Theoretical values of Rayleigh scattering amplitudes for the $K$ shell and for higher atomic shells are calculated from the second-order $S$ matrix of quantum electrodynamics. The calculations are combined with nuclear Thomson scattering to predict elastic scattering differential cross sections for photon energies between 100 keV and 1 MeV, and for elements with $Z$ ranging from 30 to 82. The resulting cross sections are compared with recent high-precision measurements.

We describe a new type of analysis yielding the oscillator-strength distributions in several spectral series throughout the lithium isoelectronic sequence. Individual oscillator strengths have been critically evaluated and tied together by requiring that they fulfill simultaneously four basic spectroscopic constraints: (a) regularities for individual transitions along the isoelectronic sequence, (b) regularities for the transitions of a spectral series, (c) compliance with the requirement of continuity across the spectral series limit, and (d) adherence to $f$ sum rules. With only slight modifications, the best available data fulfill closely all four requirements. The final set of $f$-value data—which are tabulated and partly presented in illustrations—possess, therefore, a very high degree of internal consistency and reliability. Relativistic corrections for very highly charged Li-like ions are also considered.

The techniques of many-body perturbation theory with interacting resonances are used to calculate the photoionization cross section of SiII from threshold to 1 keV. The Rayleigh-Schrödinger expansion with $\mathrm{LS}$ coupled states and the Brueckner-Goldstone expansion are compared. Correlations among the $2p$, $3s$, and $3p$ electrons are shown to have a large effect on the cross section. Resonances from $3s 3p \mathrm{np}$, $2p^{5}3s^{2}3p \mathrm{ns}$, $2p^{5}3s^{2}3p \mathrm{nd}$, and $2s 2p^{6}3s^{2}3p \mathrm{np}$ configurations are included.

The Biberman-Holstein integral equation is solved for an infinite cylinder in the limit of large optical thickness and for a wide class of absorption profiles. Special attention is paid to the Voigt profile.

The assumption of complete redistribution in frequency used in the theory of transport of resonance radiation, when Doppler broadening is important, is shown to predict the correct number densities in the excited levels at large optical depth. The line shape, however, is not a Doppler distribution in the wings but decreases like $\frac{\exp (-u^2)}{u}$.

Rayleigh-Brillouin spectra of gaseous ${\mathrm{N}}_2$ is measured for light scattered at 15° over a pressure range from 1 to 661 Torr. Spectral profiles are analyzed by means of an appropriate kinetic model. A least-${\chi }^2$ method determines the optimum values of the theory parameters: total collision frequency and rotational relaxation number. These parameters are compared with values obtained by other methods. Information provided by light-scattering measurements concerning rotational relaxation in dilute systems is discussed.

The emission spectrum is found for transitions between a pair of atomic levels which are coupled by a coherent applied field, in the case in which decay to other levels of lower energy takes place during the emission process. The problem is solved by first finding the spectral distribution of radiation emitted by an atom which is initially prepared in some combination of the two coupled states, and which then decays to states of lower energy. (The solution found to this initial-value problem is valid even when many photons are emitted before the decay takes place.) It is then shown how the solution so obtained may be used to describe the case in which the atom is continually excited to the pair of levels in question by a weak, incoherent pumping mechanism. Limiting cases are discussed and compared with those which apply when no decay out of the two-dimensional subspace of coupled states takes place. The method of solution, which is based on an evaluation of the appropriate atomic correlation function by means of the quantum fluctuation-regression theorem, is compared to the method of simply adding imaginary terms, proportional to the decay constants, to the state energies. The latter method is of far more restricted validity, in that it depends on a particular strong inequality which must be assumed to hold between the decay constants.

We extend the classical Brillouin-Sommerfeld treatment of precursors in several ways: We consider oblique incidence on an interface, the vector nature of the field, the effect of different forms of the incident signal, and the different refractive indices of the media. The amplitudes, frequencies, directions of propagation, and duration of the signal at all stages of the transient process are calculated. The steepness of the form of signals is one of the crucial factors for the experimental detection of the phenomena under consideration. The different possibilities of the experimental detection of the refracted and reflected signals are discussed.

Small-signal-pulse growth in a homogeneously broadened amplifier with swept gain is discussed from a quantum-mechanical and a semiclassical treatment. The regime in which Beer's law fails is discussed analytically. There we find nonexponential growth and the formation of steady states in the small-signal regime.

A Lamb semiclassical theory is developed for multimode, mode-locked laser operation. The equality of frequency spacings between adjacent modes causes population pulsations generated in the nonlinear atomic response to be integer multiples of the adjacent-mode spacing. This allows the atomic response to be written in terms of Fourier series, thereby reducing coupled differential equations of motion to algebraic equations ultimately expressible in terms of a continued fraction. A unidirectional ring configuration is assumed so that the more complicated spatial dependence of standing waves is avoided. Reductions are made to the rateequation limit and to the special two-mode cases of bidirectional ring operation and single-mode, standing-wave, gas-laser operation. Numerical analysis of the equations is motivated. Specific results are given in the following (companion) paper.

The multimode strong-signal theory of the preceding paper is applied to a number of two- and three-mode cases. A two-mode stability analysis yields strong-signal counterparts to the third-order coupling parameter. Two-mode homogeneous-broadened operation leads to the discussion of population inversion gratings and associated Bragg scattering. Symmetry properties of the equations for the bidirectional ring are examined. Results for Doppler-broadened two-mode special cases and the three-mode locked unidirectional-ring operation are presented. The strong-signal results are compared to those of the third-order and rate-equation-approximation theories. For the bidirectional ring the historical third-order domain of relative excitation from 1.0 to 1.2 is divided into a valid third-order region and a less accurate region. The condition of bistable operation vs monostable operation is seen to be dependent on the relative excitation and the amount of homogeneous broadening. Two types of bistable operation are seen to exist, one where one mode is completely suppressed and another where both modes may have nonzero intensities.

The refractive-index change in a collisionless gas is evaluated from the Stark shifts of the rotational energy levels that arise from the polarizability anisotropy. In the limit of an extremely-short-duration excitation, a multilevel coherent effect results in delayed refractive-index bursts. Both stationary and transient responses of this birefringence to an optical field are considered for symmetric-top molecules, with particular emphasis on the special case of linear molecules.

The "superfluorescence equations" deduced in a previous paper are generalized to the case of finite temperature and translated into the atomic coherent-state representation of Arecchi et al. We get three coupled partial differential equations involving the quasiprobability distribution $Q(\theta ,t)$ of the atomic system and two suitable functions $\mathcal{F}(\theta ,t)$ and $\mathfrak{M}(\theta ,t)$ describing the stimulated processes which are relevant in oscillatory superfluorescence. Neglecting $\mathcal{F}(\theta ,t)$ and $\mathfrak{M}(\theta ,t)$, the equation for $Q(\theta ,t)$ reduces to the Fokker-Planck equation of Narducci et al. corrected by the contribution of thermal noise. The stationary solution is given by the unperturbed canonical state, which is shown to be the exact stationary solution of the single-mode model from which the superfluorescence equations have been deduced. These equations prescribe that $\mathfrak{M}(\theta ,t)$ always remain equal and opposite to $\mathcal{F}(\theta ,t)$, so that the superfluorescence equations reduce automatically to two equations. Eliminating $\mathcal{F}(\theta ,t)$ between the two remaining equations, one obtains a non-Markovian equation for $Q(\theta ,t)$ with derivatives of all orders in $\theta$. In the case of pure superfluorescence or super-radiance, such an equation reduces to a Markovian Fokker-Planck equation with derivatives of first and second order in $\theta$ which exhibits an interesting fourth-order contribution in the diffusion term. In fact, if one excludes a very small region around the north pole of the Bloch sphere, this term is much larger than the second-order contribution in the diffusion term of the equation of Narducci et al. In the case of oscillatory superfluorescence, the presence of memory effects enhances fluctuations, thus forbidding the existence of "classical" time evolutions.

The statistics and autocorrelation function of Rayleigh-scattered light mixed with coherent light were experimentally studied. The factorial moments were measured for the cases of both heterofrequencies and homofrequencies. In the case of heterofrequencies, the mean frequency of the scattered light was shifted from that of the coherent light by using a flowing scatterer. The second- and third-order factorial moments obtained were compared with the theory of Jaiswal and Mehta and were found to be experimental proof for the theory. The autocorrelation function was also measured in the case of a heterofrequency, and the result was explained well by the theoretical expression deduced here.

A theory of the density-density correlation functions of classical binary mixtures is developed to treat the problem of phase separation. Particles in the mixture are assumed to interact through pairwise potentials, and the theory is thus appropriate both to insulating fluids and to metallic systems to the extent that these may be described, for structural purposes, by effective ion-ion pair interactions. Pair potentials are separated in the form $v_{\mathrm{ij}}={\overline{v}}_{\mathrm{ij}}+v_{\mathrm{ij}}^1$, where ${\overline{v}}_{\mathrm{ij}}$ is appropriate to a reference fluid and is so chosen that $v_{\mathrm{ij}}^1$ may be regarded as a perturbation on this reference fluid. The resulting perturbation theory can be framed in terms of functions $f_{\mathrm{ij}}$ closely related to the Ornstein-Zernike direct correlation functions. Exact equations for the $f_{\mathrm{ij}}$ are obtained by treating the densities of the mixture as basic variables in a linear-response problem. Approximate solutions to these equations (and hence to the structural problem) are given. To first order in $v_{\mathrm{ij}}^1$, these solutions are exact in the long-wavelength limit, the region of interest in the phase-separation problem. By way of application, we show the effect of these first-order corrections to the simple mean-field calculations previously applied to simple metallic mixtures.

A truncated form of the Kirkwood-Salsburg equation for the hard-sphere gas is solved. An exact algebraic recurrence relation is obtained for the coefficients of an expansion of the two-particle correlation function in powers of the activity, $z$, in the neighborhood of $z=0\mathrm{}$. The methods of series analysis and Padé approximants allow this solution to be extended to all $z\in [-1, 11.6048]$. An equation of state is obtained which reproduces the first three virial co-efficients exactly and shows a phase transition at $z=11.6048\mathrm{}$, which appears to be weakly second order. It is demonstrated that minor modifications of the integral equation could substantially alter the analytic structure of the solution.

It is shown that an application of the transfer theorem of nonstandard analysis to the usual inequalities for the free energy for finite systems (for strongly tempered potentials) leads in a very direct way to the existence and uniqueness of the thermodynamic limit.

The most accurate experimental radial-distribution-function data, obtained by thermal-neutron scattering and x-ray scattering in liquids, have been considered. A number of analytical equations for the radial distribution function $g$ as a function of the interparticle separation $R$ have been studied. The formula for $g\left(R\right)\mathrm{}$ finally chosen gives completely satisfactory agreement with the experimental data, and also with some Monte Carlo simulations and with theoretical predictions of the Percus-Yevick type. It is expected that the same formula will be also applicable to at least some of the diffractometric data for amorphous solids.

We present a model which describes the interaction of a cold homogeneous plasma with a strong linearly polarized external electromagnetic wave, taking into account relativistic terms. Instabilities appear for transverse and longitudinal waves when the pump-field frequency is at the same time below the electron plasma frequency and above but very near the cutoff frequency obtained by Kaw and Dawson with a nonlinear relativistic theory.

We have observed several different satellite transitions among HeI excited states. These satellites were induced by large, near-resonant, spatially inhomogeneous electric fields. A simple theory was proposed to explain the line shape of both the satellite and the allowed line based on the rotating-wave approximation. The plasma was contained in a dielectric waveguide and the laser mode pattern inside the guide was calculated and incorporated into the line-shape function. It was shown that excellent agreement between theory and experiment could be obtained over a wide range of experimental parameters. Agreement was very good in regimes where perturbation theory fails. We also observed that at high field strengths the Stark shift of the allowed component was less than the three-level theory predicted. We attribute this error to the fact that we have ignored all of the remaining levels of the HeI system.