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 $K{\alpha }_1$ 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 $\lambda$ W $K{\alpha }_1=(0.2090100\mathrm{}\pm 5 \mathrm{ppm})$ Å. This may be used to define a new unit, denoted by Å*, such that the W $K{\alpha }_1$ wavelength is exactly 0.2090100 Å*; hence 1Å*=1ű5 ppm. The wavelengths of the Ag $K{\alpha }_1$, Mo $K{\alpha }_1$, Cu $K{\alpha }_1$, and the Cr $K{\alpha }_2$ 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 Å* 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.

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.

Three different processes in which nuclei are synthesized by the multiple capture of neutrons with energies in the keV range are reviewed and compared, namely the $s$-process and $r$-process in stars and the synthesis of very heavy nuclei in thermonuclear explosions. In the $s$-process, neutrons are captured by nuclei in the valley of $\beta$-stability. Knowledge of neutron capture cross sections of $s$-process nuclei can clarify such features of the $s$-process as elemental abundances, time scales, and temperatures. For producing the heaviest possible nuclei by neutron capture in thermonuclear explosions there are advantages in using the heaviest obtainable target nuclei, but for these there is the disadvantage of serious competition from neutron-induced fission. A channel theory of capture to fission ratios shows encouraging agreement with experiment. More data on capture to fission ratios would be valuable, particularly for odd-odd target nuclei. From capture to fission ratios, the number of open channels for fission can be deduced.

Experiments that can reveal the nature of the second- and the fourth-order terms in the interaction between charged particles and electromagnetic fields in vacuum have been discussed here. An attempt has been made to emphasize the successes of the experiments already completed and the promising lines of future investigation. The theoretical considerations relevant to the experiments under review have also been outlined. In all cases studied so far, the experimental results agree with the predictions of conventional quantum electrodynamics.

As a contribution toward the understanding of resonant scattering in general, we examine the potential-barrier resonances which occur in a variety of simple one- and two-channel $S$-wave scattering problems. The models chosen have familiar optical analogs which, together with the appealingly simple analytic properties of their scattering amplitudes, facilitate a unified discussion of such diverse phenomena as the peculiar $S$-wave "resonances" of a square potential well, the occurrence of "resonance circles" in both elastic and inelastic scattering amplitudes, and the large nonresonant maxima which the descent of the background phase (as a function of energy) can produce in total cross sections.

Rayleigh-Schrödinger perturbation theory and the variational principle combined furnish simple criteria for investigating and predicting the accuracy of arbitrary variational wave functions computed with a perturbed Hamiltonian operator. This permits the a priori classification of all perturbed variational wave functions into three broad categories which differ in their asymptotic behavior as the perturbation approaches zero. Wave functions belonging to one of these categories have the desirable characteristics of yielding variational energies correct at least through first order and other expectation values correct at least in zero order. These conditions are not fulfilled in the other two categories. The classification scheme is used to survey a wide variety of variational wave functions for atomic isoelectronic sequences. The construction of these wave functions determines their asymptotic category. A more systematic treatment of correlation energy is proposed and various methods of introducing correlation are discussed. Some recommendations are made for future progress in the calculation of accurate atomic wave functions.

The use of Rayleigh-Schrödinger perturbation theory for obtaining accurate approximate wave functions for molecules and crystals and particularly for the hydrogen molecule is discussed in some detail for a variety of approximate ${\Psi }_0\text{'}\mathrm{s}$. Emphasis is laid on the nonunique choice of the zeroth-order wave functions, and the criterion of "goodness" is suggested to include the ability to obtain higher-order corrections rather than just the expectation value of the total Hamiltonian. The calculation of electric and magnetic properties is also described and comparisons are made for the few calculated values available. Problems of symmetry and spin in the treatment of weakly interacting atoms and ions as in "molecular" crystals are discussed at length and a few related topics are treated in Appendixes.

This paper summarizes those general properties of photoproduction processes which are independent of the interactions among particles. The two types of reactions considered in detail are $\gamma +\frac{1}{2}\to s+\frac{1}{2}$ and $\gamma +\frac{1}{2}\to s+\frac{3}{2}$, where ½ and $\frac{3}{2}$ denote particles with spins ½ and $\frac{3}{2}$, respectively, and $s$ denotes a particle with an arbitrary integer spin $s$. The first of these reactions is synthesized from the reactions $\gamma +0\mathrm{}\to 0+0$, $0+\frac{1}{2}\to 0+\frac{1}{2}$, and $0+0\to s+0\mathrm{}$. First the observable structure of these constituent reactions is derived, and then the composite structure is obtained. Detailed results for the observables are given for $s=0, 1, \mathrm{and} 2$, and experiments designed to determine spin and parity are listed. The reaction $\gamma +\frac{1}{2}\to s+\frac{3}{2}$ is similarly composed of $\gamma +0\mathrm{}\to 0+0$, $0+\frac{1}{2}\to 0+\frac{3}{2}$, and $0+0\to s+0\mathrm{}$. Some results for this reaction are also tabulated. Finally, as a specific application of the above results, a method is suggested to determine the contribution of certain corrections to the static theory of pion photoproduction from nucleons. The whole paper is an application of the general nondynamical theory of particle reactions with arbitrary spins which was recently developed by the authors.

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.

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.

All of the x-ray emission wavelengths have recently been reevaluated and placed on a consistent Å* 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-Å* wavelength units, with the respective probable errors, are also included there.

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 October 1965.

Knowledge of the molecular quadrupole moments may be used to propose a better model for the structure of the molecule and could also provide a crucial test for the accuracy of the trial wave functions that are used in other molecular calculations. A general survey of the different methods available for the evaluation of molecular quadrupole moments has been made and their relative merits and demerits have been discussed. It is suggested that the quadrupole moments determined by the microwave linewidth data are probably more reliable. The quadrupole moments of different molecules known to data have been tabulated.

The thermodynamic properties of an electron plasma are investigated on the basis of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy equation. The correlation energy has been calculated up to the ${\epsilon }^2$ term ($\epsilon$ is the standard plasma parameter) in terms of an exact solution of the binary correlation function taking into account the short-range collision effect. The contribution of ternary correlations to the correlation energy has been determined in lowest order, which turns out to be of the second order in $\epsilon$. The total sum of the corrections due to the short-range collision effect and the ternary correlation effect is found to be in agreement with the result obtained by Abe, Bowers, and Salpeter within the regime of Gibbs' statistical thermodynamics. Although the appearance of the binary correlation function obtained in the present studies differs from those derived by Bowers-Salpeter and by O'Neil-Rostoker, it is shown that all of these expressions are essentially equivalent. According to the present analysis, however, it is proven that these expressions do not describe correctly the way in which correlation effects diminish at large distances and therefore the statement made by DeWitt saying that $g\left(x\right)\mathrm{}\sim \frac{1}{8}\left(\ln 3\right){\epsilon }^2\exp (-k_{D}r)$ as $r\to \infty$ is not correct.

Previous work on the Tait equation of state, usually applied to liquids, is discussed together with a review of work on a closely related simple equation arising both from the theory of finite strain and from microscopic considerations. The latter equation has been primarily used for fitting hydrostatic compression pressure-volume data for solids. A detailed discussion of methods for assessing goodness-of-fit of data to equations of state is presented along with an analysis of ways to help decide which of two similar equations is the more applicable for given data. Nonlinear least squares fitting of the above two-parameter equations of state is carried out for the first time using published $P-V-T$ data for water, a very compressible hydrocarbon liquid, zinc, lithium, sodium, potassium, and rubidium and the results compared with those of previous analyses of these data. Careful fitting of the present type can lead to new conclusions and insights not so apparent from less careful fitting of the present equations or from fitting with other equations. For most of the materials and data considered here the finite-strain equation was found to be superior to the Tait equation. Reasons are advanced, however, why both equations should be primarily considered only as empirical, with theoretical justifications for them applicable only over limited pressure-volume ranges.

The absolute measurements of nuclear reaction energies appropriate for calibration purposes are critically reviewed and recommended values are given. A sufficent number of points are now known with precision to permit the calibration of proton beams up to 60 MeV.

General characteristics of magnetic bremsstrahlung (synchrotron radiation) are derived on the basis of calculations utilizing exact relativistic matrix elements. The spectral and total energy losses may be described by compact expressions incorporating radiative and quantum corrections. Comparisons of the relative efficacy of matter and magnetic fields as bremsstrahlung radiators indicate that even under relatively conservative conditions the natural conversion rates associated with the magnetic process (i.e., 9.6×${10}^3$ BeV/mm) tends to favor this type of conversion over ordinary bremsstrahlung.

Discussions of a similar scope—including detailed calculations, and comparisons with the corresponding processes in material media—are also carried out for magnetic pair production, direct and indirect trident cascades, and magnetic photon splitting.

Circumstances peculiar to a class of external field radiative processes involving cooperative vacuum polarization corrections are illustrated by a discussion of magnetic Čerenkov radiation.

This paper is a review of experimental data and theoretical studies devoted to the rotating helium ii problem. The problem arose when helium ii appeared to be rotating as a whole in a uniformly rotated container, while dragging of its superfluid component into rotation of a cylindrical vessel seemed impossible due to the absence of viscosity and Landau's requirement curl ${\mathrm{v}}_s=0\mathrm{}$. Later it was found that imitation of a solid body rotation by helium ii is realized by means of Onsager-Feynman's vortex lines which possess quantized circulation. In a uniformly rotating vessel they are distributed approximately uniformly along its cross section, aligned parallel to the axis of rotation, and cause a complicated velocity distribution at which curl ${\mathrm{v}}_s=\left(\frac{2\pi \hslash }{m}\right){\Sigma }_v\delta (\mathrm{r}-{\mathrm{r}}_v)$, that is, Landau's requirement is valid everywhere excluding singular lines where the curl is equal to infinity (${\mathrm{r}}_v$ is a two-dimensional radius vector of a vortex line in an arbitrary plane perpendicular to the axis of rotation).

In the review the experiments are described in which both an averaged picture of the joint rotation of the superfluid and the normal components is displayed and the studies in which the existence of vortex peculiarities is manifested in a clear manner. The measurements of the meniscus of rotating helium ii and its angular momentum belong to the first group of these experiments ("integral" ones). Propagation of second sound, oscillations of solid bodies immersed into helium ii, passage of negative ions, and different relaxation processes (acceleration and deceleration of the vessel, changes of temperature, etc.) from the second group ("local" experiments). The experiments in which the superfluid performs potential rotation, which can be persistent in the vessel at rest are also described. Vortex lines are absent there and the circulation of velocity can reach the value equal to a lot of elementary circulations.

Special attention is devoted to the problem of the nature of the phase transition helium i-helium ii (and back) in the state of rotation.

The properties of rotating helium ii illustrate brightly the main peculiarity of a quantum liquid, i.e., its complete inability to perform nonpotential motions. They show that Landau's requirement curl ${\mathrm{v}}_s=0\mathrm{}$ remains valid even when the thermodynamical condition of the free-energy ($E-L{\omega }_0$) minimization gives preferrence to the velocity distribution of the type $v={\omega }_{0}r$.

A Langevin theory for linear and nonlinear, stationary and nonstationary, processes is developed and compared with Markoff methods. For short correlation times ${\tau }_c$, we find the Markoff process that is a good approximation to the Langevin process for $\Delta t>\mathrm{}{\tau }_c$. Conversely, given the diffusion coefficients $D_n$ of a Markoff process, we find the moments (to all orders) of the Langevin forces that lead to the same process—exactly for homogeneous noise, approximately for the general nonlinear case. The techniques are illustrated by applications to Fokker-Planck processes, homogeneous noise with linear damping, the one-dimensional impurity band, spin diffusion, and population fluctuations.

The mechanisms of the reorganization of atoms with inner shell vacancies are reviewed. All available experimental values of $K$-, $L$-, and $M$-shell fluorescence yields, together with $L$-shell Coster-Kronig yields are summarized and compared with available theoretical calculations. Rather good agreement exists in general between theory and experiment for the $K$ shell, but experimental values of the $L_2$ and $L_3$ subshell fluorescence yields disagree substantially with the semitheoretical results of Listengarten. Related phenomena in $\mu$-mesic atoms are included. Experimental methods of measurement are summarized and discussed, together with suggestions for future studies with high resolution techniques. Results from the literature have been included up to 10 May 1966.