Showing papers in "Reviews of Modern Physics in 1967"

Reevaluation of x-ray atomic energy levels.

Abstract: All of the x-ray emission wavelengths have recently been reevaluated and placed on a consistent \AA{}* scale. For most elements these data give a highly overdetermined set of equations for energy level differences, which have been solved by least-squares adjustment for each case. This procedure makes "best" use of all x-ray wavelength data, and also permits calculation of the probable error for each energy difference. Photoelectron measurements of absolute energy levels are more precise than x-ray absorption edge data. These have been used to establish the absolute scale for eighty-one elements and, in many cases, to provide additional energy level difference data. The x-ray absorption wavelengths were used for eight elements and ionization measurements for two; the remaining five were interpolated by a Moseley diagram involving the output values of energy levels from adjacent elements. Probable errors are listed on an absolute energy basis. In the original source of the present data, a table of energy levels in Rydberg units is given. Difference tables in volts, Rydbergs, and milli-\AA{}* wavelength units, with the respective probable errors, are also included there.

X-Ray Wavelengths

Abstract: Inconsistencies in accepted values (in x units) of x-ray reference lines have recently been demonstrated, although all are supposedly based on "good" calcite crystals. Factors supporting the selection of the W $K{\ensuremath{\alpha}}_{1}$ line as the X-Ray Wavelength Standard are critically discussed. A review is given of the experimental measurements which are used to establish the wavelength of this line on an absolute angstrom basis. Its value is $\ensuremath{\lambda}$ W $K{\ensuremath{\alpha}}_{1}=(0.2090100\ifmmode\pm\else\textpm\fi{}5 \mathrm{ppm})$ \AA{}. This may be used to define a new unit, denoted by \AA{}*, such that the W $K{\ensuremath{\alpha}}_{1}$ wavelength is exactly 0.2090100 \AA{}*; hence 1\AA{}*=1\AA{}\ifmmode\pm\else\textpm\fi{}5 ppm. The wavelengths of the Ag $K{\ensuremath{\alpha}}_{1}$, Mo $K{\ensuremath{\alpha}}_{1}$, Cu $K{\ensuremath{\alpha}}_{1}$, and the Cr $K{\ensuremath{\alpha}}_{2}$ have been established as secondary standards with probable error of approximately one part per million. Sixty-one additional x-ray lines have been used as reference values in a comprehensive review and reevaluation of more than 2700 emission and absorption wavelengths. The recommended wavelength values are listed in \AA{}* units together with probable errors; corresponding energies are given in keV. A second table lists the wavelengths in numerical order, and likewise includes their energies in keV.

Static Phenomena Near Critical Points: Theory and Experiment

Abstract: This paper compares theory and experiment for behavior very near critical points. The primary experimental results are the "critical indices" which describe singularities in various thermodynamic derivatives and correlation functions. These indices are tabulated and compared with theory. The basic theoretical ideas are introduced via the molecular field approach, which brings in the concept of an order parameter and suggests that there are close relations among different phase transition problems. Although this theory is qualitatively correct it is quantitatively wrong, it predicts the wrong values of the critical indices. Another theoretical approach, the "scaling law" concept, which predicts relations among these indices, is described. The experimental evidence for and against the scaling laws is assessed. It is suggested that the scaling laws provide a promising approach to understanding phenomena near the critical point, but that they are by no means proved or disproved by the existing experimental data.

Angular distributions of gamma rays in terms of phase-defined reduced matrix elements.

Abstract: The theory of angular distributions of $\ensuremath{\gamma}$ rays is developed systematically, aiming at a phase consistent derivation of angular distribution formulas for gamma rays emitted in the decay of an aligned initial state. The development starts from first principles, that is, the angular distribution formulas are derived directly from perturbation theory and all quantities introduced are carefully and explicitly defined. In particular the mixing ratios are phase consistently related to reduced matrix elements of interaction multipole operators which again are well defined in phase. Hence the mixing ratios become physical quantities which can be extracted from angular distribution measurements and then compared in both magnitude and sign with the predictions of nuclear models (especially the independent particle model). Critical stages in the theoretical development at which either a choice of phase convention has to be made or transformation properties enter are emphasized.

Elements of the brueckner--goldstone theory of nuclear matter.

Abstract: The basic ideas of the Brueckner-Goldstone theory of nuclear matter are presented in a simple way. The treatment is aimed at beginners and nonspecialists. It is supposed to provide the necessary background for the review article by Bethe and Rajaraman which follows this paper. Therefore, the discussion is limited to a few important topics, and these are considered in some detail.The Goldstone expansion is presented (but not derived) and the construction and evaluation of the Goldstone diagrams are explained. The reaction matrix and the correlated two-body wave function are defined, and their properties are discussed. The reference-spectrum method for calculating the reaction matrix is derived, and its use is illustrated. Finally, the related topics of convergence and the definition of single-particle energies are considered. The choice of the single-particle potential energy for occupied states is treated in detail. (Intermediate-state energies will be discussed by Bethe and Rajaraman.) The reason for the divergence of the perturbation series for the binding energy is exhibited; and this series is rearranged into a convergent expansion, for which the density plays the role of small parameter.

Three-body problem in nuclear matter

Abstract: This work reviews some recent developments in the theory of nuclear matter. Assuming familiarity with the basic Brueckner-Goldstone theory described in the preceding article by B. Day, it is first shown that the Brueckner-Goldstone series does not converge in powers of the reaction matrix, and that the perturbation series for the binding energy has to be rearranged in powers of the density $\ensuremath{\rho}$. Physical reasons and actual estimates are provided for expecting convergence in powers of $\ensuremath{\rho}$.A detailed theory is outlined for the evaluation of the three-body energy, which gives the ${\ensuremath{\rho}}^{2}$ term. Attention is paid to both the momentum dependence of the reaction matrix and the tensor nature of nuclear forces. Finally, the last section is devoted to the choice of the single-particle potential energies, suitably designed so as to absorb most of the four-body and higher cluster terms.

Scattering Theory of Absorption-Line Profiles and Refractivity

Abstract: Attenuation cross section and refractive index using collision theory, calculating resonance profiles of autoionizing lines based on scattering theory

Moderately Long-Range Interatomic Forces

Abstract: The moderately long-range interaction energies of degenerate atoms for nonresonant cases have been studied throughout the moderately long-range region. Extensive tabulation of necessary parameters and atomic properties for the calculation of the first-order quadrupole-quadrupole interaction energies has been made. Higher multipole interactions also have been considered and it has been shown that the $\frac{1}{R}$ series of the first-order Coulombic interaction energies converges very fast throughout the long-range region for atoms in the ground configuration. The effects of atomic spin-orbit splitting have been considered explicitly. It has been shown that ($\ensuremath{\Delta}, S$) coupling and intermediate coupling may be important for the interactions between B, C, O, Al, Si, and Sc atoms in the relatively short internuclear separation range. For other atoms, the (${J}_{a}, {J}_{b}$) coupling scheme will give satisfactory results throughout the long-range region. The experimental determination of the moderately long-range interatomic forces from predissociation data also has been discussed.The estimated van der Waals dispersion energies for the first-row atoms are shown to be of almost the same size as the quadrupole-quadrupole interaction energies at the separation of twice the sum of the atomic radii. It has been also shown that the leading term [$\ensuremath{\theta}(\frac{{\ensuremath{\alpha}}^{2}}{{R}^{3}})$] of the magnetic interaction energy of two degenerate atoms is 10\ensuremath{\sim}20% of the quadrupole-quadrupole interaction energy at $R=30{a}_{0}$ through the third-row atoms.

Group algebra, convolution algebra, and applications to quantum mechanics.

Abstract: The symmetry group of the Hamiltonian plays a fundamental role in quantum theory in the classification of stationary states and in studying transition probabilities and selection rules It is here shown that the properties of the group may be given a condensed and transparent description in terms of the convolution algebra, and that Schur's lemma immediately leads to the construction of the fundamental set of projection and shift operators The projection operators form a resolution of the identity which may be used to split the Hilbert space into orthogonal and noninteracting subspaces of infinite order The question of the splitting of the conventional secular equations is discussed, and the explicit form of the decomposed equation is derived in terms of the convolution algebra and the characters The theory is here discussed only for finite groups, but the results may be generalized to the compact infinite groups having a well-defined "invariant mean"

Generalized Susceptibility Theory I. Theories of Hypochromism

Abstract: A general theory is developed for the linear response of an interacting molecular system to an external field. The exact ground state of the system is expressed in terms of the uncoupled molecule (zeroth-order) state by means of adiabatic time-dependent perturbation theory. Including the external field to first order in the time-development operator leads to an infinite-order expansion of the linear response function for physical quantities such as current-charge density and electric-dipole polarization. Two basic approximations involving partial decorrelation of charge motion and spatial separability of the molecules allow the response function to be determined by a simplified Dyson-type equation, which can be put into closed form by further approximations. The approximate linear response function (susceptibility) so obtained is given in terms of the isolated molecule susceptibility and corresponds exactly to the results of classical, local field theory. Emphasis throughout is placed on the effect of molecular interaction on absorption spectra. Comparisons among theories of hypochromism show clearly that all previously reported theories are mutually compatible and are either equivalent to, or are contained in, the theory developed here. The relation of the present theory to a coupled equivalent oscillator model is discussed and the results are applied to a simple physical model.

Formal Theory of Nonlinear Response

Abstract: This paper presents the derivation of formal, exact expressions for "generalized response coefficients," quantities which characterize the response of a system to conservative forces of arbitrary strength and time dependence. The development avoids all expansions of the response in powers of the driving forces. The generalized response coefficients thus provide the basis for calculations of nonlinear effects in those situations for which expansions in powers of the forces are not suitable. It is shown how the linear and higher-order response functions obtained first by Kubo can be obtained in a relatively more compact way. The expressions corresponding to static forces are considered in some detail. Generalized response coefficients are also derived for systems in equilibrium; the lowest order of these is just the isothermal susceptibility as usually defined.

Broadening of Isolated Lines in the Impact Approximation Using a Density Matrix Formulation

Abstract: In recent years the theory of line broadening has been considerably advanced through the generalized impact approximation, which has as its starting point an expression for the quantum-mechanical autocorrelation function. In this paper the results for isolated lines in the impact approximation are rederived using a density matrix formulation. The method presented here is not particularly rigorous; however, it assists in the understanding of the approximations made in the autocorrelation function approach, and in some cases possible methods of extending the theory become much more apparent. For example, in this formulation, the broadening of an electric quadrupole transition is obvious.

Electrodynamics of a Semiclassical Free-Electron Gas

Abstract: This article presents a simplified treatment of the high density, collisionless, free-electron gas, based on the ideas of a wave number and frequency-dependent conductivity and dielectric constant. The formalism is applied to solve a number of problems: the screening of the electrostatic potential of a foreign point charge placed in the electron gas, the rate of energy loss of a charged particle moving through the electron gas, plasma oscillations, the reflection of electromagnetic waves from the electron gas, and ultrasonic attenuation in metals due to the interaction of the sound waves with the conduction electrons. In a final section it is indicated how the methods may be generalized. Explicit expressions for the conductivity of the electron gas are obtained in an appendix.

Interaction of Two Nucleons at Low Energies. I = 1

Abstract: A summary of the knowledge of the low-energy interaction of two nucleons with one unit of isobaric spin is presented. The single energy analyses of accurate low-energy proton-proton scattering data are discussed with emphasis upon the electromagnetic effects which have been included and those which should be. The phase shifts which result are presented and discussed in terms of the effective range expansion and potentials. A brief discussion of charge symmetry and independence in terms of potentials and dispersion relations is given. The equality of the neutron-neutron and proton-proton nuclear interactions, which seems to be established, may not be sufficient for understanding the problem of mirror nuclei. The question of charge independence is much more difficult.

A General Graphical Method for Angular Momentum

Abstract: A graphical method is presented allowing the representations of $``j\ensuremath{-}m''$ and $``3nj''$ coefficients, spherical harmonics, irreducible tensor operators, and rotation matrices. Rules are established which permit calculations on expressions with the above elements. As the main difficulty of using $``3j''$ Wigner coefficients is the construction of the phase, an algorithm is proposed which simplifies this problem. Concrete examples are given for this general method.

Neutron-Neutron Interaction

Abstract: The present status of the problem of charge dependence is discussed. The information about the neutron-neutron interaction derived from the two-neutron system, three-nucleon systems, final-state interactions in multiparticle reactions, and peripheral processes is critically evaluated. The experimental data indicate the breakdown of charge independence by about 3-5%. Evidence concerning the violation of charge symmetry is inconclusive, but it seems that most of the data are consistent with the assumption that charge symmetry is satisfied within 0.5-1%. The most suitable studies which might improve the knowledge of the neutron-neutron forces are indicated.

S -Matrix Theory of Electromagnetic Interactions

Abstract: This paper studies the possibility of applying the principles of the $S$-matrix theory to electrodynamics. By using an approximation scheme different from that in the strong interactions, many results of quantum electrodynamics can indeed be reproduced. While the calculations of electrodynamical problems can be carried out in a more straightforward and less laborious way in the $S$-matrix theory, there still are some basic difficulties. However, as far as scattering processes and the anomalous magnetic moment are concerned, there is complete identity (up till fourth order) between the two methods. The bound-state problem, infrared divergence, and the many-particle problem remained unsettled.