A study is made of an analytically solvable model of multichannel scattering, leading to the derivation of various properties of the solutions. In this model the intrinsic potential in each channel and the interchannel coupling operators are all taken to be separable interactions. In an especially simple version of the model an assumption is also made concerning a factorization property of the coupling coefficients that pertain to these interactions. Extremely simple explicit expressions are obtained for the elastic- and the inelastic-scattering amplitudes, as well as for the Feshbach potential, the effective interaction in the elastic channel owing to the presence of all other channels. The expressions obtained are transparent to analysis, and in the simple version of the model they do not involve the evaluation of inverses or determinants of large matrices, as is the case in general for formal explicit solutions of Fredholm equations. The simple version of the model can also be extended to the case of a continuum of channels, thus being capable of describing systems with open rearrangement or dissociation channels. Within the framework of this model, a rigorous analysis is given for the case of an infinite number of channels, and conditions are obtained for convergence of the close-coupling scheme. Error bounds are obtained for an $N$-channel truncated calculation on an infinite-channel system. Other results obtained include: (a) establishing the asymptotic behavior of the scattering amplitude at extremely high energies, and (b) deriving some especially simple features, "scaling properties," of the scattering amplitudes.

$M$-series (14 to 15 Å) x-ray spectra from lanthanum are presented for near-threshold electron excitation. Several structures in the spectra which are not the ordinary characteristic-x-ray lines exhibit very strong intensity and wavelength (energy-position) dependence on the energy of the exciting electrons. Consideration of a sequence of spectra recorded for closely spaced ($\Delta E=1\mathrm{}$ eV) incident electron energies and of the shape of the continuum-limit spectrum from lanthanum for other energies show that the several structures are actually a single feature of the continuous spectrum. This linelike feature exactly follows the energy of the incident electrons and has large intensity resonances in the neighborhoods of the threshold excitation energies of the lanthanum $M_5$ and $M_4$ inner levels. The interpretation offered suggests that the observed spectral variations are due to resonances in the cross section for scattering incident electrons into vacant $4f$ levels and that these resonances exist because of the excitation of sets of excited negative-ion bound states involving $3d$ inner-shell electrons. Consequences of the observed spectral variations for valence-band x-ray spectroscopy, appearance-potential spectroscopy, and continuum-limit spectroscopy are discussed.

The cross section for elastic scattering of electrons by neutral argon atoms has been calculated from 0-300 eV. The optical potential, or self-energy of the single-particle Green's function, is used to include effects of electron correlations, which are found to be significant at low energy. The cross section is calculated by means of a partial-wave expansion. Our results are compared with previous calculations and with experiments. Good agreement is obtained with experiment over a wide energy range.

A photon-echo nuclear double resonance (PENDOR) experiment which detects nuclear magnetic resonances by monitoring the stimulated photon-echo intensity after the application of an rf pulse is described and used to determine the Cr-Al hyperfine and electric-quadrupole interaction parameters associated with the ${\mathrm{Cr}}^{3+}$ ion in the ${}_{}{}^2{}_{}{}^{}E\left(\overline{E}\right)$ optically excited state in ruby. Most of these parameters are found to be within a few percent of the values associated with ${\mathrm{Cr}}^{3+}$ in the ${}_{}{}^4{}_{}{}^{}A_2^{}{}_{}{}^{}$ ground state. Resonances are also observed associated with the latter, state, and they are in agreement with previous spin-echo ENDOR work. High resolution is obtained by using a gated cw ruby laser which generates a rapid train of three-pulse stimulated-photon echoes with a long time interval between the second and third excitation pulses. The theoretical behavior of the PENDOR effect is discussed with emphasis on the ability to disentangle complicated multiple overlapping resonances by causing whole sets of resonances to disappear in a controlled way.

The classical equilibrium statistical mechanics is presented for a one-dimensional system of $N$ identical hard rods and to an interval of length $L$. An attractive force acts between nearest-neighbor rods and is inversely proportional to their center-of-mass separation. The equation of state is explicitly found in terms of an incomplete $\Gamma$ function $\Gamma (\alpha ,x)$, and it relates the density $\rho$ to the pressure $p$, the temperature $T={\left(k_B\beta \right)}^{-1\mathrm{}}$, the rod lenght $a$, and the attractive interaction strength $\sigma$: $\rho =\beta p\frac{\Gamma (1-\beta \sigma ,\beta pa)}{\Gamma (2-\beta \sigma ,\beta pa)}$. For $\beta \sigma >2\mathrm{}$ the pressure goes continuously to zero along any isotherm as $\rho$ approaches a critical density ${\rho }_c$ given by ${\rho }_{c}a=\frac{(\beta \sigma -2)}{(\beta \sigma -1)}$. The compressibility diverges as ${\rho }_c$ is approached along the isotherms provided that $2<\mathrm{}\beta \sigma <3\mathrm{}$. The form of the divergence is $(\beta \sigma -2){(\rho -{\rho }_c)}^{-\nu }$ where $\nu =\frac{(3-\beta \sigma )}{(\beta \sigma -2)}$.

The spectral distribution of light scattered by a two-species chemically relaxing fluid is considered, and a normal-mode analysis is carried out for a range of $k$ values for which the pressure fluctuations are decoupled from those in entropy and concentration. Under these conditions the damping of the propagating modes is a sum of a term due solely to chemical relaxation and terms due solely to transport, whereas the damping of the nonpropagating modes contains mixed terms. The intensity factors of the propagating modes are constants, whereas those of the nonpropagating modes exhibit a $k$ dependence. This $k$ dependence of the nonpropagating modes is of some significance for the analysis of the Rayleigh line.

The ratio of dc electric-field-induced optical second-harmonic coefficients $R\equiv \frac{X_{\mathrm{yyyy}}(-2\mathrm{}\omega ;0,\omega ,\omega )}{X_{\mathrm{yyxx}}(-2\mathrm{}\omega ;0,\omega ,\omega )}$ has been measured for each of the inert gases. The results at $\omega =0.066\mathrm{}$ a.u. (the ruby-laser frequency) are: helium, 2.97 ± 0.03; neon, 3.01 ± 0.05; argon, 2.99 ± 0.04; krypton, 2.96 ± 0.04; and xenon, 3.03 ± 0.04. In the limit $\omega =0\mathrm{}$, symmetry considerations require $R=3\mathrm{}$. A theoretical estimate of $R$ at $\omega =0.066\mathrm{}$ a.u. predicts deviations from 3 which are significantly larger than those observed experimentally.

Recently published data on the resistance anomaly in a critical isobutyric acid-water mixture near the phase-separation critical point are analyzed. It is shown that the data below a value of $t\equiv \frac{(T-T_c)}{T_c}\simeq 5\times {10}^{-5\mathrm{}}$ may be influenced by gravitational effects which were not accounted for in the original analysis. In the temperature range above this value of $t$, a fit of the form $\frac{(R_c-R)}{R_c}\approx A{t}^{1-\alpha }+Bt$ (with $\alpha$ the specific-heat exponent) is shown to be satisfactory, and a semiquantitative theory which yields this result is presented.

The fully renormalized kinetic theory (FRKT) previously discussed by the author in the case of self-diffusion is extended to the case of density fluctuations. In the theory techniques are developed for calculating classical phase-space time-dependent correlation functions. The method centers around the development of an exact expression for the memory function associated with the phase-space-correlation function. This exact expression is written in a compact and symmetric form that is convenient for making approximations. It is shown explicitly how one can make contact with the Boltzmann-Enskog approximation for the memory function which is valid for moderate densities, as well as the terms which lead to the much-discussed $t^{-\frac{2}{3}}$ long-time behavior. The most striking difference between self-diffusion and the case of density fluctuation arises in the region of the critical point. It is shown in the theory how to make contact with the mode-mode coupling results of Kawasaki. From a formal point of view the development of the FRKT is carried forward three important steps in this paper. It is shown how to relate the memory function to the two-particle source function in a more powerful and direct method than the projection-operator approach developed in FRKT I. After a simple rearrangement, the memory function is written in a far more symmetric form than in FRKT I, and finally it is shown how the ideas of connectedness and cumulants can be successfully introduced into the theory.

An explicit analogy is developed between the long-range fluctuations near a critical point and the long-range fluctuations in $\overrightarrow{\mathrm{k}}$ space in a turbulent fluid in the limit of large Reynolds number. Mathematically the analogy is between the probability functional of the spins on a lattice and the probability functional for the Fourier-transformed velocity fluctuations. The latter is given by the stationary solution of an exact Fokker-Planck equation originally derived by Edwards. This functional, which is not explicitly known, plays the role of a Hamiltonian for stationary turbulence. The limit of zero viscosity is equivalent to the temperature approaching the critical temperature from above. The order parameter is the vorticity-vorticity-correlation function in $\overrightarrow{\mathrm{k}}$ space, which in three dimensions is the same as the energy spectrum $E\left(k\right)\mathrm{}$. The mechanism of energy dissipation constrains the critical exponent $\gamma$ to satisfy $\gamma =1\mathrm{}$. The fluid is stirred at an external length scale $k_0^{-1\mathrm{}}$. If effects due to the nonzero value of $k_0$ are ignored, the scaling properties of the full probability functional can be explicitly calculated, and lead to the original Kolmogorov theory. This can be characterized by the critical exponents $\nu =\frac{3}{4}$, $\eta =\frac{2}{3}$, indicating that the equivalent Hamiltonian is of long range. For $k_0\ne 0$, the range of fluctuations in $\overrightarrow{\mathrm{k}}$, space still diverges for zero viscosity. This limit no longer leads to a probability functional with the scale symmetry of a fixed point. We examine the cascade process under the assumption that $\overrightarrow{\mathrm{r}}$-space correlation functions go as some power of $r$. The original Kolmogorov theory is recovered by neglecting fluctuations in the cascade process. When these fluctuations are included, correlation functions of different order do not scale in the same way. This is in accord with phenomenological theories of intermittency and with experiment. An effective critical exponent ${\eta }^{\prime }=\frac{2}{3}+\zeta$ is still defined by the energy spectrum $E\left(k\right)\mathrm{}$. The parameter $E\left(k\right)\mathrm{}$ is one of a family of intermittency exponents which are intrinsic properties of the Navier-Stokes equations. Their calculation from the dynamical equations remains an objective for future work.

A variational method for calculating the thermodynamic properties of fluids and fluid mixtures, based on hard-sphere perturbation theory, is applied to the screened Coulomb gas and compared with recent Monte Carlo calculations. Agreement is good and indicates that the properties of liquid metallic hydrogen and liquid metallic hydrogen-helium mixtures may be adequately simulated by such a model.

A new theoretical treatment of the problem of orbital-electron ejection during $\alpha$ decay is described. The internal-ionization probability appears as the special zero-impact-parameter trajectory in a generalized impact-parameter formulation of the binary-encounter approximation described recently by the author. The probability of internal ionization during α decay of ${}_{}{}^{210}{}_{}{}^{}\mathrm{Po}$ is calculated for the $1s$, $2s$, $2p$, $3s$, $3p$, and $3d$ subshells in the daughter Pb atom. Markedly improved agreement with experimental results is found.

The structure factor of supercooled liquid gallium and mercury has been measured as a function of temperature with $\mathrm{Cu}-K\alpha$ x radiation. The use of samples consisting of a planar array of a large number of small droplets of about 10-μm diameter allowed supercooling down to 70°C and 40°C below the melting points of gallium and mercury, respectively. The experimental data are analyzed and discussed in view of recent theories of the liquid-solid transition.

We have studied the simple shear flow, in the laminar regime, of a nematic (uniaxial) liquid film between two glass plates with carefully imposed boundary conditions. In the case considered here, the optical axis at rest is normal both to the flow velocity and to the velocity gradient. Using various methods of optical observation, we find the following facts: (a) When the shear rate s is below a certain threshold $s_c$, the optical axis is unperturbed everywhere. When $s>\mathrm{}{s}_c$, it becomes distorted. $s_c$ is inversely proportional to the sample thickness. (b) When a stabilizing field $H$ is applied, $s_c$ increases. Furthermore, above a certain limiting field $H_L$, the nature of the instability changes: a pattern of rolls appears, the rolls being parallel to the (average) flow lines. These effects are then explained in terms of the Ericksen-Leslie-Parodi equations describing the couplings between orientation and flow in a nematic fluid. This analysis has led in turn to the prediction and observation of other remarkable effects occurring when the shear rate is modulated (at low frequencies) and when two fields $H$ (magnetic) and $E$ (electric) are applied at right angles.

The effect of the antiresonant atom-field interactions on the phase-transition property of the Dicke model is studied. A phase transition is shown to exist when the following inequality is satisfied between the coupling constant $\lambda$ and the ratio of the atomic transition frequency to the field frequency $\epsilon$: ${\lambda }^2>\frac{\epsilon }{4}$. Relative to the Dicke-model result the necessary value of ${\lambda }^2$ is thus smaller by a factor of ¼. In addition it is established that the critical temperature is higher than in the Dicke model. A further significant difference between the two models is reflected in the properties of the "photon statistics." In the present model an additional polynomial temperature dependence in the powers of the average number of photons per atom is found. It is concluded that the effect of the antiresonant terms is not negligible concerning the thermodynamic properties of the system, in contrast to the evidence for many time-dependent properties studied previously.

Various coherent-optical-pulse profiles are known to exhibit lossless propagation during transmission through a resonant atomic medium. It is shown that the requirement of lossless propagation provides asymptotic information that is sufficient to determine these profiles by the inverse method. Pulse profiles are obtained as solutions to the Marchenko equation. The analysis is applied explicitly to the propagation of $2\pi$, $4\pi$, and $0\pi$ pulses in an inhomogeneously broadened medium.

A relation between the maximum number of propagating hydrodynamic modes and the number of variables that are odd under time reversal and which are coupled to variables that are even under time reversal, is obtained using the Onsager symmetry relations.

We investigate, using neutron-diffraction data, several properties of the local atomic structure in liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$, both above and below the superfluid transition. Distinguishing features of the pair-correlation function are summarized and the diffraction data are employed to investigate a proposed form for the condensate fraction as a function of temperature. The pair-correlation function is used to examine an inequality that gives an upper bound that is close to the observed values, and also to examine the use of approximate integral equations for the ground-state wave function. Triplet correlations and closure approximations are studied from the isothermal density derivative of the pair-correlation function. The results suggest that, analogous to the pair-correlation function, the triplet-correlation function shows a temperature dependence that is not observed in simple classical fluids.

Measured and calculated profiles are compared for the two neutral helium lines, He I ($\lambda =4471\mathrm{}$ and $\lambda =6678\mathrm{}$ Å) and their forbidden components at electron densities between 1.6 × ${10}^{17}$ and 4.4 × ${10}^{17}$ ${\mathrm{cm}}^{-3\mathrm{}}$. The partially ionized helium plasma was created in the reflected shock of an electromagnetically driven $T$ tube. The electron temperature was measured as ∼ 25 000 °K. The calculated profiles include a random-phase approximation for the electrons to second-order in the frequency-dependent shift and width operators, and the quasistatic approximation for the ions. Calculations were done at the measured plasma conditions and are compared directly with the experimental profiles. Agreement between theory and experiment is good even though in this density range the line He I ($\lambda =6678\mathrm{}$ Å) passes from the "isolated-line" to the "overlapping-line" regime, while the line He I ($\lambda =4471\mathrm{}$ Å) passes from the "overlapping-line" to the "degenerate-hydrogenic" regime.

Structure factors of liquid helium have been determined from neutron-diffraction measurements of high statistical accuracy. Diffraction measurements were performed out to momentum transfers of 7 ${\mathrm{\AA }}^{-1\mathrm{}}$ for three different densities of liquid helium at a constant temperature above the helium $\lambda$ transition and at a nearly constant temperature below the $\lambda$ transition for the same three densities. Statistically significant differences in the structure factors are observed as the density is varied at constant temperature and for temperatures above and below the $\lambda$ transition at constant density. The radial pair-correlation functions have been calculated from the liquid-structure factors. The structure factors or the related radial pair-correlation functions can be used to obtain information about three-atom correlations in liquid helium above and below the $\lambda$ transition from a construction of their isothermal density derivative. The temperature dependence of the constant-density structure factors or their derived pair-correlation functions can also be used to test a current theoretical estimate of the condensate fraction in liquid helium.