a time-varying field of tensile stresses. If at some moment the peak stress exceeds the strength of material, a layer of the order of the heating layer thickness d T breaks away. Since such mechanical separation leads to the carry-over of matter from the surface of the irradiated target, this way of the matter carry-over is also referred to as ablation (the process of mass loss is meant by ablation). The

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emergence of tensile stresses is associated with rapid heating, so the term “thermomechanical ablation” is used. Moreover, this term allows dividing evaporative and thermomechanical ablation. It is clear that since thermomechanical ablation is related with reaching a fixed value—the material strength, such ablation is of a threshold nature. Indeed, if the tensile stresses are even slightly less than the strength, then there is no thermomechanical carry-over of matter. Calculations for aluminum, gold and nickel show that for metals at the thermomechanical ablation threshold, the absorbed laser energy is F abs ≈ 0.1 J/cm2 . In this case, the peak electron temperature is ≈ 2 eV, the maximum surface temperature on completion of two-temperature (2 T) relaxation is ≈ 2000– 3000 K, the maximum pressure is ≈ 10 GPa. If the threshold is exceeded, a layer with a thickness of 20(Ni)–50(Al)–90(Au) nm is separated near the threshold depending on metal. Let us estimate the amount of evaporated matter according to the kinetic theory of gases. A rough estimate follows from the Hertz Knudsen formula. In case of vapor outflow into vacuum, the maximum flow and thickness of the layer evaporated per ˙ cool /ρ0 , where mi is the atom pulse are limited by m˙ ∼ m i n sat c and h evap ∼ mτ mass, psat (T ) = nsat (T ) k B T is the saturated vapor pressure, c(T ) is the velocity of sound in vapor, τ cool ≈ 0.1–1 ns is the irradiated spot cool-down time, ρ 0 is the condensed phase density. In accordance with the Clapeyron Clausius formula, the saturated vapor pressure at temperatures below the critical temperature is an abrupt (exponential) function of temperature. For metals, temperatures of 2000–3000 K are the temperatures comparable with metal boiling temperatures. Hence, the saturated vapor pressure of metal near the threshold of thermodynamic ablation is about one atmosphere. For such pressure, the concentration of saturated vapor atoms is ≈ 3 × 1018 cm−3 . This concentration is four orders higher than the concentration of atoms in the condensed phase. The evaporated layer thickness is hevap ∼ 0.2α = 0.6 nm, where α ≈ 0.3 × 10−7 cm is the distance between atoms. The following maximum values of the parameters are adopted for calculations: τ cool = 1 ns, the velocity of sound in vapor c = 1 km/s. As we can see, ≈ 102 times more matter is carried away above the thermodynamic ablation threshold due to the mechanical breakaway than the maximum evaluation gives in case of evaporation into vacuum. Let us consider a shape of the acoustic wave going deep into the target. For times of about acoustic time t s = d T /cs , the hydrodynamic flow in the target is determined by the summed action of three waves (right, left and reflected). For the time ∼ t s , the left wave disappears. Accordingly, further in time, the flow is composed of the right and reflected waves (ref. Fig. 5.17). The right and reflected waves are divided by “zero” characteristic 4 that goes from the coordinate origin at the time when USLP starts acting. The target boundary trajectory is designated with 8, 9 in Fig. 5.17. The boundary accelerates and takes the maximum velocity over the USLP action time. Then, the boundary slows down during the reflection of the left wave. The areas of acceleration and deceleration are designated with the figure 8 in Fig. 5.17. The deceleration is caused by the fact that the left wave amplitude decreases in the direction from the

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Fig. 5.17 Propagation of compression and rarefaction waves into the target: 1 the position of the heating area due to USLP absorption. The absorption depth is about a skin-layer thickness of 10– 20 nm, the heating time is determined by the USLP duration, usually ≈ 0.01–1 ps; 2 is the supersonic and subsonic areas of the electron heat conductivity wave (EHCW); 3 is the transonic area where the mode of heat wave propagation changes. This is the area where the front characteristics of the compression wave 4, 5 start propagating into the target depth (wave running rightwards); 6 is the last characteristics of the wave running leftwards. As we can see, this wave propagates towards the vacuum-target boundary; 7 is the last characteristics of the reflected wave running rightwards 4–7; 8, 9 the target boundary trajectory. The reflected wave propagates after the compression wave 5–4. The pressure profile composed of a sum of the right and reflected waves takes a “zigzag” shape shown in the inset

boundary to the target volume (ref. the lower panel in Fig. 5.14). After full reflection of the left wave, the boundary displacement to the left stops. The trajectory of the boundary after the stop is shown in Fig. 5.17 with vertical straight line 9. Papers [130–133] are dedicated to the numerical simulation of the problem using a two-temperature hydrodynamic code and molecular dynamics. USLP sends a characteristic wave into the target depth with an advance compression bump followed by a tension well. This shape is shown in the inset in Fig. 5.17. In linear acoustics, the amplitudes of positive and negative pressures are half the amplitude of the initial pressure distribution profile in Fig. 5.14. In case of energy contributions of about the threshold value, the initial pressures ≈ 10 GPa are a significant share of the volumetric modulus K ≈ 100 GPa. Nonlinear effects most strongly affect the shape of the reflected wave the amplitude of which in modulus is much lower than half of the maximum initial pressure. The wave shape deforms with time due to nonlinearity (ref. Fig. 5.18 and Chap. 1). Outside the heating layer, the wave moves through homogeneous matter at a constant temperature. Inside the heating layer, the wave propagation is affected, first, by the presence of the melting layer and, second, by the temperature profile in the heated area. The wave running into the target depth behind the temperature profile of the heated layer with a thickness ∼ d T is a simple Riemann wave. Phenomena of

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Fig. 5.18 Breaking of compression and rarefaction waves. On the (x, t)-diagram, stripes 5–4 and 4–7 correspond to compression and rarefaction waves (Fig. 5.17)

compression wave steepening (gradient modulus growth) in its front part and wave profile tensioning in the area of pressure drop are well known (Chap. 1). The profile of the rarefaction wave is steepened and stretched in a similar way (p < 0). But now the steepening takes place in the rear part of the rarefaction wave where the pressure builds up [130, 132, 133]. The arrows on the upper panel of Fig. 5.18 show bending points in the front part of the compression wave and in the rear part of the rarefaction wave. Due to the convergence of the characteristics of the Riemann wave (Chap. 1), the wave profile steeping occurs. There is a point in time when a focal curve is formed, after which some characteristics start crossing each other. Compression and rarefaction waves have different points in time for such crossing. The arrows on the lower panel of Fig. 5.18 indicate the positions of the shock waves formed after breaking. The wave evolution is a curious process. At short times, the wave area near the “zero” characteristic is steep since this area corresponds to the fan of characteristics at the pressure drop in the rarefaction wave. This is area 4 in Fig. 5.16. This is an image of a section of an abrupt drop in the initial pressure at the vacuum-metal boundary in Fig. 5.14. The length of the front part in the compression wave and the rear part in the rarefaction wave is about the initial profile thickness in Fig. 5.14. Then, due to nonlinear processes, the situation changes. The zone near the zero characteristic is stretched, the pressure profile steepness in this zone decreases. On the opposite, the pressure rises zones become steeper. The maximum and minimum pressures propagate according to their characteristics. Therefore, the wave amplitudes in maximum and minimum do not change with time until the maximum characteristic comes to the front shock wave and the minimum characteristic comes to the rear shock wave. After that, the maximum and minimal amplitudes start slowly changing with time [1, 2].

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The described evolution is faster if the initial pressure is higher. The profile of the rarefaction wave going to the target depth significantly changes above the thermomechanical threshold after layer separation. The well depth becomes smaller. The higher the initial pressure, the greater the ratio between the maximum positive pressure and the modulus of the maximum negative pressure in the complex wave from the right wave p > 0 and the reflected wave p < 0. Indeed, as the absorbed energy F abs rises, the initial pressure shown in Fig. 5.14 also rises. The negative pressure modulus is limited by the tensile strength of the substance. Melting and Nucleation. The calculations show that in metals the USLP melting threshold is 2–4 times lower than the thermomechanical ablation threshold. This is a very substantial circumstance. Then, for energies being about the thermomechanical ablation threshold F abl , melting occurs within the time interval − τ L < t < t eq of equalizing the electron T e and ion T i temperatures. The thing is that for such cases when the melting threshold is exceeded, the temperature t rises above the temperature T i within this time interval on the spinodal of crystal-melting transition [122]. The temperature on the spinodal is ≈ 20% higher than the melting temperature at fixed density. The melting layer thickness is comparable with the heating layer thickness d T . The hydrodynamic decay of the high-pressure near-surface layer in Fig. 5.14 occurs after melting formation. Therefore, the above-mentioned right, left and reflected waves propagate along the melt. The growth of tensile stresses in time (the onset of this process is shown in Fig. 5.16, pressure deviation downwards, profile 5) occurs in liquid matter. It appears that the attainment of the maximum tensile stress in the studied metals (Al, Au, Ni) takes place in liquid; Al—[132]; Au—[132, 133]; Ni—[134]. Consequently, the fragmentation of the initially continuous matter starts with the nucleation of bubbles in the liquid phase when the melting ultimate strength is reached. The formation of bubbles containing vapor in the liquid is called cavitation. Let us emphasize that the effect of vapor pressure on the dynamics of the vapor–liquid mixture in the considered problem of thermomechanical ablation under the action of ultrashort laser pulses is negligible. Indeed, as it was noted above, the saturated vapor pressure within the temperature range of 2000–3000 K typical of the ablation threshold is about several atmospheres ≈ 10−4 GPa. Whereas the dynamic strength of the melt in the discussed experiments is several gigapascals. Let us note that the attainment of the maximum tension and nucleation take place at the single-temperature stage T i = T e = T. The thing is that the maximum tension requires time of about the acoustic time t s = d T /cs , and this time exceeds the temperature equalization time t eq . However, there are papers that consider cavitation inside a two-temperature time interval when the role of electron pressure in the dynamics of expansion of heated two-temperature matter is significant [135]. Let us also note that the temperature profile has a great effect on the depth when nucleation starts on the ablation threshold. This is true because strength rises as temperature falls down. There is a competition between an increase in the amplitude of the maximum instantaneous tensile stress as the reflected wave moves into the target depth, on the one hand, and a decrease in temperature in this case, on the other hand. In Fig. 5.16,

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section 5 of the pressure profile, it is shown how the amplitude grows when the reflected wave propagates. The vapor pressure inside the cavitation bubbles is insignificant. The surface tension is important. The situation when the surface tension of bubbles affects the spallation plate deceleration is unusual. This specific feature is typical of USLP with its extremely low heated layer thickness d T ≈ 100 nm—just about several skin-layer thicknesses. In case of standard experiments with spallation in a liquid (Chap. 4), this effect is negligible for a spallation layer thickness of the order of a fraction of a millimeter or more. In experiments with USLP, the spallation plate inertia is negligible due to its extreme thinness. Therefore, the surface tension plays an important role. The melt-vapor mixture has three stages when the mixture is stretched. At the first stage, nucleation takes place and individual nuclei of the vapor phase are formed with a size comparable with the interatomic spacing [136–138]. This is true because stresses are great (of the order of several gigapascals) and the diameter of a viable nucleus is very small. During nucleation, the distances between nuclei far exceed the diameters of viable nuclei. That is, individual nuclei are formed independently. An interaction between them appears short after nucleation of the first group of bubbles. The amplitude of tensile stress around the bubble decreases from a high value during nucleation to a value that slightly exceeds 2σ /r. The value 2σ /r is determined by the current bubble radius and the surface tension coefficient. The drop in the stress amplitude occurs inside the acoustic sphere around the bubble. The radius of this sphere rises together with the velocity of sound. After a short time, the spheres of neighboring bubbles are overlapped, the stress decreases and the process of nucleation of new nuclei stops. Let us note that a near-threshold situation is discussed at this point. In case the ablation threshold is significantly exceeded (for example, 2–3 times) after the first group of nucleation bubbles formed at the depth h1 under the target surface, tensile stresses are first reduced and then grow again [132, 133]. As a result, a new group of bubbles appears already at a greater depth h2. At the second stage, the formed bubbles expand. At the same time, the volume fraction ζ = V vap /(V vap + V liq ) of the vapor phase in the liquid and vapor phase mixture increases. This fraction is very low during nucleation of the first nuclei. At the second stage, percolation of the liquid phase takes place; vapor is isolated in bubbles and there is no vapor percolation. The second stage goes on until the fraction ζ reaches the value of 0.5. Then the third stage begins at which the vapor– liquid mixture forms a foam with a rather low volumetric content of melt [130]. The vapor still remains isolated as in the first and second stages. Liquid films block the flow of vapor from one cavity to another. Resistance to tensile stress of the mixture and, hence, the deceleration of the spallation plate continues due to the surface tension of films in the foam. So, first the deceleration of the plate is related with the tension of the continuous melt to densities lower than the equilibrium density at low (almost zero) pressures. This gives the main contribution to deceleration. At the second stage, deceleration is

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157

х, nm m/s

m/s

m/s

nm ps

ps

ps

Fig. 5.19 Simulation of USLP effect on a metal target by the example of aluminum. The value d T is artificially reduced six times. This was done in order to simulate the process of foam formation until its rupture in case of limited computational capabilities [134], Chap. 4. The embedded energy slightly exceeds the thermomechanical ablation threshold. The spallation plate is clearly seen from above. The arrow designates the plate velocity and the magnitude of this velocity. We can see that a substantial reduction of this velocity continues after nucleation. This position is impossible in case of high thicknesses of the spallation plate and, therefore, a large mass of the plate per unit of area. The smooth distribution of shading and the grid structure from below are divided by the melting front. The liquid phase is above (area of smooth shading), while the crystal lattice is below. The left and middle frames belong to the second and third stages of vapor–liquid mixture existence. In the right frame, films are ruptured, a vapor-droplet cloud with a low volumetric liquid content is formed under the spallation plate. After foam rupture, the plate keeps its momentum in further motion

caused by the surface tension of bubbles 2σ /r. This stage gives a significant contribution to deceleration in case of super-thin spallation plates and their low inertia. Finally, a small but noticeable contribution to deceleration is related with the third stage when the plate velocity decreases due to the tension resistance of liquid films in the foam. The third stage ends when the liquid films become thinner to a thickness of several interatomic spacings. Film rupture begins, and the plate deceleration stops. The film rupture stage continues for some time. Melt percolation disappears during this time. Accordingly, in the case of metals, the specific resistance of the vapor–liquid mixture rapidly grows. The mixture is converted from a foam state to the state of drops in vapor with a low volumetric content of drops. The value ζ monotonously tends to zero during further expansion of the mixture. The second and third stages of the development of the bubble ensemble are shown in Fig. 5.19. Newton’s Rings. Interference Newton’s rings were the first experimental [1, 2] indication of the determinant role of mechanical effects under the action of ultra-short laser pulses. However, to understand it, huge theoretical efforts have been required. We will briefly describe why interference rings appear. The reasons are the break-off of the spallation layer and the intensification of laser radiation on the beam axis (Fig. 5.20). Let the laser beam shown in Fig. 5.20 fall along the normal to the target surface. Let the intensity distribution in the beam be symmetric relative to the beam axis. Then, the illuminated area on the surface of the target in which the energy density F exceeds

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Reflected beam 1 Passed beam

Incident beam Reflected beam 2

Incident beam Ablation threshold

Fig. 5.20 Scheme explaining the principle of the formation of Newton’s rings. A laser radiation beam falls on the target surface. The intensity distribution in the transverse section of the beam is shown on the right. The maximum intensity is achieved on the beam axis. The target is on the left. Before the USLP comes, the target boundary is the plane shown with a vertical dashed straight line. Initially, target matter is to the left from this plane. The USLP sets target matter in motion. The beam transverse size r beam ≈ 10–100 µm is high as compared with the thickness d T ≈ 100 nm of the heated USLP layer. Therefore, locally, it is possible to apply the model of a flow which is unidimensional along the normal to the target surface over small pieces of the plane as compared to the scale r beam . An example of such pattern for the near-threshold area F is shown in Fig. 5.19. An example with the addition of five flows with different values of the ratio F/F abl is given in Fig. 5.21. Explanations with respect to the number of half-waves and interference are given in the text

the thermomechanical ablation threshold F abl forms a circle on the target surface. The threshold value is marked with an arrow in Fig. 5.20 on the intensity distribution. Nucleation and spallation plate formation begin above the ablation threshold. The motion velocity of this plate depends on the local value of the absorbed energy—the velocity is higher where the energy is higher. Therefore, the spallation plate bends to the center taking a dome shape (ref. Fig. 5.20). The bottom of the future crater remains under the plate. A discontinuity forms between the crater and the dome. The dome thickness decreases towards the center since the plate thickness depends on the local value of the energy density F—as F grows, the thickness decreases. There is the value F at which the plate thickness turns zero [129]. If the central fluence in the distribution shown in Fig. 5.20 exceeds this value, a hole is formed in the dome (ref. Fig. 5.21). It is clear that there are no Newton’s rings inside the hole. An example of experimental and calculational patterns is given in Fig. 5.22. A distinctive feature of the spallation plate formed under the action of ultrashort laser pulses is its extreme thinness. Its thickness almost equals a skin-layer thickness of 10–20 nm. Therefore, we can speak about nano-spallation [129]. The existence of the spallation is well known, but it has never been possible to get such thin spallation layers. Thanks to the dome thinness shown in Fig. 5.20, the light of probing laser

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159

Fig. 5.21 An example of a two-dimensional flow composition with a dome. The flow is composed of five unidimensional flows with the fluence ratio F/F abl growing from right to left in each specific frame. We can see first that in case of two insufficiently large fluences on the right, there is no nucleation, spallation and spallation plate. Second, the plate thickness decreases as the fluence grows; compare the 2nd and 3rd frames counting from the left. In the leftmost frame, the fluence ratio is such that the spallation plate disappears

Fig. 5.22 Pattern with Newton’s rings obtained in the experiment with gold (central frame) and two options of theoretical calculations of interference

pulses penetrates under the dome. The passed beam is reflected from the crater bottom, goes from under the dome and interferes with reflected beam 1 that is formed at the first reflection of the incident beam from the dome. If an integer number of half-waves of laser radiation can be placed in the discontinuity under the dome, a Newton’s ring is formed.

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Let us note that we can easily adjust the dome shape by changing the intensity distribution over the laser spot. We shall also note that the pattern with Newton’s rings in Fig. 5.22 changes with time since the dome shown in Figs. 5.20 and 5.21 is in motion. As time goes by, the distance from the dome to the crater bottom rises. At first, there are no rings in the image of the irradiated spot. For the first ring to appear, one half-wave must be placed in the discontinuity under the central point of the dome. This requires a certain amount of time after the heating effect of the heating ultrashort laser pulse. For example, with a laser wavelength of 1 µm and a velocity at the center of 1 km/s, this time is 500 nm = 0.5 ns. 1 nm/ps The higher is the input energy, the shorter is this time. The second ring appears when two half-waves are placed under the dome. The new ring appears from the center. Previous rings are shifted to the circle edge in Fig. 5.20 where the ablation threshold was reached.

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95. McWilliams RS, Spaulingg DK et al (2012) Science 338:1330–1333 96. Hicks DG, Boehly TR et al (2009) Phys Rev B79:014112 97. Benuzzi-Mounaix A, Koenig M, Ravasio A et al (2006) Laser-driven shock waves for the study of extreme matter states. Plasma Phys Control Fusion 48(12B):B347–B358 98. Jeanloz R, Gelliers PM et al (2007) PNAS 104(22):9172–9177 (2007) 99. Eggert J, Brygoo S et al (2008) PRL 100:024503-1-24 100. Gelliers PM, Loubeyze P et al (2010) PRL 104:184503-4 101. Lindl J, Landen O et al (2014) Phys Plasmas 21:020501 102. Teller E, Campbell EM et al (1995) LLNL-pepost, Aug 1995 103. Swift DC, Hawreliak JA et al (2012) Proceedings of shock compression of condenses matter, 2011. In: AIP conference proceedings, vol 1426, pp 477–480 104. Cauble R, Phillon DW et al, PRL 70(14):2102 105. Cauble R, Rekky TS et al (1998) 80(6):1248–1251 106. Rothman SD, Evans AM et al (2002) Phys Plasma 9(5):1721 107. Boehly TR, Goncharov VN et al (2011) PRL 106:195005 108. Higcks DG, Meezman NB et al (2012) Phys Plasma 19:122702 109. Kritcher AL, Doppner T et al (2014) High Energy Density Phys 10:27–34 110. Betti R, Zhou CD, Anderson KS, Perkins JL, Theobald W, Solodov AA (2007) Phys Rev Lett 98:155001 111. Perkins IJ, Betti R, LaFortune KN, Williams WH (2009) Phys Rev Lett 103:045004 112. Ribeyre X, Schurtz G, Lafon M, Galera S, Weber S (2009) Plasma Phys Control Fusion 51:015013 113. Lafon M, Ribeyre X, Schurtz G (2013) Phys Plasma 20:022708 114. Schmitt AJ, Bates JW, Obenschain SP, Zalesak ST, Fyfe DE (2010) Phys Plasma 17:042701 115. Atzeni S, Ribeyre X, Schurtz G, Schmitt AJ, Canaud B, Betti R, Perkins L (2014) Nucl Fusion 54:054008 116. Batani D, Bataon S, Casner A, Depierreux S, Hohenberger M, Klimo O, Koenig M, Labaune C, Ribeyre X, Rousseaux C et al (2014) Nucl Fussion 54:054009 117. Inogamov N, Ashitkov S, Zhakhovsky V et al (2010) Acoustic probing of two-temperature relaxation initiated by action of ultrashort laser pulse. Appl Phys A 101:1–5 118. Nora R, Theobald W et al (2015) PRL 144:045001 119. Theobald W, Nora R et al (2012) Phys Plasmas 19:102706 120. Anisimov SI, Kapelovich BL, Perelman TL Electron emission from surface of metals under the action of ultrashort laser pulses. JETP 66(2):776–779 [Anisimov S. I., Kapelovich B. L., Perelman T. L. Elektronnaya emissiya s poverkhnosti metallov pod deystviyem ul’trakorotkikh lazernykh impul’sov // ZHETF. 1974. T. 66, №2. S. 776–779 (in Russian)] 121. Inogamov NA, Zhakhovsky VV, Ashitkov SI et al (2011) Laser acoustic probing of two temperature zone created by femtosecond pulse. Contrib Plasma Phys 51(4):367–374 122. Inogamov NA, Zhakhovskii VV, Ashitkov SI et al (2009) Two-temperature relaxation and melting after absorption of femtosecond laser pulse. Appl Surf Sci 255(24):9712–9716 123. Zhigilei LV, Lin Z, Ivanov DS (2009) Atomistic modeling of short pulse laser ablation of metals: connections between melting, spallation, and phase explosion. J Phys Chem C 113(27):11892–11906 124. Landau LD, Lifshits EM (1986) Hydrodynamics. Nauka, Moscow [Landau L. D., Lifshits E. M. Gidrodinamika. - M.: Nauka, 1986 (in Russian)] 125. Zel’dovich YB, Raizer YP (1966) Physics of shock waves and high-temperature hydrodynamic phenomena. Nauka, Moscow [Zel’dovich Ya. B., Raizer Yu. P. Fizika udarnykh voln i vysokotemperaturnykh gidrodinamicheskikh yavleniy. - M.: Nauka, 1966 (in Russian)] 126. Inogamov NA, Anisimov SI, Retfeld BZ (1999) Rarefaction wave and gravity equilibrium in two-phase liquid-vapor medium. JETP 115(6):2091–2105 [Inogamov N. A., Anisimov S. I., Retfeld B. Zh. Volna razrezheniya i gravitatsionnoye ravnovesiye v dvukhfaznoy srede zhidkost’–par // ZHETF. 1999. T. 115, №6. S. 2091–2105 (in Russian)] 127. Anisimov SI, Inogamov NA, Oparin AM et al (1999) Pulsed laser evaporation: equation-ofstate effects. Appl Phys A 69:617–620

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Chapter 6

Shock Waves in Nuclear Explosions

People have always considered the search and development of new sources of energy to be among the priority areas of fundamental and applied research [1]. In the first third of the twentieth century, research into the structure of the atom and atom nucleus was rapidly developing. Based on the achievements of a number of leading world physicists in this area, as early as in 1939, E. Fermi, L. Szilárd, F. Joliot-Curie, Ya. B. Zel’dovich, Yu. B. Khariton, N. N. Semenov and others substantiated the possibility of fission chain reaction in uranium and, therefore, the practical use of a principally new—nuclear source of energy, which is million times more powerful than traditional chemical reactions. The conditions of implementing an explosiontype fission chain reaction (earlier discovered by the academician N. N. Semenov for chemical reactions) with the release of unprecedentedly high amount of energy were formed soon afterwards [2]. This meant that scientists have approached the invention of a weapon of an unusually destructive power [3]. For the practical use of nuclear energy, scientists had to solve a high number of complex scientific and technical tasks. They created a new science—high energy density physics the primary postulates of which are given in a large number of publications and summarized in the review [4] and proceedings of the scientific session of the Division of Physical Sciences of the Russian Academy of Sciences [5]. It became clear at once that shock waves would play a decisive role in this problem. In 1940s, the USA and the USSR made the first steps in solving the problems of ensuring the conditions for the development of an explosion-type fission reaction and the creation of nuclear weapons. Scientific and technological achievements in these areas are still the military and state secrets of the countries which own nuclear weapons. International agreements strictly govern publications and prohibit the transfer of the technologies of nuclear weapons production to other countries. However, a nuclear explosion is also a unique physical phenomenon the study of which is of interest for scientists from various areas of knowledge. Including for scientists studying the physics of intense shock waves and physics of high energy densities. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Fortov, Intense Shock Waves on Earth and in Space, Shock Wave and High Pressure Phenomena, https://doi.org/10.1007/978-3-030-74840-1_6

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It is well known [6, 7] that the study and large-scale practical application of extreme states of matter are inseparably associated with the entry of our civilization into the atomic area and the first step made in this direction—the creation of nuclear and thermonuclear weapons [7–16]. Powerful spherical pressure shock waves of dozens and hundreds of megabars, created in nuclear charges multiply compress nuclear fuel initiating in it chain fission reactions of pulsed energy release. In its turn, this nuclear energy release is the main tool for the compression and heating of thermonuclear fuel to billion atmospheres and ten million degrees in order to initiate in it the thermonuclear fusion reactions of the deuterium and tritium mixture [7]. A nuclear explosion in a broad sense starts with the creation of conditions for conducting an explosion-type chain fission reaction that lasts for dozens of nanoseconds in a small volume of uranium or plutonium and ends with a long-term decay of radioactive explosion products distributed over the atmosphere and earth surface. A nuclear explosion (if by this term we mean primary energy release) develops for no more than several microseconds—at first in the chain nuclear fission reaction of uranium or plutonium, and then (in case of a thermonuclear explosion device) in some sequence of nuclear reactions of tritium and deuterium fusion [2–6, 17]. Most frequently, when speaking about a nuclear explosion, we mean an external image (Fig. 6.1 [3, 17])—the appearance of intense shock waves and a luminous area, arrival of the shock wave at the observation point, the rise of an explosion cloud, etc. In other words, the notion of a nuclear explosion is usually related with the formation and development of physical processes that are usually called effects from a nuclear explosion [17] among which shock waves are ranked first. In this chapter, we will consider the first of these stages (nuclear explosion initiation) followed by its extreme actions. The last section of the chapter is dedicated to experiments on the generation and study of extreme states of matter using intense shock waves excited by a nuclear explosion. High Energy Densities for Explosive Nuclear Reactions. It is astonishing how quickly the humanity mastered quasi-controlled nuclear and thermonuclear energy. The first nuclear explosion was set off six and a half years after discovering chain fission reactions. In 1938/1939, O. Hahn and F. Strassmann implemented the idea of E. Fermi on the fusion of super-heavy elements by exposing uranium to neutron radiation and were surprised to find not super-heavy but lighter elements that were fragments of the nuclei of exposed targets. • In 1942, the Manhattan project was launched in the USA. In the same year, E. Fermi started the first stationary fission reactor. • A month and a half later, on September 28, 1942, uranium researches suspended by the war were resumed in the USSR. • In 1943, the Los Alamos scientific laboratory was established. • In July 1945, the USA tested the first successful nuclear explosion. • In August 1945, atomic weapons were used against Japan.

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Fig. 6.1 General view of a nuclear explosion [17]

• In December 1946, the academician I. V. Kurchatov launched the first fission reactor in Europe. • In August 1949, the USSR exploded its first atomic bomb created 4 years after the large-scale stage of the Soviet Atomic Project. • Works intended to create thermonuclear weapons (a super bomb) were started in the USA. • The USA exploded a non-transportable thermonuclear device in 1952 and a transportable hydrogen bomb in 1954. • The USSR tested the first mobile hydrogen bomb (named “Sloika”) [8] in August 1953 and a more advanced mobile charge in two years (3rd idea of A.D. Sakharov) [8]. According to the research advisor of the Arzamas-16 Russian Nuclear Center (currently—the Russian Federal Nuclear Centre—All-Russian Scientific Research

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Institute of Experimental Physics (RFYaTs-VNIIEF) and academician Khariton [18], “so fast pace and final success became possible because scientists of both countries were people of high scientific qualification and culture and relied in their work on the first-class European scientific physical school. At that time, scientists from the USA and the USSR devoted all their energies to defeat Germany.” Moreover, E. Fermi said in his work on a nuclear project: “All in all, this is an interesting physics.” The creation of nuclear and thermonuclear weapons became a prominent scientific achievement of the humanity and ensured geopolitical stability and peace on Earth for 75 years already. With the exception of a one hundred year period (1812–1914) after the Congress of Vienna that brought the Napoleonic wars to a close, the humanity has never lived without large-scale battles for so long. Further, we will only discuss the issues of the generation and application of shock waves and extreme states initiated by them in case of nuclear explosions by using only open access publications [8–16, 18–21]. Most of these materials were declassified [10–16, 18–47] and became widely available during the work of a group of specialists of the Ministry of Nuclear Energy of Russia over the collection called the USSR Atomic Project published by the Ministry and edited by L. D. Ryabov. These materials became publicly available in accordance with Order of the President of the Russian Federation B. N. Eltsin No. 160, dated February 17, 1995 “On the Preparation and Publication of the Official Collection of Archive Documents for the History of Creating Nuclear Weapons in the USSR.” Some of these documents have already been published in volumes of the collection [10–12], while others are being prepared for publication. Explosive Fission Reactions. Ya. B. Zel’dovich and Yu. B. Khariton in their article Kinetics of the Chain Decay of Uranium [2] published in 1940 described the conditions necessary for a nuclear explosion as follows: “The explosive use of chain decay requires special tools for a very fast and deep transition to the supercritical region.” They noted the extremely high rate of the exponential growth of neutron concentration in such a system in case of high supercriticality (increase by e times for 10–7 s) and the challenges related, as they believed, with that: “In case of such rapid development of chain decay, we have no right to get distracted from considering the creation of supercritical conditions under which the chain decay is only possible. The time of processes providing the transition of critical conditions, for example, the time of the approach of two uranium masses each separately being in the precritical region (relative to the chain decay) will not be able to become at least comparable with the reaction acceleration time.” Ya. B. Zel’dovich and Yu. B. Khariton underlined that “the kinetics of the development of chain decay is decisive for judging various ways of practical, energetic or explosive use of uranium decay [13].” Now it is important that the necessity to use and, therefore, study intense shock waves and states of matter with extremely high energy densities was already understood at that time [10–13]. On March 20, 1943, almost a month after his appointment as the research advisor of uranium works, I. V. Kurchatov appealed to M. G. Pervukhin with a letter which said [10, 11]: “At the beginning of the development of a uranium bomb explosion, a

6 Shock Waves in Nuclear Explosions

Usual explosion Uranium-235 Steel

Uranium-235

14 cm

Fig. 6.2 Layout of the Little Boy bomb dropped over Hiroshima. A nuclear explosive in the bomb was uranium-235 divided into two parts the critical mass of which was less than critical. The critical mass of uranium-235 required for explosion was created as a result of combining two halves by the gun method using a regular explosive [15, 16]

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7 cm

large part of matter that is still not in the reaction will be in a special state of almost complete ionization of all atoms, and the further development of the process and damage capability of the bomb will depend on this state of matter. In experiments, nothing close to that state of matter has been observed and it cannot be observed only before the bomb is implemented. The existence of such a state of matter is assumed only in stars. It seems possible to theoretically, without going into details, consider the explosion process at this stage.” At the initial stage of the development of nuclear weapons in the USA, two schemes of nuclear charges were implemented [9–13]: the gun, or ballistic (Fig. 6.2) scheme, and the implosion scheme (Fig. 6.3). The academician I. V. Kurchatov noted in his summary Design of Nuclear Bombs with Uranium-235 and Plutonium-239 [10– 13] as follows: “An atomic bomb can be activated in two ways: by the fast approach of two halves of uranium-235 and plutonium-239 charges located 0.5–1 m from each other before contact, and by densification of uranium-235 and plutonium-239 charges with a powerful explosion of trinitrotoluene surrounding these substances.” In the summary dated May 18, 1944, I. V. Kurchatov gave a layout of a gun approach-type atomic bomb with the following description of its design and operation [13]: “An atomic aviation bomb consists of a cylindrical shell with an atomic explosive on its ends – uranium-235 or plutonium-239. The bomb is actuated by exploding powder charges placed under the active explosive. The atomic bomb explodes when the halves (a) and (b) of uranium-235 or plutonium-239 are combined. The calculations show that to make a bomb equivalent in its action to 1000 tons of trinitrotoluene, 2–5 kg of uranium-235 or plutonium-239 are required. Currently, there is no absolutely valid data showing that a bomb of this design will act, but the more experiments are undertaken, the more confident are evidences. The main difficulty in making an atomic bomb is to obtain uranium-235 and plutonium-239…” The layout of the gun used in Hiroshima is given in Fig. 6.2. Pressures occurring when two uranium halves (the mass of which is 0.7 of the critical one) collide [48]

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Fig. 6.3 a Layout of implosion-type nuclear charge [14], b layout of the American nuclear charge from [7]

have a relatively low value P = ρW 2 ≈ 100 bar that is insufficient for any substantial compression of nuclear fuel. An implosion layout, which is an alternative to the gun type, used in bombing of Nagasaki (Fat Man) is represented in Fig. 6.3a, b. Plutonium 239 Pu was used as a fuel, which is an artificial element not available on the Earth and produced in stationary nuclear reactors by exposure of natural uranium 238 U to neutrons. The critical mass of a bare sphere of plutonium is ~ 10 kg, while that of uranium-235 is 50 kg [48]. As compared to uranium-235, the production of plutonium is 5 times more expensive [48]. Since the fuel critical mass is abruptly (quadratically) decreased as the compression grows, this layout permits the use of smaller amounts of expensive nuclear fuel (hundreds of grams) [6–16, 19, 38–57] and provides higher efficiency of using it in the charge [8–11, 48].

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Fig. 6.4 American atomic bombs Little Boy of gun type (15 kt TNT), based on uranium-235 and Fat Man using the implosion principle based on plutonium-239 (21 kt TNT) dropped on Hiroshima and Nagasaki [46]

The efficiency of this layout and its main characteristics significantly depend on the physical properties of used materials and substances in extreme conditions: on the shock wave compressibility of nuclear fuel within the range of pressures of dozens of Mbar, on the thermodynamic properties of the detonation products of condensed explosives providing explosion compression and generating powerful convergent spherical shock waves in the nuclear charge, on the thermophysical properties of construction materials and on many other characteristics of charge elements manifested in the initiation of explosion and as a result of explosive nuclear energy release [6–16, 19, 38–47, 49–57]. At the beginning of works intended to create an atomic bomb, scientists had uncertain knowledge and poor understanding of physics and gas dynamics of powerful shock and detonation waves and behavior of matter in exotic conditions of a nuclear explosion. According to [7], a number of specialists believed that metals in these conditions are non-compressible and high uncertainty in the equations of state of the explosion products of condensed explosives discredited the actuation of the “article” [6, 11, 12] with resulting “organizational conclusions” for Soviet scientists [11, 12]. To get necessary information in the USSR on the physics of shock waves and detonation waves, on the physical properties and gas dynamic features of the behavior of matter in the extreme conditions of a nuclear explosion, a large scope of experimental and theoretical works was completed, which gave rise to a new science—dynamic physics of intense shock waves and physics of high energy densities [5–7, 15, 50, 51]. Layouts and photographs of nuclear devices created at that time are given in Figs. 6.4, 6.5 and 6.6 [58]. All obtained fundamental information about the shock wave physics and the physical properties of substances and construction materials at ultra-high pressures and temperatures and also a large number of other original ideas and approaches were fully used in their designs. Let us give some characteristics of the American implosion atomic bomb [15, 16, 19]. The bomb was a pear-shaped shell with a maximum diameter of 127 cm, length of 325 cm (with the stabilizer) and weight of about 4500 kg. The neutron initiator was a polonium-beryllium system with a radius of 10 mm. The total amount of polonium was 50 Cu.

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Fig. 6.5 RSD-1 nuclear charge first tested on August 29, 1949 at the Semipalatinsk test range. The charge power is up to 20 kt TNT. Museum of the Russian Federal Nuclear Centre—All-Russian Scientific Research Institute of Experimental Physics (RFYaTs-VNIIEF), Sarov [16]

Fig. 6.6 Principal diagram of the first Soviet implosion atomic bomb being an equivalent to the American Fat Man [58]

The bomb fissile material was δ-phase of plutonium with a specific weight of 15.8 g/cm3 . The external diameter of the plutonium sphere was 80–90 mm. The plutonium core was located inside a hollow sphere consisting of uranium metal and having an external diameter of 230 mm [15, 16, 19]. Uranium metal was located inside the aluminum shell being a hollow sphere with an external diameter of 460 mm [15, 16]. A layer of explosive with a detonation focusing lens system of 32 special-shaped blocks was located behind the layer of aluminum. The total weight of the explosive was about 2 tons [15, 16]. While working with a uniform charge, the group of professor L. V. Al’tshuler found a more successful design combining gun-type and implosion ideas (shellnuclear design) [10–14], when the explosive charge does not compress the uniform spherical charge directly but accelerates the spherical shell of a fissile material impacting a sphere of a smaller size [10–13, 49, 51]. In this case, the energy release density increases multiple times [59]. This made it possible to double the efficiency of the device with significant reduction of its dimensions and weight.

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More advanced systems (as compared to polonium-beryllium) of external neutron initiation were applied for the controlled start of the chain reaction. The actuation time of this “power plant” in the bomb [20] is calculated with the help of the perfect gas dynamic calculations of shock wave compression using equations of state of matters at extremely high energy densities. These ideas were fully confirmed in tests of 1954 [20]. According to [49], another method to achieve the automatic synchronization of atomic explosion initialization processes is as follows. A small amount of solid matter containing deuterium is located in the center of the charge. The shock wave coming from the chemical explosive charge initiates the deuterium-deuterium nuclear reaction with the emission of neutrons that stimulate the chain fission reaction [49]. Further development of nuclear weapons quickly revealed the limiting capabilities (dozens of kilotons) of purely fission schemes of explosion related with the fast expansion of nuclear fuel during the fission reaction and, as a consequence, the limited efficiency of its use (20% from 6.2 kg of the plutonium charge in Fat Man and 1.4% from 64 kg of enriched uranium in Little Boy). Further development perspectives were related with thermonuclear reactions of fusion that, according to the academician Sakharov [8], represent “a source of energy of stars and Sun, a source of life on the Earth and possible reasons for its end.” Pulsed Thermonuclear Fusion. The role of nuclear sources in energy for powering stars was already clear early in the twentieth century. In 1920, A. Eddington said at the meeting of the British Association for the Advancement of Science [41]: A star is a shell wrapped on a huge reservoir of energy by methods unknown to us. This reservoir hardly represents anything but internal atomic energy that, as we know, is present in any matter... This reserve is almost inexhaustible if measurable at all... If internal atomic energy in stars is really used at a large scale to maintain such a huge hearth as a star, it seems that this brings us a little closer to making true a dream to control this hidden energy for human wellbeing or suicide.

The leading role of thermonuclear reactions as a source of star energy was discovered by the Nobel laureate H. Bethe (1926) who afterwards headed the theoretical department of the Los Alamos scientific laboratory where nuclear and thermonuclear weapons were created (USA). Although by the mid-forties, the respective schemes of thermonuclear reactions and the required sections of these processes were well known [8–16, 18–38], the practical implementation of an explosive thermonuclear reaction under terrestrial conditions encountered great difficulties caused by the need to achieve ultra high compressions and temperatures of dozen million degrees in thermonuclear fuel. Huge efforts to search acceptable physical schemes and techniques for creating a thermonuclear superbomb are described in details in a number of publications [8, 9, 13, 14, 21, 49] reflecting the history of the thermonuclear project full of drama, despair and flashes of genius. According to [47], the primary task was to use the energy released during the explosion of an atomic bomb to heat and set fire to a mixture of heavy hydrogen isotopes—deuterium and tritium, i.e. to execute thermonuclear reactions

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D + D = H + T + 4 MeV, D + D = n +3 He + 3.3 MeV with the release of energy, which are capable of self-supporting. To increase the share of burnt deuterium, A. D. Sakharov proposed to surround deuterium with a heavy shell made of simple natural uranium that must decelerate the expansion and substantially increase the deuterium density [8, 15, 16, 19, 47]. An increased rate of DD-reaction leads to a significant formation of tritium that immediately reacts with deuterium in a thermonuclear reaction. D + T = n +4 He + 17.6 MeV with a section 100 times more than the DD-reaction section and with a five times higher energy release. Moreover, the nuclei of the uranium shell fission under the action of fast neutrons occurring in the DD-reaction and the nuclei significantly increase the explosion power. This is why uranium was selected as a shell. The thermonuclear process power in deuterium could be significantly increased if some deuterium is replaced with tritium at the beginning. Indeed, the use of a thermonuclear charge in the form of deuteride-tritide lithium6 led to a radically increased power of the thermonuclear process and to the release of energy from the uranium shell, which is several times more that the thermonuclear energy release during DD-reaction. These are physical ideas included in the first version of our thermonuclear weapon. The temperature dependence of the rate of thermonuclear reactions makes the deuterium–tritium fusion reaction most preferable; a temperature of ≈2–10 keV is required for the reaction to be efficient. For the expansion velocity of thermonuclear plasma equaling 108 cm/s, this results in a characteristic time of the order of 10–8 s in case of a spherical target with a radius of about 1 cm. The condition of energy balance for such thermonuclear reaction (Lawson criterion [1, 60]) looks as nτ ∼

ρr ≈ 2 · 1014 s/cm3 , 4cs m i

which corresponds to ρr ≈ 0.1–3 g/cm2 and requires the thermonuclear fuel to be compressed to densities of dozens of grams per cubic centimeter. At 10 keV, this is responded by a huge pressure of several gigabars [49]. A high density of thermonuclear plasma is required to increase the reaction rate and decrease the run of thermonuclear alpha particles that are stuck [6, 7, 49] in a dense target and form a wave of thermonuclear burning propagating from the sphere center to its periphery. Hydrodynamic Thermonuclear Fusion. Achievement of the necessary superextreme conditions of thermonuclear fuel using powerful shock waves from chemical explosives (similar to nuclear fission charge) failed to live up to expectations [52]

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due to the insufficient explosive power and the development of some instabilities, in particular, the Rayleigh–Taylor instability in case of deep degrees of compression. Despite significant efforts on “hydrodynamic” ignition using this method, by now, scientists succeeded in obtaining a yield of ~ 5 × 1013 [52–54] of thermonuclear neutrons. The research advisor of the All-Russian Scientific Research Institute of Experimental Physics, the academician R. I. Il’kaev names this problem as one of the most important unsolved problems of explosive nuclear energetics [4]: “It seemed very tempting to “ignite” thermonuclear fuel in the conditions of cumulation in it of the energy of the explosion of chemical explosive. Since then, more than one hundred large experiments have been conducted but the problem is not solved and apparently is far from solution. Compression levels of central metal shells reach ≥ 50 in these conditions and the density of thermonuclear fuel exceeds 102 g/cm3 . Probably, we deal here with the principal development of instabilities that we cannot reduce in practice.” When discussing this area of thermonuclear studies, we will follow the review in [52]. Works on hydrodynamic thermonuclear fusion (HDTF) were apparently started in Germany during the Second World War due to the German nuclear project [54–56]. An idea to implement thermonuclear fusion at the focus of a spherical charge with an embedded shell [57] appeared in the development of the principle of hollow charges with jacketing (faust cartridges), also proposed for the first time in Germany. These works were supported by the development of the classical theory of a convergent spherical shock wave by Huderley. These works were unknown in the USSR at that time. It was only 1945 when L. D. Landau and K. P. Stanyukovich, independently from each other, proved the fact of pressure rise, i.e. cumulation at the front of a shock wave converging into continuous matter [61]. Work on HDTF was initiated in the USSR by the proposal of A. S. Kozyrev who expressed an idea that it is possible to reach such pressures and temperatures at the focus of a spherical charge of a chemical explosive that are sufficiently high for thermonuclear reactions in heavy isotopes of hydrogen [62]. Moreover, Kozyrev proposed a design of an explosion thermonuclear reactor based on this idea. Later numerical assessments showed that to reach high temperatures (~ 2 keV) necessary to excite a self-supporting thermonuclear reaction (ignition) in an explosive charge ~ 50 cm in radius, spherical-symmetric “convergence” of the shell up to the radius of several tenth fractions of a millimeter is required. In this condition, a thermonuclear flash would be excited in a DT-mixture having a weight of ≈ 10–3 g [63]. Further intensive experimental efforts (see review [52]) were intended to search optimal spherical cumulations, to study variabilities destroying deep implosion, to find suitable highly homogeneous explosives and to study other complex gas dynamic processes.

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The most advanced of the HDTF charges developed consists [52] of alternating layers of light and heavy materials, which efficiently extract energy from chemical explosives. In particular, a liquid explosive was used, which made it possible to get rid of explosive density variations occurring due to the compositional inhomogeneity of the explosive and to exclude technological joints and gaps between structural elements. The transition to a multipoint highly synchronous system of initiation with a maximum uniform location of 1000 points of initiation on the external spherical surface of the charge with the symmetry of a pentagondodecahedron would allow reaching high precision of focusing differing from the calculated geometry by no more than 0.1 mm [52, 63]. At the same time, the manufacturing accuracy of the elements of such a cumulating system was brought to a level of ≈ 0.03 mm for the external layers of the system and ≈ 0.003 mm for its internal elements [64]. The maximum, currently record-high neutron yield of 5 × 1013 neutrons in similar advanced systems was recorded on December 10, 1982 from the center of a target containing DT-gas with the initial radius r 0 ≈ 1 mm and the initial density ρ 0 ≈ 0.1 g/cm3 . In this experiment, according to estimates [65], the temperature T ≈ 0.65 keV and the maximum density ρ ≈ 80 g/cm−3 were reached. The value ρr ≈ 0.8 g/cm2 is record-high today for the internal thermonuclear fusion and sufficient for the ignition of DT-gas if temperature is increased by about 3–4 times. However, the experimental yield in this experiment was below the calculated one by 2–3 orders of magnitude [52, 64]. To explain this difference, a hypothesis was advanced concerning a vertical systematic component of asymmetry being present in the liquid explosive charge which is possibly caused by the influence of gravity on the liquid explosive density and with the target density exceeding random oscillations in amplitude between various experiments (these oscillations lead to the instability of the neutron yield in several dozens percent). This hypothesis has been partially proved in specially set-up experiments [52]. The gap between the idealized calculational and experimental values of the neutron yield in the HDTF systems can be explained [52] by the effects of asymmetry and mixing at the boundaries of matters of various density, not considered in calculations. The degree of taking into account these effects by modern calculational methods does not allow an adequate description of the obtained experimental results so far. Recently, specialists started, using calculations, to take into account an intake of heavy matter of the compressing shell into central gas in the form of jets. Jets are formed by means of increased low-scale perturbations the primary reason of which is structural specific features of the materials of central shells and their surfaces. Calculations that take into account this mechanism of gas clogging and, therefore, reduction of its maximum temperature give a much stronger effect than calculations using the existing theories of developed turbulence. Calculations completed so far and intended to take into account jet mixing give encouraging results for both explaining the experimental neutron yield and reducing

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this effect (and, hence, increasing the neutron yield) for the existing structurally and technologically advanced HDTF systems. The Rayleigh–Taylor instability and turbulent mixing occur in case of the tangential discontinuity of velocities at the contact boundary of media with different densities when it is accelerated on the side of light-weight matter. In particular, when the compressing heavy shell is decelerated by central light-weight gas and also when the heavy shells of the layered structure are decelerated and accelerated by lightweight layers [58, 66–68]. The instability of the boundary surface between two media with different densities occurs also after a shock wave front goes through it: the Richtmyer-Meshkov instability [58, 69]. It’s also worth noting a substantial dependence of instability and turbulent mixing on the physical viscosity and strength of materials, which is especially important for the external areas of HDTF systems. A positive effect of the high strength and refractoriness of used materials has been experimentally proved [65]. Other possible sources of low-scale perturbations are related with the matter mesostructure [48] (ref. Chap. 4). The external layers where matter has not yet melted are partially deformed. According to modern concepts, the plastic flow of a continuous medium occurs as the movement of plasticity centers at the meso-level 1–100 µm in size rather than continuum deformation. These centers apparently have a vortex nature. The dispersion of the free boundary velocity is experimentally recorded and measured when a shock wave arrives at that boundary. In other words, the nonuniformity of the plastic flow at the meso-level can serve as a source of low-scale velocity perturbations. We can figuratively compare such flow with the movement of a freezing river when brash ice moves over it. Thus, here it is necessary to use and improve the models of elastic–plastic flow (ref. Chaps. 2–4). We can use the exhaustive materials of theoretical and experimental studies in this area [48]. The plastic flow of molten internal shells located at low radii can serve as such source of perturbations. A cumulative flow converging to the center is always accompanied by a shift that, in its turn, is a source of low-scale turbulence. The conditions for the ignition of thermonuclear reactions can be achieved by reducing the negative effect of mixing and improving asymmetry (compensation or elimination of a regular component of asymmetry, etc.) [52]. If the achieved level of asymmetry and mixing is preserved, it is possible to approach the conditions of ignition by increasing the size of the charge. The difference between HDTF and other types of inertial thermonuclear fusion lies in the energy cumulation degree that was evaluated [52] relative to temperatures: k = T eff /T ex , where T eff = 2 keV is the effective ignition temperature of the DT-mixture, T ex ≈ 1 eV is the temperature of detonation products. Hence, k ≈ 10–3 for HDTF. For projects of laser and ion thermonuclear fusion, the temperature in the evaporated part of the shell ≈ 100 eV, i.e. k ≈ 10. This difference causes the main difficulties related with the possibility of implementing the hydrodynamic thermonuclear fusion. Therefore, any information obtained in the studies of laser-induced thermonuclear fusion can hardly be directly transferred to the HDTF problem. On the contrary, the results of HDTF studies obtained in more complicated conditions can be used for other types of inertial thermonuclear fusion.

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Fig. 6.7 Scheme of experimental setup for the dynamic compression of thermonuclear plasma in the conditions of acute-angled geometry: 1 striker (liner); 2 explosive charge; 3 detonation lens; 4 target cover; 5 target; 6 steel protection; 7 neutron recording unit; 8 striker flight velocity measurement unit [53]

The method of igniting the gas preliminarily compressed to moderate densities (several dozens of grams per 1 cm3 ) using a focusing shock wave can be more promising since the effects of the Rayleigh–Taylor instability and mixing are not so great in it with the maximum gas compression by the shell required for ignition. However, the issue remains concerning the creation of a preliminary moderate cold compression of gas with its further compression and heating by a sufficiently symmetric igniting shock wave. Unfortunately, such experimental studies of the operation of systems (not only HDTF systems), for example, laser targets under the action of a double pulse ensuring the pre-compression and formation of a powerful focusing shock wave that can eventually ignite the target, are still lacking. A different HDTF scheme is given in Fig. 6.7 [53]. In this scheme, a metal striker accelerated by the detonation products of condensed explosive was used for the quasi-spherical compression of deuterium and its mixture with xenon at the end of a conic target. The use of a flat version of the layered structure to increase the striker velocity to 6 km/s and a small admixture of 1–3% of xenon to increase the deuterium plasma temperature allowed recording a neutron yield in these experiments of about 106 thermonuclear neutrons per explosion. To analyze the processes in targets and optimize experimental parameters, a number of calculations was made for deuterium plasma compression under the action of laser. Calculations were made within the simplest spherically symmetric twotemperature model taking into account transfer process, radiation losses and thermonuclear reactions. This model was previously used to describe the dynamics of conic-shaped targets in laser experiments. The calculations have shown that compression occurs in two stages. First, a series of waves successively reflected from the liner and the center heats and ionizes the gas, setting profiles of hydrodynamic parameters that are quasi-homogeneous in radius, and then plasma adiabatic additional compression occurs with the entropy determined by irreversible processes at the first stage. Estimates show that the effect of striker’s non-sphericity is negligibly small. Shock waves occurring in lead when the striker collides with the target do no affect the compression process. The initial stage of real compression is thus adequately

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described by a uni-dimensional model. Notable effects of non-unidimensionality can be expected at the final stage when the cone deformation and thermal conductivity become significant. Before that moment, the compression stage can be deemed adiabatic and the limiting parameters necessary for the calculation of the neutron yield can be estimated if we know the plasma specific entropy, using the semi-empirical equation of state for lead. The calculation gives P ≈ 50 to 100 Mbar, T ≈ 0.3 to 0.5 keV, ρ/ρ 0 ≈ 103 , which corresponds to the neutron yield within 104 to 108 . In conclusion of the section on HDTF, we note that explosion compression experiments give an interesting approach to the study and optimization of targets for pulsed thermonuclear fusion. In addition to a significant expansion of the energy range, an explosion method enables studying compression hydrodynamics in a pure form, without considering the complex processes of interaction of corpuscular beams or light with plasma. Nuclear Explosions for Studying Extreme States of Matter. The creation of nuclear weapons gave experimenters a source of high energy concentration to generate intense shock waves in condensed media [51, 70]. Therefore, along with the measurements of the integral characteristics of a nuclear explosion and its effects, already in the first experiments a start was made to study the physical properties of matter at extreme pressures and temperatures. Physicists and, first of all, nuclear weapon developers immediately fully appreciated the possibilities of nuclear explosions as a tool to generate super-powerful shock and radiation waves and to study them using ultra-extreme states of matter [6, 7, 50, 51, 70]. In addition to the possibility of getting super-powerful shock waves and superhigh energy densities, these experiments have a number of other unique features. This is the widest range of pressure change in experiments, unidimensionality and high symmetry of measurement behavior using large samples the dimensions of which exceed laboratory dimensions by orders of magnitude. This resulted in a significant volume of invaluable experimental information in the area of ultra-high pressures the lower range of which is now approached by the technique of powerful lasers [51] (ref. Chaps. 3, 5). Plasma energy densities that are record-high for terrestrial conditions were obtained precisely in the near zone of a nuclear explosion. The physical setups and the main physical results of such experiments are contained in an exhaustive review [51] that we will follow in this discussion. First works with shock waves included measurements of the relative compressibility of matters with the use of the reflection method [50, 51] (Fig. 6.8). The method [50, 51] (Chap. 3) is based on recording the shock wave velocity successively going through the layers of investigated matters one of which is deemed reference the equation of state of which is assumed to be known. Its Hugoniot adiabat is set by interpolation between the experimental region and the calculational area where the respective theoretical models are applicable. The shock wave velocity in the reference standard is used to find the parameters of the initial states of reference matter, and the sought characteristics in the samples under study are determined by building pressure/mass velocity diagrams [50].

182 Fig. 6.8 Experimental setup for studying comparative compressibility of Fe (screen)–Pb–Cu–Ti [51]

6 Shock Waves in Nuclear Explosions

Polyethylene

Concrete Energy source Samples Ph el oto de ectr tec on to ic r

The first experiments using this method were done in accordance with [51] late in 1965 and published somewhat later [50, 59]. The limit of 1 TPa was then overcome using the system of Fe-Pb-U (3.8 TPa for Fe and 4.0 TPa for U). Data were then obtained for the comparable compressibility of water (for details, refer to [51, 70]) at pressures of 1.4 TPa, quartz with an initial density of 1.75 and 1.35 g/cm−3 at pressures of 2.0 and 1.8 TPa, plexiglass (C5 H8 O2 )n at pressures of 0.6 TPa, graphite, rutile (TiO2 ), rock salt, aluminum, a number of rocks (graphite, slate, dolomite) and a number of other continuous and porous matters below 1 TPa. The comparable compressibility of metals in the Pb-Cu-Cd system was studied in 1968 at pressures of 1.5 TPa [119] and then at 5 TPa [120]. In [120], a pressure of 5.2–5.8 TPa [121, 122] was obtained for the Fe-Pb system and the compressibility of porous metals was measured: coper, iron, tungsten [51, 121, 122] and a number of other elements at terapascal pressures. Since the energy of tested nuclear charges was not limited at the time, it became possible [51] to measure shock wave parameters at pressures of several terapascals at relatively long (up to 10 m) distances from the explosion center, which simplified the interpretation of results and made measurements with attenuation and low drop more reliable, since for these distances a shock wave has high symmetry and comparatively low pressures beyond the horizon. The limit of 10 TPa was overcome in [123] and then brought to 700 TPa [124, 126]. The world record in pressure at which the parameters of the equation of state of heavily compressed and multiply ionized plasma were measured is P ≈ 4 bln atmospheres and it was obtained in the near zone of a nuclear explosion [125–127]. The Los Alamos data for recording the absolute compressibility in the conditions of underground nuclear explosions are contained in [59, 65], and the data of the Lawrence Livermore National Laboratory—in [76]. Chinese scientists asserted themselves in 1990 [77].

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Experiments intended to record the relative compressibility of matters contain an uncertainty related with the inaccuracy of the equation of state for the standard, which grows as the studied pressures are increased. The method of “absolute” measurements of impact compressibility of materials lacks this disadvantage (Fig. 6.9) [51, 70]. This scheme records the flight velocity of an iron striker that is accelerated by a nuclear device to velocities W and that impacts an iron target. The experiments measure approach velocity W and shock wave velocity in the target DFe . Due to symmetry, the mass velocity of matter in the shock wave U Fe = W /2, which gives pressure PFe = g0 DFe U Fe and energy E Fe = 2 /2 in accordance with the laws of mass and energy conservation. UFe The data obtained using this method do not depend on the properties of the reference standard but require a number of conditions to be met: the striker and the target must have the well-known initial conditions ρ 0 , P0 , T 0 before an impact (heating by the radiation of a nuclear explosion is low), have a constant and well-known velocity W = const of striker approach at the time of impact, have good symmetry of the striker in flight and good symmetry of the shock wave in the target. The strict equality U = W /2 can be reached only in case of smooth, no-heating acceleration of the striker with its integrity being preserved. Deviations from this equation in reality are caused by the shock wave heating of the striker and its rear backing on the side of evaporated matter. According to [51] in the scheme (Fig. 6.9a), the striker gains 80% of the velocity W when it covers 20% of its path to the target. The parameters of the experiments (Fig. 6.9a, b) are given in Table 6.1 [51]. Further development of dynamic methods in the area of high pressures is related with the use of penetrating physical fields [6, 7, 51, 70]. American researchers [59] proposed a measurement scheme for the mass velocity in the field of high pressures, which is based on the shift of resonances of interaction of neutrons with the nuclei of moving matter relative to their position near resting nuclei (Doppler shift). To get high pressures, the energy of uranium nuclei fission by neutrons formed in a nuclear explosion is used. According to [65], a sample of the material under study adjoins the flat layer of uranium. The shock wave front velocity is determined by the time when light flashes appear on the test surfaces in the sample. To measure the mass velocity, neutrons with energies within 10–103 eV where resonances are usually located are used. Matters with prominent nuclear resonances are of interest, which ensures the significant attenuation of a flux of neutrons coming through the sample and recorded in the experiment. If the resonance in the sample is observed at the neutron energy εn (or at the neutron velocity vn ) and at the same time some part of the sample substance moves to the recording direction at the velocity u, the flux of neutrons the velocity of which is vn = vn + u will be additionally attenuated. In the spectrum of the neutron flux coming through the sample there will be two minimums owing to the separated resonance. The experimental task consists in recording the spectrum of the neutron flux. If the source of neutrons can be deemed instantaneous, the spectrum of the neutron flux is measured by the time t of neutrons passing the known base with size L. In this case, the time resolution of resonances will be determined by the difference

Energy source

Rock Target (Fe)

Foam plastic Striker (Fe) Screen (Fe)

b

Energy source

Rock

Target (Fe)

Striker (Fe) Screen (Fe) Foam plastic

Fig. 6.9 Two versions of an accelerating device (a) and (b) with similar parameters for determining the impact compressibility of iron by deceleration method [51]

a

184 6 Shock Waves in Nuclear Explosions

W 1 (km/s)

36.5

42.7

48.6

Experiment scheme

11a

11b

11c

58.8

42.7

36.5

W 2 (km/s)

D (km/s) 28.85 ± 0.7 32.4 ± 0.8 43.5 ± 1.0

W av (km/s) 36.5 ± 1.0 42.7 ± 1.2 60.8 ± 2.54 30.60

21.35

18.25

U (km/s)

Table 6.1 Experimental measurements obtained with the accelerating device shown in Fig. 6.9a, b

10.50

5.42

4.13

P (TPa)

26.50

22.99

21.34

ρ (g/cm3 )

3.37

2.93

2.72

σ = ρ/ρ 0

6 Shock Waves in Nuclear Explosions 185

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Fig. 6.10 Scheme of experiments for the generation of intense shock waves using an underground nuclear explosion [59], A nuclear device: 1 B4C neutron absorber; 2 experimental assembly of uranium-235 and molybdenum; 3 waveguides (12 m long); 4 optical radiation recorders; 5 time-of-flight neutron spectrometer; 6 solid-state detectors; 7 foils of lithium and plutonium

in the moments of approaching the neutron sensors at velocities vn and vn . The value u/vn < 1, so t ~ u. If the resonance in the sample is observed at the neutron energy εn = 103 eV and the shock wave velocity u ≥ 10 km/s, which is acceptable for measuring t, the base size L ≈ 20 m. Neutrons will cover this base for t ≈ 5 × 103 µs. The duration of the source of neutrons is 0.5–0.8 µs according to [59]. The sample thickness is selected such that a significant part of the sample is involved in motion over the source operation time. A significant role for the experimental support of the method is played by the spectral flux of neutrons in the resonance field εn = 10–103 eV. The number of neutrons in this spectrum is rather low. To increase it, a thin layer of hydrogen-containing matter (plexiglas) was placed between uranium and the sample in [59] (Fig. 6.10). This method makes it possible to increase the neutron flux at the boundary by about an order of magnitude. The discussed method of measuring the mass velocity is not universal. For matters under study, the cross-sections in resonances must ensure the well recordable attenuation of the neutron flux by both fixed and moving matters on thicknesses comparable with measurement bases. Molybdenum, iron, copper, etc. have such properties. Moreover, there are elements the nuclei of which have abnormally high resonance

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187

Fig. 6.11 Scheme of experiments for the absolute recording of aluminum compressibility: 1 optical channel; 2 fiducial layers; 3 matter under study; 4 shock wave formation channel; 5 collimating system; 6 collimating slits; 7 detectors of γ -radiation; 8, 9 radiation signals of fixed and mobile reference points [71, 72]

cross-sections (tungsten, gold, cobalt). Placing thin layers of these elements in the sample can also be used to measure the mass velocity. Moreover, in some cases, the energy widening of individual resonances can be used to evaluate matter temperature in front of and behind the shock wave front. Molybdenum studied in [59] has the most prominent resonances. Pressure P ≈ 90 Mbar was implemented in uranium, D = 18.7 km/s (± 5%) and u = 10.2 km/s (± 5%) were recorded in molybdenum. The accuracy achieved in measurements complicates the use of the obtained experimental point for calibrating the molybdenum equations of state. The main sources of measurement error are related with the uncertainty of the neutron source duration and various smears of resonances. However, the contribution to the error of many of these factors attenuates as the mass velocity rises, so that at u ≈ 100 km/s, there is a fundamental possibility to reach the accuracy Δu/u ≈ 1%. This stimulates further works to improve the method. A method for the simultaneous measurement of D and u values using gammaactive fiducial layers embedded into matter under study is proposed in [71]. The fiducial layers are carried away by moving matter during gas dynamic motion. The moments of passing test positions by them are recorded using a system of collimating slits (Fig. 6.11). The “flat” geometry of the shock wave front, fiducial layers and collimating slits when the planes of the respective surfaces are parallel to each other is most simply implemented in experiments (Fig. 6.11). For this purpose, a cylindrical channel made of matter (magnesium, organics, etc.) the density of which is less than that of a material where the experimental setup is located is installed along the wave travel path. The results of two-dimensional gas dynamic calculations show that the setting of such a channel ensures the sufficient advance of the shock wave front relative to the front in the environment, so that the wave front will be flat in the central part of the cylinder the diameter of which is ≈ 2/3 of the external diameter. The measurement unit is installed at the cylinder end. Protection of the collimating system that excludes

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its damage until the completion of recording the moments when the fiducial layers pass the test positions is provided by placing the layers of dense matter (lead, steel) along the wave travel path. To measure the mass velocity u, it is sufficient to record the moments when one fiducial layer passes the planes of two collimating slits with the distance d between them being a base, then u = d/(t 3 – t 1 ) (ref. Fig. 6.11). The placement of two fiducial layers near two collimating slits allows finding D by comparing the moments when their movement starts. D = d/(t 2 – t 1 ). In this case, shock compression on the wave front δ is measured only based on measured time intervals δ = (t 3 – t 1 )/(t 3 – t 2 ). Usually, the number of fiducial layers and collimating slits exceeds the indicated minimum in experimental units. This allows getting information about the transiency of the recorded shock wave phenomenon and about the behavior of neutron and gamma processes. The main element in the proposed simple scheme that allows for time measurements is the fiducial layer. Its radiation should pass through the peripheral layers of matter under study, where the gas dynamic motion significantly differs from the used “flat” shock wave flow. Therefore, the sources of rigid gamma-radiation are used in the fiducial layer. Due to the transiency and high velocity of processes, the layer’s radiation intensity should provide the possibility of recording the moments when they cross the planes of collimating slits, using photoelectronic detectors of PMT (photomultiplier tube) or PEK (photoelectric colorimeter) type in the analog mode. Therefore, pulse sources were used in [71, 72], which existed at the separated stage of the gas dynamic process. In particular, an intensive gamma-ray source was obtained in case of pulse neutron irradiation of matter whose nuclei have a radiativecapture cross-section that is ≈ 103 times greater than that of aluminum. In the existing pulse sources, fast neutrons are usually generated (E n ≈ 1 meV). The reactions of radiation capture are efficient at lower E n . Therefore, a neutron pulse must be in advance of the recorded gas dynamic motion for the time interval required to decelerate neutrons in matter under study to optimal energies. In some cases, europium can be used in fiducial layers, for which the cross-section of the (n, γ )-reaction is σ = 220–80 barn at E n = 10–100 eV. As seen above, aluminum is widely used as reference matter in using the reflection method. There are substantial uncertainties in the equation of state of this matter in the range of 5–150 Mbar. This attracted interest to aluminum in method applications (Fig. 6.12) [71, 72]. Other combinations of fiducial layer and sample matters are also possible. In some cases, it is apparently possible to use pulse neutron sources (converters) in fiducial layers. The results of measurements of transiency of movement and radiation of fiducial layers, conducted taking into account the latest methodological achievements in the processing of oscillograms are given in [51, 71, 72]. Here, we shall confine ourselves to giving shock adiabats for aluminum and lead, and the Hugoniot adiabat for aluminum [51] in comparison with the obtained experimental data (Fig. 6.13). It is seen that the Hugoniot adiabat reasonably describes the experimental data on average but some points go beyond regular 1–1.5% in terms of the error velocity. Possible reasons for that are considered in [51] (Table 6.2).

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189

D, km/s

D, km/s

U, km/s

U, km/s Fig. 6.12 D-U-diagram for aluminum according to laboratory measurements (1–6) and in the conditions of underground explosions: 7–10 absolute measurements; 11, 12 relative measurements. The ellipses of possible processing options of experiments [71] are dashed. The dash-dotted line shows the interpolation under [73], the dashed line shows the calculation under the Tomas-Fermi model [1, 60] P, TPa

Fig. 6.13 Shock adiabats of aluminum and lead at super-high pressures [140]. The arrows show the shift of the experimental data in the transition of the adiabat of the iron standard from curve 1 to curve 2

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6 Shock Waves in Nuclear Explosions

Table 6.2 Results of absolute measurements in aluminum [50, 71] Experiment

D, km/s

U, km/s

P, TPa

σ = ρ/ρ 0

1

24.2 ± 0.7

15.1 ± 0.4

0.99 ± 0.03

2.65 ± 0.1

2

23.4 ± 0.6

14.5 ± 0.3

0.93 ± 0.02

2.63 ± 0.7

3

40.0 ± 5.0

30.0 ± 2.0

3.20 ± 0.5

3.90 ± 1.2

4

30.5 ± 0.7

21.0 ± 0.6

1.73 ± 0.07

3.21 ± 0.2

Intense gamma-radiation of fiducial layers leads to an increase in the internal energy of aluminum to the values ≈ 1.5 kJ/g [71]. The resulting pressure leads to a decrease in the aluminum density to ≈ 2.55 g/cm3 and compression of a porous material of the fiducial layer, i.e. the wave enters the sample characterized by the initial state ρ 0 = 2.71 g/cm3 ; E 0 = 1.5 kJ/g. The experience of using a reference method for measuring shock compressibility shows [71] that there are wide opportunities to improve it in using various combinations of the reference standard and the material under study and also for various organization of experiments. The combination of experimental data on shock-compressed aluminum plasma corresponds to record-high and most extreme parameters in terrestrial conditions. The density of the internal energy of such plasma is E ≈ 109 J/cm3 , which is close to the energy density of nuclear matter, and the pressure P ≈ 4 Gbar is close to the pressure in the internal layers of the Sun. Plasma in these conditions n ≈ 4 × 1024 cm−3 , T ≈ 8 × 106 [74, 75] is not degenerated, nλ3 ≈ 0.07, 12-times ionized and the nonideality parameter G ≈ 0.1 is low. This is an experimental illustration of the assertion of the academician Sakharov [8] on the “heaven” for theorists in analyzing super-extreme states and on the simplification of the physical properties of plasma in the limit of ultra-high energy densities. It can be seen that the parameter range in question is adjacent to the region where the energy and pressure of equilibrium light radiation make an appreciable contribution to the thermodynamics of the system: En ∼

4σ T 4 ER , PR ∼ , c 3

i.e., the mode of matter dynamics is implemented which is required to implement the “third” idea [8]. In conclusion of this chapter, we give the words of the academician Yu. B. Khariton, chief designer of the Arzamas-16 Nuclear Center (currently—the Russian Federal Nuclear Centre All-Russian Scientific Research Institute of Experimental Physics): “I am astonished of and bow to what was done by our people in 19461949. But it was also hard thereafter. In terms of stress, heroism, artistic flashes and devotion, this period is indescribable. Four years after the deadly battle with fascism, my country eliminated the USA monopoly for the possession of an atomic bomb. Eight years after the war, we were the first to create and test a hydrogen bomb, and in twelve years we launched the first Earth satellite; another four years passed and

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we opened the road to space for the humanity. You see that these are the milestones of transcendent importance in the civilization history… The creation of missile nuclear weapons required the utmost exertion of human intelligence and powers. Perhaps, an excuse here is that for almost fifty years nuclear weapons, with their unprecedented destructive power, the use of which threatens life on Earth, restrained the global powers from war, from an irretrievable step leading to a worldwide catastrophe. Probably, the main paradox of our time is that the most sophisticated weapons of mass destruction still supports peace on the planet being a very powerful restraining factor…” [13, 38, 67, 68].

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Chapter 7

Cosmic Shock Waves

Almost all observed astrophysical phenomena and objects excite powerful shock waves during their inception, evolution and death [1–20], and these waves bear unique information on these extreme processes and explain the generation of nonthermal ultrahigh-energy particles. The nature, characteristics and physical reasons of shock waves generated in space can catch even the most daring imagination and shock wave processes have characteristic sizes from several kilometers to several megaparsecs—the range of 10 orders of magnitude! The lower part of this range includes shock waves of solar and stellar winds with velocities of 500–3000 km/s when they go around planets, their satellites and other compact objects [8, 10, 18, 20] (ref. Figs. 7.1 and 7.2), and the upper part has shock waves in such extended formations as galactic clusters and active cores of galaxies [4, 5], remnants of supernovas and pulsar nebulae [6–8], solar and stellar winds [8, 10, 18, 20] (Fig. 7.3). Shock waves formed during fast energy release near compact relativistic objects— black holes and pulsars can have velocities close to the velocity of light [6–8]. They are called relativistic unlike relatively slow non-relativistic shock waves. The front width of cosmic shock waves in a rarefied plasma is many orders less (∼ 106 times) than the classical size determined by any binary (Fig. 7.3, [21]) collisional (see below) paths and is set by collective interactions of electromagnetic field fluctuations and charged particles. Let us highlight that the presence of charged particles (plasma) and magnetic field is fundamentally important for the implementation of this beautiful nonlinear mechanism. Despite such huge difference in scales, it has been found (ref. for example, Fig. 7.4, [21]) that shock wave thickness in many astrophysical objects is much less than the length of the free path of particles relative to binary impacts. Currently, the study of shock waves in space plays a increasingly important role in our understanding of principal issues of the Universe evolution, the formation of stars and galaxies, the heating of interstellar and galactic matter, the generation of super-energetic particles, the evolution of stars and the appearance of heavy (heavier

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Fortov, Intense Shock Waves on Earth and in Space, Shock Wave and High Pressure Phenomena, https://doi.org/10.1007/978-3-030-74840-1_7

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Fig. 7.1 Balogh and Treumann [10]. Scheme of shock waves when a supersonic solar wind of the Earth’s dipolar geomagnetic field flows around the Earth (Tsurutani and Stone, 1985, courtesy of the American Geophysical Union). Inclined blue lines are the solar wind magnetic field (interplanetary magnetic field)

Fig. 7.2 Schematic representation of the entire magnetosphere of the Earth. The color indicates plasma temperature

7 Cosmic Shock Waves

199

Fig. 7.3 Gas ring heated by a powerful shock wave around supernova 1987A (SN1987A) [22]

than iron and nickel) chemical elements in case of explosions of supernovas, and, finally, helps finding a key to understanding the advent of life. Modern progress in understanding the processes of inception and collision of galactic clusters each containing thousands of galaxies that generate intense shock waves lies in that these phenomena are considered as an important and one of the few sources (particles, radio-frequency radiation) of information on these extreme stages of the Universe evolution and astrophysical phenomena at distances above 200 kiloparsecs. Finally, shock wave phenomenon are used to judge about dark matter—its density, distribution and effect on the structure of the Universe in the range invisible to us. In particular, it has been found that large-scale cosmologic shock waves are born in the external part of the forming halo of dark matter. Shock waves propagate from the place of their origin in galaxies to huge (more than 100 kilopascals) distances creating turbulence and heating the intergalactic medium to the temperatures of the kiloelectronvolt range accelerating cosmic rays to energies of ∼ 2 × 1012 –1020 eV and more and generating magnetic fields in turbulent plasma. The main sources of energy in astrophysical objects are the gravity of massive bodies and nuclear reactions in stars. The role of shock waves is often reduced to the transformation of part of the gravity field energy into the kinetic energy of supersonic motions and into the internal energy of the space environment, to the amplification of

200

7 Cosmic Shock Waves

Cluster 1

Cluster 2

Cluster 3

Cluster 4

Fig. 7.4 Scopke et al. [21]. Profiles of plasma and magnetic field parameters recorded by ISEE-1, 2 space probes (Courtesy Amer. Geophys. Union) at the intersection of a detached shock wave in the vicinity of the Earth. N e , N i is the electron density in cm−3 , T p and T e is the temperature of electrons and protons in K, V p is the velocity of protons in km/s, Pe is the pressure of electrons in 10−9 nm−2 , B is the magnetic field intensity in nTl

magnetic fields and the acceleration of relativistic particles [4]. This is an important and sometimes the only way to transfer the gravitational energy into the energy of visible matter. According to [4, 5], hot gas in galactic clusters was primarily heated by shock waves. As a result, in accordance with existing estimates, about 40% of intergalactic baryonic matter was heated by shock waves to temperatures of 105 –106 K [4, 5]. The radiation of plasma having such temperature is in the ultraviolet band

7 Cosmic Shock Waves

201

where observations are significantly hampered by strong absorption of radiation by gas. According to [1], radiation in this band indicates that the Universe consists of ordinary matter, photons, relic radiation, hidden mass, and “vacuum-like” matter, which manifests itself as a nonzero cosmological constant [23, 24]. Ordinary matter is considered to mean mainly protons, electrons, and neutrons. Hydrogen is the dominant element. There are also helium and a small amount of lithium. Heavy atoms are found in very small amounts in the Universe. The number of protons in our Universe of radius 1028 cm is estimated to be N = 1080 , in agreement with the Eddington–Dirac number. The density of matter in the Universe is ρ matt = 10–31 g/cm3 . Ordinary matter is found in stars, planets, comets, interstellar gas, meteorites, and cosmic rays. Antigravitating “dark” energy makes up about 74% of all energy–mass [1]. The gravitating mass (“dark”, or hidden, mass) accounts for about a quarter of the average density of the Universe, approximately 22% is occupied by “dark” matter, and only about 4% is accounted for by ordinary baryonic matter presented in the Mendeleev’s Periodic Table. These 4% are found in stars, planets, and the interstellar medium. The interstellar medium accounts for 4/5 of the mass of baryonic matter and only 0.5% of the average density of the Universe is concentrated in stars. They occupy only a 10−25 part of the total volume of the Universe [1, 2]. Despite these modest “average” figures, stars play a truly outstanding role in our Universe, since their bright radiation reaches us from enormous distances and is the main source of information about the various processes of transformation of matter, radiation and energy, taking place in the Universe. In stars, there occur irreversible thermonuclear transformations, the production of heavy elements, the generation of exotic, inaccessible for us, forms of matter, i.e. neutron matter, quark-gluon plasma, etc. While the information about the first hundred thousand years of the evolution of the Universe reaches us in the form of relic radiation, we judge the history of the next billions of years by observing stars. The range of variation of the parameters of matter in the Universe is extremely wide: from cosmic vacuum and rarefied intergalactic gas with a density of 1030 g/cm3 to extremely high densities of 1014 –1017 g/cm3 of neutron stars (Table 7.1). The temperature of the intergalactic gas with the density n ≈ 10−4 –10−3 cm−3 amounts to 107 –108 K and can reach a billion degrees under heating by shock waves (as a result of the shedding of the outer stellar layers, stellar collisions and explosions, collisions of gas clouds, etc.). Inside the neutron stars, the temperature is 108 –1011 K [1]. The most part of visible matter (99%) is heated to a temperature exceeding 105 K. While the magnetic fields are of the order of 10−9 Gs and 10−6 Gs in the intergalactic space and near the Galactic plane, respectively, at the surface of neutron stars this field is 22 orders of magnitude higher. The record here belongs to magnetars— neutron stars formed after supernova explosions. Magnetars have a giant magnetic field of up to 1015 Gs, which corresponds to the densities of the order of 108 g/cm3 , approaching the density of nuclear matter [1].

202

7 Cosmic Shock Waves

Table 7.1 Characteristic parameters of matter in nature and in the laboratory Object

T (K)

ρ (g/cm3)

P (bar)

Intergalactic gas

107 –108

10−30 –10−3

10−17 –10−7

Earth, center

5 × 103

10–20

3.6 × 106

Jupiter, center

(1.5–3) ×

5–30

(3–6) × 107

Exoplanets

103 –105

1–30

107 –108

Diamond anvils

4 × 103

5–20

5 × 106

Shock waves

107

13–50

5 × 109

Controlled thermonuclear fusion, magnetic confinement

108

3×

Controlled thermonuclear fusion, inertial confinement

108

150–200

2 × 1011

Sun

1.5 × 107

150

1011

Red giant

(2–3) ×

103 –104

5 × 1012

White dwarf

107

106 –109

1016 –1022

Relativistic collisions of gold nuclei, 100 GeV per nucleon, Brookhaven

2×

1015

1030

Neutron star, black hole, γ -bursts

108 –1011

1014 –5 × 1015

1025 –1027

Early Universe (Planckian conditions)

1032

1094

10106

104

107

107 –7

×

1013

10−9

50

Giant black holes “devour” entire star systems and hot galactic nuclei. The recently discussed magnetic tunnels (“wormholes”) [24] probably connect our and other universes, if they exist. The gravitational accretion of matter generates highly collimated jets, beams of charged particles accelerated to ultrahigh energies. Explosions of supernovae generate shock waves, plasma radiations, turbulent plasma, and dust clouds, providing the raw material for the formation of new stars. Neutron stars with a size of several kilometers rotate at kilohertz frequencies and affect plasma by their giant magnetic fields generating power X-ray radiation [1, 2]. Figure 7.5 [25] shows the Hillas diagram with characteristic magnetic fields and sizes of astrophysical objects. It also demonstrates their capabilities as particle accelerators. Solid lines show the estimates of the sizes (in parsecs) and the values of magnetic field induction (in gausses) of objects capable of accelerating protons to energies of 1020 and 1021 eV at a shock wave velocity 300 times less than the velocity of light. Dashed line is the same but for iron cores. Dark spots indicate the observed sizes and magnetic fields of various astrophysical objects. It seems that in the Universe known to us, there are no, under the assumptions made about the particle acceleration mechanism (Fermi) itself, an obvious candidate for the role of a Tevatron accelerator—an accelerator of observable-energy cosmic particles. The characteristics of the Tevatron and LHC accelerators are also shown in Fig. 7.1 for comparison.

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В, Gs Neutron stars

Active galactic nuclei (AGN) 1020 eV protons

White dwarfs

Gamma-ray bursts

1020 eV iron

Interplanetary medium

Radio-galaxies

1012 eV

Galactic clusters Galaxy: 1 km

106 km

1АЕ

Disc Halo 1 pc 1 kpc 1 Mpc

Fig. 7.5 Hillas diagram with characteristic magnetic fields and sizes of astrophysical objects [25]

The Fermi mechanism plays a specific role in problems concerning the formation of spectra of relativistic particles and, in particular, observed space particles. This smart mechanism of acceleration was proposed by Fermi [26] in 1949 to explain the origin of space particles of superhigh energies from several kilovolts to 1020 eV [1]. The thing is that at the beginning of the last century it was established [27] that a flux of charged particles falls to the Earth from space, and their energy is within a wide range: from several kiloelectronvolts up to 1020 eV (Fig. 7.6). Cosmic rays of giant energies up to 3 × 1020 eV (which corresponds to a tennis ball energy of ∼ 50 J) were first recorded in 1991. Since then, dozens of cases of energy of about 1020 eV were recorded at various facilities [27]. The events were detected by recording the fluorescence of atmospheric cascades in the atmosphere or in specially created detecting gratings at the Earth’s surface. A surprisingly isotropic spatial distribution of incident particles is noted. If we are talking about protons, then the effect of the magnetic field on them is little and most likely they are not of galactic origin. Cosmic rays are a strongly rarefied relativistic gas the particles of which slightly interact with each other, with a power law spectrum (Fig. 7.6) rather than Maxwellian energetic spectrum. At the same time, cosmic rays collide with interstellar medium particles and interact with interstellar and intergalactic magnetic fields. The flow of cosmic rays near the Earth is low and amounts to about 1 particle/(cm2 s). This refers to the integral energy flow of particles in which cosmic protons are predominant,

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Fig. 7.6 Observed spectrum of cosmic ray energies [1, 27]

Stratospheric and extraterrestrial observations

m-2·sr-1·eV-1

Giant air shower measurement

Knee

Second knee Ankle Expected GZK-limit eV

with energies of the order of GeV. However, the energy density of about 1 eV/cm3 is comparable with the density of electromagnetic radiation of all stars in the Galaxy or with the energy density of the thermal motion of interstellar gas and with the kinetic energy of turbulent motion and also with the energy density of the Galaxy magnetic field [1]. A flow of particles of superhigh energies is extremely low, about 1 particle/km2 per 100 years, however, their origin and propagation are of greatest interest. Data on synchrotron radio-frequency radiation in supernova remnants indicate the presence of electrons with energies of 50 meV, 30 GeV [1]. High-energy synchrotron radiation was recorded in the infrared band, which indicates the presence of electrons with energies up to about 200 GeV. The detection of nonthermal X-radiation with a characteristic power law spectrum and energies up to several dozens of kiloelectronvolts in about ten young galactic supernova remnants is explained by the synchrotron radiation of electrons with very high energies up to 10–100 TeV. The Compton back-scattering of background photons by electrons with such a high energy and the generation of gamma radiation during the interaction of protons and nuclei with energies up to ≈ 100 TeV with gas nuclei explain the presence of teraelectronvolt gamma-radiation recorded for a number of young supernova remnants. The spatial distribution of nonthermal radiation in all bands indicates that the acceleration of particles in shell-type remnants of supernovae occurs directly on the shock wave generated by a supernova explosion. The natural condition that the sources must comply with is the geometric one: a particle must not leave the accelerator until it reaches the required energy. It is usually assumed that the particle is accelerated by an electric field and retained by a magnetic field; then, the geometric criterion is expressed in the fact that the particle’s Larmor radius must not exceed the linear size of the accelerator [28].

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In paper [28], the following criteria are given that must be satisfied by the superhigh energy particle accelerators: • the accelerated particle must remain inside the accelerator during acceleration (geometric criterion); • each source must have a sufficient energy reserve for transfer to particles (individual power); • radiation losses must not exceed the energy acquired; • interaction losses must not exceed the energy acquired; • the total number and total power of the sources must be sufficiency high to ensure the observed flow of cosmic rays; • the accompanying radiation of photons, neutrino and cosmic rays of lower energies must not exceed observational restrictions both for an individual source and for a diffuse flow. Only some galaxies satisfy these criteria and can be accelerators of particles of observed energies while most do not. Cosmic rays must be apparently accelerated by some nonthermal mechanism since the temperature in the center of the most massive stars is below tens of kiloelectronvolts. Therefore, the studies of such particles involve classical inductive and stochastic mechanisms (for example, the Fermi mechanism of the first and second orders [26] (Fig. 7.7) when particles are accelerated in the stochastic interaction of particles with magnetic areas or by collisionless shock waves generated by the explosions of supernova or radiations of matter from the active nuclei of galaxies in the turbulent area of which various instabilities are developed. However, in case of high energies of accelerated electrons and positrons, significant radiation energy losses occur [26, 29]. The data on the composition of cosmic rays also confirm [29] that particles are accelerated on the shock wave propagating through the interstellar medium or in the pre-supernova wind. According to [29], the plasma-pinch mechanism of acceleration must be also taken into account. In particular, after taking into account such atomic properties as the first potential of ionization or volatility, the chemical composition of cosmic ray sources appears to be close to the normal composition of the local interstellar medium and solar photosphere. Probably, the acceleration of ions and dust particles occurs in partially ionized interstellar gas and/or hot “bubbles” of interstellar gas with a high frequency of supernova explosions. The acceleration mechanism of cosmic rays in supernova remnants is a version of the first order Fermi acceleration mechanism [29]. Fast particles are accelerated in a flow of gas compressed on the shock wave due to multiple intersections of the shock-wave front by diffusing fast particles (Fig. 7.7). Particles are diffused by their scattering on magnetic field irregularities. The acceleration mechanism in the wakefield excited by the magnetic hydrodynamic shock waves of gamma-ray burst is proposed in paper [30]. A consideration gives high (about 1019 eV) wakefields in a relativistic moving plasma, which are close to Schwinger wakefields and used for the description of the gamma-particle spectrum of these relativistic sources.

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Accelerated articles

Shock wave Interstellar magnetic field Fig. 7.7 Remnants of SN 1987 supernova explosion. The figure shows a qualitative picture of the mechanism of the first order Fermi particle acceleration on the shock wave. X-ray telescope image from the Chandra satellite [31]

The wakefield acceleration mechanism can be implemented when electrons are accelerated in jets of massive black holes that emit powerful X-radiation occurring by the braking and synchrotron mechanisms in relativistic jets from well collimated flows of electrons (or positrons) with an energy of the order of 1 GeV. The role of Alfven waves in the generation of wakefields and their possibilities for particle acceleration are discussed in [30]. Supernova remnants are the main ones [1, 2] but not the only sources of relativistic particles in the interstellar medium. In particular, pulsars generating high-energy electron–positron pairs can be responsible for the presence of positrons observed in cosmic rays. The measured flow of positrons with energies above 10 GeV appeared to be higher than the expected flow of secondary positrons arising from the interaction of cosmic rays with atoms of interstellar gas and the contribution of pulsars explains this discrepancy in principle. The final clarification of the nature of a high flow of positrons in cosmic rays is very important because an alternative explanation suggests that these positrons are the decay products of “dark” matter. The central problem of cosmic-ray astrophysics is the origin of particles of the highest energies, E > 1019 eV. The observed abrupt decrease in the flow of particles at energies exceeding 5 × 1019 eV shows that these particles withstand interaction with

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photons of the Universe background radiation for more than 3 × 109 years and have an extragalactic origin. That being the case, protons of superhigh energies lose energy for the creation of electron–positron pairs and pions (Greisen-Zatsepin-Kuzmin effect— Fig. 7.6) [1, 2], while nuclei are additionally subject to photodisintegration. Cosmic rays with energies below 1017 eV observed near the Earth have a galactic origin and are accelerated in supernova remnants. The electromagnetic acceleration mechanism is related with the generation of nonstationary electric fields, for example, in the magnetospheres of pulsars and neutron stars where magnetic fields near the surface reach 1012 Gs and in the magnetospheres of magnetars where magnetic fields reach 1014 Gs at a scale of several kilometers. With a rotation frequency of just 10−3 s−1 , this is enough for the acceleration of particles to 1019 eV. Let us note that the direct (though not conventional) method for measuring induction B in neutron stars is based on cyclotron lines. However, the observed radiation of superhigh-energy particles is isotropic and not explicitly correlated with the position of the radiating object. Particles can be accelerated both in the central areas of active galaxies (in the immediate vicinity to the black hole of accretion disk) and in extended structures (jets, radio ears, hot spots and nodules) [1, 2]. Simple evaluations [1, 2] show that in terms of energetic characteristics, galactic jets with active nuclei can also be sources of observed cosmic rays of superhigh energies. To maintain the intensity of cosmic rays in the intergalactic medium, observed at energies above 1019 eV, a source power of the order of 3 × 1036 erg/(s Mpc3 ) is required. At the same time, galactic jets with active nuclei emit a kinetic energy of 1040 erg/(s Mpc3 ) and even 1044 –1046 erg/s. It was found that the nuclei of active galaxies feed large-scale (from subparsec to megaparsec), more or less rectilinear jets [1, 2]—relativistic (in powerful radiogalaxies, blazars and quasars) or non-relativistic (in Seyfert galaxies). Black holes in the active nuclei of galaxies have masses usually exceeding 108 solar masses. It is assumed that all jets are fed from the central black hole. At scales below 1 pc, the main contribution to the energy flow is determined by the magnetic field, but at scales of the order of 1 pc, the energy of relativistic particles can predominate. In relativistic jets, internal shock waves (nodules) and terminal shock waves (hot spots), extended areas of extragalactic space fed by a jet can be observed. Magnetic fields in such objects 10−2 –102 cpc in size are estimated to be 10−6 –10−3 Gs. Cosmic gamma-ray bursts related with the relativistic jets of compact objects of star masses (most probably—black holes) form magnetic fields of 10−3 –108 Gs at scales of 10−5 –1 pc. Comparison of observations in radio and X-ray bands indicates, with sufficient confidence, that the radiation of a number of jets is related with accelerating particles, and such acceleration takes place not only in the finite number of shock waves but also using some distributed mechanism. In some cases, the presence of ordered fields in the jets has been proved, which allows the possibility of inductive acceleration. Plasma of high energy densities plays a significant role in understanding such phenomena and the study of cosmic-ray showers possibly will be carried out in plasma and accelerator laboratories. The most important are shock waves occurring

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from multiple galactic sources, such as nuclei of active galaxies and relativistic jets, gamma-ray bursts, collisions, galactic clusters, etc. An alternative shock-wave mechanism of generation by magnetic reconnection plays an important role in flows with the dominating role of magnetic energy. Let us briefly discuss the situation beyond compact galactic objects [1]. Gigantic momentum energy releases in astrophysical objects are closely related with the generation of intense shock waves, soliton waves and contact surfaces in space plasma, which can be caused by an explosion of supernova, stellar wind, galactic spiral wave, mutual collision of clouds and stars and other reasons. When an explosion wave leaves dense medium, the time for its radioactive cooling becomes less than the hydrodynamic time. Such radiating shock waves can occur in the final phases of supernova evolution, at the stage of the afterglow of gamma-ray bursts and supernova. The temperature of shock-compressed plasma in dense molecular clouds reaches dozens of million degrees. These areas are subject to radial oscillations if the cooling time grows slowly with temperature. Moreover, dense cooled areas are also unstable in the longitudinal direction. Detailed computer calculations have shown that the propagation rate of such radiation waves in the interstellar medium n = d ln R/γ ln t ≈ 1/ 3, which is lower than the Sedov-Taylor rate equal to 2/5 but higher than the simple self-similar analytical limit equal to 2/7. Therefore, the study of radiating explosion waves in astrophysics is of great importance, linking complex hydrodynamic processes and stimulating large-scale ejections of matter. Observations show that the space between stars is not empty or homogeneous (Figs. 7.8, 7.9). It is filled with matter of low density, radiation and magnetic field with the respective characteristic energy density of the order of 1 eV/cm3 . On the average, one cubic centimeter of interstellar space contains no more than one atom of hydrogen and much less atoms of other chemical elements. No more than ten dust particles one micron each are found in one cubic kilometer of such space. The temperature of matter varies here widely—from 10 to 106 K. The electron temperature is within 8000–17,000 K with an average temperature of 12,000 K, and the electron density in planetary nebulae is 102 –105 cm−3 in young compact objects [32]. Dense regions of gas and dust (Figs. 7.9, 7.10, 7.11, 7.12, 7.13 and 7.14) with sizes of 100–300 pc and masses up to 107 M  are called clouds (or nebulae) and are divided into diffuse (T ≈ 102 –103 K, n ≈ 1–102 cm−3 ), dark (10–102 K, 102 –104 cm−3 ), molecular (5–50 K, 4 × 102 –106 cm−3 ) and globules (10–30 K, 103 –106 cm−3 ). Probably, some of them occurred in the regions of active star formation, and the occurrence of molecules, in turn, shows that the medium has cooled down, condensed and is ready for star formation [1]. A wide spectrum of molecules has now been discovered, up to structures containing 13 atoms (cyano-decapeptin) formed in gas-phase ion reactions and on the surface of dust particles having catalytic properties. It has been shown [1] that the simplest molecules are formed during 104 –105 years, and a 10–13-atom molecule requires dozens million years. Dust 0.01 to 0.2–0.3 μm in size is present in the interstellar medium taking part in the catalytic synthesis of molecules, retains star radiation and transfers its

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Fig. 7.8 The Milky Way in the direction of the Norma constellation in the infrared band. The central strip: Milky Way segment 9 long and 2 wide. The upper and lower rows: enlarged fragments of the central image, demonstrating typical objects and structures of the Milly Way—star-forming regions, dense and diffuse interstellar clouds, planetary nebulae. The images were made by the Spitzer (NASA) infrared space telescope in 2005 within a spectrum band of 3.6–8.0 μm (https:// photojournal.jpl.nasa.gov/catalog/PIA03239)

momentum to the interstellar gas, taking part in the radiation balance and, surely, is an important factor of the star and planet formation. Although the dust mass is only 0.03% of the total Galaxy mass, its luminosity is 30% of the luminosity of stars and completely determines the galactic radiation in the infrared band. The radiationinduced expulsion of dust from the atmosphere of red giants and explosions of nova and supernova are the main dust producers in space ((3–4) × 10−3 M  per year). According to estimates, our Galaxy has 8000–40,000 planetary nebulae located up to dozen parsecs from us [25]. Molecular Clouds. The equilibrium in giant molecular clouds is maintained by gravity forces and dynamic pressure of large-scale flows of such matter as in collimated jets with a power of up to 1037 erg/s, in shock waves, collisions with other clouds and in star formation processes [1]. In the course of time, star formation destroys molecular clouds that live on the average up to 107 –108 years. On the other hand, massive interstellar clouds themselves can have a destructive effect on star clusters. The precursors of planetary nebulae belong to stars with intermediate masses of 0.8–8 M  . Stars with the initial masses above 8 M  on the main sequence do not go through the stage of planetary nebula but experience core collapse, flare up in this case as supernovae and release their massive shell, and their core is transformed

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0.44 μm

0.55 μm

0.90 μm

2.16 μm

1.65 μm

1.25 μm

Fig. 7.9 Gas-dust cloud B68 0.2 pc in size in the Serpentarius constellation distanced 150 pc from the Earth. In the optical band, no star can be seen behind it (first photo). However, interstellar dust absorbs infrared rays weaker and as the wavelength increases (given in micrometers), the cloud becomes more and more transparent. Photo of VLT (ESO) telescope (https://www.eso.org/public/ outreach/press-rel/pr-1999/phot-29-99.html)

into a neutron star or a black hole. Stars with masses below 0.8 M  cannot create a planetary nebula: they are transformed into carbon–oxygen white dwarfs. When analyzing the formation of interplanetary clouds, several models are considered [1, 2]: agglomeration in random collisions, Parker or magnetic Rayleigh–Taylor instabilities, gravitational (Jeans) instability and gas compaction by expanding gas shells around the regions of star formation. The amount of interstellar gas in galaxies depends on many factors [24, 25], among which there are star formation, gas release by post-evolution stars (Figs. 7.9, 7.10, 7.11, 7.12, 7.13, 7.14, 7.15 and 7.16), galaxy accretion, etc. The bulk of the gas in disc galaxies is atomic and molecular gases. About 97% by mass are hydrogen (70–75%) and helium (20–25%). The interstellar gas medium is heterogeneous at multiple scales due to density waves, shock waves, soliton waves, heated or cold giant clouds. Hot or coronal gas has a temperature of about 105 –106 K and dimensions of hundreds of parsecs. Its origin is related with the activity of young stars—stellar wind, star formation, supernova explosions, etc. Hot gas in giant galaxies is a result of the ejection of gas shell by red giants during their evolution or collision (with a white dwarf). Since a significant fraction of the energy of a supernova explosion is transferred into the environment, expanding regions of hot gas (Fig. 7.11, [33]) and shock waves moving through the interstellar gas occur. They stimulate star formation by compressing dense gas areas, accelerating their cool-down and further

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Fig. 7.10 Cone Nebula (NGC 2264) [22]

Fig. 7.11 Omega Nebula (M17) [24]

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Fig. 7.12 Galaxies M81 (below) and M82 (above) in UV-band. An explosive nature of star formation in M82 demonstrates the release of gas heated by stars from the central part of the galaxy [33]

Shock-wave fronts

Fig. 7.13 Interacting Antennae galaxies (NG 4038 and NG 4039). The right photo of the “Hubble” space telescope demonstrates active star formation in the central regions of collided galaxies [33]

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Fig. 7.14 Red Rectangle Nebula (HD 44179). Jets of ejected gas can be seen [34]

Fig. 7.15 A new-born star cluster (region N90) surrounded by the residual gas of which it was formed [1]

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Fig. 7.16 Galaxy NGC 3079. A photo obtained by the “Hubble” space telescope demonstrates the high activity of the core. Hot gas ejections into the intergalactic space serve as a material for the formation of young stars

compression thereby assisting star formation (Figs. 7.11, 7.12 and 7.13, [33]). Shock waves generated by strong tidal interactions in tight galaxy systems play the same role. Observations using “Hubble” telescope [22] and BIMA ground-based millimeter interferometer provided valuable data on the distribution of velocities and dynamics of matter inside clouds enabling to check many hypothesis on the radiation dynamics of matter, the role of cloud matter in star formation, etc. (Figs. 7.15, 7.17). Molecular gas is observed not only in the CO line, but in OH, H2, NH3 , HCN, HCO+ lines as well. This means that molecules are preserved long after nebula ionization. Dust that is heated in this case and becomes a source of the IR-radiation protects them from the severe radiation of the core. In many astrophysical objects, there is a significant admixture of neutral atoms and molecules along with the charged component [33, 34]. In addition to hot rarefied and fully ionized caverns (the temperature T ≈ 106 K, the concentration n ≈ 2 × 10−3 ions/cm3 ), there is a “warm phase” in the galactic disc (T ≈ 104 K, n ≈ 0.2 particles/cm3 ) the ionization degree of which is about 0.1 and which occupies dozen percent of the volume. In cold neutral clouds being the main centers of star formation and often containing young, rapidly evolving active stars, the concentration of matter is another 2–3 orders of magnitude higher and temperature and ionization degree are lower. The degree of ionization of matter is also low in the photospheres of the Sun and many stars. It is only about 10−3 in the Sun photosphere. In all the above and many other objects, apart from partially ionized background plasma, there are also relativistic (cosmic

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Fig. 7.17 The central star of the Bug nebula (NGC 6302) surrounded by dense, dusty and carbon-saturated torus (top right) is one of the hottest [34]

rays) and non-relativistic super-thermal particles the sources of which can be located both in the considered object and far beyond it. Obviously, similar conditions are also possible in objects of terrestrial origin (ionosphere) [35]. Planetary Nebulae. A planetary nebula is, in the first approximation, an ionized gas shell surrounding a hot star (core) located in its center. The external areas of the shell may contain neutral gas. Dust is not only in the interstellar medium and molecular clouds but it can also form its own dust clouds identified according to excessive IR-radiation. Today, we know more than 100 young planets with gas dust discs, which contain the matter of the planetary system being formed [36]. The analysis of IR-spectra of dust discs gives large sizes of particles—from 0.5 μm to several centimeters. The jet dynamics of dust clouds and dust discs is influenced by the fact that particles can be charged (due to photoionization) forming the so-called “dusty” plasma [37] having unusual properties and, as experiments have shown, increased viscosity (for details, Refs. [37, 38]). Planetary nebulae play an important role in the evolution of the interstellar medium of galaxies. Along with supernovas, they supply matter to this medium for the formation of the following generations of stars, so that the medium is enriched by nucleosynthesis products from the stellar interior that gave birth to that nebula [1]. In our Galaxy, a new planetary nebula is formed approximately every year. The mass returned to the interstellar medium by stars in the form of stellar winds and a scattering planetary nebula is on the average 3.3 M  per year. About 95% of all stars in the Galaxy end their evolution as a white dwarf. Among them, about 60% pass this stage of evolution fast enough to ionize their shell and

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form a planetary nebula. The data on these nebulae allow us to evaluate the rate of star death in the Galaxy in the modern era and judge of the star formation history in it [33, 34]. To describe the initial phase of spontaneous star formation, various models are used [36]: gravitational (Jeans) model, generalized model in case of general relativity [2], magnetodynamic model, thermal instability model, and interstellar matter shockwave compression model. Along with a spontaneous star formation, we consider the induced formation stimulated by an explosion when a neighboring star is born, which gives rise to compression waves and causes a chain reaction of star formation [1]. A confirmation of such a stimulated mechanism can be a region around the Omega emission nebula (M17), with a space free of gas on the one side of it, with a group of young stars (even younger stars are located in the emission nebula), and a molecular cloud on the other side, with signs of star formation. Paper [39] gives other examples of the waves of stimulated star formation, propagating along a single molecular cloud at a scale of about 100 pc. In this case, an analogy with the combustion and detonation wave propagation in chemically active media seems possible. Paper [40] was apparently the first to express an idea of a shock wave from a supernova explosion as of the initiating reason of star formation. The role of radiation losses and cloud fragmentation has been analyzed for two velocities of shock waves. The mechanisms of compression of interstellar clouds by external radiation pressure were considered as a trigger mechanism to launch a self-propagating or “epidemic” wave of star formation. The idea of the instability of shells of gas compressed by a flow of stellar wind was also considered. At the same time, the production of a star and a system of stars in the mother cloud often leads to its destruction and can stimulate the production of new interstellar clouds. The scheme of matter exchange between stars and interstellar medium is given in Fig. 7.18 [36]. The interstellar medium is constantly replenished with the material of stars in case of their explosions and the stars themselves are born from the interstellar medium. At the same time, the remnants of evolution, which are then weakly destroyed, constantly fall out of this eddy: white dwarfs, neutron stars, black holes and planets. In general, stars with a mass of 1 M  return 40% of their matter into the interstellar medium and a star with a mass of 9 M  returns more than 90% [36]. The total flow of star mass into the interstellar medium is about 1 M  /year, while gas inside the interstellar medium goes from the diffuse medium to molecular clouds and back at a rate of 102 M  /year compared with which efforts for star formation are just 1 to 3%. However, not all this baryonic matter can be as easily observed as the matter that gets into bright stars, nebulae and other visible objects. The thing is that if we collect all the matter visible through telescopes, the result will be only tenths of the total amount of matter occurred as a result of the Big Bang. This matter deficiency is related with “dark” energy and “dark” matter (96% of the observable Universe) and belongs to remaining 4% of baryonic matter [1, 23]. Based on the above, let us make rough evaluations of the shock-wave front thickness in the cosmic space [4–13].

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Intergalactic medium Accretion (0.2-1)M/year Interstellar medium

≈200M/year Diffuse gas Giant molecular cloud ≈200M/year

Star formation process

1M/year

(3-5)M/year

Stars

Red dwarfs Neutron stars Black holes Fig. 7.18 Matter exchange between stars and intergalactic medium [36]

In the interplanetary plasma, the characteristic concentration of electrons is N e ∼ 101 –102 cm−3 , and in the interstellar medium it amounts to N e ∼ 0.1 cm−3 . The temperature of charged particles is 0.1 < T i < 30 kW. The magnitude of the stellar wind velocity is 500 < V VS < 3000 km/s (2 < M A < 12) and reaches for young stars a value which is 2–3 orders of magnitude higher. At √ the typical magnetic field strength B ∼ 0.5 nT, the Alfven velocity v A ∼ B/ μ0 Ne m i will be 5–10 km/s. If we assume, according to ordinary gas dynamics (ref. Chap. 1), that the compression shock wave thickness  is of the order of the path for the binary collision path of particle λB ∼ (σ B N)−1 at σ B ∼ 5 × 10−15 cm3 , we will obtain a giant shock wave thickness as compared to the distance from the Earth to the Sun – B ∼ 2 × 1014 cm. −1 The estimate based on the Coulomb path of λB ∼ (Nσ c ) − 1; σc = (16π N 2 λ D4 ) ; (λD is the Debye screening length) leads to the same front thickness.

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This huge size of the shock wave discontinuity must result in an extreme slow change in shock-compressed plasma parameters observed from the Earth with a typical gain time of several hours. On the other hand, the first terrestrial measurements of space plasma parameters showed the presence of abrupt (at the minute scale) fronts of plasma shock waves that were later confirmed by multiple records from spacecraft [41] (ref. Fig. 7.4). A way out was found by taking into account much more powerful (as compared to classical) binary plasma collective magnetodynamic mechanisms of energy dissipation in the wave front [12, 13]. This, the accounting for ion-cyclotronic frequency W ci = eB/mi strongly corrects the contradiction revealed. R. Sagdeev developed a theory [12] of a finite-amplitudemagnetoacoustic wave in plasma. The wave front thickness δ B ∼ vA /ωpi , ωpi ∼ 4π e2 n/M in a solar wind plasma gave δ B ∼ 6 × 103 cm, which is many orders less than the Coulomb size. Thus,1 strong plasma perturbations in the processes of intensive energy release in space objects lead to the formation of shock waves. As a rule, space plasma is an interplanetary, interstellar and intergalactic medium, plasma of stellar winds and star shells, which is heavily rarefied and penetrated by large-scale magnetic fields. Both structure and dynamics of shock waves in such rarefied plasma with a large-scale field have substantial specific features. They are caused not only by a large action radius of Coulomb forces, but by collective processes with the participation of many particles induced by plasma electromagnetic fields. These collective processes and the interaction of particles with large-scale magnetic fields determine the diversity of the dispersive properties of plasma and the specific features of the dissipation of perturbations propagating in it. In particular, the linear modes of perturbations can have substantially different phase velocities, polarization and excitation mechanisms. The latter may occur both through the wave-particle interaction and macroscopic plasma instabilities. Such instabilities are often related with the currents caused by strongly anisotropic distributions of particles in the conditions of rare Coulomb collisions in a rarefied plasma. A shock wave propagating at the velocity vsh in a plasma with the standard (solar) abundance of chemical elements has the sound Mach number M s that can be found from the following relation Ms =

vsh ≈ 85v8 ≈ [T4 · (1 + f ei )]−1/2 , Cs

(7.1)

where T 4 is the temperature of ions T i in the plasma, in units of 104 K, and the multiplier f ei describes the ratio between temperatures of ionic and electronic components of the plasma: f ei = T e /T i . The Alfven Mach number of such shock wave is

1 The

next part of this chapter is written by A. M. Bykov, to whom the author expresses his acknowledgment.

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Ma =

vsh (4πρa )1/2 ≈ 460v8 n 1/2 a /B−6 , B0

(7.2)

where na and ρ a are the concentration of ions and plasma density, v8 is the plasma velocity measured in 1000 km/s, and B6 is the magnetic field strength in the incident flow measured in gauss. To characterize the parameters of space plasma, we shall remind of Debye screening length l D characterizing the screening effect of the Coulomb interaction and of the inertial ion length li = c/ωpi characterizing the dispersion properties of cold plasma:  T 1/2 ≈ 6 × 102 n −1/2 T4 CM, e 4π n e c2  m p c2 1/2 ≈ 2 × 107 n i CM. li = 4π n i e2

lD =

(7.3)

(7.4)

Here, the concentration of electrons ne and ions ni is expressed in cm−3 . In a quasi-neutral non-relativistic plasma, the Debye screening length is always less than the inertial length: lD = li



T < 1. m p c2

(7.5)

In dense plasma, Coulomb collisions of particles provide the relaxation of non-equilibrium distributions and the associated processes of transfer of particles, momentum and energy. A detailed consideration of transfer phenomena in nonrelativistic plasma can be found, in particular, in classical articles [42, 43] that discuss characteristic Coulomb relaxation times—the time of particle deceleration in plasma, the time of deviation of particle velocity direction and the time of energy exchange between various plasma particles. To characterize the role of Coulomb collisions in the incident flow of nonrelativistic shock waves where the kinetic energy is mainly concentrated in ions having supersonic velocities, one can use the time τ si of deflation of a test ion with mass M, charge Z and energy E in scattering by electrons and ions of plasma with the concentration ni and the temperature T. Scattering by electrons of background plasma for high-energy ions, E  (M/m) T is the fastest and is rated as τsi ≈ τsie =

m e E 3/2 1/2

21/2 Mi π n i Z 2 e4 ln c

(for E  T M/m),

(7.6)

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where c is the Coulomb logarithm describing the contributions of small-angle scattering due to the long-range nature of the Coulomb interaction. When the ion energy falls down to about E ∼ (9π M/ 4me )1/ 3 T, contributions of electrons and ions of plasma into the Coulomb scattering of a test ion become comparable. Finally, for an ion with the energy E just slightly exceeding the plasma temperature, the Coulomb scattering time is primarily determined by the interaction with ions. In particular, for E = (3/ 2)T, we have: 1/2

τsi ≈ τsie =

0, 82Mi E 3/2 (at E = T · 3/2) 21/2 π n i Z 2 e4 ln c

(7.7)

The free path of an ion relative to the Coulomb scattering can be determined as λc ≈ vτsii , where v is the particle velocity. Let us give estimates for the length of the Coulomb path of a proton of energy 100 eV in a cold plasma flow coming on the shock-wave front:  λc ≈ 1.2 × 1012

E 100  B

2

n i−1



c 40

 cm

(7.8)

In dense plasma, the structure of a collisional shock wave is greatly affected by the interaction of particles with photons and the pressure of radiation locked in the optically dense medium [3]. Optical and multi-wave observations allow measuring the supernova luminosity curve, the time dependence of radiation intensity of supernova in a specific range of wavelengths. The supernova luminosity curve is determined by the dynamics of shock wave coming to the star surface, which was formed in collapse of the mother star core. A shock wave formed at high, ∼ 1051 –1052 erg, and fast,  1 s, generation of the kinetic energy in case of collapse propagates in the medium with a dominant radiation pressure. Under typical conditions of massive stars of red or blue giants with an external layer density of about 10−8 –10−9 g /cm3 , at a distance of ∼ 108 km from the star center and a shock wave velocity of about 10,000 km/s, the plasma becomes optically transparent for radiation and the observer sees a shock wave egress and a supernova explosion—one of the most superb phenomena in the Universe [1]. Observational manifestations of supernovae are very diverse, which indicates different mechanisms of energy release, accompanying the gravitational collapse of massive stars and thermonuclear instabilities of dwarf stars [1, 2]. The probable sources of the kinetic energy of release usually include a decay of significant masses of 56 Ni produced in Si combustion and the acceleration of star external layers by neutrino flows. Neutrino carry away almost all huge energy of the collapse (∼ 1053 erg) but they can transfer a small share (∼ 1%) of its luminance to the star shell motion. Magnetars are considered as an energy–momentum source for some types of supernova, which are the remnants of the collapse with a strong (∼ 1015 gauss) magnetic field and fast rotation with a period of about several milliseconds [1, 2]. Such remnants can form relativistic releases in the form of jets or wind that interact with star external layers and sometimes penetrate through its shell.

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221

Further evolution of supernova remnants in the rarefied near-star medium involves the formation of collisionless shock waves. These shock waves in plasma are today an object of intensive studies, and below we will briefly discuss some of their specific features. Let us begin, however, by considering some general properties of nonrelativistic shock waves in a plasma with a magnetic field. In the plasma of a collisional shock wave, the Coulomb scattering of ions promotes isotropization of the ion velocity in the flow and forms the structure of a viscous ion jump in the velocity and density of the plasma. The characteristic sizes of this jump are determined by the length of the Coulomb ion path, λi . In case of no collissionless sources of heating, the ion temperature right after the viscous ion jump, as a rule, significantly exceeds the electron temperature. The time of the Coulomb relaxation of these temperatures τ ei in the plasma flow behind the collisional shock wave front is much longer than the time of the Coulomb scattering of the ion: τei ∼



M/m e τSii > τSi

The equalization of electron√and ion temperatures in a collisional shock wave occurs quite slowly at scales  M/m e λc that are much larger than the length of the Coulomb path [42, 43]. Taking into account the effects of electron conductivity, this determines the structure of the collisional shock wave in case of a weak magnetic field. The magnetic field B characterizes the parameter of electron magnetization τsie ωce , where ωce = eB/cme is the cyclotron frequency of electrons. The role of the magnetic field role is low if τsie ωce < 1. Due to a rather slow mutual relaxation of the temperatures of ions and electrons, the description of the structure of the dissipative area of the collisional shock wave requires at least a two-component description of the plasma (in the simplest case—a model of electronic and ionic liquids). In space plasma, the dimensions of objects can exceed the relaxation length of the temperatures of electrons and ions. Therefore, to describe plasma motions that are slow as compared with the relaxation time of the temperatures of electrons and ions, we can use the approximation of single-fluid magnetohydrodynamics (MHD). In this approximation, the mass density and the medium velocity (non-relativistic components) u are determined by ions, and the electrical current density is determined by electrons. The high conductivity of plasma ensures, in case of slow processes, the small potential part of the electric field and the absence of the electric field in the local frame of reference related with the plasma at rest. Thus, in the MHDapproximation, the electric fields in the laboratory frame of reference are induced by medium motions: E = − (u/c)xB. In this case, they say that the magnetic field is frozen into the plasma. For non-relativistic MHD-flows, the electric field is much less in magnitude than the magnetic field B. The evolution of the electromagnetic field is described in the MHD-approximation by the induction equation. In case of low dissipation caused by magnetic viscosity, we deal with ideal MHD. Single-liquid MHD equations are frequently used to simulate the dynamics of collissionless plasma. In particular, to describe plasma motions transverse to the magnetic field.

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7 Cosmic Shock Waves

The single-liquid magnetohydrodynamic description of shock waves in plasma makes it possible to obtain convenient relations connecting the macroscopic characteristics of the flow before and after the transition layer (dissipation areas) and to classify stable discontinuities in MHD [32, 44, 45]. Inside the shock wave transition layer where the kinetic energy of the incident flow dissipates, the applicability of the single-liquid model is limited. Single-Liquid Description of Magnetohydrodynamic Shock Waves. Let us consider a shock MHD-wave as an abrupt transition between a macroscopic supersonic and super-Alfven flow (state 1—ahead of the shock wave) and a flow slowed down to subsonic velocity (state 2—behind the shock wave) provided there is mass flow J n through the shock wave surface. It is assumed that an elementary macroscopic particle of plasma is located at each point in time in the local thermodynamic equilibrium corresponding to the instantaneous local values of macroscopic parameters. It is deemed that the Maxwell distribution is set for all components of the magnetic liquid as a result of collisions. Macroscopic parameters characterizing the plasma state, such as density, specific internal energy or temperature, change slowly as compared to the relaxation process velocities bringing the system to thermodynamic equilibrium. Under these conditions in the frame of reference related with the shock-wave front, in the presence of a mass flow through the front surface, the equations for the laws of conservation of mass (in non-relativistic flows), momentum and energy can be written as follows:  jn

 Bt = Bn [u t ], ρ

Bn [Bt ], 4π jn   2 B2 jn + P + t = 0, ρ 8π jn [u t ] =

  jn2 Bt2 Bn u 2t w+ 2 + + − Bt u t = 0. 2ρ 2 4πρ 4π jn

(7.9) (7.10)

(7.11)

(7.12)

Here, un and ut are the normal and tangential to the wave front components of the mass plasma velocity, u = (un , ut ), w = ε + P/ρ is the mass unit entalpy, and the values ε, P, ρ are the energy of the unit of gas mass, pressure and density, respectively. The indexes n and t designate the normal component and the component tangential to the front surface. The standard designations [A] = A2 − A1 are used for the jump of the function A at the shock-wave front. A characteristic feature of shock wave MHD is the so-called coplanarity theorem [32]. It states that the magnetic fields ahead and behind the shock wave, B1 and B2 , and the normal to the shock-wave front N lie in the same plane (ref. Fig. 7.19). It should be noted that if the field has a normal component, Bn = 0, there is a specific

7 Cosmic Shock Waves

223

Fig. 7.19 Scheme illustrating the coplanarity theorem for a plane shock wave in ideal MHD. The magnetic fields ahead and behind the shock wave, B1 and B2 , and a normal to the shock-wave front N lie in the same plane. A frame of reference can be selected in which the mass velocities u1 and u2 are in the same plane and the flow in the shock wave MHD is plane. In the frame of reference where the front is at rest, also ut1 = 0. In this scheme, a shock wave is infinitely thin

frame of reference in which the local velocity and the magnetic field are parallel on both side of the front, whence it follows that E = 0 in this frame of reference. Equations (7.9)–(7.12) can give an expression similar to the Rankine Hugoniot adiabat for a plasma with a magnetic field: 1 ε2 − ε1 + · 2



1 1 − ρ2 ρ1

   1 2 · (P2 + P1 ) + (Bt2 − Bt1 ) = 0. 8π

(7.13)

The Rankine Hugoniot adiabat (ref. Chap. 1) gives a link between the macroscopic parameters of the flow after it passes a shock wave via the flow parameters assumed to be known before it passes the shock wave. In case of a parallel shock wave (Bt = 0), we have

T2 = T1

 γg + 1 Ms2 ρ2 r= = , ρ1 γg − 1 Ms2 + 2    2γg Ms2 − γg − 1 · γg − 1 Ms2 + 2 (γ + 1)2 Ms2

(7.14) ,

(7.15)

where γ g is the magnetic liquid adiabat indicator. Here, we confine ourselves to the case of a shock wave fast mode for which cs1 < u1 and va2 < u2 < cs2 at va1 < cs1 . Phase velocity va2 is the Alfven velocity behind the shock wave, cs1 and cs2 are the velocities of sound ahead and behind the shock wave, respectively. For a transverse shock wave, (Bn = 0), the compression ratio is as follows

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7 Cosmic Shock Waves

 2 γg + 1 ρ2 r= = 1/2 ,    ρ1 ψ + ψ 2 + 4 γg + 1 2 − γg Ma−2   ψ = γg − 1 + 2Ms−2 + γg Ma−2

(7.16)

We can note that a magnetic field transverse to the front normal that ensures the finite values of the Alfven Mach number M a decreases the ratio of plasma compression by a strong shock wave in the limit M s  1 as compared with (γ g + 1)/ (γ g − 1). In the single-liquid strong shock wave with M s  1 and M a  1, we can obtain the relation  γg − 1 2 7 2 (7.17) T2 ≈ 2 ·  2 μvsh = 1.38 × 10 vs8 K, γg + 1 which is true for any magnetic field slope relative to the normal to the front. For applications to plasma with many elements close to the solar one, the ion mass μ is here assumed to be equal to (1.4/ 2.3)mp , and vs8 is the shock wave velocity in units of 1000 km/s characteristic of waves in the interstellar and intergalactic media. The Rankine–Hugoniot adiabat connecting the thermodynamic states of plasma before and after the discontinuity does not depend on the nature of dissipative mechanisms that determine a transition between states 1 and 2. It assumes a single-liquid motion in regular electromagnetic fields. However, dissipative effects determine the thickness of the transition layer near the shock-wave front. In case of a weak shock wave with the Mach number M s – 1 1, the transition layer thickness can be quite large; in collisional plasma, this allows for the macroscopic hydrodynamic description of the liquid inside it; ref., for example [45]. The shock wave width  in collisional gas dynamics without magnetic fields is given by the expression =

8aV 2 .  (P2 − P1 ) ∂ 2 V /∂ P2

(7.18)

In gas without a magnetic field, the value a can be expressed through viscosities η, ζ and heat conductivity κ: a=

1 2ρvs3



   4 1 1 , η+ς +κ · − 3 cv cp

(7.19)

here cv and cp are the specific heat capacities at constant volume and pressure, respectively. Extrapolating expression (7.18) for the case of a shock wave with finite intensity when P2 − P1 ∼ P2 , we can see that the width of the shock wave in gas appears to be about the average length of the free path of particles λ [3].

7 Cosmic Shock Waves

225

It is useful to note that the entropy behaves non-monotonously inside the transition layer of the finite width of a weak shock wave in gas with M s − 1 1 and the total entropy jump s when passing through this layer determined in accordance with the Rankine–Hugoniot relation is proportional to the third power of (M s − 1): 1 · s = 12T1



∂2V ∂ P2

 · (P2 − P1 )3 ∝ (Ms − 1)3 ,

(7.20)

s

At the same time, jumps of density, temperature and pressure are proportional to the first power ∝ (M s − 1) [3, 45]. Let us note that in case of multifluid shock waves in plasma with a magnetic field, the entropy growth can be determined by nonthermal particles reflected from the wave front [7, 41]. Collisionless Shock Waves in Space Plasma. In a rarefied plasma of extended space objects, the free path of an ion relative to the Coulomb scattering λc often exceeds the characteristic dimensions of the system. In this case, plasma is considered as collisionless. For example, in a still solar wind near the Earth’s orbit, the plasma concentration is ni  10 cm−3 , and the temperature is T ∼ 105 K, which gives λc ∼ 1012 cm, a value of about 0.1 of the distance from the Earth to the Sun. Such a significant length of the Coulomb path of solar wind ions allowed Thomas Gold in 1955 to relate the observed effects of a sudden onset of a geomagnetic storm on the Earth with the motion of a shock front that compresses the geomagnetic field in the near-earth space and has a width below λc . This was one of the first observational indications to the presence of collisionless shock waves in space plasma. The first direct observations of the “solar corpuscular radiation”, i.e. solar wind plasma, made in 1959–1961 by Soviet experiments in rockets and by spacecraft Luna −1, −2, −3 and in 1961 by Venera-1 demonstrated a real perspective of measuring the characteristics of space plasma [46]. The first studies of collisionless shock waves in solar wind were carried out by magnetometers of NASA spacecraft Pioneer-1 [41] and IMP-1 (The Interplanetary Monitoring Platform). The latter found and studied a collisionless magnetohydrodynamic shock wave at a distance of ∼ 13.4 of the Earth radius [47]. Observational programs for studying the structure and properties of collisionless shock waves by directly detecting plasma characteristics and electromagnetic fields in solar wind and Earth magnetosphere are carried out at spacecraft of all primary space agencies—ISEE-1, IMP-8, ACE, THEMIS, Prognoz-7 and 8, INTERBALL-1, Cluster, etc. These programs are an important part of the new projects Solar Orbiter and Intergeliozond. Shock waves observed in the interplanetary medium are related with the nonstationary processes of rapid energy release in solar bursts and with the stationary flow around the magnetospheres of the planets by supersonic solar wind. The structure of the head shock wave that flows around the Venus with solar wind and the specific features of which are related with the own low magnetic field of the Venus was studied by Venus Express spacecraft in 2006–2009. In 1958, R. Z. Sagdeev constructed [12] a model of a magnetosonic finiteamplitude wave propagating across the magnetic field in rarefied plasma. The theory predicted the width of the front of a magnetic field single pulse to be about δ v =

226

7 Cosmic Shock Waves

 √ va /wpi , where va = B/ 4π Mn is the Alfven velocity, and wpi = 4π e2 n/M is the ion plasma frequency. In solar wind plasma δ v ∼ 6 × 103 cm, which is much less than the Coulomb path of solar wind ions. First papers in physics of collisionless shock waves [12, 13] served as the basis for a large number of studies in this area [31, 48]. In the 70s and 80s of the previous century, analytical methods were developed to describe the dynamics of weak-linear waves in media with dispersion. Nonlinear breaking of the wave crest in plasma balanced by dispersion effects can lead to the formation of stable localized perturbations and, in case dissipation is present, to the formation of weak shock waves. The propagation of waves of finite but low amplitude in plasma along the magnetic field can be described by the Schroedinger nonlinear equation and across the magnetic field—by Korteweg–de Vries–Burgers nonlinear equations. Let us note that the derivation of these equations in case of hot plasma with the parameter β = P/(B2 / 8π )  1 and rare Coulomb collisions requires the kinetic description of plasma, which greatly complicates the substantiation of these equations. The details of the derivation can be found in paper [9]. According to the models, collisionless shock waves have an oscillatory structure of fronts. The correct description of dissipative effects in plasma is nontrivial. At sufficiently large Mach numbers, the classical electrical conductivity does not provide the required rate of kinetic energy dissipation in a shock wave. Analysis of the role of dissipation related with the classical electrical conductivity, thermal conductivity and viscosity leads to the concept of critical Mach numbers [10, 48, 49]. This concept arises in the transition from the classical mechanisms of dissipation to abnormal ones, such as reflection and instability of ion flows. According to [48], the critical Mach number of the shock wave can be determined from the condition of equality behind the wave front of the normal component of the plasma velocity and the local velocity of sound (in the frame of reference associated with the wave front). Hereinafter, we will confine ourselves to considering supercritical shock waves with Mach numbers > 2.8. To conclude, we shall note that space plasma has been almost the only one “experimental laboratory” for studying collisionless shock waves for more than 60 years. Currently, experimental possibilities have been expanded due to the development of ground-based laser facilities in which collisionless shock waves are initiated by large magnetic pistons [50, 51]. An analytical description of a collisionless plasma is very difficult because of the highly nonlinear nature of the processes occurring in it. For the purpose of a consistent description of strongly nonlinear flows of collisionless plasma, numerical simulation methods were developed, which are considered below. Microscopic Simulation of Shock Waves in Plasma with Magnetic Field. With the growth of computer performance in the recent decades, it became possible to use direct microscopic simulation of plasma by the “particle-in-cell” method (PIC) to study the structure of the front of collisionless shock waves, heating of ions in plasmas of complex chemical composition and charge states as well as injection of particles into the Fermi acceleration mode. The most detailed description of shock wave properties is achieved in the full version of the PIC method including the combined integration of dynamics equations for particles and electromagnetic fields

7 Cosmic Shock Waves

227

Fig. 7.20 Model of the structure of a magnetic field perpendicular to a relativistic shock wave running in an electron–positron plasma. Wave Lorentz factor  sh = 40. Initial plasma magnetization, σ 0 = 0.3. Time after wave launch t = 200we −1 . Calculation made by PIC method

in plasma (ref. method basics in [52, 53]) and its application in review [54]. This method is efficiently used in the simulation of the structure of relativistic waves in electron-proton and electron–positron plasmas. The nature of plasma instabilities responsible for the formation of shock waves in relativistic plasma is determined by a number of plasma parameters. In particular, by the composition of plasma (electron–positron or electron–ion plasma), the slope angle of the shock wave velocity towards the magnetic field, the values of the shock front Lorentz factor  sh and the parameters of the initial plasma magnetization: σ0 =

B⊥2 . 2 4π m e c2 n e sh

(7.21)

Here, the magnetic field B⊥ is measured in the frame of reference related with the wave front while the concentration of electrons ne is measured in the rest frame of the plasma flowing onto the front. The parameter σ 0 describes the ratio of energy flow densities of the electromagnetic field and the kinetic energy of plasmas in the relativistic flow. In astrophysical applications, the magnetization parameter playing an important role in the wave structure formation varies within a very wide interval— from σ ∼ 10−9 in the interstellar medium to σ 0  1 in e± -plasma of pulsar wind. The main mechanisms determining the front structure of a relativistic shock wave include the Weibel instability, filamentary and current instabilities. Figure 7.20 shows the results of simulating the structure of a relativistic shock wave propagating in an electron–positron plasma with the initial magnetization σ 0 = 0.3. Such values σ 0

228

7 Cosmic Shock Waves

can be characteristic for the relativistic jets of black holes forming internal black holes in the sources of gamma-ray bursts and active galactic nuclei. The spatial resolution of the PIC method is characterized by the small value c/wpe , which is optimal for simulating processes in an electron–positron plasma and also for simulating the structure of relativistic shock waves. In problems of non-relativistic shock waves in an electron–ion plasma, the simulation by the PIC method with the realistic mass ratio mp /me requires very high computer capacity. Since ions in such waves make the main contribution to the energy–momentum flow ahead of the wave front and the resolution of about c/wpi is usually sufficient for simulation, a simplified version of the PIC method is usually used—a hybrid code. In the hybrid PIC-code, ions are still described as particles, and electrons—as a massless neutralizing liquid [30, 54–57]. Calculations with the hybrid model make it possible to calculate the flow structure with a spatial scale of about 10 000li and more (l i is the inertial length of ions). Below we consider the results of simulating the structure of non-relativistic collisionless shock waves using the hybrid PIC-method. Hybrid Simulation of Shock Waves in Plasma. The equation system for describing collisionless plasma and the operating algorithm of the hybrid PIC-method involve the solution of the following system of equations [58]:

E=

drk = Vk , dt

(7.22)

dVk Zk = (E + Vk × B), dt mk

(7.23)

∇ × B = J,

(7.24)

∂B = −∇ × E, ∂t

(7.25)

1 1 (∇ × B) × B − (J × B)k , n n

(7.26)

where rk , Vk , Z k , mk are the coordinates of velocity, charge and mass of the particles of k type; E, B are the electric and magnetic fields, n and J—are the charge density and ion current. Equation (7.26) is conveniently represented in the following form 1 ∂ P jk 1 − (J × B)k , n ∂x j n B

Ek = −

where P jkB =



B2 2

 δ jk − B j Bk —is the magnetic pressure tensor.

(7.27)

7 Cosmic Shock Waves

229

Hereinafter all variables are normalized to li , i , M 0 , Z 0 (inertial length, gyrofrequency, ion mass and charge with the smallest spatial and time scale) and to B0 , ρ 0 (the average magnetic field and density of steady-state plasma). Such normalization allows applying the obtained results to plasmas of various composition by spatial-time scaling. As we can see from Eqs. (7.22)–(7.26), the fields and particles are mutually dependent, which makes the problem self-consistent. Problems of such type can be solved using a famous numerical method leapfrog having the second order of accuracy in time [59, 60]. In this method, the positions and velocities of the particles are known at step n, and the values of the electromagnetic field—at step n + 1/ 2. Figures 7.21 and 7.22 illustrate the results of the 3D hybrid simulation of the structure of collisionless shock waves with different slope angles of the magnetic field to the normal to the front: α = 0°, 50°, 80°, 90°; ref. examples a, b, c, d, respectively. The calculations were performed [58] for a shock wave with the Alfven Mach number M a = 9 in plasma with the ratio of plasma thermal pressure to magnetic pressure being β = 1. The calculations assumed that the plasma consists of oxygen with charge + 3 (O IV) and a small admixture of silicone with charge + 1 (Si II). A similar chemical composition, charge states of ions and shock wave parameters simulate the conditions in the shells of certain types of supernova remnants. The shock wave was initialized by the method of reflection of a supersonic particle flow from a stationary conducting wall located in the plane x = 0, which resulted in the formation of the shock wave front moving in the positive direction of the x-axis. Dimensions of the simulation domain (x, y, z) were (2400 × 1 × 100) · li , where li = V A / i is the inertial length of predominant plasma ion (O IV), V A is the Alfven velocity, and i is the gyrofrequency. At the opposite end of the simulated space, an open boundary was located where the supersonic flow of particles with the Maxwell distribution was constantly injected. In this case, the thermal velocities of particles of different types coincided, i.e. the ratio of ion temperatures amounted to the ratio of their masses. Periodic boundary conditions were set along the y, z axes. The calculations showed that in collisionless shock waves, the width of the magnetic field profile in quasi-transverse shock waves (models b in Fig. 7.22) is much smaller than in quasi-longitudinal waves (models a in Fig. 7.22). The latter are characterized by wide profiles of the shock front with the characteristic width  100 l i . The simulation of collisionless shock waves in a two-component plasma with different slope angles α of the magnetic field relative to the normal to the wave front illustrated the emergence in quasi-longitudinal shock waves of a clearly defined precursor of an area ahead of the wave front perturbed by reflected and accelerated particles. In Fig. 7.22, one can see that there is an admixture of reflected and heated heavy particles of a doping ion ahead of the wave front with the small angle α. For the predominant ion, the precursor effect is also present, but it is far less pronounced. The transverse velocities of reflected particles of both types in the precursor experience quasi-periodic perturbations close to harmonic ones. The suprathermal particles in the precursor can subsequently be injected into the Fermi acceleration mechanism and, being accelerated, can form nonthermal radiation of supernova remnants, which

230

7 Cosmic Shock Waves

а1

а2

b1

а3

b2

а4

b3

а5

b4

а6

Fig. 7.21 Magnetic field profiles By and Bz behind the front and in the precursor of quasilongitudinal and quasi-transverse shock waves [58]

7 Cosmic Shock Waves

231

Fig. 7.22 Types of phase spaces of a small Si II admixture against the background of O IV in collisionless shock waves with different slope angles of the magnetic field to the normal to the wave front: a − α = 0°, b −α = 50°, c −α = 80°, d −α = 90° (the scale along the x-axis differs for quasi-longitudinal and quasi-transverse shock waves) [58]

232

7 Cosmic Shock Waves

is observed in the range from radio-waves to gamma-rays and can also serve a source of cosmic rays. In late 70s of the previous century, an efficient mechanism was discovered to accelerate nonthermal particles near collisionless shock waves propagating in turbulent plasma. Particles gain energy under the Fermi mechanism due to the multiple intersection of the shock-wave front by fast suprathermal particles injected into the acceleration process [9, 61, 62]. Acceleration of High-Energy Particles by Collisionless Shock Waves. In accordance with the Fermi mechanism, reflected ions with a gyroradius exceeding the width of the shock wave transition layer can be then efficiently accelerated by converging plasma flows conveying magnetic field inhomogeneities in the form of MHD-waves. Particles are scattered on the fluctuating magnetic fields of MHDwaves that are carried by the super-Alfvenic plasma flow coming to the front at the velocity u. When a particle with the momentum p crosses the front each time, it obtains a momentum increment (in the rest frame of the shock-wave front) p ≈ p ·

 u + p (u/v)2 . v

(7.28)

Let us note that in quasi-transverse shock waves, an electric field perpendicular relative to the normal to the front exists in all frames of reference. Therefore, acceleration in quasi-transverse shock waves can be caused not only by the Fermi scattering by inhomogeneities but also by the work of the electric field during the drift motion of particles. This field can significantly increase the momentum transverse relative to the normal to the front and gained by the particle during acceleration. The spectra of accelerated particles in the scattering medium can be calculated by using the diffusion approximation of the kinetic equation. In the diffusion approximation, the scattering medium is characterized by the diffusion coefficients of particles k 1 (p) and k 2 (p) depending on momentum. In the approximation of test particles, a shock wave is considered as a jump of mass velocity (shown with a short-dash line in Fig. 7.23), the reverse effect of the pressure of accelerated particles on the velocity of plasma coming onto the front is neglected, and the test particles are injected at some suprathermal energy (so that they could be involved into the acceleration process). To do this, particles are imparted with some initial momentum p = p0 so that their gyroradius exceeds the width of the shock front. For a weakly anisotropic distribution of test particles in the phase space, the solution of the kinetic equation is power-series distribution. f ( p, x) ∝ ( p/ p0 )−b ,

(7.29)

where p  p0 , and the indicator b = 3r/r − 1 depends on the compression ratio r [61–64]. For a strong shock wave with M s  1 and M a  1, the compression ratio given by formulas (7.14) and (7.16) is close to 4 for a non-relativistic plasma with γ g = 5/ 3, or above 4, if the dominant contribution to the equation of state is made by the relativistic component of the plasma with γ g = 4/ 3. The pressure of accelerated

7 Cosmic Shock Waves

233

particles is determined by the formula

PCR

4π = 3

∞ pv f ( p, x) p 2 dp.

(7.30)

p0

Let us note that for index of power b = 4, this formula gives PCR ∝ ln (pmax /p0 ), which corresponds to a sufficiently high pressure of cosmic rays if particles are accelerated to the energy pmax  p0 . The spatial distribution of cosmic rays obtained in the model is shown with a dash line in Fig. 7.23. The maximum energy pmax , to which particles can be accelerated, depends on the diffusion coefficient of the system. The final spatial dimension of the shock wave is usually taken into account by introducing a free-escape surface located ahead or behind the shock wave. For electrons, the value pmax can also be limited by energy losses of relativistic particles to synchrotron or reverse Compton radiation. The acceleration time of test particles on the shock wave τ a (p) can be evaluated as τa ( p) =

u 1n

3 − u 2n

p p0

 k1 ( p) k2 ( p) dp , + u 1n u 2n p

(7.31)

where the normal components of mass velocities of flows ahead and behind the shock wave, u1n and u2n , are measured in the frame of reference associated with the wave front. In a collisionless plasma, the distribution of particles perturbed by field variation does not return to the Maxwell distribution in time intervals characterizing the process. The slowness of the Coulomb relaxation results in the formation of strongly non-equilibrium distributions of plasma particles with the injection into the mode of the efficient acceleration of particles by the Fermi mechanism. In extended cosmic shock waves, this forms the spectra of particles which, along with quasi-thermal peaks, have piecewise-power areas extending to plasma energies that are orders of magnitude higher than thermal energies. This results in a possibility of transforming a significant fraction of the kinetic energy density of the incident plasma flow into the pressure of accelerated high-energy particles. In its turn, the high-energy particles penetrate far to the area of the incident plasma flow. Ponderomotive forces caused by the pressure gradient of accelerated particles decelerate the incident plasma flow before the front, and instabilities related with the anisotropy of nonthermal particles can efficiently intensify fluctuating magnetic fields in the wave precursor (ref. Fig. 7.21). Accurate simulation of the structure of a collisionless shock wave taking into account the acceleration effects of nonthermal particles requires the use of the perturbation theory not limited by various approximations, but rather the approximation of the self-consistent description of the multi-component system with a wide range of

234 Fig. 7.23 Model structure of flow near a strong shock wave in collisionless turbulent space plasma

7 Cosmic Shock Waves Precursor

Generation of MHD-perturbations Cosmic rays Viscous jump -V Рсr

scales where all physical processes are developed, including strong MHD-turbulence with its dynamics. At present, PIC direct simulation methods cannot fully take into account the effects of shock wave modification with nonthermal particles due to the very large dynamic range of fluctuation scales and particle energies. A simplified description of the structure of a multi-component strong shock wave can be implemented with the acceptable parametrization of the mechanism of transfer of high-energy particles in an extended shock wave precursor and the structure of a viscous jump in plasma. In the following section, we will discuss the results of the nonlinear macroscopic simulation of collisionless shock waves in a strongly nonequilibrium turbulent plasma with relativistic components, obtained by the Monte Carlo method. Models of Shock Waves Modified by Accelerated Particles. The large-scale structure of a collisionless shock wave modified by accelerated particles can be simulated within the framework of both a two-liquid approach with the kinetic description of nonthermal particles [65] and using the Monte Carlo method [57, 66]. Both approaches require a priori parametrization of particle scattering processes. Simulation using the Monte Carlo method, however, does not suggest separation of plasma particles into thermal (background) and suprathermal accelerated particles. A weak anisotropy of the particle distribution is not suggested either, which allows taking into account the development of instabilities of anisotropic distributions and studying the injection of thermal particles. The Monte Carlo method eliminates a free injection rate parameter present in all models based on the diffusion approximation and used for selecting injection efficiency; instead, injection is defined by the assumption about the nature of particle scattering. A strong nonlinear link between injection, shock wave structure and magnetic field intensification makes the use of the Monte Carlo method especially useful. The method allows obtaining, in an iterative manner, a shock wave velocity profile and the particle distribution function (consistent with the laws of conservation of mass-energy–momentum flows) and taking into account their nonlinear reverse link with accelerated high-energy particles. Figure 7.24 gives calculation results for the macroscopic structure of an extended front of a strong collisionless shock wave in turbulent plasma.

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235

Fig. 7.24 Macroscopic profiles of plasma velocity u(x) and magnetic field magnitudes B(x) in the precursor and behind the front of a strong quasi-longitudinal shock wave modified by the pressure of relativistic particles accelerated by the Fermi mechanism. The calculation was performed by the Monte Carlo nonlinear method [66]. A velocity jump at x = 0 may have a thin structure corresponding to that shown on the left panel in Fig. 7.22, which is not solved by calculation using the Monte Carlo method

The ratio of the scales resolved  by Monte Carlo models and hybrid calculations is given by the relation r g0 /li = βγg /2Ms  1, where rg0 = mp vsh c/eB0 , and l i is the ion inertial length. As follows from the calculations within the framework of the hybrid model (Fig. 7.24), the width of the viscous jump is the value of about several lengths of li . Therefore, a viscous jump in the Monte Carlo model is not resolved and has a negligibly low width. The proton spectrum behind the shock wave is shown on the left panel of Fig. 7.25. The spectrum was obtained by simulation using the Monte Carlo method, which is described by Bykov et al. in paper [66]. The calculations were performed in the frame of reference related with the viscous jump for a flow depending on a single coordinate; particle momenta in this case are three-dimensional. To take into account the finiteness of the system size, a boundary condition of free escape of particles from the surface located 106 r g0 from a viscous jump is implied. Simulation was carried out for a supernova remnant shock wave propagating at the velocity vsh = 5000 km/s in an interstellar medium with the proton concentration n0 = 0.3 cm−3 and the unperturbed magnetic field B0 = 3 uGs. The efficiency of converting the kinetic energy of the incident plasma flow into the energy of nonthermal particles can be quite high giving birth to a hard spectrum of nonthermal particles up to a certain maximum energy. This maximum energy depends on the spatial scale of the system. At sufficiently high energy conversion efficiency, an extended shock wave precursor arises, caused by the deceleration of the

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km/s

Fig. 7.25 On the left: distribution function in phase space for protons accelerated by the Fermi mechanism on a strong shock wave. The distribution function is multiplied by (p/ (mpc))4 . The spectra of particles clearly show a quasi-thermal peak and extended piecewise-power areas. The spectra are calculated using the numerical nonlinear Monte Carlo model. The model takes into account the intensification of a fluctuating magnetic field by the instabilities of the anisotropic distribution of fast particles in the precursor and the nonlinear modification of the plasma velocity profile. Different curves correspond to different model hypotheses on the transfer of turbulent magnetic fields; ref. paper [66] by Bykov et al. On the right: the conversion efficiency Pm of the energy density of a plasma flow coming to the front into the energy density of magnetic fluctuations

plasma incident flow due to the pressure of accelerated particles. The characteristic scale of the shock wave precursor, L (c/vsh ) λ∗ exceeds the width of the transition layer by orders of magnitude (ref. Fig. 7.23). Here λ∗ is the maximum free path length of particles in the nonthermal component of the spectrum, containing the most part of the energy, and vsh is the shock wave velocity. Hereinafter, we will talk about these high-energy particles as of cosmic rays. To conclude, we shall note that in the prediction of the nonlinear Monte Carlo model, the front of a strong collisionless shock wave consists of an extended precursor and a viscous velocity jump with a local Mach number less than the total Mach number of the shock wave. Accordingly, matter compression in a viscous ionic jump can be much less than the total medium compression by a shock wave due to strong compression in the precursor. Compression and Heating of Ions in Strong Modified Shock Waves. The efficient acceleration of high-energy particles by the Fermi mechanism is accompanied by their escape from the vicinity of a collisionless shock wave, in particular, across the free escape boundary. The energy flow Qesc carried by escaping high-energy particles must be taken into account in energy balance equations. Energy losses lead to a decrease in the efficient adiabatic exponent and make it possible to increase the total gas compression behind the shock wave. The total compression ratio r tot of a strong MHD shock wave modified by the efficient acceleration of nonthermal particles can be estimated according to the formula

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237

Fig. 7.26 Total compression r tot for a strong MHD shock wave modified by the efficient acceleration of particles as a function of the energy flow carried away by high-energy particles Qesc /kin , where 3 /2. The upper dotted curve corresponds to the efficient adiabatic exponent γ = 4/3 kin = ρ0 vsh (relativistic gas), and the lower continuous curve corresponds to the efficient indicator γ = 5/3 (non-relativistic gas)

rtot =

γg −

γ +1

 3 1 + 2 γg2 − 1 Q esc /ρ0 vsh

(7.32)

on the assumptions that the energy density in the incident flow is determined primarily by the kinetic energy of particles and that the escape of relativistic particles from the system occurs at energies close to the maximum energies of accelerated particles. Here γ g is the efficient adiabatic exponent. 3 for γ g Figure 7.26 shows the dependence of the compression ratio on Q esc /ρ0 vsh = 4/ 3 (relativistic ideal plasma) and 5/3 (non-relativistic plasma) on the assumption that the efficient adiabatic exponent is between these values and depends on the spectrum of accelerated relativistic particles. The distribution function of nonthermal particles and the plasma velocity profile in the area of incident flow are sensitive to the total compression ratio r tot . Accordingly, an accurate calculation of the energy flow carried away by particles Qesc can be performed only within the framework of completely nonlinear kinetic simulation. Nevertheless, using the approximated approach (for example, within the framework of the Monte Carlo method discussed above) a stationary solution can be obtained for the distribution function consistent with the compression of the shock wave and in the assumption of a certain model of diffusion of nonthermal particles. For such a simplified two-liquid model of a strong shock wave modified by accelerated nonthermal particles, the efficient ion temperature behind the wave front T i2 can be evaluated for a shock wave at the defined velocity if the total compression ratio and compression ratio on the viscous jump are known (r tot and r sub ),

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 2 2 2γg Msub − γg − 1 μvsh 2 Ti2 ≈ ϕ(Msub ) · , ϕ(Msub ) =  . 2 γgrtot (vsh ) γg − 1 Msub +2

(7.33)

Here M sub is the velocity jump Mach number caused by ion viscosity. Plasma heating behind the front of a strong single-liquid shock wave corresponds to the limit M sub = M s  1; since there is no precursor in this case, we have a result described by expression (7.17). In single-liquid systems, the compression ratio r to = r sub → (γ g + 1)/ (γ g − 1) does not depend on the shock wave velocity and expression (7.33) is also reduced to (7.17). However, in multifluid shock waves, the total compression ratio depends on the shock wave velocity and can be much higher than in the case with single-liquid shock waves. This assumes lower ion temperatures behind the front of a strong multifluid shock wave as compared to a single-liquid wave of the same velocity. This assumption can be verified by observations. To describe various cases of heating by intense shock waves, it is convenient to ξ . Then, introduce the following approximation of total compression: rtot (vsh ) ∝ vsh from Eq. (7.33), we can obtain that the ion temperature behind the front is Ti2 ∝ 2(1−ξ ϕ(Msub )vsh ) . Mach number for a viscous jump M sub depends in general on M s and M a . If we express the dependences of the function ϕ and the ion temperature ξ behind the wave front on its velocity in the power-mode form: ϕ(Msub ) ∝ vsh and ξ Ti2 ∝ vsh , then, the index of power α = 2 (1 − ξ ) + σ . For the case when a shock wave precursor is heated by Alfven waves amplified by resonance interactions with the unstable distributions of accelerated particles, the index α = 1.25. The role of plasma entropy behind the front of a strong collisionless shock wave with the efficient acceleration of particles is proportional to the value 2 × (rtot (vsh ))−(γg +1) · ϕ(Msub ), which gives a significantly lower entropy growth vsh as compared to the case of a single-liquid shock wave having the same velocity. The described phenomena are related with wave modification under the action of accelerated particles and with the amplification of fluctuating magnetic fields. Amplification of Magnetic Fields in Modified Shock Waves. An important predicted property of intense shock waves with effective acceleration of cosmic rays is their ability to strengthen the initial seed fluctuating magnetic field [67–69]. A source of energy for the field amplification is the high energy density of accelerated relativistic particles in the prefront of a collisionless shock wave. Plasma instabilities initiated by the anisotropic distributions of accelerated particles and processes of resonance interaction of waves with particles lead to the intensification of magnetic field fluctuations of various scales. High energy particles penetrate into the area of the plasma incident flow. They are related with the incident flow through fluctuating magnetic fields (including Alfven waves generated by high energy particles). Magnetic field dissipation generates preliminary plasma heating and increases the entropy in the extended shock wave precursor. Electric current and pressure gradient of energetic particles penetrating into the plasma incident flow ahead of the viscous jump of a strong shock

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wave efficiently amplify the magnetic field fluctuations at the scale of about the size of the shock wave precursor. The size of the precursor of a modified shock wave L ∼ (c/vsh ) λ∗ , where λ∗ is the free path of energetic particles in that part of the spectrum, which contains the main pressure of the nonthermal component. The path value λ∗ is determined by the scattering of energetic particles on magnetic fluctuations and is many orders of magnitude smaller than the Coulomb path of relativistic particles. The nonlinear simulation of the macroscopic structure of a strong shock wave with the efficient acceleration of relativistic particles by the Monte Carlo method demonstrates a possibility of strong non-adiabatic amplification of fluctuating magnetic fields. The right panel in Fig. 7.25 shows the efficiency of conversion Pm of the energy density of the plasma flow coming to the front (0.5ρ 0 vsh ) into the energy density of magnetic fluctuations B2 / 8π, calculated by the Monte Carlo method. The conversion efficiency depends on the shock wave velocity and reaches 10% [66]. X-ray images of young supernova remnants with high angular resolution (about a second of arc) are obtained by the Chandra space telescope and allowed making a conclusion on strong super-adiabatic amplification of magnetic fields by supernova shock waves; ref. review [70]. Narrow extended structures of synchrotron X-ray radiation were found in the vicinity of supernova shock waves the origin of which is related with the acceleration of electrons by shock waves to energies above 1012 eV. The interpretation of the observations indicates the 10X amplification of the magnetic field of supernova remnants by strong shock waves. Heating of Electrons by Cosmic Shock Waves. The observed optical and X-ray radiations of cosmic shock waves in most cases depend on the electron temperature behind the shock-wave front. The kinetic energy of the incident flow in a strong shock wave in case of non-relativistic electron–ion plasma is concentrated in the ion component. Ions are heated on a viscous ion jump of a shock wave [5, 71, 72]. Heating of electrons due to Coulomb collisions within the viscous ion jump is low. Electrons behind the wave front must go through the plasma depth of about N eq = n2 u2 τ eq , so that the electron temperature T e would reach the equilibrium values T eq . The equilibrium temperature in the electron-proton plasma behind the viscous jump is estimated as 2T eq ≈ T eo + T po [43]. Figure 7.27 gives the calculation results for the profile of the minimum electron temperature T e behind the shock-wave front on the assumption of Coulomb relaxation of the ion temperature [5]. Analysis of the observational data obtained both in interplanetary shock waves and collisionless shock waves in supernova remnants [73, 74] showed that the efficiency of collisionless heating of electrons, i.e. T e /T i behind the viscous jump is a descending function of the shock wave velocity. Quantitative simulation of electron heating in a collisionless shock wave is an urgent problem not studied to the full extent. Gas Heating and Entropy Generation in Weak Shock Waves. The efficiency of heating and acceleration of particles by shock MHD-waves of weak and moderate power can be estimated by calculating the energy dissipation rate of directed gas motion per unit of area of the front of a weak shock wave,  h . Determining the dissipation rate as  h = vsh ρT s , where s − is the difference in specific entropies behind and ahead of the shock-wave front, we can calculate the gas heating rate:

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Fig. 7.27 Relaxation of electron temperature behind the shock-wave front due to Coulomb scattering. The electron temperature T e behind the shock-wave front is shown as a function of plasma depth expressed in units N eq = n2 u2 τ eq (Ref. [5]). The electron temperature values represent the minimum heating of electrons. Additional sources of heating are associated with dissipation of plasma modes and absorption of radiation

˙h = (5/4)(Ms − 1)3 vsh T ,

(7.34)

where  T is the internal energy of a gas volume unit. The rate of energy transfer to reflected nonthermal particles can be estimated as follows: ˙CR = (Ms − 1)2 vsh  B ,

(7.35)

where  B is the magnetic field energy density. Expressions (7.34 and 7.35) show that the gas heating rate has the third order of magnitude M s − 1 1 (ref. Eq. (7.20) and [45]), while the wave attenuation due to particle acceleration is a second-order effect. It’s worth noting that for the outer regions of galactic clusters, the ratio  T   B is typical. At the same time, in the central regions of the clusters  T can be comparable with  B , as in case of the Milky Way. Thus, weak shock waves can efficiently accelerate nonthermal particles, reducing gas heating. Acceleration of particles by an ensemble of large-scale shock waves can create a population of nonthermal particles having a significant pressure. To illustrate cosmic shock waves in Fig. 7.28, let us give an example of cosmological structures that occur when two galactic clusters, the most massive gravity-related objects in the Universe, merge. The mass of large clusters can reach 1014 –1015 M  (where M  = 2 × 1033 g is the Sun’s mass) and be distributed between dark matter and hot plasma filling the space with a size of several megaparsecs (3 × 1024 cm). The link energy released during the formation of mass clusters can reach ∼ 1064 erg. Shock waves in the galactic cluster CIZA J2242.8 + 5301 the image of which in Fig. 7.28 is borrowed from paper [75] have velocities of about 2000 km/s and heat

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241

Fig. 7.28 X-ray and radio images of the cosmological massive gravity-related object—CIZA J2242.8 + 5301 galactic clusters (red shift z ≈ 0.19). Thin yellow lines depict the restored positions of cosmological shock fronts with dimensions exceeding Mpc (Z × 1024 cm). The blue color designates the radio-structures observed at a frequency of 14 GHz with WSRT Westerbork Synthesis Radio Telescope probably caused by electrons accelerated in the shock wave. The red color designates a smoothed X-ray image of the cluster obtained with the Suzaku space telescope within the photon energy range of 0.5 to 2.0 keV [75]. Hot gas radiating in the X-ray band is heated by shock waves related with the process of merge of massive clusters

gas up to X-ray temperatures. The authors of the observations identified a jump in the plasma temperature at the front from ∼ 3 to ∼ 8 keV corresponding to a shock wave with the Mach number ∼ 3. A radio-image of the cluster at a frequency of 1.4 GHz obtained with the Westerbork telescope revealed the presence of narrow extended structures of polarized radio-frequency radiation. The origin of the radio structures is probably related with the acceleration of relativistic electrons by a shock wave. The energy density of the magnetic field is estimated as 9% of the energy density of heated plasma. To conclude this chapter, let us enumerate the main specific features of collisionless shock waves in space plasma.

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• A viscous ion jump in a strong (super-critical) shock wave is implemented through microturbulence formed by the instabilities of interpenetrating flows of ions and reflection of ions from magnetic barriers. The viscous jump has a width up to several dozens of inertial lengths of incident ions, which is much less than the Coulomb scattering length of these particles. • Along with ion heating, a relatively small share of the incident flow of ions appears to be suprathermal and injected into the Fermi acceleration mode. In cosmic plasma with its strong extended shock waves, it is possible to efficiently accelerate ions and electrons with the conversion of a significant share of wave energy (dozens of percent) into the energy of nonthermal, often ultrarelativistic particles and fluctuating magnetic fields. Accelerated particles penetrate into the region ahead of the viscous ion jump and form an extended prefront of a shock wave in which the velocity of the incident flow is decelerated by the macroscopic ponderomotive force caused by the pressure gradient of the accelerated particles. • The effect of the formation of a prefront is partially similar to the previously studied radiation shock waves. An important difference is the presence of instabilities of nonthermal distributions of accelerated ions: they efficiently amplify magnetic fluctuations ahead of the shock wave front, which, in their turn, have a substantial effect on the acceleration of particles. The effect is highly nonlinear and multiscale since injection takes place at the scale of a viscous jump but it affects the prefront structure with the scale six to eight orders of magnitude larger than the scale of the jump. A self-consistent account of the effects of the generation of strong magnetic fluctuations is necessary, because they play a key role in the Fermi mechanism. • Efficient acceleration and escape of energetic particles from the system can result in a substantial increase in the ratio of plasma compression in a strong shock wave as compared to the standard compression in an adiabatic wave. As a consequence, the temperature of the thermalized ions behind the front of the modified wave decreases substantially.

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Solntsa pri pomoshchi trekhelektrodnykh lovushek zaryazhennykh chastits na vtoroy sovetskoy kosmicheskoy rakete // Dokl. AN SSSR. 1960. T. 131. S. 1301–1304 (in Russian)] Ness NF, Scarce CS, Seek JB (1964) Initial results of the IMP-1 magnetic field experiment. J Geophys Res 69:3531 Kennel CF, Edmiston JP, Hada T (1985) A quarter century of collisionless shock research, collisionless shocks in the heliosphere: a tutorial review (A87-25326 09–92). American Geophysical Union, Washington, DC, p 1 Burgess MD (2015) Scholer collisionless shocks in space plasmas. Cambridge University Press Schaeffer DB, Winske D, Larson JD, Cowee MM, Constantin CG, Bondarenko AS, Clark SE, Niemann C (2017) On the generation of magnetized collisionless shocks in the large plasma device. Phys Plasmas 24:041405 Huntington CM, Manuel MJ-E, Ross JS, Wilks SC, Fiuza F, Rinderknecht HG, Park H-S, Gregori G, Higginson DP, Park J, Pollock BB, Remington BA, Ryutov DD, Ruyer C, Sakawa Y, Sio H, Spitkovsky A, Swadling GF, Takabe H, Zylstra AB (2017) Magnetic field production via the Weibel instabllity in interpenetrating plasma flows. Phys Plasmas 24:041410 Bedsel C, Landgdon A (1989) Plasma physics and numerical simulation. Energoatomizdat, Moscow [Bedsel C., Landgdon A. Fizika plazmy i chislennoye modelirovaniye. - M.: Energoatomizdat, 1989 (in Russian)] Hockney RW, Eastwood JW (1989) Computer simulations using particles. Adam Hilger, Bristol Marcowith A, Bret A, Bykov A et al (2016) The microphysics of collisionless shock waves. Rep Progr Phys 79:046901 Caprioli D, Pop AR, Spitkovsky A (2015) Simulations and theory of ion injection at nonrelativistic collisionless shocks. Astrophys J Lett 798(2):L28 Winske D, Quest KB (1988) Magnetic field and density fluctuations at perpendicular supercritical collisionless shocks. J Geophys Res 93:9681–9693 Jones FC, Ellison DC (1991) The plasma physics of shock acceleration. Space Sci Rev 58:259– 346 Kropotina YA, Bykov AM, Krasil’shchikov AM, Levenfish KP (2016) Relaxation of heavy ions in collisionless shock waves in space plasma. J Techn Phys 86(4):40–47 [Kropotina Yu. A., Bykov A. M.., Krasil’shchikov A. M., Levenfish K. P. Relaksatsiya tyazhelykh ionov v besstolknovitel’nykh udarnykh volnakh v kosmicheskoy plazme // Zhurnal tekhnicheskoy fiziki. 2016. T. 86, vyp. 4. S. 40–47 (in Russian)] Lipatov AS (2002) The hybrid multiscale simulation technology. Springer, Berlin, 403 p Matthews AP (1994) Current advance method and cyclic leapfrog for 2D multispecies hybrid plasma simulations. J Comput Phys 112(1):102–116 Bell AR (1978) The acceleration of cosmic rays in shock fronts. Monthly Notices R Astron Soc 182:147 Blandford RD, Ostriker JP (1978) Particle acceleration by astrophysical shocks. Astrophys J Lett 221:129 Krymsky GF (1977) Regular mechanism of charged particle acceleration at the shockwave front. Sov Phys Dokl 234:1306 [Krymsky G. F. Regulyarnyy mekhanizm uskoreniya zaryazhennykh chastits na fronte udarnoy volny // DAN SSSR. 1977. T. 234. S. 1306 (in Russian)] Axford WI, Leer E, Skadron G (1977) The acceleration of cosmic rays by shock waves. Proc 15th Int Cosmic Ray Conf II:132 Malkov MA, Drury LOC (2001) Nonlinear theory of diffusive acceleration of particles. Rep Progr Phys 64:429–481 Bykov AM, Ellison DC, Osipov SM, Vladimirov AE (2014) Magnetic field amplification in nonlinear diffusive shock acceleration including resonant and non-resonant cosmic ray driven instabilities. Astrophys J 789(2):137 Bykov AM, Brandenburg A, Malkov MA, Osipov SM (2013) Microphysics of cosmic ray driven plasma instabllities. Space Sci Rev 178:201 Bell AR, Lucek SG (2001) Cosmic ray acceleration to very high energy through the nonlinear amplification by cosmic rays of the seed magnetic field. Monthly Notices R Astron Soc 321:433

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69. Schure KM, Bell AR, Drury LOC, Bykov AM (2012) Diffusive shock acceleration and magnetic field amplification. Space Sci Rev 173:491 70. Helder EA, Vink J, Bykov AM, Ohira Y, Raymond JC, Terrier R (2012) Observational signatures of particle acceleration in supernova remnants. Space Sci Rev 173:369–431 71. Artsimovich LA, Sagdeev RZ (1979) Plasma physics for physicists. Nauka, Moscow [Artsimovich L. A., Sagdeev R. Z. Fizika plazmy dlya fizikov. - M.: Nauka, 1979 (in Russian)] 72. Velikovich AL, Liberman MA (1987) Shock wave physics in gases and plasma. Nauka, Moscow [Velikovich A. L., Liberman M. A. Fizika udarnykh voln v gazakh i plazme. - M.: Nauka, 1987 (in Russian)] 73. Raymond JC, Edgar RJ, Ghavamian P, Blair WP (215) Carbon, helium, and proton kinetic temperatures in a cygnus loop shock wave. Astrophys J 805:152 74. Vink J, Broersen S, Bykov A, Gabici S (2015) On the electron-ion temperature ratio establish by collisionless shocks. Astron Astrophys 579:A13 75. Akamatsu H, van Weeren RJ, Ogrean GA, Kawahara H, Stroe A, Sobral D, Hoeft M, Rottgering H, Bruggen M, Kaastra JS (2015) Suzaku X-ray study of the double radio relic galaxy cluster CIZA J2242.8+5301. Astron Astrophys 582:A87

Chapter 8

Electromagnetic Shock Waves

When studying electromagnetic waves, the object of interest is primarily the oscillatory process at various frequencies. However, oscillations do not exhaust the whole variety of electromagnetic processes in a wave. This chapter will focus on electromagnetic shock waves considered as solitary pulses of an electromagnetic field. Today’s developers of powerful pulse magnetic explosion generators [1, 2] whose actions are based on the explosive compression of the magnetic field by a charge of condensed explosive take interest in such pulse processes. Moreover, a violent process of laser technology and nonlinear optics [1, 2] makes it possible to obtain the generation of light pulses of the attosecond range and study their propagation. Since works in the field of physics of electromagnetic shock waves are few and contained in difficult-to-access publications, we will follow papers [3–5]. Let us start with considering the dynamics of electromagnetic waves in vacuum. Electromagnetic Shock Waves in Vacuum. As shown in [3], an electromagnetic shock wave occurs when a conductor in the magnetic field suddenly starts moving in the direction perpendicular to power lines (ref. Fig. 8.1). At the start of conductor movement from point A, a wave goes in the field at velocity c, and the initial field H 0 occupying space c will become compressed to size c − u, its strength will be increased to H 1 = H 0 c/(c − u). From H 0 to H 1 , the field strength grows step-wise on the wave front, and an electrical field E = H 1 − H 0 = H 0 u/c occurs here in a step-wise manner as well. An ordinary shock wave emerging from its depth to the surface may suddenly push the conductor surface. Such push, however, will not be sudden, since the shockwave front in the conductor in the magnetic field is accompanied by an advance electromagnetic disturbance H = H0 (δ − 1)e−x/l (and similar for E), where δ = ρ 1 /ρ 0 is the material compression in a shock wave, x is the distance from the front, l = (c2 − D2 )/(4π λD) is the size of the electromagnetic wave front blurring zone, D is the wave velocity, λ is conductivity ahead of the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Fortov, Intense Shock Waves on Earth and in Space, Shock Wave and High Pressure Phenomena, https://doi.org/10.1007/978-3-030-74840-1_8

247

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Fig. 8.1 Formation of an electromagnetic shock wave when a conductor in the magnetic field suddenly starts its movement [3]

Conductor

Field

wave. When the wave reaches the boundary of the conductor with vacuum, the front velocity rises   from D to c and the blurring value I transforms into L = cl/D =  c c2 − D 2 / 4π λD 2 . For a strong wave in copper (D = 10 km/s), we obtain L = 3.7 cm (λ = 5.8 × 1017 −1 s ); when conductivity is increased, blurring falls down, and we obtain L = 0.2 mm for the same copper cooled down to 20 K. In this manner, the wave front in the vacuum can be rather narrow and it is convenient to be considered as a mathematical discontinuity. If the wave is plane, it propagates with constant amplitude, since the energy contained in the near-front layer x is transferred together with it and the volumetric density is preserved. The field perturbations behind the wave move with the same speed c, do not catch up with the front and cannot influence it. If the surface is curved, the energy density and amplitude change for the same reasons. The simplest example of a converging wave is a cylindrical wave with the magnetic field √ parallel to the axis. Here, the energy density rises as 1/r, and the amplitude as 1/ r . In this manner, the electromagnetic shock wave is capable of unbounded cumulation, and the obtained law of amplitude growth does not encounter any physical limitations so far as the Maxwell equations are true. The described phenomenon also shows another unexpected specific feature that is not easily found and obtained only by solving equations. Let us discuss this in detail. Let us consider a cylindrical wave converging in vacuum at the focusing stage, i.e., let us find a self-similar solution. The phenomenon scheme is shown in Fig. 8.2 [3], where AO and OB are convergent and reflected waves, AFB is the law of motion of a cylindrical piston compressing the field. The Maxwell equations for this case look as follows ∂H 1 ∂E =− , c ∂t ∂r

1 ∂H 1 ∂(r E) =− . c ∂t r ∂r

8 Electromagnetic Shock Waves

249

Fig. 8.2 Cylindrical electromagnetic shock wave (on the converging wave AO, the√field is intensified as 1/ r ; on the reflected wave OB, there is logarithmic divergence for all r) [3]

Here, the magnetic field H is parallel to the cylinder axis, and the electrical field E is circular. Excluding H and E from these equations, we obtain, respectively   ∂ 1 ∂(r E) 1 ∂2 E = 0, − c2 ∂t 2 ∂r r ∂r   1 ∂2 H ∂H 1 ∂ r = 0. − c2 ∂t 2 r ∂r ∂r We will find the self-similar solution as follows   r0 r0 E = E0 e(τ ), H = H0 + H0 h(τ ), r r where τ = ct/r, t is the time counted from the moment of focusing, E 0 and H 0 are wave front amplitudes on the radius r 0 . On the front τ = − 1 and e(− 1) = h(− 1) = 1. The substitution of self-similar formulas into the equation gives as follows     3 h e 1 − τ 2 − 2τ e + e = 0, h 1 − τ 2 − 2τ h  − = 0. 4 4 If a solution is sought as a series in powers of 1 + τ for − 1 < τ < 1 (between incident and reflected waves), we will obtain [3] e(τ ) = 1 −

∞ (2n + 1)!!(2n − 3)!! 1

23n (n!)2

(1 + τ )n ,

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Fig. 8.3 Self-similar solution for converging cylindrical shock wave of the field [3]

h(τ ) =

 ∞  (2n − 1)!! 2 (1 + τ )n 0

n!

23n

.

For the zone behind the reflected wave (1 < τ < ∞), we additionally use the conditions of finiteness E and H on the axis for t = 0, as well as conservation E − H when passing through the reflected wave front. Finding the solution in this field as a series in powers of 1/τ, we obtain ∞

1 (4n + 1)!! 1 , (2τ )3/2 0 24n n!(n + 1)! τ 2n  

∞ 2 (4n − 1)!! 1 h(τ ) = . 1+ τ 24n (n!)2 τ 2n 1

e(τ ) = −

For τ → 1, all these series diverge, i.e. the fields on the reflected wave front are infinitely large. The asymptotic expressions for them near τ = 1 are as follows: e∼

1 1 ln|1 − τ | + 0.17, h ∼ − ln|1 − τ | + 1.10. π π

Dependence diagrams of e and h versus τ are shown in Fig. 8.3 [3]. We shall note that if a cylindrical piston suddenly starts expanding rather than compressing (i.e. generates an rarefaction wave), the wave amplitude will behave in the same manner and is described by the same solution near the focusing. Thus, it appears that the reflected wave amplitude is infinitely large not only on the cylinder axis but also at all points of space (at different times). This phenomenon was the first example of such type and it appeared to be a paradox until this problem was solved in a different manner.

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251

In 1957, Ya. B. Zel’dovich considered a cylindrical wave as superposition of plane waves and built a family of self-similar solutions for converging waves which included a solution for a shock wave. For each component of plane waves, the passage of the axis has no physical or formal specific features. The summation of these waves leads to a divergence at the front of the wave reflected from the axis, i.e. the seemingly unexpected result is confirmed. Let us note that the discovered specific feature is not related only with the electromagnetic nature of the wave but is specific for the cylindrical shape: the equation for H is just a wave equation to which the problem of a weak shock wave in matter, or acoustic wave, is also reduced but in this case the equations are correct for low amplitudes only. The convergence of an electromagnetic wave is hard to reproduce in an experiment due to the extreme requirements to the accuracy of the shape of the shock wave giving rise to it in the conductor, so we discussed it in details only because of its principal specific features. Stationary Unbounded Cumulation. All examples of unbounded cumulation in [3] are related to the cases of nonstationary motions where density divergence is attained at a point (spherical cumulation) or on a line (cylindrical cumulation) just for a single moment. However, the following example shows that this condition is not compulsory and, in principle, unbounded cumulation can be stationary, i.e. in some point of the flow, the energy density can be maintained indefinitely large all the time. At first glance, it seems that to build such an example, it is sufficient to consider a converging conical shock wave at the top of which an unlimited amplitude can be expected. However, this does not occur since the amplification of the wave as it approaches the axis is accompanied by an increase in its velocity whereby the cone becomes blunted and a wave section occurs instead of a pointed top, which is normal to the axis and moves at a finite velocity (phase velocity of the process), i.e. bearing the final pressure. There is no such obstacle for an acoustic wave moving at a permanent velocity, but the solution for it near the axis is physically contradictory and therefore incorrect: particles move faster than the wave that pushed them and cross the cone axis, which contradicts the continuity property of the medium. These challenges disappear if we consider an electromagnetic shock wave whose velocity always equals c and does not depend on the amplitude. Let us assume that there is a longitudinal magnetic field H 0 inside a cylindrical cavity in the conductor, and a conical converging shock wave comes from the conductor to the cavity surface. This shock wave pushes the surface inwards and the point of exit of the wave moves at superluminal velocity D. A converging electromagnetic shock wave will come to the cavity axis. This phenomenon is schematically shown in Fig. 8.4 [3]. The magnetic field has two components, H x and H r , but the electrical field still has only E ϕ . They depend only on r and x + Dt and the Maxwell equations look as follows:

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8 Electromagnetic Shock Waves

Fig. 8.4 Conical electromagnetic shock wave (on the converging wave, √ the field is amplified as 1/ r ; on the reflected wave, there is logarithmic divergence for all r) [3]

 ∂ Hx ∂  Hr − M E ϕ = , ∂x ∂r  ∂  E ϕ − M Hr = 0, ∂x

∂ Hr Hr ∂ Hx + + = 0, ∂r r ∂x

Eϕ ∂ Hx ∂ Eϕ + +M = 0, ∂r r ∂x

where M = D/c = 1/sin θ. Let us exclude E ϕ to obtain as follows   ∂ 1 ∂(r Hr ) = 0, ∂x2 ∂r r ∂r    2  ∂ 2 Hx 1 ∂ ∂ Hx M −1 − r = 0. ∂x2 r ∂r ∂r



M2 − 1

 ∂ 2 Hr

−

These equations fully coincide with the equations describing a cylindrical wave (non-stationary) with the only √difference

that the role of H, E and t is now played by √ 2 2 Hx , Hr / M − 1 and x/ c M − 1 . In this manner, one problem is reduced to another and we can use the already discovered self-similar solution for the vicinity of the focus, overwriting it as applicable to the designations of the new problem:  Hx = H0 + H0x  Hr = H0r

R h x (τ ), r

R h r (τ ), r

8 Electromagnetic Shock Waves

253

  where τ = x/r M 2 − 1 , H 0x and H 0r are the components of the wave amplitude on radius R. Functions hr (τ ) and hx (τ ) coincide with h(τ ) and e(τ ), respectively, as shown in Fig. can also see from the equations that E ϕ = M Hr and  8.3. (One  H0x /H0r = − M 2 − 1 .) √ Thus, the wave amplitude grows as 1/ r as it approaches the axis and unbounded cumulation is stationary at the vertex of the cone. Let us remind that e and h diverge at τ → 1, i.e. for a conical wave just as for a cylindrical one, the unlimited amplitude on the reflected wave is preserved also at a finite distance from the axis, and, therefore, this property is specific not only for a cylindrical wave as it seemed at the beginning. Let us note that in case of the superluminal velocity of the cone vertex for a shock wave in matter, it will not be blunted, since its normal velocity cannot exceed c; in this case, stationary unbounded cumulation is also possible. Electromagnetic Shock Waves in Matter. Electromagnetic shock waves can be not only in vacuum but also in matter. If the electromagnetic “constants” of matter depend on the field strength, then the propagation velocities of electromagnetic signals of various amplitude differ. As a result, some waves can catch up with other waves, and, as in gas dynamics, a discontinuity can spontaneously appear in a wave with a smooth profile (ref. Chap. 1), i.e. an electromagnetic shock wave. This interesting physical object and its theory are given in [4–6] (ref. Chap. 1). Reports appear in some papers, for example, by Rosen [6] who considered a specific type of nonlinear electrical properties. It should be expected that such a shock wave, which is cylindrical and converging, will be amplified infinitely before focusing. In other words, the fact of wave discontinuity apparently opens up a principal possibility of its unlimited amplification. The physical specific feature of this suggested case of cumulation will be the occurrence of a very strong field inside matter, which may be interesting for various applications [1, 2]. Considering electromagnetic shock waves in matter, we will follow the classical works of the academician Gaponov-Grekhov given in the review [4]. The study of the phenomena occurring in the propagation of waves of finite amplitude in media with nonlinear parameters is currently the content of numerous sections of mechanics and electrodynamics of continuous media [7–10]. Studies taking into account the specific problems of gas dynamics and physics of plasma, electrodynamics of nonlinear media and nonlinear optics, carried out in recent years showed that the qualitative specific features of nonlinear wave processes are primarily defined by two characteristics of the medium: nonlinearity leading to the local distortions of the wave profile due to the enrichment of its spectrum with higher harmonic components and dispersion causing wave spreading and decreasing the interaction between individual harmonics (Chap. 1). This chapter considers media with low dispersion and relatively strong nonlinearity when areas with high (as compared to the initial scale) gradient of fields

254

8 Electromagnetic Shock Waves

describing the wave1 may appear in the wave profile. The most characteristic nonlinear effect for such media is the shock wave formation. Here, a shock wave should be understood as a process of quick change (“jump”) in fields in some generally moving area with size δ. The change of the field in this area is described by equations of a higher (as compared with the equations in the area beyond δ) order the solution of which on both sides of δ asymptomatically tends to various values (constant in δ scale) interrelated by homogeneous boundary conditions not depending on the field structure inside the jump. As we know, energy dissipation always takes place in a shock wave (Chap. 1). Theoretical and experimental studies of shock waves started with the research into the dynamics of a neutral compressible gas [7–10] (ref. Chap. 1). The late development of plasma physics, in particular, magnetohydrodynamics [11–13], led to the study of shock waves in a conductive medium (ref. Chap. 7). Such waves represent “jumps” not only of mechanical and thermodynamic values but also of the electromagnetic field. However, although the presence of the electromagnetic field plays an important role, the existence of magnetodynamic and plasma waves is principally related with the macroscopic motion of the medium (Chap. 7). In the equations describing such processes and including the equations of gas dynamics and electrodynamics (the latter are frequently considered in quasi-stationary approximation), the presence of nonlinear members is caused by hydrodynamic effects. For the “electromagnetic branch” of waves, the nonlinearity of such media is usually small, and the formation of shock wave is impossible due to dispersion. The appearance of solid materials (ferrites, ferrielectrics, semiconductors) with a prominent nonlinearity of electromagnetic characteristics and dispersion having a substantial value only in the area of comparatively high frequencies attracted the attention of researchers to “purely electromagnetic” shock waves (EMSW). First papers in this field appeared in 1958–1960 [14–18]. Macroscopic movements and changes in the thermodynamic condition of the medium (tough unavoidable in principle) play the secondary role in the formation of EMSW and, for the most part, may not be taken into account. Processes of such kind can be described by equations of electrodynamics supplemented by phenomenological constitutive equations characterizing the electromagnetic properties (nonlinearity, dispersion) of the medium.2 The typical types of nonlinear constitutive equations can be illustrated by the example of ferrite. With a sufficiently slow (as compared to the relaxation time of 1 In

another extreme case (strong dispersion and low nonlinearity), nonlinearity has a significant effect on large-scale processes covering many periods of a quasi-harmonic wave. Thus, the shape of the “envelopes” of the amplitude and frequency of the modulated wave in such a medium can experience strong distortions; in particular, Riemann and shock waves of envelopes are possible. In nonlinear optics, such cases of special dispersion are intensively studied when only individual spectral components of the wave efficiently interact. 2 In a general way, the constitutive equations include the equations of mechanics of deformable media and thermodynamics and the complete system of equations describes not only electromagnetic processes. Henceforth, we will consider quasi-electromagnetic waves and spatial dispersion will be taken into account only when it is also associated with electromagnetic processes.

8 Electromagnetic Shock Waves

255

Fig. 8.5 Typical characteristics of magnetization (dependence B(H) or (I)) [4]

b

a

b

magnetization) change in the field, the bond between the magnetic induction B and the magnetic field strength H in ferrite can be considered independent of the field change rate (quasi-static). If we neglect the anisotropy and hysteresis phenomena, the vectors B and H are collinear and dependence between their values is unique and ungerate (Fig. 8.5, curve a). When the field changes quickly, various dynamic models are used to describe the magnetic polarity reversal process. Thus, to describe small fields, the equations describing the motion of the domain walls can be used [9]. In a strong and fastchanging field specific for shock waves, the domain structure is insignificant, and the primary role is played by the precession of the magnetization vector M. In saturated ferrite, this process is described by the uniform precession equation   α ∂M ∂M = −γ [M H ] + M , ∂t M ∂t

(8.1)

where M is the constant saturation magnetization, γ is the hydromagnetic ratio, α is the dispersion coefficient. Value H is the sum of the external field H e , the anisotropy field H an , the exchange field H ex , etc.; however, in many problems of the theory of EMSW in ferrite, one can neglect all internal fields and set H = H e . In case of slow processes (∂/∂t → 0) from (8.1), we obtain the quasi-static equation [MH] = 0. In this case, for example, in a wave propagating along a constant magnetizing field H 0 , the relationship between the transverse components H ⊥ and B⊥ = H ⊥ + 4π M ⊥ qualitatively corresponds to Fig. 8.1. If the rate of change in the field is comparable with the ferrite relaxation time, then, due to the differential nature of the constraint Eq. (8.1), the order of the system of equations describing wave propagation in ferrite increases. Physically, this means the appearance of time dispersion related both with a purely reactive process (the first member in the right

256

8 Electromagnetic Shock Waves

part) (8.1) and with dissipation (the second member). Let us note that both processes are nonlinear. Uniform precession of magnetization is usually possible only in the presence of a strong (as compared with the wave field) constant field saturating ferrite. If there is no such field, incoherent magnetic polarity reversal occurs, which is related with the excitation of spine and magnetostatic oscillations [19]. In this case, the precession M is locally described by Formula (8.1) and the constitutive equation neglecting spatial dispersion can be obtained by averaging (8.1) over a sufficiently large (on the scale of spin wave length) element of volume. As a result, the averaged magnetization vector is parallel to the magnetic field H and the change in its value M h is described by the equation   Mh2 αγ Mh ∂ Mh 1 − 2 H. =− ∂t 1 + α2 M

(8.2)

A specific feature of Eq. (8.2) is that all nonlinear effects described by it are related with dissipation: in case of α → 0, the nonlinearity (and dispersion) disappears. Let us also note the saturation property typical of ferrite and a number of other nonlinear electrodynamic materials: irrespective of a specific mechanism of nonlinearity, a medium becomes linear in a very strong field; in particular, this results from Eqs. (8.1), (8.2). This property makes it possible to simplify the solution of some problems on the propagation of intense shock waves, digressing from a specific type of constraint equations in the area of small fields and approximating the dependence B(H) with a piecewise-linear characteristic (Fig. 8.5, curve b [4]). Similar consideration can be made for other nonlinear media. For example, the equations of the motion of charge carriers in semiconductors allow for identifying mechanisms of nonlinearity, dispersion and dissipation in that medium and making a phenomenologic description of its electrodynamic properties (which is in simplest cases reduced to finding an equivalent scheme of p–n-transition [20]). A nonlinear system consisting of Maxwell equations and phenomenologic constraint equations of the type (8.1) or (8.2) is extremely complicated and its study is related with insuperable difficulties in a general case. Many phenomena specific for shock waves in free space can be studied by considering one-dimensional processes (homogeneous plane waves) where the field depends on the time t and one longitudinal coordinate z only. However, the practical applicability of such one-dimensional idealization in the EMSW theory is rather limited since all experimental studies are currently related to a limited space—in particular, to doubly coupled waveguides (transmission lines) filled with a nonlinear medium. In such systems, the field dependence on transverse coordinates is substantial. Along with that, waves in two-wire lines with low (as compared to a characteristic wave length) transverse dimensions in many cases can be described by telegraph equations for integral values: ∂ ∂U + = 0, ∂z ∂t

∂I ∂Q + = 0. ∂z ∂t

(8.3)

8 Electromagnetic Shock Waves

257

Here I, U, , Q are respectively current, voltage, linear induction flux and linear charge in the line. The relations between these values (constitutive equations) in a general case are given by nonlinear integral–differential operators  = {I, U }, Q = Q{U, I },

(8.4)

the type of which depends both on the configuration of line conductors and properties of medium filling it. Nonlinearity and dispersion of the waveguide take place in those cases when both operators,  and Q, are nonlinear. The system (8.3) is also approximately applicable to waves in “artificial” lines (nonlinear filters) consisting of identical links with lumped parameters; such lines were used in many experiments with EMSW. Nonlinear differential-difference equations describing processes in systems with discrete parameters under certain conditions (small changes in the IU values on a single link) can be replaced with differential equations (8.3).3 Thus, there is a class of important wave systems having a high practical value, the processes in which can be considered as one-dimensional and are described by Eq. (8.3) (similar to the equations of a plane linear-polarized wave in an unbounded medium). Therefore, in further description of general topics (formation, development, interaction, structure of EMSW) we use these equations (sometimes with some additional members). As a rule, we will assume that the flux  is related with the current I, and the charge Q is related with the voltage U by differential equations containing partial derivatives for t (time dispersion) and for z (spatial dispersion). In this case, nonlinearity will be taken into account only in one of the constraint equations. These assumptions are consistent with most experiments. Though many of the obtained results can be applied in those cases when both operators {I}, Q{U} are nonlinear. It’s worth noting that the dynamics of nonlinear wave processes is characterized by the general principles and methods of research not related with the specific physical nature of the phenomena under consideration. In particular, the initial information on electromagnetic waves in media and waveguides with nonlinear parameters can be obtained based on an analogy between the equations of electromagnetic and gas dynamic waves (Chap. 1), which takes place in the simplest cases. On the one hand, this analogy makes it possible to immediately write a solution for some problems of the EMSW theory, but on the other hand, it can be used for modeling the gas dynamic processes of electromagnetic phenomena the experimental implementation of which often turns out to be much simpler. The above is true, for example, for simple waves, the interaction of shock waves, etc. Finally, individual theoretical methods and results first obtained specially for EMSW are already applied in the problems of ordinary and magnetic hydrodynamics. The analogies, however, end where it is required to specify the type of nonlinear dependences. In electrodynamics, the latter have another nature and are more diverse. 3 In

this sense, a lumped-parameter network can be considered as an equivalent scheme of the respective distributed line.

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8 Electromagnetic Shock Waves

To an even greater degree, this refers to the dispersion properties of media and transmission lines: though the dispersion is low within a slow process, its effect on the parameters (duration, shape) of the impact front is determinant. All this leads to the unique processes of the formation and specific types of EMSW structures. The use of various transmission lines, in particular, lines with discrete parameters, opens up wide possibilities for the detailed study of nonlinear wave processes, which is hardly realizable for compressible media. In its turn, this promotes the development of the general theory of waves in nonlinear media. Formation and Propagation of Shock Waves in Nondispersive Waveguides (Transmission Lines). Let us start considering rather slow processes for which the relationship between  and I can be deemed quasi-static. In this case, (8.3) is a quasilinear system of hyperbolic type which is invariant with respect to such changes in the scale of independent variables for which z/t = const. Methods for studying such equations are based on the theory of characteristics [7–10] (Chap. 1). From the physical standpoint, here we can speak of no dispersion: the equations have no independent time or spatial scales; in particular, small (linear) perturbations propagate along the characteristics at the rate that does not depend on their frequency. Simple Waves and Formation of Discontinuities [7–10]. Important particular integrals (8.3) allowing for tracing the formation of shock waves are Riemann solutions (simple waves) of type (ref. Chap. 1):  I =F t∓

  z , U = ∓ ρ(I )d I, v(I )

(8.5a)

or t∓

z = ψ(I ), v(I )

(8.5b)

where F, ψ are arbitrary functions, v(I) = (LC)−1/2 , ρ(I) = (L/C)1/2 , L(I) = d/dI. The properties of such a wave (8.5a) are well studied [7, 10] (Chap. 1). Each point of its profile moves along the characteristic (8.5b) at a constant velocity v(I) which depends on the current magnitude at this point and coincides with the velocity of small perturbations at the corresponding values of l. It is obvious that in the areas where ∂v/∂z < 0, the wave profile steepness grows and at some moment the dependence of I from z, t becomes ambiguous (Fig. 8.6a). This ambiguity is devoid of physical sense (Chap. 1) and further solution (8.5a) loses its validity. Since the system (8.3) has no other continuous solutions under the given initial conditions, then, within the limits of the assumption on a quasi-static link (I), a break of the continuity of the values I, U (a shock wave) should appear. The moment t and coordinate z of discontinuity appearance as well as the respective current magnitude I can be determined from the equations connecting these values at the function breaking point (8.5a) with a vertical tangential line [3]:

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259

Fig. 8.6 a Simple wave profile evolution; b oscillograms of voltage U(t) in various sections of a coaxial line with longitudinally magnetized ferrite (timestamps every 10 ns) [4]



∂t ∂I



 = 0, z,I

∂ 2t ∂I2

 = 0. z,I

Substituting (8.5a) to (8.6a), we will obtain as follows

(8.6a)

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8 Electromagnetic Shock Waves

Fig. 8.7 Pulse oscillogram with two shock waves (timestamps every 100 μs) [4]

d(v − 1) dψ + = 0, dI dI d 2 (v − 1) d 2 ψ z + = 0. dI2 dI2

z

(8.6b)

The sought values z, t, I are given by the solution of equations (8.6b) together with (8.6a). It’s worth noting that although the solution (8.5a) belongs to an unbounded line, a simple wave in the form of a finite duration pulse is possible (within a finite amount of time) in a bounded system as well; obviously, the length of the latter must exceed z for a discontinuity to occur. The process of the formation of discontinuities as a result of the evolution of simple waves is well known in gas dynamics (ref. Chap. 1) and magnetohydrodynamics, but its experimental observations are apparently few. The respective experiment in electrodynamics is relatively simple. Its result for waves in a coaxial line with ferrite [21] is represented in Fig. 8.6 [4] (similar results are obtained for a line with semiconductor diodes [22]). As follows from (8.5a), in case of pulse deformation, its duration is preserved at any fixed level. A specific nature of nonlinearity in electrodynamics generates a number of specific features of the process of EMSW development from simple waves. A picture given in Fig. 8.6 [4] takes place only in the area of monotonically decreasing dependence L(I). If the current in the pulse starts from the value I = I  = 0 and changes the sign (“operating point” I  can be easily changed by passing direct current through the line), the curve (I) has an inflection point in the area of current change in the wave and L(I)—maximum. Then two shock waves appear—in the rising and falling parts of the pulse (Fig. 8.7) [4, 21]. Boundary Conditions in Discontinuity and Its Stability [5, 23]. When considering a shock front as a moving discontinuity, we can obtain the boundary conditions connecting the points and stresses on its both sides. Integrating Eq. (8.3) for a fixed time moment with respect to z near the discontinuity, we will find 

I2 − I1 = v p (Q 2 − Q 1 ), U2 − U1 = v p (2 − 1 ).

(8.7a)

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261

Here, vp is the velocity of discontinuity movement; indexes 1 and 2 belong to the values before and after the discontinuity, respectively. For given I 1 , U 1 , the ratios of (8.7a) determine the relationship between I 2 and U 2 similar to the Hugoniot adiabat in gas dynamics (while the link between I and U in a simple wave corresponds to the Poisson adiabat). If the link Q = CU is linear as we can assume, the velocity vp can be represented by the function of only the boundary values of current: v 2p =

I2 − I1 . C(2 − 1 )

(8.7b)

The relation (8.7b) allows a simple graphical interpretation: slope of the straight   line connecting points 1 and 2 on the curve (I) equals 1/ v 2p C (Fig. 8.5) [4]. The dependence (8.7b) is well correlated with the experiment [4]. The propagation of EMSW is always accompanied by energy dissipation. To show this, it is sufficient to write the energy balance equation for the area containing a discontinuity and use the boundary conditions (8.7a). As a result, it is easy to find that the power P dissipated at the discontinuity equals ⎡ (2 − 1 )(I2 + I1 ) P = vp⎣ − 2

2

⎤ I d⎦.

(8.8)

1

If the dependence (I) is not linear, the power P is nonzero and positive for a stable discontinuity. Hence it follows that to form EMSW, the presence of losses of “high frequency” type is required, which affects the area of fast changes in current and voltage. It is substantial that P does not depend on the specific dissipation mechanism, but only on the value of the jump. Not all discontinuities satisfying the relations (8.7a) may really exist (Chap. 12). The condition of shock wave stability with respect to small perturbations is defined by the requirement that the number of parameters describing these perturbations on both sides of the discontinuity be equal to the number of boundary conditions connecting them [7, 9]. This condition leads to the inequality v1 < v p < V2 ,

(8.9)

where v1,2 = (CL 1,2 )−1/2 are velocities of small perturbations ahead and behind the discontinuity. It follows from (8.9) that a jump changing the field from H 1 to H 2 in Fig. 8.5 [4] is stable, and the reverse jump (from H 2 to H 1 ) is unstable. Formation and Development of Shock Waves in Lines with Low Linearity [4, 23, 24]. Since the shock wave is a jump of line parameters, reflections from its front are unavoidable. Therefore, after the formation of discontinuity in a travelling wave, the latter stops being Riemann in the area behind the discontinuity. The corresponding problem is reduced to the integration of Eq. (8.2) together with the conditions (8.7a). The simplest case of low nonlinearity when the relation (I) looks as follows

262

8 Electromagnetic Shock Waves

˜ ),  = L 0 I + (I

˜ L 0 I. L 0 = const, ||

(8.10)

In this case, the jump of line parameters is small and, in the first approximation, the solution (8.5a) remains in force on both sides of the discontinuity. Then, “seaming” (8.5b) and (8.7a) along the discontinuity trajectory, we can find the dependence of the ˜ = −x L I 2 (quadratic nonlinearity) jump on the coordinate [4]. If for definiteness  and current before a jump equals zero, then 2v0 z p = z0 + 2 x Ip

I p I

dψ d I, dI

(8.11)

I0

where υ 0 = (LC)−1/2 , I p is the jump magnitude, z is its initial position, I 0 = I p (z0 ). Substituting (8.11) into the expression for vp , we can determine the trajectory of the discontinuity zp (t). If the wave has the form of a pulse of finite length, then the integral in (8.11) tends to some finite value depending on the initial pulse shape at high z. Hence, it follows that at large distances I p decreases in proportion to z−1/2 . It is also easy to determine the change in the pulse duration τ with distance. Since the front propagates at vp , at the end of decline v = v0 , then z  τ = τ0 + z0

 z x Ip 1 1 − dz. dz = τ0 − v0 vp 2v0

(8.12)

z0

Here, I p is determined by Formula (8.11). At large distances τ ~ z1/2 , i.e. the pulse decays and becomes stretched. The obtained expressions have a clear geometric meaning. If, along with the true (discontinuous) solution, we build an ambiguous function that continues a simple wave at time intervals after the occurrence of a discontinuity, then, it is simple to show [4] that the discontinuity position is determined by the condition of preserving the area bounded by the curve I(z) (equality of the areas abc and cde in Fig. 8.6a). A more general method for studying small-amplitude travelling waves is associated with the consideration of solutions and characteristics that change slowly along the direction of wave motion. These solutions (quasi-simple waves) are locally close to (8.5a) and satisfy the following equations I ρd I + ζ,

(8.13)

∂I ∂I +v = ϕ, ∂t ∂z

(8.14)

U= 0

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263

where φ, ζ are small functions (equal to zero for the wave (8.5a)). Substitution of (8.14) into (8.3) reduces the problem to seeking the function ϕ and further integration of the first-order Eq. (8.14). This allows taking into account a number of additional factors affecting the wave deformation. Thus, for a line with a sufficiently smooth non-homogeneity of the parameters (v, ρ depend on the coordinate), we 0 I and the integral (8.13) looks as follows in case of low obtain that ϕ = ρ20 ∂ρ ∂z quadratic nonlinearity z t− z0

dz 1√ − ρ0 v0 2



−1/2

xρ0 v0

 √  dz = ψ I ρ0 ,

(8.15)

where ρ, v is the value of ρ(I, z), v(I, z) in the linear approximation. For a homogenous line, the expression (8.15) transforms into (8.5b), and for a linear but inhomogenous line, it is equivalent to the one-dimensional approximation of geometric optics I = −1/2 ρ0 F(t − ∫ dz/v0 ). Therefore, Formula (8.15) can be considered as one of the generalizations of this approximation to nonlinear problems. The solution (8.15) can be generalized to the stationary case when the parameters explicitly depend on t; similar solutions were also obtained for gas dynamics [4]. As above, from (8.15) we can obtain the condition for the formation of a discontinuity in a heterogeneous nonlinear line and track its development in a weak wave of arbitrary shape. It appears that for some laws of ρ 0 , v0 change (for example, if ρ 0 grows exponentially), the discontinuity cannot be formed even in the area with ∂v/∂z < 0; its occurrence requires the wave amplitude to exceed some value at the initial point in time. Moreover, the pulse duration with a discontinuity at the front in a heterogeneous line does not always rise indefinitely but can tend to a finite value. Let us note another interesting practical case when the solution for a travelling wave is known in a line with an arbitrary explicit dependence of the parameters on the coordinate and with high nonlinearity. This is a case of “correlation” when the impedance ρ(I) = [L(I, z)/C(z)]1/2 does not clearly depend on z. In this case, the initial system (8.3) has an exact solution in the form of a quasi-simple wave propagating without reflections. Another area of applying the method of quasi-simple waves refers to lines with dissipation considered below. Propagation of Strong Discontinuities [4, 25, 26]. If a strong discontinuity develops in the line, the problem of its propagation in a wave of an arbitrary shape is extremely complicated in the general case. However, in electrodynamics this problem is simplified thanks to the aforementioned property of nonlinearity saturation that allows assuming that the properties of the transmission line behind the front of a strong EMSW are linear. The dependence (I) can be then approximated by the piece-linear function [4]. Presenting current behind the discontinuity as a superposition of the incident (I + ) and reflected (I − ) waves and using the boundary conditions, it is easy to express all values at the discontinuity through I+ . In particular, the discontinuity trajectory equation zp (t) looks as follows [25]:

264

8 Electromagnetic Shock Waves

Fig. 8.8 Change in the pulse shape in a line with discontinuous dependence (I) [4]

2 zp +  s − 1

z p −v2 t

I + (ξ )dξ = 0,

(8.16)

z0

where v2 is the perturbation velocity behind the discontinuity, 1 is the value  in front of it, s is the saturation flux at I = 0. The function I + (z − v2 t) near the discontinuity is set directly by the initial and boundary conditions. Using (8.16), we can fully track the shock wave development. Figure 8.8 [4] shows the pattern of the propagation of a pulse having a linear wave front l (I = −I m z/l, −l ≤ z ≤ l) at the initial point in time and a flat top (I = I m , z < −l); the same result is obtained both for a wave in a homogeneous nonlinear line and for a pulse passing the boundary of linear (z < 0) and nonlinear (z > 0) lines, if their parameters L 0 at I = 0 coincide (there are no reflections on the line boundary (z = 0)). Used here are nondimensional variables z =

2L 0 Im z 2L 0 Im v2 t , t = . l(s − 1 ) l(s − 1 )

The maximum amplitude of a shock wave I 2 max = I m /[1 + (s − 1 )/4L 0 I m ]; the distance zopt where this value is reached is lL 0 I m /(s − 1 ). A similar calculation can be also carried out for the dependence (I) approximated by a large number of linear sections [4, 25]. However, in a pulse with the leading edge of a finite duration, the current ahead of the discontinuity can differ from zero for a finite time only after which the propagation will take place in the same manner as in the case just considered. Interaction of Shock Waves [4, 27, 28]. If two shock waves propagate at the initial point in time (t = t 0 ) in the line, then after their fronts contact (at point z = z0 ) the propagation pattern qualitatively changes due to nonlinear interaction. Since the problem has no parameters depending on z and t individually (only on z/t), the interaction can result in the formation of those waves only the current of which depends on the ratio z/t (self-similar). In case of dependence (I) shown in Fig. 8.5 [4], three types of such waves are possible [4]: a stationary discontinuity, a

8 Electromagnetic Shock Waves

265

Fig. 8.9 Graphical analysis of the collision process of shock waves

simple wave with characteristics diverging from the point z0 and a combined wave consisting of a simple wave and its trailing shock wave that adjoin each other at the point where v = vp (the last type is specific for electromagnetic waves). Moreover, it is easy to show that only one of the described waves propagates to each direction from the interaction point. If the type of each of them is known, the problem lies in determining the values of IU in the area between diverging waves, which, in its turn, requires the solution of equations binding the quantities in various areas (an equation of (8.7a) type for shock waves). The general problem of shock wave interaction, just as in gas dynamics [1] (Chap. 1), can be solved by semigraphical method [4]: intersection points of curves (adiabats) are located on the plane of variables IU, which bind I and U in self-similar waves and passing through points III corresponding to the values of IU at z → ±∞ (two curves corresponding to two propagation directions go through each point). Figure 8.9 shows such a construct for counter interaction of shock waves when only shock waves occur as a result. The points of intersection of the adiabats determine the  possible values of IU in area III between the discontinuities. Point III gives the state  in the given area before interaction, and point III —the sought result of interaction. All the above-said refers also to the incidence of a shock wave on the interface between two different media (this interface is an interaction point) taking into account the fact that the functions (I) on both sides of the interface differ. An analytical solution of such problems has been obtained in a few cases only: for lines with a piecewise-linear characteristic of magnetizing [4] and for an arbitrary line in a partial case of collision of two similar discontinuities or, which is the same,

266

8 Electromagnetic Shock Waves

Fig. 8.10 Oscillogram of the current at a finite distance from the short-circuited end of the transmission line for nonlinear dependence (I) [4, 27]

shock wave reflection from a closed end of the transmission line [27]. We shall note that as a result of reflection, the current I at the end of the line can increase greatly: its ratio to the current of an incident shock can be significantly greater than the value 2, which is maximum for the linear case. This possibility was checked experimentally (ref. Fig. 8.10) where the first jump corresponds to the incident EMSW and the second jump—to the reflected EMSW at some distance from the end [27]. Let us now touch upon boundary value problems for nonlinear nondispersive lines [4]. Above, we mainly considered cases above when the line is either infinite (semi-infinite) or so long that the perturbation does not affect its boundaries within the time interval of interest. In the problems intended to find continuous solutions of the system (8.3), a significant role is played by Riemann invariants—J ∓ functions  preserved in simple waves (according to (8.5a), for the system (8.3) J ∓ = U ∓ ρdI). The use of Riemann invariants permits determining the number of values and the range of their assignment, necessary for the correct setting of the boundary value problem [3]. Its solution can be reduced to the integration of a linear equation with variable coefficients [3]. To do this, we need to “invert” Eq. (8.3) by multiplying them by the Jacobian ∂(z, t)/∂(I, U). Considering now I, U as independent variables and introducing the function χ (IU), according to the formulas t = ∂ 2 χ /∂I, C z = − ∂χ /∂U, we will obtain the wave equation t=

1 ∂ 2χ ∂χ − = 0. ∂U 2 ρ 2 (I ) ∂ I 2

(8.17)

In another variant of the method, the Riemann variables J ∓ are deemed independent, then (8.3) results in the canonical form [4]:   1 ∂ L ∂t ∂t ∂ 2t =0 − + √ ∂ J+ ∂ J− ∂ J− 8L L ∂ I ∂ I+

(8.18)

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267

(or similar for z). For some approximations of the functions L(I), Eqs. (8.17), (8.18) can be integrated for various boundary and initial conditions. In particular, the Cauchy problem is solved for (8.18) [4]. Equation (8.17) or (8.18) contains all the integrals of the initial system except for simple waves. Indeed, according to (8.5a), I and U in a simple wave are bound by one-to-one relationships; in this case, ∂(z, t)/∂(I, U) = 0 and the transformation given here is impossible. Consequently, simple waves represent a class of special integrals of the system dividing the areas of analytically different solutions. In particular, in case of no discontinuities, only a simple wave can border on the area of permanent values I, U [7]. Let us analyze the effect of losses on the nonlinear deformation of waves. As indicated in Chap. 1, energy always dissipates at the shock wave front. Therefore, loss accounting, generally speaking, is relevant for the researching shock waves and in any event is compulsory for abandoning the “discontinuous” idealization and studying the shock front structure. Allowance for dissipation leads either to the addition of new members to the system (8.3) or to the substitution of quasi-static link (I) with a more complex differential or integral–differential dependence. In this case, independent parameters appear in Eq. (8.3) with time dimensionality, and the wave process nature depends on the relationship between its characteristic time scale and the value of the given parameters. In other words, losses inevitably introduce some dispersion. From this point of view, we can identify the various types of dissipative processes: the specific features of their effect on the nonlinear deformation of a wave are explained in the review [4]. Structure of Stationary EMSW. Shock waves with an infinitely low front duration, described by discontinuous solutions of the telegraph Eq. (8.3) can be formed only if the transmission line has a vanishingly low dispersion within the range of very high (no matter how high, to be more precise) frequencies. The absence of dispersion corresponds, in particular, to the assumption that can be made when studying the problem of discontinuity occurrence in the profile of a simple wave. According to this assumption, the operators (8.4) binding the linear flux Φ and the linear charge Q with the current I and the voltage U in Eq. (8.3) can be replaced with quasi-static constraint equations that can be written as one-to-one functions  = (I), Q = Q(U). However, due to the inertance of polarization and magnetization (for example, due to the fading precession of the magnetization vector occurring at high rates of magnetic polarity reversal (8.1)) and in some cases due to the structural specific features of the transmission line (the discreteness of the parameters of the multilink “artificial” line manifested, for example, at high frequencies), there is always a certain limiting frequency starting from which the derivatives play a significant role in voltage and current dispersion in operators (8.4) with respect to the longitudinal coordinate z and time t. The telegraph equations in this approximation no longer have discontinuous solutions and the total duration and specific features of the EMSW structure are defined by the specific form of operators (8.4), i.e., at long last, by

268

8 Electromagnetic Shock Waves

the nonlinearity and dispersion (in particular, dispersion related with dissipative processes) of the transmission line. Allowance for dispersion in the area of high frequencies increases, generally speaking, the order of the equations describing wave propagation and extremely complicates their general study. Only in case of special assumptions on the nature of dispersion and nonlinearity, we can keep track of the deformation of a travelling wave and the formation in its profile of relatively abrupt drops which correspond to shock waves. However, in a general case, a solution of the problem is substantially facilitated by the fact that the shock wave duration is low, by definition, as compared with the scales characterizing the change in voltage and current beyond an abrupt drop. This makes it possible to consider the process of formation and development of a shock wave in the “discontinuous” approximation, neglecting high-frequency dispersion and using boundary conditions (8.7a) excluding the area of fast change in voltage and current and then to study the structure of that area, i.e. the structure of the shock wave itself, considering the process to be stationary (the same method was already used, in particular, in the previous section in solving the problem of the effects of high-frequency losses). In a stationary wave whose velocity vp is constant and determined by the boundary conditions (8.7a), all values depend on only a single variable ξ = t − z/vp ; this does not only facilitate the study of the solutions of telegraph equations at relatively general assumptions, but also makes it possible to judge of the structure and duration of shock waves even in those cases when the problem cannot be reduced to the equations of types (8.3), (8.4). For transmission lines with time dispersion and in cases when stationary EMSW in the transmission line with time and spatial dispersion are described by the system of second-order equations, the conditions (8.15) are sufficient if the positions of equilibrium corresponding to the “foot” and top of the shock wave are closest to the shock adiabat, i.e. not divided by other equilibrium positions [4]. This conclusion is apparently also true in the general case, since the requirement for the proximity of special points complying with the initial and final values of voltage and current in a shock wave correspond to the condition of stability (Chap. 12) of a discontinuous solution (in the approximation that does not take into account dispersion) relative to perturbations of finite amplitude. The discontinuity in which the initial and final states are not closest in the above sense splits into the respective number of jumps satisfying this requirement (Fig. 8.11) [4]. Although the conditions for the existence of a stationary shock wave, as noted above, do not depend on the dispersion nature in the area of high frequencies (short wave lengths), its overall duration and structure are determined by operators , Q the specific form of which is directly related with the dispersion properties of the transmission line. A complete qualitative and even more so quantitative study of the EMSW structure is possible only in the simplest cases when the system is reduced to one or two differential equations of first order; the application of numerical methods to study stationary EMSW is also related with certain difficulties since they are described by the special trajectories of these equations. However, some considerations about the structure of the “foot” and “top” of a shock wave can be expressed by analyzing the equations linearized in the vicinity of

8 Electromagnetic Shock Waves

269

Fig. 8.11 Breakdown of discontinuity the initial and final points of which are not closest on the curve (I) [4]

the respective special points. The characteristic equations for the complex frequencies ω that define the phase trajectories near these points can be written as a condition of () equality between the shock-wave propagation velocity and the phase velocities v1,2 of low-amplitude waves (with complex wave numbers β 1,2 and complex frequencies ω) ahead of the EMSW front (index 1) and behind it (index 2):  = v1,2

ω = vp. β1,2 (ω)

Let us note that since the phase trajectory defining the stationary shock wave must leave point 1 and come to point 2, it can be defined by those roots only for which Imω1(k) > 0, Imω2(k) < 0. In other words, the waves excited at the EMSW front must decay when moving away from it. The given considerations on the EMSW front duration and structure of its “foot” and “top” can be extended not only to complex systems (consisting of several transmission lines with the finite critical frequency related with the main line) but also to waveguides in which changes in linear flux and charge are described by nonone-dimensional equations (for example, in lines with dispersion conditioned by a dielectric or magnetic heterogeneous in cross-section) as well as waveguides that are not described at all by telegraph equations (for example, lines with discrete parameters). In such transmission lines, propagation of waves of several types is possible along with excitation at the initial and final sections of the EMSW oscillations with complex oscillation frequencies. Main Types of EMSW Structures [4]. For transmission lines with rather simple dispersion laws, the systems of differential equations [4] describing stationary shock waves have a low order (one or two first-order equations). A complete qualitative

270

8 Electromagnetic Shock Waves

Fig. 8.12 Scheme of the simplest line with time dispersion [4]

study of such system is quite possible. Consideration of such particular examples allows for not only illustrating the main types of EMSW (classified according to the type of “foot” and “top”) and their relation with the dispersion characteristics of the transmission line but also for studying the details of the structure and the duration of the shock wave front. Let us consider briefly transmission lines with time dispersion. The simplest example of a transmission line with time dispersion can be a line the scheme of which is given in Fig. 8.12 [4]. When decay is decreased (resistance value R), the area of substantially nonharmonic oscillations grows, and they approach the set of solitary waves (solitons) being a precise solution in lines without dissipation (separatrix in Fig. 8.13 [4]). Lines with spatial dispersion are of specific interest. In case of a non-local link of the linear flux  and the charge Q with current and voltage, the group velocity of perturbations excited in the area of EMSW front can be higher than the velocity vρ of the shock wave itself as a result of which oscillations occur at the leading section of its front (at the “foot”). This situation can take place in the transmission line depicted in Fig. 8.12 [4] provided there is an inductive coupling between its cells. Another example of the transmission line with spatial dispersion can be multilink “artificial” lines most frequently used in experimental works on EMSW. In such lines (described by differential-difference equations), the spatial dispersion is caused by the structure periodicity; such dispersion can be approximately (for not too fast processes) taken into account by replacing difference equations with differential ones containing finite-order derivatives with respect to the longitudinal coordinate [4]. The study of these approximate equations describing the structure of EMSW “foot” and “top” shows that in this case as in the lines with time dispersion, oscillations can be excited only near the “top”, i.e. behind the EMSW front. The solution of the problem

8 Electromagnetic Shock Waves

271

Fig. 8.13 The phase plane of the equations for stationary waves and the EMSW structure in the line with time dispersion (Fig. 8.12): a in case of strong decay; b in case of weak decay; c in the line without dissipation; d the EMSW front structure corresponding to case b [4]

using numerical methods [4] and a number of experiments [4] confirm that the effect of the spatial dispersion related with the discreteness of the line parameters on the formation of the EMSW structure is similar to the effect of time dispersion. Lines with slow excitation of internal degrees of freedom have their interesting specific features. If the transmission lines have elements characterized by substantially different time constants, the constraint equations may contain a small parameter μ with the highest derivative. The processes occurring at the shock wave front in such lines can be in some cases divided into fast and slow. In this case, fast processes manifest themselves as discontinuities in the solution of the approximate (μ = 0)

272

8 Electromagnetic Shock Waves

equations of slow movements, and the change in slow variables in the study of fast processes can be neglected. A similar in this sense situation is known in gas dynamics (ref. Chap. 1) and plasma dynamics (ref. Chap. 7). Let us note that the use of one or another idealization in the generation of equations is in most cases related with the neglect of higher derivatives. Therefore, it is important to set a criterion that allows judging the appearance of fast processes based on the analysis of “slow” (μ = 0) equations and clarifying the position of jumps in the shock wave profile described in this approximation. It is shown in [4] that, in the general case, “fast” processes (jumps in discontinuous approximation) can be excited on a comparatively wide shock wave front if there are the joint surfaces of the phase trajectories of slow movement equations are present between the equilibrium positions corresponding to the initial and final EMSW states. Depending on the nonlinear characteristics of the line parameters, there can be several such surfaces. But always (for stable shock waves) (ref. Chap. 12) it turns out to be possible to build the unique discontinuous solution of the approximate equations corresponding to a stationary shock wave. It is essential that the minimum number of jumps in this solution and some specifics of the location of their initial and final points can be defined if only initial equations and the instability order of the equilibrium positions of the slow-motion equations are known [4]. Systems with active elements [29–36] the presence of which can lead to an increase in the wave energy are also an interesting class of objects. Let us consider the line shown in Fig. 8.14 [4] and containing a nonlinear active element along with nonlinear inductance. Voltage-current characteristic of the latter I R (U R ) (corresponding to a tunnel diode, for example) has a falling part (Fig. 8.14b). It should be noted that processes which are qualitatively different from EMSW are possible in the active system, in particular, processes of self-oscillatory type [4]. However, shock waves can also propagate in such systems under certain initial and boundary conditions. A specific feature of shock waves in this case is related with the ambiguity of the dependence Q(U ) = C1 (U − U Ri ) + C2 U, where U Ri (i = 1, 2, 3) are U R values for which I R = 0. Let us now consider waves in an unbounded nonlinear medium. Problems that take into account a vector nature of the EMSW field in free space are so far poorly related to the experiment due to no weak-dispersing nonlinear materials of a sufficiently high volume. However, such problems are of certain physical interest. Moreover, many of them permit modeling in bounded lines, and the situation with nonlinear materials constantly changes. Below, we consider waves in an unbounded nonlinear magnetic; if it is required to specify the form of constitutive equations, Eq. (8.1) is used, which describes the precession of magnetization in saturated ferrite. Simple waves are considered in [4, 32]. First let us consider the one-dimensional process—a plane wave propagating along the z axis. Maxwell equations for such a wave in ferrite lead to the following:

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273

Fig. 8.14 a Scheme of a line with active element; b voltage-current characteristic of the active element (dependence IR(UR))

∂ 2 Hτ ∂ 2 Bτ = , ∂z 2 ∂t 2

Bz = const,

(8.19)

where H τ , Bτ are field components in the xy plane. If we consider the change in the field to be slow, let us use a quasi-static constraint equation for saturated ferrite: |MH| = 0, |M| = M s = const. Let us find simple waves. Using the well-known methods of analysis of quasi-linear equations, we can show [4] that in this case there are four families of characteristics (two for each of opposite directions of propagation along the z axis). The respective propagation velocities for one pair of families of characteristics are as follows C dz =± , dt + 4π Ms /H )1/2 (1

(8.20a)

 1/2 1 + 4π Ms Hτ2 /H 3 dz = ±C , dt 1 + 4π Ms H

(8.20b)

and for the other pair

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8 Electromagnetic Shock Waves

EMSW

RD (

)

RD (

)

EMSW

Fig. 8.15 Discontinuities resulted from the “collision” of shock waves with various polarization [4]

where H = (H 2 + H 2 )1/2 . In this case, a simple wave propagating along z τ of the characteristic (8.20a) is linearly polarized (H x /H y = const), and a simple  wave complying with (8.20b) has circular polarization Hx2 + Hy2 = Hτ2 = const . Note that the values Hz2 and Hτ2 are related by the ratio   4π Ms = const. Bz = Hz + 4π Mz = H 1 + H

(8.21)

Therefore, the waves of the first type do not differ from those considered above; their deformation leads to the appearance of discontinuities. In a wave of the second type, the velocity is constant and, therefore, the wave whose field arbitrarily rotates around the axis with H 2 = const is not deformed in the quasi-static approximation. In case of small nonlinearity (|Hτ | 4π Ms ), it is possible to solve the problem of wave propagation with the homonymous change in the modulus and polarization direction H. In this case, |H τ | always propagates along characteristics (8.20a, 8.20b), and the change in the rotation angle ϕ of the vector H τ parametrically depends on |H τ | (the propagation velocity corresponds to (8.20b)). The decay and interaction of discontinuities are discussed in [33]. The problem of the interaction of two flat discontinuous fronts parallel to each other is reduced (as for the transmission line) to the problem of the decay of some initial discontinuity occurring when the fronts contact. Let us immediately note a specific feature of a shock wave in free space [4]. The EMSW with a change in the sign H τ is non-evolutionary (taking into account perturbations polarized other than H 1,2 ) and decomposes into a rotational discontinuity (RD) with the rotation of the field by π and an advancing evolutionary shock wave on which the sign Hτ is preserved (in a two-wire line, the number of perturbations is less and the EMSW is always evolutionary). Let us discuss the interaction of discontinuities with various directions of the field using the example of a “collision” of two counter propagating EMSWs with the same moduli Hτ [4]. If the angle ϕ between the vectors Hτ in converging waves is below 120°, then EMSWs go to each direction from the point of interaction and the following rotational discontinuity has the rotation angle ϕ/2 (Fig. 8.15 [4]). Due to symmetry, the problem is fully defined if the values of H and B of the field remaining

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within the area between diverging EMSWs are known (obviously, H, B are the same in areas II and III). If the quasi-static function B(H) is known, then in case of ϕ > 120°, Eq. (8.21) solves the problem unambiguously. If ϕ > 120°, then the interaction pattern differs in that only simple waves of the type (8.20a, 8.20b) propagate ahead of the rotational discontinuities instead of EMSWs. The EMSW structure in saturated ferrite was studied in [5, 15–17]. Just as in transmission lines, the shock wave front structure depends on the dissipative and dispersive mechanisms acting in the medium. The respective problem is relatively simple for a wave propagating in ferrite along the constant field H 0 magnetizing ferrite up to a saturation level [4, 18]. Finding out a stationary (depending on ξ = t − z/vp , where z0 || H 0 ) solution of Maxwell equations supplemented by the equation of homogeneous precession, we obtain a shock front where the magnetization vector M rotates and passes from the longitudinal orientation to some other, which coincides with the direction H 2 = H 0 + H τ 2 (H τ 2 is the transverse field of the wave behind the EMSW front), ω is related with the angle θ of M deviation from the z axis by the ratio:   cos θ −1 , (8.22) ω = γ B0 cos θ2 θ 2 = θ (ξ → ∞). According to (8.22), ω decreases from γ B0 (1 − cos θ 2 )/cos θ 2 ahead of the wave front to zero behind its front. The EMSW front duration τ has the following order τ τ0 M cos

θ2 sin2 θ2 , 2B0

(8.23)

where τ 0 is the ferrite relaxation time. Vector H also rotates; the profile of each transverse components H has oscillations ahead of the front (noticeable at τ 0 γ M > 1). A qualitative analysis of the specific features of the structure can be done for EMSWs propagating at an arbitrary angle to the magnetic field ahead of them.4 In particular, we can trace the relationship between the uniqueness of the EMSW structure and their evolutionism [4]. The oblique incidence of electromagnetic shock waves on the conducting plane was studied in [16]. One-dimensional waves were considered so far (all values depended on one spatial variable). Two-dimensional nonlinear processes that are very interesting physically are less available for experimental observation. We will consider here only one two-dimensional problem—a reflection of an incident shock wave from a conducting plane [16]. Two cases can be naturally distinguished here. 4 Let

us note that the influence of the crystalline anisotropy of ferrite as well as exchange and mangetoelastic effects apparently does not lead to substantial differences in the shock wave structure [4].

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Fig. 8.16 Oblique incidence of a shock wave on the reflecting plane (H is perpendicular to the plane of incidence) [4]

The magnetic field is polarized perpendicular to the plane of incidence. Assuming that the front of the reflected wave remains plane and using the boundary conditions (8.19) and the condition E τ = 0 at the metal boundary, we will obtain the relationship between the absolute values of the fields in areas I and II (Fig. 8.16 [4]):   tgα2 , H2 = H1 1 + tgα1 sin(α1 + α2 ) E2 = E1 cos α2

  2α1 B2 = B1 1 + sin , sin 2α2

(8.24)

(there is no field in area III). For a given relationship B(H), the ratios (8.24) define the angle of reflection α 2 . So, if the field H 1 of the shock wave saturates ferrite, then B = Bs + H and   Bs 1/2 . sin α2 = sin α1 1 + H1

(8.25)

Therefore, α 3 > α 1 . Formula (8.25) loses its meaning at α 1 > α cr , where sin α cr = 1/(1 + Bs /H 1 )1/2 . In case of a larger angle of incidence, the considered reflection pattern is impossible: it can be shown that any configuration of the finite number of plane shock waves is impossible. According to (8.24), with α 1 → α cr the field H 2 grows infinitely. This circumstance can be used to obtain strong magnetic fields. The magnetic field is polarized in the plane of incidence. Then, obviously, H 2 is parallel to the conducting plane and E 2 = 0. When there is no field ahead of the incident wave, the boundary conditions are given for the absolute values of the fields   tgα2 , H2 = H1 cos α1 1 + tgα1

B2 = B1

sin(α1 + α2 ) . sin α2

(8.26)

If the fields H 1 i H 2 saturate ferrite, the reflection angle satisfies the following relation   Bs sin(α1 + α2 ) sin(α2 − α1 ) sin(α2 + α1 ) = −1 . (8.27) sin α1 sin α2 2H1 sin α2

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Fig. 8.17 Oscillogram U(t) of the EMSW front in the line with ferrite [34]. The period of calibration sine wave is 2 ns [4]

Relations (8.26) and (8.27) are true as long as the reflected wave remains shock wave. This takes place if α 1 < α cr and α 1 + α 2 ≤ π /2 (then the tangent components H on the reflected front have the same sign and H 2τ > H 1τ ). If α 1 > α cr , then a rotational discontinuity with the rotating angle π will appear in case of reflection in addition to a shock wave. Let us note that in the area of angles where both Formulas (8.25) and (8.27) are true, the value α 2 for the given α 1 always exceeds the case of (8.25). The interest in electromagnetic shock waves is related not only with the above physical specific features of EMSW or with the possibility of modeling the compressibility motions of medium in electrodynamic systems but also with the prospects for technical applications. EMSWs are widely used in the technique of nanosecond pulses. The maximum current and voltage amplitude and pulse front duration depend on the used nonlinear materials (semiconductors, ferrites, ferrielectrics). For example, the EMSW duration at the front edge of strong pulses currently obtained in lines with ferrite reaches 5 × 10−10 s and less (Fig. 8.17 [34]). However, technical problems associated with EMSW go beyond the scope of this book and their discussion can be found in a number of articles and also in monographs [1, 2]. The paper [35] shows that it is possible to observe the reverse Doppler effect in an electrodynamic system, such as coupled power transmission lines with various types of dispersion. The conditions are found when a radio pulse reflected from the escaping front of the electromagnetic shock wave will have a higher filling frequency than in the radio pulse being incident on it, which is expressed as a reverse Doppler effect (Fig. 8.18) [36]. Systems are proposed where the reverse Doppler effect should manifest itself. Almost 60 years after the theoretical predictions of this effect, [37] presents the results of experimental recording of the reverse Doppler effect (Fig. 8.19 [37]). This astonishing and unexpected phenomenon was recorded in electrical transmission lines with dispersion and strong nonlinearity.

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8 Electromagnetic Shock Waves a

Incident wave

Reflected wave

b

Moving boundary

Incident wave

Reflected wave

Moving boundary

Fig. 8.18 Wave advancing beyond the receding discontinuity a wave in a medium with normal dispersion, b wave in a medium with anomalous dispersion, ω—frequency, vp —group velocity [36]

Fig. 8.19 Normal and abnormal Doppler signals [37]

References

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References 1. Fortov VE (2013) High energy density physics. FIZMATLIT, Moscow [Fortov V.E. Fizika vysokikh plotnostey energii. — M.: FIZMATLIT, 2013 (in Russian)] 2. Fortov VE (2016) Extreme states of matter. High energy density physics, 2nd edn. Springer, Heidelberg, New York, London 3. Zababakhin EI. Unbounded cumulation phenomena. In: Sedov LI (ed) In the collection Mechanics in the USSR for 50 years, vol 2, p 314 [Zababakhin E.I. Yavleniya neogranichennoy kumulyatsii. V sb. «Mekhanika v SSSR za 50 let». T. 2 / Pod. Red. L. I. Sedova. S. 314 (in Russian)] 4. Gaponov AV, Ostrovsky LA, Freidman GN (1967) Radiophysics X(9–10) [Gaponov A.V., Ostrovsky L.A, Freidman G.N. / Radiofizika. 1967. T. Kh, № 9–10 (in Russian)] 5. Gaponov-Grekhov AV, Freidman GN (1949) JETP 36:957 [Gaponov-Grekhov A.V., Freidman G.N. / ZHETF. 1949. T. 36. S. 957 (in Russian)] 6. Rosen J (1965) Phys Rev 139(2A):539–543 7. Landau LD, Lifshits EM (1986) Hydrodynamics. Nauka, Moscow [Landau L.D., Lifshits E.M. Gidrodinamika. — M.: Nauka, 1986 (in Russian)] 8. Zel’dovich YaB, Raizer YuP (2008) Theory of shock waves and high-temperature hydrodynamic phenomena, 3rd edn., corrected. FIZMATLIT, Moscow [Zel’dovich Ya.B., Raizer Yu.P. Teoriya udarnykh voln i vysokotemperaturnykh gidrodinamicheskikh yavleniy. 3-ye izd., ispr. — M.: FIZMATLIT, 2008 (in Russian)] 9. Zel’dovich YaB, Raizer YuP (1946) Theory of shock waves and introduction into gas dynamics. Ed. of the USSR Academy of Sciences, Moscow-Leningrad [Zel’dovich, Ya.B., Raizer, Yu.P. Teoriya udarnykh voln i vvedeniye v gazodinamiku. — M.–L.: Izd. AN SSSR, 1946 (in Russian)] 10. Raizer YuP (2011) Introduction into fluid and gas dynamics and theory of shock waves for physicists. Intellect Publishers, Dolgoprudny [Raizer Yu.P. Vvedeniye v gidrogazodinamiku i teoriyu udarnykh voln dlya fizikov. — Dolgoprudnyy: ID «Intellekt», 2011 (in Russian)] 11. Landau LD, Lifshits EM (1959) Electrodynamics of continuous media. Fizmatgiz, Moscow [Landau L.D., Lifshits E.M. Elektrodinamika sploshnykh sred. — M.: Fizmatgiz, 1959 (in Russian)] 12. Kulikovsky AG, Lyubimov GA (1962) Magnetohydrodynamics. Fizmatgiz, Moscow [Kulikovsky A.G., Lyubimov G.A. Magnitnaya gidrodinamika. — M.: Fizmatgiz, 1962 (in Russian)] 13. Polovin RV (1960) Shock waves in magnetohydrodynamics. Phys Usp 72:33 [Polovin R.V. Udarnyye volny v magnitnoy gidrodinamike / UFN. 1960. T. 72. S. 33 (in Russian)] 14. Ostrovsky LA (1959) Interaction of weak signals with electromagnetic shock waves. Gazette of Universities. Radiophysics 2:833 [Ostrovsky L.A. O vozdeystvii slabykh signalov s elektromagnitnymi udarnymi volnami / Izvestiya vuzov. Radiofizika. 1959. T. 2. S. 833 (in Russian)] 15. Yulpatov VK (1959) Graduation thesis. Gorky University [Yulpatov V.K. / Diplomnaya rabota. — Gor’kovskiy universitet, 1959 (in Russian)] 16. Gaponov A, Ostrovsky L, Freidman G (1960) Shock electromagnetic waves. In: XIII general assembly URSI, London 17. Gaponov AV, Freidman GN (1960) To the theory of electromagnetic shock waves in nonlinear media. Gazette of Universities. Radiophysics 3:79 [Gaponov A.V., Freidman G.N. K teorii udarnykh elektromagnitnykh voln v nelineynykh sredakh / Izvestiya vuzov. Radiofizika. 1960. T. 3. S. 79 (in Russian)] 18. Landauer R (1960) Shock waves in nonlinear transmission lines and their effect on parametric amplification. IBM J Res Dev 4:391 19. Gyorgy TM (1958) Modified rotational model of flux reversal. J Appl Phys 29:1709 20. Gorodetsky AF, Kravchenko AF (1967) Semiconductor devices. Higher School, Moscow [Gorodetsky A.F., Kravchenko A.F. Poluprovodnikovyye pribory. — M.: Vyssh. shkola, 1967 (in Russian)]

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21. Belyantsev AM, Gaponov AV, Daume EYa, Freidman GI (1964) Experimental study of the propagation of electromagnetic waves of finite amplitude in waveguides filled with ferrite. JETP 47:1699 [Belyantsev A.M., Gaponov A.V., Daume E.Ya., Freidman G.I. Eksperimental’noye issledovaniye rasprostraneniya elektromagnitnykh voln konechno amplitudy v volnovodakh, zapolnennykh ferritom / ZHETF. 1964. T. 47. S. 1699 (in Russian)] 22. Belyantsev AM, Ostrovsky LA (1962) Propagation of pulses in lines with nonlinear semiconductor capacitances. Gazette of Universities. Radiophysics 5:183 [Belyantsev A.M., Ostrovsky L.A. Rasprostraneniye impul’sov v liniyakh s nelineynymi poluprovodnikovymi yemkostyami / Izvestiya vuzov. Radiofizika. 1962. T. 5. S. 183 (in Russian)] 23. Katayev IG (1963) Electromagnetic shock waves. Soviet Radio, Moscow [Katayev I.G. Udarnyye elektromagnitnyye volny. — M.: Sov. radio, 1963 (in Russian)] 24. Ostrovsky LA (1961) Electromagnetic waves in heterogeneous nonlinear medium with low losses. Gazette of Universities. Radiophysics 4:955 [Ostrovsky L.A. Elektromagnitnyye volny v neodnorodnoy nelineynoy srede s malymi poteryami / Izvestiya vuzov. Radiofizika. 1961. T. 4. S. 955 (in Russian)] 25. Ostrovsky LA (1963) Formation and development of electromagnetic shock waves in transmission lines with non-saturated ferrite. Tech Phys J 33:1080 [Ostrovsky L.A. Obrazovaniye i razvitiye udarnykh elektromagnitnykh voln v liniyakh peredachi s nenasyshchennym ferritom / ZHTF. 1963. T. 33. S. 1080 (in Russian)] 26. Gutzwiller MC, Miranker WL (1963) Nonlinear wave propagation in a transmission line loaded with thin permalloy films. IBM J Res Dev 7:278 [kcpp. infopm. vyqiclit. texn. 1964. № 14] 27. Ostrovsky LA (1963) Reflection of electromagnetic shock waves from the short-circuited end of a transmission line with ferrite. Gazette of Universities. Radiophysics 6:413 [Ostrovsky L.A. Otrazheniye udarnykh elektromagnitnykh voln ot korotkozamknutogo kontsa linii peredachi s ferritom. Izvestiya vuzov. Radiofizika. 1963. T. 6. S. 413 (in Russian)] 28. Whitham GB (1965) Nonlinear dispersive waves. Proc R Soc Ser A 283:238 29. Gaponov AV, Ostrovsky LA, Rabinovich MI (1965) Electromagnetic waves in non-linear transmission lines with active parameters. In: URSI symposium, Delft; Electromagnetic wave theory, Pergamon Press, 1967 30. Ostrovsky LA (in print) Electromagnetic shock waves in nonlinear active lines. Gazette of Universities. Radiophysics [Ostrovsky L.A. Udarnyye elektromagnitnyye volny v nelineynykh aktivnykh liniyakh // Izvestiya vuzov. Radiofizika (v pechati) (in Russian)] 31. Bogatyrev YuK, Ostrovsky LA, Papko VV (in print) Studies of electromagnetic shock waves in nonlinear active lines. Gazette of Universities. Radiophysics [Bogatyrev Yu.K., Ostrovsky L.A., Papko V.V. Issledovaniya udarnykh elektromagnitnykh voln v nelineynykh aktivnykh liniyakh // Izvestiya vuzov. Radiofizika (v pechati) (in Russian)] 32. Hatfield WB, Auld BA (1963) Electromagnetic shock waves in gyromagnetic media. J Appl Phys 34:2941 33. Ostrovsky LA (1965) Rotational explosions in the electrodynamics of nonlinear media. Gazette of Universities. Radiophysics 8:738 [Ostrovsky L.A. Vrashchatel’nyye vzryvy v elektrodinamike nelineynykh sred / Izvestiya vuzov. Radiofizika. 1965. T. 8. S. 738 (in Russian)] 34. Belayntsev AM, Bogatyrev YuK, Solovyeva LI (1963) Stationary electromagnetic shock waves in transmission lines with unsaturated ferrite. Gazette of Universities. Radiophysics 6:561 [Belayntsev A.M., Bogatyrev Yu.K., Solovyeva L.I. Statsionarnyye elektromagnitnyye udarnyye volny v liniyakh peredachi s nenasyshchennym ferritom / Izvestiya vuzov. Radiofizika. 1963. T. 6. S. 561 (in Russian)] 35. Belyantsev AM, Kozyrev YuK (2002) Tech Phys J 72(11):133–136 [Belyantsev A.M., Kozyrev Yu.K. / ZHTF. 2002. T. 72, vyp. 11. S. 133–136 (in Russian)] 36. Seddon N, Bearpark T. Science 302:1489–1537 37. Belyantsev AM, Kozyrev AB (2001) Tech Phys J 71(7):79–82 [Belyantsev A.M., Kozyrev A.B. ZHTF. 2001. T. 71, vyp. 7. S. 79–82 (in Russian)]

Chapter 9

Nuclear Shock Waves

Energy densities and their corresponding extremely high intensities of shock waves which are maximum of those available in terrestrial conditions are reached today in the collision of relativistic heavy ions [1, 2]. The accelerators [3] required for this purpose operate in several laboratories throughout the world and are well known as the main experimental tool in research in the field of nuclear physics, elementary particle physics, quantum chromodynamics, and superdense nuclear matter physics [4–9], i.e. in the areas which have always been at the forefront of modern natural science. At the same time, progress in research requires constant advance into the domain of ever-increasing energies, increase in the phase density of beams of accelerated particles and, therefore, increase in the power of shock waves. Research in the field of high energy physics and relativistic nuclear physics makes it possible to establish that the principal laws controlling the motion and interaction of elementary particles are unusual and simple [1, 2]. These laws are based on considerations of symmetry asserting that everything that does not contradict symmetry is actually permitted, can and must occur. To the fullest extent, this is implemented in gravitational, electromagnetic, weak and strong interactions of particles. The science and technology of particle accelerators have come a long way from the first cyclotron with a proton energy of 1.2 meV, created by E. Lawrence in 1932 to the Large Hadron Collider (LHC) (Fig. 9.1), built at the European Centre for Nuclear Research (CERN), accelerating the protons to a velocity of only a millionth of a percent lower than the light velocity and having a colliding-beam energy of 7,000,000 meV each, which is 7000 times greater than the rest energy of a proton, mp c2 . In the frame of reference associated with the center of mass this corresponds to an energy of colliding protons of ≈ 14 TeV. Since then, the world has seen the construction of dozens of accelerators of various types, which are giant electrical facilities incorporating cutting-edge engineering ideas and exhibiting a high degree of reliability. Today, they are unsurpassed recordbreakers in high energy density physics [1, 2].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Fortov, Intense Shock Waves on Earth and in Space, Shock Wave and High Pressure Phenomena, https://doi.org/10.1007/978-3-030-74840-1_9

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Fig. 9.1 Schematic representation of the Large Hadron Collider (LHC) at the European Centre for Nuclear Research (CERN). Its underground tunnel measures about 27 km in diameter. Shown at the top are the main LHC detectors: ALICE, ATLAS, CMS, and LHC-B [1]

The LHC accelerator complex is being employed to collide two proton beams with an energy of 7 × 7 TeV to reach the new domain of distances of 10−16 cm and energies on the 1 TeV scale, which are sufficient, in particular, for the production of the particles of dark matter (their mass mDM ≈ 10 GeV–1 TeV), the Higgs boson, obtaining of a quark-gluon plasma, perhaps for discovering new dimensions, and for the solution of other intriguing problems of high energy physics [10, 11]. The main goal of these experiments [12] is to reveal the mechanisms of violation of electroweak symmetry by recording the Higgs boson and other new particles associated with the possible expansion of the Standard Model. Protons on LHC move as 3000 bunches distributed along the entire 27 km circumference of the collider. Each bunch containing 100 billion protons have a length of several centimeters and a diameter of 16 μm (as a thin human hair) at the collision points. The total of 2808 bunches, 100 billion protons each, collide in detector locations and generate more than 600 million proton collisions of particles per second (up to 20 collisions in case of crossing). These collisions take place between the particles the protons are made of—quarks and gluons. In case of the maximum energy of the particles, approximately one seventh of the energy contained in initial protons, or about 2 TeV, is released. Four systems of giant detectors—the largest one would occupy half of the NotreDame Cathedral in Paris and the heaviest one contains more iron than the Eiffel Tower—will measure the parameters of the thousands of particles that separate at

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Fig. 9.2 RHIC accelerator of Brookhaven National Laboratory [1]

each collision. Despite the huge size of the detectors, the individual elements must be assembled with an accuracy of 50 μm. Later on, the detectors will be used to study the processes in the collision of highly ionized lead ions (Pb82 +) with an energy of up to 155 GeV per nucleon. The Stanford linear accelerator (USA) generates a 5 ps pulse of 10 electrons with a kinetic energy of 50 GeV, which is focused to a spot 3 μm in size to provide a power density of 1020 W/cm2 . The actively operating Relativistic (99.99% of the velocity of light) Heavy Ion Collider (RHIC) (Fig. 9.2) at Brookhaven National Laboratory (USA) provides the energy of colliding gold ions up to 100–500 GeV per nucleon, 39 TeV for Au + Au, 13 TeV for Cu + Cu in the frame of reference associated with the center of mass [12–14]. About 5000 elementary particles are produced at each frontal collision. But only a few of them carry the necessary information. The new experimental data obtained on this accelerator are discussed in [13, 14]. A unique ion and antiproton accelerator FAIR providing an energy of 1.5–34 GeV per nucleon, with the number of accelerated ions U92+ ≈ 5 × 1011 and antiprotons ≈ 4 × 1013 operates in Darmstadt (Germany). The cost of building of each of these largest ultra-relativistic hadron accelerator complexes (LHC, RHIC, FAIR) is several billion dollars and is on the verge of economic opportunities for the world’s richest countries and even for such an international community as the European Union.

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The scientific programs implemented at these complexes involve an experimental research into the fundamental problems of high energy physics in hadron collisions, which are accompanied by the formation of intense shock waves and superdense nuclear matter, i.e. a quark–gluon plasma. In accordance with modern concepts, this was precisely the state of the Universe’s matter during the first microseconds after the Big Bang, and also the state of the matter of such astrophysical objects as gamma-ray bursts, supernova neutron stars, and black holes. For our consideration, it is important that these acceleration experiments are aimed at producing the particle beams of ultra-relativistic energies not only to investigate individual hadron collision events [14], but also to generate super-power shock waves and macroscopic heating of matter [1, 15, 16]. Generation of Macroscopic Hot Plasma Volumes. The methods of gas-dynamic acceleration of condensed strikers, as described in Chaps. 3 and 4, have a significant drawback arising from the limited value of the velocity of sound in the driving gas, as a result of which the acceleration efficiency sharply (exponentially) decreases when the accelerated striker reaches the velocity of sound. Methods for generating high energy densities based on the use of high-intensity fluxes of charged particles—electrons, light or heavy ions—as well as electrodynamic acceleration methods, where the velocity of light fulfills the role of the sonic barrier, are devoid of these limitations. An important positive feature of the beams of charged particles is the volume character of their energy release [16, 17]. This distinguishes them from laser irradiation (Chap. 5), where the main energy release ofradiation with the frequency ωl occurs in a narrow critical zone [18, 19], ωl ~ ωa ~ 4π e4 n e /m e , and then is transferred into the depth of the target by electronic thermal conductivity [18, 19]. As a result of deceleration of charged particles, a layer of isochorically heated plasma emerges, which subsequently expands to generate a shock wave traveling into the depth of the target or a cylindrical shock wave diverging from the beam axis. Modern studies in the field of high energy density physics take advantage of both of these methods, i.e. isochoric heating and compression by shock waves generated by corpuscular beams. Either cyclotrons developed for the study of high energy physics and nuclear physics [17], or high-current diode systems [20, 21] are used as generators of corpuscular beams. In the latter case, we are dealing with subnanosecond current pulses of the megaampere range with a kinetic particle energy of 1–20 meV [22, 23]. Relativistic electron beams with an energy of the order of megaelectronvolt [24] were used to excite shock waves in aluminum targets in order to study the features of electron absorption in a dense plasma and to elucidate the effect of the intrinsic magnetic fields of the beam on its stopping power in a magnetized plasma (“magnetic stopping” effect) [1, 2]. Owing to the substantially shorter paths of ions in comparison with electrons, ion beams make it possible to obtain higher energy densities in matter. In a series of papers [20], the high-current pulsed accelerator KALIF generated a specific power density on a target of 1012 W/cm2 in a proton beam with an energy of about 2 meV and a current of the order of 400 kA. This made it possible to accelerate thin (50–100 μm) strikers to velocities of 12–14 km/s, to perform the informative measurements of

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285

Spallation strength, kbar

Theoretical limit

Molecular dynamics

Statistics Protons Explosive

Lasers

Deformation rate, s-1

Fig. 9.3 Spallation strength of aluminum-nickel alloy as a function of deformation rate [1, 3, 17]

the stopping power of fast protons in a dense plasma, to record the thermodynamic parameters and viscosity of shock-compressed plasma and to determine the spallation strength of metals at record-high deformation rates. It turned out (ref. Chap. 4, [25]), for example, that the spallation strength of metals noteworthily increases (by one or two orders) with the increase in the deformation rate to approach its theoretical limit (ref. Fig. 9.3), which is related to the propagation kinetics of dislocations and cracks in the field of pulsed stresses. Previously, the high-current pulsed generator BPFA-II was used to generate megavolt beams of lithium ions with an intensity of the order of 1012 W/cm2 for the inertial confinement fusion program [1, 2]. Now this unit is successfully used in the mode of a high-current Z-pinch for the generation of “soft” X-rays and for the electrodynamic generation of shock and adiabatic compression waves [25–28]. The relativistic heavy ion accelerators constructed for experiments in high energy physics turned out to be promising candidates for controlled thermonuclear fusion with inertial plasma confinement and for experiments on the compression and heating of dense plasma (28, 31) [1, 15]. The Large Hadron Collider (CERN) (Fig. 9.1) constructed to study the collisions of two proton beams with an energy of 7 TeV each, generates 2808 proton bunches having a duration of 0.5 ns, with 1.1 × 1011 protons in each bunch spaced at 25 ns, so that the total beam duration is 89 μs, and the energy is 350 MJ, sufficient for evaporation of 500 kg of copper. The energy density in one beam is 1010 J/cm3 . The characteristic kinetic energy of one heavy relativistic ion is comparable to the kinetic energy of a metal liner accelerated by the explosive products of the explosive launching system described in Chap. 3.

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PHELIX petawatt laser

Beam of heavy ions

Fig. 9.4 Schematic representation of the HIHEX experiment of the FAIR project using a relativistic heavy ion beam and a petawatt laser [3, 17, 28, 29]

Heavy-ion beams with a kinetic energy of 3–300 meV per nucleon were used in experiments on heating condensed and porous targets, measuring the stopping power of ions in a plasma, and also on the interaction of charged beams with shock-compressed plasma obtained with the use of miniexplosive-driven shock tubes [1–3, 28–32]. Of particular interest is the use of the heavy ion accelerator at GSI in combination with the petawatt high-power laser system PHELIX (Fig. 9.4), which qualitatively extends the experimental capabilities of such a facility. The potential of and prospects for the application of accelerator complexes at GSI in Darmstadt (Germany) are shown in Figs. 9.5 and 9.6. One can see that highintensity relativistic heavy-ion beams are interesting candidates for the generation of high energy density plasma and for pulsed thermonuclear fusion [1, 16]. Relativistic Collisions of Nuclei. As we noted above, record-high pressures in shock wave and energy densities are achieved today in case of the frontal collision of heavy ions (Figs. 9.7 and 9.8 [33]) accelerated in accelerators to sublight velocities. These collision experiments form the basis of experimental studies of fundamental problems in high energy physics, elementary particle physics and relativistic nuclear physics. These collisions of hadrons are accompanied by the formation of a superdense nuclear matter—quark-gluon plasma (QGP). In CERN and Brookhaven, accelerators were used to carry out unique experiments for the generation (in individual acts) of collisions of heavy nuclei Cu–Cu and Au–Au of ultra-extreme baryonic matter in a superdense and heated state with a density of about 1015 g/cm3 , a pressure of 1030 bar, a temperature of about 200 meV and a specific energy density of 1 GeV/cm3 (Figs. 9.2, 9.4 and 9.5). Experiments at RHIC included collisions of iron nuclei with a specific energy of 100 GeV/nucleon (nucleon weight of 1 GeV), which resulted in a heating temperature of 300–600 meV.

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287 Limit for explosives

Current limit for Z-pinches

Nuclear explosions and lasers

Limit of light-gas guns

Pressure, GPa

Melting

Liquid + Gas

Entropy, J/g K

P, GPa

Fig. 9.5 Possibilities of SIS18, SIS100, SIS300 heavy ion accelerators for the generation of shock waves and high energy densities in lead [3, 17, 28, 29]

Plasma

Gas

Fig. 9.6 Sections of the zinc phase diagram, accessible to heavy-ion generators [13, 17, 28, 29]

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9 Nuclear Shock Waves

Collision

Formation of quark–gluon plasma

Recombination

Expansion at almost the velocity of light

Formation of hadrons

Detector

Fig. 9.7 Dynamics of collisions of relativistic heavy nuclei on accelerators [11]

The emergence in the laboratory of an area with extremely high parameters during ion collisions and its further expansion is called a ‘Small Bang’ the analogies of which with the Big Cosmologic Bang are the subject of special analysis [34]. The internal structure and interaction of particles and nuclei cause a heightened interest and are the main objects of study at modern accelerators [34]. Let us emphasize the idea-driven affinity of experiments with the relativistic collisions of nuclei and shock waves discussed in other sections of the monograph and occurring from “ordinary” non-relativistic collisions. In both cases, strikers (ions, nuclei) are relatively slowly accelerated and then impact the target at high velocity. Intense shock waves formed in this case cause the compression and irreversible heating of matter in the front of a viscous compression shock wave. In accordance with modern ideas, strong interactions are described by quantum chromodynamics that considers quarks and gluons as elementary objects (partons) responsible for interaction. In case of low energies, partons are held inside strongly interacting particles (hadrons). The matter formed in collisions (Figs. 9.7 and 9.8) is assuredly different from the medium that occurs in relativistic interactions. Strong internal fields and highly coherent parton configurations become especially important. In case of collision of two heavy nuclei at ultra-relational energies, we expect a hot and dense colored medium to occur. This medium must exhibit some collective properties that are different from those predicted in static conditions. Studying matter properties in relativistic nuclear collisions will always be the most advanced area of natural science [34–36]. According to H. A. Bethe, in the twentieth century, humanity spent more intellectual efforts for studying nuclear forces and the respective processes than for studying all other scientific disciplines taken together. According to [13, 37, 38], let us briefly give the primary characteristics of modern accelerators. Leading global scientific centers that use accelerators: • GSI, Germany, E lab ≈1–2 AGeV, which corresponds to nucleon-nucleon energy in the center of mass s N N ≈ 2–2.5 GeV;  • CERN, SPS, Switzerland/France, E lab ≈ 20–158 AGeV, s N N ≈ 6–17 GeV;  • BNL, RHIC, USA, s N N ≈ 20–200 GeV;

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289

Fig. 9.8 Relativistic collisions of gold nuclei. On the left is the energy density, the hadronization region is shown in light color. On the right is the time evolution of the total baryon energy density in the center and the trajectory of parameters change. The region of phase existence is shown in yellow, the region of phase transition is shown in green [33]

• JINR, Nuclotron, Russia, E lab ≈ 1–6 AGeV,



sNN

≈ 2.5–3 GeV.

More detailed information about them is given in [1, 2]. Among the large number of new intriguing results, we note only the study of the phase diagram of compressed and heated baryonic matter. Theoretical and experimental efforts in studying the phase diagram of compressed baryonic matter and the search for new and ordinary phases of strongly interacting matter at high energy densities have been intensively undertaken for about twenty five years. These studies will help understand the hadron behavior in hadron medium,

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9 Nuclear Shock Waves

Fig. 9.9 On the left: water–vapor transition (first-order phase transition with latent heat of evaporation) ends at the critical point (second-order transition). There is no difference between vapor and water above the critical point. On the right: according to modern theoretical models, the structure of the ‘quarks-hadrons’ phase transition is similar to the ‘water–vapor’ transition with no differences between phases above the critical point (μB is the baryon chemical potential related with baryon density, T is the temperature)

get the equation of state for hadron and nuclear matter and record the observed manifestations of quark deconfinement and chiral symmetry breaking. The primary trend in the studies of relativistic collisions is related with a tendency to move up the energy scale of relativistic particles. This growth of collision energy surely leads to the appearance of principally new interesting high-energy effects but decreases the interaction time in case of nuclei collisions making unbalanced processes more probable at ultrarelativistic interaction. This circumstance explains a “reverse” trend in studies—interest in a reduced (as compared to the record one) energy range of 8–40 GeV within the area of the phase boundaries of the formation of new hadron phases [37–39]. According to modern physical concepts, nuclear matter can withstand a number of phase transformations as the temperature and/or density of baryons grow (Figs. 9.9 and 9.10). One of them is a chiral symmetry breaking transition caused by strong interaction. This interaction breaks chiral symmetry at low temperatures and/or low baryonic densities, which is restored at high parameters. That being the case, the area of existence of phases with broken and restored symmetry is possible just as this takes place in water vapor under subcritical conditions. Since pairing of quarks can take place, the appearance of the effects of light supervisibility with system crystallization is possible. Another example of exotic phase transition is a transition of quark-gluon plasma [40] that we will consider in detail below. The presence of such phase transitions qualitatively changes the usual form of the baryonic matter phase diagram (Fig. 9.10), leading to new phase boundaries of

291

Quarks and gluons

Early Universe

Temperature Т, MeV

9 Nuclear Shock Waves

Critical point? Deconfinement transition Hadrons QCD lattice

Chiral transition, quarkotonic phase

Ideal liquid Super-conductive state of chrome conductivity Protoneutron stars

Nuclei

Baryon density

Fig. 9.10 Phase diagram of strongly interacting baryonic matter. Presented on the NICA website [41, 44]

chiral and quark-gluon transitions that, in their turn, can mutually overlap or even coincide [40, 41]. In case of high hadron densities and low temperatures, color superconductivity may disappear. This phenomenon is now intensively studied by specialists in neutron stars. This phenomenon can result in the appearance of a new baryon phase called quarkonic. Thus, we see that the studies of the physical properties of a strongly compressed baryonic matter, even within the range of moderate energies, is of significant interest, allowing for the discovery of new exotic states of matter. However, a desire to work with the maximum possible collision energies remains the main focus of work in high energy physics. The hypothetical (so far) possibility to reach short distances into the Grand Unification region is discussed [42, 43]. At these distances, the electromagnetic and weak interactions will unite even stronger in the sense that they will be described by a single interaction constant. Can we expect that in this case the strongly chromodynamic interaction will also join them? I.e. that leptons and quarks will form a unified group at short distances. The hope to get a positive answer to this question is based on the fact that three constants at distances of about 10–19 m do not differ so much from each other and show a tendency towards further approach. A number of theoretical models of such unification is proposed. Quark-Gluon Plasma. Among the large number of interesting physical results obtained with the use of accelerating facilities, of special interest is the generation of quark-gluon plasma (QGP) arising during quark deconfinement at energies ≥ 200 meV [37, 45]. In experiments on the collision of two nuclei, the kinetic energy of motion is converted into the internal energy of nucleons, which, in accordance with the predictions of the theory of quantum chromodynamics (QCD [41]), leads to the appearance of the so-called “colored glass condensate”, and then, after subsequent

292

9 Nuclear Shock Waves Tau 1,777 MeV

Muon 106 MeV

Charged leptons

Electron 0.511 MeV с 1,270 MeV t 171,000 MeV

u 2 MeV

“Upper” quarks

d 5 MeV s 104 MeV

“Lower” quarks

b 4,200 MeV

Fig. 9.11 Masses of charged Standard Model fermions. The area of the circle is proportional to the mass of the particle [46]

thermalization, to the formation of a new state of matter, i.e. a quark-gluon plasma or “quark soup” [14, 41]. In the RHIC, gold nuclei flying almost at the velocity of light collide. Each collision (mini-explosion) occurs at several stages (Fig. 9.8). It starts with a short-lived expanding “fireball” consisting of gluons (green), quarks and antiquarks, primarily upper, lower and strange (blue) with a small number of heavier charmed and beautiful ones (red). In the end, the fireball explodes and disintegrates into hadrons (silver) detected together with photons and other disintegration products. The physical properties of quark-gluon plasma are defined by the properties of recorded particles. In this case, it is assumed that the collision time is sufficient for matter thermalization, so the kinetic energy has time (this is the subject of separate consideration) to pass into the internal energy of the formed plasma. The QGP arising during such collisions consists of quarks, antiquarks and gluons [7, 8, 44, 46, 47]. The masses of quarks and other fermions are shown in Fig. 9.11 [46]. This plasma is sometimes called the “oldest” form of matter, because it existed even in the first microseconds after the Big Bang; hadrons were formed in the course of expansion and cooling of this matter. QGP has the highest density, approximately 9–10ρ 0 (ρ 0 = 2.5 × 1014 g/cm3 is the nuclear density), and can emerge in the center of neutron stars, black holes, or in the collapse of ordinary stars [1, 2]. Large-scale experimental programs have been launched to study the QGP properties in collisions of ultrarelativistic ions at HERA and RHIC accelerators in Brookhaven, at the GSI accelerator facility in Darmstadt, and at the Super Proton Synchrotron (SPS) and LHC at CERN. The first experiments with QGP in Brookhaven (RHIC) and CERN (SPS) showed a more diverse behavior of such a plasma than it was previously assumed (quark and gluon gas). It was found that special attention should be paid to the energy range

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293

Quark-gluon plasma

Temperature Т, MeV

QCD-structure Critical point

Condensed baryonic Discharged hadron medium

ne = 0.12 fm-3

Atomic nuclei

Neutron stars

Baryonic chemical potential µВ, GeV

Fig. 9.12 Phase diagram of nuclear matter



≈ 2–10 GeV, where a strongly interacting (nonideal) QGP was expected to emerge. In any case, we are dealing with the collision of heavy nuclei with energies of the order of 100 GeV and higher per nucleus in the frame of reference associated with the center of mass or with energies of 20 TeV per nucleus in the laboratory frame of reference. The conditions attainable on modern accelerators are shown in the phase diagram of nuclear matter (Fig. 9.12). The region of low temperatures and baryonic densities is occupied by hadrons (nuclei and mesons). The limiting case of high densities (5–10 times higher than the nuclear density: ρ 0 = 2.8 × 1014 g/cm3 ) and high temperatures (T > 200 meV ≈ 1012 K) corresponds to quarks and gluons, which under these conditions are not bound in hadrons, but form a quarkgluon plasma. The transition between these states can be either nonabrupt or abrupt, like a first-order phase transition with a critical point (Fig. 9.10). To describe the behavior of compressed baryonic matter in the corresponding domain of the phase diagram, use is made of the methods of quantum chromodynamics, which are also the object of experimental verification. The phase diagram taking into account quantum chromodynamic calculations is given in Fig. 9.13 [33]. QGP is a substantial element of matter transformation after the origination of our Universe. During the first microseconds [7, 8] after the Big Bang, the temperature √ falls down as T (MeV) ~ 1/ t, where t is the time in seconds, so that QGP with a temperature of hundreds MeV could exist during the first 5–10 μs after the Big Bang. The baryon density was not so high. As the Universe expanded, plasma cooled down with matter hadronization and further formation of pions. If the first-order transition sNN

294

9 Nuclear Shock Waves

Fig. 9.13 Phase diagram of hadron and quark-gluon matter. Circles are experiments and lattice QCD-calculations. A light triangle is a critical point, a dark triangle is an atomic nucleus [33]

took place, then hadron bubbles probably formed inside plasma: neutrons, protons and pions. QGP is a superdense and superhot form of nuclear matter with unbound quarks and gluons that are bound in hadrons at lower energies [48–51]. We know that ordinary matter consists of atoms. An atom consists of an atomic nucleus and electrons. The atomic nucleus consists of protons and neutrons. Proton and neutron, in their turn, consist of quarks, u and d. Therefore, electron and two types of quarks, u and d, are the fundamental particles that make up ordinary matter. Quantum chromodynamics is created to describe strong interactions [34, 45, 52]. The fundamental QCD particles, unlike other Standard Model particles, cannot be directly observed—these are quarks and gluons. Instead, we see the bound states of hadrons. One of the fundamental QCD concepts lies in the principle of asymptotic freedom. At low energies, the coupling constant is high and it decreases approaching zero as energy rises. A high magnitude of the coupling constant at low energies does not allow using the perturbation theory for this case. This theory works and gives high results in the limit of high energies. An important property of the principle of asymptotic freedom lies in the fact that as energy (and, therefore, temperature) grows, the primary contribution is made not by bound hadron states but free quarks and gluons (Stefan-Boltzmann limit). Thus, two phases arise (hadron and quark-gluon) with a phase transition between them, with a critical temperature T c ≈ 200 meV and an energy density of ≈ 0.7 GeV/fm3 . The same effect is caused by nuclear matter compression [45, 52]. At high baryon densities, the coupling constant is low since the Fermi surface corresponds to high energies, which also leads to quark deconfinement. The corresponding phase diagram

9 Nuclear Shock Waves

295

of baryonic matter has a non-trivial form (Figs. 9.9 and 9.10). Many papers [34, 45, 52–61] describing not only the region of the first-order and second-order phase transition, thermodynamics and kinetics of a mixture of hadron and quark-gluon phases, but also analytical crossover in supercritical conditions are devoted to the calculations for the phase diagram. Let us note (ref. Fig. 9.11 [46]) that in accordance with the principle of asymptotic freedom, fermions will poorly interact at high densities (chemical potential). Since quarks are drawn to each other, they will form Cooper pairs, leading to color superconductivity [52] (ref. Fig. 9.10). The appearance of quark-gluon plasma manifests itself as an increase in the number of degrees of freedom from hadronic, equal to 3–8 for gluons, multiplied by 2 spin, plus 2–3 light quark flavors which, in their turn, have 2 spins and 3 colors. Thus, according to quantum electrodynamics, quarks have 24–26 degrees of freedom, and 40–50 degrees of freedom are excited in quark-gluon plasma at T ≈ (1–3) T c versus 3 degrees of freedom in pion gas of low temperatures T < T c . Since the energy density, pressure and entropy are approximately proportional to the excited degrees of freedom of the system, and a sharp change in these thermodynamic parameters within a narrow range of temperatures around T c explains such a high (several fold) difference between the energies of ordinary nuclear matter and QGP. This leads to the specific features of the shock adiabats of nuclear matter and, as a consequence, to the possible instability of nuclear shock waves (ref. Chap. 12). Now it is hard to say unambiguously whether the transition to a quark-gluon plasma is a true thermodynamic phase transition with an energy density jump or a sharp and yet continuous transition [1, 2]. It is possible that at low values of baryonic chemical potential μB is a continuous transition, and at high values μB is a firstorder phase transition (Fig. 9.14). In any case, the theory predicts a low value for the velocity of sound in the transition region, which is reflected in the hydrodynamic anomalies observed in the relativistic collisions of heavy nuclei. These specified features of the adiabatic compressibility of a quark-gluon plasma testify to a “softer” equation of state for T ≈ T c and a “stiffer” one at high temperatures as well as for T ≤ T c [1, 2]. In the limitT 4 GeV). The measurements overlap the pT interval up to a value of 20 GeV. The measured large deficit of particles with a high transverse momentum is indicative of partons’ energy loss in the medium. It corresponds to the effect of the so-called jet suppression (quenching) effect, which manifests itself in the softening of the hadronic spectrum obtained from in-medium partons in comparison with the spectrum in vacuum. Therefore, the suppression factor is a powerful means for determining the density of the medium. We see that the effects of suppression of jets produced in relativistic collisions of nuclei contain information about the properties of shock-compressed matter [75–77] and about the emergence of QGP. By the order of magnitude, this suppression is determined by the radiation losses of gluons, while the contribution of elastic losses is insignificant. The results of such “tomography” for PHENIX experiments [78] show that the initial reduced gluon density must be equal to dN g /dy ≈ 1000 ± 200 to explain the observed jet suppression. These values are in reasonable agreement with another set of independent measurements [12]: (a) (b) (c)

with the values of initial entropy determined from the plasma expansion after the collision; with the initial plasma parameters that follow from the hydrodynamic calculations of “elliptical” flows; with density variation rates calculated by the methods of quantum electrodynamics.

These data sets make it possible to find the initial energy density in relativistic collisions: E 0 = E(1/ρ0 ) ≈ ρ02 /π R 2 · d N g /dy ≈ 20 GeV/fm3 ≈ 100E a

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9 Nuclear Shock Waves

at a characteristic gluon momentum (P0 ≈ 1.0–1.4 GeV), which in its turn defines the formation time, è/P0 ≈ 0.2 fm/c, of the primary nonequilibrium QGP. Under these conditions, the local thermodynamic equilibrium necessary for the application of hydrodynamics occurs at τeq ≈ (1−3)B/P0 < 0, 6 fm/s. By this point in time, the temperature becomes   T τeq ≈



ε0 · 12 1−3

 14

≈ 2Tc .

According to one of the models [12], at P0 ≈ 2–2.2 GeV, the number of minijets must be of the order of 1000. More detailed data on the properties of plasma can be obtained from the study of the correlation between secondary particles by recording correlated double jets in nuclear collisions. It was shown [34] that both two- and three-particle correlations exhibit two contra-directional jet-like peaks (two-jet transitions). From a theoretical point of view, at a first approximation, jets are considered as the residual manifestations of the hard scattering of quarks and gluons. The consistency of jet characteristics obtained under different conditions (central and peripheral collisions, protons and gold) is considered as a powerful argument in favor of the formation of quark-gluon plasma and the applicability of quantum chromodynamics methods to the description of observed phenomena. Thus, the observed phenomena of jet suppression in nuclear collisions allow determining the energy density of nuclear matter and drawing conclusions about the strong collective interaction (nonideality) of this plasma, proceeding from the analysis of the energy losses of jets in their motion through the quark-gluon plasma. Multiparticle correlations and totally (calorimetrically) reconstructed jets are the main focus of the recent efforts in the study of proton-proton and nuclear-nuclear collisions. These features are observed both for two-particle and three-particle correlations. The existence of such features is undoubtedly associated with the collective properties of the medium. Now, observational manifestations of QGP are being intensively analyzed [1, 2], among them: • barometric manifestations—by the parameters of collective flows (elliptical, longitudinal, radial) caused by internal pressure in the quark-gluon medium; • thermodynamic manifestations—photons, lepton pairs, vector mesons; • critical phenomena—hadron fluctuations and densities; • tomography—short jets, flows of heavy quarks; • exotics—multi-quark states, femto-size fullerenes. Among the interesting hydrodynamic phenomena, special mention should be made of Stöcker’s elegant and beautiful idea [62] about the formation of conical

9 Nuclear Shock Waves

a

305

c

b

Implementation stage

Piercing stage

Final stage

Fig. 9.26 Formation of Mach shock waves in nuclear matter [62] in the collision of a light nucleus (on the left) with a “heavy” one

Mach shock waves in relativistic nuclear collisions (Fig. 9.26), the properties of which make it possible to judge the characteristics of compressed nuclear matter. It is clear that the manifestation of various physical phenomena during compression and heating of dense baryonic matter leads to nonmonotonicity of shock adiabats and possible instabilities (ref. Chap. 1) of nuclear shock waves (Fig. 9.26). On the Behavior of Relativistic Shock Waves in Nuclear Matter. As we have seen [1, 2], hot matter occurring during the collision of ultrarelativistic heavy ions demonstrates a number of collective phenomena that can be described within the framework of almost ideal (low viscosity) relativistic hydrodynamics (ref. Chap. 1), ref. [65, 66, 79]. Such description assumes a large number of particles in the system (thousands or even dozen thousands) being in the conditions (at least approximately) of local thermodynamic equilibrium [63]. Only in this case we can speak about the equation of state establishing the relationship between thermodynamic parameters required for hydrodynamic calculations. Currently, there are two approaches to building a phase diagram of nuclear matter. 1.

2.

Quantum chromodynamic (QCD) lattice calculations (method proposed in [80], a review of further modifications is given in [73]) the results of which in the ideal case would be sufficiently accurate, but unfortunately, as given in [63], the provision of the required (from the physical standpoint) number of cells (nodes), correct ratio between the masses of current u- and d-quarks, on the one hand, and s-quarks, on the other hand, as well as the counting stability at a non-zero chemical potential causes serious challenges. The use of phenomenological models of the MIT-bag-type (Massachusetts Institute of Technology where the first such model was developed) based on an assumption that in case of deconfinement quarks are locked in the hadron boundaries under the action of excessive pressure occurred due to the displacement of fields forming a physical vacuum [63]. The value of that pressure B (bag constant) is a free parameter of the model. As a rule, in describing the quark-gluon phase, bag models are based on the approximation of ideal gas, while for hadron gas, Van-der-Waals models are used with excluded volume; various phase equilibrium conditions are used in describing a mixed phase. The

306

9 Nuclear Shock Waves

detailed comparison of various bag models is given in [81]. The opportunity to consider media having variable chemical potential keeping all advantages of equations of state built on the basis of lattice QCD-calculations predetermines the predominant use of bag models in hydrodynamic calculations. An analysis of the data contained in [81] shows that anomalous hydrodynamic effects can occur in the region of quark-hadron phase transition, in particular, the splitting of compression shock waves caused by their instability and the occurrence of rarefaction shock waves. The stability of shock waves in nuclear matter will be considered in Chap. 12 [74, 82–86]. A decay of a compression shock wave with the formation of a two-wave structure was found and an assumption was made that this circumstance can be used as a “marker” to identify a phase transition from hadron matter to quark-gluon plasma. An analysis of the results obtained in [7–12] (with somewhat different equation of state) confirms the splitting of a shock wave in the region of the first-order phase transition from hadron matter to quark-gluon plasma. The characteristics of the decay configuration are evaluated in terms of the possibilities of its experimental detection. The possibility of the emergence of neutrally stable shock waves in nuclear matter is considered. Fireball expansion with a reverse phase transition from QGP to hadron matter is modelled. The characteristics of a combined rarefaction wave are analyzed from the same point of view as the shock wave splitting (Chap. 12). Equation of State and Phase Diagram. To describe shock wave processes in nuclear matter taking into account the deconfinement phase transition, an equation of state is required that would adequately describe a relationship between the thermodynamic parameters of such a medium. As a result of the analysis of existing models for describing the thermodynamic properties of quark-gluon plasma and hadron matter resulted, a bag model was chosen [87] which is an approximation of ideal gas with the confinement parametrization by constant B. Within the framework of this model for gas of quarks and gluons with N f = 2 aromas, N c = 3 colors and N g = 8 color states of gluon, the expressions for pressure, energy density and density of baryonic number look as follows: Pp =

μq4  37 2 4 1 ε p − 4B = − B, π T + μq2 T 2 + 3 90 2π 2 n Bp =

 μq3 2 μq T 2 + 2 , 3 π

where μq is the chemical potential assumed equal for u- and d-quarks. Taking into account that nucleons consist of three quarks, the Gibbs conditions of phase equilibrium look as Pp = Ph , T p = T h , μ = 3μq . The hadron phase is considered as an ideal relativistic gas of pions, nucleons and antinucleons. The pressure and energy density of the hadron phase contain the contribution of pions and nucleons Ph = Pπ + PN , εh = επ + εN . Contribution of pions into pressure and energy is determined by the following relations

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307

Pπ (T ) =

∞ 3m 2 T 2  K 2 (km/T ) , 2π 2 k=1 k2

επ (T ) = 3Pπ (T ) +

∞ 3m 3 T  K 1 (km/T ) . 2π 2 k=1 k

Here, K 1 i K 2 are modified Bessel functions of the second kind, m is the pion mass. Pressure, energy density and baryonic number for gas of nucleons and antinucleons are determined as per [87]: 1

2M 4 PN (T, μ) = 3π 2

0

2M 4 ε N (T, μ) = 2 π

n h (T, μ) =

2M 3 π2

1 0

1 0

u 4 du  3 [ f (u; T, μ) + f (u; T, −μ)], 1 − u2 u 2 du  3 [ f (u; T, μ) + f (u; T, −μ)], 1 − u2

u 2 du  5/2 [ f (u; T, μ) + f (u; T, −μ)], 1 − u2

where f (u; T, μ) = 1 + exp



M

 1/2 1 − u2



μ − T

−1 .

In the calculations, nucleon mass M was assumed to be 940 meV, the pion mass— 139.6 meV. The dependence of pressure p = p(n, T ) as a function of temperature and density of baryonic number is shown in Fig. 9.27. Stability of Compression and Rarefaction Shock Waves. The stability of relativistic shock waves (ref. Chap. 12) within the framework of linear theory was studied in [88, 89] where the respective criteria were obtained. Shock waves that are unstable according to these criteria belong to the segment of the shock adiabat with an ambiguous representation of the shock wave discontinuity and are broken down with a configuration of several wave elements being formed. The analysis shows that in the collision of ultrarelativistic nuclei, one can expect the fulfilment of the criterion of shock wave instability (L < − 1, ref. Chap. 12) along with the condition of neutral stability (spontaneous sound emission). In the first case, instead of a single shock wave, a complex compression wave can be observed, which includes two waves following in the same direction (but not overtaking each other)—the socalled two-wave structure. This case was discussed in [74, 82–86]. In the second case (neutrally stable shock wave), secondary waves are formed, the wavevector of which makes a specific angle with the direction of shock wave propagation. It is

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Fig. 9.27 Dependence of pressure p (in MeV4 ) on temperature T and baryon charge density in the region of deconfinement phase transition, built on the basis of the MIT-bag model [87] at B1/4 = 153.8 meV

also important to note that in case of a first-order phase transition, the formation of rarefaction shock waves is possible when a shock-compressed matter is unloaded. The condition of neutral stability (spontaneous sound emission) of a relativistic shock wave looks as follows [89] 1−u − 2

     ∂p < 0, −1 M 1+M u ∂ε n

u

0

(9.1)

where u0 , u are velocities before and after shock wave, respectively, as recorded in the coordinate system associated with the discontinuity, M = |u|/cs , cs is the velocity of sound defined by the ratio cs2

   ∂ p  ∂ p  n ∂ p  = = + . ∂ε s/n ∂ε n p + ε ∂n ε

The velocity of sound is an increasing function of temperature and baryon charge density in the region of the hadron phase, which is constant in the region of the quark-gluon phase and takes low values in the two-phase region (the square of the velocity of sound is shown in Fig. 9.28). Condition (9.1) permits an equivalent formulation in terms of thermodynamic functions of the final state and intensity of the shock wave p/p0 :

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309

Fig. 9.28 Dependence of the square of the velocity of sound in nuclear matter in the region of deconfinemnt phase transition on temperature and density of baryon charge

n ∂p + p ∂n ε



ε ∂p p ∂ε

 −1+ n

p0 < 0. p

(9.2)

Verification of the fulfillment of this condition for the selected equation of state showed that condition (9.2) is not fulfilled in the region of the phase diagram that corresponds to quark-gluon plasma. Indeed, the relationship between pressure and energy density in this case looks as p = 13 (ε − 4B) and the left part (9.2) is positive at any value of the initial pressure p0 . For shock waves with the final state in the region of the quark-gluon plasma, shock waves considered as a structureless discontinuity are stable (nevertheless, they can be unstable relative to decay). The left part of (9.2) is also positive in the hadron phase region, i.e. shock waves with the final state in the hadron phase region are absolutely stable. Value     ε ∂p n ∂p + − 1, F= p ∂n ε p ∂ε n included in the left part of (9.2) takes negative values in the two-phase region of the phase diagram, which shows the potential for the implementation of neutrally stable

310

9 Nuclear Shock Waves b P, eV4

а P, eV4

X, MeV-2

Fig. 9.29 Taub shock adiabat in the region of quark-hadron phase transition (a) and a shock wave decay in the region of ambiguous representation of shock wave discontinuity (b) (a self-similar solution is shown for pressure in the shock adiabat segment between points 2 and 3)

shock waves with the final state in the two-phase region in case of sufficiently high shock wave intensity p/p0 . However, such shock waves, including those associating the primary state of nuclear matter and the state belonging to the two-phase region of the phase diagram are unstable (ref. criterion (9.3) below, belong to the region of ambiguous representation of shock wave discontinuity and are not implemented, with multiwave configuration occurring instead of them). The Taub shock adiabat [90] n 2 X 2 − n 20 X 02 − ( p − p0 )(X + X 0 ) = 0, where X = (ε + p)/n2 is the generalized specific volume, built on the basis of this equation of state is shown in Fig. 9.29. It looks typically for materials with firstorder phase transition [91] characterized by breaks at the two-phase region boundary (points 24). The instability conditions of a relativistic shock wave with the exponential growth of low disturbances looks as follows according to the linear theory [88, 89]  L=m  m2

∂X ∂p

2



∂X ∂p

> H

 < 1,

(9.3)

H

1 + 2M + u 0 u . 1 − u0u

(9.4)

 1/2 Here m = nu/ 1 − u 2 is the density of the baryonic number flux through the shock-wave front, which can be expressed as m2 = p − p0 /X 0 − X. Condition (9.4) is not fulfilled on Taub shock adiabats built using the equation of state in the bag model (a shock adiabat represented in Fig. 9.29 is typical for such

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311

equations of state). Condition (9.4) is fulfilled at final states belonging to the twophase region of the phase diagram. However, when it is fulfilled, a straight line drawn through the points depicting the initial and final states in the plane (p–X) crosses the shock adiabat in an intermediate point with necessity. This means that the shock wave discontinuity can be represented as a combination of wave elements including a two-wave structure. Thus, the solution of the Cauchy problem with initial data in the form of a shock wave discontinuity is not the only one. In this case, the problem of the shock wave viscous structure has no solution. Numerical solutions of the problem of shock wave discontinuity show a shock wave decay in the region of its ambiguous representation. The calculation was done based on the equations of relativistic hydrodynamics [92] (Chap. 1) expressing preservation of the baryon charge: ∇ β (nuβ ) = 0, as well as preservation of energy and momentum: ∇ β T αβ = 0, where the energy–momentum tensor looks as T αβ = (ε + p) uα uβ − gαβ p, and thermodynamic functions are determined using the equation of state of nuclear matter. As initial data, the solution was specified as corresponding to the smoothed shock wave discontinuity in the coordinate system where the discontinuity is at rest. Calculations studied convergence to the self-similar solution, and the calculation stopped after fulfilling the condition u n−1 − u n ≤ ε, where un = un (x/t) is the function of self-similar variables interpolating the solution vector on the time layer n, ε is the parameter controlling the accuracy of convergence to the self-similar solution. Figure 9.29 gives solutions for the Cauchy problem with initial data corresponding to the shock wave discontinuity in the conditions of its ambiguous representation (the segment of the Taub adiabat enclosed between points 1 and 2). Final states are characterized by temperatures of 28.5 meV (a), 29 meV (b), 29.5 meV (c), 30 meV (d). Pressure is plotted on the X axis and self-similar variable ξ = x/(ct) is plotted on the Y axis. As the chart shows, the velocity difference of the first and second shock waves in the decay configuration is much less than the relativistic velocity of a shock wave relative to the matter. This means that the shock wave decay can be hidden by viscosity and non-equilibrium effects and its consequences do not have a significant effect on the dynamic pattern of collision. Figure 9.30 shows the Poisson adiabat crossing the region of quark-hadron phase transition. The adiabat has breaking points at the two-phase area boundary (points 2, 3). Since the adiabat is not convex, a complex rarefaction shock wave is implemented instead of a simple rarefaction shock wave from initial state 1 to state 5 in the numerical solution. Such wave includes a rarefaction wave in quark-gluon phase 1– 2, a rarefaction shock (phase transition wave 2–4) and a rarefaction wave in hadron phase 4–5 (Fig. 9.30b). The complex rarefaction wave structure includes a region of constant solution, enclosed between the rarefaction wave in quark-gluon plasma and the rarefaction shock. The thermodynamic state of matter in this region corresponds to a point located at the two-phase region boundary. The rarefaction shock is a wave of quarkhadron phase transition. The hydrodynamic calculation allows evaluating the energy density and temperature in the formed hadron phase in the approximation of instantaneous kinetics of phase transformation, which potentially provides a link between

312 a

9 Nuclear Shock Waves P, MeV4

b

P, MeV4

x, MeV-2

Fig. 9.30 Poisson adiabat

the equation of state of nuclear matter and the experiment. Finding elements of a complex rarefaction wave in the experiment (plateau and rarefaction shock) will not only be a clear marker of phase transition from quark-gluon plasma to hadron matter but can also be used for the quantitative specification of the equation of state. The completed calculations show that the difference between the velocities of the precursor shock wave and the closing wave of the phase transition are much less than the relativistic velocity of matter. Hence, it follows that the splitting of shock waves in experiments with collision of heavy ions can be masked by viscous and non-equilibrium effects and its identification is difficult. A complex rarefaction wave emerging in the phase transition from QGP to hadron matter during fireball expansion presents much higher possibilities for experimental detection. A condition of neutral stability is fulfilled only for shock waves with the final state in the region of mixed phase. However, such shock waves are not implemented due to their splitting. It seems that only phase transition shock waves closing the complex compression wave can be neutrally stable.

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29. Tahir NA, Deutsch C, Fortov VE et al (2005) Proposal for the study of thermophysical properties of high-energy-density matter using current and future heavy-ion accelerator facilities at GSI Darmstadt. Phys Rev Lett 95(3):035001 30. Rosmej ON, Blazevic A, Korostiy S et al (2005) Charge state and stopping dynamics of fast heavy ions in dense matter. Phys Rev A 72(5):052901 31. Efremov VP, Pikuz Jr SA, Fayenov AYa et al (2005) Study of the energy release region of a heavy-ion flux in nanomaterials by X-ray spectroscopy of multicharged ions. JETP Lett 81(8):468 [Efremov V.P., Pikuz Jr. S.A., Fayenov A.Ya. i dr. Issledovaniye zony energovydeleniya potoka tyazhelykh ionov v nanomaterialakh metodami rentgenovskoy spektroskopii mnogozaryadnykh ionov // Pis’ma ZHETF. 2005. T. 81, №8. S. 468 (in Russian)] 32. Mintsev V, Gryaznov V, Kulish M et al (1999) Stopping power of proton beam in a weakly non-ideal xenon plasma. Contrib Plasma Phys 39(1–2):45–48 33. Friman B, Höhne C, Knoll J et al (eds) (2010) The CBM physics book, 1st edn. In: Lecture notes in physics, vol 814. Springer 34. Dremin IM, Leonidov AV (2010) The quark-gluon medium. Phys Usp 180(11):1167–1196 [Dremin I.M., Leonidov A.V. Kvark-glyuonnaya sreda // UFN. 2010. T. 180, №11. S. 1167–1196 (in Russian)] 35. Gyulassy M. Relativistic heavy ions and QGP at FAIR. https://www-win.gsi.de/FAIR-sti/fairsymposium-2007/Gyulassy_FAIR110807final.pdf 36. Ginzburg VL (2004) On superconductivity and superfluidity (what I have and have not managed to do), as well as on the “physical minimum” at the beginning of the XXI century. Phys Usp 174(11):1240 [Ginzburg V.L. O sverkhprovodimosti i sverkhtekuchesti (chto mne udalos’, a chto ne udalos’), a takzhe o «fizicheskom minimume» na nachalo XXI veka // UFN. 2004. T. 174, №11. S. 1240 (in Russian)] 37. Sissakian AN, Sorin AS (2009) The nuclotron-based ion collider facility (NICA) at JINR: new prospects for heavy ion collisions and spin physics. J Phys G Nucl Part Phys 36(6):064069 38. Sissakian A, Sorin AS (2009) The QCD phase diagram NICA, JINR communication. JINR, Dubna 39. Sissakian A et al (2009) The multi-purpose detector—MPD to study heavy ion collisions at NICA. Conceptual design report. JINR, Dubna 40. Randrup J, Ruuskanen PV (2004) Thermodynamic consistency of the equation of state of strongly interacting matter. Phys Rev C 69:047901 41. Mrowczynski S, Thoma MH (2007) What do electromagnetic plasmas tell us about the quarkgluon plasma? Annu Rev Nucl Part Sci 57(1):61–94 42. Blaschke D et al (2009) Searching for a QCD mixed phase at the nuclotron-based ion collider facility (NICA white paper). https://theor.jinr.ru/twiki/pub/NICA/WebHome/Wh_Paper_dk6. pdf 43. Okun’ LB (2012) The basics of physics. A very brief guide. FIZMATLIT, Moscow [Okun’ L.B. Azy fiziki. Ochen’ kratkiy putevoditel’. — M.: FIZMATLIT, 2012 (in Russian)] 44. Hands S (2001) The phase diagram of QCD. J Contemp Phys 42(4):209–225 45. Shuryak E (2009) Physics of strongly coupled quark-gluon plasma. Prog Part Nucl Phys 62(1):48–101 46. Troitsky SV (2012) Unsolved problems in particle physics. Phys Usp 182(1):77–103 [Troitsky S.V. Nereshennyye problemy fiziki elementarnykh chastits // UFN. 2012. T. 182, №1. S. 77–103 (in Russian)] 47. Glendenning N (2000) Compact stars, nuclear physics, particle physics and general relativity. Springer, New York 48. Shuryak EV (1980) Quantum chromodynamics and the theory of superdense matter. Phys Rep 61(2):71–158 49. Shuryak EV (1978) Quark-gluon plasma and hadronic production of leptons, photons and psions. Phys Lett B 78(1):150–153 50. Kalashnikov OK, Klimov VV (1979) Phase transition in the quark-gluon plasma. Phys Lett B 88(3–4):328–330

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75. Gyulassy M, Plumer M (1991) Jet quenching as a probe of dense matter. Nucl Phys A 527:641– 644 76. Gyulassy M, Plumer M, Thoma M, Wang XN (1992) High PT probes of nuclear collisions. Nucl Phys A 538:37–49 √ 77. Wang X-N, Gyulassy M (1992) Gluon shadowing and jet quenching in A+A collisions at s=200 A GeV. Phys Rev Lett 68(10):1480–1483 78. Vitev I, Gyulassy M (2002) High-PT tomography of d+Au and Au+Au at SPS, RHIC, and LHC. Phys Rev Lett 89(25):252301 79. Shuryak E (2009) Physics of strongly coupled quark-gluon plasma. Prog Part Nucl Phys 62:48 80. Wilson KG (1974) Confinement of quarks. Phys Rev D 10:2445 81. Tiwari SK, Singh CP (2013) Particle production in ultrarelativistic heavy-ion collisions: a statistical-thermal model review. Adv High Energy Phys 27 pages. Article ID 805413. https:// doi.org/10.1155/2013/805413 82. Barz HW, Csernai LP, Kampfer B, Lukács B (1985) Stability of detonation fronts leading to quark-gluon plasma. Phys Rev D 32:2903 83. Gorenstein MI, Zhdanov VI (1987) Shock stability criterion in relativistic hydrodynamics and quark-gluon plasma hadronization. Z Phys C Part Fields 34:79 84. Bugaev KA, Gorenstein MI (1987) Relativistic shocks in baryonic matter. J Phys G Nucl Phys 13:1231 85. Bugaev KA, Gorenstein MI, Zhdanov VI (1988) Relativistic shocks in the systems containing domains with anomalous equation of state and quark baryonic matter hadronization. Z Phys C Part Fields 39:365 86. Bugaev KA, Gorenstein MI, Kämpfer B, Zhdanov VI (1989) Generalized shock adiabatics and relativistic nuclear collisions. Phys Rev D 40:2903 87. Cleymans J, Gavai RV, Suhonen E (1986) Quarks and gluons at high temperatures and densities. Phys Rep 130(4):217 88. Kontorovich VM (1968) Shock wave stability in relativistic hydrodynamics. JETP 34(1):186 [Kontorovich V.M. Ustoychivost’ udarnykh voln v relyativistskoy gidrodinamike // ZHETF. 1968. T. 34, №1. C. 186 (in Russian)] 89. Russo G (1988) Stability properties of relativistic shock waves: applications. Astrophys J 334:707 90. Taub AH (1978) Relativistic fluid mechanics. Annu Rev Fluid Mech 10:301 91. Menikoff R, Plohr BJ (1989) The Riemann problem for fluid flow of real materials. Rev Mod Phys 61:75 92. Landau LD, Lifshits EM (1988) Theoretical physics, vol VI. Nauka, Moscow, 736 p [Landay L.D., Lifxic E.M. Teopetiqecka fizika. T. VI. — M.: Hayka, 1988. — 736 c. / Landau L.D., Lifshits E.M. Teoreticheskaya fizika. T. VI. — M.: Nauka, 1988. — 736 s (in Russian)] 93. Anisimov SI, Prokhorov AM, Fortov VE (1984) Application of high-power lasers to study matter at ultrahigh pressures. Phys Usp 142(3):395 [Anisimov S.I., Prokhorov A.M., Fortov V.E. Primeneniye moshchnykh lazerov dlya issledovaniya veshchestva pri sverkhvysokikh davleniyakh // UFN. 1984. T. 142, №3. S. 395 (in Russian)]

Chapter 10

Shock Waves in Traffic Flows

In this chapter, we will take a brief look at the generation of shock waves in an exotic situation of traffic flows (Fig. 10.1). In doing so, we will follow the paper [1] where an expert analysis of this issue is carried out and a detailed list of Refs. [1, 2]. The problem of optimal organization of automobile traffic is one of the most acute problems in the development of metropolises and has always attracted the attention of specialists in various fields of knowledge—physicists, mathematicians, operation research specialists, transport specialists, economists. A huge experience has been accumulated in the study of traffic processes. However, the total level of studies and their practical use is obviously insufficient today for the following reasons: • a traffic flow is unstable and diverse, getting objective information thereof is the most challenging and resource-intensive element of the management system; • quality criteria of road traffic management are controversial: it is required to ensure smooth traffic and reduce damage from traffic by implying speed and direction restrictions; • even if stable, road conditions have unpredictable specific features in terms of changes in weather and climatic parameters and impaired quality of road pavement; • fulfillment of decisions for road traffic management is always not clear and, taking into account the road traffic process, leads to unforeseeable effects. Challenges of traffic process formalization thus became a serious reason for the lag of research results from the requirements of practice that is directly related with the well-being and life of people. The largest road accidents receive the same response in mass media as railroad accidents and aircraft crashes. Road accidents cost on average 2–3% GDP. In 1999, economic losses from road accidents were $500 bln globally. The amount of annual damage in Russia exceeds $1 bln according to the Rambler web portal. According to statistics [1, 2], more than 180,000 vehicles simultaneously are present in the road network of Moscow (if this number rises to 230,000–250,000, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Fortov, Intense Shock Waves on Earth and in Space, Shock Wave and High Pressure Phenomena, https://doi.org/10.1007/978-3-030-74840-1_10

317

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10 Shock Waves in Traffic Flows

Fig. 10.1 Cartoon illustration of the formation of a shock wave in traffic

traffic jams occur), with up to 300,000 parked temporarily. These numbers are 25,000 and 75,000 in the central part of the city, including the Garden Ring. An average traffic speed in the city is 33 km/h, an average speed in the central part is less than 18 km/h [1, 2]. These today’s and perspective problems can be fully solved only based on methods of mathematical modeling [1, 2] that were developed in 1912 by the Russian scientist, professor G. D. Dubelir. The first-priority task that promoted the development of traffic flow modeling was the throughput capacity analysis of main roads and crossings. Throughput capacity means the maximum possible number of vehicles that can get through the road cross-section per unit of time. The first macroscopic model when the traffic flow was considered in terms of continuous medium mechanics was proposed in 1955 by Lighthill and Whitham [3, 4]. They showed that description methods for transfer processes in continuous media can be used for traffic jam modeling. Hate was the first to distinguish mathematical studies of traffic flows as an individual section of applied mathematics [5]. Traffic system research became interesting again in the 60–70s. This interest was shown also in funding multiple contracts and appealing to prestigious scientists, specialists in the field of mathematics, physics, management processes such as the Nobel laureate Prigogine [6–8], a specialist in automatic management M. Atans, the author of fundamental papers in statistics L. Breiman. In Russia, motor traffic was actively studied in late 1970s due to the preparations for the 1980 Olympic Games in Moscow. In late 1980s and early 1990s, the research of transport systems in the USA became the matter of national security. This problem was solved by the best physical minds and computer equipment of the Los Alamos National Lab (LANL). There are two main approaches in traffic modeling: deterministic and probabilistic (stochastic). Deterministic models are based on functional dependence between individual indicators, for example, speed and distance between vehicles in the flow. In stochastic models, a traffic flow is considered as a probabilistic process. All traffic flow models can be divided into three classes: analog models, leaderfollowing models and probabilistic models.

10 Shock Waves in Traffic Flows

319

In the analog models, a vehicle motion is assimilated to any physical flow (hydroand gas-dynamic models). Models of this class are called macroscopic. In the leader-following models, an assumption of a link between the motion of a led and leading vehicles is substantial. As the theory developed, the models of this group considered driver’s response time, studied traffic at multi-lane roads and traffic stability. Models of this class are called microscopic. In probabilistic models, a traffic flow is considered as an interaction result of vehicles on transport network elements. Due to rigid restrictions of the network and mass traffic, clear principles of queues, intervals, lane loads, etc. are formed in the traffic flow. These principles are substantially of a stochastic nature. Recent studies of transport flows started using inter-disciplinary mathematical ideas, methods and algorithms of nonlinear dynamics. The expediency of their application is substantiated by stable and unstable traffic conditions in a traffic flow, stability losses in case change in traffic conditions, nonlinear reverse links, a need for a higher number of variables to adequately describe the system. The highest interest here is demonstrated by a hydrodynamic approach when a traffic flow can be considered as a flow of one-dimensional compressible fluid, assuming that the flow is preserved and there is a mutually unambiguous dependence between the speed and density of a traffic flow. The first assumption is expressed by the continuity equation. The second one is expressed by a dependence between a flow speed and density to take into account traffic speed drop as the flow density rises. This intuitively correct assumption can theoretically result in a negative density or speed. It is obvious that one density value can correspond to several speed values. Therefore, for the second assumption, the average flow speed at each point in time must correspond to an equilibrium value for this density of vehicles on the road. The equilibrium situation is a purely theoretical assumption and can be observed only on road segments without crossings. Therefore, some researchers abandoned continuous models and some find them too rough. Hydrodynamic models include models with and without regard to the inertia effect. The latter can be obtained from the continuity equation if the speed is regarded as a function of density. Models taking into account inertia are represented by the Navier–Stokes equations with a specific member describing the drivers’ desire to drive at a comfortable speed. Traffic Flow Conservation Law. Let us consider a traffic flow on a single-lane road, i.e. without overtaking. A density of vehicles (number of vehicles per unit of road length) ρ(x, t), x ∈ R, at the point in time t ≥ 0. The number of vehicles in the interval (x 1 , x 2 ) at the point in time t is as follows: x2 ρ(x, t)d x.

(10.1)

x1

Let us assume that v(x, t) is the vehicle speed at point x at time t. The number of vehicles passing at speed v(x, t) over x (unit of length) at time t is ρ(x, t), v(x, t). Let us find the equation of density change. The number of vehicles within interval (x 1 ,

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x 2 ) over time t changes as per the number of entering and leaving vehicles: d dt

x2 ρ(x, t)d x = ρ(x1 , t)v(x1 , t) + ρ(x2 , t)v(x2 , t).

(10.2)

x1

Integrating over the time and assuming that ρ and v are continuous functions, we obtain t2 x2 t1 x1

∂ρ(x, t) d x dt = ∂t

t2 (ρ(x1 , t)v(x1 , t) + ρ(x2 , t)v(x2 , t))d x dt t1

t2 x2 =− t1 x1

∂(ρ(x, t)v(x, t)) d x dt. ∂x

(10.3)

Since x 1 , x 2 ∈ R, t 1 , t 2 > 0 are arbitrary, ρt + (ρv)x = 0, x ∈ R, t > 0.

(10.4)

Let us supplement this equation with initial conditions ρ(x, 0) = ρ0 (x), x ∈ R.

(10.5)

Let us find an equation for speed v. Let us assume that v depends on density p only. If the road is empty (ρ = 0), vehicles drive at the maximum speed v = vmax . When the road is filled with vehicles, the speed falls down up to a complete stop (v = 0) when vehicles are placed bumper to bumper (ρ = ρ max ). This simplest model is expressed by the following linear relation:   ρ , 0 ≤ ρ ≤ ρmax. v(ρ) = vmax 1 − ρmax

(10.6)

Then Eq. (10.4) looks as follows    ρ = 0, x ∈ R, t > 0. ρt + vmax ρ 1 − ρmax x

(10.7)

This is obviously the law of conservation of the number of vehicles. Indeed, integrating (10.7) with respect to x ∈ R, we obtain d dt



 ρ(x, t)d x = −x R

R

   ρ(x, t) ∂ d x = 0, vmax ρ(x, t) 1 − ∂x ρmax

(10.8)

10 Shock Waves in Traffic Flows

321

and, therefore, the number of vehicles in R is constant for any values t ≥ 0. We can build a macroscopic model where Eq. (10.6) is a special case [9]. Let us consider the relationship between the speed v and the density p of vehicles on the road. In a general case when density p rises, drivers reduce the speed and vice verse, therefore v = v{ρ(x(t), t)},

(10.9)

where x(t) is the flow element movement coordinate. Let us trace the time variation of the speed for some moving element of the flow, which is determined as the total time derivative dv ∂ρ dv ∂ρ d x dv = + . dt dρ ∂t dρ ∂ x dt

(10.10)

From (10.4), follows the below relation dρ dv ∂ρ = −ρ −v , dt dx ∂x

(10.11)

that looks as follows after its substitution into (10.10)         dv db dv ∂ρ dv ∂ρ dv dv dv = −ρ −v +v = −ρ . dt d x dρ ∂t dρ ∂t dρ d x dρ

(10.12)

According to (10.9), dv ∂ρ dv = , dt dρ ∂ x

(10.13)

the relation (10.12) can be rewritten as  2  2 ∂ρ dv dv ∂ρ = −ρ = −ρ v  , dt dρ ∂ x ∂x

(10.14)

 

2 where v = dv/dρ, and negative proportionality coefficient −ρ v  can be interpreted as viscosity in fluid. For a classical compressible fluid, Eq. (10.14) is called the Euler equation (ref. Chap. 1), in which case ∂ρ dv = −C 2 ρ −1 , dt ∂x where C is a non-negative constant with speed dimensionality. It is customary to consider a more general class of models in which

(10.15)

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10 Shock Waves in Traffic Flows

dv ∂ρ = −C 2 ρ n . dt ∂x

(10.16)

Equation (10.15) corresponds to case n = −1, so from Eqs. (10.14) and (10.15) it follows that v = C 2 p (n − l)/2. The solution for this equation will be v = C ln

ρmax , ρ

(10.17)

at n = −1, and v=

C  n+1/2 ρ − ρ n+1/2 n + 1 max

(10.18)

at n = −1. The model (10.17) was first obtained by Greenberg [10]. Denoting v0 as the speed at ρ = 0, we can write for values n ≤ 0



v = v0 1 −

ρ

n+1/2 

ρmax

.

(10.19)

Equation (10.6) first obtained by Greenshields [11] is a special case of Eq. (10.19) for n = 1. When building the model of kinematic waves of Lighthill-Whitham, the following assumptions were made [4]: • traffic flow is continuous, its density ρ(x, t) is the number of vehicles occupying a unit of road length; • flow value q(x, t) is the number of vehicles crossing the line x per unit of time and is defined by local density ρ: q = Q(ρ).

(10.20)

The flow speed is v(ρ) = Q(ρ)/ρ, i.e. the average speed is the function of density v(x, t) = ve (ρ(x, t)); • the number of vehicles on the road segment without exits/entrances is preserved (10.7). Equations (10.7) and (10.20) form a complete system. After substitution, we obtain ρt + c(ρ)ρx = 0, 

(10.21)

where c(ρ) = Q  (ρ) = ve (ρ) + ρve (ρ) is the perturbation speed. The relation Q(ρ) = ρve (ρ) plays an important role in the theory of traffic flows and is called a fundamental diagram (Fig. 10.2). In the Lighthill-Whitham model,

10 Shock Waves in Traffic Flows

323

Fig. 10.2 Fundamental diagram of traffic flow [1]

this dependence is continuous, so the maximum throughput capacity of the road segment is defined by the flow density. General view of the solution of nonlinear Eq. (10.21): c(x, t) = F(x − vt),

(10.22)

where F is an arbitrary function. Relation (10.22) describes a traveling wave considered as a compression wave in the medium. Waves of type (10.22) are called kinematic waves, which underlines their kinematic origin in contrast to the dynamic nature of acoustic and elastic waves. Shock Waves in Traffic Flow. An analysis of the models considered above has shown that there is an area of instability on curves q(v) [12]. Let us consider the Greenshields model (10.6) (case n = 1). Let speed v lies within 0 < v ≤ v0 ,

(10.23)

so that dq/dv > 0. If for any reason the speed of some portion of the flow falls down by v, the traffic intensity will decrease by ρ j (1 − 2v/v0 ) v. The density ρ of this flow portion will rise and the speed will further decrease. The speed perturbation is nonattenuating, which is demonstrated by the instability of the traffic flow behavior. In these cases, vehicles in the flow must frequently take off and stop. This phenomenon is called a shock wave (ref. Chap. 1). Equation (10.4) also demonstrates the presence of shock waves. Its solution was first proposed by Lighthill and Whitham (1955) and independently by Richards (1956). The analytical solution of Eq. (10.4) in a general case is complex and is not used in practical calculations. For a special case, on the road segment without exits/entrances, we can assume that q = f (ρ) or v = f (ρ) (equilibrium flow), i.e. v(x, t) = ve (ρ(x, t)).

(10.24)

Let us rewrite Eq. (10.4) as 

 d f (ρ) ∂ρ ∂ρ f (ρ) + ρ + = 0. dp ∂x ∂t

(10.25)

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10 Shock Waves in Traffic Flows

The function f (ρ) is, generally speaking, arbitrary. If we assume the relationship between speed and density to be linear [11], then Eq. (10.25) will look as follows   ∂ρ ρ ∂ρ v0 − 2v0 + = 0. ρj ∂x ∂t

(10.26)

Equation (10.25) is solved by the method of characteristics (ref. Chap. 1). Analysis of the solution to Eq. (10.26) leads to the following conclusions [12]: • density p is constant along the family of characteristics; • slope of characteristics dx dq = f (ρ) + ρ[ f (ρ)] = , dt dρ

(10.27)

equals the slope tangent of the flow density curve at the point representing the flow state at the boundary from which these characteristics emerge; • density at any point of the phase area (x, t) is found by drawing the inherent characteristics through this point. The intersection of the characteristics is explained by the presence of shock waves (ref. Chap. 1) since density has two values at the point of intersection, which is physically impossible. Mathematically, a shock wave is discontinuity of ρ, q or v. The shock wave velocity is defined by the slope of the line connecting two flow states (ascending and descending) vw =

qd − qu , ρd − ρu

(10.28)

where ρ d , qd represent the downward flow, and ρ u , qu —the upward flow. The shock wave moves downwards relative to the road when vw > 0, and it moves upwards if vw < 0. The application of the methods of mechanics of a continuous compressible medium to the study of traffic flows has been fruitfully developed in the creation of the so-called second-order hydrodynamic models [1, 2]. The matter is that the above models have the following constraints [1, 2]: • stationarity of the speed/density relation (the average movement speed at a certain density is established instantaneously); • oscillatory solutions describing the onset of instability in the form of regular startstop waves with amplitude-dependent oscillation time cannot be deduced from the equations of kinematic waves; • prevent from describing the phenomenon of hysteresis—return of the flow to the stable state at lower densities [13]. In a real flow, the density does not change intermittently. Drivers usually reduce speed when the density of vehicles in front increases, and vice versa. So q also

10 Shock Waves in Traffic Flows

325

depends on density gradient ρ x [3, 4]: q = Q(ρ) − vρx ,

(10.29)

where v is some positive constant value. Due to (10.21) and (10.29), we have ρt + c(ρ)ρx = vρx x , c(ρ) = Q  (ρ).

(10.30)

Multiplying (10.30) by c (ρ), we will rewrite it as follows ct + ccx = vc (ρ)ρx x = vcx x − vc (ρ)ρx2 .

(10.31)

In case of Q(ρ) approximation by linear function, c(ρ) will be linear in ρ, and c (ρ) = 0. Thus, Eq. (10.31) takes the form of the Burgers’s equation: ct + ccx = vcx x ,

(10.32)

where ccx member describes the formation of traffic jams—fast vehicles overtake slow ones, and a density discontinuity occurs. The vcxx member sets the final width of this discontinuity. Burgers’s Eq. (10.32) can be considered as a onedimensional Navier–Stokes equation for a compressible fluid with unit density. Nonlinear Eq. (10.32) is reduced to a linear equation of heat conductivity by Cole–Hopf substitution: c = 2v

ϕx ∂ ln ψ(x, t) = −2v . ∂x ϕ

(10.33)

When studying the properties of traffic flow, other versions of the Burgers equation are also interesting [14, 15]. A disadvantage of the Lighthill-Whitham model is an assumption of the equilibrium value of the speed V e for a given vehicle density. This does not allow adequately describing the situation near road irregularities (entrances, exits and narrowings). To describe nonequilibrium situations, it was proposed to use a differential equation for modeling dynamics of average speed instead of the deterministic equation V (x, t) = Ve (ρ(x, t)). The speed equation proposed by Payne in 1971 looks as follows [16] vt + vvx = −

c(ρ) 1 ρx + (Ve (ρ) − ρ), ρ τ

(10.34)

where c(ρ) = −

1 d Ve . 2τ dρ

(10.35)

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10 Shock Waves in Traffic Flows

Equation (10.34) was derived from the microscopic description of the movement of individual vehicles according to the drag race model. The addend vvx is called convectional and describes the change in the flow speed at a given place of the road due to the kinematic transfer of vehicles with the average flow speed from the previous segment of the road. The first addend in the right part is called pro-active and describes the tendency for speed reduction as the density increases. The most general form of the pro-active member looks as follows [17] −

c02 ∂x ρ. ρ

(10.36)

The second addend in the right part is called relaxational and describes the tendency for the average speed v to approach the equilibrium value V e (ρ) at a given density (τ is the characteristic relaxation time). The analysis of empirical data shows that at high densities, the laminar movement of the traffic flow becomes unstable and small perturbations lead to the emergence of start-stop waves. It is the stability in the linear approximation to small perturbations for all values of the density of the stationary homogeneous solution ρ(x, t) ≡ ρ 0 , V (x, t) = V e (ρ 0 ) of the Payne equation that is its substantial disadvantage. This disadvantage can be eliminated by the following change in the pro-active member of the equation: c(ρ) =

d Pe (ρ), dρ

Pe (ρ) = ρe (ρ).

(10.37)

Here, P is the internal pressure of the traffic flow, expressed through the variation of speeds in flow θ. Then the speed equation in case of such substitution will look as follows 1 1 vt + vvx = − ∂x Pe + (Ve (ρ) − v). ρ τ

(10.38)

Equation (10.38) describes drivers’ behavior depending on the pressure of the flow in front—deceleration when it rises and acceleration when it falls down. To evaluate the variation  as a function of density, various approximations obtained in the analysis of empirical data are used. For example, in Kühne and Kerner–Conhauser models [1, 2], a positive constant [5] e (ρ) = 0 is used as the first approximation. Equation (10.38) also predicts the emergence of shock waves. To prevent discontinuities, the right part is supplemented by the diffusive member vvxx —an analog of viscosity in hydrodynamics equations: vt + vvx = −

0 1 ρx + vvx x + (Ve (ρ) − v). ρ τ

(10.39)

10 Shock Waves in Traffic Flows

327

The analysis of the stability of a stationary homogeneous solution shows that for densities exceeding the critical value, the solution becomes unstable to small perturbations. This property allows modeling the occurrence of phantom jams— start-stop modes of waves in a homogeneous flow occurring as a result of small arbitrary perturbations. The standard model suggests an equation in the following form ∂v 1 ∂ L(ρ) μ ∂ 2 v ∂v +v = (V (ρ) − v) − c02 + . ∂t ∂x τ ∂x ρ ∂x2

(10.40)

The right part (10.40) contains three coefficients concerning the traffic flow speed. The first member reflects the flow tendency at a given density p to reduce the average speed V (ρ) to some natural value. At low densities, this speed is determined by road conditions and movement speed limitations and slightly depends on ρ. At high densities, V (ρ) approaches zero and slightly depends on ρ. At medium densities, it rapidly falls down and is greatly caused by the fact that drivers find it difficult to overtake in case of high flow density. Thus, we assume that V (p) will be a descending function with a small derivative at high and low ρ. The second advance factor means that drivers reduce speed if the traffic flow ahead has a higher density. The dimensionless function L(ρ) must be monotonously rising in this case. Usually, it is assumed to be equal ln ρ, and the value c02 ρ plays the role of pressure. The last member, viscosity or diffusion, reflects the tendency of movement speed coordination with the speed of surrounding vehicles in the flow. This is where this chapter is finished. In it, we made an effort to illustrate the efficiency of hydrodynamic methods (first of all, the shock wave model) to a seemingly irrelevant problem of traffic flows. Therefore, many other models [1, 2] used to solve this problem are not discussed. These are the so-called stochastic and macroscopic models, models of cellular automata, neural networks and other models [1, 2, 5] which allow describing many characteristic phenomena such as traffic jams, single, correlated and moving jams, start-stop movements, multiple stable states and chaos. Taking into account these modern ideas of traffic flow stages, the following situation is observed [1, 2, 5]: • traffic flow behavior is assimilated to phase transitions gas → fluid → freezing fluid → ice [19]; • phase transitions are caused by the properties of the traffic flow—the approach proved to be fruitful and led to the discovery of clusters on highways; • the effects of external factors on the flow behavior are considered as inevitable evil and acts as fluctuations—generalizations of various disturbing actions on the traffic flow; • researchers pay special attention to modeling the transition fluid → freezing fluid as the hardest to reproduce due to its nonlinearity; • the transport flow behavior is determined by the narrowings and widenings of the road, these concepts are interpreted in an extremely general way;

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10 Shock Waves in Traffic Flows

• the main object of the study must be queues (their proprieties, structures, behavior, etc.) occurring near narrowings. Despite serious efforts, the problem remains incompletely solved. According to [20], “the situation in this field is such that, despite significant progress, there is no complete understanding of the nature of traffic jams. Scientists say that they are closer to understanding the Universe origin processes than to the reduction of traffic jams.” One more thing: “Physics proposes a wide variety of methods to explain movement. There are still many open problems”, says the German physicist Kay Nagel, a key figure in the Los Alamos project (LANL) [1, 2, 17].

References 1. Semenov VV (2004) Mathematical modeling of traffic flow dynamics. Pre-Print of the Keldysh Institute of Applied Mathematics [Semenov V.V. Matematicheskoye modelirovaniye dinamiki transportnykh potokov, Preprint IPM im. M. V. Keldysha, 2004 (in Russian)] 2. Shvetsov VI (2003) Mathematical modeling of traffic flows. Autom Telemech No. 11 [Shvetsov V.I. Matematicheskoye modelirovaniye transportnykh potokov / Avtomatika i telemekhanika. 2003. № 11 (in Russian)] 3. Lighthill MJ, Whitham FRS (1995) On kinetic waves II. A theory of traffic flow on crowded roads. Proc R Soc Ser A 229(1178):317–345 4. Whitham J (1977) Linear and nonlinear waves. World, Moscow [Whitham J. Lineynyye i nelineynyye volny. — M.: Mir, 1977 (in Russian)] 5. Hate F (1966) Mathematical theory of traffic flows. World, Moscow, p 286 [Hate F. Matematicheskaya teoriya transportnykh potokov. — M.: Mir, 1966 — S. 286 (in Russian)] 6. Prigogine I, Andrews FC (1960) A Boltsman-like approach for traffic flow. Oper Res 8:789–797 7. Prigogine I, Herman R (1971) Kinetic theory of vehicular traffic. Elsevier, New York 8. Prigogine I (1961) A Boltsman-like approach to the statistical theory of traffic flow. In: Herman R (ed) Theory of traffic flow. Elsevier, Amsterdam 9. Inose H, Hamada T (1983) Road traffic management. Transport, Moscow [Inose H., Hamada T. Upravleniye dorozhnym dvizheniyem. — M.: Transport, 1983 (in Russian)] 10. Greenberg H (1959) An analysis of traffic flow. Oper Res 7:79–85 11. Greenshields BD (1934) A study of traffic capacity. In: Proceedings of (US) highway research board, vol 14, pp 448–494 12. Nagel K, Wagner R, Woesler R (2003) Still flowing: approaches to traffic flow and traffic jam modeling 13. Treiterer J, Myers JA (1974) The hysteresis phenomenon in traffic flow. In: Buckley DJ (ed) Proceedings of 6th ISTT. Artarmon, New South Wales, p 13 14. Krug J, Spohn H. Phys Rev A 83:4271 15. Binder PM, Paczuski M, Barma M. Phys Rev E 49:1174 16. Payne HJ (1971) Models of freeway traffic and control. In: Bekey GA (ed) Mathematical models of public systems, vol 1. Simulation Council, La Jolla, pp 51–61 17. Nagel K (1995) Particle hopping models and traffic flow theory. Los Alamos National Laboratory. Received 12 Sept 1995 18. Philips WF (1979) A kinetic model for traffic flow with continuum implications. Transp Plan Technol 5:131–138 19. Traffic jam dynamics in traffic flow models. In: 3rd Swiss transport research conference, STRC 03 conference paper, Monte Verita, Ascona, 19–21 Mar 2003

References

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20. Sipress A (1999) Studying the ebb and flow of stop-and-go; Los Alamos Lab using cold war tools to scrutinize traffic patterns Washington post staff writer, Thursday, 5 Aug 1999. www. science.com

Chapter 11

Fermi-Zeldovich Problem

Construction of Thermodynamically Complete Equation of State of Matter Based on the Results of Shock-Wave Measurements. Three equations of gas dynamics (1.4–1.6, Chap. 1) associate four continuum flow parameters ρ, P, ε, u [1, 2] as functions of coordinates and time. Introduction of the equation of state (EoS) E(p, ρ) obtained experimentally or from the statistical theory [1, 2] closes the system of equations of gas dynamics and makes it possible to calculate the flow field for given initial conditions. As we have already noted in Chap. 1, a reverse problem is also possible, when the flow field (ρ, P, ε, u) found from the experiment is used to find the equation of state of matter at extremely high pressures and temperatures—i.e. in the area of states where an ordinary thermophysical experiment is complicated, and the methods of theoretical physics are inefficient due to the absence of small parameters in perturbation theory [1, 3]. In a general form, this reverse problem of gas dynamics based on the reversal of the system of differential equations (Chap. 1) is currently unavailable even for most advanced computers and most sophisticated methods of computational mathematics. Instead of differential equations, self-similar solutions such as a plane stationary shock wave or a Riemann simple rarefaction shock wave are used in practice. For these self-similar solutions, conservation laws are written in simpler algebraic and integral forms (Chap. 1). In this case, in shock wave experiments, the experimenter must ensure the conditions of self-similarity [3–7]. The generated shock wave must be plane and stationary [1, 4, 5, 8]. In [3, 9], a method is presented for determining the equation of state (EoS) of deuterium plasma, using conservation laws in cylindrical and spherical geometries. However, there is one important specific feature related with the thermodynamic incompleteness of the equation of state obtained in this manner. This thermodynamic incompleteness of the caloric equation of state in the form of E(p, v) results from the fact that P, V are not conjugate thermodynamic variables for the internal energy E.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Fortov, Intense Shock Waves on Earth and in Space, Shock Wave and High Pressure Phenomena, https://doi.org/10.1007/978-3-030-74840-1_11

331

332

11 Fermi-Zeldovich Problem

Fermi [10] and then Zeldovich [11] proposed a smart method to solve this problem, based on the use of a thermodynamic identity—the first law of thermodynamics [1, 4, 5, 8, 10–14]. Following [1, 4, 5, 8, 11–14], we shall note that a specific feature of shock wave (dynamic) experiments for determining the equation of state consists in the fact that experimental data make it possible to obtain an equation of state of matter only in a thermodynamically incomplete form E = E(p, V ), since the internal energy is not a full thermodynamic potential relative to the variables p and V. Therefore, to construct closed thermodynamics, additional information about the temperature T = T (p, V ) [10–14] or entropy S = S(p, V ) is required, and it is extremely substantial to develop adequate equations of state of the shock wave medium. The thing is that dynamics research methods [3, 5] are adiabatic (ref. Chap. 1) and based on recording the mechanical parameters of hydrodynamical motion. They do not give information about the thermal or entropy characteristics of shocked matter. When a stationary shock discontinuity propagates through matter, the laws of conservation of mass, momentum and energy [5] are fulfilled at its front, which, in the most general form, relate the kinematic parameters—the velocity D (in the laboratory system of coordinates) of the shock-wave front and the mass velocity u of the matter behind its front—with thermodynamic parameters, specific internal energy E, pressure ρ and specific volume V: D−u Du V , p = p0 + = , V0 D V0 1 E = E 0 + ( p + p0 )(V0 − V ), 2

(11.1)

where index 0 indicates the parameters of matter at rest ahead of the shock-wave front. These equations help finding the hydro- and thermodynamic characteristics of the matter by recoding any two of the five parameters E, p, V, D, u describing the shock wave discontinuity. The shock wave velocity D can be easily and precisely measured by basic methods. The selection of the second measured parameter depends on specific experimental conditions. The analysis of the errors in relations (11.1) shows [3, 6, 7, 15] that in case of strongly compressible (gas and plasma [4, 5, 8]) media, it is reasonable to record the density ρ = V −1 of shocked matter. Currently, a technique for such measurements has been developed based on recording the absorption of cesium, argon and air of soft X-ray radiation by plasma [3, 7]. In case of lower system compressibility (condensed media), acceptable accuracies are ensured [7, 8] by recording the mass velocity u of the matter motion. Thus, the states of a degenerate plasma of metals and a dense Boltzmann plasma of argon and xenon were studied [1–3, 6, 7, 15]. In experiments for recording the curves of shocked matter isentropic expansion [3, 5–7, 15], the states in a centered unloading wave are described by Riemann integrals [3, 5–7, 15] (Chap. 1):

11 Fermi-Zeldovich Problem

p H V = VH +

333

du dp

2 dp,

p H  2 du E = EH − p dp, dp

p

(11.2)

p

that are calculated along the measured isentropy ps = ps (u). Conducting records under various initial conditions and shock wave intensities, we can determine the caloric equation of state E = E(p, V ) in the field of the p–V-diagram covered by the Hugoniot and/or Poisson adiabats. In the experiments carried out by now for dynamic effect on plasma, the change in shock wave intensity was done by varying the power of excitation sources—driving gas pressure, types of explosives, throwing devices and targets. Moreover, various methods were used to change the parameters of initial states: changing the initial temperatures and pressures (plasma of inert gases, cesium, fluids), using fine targets for the purpose of increasing the effects of irreversibility [1, 2, 5, 6, 8, 15]. Thus, the dynamics methods of diagnostics based on general conservation laws make it possible to reduce the problem of finding the caloric equation of state E = E(p, V ) to measuring the kinematic parameters of shock wave motion and contact surfaces, i.e. to record distances and times, which can be done with a high accuracy. As we noted, the internal energy is not a thermodynamic potential with respect to the variables p, V, so to construct the closed thermodynamics of the system, the additional dependence of the temperature T (p, V ) is required. In optically transparent and isotropic media (gases, ion crystals, etc.), the temperature is measured together with other parameters of shock compression. Condensed media and, first of all, metals are usually non-transparent for optical radiation, so the light effect of a shocked medium is inaccessible for recording. As per the proposal of Fermi [10] and Zeldovich [11], the thermodynamically complete equation of state can be constructed directly following the results of dynamic measurements without introducing a priori limiting assumptions on the properties and nature of matter under study [5, 11, 12], proceeding only from the first law of thermodynamics only and the dependence E = E(p, V ) known from the experiment. A linear homogeneous differential equation for T (p, V ) can be easily obtained:       ∂T ∂E ∂T ∂E − = T, (11.3) p+ ∂V p ∂p ∂p V ∂V the solution of this equation is constructed by the method of characteristics: p + (∂ E/∂ V ) p ∂p = , ∂V (∂ E/∂ p)V or

T ∂T =− ∂V (∂ E/∂ p)V

(11.4)

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11 Fermi-Zeldovich Problem

⎧ V ⎫ ⎨  ⎬ E = E 0 exp − γ (V, E)d ln V , ⎩ ⎭ V0

⎫ ⎧ V ⎬ ⎨   ∂ ln γ (E, V )  pV d ln V . ex p − T = T0 ⎭ ⎩ p0 V0 ∂ ln V

(11.5)

V0

Equations (11.3)–(11.5) are supplemented by the initial conditions: the temperature is set in the region of low densities where its reliable theoretical calculation is possible (cesium plasma) or it is known from a static experiment [12, 13]. The dependences E(p, V ) necessary for the calculation of right parts (11.4)–(11.5) are determined from experimental data in the form of power polynomials: E( p, V ) =

i

ai j pi V j ,

(11.6)

j

and for γ (E, V ) = PV /E(p, V )—in the form of fractional rational functions. Construction of Energy Surface E= E(p, V). Following [12] and abstracting from a specific experimental technique for determining the parameters of a medium under shock compression, we will assume that there is a certain number N of experN imental points {E i , Vi , Pi }i=1 , arbitrarily located in the P–V-plane. To construct the analytical dependence E = E(P, V ) using this data, we must solve the problem of regressive analysis for the functions of two independent variables. According to regular considerations, we use the least square method to construct a regression surface. Considering the case of two-dimensional parabolic approximation, we construct the equation of state (11.6) as follows: E(P, V ) =



akl V k P l .

k+l≤q

The degree of polynomial q must satisfy the condition 21 (q + 1)(q + 2) ≤ N . To find polynomial coefficients akl , we must solve the system of normal equations (Appendix, [13, 14]). Ibn so doing, difficulties emerge that are well known from the one-dimensional analog of the problem and related with the ill-conditioning of the matrix of the system of normal equations at larger N and q [14]. In this case, there are no efficient numerical methods to solve the system of linear equations with a determinant close to zero, since round-off errors in computer calculations substantially distort the results of the solution that is defined with a high loss of accuracy. These difficulties can be avoided when passing to the Chebyshev orthogonal polynomials (Appendix, [14]) by rearranging the members so that the orthogonality condition is fulfilled between individual groups. The normal system matrix is converted

11 Fermi-Zeldovich Problem

335

into a diagonal matrix and can be easily and quickly reverted without a high loss of accuracy. The calculation of the approximating polynomial by this algorithm is reduced to the multiple derivation of the scalar product with the following reduction of similar terms, which gives a solution in the form of (11.6). The degree of polynomial q is selected from the analysis of experimental data. An increase in the degree of the approximating polynomial leads to a decrease in the element of best approximation, which is achieved at the cost of an increase in the variance of the respective polynomial. It is necessary to successively increase the degree q by evaluating the newly emerging members each time according to Fisher’s statistical significance tests [13, 14]. This is especially convenient for orthogonal polynomials due to their property of inclusion by degrees according to which the results obtained in the previous step for q − 1 can be used to calculate the polynomial of degree q. Constructing the equation of state in the form of (11.6) with the use of experimental data, we can then calculate the right parts of the system (11.4) and integrate it using a particular numerical scheme. Usually, the system was integrated according to the Adams scheme; the initial sections of isentropies were calculated using the Runge–Kutta method [16]. When using the Monte Carlo method, determinant is the issue of accuracy with which the temperature can be found from the relationship E = E(P, V ) known from the experiment. Due to the complex dependence of the solution of Eq. (11.3) on experimental data (approximation (11.6) and solution of the characteristic system (11.4)), it seems reasonable to use the method of statistical testing to evaluate the accuracy. It should be noted that the maximum evaluation would be exaggerated, since errors in computer calculations are random and partially compensated. In essence, similar considerations are used in calculating queueing systems when determining the quality and reliability of complex systems [16–18]. Each measurement result is affected by a large number of hardly recorded random facts acting independently, so that the result itself is a random value that can be described by the respective distribution given on the multitude of implementations. For usual reasons, we come to the normal law for error distribution based on the Lyapunov theorem. According to the principles of mathematical statistics [18], speaking of experimental data, we should not consider only those results that are obtained in this experiment for a certain combination of random factors but also take into account the entire set of possible results that could have been obtained for a different combination of reasons causing random errors. It is accepted to consider the results of the first type as a sample (implementation) from the general aggregate of all possible results in the fixed set of external conditions. In the case under consideration, the Monte Carlo method [14, 16, 18] consists in modeling the probable structure of the measurement process by exhaustively searching for possible combinations of random factors leading to an experimental error and in determining the effect of this error on the considered solution using a

336

11 Fermi-Zeldovich Problem

N computer. The given experimental array {E i , Vi , Pi }i=1 is associated with the statisN  , where E˜ i are random values with the normal distribution tical array E˜ i , Vi , Pi i=1   density f E˜ i , having mathematical expectation E i and variance σ i = i /3 (99.9% is the confidence coefficient, i is the error of the ith experiment):



f E˜ i



⎧  2 ⎫ ⎪ ˜i ⎪ ⎨ ⎬ E − E i 1 =√ exp − . ⎪ ⎪ 2σi2 2π σi ⎩ ⎭

(11.7)

Such restructuring of arrays is made by the sensor of pseudorandom numbers distributed according to the law (11.7). N {E i , Vi , Pi }i=1 is used to construct an equation of state as (11.6) that is applied in solving the system of characteristic equations (11.4). This results in determining isentropies P˜s = P˜s (V ) and T˜s = Ts (V ) that are also random values with some distribution function. The mathematical expectation μξ (V ) and the variance σ ξ (V ) are calculated using formulas (ξ = P, T ) [14, 18]: μξ (V ) =  σξ (V ) =

a 1 ξi (V ), a i=1

  1/2 1  ξ˜i (V ) − μξ (V ) . a − 1 i=1

(11.8)

If the number of implementations a used for the evaluation of sought values is large enough, then, due to the law of large numbers, evaluations (11.8) obtain statistical stability (the order of variance of relations (11.8) is 1/a). As a result of calculations, we get the following dependences:   N , μ P,T = μ V, i , {E i , Vi , Pi }i=1   N σ P,T = 0 V, i , {E i , Vi , Pi }i=1 , describing the effect of experimental errors on the accuracy of determining the temperature along isentropy. We must note that the solution is not only affected by the experimental errors but also by the location of experimental points in the P– V-plane. This evaluation method allows finding the solution accuracy for any nature of the location of experimental points in the P–V-plane. It appears that the described method for constructing a thermodynamically complete equation of state is of general nature, since it uses the first principles of thermodynamics and continuum mechanics—conservation laws (11.1, 11.2) and primary thermodynamic identity (11.3)—and, hence, it is free from constraining assumptions about properties, nature and phase composition of the medium under

11 Fermi-Zeldovich Problem

337

p, Mbar

Fig. 11.1 Shock adiabats of various initial porosity m and cold curve (p0 ) of tungsten [1, 2, 6, 15]

study. This thermodynamic universality allows constructing equations of state using a single method for a wide number of condensed media and applying it to describe phase transformations [1–3, 6, 7, 10, 15, 19]. The method appeared to be especially efficient for studying thermodynamics of non-ideal plasma of cesium [1, 2, 6, 15] based on experiments for shock and adiabatic compression of saturated vapors of metals of base and transition groups, ion crystals, silicone oxide. The results of such calculation carried out for tungsten are given in Fig. 11.1.

Appendix Two-dimensional approximation of the caloric equation of state E= E(P, V) [14]. N with an accuracy determined According to [12] and knowing the set { f i , xi , yi }i=1 by the weights pi = 1/σ i (σ i is the absolute error), let us consider a model that is linear relative to the parameters: Pm =

m

Ck ϕk (x, y),

(11.9)

k=1

where {ϕk (x, y)}m k=1 is some selected system of linearly independent functions. The system of conditional equations looks as follows: Si = f i − Pm (x, y), i = 1, 2, . . . , N . Let us evaluate the parameters C R in (11.9). To do it, let us require the element of best approximation with respect to the variables C k to be minimal:

338

11 Fermi-Zeldovich Problem

S=

N

pi2 Si2 .

(11.10)

i=1

The condition is ∂S/∂C k = 0, which leads to a system of normal equations: [ϕ1 , ϕ1 ]C1 + [ϕ1 , ϕ2 ]C2 + · · · + [ϕ1 , ϕm ]Cm − [ϕ1 , f ] = 0, .......................................................................................... [ϕm , ϕ1 ]C1 + [ϕm , ϕ2 ]C2 + · · · + [ϕm , ϕm ]Cm − [ϕm , f ] = 0,

(11.11)

where [ϕk , ϕl ] =

N

pi ϕk (xi , yi )ϕl (xi , yi )

i=1

means the scalar product of the functions ϕk and ϕl on the set N. The solution (11.11) is the required values C k (11.9). Due to symmetry, the system (11.11) is solved on the computer using the square-root method. The Gramm determinant (11.11) differs from zero due to the linear independence ϕk (x, y). The following power functions were selected as ϕk : ϕ1 = 1, ϕ2 = x, ϕ3 = y, ϕ4 = x2 , . . . ; C1 = C00 , C2 = C10 , C3 = C01 , C4 = C20 , . . . . To pass to orthogonal polynomials, let us consider, along with {ϕk (x, y)}m k=1 , their linear combinations ψi =

m

ai j ϕ j (x, y)

(11.12)

j=1

The problem solution will be sought as follows Pm (x, y) =

m

bi ψi (x, y),

(11.13)

i=1

which leads to the system (11.11). Let us select the functions ψ i as mutually orthogonal:   ψi , ψ j = Ai j δi j where δ ij is the Kronecker symbol, and Aij is a non-negative number. The system (11.11) will have a diagonal matrix

Appendix

339

 2  ψ1 0   2  ψ 2   ..  .    0 ψm2

        

The expression (11.13) takes the form Pm(x,y) =

[ψi , f ] ψi (x, y). [ψi , ψi ]

To find a link between ψi and ϕi , we use (11.12) further defining aij by the formula  1, i = j, . We obtain the recurrence relation ai j = 0, i > j.   ϕk , ψ j . ak j ψ R , ak j = −  ψk = ϕk + ψj, ψj j=1 R−1

Thus, all coefficients are defined in the formula (11.12). These algorithms were used to compile computer programs that permit finding N the coefficients aij in (11.6) in accordance with the given data { f i , xi , yi }i=1 . The maximum degree of polynomial is determined by the computer random access memory and is q = 10 (66 coefficients).

References 1. Fortov VE (2012) Equation of state of matter. From ideal gas to quark-gluon plasma. FIZMATLIT, Moscow [Fortov V.E. Uravneniye sostoyaniya veshchestva. Ot ideal’nogo gaza do kvark-glyuonnoy plazmy. — M.: FIZMATLIT, 2012 (in Russian)] 2. Fortov VE (2016) Thermodynamics and equations of states for matter. From ideal gas to quark-gluon plasma. World Scientific, New York, London, Tokyo 3. Fortov V, Yakubov I, Khrapak A (2006) Physics of strongly coupled plasma. Clarendon Press, Oxford 4. Landau LD, Lifshits EM (1986) Course of theoretical physics. In: Hydrodynamics, vol 6. Nauka, Moscow [Landau L.D., Lifshits E.M. Kurs teoreticheskoy fiziki. T. 6. Gidrodinamika. - M .: Nauka, 1986 (in Russian)] 5. Zel’dovich YaB, Raizer YuP (2008) Theory of shock waves and high-temperature hydrodynamic phenomena, 3rd edn., corrected. FIZMATLIT, Moscow [Zel’dovich Ya.B., Raizer Yu.P. Teoriya udarnykh voln i vysokotemperaturnykh gidrodinamicheskikh yavleniy. 3-ye izd., ispr. — M.: FIZMATLIT, 2008 (in Russian)] 6. Fortov VE (2016) Extreme states of matter. High energy density physics, 2nd edn. Springer, Heidelberg, New York, London 7. Fortov V, Khrapak A, Yakubovich I (2004) Physics of non-ideal plasma. FIZMATLIT, Moscow [Fortov V., Khrapak A., Yakubovich I. Fizika neideal’noy plazmy. — M.: FIZMATLIT, 2004 (in Russian)]

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8. Al’tshuler LV (1965) Use of shock waves in high-pressure physics. Phys Usp 85(2):197 [Al’tshuler L.V. Primeneniye udarnykh voln v fizike vysokikh davleniy / UFN. 1965. T. 85, № 2. S. 197 (in Russian)] 9. Mochalov MA, Il’kaev RI, Fortov VE et al (2010) JETP Lett 92:336–340 [Mochalov M.A., Il’kaev R.I., Fortov V.E. i dr. Pis’ma ZHETF. 2010. T. 92. S. 336–340 (in Russian)] 10. Fermi E (1927) Rend Accad Nazl Lincei 6:602 11. Zel’dovich YaB (1957) JETP 32:1577 [Zel’dovich Ya.B. / ZHETF. 1957. T. 32. S. 1577 (in Russian)] 12. Fortov VE, Krasnikov YuG (1970) JETP 59(1):1645–1656 [Fortov V.E., Krasnikov Yu.G. / ZHETF. 1970. T. 59, № 1. S. 1645–1656 (in Russian)] 13. Fortov VE (1972) J Appl Mech Tech Phys 156 [Fortov V.E. / PMTF, 1972. S. 156 (in Russian)] 14. Buslenko NP (1968) Modeling complex systems. Nauka, Moscow [Buslenko N.P. Modelirovaniye slozhnykh sistem. — M.: Nauka, 1968 (in Russian)] 15. Fortov VE (2013) High energy density physics. FIZMATLIT, Moscow [Fortov V.E. Fizika vysokikh plotnostey energii. — M.: FIZMATLIT, 2013 (in Russian)] 16. Shchigolev BM (1969) Mathematical processing of measurements. Nauka, Moscow [Shchigolev B.M. Matematicheskaya obrabotka izmereniy. — M.: Nauka, 1969 (in Russian)] 17. Hemming RB (1968) Numerical methods. Nauka, Moscow [Hemming R.B. Chislennyye metody. — M.: Nauka, 1968 (in Russian)] 18. Hudson D (1967) Statistics for physicists. World, Moscow [Hudson D. Statistika dlya fizikov. — M.: Mir, 1967 (in Russian)] 19. Mochalov MA, Il’kaev RI, Fortov VE et al (2017) JETP Lett 151:592–620 [Mochalov M.A., Il’kaev R.I., Fortov V.E. i dr. Pis’ma ZHETF. 2017. T. 151. S. 592–620 (in Russian)]

Chapter 12

Shock Wave Stability

Hydrodynamic phenomena observed in nature must not only satisfy the conservation laws (Chap. 1) but also be stable towards small perturbations, ref. Chap. 1, as well as [1–5]. The stability of flat standard discontinuities in a medium with an arbitrary equation of state towards weak irregular periodical perturbations of the front were first studied by Dyakov [6] and then in [7–15]. By linearizing the hydrodynamic equations describing the motion behind the discontinuity front and taking into account the boundary conditions at the front, a characteristic equation was obtained the study of which allowed revealing various types of instabilities of shock wave discontinuities. The results of the performed analysis show that two types of instability are possible: absolute (or ripple) consisting in the exponential growth of the amplitude of periodic perturbations of the shock-wave front with time, and acoustic when front perturbations can exist infinitely long without damping and without increasing. In the latter case, the shock-wave front is a source of sound and entropy perturbations propagating downstream. The results of these papers were named, after the names of their authors, the Dyakov–Kantorovich linear theory. This theory helped formulating the respective shock wave instability criteria as per the stability parameter value  L= j

2

∂V ∂p

 , H

where j2 = (p − p0 )/(V 0 − V ) is the flow of matter through the shock-wave front, V = 1/p is the specific volume, p is the pressure, and the partial derivative is taken along the shock adiabat; the following areas differing in the quality nature of the evolution of small periodic perturbations of the shock-wave front are distinguished: (1)

stability area, where −1 < L < L s ;

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Fortov, Intense Shock Waves on Earth and in Space, Shock Wave and High Pressure Phenomena, https://doi.org/10.1007/978-3-030-74840-1_12

341

342

(2)

12 Shock Wave Stability

acoustic instability area, where L s < L < 1 + 2M;

(3)

absolute instability area, where L−1, L1 + 2M. The following designations are used here: Ls = θ=

1 − θ M2 − M2 , 1 + θ M2 − M2 V0 , V

M=

u . c

In more common variables, the conditions of absolute acoustic instability can be written as follows:   ∂V (V0 − V ) , (12.1)

∂P H c ( p − p0 )     ∂V V0 − V (V0 − V ) 1 − θ −2 (D/c)2 − θ −1 (D/c)2 −1 D 1 + 2θ . < < ∂P H p − p0 c ( p − p0 ) 1 − θ −2 (D/c)2 + θ −1 (D/c)2 (12.3) Here D is the modulus of the shock wave propagation velocity in a stationary medium, e.g. D = u0 . Thereafter, it was shown (ref. for example, [7, 8]) that the shock adiabat segments corresponding to the absolute instability lie within the areas of the ambiguous representation of the shock wave discontinuity. In these areas, the initial shock wave can decay forming an aggregate of stable elements (shock waves, isentropic noncollapsing rarefaction and compression shock waves, contact breaks) that do not catch up with each other. In this connection, an assumption was made in [7, 8] that the absolute instability is not implemented in practice due to the decay (decay instability) of the initial shock wave forming one of the possible wave configurations. It is now useful to give some facts referring to the problem of shock wave stability. As given in [1], the point of tangency between the shock adiabat and the straight line passing via the initial point on the p–V diagram is acoustic, i.e. a point where the Mach number behind the shock wave in the coordinate system associated with the front turns into one:

12 Shock Wave Stability

343

M=

|u − D| D−u = = 1. c c

Correspondingly, the following relation is fulfilled for the tangent of the tangential line slope angle to the Poisson adiabat: 

∂p ∂V

 =− s

j2 = − j 2. M2

By definition, at the point of tangency 

∂p ∂V

 = − j 2, H

where from it follows that the point of tangency between the Rayleigh straight line and the shock adiabat is also the point of tangency between the shock adiabat and isentropy: 

∂p ∂V



 = −j = 2

S

∂p ∂H

 , H

and the stability parameter L that characterizes the evolution of small perturbations in the Dyakov–Kantorovich theory takes a value equal to −1 at the point of tangency between the Rayleigh straight line and the shock adiabat:  L= j

2

∂V ∂p

 = −1. H

Thus, the points tangency of the Rayleigh straight line and the shock adiabat are the boundaries of the region of absolute instability. The inequality L < −1 is fulfilled in a gap between these points. The point of tangency corresponding to the lowest pressure is also the lowest boundary of the region of ambiguous representation (decay instability) of the shock wave after which the initial wave disintegrates into a combination consisting of shock and isentropic compression waves. On smooth adiabats, the condition of thermodynamic abnormality (Bether–Weil condition) 

∂2 p ∂V 2

 0. This form of the shock adiabat in the p–V-plane is typical for plasma where the internal degrees of freedom are excited as the temperature rises (ionization, dissociation, oscillatory, rotational, electronic excitation, etc.) [2, 3]. Figure 12.2 gives a shock adiabat of tungsten H TF (in the megabar region of pressures) from [19], calculated under approximation formulas of the Thomas–Fermi

12 Shock Wave Stability Fig. 12.3 Shock adiabat of copper in the two-phase region calculated using the semi-empirical equation of state [3–5]: cross-hatching—instability region (12.3); continuous bold curve—phase boundary

347

p, 105 Pa

v0 = 5 cm3/g

v0 = 150 cm3/g

v, cm3/g

quasiclassical theory. The wave line in Fig. 12.2 designates the lower boundary of the instability region (12.3) while the upper boundary lies beyond the applicability limit of the quasiclassical model of matter due to the spontaneous creation of electron– positron pairs in such plasma [4, 5]. It’s worth noting that taking into account the energy and pressure of equilibrium radiation gives shock compression stable within the entire range of pressures. The shock adiabats of a two-phase liquid–metal vapor mixture calculated under the semi-empirical equation of state [4, 5] of copper are given in Fig. 12.3. The reverse travel of shock adiabats and acoustic instability (12.3) caused by it are determined by phase transformation processes. The same region of instability was later found for water and magnesium. A bend of shock adiabats, determined by thermal ionization turns out to be especially significant for a non-ideal cesium plasma [21] since the first and second potentials of ionization differ greatly in this case—3.89 and 25.1 eV, respectively. Experiments [22, 23] for the compression of gaseous Ar, Xe, CO2 by plane shock waves using the interferometer technique recorded the development of instabilities in plasma behind the wave front. The effects observed in these papers are related with the reverse travel of shock adiabats, though the criterion of acoustic instability was achieved in none of the studied cases (12.3). It is possible that the irregularities observed in [22, 23] are related with the kinetic phenomena of plasma dissociation and ionization. It is shown in [17] that shock waves can lose their stability in case when the Hugoniot adiabat crosses the phase boundary so that condensate forms behind the shock-wave front. Currently, there is no understanding what the occurrence of possible sound emission by the discontinuity means. Probably, this is a specific type of instability that can lead to front deformation or plasma “plug” decay behind the shock-wave front. In this case, a solution of a higher order than obtained in [19–26] is required. Below

348

12 Shock Wave Stability

we present the results of numerical modeling of the dynamics of a plane stationary shock wave propagating through matter and described by an equation of state of a special form. The parameters of this equation of state were selected so as to ensure the fulfillment of criteria (12.1)–(12.3) for the upper segments of the shock adiabats, provided the conditions of matter thermodynamic stability are fulfilled within the entire region of parameters. The calculations of a strictly nonlinear stage showed the development of perturbations on the wave front in case of situation (12.3) and the formation of a two-wave structure when criterion (12.1) is fulfilled while the shock wave was rather stable beyond these criteria. The analytical consideration of region (12.3) is given in [7, 8] where the possibility of shock-wave front breaks is noted. Paper [1] presents a smart analysis of the occurrence of the so-called ripple (absolute) instability of a plane shock wave propagating via a long tube of variable crosssection the area S of which slightly changes along the x length. This model of wave propagation allows using the so-called hydraulic approximation. That is to say, this model allows considering all values in the flow to be constant along each crosssection of the tube, and the velocity—to be orientated along its axis. In other words, the flow is considered as quasi-one-dimensional. This approach leads to a simple link between the change in the velocity δv1 of the shock wave and the rate of change δS in the tube cross-section area:   1 δs (v1 − v2 + c2 ) 1 + 2v2 /c2 − h = − , (12.4) s δv1 1+h (v1 c2 )  2 2 j / p2 d p¨2 d V2 = j2 . (12.5) h=− dp2 dp2 The coefficient in square brackets of this equation is positive. Therefore, the sign of the ratio δv1 /δS is determined by the sign of expression (12.4). For all stable shock waves, this sign is positive and δv1 /δS < 0. In case any condition of ripple instability is fulfilled, the expression in brackets becomes negative: δv1 /δS > 0. This result allows for the illustrative explanation of the origin of the instability. Figure 12.4 shows a ripple surface of the shock wave moving to the right; arrows schematically show the direction of flow lines. When the shock wave moves, the area δS grows on the surface areas protruding forward and falls down on lagging areas. For δv1 /δS < 0, this results in a deceleration of the protruding areas and an acceleration of the lagging ones so that the surface tends to be smoothed out. On the opposite, in case of δv1 /δS > 0, surface shape perturbation will increase: the protruding areas will go father, and the lagging ones will lag even more. Papers [24, 25] give a quasi-one-dimensional analytical analysis of ripple instability referred to by Landau and Lifshits [2] in Vol. VI of the course of theoretical physics [1]. Let us carry out a quasi-one-dimensional analysis of the shock wave stability and consider, following papers [24, 25], the problem of the time-dependent behavior of an inhomogeneity at the front of an initially plane stationary shock wave the parameters of which will be designated by index 1. Considering that the deviations

12 Shock Wave Stability

349

Fig. 12.4 Scheme of ripple instability [1]

of the shape of the shock-wave front from the plane are small, let us write the motion equations of a non-viscous non-heat-conductive matter behind the shock wave in the quasi-one-dimensional approximation: A (x) 1 = 0, u t + uu x = − px , A(x) ρ pt + upx − a 2 (ρt + uρx ) = 0,

ρ1 uρx + ρu x + ρu

(12.6)

where A is the cross-section area of the considered flow tube; the lower index indicates differentiation. Let us assume that the shock wave propagates in the positive direction of the x-axis. Then, linearization (12.6) relative to unperturbed values and integration of the obtained equations give p − p1 + p1 a1 (u − u 1 ) = − p1 a12 u 1

A(x) − A + F[x − (u 1 + a1 )]. A(u 1 + a1 )

Let us assume the invariant F = 0. This corresponds to the fact that perturbations do not come to the shock wave. Then, the first of the obtained equations can be used to close the Rankine Hugoniot relations that look as follows in the linearized form ρ0 δ D = −ρ1 δu 1 − u 1 δρ1 + ρ1 δ D + Dδρ1 , 2ρ0 Dδ D = δp1 + 2ρ1 (D − u 1 ) + (D − u 1 )δρ1 ,

350

12 Shock Wave Stability

where δD, δu, δp, δρ are the variations of the shock wave velocity; velocity, pressure, density behind the shock-wave front. The inequality δD/δA > 0 complies with the unstable situation when the shock wave velocity grows along the flow tube. On the other hand:   2 j − (1 + 2M)(∂ p/∂ V ) H 1 ρ1 a1 u 1 δD = , δA j 2 + (∂ p/∂ V ) H A (ρ0 − ρ1 )(u 1 + a1 )

where j = ρ1 (D − u 1 ); ∂∂Vp = dpdH − V ; p H = p H (ρ) is the Hugoniot adiabat H equation; M = (D − u1 )/a1 is the Mach number. Thus, the shock wave is unstable when   ∂p j 2 > (1 + 2M) , (12.7) ∂V H   ∂p 2 −j < . (12.8) ∂V H Criteria (12.7) and (12.8) coincide with the criteria of absolute instability [19, 27, 28]. In this case, criterion (12.8) is obtained in a one-dimensional analysis of stability and corresponds to an abrupt rise of compressibility (for example, as a result of a phase transition), which leads to the emergence of several shock waves separated by areas of constant flow or continuous compression waves [2, 4, 5]. Condition (12.7) corresponds to a strong bend of the shock adiabat and is rather hard to be implemented in practice. It is stronger than the condition 

∂V ∂p

 > H

V0 − V , p − p0

(12.9)

corresponding to shock compression ambiguity [3]. Let us now use the Whitham equation for the angle of inclination of the shock wave normal θ to the x-axis ∂ 2θ A d D ∂ 2θ + = 0. ∂x2 D d A ∂ y2 It is seen that we have the d’Alembert equation in a stable situation (dD/dA < 0). In the unstable case, the problem of the evolution of the shock wave shape is equivalent to the Cauchy problem for the Laplace equation being an incorrect problem of mathematical physics. To analyze shock wave stabilities in media with an arbitrary equation of state, it would be natural to go beyond the frames of analytical approaches and employ numerical modeling. It seems even more natural because the instabilities considered in this case are proved by the analysis presented in [25] to have no threshold values

12 Shock Wave Stability

351

Fig. 12.5 Shock wave with perturbation at its front [24, 25]

p1, ρ1, u1

p0, ρ0, u0 = 0 D + δD

along the lengths of perturbation waves and are implemented for all wave lengths (including those that exceed the calculating spatial grid in size). The numerical modeling of the stability of plane shock waves was first done in papers [24, 25] (ref. Fig. 12.5) and we will follow them below. As we noted in Chap. 1, a general description of continuum mechanics is based on three laws of conservation supplemented by the equation of state that contains all information on the properties of the medium under study. Therefore, to reproduce the instability criteria in calculations, the construction of an adequate equation of state reproducing the criteria (12.1)–(12.3) is of crucial significance. Let there be a functional dependence of the form E = E(p, V ), where E is the internal energy, p is the pressure, V is the specific volume, and E, together with its first and second derivatives, is limited in any final subregion of the first quadrant of the plane (p, V ). It is asserted that if Ep > 0

(12.10)

and the square of the velocity of sound a2 =

V 2 (E v + p) , Ep

(12.11)

then, we can always suggest such function S = S(p, V ) having the sense of entropy that the following inequations are true  T > 0,

∂T ∂S

 v

 > 0,

∂T ∂V

 < 0,

(12.12)

T

where T is the temperature responsible for the thermodynamic stability of matter [26]. The entropy S is constant along each Poisson adiabat that, by definition, has a slope dp/dV = −a2 /V 2 at an arbitrary point (p, V ) in space. Assume that S(0, V ) = f (V ) at p = 0. Let us formulate the requirements imposed on the function f (V ) the fulfillment of which makes inequations (12.10), (12.11) true.

352

12 Shock Wave Stability

Fig. 12.6 Hugoniot adiabat constructed on the basis of the equation of state (12.14). Regions in p–V-coordinates: 0–1—stability; 1–2—two-wave structure; 2–3—acoustic instability; 3–4—absolute instability meeting criterion (12.8) [25]

By definition of temperature  T =

∂E ∂S

 v

=

Ep ; Sp

hence 

∂T ∂S

 v

=

  E pp s p − E P S pp E pv S p − E p S pv ∂p , = . 3 Sp ∂V t E pp S p − E p S pp

Let us designate ξ as the crossing point between an arbitrary Poisson adiabat and the V axis. As noted above, for any point of a given adiabat S = S(ξ ); then S p = Sξ ξ p , S pp = Sξ ξ ξ p2 + Sξ ξ pp , S pv = Sξ ξ ξ p ξv + Sξ ξ pv .

(12.13)

Let g(ξ ) be such that g(ξ ) > 0, gξ ξ < 0. Then it is easy to show that by assuming f (ξ ) = g(λξ ) and selecting a sufficiently high value of λ, we can always satisfy inequalities (12.12). We can make sure that if we take



2 2 4 − e−(V −4) , (12.14) E( p, V ) = 1 − e−P then conditions (12.12) are met, and the functional dependence E(p, V ) can be used as one of the equations of state of a thermodynamically stable system. Figure 12.6 gives a Hugoniot adiabat in the p–V coordinates, constructed numerically with the use of the equation of state (12.14) and expression for a shock adiabat. Charts show the regions of stability, two-wave structure, acoustic instability and absolute instability.

12 Shock Wave Stability

353

Fig. 12.7 The same as in Fig. 12.6 in p–u-coordinates

Fig. 12.8 Profiles of velocity u, pressure p and internal energy E obtained during piston retraction into the medium with the equation of state (12.14) for U = 0.58

Numerical Modeling of Shock Wave Instability. The calculation of the nonlinear stage of development of perturbations in a medium with the equation of state (12.14) was done by the numerical solution of two-dimensional unsteady-state equations of conservation of mass, momentum and energy. The situation was considered when a shock wave was created by a sudden pushing of a piston into a tube at constant velocity U. To model such process, a through finite-difference method was used, which is based on the fully conservative scheme “predictor–corrector” having the second order of approximation on a rectangular grid. A quadrilateral was used as a discrete element. In this case, the velocities and displacements were calculated for the grid nodes and the density, pressure and internal energy—for the cell center. The counting oscillations were suppressed using quadratic and linear bulk artificial viscosities, which gave smooth profiles of parameters behind the shock wave with front blurring for about three counting cells. According to the selected piston speed U based on the Rankine–Hugoniot equations of preservation and equation of state (12.13), we can determine the flow parameters behind the shock-wave front and find the shock adiabat in p–u-variables (Fig. 12.7). The profiles of longitudinal velocity, pressure and internal energy are given in Figs. 12.8, 12.9 and 12.10, where the following values were given for the piston

354

12 Shock Wave Stability

Fig. 12.9 The same as in Fig. 12.8 (U = 1.65)

Fig. 12.10 The same as in Fig. 12.8, but with a piston speed of U = 2.56

speed U: 0.58—stability region; 1.65—region of two-wave structure (12.1); 2.56— region of acoustic instability (12.4). The graphs of the distribution of the parameters of the flow field occurring during piston retraction show that the computational algorithm quite well simulates the real process of compression; the physical values behind the shock-wave front, calculated by the through method are agreed with the values calculated directly from the Hugoniot relations and equation of state (12.13). For a piston speed of 1.65 (Fig. 12.9), a two-wave structure [2, 21] of the shock wave is seen, which occurs thanks to the shock adiabat break at V  3.5 located in the p–V-plane below the Rayleigh straight line. Figure 12.7 shows that for piston speeds from 2.4 to 2.8, three values of p correspond to the given value of u. However, the calculations implement the lowest value of p, which is well seen in Fig. 12.10. To study the shock wave stability with respect to small perturbations in a lateral direction to the piston motion line, a density perturbation was set at the initial instant of time as follows    2π y , ρ = ρ0 1 + ε sin L

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355

Fig. 12.11 Development of density perturbation at a certain instant of time: 1—U = 0.58; 2—1.65; 3—2.0; 4—2.4

where ρ 0 is the unperturbed value of density; L is the tube length in the lateral direction; ε is the perturbation amplitude. Figure 12.11 shows a development of density perturbation at a specific instant of time for piston speeds of 0.58; 1.65; 2.0; 2.4. Small perturbations of density (ε = 0.1) set at the initial instant of time for a piston speed of 0.58, which corresponds to the stability region, do not rise with time. For piston speeds of 2 and 2.4 giving states from the region of acoustic instability on the p–u-curve (Fig. 12.7), the rise of density perturbations is observed. The rate of perturbation rise increases when approaching to the region of absolute instability (Fig. 12.11). Thus, a method [25] has been proposed for constructing thermodynamically stable equations of state (12.14) permitting unstable shock waves. Shock adiabats are constructed in p–V and p–u coordinates for the obtained equation of state, where the regions of stability, two-wave structure, acoustic and absolute instability are determined. Numerical calculations [25] show that a shock wave for the equation of state (12.14) is unstable with respect to two-dimensional perturbations in the region of acoustic instability. On the basis of the consideration of the time-dependent behavior of the nonhomogeneity of the initially plane shock wave at the front, conditions for the instability of the shock wave were obtained which coincide with conditions (12.1). Numerical calculations of the piston motion in the medium described by the obtained equation of state (12.14) show the formation of a two-wave structure if the piston speed gives states falling into the region of absolute instability (12.7). The superposition of small perturbations in the transverse direction to the piston motion line shows that density perturbations in the region of acoustic instability grow with time, and the growth rate of perturbations increases when approaching the region of absolute instability. Numerical Study of Shock Wave Instability in Thermodynamically Non-Ideal Media. The authors of paper [29]b, based on the Euler equation, studied the problem of shock wave stability in hard deuterium (the equation of state was taken from SESAME [30, 31] library) and in the model medium with the equation of state (12.13)

356

12 Shock Wave Stability



2 2 e( p, ρ) = 1 − e− p 4 − e−(4−1/ρ) , used in [25]. This equation of state is thermodynamically correct and has the property that the shock adiabats constructed on its basis contain regions with all known types of instability. The real three-phase SESAME equation of state given in a tabular form describes the first-type phase transition from the molecular phase of hard deuterium to the metallic state. The shock adiabat constructed with the use of the model equation of state is characteristic for shock compression processes accompanied by phase transitions or chemical reactions with heat absorption. This shock adiabat is characterized by two types of instability: decay and acoustic. The section of this adiabat corresponding to comparatively low pressures is shown in Fig. 12.1a. The shock adiabat given in Fig. 12.1b corresponds to the shock compression of low-temperature deuterium from the initial state p0 = 1.6 × 103 GPa, V 0 = 0.7 cm3 /g and has two break points at which the adiabatic compressibility changes abruptly, first it decreases (point 1) and then increases (point 3). Considering a curve with a break as a limit case of a smooth curve, we can say that the region of thermodynamic abnormality extended in the first case turns out to be contracted to a point in the case of adiabat with a break. In the same sense, we will discuss tangency at the break point. Numerical Method. For the numerical integration of the Euler equations, a generalization of the Roe method [32] is used for an arbitrary equation of state of the form ε = (p, ρ). This method has the following properties: (1) (2) (3) (4)

it is conservative; it uses a precise characteristic splitting of flows; a stationary discontinuity where the Rankine Hugoniot conditions are met is a precise solution of finite-difference equations and the computationally economic formulation of the method is applied using the vector Y = (p, p1/2 , u)T [33].

In accordance with approach [33], to calculate the solution vector on the time layer (n + 1), integral laws of conservation are used that are resolved relative to the grid functions of pressure, velocity, p1/2 :

1/2   (z n )2 u n + ξ2 2 z n + ξ1 , u n+1 =  2 , z n+1    1  n n 2  n+1 n+1 2

 z u + ξ3 , − z u ε p n+1 , ρ n+1 = ε p n , ρ n + 2 z n+1 =

where z = p1/2 , ε is the internal energy per unit of volume, ξ =−

 τ Fi+1/2 − Fi−1/2 ; h

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357

• the total flow of conservative values through the faces of the computational cells, τ and h are integration steps in time and space, respectively. The approximation of the flow vector on the faces using the pressure-density-velocity variables is written in a more compact manner than using conservative variables:

Fi+1/2

⎛

⎞ ρu F = ⎝ p + ρu 2 ⎠, ρu H ⎛ ⎞ ⎞

p ρ

ρ − 2 c 2c ⎜ ρ ⎟ ρ(u+c) ⎠ , α = ⎝ ρc − u ⎠, 2c ρ(H +uc)

ρ + u 2c ρc

1 = Fi + Fi+1 + Ri+1/2 i+1/2 , 2 ⎛

R=⎝

1 u ερ + 21 u 2

ρ 2c ρ(u−c) 2c ρ(H −uc) 2c

λ = (u, u − c, u + c)T , λ± =

λ ± |λ| . 2

The elements ϕ l of the vector j+1/2 look as follows  l ϕi+1/2

=

gi1

+

1 gi+1

−ψ

1 λi+1/2

 1 − gi1 1 gi+1 + αi+1/2 ; 1 αi+1/2

– TVD1 (first-order approximation method):  1  1 l αi+1/2 ; = −ψ λi+1/2 ϕi+1/2 – TVD2 (Harten TVD-scheme of the second-order approximation [34]):

gii =

  1 1 − |σ |  1 1 ψ λi+1/2 m αi+1/2 , αi−1/2 2

• UNO3-scheme using ENO-interpolation of the third-order approximation (RP3 [35]):

  1 − |σ |  1 ψ λi+1/2 m a 1j+1/2 a 1j−1/2 2 ⎧

m  a 1 1

, − j−1/2 + a j−1/2 ⎪ 2−3|σ |+σ 2 1+ 1− σ 2 −1     ⎪ λ + λ ⎨ i+1/2 1 i+1/2 6 6  ≤ a 1 , if ai−1/2 i+1/2 +

m a1 1 ,

⎪ − j+1/2 + a j+1/2 2−3|σ |+σ 2 1− 1+ σ 2 −1 ⎪     ⎩ λi+1/2 + λi+1/2 6 1 6  ≤ a 1 , if ai−1/2 i+1/2

gil =

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12 Shock Wave Stability

σ =

τ λ1j+1/2 h 

m=

 , m[x, y] = x, |x| ≤ |y| y, |x| > |y|

0, x y ≤ 0, min (|x|, |y|)sign (x), x y > 0,  |x|, |x| ≥ ε, ψ(x) = x 2 +ε2 , |x| < ε, 2ε

(12.15)

where ε is the entropy correction parameter [34]. For two states designated by indexes 1 and 2 and corresponding to adjacent cells, the approximated solution of the problem of the decay of discontinuity is determined by the relations ρ = z1 z2 , u = H=

z1u 1 + z2 u 2 , z1 + z2

z 1 H1 + z 2 H2 h − ερ , c2 = . z1 + z2 ερ

The derivatives of the equation of state are determined by numerical differentiation using states2 1 and 2: ερ =

0.5 (ε( p˜ 2 , ρ˜2 ) + ε( p1 , ρ˜2 ) − ε( p˜ 2 , ρ1 ) − ε( p1 , ρ1 )), ρ˜2 − ρ1

ερ =

0.5 (ε( p˜ 2 , ρ˜2 ) + ε(ρ˜2 , p1 ) − ε( p1 , ρ˜2 ) − ε( p1 , ρ1 )), ρ˜2 − ρ1  | p2 − p1 | ≥ p1 δ, p2 , p˜ 2 = p1 + p1 δ, | p2 − p1 | < p1 δ,  |ρ2 − ρ1 | ≥ ρ1 δ, ρ2 , ρ˜2 = ρ1 + ρ1 δ, |ρ2 − ρ1 | < ρ1 δ,

where δ is a small positive value. This calculation method for derivates guarantees the following relation to be fulfilled at a strong discontinuity ε2 − ε1 = ερ (ρ2 − ρ1 ) + ε p ( p2 − p1 ) and, therefore, the relation

F = A U,

2 An approach similar to the solution of Glaister [36] who was the first to apply a method of introducing mixed states to calculate pressure derivatives for the extension of the Roe method to the arbitrary equation of state.

Fig. 12.12 Change in the deviation modulus for solutions in the region of decay instability. Along the x-axis—number of the time step

359

Deviation modulus

12 Shock Wave Stability

where A is the Jacobi matrix of the flow vector for an averaged state. In this case, an arbitrary stationary jump satisfying the Hugoniot conditions at the discontinuity, i.e. determined by the grid function having the form  Uin

=

U1 , i ≤ i 0 , U2 , i > i 0 ,

where F(U 2 ) − F(U 1 ) = 0, i0 is a certain value of grid index, which is a precise solution of difference equations3 . A weak solution of the Euler equations is not the only one in the region of ambiguous representation. Since the initial shock wave considered in the coordinate system related with the front is a precise solution of the discrete model, low perturbation of initial data either attenuates, which corresponds to a stable shock wave, or grows exponentially, which corresponds to the development of decay instability, i.e. a transition to an alternative self-similar solution. The solution of discrete model equations is not unique in this case. An example of the change in the deviation modulus in the region of instability for all used schemes is given in Fig. 12.12. Saturation corresponds to reaching the decay solution. Studying Shock Wave Instability Relative to Decay. The Cauchy problem for Euler equations was solved in the coordinate system moving at the front velocity, and the initial shock wave is stationary on the calculational grid. Solutions for various points of the shock adiabat are presented in Fig. 12.13. For each point on the shock adiabat that is characterized by pressure in the finite state P, a self-similar solution 3 Only in case if the Harten entropy correction parameter (12.15) equals zero. The entropy correction

procedure is intended to exclude a solution corresponding to the rarefaction shock wave, which is permitted in the Roe method. In the context of the task under consideration, the problem of occurrence of rarefaction shock waves in the numerical solution does not arise in the region of

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12 Shock Wave Stability

Fig. 12.13 On the left: results of one-dimensional calculations of the shock wave decay—pressure distribution depending on the variable x/t for the shock waves corresponding to different points (P values) on the shock adiabat given in the figure on the right. On the right: shock adiabat (continuous line) for the model equation of state [16]

for pressure is represented in the x/t variables. The following points are marked on the adiabat: 0—initial state; 1—the first point of tangency of the Rayleigh straight line, the lower boundary of the region of absolute instability and the lower boundary of the region of instability with respect to decay; 2—the upper boundary of the region of thermodynamic abnormality; 3—the second point of the tangency of the Rayleigh straight line, the upper boundary of the region of absolute instability and the lower boundary of the region of Dyakov–Kantorovich acoustic instability; 4—crossing point between the shock adiabat and a straight line drawn from the initial state and tangential to the shock adiabat at point 1, the upper boundary of instability of the shock wave with respect to decay. The solution in the plane (P, x/t) is visualized by equal pressure lines. Contour pressures lines can be determined from the P-axis values. normality of the thermodynamic properties of the medium. There is another symmetric problem of compression shock waves in the region of abnormality.

12 Shock Wave Stability

361

Below, for the sake of brevity, we will call such method of presenting the results of studying the shock adiabat as the decay diagram. The respective section of the shock adiabat with the marked characteristic points is shown on the right. The position of the lower boundary of the region of ambiguous representation in the calculation coincides with the beginning of the region of absolute instability L = −1 (point 1). The shock wave is stable up to this point. In interval 1–2, there is a decay of the initial shock wave into a two-wave configuration that is a shock wave and an isentropic compression wave adjoining it:

Below point 2, we have a decay into a configuration of two shock waves which are divided by an isentropic compression wave:

Point 3 corresponds to the end of the region of absolute instability L = −1 and is simultaneously a starting point of the region of acoustic instability L = L s of the initial shock wave; respectively at this point M = 1. The difference in the shock wave propagation velocities can be defined as the difference in the positions of the fronts on the x/t axis. When moving along the shock adiabat towards increasing P, the intensity of the first shock wave remains unchanged while the intensity and velocity of the second shock wave increase and the pressure amplitude in the isentropic compression wave decrease. At point 4, the velocities of the first and second waves are equal, the amplitude of the isentropic compression wave becomes zero. Consequently, point 4 is the upper boundary of the region of shock wave instability relative to decay. In contrast to the decay configurations for the shock adiabat having a lengthy area of thermodynamic abnormality, the decay scheme for the shock adiabat with a break point (Fig. 12.14b) contains no isentropic compression wave (ICW) and, within the entire region of instability relative to decay, looks as follows

For comparison, Fig. 12.14 gives self-similar solutions for melting at point P = 0.54 for the model equation of state (a) and for deuterium at P = 3.6 × 103 GPa (b). A pressure range corresponding to the isentropic compression wave is observed. The difference in the velocities of the first and second shock waves after initial shock wave decay in deuterium is ~ 2.8 × 103 m/s. The calculations were checked using various numerical methods apart from those described above, including Eulerian and Lagrangian approaches. Good agreement was obtained between all these calculations and the calculations made earlier [37]. Comparison of Results for Several Finite-Difference Schemes. Need for Entropy Correction. Methodological calculations using TVD schemes of the

362

12 Shock Wave Stability

a

b

ICW

for adiabat Fig. 12.14 On the left: shock wave decay according to the scheme without a break point (model equation of state). On the right: shock wave decay according to the scheme for adiabat with a break (deuterium, SESAME) [16]

first and second orders of accuracy and a scheme using one-dimensional ENOinterpolation of the third order of accuracy (RP3 [35]) to calculate flows on the faces of cells showed that it is required to introduce entropy correction into the TVD schemes to get a physically correct solution satisfying the entropy increase law in the region of the abnormality of the thermodynamic properties of the medium. Only the UNO3 scheme out of all tested schemes gives a right solution without introducing entropy correction. For example, in case of no entropy correction, the first-order scheme gives an erroneous decay solution with two shock waves without isentropic compression . wave instead of decay according to the scheme On the other hand, high order approximation schemes give a reduced value of the experimentally found upper boundary of the shock wave instability relative to decay. The situation changes for the better when additional diffusion is introduced into the scheme. To do it, the Harten entropy correction mechanism was used (12.15). For the entropy correction parameter of 0.125ω, where ω is the spectral radius of the Jacobi matrix of the flow vector, the solution quality in the region of the upper boundary of instability is satisfactory. The decay diagrams constructed using various schemes for the shock adiabat, calculated on the basis of the model equation of state for the above value of the entropy correction parameter (Fig. 12.13) are given in Fig. 12.15. Modeling of Acoustic Instability Inside the Region of Shock Wave Instability Area Relative to Decay. Acoustic instability was observed in the calculations in the following cases: (1)

as the instability of the initial wave relative to decay with the finite state on the shock adiabat above point 4 (ref. Fig. 12.1);

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363

Fig. 12.15 Comparison of decay diagrams obtained with the use of various finite-difference schemes

(2)

(3)

during a limited time period inside the region of decay instability if the acoustic instability develops faster than the decay instability. In this case, after waves I and II diverge to a distance exceeding the length of the sonic waves, residual perturbations attenuate and two stable shock waves are formed; as the acoustic instability of the second shock wave in the decay configuration.

Let us consider the results of numerical analysis corresponding to the latter case of those listed above. The initial shock wave corresponds to point P = 0.82 for the shock adiabat given in Fig. 12.13. At this point, the conditions of the Kantorovich acoustic instability and the conditions of instability relative to decay are fulfilled simultaneously for the initial wave. Figure 12.16 presents the dependences of the stability parameter L and the critical value L s for two shock adiabats: the initial shock adiabat with the origin at point 0 and the shock adiabat constructed for waves I and II. In Fig. 12.16a, the vertical lines designate the pressures corresponding to the initial shock wave and wave I, in Fig. 12.16b—those corresponding to wave II. As the figure shows, for the initial shock wave and wave II formed after the decay of the initial shock wave, the acoustic instability condition is fulfilled: L s < L < 1 + 2M. The block on the left part of the figure shows that there are two close solution roots of the equation L(P) = L s (P), which means the presence of a small interval of acoustic instability adjoining point 1 from in Fig. 12.16b. The decay of a shock wave into a configuration including stable and unstable shock waves is implemented in a computational experiment. The problem of shock wave decay is solved in a two-dimensional setting in the coordinate system related with the front of the initial shock wave on a computational grid 400 × 300. The initial data approximate the front segment of the unstable shock wave under study

364

12 Shock Wave Stability

Fig. 12.16 Dependence of the stability parameter L and the critical value Ls on the pressure behind the shock-wave front P: a for the shock adiabat of the initial wave, the points corresponding to the initial shock wave (0) and the first wave in the decay configuration (I) are marked; b for the shock adiabat of the second wave, a point corresponding to the amplitude of the second shock wave (II) is marked

and the adjacent regions of homogeneous flow. A decrease in velocity by 1% was used as a perturbation in the center of the integration area in one cell adjoining the shock-wave front. Neither selection of the perturbed component nor the perturbation amplitude have a qualitative effect on the further flow pattern. The numerical solution at P = 0.82 for several instants of time with the integration step corresponding to the Courant number of 0.3 is shown in Fig. 12.17 that presents the fields of the absolute value of the pressure gradient (darker areas correspond to a higher value of the pressure gradient at this point). The times given in the figure are conditional since the problem has no specific scale. The surface of the second (right) wave provides the nodal points of a three-wave configuration—the break points of the shock-wave front with a downstream weak perturbation wave, which move along the front. The values the define the acoustic instability—stability parameter, Mach number behind the shock-wave front and the ratio of specific volumes—are as follows for the calculation variant: L = −0.59, M = 0.83, θ = 2.8. Equation for the cosine of the angle between the perturbation wave vector and the positive direction of the x-axis [1]

12 Shock Wave Stability

365

Fig. 12.17 Field of the absolute value of the pressure gradient. Darker regions correspond to the higher values of the pressure gradient

366

12 Shock Wave Stability



   4 3 + M2 2 + θ − 1 cos α + 2M − 1 cos α M 1+L 1+L     2 1 + M2 − 1 + θ M2 = 0 + 1+L 2

in this case has solutions (−1, −0.66). The second root lies in the range −M 2 < cos2 α < 1, which corresponds to the diverging sonic wave the wave vector of which makes an angle of ±131° with the positive direction of the x-axis. Three close consecutive instants of time depicted in the right part of Fig. 12.17 illustrate the motion of sonic wave fronts and show good agreement with the theoretical value. Thus, in [16], for the first time in a numerical calculation, the shock wave decay was obtained in the region of ambiguous representation of a configuration containing stable and unstable (acoustic instability) shock waves. The direction of the propagation of perturbation sonic waves corresponds to the linear theory results. The analysis of self-similar solutions using a decay diagram in variables (P, x/t) confirms the theoretical results [7, 8] for the solution structure and decay boundaries under the schemes

A computational experiment for modeling the shock wave instability relative to decay using the generalized Roe method (the approximated solution of the problem of discontinuity decay for an arbitrary equation of state) identified the need of introducing an entropy correction to obtain physically valid solutions near the upper boundary of the instability region as well as solutions satisfying the condition of entropy increase in case of isentropic compression waves in the decay configuration. The thermodynamic analysis carried out in [3] shows that the stability criteria of plane shock waves (12.1)–(12.3) can be disturbed for a medium described by the van der Waals equation of state in the region of high-temperature boiling. The evolution of neutrally stable shock wave was modeled in [38] in the van der Waals polytropic gas (cυ = 30R, where cυ is the specific heat capacity at constant pressure, R is the individual gas constant). The propagation of shock waves in a channel with a small sudden narrowing of cross-section was considered. When passing through the narrowing region, the shock wave was perturbed and its front became a source of acoustic perturbations. The calculated value of the angle of slope of sonic waves to the channel axis (shock wave motion direction) matched the prediction of the linear theory with high accuracy. In general, this paper was of a demonstration nature and contained no analysis of the effect of nonlinearity on the shock wave behavior in the neutral stability region. Thus, the results of theoretical analysis and computer modeling of the behavior of neutrally stable shock waves were obtained within the framework of Euler equations, using the equation of state of real media (van der Waals gas, magnesium) and an approach was developed, which was used as a basis to find the region of shock wave neutral stability for each pressure value in front of it from the analysis of the equation

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367

of state. A simple algorithm has been created that allows finding a source of acoustic perturbations (shock-wave front or external source) directly from the flow pattern. It is noted that, unlike linear theory predictions, the perturbation amplitude of a neutrally stable shock wave decreases with time, though this process occurs much slower than in case of an absolutely stable shock wave. Realizability of neutral stability condition (12.3) in a specific medium can be revealed by direct verification of its fulfillment on selectively constructed shock adiabats, but this way is rather time-consuming and has no required generality. Meanwhile, the presence on the phase diagram of a region of shock wave neutral stability for a family of shock adiabats with a certain value of the initial pressure p0 and its boundary (if it exists at all) can be determined beforehand from the analysis of the equation of state. Let us describe an algorithm of such analysis using the example of a generalized equation of state of the van der Waals type (ref. for example, [39]):

p+

a

(V − b) = RT, Vn

(12.16)

where T is the temperature, a, b are constant values, n is the parameter. The medium is assumed to be polytropic with constant molecular mass, i.e. cv = const, R = const. For n = 2, Eq. (12.16) goes over into the standard van der Waals equation; for n = 5/3, into the second Dieterici equation. Transforming (12.16) into caloric form e=

a

a cυ p + n (V − b) − R V (n − 1)V n−1

(12.17)

and obtaining the shock adiabat equation and expression for the velocity of sound, let us express the shock wave stability parameter L in the form L = −1 +

1 − M12 +

1

2 (γ

1 − M12 + 1)V1 + 21 (γ − 1)V0 − b

−1

M12

where γ = 1 + R/cv . For M 1 < 1 (one of the conditions of shock wave evolutionism), L < 0, i.e. the right inequality in the van der Waals gas in condition (12.1) is always fulfilled. After simple transformations, the left part of condition (12.1) can be written as a γ p0 V1 p1 < p0 + (n − γ ) V11−n − naV11−n − b b which allows, for each family of shock adiabats with the initial pressure p0 , identifying a region of the p–V-diagram with the areas of shock adiabats meeting the condition of the shock wave neutral stability. As an example, Fig. 12.18 presents the case of van der Waals polytropic gas with n = 2, γ = 1.2 (the pressure in front of the shock wave p0 = 0.1pc —the initial point 0 of the shock adiabat is located at the boundary of the two-phase region).

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12 Shock Wave Stability

Fig. 12.18 Position of the region of the neutral stability of shock waves in van der Waals polytropic gas for n = 2, γ = 1.2, p0 /pc = 0.1: 1—boundary of the hyperbolicity region of Euler equations; 2—boundary of the neutral stability region; 3—spinodal; 4—binodal. The A–B region of the shock adiabat corresponds to neutrally stable shock waves [40]

Hugoniot adiabat

Due to the problem of identifying a neutral stable shock wave in the experiment, let us recall some conclusions of the linear theory of shock wave stability based on the search for a solution in the form of acoustic and entropy-vortex waves consistent with the conditions at the discontinuity. The perturbation of entropy and vorticity is generated at the shock-wave front at the nodal point and drifted to the region of the behind-the-front flow. The acoustic component of this solution is a plain sonic wave of an arbitrary frequency the slope of which to the unperturbed initial shock wave makes the angle α (ref., for example, [1]):

cos α =

L − 2 − M12 +



1 − M12

2

1/2    + L 2 − 1 1 − M12 θ + M12 θ 2 (L + 1)2

M1 (4 + (L + 1)(θ − 1)) (12.18)

In case of cos α < −M 1 , the sonic wave energy flow in the coordinate system associated with the discontinuity is directed towards the shock wave, i.e. stationarity is maintained from the outside. For cos α > −M 1 , the source of sound is a shock wave. The latter condition defining the range of spontaneous sound emission can be transformed into the form 1 − M12 (1 + θ ) = L A < L. 1 − M12 (1 − θ ) From (12.18), it follows that in the vicinity of a sound point (L → −1, M 1 → 1), sound is emitted in the direction of the shock wave motion (cos α → −1). In case of cos α > 0, the following condition is fulfilled

12 Shock Wave Stability

369

1 + (2 − θ )M12 = LB < L. 1 + θ M12 The range L B < L < 1 + 2M 1 is interesting because in this case the nodal point from which the acoustic and entropy-vortex waves emerge along the shock-wave front at a velocity vD = c1 (1 + M 1 cos α)/sin α exceeding the velocity of sound on both sides of the discontinuity. The position of the fronts of the acoustic and entropy-vortex waves is related with a simple criterion that makes it possible to identify the reason for acoustic perturbations (shock-wave front or external source) directly from the photograph of the flow structure, i.e. from the distribution of density the gradient zones of which are present in the sound and entropy-vortex waves. Let us consider an acoustic wave propagating in the positive direction of the y-axis in a coordinate system in which the three-wave configuration with a weak outgoing wave is stationary. Let us assume that (u, v) are the components of the gas flow velocity after the shock wave in the system of coordinates under consideration. If the vector n with components (nx , ny ) is a unit normal to the acoustic wave, then the condition nz c1 + u1 > 0 at which the acoustic wave is outgoing, taking into account the condition of stationarity (in the coordinate system related with the nodal point) nx u1 + ny v1 + c1 = 0 is written as ny u1 − nx v1 > 0. Consequently, a weak acoustic wave is outgoing (the shock wave emits sound) if and only if the angle β between the wave vectors of acoustic and entropy-vortex waves (or between waves as such) is below 90° (components of the normal to the entropy wave (−v, u)). Let us consider the behavior of a perturbed neutrally stable shock wave in van der Waals gas (12.16) for n = 2, γ = 1.4. Here and in the following calculations, it is assumed that the discontinuity is a mathematical surface and the effect of the structure and thermodynamic equilibrium are neglected. The numerical method of finite volumes is used as a discrete analog of the weak solution of the Euler equations. The flows on the cell faces are approximated using the Yang ENO-scheme [41]. The Roe method generalized for an arbitrary equation of state ensures the exact fulfillment of conditions on a stationary discontinuity [32, 36]. To decrease the variance error and select thermodynamically acceptable discontinuities, the Harten entropy correction is used [34]. Time integration is carried out using the Runge–Kutta method of the third-order approximation [42]. The degree of discretization of the computational region was selected from the condition of ensuring the convergence of the solution for all relevant elements of the flow structure. The calculations were carried out on a rectangular grid sizing 800 × 600 with a permanent step along both spatial coordinates. The initial conditions comply with the harmonic perturbation of the stationary shock wave shape ξ (y) = ξ 0 cos (ky), ξ 0 = 0.1. Caloric equation of state (12.17) was reduced to the dimensionless form    n + 1 1−n cv n + 1 −n n−1 Vr − pr + − Vr V2 , er = r n−1 n+1 (n − 1)2 where pr = p/pc , V r = V /V c , and the specific internal energy refers to the value pc V c .

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12 Shock Wave Stability

Fig. 12.19 Decay of shock wave 2D perturbation in ideal gas t = 0; 2; 5 [40]

The following medium parameters before and after the shock wave were given:  ( p, V, u) =

(0.005, 40, 0.933), x < ξ (y), (0.02, 12.45, 0.290), x ≥ ξ (y).

For comparison, a shock wave in an ideal gas with the adiabat indicator γ = 1.4 and the following parameters was considered  ( p, V, u) =

(0.005, 40, 1.4), x < ξ (y), (0.04, 11.43, 0.4), x ≥ ξ (y).

The following boundary conditions were given: supersonic conditions at the inlet boundary, non-reflective conditions at the outlet boundary, periodicity conditions at side boundaries y = const. In both cases, the decay of the initial perturbation initiates a perturbation of the shock-wave front and behind-the-front flow. Solution components (pressure, density, etc.) at each point behind the shock-wave front experience the underdamping oscillations caused by the interference of the acoustic waves propagating in the positive and negative directions of the y-axis. The oscillation period equal to λ/2vD , where λ is the perturbation wave length in the initial data, is a natural time scale for the comparison of shock wave decay characteristics (it should be noted than the velocities of the compared shock waves differ). Figures 12.19 and 12.20 present a flow pattern at successive times for an ideal gas and van der Waals gas, respectively. The pressure maximum in case of the ideal gas is located at the nodal point on the shock-wave front. In case of the van der Waals gas, the pressure maximum is located on the acoustic wave, and the minimum is implemented in the centers of the vortexes formed at the initial stage of the flow evolution. Figures 12.21 and 12.22 show the dependence of the acoustic perturbation intensity in the flow behind the front for absolutely stable (ideal gas) and neutrally stable (van der Waals gas) shock waves. As a characteristic of the decay process, the value

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371

Fig. 12.20 Decay of the 2D-perturbation of a neutrally stable shock wave in van der Waals gas t = 2; 5; 8

Fig. 12.21 Decay of shock wave perturbations for L > L s (upper curve) and L < L s (lower curve) [40]

of (pmax − p1 )/p1 is taken where pmax is the maximum pressure in the flow behind the front. It can be seen that the perturbations behind the neutrally stable shock wave decay but much slower than after an absolutely stable shock wave. The authors of paper [43] constructed a wide-range equation of state for magnesium and established the existence of the shock adiabats on which the condition of neutral stability is fulfilled. The analysis of this equation of state using the above algorithm has found the following specific features of the implementation of condition (12.1) (Fig. 12.22). There are two families of embedded areas (L L A ) first of which corresponds to shock waves with the finite state in the two-phase “liquid–gas” region and is located in Fig. 12.22 below the curve of phase coexistence. Some boundaries of this family

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Fig. 12.22 Boundaries of the regions L > L A in magnesium for different values of the initial pressure p0 . Shock adiabats with the initial state in the two-phase (A) and one-phase (B) regions having the sections L > L A (A, B) and L > L B [40]

coincide with the boundary of the two-phase area. The pressure maximum p0 (the common point of regions L L A for various p0 ) is achieved in the near-critical region on the side of the liquid phase and amounts to ~ 0.05 GPa. In the two-phase region, the implementation of the L > L B condition is possible at which the nodal points propagate along the shock wave surface at the velocity exceeding the velocity of sound after the shock wave. The respective section A1 − A2 is shown on the shock adiabat with the initial state p0 = 24 bar, v0 = 55 cm3 /g (point A). It should be noted that the length of the regions L > L A of shock waves with the finite state in the one-phase region in the given variables is much longer than for the simple models of real gas. Thus, in the van der Waals gas model, the pressure p1 behind a neutrally stable shock wave at any heat capacities and initial pressures does not exceed 27/8pc , and the maximum temperature does not exceed the Boyle temperature T b = 27/8T c . At the same time, the respective values in magnesium are dozen times higher (Fig. 12.22). This high difference is apparently explained by the effect of the degeneracy of the gas of thermal electrons. A common point of the regions L > L A of the second family (shock waves with the finite state in the one-phase region) is characterized by the initial pressure p0 ~ 1 GPa. The behavior of a neutrally stable shock wave in magnesium (Fig. 12.22) was modelled using the same model as for the van der Waals gas. Point B (Fig. 12.22) with parameters p0 = 0.5 kbar, V 0 = 35 cm3 /g located in the one-phase region was selected as the initial point of the shock adiabat. The finite state characterized by a pressure of 10 kbar belongs to the neutral stability region the boundary of which is shown in Fig. 12.22a with a short-dash line. As we can see, the decay of the initial perturbation resulted, as in the previous solutions, in two groups of the secondary waves propagating in the opposite directions. Figure 12.23

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Fig. 12.23 Identification of a neutrally stable shock wave by the position of acoustic and entropyvortex waves: a ideal gas, L < L s , pressure visualization; b magnesium, L > L s , pressure visualization; c magnesium, L > L s , density visualization (here sw is a weak shock or acoustic wave)

illustrates the above-formulated rule of angles making it possible to find the source of acoustic perturbations of the behind-the-front flow. The position of the front of entropy perturbations, determined from the nodal point velocity and flow velocity behind the shock wave is in good agreement with the calculated position of the entropy-vortex wave (Fig. 12.23). The short-dash line shows the position of the entropy wave front, while the continuous line shows a perpendicular to the entropy-vortex wave. As these pictures show, the acoustic waves related with the nodal point are located within the angle π /2 counted from the position of the front of entropy perturbations, which indicates that the sonic waves are outgoing. For comparison, Fig. 12.23a shows a solution of a similar problem for ideal gas. The value of angle β formed by the fronts of acoustic perturbations (dash-dotted line) and entropy perturbations (short-dash line) exceeds π /2. This means that sound falls on the shock wave, which must take place at L < LA . Thus, the authors of paper [40] made a number of computational experiments for modeling the behavior of neutrally stable shock waves. The evolution of a neutrally stable shock wave in van der Waals gas under a periodic perturbation of its front was numerically studied within the 2D formulation. In the course of time, a characteristic lattice of sonic waves is formed behind the front of the perturbed shock wave. The sonic waves propagate at the angles ±α predicted by the linear theory to the direction of gas motion in the shock wave. Calculations using a wide-range equation of state

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Fig. 12.24 Results of parametric one-dimensional calculations at the weak structural perturbation of the shock-wave front [45]

for magnesium confirmed the results of the theoretical analysis carried out in this paper and in paper [44] concerning the possible existence of neutrally stable shock waves in this metal. In general, in all the considered cases, the sound emission by a shock wave was stimulated, and the spontaneous (not caused by external actions) generation of acoustic waves was not observed. At the same time, the decay of acoustic perturbations in the behind-the-front flow of such shock waves takes place much more slowly than in the case of absolutely stable shock waves, which may be practically important in many applications. The authors of paper [45] made a numerical study of the stability of plane shock waves based on the equations of viscous and heat-conducting gas in the two-dimensional and three-dimensional formulations. The authors used a modification of the equation of state from [25], which reproduced the absolute instability (12.2)       e( p, V ) = 1 − exp − p 2 + εV p 2 4 − exp −(4 − V )2 ,

(12.19)

and made it possible to overcome the mentioned challenges to a certain degree (ε is a small positive parameter the variation of which allows modeling the S-shaped bend of the shock adiabat) (in the p–u variables) and modifying the length of the section (L > 1 + 2M). Considered was the shock adiabat with the coordinates of the initial point p0 = 0.1, V 0 = 5.49 constructed on the basis of (12.19) for ε = 10−3 (Fig. 12.24). At the EF

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375

section, the condition of shock wave instability towards small periodic perturbations is fulfilled. At the boundaries of this section L = 1 + 2M, which means that (∂u/∂p)H = (∂u/∂p)8 , i.e. points E and F are the tangency points of the shock adiabat and the respective Poisson adiabats. Point D is characterized by the fact that the shock adiabat drawn from it (shown with a dotted line in the figure) touches the initial shock adiabat in point F. The Poisson adiabat touching the initial shock adiabat in point E crosses it in point G. It should be noted that the shock adiabats having the initial points at the section DG and isentropies in this region are geometrically close, so the shock adiabat coming from point E crosses the initial shock adiabat near point G and the isentropy passing through point F crosses the initial shock adiabat near point D. From the analysis of crossings of secondary shock adiabats and isentropies with the shock adiabat under consideration, it follows that the solution of the problem of shock wave discontinuity is ambiguous inside the fragment of the shock adiabat DG including the EF section of the shock wave instability L 1 + 2M. Except for a ←

→

single shock wave, this solution can be represented in the form of S T S at the DE ← → ← ← → → section, S T S or S T R at the EF section and S T R at the FG section (here S is the shock wave, T is the contact discontinuity, R is the rarefaction wave, the arrow indicates the wave motion direction relative to the contact discontinuity). The issue of realizability of a particular solution in the region of its nonuniqueness was studied numerically within the model of viscous heat-conducting gas using the equation of state (12.18). The problem was solved in a non-dimensionless form in 1D, 2D and 3D formulations. The viscous stress tensor and heat flow density were determined by the expressions   2 1 T ∇ ⊗ v + (∇ ⊗ v) − ∇ · v , τ= Re 3 q=−

1 ∇i, RePr

(12.20) (12.21)

where i is enthalpy. (12.19) neglects the second viscosity, the model nature of the expression (12.20) is related with the caloric form of the equation of state (12.18). To determine nonviscous flows through cell boundaries, a TVD-scheme of the second order of accuracy was used with differences against the flow, constructed on the basis of [32] with extension [36] in the event of an arbitrary equation of state. The approximation of the members of the equations related with accounting for viscosity and heat conductivity was done within the framework of the standard symmetric scheme of the second order of accuracy. The introduction of physical diffusion and dissipation allows suppressing the influence on the solution of numerical effects (to a certain degree) related with the shock wave motion along the grid but does not define, as noted in [27], the selection of a specific wave configuration in the region of solution nonuniqueness. In the calculations, irrespective of the dimensionality of the problem, grids with constant step h were used. The size of the step h = 0.002 and the grid Reynolds number Reh = 3.5 were selected from the condition of the shock

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Fig. 12.25 Results of individual one-dimensional calculations for a weak structural perturbation of the shock-wave front [45]

wave structure resolution (at least 10 grid steps), the Prandtl value Pr was assumed to be 1. Within the framework of a one-dimensional formulation, two problems are considered with regard to the behavior of the shock wave discontinuity in the region of its ambiguous representation, differing by the form of the specified perturbation. In the first of them, the initial conditions looked as follows:  (ρ, u, p) =

(ρ0 , u 0 , p0 ), x ≤ −δ, (ρ1 , u 1 , p1 ), x ≥ δ,

(12.22)

where the parameters with indexes 0 and 1 meet the Hugoniot relations. A shock wave moves through a fixed medium in the negative direction of the x-axis. A plane monotonous smoothing of the discontinuity is set within the interval δ < x < δ, which can be considered as a weak structural perturbation of the shock-wave front at the initial instant of time. The value δ was set equal to 10−2 (five steps of the grid). Nonreflective boundary conditions were set at the input and output bounds of the calculational region. The results of the parametric calculations made in this formulation are given in Fig. 12.25 that shows self-similar solutions for pressure at different points of the shock adiabat. The self-similar coordinate ξ = x/Dt is plotted at the x-axis, where D is the propagation velocity of a unperturbed shock wave relative to a stationary medium. The pressure behind the shock-wave front is plotted at the y-axis. The pressure value in the flow field is shown by color as per the attached color scale. Figure 12.25 presents the results of individual calculations at different intensities of the initial shock wave.

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377

Fig. 12.26 Results of one-dimensional calculations in case of parameter perturbation behind the shock wave [45]

It can be seen that a decay of the initial shock wave discontinuity is implemented at the DF section with the formation of two diverging shock waves (it is interesting that this exact configuration is assumed to be more probable in the private Wendroff statement cited in [27]). The shock wave does not decay and remains stable at the FG section. In the second of the considered one-dimensional problems, the initial conditions differed from those described above with a weak (but final) perturbation of parameters behind the shock wave specified in the form of a weak compression or rarefaction wave coming to the shock-wave front from the side of the shock-compressed gas:  (ρ, u, p) =

x ≤ −δ, (ρ0 , u 0 , p0 ), (ρ1 + ρ, u 1 + u, p1 + p), x ≥ δ.

The set perturbation of pressure | p/p1 | ≤ 0.05, density and velocity was determined in the acoustic approximation:

p =

p

p , u = − − 1, p1 c1 c12

where c1 is the isentropic velocity of sound behind the front of an unperturbed shock wave. The obtained results are given in Fig. 12.26. For p > 0, the solution looks the same as in the above case: At the DF section, the shock wave decays forming the ←

→

configuration S T S and remains unchanged at the FG section (Fig. 12.26a, b). For p < 0 within the EG interval, the shock wave decays into diverging shock wave and rarefaction wave (Fig. 12.26b, c), and at the DE section, no decay occurs (Fig. 12.26a). The short-dash line corresponds to the initial shock wave, and the

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deviation of continuous curves from the short-dash line characterizes the perturbation intensity. The obtained results show that the reflection coefficient of finite perturbations coming to the shock wave from compressed matter is positive, including the range of L > 1 + 2M. This contradicts with the results of the linear theory [6–13] according to which the reflection coefficient of acoustic waves falling normally on the shock wave surface must be negative in this range (ref. for example [8]). The discrepancy is due to the fact that of the three possible solutions of the discontinuity decay problem, it is not the initial shock wave that is implemented, as implied in the linear theory, but one of the other two alternative solutions. A compression wave is reflected by the compression wave with the transition of the point corresponding to the initial shock wave to the FG section, while the rarefaction wave is reflected by the rarefaction wave with the transition to the DE section. In both cases, the reflection coefficient is much greater than unity and the solutions can be interpreted as forced shock wave decay initiated by the finite perturbation coming to the shock wave. It should be noted that large values of the reflection coefficient allow for one-dimensional oscillations of shock wave parameters with the alternative transition of a shock wave from the lower stable section DE to the upper section FG and back provided there is a reflective surface behind the shock-wave front, in particular, a piston. However, as shown below, such transitions can be related not only with normal waves but also with transverse waves propagating along the shock wave surface from the side of compressed matter. Let us consider the behavior of the shock wave in the region of its ambiguous representation in 2D and 3D formulations. The initial conditions of the problem were set in the following form:  (ρ, u, p) =

(ρ0 , u 0 , p0 ), x ≤ ξ − δ, (ρ1 , u 1 , p1 ), x ≥ ξ + δ,

(12.23)

where ξ is the spatial perturbation of the shock-wave front set as a periodic function of the transverse coordinates: ξ = ξ (y) = A cos (2π y) (2D setting) or ξ = ξ (y, z) = A cos (2π y) cos (2π z) (3D formulation), the perturbation amplitude A = 0.02. As in the onedimensional case, the discontinuity was smoothed by a plane monotonous function in the interval ξ − δ < x < ξ + δ. Non-reflective boundary conditions were set at the input and output bounds of the calculational region, and periodic conditions were set at side bounds. The calculations were carried out in the rectangular computational region. The Reynolds value Reλ = 1750 measured along the period length. A solution example is shown in Figs. 12.27 and 12.28. The shock wave parameters correspond to a point on the shock adiabat located at the DE section in the region of the ambiguous representation of the shock wave discontinuity with the theoretical probability of decay into two diverging shock waves. Figure 12.27 shows the initial stage of the solution (left scale in pressure terms, right scale in density terms). At concavity points of the front (relative to the direction of shock wave propagation), the shock-wave front starts decaying under the action

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379

Fig. 12.27 The initial stage of two-dimensional solution in case of periodic shock-wave front perturbation (in pressure terms—on the left, in density terms—on the right) [45]

of the weak perturbations coming from the adjacent sections of the front, into the ←

→

configuration S T S . The decay is diagnosed by an increase in pressure behind the shock-wave front to the values corresponding to the FG section. No decay at the initial stage is observed at the sections of front concavity where weak rarefaction waves come. This results in the formation of the transverse waves propagating along the shock-wave front from the side of shocked matter. An abrupt “switching” of local parameters occurs at the nodal point of the three-wave configuration, behind the shock-wave front from the DE section to the FG section bypassing the range L > 1 + 2M. In the rear part of the transverse waves corresponding to

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Fig. 12.28 Two-dimensional solution with periodic perturbation of the shock-wave front at the saturation stage (on the left—in pressure terms, on the right—in density terms) [45]

the convex sections of the front, a reverse transition is observed as a result of which the local parameters of the shock wave take values close to the lower boundary of the ambiguous representation of the discontinuity. The intensity of transverse waves monotonously increases with time reaching saturation for the time amounting to several cycles of oscillations (Fig. 12.28 [45]: on the left—pressure, on the right—density). The identified self-oscillation mode of the shock wave is characterized by the fact that it is periodic, non-decaying and has a finite amplitude of the change in parameters, covering the EF range in which the condition of the shock wave instability L > 1 + 2M is fulfilled. The shock wave averaged characteristics in self-oscillation mode differ greatly from the parameters of an unperturbed shock wave. The difference in the velocity of the shock wave along its perturbed front means a difference in the velocity of the unperturbed shock wave on the lower and upper boundaries of the region of ambiguous representation of FG. It is interesting that in the lower part of the ambiguous representation region (below the point p1 = 3), multidimensional front oscillations lead to an increase in the average velocity of shock wave propagation as compared to the unperturbed state and in the upper part, on the contrary, to a decrease. We should also note a low Mach number of the flow after the shock wave (in the counting system related with the front), the weak adiabatic compressibility of matter in the state behind the shock-wave front, large density variations with a characteristic pattern of entropy waves. Matter particles crossing the front with the local shock wave parameters corresponding to the FG section have a lower density than matter particles crossing the shock-wave front at lower pressures behind the front. This is caused by the positive slope of the shock adiabat in the p–V variables at the section L > 1 + 2M. The low Mach number after the shock wave is also related with the realizability conditions for the inequality L > 1 + 2M. We can assume that the observed specific features of the solution are related not with the form of the used

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381

Fig. 12.29 Three-dimensional solution for the periodic perturbation of the shock-wave front in terms of equal pressure surfaces: three consecutive instants of time [45]

model equation of state but with specific features of the shock adiabat section under consideration. The results of 3D calculations shown in Fig. 12.29 [45] show that the spatial nature of perturbation does not lead to the qualitative changes in the solution. The calculations made for the same grid Reynolds number Reh = p0 Dh/η = 3.5 and different values of the physical Reynolds number calculated for the period Reλ = 875, Reλ = 1750, Reλ = 3500 show convergence to a single solution. Thus, the numerical modeling of the behavior of a shock wave with a viscous structure using the model equation of state shows that in the region of the ambiguous representation of discontinuity, which overlaps the section L > 1 + 2M, persistent multivariable shock wave oscillations are implemented with the formation of a cellular structure of the front, instead of one of the wave configurations obtained in the one-dimensional solutions. The oscillation amplitude of the local shock wave parameters overlaps the shock adiabat section where the condition of shock wave instability L > 1 + 2M is fulfilled. The waves observed on the shock wave surface in the region of its ambiguous representation are apparently explained by the “switching” of local shock wave parameters between the permitted wave configurations. In this connection, there is an analogy with the cellular structure of detonation waves in the limit case when transverse waves initiate the “switching” of local shock wave parameters between the values corresponding to the equilibrium shock adiabat and the shock adiabat constructed under the condition of frozen chemical reactions. In contrast to detonation waves, the “switching” in the case under consideration takes place between different areas of the equilibrium shock adiabat. Thus, in the framework of the models used, the unstable shock waves predicted by the linear theory [6–12] are not implemented due to the fact that they belong to the region of nonuniqueness of the solution and selection of other (stable) wave configurations. This conclusion corresponds to assumptions [7, 8] and can be implemented in practice [3, 4]. Paper [28] considers a practically important problem of vortex interaction with an absolutely unstable shock wave forming a two-wave configuration. The problem of shock wave interaction with the vortex has long been drawing attention of specialists in the field of aerodynamics and air acoustics (ref. for example, a review in [46]). On the one hand, this phenomenon leading to the curvature of the shock-wave front, the appearance of the local zones of compression and rarefaction in the flow field behind

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Table 12.1 Shock wave mode in terms of linear stability criteria, the condition on the shock wave parameters and the structure of secondary waves Shock wave mode in terms of linear criteria of stability

Ls =

1−M 2 (1+θ) 1−M 2 (1−θ)

L2 =

1+(2−θ)M 2 1+θ M 2

Structure of secondary waves (ref. Fig. 12.1)

References

−1 < L < L s

−1 < cos α < −M β > π/2

[3]

Subsonic ripple

Ls < L < L1

−M < cos α < 0

[4]

Supersonic ripple

L1 < L < 1 + 2M

0 < cos α < 1

L < −1

cos α < −1

–

L > 1 + 2M

cos α > 1

–

Absolute stability Neutral stability

Condition for shock wave parameters

Instability

Fig. 12.30 Interaction of sonic wave with shock wave discontinuity [28]

0 < π/2

[3, 4] [3]

SW Sonic wave

Entropyvortex wave

the shock wave and, therefore, the generation of acoustic noise, plays an important role in applications. On the other hand, this problem is frequently considered as a simplified model of a more general problem of the interaction of a shock wave with coherent structures in a turbulent flow [47]. As a medium where the shock wave interacts with the vortex, a normal gas is usually considered where the shock wave is absolutely stable and the perturbations of its front quickly decay with time (exponentially at the linear stage). When a shock wave propagates in a thermodynamically non-ideal medium, the problem becomes more complicated since the shock wave in it can lose its absolute stability. A very interesting case is the interaction of a vortex with a neutrally stable shock wave. The relations that classify the behavior of shock waves in terms of the linear theory of stability are given in Table 12.1. The following designations are used in the table in addition to the previously introduced degree of compression B: α—the angle between the sonic wave and the front of the unperturbed shock wave, β—the angle between the sonic and entropy-vortex wave (ref. Fig. 12.30). The angle α is determined from

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383

the equation which is a consequence of the condition of compatibility of linearized relations on the shock wave in case of the weak periodic perturbation of its front [5]: 

   4 3 + M2 2 M + θ − 1 cos α + 2M − 1 cos α 1+L 1+L 1 + M2 − 1 − θ M 2 = 0. +2 1+L 2

The condition of shock wave neutral stability: L s < L < 1 + 2M is obtained by substituting the inequality cos α > −M into this equation meaning that the sonic wave is outgoing. In paper [16], the behavior of an unstable shock wave was studied provided that the inequality L < −1 is fulfilled. The calculations used a model equation of state (12.13) proposed in [25] typical of shock compression processes accompanied by phase transitions, developed ionization or chemical reactions with energy absorption. This equation of state is thermodynamically correct and is characterized by the fact that the shock adiabats constructed on its basis contain regions with all known types of instability. The calculations showed that an unstable shock wave decays forming a complex compression wave so that the section of the shock adiabat where the initial shock wave decays significantly overlaps the instability section on the side of high pressures. The term “complex compression wave” (CCW) was introduced in [20] where it was noted that in media with a variable sign (∂ 2 p/∂V 2 )s (the derivative is taken for constant entropy), simple irreversible compression waves can emerge, one or two boundaries of which are strong discontinuities (incomplete and complete CCW, respectively). In one of the solutions obtained in [16], the decay configuration (complete CCW) contains the shock waves identified in terms of criteria of the linear theory of stability. The condition L = L s = −1 is fulfilled on the first shock wave (on the side of the incoming flow), and the condition of neutral stability L > L s is fulfilled on the second shock wave. Modeling of vortex passage through a CCW of such type provides an interesting opportunity to consider, within a single computational experiment, the interaction of a vortex with both absolutely stable and neutrally stable shock wave. The obtained solution is analyzed from the point of view of the compliance of the structure of secondary waves with the theoretical structure (ref. Table 12.1). Problem Formulation. A model equation of state (12.18) is taken as the equation of state [25]. Let us consider a complex compression wave with the initial parameters p1 = 0.1, p1 = 0.1821 and pressure in the finite state p2 = 0.82. This wave includes a shock wave precursor (L = L s = −1), an isentropic compression wave of small amplitude (gradients in this wave can be neglected) and a rear shock wave the parameters of which comply with the condition of neutral stability in linear approximation L > L s .

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Such a multiwave structure was observed in a number of experiments for the shock compression of heavily non-ideal media [31]. The interaction of this complex compression wave with perturbation in the form of a transverse vortex is considered. Perturbations of the components of velocity (u  , v  ), density ρ  and pressure p  look as follows: 



   1 −y 2 = vm exp 1−r , x 2       1 p  (r ) = ρ0 exp − M 2 exp 1 − r 2 − 1 , p  (r ) = c02 ρ  . 2 u v



Here, r = (x 2 + y2 )1/2 is the distance from the vortex center, M = vm /c0 is the Mach number calculated using the maximum linear rotation velocity in the vortex vm and the velocity of sound c0 , or vm = 0.5, M ≈ 1/4 in the calculation case. The characteristic vortex size is taken as the length size—the distance from the center of the vortex at which the maximum linear rotation velocity is reached. The problem is solved in the region of rectangular form. In adopted units, the calculation area width is 20, the distance from the vortex center to the first shock wave in the structure of the complex compression wave at the initial instant of time is 7.5, the distance between shock waves is 5. The presented results were obtained on the computational grid sizing 900 × 600. The parameters of the incoming flow are recorded at the input bound. The conditions of periodicity are implemented at the lower and upper boundaries of the calculational area (thus, an infinite chain of vortexes is modelled in reality), with non-reflective boundary conditions used at the output bound. The pattern of pressure distribution at the initial instant of time is shown in Fig. 12.31a. The pressure minimum corresponds to the vortex position, pressure jumps show the initial position of the first and second shock waves in the structure of the complex compression wave. In the process of flow evolution, the vortex successively goes through shock-wave fronts the reaction of which to this perturbation is a subject of consideration. Numerical Method. As a discrete analog of the weak solution of the Euler equations, a numerical solution of laws of mass, momentum and energy conservation is used based on the finite volume method. The flows on the cell faces are approximated using the Yang ENO-scheme [41]. The Roe method generalized for an arbitrary equation of state ensures the exact fulfillment of conditions on a stationary discontinuity [12, 13]. To decrease the variance error and select thermodynamically acceptable discontinuities, the Harten entropy correction is used [34]. Time integration is carried out using the Runge–Kutta method of the third order of approximation [42]. The used approach is an alternative to the method of the explicit selection of discontinuity surface [48]. Calculation Results. A vortex perturbation successively crosses the shock waves within a complex compression wave (Fig. 12.31c). The result of the interaction is the development of acoustic and entropy-vortex perturbations in the space after the shock wave. The pressure distribution pattern in the flow field at different times

12 Shock Wave Stability

Fig. 12.31 Vortex interaction with two-wave shock structure

385

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12 Shock Wave Stability

allows tracking the perturbation evolution. As follows from pressure distributions, the short-term response of the first (L > −1) and second (L > L s ) shock waves is comparable in the amplitude of shock-wave front deviations from the unperturbed state and in the maximum amplitude of the acoustic perturbations arising from the interaction. On the other hand, the long-term consequences of the interaction are such that in the first case, the shock-wave front comparatively quickly restores its formula, and acoustic and entropy vortex perturbations behind the shock-wave front decay. In case of a neutrally stable shock wave, on the contrary, slowly decaying secondary waves are observed. The slowly decaying quasi-stationary secondary waves behind the second shockwave front present in the solution are characterized by the fact that the angles formed by the wave vectors of the acoustic and entropy vortex components of secondary waves are less than π /2. This, in its turn, means that these acoustic perturbations are lagging according to Landau, i.e. the acoustic energy flow is directed from the shock wave (as per the linear theory of shock wave stability). Thus, the numerical solution of the problem of interaction with a vortex at different dimensionless parameters of shock waves L = −1 (sonic wave), L > L s (region of spontaneous sound emission) shows that in the latter case, the passage of shock waves through the vortex leads to the formation of slowly decaying secondary waves. In the conditions of initially turbulized medium, this phenomenon can result in a sharp increase in the level of acoustic noise after shock wave passage. The structure of secondary waves observed in the calculation corresponds to the theoretical one. The angles formed by the wave vectors of acoustic and entropy vortex perturbations show that weak secondary waves are lagging, i.e. the energy flow of acoustic waves is directed from the shock wave. Thus, the problem of shock wave stability in media with an arbitrary equation of state is apparently the most fascinating and hard in modern shock wave physics. It is still far from its complete solution. In particular, the experimental discovery and study of such nonlinear phenomena would be very interesting.

References 1. Landau LD, Lifshits EM (1986) Course of theoretical physics. In: Hydrodynamics, vol 6. Nauka, Moscow [Landau L.D., Lifshits E.M. Kurs teoreticheskoy fiziki. T. 6. Gidrodinamika. — M.: Nauka, 1986 (in Russian)] 2. Zel’dovich YaB, Raizer YuP (2008) Theory of shock waves and high-temperature hydrodynamic phenomena, 3rd edn., corrected. FIZMATLIT, Moscow [Zel’dovich Ya.B., Raizer Yu.P. Teoriya udarnykh voln i vysokotemperaturnykh gidrodinamicheskikh yavleniy. 3-ye izd., ispr. — M.: FIZMATLIT, 2008 (in Russian)] 3. Fortov VE, Yakubov IT (1994) Physics of non-ideal plasma. Energoizdat, Moscow [Fortov V.E., Yakubov I.T. Fizika neideal’noy plazmy. — M.: Energoizdat, 1994 (in Russian)] 4. Fortov VE (2013) High energy density physics. FIZMATLIT, Moscow [Fortov V.E. Fizika vysokikh plotnostey energii. — M.: FIZMATLIT, 2013 (in Russian)]

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