We have measured with a drift-tube mass spectrometer the mobilities of C${\mathrm{O}}^{+}$, C${\mathrm{O}}^{+}$·CO, and ${\mathrm{C}}^{+}$ ions in carbon monoxide gas at 300 °K. The measurements were made over a substantial range of $\frac{E}{N}$, where $E$ is the drift-field intensity and $N$ is the gas number density. Zerofield reduced mobilities were determined for C${\mathrm{O}}^{+}$·CO and ${\mathrm{C}}^{+}$ ions. Their values are 1.90 ± 0.03 and 2.7 ± 0.1 ${\mathrm{cm}}^2$/V sec, respectively. Measurements were also made of the longitudinal-diffusion coefficient of C${\mathrm{O}}^{+}$ ions in CO at 300 °K as a function of $\frac{E}{N}$.

The rate coefficient for the ion-molecule reaction C${\mathrm{O}}^{+}$ + 2CO → C${\mathrm{O}}^{+}$ · CO + CO was measured with a drift-tube mass spectrometer at a gas temperature of 300 °K. The range of $\frac{E}{N}$ which was covered extended from 75 to 150 Td. [Here $\frac{E}{N}$ is the ratio of the electrostatic-driftfield intensity to the gas number density and is measured in Townsend (Td), where 1 Td = ${10}^{-17\mathrm{}}$ V ${\mathrm{cm}}^2$.] The technique used to evaluate the rate coefficient was the comparison of detailed "arrival-time" profiles for the product ions with analytical solutions to the differential equation governing the motion of the ions. The measured values of the rate coefficient varied from a high of 1.35 × ${10}^{-28\mathrm{}}$ ${\mathrm{cm}}^6$/sec at an $\frac{E}{N}$ of 75 Td to 1.1 × ${10}^{-28\mathrm{}}$ ${\mathrm{cm}}^6$/sec for values of $\frac{E}{N}$ between 110 and 150 Td.

It has been proposed that the infrared absorption spectra of diatomic molecules in simple liquids should exhibit effects of the collective excitations of liquid. On the other hand, it was suggested that the absorption line shapes for the ${\mathrm{H}}_2$-Ar system are essentially determined by the diffusive motion of the hydrogen impurity. We have made a detailed study of the ${\mathrm{H}}_2$-Ar system and find that both effects must be included to accurately describe the $S_1$ (1) vibrational absorption line shape. Numerical calculations based on a model that does not include diffusive effects yield a line of the right qualitative shape, but that is narrower than the observed line. Extension of the model to include these diffusive effects leads to a line shape in good agreement with experiment.

The energy spectrum of a particle in the presence of random fixed finite-ranged repulsive potentials is investigated at low energy. The resulting asymptotically exact density of states is derived. It exhibits an exponential dependence on energy. This result is shown to be equivalent, for the case of $\delta$-function potentials, to that of Frisch and Lloyd (1960) in the limit of low energy.

We study the thermodynamics of a one-dimensional system of noninteracting bosons in the presence of random fixed finite-range repulsive potentials. It is shown that a Bose—Einstein condensation into the lowest state occurs. The critical temperature increases with the number of impurities. The specific heat exhibits a discontinuity and is infinitely differentiable from both sides of the critical point. For a suitably defined pressure the isothermals exhibit a flat portion below the "transition length."

It is shown that the Frank elastic constants may be expressed in terms of even-order Legendre polynomials averaged over the one-molecule orientational distribution function. In particular, it is found that $\frac{(K_{11}-\overline{K})}{\overline{K}}=C-3\mathrm{}{C}^{\prime }\frac{{\overline{P}}_4}{{\overline{P}}_2}+\cdots ,\frac{(K_{22}-\overline{K})}{\overline{K}}=-2C-C^{\prime }\frac{{\overline{P}}_4}{{\overline{P}}_2}+\cdots ,\frac{(K_{33}-\overline{K})}{\overline{K}}=C+4\mathrm{}{C}^{\prime }\frac{{\overline{P}}_4}{{\overline{P}}_2}+\cdots$, where $\overline{K}=\left(\frac{1}{3}\right)(K_{11}+K_{22}+K_{33})$, $C$ and $C^{\prime }$ are constants, which depend on the details of the system, and ${\overline{P}}_m$ is the weighted average of the $m$th Legendre polynomial. Higher-order terms in these series involve ${\overline{P}}_6$, etc. The constants $C$ and $C^{\prime }$ are calculated for the case of rodlike molecules interacting via a hard-core repulsion. The results are in good agreement with experiments on the substance $p$-azoxyanisole.

Several approximate integral equations for the correlation function $u\left(r\right)\mathrm{}$ are examined for the ground state of liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ described by the Bijl—Dingle—Jastrow type of variational wave function. The radial distribution function computed by Schiff and Verlet using a trial form of $u\left(r\right)\mathrm{}$ is used in the present study to calculate approximate values of $u\left(r\right)\mathrm{}$ from the integral equations, and the resulting values of $u\left(r\right)\mathrm{}$ are then used to obtain corresponding energy expectation values. Accuracy of these equations for $u\left(r\right)\mathrm{}$ is estimated by comparing the obtained values of $u\left(r\right)\mathrm{}$ and $〈H〉$ with the exact values which do not involve any approximation for $u\left(r\right)\mathrm{}$.

Self-focused laser filament tracks in various glasses and polymers were chemically etched to produce conical voids in a manner similar to the etching of charged-particle tracks in these same materials. The etched filament cones were related to the sharpest ${\mathrm{Cf}}^{252}$ fission-fragment cones by roughly the same ratio of cone angle for all the different materials tested. One interpretation of this observation is that the same damage mechanism operative in the charged-particle-track formation, ionization, is also operative in the formation of filament tracks. The sharper cone angle indicates a higher damage intensity for filament formation than for particle-track formation consistent with the theoretical estimates for energy deposition in the two processes. Annealing studies of the filament tracks show that filament-track fading occurs at much higher temperatures than required for the fading of fission tracks for several of the materials. This is qualitatively in agreement with the observation that higher-energy deposition rates from highly ionizing particles result in tracks which require much higher temperatures for track fading than for tracks formed by less-ionizing particles. Etching reveals an important difference of the filament tracks from charged-particle tracks: The surface of the etched cones is not smooth, but varies periodically in diameter along the track (approximately with cylindrical symmetry). Examination of unetched filament tracks with high magnification and reflected light reveals that the filaments contain a rapidly modulated damage component along the length of the track in all the materials tested. The observed periodicity is quite uniform and seems to be characteristic of the material irradiated. The characteristic period length in fused silica, for example, was measured to be 1.5 $\mu$. This small-scale periodicity in solids seems to be similar to the periodicity observed by Brewer and Townes for filaments formed in liquids.

Calculations of gravitational effects on the small-angle x-ray scattering from argon in the critical region have been carried out for conditions simulating recent experiments. The calculations consider densities different from the critical density and thus extend earlier calculations which dealt only with the critical density. The magnitude of gravitational effects on the scattered intensity is evaluated. The influence of gravity on the coexistence curve and on the measured temperature dependence of the isothermal compressibility is considered. Further study has also been made of the result that for scattering with slit collimation in the presence of gravitational effects, the maximum scattered intensity at a given scattering angle is found to occur for temperatures slightly above the critical temperature. This effect is found to be due to a combination of gravitational and collimation effects.

We consider in detail the critical behavior of a $d$-dimensional system that is finite in one of its dimensions. Such a system can exhibit four different types of critical behavior: $d$-dimensional mean field, $d$-dimensional critical, ($d-1\mathrm{}$)-dimensional mean field, ($d-1\mathrm{}$)-dimensional critical. By matching its behavior in the various regions we obtain equalities connecting the critical indices in $d$ and $d-1\mathrm{}$ dimensions. Assuming the usual two-dimensional critical indices of the Ising model, these relations lead to the following indices for the three-dimensional Ising model: $\nu =\frac{2}{3}$, $\alpha =0\mathrm{}$, $\beta =\frac{3}{8}$, and $\gamma =\frac{5}{4}$.

The effect of departures from Ornstein—Zernike correlations on the decay rate of order-parameter fluctuations in fluids near the critical point is considered here in the framework of the Kawasaki and Ferrell theories. For reasonable forms for the correlation function, the effect of departures from Ornstein—Zernike behavior varies from 1 to 11%, which is measurable using currently available light-scattering spectroscopic techniques.

A previously constructed laser model with quantum (noncommuting) noise sources was shown to lead near threshold to a quantum rotating-wave Van der Pol oscillator. A full dynamical correspondence between quantum and classical random processes allows one to compute the average of any time-ordered, normal-ordered operator function by averaging the associated function of classical random variables. Numerical calculations for the associated classical Van der Pol oscillator of the steady-state distribution, the total intensity fluctuations, and the linewidth versus operating point were amply confirmed experimentally. Measurements (and calculations) of higher than two-time correlations were sparse and contradictory. Photocount distributions, at times short compared to the intensity correlation time, confirm only the steady state of the laser. Photocount distributions at intermediate and longer times are difficult to compute because they involve multitime correlations of high ($\infty$) order. By providing an exact solution for photocount distributions and their moments for all times, we expected to stimulate measurements near threshold which would provide an adequate test of the Van der Pol laser model. Comparison of the results reported here with recent photocount experiments of Meltzer, Davis, and Mandel and of Jakeman, Oliver, and Pike provides gratifying agreement and confirmation of our statistical understanding of laser fluctuations near threshold.

Taking the viewpoint that a complete solution of the Liouville equation contains much more information than is necessary in response theory (wherein one is concerned with a particular response), projection techniques are employed to introduce a shortcut in the problem. The treatment supplements and completes the formalism initiated in a previous publication and yields a general equation which is more relevant to response theory in the usual sense. The most natural application is to the theory of transport. The general transport equation is analyzed in the context of electrical conduction. Meaningful similarities to the simple classical Drude model of conduction are exhibited; the relationship of this exact formalism to the Kubo theory of linear response is shown, an exact solution for a step-function stimulus and approximation procedures which go beyond the linear regime are presented, and an expression for the electrical resistivity is explicitly calculated thus making contact with an earlier paper by Kenkre and Dresden. The response equation is also analyzed in the context of electric and magnetic polarization.

Simultaneous measurements of the time dependence of the electron number density and the ion wall current in a helium afterglow are reported. The transition from ambipolar-to-free diffusion is investigated as a function of gas pressure and discharge-tube size. The onset of the transition regime occurs when $\frac{\Lambda }{〈{\lambda }_D〉}=86\mathrm{}$, where $〈{\lambda }_D〉$ is the Debye length corresponding to the average electron or ion density, and $\Lambda$ is the characteristic diffusion length of the vessel. The time dependence of the transition regime was found to depend only on $\frac{\Lambda }{〈{\lambda }_D〉}$ if times are scaled as $\frac{t{D}_{+}}{{\Lambda }^2}$. This latter result is in agreement with high-pressure theories which assume mean free paths short with respect to experimental dimensions. The ion current changed by a factor of (1.2 ± 0.5) × ${10}^5$ during the transition. The ions diffuse free by space-charge effects when $\frac{\Lambda }{〈{\lambda }_D^i〉}<~0.25$, where $〈{\lambda }_D^i〉$ is the Debye length corresponding to the average ion density.

Accurate measurements are presented of the thickness of helium films on cleaved surfaces of alkaline-earth fluoride crystals at 1.38 ${}_{}{}^{\mathrm{o}}{}_{}{}^{}\mathrm{K}$. The films were measured between 10 and 250 Å using an acoustic interferometry technique. The data are in excellent agreement with calculations based on the Lifshitz theory of van der Waals forces. Calculations of the film thickness on a variety of other substrates are also given.

It was suggested by Gusinow et al. that the triatomic helium ion $\mathrm{He}_3^{}{}_{}{}^{+}$ could account for a large portion of the total electronic recombination in 300°K helium plasmas dominated by molecular ions. Their conclusion was based on some reported values of the electronic-recombination coefficient $\alpha$ which are now known to be incorrect. Based on some recent measurements of $\alpha$ and some experimental results reported here, we show that $\mathrm{He}_3^{}{}_{}{}^{+}$ ions can evidently account for only a small fraction of the net recombination loss of electrons at gas pressures less than 50 Torr and temperatures near 300°K.

Recent measurements on the pressure dependence of the sound velocity $c\left(P\right)\mathrm{}$ in liquid ${\mathrm{He}}^3$ and liquid ${\mathrm{He}}^4$ are employed for deriving the ground-state energy of these liquids as power series in the parameter $\frac{(\rho -{\rho }_0)}{{\rho }_0}$, where $\rho$ is the density of the liquid while ${\rho }_0$ is the value of $\rho$ at saturated vapor pressure. Results thus obtained are compared with the corresponding ones derived theoretically. The existence of an almost linear relationship between the variables $c$ and $\rho$, as observed by Aziz et al. on one hand and by Abraham et al. on the other, is discussed.

Necessary and sufficient conditions are derived for the validity of the bound, $\mathrm{tr}\rho H>~E_g\left(N_0\right)$, recently assumed by Abdulnur et al. in their paper [Phys. Rev. A 6, 889 (1972)]. Here $\rho$ runs over all density matrices for which $\mathrm{tr}\rho N=N_0$, the average number of electrons. $H$ is a Hamiltonian commuting with the operator for the number of particles $N$, with ground-state energy $E_g\left(N_0\right)$ when the number of particles is $N_0$. An example is given showing that the above inequality is not true for every pair of commuting Hermitian operators $H$ and $N$; thus this inequality differs in a fundamental way from the familiar energy variational principle. The proof of the inequality requires the property of $H$ that the first ionization potential is greater than the electron affinity, essentially justifying the application by Abdulnur et al. The failure of the inequality implies that in the grand canonical ensemble there are nonzero fluctuations in $N$ at zero temperature, with zero probability of finding $N=N_0$.

The frequency shift of the stimulated line of the Rayleigh wing (SRL) has been found to be highly dependent upon the focal position of the exciting laser light within the scattering sample, unexplained by existing theory. Light from a giant-pulse ruby laser, when focused deeply into a liquid composed of optically anisotropic molecules, stimulates a scattering whose frequency shift is in agreement with previously measured values. However, when the light is focused less than 5 cm from the entrance window of the scattering cell, the frequency shift increases dramatically with decreasing distance and deviates from the expected value by as much as a factor of 4. Various physical parameters which may affect the frequency shift are investigated. The SRL frequency shift in nitrobenzene is found to be independent of the incident laser power, the focal length of the focusing lens, and the path length in the liquid beyond the focus. The SRL dependence upon focal position is also observed in 1-chloronaphthalene.

The ion-acoustic instability of a uniform current-free two-temperature plasma requires heat conduction by electrons in addition to electron-ion energy transfer.