Topics in Current Physics – serie

In the seven years since this volume first appeared. there has been an enormous expansion of the range of problems to which Monte Carlo computer simulation methods have been applied. This fact has already led to the addition of a companion volume ("Applications of the Monte Carlo Method in Statistical Physics", Topics in Current Physics. Vol . 36), edited in 1984, to this book. But the field continues to develop further; rapid progress is being made with respect to the implementation of Monte Carlo algorithms, the construction of special-purpose computers dedicated to exe­ cute Monte Carlo programs, and new methods to analyze the "data" generated by these programs. Brief descriptions of these and other developments, together with numerous addi­ tional references, are included in a new chapter , "Recent Trends in Monte Carlo Simulations" , which has been written for this second edition. Typographical correc­ tions have been made and fuller references given where appropriate, but otherwise the layout and contents of the other chapters are left unchanged. Thus this book, together with its companion volume mentioned above, gives a fairly complete and up­ to-date review of the field. It is hoped that the reduced price of this paperback edition will make it accessible to a wide range of scientists and students in the fields to which it is relevant: theoretical phYSics and physical chemistry , con­ densed-matter physics and materials science, computational physics and applied mathematics, etc.

Deals with the computer simulation of complex physical sys- tems encounteredin condensed-matter physics and statistical mechanics as well as in related fields such as metallurgy, polymer research,lattice gauge theory and quantummechanics.

The characterization of thin films and solid interfaces as well as the determina­ tion of concentration profiles in thin solid layers is one of the fields which re­ quire a rapid transfer of the results from basic research to technological applica­ tions and developments. It is the merit of the Dr. Wilhelm Heinrich and Else Heraeus-Stiftung to promote such a transfer by organizing high standard seminars mostly held at the "Physikzentrum" in Bad Honnef near Bonn. The present book has been stimulated by one of these seminars assembling most of the invited speakers as co-authors. The editor appreciates the cooperation of his colleagues contributing to this book. H. Oechsner Kaiserslautern, April 1984 v Contents 1. Introduction. ByH. Oechsner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 1 Requirements for Thin Film and In-Depth Analysis . . . . . . . . . . . . . . . . . . . 1 1. 2 Object and Outl i ne of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4 References 2. The Application of Beam and Diffraction Techniques to Thin Film and Surface Micro-Analysis. By H. W. Werner (With 25 Fi gures) . . . . . . . . . . . . . . . . 5 2. 1 Methods to Determine Chemical Structures in Material Research 5 2. 2 Selected Analytical Features Used to Determine Chemical Structures 9 2. 2. 1 Depth Profi 1 ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 9 a) Destructive Depth Profiling b) Nondestructive Methods for Depth and Thin Film Analysis 15 19 2. 2. 2 Microspot Analysis and Element Imaging 2. 3 Determining Physical Structures in Material Research . . . . . . . . . . . . . . . 27 2. 3. 1 X-Ray Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 27 2. 3. 2 X-Ray Double Crystal Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2. 3.

Methods of scientific investigation can be divided into two categories: they are either macroscopic or microscopic in nature. The former are generally older, classical methods where the sample as a whole is studied and various local prop­ erties are deduced by differentiation. The microscopic methods, on the other hand, have been discovered and developed more recently, and they operate for the most part on an atomistic scale. Glancing through the shelves of books on the various scientific fields, and, in particular, on the field of physical metallurgy, we are surprised at how lit­ tle consideration has been given to the microscopic methods. How these tools provide new insight and information is a question which so far has not at­ tracted much attention. Similar observations can be made at scientific confer­ ences, where the presentation of papers involving microscopic methods is often pushed into a far corner. This has led users of such methods to organize their own special conferences. The aim of this book is to bridge the present gap and encourage more interaction between the various fields of study and selected microscopic meth­ ods, with special emphasis on their suitability for investigating metals. In each case the principles of the method are reviewed, the advantages and successes pointed out, but also the shortcomings and limitations indicated.

During the last decade, surface research has clearly shifted its interest from the macroscopic to the microscopic scale; a wealth of novel experimental techniques and theoretical methods have been applied and developed successfully. The Topics volume at hand gives an account of this tendency. For the understanding of surface phenomena and their exploitation in tech­ nical applications, the theoretical and experimental analysis at the microscopic level is of particular interest. In heterogeneous catalysis, for example, a chemical reaction takes place at the interface of two phases, and the process occurring at the surface is composed of a sequence of individual microscopic steps. These individual steps include adsorption, desorption, surface diffusion, and reaction on the surface. These elementary steps are greatly influenced by the structure and the dynamics of the surface region. Especially the catalytic activity may strongly depend on the structure of the catalyst's surface. The necessity of per­ forming surface investigations on a microscopic scale is also reflected clearly in research work relating to metal-semiconductor interfaces which determine es­ sentially the properties of electronic device materials. The experimental probe on the atomic scale, coupled with parallel theoretical calculations, showed that the electronic properties of a metal-semiconductor interface strongly depend on the crystallographic structure of the semiconductor; in particular, it is im­ portant to know in this context the modification of the atomic arrangement in the surface region caused by the termination of the crystal by the surface.

Structural and dynamical properties of surfaces are of considerable importance for the elucidation of surface phenomena. For example, the adsorption of particles and chemical reactions at surfaces are influenced by these properties. Also, the electronic surface states of electronic device materials (e.g., Schottky barrier diodes and metal-insulator-semiconductor devices) depend critically on the structure of the semiconductor surface and the overlayers. As in Structure and Dynamics of Surfaces I, this Topics volume considers the problems at the mircoscopic level. It deals with ordered and disordered as well as harmonic and anharmonic surface effects. Novel experimental techniques and theoretical developments which have been applied successfully during the last decade are integrated. The contributions of this volume also include the background of the subject, typical results, and, in most cases, trends of future developments.

This book is devoted to the systematic treatment of nonequi 1 ibrium vibrational kinetics in molecular systems. Particular emphasis is given to the vibrational excitation of diatomic molecules by low-energy electrons in a discharge and by IR photons in laser-pumped systems. The book follows the different steps of the introduction, redistribution, loss, and chemical conversion of the vibrational quanta, from the points of view of the overall kinetics and the dynamics of elementary processes. These two aspects are balanced in a multidisciplinary approach. The different chapters give the basic instruments (theoretical and experimental) which are needed to understand the ki­ netics of nonequilibrium systems. The book will introduce the reader to different areas such as plasmachemistry, laser chemistry, IR and Raman spectroscopy, and relaxation phenomena, emphasizing how the vibrational energy affects such research fields. The chapters dedicated to collisional dynamics involving vibrational excited molecules provide an introduc­ tion to the modern techniques uti 1 ized in the scattering theory of inelastic and reactive collisions. The extension of the vibrational kinetics to polyatomic mole­ cules, discussed in Chap. 10, is the natural bridge between coll ision and coll i­ sionless regimes. In conclusion, we hope that the approach followed in this book will stimulate the collaboration of researchers coming from different research fields, which are too often completely separate.

The magnetism of iron and other transition metals had been a subject of inten­ sive research for a long time, but the understanding of the microscopic origin of "metallic magnetism" was quite limited until the early 1970's. During the last 10 to 15 years both theory and experiment contributed towards signif­ icant progress in this field, such that today a qualitative understanding has been achieved. The word "qualitative" indicates that the knowledge is still not complete; although many properties, the ground state as well as the finite­ temperature behaviour and the phase transition from magnetic order at low temperatures to the paramagnetic state at high temperatures, can be explained in a coherent way, a quantitative description still is not fully achieved. It is certainly appropriate to summarize the developments of the last 15 years and the present-day understanding of the field, this is the aim of this Topics volume. The form chosen is a collection of reviews, written by prominent scientists who themselves contributed decisively to the progress. Scientists with a general interest in the field as well as specialists and active researchers in metallic magnetism should be able to profit from the two-volume treatment. The subjects not covered extensively in the present first volume (in particular neutron scattering and electronic structure properties) will make up the second volume.

Energetic electromagnetic radiation finds frequent uses in science (e. g. , for expe­ riments in nuclear and elementary - particle physics), in technology (for materials testing), and in medicine (for medical X-rays). The most common method of genera­ ting such radiation is via the process of "bremsstrahlung" (a German term coined by A. Sommerfeld, meaning "braking radiation") in which a beam of electrons is direc­ ted into matter (e. g. , a metal target), losing energy during its collisions with the atoms and releasing this energy in the form of emitted radiation. The character of such radiation may be drastically changed by the use of a tar­ get with periodic structure (most commonly, a crystal target). The coherent waves emitted from individual crystal atoms interfere with each other, monochromatizing and polarizing the radiation and often increasing its intensity manifold, thereby creating a powerful radiation source of high quality for purposes of scientific and technical applications. This is true both for the well - established "coherent bremsstrahlung" process in which the interfering radiation is emitted while the electrons cross a succession of crystal planes, as well as for the more recently discovered process of "channeling radiation" (generating radiation of even higher intensity, but lower energy) in which the radiation is emitted while the electrons propagate along a crystal plane, or a crystal axis, in an oscillatory fashion.

Stanley Bashkin Beam-foil spectroscopy has enjoyed a rapid growth since the publication of KAY's first experiment [I.1J and my own first formal discussion of the possibilities inherent in a foil-excited particle beam [1.2J. In addition to fulfilling a number of the impor­ tant promises, the beam-foil source has been found to hold substantial surprises, the unearthing of which has contributed to our knowledge of basic atomic physics. Since the early days, major extensions have been made in the range of wavelength and par­ ticle energies which have been used, but only the bare beginnings have been made in exploiting the potential of the beam-foil source. Since there are many people who would like to turn their accelerator facilities to beam-foil problems or apply their theoretical techniques to calculations which bear on the beam-foil field, it seemed appropriate to assemble a discussion of the present status of beam-foil spectroscopy. The present volume attempts to summarize what has been learned and outlines a number of studies which remain to be made.

The model calculations considered here test the DWBA with a wide range of three-body models. Each of the calculations, however, considers only one or two aspects of the accuracy of the DWBA. The lack of overlap of the testing parameters limits the con­ clusions which can be drawn. This is particularly true with reference to the nucleon­ core potential where comparable parameters are rarely used. In spite of this limitation, we may make a few observations about the sensitivity of the model cross sections and the mechanism which produces an accurate DWBA. Specifically we may summarize the most important results as follows: 1) In these models the exact DWBA at low energies (E

The development of surface physics and surface chemistry as a science is closely related to the technical development of a number of methods involving electrons either as an excitation source or as an emitted particle carrying characteristic information. Many of these various kinds of electron spectroscopies have become commercially available and have made their way into industrial laboratories. Others are still in an early stage, but may become of increasing importance in the future. In this book an assessment of the various merits and possible drawbacks of the most frequently used electron spectroscopies is attempted. Emphasis is put on prac­ tical examples and experimental design rather than on theoretical considerations. The book addresses itself to the reader who wishes to know which electron spectroscopy or which combination of different electron spectroscopies he may choose for the particular problems under investigation. After a brief introduction the practical design of electron spectrometers and their figures of merit important for the different applications are discussed in Chapter 2. Chapter 3 deals with electron excited electron spectroscopies which are used for the elemental analysis of surfaces. Structure analysis by electron diffrac­ tion is described in Chapter 4 with special emphasis on the use of electron diffrac­ tion for the investigation of surface imperfections. For the application of electron diffraction to surface crystallography in general, the reader is referred to Volume 4 of "Topics in Applied Physics".

Inelastic neutron scattering is a well established and important technique for studying the dynamical properties of condensed matter at the atomic level. Often, as is the case of experiments designed to study motions of hydrogen atoms, or magnetic excitations, it may yield information obtainable in no other way. Our aim in assembling this book is to produce an overview of some research topics which have come to the fore recently with the development of high neutron fluxes and high performance inelastic scattering spectrometers. The topics dis­ cussed here are, by and large, developing rapidly and have not reached the stage at which definitive accounts are always possible. Authors have not therefore attempted to make an extensive review of their topic, and the papers quoted in the text are, in general, those which are seen as having been important in its develop­ ment (they date, roughly, from the 1971 IAEA conference on neutron scattering held in Grenoble). Basic phenomena are illustrated for the most part by the discussion of one, or two, typical examples. The authors hope that the book will be useful to researchers who are not yet fully aware of the diverse range of problems to which the technique can be applied, and to students beginning research work. For this reason, the first chapter by S. w.

The central subject of this volume is the atomic and molecular physics of heavy par­ ticles as investigated with charged particle accelerators. The natural division be­ tween atomic structure and ion-atom collision studies, and the similar division be­ tween the theoretical and experimental branches of these subjects, are reflected in a parallel subdivision into corresponding chapters. In addition, one chapter is de­ voted to the important interface between atomic and molecular physics with condensed matter physics. A principal aim of the present volume is to provide a compact de­ scription of a number of current interests and trends within the heavy particle structure and collisions field in a sufficiently general, non-specialized way that interested scientists who wish to become acquainted with such interests and trends can do so without becoming bogged down in excessive archival detail. It is, therefore, hoped that the book will be of some use to advanced students who seek a general in­ troduction to these subjects. Numerous, more specialized, archival review articles are frequently referred to in each chapter for the benefit of those who seek more detailed knowledge about particular topics discussed. The editor wishes to acknowledge the support of two U. S. government agencies: the Office of Naval Research and the National Science Foundation, during the preparation of this volume. Sincere thanks are due Mrs. Betty Thoe for her excellent editorial work on the various manuscripts and Mrs.

Since the great discovery made by Laue, x-ray diffraction has become the most im­ portant method for the investigation of atomic structure in condensed matter. Cer­ tain investigations, however, are difficult or totally impossible to conduct using x-rays, for example, the localization of atomic nuclei or atoms having only a few core electrons, and the observation of magnetic moments. The investigation of these important areas is made possible by neutron diffraction. Thus this method has devel­ oped into an important supplement to x-ray investigations. An ever-increasing de­ mand is put on the method by research areas, old and new. Neutron diffraction has completely reformed the subject of solid-state magnetism; especially the area of chemical binding has received a new impetus by the union of x-ray and neutron dif­ fraction. An exchange of different isotopes of the same element gives rise, as a rule, to a change in the strength of the neutron diffraction. Due to this effect itls possible, by means of deuteration, to make visible a single chain in a solid high polymer. Thus neutron small-angle scattering is important in protein research and for the biology of macromolecules. Of equal importance is the application of neu:ron diffraction in metallurgy. There already exist several excellent books which discuss the results obtained by neans of neutron diffraction.

H. P. Baltes We begin the introductory chapter with a general definition of the inverse optical problem. Next, we discuss the role of prior knowledge and the questions of uniqueness and stability. We then review the various specific inverse problems in optics as well as the contents of Chapters 2 to 6. Finally, we summarize the notation in co­ herence theory. 1. 1 Direct and Inverse Problems in Optical Physics The "direct" or "normal" problem in optical physics is to :Jredict the emission or propagation of radiation on the basis of a known constitution of sources or scat­ terers. The "inverse" or "indirect" problem is to deduce features of sources or scatterers from the detection of radiation. An intuitive solution of the optical inverse problem is commonplace: we infer the size, shape, surface texture, and ma­ terial of objects from their scattering and absorption of light as detected by our eyes. Intuition has to give way to mathematical reconstruction as soon as we wish to analyze optical data beyond their visual appearance. Examples are the extrapola­ tion and deblurring of optical images, the reconstruction from intuitively inacces­ sible data such as defocused images and interferograms, or the search for information that is "lost" in the detection process such as the phase. Following CHADAN and SABATIER [1. 1], a general definition of inverse optical problems can be attempted as follows. We describe the sources and scatterers by the set (1.

The Raman effect is a most useful tool for the study of molecular vibrations and molecular structure. Information about the structure and symmetry of molecules, as well as about their vibrational energies can be obtained to a reasonable degree of satisfaction from their infrared and Raman vibrational spectra. The body of knowl­ edqe of the vibrational infrared and Raman spectra of molecules is immense and is now so well organized and understood that it is found to be represented in any stan­ dard upper level undergraduate curriculum in chemistry. The rotational energies of a molecule and quantitative details about its structure can only be obtained through the techniques of microwave, and high-resolution infrared and Raman spectroscopy of low pressure gases and vapors. The results of such investigations are of interest . not only to the academic scientists, but also to scientists and engineers who are active in applied fields of chemistry and physics, as well as the atmospheric sciences. This book deals with basic investigations of the Raman scattering of light by gases, with some attention also being given to liquid substances. After a brief in­ troductory chapter that delineates the historical development of Raman spectroscopy of gases, high-resolution rotation-vibrational and pure rotational Raman spectros­ copy is described in Chapters 2 and 3. The all-important intensity parameter, the Raman scattering cross section, is treated in Chapter 4, while the broadening of Raman lines due to the effects of intermolecular forces is taken up in Chapter 5.

This Topics volume is devoted to a study of sound propagation in the ocean. The effect of the interior of the ocean on underwater sound is analogous to the effect of a lens on light. The oceanic lens is related, as in light propagation, to the index of refraction of the medium. The latter is giv~n by the ratio of the sound frequency to the speed of sound in water, typi ca lly about 1500 m s -1. It is the vari­ ation of the sound speed due to changing temperature, density, salinity, and pres­ sure in the complex ocean environment which creates the lens effect. Many oceanic processes such as currents, tides, eddies (circulating, translating regions of wa­ ter), and internal waves (the wave-like structure of the oceanic density variabil­ ity) contri bute in turn to the changes in sound speed'. The net effect of the ocean lens is to trap and guide sound waves in a channel created by the lens. The trapped sound can then propagate thousands of miles in this oceanic waveguide. In addition to the propagation in the interior of the ocean, sound can propagate into and back out of the ocean bottom as well as scatter from the ocean surface. Just as the sound produced by a loudspeaker in a room is affected by the walls of the room, so the ocean boundaries and the material properties below the ocean bottom are essential ingredients in the problem.

Synchrotron radiation as a spectroscopic research tool has undergone a most inter­ esting and astonishing historical development and has now come to the stage of an exciting boom. The machines which produce synchrotron radiation were built and de­ veloped exclusively for other purposes in the past, namely high-energy physics. At the same time, however, they involuntarily became better and better light sources for the spectral range from the visible to the hard x-ray region. Now we are at the point that the first few storage rings have gone into operation as machines dedicated to synchrotron radiation and several more are in the stage of construction and planning. All this was brought about by the successful research performed during the past fifteen years in which several groups allover the world haVe participated at dif­ ferent accelerator centers mostly symbiotic with high-energy physics. As it happens with a young and rapidly developing field, the number of reviews and monographs is still minute. The objective of this book is to fill an apparent gap and to provide a sound basis for those who are interested in synchrotron radiation and its applica­ tions.

In condensed matter initially fast positrons annihilate after having reached equi­ librium with the surroundings. The interaction of positrons with matter is governed by the laws of ordinary quantum mechanics. Field theory and antiparticle properties enter only in the annihilation process leading to the emergence of energetic photons. The monitoring of annihilation radiation by nuclear spectroscopic methods provides valuable information on the electron-positron system which can directly be related to the electronic structure of the medium. Since the positron is a positive electron its behavior in matter is especially interesting to solid-state and atomic physi­ cists. The small mass quarantees that the positron is really a quantum mechanical particle and completely different from any other particles and atoms. Positron physics started about 25 years ago but discoveries of new features in its interac­ tion with matter have maintained continuous interest and increasing activity in the field. Nowadays it is becoming part of the "stock-in-trade" of experimental physics.

Superionic conductors are solids whose ionic conductivities approach, and in some cases exceed, those of molten salts and electrolyte solutions. This implies an un­ usual state of matter in which some atoms have nearly liquidlike mobility while others retain their regular crystalline arrangement. This liquid-solid duality has much appeal to condensed matter physicists, and the coincident development of powerful new methods for studying disordered solids and interest in superionic conductors for technical applications has resulted in a new surge of activity in this venerable field. It is the purpose of this book to summarize the current re­ search in the physics of superionic conduction. with special emphasis on those aspects which set these materials apart from other solids. The volume is aimed to­ wards the materials community and will, we expect, stimulate further research on these potentially useful substances. The usual characterization of the superionic phase lists high ionic conductivity; low activation energy; and the open structure of the crystal, with its interconne­ ted network of vacant sites available to one ionic species. To these, as we demon­ strate in this volume, should be added important dynami~ and collective effect~: the absence of well-defined optical lattice modes, the presence of a pervasive, low-energy excitation, an infrared peak in the frequency-dependent conductivity, unusual NMR prefactors, phase transitions, and a strong tendency for the mobile ion to be found between allowed sites.

Contents: Cho, K.: Introduction. - Cho, K.: Internal Structure of Excitons. - Dean, P. J. /--01Herbert, D. C.--02: Bound Excitons in --Semiconductors. - Fischer, B. /Lagois, J.: Surface Exciton Polaritons. - Yu, P. Y.: Study of Excitons and Exciton-Phonon Interactions by Resonant Raman and Brillouin Spectroscopies.

Towards the end of the 1960s, a number of quite different circumstances combined to launch a period of intense activity in the digital processing of electron micro­ graphs. First, many years of work on correcting the resolution-limiting aberrations of electron microscope objectives had shown that these optical impediments to very high resolution could indeed be overcome, but only at the cost of immense exper­ imental difficulty; thanks largely to the theoretical work of K. -J. Hanszen and his colleagues and to the experimental work of F. Thon, the notions of transfer func­ tions were beginning to supplant or complement the concepts of geometrical optics in electron optical thinking; and finally, large fast computers, capable of manipu­ lating big image matrices in a reasonable time, were widely accessible. Thus the idea that recorded electron microscope images could be improved in some way or rendered more informative by subsequent computer processing gradually gained ground. At first, most effort was concentrated on three-dimensional reconstruction, particu­ larly of specimens with natural symmetry that could be exploited, and on linear operations on weakly scattering specimens (Chap. l). In 1973, however, R. W. Gerchberg and W. O. Saxton described an iterative algorithm that in principle yielded the phase and amplitude of the electron wave emerging from a strongly scattering speci­ men.

The suggestion by Dr. Franklin S. Harris, Jr. , that these books be written arose pursuant to the editor's plaints that despite the implicitly or explicitly ack­ nowledged importance of both aerosols and particulate matter in innumerable domains of technology and human welfare, investigations of these subjects were generally not supported independently of the narrowest conceivable domains of their appli­ cations. Frank Harris, who has long been a contributor in one of the important domains of aerosol macrophysics, atmospheric optics, challenged the editor to elaborate his views. Ideally, they would have taken the form of a monograph; however, there is as yet an insufficient body of information to present a unified treatment. At the same time, substantial efforts are in progress in the component fields to hold the promise for the emergence of unifying elements which will even­ tually facilitate their presentation to be made with a high degree of integrity. There are numerous pertinent and systematic tie-ins between project-oriented aerosol work and basic physical investigations which are themselves quite closely akin to much classical and current work in physical science. The most significant aspect of these tie-ins is their potential for making substantial contributions to the functional needs of the applications areas while stimulating significant questions of basic physics. For this to be possible, it is necessary that the most relevant areas of physics be identified in such a manner as to make clear their re­ levance for aerosol-related studies and vice versa.

The theory of the chemical interaction of molecules with surfaces has advanced handsomely in the last few years. This is due in part to the application of the entire arsenal of bulk solid-state theory and molecular quantum chemistry methods. This considerable activity was stimulated by an outpouring of experimental data, particularly of photoemission spectra. In many cases the theoretical techniques are now such that accurate, atomistic pictures of chemisorption phenomena are computed from first principles. This level of capability has been reached only recently, and has not been described anywhere in a comprehensive manner. The purpose of this monograph is to review these recent advances and, at the same time, to indicate a number of important questions which have not been answered. We discuss chemisorption on oxides, semiconductors, and both simple and transition metals. Solid surfaces as well as clusters are considered. While the review should be valuable to workers in the field, care has been taken to make the chapters understandable to the nonspecialist.

Until the beginning of this decade the number of significant exactly soluble problems in physics was limited to a very few: the classical or quanti sed harmonic oscillator, the linearised many-body problem, the quanti sed hydrogen atom, Newton's solution of the planetary orbit problem, Onsager's solution of the two-dimensional Ising problem, almost exhaust the list. Now the situation is quite different. We have a large number of exactly soluble nonlinear systems of physical significance and the number of these is growing steadily. Recent examples include a limited solution of Ein- stein's field equations!, an apparently exact solution of the quantised sine-Gordon 2 equation u - U = sin(u) which establishes connections with the Ising models , xx tt 3 and a solution of the equations of motion of a rigid body in g dimensions . This book is concerned with problems such as these. But more specifically it is concerned with solitons. These mathematical objects are exact, analytical, solutions of nonlinear wave or evolution equations like the sine-Gordon equation (s-G) or the Korteweg-de Vries equation (KdV) u + 6uu + u = O. The discovery by Gardner, t x xxx .

When, in the spring of 1979, H.P. Baltes presented me with the precursor of this vo 1 ume, the book on "Inverse Source Problems in Opti cs", I expressed my gratitude in a short note, 11hich in translation, reads: "Dear Dr. Ba ltes, the mere titl e of your unexpected gift evokes memori es of a period, which, in the terminology of your own contribution, would be described as the Stone Age of the Inverse Problem. Those were pleasant times. Walter Kohn and I lived in a cave by ourselves, drew pictures on the walls, and nobody seemed to care. Now, however, Inversion has become an Industry, which I contemplate with as much bewilderment as a surviving Tasmanian aborigine gazing at a modern oil refinery with its towers, its fl ares, and the confus i ng maze of its tubes." The present volume makes me feel even more aboriginal - impossible for me to fathom its content. What I can point out, however, is one of the forgotten origins of the Inverse Scattering Problem of Quantum Mechanics: Werner Heisenberg's "S-Matrix Theory" of 1943. This grandiose scheme had the purpose of eliminating the notion of the Hamiltonian in favour of the scattering operator. If Successful, it would have done away once and for all with any kind of inverse problem.