Detailed studies of the time dependence of the spectral emission and that of the number density of ions have been made during the decay period of plasmas produced in helium containing various concentrations of neon atoms. The occurrence of the recombination of ${\mathrm{Ne}}^{+}$ via the collisional recombination process with a recombination coefficient proportional to the electron density is established for the first time. The studies confirm the occurrence of recombination processes previously reported for $\mathrm{He}_2^{}{}_{}{}^{+}$ and $\mathrm{Ne}_2^{}{}_{}{}^{+}$. Evidence has also been found for the population of the $4d$ levels of neon by $\mathrm{He}\left(2\mathrm{}{}_{}{}^1{}_{}{}^{}S\right)$ metastable atoms. New values for the rate constants of the following processes were obtained: ${\mathrm{Ne}}^{+}+2\mathrm{}\mathrm{He}\to {\left(\mathrm{HeNe}\right)}^{+}+\mathrm{He}$ (2.1 × ${10}^{-32\mathrm{}}$ ${\mathrm{cm}}^6$ ${\sec }^{-1\mathrm{}}$); ${\mathrm{Ne}}^{+}+\mathrm{Ne}+\mathrm{He}\to {\left(\mathrm{HeNe}\right)}^{+}+\mathrm{Ne} \mathrm{and} \to {{\mathrm{Ne}}_2}^{+}+\mathrm{He}$ (3.0 × ${10}^{-31\mathrm{}}$ ${\mathrm{cm}}^6$ ${\sec }^{-1\mathrm{}}$); ${\left(\mathrm{HeNe}\right)}^{+}+\mathrm{Ne}\to {{\mathrm{Ne}}_2}^{+}+\mathrm{He}$ (3 × ${10}^{-11\mathrm{}}$ ${\mathrm{cm}}^3$ ${\sec }^{-1\mathrm{}}$) — all at a gas temperature of 300 °K.

Using high-resolution semiconductor detectors, a study was made of the large $K$-ionization cross sections of atoms over a wide range of atomic numbers ($Z=29-83\mathrm{}$) and of incident electron energies ($E=150-900\mathrm{}$ MeV). For the higher-$Z$ elements ($Z>\mathrm{}60\mathrm{}$), the $L$ x-ray cross sections were also measured over the incident electron energy range. Ratios of the relative intensities of various distinct lines were determined and compared to recent low-energy work. The results of the $K$-ionization experiment are compared to the recent relativistic calculations of Kolbenstvedt. The experimental $K$-ionization cross sections for each incident electron energy appear to fit a power-law dependence on the atomic number. The Kolbenstvedt theory also exhibits a power-law dependence and is in good agreement with this experiment. All differential x-ray cross sections were found to be isotropic. The $L$ x-ray cross sections measured for four high-$Z$ elements increase with atomic number and incident electron energy. The $\frac{L\beta }{L\alpha }$, $\frac{L\gamma }{L\alpha }$, and $\frac{K}{L}$ intensity ratios are also given. A search for nuclear $\gamma$ rays produced by high-energy electron scattering was also undertaken in this work. No nuclear $\gamma$-ray lines were detected. Recent theoretical calculations show that the expected cross sections are below the minimum observable levels in these experiments. However, a 511-keV $\gamma$ line was observed. This line was associated with positron annihilation and was apparently produced in the target.

The influence of molecular structure on the bremsstrahlung process at low energies has been studied for diatomic molecules. Interference effects give rise to an increase in the total radiation cross section which is large for low electron energies. Although the various cross sections are calculated in Born approximation, the results are expected to be qualitatively correct.

The convergence of the separable-expansion method for solving the Faddeev equations is investigated for systems with pure Coulomb interactions between particles in each pair. Two alternative separable expansions for the off-shell two-body amplitudes are utilized. It is observed that the expansion method is capable of predicting qualitatively all the characteristic features such as the three-body bound-state and resonance poles, the scattering length, and the energy dependence of the phase shift in atomic systems such as the ($e$, H) system. Although some convergence behavior is observed, this method in the present form does not provide the desired accuracy when reasonable numbers of terms are included in the expansion for the attractive ($e-p$) scattering amplitudes. This behavior is interpreted as due to the failure of the two alternative expansions for the off-shell two-body amplitudes to converge monatonically.

Precise thermodynamic measurements have been made to determine the thermal conductivity of C${\mathrm{O}}_2$ near its critical density using a horizontal parallel-plate apparatus. Tests were conducted for $0.89<T-T_c<5.48\mathrm{}$ °C. The exponent for the thermal-conductivity divergence on the critical isochore was found to be 0.67 ± 0.02, in good agreement with recent predictions from theory and from light-scattering experiments.

The liquid-gas critical line in binary mixtures is discussed in terms of a theory based on the assumption that the phase transition may be described in terms of a single ordering field and an effective critical temperature. The theory, an application of a theory of Fisher to a case not considered by him, makes a number of predictions, among them that the specific heat at constant pressure and the isothermal compressibility, both at constant concentration, are weakly singular at the critical line. Azeotropic critical points, as well as points where either the critical pressure or critical temperature shows an extremum, are special cases in which the transition differs from that at a general point on the critical line. The theory is consistent with the few available experimental results.

A complete set of time-independent orthogonal phase functions $\left\{{\Psi }_s\right\}$, $s=0,1,2,\dots$, is generated via the Schmidt process and used to represent the Fourier coefficient $R_{\overrightarrow{\mathrm{k}}}\mathrm{}\left(t\right)\mathrm{}$ of the time-dependent microscopic density function. The projection of $R_{\overrightarrow{\mathrm{k}}}\mathrm{}\left(t\right)\mathrm{}$ on ${\Psi }_0$ is essentially the density autocorrelation function. The equation of motion of the coefficients of this expansion is found and formally solved to yield the Laplace-Fourier transform of the density autocorrelation function as a ratio of infinite determinants, closely related to Mori's continued-fraction expansion. A non-Markovian memory function is then readily defined in the same terms. These exact results are illustrated by explicit calculations for the ideal gas. Finally, a perturbation expansion of the memory function is developed, leading to practical approximations.

Time resolved spectra have been taken of a pulsed arc operating in 2 Torr of helium, C${\mathrm{O}}_2$, air, or argon. Wall confinement aided by close-fitting return conductors produce a 3.5-eV arc plasma in local thermodynamic equilibirum (LTE) during the first 10 μsec. The electron density is ${10}^{17}$ ${\mathrm{cm}}^{-3\mathrm{}}$. Thereafter, the plasma cools and the silicon impurity rises. The arc temperature is limited by radiation losses. LTE is found among eight species, as expected at these values of $N_e$; but neutral helium is found to be about 1.5 eV cooler than the rest of the plasma. Such a result could cause large temperature errors in plasma diagnostics using He II/I line ratios.

The behavior of a quantum-mechanical harmonic oscillator under a random perturbation of the form $f\left(t\right)\mathrm{}{q}^2$ is discussed. Equations are obtained governing the time evolution of the Wigner function and the weight function for the $P$ representation. The latter function satisfies a fourth-order differential equation, in contrast to the simpler second-order equation obeyed by the Wigner function.

We report the results of experiments done attempting to observe the periodic ion-mobility discontinuities in He II reported by Careri et al. and others. Using a less ambiguous technique, we obtain results that indicate the absence of these discontinuities over the temperature range of the original experiments. Schwarz reported results similar to ours at a single temperature.

The time dependence of the order-parameter magnitude in superfluid helium is examined, and a simple conservation law is proposed for the hydrodynamic regime. The conservation law for the square of the order-parameter magnitude $\Psi$ has the form $\frac{\partial \Psi }{\partial t}=-\overrightarrow{\nabla }·[n_0{\overrightarrow{\mathrm{v}}}_s+a_1({\overrightarrow{\mathrm{v}}}_n-{\overrightarrow{\mathrm{v}}}_s)-a_2(\frac{\kappa }{T\sigma }\left)\overrightarrow{\mathrm{\nabla }}T\right]$, where $a_1$ and $a_2$ depend on thermodynamic derivatives of the equilibrium condensate density $n_0$. We find that $\Psi$ cannot be determined entirely from the local temperature and pressure except for infinitely slow disturbances, and the latter condition of "static local equilibrium" is used to develop the conservation law. Specifically, the existence of some conservation law $\dot{\Psi }=-\overrightarrow{\nabla }·\overrightarrow{\Lambda }$ together with static local equilibrium and a hydrodynamic structure for the current $\overrightarrow{\Lambda }$, are necessary assumptions in this development.

Quantum-mechanical phase coherence in He II produces a coupling between elements of the superfluid that places restrictions upon the flow. This phase coupling was utilized in the flow of superfluid through wide (> ${10}^{-3\mathrm{}}$ cm) parallel channels to detect the first sign of flow impedance. A sensitive pressure-measuring device located in one of the channels was used to monitor the velocity and detect the superfluid critical velocity. Isothermal, as well as heat current flow, was investigated in the temperature range 1.71-2.15°K. The results substantiate the presence of phase coupling and indicate that a superfluid critical velocity exists, which agrees with the concept of vortex generation and is not influenced by normal fluid flow. Channel-size dependence was found to be approximately $d^{-1\mathrm{}}$ with only a small temperature dependence. At velocities well above critical, flow transitions were observed which indicated the onset of superfluid turbulence. The values of these turbulent critical velocities compare favorably with past data which give a channel-size dependence of $d^{-\frac{1}{4}}$.

This paper examines the "runaway solutions" (exponentially diverging solutions) which often arise in slow-motion approximations such as classical electron theory. Detailed study of a typical runaway shows that it is a spurious solution attributable to faulty approximation technique rather than a physical result connected with, for example, the point-electron assumption. Methods for consistently using the slow-motion approximation while avoiding these spurious solutions are given.

Precision heat-capacity measurements at constant volumes for hcp ${\mathrm{He}}^4$ are presented for molar volumes between 13.7 and 20.8 ${\mathrm{cm}}^3$ and for temperatures between 1.3 °K and the melting temperature. These data are used to calculate the deviations of the equation of state for hcp ${\mathrm{He}}^4$ from the Grüneisen equation of state. It is found that the reduced Debye temperature $\frac{\Theta }{{\Theta }_0}$ is not a volume-independent function of the reduced temperature $\frac{T}{{\Theta }_0}$ and that the Grüneisen parameter $\gamma$ is both volume and temperature dependent. It is observed that $\gamma$ and $\frac{\Theta }{{\Theta }_0}$ at a given volume and $\frac{T}{{\Theta }_0}$ are the same for hcp ${\mathrm{He}}^4$ and hcp ${\mathrm{He}}^3$, and it is suggested that the observed volume and temperature dependence of $\gamma$ and $\frac{\Theta }{{\Theta }_0}$ are typical for close-packed van der Waals solids in general. The temperature-dependent contributions to other thermodynamic functions are given as well.

A naive collective coordinate analysis leads to no damping of the acoustic modes of a classical fluid. A similar analysis of the linearized Vlasov equation leads to the well-known phenomenon of Landau damping. The qualitative form of Landau damping is, however, inaccurate for nonsingular forces and probably for low-$k$ propagation in plasmas. Comparison of the two approaches shows that retention of the velocity variable in the Vlasov equation allows for an accurate description of positional fluctuations due to velocity dispersion by itself, but a poor assessment of collective forces. If, instead, the phase mixing of collective modes due to velocity dispersion is taken into account, a damping mechanism is introduced with the anticipated hydrodynamic form in fluids at low $k$, and with a $k^4$ dependence for plasmas.

When a chemical potential difference exists between the superfluid helium on two sides of an orifice, superfluid flow takes place at a critical velocity accompanied by the creation of quantized vortices. The frequency with which vortices are created is expected to be equal to the chemical potential difference (the analog of the ac Josephson frequency in superconducting tunneling). An experiment is described in which this frequency condition is verified by a method analogous to that used in superconductors. Experimental conditions were selected such that the gravitational head difference $\mathrm{mgZ}$ was the dominant contribution to the chemical potential difference. $Z$ was measured by a capacitance technique. An ultrasonic transducer was used to modulate the flow through the orifice at the frequency $\omega$ so that the rate of creation of vortices was synchronized with the modulation frequency. The system was observed to exhibit dynamic stability at values of head difference $Z=\frac{n\hslash \omega }{n^{\prime }\mathrm{mg}}$ corresponding to $n$ quantized vortices created every $n^{\prime }$ cycles of the modulation. The correspondence between theory and experiment is most convincing under experimental conditions such that the strength of the stability decreases rapidly with increasing values of the integer $n^{\prime }$. This was found to occur with small orifices and with moderate modulation amplitude, in agreement with the results of the analogous experiments in superconductors.

The description of the statistical dynamics of quantum oscillators is formulated in terms of the Wigner distribution, analogous to the more commonly used $P$ distribution, with explicit formulas being obtained for its time evolution and for average values. This formulation is desirable because, e.g., the Wigner distribution always exists whereas the $P$ distribution does not. The formalism is applied to the process of parametric amplification in a single mode, which may be considered as the degenerate form of the well-known two-mode case. This degeneracy gives rise to significantly different properties; for example, the $P$ distribution for the single mode of interest evolves from a circularly symmetric two-dimensional Gaussian into an elliptically symmetric form and ceases to exist after a finite time, even for amplification in the presence of losses. This is contrary to the two-mode case. The corresponding Wigner distribution is found to exist as a well-behaved function for all time as expected, regardless of the amount of losses, and is used to calculate average values of various quantities of interest. It is found, e.g., that in the lossless case the average number of photons in the signal mode always becomes infinite as $t\to \infty$. This is in contrast to the corresponding classical result for the lossless case which allows the signal to decay rather than to grow with time, depending on the relative phase between the signal and the pump. Field fluctuations are discussed and found to have some unusual properties. The combination of frequency up-conversion with single-mode amplification is also described briefly. The effect of the quantization of the pump oscillator is considered in an Appendix.

A consistent set of hydrodynamic equations for liquid crystals is derived from the necessary conservation laws and the requirements of Galilean invariance. In the stationary case, the equations reduce to the Oseen-Frank hydrostatic theory. The equations should be useful in discussing the hydrodynamics of cholesteric and smectic crystals. Linear dissipative effects are also considered.

We present measurements of the equation of state of ${\mathrm{He}}^3$ with an impurity of 250 ppm ${\mathrm{He}}^4$ near the critical point. In these experiments the density is measured by a dielectric-constant technique. The density cell is a horizontal capacitor to reduce the gravitational effects. The temperature gradient between the cell and 4 °K occurs along a horizontal capillary. The results are analyzed in terms of the critical exponents which are found to have values similar to those of other simple fluids. From the coexistence curve and the compressibility on the critical isochore above $T_c$, we get $\beta =0.\mathrm{}361$, $B=1.\mathrm{}31$, $\gamma =1.\mathrm{}18$, $\Gamma =2.\mathrm{}48\times {10}^{-4\mathrm{}}$ ${\mathrm{Torr}}^{-1\mathrm{}}$, $T_c=3.\mathrm{}3105$ °K, ${\rho }_c=0.04145\mathrm{}$ g/${\mathrm{cm}}^3$, and $P_c=860.5\mathrm{}$ Torr. From the compressibility along the coexistence curve, we obtain ${\gamma }^{\prime }=1.12\mathrm{}$ and ${\Gamma }^{\prime }=6.9\mathrm{}\times {10}^{-5\mathrm{}}$ ${\mathrm{Torr}}^{-1\mathrm{}}$ for both the liquid and the vapor sides. An analysis of the critical isotherm using the conjugate variables $\mu$ and $\rho$ gives $\delta =4.21\mathrm{}\pm 0.10$ and $\nabla =2.77\mathrm{}$. The critical isobar gives the exponents ${\pi }_{+}(\rho >{\rho }_c)=4.21\mathrm{}\pm 0.10$ and ${\pi }_{-}(\rho <{\rho }_c)=4.35\mathrm{}\pm 0.10$. The study of the critical isochore shows $\frac{d^{2}P}{d{T}^2}$ to be always positive. Combination of these data with specific-heat results by Moldover indicate no discontinuity in $\frac{d^2\mu }{d{T}^2}$ within experimental error. We find at the critical point ${\left(P_c{V}_c\right)}^{-1\mathrm{}} \frac{d^2\mu }{d{T}^2}=-0.22\mathrm{}$ ${\left(\mathrm{°}\mathrm{K}\right)}^{-2\mathrm{}}$. The isotherms expressed by the conjugate variables $\mu$, $\rho$ are fitted to the scaling equation of state proposed by Missoni et al. A good fit was obtained with the following parameters: $E_1=2.53\mathrm{}$, $E_2=0.44\mathrm{}$, $\delta =4.23\mathrm{}$, $x_0=0.475\mathrm{}$, and $\gamma =1.17\mathrm{}$. Results of a less extensive study of ${\mathrm{He}}^3$ with an impurity of 10 ppm are presented. They give critical indices and $P_c$, $T_c$, and ${\rho }_c$ in substantial agreement with the less pure ${\mathrm{He}}^3$. A comparison of the present data with previous work is presented.

Virtually all measurable properties of a classical fluid may be determined from the expectation value of the phase-space density operator $f\left(\overrightarrow{\mathrm{r}}\overrightarrow{\mathrm{p}}t\right)=\Sigma {\alpha }^{}\delta (\overrightarrow{\mathrm{r}}-{\overrightarrow{\mathrm{r}}}^{\alpha }\mathrm{}(t\left)\mathrm{}\right) \delta (\overrightarrow{\mathrm{p}}-{\overrightarrow{\mathrm{p}}}^{\alpha }\mathrm{}(t\left)\mathrm{}\right)$, and the phase-space density correlation function $〈f\left(\overrightarrow{\mathrm{r}}\overrightarrow{\mathrm{p}}t\right)f\left({\overrightarrow{\mathrm{r}}}^{\prime }{\overrightarrow{\mathrm{p}}}^{\prime }{t}^{\prime }\right)〉-〈f\left(\overrightarrow{\mathrm{r}}\overrightarrow{\mathrm{p}}t\right)〉 〈f\left({\overrightarrow{\mathrm{r}}}^{\prime }{\overrightarrow{\mathrm{p}}}^{\prime }{t}^{\prime }\right)〉$, a matrix with indices ($\overrightarrow{\mathrm{r}}\overrightarrow{\mathrm{p}}t$). Systematic procedures for approximating this matrix, unhindered by secular effects, are always based on approximations to its inverse. For a weakly coupled fluid, the inverse can be expanded in powers of $\lambda$, the ratio of potential to kinetic energy. The leading term in this expansion gives rise to a Vlasov equation for the phase-space correlation function. The next term is the first that includes collisions, and results in relaxation towards equilibrium. This paper is concerned with the detailed study of the resulting fundamental nontrivial approximation. It is not Markovian and is perfectly reversible. Although the approximation is complicated, it is tractable analytically in various limits, and numerically for all wavelengths and frequencies. In this paper, only the behavior in certain limits is evaluated. Particular attention is directed toward its contractions - the density correlation function, which is measured by inelastic neutron and light scattering, and the momentum correlation function. Calculation of the former at long wavelengths corroborates the Landau-Placzek expression for light scattering, and therefore demonstrates that the kinetic equation predicts hydrodynamic behavior at long times. Since the correlation function is correct to order ${\lambda }^2$, it has, in contrast to a solution to the Boltzmann equation, the correct long-wavelength velocity of sound, $c^2={\left(\frac{\mathrm{dp}}{\mathrm{dmn}}\right)}_s\ne \frac{5}{3}\frac{k_{B}T}{m}$. It also predicts different transport coefficients than those deduced from a Boltzman equation. These include a nonvanishing bulk viscosity. The transport coefficients reduce to those derived from the Boltzmann equation at low densities. Some aspects of the short-time behavior are also discussed.