We have calculated the contribution of the triple-dipole three-body interactions to the ground-state energy of liquid ${\mathrm{He}}^4$ and solid He. The liquid calculation was done by a Monte Carlo method using a variationally determined product wave function, and the solid calculation by evaluating the triple-dipole sum in the static lattice for several crystal lattices. The triple-dipole term is found to be quite small, of the order of 2% of the ground-state energy of liquid ${\mathrm{He}}^4$.

All of the published experimental values of longitudinal and transverse diffusion coefficients of gaseous ions in electric fields, except those cases dominated by resonant charge transfer, are compared with the predictions of Wannier's theory. The products of the measured coefficients and the gas number density appear to vary as $v_d^{{}^3}{\left(\frac{E}{N}\right)}^{-1\mathrm{}}$, where $v_d$ is the ionic drift velocity, in agreement with Wannier's predictions. Here $\frac{E}{N}$ is the ratio of the electric drift field intensity to the gas number density. In addition, the absolute agreement between experiment and theory is quite satisfactory.

Spontaneous emission by a "two-level atom" interacting with a highly excited field mode is described using quantum electrodynamics without time-dependent perturbation theory. It is found that both the line shape and the Lamb shift are affected by the "applied field." Detailed solutions are presented as a function of the population of the "applied-field mode" and as a function of the detuning.

Results are presented on the propagation of excitations generated by a small (∼ 1 mm square) pulsed heater immersed in a large (∼ 300 cc) sample of He II maintained at temperatures below 300 mK. Small carbon-film detectors placed at 1, 2, 3, and 4 cm from the heater are used to measure the flux of radiation over a wide range of heater-power densities ($W_H$). The observed signals are free from wall reflections. At the lowest temperatures and small $W_H$, the fastest excitations propagate without dispersion or attenuation at a velocity of 234 ± 4 m/sec. Spatial attenuation which is observed at higher temperatures corresponds to effective mean free paths for large-angle scattering of 1, 2, and 3 cm at 306 ± 6, 272 ± 6, and 254 ± 8 mK, respectively. When $W_H$ exceeds 0.4 W/${\mathrm{cm}}^2$, signal shapes reflecting appreciable interactions between radiated excitations are observed. For $W_H>2.9\mathrm{}$ W/${\mathrm{cm}}^2$, the observed signal acquires another component which propagates without appreciable dispersion at a velocity of 200 ± 10 m/sec. In these experiments, no evidence has been found of any excitations associated with any upward bend in the He II phonon-dispersion curve resulting in signal velocities in excess of the first-sound velocity.

Measurements of the spin-diffusion coefficient in dilute ${\mathrm{He}}^3$-He II mixtures are presented. The thermodynamic parameters of the samples are varied within the following limits: $0.3\le T\le 1.15$ °K, $\mathrm{saturated}\mathrm{vapor}\mathrm{pressure}\le P\le 24$ atm, $0.322\times {10}^{-4\mathrm{}}\le n_3\le 4.54\times {10}^{-4\mathrm{}}$ mole/${\mathrm{cm}}^3$. It is shown that all samples considered are within the limits of the semiclassical regime. The data are analyzed by considering the reciprocal of the diffusion to be the sum of two parts: $\frac{1}{D}(T,P,n_3)=\frac{1}{D_{3\rho }}\mathrm{}(T,P)+\frac{1}{D_{33}}(T,P,n_3)$, where $D(T,P,n_3)$ is an experimental measurement at ($T,P,n_3$), $D_{3\rho }\mathrm{}(T,P)\mathrm{}$ is the $n_3$-independent spin diffusion of a ${\mathrm{He}}^3$ quasiparticle as limited by interactions with rotons, and $D_{33}(T,P,n_3)$ is the spin diffusion of a ${\mathrm{He}}^3$ quasiparticle limited by interactions with other ${\mathrm{He}}^3$ quasiparticles. Since $D_{3\rho }$ is independent of $n_3$, and $D_{33}$ is shown to be proportional to $n_3^{-1\mathrm{}}$, a least-squares fit of the data to the above equation gives the experimentally determined values of $D_{3\rho }$ and $D_{33}$. A comparison of $D_{33}$ with the theoretical expression gives directly the moment of the effective ${\mathrm{He}}^3$-${\mathrm{He}}^3$ interaction measured by spin diffusion in the semiclassical regime: $〈V\left(k\right)\mathrm{}〉{|}_{T,P}=A\int e^{-ak2\mathrm{}}{k}^3{\left|V\right(\mathrm{bk}\left)\mathrm{}\right|}^2\mathrm{dk}$, where $A$, $a$, and $b$ are constants. The measurements are insufficient to determine $V\left(k\right)\mathrm{}$ uniquely. Calculations using an assumed interaction (which reduces to the interaction used by Ebner at saturated vapor pressure) show a remarkable similarity to the experimental data.

As a model of a helium monolayer a system of hard-core bosons of mass $m$ and diameter $a$ constrained to motion in two dimensions is considered at absolute zero. In the low-density limit, the ground-state energy per particle and condensate depletion are found to be $\frac{E}{N}=-\frac{2\pi {\hslash }^{2}n}{m\ln n{a}^2}$ and $n_0=n(1+\frac{1}{\ln n{a}^2})$, where $n$ is the areal density of the system. The expansion parameter $-\frac{1}{\ln n{a}^2}$ is approximately equal to unity for real helium monolayers. The variation of the above results with temperature is discussed for a system of finite size.

Differential neutron scattering cross sections for liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ have been obtained for momentum transfers along the free-particle excitation curve up to 20.3 ${\mathrm{\AA }}^{-1\mathrm{}}$. Momentum transfers in this range are much higher than in previously reported work. The measured recoil energies as well as the angular dependence of the scattering confirm that the neutron scattering, at these high values of $\kappa$, is sensitive to the motions of single helium atoms. A temperature dependence in the widths of the cross-section peaks, as well as shape changes in these peaks as the temperature is decreased below $T_{\lambda }$ is consistent with the idea that a narrow condensate component contributes to the scattering for $T<\mathrm{}{T}_{\lambda }$. Results from a detailed comparison of the measured cross sections with a theory for high-energy neutron scattering are reported. The kinetic energy per liquid-helium atom is deduced from this theory-experiment comparison and is found to be in agreement with computations of the kinetic energy based on thermodynamic phase equilibria considerations. A fractional occupation of the zero-momentum condensate state of (8.8 ± 1.3)% at 1.27 °K is also deduced from this theory-experiment comparison.

Measurements are reported on low-frequency self-excited oscillations in an inhomogeneous afterglow plasma, in which there is a radial density gradient perpendicular to an axial homogeneous magnetic field. The plasma was produced either by using a short pulse of electrons emitted from a hot cathode to ionize the background gas, or by pulsing off an rf discharge. The plasma so formed had an initial density of ∼ ${10}^{10}$ ion ${\mathrm{cm}}^{-3\mathrm{}}$ and a temperature $T_e\sim 1000$ °K. The experiment was carried out using tubes of 2.5 and 5.0 cm diam, giving inverse density scale lengths for the radial density gradient of ∼ 2.3 and ∼ 1.3 ${\mathrm{cm}}^{-1\mathrm{}}$, respectively. The oscillations occurred mainly as $m=+1\mathrm{}$ azimuthal modes although some results for $m=+2\mathrm{}$ were obtained. The oscillations were shown to be standing waves in the axial direction $z$ with a wavelength ${\lambda }_z$ approximately equal to the column length. It was found that the frequency of oscillation $\omega$ was dependent on the column length and by varying this, a dispersion relationship $\omega$ vs $k_z=\left(\frac{2\pi }{{\lambda }_z}\right)$ could be plotted under various conditions. It was found that the frequency was independent of time (or density) in the afterglow and that the frequency of oscillation tended to zero as $k_z\to 0$ and to ${\omega }^{*}$ as $k_z$ assumed larger values. The simple theory of drift dissipative instabilities has been extended to include effects due to electron-neutral and ion-neutral collisions and the $\mathrm{Re}\left(\omega \right)$, and growth rates $\mathrm{Im}\left(\omega \right)$, vs $k_z$ have been calculated for the relevant experimental cases. Comparison of theory and experiment shows remarkably good agreement considering the possible errors in some of the experimental quantities and the assumptions present in the theory.

A general theory is presented for sound attenuation and dispersion near the critical point in the region of frequencies which are much greater than the characteristic frequency of order-parameter dynamics, but are much smaller than the frequency of sound with the wavelength equal to the range of correlation of local order-parameter fluctuations. The method used is the most general version of the mode-mode coupling theory plus the static and dynamic scaling hypotheses and some general properties of time-correlation functions. If we ignore the small critical exponent $\alpha$ associated with the heat-capacity singularity, we find that the attenuation behaves as $f{\epsilon }^0$ and the dispersion (relative sound-velocity change with frequency) as $f^0{\epsilon }^0$, where $f$ is the dimensionless frequency and $\epsilon$ the dimensionless temperature distance from the critical point.

It is proven that for Bose-Einstein condensation, in the sense of quasiaverages, condensation into any one-particle state is forbidden at nonzero temperatures in any two-dimensional system, whether under rotation or not, regardless of the external potential, so long as the particle density is bounded everywhere. The proof is based on Bogolyubov's inequality.

The transient statistical properties of a single-mode laser field are investigated both experimentally and theoretically. The physical system is a $Q$-switched He-Ne gas laser oscillating on the 6328-Å transition. The measurements are performed by photon-counting techniques, using the so-called linear method which enables resolving times to be in the nsec region. The effects of the photomultiplier and of the attenuators on the statistics are taken into account. The measurements are interpreted in terms of the available laser theories as well as in terms of a simple phenomenological model which attributes the statistical spread to the initial photon distribution, neglecting the stochastic forces in the course of the buildup. Finally, some of the approximations of the theories are discussed, and the sensitivity of a transient measurement is compared with that of a stationary-intensity-correlation measurement.

An $n$-dimensional linear Markov process with parameters ${\alpha }_i(i=1,\dots ,n)$ and $\dot{\alpha }=-\mathrm{M}\mathit{\alpha }$ is considered. Criteria for stationarity of the process and spectral properties of the fluctuations around the stationary state $\mathit{\alpha }=0$ are derived. When the stationary state is a thermodynamic equilibrium state, the spectra are proven to be monotonic functions of the frequency. Positive-definiteness of the matrix ${\mathrm{M}}^3〈\mathit{\alpha }{\mathit{\alpha }}^T〉$ turns out to be the necessary and sufficient condition for the absence of local maxima in the spectra. A process with ${\mathrm{M}}^3〈\mathit{\alpha }{\mathit{\alpha }}^T〉$ positive-definite also has the property that ${\mathrm{M}}^2〈\mathit{\alpha }{\mathit{\alpha }}^T〉$ is positive-definite, which guarantees the absence of external driving forces, and corresponds to a pure relaxation process. $\mathrm{M}〈\mathit{\alpha }{\mathit{\alpha }}^T〉$ positive-definite is a necessary condition for stationarity. Moreover, it is a necessary and sufficient condition for the existence of spectra and transport coefficients. Positive-definiteness of $\mathrm{M}〈\mathit{\alpha }{\mathit{\alpha }}^T〉$, ${\mathrm{M}}^2〈\mathit{\alpha }{\mathit{\alpha }}^{\mathrm{T}}〉$, and ${\mathrm{M}}^3〈\mathit{\alpha }{\mathit{\alpha }}^{\mathrm{T}}〉$ is linked to properties of the excess-entropy production.

The interaction of two noble-gas atoms on a solid surface is studied with a view to finding conditions under which they form a bound state. Two limiting models for the surface are considered - a flat surface which constrains the atoms to move in two dimensions and imposes no further restriction on their motion, and a surface consisting of a periodic arrangement of potential wells separated by large barriers, to which the tight-binding method is applicable. A curious feature of the result for the binding energy is shown to be a consequence of the fact that the problem is two dimensional.

Nonlinear effects involving x-ray and optical photons are described with particular emphasis on the generation of sum and difference frequencies. Efficiencies for sum and difference frequency generation are calculated and found to be large enough to be observable. The expected advent of x-ray lasers should enhance the usefulness of such mixing techniques in the measurement of excited-state wave functions. Under favorable circumstances, the mixing technique may provide a means of efficiently tuning x-ray laser outputs.

For a special model, it is possible to compute the self-consistent confinement and penetration of an energetic plasma by an rf field of large amplitude for arbitrary values of the quantity $\left(\frac{v}{c}\right) \left(\frac{{\omega }_P}{\omega }\right)$, where $v$ is a characteristic electron thermal speed, ${\omega }_P$ is the plasma frequency deep in the plasma, and $\omega (<\mathrm{}{\omega }_P)$ is the rf frequency. Previous theories concerning nonlinear behavior of electromagnetic waves in plasmas have required this quantity to be small compared to unity.

Pulse-time-of-flight measurements have been used to determine the velocities of 10-MHz transverse and longitudinal sound in the hcp phase of solid helium-four at a molar volume of approximately 21 ${\mathrm{cm}}^3$ per mole. The measurements have been made on single crystals grown from superfluid helium. The orientations of these crystals have been determined by an optical birefringence technique. The five elastic constants for this molar volume of hcp solid helium have been calculated using the data obtained in these experiments.

A kinetic theory has been proposed by several authors with the goal of eliminating the divergences which appear in the density expansion in nonequilibrium systems. Here, it is shown that for a two-dimensional simple gas the theory presents a new divergence, resulting from the fact that correlations propagate over long distances as a result of hydrodynamic transport. This divergence is discussed explicitly for a gas model: the Maxwell model. It will be indicated why the kinetic theory for a perfect Lorentz gas does not exhibit this new divergence.

It is shown that in using the two-frequency technique of Prior et al. for investigation of the hyperfine structure of atoms with integral $J$, any resonance observed may arise from a two-quantum transition (${\nu }_{\alpha }+{\nu }_{\beta }=\frac{\Delta E}{h};{\nu }_{\alpha }\ne {\nu }_{\beta }$) rather than from two successive single-quantum jumps. Although this feature requires that care be used in interpreting the observed resonance frequencies, it appears to extend the usefulness of the technique as a spectroscopic tool.

A recent statement that significant differences in the structure of a vortex line appear between the simple Hartree calculation and the variational Jastrow approach is shown to have no basis in reality.