A previously developed theory of dynamic nuclear polarization (DNP) in liquids is extended and applied to viscous liquids with anisotropic electron-spin relaxation ($T_{1e}\gg T_{2e}$). It is based on a Schrödinger equation for two spins, $S$ and $I$, in which the influence of the "lattice" motion is incorporated by a randomly time-dependent field acting on the spin $S$ and a random time dependence of the dipole interaction between $S$ and $I$. The quantum-mechanical equations of motion in an applied rotating field near $S$-spin resonance are solved and the DNP of the $I$ spin is expressed in terms of the stochastic parameters which characterize the liquid motion. Using values of these parameters as obtained from EPR and NMR experiments on two dilute organic radical solutions, one finds excellent agreement with our DNP measurements on these systems as reported in the preceding paper. It is characteristic of these solutions that the NMR requires a distribution of relaxation times. The analysis of DNP measurements confirms this; it gives a somewhat different distribution and permits a more specific correlation between it and the structure of the liquid. The range of applicability of the theory is discussed, and in an appendix a completely general set of equations for the two-spin relaxation in our model is derived. Using these, the only restrictions on the application of our theory would be that the $S-S$ as well as the $I-I$ interactions must be small and that the amplitude of the rotating field must be substantially smaller than the constant external field.

A description of the stability properties of a plasma under the influence of an external electric field oscillating near the plasma frequency is presented.

A plane slab of current-carrying plasma is generally shown to be stable against ideal magnetohydrodynamical modes. It is shown that some of the stability criteria appearing in the literature and contradicting this fact can be brought to bear on the stability of the tearing mode. In this connection, simple stability diagrams are shown for the tearing mode in plane and cylindrically symmetric force-free magnetic fields.

A "factor method" of evaluating the double-scattering correction for inelastic-neutron-scattering experiments with an infinite-plane-slab specimen has been developed. Using appropriate dynamical models, calculations have been made both for highly compressed room-temperature gaseous methane and for liquid methane at 98 °K. As a result, the agreement between analytical results with measurements is remarkably improved. In addition, the model sensitivity of the correction and the advantages of reflection-type experimental geometry over transmission-type are investigated.

Various lower and upper bounds are derived for the energy, free energy, and pressure of particles interacting via an $r^{-n}$ pair potential in $\nu$ dimensions. The results include (i) lower bounds for mixtures with $n<\mathrm{}2\mathrm{}$, of the form $\mathrm{const}{\rho }^{\frac{1+2}{\nu }}$ as $\rho$ increases either isentropically or isothermally, complementing an upper bound of the same form in the isentropic case, recently derived by Kleban and Puff, and (ii) upper and lower bounds for the case $n>\mathrm{}\nu$, $n>\mathrm{}2\mathrm{}$, which for the pressure are both of the form $\mathrm{const}{\rho }^{\frac{1+n}{\nu }}$ as $\rho$ increases isentropically.

A new equation of motion, which describes the time evolution of the $P$ representation in the Schrödinger picture, is derived. The equation applies to a very general class of Hamiltonians, and thus should be useful in many problems. As a demonstration of the usefulness of this equation, some known results are recovered by its direct application.

A new model description for many-component mixtures is proposed. For the mathematical treatment of the model, a cluster approximation is developed. As an application, activity coefficients are evaluated. Furthermore it is shown that the method is exact for one-dimensional systems with nearest neighbor interactions. The implications of this result for magnetic systems are discussed.

An exact test-particle propagator of the Lenard-Bernstein equation in the presence of an external magnetic field is obtained as a generalization of the well-known bivariate Gaussian distribution given by Chandrasekhar. This propagator is applied to the problem of temporal transverse-plasma echoes. The free-particle model used below can predict a transverse echo when one of the two external signals is transverse and the other longitudinal. When both of two external excitations are transverse, it does not explain a plasma wave echo, unless velocity increments of particles along the external magnetic field are explicity given as a function of their transverse energy increments. The reason is that transverse velocity increments, as are described in this paper, can not properly account for the circularly polarized nature of exciting signals.

I point out a scaling symmetry law in thermodynamical systems and discuss its various manifestations, particularly in magnetic systems.

Measurements of the nuclear susceptibility of pure solid ${\mathrm{He}}^3$ from 0.4 to 0.04 °K are described. The ${\mathrm{He}}^4$ content of the sample was less than 10 ppm. The susceptibility is shown to obey a Curie-Weiss law $\chi =\frac{c}{(T-\Theta )}$ with $\Theta =-1.5\mathrm{}\times {10}^{-3\mathrm{}}$ °K for a molar volume $v$ of 23.3 ${\mathrm{cm}}^3$/mole, $\Theta =-4\mathrm{}\times {10}^{-3\mathrm{}}$ °K for $v=23.6\mathrm{}$ ${\mathrm{cm}}^3$/mole, and $\Theta =-5.4\mathrm{}\times {10}^{-3\mathrm{}}$ °K for $v=24.2\mathrm{}$ ${\mathrm{cm}}^3$/mole. The negative values for $\Theta$ mean that solid ${\mathrm{He}}^3$ nuclei should order antiferromagnetically at very low temperatures and low magnetic fields. The experimental data are compared with current theories and other experiments.

We consider the equilibrium statistical mechanics of classical fluids in which the potential energy is decomposable into repulsive pair interactions. A generalized cluster expansion is derived relating the thermodynamic and structural properties of such systems to those of the hard-sphere fluid. The expansion is ordered by a softness parameter $\xi$ which is essentially the range of intermolecular distances in which the difference between the Mayer $f$ functions for the repulsive potential and an appropriate reference hard-sphere potential is nonzero. The first (lowest-order) approximation generated by the expansion equates the free energy and $y\left(r\right)\mathrm{}$ for the fluid to the respective functions appropriate to a system of hard spheres with diameter $d$. Here $y\left(r\right)=g\left(r\right)\mathrm{} e^{+\beta u\left(r\right)\mathrm{}}$, where $g\left(r\right)\mathrm{}$ and $u\left(r\right)\mathrm{}$ denote the radial distribution function and repulsive pair potential, respectively. A prescription is given for choosing a temperature- and density-dependent diameter $d$ in the reference hard-sphere fluid so that the first approximation for the free energy contains errors of order ${\xi }^4$ only, and the corrections to the first approximation for $g\left(r\right)\mathrm{}$ are of order ${\xi }^2$. The method is used to calculate the properties of a fluid whose intermolecular potential varies as $r^{-12\mathrm{}}$. The repulsive potential that produces the repulsive forces in the Lennard-Jones potential is also studied. Since the properties of the hard-sphere fluid are known from the results of computer calculations and conveniently summarized by analytic equations, the application of the first approximation is numerically very simple. With this approximation, the results obtained for both model systems agree closely with those obtained by Monte Carlo calculations.

Highly accurate x-ray diffraction measurements are presented for the static structure factor $a\left(q\right)\mathrm{}$ for liquid Na (at 100 and 200 °C) and liquid K (at 65 and 135 °C). A detailed error analysis is presented showing that the over-all root-mean-square error in $a\left(q\right)\mathrm{}$ never exceeds 2.5% for any value of the momentum transfer $q$ and the relative root-mean-square error in $a\left(q\right)\mathrm{}$ between different temperatures is always less than 1.5%. We discuss and demonstrate the reliability of the tabulated values for the atomic form factor and the Compton-scattering correction. A brief discussion is included of the relative merits of x-ray vs neutron diffraction for obtaining the static structure factor.

A generalization of the Navier-Stokes equation, valid for wavelengths and times of a molecular order of magnitude, is discussed on the basis of viscoelastic behavior of simple classical liquids. In this theory, transport coefficients are replaced by appropriate viscoelastic memory functions. The theory is verified by analyzing the data on current-correlation functions obtained from computer experiments. Three different models for the time dependence of the viscoelastic memory are investigated, namely, a single-exponential decay, a modified-exponential decay, and a Gaussian decay. It is obsrved that the memory functions are approximately Gaussian, at least for times of the order of one or two relaxation times. This is in agreement with a conjecture of Forster, Martin, and Yip. The wave-number dependence of the half-width of the Gaussian decay, and of the longitudinal- and shear-viscosity coefficients, are found from computer experiments. The extrapolated values of these transport coefficients, in the limit $k\to 0$, are in good agreement with experiments on liquid argon.

A comprehensive theoretical description is given for the effects of magnetic resonance on the angular distribution of radiation emitted from oriented nuclear states. The formulation is made in a general way. It may be applied to an ensemble of nuclei oriented by any method: For example, nuclear reactions, angular correlations, and low-temperature nuclear orientation may be treated. In fact the theory can also be applied to optical-double-resonance experiments. Statistical tensors are defined to describe nuclear crientation in the "resonant" state. Interactions of the oriented ensemble with extranuclear fields are then considered, and the effect of a radio-frequency (rf) field on the angular distribution of radiation is given. Two formulations are given for the "pure magnetic" case, for which numerical calculations were done. One employs an angular correlation formalism, following the evolution of the density matrix in the laboratory frame $S$, while the other is more closely related to conventional NMR. In the latter approach the transformation into the frame $S^{\prime \prime \prime }$, wherein the statistical tensors are time-invariant, is described in terms of a "generalized torque equation" governing the motion of a unit vector along the symmetry axis in $S^{\prime \prime \prime }$. Both formulations are exact. Time-dependent distribution functions are worked out in detail, with both fixed and random-phase angles between the rf field and the initial symmetry direction. Fast oscillations due to the constant magnetic field are modulated by slow oscillations due to the rf field. Time-integral curves were calculated. These show great sensitivity to the rank of the relevant statistical tensor, to geometry, and to the phase of the rf field. Multipole structure is predicted for certain geometries, with the resonance line showing a number of maxima equal to the rank of the statistical tensor. Under certain conditions two types of asymmetry are observable. A "transient" asymmetry appears for low-rf-field values: This asymmetry is sensitive to the sign of the nuclear moment, but it disappears in high-rf fields. Odd-rank statistical tensors can also give response functions with "persistent" asymmetry that remains at high-rf fields. This is a parity effect and is not sensitive to the sign of the nuclear moment. Effects of relaxation are also discussed briefly.

A new experimental technique has been utilized for the measurement of rate constants for two- and three-body electron-attachment reactions. This technique is based on the observation of positive-ion and/or negative-ion decay(s) in a stationary afterglow. The transition of the decay from positive-ion electron ambipolar-diffusion domination at early times to positive-ion-negative-ion ambipolar-diffusion domination at late times is related to the disappearance of electrons from the decaying plasma. The time of occurrence of the transition in the afterglow is used to relate the attachment rate constants to other experimental observables. The two-body rate constant $k_2$ for attachment to N${\mathrm{O}}_2$ is determined to be ${1.4}_{-0.9\mathrm{}}^{+2.0\mathrm{}}$ × ${10}^{-11\mathrm{}}$ ${\mathrm{cm}}^3$/sec; similarly, the three-body rate constant $k_3$ for attachment to NO is (6.8 ± 3.4) × ${10}^{-32\mathrm{}}$ ${\mathrm{cm}}^6$/sec, both at 296°K. The only negative ion observed in this study is N${\mathrm{O}}_2^{-}$, even under conditions in which three-body attachment in NO dominates the electron-loss processes; the absence of N${\mathrm{O}}^{-}$ under these conditions is investigated.

Resonant two-photon absorption is discussed for the case in which the exciting field may be intense enough and near enough to resonance to cause appreciable saturation. A single pair of levels in an otherwise quite general atom is assumed to satisfy the appropriate resonance condition, and equations are derived governing the time evolution of the associated 2 × 2 submatrix of the full atomic density matrix. Atomic relaxation is treated explicitly. The analysis, which treats both inversion-symmetric and inversion-nonsymmetric cases within the same general formalism, is developed systematically, the necessary approximations receiving careful discussion. The results are analogous to those for the corresponding one-photon process, but with a shift in the resonance frequency proportional to the intensity of the exciting field. This frequency shift is not identical to the one calculated by time-independent (second-order) perturbation theory, and becomes large when the energy of an intermediate state of the atom approaches the mean of the energies of the active states.

The ground state of a many-particle boson system is studied for two closely related limits: the uniform limit and the weak coupling limit. The former is defined by $\alpha =1-g\left(0\right)\mathrm{}\ll 1$ and the latter by $\beta =\frac{(N-N_0)}{N}\ll 1$, where $g\left(r\right)\mathrm{}$ is the radial distribution function and $N_0$ is the mean occupation number of the zero-momentum state. In the uniform limit the variation-perturbation approach based on (a) the method of correlated basis functions and (b) the series expansion in powers of $\alpha$ is found to be equivalent to the field-theoretic treatment given by Brueckner (for the charged-boson gas) in the weak coupling limit. In particular, it is shown that the variation-perturbation energy obtained for the uniform limit in the momentum representation is identical through second order to the ground-state energy evaluated for $\beta \ll 1$ by summing one- and two-ring diagrams in the Bogoliubov occupation-number representation. The charged-boson gas and the one-dimensional boson system with a $\delta$-function interaction are considered to examine some of the interesting features of the uniform-limit procedure.

Results for the nuclear longitudinal relaxation time $T_1$ in liquid ${\mathrm{D}}_2$ are presented as a function of the mole fraction $X$ of the molecules with rotational angular momentum $J=1\mathrm{}$ at $T=19.9\mathrm{}$ °K. A composite signal with two relaxation times was observed: a long one ($T_1\sim {10}^3$ sec)and a short one ($T_1\le 2.3$ sec)—attributed, respectively, to the molecules with $J=0\mathrm{}$ and $J=1\mathrm{}$. A discussion of the $T_1\mathrm{}(J=1\mathrm{})\mathrm{}$ results is made in terms of the prediction by the theory of Deutch and Oppenheim, and uses the $T_1$ results for liquid ${\mathrm{H}}_2$. A relaxation mechanism is proposed to account for the $T_1\mathrm{}(J=0\mathrm{})\mathrm{}$ results which uses the admixture of higher-energy states into the $J=0\mathrm{}$ state due to interaction with neighboring $J=1\mathrm{}$ molecules. The relaxation rate for this mechanism had been calculated by Harris for solid ${\mathrm{D}}_2$ and was adapted to the liquid. Reasonable agreement between theory and experiment is found for both $T_1\mathrm{}(J=1\mathrm{})\mathrm{}$ and $T_1\mathrm{}(J=0\mathrm{})\mathrm{}$.

Numerical computations of the thermal conductivity of simple two-dimensional disordered lattices with two distinct phonon branches have been made. Results similar to those for a single-branch lattice were found: primarily that the addition of anharmonicity to a disordered harmonic lattice increases, rather than diminishes, the thermal conductivity.

The proof of Jasnow and Fisher that no spontaneous ordering can occur in one- and two-dimensional systems with continuous symmetry is shown to fail for nonlocal potentials of sufficiently long range. The occurrence of Bose-Einstein condensation for finite temperature (in the infinite-volume limit) is, therefore, not forbidden for such potentials. This result also follows from the postulate of the completeness of the noninteracting equilibrium states. Thus, the completeness postulate represents a weaker assumption on the nature of an interacting equilibrium state (or, equivalently, on the basic interactions between particles) than is usually found in the literature.