The relaxation times $T_1$ and $T_2$ of ${\mathrm{He}}^3$ nuclei in pure liquid ${\mathrm{He}}^3$ and in liquid ${\mathrm{He}}^3$ - ${\mathrm{He}}^4$ mixtures have been measured in the temperature range $0.9°\mathrm{K}<T<\mathrm{}2.6\mathrm{}°\mathrm{K}$. Reproducible relaxation times longer than those previously reported have been obtained by employing an extensive cleaning procedure. The Pyrex sample chamber and filling capillary were repeatedly cleaned in the presence of several mm of pure ${\mathrm{He}}^4$ by heating the glass to almost melting and, at the same time, touching the glass surface with a Tesla coil. This coil creates a high-frequency discharge which drives absorbed gases off the wall. $T_1$ for pure ${\mathrm{He}}^3$ was found to have the value and the temperature dependence predicted by the modified Bloembergen-Purcell-Pound theory to within ±3%. $T_1$ is proportional to the diffusion coefficient, and increases from 350 sec at 1.0°K to 450 sec at 1.8°K. $T_2$ was found to be approximately equal to $T_1$. These long relaxation times were significantly shortened by contamination of the clean chamber with air or by inadequate purification of the ${\mathrm{He}}^3$ sample. Since the work on pure ${\mathrm{He}}^3$ showed that wall effects could be minimized by cleaning, it was considered of interest to measure $T_1$ in ${\mathrm{He}}^3$ - ${\mathrm{He}}^4$ mixtures. The relaxation times of a 33% ${\mathrm{He}}^3$ solution were determined in an extensively cleaned system. Above the $\lambda$ point, $T_1$ is about 50% greater than for pure ${\mathrm{He}}^3$, while just below this temperature it increases sharply by about 15%. Measurements of $T_1$ of other mixtures were made in partially cleaned systems. The results were affected by wall relaxation and bulk impurities. Some conclusions about $T_1$ are drawn from these data.

The possibility of a perturbation treatment for quantum crystals is investigated. It is found that the usual form of the unperturbed Hamiltonian, being a single-particle operator, is not suitable for this situation, since it does not produce correlations necessary in a theory of quantum crystals. These correlations prevent the unperturbed ground state or low excited states from containing multiply occupied lattice sites. An unperturbed Hamiltonian is proposed which generates these correlations but is no longer a single-particle operator. The rules for a diagrammatic representation of a perturbation expansion starting from this unperturbed Hamiltonian are given. They differ from the usual rules for Goldstone diagrams. In particular, the linked-cluster theorem is not valid. Adding and subtracting pairs of mutually cancelling diagrams, in a manner similar to that in which Pauli-principle-violating diagrams are treated in the standard perturbation theory for fermions, we find new rules which, at least at low temperatures, are identical to those for spinless fermions. This result is essentially independent of the statistics of the actual lattice particles. In the case where the lattice particles are actually fermions with spin ½, e.g., ${}_{}{}^3{}_{}{}^{}\mathrm{He}$, we find an expansion formally identical to the problem of spinless fermions interacting among themselves and with a set of spins ½ localized at lattice sites.

Using the perturbation expansion for quantum crystals proposed in an earlier paper, we investigate here the ladder approximation. A summation of ladder diagrams to all orders is necessary since the interaction, e.g., a Lennard-Jones potential, is singular at short distances. The Bethe-Goldstone equation, describing the motion of two particles in a mean field due to the remaining lattice particles, is derived, and further approximations are discussed, leading to a simplified equation tractable by numerical methods. In this approximation, we calculate the ground-state energy, pressure, and compressibility of the isotopes of helium at various densities in the solid phase. The results are presented and agree quite well with experiments.

The unabbreviated Lamb semiclassical equations for the case of three interacting modes are numerically solved for a variety of laser parameters. Steady-state solutions are obtained for the amplitudes, beat frequencies, and time development of the modes by using the Kutta-Merson method of integration. It is found that for some solutions the relative phase angle $\psi$ becomes constant, and so it is clear under what conditions self-locking is possible. When the modes are unlocked, the value of $\psi$ varies with time even after the steady-state amplitudes have been achieved. Comparison is made with the locking criteria predicted by approaches to the theory where approximations have been made. A physical interpretation is given for the presence or absence of self-locking in terms of competition and of the ratio $\frac{\Delta }{\Delta {\nu }_D}$. Some general rules are established for the behavior of the rise time and delay of the onset of oscillation of the three modes.

Nonlinear coupling of a modulated wave and a low-frequency mode is shown to produce a resonant interaction and instability when the Cherenkov condition $v_p=v_g$ $cos\theta$ is satisfied for the phase velocity $v_p$ of the low-frequency mode, the group velocity $v_g$ of the modulated wave, and the angle $\theta$ between the two wave vectors. This effect stimulates the self-trapping (modulational instability) or self-focusing of the modulated wave. Examples are shown for the cases of couplings between plasma cyclotron waves and magnetohydrodynamic (MHD) modes.

The attenuation and dispersion of sound near the gas-liquid critical point are studied theoretically using the author's extended mode-mode coupling theory. The results differ in the different regions of the sound-wave frequency $f$ expressed in a dimensionless unit and of $\epsilon$, the dimensionless temperature distance from the critical point. The attenuation behaves as $f^2{\epsilon }^{-3\mathrm{}\nu -\frac{\alpha }{2}}$ for $0\le f\ll {\epsilon }^{3\nu }$, and as $f^{2-\frac{2p}{3}}{\epsilon }^{-\frac{3\alpha }{2}}$ for ${\epsilon }^{3\nu }\ll f\ll {\epsilon }^{\nu }$, where $p$ is the exponent which appears in the wave-number ($\overrightarrow{\mathrm{k}}$)-dependent correlation of the order parameter expressed as $A_1{k}^{-2+\eta }+A_2{\epsilon }^{1-\alpha }{k}^{-2+\eta -\rho }$, when $k$ is much greater than the inverse correlation range of critical fluctuations. The relative sound-velocity change with $f$ behaves as $f^{\frac{3}{2}}{\epsilon }^{-\frac{9\nu }{2}}$ for $0\le f\ll {\epsilon }^{3\nu }$, as $f^{1-\frac{2p}{3}}{\epsilon }^{-\alpha }$ if $p\le \frac{3}{2}$, and as $f^0{\epsilon }^0$ if $p>\mathrm{}\frac{3}{2}$ for ${\epsilon }^{3\nu }\ll f\ll {\epsilon }^{\nu +\frac{\alpha }{2}}$. The explicit expressions for the attenuation and dispersion are given for $f\sim {\epsilon }^{3\nu }$.

In a previous paper, a Hartree-Fock theory of the quantum Ursell expansion for Fermi fluids was developed. The same techniques are used in this paper to derive a theory of degenerate Bose fluids from the $x$-ensemble theory of Lee and Yang. It is shown that this theory is equivalent to the $\Lambda$-transformed theory recently derived by Mohling, RamaRao, and Shea. The present approach considerably simplifies the formalism and eliminates the need for the $\Lambda$ transformation. By working entirely in a quasiparticle framework, we also avoid the problems with unphysical line factors which have appeared in earlier theories. Finally, a greatly simplified $\Lambda$ transformation is obtained for use in problems in which the potential contains an infinitely repulsive core.

Superluminous waves propagating transverse to the direction of relative streaming (relativistic or nonrelativistic) of a plasma are investigated in the absence of any external field. It is shown that these waves do not excite any instability, that they are essentially stable.

The method of point transforms, assuming pairwise additivity, as developed by Eger and Gross, is applied to the many-body hard-core Fermi system. The ground-state energy and Landau parameters are found to second order in the hard-core radius. The dilute-gas-limit results are in agreement with those of Huang and Yang and of Abrikosov and Khalatnikov. Methods of extending the calculation to denser systems via the determination of the cutoff function of the transformation are discussed.

The effect of particle correlations in a classical system of charged particles moving in a static uniform background is investigated by means of a dielectric theory developed by Singwi et al. for a completely degenerate electron gas. In this theory, the short-range correlations arising from the Coulomb interaction is taken into account through a local-field correction depending on the pair correlation function. In a first approximation (I), the pair correlation function is chosen to be time independent. The extension of allowing also the pair correlation function to adjust itself to the external field results in a screening of the Coulomb potential entering the local field. The effect of this screening has been investigated for the cases of a static random-phase-approximation (RPA) dielectric function (II) and a static fully self-consistent dielectric function (III). Numerical self-consistent calculations have been carried out for the cases I-III in order to determine the static structure factor $S\left(\overrightarrow{\mathrm{q}}\right)$. From $S\left(\overrightarrow{\mathrm{q}}\right)$ thus obtained, the Helmholtz free energy, the correlation energy, the pair correlation functions, and the isothermal compressibility have been calculated numerically, The plasma dispersion and the compressibility sum rule have also been investigated. The present method can be regarded as a rather natural extension of the RPA. Considerable improvement upon the simple Debye-Hückel (or, equivalently, classical RPA) is also found. The results of the present calculations are also compared with the results of other elaborate methods.

A new formulation of the second-virial-coefficient problem, particularly useful for the computation of the direct second virial coefficient $B_{\mathrm{dir}}$ at high temperatures and as good as the phase-shift formulation at low temperatures, is given. High-temperature asymptotic expansions of $B_{\mathrm{dir}}$ for hard-core and hard-core-plus-square-well potentials are calculated. The exchange second virial coefficient for a hard-core-plus-square-well potential at high temperatures is investigated.

Intensity-correlation measurements, particularly by the two-photon fluorescence (TPF) technique, have become very popular for determining the temporal behavior of laser emission on a picosecond time scale, although they are subject to considerable difficulties of interpretation. The characteristics of the fluorescent intensity in the TPF technique are analyzed theoretically for two different physically interesting and tractable models of imperfectly mode-locked lasers: (i) a partial-locking model where some of the modes are locked and the rest unlocked, and (ii) a "domain model" where the set of oscillating modes is composed of several perfectly locked subsets randomly phased relative to one another. For each case, the dominant features of the spatial dependence of the fluorescence are calculated analytically, using approximations which are valid for realistic cases involving large numbers of oscillating modes. Computer results for the intensity of the laser field and of the fluorescence are also presented for the domain model. It is shown that the partial-locking model is incompatible with experiment, whereas the domain model is in reasonably good agreement with it. Finally, the dynamical basis of the domain model is discussed.

The Glauber approximation has been applied to excitation of the $2s$, $2p$, $3s$, and $3p$ levels of the hydrogen atom by electron impact. The differential and integrated excitation cross sections predicted by the Glauber theory have been compared with experiment and with other calculations. The Glauber approximation is a considerable improvement over the Born approximation at energies < ∼ 100 eV. At energies > ∼ 100 eV, the Glauber total excitation cross sections approach the Born approximation, even though at large scattering angles (>40°) the Glauber differential cross sections may be very different from the Born approximation. At intermediate energies (∼30-100 eV), the Glauber predictions are surprisingly good; at energies < ∼ 20 eV, the Glauber integrated cross sections are smaller than those observed experimentally.

A previously unpublished analysis of nuclear-magnetic-susceptibility data on high-molar-volume bcc solid ${\mathrm{He}}^3$ obtained by Anderson, Reese, and Wheatley shows that nuclear antiferromagnetism is to be expected in the ordered solid at low enough temperatures.

The fine structure of the $2{}_{}{}^3{}_{}{}^{}P$, $3{}_{}{}^3{}_{}{}^{}P$, and $4{}_{}{}^3{}_{}{}^{}P$ states of ${\mathrm{Li}}^{+}$ has been computed, and the results are given in a table.

We note that a divergence predicted for the contribution of capillary waves to the surface-layer thickness disappears if the effect of gravity is included. The resulting finite value for ${\mathrm{He}}^4$ is computed as a function of temperature, and a comparison is made with the static contribution to the thickness. Application to other liquid systems is discussed.

The equation of motion for the electronic polarization of a ground-state alkali atom undergoing spin-exchange collisions is derived. It is shown that the equation of motion for the electronic polarization is unaffected by the inclusion of nuclear spin. A simple interpretation of the spin-exchange frequency shift is also given.

The configuration-interaction (CI) method is employed in an exhaustive investigation of the ground state of Ne. A 231-term CI expansion (formed from a determinantal space of dimension 5343), obtained with an optimized $6s6p5d4f$ basis, gives an energy $E=-128.\mathrm{}8868$ a.u. (Ne). The Ne wave function is compared with those of ${\mathrm{Ne}}^{6+}$ and ${\mathrm{Ne}}^{8+}$. A quantitative assessment of all sources of truncation errors is given. The nonrelativistic energy is estimated to be -128.932±0.008 a.u. (Ne), in agreement with previous estimates obtained by different methods. Three- and four-excited determinants are shown to contribute 1% of the correlation energy. The $L$-shell correlations are shown to be nonadditive. Also, about 8% of the correlation energy is given by configurations containing $g$, $h$, and higher harmonics.

Using the WKB approximation in the spirit of Miller and Good, we have obtained an expression which connects the energy eigenvalues of two parts: One is supposedly known and the other is to be solved. We use the Balmer energy formula of the one-electron bound state as the known part and solve for its relativistic counterpart, the Sommerfeld-Dirac fine-structure formula. An exact form of the relativistic energy-level formula can be obtained through this modified WKB expansion to order ${\hslash }^2$.