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\(R\)-matrix theory of atomic collisions. Application to atomic, molecular and optical processes. (English) Zbl 1223.81004
Springer Series on Atomic, Optical, and Plasma Physics 61. Berlin: Springer (ISBN 978-3-642-15930-5/hbk; 978-3-642-15931-2/ebook). xvii, 745 p. (2011).
\(R\)-matrix theory was first introduced by E. P. Wigner and L. Eisenbud in the late 1940s in their analysis of nuclear resonance reactions. The fundamental idea in \(R\)-matrix theory is to partition the configuration space describing the collision process into two or more regions with the processes in each of these regions having distinctly different physical properties. A different representation of the wave function describing the process is then adopted in each region in which these wave functions are connected by the \(R\)-matrix defined on the common boundaries. Nowadays the \(R\)-matrix method, arguably, is established as the most versatile and efficient method for the modeling of scattering by electrons (and positrons) by atomic and molecular targets.
This work is a fundamental monograph devoted to \(R\)-matrix theory (it is not easy to report within a short review the exact content of this book). The author describes a generalized \(R\)-matrix theory of atomic collisions and its application to the ab initio study of atomic, molecular and optical collision processes. It is written for physicists rather than for mathematicians.
The book is conceptually divided into two parts. The monograph commences by presenting an overview of the collision theory in Part I. As well as giving a self-contained summary of this theory, it also provides an introduction to the basic concepts and notation required in Part II.
Chapter 1 introduces the basic concepts of atomic collision theory by considering potential scattering. The author considers the solution of the non-relativistic time-independent Schrödinger equation for a short-range spherically symmetric potential, defines the scattering amplitude and various cross sections, obtains explicit expressions for these quantities in terms of the partial wave phase shifts, and defines the \(K\)-matrix, \(S\)-matrix and \(T\)-matrix in terms of the partial wave phase shifts. The discussion is extended to the situation in which a long-range Coulomb potential is present, in addition to a short-range potential. Then the author turns his attention to the analytic properties of the partial wave \(S\)-matrix in the complex momentum plane and discusses the connection between poles in the \(S\)-matrix and bound states and resonances. The chapter includes a discussion of analytic properties to consider the analytic behaviour of the phase shift and the scattering amplitude in the neighbourhood of threshold energy. Variational principles for the partial wave phase shift and for the \(S\)-matrix are derived. The author concludes this chapter by considering relativistic scattering of an electron by a spherically symmetric potential.
In Chapter 2 the author introduces the basic concepts of multichannel collision theory and applies this theory to non-relativistic electron collisions with multi-electron atoms and atomic ions. This chapter provides an introduction to a discussion of resonances, threshold behaviour, and \(R\)-matrix theory presented in the subsequent chapters in this monograph. It mainly concerns low-energy elastic scattering and excitation processes.
The author initiates a discussion of the multichannel collision theory by considering the solution of the time-independent Schrödinger equation describing low-energy electron collisions with multi-electron atoms and atomic ions, then he defines the scattering amplitude in terms of the asymptotic form of the solution of the Schrödinger equation. The differential and total cross sections are defined in terms of this scattering amplitude. Then the author considers the atomic or ionic target eigenstates which take part in the collision process, and introduces the concept of pseudostates. Finally, the author defines the multichannel \(S\)- and \(T\)-matrices in terms of the \(K\)-matrix and summarizes the angular momentum transfer formalism.
Chapter 3 is devoted to the theory of resonance reactions and the closely related behaviour of cross sections near threshold. This chapter includes:
a discussion of the analytic properties of the \(S\)-matrix by defining multichannel Jost functions in terms of the solutions of coupled second-order integrodifferential equations;
a discussion of bound-state and resonance poles in the \(S\)-matrix for multichannel collisions by considering their distribution in the multi-Riemann-sheeted complex energy plane;
a derivation of an explicit expression for the multichannel \(K\)-matrix and \(S\)-matrix in the neighbourhood of an isolated resonance pole;
a description of the threshold behaviour of excitation and ionization cross sections.
In Chapter 4 the author considers first the solution of the radial Schrödinger equation in potential scattering. The collision wave function in the internal region is expanded in terms of a complete set of basis functions satisfying homogeneous boundary conditions on the surface of this region. The analytic properties of the \(R\)-matrix enable the analytic behaviour of the phase shift and scattering amplitude to be determined if the potential interaction in the external and asymptotic regions can be neglected except for the long-range Coulomb potential. Next, the author generalizes the Wigner-Eisenbud theory to derive expressions for the collision wave function and the \(R\)-matrix in terms of basis functions satisfying arbitrary boundary conditions on the surface of the internal region. In this generalized theory, a surface or Bloch operator is introduced (which, if added to the Hamiltonian operator, makes their sum Hermitian over the internal region in the arbitrary boundary condition basis). A special section is devoted to the discussion of variational principles for the \(R\)-matrix in potential scattering. This chapter includes also a discussion of approximation methods which have been used in practical \(R\)-matrix calculations and \(R\)-matrix propagator methods appropriate for the case of the wavelength of the scattered particle being small comparatively to the range over which the potential interaction varies appreciably. Finally, the author considers the extension of the \(R\)-matrix theory to treat the relativistic potential scattering described by the Dirac equation.
Chapter 5 provides an introduction to the basic concepts of multichannel \(R\)-matrix theory which will be applied in later chapters to a wide range of other atomic, molecular and optical collision processes. Consideration in this chapter is restricted to low-energy electron collisions, where only elastic scattering and excitation processes are energetically allowed or play a significant role in the collision process. The multichannel \(R\)-matrix theory is introduced by considering first electron collisions with light multi-electron atoms and atomic ions where an accurate representation of the collision process can be obtained by solving the time-independent non-relativistic Schrödinger equation. The author starts with a general introduction to the \(R\)-matrix theory describing the partitioning of the configuration space adopted in this theory, and then gives a brief overview of the computer programs that have been developed to implement this theory. He also describes in detail the solution of the Schrödinger equation, first in an internal region, then in an external region, and finally in an asymptotic region, yielding the \(K\)-matrix and \(S\)-matrix from which the collision cross sections can be determined. The author derives a variational principle for the \(R\)-matrix defined on the boundary of the internal region.
Let us briefly list other topics of this chapter:
methods for determining zero-order radial continuum basis orbitals which represent the scattered electron in the expansion of the total wave function in the internal region;
methods for calculating corrections to the \(R\)-matrix and wave function;
a partitioned \(R\)-matrix method where the calculation of the \(R\)-matrix is sub-divided into two parts: a low-energy part which is accurately determined and a high-energy part for which an approximation is derived enabling much larger problems to be treated;
electron collisions with atoms and ions with higher nuclear charge number \(Z\), where relativistic effects must be included in the calculation;
a frame-transformation theory approach where relativistic effects are omitted, or only partly included, in the internal region.
In Chapter 6 Burke considers electron collisions with atoms and atomic ions at intermediate energies which are defined to range from close to the ionization threshold to several times this threshold. The author describes the intermediate-energy \(R\)-matrix method, analyzes the analytic behaviour of the scattering amplitude and cross sections at intermediate energies, and describes distorted wave and Born series \(R\)-matrix methods for electron impact ionization. Finally, results from some recent electron-atom and electron-ion collision \(R\)-matrix calculations at intermediate energies are presented.
Chapter 7 studies the multichannel \(R\)-matrix theory of electron collisions with atoms and atomic ions. It begins with a general discussion of the processes that can occur in positron and positronium collisions with atoms and atomic ions. Then the author considers the new extensions to the multichannel \(R\)-matrix theory of electron collisions with atoms and ions, in order to enable the channels corresponding to positronium collisions to be included in the theory.
Chapter 8. In this chapter the author considers the development of the \(R\)-matrix theory to treat the atomic photo-ionization and photo-recombination processes (the consideration is restricted to single photon processes) and considers also an extension of the \(R\)-matrix theory to describe the spectra of atoms in external fields. In particular, this chapter includes:
derivation of a general expression for the differential cross section for photo-ionization of an unpolarized atom or atomic ion by a polarized beam of photons;
a discussion of \(R\)-matrix methods for calculating both the initial bound state of the target atom and the final continuum state;
description of methods for carrying out calculations in the neighbourhood of \(R\)-matrix poles;
extension of the usual Schrödinger equation describing electron-ion collisions by adding a “radiation damping potential”;
description of a computational approach which describes the spectra of atoms in external fields (this approach combines the complex coordinate rotation method with a new external region \(R\)-matrix method).
In Chapter 9 the author discusses multiphoton processes by considering the atomic \(R\)-matrix-Floquet theory and applications. This is an ab initio theory which is fully non-perturbational and is applicable to arbitrary multi-electron atoms and atomic ions, allowing an accurate description of electron-electron correlation effects. In principle this theory is confined to treating laser pulses involving many cycles of the field, typically exceeding tens of femtoseconds. There is also a discussion of an extension of this theory using multistate non-Hermitian Floquet dynamics, which has allowed detailed calculations to be carried out for shorter laser pulse interactions, well agreeing with fully time-dependent calculations.
In Chapter 10 the author considers the interaction of ultra-short laser pulses with atoms and atomic ions, where the laser pulses may involve only a few cycles of the field. The interaction of intense ultra-short laser pulses with atomic targets cannot be accurately treated by the \(R\)-matrix-Floquet theory or by using multistate non-Hermitian Floquet dynamics, considered in the previous chapter. Instead, the full time-dependent Schrödinger equation describing the laser-atom interaction must be solved.
As well as providing a unique way of studying ultra-short laser pulse interactions with atomic targets, time-dependent theory complements Floquet theory by emphasizing the time domain instead of the energy domain of the process. The time-dependent approach can accurately model laser-atom interactions with arbitrary laser pulse profiles and is the natural description of atomic processes in the intense field limit when ionization may occur in the order of a few field periods. However, new phenomena may be difficult to interpret using the time-dependent theory, while the Floquet approach provides a framework for achieving this by describing the behaviour in terms of atomic states dressed by the laser field. The energies of dressed states are directly calculated in the Floquet approach, and in a wide range of multiphoton processes the dynamics can be understood very simply in terms of just a few dressed states. Thus, for longer laser pulses, where both theories are applicable, they provide two ways of looking at the same problem. In this chapter the author describes ab initio non-perturbative time-dependent \(R\)-matrix theories of atomic multiphoton processes which make it possible to calculate the interaction of few femtosecond and attosecond laser pulses with arbitrary multi-electron atoms and atomic ions. Computational methods for solving these equations are also discussed.
In Chapter 11 the \(R\)-matrix theory is extended to treat electron and positron collisions with molecules and to treat molecular multiphoton processes. The processes that occur in \hboxelectron-, positron- and photon-molecule collisions are considerably more varied and challenging than those that arise in electron collisions with atoms and atomic ions, partly because of the loss of spherical symmetry and partly because of the possibility of exciting degrees of freedom associated with the motion of the nuclei in the molecule.
Chapter 12 includes two extensions of the \(R\)-matrix theory describing interactions of electrons in solids. The first extension describes low-energy electron collisions with transition metal oxides and the second extension describes electron transport in two-dimensional semiconductor devices in the presence of an external field.
The monograph concludes with six appendices which summarize basic mathematical results (Clebsch-Gordan and Racah coefficients, Legendre polynomials, Bessel functions, …) and computational methods which are used in Parts I and II.

81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
81V35 Nuclear physics
81V45 Atomic physics
81Uxx Quantum scattering theory
34L25 Scattering theory, inverse scattering involving ordinary differential operators
35P25 Scattering theory for PDEs
47A40 Scattering theory of linear operators
35B34 Resonance in context of PDEs
47A10 Spectrum, resolvent
81V80 Quantum optics
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