Classical binary-encounter calculations using a Hartree—Fock velocity distribution for the bound electrons have been carried out for electron-impact ionization of alkali metals. The effect of inner-shell ionization, as well as the excitation of some autoionizing levels, has been included. A satisfactory agreement with recent experimental observations has been achieved.

A general theory is developed to treat two-photon ionization through a single intermediate state in which an arbitrary degree of relaxation may occur. The method is set up to probe the parameters governing the relaxation, by determination of the angular distribution of photoelectrons. The method is applied to the ionization of Na atoms, through the $3p{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}$ and $3p{}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}$ intermediate states.

The transient behavior of a laser is discussed using the quantum theory as did Scully and Lamb. The formal solution of the density-matrix equation is expressed in terms of exponentially decaying eigenmodes. Some of the lower decay constants are obtained numerically. The equations for the moments of the density matrix are then derived and solved by a truncation method. The equations of motion are integrated numerically for the case where the average number of photons in a laser cavity has the realistically large value 1.3 × ${10}^5$. An alternative Fokker-Planck-equation approach is discussed.

Using semiclassical theory, we investigate the shot- and thermal-noise effects on the behavior of a laser. The Fokker-Planck equations for the probability distribution of the laser field are derived. These equations are approximately solved, using a Gaussian function, from which we calculate the spectral profile of the laser field. The width constant for the thermal noise is related to the temperature of the cavity.

The probability distribution of electric fields, $P\left(E\right)\mathrm{}$, in stable and unstable plasmas is investigated by numerical simulation techniques. The usefulness of this approach is demonstrated by comparing the present results with both new and previously published analytic calculations. The results have implications for experiments which use spectral line shapes [governed by $P\left(E\right)\mathrm{}$] as a diagonostic tool to determine the plasma density and temperature.

We investigate the dominant corrections in the critical region to previously derived expressions for the index of refraction $n$ of a nonpolar fluid. We show that the expression for $n$ has a form that can be used to test the assumption that the density-density correlation function is an asymptotically homogeneous function, and to measure the critical exponent $\eta$. We find that at the critical point both the real and imaginary parts of $n$ are finite under quite general assumptions, the change in $n$ owing to critical behavior remaining small. A simple but highly accurate approximate expression is given for the difference between real $n$ and its slowly varying background contribution $n_B$ in the critical region.

In several different theoretical models, the critical velocity for onset of dissipation in superfluid flow through a packed bed of particles appears as a condition for escape of segments of quantized vortex line pinned between particles. This condition has essentially the form of the Feynman critical velocity based on particle diameter $d$; thus the predicted critical velocity is of order $\frac{h}{\mathrm{md}}$ and is independent of temperature. This is in accord with the well-known increase in critical velocity when a tube is packed with fine powder, but predicts a decrease for coarse powder. Measurements of superfluid flow through a glass tube constricted to 390-μm diam and packed with 35-μm silicon-carbide grains give a critical velocity of 1.6 cm ${\sec }^{-1\mathrm{}}$, nearly constant over the temperature range $1.35<T<\mathrm{}2.13\mathrm{}$ K. This is considerably slower than the flow observed in the empty tube (at least for $\Delta p\gtrsim 0.1$ dyne ${\mathrm{cm}}^{-2\mathrm{}}$). The results are not sensitive to the vortex nucleation or regeneration processes, which would distinguish between theoretical models.

An approximation is developed for nuclear scattering of neutrons from a target of spinless particles of a single isotope. It has the correct limit for large momentum transfer, the impulse approximation, and is calculable in practice if the single-particle momentum distribution, the off-diagonal two-particle density matrix, and the two-body interaction potential of the target atoms are known. An intended application of the theory is to neutron scattering from liquid helium in the moderately-high momentum-transfer region ($\overrightarrow{\mathrm{k}}\approx 15$ ${\mathrm{\AA }}^{-1\mathrm{}}$) presently accessible with reactor neutrons.

Short-pulse amplification in gaseous molecular amplifiers is complicated by many aspects of atomic and molecular interactions. Among these are the various vibrational, rotational, and reorientational relaxational processes which influence the coupling of the energy stored in the molecular radiators to the electromagnetic field. The properties of plane-wave pulse amplification, especially in the saturated regime, are examined numerically in order to quantitatively determine the detailed effects of these relaxational phenomena. As expected, the results for saturated amplification show that the amount of extracted energy decreases significantly when the rotational relaxation time is sufficiently long in comparison to the pulse width. We also observe the development of pulse-shape variations which are a direct result of the collisional phenomena and differ qualitatively from the results obtained for amplifying media without an energy reservior. There is a tendency for the pulse lengths to increase owing to the energy transfer, in contrast to the strong narrowing effects which occur in the absence of the collisional processes. The influence of reorientational collisions is found to be small, accounting for less than a 20% effect on the over-all conclusions. Finally, we present results concerning the development of an asymptotic pulse shape in high-gain amplifiers. In this case, the pulse shape clearly exhibits the competition between the stimulated rate, which scales with the optical flux, and the collisional rates which are determined by the particle density. Calculations of this nature may be applied directly to C${\mathrm{O}}_2$, CO, and HF molecular amplifiers for both the electrically and chemically driven systems.

Helium-II film transfer rates have been measured for filling and for emptying beakers of solid Ar. Liquid Ar was frozen slowly in a mold which was later removed, leaving free-standing transparent solid-Ar beakers. The beakers used were of 3-mm i.d., 5-mm o.d., and 3.5 cm in height. Experiments were carried out in the temperature range 1.45-2 K. The dependence of the transfer rate $\sigma$ (${\mathrm{cm}}^3$/s cm of circumference) on the difference between inner and outer He levels $Z$, height of the beaker lip above the level of the liquid source $H$, and temperature was studied. The transfer rates were lower than those observed with glass beakers, in agreement with theory. For beaker fillings with a level difference of 1 mm and at a temperature of 1.66 K, the measured transfer rates may be described by the relation $\sigma =4.8\mathrm{}{H}^{-0.21\mathrm{}}\times {10}^{-5\mathrm{}}$ ${\mathrm{cm}}^2$/s, where $H$ is in cm.

The autocorrelation function of the density of a tagged particle is studied using the Mori formalism. The variables used are the collective conserved variables, the tagged-particle density, and bilinear products thereof. The case of point particles is considered in two dimensions, and, in three dimensions, self-diffusion by a particle of arbitrary size is treated. It is found that the bilinear-hydrodynamic approach automatically separates the self-diffusion coefficient of the tagged particle into a nonhydrodynamic part, and a hydrodynamic part which resembles the Stokes-Einstein law. In two dimensions, it is found that the mean-square displacement of a particle increases as $t\ln t$, and that certain natural redefinitions of the diffusion and friction coefficients leave Einstein's law invariant. In three dimensions, for a large particle, the Stokes-Einstein law is reproduced. The relation between the well-known $t^{\left(\frac{-3\mathrm{}}{2}\right)}$ "tails" on correlation functions, and the Stokes-Einstein law, is discussed.

The ground-state energies of solid ${\mathrm{He}}^3$ and ${\mathrm{H}}_2$ at various densities are calculated using a self-consistent method in the $t$-matrix formulation. The two-body equation of motion is solved by expanding the two-body wave function in terms of partial waves. The partial-wave expansion gives rise to a set of coupled differential equations which are solved numerically for the ground-state eigenfunctions. The calculations for ${\mathrm{He}}^3$ are done using three different two-body potentials, the Lennard-Jones potential, the Beck potential, and the Frost-Musulin potential. The calculations for ${\mathrm{H}}_2$ are done using the Mason-Rice two-body potential. A bcc structure is assumed for solid ${\mathrm{He}}^3$, while an fcc structure is assumed for solid ${\mathrm{H}}_2$. Exchange effects are neglected. Figures and tables are given which compare the present results with those of other authors.

A self-consistent, microscopic theory of scattering of light by molecular aggregates is evolved from the standpoints of random-phase-modulation theory and stochastic theory. Contemporary theory is founded on the premise that the scattered light spectrum is proportional to a four-dimensional Fourier transform of the molecular density-correlation function. This premise is justified only in a continuum representation of density, but it breaks down when the motion of discrete molecules is taken into account. The Rayleigh spectrum is shown to be a manifestation of translational degrees of freedom of molecules, much as the Raman spectrum is of internal degrees. Both the central and shifted components are attributed to propagating waves representing probability densities. Theoretical spectra are in agreement with experimental data in both the kinetic and hydrodynamic regimes, and the shifted frequencies are simply related to the rms speed of typical molecules. This theory also provides a mechanism which could account for deviations of total integrated intensity from that predicted by incoherent scattering. Such deviations are simply related to the ratio of intensities of the shifted and central components of the spectrum. Success of this theory, however, is not achieved without complete departure from conventional approaches of kinetic theory. By necessity, statistical aspects of the problem are approached through the use of a set of partial differential equations for the probability densities of continuous, differentiable stochastic processes. Statistical trajectories of molecules are characterized by a single function $h\left(\tau \right)$ defined as the logarithmic derivative of the conditional expectation value of velocity. Solutions based on the asymptotic behavior of $h\left(\tau \right)$ suggest the possibility for existence of other lines at the high-frequency end of the Rayleigh spectrum.

The ground-state correlation energy of the electron gas is calculated in the region of intermediate densities using the variational method of Becker, Broyles, and Dunn for two trial wave functions. Each trial function is taken to be a product of two factors, one factor being the ground-state wave function for the ideal gas of spin-1/2 particles and the other being a product of pair functions in the relative coordinates of the electrons. In one trial function a single pair function is used; in the other, the pair functions between parallel and antiparallel spins are allowed to differ. The pair functions are parametrized and approximations to the energy minimized. The three-particle correlation functions appearing in the kinetic energy are replaced by either the Kirkwood superposition approximation (KSA) or the convolution approximation (CA) to give two approximate energy functionals for each wave function. The ideal-gas $N$-particle probability density is approximated by a Boltzmann factor with an effective pair potential. This effective potential is obtained by inverting the hypernetted chain equation for the known pair-correlation function of the ideal Fermi gas. The pair-correlation functions for the interacting system are then calculated by means of the hypernetted-chain equation. The CA correlation energies join smoothly with both the high- and low-density expansions. The CA and KSA correlation energies differ by less than 4% everywhere in the intermediate-density region. The pair-correlation functions exhibit generally reasonable physical behavior.

Hylleraas-type wave functions have been calculated for the ($1s {}_{}{}^2{}_{}{}^{}2p$) ${}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}$ state of the Li atom. With a 78-term function an energy of -7.40999 a.u. was obtained as compared to the relativistically corrected experimental value -7.41016 a.u. Calculated hyperfine-structure expectation values and oscillator strengths for transitions to the four lowest ${}_{}{}^2{}_{}{}^{}S$ states are in good agreement with available experimental results.

The $X\alpha$ method for the calculation of atomic and molecular orbitals is compared to a method employing ensemble averages. The formal theory as well as calculated results are discussed and the choice of $\alpha$ parameter values for the local exhange approximation is commented upon.

Upper and lower bounds were obtained previously on matrix elements of the form $W_{\mathrm{nm}}=({\psi }_n,W{\psi }_m)$, where $W$ is a Hermitian operator and ${\psi }_n$ and ${\psi }_m$ are the wave functions of the $n$th and $m$th states of the system. The bounds are variational but nonstationary; they are expressed in terms of trial wave functions ${\Psi }_{\mathrm{nt}}$ and ${\Psi }_{\mathrm{mt}}$ containing variational parameters, but the error in the bound is of first order in the errors in the ${\Psi }_{\mathrm{nt}}$ and ${\Psi }_{\mathrm{mt}}$. The results have been either subject to rather restrictive conditions (for example, only for certain specific choices for $W$ and only for real wave functions) or have been very conservative. We remove most of these restrictions ($W$ need not be positive or negative definite, the wave functions may be complex, the system may not even be invariant under time reversal) but maintain rigorous bounds of good quality. The method of using Gram-determinant inequalities, which has been employed previously, especially by Weinhold, and which we adopt, leads to variational but nonstationary bounds on $W_{\mathrm{nm}}$ in terms of "simple" upper bounds (which may be poor) on ${W^2}_{\mathrm{nn}}$. Here again, only for a very few particular choices of $W$ have such simple bounds on ${W^2}_{\mathrm{nn}}$ been given previously (for example, $n$ restricted to be the ground state, and $W$ the operator $z_i$, the coordinate of the $i$th electron). The main result of this paper is to show that such simple upper bounds can be obtained for a very wide class of operators $W$ in terms of the energy eigenvalues of the Hamiltonian. (They can be improved if given additional experimental information on oscillator strengths, for example). These simple bounds on ${W^2}_{\mathrm{nn}}$ do not involve any trial wave functions. The method of variational but nonstationary bounds is illustrated for diagonal matrix elements of $r_1$ and $r_1^2$—we, therefore, require simple bounds on $r_1^2$ and $r_1^4$—for the states $1s^2 {}_{}{}^1{}_{}{}^{}S$ and $1s2s {}_{}{}^3{}_{}{}^{}S$ of the helium atom, with rather good results.

Rigorous upper and lower bounds on the diagonal and off-diagonal bound-state matrix elements of an arbitrary Hermitian operator $W$ are derived which represent a significant improvement over previous results in that all first-order error terms are eliminated. These bounds do, however, contain error terms of the three-halves power of the errors in the trial functions used; they are therefore termed "quasistationary bounds." In the particular but important case of the diagonal bound-state matrix elements of non-negative (nonpositive) Hermitian operators, true stationary lower (upper) bounds are obtained; the errors are of second order in the errors of the trial functions used. All trial functions can contain variational parameters. The results obtained are tested by computing upper and lower bounds on the expectation value of $r_1+r_2$ for the ground state of helium, and a substantial improvement over previous bounds is obtained.

The formal exact solution to the problem of the radiation of a charged particle traveling with a constant velocity through a dielectric sphere is obtained. The electromagnetic field may be expressed in terms of an infinite number of spherical normal modes. It is found that radiation may be emitted for arbitrarily small particle velocities. For larger spheres (i.e., $ka\gg 1$ when $k$ is the wave number and $a$ is the radius of the sphere) the radiated field is predominantly Čerenkov-type radiation when the velocity of the particle is above the Čerenkov threshold velocity. For small spheres ($ka\gg 1$) and low particle velocities the radiation is shown to be mainly of transition type. Numerical results are presented to illustrate the behavior of the spectra of the various lower-order radiative modes as the velocity of the particle is varied.

There has existed a continuing dialogue concerning the proper identification of radiation forces and momenta in dielectric media. I argue herein that a sensible and consistent picture of these forces and momenta is available. That is, the density of electromagnetic momentum $\overrightarrow{\mathbf{G}}$ is given by $\overrightarrow{\mathbf{G}}=\frac{\overrightarrow{\mathbf{S}}}{c^2}$, where $\overrightarrow{\mathbf{S}}$ is Poynting's vector. The forces which are exerted on material objects in dielectric media are associated with changes in both the electromagnetic and mechanical momenta of the media. In fairly broad circumstances, such forces may be found from the rate of change in a pseudomomentum $\overrightarrow{\mathbf{K}}$ given by $\overrightarrow{\mathbf{K}}=\epsilon \overrightarrow{\mathbf{G}}$, where $\epsilon$ is the dielectric constant of the medium.