Given a vector field defined in a volume enclosed by a boundary, it is possible to predict the total index of point singularities of the vector field within the volume if it satisfies specified conditions on the boundary. This is made precise in a mathematical theorem, and its applications to physical systems, especially nematic liquid crystals, are discussed.

Exact lower bounds for the correlation energy of the one-component plasmas (OCP) with the $r^{-1\mathrm{}}$ potential are given in $d$ dimensions $1<d\le 3$. For classical OCP's with $d=2\mathrm{} \mathrm{and} 3$, this lower bound contains most of the correlation energy in the high-density domain, and the residue is given by two integrals of positive-definite functions, including the pair correlation function and the structure factor. For the classical OCP with $d=3\mathrm{}$, a lower bound which improves upon known ones due to Mermin and to Lieb and Narnhofer in the domain $\Gamma <15.3\mathrm{}$ is given.

Assuming a one-parameter model for the two-particle distribution function of a surface, one can choose a value of the "width parameter" to yield a correct surface tension. In order to then identify the width of the surface layer with the value of the width parameter, as is often done, one can check the validity of the model by verifying that other properties (e.g., surface energy) are correctly calculated, or, as proposed herein, by demonstrating that different formulas for the surface tension give identical results. A new formula is derived, and the relation between the different formulas discussed. Calculations are performed for the Ar system. For the model used, no choice of the parameter can yield identical results for both ways of calculating surface tension. This points to problems in the interpretation of the width parameter. Misleading results may be obtained if one- and two-particle distributions not related by the Born-Green-Yvon equation are used together.

The effect of a constant electric field $\overrightarrow{\mathrm{E}}$ on the low-temperature Johnson noise of a metallic resistor is calculated by nonequilibrium diagrammatic perturbation theory. The expansion, in powers of $E^2$, is shown to diverge, and the divergence is associated with accumulating Joule heat. An insight into how to treat a steady state properly when there is heating is provided by the Boltzmann equation. A vertex function that corresponds to a steady-state solution of this equation is identified. With this vertex function it is possible to evaluate not only one-particle operators, for which the Boltzmann equation is sufficient, but also troublesome contributions to the current fluctuations and other two-body observables for which a many-body approach is required. The calculated corrections to the current correlations agree qualitatively, but not quantitatively, with simple phenomenological agruments. In particular, it is not possible to describe the leading effects of the electric field solely in terms of a variable local temperature. The model calculation illustrates and sheds some light on several problems that only appear when nonlinear deviations from thermal equilibrium must be considered: the complicated effects to which heat generation leads, the identification and treatment of these effects in perturbation theory, and the difficulties these effects pose for hypothetical generalizations of the fluctuation dissipation theorem. Some possible connections between the calculation and $\frac{1}{f}$ noise are also briefly explored. In particular, recent experiments that may confirm the conclusion of this calculation, that systems in which there is only impurity scattering do not exhibit $\frac{1}{f}$ noise, are noted.

Statistical consequences of the subdivision of the phase space of a classical nonlinear system (a Lennard-Jones chain) into regions of ordered and stochastic motions are investigated by numerical computations on two parameters, which are related to the average fluctuations and to the equipartition of the kinetic energy. While the former parameter behaves as if it were an ergodic function (i.e., its time average is correctly given by its canonical average), the time average of the latter cannot be so computed, and furthermore it exhibits a transition consistent with previous computations. Comparisons are made with a harmonic chain and with a Toda chain.

The paper reexamines the relationship between fluctuations and nonlinear irreversible processes. The deterministic equations for nonlinear irreversible processes are shown to be derivable from a minimum principle, which permits the introduction of a set of variables $\eta$ canonically conjugate to the macroscopic variables $a$. In terms of the action integral of the minimum principle and the conjugate variables $\eta$, we are able to construct a covariant expression for the conditional probability of the fluctuations for a small interval of time. The short-time conditional probability is used to construct the conditional probability for finite times as a path integral. This path integral is a generalization of a corresponding expression of Onsager and Machlup for the linear regime. An explanation for a difference with Graham's recent calculation is given. The conditional probability is shown to satisfy a Fokker-Planck equation, which has the form derived from statistical mechanics by one of us.

Weak-beam amplification and diffracted-beam production are predicted under a variety of conditions, in which thin nematic-liquid-crystal samples maintained in an external magnetic field near Freedericks transitions are simultaneously illuminated by strong and weak beams derived from the same light source. It is shown that for 4-methoxybenzylidene-4′-butylaniline, substantial weak-beam amplification should occur under continuous illumination for modest (∼ 0.1 W/${\mathrm{cm}}^2$) strong-beam intensities, for certain cell and beam geometries. If a small positive frequency difference between the strong and weak incident beams is maintained, weak-beam amplification is further enhanced, and the conditions on the angles of incidence for amplification are considerably relaxed.

A curved-crystal Bragg x-ray spectrometer has been used to measure $K\alpha$ or $1s-2p$ radiation from highly stripped Fexviii-Fexxv impurity ions in the Princeton large torus tokamak. The spectrometer has sufficient energy resolution (≲4 eV at 6400 eV) to distinguish between the different ionization states of iron by measuring the energy shift of the $K\alpha$ x rays. The measured wavelengths agree well with theory and with spectra from solar flares and from laser-produced plasmas. The distribution of Fe charge states in the center of the discharge has been inferred from a comparison of the measured x-ray spectrum with theory. The shape of the spectrum depends strongly on electron temperature ($T_e$) in the range $T_e=800-1500\mathrm{}$ eV. Within the factor of 2 uncertainty in $L$-shell ionization cross sections, measured intensities agree with theory, which is based on coronal equilibrium, indicating that the ion lifetime in the center of the plasma is approximately equal to or greater than the equilibration time.

Numerical evidence is presented which indicates that the $N$-degree-of-freedom Hamiltonian that describes a set of pairs of sinusoidal waves interacting with a single, "test" wave is completely integrable. Computations of the Poincaré surface of section and Greene's residues provide the evidence for two background wave pairs. Linear separation of neighboring orbits for up to four wave pairs also suggests integrability for more degrees of freedom.

Numerical self-consistent calculations of the static and dynamic properties of a two-dimensional classical electron plasma have been carried out on the basis of the theory derived by Singwi etal. In this theory the short-range correlations are present through a local-field correction depending on the pair correlation function. The static structure factor and the density-density response function are determined in a self-consistent scheme. The pair correlation function and the correlation energy obtained are in very good agreement with the numerical experiments carried out by Totsuji. It is found that the onset of short-range order appears at the plasma parameter in the range $14<\mathrm{}\alpha <15\mathrm{}$. The plasmon dispersion relation is determined and is much more accurate than previous theories.

The effect of Coulomb screening on the inverse-bremsstrahlung heating process in a plasma illuminated by a laser beam is discussed. It is shown that, although the screening effect actually lowers the electron-nuclei interaction, a situation may arise, namely, when the laser frequency approaches the plasma frequency, in which an enhancement of the collisional plasma heating can be expected.

The molecular-dynamics method is used to determine the equation of state and the viscosities of systems of particles interacting through potentials assumed to represent the potential between krypton atoms. Three potentials are considered: two Lennard-Jones potentials (12-6, 9-6) and a potential reproducing the properties of krypton at very low density and in the solid phase. The equation of state and the viscosity are compared with the values measured on krypton in the fluid phase at 50°C. It appears from this comparison that none of the potentials used are compatible with experiment. For the Lennard-Jones potentials, it seems unlikely to obtain any agreement between the calculated and experimental values for dilute and dense gases by varying the values of the parameters defining them. For the third potential such an agreement is obtained by varying the values of the parameters defining it.

Cross sections for the production of $K$ vacancies by relativistic projectiles are calculated using the plane-wave Born approximation. Electronic $1s$ and continuum states are represented using semirelativistic Darwin wave functions. The calculated cross sections are in good agreement with measurements using 4.88-GeV protons, and also very high-energy (50-900-MeV) electrons. Simple formulas are given to calculate the cross section for projectile energies greater than $\sim 3M{c}^2$.

An algebraic variational procedure has been developed for the calculation of the phase shift $\rho$ of the radial wave function for a particle undergoing single-channel scattering. The method is essentially an optimization of Kohn's theory with respect to the phase parameter $\theta$ as involved in Kato's wave function. Specifically, the basis set has been transformed linearly so as to permit distinction between avoidable (spurious) and unavoidable (innate) singularities. On this basis, two new optimization procedures, which are termed the minimum-basis-dependence (MBD) and minimum-error (ME) methods, have been proposed. Various standard variational theories have also been reformulated in a unified manner. Sample basis-set calculations of $\rho$ have been carried out for the Hazi-Taylor model potential in order to demonstrate the relative merits of the MBD and ME methods.

The minimum-basis-dependence method which has previously been proposed for the single-channel scattering is extended to the multichannel case. The extension entails solutions of a set of simultaneous equations in two matrices which directly give the reactance matrix. A general variational functional is proposed explicitly for any wave function which is determined by an algebraic method. The functional can be made use of as a correction formula for a reactance matrix which is obtained as the solution of the algebraic equation.

It was shown recently that the classical treatment of Thomas for the double-scattering contribution to capture by a fast-moving ion of an electron from a hydrogenlike atom in its ground state—a state for which the classical description is surely not precise—becomes exact when applied to capture from a high-Rydberg state: That is, summed over all final bound states, the classical cross section becomes equal to the quantum-mechanical cross section when the initial state is a high-Rydberg state. It is here shown that the classical treatment of Bates and Mapleton for "knock-on" capture from the ground state also becomes exact (in the same sense as above) when applied to capture from a high-Rydberg state. The radiative and Brinkman-Kramers mechanisms, which are important for capture from the ground state at high impact velocities, vanish in the classical limit.

Absolute charge-state yields of 20-MeV Fe ions emerging within a large acceptance angle from a differentially pumped gas cell of length 9.4 cm have been measured using ${\mathrm{N}}_2$, Ar, Kr, Xe, and S${\mathrm{F}}_6$ targets. The gas pressures ranged from 2 to 400 mTorr, thereby including target thicknesses where single events dominate the charge change, as well as target thicknesses where charge-state equilibrium is established. From the low-pressure yields total cross sections, ${\sigma }_{4q}$ with $q=5-15\mathrm{}$ were obtained using an improved version of the initial growth method.

We have measured the impact-parameter ($b$) dependent probabilities ($P$) for the production of Ni (and Cu) $K$ x rays in the bombardment of thin nickel foils by 50- and 65.5-MeV Cu beams. The shape of $P\left(b\right)\mathrm{}$ is found to be independent of nickel-foil thickness ($t$) and of the presence of a pre-exciting carbon layer, but the magnitude scale of both $P\left(b\right)\mathrm{}$ and of the total $K$-x-ray cross section is quite dependent upon $t$ and on the presence of the carbon. The data suggest that $2p\sigma$ vacancies are being produced via $2p\pi -2p\sigma$ rotational coupling preceded by preparation of the necessary projectile $2p$ vacancies in both single- and multiple-collision processes. The shape of $P\left(b\right)\mathrm{}$ is only qualitatively similar to that expected for the rotational-coupling process, however.