An experimental total cross section for single-electron capture by heliumlike ${\mathrm{B}}^{3+}$ from He has been measured at relative velocities between 0.7 and 7.4 × ${10}^7$ cm/sec (300-eV to 31.5-keV ${\mathrm{B}}^{3+}$ laboratory energy), and has been compared with a cross section calculated via the Landau-Zener theory. In the velocity range considered, the dominant crossing of the (B + He${)}^{3+}$ system is between $s$ states and the often-neglected conditions for applicability of the theory are met. The velocity positions of the cross-section maxima are in agreement, but the experimental magnitude at maximum is greater by a factor of 2.

The analogy between a vortex line and a stretched string is examined by considering the equation of motion of an elastic filament surrounded with circulating irrotational fluid. A generalization to filaments with discontinuities in density provides a model of an impurity trapped on a vortex line. The corresponding long-wavelength reflection probability reproduces that found previously.

A canonical theory is developed for a collisionless plasma in which the electrons interact with quasistationary ions by the self-consistent electric field. The nonlinear equations of motion of the electron fluid and continuity for the electric current density are obtained as functional derivatives of the Hamiltonian of the plasma with respect to the canonical conjugate fields (which are proportional to the mean mass velocity and electric field, respectively). By means of a Fourier analysis, it is shown that Hamiltonian equations exist for the complex Fourier amplitudes of the canonical-conjugate fields, which correspond to the Fourier transforms of the nonlinear Eulerian equations of motion and continuity of the electron gas. The Liouville equation is derived for the distribution function in the multidimensional phase space formed by the complex Fourier amplitudes of the canonical-conjugate fields. A statistical definition of the entropy contained in the wave modes is given in form of an integral functional of the phase-space density. The theoretical considerations indicate a quasiparticle behavior of the wave modes of the classical plasma continuum.

We consider the anisotropic one-dimensional $X-Y$ model of Lieb, Schultz, and Mattis and study the three zero-temperature correlation functions ${\rho }_{\mu \nu }\mathrm{}(R,t)=〈S_1^{\mu }\mathrm{}\left(0\right)\mathrm{}{S}_{R+1\mathrm{}}^{\nu }\mathrm{}\left(t\right)\mathrm{}〉$, where $\mu \nu =x,y,z$ but $\mu \ne \nu$. We obtain the first-order term of ${\rho }_{\mathrm{xy}}$ for $R\to \infty$ or $t\to \infty$ and exact expressions for ${\rho }_{\mathrm{xz}}$ and ${\rho }_{\mathrm{yz}}$.

We show that the circle theorem on the distribution of zeros of the partition function breaks down for the ferroelectric potassium dihydrogen phosphate (KDP) model if the field lies outside the first quadrant. We also use a recent result by Suzuki and Fisher to establish the circle theorem for the antiferroelectric $F$ model with a staggered electric field. Numerical results on the distribution of zeros for a 4 × 4 lattice are given.

The problem of response theory in statistical mechanics involves the determination of the density matrix $\rho$ from the Liouville equation and the subsequent computation of the response $r$ from this $\rho$. Projection techniques are applied to avoid the entire complicated problem of the full dynamics of $\rho$ and to select only that part of $\rho$ which is relevant to the response $r$. The procedure replaces an inhomogeneous equation by a linear homogeneous integrodifferential equation for response theory. This is a very general equation which can be analyzed in different ways to yield a variety of results. It is shown that the Kubo theory of linear response emerges as the lowest-order approximation. The general equation is solved without approximations for a step-function stimulus, and it is discussed in the context of the steady state.

We compute the correlation functions $〈S_0^x\mathrm{}\left(t\right)\mathrm{}{S}_R^x\mathrm{}\left(0\right)\mathrm{}〉$ and $〈S_0^y\mathrm{}\left(t\right)\mathrm{}{S}_R^y\mathrm{}\left(0\right)\mathrm{}〉$ at $T=0\mathrm{}$ for the one-dimensional $\mathrm{XY}$ model in the presence of a magnetic field.

We report here the first determination for a simple liquid (specifically liquid C${\mathrm{Cl}}_4$) of the fraction of its Kerr effect that arises from the (nearly instantaneous) nonlinear response of its electronic currents, and hence would exist even if the nuclei were frozen in position. To do this, we have remeasured the power dependence of the rotation of the polarization ellipse of a monochromatic beam in C${\mathrm{Cl}}_4$ with greatly improved accuracy (± 10% absolute) using a single-Gaussian-mode ruby (giant pulse) laser. We then compare the results of this ellipse rotation measurement with existing Kerr data, and, using a general relation between the relative electronic contributions to both effects which we demonstrate, we show that (54 ± 16)% of the Kerr effect in C${\mathrm{Cl}}_4$ arises from nonlinear electronic response. The method should be useful for any isotropic material.

The resistive magnetohydrodynamic "tearing" and "gravitational-interchange" instabilities are investigated in the linear incompressible limit in the absence of heating. A periodic model is used for the initial magnetic field. The requirement that the spatial Fourier series of the perturbation must converge uniquely determines the growth rate of the instability. At the most unstable wavelength the tearing mode has a growth rate $\omega {\tau }_r=3.4\mathrm{}{S}^{0.57}$, where $S$ is the magnetic Reynolds number at the Alfvén velocity and ${\tau }_r$ is the resistive diffusion time. The interchange mode exhibits behavior, at small wavelengths and large gravitational fields, similar to that of the magnetohydrodynamic Rayleigh-Taylor instability. The resistive mode grows for gravitational fields smaller than the equivalent Suydam threshold for infinite-conductivity instability in a sheared magnetic field. Application of these results to the solar-flare problem is briefly discussed, in the context of the temporal behavior exhibited.

The specific heat of liquid ${\mathrm{He}}^3$/${\mathrm{He}}^4$ mixtures has been measured at saturated vapor pressure for eight values of the ${\mathrm{He}}^3$ mole fraction $x$ in the range $0.53\le x\le 0.73$ at temperatures from ∼0.6 to ∼1.4°K. For purposes of comparison, the specific heat of pure liquid ${\mathrm{He}}^3$ has also been measured under similar conditions. The measured specific heat, obtained with a calorimeter almost completely full of liquid, has been corrected to yield the specific heat of the liquid $c_{P,x}$ at constant pressure and mole fraction. For the mixtures, the correction required was everywhere ≲ 1%. To a resolution in temperature which varies from 0.1 to 3 m°K from one value of $x$ to another, the specific heat at the $\lambda$ peak appears to be finite, continuous, and cusped. Both the $\lambda$ peak and the discontinuity in $c_{P,x}$ as the system goes from the two-liquid-phase region to the one-liquid-phase region tend to disappear as the junction of phase boundaries is approached.

The set of differential equations describing the time-dependent decay of a singly ionized optically thin monatomic gas has been solved numerically for constant-temperature-constant-pressure helium afterglows. In the range of conditions studied ($6000\le T_e\le 14000$°K, $1.5\times {10}^{12}\le n_e\le 5.38\times {10}^{15}$ ${\mathrm{cm}}^{-3\mathrm{}}$, $1.5\times {10}^{14}\le n_0\le 1.63\times {10}^{17}$ ${\mathrm{cm}}^{-3\mathrm{}}$) it was found that collisional-radiative recombination can be applied to the electron-density decay before the plasma reaches the quasisteady state. The initial condition for each plasma studied was a Boltzmann distribution from the $2{}_{}{}^3{}_{}{}^{}S$ state throughout the higher states. The mechanism by which the quasisteady state is obtained is examined and the transient coupling between states is illustrated. Times for the plasma to reach the quasisteady state ranged from $t\approx {10}^{-8\mathrm{}}$ sec for high-density plasmas to $t\approx {10}^{-4\mathrm{}}$ sec for low-density plasmas.

The intensity of $\alpha$-particle-produced scintillations of liquid helium has been measured in the 1.3-4.2 °K temperature range and at pressures varying from the vapor pressure at 1.3 °K to 28 atm.

Measurements of the $\alpha$-particle-produced scintillation intensity of liquid ${\mathrm{He}}^4$ under pressure have been performed at temperatures ranging down to 0.3 °K. The data show a strongly temperature-dependent component of the pulse intensity which drops to zero for $T\lesssim 0.75$ °K. The principal pressure dependence of the intensity is in the component remaining below 0.75 °K. A physical interpretation of the results is presented which invokes metastable systems and the superfluidity of He II to explain the temperature dependence, and ionic recombination to explain the pressure dependence.

The joint density-of-states integral for production of a pair of elementary excitations in an isotropic fluid is studied as a function of energy and momentum. Isotropy permits one to take advantage of the essentially two-dimensional character of the system. Specializing to liquid ${\mathrm{He}}^4$ at $T=0\mathrm{}$ °K, we show that certain abrupt variations in the density of states are associated with thresholds and saddle points in the energy function. The expected influence of these phenomena on $S(\overrightarrow{\mathrm{k}},\omega )$ is described. Resonant pairs of excitations, as judged by their linesharpening effects, are shown to exist at special values of $\overrightarrow{\mathrm{k}}$ and $\omega$.

Results were recently reported on statistical properties of phase-modulated laser light. A detailed analysis is presented here of random-phase modulation. In particular it is shown that there are two kinds of such modulation: modulation by phase fluctuations, in which the instantaneous phase is a stationary stochastic process, and modulation by phase diffusion in which the phase is a diffusion process with an always increasing variance. In the case of a Gaussian modulation we show that the optical fields in the two preceding cases are very different, and we calculate their coherence functions. This difference can be detected by an interference experiment, but more easily by photon coincidence or intensity-correlation experiments on a beat between a reference beam and the modulated beams. We present a theoretical analysis of such experiments in the two cases of modulation.

Recent theoretical evidence has suggested that in general the temperature derivative of the diameter of the liquid-vapor coexistence curve should diverge at the critical point at least as fast as the constant-volume specific heat. This evidence implies with equal force that the difference across the coexistence curve in the density fluctuations in the two phases, $\Delta {\left(\frac{\partial \rho }{\partial \mu }\right)}_T={\left(\frac{\partial \rho }{\partial \mu }\right)}_T^L-{\left(\frac{\partial \rho }{\partial \mu }\right)}_T^V$, should in general diverge at least as fast as the temperature derivative of the liquid-vapor density difference: $\Delta {\left(\frac{\partial \rho }{\partial \mu }\right)}_T\sim \left(\frac{d}{\mathrm{dT}}\right)\Delta \rho \sim {(T_c-T)}^{\beta -1\mathrm{}}$. Experimental estimates of this quantity would provide an alternative test of those hypotheses that currently predict a singular diameter.

The probabilities of reflection and transmission of vortex waves by a spherical impurity are derived in the long-wavelength limit. The results are exactly the same as those for a cylindrical impurity, showing that the shape of a trapped ion by a vortex does not affect the mobility calculation given previously.

A comparison of the recent experimental data for He II of Cowley and Woods for the shift of the peak of the distribution of the inelastically scattered neutrons from the free-particle value appropriate for large momentum transfer is made with the phenomenological theory of Kerr etal.

An approximate solution is given to the master equation describing the cooperative spontaneous emission from a large number of excited two-level atoms. Our solution allows an easy analytical computation of the statistical properties of the radiated optical field. Comparison with previously reported computer results shows a satisfactory agreement. Furthermore, a formal justification is provided to a recent heuristic model of superradiant pulses.