The $\mathrm{SU}\left(2\right)\mathrm{}$ crossing matrices for the scattering of $I=0, \frac{1}{2}, 1, \frac{3}{2}$ particles and antiparticles, and the $\mathrm{SU}\left(3\right)\mathrm{}$ crossing matrices for the scattering of singlets, octets, and decimets are listed. The $s-t$, $s-u$, and $t-u$ crossing matrices and their inverses are given for each case. The relative phases of the crossing matrix are discussed in detail.

Recent developments in the concepts and models used to describe high-energy collisions of fundamental particles are reviewed. The areas of appreciable activity in research in high-energy physics are surveyed briefly and the general framework for the description of processes at high energies is outlined. There follows a sampling of recent experimental data designed to show the extent and detail of present-day experimental results. Recent applications of Regge pole models are reviewed with emphasis on the difficulties as well as the successes of models employing only Regge poles. The various multiple-scattering models are then discussed and correlated by means of the methods of Glauber. The connection of these models with amplitudes having branch cuts in the angular-momentum plane (Regge cuts) is described, as well as some comparisons with experiment. Finite energy sum rules as a means of relating the low-energy and high-energy domains are discussed in some detail. Next, the far-reaching concept of duality whereby the direct-channel resonances are (in some sense) the crossed-channel Regge exchanges is described, along with the related ideas of exchange degeneracy, the special role of the Pomeranchon Regge pole, and duality diagrams. An explicit realization of duality is achieved in the Veneziano model. This model is discussed in some detail for the relatively simple and interesting example of pion-pion scattering. Brief mention is also made of the extensions of the Veneziano model to the $n$-particle amplitude, attempts at unitarization, and various applications. The topic of multiparticle final states is covered relatively briefly with emphasis on the applications of double Regge pole exchange to three-body final states, the calculation of proton and pion energy and angular distributions in proton-proton collisions by means of the multi-Regge exchange model, and the generation of self-consistent Regge singularities with the multi-Regge exchange model and unitarity. The final section of the review concerns recent results on pion-pion scattering phase shifts and also $K-\pi$ phase shifts, and some remarks on theorems connecting decay correlations of resonances with the mechanism of production. The literature survey ended, with a few exceptions, in May 1969.

This paper deals with ordinary material systems whose elementary constitutents are fermions. It is pointed out that in such systems there can occur two kinds of bosons with quite different physical and mathematical characteristics. Type I bosons are bound complexes of an even number of fermions (such as ${}_{}{}^4{}_{}{}^{}\mathrm{He}$); and type II bosons are elementary excitations which are bound complexes of fermions and their holes (such as excitons). When the first type condenses, a superfluid state results with so-called off-diagonal, long-range order; while when the second type condenses, there is no superfluidity, but a change in spatial order. Thus both kinds of long-range order are related to Bose condensation.

Two new independent mass relations are derived and are shown to be consistent with several existing nuclear models. The most general functional dependence on proton number, neutron number, and mass number (or isospin value) of masses which satisfy these relations exactly is discussed, and a procedure for determining the values of these functions which give a best least-squares fit to the body of known masses is developed. The functions which give the best over-all fit are listed together with the resulting theoretical mass table which shows the discrepancies to known masses and the theoretical values for proton, neutron, and alpha-particle decay energies.

The recent proliferation of literature on the Regge pole model has made it evident that a general review of the model is needed. This paper summarizes work done in the field prior to October 1968. It reviews the types of singularities thought to exist in the complex angular-momentum plane, the symmetries of Regge amplitudes and trajectories with special emphasis on four-dimensional symmetry associated with zero momentum transfer, properties of both meson and baryon trajectories, predictions of the model, and suggested experimental tests of various assumptions made in the model. The paper is written for the general reader with an interest in the field and for those in the field interested in areas in which they are not directly involved. There is as little use of mathematics as possible, with emphasis essentially on the results and conclusions of recent papers.

This review is divided into three parts. First, we survey the general features of nonforward scattering. We comment on two large-angle scattering models (the statistical model and the proposal for explaining the "breaks" in the differential cross section in terms of production cross sections) and discuss the axiomatic results on lower bounds and the asymptotic angular dependence. Second, we review the motivation and the general formalism of the multiple-scattering model and the qualitative success of the model—in particular, its application to the nonforward scattering. Third, we discuss the impact of the multiple-scattering model on high-energy phenomenology. Here, we emphasize that, in addition to providing a concrete model for large $\mathrm{}\left|t\right|\mathrm{}$ scattering, the multiple-scattering model contains virtues from both the optical model and the Regge-pole model. Similar to the absorption model, it helps to explain problems encountered in a pure Regge-pole model. On the other hand, it preserves the past success of the Regge-pole model and opens new possibilities for it.

This paper reviews, analyzes, and integrates the various quantitative results that have been obtained for the process of pair production by photons. Included in this summary is a detailed review of total and differential cross sections for pair production in an atomic and in an electron field, with a critical evaluation of the conditions of validity and the accuracy of the results. In addition, a summary is given of the important kinematic relations, theoretical considerations, and the polarization effects that occur in pair production. The paper does not include a treatment of radiative corrections to pair production, thick-target pair production, or pair production by electrons in the field of a nucleus (trident production). Otherwise, the review is intended to include results on pair production available up to January 1969.

We survey some recent progress in constructing models of local, relativistic quantum mechanics. These field-theory models have nontrivial scattering.

This paper reviews recent developments of effective Lagrangians and field algebras as means of treating chiral symmetry and partially conserved axial current (PCAC) for the study of elementary particle physics. The techniques employed are developed in considerable detail. As examples, we concentrate primarily on spin 0 and 1, linear and nonlinear realizations of $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$ and $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$ and some of the significant predictions of the theory are derived. The paper contains an extensive discussion of an effective Lagrangian with nonets of real scalar, pseudoscalar, vector, and axial-vector mesons that illustrate the problems of broken $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$.

The emphasis in this review is on the extent to which theoretical studies have accounted for the empirical data on the three-nucleon systems. The ground, excited, and continuum states of the nuclear three-body systems are discussed, and a summary is given of the interpretation of the electromagnetic- and weak-interaction properties.

The implications of the new determination of $\frac{e}{h}$ using the ac Josephson effect in superconductors for both quantum electrodynamics (QED) and our knowledge of the fundamental physical constants are analyzed in detail. The implications for QED are investigated by first deriving a value of the fine structure constant $\alpha$ from experimental input data which do not require the use of QED theory for their analysis. These include the Josephson-effect value of $\frac{e}{h}$, the Faraday constant, the gyromagnetic ratio of the proton, the magnetic moment of the proton in units of the nuclear magneton, the ratio of the ampere as maintained by the United States National Bureau of Standards to the absolute ampere, and certain accurately known auxiliary constants. This is done by critically reevaluating all of the experimental data presently available on these quantities and applying the standard techniques of a least squares adjustment, including tests for imcompatibility. The value of $\alpha$ so obtained is then used to evaluate the theoretical expressions for the Lamb shift and fine structure splitting in hydrogen, deuterium, and ionized helium, the hyperfine splitting in hydrogen, muonium, and positronium, and the anomalous magnetic moment of the electron and muon. These theoretical values are compared with critically reexamined experimental values, thus providing a test of QED in which a priori information from QED itself is not essential. The consequences of the new measurement of $\frac{e}{h}$ for our present knowledge of the fundamental physical constants are demonstrated by deriving new "best" values for the fundamental constants from a critically selected subset of all the available data. In addition to providing a consistent set of constants, this analysis focuses attention on areas in which there remain important questions which require clarification. The experimental and theoretical work necessary for the resolution of these questions is discussed, with emphasis on ways in which the study of quantum phase coherence effects in low temperature superfluid systems can make significant contributions.

The present work is an attempt to re-evaluate the nuclear corrections to the energy levels of hydrogen by using an effective potential approach. The basic idea is to infer from electron-proton scattering a potential which may then be applied to the bound-state problem. In lowest order, the potential is chosen from the first-order Feynman diagram for the scattering. With this choice the Breit equation is obtained. It is then solved in an approximate way, in the non-relativistic limit of the proton, to obtain wave functions which are accurate enough for use in evaluating the effects of perturbations of the potential. The reduced mass corrections to the fine structure and the hyperfine structure levels are readily found. The effect on the hyperfine splitting of the distribution of the proton charge and magnetic moment is obtained by correcting the lowest-order potential to include the proton form factors. A further modification is needed in evaluating additional recoil corrections, of relative order $\frac{\alpha m}{M}$, to the fs and the hfs. This additional term accounts for the failure of the iteration of the lowest-order potential to reproduce the scattering obtained from the second-order Feynman diagrams. The $\frac{{\alpha }^{2}m}{M}$ contribution to the state-dependent mass corrections to the hfs is also analyzed within the context of this approach. All the corrections found are in complete agreement with previous results obtained by the Bethe-Salpeter (BS) equation, but the present method has the virtue of conceptual simplicity.

Four different polynomial equations and seven nonlinear equations, all applicable to both solids and liquids, are compared theoretically and statistically. Detailed curve-fitting results are presented for recent water and Hg isothermal data. Uncommonly used methods of statistical analysis and comparison, including generalized least squares, are described, justified compared to usual methods, and applied. In general, certain polynomial equations are found to yield significantly better fits of many different water and Hg data sets than any nonlinear equation considered. The Tait and Murnaghan equations, in particular, lead to strong systematic behavior of all residuals calculated herein with them, showing that they are inadequate models for all the data considered. Even a nonlinear equation derived from a second-order expansion of the bulk modulus $K$ in powers of the pressure, which is shown to include several frequently used equations as special cases, is inferior to selected polynomial equations but is still the best equation examined when appreciable extrapolation is necessary. The method of volume normalization almost always used heretofore in statistical fitting of equations of state to $P-V$ data is shown to be inadequate and two alternative approaches are proposed and employed herein. Critical comparison of previous analyses of water and Hg data is made with the results of the present, more refined approach. The likelihood of important systematic errors in $P-V$ data, particularly data derived from ultrasonic measurements on liquids under pressure, is pointed out and high probability of their occurrence in some of the data analyzed is demonstrated. Even the combination of the best data apparently available and the use of better statistical-analysis methods than have been employed before does not yet allow one to obtain highly accurate values of the ${K_0}^{\prime }$ parameter of water or Hg, and only an order-of-magnitude estimate of the ${K_0}^{\prime \prime }$ parameter seems currently possible. Nevertheless, it appears that near room temperature ${K_0}^{\prime \prime }$ is positive for water and probably negative for Hg and that its appreciable magnitude for both materials renders a second-order expansion of $K$ inadequate.

A general introduction to and bibliography for transport phenomena in gases is provided. Methods for obtaining density expansions of transport coefficients from time-correlation functions in a moderately dense gas with short-range repulsive intermolecular forces are considered. A unified treatment of the two methods appearing in the literature (the $t$ method due to Cohen, Dorfman, and Ernst and the $\epsilon$ method due to Zwanzig) is given. Both of these methods lead to integral equations from which the first two terms in the density expansion of transport coefficients can be computed. However, because of many-body effects in the gas, both methods diverge when used to compute terms beyond the first two in these density expansions. Because of this divergence, it is necessary to prove that the $t$ and $\epsilon$ methods give the same results for the first two terms in the density expansion of transport coefficients. The required proof is provided, and we conclude that either the $t$ or $\epsilon$ method can be used to compute the first two terms in the density expansion of transport coefficients provided one assumes that the remaining (divergent) terms, which are neglected, do not contribute to the first two terms.

Although the theory of the propagation of electron waves in periodic solids and the theory of the elastic diffraction of electron waves by periodic solids have developed independently into band theory and dynamical electron-diffraction theory, respectively, they are in fact formally identical. The electron wave functions which can exist in the crystal are determined by a seven-dimensional hypersurface in energy-complex K space which defines the totality of solutions to the wave equation in the infinite crystal. In the diffraction problem, the introduction of the crystal surface together with the magnitude and direction of the external electron wave vector selects the particular set of eigenfunctions which are excited during a given experiment and which correspond to the allowed electron states in the crystal. This set constitutes the wave field in the self-consistent multiple-scattering approach. It is demonstrated that the energy band diagram of band theory and the constant energy dispersion surface of dynamical theory are in fact sections of the same hypersurface. The complex nature of the dispersion surface leads to the excitation of evanescent waves both in the crystal and in the vacuum. The diffraction boundary conditions, notably conservation of total energy and of momentum parallel to the crystal surface, can easily be introduced, geometrically, by means of a constraint surface which contains the crystal normal. In a given experiment the excited wave functions are determined by the intersection of the hypersurface with the appropriate constraint surface. It is shown that the most useful constraint surfaces are those at constant energy and furthermore that the dispersion hypersurface is an ameniable method for the discussion of low-energy electron-diffraction (LEED) cases of high order and high symmetry, several of which are outlined in detail. The variation of the reflected intensities observed in electron-diffraction experiments is discussed in terms of the changes in the allowed electron wave functions as calculated by three-dimensional band-structure and/or dynamical diffraction theory. In particular, we predict zeros in the reflected Bragg intensities for certain special geometries in the case of two simultaneous reflections (mixed Bragg-Laue case).

Inhomogeneous broadening has been observed in resonance lines in solids over the wide range of energies spanned by nuclear magnetic resonance, electron spin resonance, optical, and Mössbauer methods. The broadening arises from random strains, electric fields, and other perturbations from the defects in the lattice containing the centre whose transitions are studied. This paper reviews the calculation of the shapes of such resonance lines. The most important method used is the so-called statistical method. This method determines the line shape as a function of the distribution of the defects with respect to the centres studied, the density of the defects, and the perturbation fields of the individual defects. Emphasis is laid on the physical assumptions and approximations in this method and on its relation to other approaches. The theory is applied to a variety of broadening mechanisms, both in the widely used continuum approximation for the lattice containing the defects and in the more realistic discrete-lattice model. Two classes of experimental work are reviewed. The first deals with the ways in which resonance lines are recognised as being inhomogeneously broadened. These methods show that a wide range of phenomena can be used to check the theory of the line shapes. The second discussion of the experimental work compares theory and experiment for each of the various broadening mechanisms. These mechanisms include broadening by the strains from dislocations and point defects, by the electric fields and field gradients from charged defects, and by unresolved hyperfine structure. In each case theory and experiment are compared in detail for the system for which the most complete data are available. The conclusion is that the statistical method provides a satisfactory approach in all cases for which there are adequate data.

Kapitza resistance ($R_K$) is the thermal boundary resistance which occurs at the interface when heat flows from a solid into liquid helium. This review attempts a comprehensive presentation of experimental and theoretical knowledge since the discovery of Kapitza resistance in 1941. The experiments discussed and data presented include measurements of $R_K$ at interfaces between liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ and: copper, lead, mercury, tin, indium, nickel, constantan, gold, silver, platinum, tungsten, silicon, quartz, lithium fluoride, and sapphire. The experiments between solids and liquid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ are also discussed. The treatments include discussion of the dependence of $R_K$ on these variables: temperature, pressure, surface structure and preparation, and elastic properties of the solid. The principal experimental problems are associated with the surface properties, so these are discussed in detail. The principal theoretical discussion is of the acoustic impedance theory, following Khalatnikov and Mazo and Onsager, and the results are compared with the experiments. Modifications of the theory connected with improved matching at the interface, due, for example, to condensed He are also considered. When the theory is applied to interfaces between metals and liquid He, then it must be modified to take into account phonon electron interactions. The theory gives a temperature dependence $R_K\propto T^{-3\mathrm{}}$, which is approximately what is experimentally, observed. However, the observed $R_K\text{'}\mathrm{s}$ are usually an order of magnitude or more smaller than theoretical values. The source of the disagreement lies either in a lack of knowledge of the surface physics or in another, dominant, mechanism for thermal energy exchange across the interface. Evidence for each possibility and some suggestions aimed at resolving the discrepancies are offered.