We present a rigorous technique, which should be applicable to both atoms and molecules, for calculating photoabsorption cross sections using square-integrable basis functions. The technique is based on the method of complex coordinates as developed by Nuttall and co-workers. In contrast to some other $L^2$ methods, the calculations converge directly at real energies. The method is illustrated by application to the case of atomic hydrogen.

Angular correlations between $L$ and $M$ x rays in ${}_{}{}^{239}{}_{}{}^{}\mathrm{Pu}$ obtained from a radioactive source of 32-yr ${}_{}{}^{243}{}_{}{}^{}\mathrm{Cm}$ were measured using Si(Li) and Ge(Li) detectors. The measured angular-correlation coefficients $A_{22}=(8.5\mathrm{}\pm 1.5)\times {10}^{-2\mathrm{}}$ for the $L{\beta }_1-M$ cascade and $A_{22}=(7.7\mathrm{}\pm 2.0)\times {10}^{-2\mathrm{}}$ for the $L{\alpha }_{1+2}-M$ cascade are both significantly higher than theoretical predictions. The intensity of the Pu $L_1-L_3$ x ray per $L_1$ vacancy was found to be $\frac{I(L_1-L_3)}{\Gamma \mathrm{}\left(L1\mathrm{}\right)\mathrm{}}=(6.6\mathrm{}\pm 0.8)\times {10}^{-2\mathrm{}}$. The angular correlation coefficient of the $L\alpha -(L_1-L_3)$ x-ray cascade was measured and found to be $A_{22}=(3\mathrm{}\pm 7)\times {10}^{-2\mathrm{}}$ and is in agreement with theoretical predictions.

A solution of the Dirac equation for an electron in a point-Coulomb potential is given in the form of an expansion in the parameter $\frac{1}{r}$. The main part of the solution is the Sommerfeld-Maue wave function. We introduce a correction term. All orders in $\alpha Z$ are kept. The given solution (Sommerfeld-Maue plus correction term) contains the exact contributions of orders 0 and 1 in the expansion in $\frac{1}{r}$. This modified Sommerfeld-Maue wave function is used to get the cross section for the pair-production process by $\gamma$ rays in the field of a nucleus. The cross section is written in the form of the Fink-Pratt cross section plus a correction term. The formula is valid for all $\alpha Z$. The energetic relative accuracy should be of order $\frac{1}{E^2}$. The small-angle approximation is not introduced.

The fine-structure intervals of the lowest two triplet $D$ states of ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ have been obtained by a new method: Ground-state atoms are excited by a discharge to the triplet metastable state, which then undergoes a two-step optical excitation to the triplet $D$ states, where level-crossing measurements are made. The experimental results for ${}_{}{}^3{}_{}{}^{}D_1^{}{}_{}{}^{}-{}_{}{}^3{}_{}{}^{}D_3^{}{}_{}{}^{}$ (MHz) are 1400.67 ± 0.29 and 591.25 ± 0.14 for the $3{}_{}{}^3{}_{}{}^{}D$ and $4{}_{}{}^3{}_{}{}^{}D$ states, respectively; for ${}_{}{}^3{}_{}{}^{}D_2^{}{}_{}{}^{}-{}_{}{}^3{}_{}{}^{}D_3^{}{}_{}{}^{}$ (MHz) they are 75.97 ± 0.23 and 36.15 ± 0.24 for $3{}_{}{}^3{}_{}{}^{}D$ and $4{}_{}{}^3{}_{}{}^{}D$. These values are much more precise than the previously available data, obtained by optical, particle-bombardment level crossing, or beam-foil quantum-beat spectroscopy, and are consistent with the previous data except for one case, where a previous measurement of $3{}_{}{}^3{}_{}{}^{}D_2^{}{}_{}{}^{}-3\mathrm{}{}_{}{}^3{}_{}{}^{}D_3^{}{}_{}{}^{}$ by electron bombardment level-crossing spectroscopy is believed to be in error. These new experimental values for the fine structures of the triplet $D$ states of ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ clearly indicate the inadequateness of existing theoretical calculations of these fine structures.

The techniques we developed previously for calculating $L$-shell Compton profiles within the Born approximation are applied to the calculation of improved core Compton profiles for real atoms. The method is referred to as the exact hydrogenic (EH) method. We introduce a new method for determining effective nuclear charges $Z^{*}$ to use in hydrogeniclike wave functions and our EH method. These nuclear charges $Z^{*}\mathrm{}\left(0\right)\mathrm{}$ are determined by imposing the requirement for each atomic orbital that the impulse hydrogenic (IH) Compton profile at its center $q=0.0\mathrm{}$ match the impulse Hartree-Fock (IHF) Compton-profile value at $q=0.0\mathrm{}$. It is then demonstrated that the two $L$-shell impuse profiles are found to lie very close to one another as we go away from the profile center, the agreement improving as we go to atoms of higher atomic number. The wave functions for the $2s$ and $2p$ orbitals are studied and it is observed that hydrogenic wave functions with our values of $Z^{*}\mathrm{}\left(0\right)\mathrm{}$ fit the Hartree-Fock wave functions much more accurately than if a $Z^{*}$ associated with the binding energy is used. We demonstrate in the case of Mo $K\alpha$ radiation scattering from neon and aluminum that our EH method using $Z^{*}\mathrm{}\left(0\right)\mathrm{}$ gives results in better agreement with experiment than any impulse calculation. In the neon case, the EH correction to the IHF result is ∼2.4% at the profile center. Dependence of the Compton profile on incident photon energy is demonstrated theoretically and observed to be in agreement with experiment.

The impact broadening due to He, Ne, Ar, Kr, and Xe of the Na $3p{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}-4d{}_{}{}^2{}_{}{}^{}D_{\frac{3}{2}}^{}{}_{}{}^{}$ (5683 Å) line has been measured at a temperature of 500 °K. In addition, impact shifts of this line and other diffuse series lines up to the transitions from $6d{}_{}{}^2{}_{}{}^{}D$ have also been determined. The observed broadening and shift rates are in agreement with Lindholm-Foley calculations using semiempirical alkali-metal-noble-gas potentials reported previously. A comparison of calculated and observed shifts show that alkali-metal-helium potentials are somewhat more repulsive than previously thought, and also that the collision-averaged potentials corresponding to $3p{}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}$ are less attractive than those for $3p{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}$. We conclude that the use of these potentials in the impact approximation will predict the broadening and shift of alkali-metal spectral lines by noble gases with an accuracy of 10-20%.

The phenomenon of molecular optical activity is examined in the region of continuous absorption. When the "excited" state of the molecule describes an infinite (ionized) system, then the angular distribution of photoelectrons is the sum of coherent contributions corresponding to different magnitudes and interferences of $\hslash \overrightarrow{1}$, the angular momentum of the photoelectron. The amplitude for such a process is the sum of terms for each $l$; thus, since both even and odd values of $l$ can coexist at a single energy in the continuous spectrum, the electric and magnetic dipole matrix elements can coexist in this amplitude, making possible the existence of electric-dipole-magnetic-dipole interference in the angular distribution even for a molecule with a center or plane of symmetry. For discrete absorption, in which the intensity is the sum of incoherent contributions corresponding to the intensities for populating the fine-structure levels of a given excited state, the coexistence of the electric and magnetic dipole matrix elements in the amplitude is possible only for a molecule with a site which is asymmetric with respect to inversion or reflection; otherwise both even and odd values of $l$ could not coexist at a single energy in the discrete spectrum. The signs of the electric-dipole-magnetic-dipole interference terms are opposite for left and right circularly polarized light; thus there exists a signal for the angular distribution difference for absorption of left and right circularly polarized light of order $\alpha$ relative to the angular distribution for absorption of light of either polarization. This is just the phenomenon of "circular dichroism" which characterizes molecular "optical activity" in the region of absorption. It exists for the angular distribution of photoelectrons ejected from an oriented molecule with a center or plane of symmetry, but vanishes for isotropic systems (atoms) owing to the independence of the radial wave functions from the magnetic quantum number. This ensures the orthogonality of atomic radial wave functions belonging to states of different $m$ and is responsible for the selection rule in atomic spectroscopy that magnetic-dipole transitions are possible only between the fine-structure levels of a given multiplet. Measurement of the angular distribution characteristic for this process would provide a sensitive probe of the parameters of the initial molecular orbital. The existence of even-odd-type interferences of the partial waves of the photoelectron would provide a test of the time-reversal invariance of the wave function for the ionized system, since these interferences depend on the sine rather than the cosine of the phase-shift difference and hence on the normalization of the wave function to satisfy incoming boundary conditions. Calculations are carried out to illustrate these and other points.

Adopting the so-called "genealogical construction," one can express the eigenstates of collective operators corresponding to a specified mode for an $N$-atom system in terms of those for an ($N-1\mathrm{}$)-atom system. Using these Dicke states as bases and using the Wigner-Eckart theorem, a matrix element of a collective operator of an arbitrary mode can be written as the product of an $m$-dependent factor and an $m$-independent reduced matrix element (RME). A set of recursion formulas for the RME is obtained. A graphical representation of the RME on the branching diagram for binary irreducible representations of permutation groups is then introduced. This gives a simple and systematic way of calculating the RME. This method is especially useful when the cooperation number $\mathcal{r}$ is close to $\frac{N}{2}$, where almost exact asymptotic expressions can be obtained easily. The result shows explicitly the geometry dependence of superradiance and the relative importance of $\mathcal{r}$-conserving and $\mathcal{r}$-nonconserving processes. This clears up the chief difficulty encountered in the Dicke-Schwendimann approach to the problem of $N$ two-level atoms, spread over large regions, interacting with a multimode radiation field.

In this paper we discuss the quantum features of cooperative radiation processes in a two-level system taking into account non-Markovian effects. If the initial condition is spatially uniform, the many-mode atom-field master equation which has been derived and semiclassically discussed in a previous paper, reduces to the well-known single-mode model laser master equation. Hence, using a suitable projection operator, we derive a generalized non-Markovian master equation which turns out to be correct already in the first Born approximation. This master equation incorporates the effects both of stimulated emission and absorption and of spontaneous emission. When stimulated processes can be neglected, it reduces to a previously known Markovian superradiance master equation, whereas in general it describes non-Markovian oscillations of the radiated intensity. Finally we derive a linear system of equations for the probability of occupation of the Dicke states which generalizes a previously discussed system, taking into account the contribution of stimulated processes.

A derivation is given of free-space Gaussian modes described by six parameters. For a beam traveling in the $z$ direction, three parameters describe the beam properties in the planes of constant $y$ and three parameters describe the beam properties in the planes of constant $x$. It is further shown that the beam waist for the properties in the planes of constant $y$ need not coincide with that for the properties in the planes of constant $x$. For special values of the parameters the six-parameter modes reduce to the familiar three-parameter modes.

A derivation is given of free-space Gaussian modes described by six parameters. For a beam traveling in the $z$ direction, three parameters describe the beam properties in the planes of constant $y$ and three parameters describe the beam properties in the planes of constant $x$. It is further shown that the beam waist for the properties in the planes of constant $y$ need not coincide with that for the properties in the planes of constant $x$. For special values of the parameters the six-parameter modes reduce to the familiar three-parameter modes.

It is suggested that coherent soft x rays can be produced by inverting the electron population in a suitable target, such as Li, through irradiation with x rays generated by fast electrons traversing an electromagnetic field (as in a storage ring). Conditions to be satisfied by target and radiation parameters are stated, and examples given.

The ultrasonic velocity and attenuation in magnetically aligned liquid crystals has been measured. All well-known liquid-crystalline symmetries (nematic, cholesteric, and smectic $A$, $B$, and $C$) were studied. Sound anisotropy measurements in the smectic-$B$ and -$C$ and cholesteric materials are reported here for the first time; the measurements in the smectic-$C$ and cholesteric liquid crystals are still in a preliminary stage. We unambiguously attributed the sound velocity anisotropy to the existence of a repeated structure (broken translational symmetry) in the hydrodynamic (low-frequency) limit by measuring the anisotropy across the nematic-smectic-$A$ transition in $N-p$-cyanobenzylidene-$p$-octyloxyaniline (CBOOA); a sudden increase of the anisotropy extrapolated to zero frequency was observed at the transition despite a large dispersion. The smectic-$A$— smectic-$B$ transition in ethyl-$p$-[($p$-methoxybenzylidene)amino] cinnamate shows a two-dimensional liquid-to-solid transition. A distinct minimum was found at the nematic-smectic-$C$ transition in $p-p^{\prime }$ heptyloxyazoxybenzene which was not accompanied by an attenuation peak. The velocity anisotropy in a cholesteric mixture of cholesteryl chloride and cholesteryl myristate was tentatively assigned to the dispersion.

In bistable systems the transition kinetics between the two locally stable states can be altered without changing the behavior in the immediate vicinity of the two favored steady states. It follows that quantities which characterize only the vicinity of the favored states cannot determine the most probable state. A second point: Even in monostable linear circuits the steady state need not correspond to minimum entropy production.

A stochastic evolution following a nonlinear Langevin equation with one degree of freedom was examined by a machine computation. The autocorrelation function of macroscopic fluctuations fixed by the time average instead of by the ensemble average is observed to have a long-time tail exhibiting a divergence of their power spectrum in the low-frequency limit. Such a divergence prevents the establishment of strongly steady stochastic process of fluctuations. The present results confirm that nonlinear Markovian kinetics keeps everlasting macroscopic fluctuations.

The slowly decaying hydrodynamic modes in many-body systems are obtained from a theory which avoids perturbation methods. The theory is based on a projection-operator technique which is applied to quantities characterizing the state of the system at various levels of its description. The main result is that a true transport coefficient is shown to consist of two contributions (one which decays rapidly in time and another one which decays slowly), is non-Markoffian, and arises from the nonlinear interactions among the collective modes. This contribution is obtained in a closed analytical form which is further analyzed within the context of our present-day information derived from computer and laboratory experiments. The relationship between this work and similar earlier work on this problem is also discussed.

The effect of spatial background turbulence on parametric interactions in an infinite, inhomogeneous plasma with a linear density profile is considered. It is shown that the wellknown convective saturation arising from the density gradient can disappear in the presence of turbulence. Absolute instability then occurs. The criteria for the existence of temporally growing modes are presented.

In the linear-response regime, a normal Bose or Fermi fluid can be described by an exact kinetic equation whose kernel is nonlocal in space and time. We derive a general expression for this kernel and evaluate it explicitly to second order in the interparticle potential. The result is a wave-vector- and frequency-dependent generalization of the linear Uehling-Uhlenbeck kernel with the Born-approximation cross section. Our theory is developed in terms of a second-quantized form of the Wigner representation. Convenient expressions are obtained for the commutators and anticommutators of the phase-space density operators, and the equilibrium averages of these operators are analyzed in terms of momentum-dependent generalizations of the classical pair distribution function $h\left(k\right)\mathrm{}$ and direct correlation function $c\left(k\right)\mathrm{}$. The central quantity in this study is a two-particle equilibrium correlation function, the phase-space density-density anticommutator, whose Fourier transform $S\left(k\omega p{p}^{\prime }\right)$ gives the symmetrized scattering function $S\left(k\omega \right)$ by integration over the momenta. The kinetic equation is obtained by a formal closure of the quantum BBGKY hierarchy, with the nonlocal kernel expressed in terms of correlation functions involving two, three, and four particles. We show that our method for approximating the kernel and initial condition by a second-order expansion preserves all the sum rules of $S\left(k\omega p{p}^{\prime }\right)$ to the same order and that the result satisfies the appropriate positivity and symmetry conditions.

The self-consistent reservoir (SCR) model of lattice thermal conductivity is combined with the quantum Langevin equation formulation of Ford, Kac, and Mazur to obtain a quantum SCR model for lattice conductivity. Application is made to simple cubic monatomic lattices with harmonic nearest-neighbor interactions. A general expression for the thermal conductivity is obtained, and its high- and low-temperature limits examined.

Raman investigation on the lattice modes (8-200 ${\mathrm{cm}}^{-1\mathrm{}}$) of several smectic phases in terephtal-bis-butylaniline (TBBA) reveals significant spectral differences. Softening of the lowest lattice 19-${\mathrm{cm}}^{-1\mathrm{}}$ mode, observed very near to the solid-smectic-$B$ phase transition, is correlated with the gradual release of the smectic inter-layer glide motion. The Raman spectra add new insight into the ordering within the layers of the metastable smectic phases and also indicate the existence of molecular dynamic rotations in the smectic-$B$ phase.