The current understanding of the theory of the atomic photoelectric effect is reviewed for incident photon energies above 10 keV, complementing the earlier review of Fano and Cooper (1968) at lower energies. The theoretical developments of the last two decades are of two types: (1) analytic results giving insight into many aspects of the photoelectric process and (2) exact numerical cross sections calculated with high speed electronic computers. The basic assumptions underlying the photoeffect calculations are described, and pertinent atomic models are discussed. In the energy range considered, satisfactory results are obtained with the process described as the ejection of an electron moving in a relativistic Hartree-Fock-Slater potential. Exchange, correlation, and other effects are discussed. Many features of the process can be understood with the realization that the important regions in configuration space in the photoeffect matrix element are of the order of an electron Compton wave length. This leads to the predictions that the photoeffect cross section in a screened potential can be obtained from a point-Coulomb result by a simple normalization, that angular distribution shapes and polarization correlations are the same in the two cases, and that results for photoeffect from different subshells of the same angular momentum are similarly related. The numerical methods, achieving total cross sections accurate to 1%, are described and compared with experiment. Different self-consistent atomic models yield cross sections which differ by 3%-8%; these agree with experiments of similar accuracy. New and more accurate cross section tabulations that have recently become available are discussed and recommendations are made concerning their use.

Thermal neutron-capture $\gamma$ rays have found applications in a variety of nuclear reaction studies requiring $\gamma$ rays in the energy region up to 11 MeV. The importance of these $\gamma$-ray sources is that they provide very high intensities, which exceed by two orders of magnitude the intensities per unit energy interval obtainable from other sources. The experiments performed with these sources are reviewed in this article, and the advantages and disadvantages of the method are discussed.

Charge changing processes of hydrogen beams in gases (${\mathrm{H}}_2$, He, ${\mathrm{N}}_2$, ${\mathrm{O}}_2$, Ne, Ar, Kr, Xe), atomic hydrogen, alkali metal vapors (Li, Na, Mg, K, Cs), and gaseous carbon are reviewed primarily from an experimental point of view. Following a simple description of charge changing phenomena and typical techniques of measurement, problems associated with the experiments are discussed. Experimental cross-section results for the various charge changing processes are presented in figures with critical comments. A brief review of the theoretical developments relevant to the charge changing processes of hydrogen beams is also presented.

Theoretical and experimental work relevant to the creation of atomic inner-shell vacancies in collisions of ions and atoms is reviewed. The experimental data on total excitation cross sections and electron and x-ray emission spectra are discussed in some detail. Energy loss data from inelastic scattering experiments involving heavy ions are also reviewed. An attempt is made to relate the different kinds of data to one another and to the available theoretical models. Excitation by the light ions (protons, alpha particles) has been well described theoretically in terms of perturbation by an incident point charge. However, no comprehensive model yet exists for the case of incident heavier ions, although the heavy-ion data for low collision velocities support an interpretation based on the formation of a transient quasimolecule. The literature has been reviewed to about April 1972.

This complement of an earlier paper [Rev. Mod. Phys. 38, 121 1966] reviews experimental and theoretical progress made since 1965 in the knowledge of quasi-free ($p$, $2p$) and ($e$, $e^{\prime }p$) processes in nuclei. Some aspects of the reaction mechanism are considered and a new method to present the experimental data is suggested. The character of the final hole states is discussed. New experimental results are presented and the available information on "hole-spectroscopy" is summarized. Several open problems are pointed out.

The theory of the treatment of particle hole states in the continuum is reviewed with emphasis on the eigenchannel method. The results of actual calculations are compared with experimental data. Some aspects of the experiments are rather well reproduced by the calculations. However, in certain other aspects a qualitative disagreement exists between theory and experiment, pointing to an essential inadequacy of the underlying nuclear model.

The applications, most of which have been developed in the last decade, of the Hellmann-Feynman (H-F) theorem in molecular quantum mechanics are reviewed. In general, the forces (on the nuclei of molecules) calculated with the use of this theorem provide great qualitative insight into the nature of the phenomena investigated; outstanding examples of these are in the concepts of chemical binding and molecular shapes. However, there are serious limitations in quantitative applications of the H-F theorem with approximate wave functions, since the calculated forces are extremely sensitive to small inaccuracies in the wave functions, especially near the nuclei of interest. Nevertheless, in view of the fact that it is difficult to discern general qualitative features in very accurate or ab initio molecular calculations, the H-F theorem is likely to be a highly useful tool for developing much needed qualitative chemical models which will be based on firm quantum mechanical foundations and will also remain open to quantitative extension, at least in principle.

The conflict between observation and theoretical prediction of the flux of electron neutrinos from the sun has advanced in the past year from being merely difficult to understand to being impossible to live with. We review here attempts to explore the nature of the conflict, to seek possible ways out of it, and to inquire into additional experiments that have the capability either of resolving the conflict or at least of deciding which branch of physics or astrophysics is responsible for it.

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.

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.

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.

The Dirac variation-of-constants method has long provided a basis for perturbative solution of the time-dependent Schrödinger 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 Schrödinger 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.

Methods for constructing states of good total orbital angular momenta of $N$ identical, free, structureless particles through the use of the orthogonal and unitary groups are developed. The first part of the paper reviews the existing literature, particularly for the three-particle problem. New results include the discrete symmetry properties of the SU(3) states vectors of the three-particle problem. The general $N$-particle problem is approached through the use of the subgroup property $O\left(n\right)\mathrm{}\subset U\left(n\right)\mathrm{}$. An imbedding of $O\left(n\right)\mathrm{}$ in $U\left(n\right)\mathrm{}$ is given which greatly simplifies the study of the $O\left(n\right)\mathrm{}$ subgroup of $U\left(n\right)\mathrm{}$. Particular applications of this imbedding are: (1) an explicit constructive procedure for obtaining all the single-valued irreducible representations of $O\left(n\right)\mathrm{}$, and (2) an explicit constructive procedure for obtaining all $N$-particle states of good angular momenta up through the degree four solid harmonics.

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 $v_0<v<Z{v}_0$ ($v_0=\frac{e^2}{\hslash }=2.188\mathrm{}\times {10}^8$ cm/sec). Particular emphasis is given to ions with atomic numbers in the range $16\le Z\le 92$. 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 ∼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.

Natural orbitals are a particularly efficient choice of single-particle states for use in construction of electronic wave functions. This review discusses the general properties of the natural orbitals. Natural orbitals are compared to Hartree-Fock orbitals and relative merits are presented. Methods of determining and using natural orbitals in construction of electronic wave functions are presented.

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.

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.