Total (elastic + inelastic) differential photon-scattering cross sections have been measured for ${\mathrm{H}}_2$ gas over a range of $2.5\le S=\left(\frac{4\pi }{\lambda }\right)sin\left(\frac{\theta }{2}\right)\le 6.7$ ${\mathrm{\AA }}^{-1\mathrm{}}$ and for He over $3\le S\le 11.2$ ${\mathrm{\AA }}^{-1\mathrm{}}$, using an x-ray beam in the Stanford Synchrotron Radiation Laboratory. Absolute measured cross sections agree with theory within the probable errors. Relative cross sections (normalized to theory at large $S$) agree to better than 1% with theoretical values calculated from wave functions that include the effect of electron-electron Coulomb correlation, but the data deviate significantly from theoretical independent-particle (e.g., Hartree-Fock) results. The ratios of measured absolute He cross sections to those of ${\mathrm{H}}_2$, at any given $S$, also agree to better than 1% with theoretical He-to-${\mathrm{H}}_2$ cross-section ratios computed from correlated wave functions. It appears that photon scattering constitutes a very promising tool for probing electron correlation in light atoms and molecules. The degree of polarization of the synchrotron radiation beam has been measured by rotating the scattering plane about the beam axis; results are compared with theory.

We consider the infinite-order set of coupled, integral equations for the states of an atom in a very-narrow-bandwidth classical radiation field. In the vicinity of a one-photon resonance for two-photon ionization, this set is reduced to a pair of coupled equations describing the response of the atom to the radiation field. This set is derived by dropping terms of the original set whose lowest-order contributions to the radiation-induced shift and width of the resonance are quadratic in the intensity. In the absence of simultaneous absorption and emission, impossible except for two-photon processes from excited states, this pair can be decoupled. The remaining uncoupled integral equation describes the ac Stark effect based on contributions to the shift and width linear in the intensity. These contributions derive from radiative corrections to the intermediate atomic state and depend on the virtual processes of ionization from and recombination into, as well as emission from and reabsorption into, this state. A method for obtaining the exact numerical solution is discussed. An approximate solution is obtained by use of the single-eigenfunction approximation to the Green's function belonging to the response function. This Green's function has a pole at the static (unshifted) position of the resonance; thus, this approximation is quite accurate very near the pole. The order of nonlinearity is calculated as a function of the static detuning through the resonance and is shown to undergo rapid fluctuations about the nonresonance value of 2. The error inherent in the single-eigenfunction approximation to the Green's function is estimated and found to range from about 1% to 27% corresponding to static detunings ranging from about 1 to 25 Å, respectively.

A precise measurement has been made of the lifetime of ${\mathrm{He}}^{+}$ in the metastable $2{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}$ state. The result, ${\Gamma }_{2S}=525\mathrm{}\pm 5$ ${\sec }^{-1\mathrm{}}$, is consistent with the theory of spontaneous two-photon decay and places an upper limit $\left|\delta \right|<\mathrm{}2.4\mathrm{}\times {10}^{-5\mathrm{}}$ on the amplitude of any parity-nonconserving $2P$ admixture.

Expressions for the fluctuations and correlations of the light emitted by a ring laser with counter-rotating traveling wave modes are derived theoretically. It is shown that the competition of the two modes for the excited atomic population leads to negative correlations between their intensity fluctuations, whose magnitude depends on the detuning of the laser cavity from the atomic line center. As a result of the mode competition, the relative intensity fluctuations do not die out in general as the working point of the laser is raised above threshold at line center, and the emitted light does not become fully coherent as in a conventional laser. The two-time amplitude and intensity correlation functions of the light are calculated at line center, and are shown to be expressible in terms of the eigenfunctions and eigenvalues of a certain Schrödinger equation, in complete analogy with the theory of the single-mode laser. However, the intensity correlation time increases with pump parameter above threshold, unlike that for the single-mode laser.

The light intensities of the two counter-rotating modes of a ring laser have been investigated by photoelectric counting measurements near the laser threshold and for various detunings. The results are found to be in good quantitative agreement with the theory. They confirm that, at the line center, competition effects prevent the light output in one mode from growing and becoming coherent as the pump parameter is increased. Instead the relative intensity fluctuations of this mode become unity, as for thermal light. Negative cross correlations between the intensity fluctuations of the two modes are also measured. The competition effects virtually disappear when the laser is detuned three natural linewidths from line center, and an intermediate situation is found for smaller detuning.

The amplification of noise in a laser amplifier is treated theoretically. The model for the active medium and its description using density-matrix techniques, are taken from the theory of laser operation. The fundamental concern of this investigation is the spectral behavior of the radiation in the nonlinear regime; hence, the formalism is written from the onset in the frequency domain. The statistics of the light are gradually modified by the nonlinear amplification process, and expressions are derived for the rate of change of fluctuations in intensity as a measure of statistical changes. These expressions should be easily susceptible to detailed experimental observations. In addition, the range of validity of Litvak's Gaussian-statistics approximation is discussed with some detail. In the homogeneous-broadening case, the evolution of initially broadband Gaussian radiation toward quasimonochromatic oscillations with laserlike statistics is extensively explored in several numerical examples. The connections of this study with the time-domain work of Risken and Nummedal, on self-pulsing in a ring-laser configuration, are clearly established. Finally, spectral-narrowing and -rebroadening effects in Doppler-broadened media are discussed both analytically and with numerical examples. These examples show the distinct contribution of pulsations in the population ("Raman-type terms"), and saturation phenomena. The predicted narrowing-rebroadening rate is compared with expressions found in previous literature.

The optical anisotropy induced by a saturating stationary field in a gas laser which is initially isotropic is described theoretically. The calculation of the polarizability tensor shows that linear or circular anisotropies can be induced depending upon the electromagnetic field polarization. The Jones matrix formalism is extended to the nonlinear case. The experimental verification of the predicted linear gain anisotropy in a He-Ne laser is obtained at 3.39 μm. The Jones matrix for a probe field in this particular case is also calculated. The corresponding linear anisotropy is attributed to Zeeman coherences between sublevels. Moreover, we find that the maximum of this anisotropy is slightly shifted from the center of the line.

$d$-dimensional homogeneous isotropic incompressible turbulence is defined, for arbitrary nonintegral $d$, by analytically continuing the Taylor expansion in time of the energy spectrum $E_k\mathrm{}\left(t\right)\mathrm{}$, assuming Gaussian initial conditions. If $d<\mathrm{}2\mathrm{}$, the positivity of the energy spectrum is not necessarily preserved in time. For $d\ge 2$ all steady-state and initial-value calculations have been made with a realizable second-order closure, the eddy-damped quasi normal Markovian approximation. Near two dimensions the enstrophy (mean square vorticity) conservation law is weakly broken, enough to allow ultraviolet singularities to develop in a finite time but not enough to prevent energy from cascading in the infrared direction. A systematic investigation is made of zero-transfer (inertial) steady-state scaling solutions $E_k\propto k^{-m}$ and of their stability. Energy-inertial solutions with $m=\frac{5}{3}$ exist for arbitrary $d$; the direction of the energy cascade reverses at $d=d_c\simeq 2.05$. For $d<\mathrm{}{d^{\prime }}_c\simeq 2.06$ there are in addition, as in the cascade model studied by Bell and Nelkin, inertial solutions with zero energy flux; their exponents $m\left(d\right)\mathrm{}$ are given by a roughly parabolic curve in the ($m, d$) plane, linking enstrophy cascade ($m=3\mathrm{}$, $d=2\mathrm{}$) to enstrophy equipartition ($m=1\mathrm{}$, $d=2\mathrm{}$) For any point in the ($m, d$) plane such that the transfer integral is finite and negative, a steady-state scaling solution $E_k\propto k^{-\mathrm{m}}$ is obtained when the fluid is subject to random forces with spectrum $F_k\propto k^{\frac{3(m-1\mathrm{})\mathrm{}}{2}}$. A special case is the "model B" [$m=-1=\frac{2}{3}\epsilon +O\left({\epsilon }^2\right)$, $d=4-\epsilon$] obtained by Forster, Nelson, and Stephen using a dynamical renormalization-group procedure. Forced steady-state solutions are actually not resticted to the neighborhood of $m=-1\mathrm{}$, $d=4\mathrm{}$; they are amenable to renormalization-group calculations on the primitive equations for arbitrary $d>\mathrm{}2\mathrm{}$ when $m$ is close to the crossover -1 and, perhaps, also near the crossover +3.

The dynamics of concentration fluctuations in a gel near phase separation is described by a mode-coupling theory. The scaling relations, $E\propto \frac{1}{{\xi }^2}$ and $f\propto \frac{1}{\xi }$, where $\xi$ is the correlation length of the concentration fluctuations in the network, are obtained for the elastic constant of the network, $E$, and the frictional coefficient between the network and the fluid medium, $f$. The theory fully explains the experimental results of Tanaka, Ishiwata, and Ishimoto on a polyacrylamide gel.

We present an orthonormal set of single-particle orbital functions explicitly satisfying the Hartree-Fock (HF) equations for occupied states, and which give low-density few-particle clustering effects in a many-fermion system. The initial study given here is in one and three dimensions for attractive $\delta$ pair interactions. In one dimension the numerical results compare favorably with the exact (lowest) HF state at zero density. The results turn out to be superior to the classic Overhauser HF orbitals, as well as to some more recent ones.

We numerically investigate the behavior of a simple dynamical system, a plane billiard which by a continuous deformation of the border passes from a completely integrable system to a well-defined type of completely stochastic system, namely a $K$ system. A stochastic transition is observed, with the usual coexistence of ordered and stochastic regions. Moreover, the stochastic region at certain intermediate stages appears to be separated into several invariant components. An estimate of the Kolmogorov entropy is presented.

We have performed a detailed numerical investigation around the region of transition from quasiperiodic to stochastic motions for a chain of particles with Lennard-Jones interaction. We have found that the curve for the stochastic parameter (or maximal Lyapunov characteristic number) as a function of energy presents successive bifurcations in that region, a phenomenon not yet observed in this field. This phenomenon is interpreted as indicating that the so-called stochastic region is in general subdivided into disjoint invariant components, which merge into a unique stochastic region as energy increases.

The Debye electron-proton sum over states $B\left(T\right)\mathrm{}$ is shown to have a continuous temperature derivative. $B\left(T\right)\mathrm{}$ is therefore expected to be analytic with respect to the interaction strength.

Calculations of the collision strength for excitation of the $2p$ state of ${\mathrm{Be}}^{+}$ are performed at incident energies of 2, 5, and 8 Ry. A modified five-state close-coupling expansion is used in which three pseudostates $\overline{3}\overline{s}$, $\overline{3}\overline{p}$, and $\overline{3}\overline{d}$ are retained in addition to target states $2s$ and $2p$. Pseudostates are chosen so that, for a given incident electron energy, there is a minimum in the dominant partial-wave collision strength with respect to variation of the range parameter. Collision strengths are obtained which lie a few percent closer to the measurements of Taylor, Phaneuf, and Dunn than do the five-state calculations of Hayes et al.

Previous studies showed the configuration-mixed excitation channel structure of two-electron atoms to be classified by two quantum numbers $K$ and $T$. Three new aspects of the classification are considered here: (i) $K$ and $T$ are shown to be limiting cases of two exact invariants for channels in the asymptotic regime $r_2\gg r_1$. The invariants classify channels for the entire isoelectronic series. Sum rules, quantum number correlation diagrams for isoelectronic series, and perturbation formulas for the invariants are described. (ii) The $K$, $T$ spectrum is shown to be contained in a chain SU(4) ⊃ SU(3) ⊃ SU(2) defined mathematically on the two-electron channels. Possible physical connection with O(6) quantum numbers for noninteracting particles in hyperspherical coordinates is discussed. (iii) An empirical link is established between autoionization stability of heliumlike atoms, and unitarity of irreducible representations for a SU(2, 1) channel scheme. The noncompact classification also includes channels for double ionization.

Electron-photon angular correlation measurements have been made for scattered electrons which have excited the $2{}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}$ state of helium and photons from the $2{}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}\to 1{}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ transition at an incident electron energy of 80 eV for a range of electron scattering angles ${\theta }_e$ from 5° to 155° and for a photon angle ${\theta }_{\gamma }$ of 90° in the scattering plane. These measurements have been analyzed to obtain relative values of the ratio $\lambda$ of the differential cross section for exciting the $m_j=0\mathrm{}$ sublevel ${\sigma }_0$ to the total differential excitation cross section $\sigma$, for $J=1\mathrm{}$ ($\lambda =\frac{{\sigma }_0}{\sigma }$). The results extend the measurement range to both larger and smaller values of ${\theta }_e$ at this energy and are consistent with previous measurements and the most recent calculation for the process.

As shown in papers I and II of this series, far from symmetry the $K$-vacancy production cross section of the lower-$Z$ ($Z_L$) partner in heavy-ion collisions varies rapidly with the atomic number $Z_H$ of the higher-$Z$ partner. We examine our and other data for $K$ and $L$ x-ray production with 45-94-MeV Ni, 45-110-MeV Br, 47- and 62-MeV I projectiles and interpret the $Z_L$ $K$-vacancy production far from symmetry in terms of the $K-L$ level matching mechanism of Barat and Lichten. Sharing of $3d\sigma$ vacancies on the outgoing part of the collision provides a semiquantitative understanding of the dependence of $K$ and $L$ cross sections on $Z_L$, $Z_H$, and bombarding energy. Some remarks about the general role of vacancy sharing in inner-shell vacancy formation in heavy-ion collisions are made.

We define a class of primitive functions which are least distorted from the unsymmetrized function $F^0$, a product of atomic or other group functions, in the limit that the interactions between the groups have been turned off. These primitive functions have the property that at least one Schrödinger eigenfunction may be obtained from them by symmetry projection. It has been shown elsewhere that these functions satisfy a transformed Schrödinger equation in which the interactions between groups are screened. The screened potential is regarded as a perturbation and the corresponding Rayleigh-Schrödinger perturbation equations are derived. It is shown that a number of inequivalent, but equally valid energy expressions may be defined in terms of the primitive functions. Only when the primitive function is calculated exactly to infinite order will the different energy expressions all yield the same numerical value. It is suggested that this provides a check on the accuracy of approximate primitive functions.

A simple tractable asymptotic expression is obtained for a continuum Dirac electron in a Coulomb field. It is basically a WKB expression, which is to be matched to a solution valid for small values of $r$. Simple methods for estimating the error in the solution are also obtained. The expression is shown to be very accurate down to small values of $r$ (of approximately 1 Bohr radius). It is also shown to be accurate even where the ratio $\frac{Z}{k}$ is very large and in regions where the local kinetic energy far exceeds the total energy. Methods of employing this work in the normalization of numerical solutions and obtaining phase shifts are pointed out.