Data on the properties of leptons, mesons, and baryons are listed, referenced, averaged, and summarized in tables and wallet cards. This is an updating of the Reviews of Modern Physics article of January 1967.

This paper supplements the preceding paper by Maradudin and Vosko on the same subject. Extensions and additions include discussions of (1) the group of the bond, (2) tabulations of the time-reversal invariance criterion, (3) the labeling of experimental dispersion curves, (4) compatibility relations, (5) accidental degeneracies, (6) the use of computers for extracting symmetry information, and (7) application to magnetic normal modes. The chief results of the preceding paper are summarized. Group theory is used to predict the connectivity of the phonon dispersion relations in CsCl and Mg. A collection of corrections to primary references useful in this field is included.

The relation between anomalous charging or discharging currents and ac dielectric behavior of a material is discussed together with the use of the Hamon approximation for calculating the loss factor from these currents.

The validity of the assumption used in deriving the Hamon approximation is discussed with reference to the behavior of a material with a Cole-Cole distribution of relaxation times. An analytical derivation of the Hamon approximation which has appeared in the literature is described together with a discussion of the limits of its validity. A description of the experimental methods which have been used to measure these anomalous currents (dc transient technique) is given, and experimental data in the literature showing the continuity of the loci of relaxation processes and values of loss factor from the audio frequency region into the very low-frequency region (e.g., ${10}^{-4\mathrm{}}$ to ${10}^{-2\mathrm{}}$ cps) is discussed.

Solids often have long relaxation times at ordinary temperatures and are then suitable for examination with the dc transient technique. Moreover, the components of loss due to relaxation and conduction may be separately determined using this technique, and this is an essential prerequisite to an understanding of the different mechanisms involved. When peaks in loss factor are obtained, all the usual dielectric information may be obtained—including distribution of relaxation times, activation energy and entropy for dipole orientation, and dipole-moment values for the structural units involved.

The significance of chemical relaxation experiments (e.g., temperature jump measurements) for kinetic investigations of reaction systems involving any degree of complexity is examined. On the basis of a general solution of the relaxation problem, the properties of experimentally measurable relaxation curves—characterized by a relaxation spectrum—are discussed. Usually, actual theoretical or experimental determinations of complete relaxation spectra and their use for kinetic analyses are practicable only for fairly simple reaction systems. In the case of more complex systems an evaluation of relaxation experiments in terms of mean relaxation times is suggested. The mean reciprocal relaxation time is of particular significance. This quantity is shown to be fairly easily accessible experimentally. On the other hand, it can be computed without a complete solution of the relaxation problem. Examples are given to demonstrate the procedure and its potentialities with respect to the investigation of complicated reaction processes.

With the availability of modern lasers, light scattering can now be used as a probe of the energy, damping, and relative weight of the various hydrodynamic collective modes in anharmonic insulating crystals. We first give a general review of the distinction between hydrodynamic and high-frequency (or dynamic) vibrational modes. We then express the intensity and spectral distribution of scattered light in terms of the Fourier transform of the displacement-displacement correlation function ${\chi }^{\prime \prime }\left(\mathrm{K}\Omega \right)$, which is the spectral weight of the phonon propagator. This follows some work of Loudon, who has discussed first-order Raman (or Brillouin) scattering using standard quantum-mechanical perturbation theory. Next we summarize what can be said about the spectral weight ${\chi }^{\prime \prime }\left(\mathrm{K}\Omega \right)$ in the region of low frequencies (or small energy transfer). We use the model calculations of Kwok and Martin, as well as the standard theory of an elastic medium with a nonlocal form of Fourier's law of heat diffusion. In the case of pure isotropic anharmonic crystals ${\chi }^{\prime \prime }(\mathrm{K}, \Omega )$ has resonances corresponding to first-sound (pressure) and second-sound (temperature) waves, in addition to that from transverse or shear elastic waves. Unless fairly restrictive conditions are met, the second-sound wave does not propagate and reduces to the ordinary thermal diffusion mode of Landau and Placzek. The special nature of second sound in He II is discussed in an Appendix.

This paper presents the equations-of-motion method as a useful and flexible tool in the study of nuclear spectroscopy. It is partly a review, but also it introduces a new and much more powerful equations-of-motion technique which supercedes the older linearization methods. The older methods worked with operator equations. To obtain closed expressions they had to be linearized in a rather arbitrary manner. The present approach works with the ground-state expectation of operator equations and thereby avoids all problems of linearization. Thus, like the Green's function method, the equations-of-motion method becomes potentially exact. It has many advantages over Green's function methods, however, among which are its greater compactness, simplicity, and the physical insight it yields.

The method is first applied to rederive the random phase approximation (RPA) and the quasi-particle RPA (QRPA) and to show precisely what terms they neglect. It is demonstrated that some of these terms have coherent phases. A higher RPA and QRPA are then derived to include these terms. The corrections have some interesting effects: notably, there is a reduction of the effective interaction strength and a stabilization of the nucleus against sudden phase transitions. The equations-of-motion method is also used to generalize, in a very simple and natural way, the Hartree-Fock (HF) and Hartree-Bogolyubov (HB) concepts of independent particles and quasi-particles to nonsimple ground states.

The equations-of-motion method is presented as a simple extension of the shell model to the treatment of excitationg of a correlated ground state. By concentrating on the quantities of direct physical interest, the complexity of workins with correlated wavefunctions is avoided.

Assuming isospin, the current-current form of the weak interactions of hadrons, and the most general matrix element for the hadron current, expressions are obtained for the decay rate, the final particle energy spectra, and the final baryon-lepton angular correlation. The results are presented in a manner that allows easier comparison between theories and significant data. In particular, a table is given which will allow comparisons between theory and experiment for decay widths. It also allows one to determine which of the many interactions are important in the energy spectra for the various decay modes.

An integrated interpretation of the fundamental particle spectrum and the phenomena observed in high-energy scattering is possible within the framework provided by the $\mathrm{SU}\left(3\right)\mathrm{}$ and Regge pole theories of strong interactions. Recent progress in this direction is reviewed with particular emphasis on qualitative predictions and experimental tests. The presence of forward and backward peaks in scattering angular distributions is associated with the existence of certain $\mathrm{SU}\left(3\right)\mathrm{}$ multiplets of mesons and baryons that mediate the reactions as peripheral exchanges. The observed baryons are tentatively classified as Regge recurrences of 1-, 8-, and 10-dimensional $\mathrm{SU}\left(3\right)\mathrm{}$ multiplets. For mesons, recurrences of only 1 and 8 representations are expected. Predictions for charge-exchange processes based on meson Regge exchanges are described in some detail. $\mathrm{SU}\left(3\right)\mathrm{}$ symmetry for the vertices of the Regge exchanges leads to relations that agree with experiment. Scattering in the intermediate momentum range (1-4 BeV/c) is described in terms of an interference model which incorporates direct channel resonances as well as Regge exchange. Structure in forward ${\pi }^{-}p\to {\pi }^{0}n$ scattering and backward ${\pi }^{-}p$ elastic scattering is interpreted by interference model calculations.

A group-theoretic study is made of the degeneracies of the normal modes of vibration of a crystal and of the manner in which the polarization vectors describing these modes transform under the operations of the space group of the crystal. To describe the effects of the spatial symmetry operations a set of $3r$-dimensional matrices is constructed, where $r$ is the number of atoms in a primitive unit cell of the crystal, each of which commutes with the Fourier-transformed dynamical matrix for each value of the wave vector labeling the modes. These matrices are shown to provide a multiplier representation of the point group of the wave vector. The reduction of this representation yields the degeneracies (due to spatial symmetry) and transformation properties of the polarization vectors corresponding to a given wave vector, while the forms of the eigenvectors are obtained by projection operator techniques. For appropriate wave vectors, the consequences of time-reversal symmetry on the degeneracies and polarization vectors are investigated by introducing an anti-unitary matrix operator which commutes with the Fourier-transformed dynamical matrix. A criterion for the existence of extra degeneracies due to time-reversal symmetry is presented. The symmetries of lattice vibrations and selection rules for two-phonon absorption processes corresponding to several values of k in the first Brillouin zone of diamond are determined to illustrate the methods developed in this paper.

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 ($\Delta , S$) 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 [$\theta \left(\frac{{\alpha }^2}{R^3}\right)$] of the magnetic interaction energy of two degenerate atoms is 10∼20% of the quadrupole-quadrupole interaction energy at $R=30\mathrm{}{a}_0$ through the third-row atoms.

The problem of liquid helium is treated in terms of the theory of collective coordinates, applying a canonical transformation similar to that used previously for plasmas. This treatment provides further physical and mathematical insights into the origin of phonons, and into the nature of the roton excitations and their interactions. In particular, starting from the many-body Hamiltonian, it directly gives the Feynman-Cohen wave function, and the dipole-dipole interaction of rotons, along with certain further predictions with regard to the roton spectrum and the correlation function. The property of superfluidity is explained by the subsidiary conditions which imply that individual particle excitations are impossible in the lowlying states of the system. The transformed Hamiltonian can thus be shown to have eigenfunctions corresponding to quantized vortices.

Many physico-chemical systems can be represented more or less accurately by a lattice arrangement of molecules with nearest-neighbor interactions. The simplest and most popular version of this theory is the so-called "Ising model," discussed by Ernst Ising in 1925 but suggested earlier (1920) by Wilhelm Lenz.

Major events in the subsequent history of the Lenz-Ising model are reviewed, including early approximate methods of solution, Onsager's exact result for the two-dimensional model, the use of the mathematically equivalent "lattice gas" model to study gas-liquid and liquid-solid phase transitions, and recent progress in determining the singularities of thermodynamic and magnetic properties at the critical point. Not only is there a wide range of possible physical applications of the model, there is also an urgent need for the application of advanced mathematical techniques in order to establish its exact properties, especially in the neighborhood of phase transitions where approximate methods are unreliable.

This paper reviews some aspects of our knowledge of the gravitational theory of fluctuations of density in homogeneous and isotropic models of the universe. Irrotational fluid motions only are considered. All perturbations are assumed to be small and a normal mode analysis is used. The nature and time dependence of the amplitude of the various modes in expanding and contracting models of the universe are considered within the framework of Newtonian and general relativity theories. The origin of celestial structure requires that a uniform universe is unstable against arbitrarily small perturbations. However, the rate of growth of the allowed modes, particularly in an expanding universe, is sufficiently slow to cast doubt on the instability of the models.

A brief selection of problems occurring in the interpretation of planetary atmospheres, the resolution of which demands a knowledge of atomic collision processes, is presented. The selection is largely arbitrary.