Showing papers in "Reviews of Modern Physics in 1972"

X-Ray Fluorescence Yields, Auger, and Coster-Kronig Transition Probabilities

Abstract: The present status of the field of fluorescence yields, radiationless (Auger and Coster-Kronig) and radiative transition probabilities is summarized. Tables of experimental and theoretical results are included, and tables of "best values" of important quantities are presented.

Funny Hills: The Shell-Correction Approach to Nuclear Shell Effects and Its Applications to the Fission Process

Abstract: This paper reviews various results related to the single-particle structure in spherical and deformed nuclei, discussed from the viewpoint of the so-called shell-correction method. This method stresses the importance of large-scale nonuniformities in the energy distribution of the individual particles especially near the Fermi energy. The way in which these nonuniformities affect in an essential way many nuclear properties, such as the shape stiffness, the spatial density distribution, the total mass of the nucleus, the mass and inertia of the nuclear shape variations, etc. is also discussed. Against this background, the behavior of the nuclear deformation energy is described, in particular for larger distortions relevant to the fission process. In this connection, some qualitative singularities of the phenomenological liquid-drop deformation energy at large shape distortions are pointed out, and their possible implications for fission are discussed. As the problems considered cover a wide range of nuclear properties, this paper is not a review in the narrow sense of the word. Comparison with other approaches as well as historic references are given mainly to clarify specific points, because a complete review would be a monumental undertaking.

Charge States and Charge-Changing Cross Sections of Fast Heavy Ions Penetrating Through Gaseous and Solid Media

Abstract: This review surveys the experimental and theoretical situation concerning charge states and charge-changing cross sections of heavy ions up to and including uranium, which penetrate through gaseous and solid targets with velocities primarily in the range ${v}_{0}lvlZ{v}_{0}$ (${v}_{0}=\frac{{e}^{2}}{\ensuremath{\hbar}}=2.188\ifmmode\times\else\texttimes\fi{}{10}^{8}$ cm/sec). Particular emphasis is given to ions with atomic numbers in the range $16\ensuremath{\le}Z\ensuremath{\le}92$. The published literature is covered through August 1971. General physical and mathematical relations are outlined which describe the composition of charge states in a heavy-ion beam which passes through matter. Recent experimental techniques and methods of data analysis are summarized. Extensive experimental results on heavy-ion equilibrium charge state distributions, average equilibrium charge states, and cross sections for capture and loss of one or more electrons in single encounters with target atoms are presented and critically examined. The data extend to ions as heavy as uranium and energies up to \ensuremath{\sim}400 MeV. Systematic trends are emphasized and generalizations are discussed which allow interpolations and to some extent extrapolations of the data to be made to ranges which have not been investigated experimentally. Attention is given to the cross sections for multiple-electron loss which are relatively large but which are poorly understood. We deal with effects of residual ion excitation on charge-changing collisions in the light of recent experimental results. It is shown that the average equilibrium charge of heavy ions can be approximated by utilizing both theoretical concepts which originate from the work of Bohr and Lamb, and semiempirical relations which are based on observed regularities of the data. Recent interpretations of phenomena associated with density effects, i.e., with the increase of projectile ionization which is observed for increasing target densities, are scrutinized and refinements of the theory by Bohr and Lindhard are explored.

Aspects of Time-Dependent Perturbation Theory

Abstract: The Dirac variation-of-constants method has long provided a basis for perturbative solution of the time-dependent Schr\"odinger equation. In spite of its widespread utilization, certain aspects of the method have been discussed only superficially and remain somewhat obscure. The present review attempts to clarify some of these points, particularly those related to secular and normalization terms. Secular terms arise from an over-all time-dependent phase in the wave function, while normalization terms preserve the norm of the wave function. A proper treatment of the secular terms is essential in the presence of a physically significant level shift that can produce secular divergences in the time-dependent perturbation functions. The normalization terms are always important, although the formulation of a simple method for including them is of greatest utility in applications requiring higher-order perturbation theory (e.g., nonlinear optical phenomena), where difficulties have arisen in previous treatments. Although the Dirac perturbation technique includes the correct secular and normalization terms when properly executed, it is convenient to reinterpret the perturbation problem so that the secular and normalization terms can be factored from the wave function to all orders. It is shown that an appropriate over-all multiplicative, time-dependent normalization and phase factor can be obtained, and that it is simply the amplitude for finding the system in the unperturbed eigenstate at any time $t$. The regular part of the wave function remaining after this factorization provides a complete description of the physical properties of the system of interest and determines the over-all normalization and phase, as well. Most important, the regular function and its perturbation expansion satisfy equations which are more convenient for computational applications than are the customary Dirac equations, and, in contrast to the latter, they reduce directly to the familiar time-independent perturbation equations in the static limit. To illustrate the general development, the model problem of a linearly perturbed harmonic oscillator and the static, harmonic, and electromagnetic perturbations of arbitrary quantum-mechanical systems are treated explicitly. In the case of an adiabatically applied static perturbation, the familiar adiabatic theorem is recovered with the over-all phase factor giving the perturbed eigenvalue, while in the case of an harmonic perturbation, the overall phase factor obtained includes the system level shift appropriate for a quasiperiodic state. For an electromagnetic perturbation, compact expressions are obtained for various nonlinear optical susceptibilities in forms suitable for computations. Time-dependent Hartree-Fock approximations are treated explicity to demonstrate that difficulties can arise when normalization and secular terms are not extracted prior to application of the perturbation formalism. Connection is also made with other methods which can be employed to eliminate secular and normalization terms from the wave function; these include a projection procedure and multiple-time-scales perturbation theory. The elimination of secular divergences from the perturbation functions is shown to be important for the construction of a valid Fourier transform. Secular and normalization terms also arise in connection with variational principles for the time-dependent Schr\"odinger equation. By employing the Frenkel variational principle and an ansatz for the total wave function that explicitly isolates the secular and normalization terms, a computationally convenient variational functional is obtained. This form of the Frenkel principle provides a bound to the system level shift induced by an oscillatory perturbation and is equivalent to the Ritz variational principle in the static limit. Explicit expressions for the variational functional in the Hartree-Fock approximations are derived in forms suitable for computational applications to the interactions of radiation and matter.

The Vibrational Properties of Disordered Systems: Numerical Studies

Abstract: A review is presented of the direct numerical approach to the study of the atomic vibrational properties of disordered systems. The basis and details of the numerical methods employed are first described. This is followed by a review of applications of the approach to two-component disordered lattices, glasses, mixed-crystal systems, orientationally disordered crystals, and random polymers.

Response Functions in the Theory of Raman Scattering by Vibrational and Polariton Modes in Dielectric Crystals

Abstract: The paper begins with a tutorial introduction to the theory of inelastic light scattering by polaritons in dielectric crystals. The treatment is based on a simple two-oscillator model which represents the ionic and electronic motions of a crystal. The model contains a third-order anharmonicity which allows an incident laser beam to mix with the oscillator fluctuations and produce scattered light of frequency different from the incident frequency. The magnitude of the oscillator fluctuations is determined by an application of the Nyquist of fluctuation-dissipation theorem, using the response functions of the oscillators for externally applied forces. The simple model gives results for light scattering cross sections which agree with more rigorous derivations in the existing literature. The response function approach is generalized to apply to crystals having many ionic resonances and of uniaxial or orthorhombic structure. The general formulas reduce in appropriate special cases to results already published. Experimental and theoretical work on light scattering by polaritons and by pure phonons is reviewed in the context of both the two-oscillator model and the general theory. Particular attention is given to resonance scattering in an attempt to achieve consistency between the differing theoretical treatments in the literature. The subject matter of the review overlaps some topics in nonlinear optics, and contact is made with the theories of the electrooptic effect and stimulated Raman scattering.

A Theory of Isobaric Analog Resonances

Abstract: A theory of analog resonances is reviewed which makes use of projection operators. The Hilbert space is divided into three parts: a continuum or open-channel space, an analog-state space, and a compound space. The phenomena are discussed in terms of the dynamical coupling of these spaces. The parameterization of the $T$ matrix is discussed in detail, and equations are presented for various cross sections. The commutator [$H$, ${T}_{\ensuremath{-}}$], where ${T}_{\ensuremath{-}}$ is the isospin-lowering operator, plays an important role in the theory, and the various terms which contribute to this commutator are discussed. The energy splitting of the isospin multiplet, i.e., the Coulomb displacement energy, is discussed in detail. The importance of the analog resonance phenomena for the extraction of spectroscopic information is stressed, and it is shown how such information may be obtained. Various processes which contribute to the escape amplitude of the analog state are classified, and some numerical estimates are given. For several regions of the Periodic Table, graphs are presented for the various theoretical escape amplitudes, continuum energy shifts, asymmetry phases, and optical phase shifts, etc. Spectroscopic factors are calculated and compared with those obtained in other experiments.

The Properties of Defects in Magnetic Insulators

Abstract: The introduction of a low concentration of defects into a magnetic insulator modifies the spectrum of the magnetic excitations. In general the spectrum consists of a set of impurity modes associated with the defect and its immediate neighbors. Impurity modes that occur outside the band of host excitations are localized in the neighborhood of the defect and at the same time perturb the host band, while modes lying within the band lead to resonant behavior of the excitations of the host. In recent years, optical, neutron scattering, and nuclear magnetic resonance techniques have been used to study mixed crystals of antiferromagnetic transition metal fluorides. Many of the features may be understood by using the molecular field or Ising model for the excitations. An improvement on this form of the theory is to use a cluster model to describe the excitations near to the defect. Some features may however be described only when the excitations of the host are treated adequately; this requires the use of Green's function theories that have been developed for antiferromagnets containing defects. A detailed comparison is presented of the predictions of the various theories with the experimental results. Although the theory is fairly satisfactory for a low concentration of defects and low temperatures, considerable complexities arise in extending it to higher temperatures and large concentrations.

High-Energy Multiparticle Reactions

Abstract: Hadronic multiparticle reactions at very high energies are reviewed with emphasis on current theoretical pictures and models.

Thermodynamic, Elastic, and Magnetic Properties of Solid Helium

Abstract: Recent experimental studies of solid helium are reviewed with special attention given to specific heat, isochoric pressure, neutron scattering, sound velocity, and magnetic susceptibility measurements. The relationships among the properties are stressed. Low temperature properties of $^{3}\mathrm{He}$, including exchange, nuclear spin ordering, the melting curve, and Pomeranchuk colling, are discussed. Phase separation in solid $^{3}\mathrm{He}$-$^{4}\mathrm{He}$ mixtures is reviewed.

The current status of the lepton g factors

Abstract: The past three years have seen new measurements of the $g$-factor anomalies of the free leptons (${e}^{\ifmmode\pm\else\textpm\fi{}}, {\ensuremath{\mu}}^{\ifmmode\pm\else\textpm\fi{}}$). At the same time there have been several major theoretical advances including a quantum electrodynamic calculation of the sixth-order coefficient of the electron anomaly. In this article we review these recent developments in detail, however we also: (1) present a historical overview of $g$-factor measurements and their interpretation since the early days of quantum theory, (2) discuss several experiments which are currently being attempted but which have not yet yielded results of high precision, (3) attempt to predict the most likely course of $g$-factor research in the near future. An extensive Bibliography and Tables summarizing both historical events and current work are included.

Rigorous constraints, bounds, and relations for scattering amplitudes.

Abstract: We present a review of rigorous results on scattering amplitudes at low, intermediate, and high energies. The emphasis is on constraints that can be compared with experiment. Some new results are presented, in particular the existence of absolute bounds for inelastic processes, constraints for odd pion-pion waves, and a lower bound for scattering amplitudes at positive t. Most of the results rest only upon unitarity and what is popularly known as axiomatic analyticity, but there are a few cases where larger analyticity domains are needed.