Recombination coefficients of electrons with ions of the hydrated hydronium-ion series ${\mathrm{H}}_3{\mathrm{O}}^{+}·{\left({\mathrm{H}}_2\mathrm{O}\right)}_n$, where $n=0,1,2,3,4,5,6\mathrm{}$, have been determined with a microwave after-glow/mass-spectrometer apparatus. Afterglow measurements of electron density decays in helium-water-vapor mixtures are correlated with the decay of mass-identified ion currents to the wall of the microwave cavity. By varying the temperature of the gas and the partial pressure of the water vapor in the mixture, different groups of hydronium-series ions are made to dominate the afterglow. The measured recombination coefficients, in units of ${10}^{-6\mathrm{}}$ ${\mathrm{cm}}^3$/sec, are given. The implications of these results concerning ionization levels in the $D$ region of the ionosphere are discussed.

The self-diffusion coefficient in Kr has been measured throughout a large range of densities ($0.35<\mathrm{}\rho <2\mathrm{}$ ${\mathrm{g}/\mathrm{c}\mathrm{m}}^3$) at temperatures near the critical one. The results are compared with molecular-dynamic calculations and with the C${\mathrm{H}}_4$ behavior. Although the qualitative agreement is good, disagreement from a quantitative point of view was found. The "normal behavior" of the self-diffusion coefficient in the critical region has been deduced.

The contribution of small-momentum-transfer processes to the $T^3 \ln T$ term in the specific heat of a normal Fermi liquid is considered using Landau theory, and is evaluated exactly in terms of Landau parameters. This contribution is shown to be due to a term in the quasiparticle interaction, $f_{\overrightarrow{\mathrm{p}}, \overrightarrow{\mathrm{p}}+\overrightarrow{\mathrm{q}}}$, which varies as ${\left(\hat{p}·\hat{q}\right)}^2$ when $\overrightarrow{\mathrm{q}}\to 0$; the physical process responsible for this behavior is the repeated scattering of quasiparticle-quasihole pairs. It is well known that for the two-body problem the interaction energy, while not, in general, equal to the real part of the forward-scattering amplitude, may be expressed in terms of the scattering amplitude. We take over the results for the two-body problem and apply them to a quasi-particle-quasihole pair. This gives an expression for $f_{\overrightarrow{\mathrm{p}},\overrightarrow{\mathrm{p}}+\overrightarrow{\mathrm{q}}}$ in terms of the quasiparticle-quasihole scattering amplitude, which, since $\overrightarrow{\mathrm{q}}$ is small, may be expressed in terms of Landau parameters. We apply our results to liquid ${\mathrm{He}}^3$ at low pressure.

The stability of electromagnetic waves propagating transverse to the direction of streaming in a system comprised of identical nonuniform plasmas contrastreaming along a nonuniform magnetic field is investigated. It is found that the low-frequency waves ${\omega }^2\ll {\Omega }_i^2$ ($\omega$ and ${\Omega }_i$ being the wave frequency and the ion cyclotron frequency, respectively) can become unstable due to the excitation of either resonant ion instability or resonant electron instability. The latter instability can be excited only in the presence of streaming. The drift cyclotron instability gets excited for some bounded values of $U_e$, $U_i$, $k$, $\beta$, and $\mu$ ($U_e$ being the electron streaming velocity, $U_i$ the ion streaming velocity, $k$ the characteristic wave number, $\beta$ the ratio of kinetic pressure to magnetic pressure, and $\mu$ the inhomogeneity in the magnetic field). The effect of ion temperature on this mode is stabilizing and the typical growth rates associated with this unstable mode are of the order of ${\Omega }_i$. The presence of even a weak inhomogeneity can render the otherwise marginally stable waves unstable, their growth rates being as high as $0.05{\Omega }_e$ (${\Omega }_e$ being the electron cyclotron frequency). However, the effect of $\mu$ and $\nabla n$ on the transversal instability (for frequencies ${\Omega }_i^2\ll {\omega }^2\ll {\Omega }_e^2$) is to increase the growth rate of the unstable waves slightly.

The method described in this paper permits the determination of reaction-rate constants for formation and loss of each type of ion through the measurement of the radial distribution of the various ion densities in the steady-state negative glow of a cylindrical hollow cathode. The method is explained with the following example: A discharge is produced in an Ar + 0.15% ${\mathrm{H}}_2$O mixture, and a quantitative examination is made of the principal reactions: $\mathrm{Ar}+e\stackrel{k_1}{\to }{\mathrm{Ar}}^{+}+2e$; ${\mathrm{Ar}}^{+}+{\mathrm{H}}_2\mathrm{O}\stackrel{k_2}{\to }\mathrm{Ar}{\mathrm{H}}^{+}+\mathrm{OH}$; $\mathrm{Ar}{\mathrm{H}}^{+}+{\mathrm{H}}_2\mathrm{O}\stackrel{k_3}{\to }{\mathrm{H}}_3{\mathrm{O}}^{+}+\mathrm{Ar}$; ${\mathrm{H}}_2{\mathrm{O}}^{+}+{\mathrm{H}}_2\mathrm{O}\stackrel{k_4}{\to }{\mathrm{H}}_3{\mathrm{O}}^{+}+\mathrm{OH}$; ${\mathrm{Ar}}^{+}+{\mathrm{H}}_2\mathrm{O}\stackrel{k_5}{\to }{\mathrm{H}}_2{\mathrm{O}}^{+}+\mathrm{Ar}$; and ${\mathrm{H}}_3{\mathrm{O}}^{+}+e\stackrel{\alpha }{\to }{\mathrm{H}}_2\mathrm{O}+\mathrm{H}$, the loss reaction. The reaction constants obtained have the following values, all in units of ${\mathrm{cm}}^3$ ${\sec }^{-1\mathrm{}}$: $k_1=1.2\mathrm{}\times {10}^{-7\mathrm{}}$; $k_2=1.3\mathrm{}\times {10}^{-9\mathrm{}}$; $k_3=4.5\mathrm{}\times {10}^{-9\mathrm{}}$; $k_4=1.3\mathrm{}\times {10}^{-9\mathrm{}}$, and $k_5=1.5\mathrm{}\times {10}^{-10\mathrm{}}$; $\alpha =1.1\mathrm{}\times {10}^{-6\mathrm{}}$. The values of $k_4$ and $\alpha$ are in good agreement with those given by other authors. The remaining values of constants are believed to be new. A rough estimate is also made of the density of the fast dark-space electrons ($E_{\mathrm{kin}}\gtrsim 50$ eV) that exist in the negative glow.

A simple model incorporating the gross features of the structure and phonon spectrum of hcp ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ can explain the very large anisotropy of the thermal-conductivity tensor. In the model, a combination of isotropic Debye and Einstein dispersion relations for the acoustic and low-lying optic-phonon branches is used within a "Debye ellipsoid" which replaces the hexagonal Brillouin zone. The arrangement of the reciprocal-lattice vectors for three-phonon umklapp processes is the same as in the true crystal. Variational calculations of the baseplane and $c$-axis conductivities give reasonable fits to experiment (at 18.3 ${\mathrm{cm}}^3$/mole) with a total of only four semiadjustable parameters (two phonon frequencies and two Grüneisen constants). Because of the arrangement of the reciprocal-lattice vectors for umklapp processes, the base-plane and $c$-axis conductivities probe the phonon spectrum in different directions, and their difference reflects two features: differences in zone-boundary phonon frequencies between the two principal-axis directions and the shape of the transverse-optic branch. No explicit quantum-crystal effects appear to be involved.

Transition amplitudes corresponding to bound-bound multiphoton transitions with the emission or absorption of one and two Raman photons in the hydrogen atom have been evaluated following a recent method of Reiss. Results show that the transition probability varies smoothly for low values of $N$, where $N$ is the number of intense photons absorbed or emitted during the transition. For larger values of $N$ the amplitude, however, has oscillatory behavior. A particular feature of the present method of calculation is that we find the "reduced" transition amplitude for induced emission of two Raman photons is enhanced compared with single-Raman-photon emission. Another distinctive feature is the appearance of many more extrema in the transition amplitude for higher transitions for multiphoton processes at high intensity.

The asymptotic expansion of the incoherent-scattering function in inverse powers of the magnitude of the momentum transfer $\kappa$, which was given in a previous article for simple classical liquids, is generalized within the Gaussian approximation to include quantum effects. Inverse-$\kappa$ expansions are also obtained for the full width at half-maximum, the mean frequency, and the frequency of the maximum of the scattering function. The results are applied in the interpretation of recent neutron-inelastic-scattering data for liquid neon, liquid rubidium, and hydrogen gas.

A molecular-dynamics study of clustering in systems of 400 and 900 hard parallel squares shows that in the high-density region the distributions $N\left(n\right)\mathrm{}$ of clusters of size $n$ depend only upon one dimensionless variable. Results are slightly different in the intermediate-density fluid region. For all densities and values of the clustering distance parameter $l$, $N\left(n\right)\mathrm{}$ is a monotonic decreasing function of $n$. A simple free-volume theory predicts relatively well the results in the high-density region. The clusters are relatively compact. From an exponential fit to the function $\frac{N\left(n\right)\mathrm{}}{N\left(1\right)\mathrm{}}$ obtained for densities near the phase-transition region, one estimates the values of the parameters of the exponential law given by the classical theory of nucleation.

Linewidth, angular dissymmetry, and hydrodynamic-shear-viscosity data of a critical binary-liquid mixture of isobutyric acid in water compares well with the modified mode-mode coupling theory of Kawasaki in the range $11\ge K\xi \ge 0.0067$ after corrections for vertex, non-local shear viscosity and deviation of the Ornstein-Zernike correlation function. Deviations between experiments and theory in the critical region ($K\xi >4\mathrm{}$) suggest needs for further improvements in theory at large values of $K\xi$ as well as measurements of the hydrodynamic shear viscosity at small temperature distances from the critical mixing point.

A discussion of the numerical performance of singular integrals appearing in the Fredholm series leads to a procedure for obtaining the Fredholm determinant over a continuous range of energies using a single fixed quadrature mesh. The same analysis shows that the singular momentum-space Lippmann—Schwinger equation may be inverted by direct-matrix methods with no error arising from the singularity.

Some consequences of the rotating-wave approximation (RWA) for spontaneous emission are explored. The Winger—Weisskopf theory of the spontaneous exponential decay of a single excited atom is shown to contain an implicit form of the RWA. However, to describe the cooperative superradiant level shift, the RWA should not be used, and care should be taken over the choice of interaction Hamiltonians.

It is maintained that in a recent comparison of quantum-electrodynamical with semiclassical treatments of the interaction between atoms and radiation by Chang and Stehle the semiclassical work was misrepresented and that the conclusions reached by these authors are invalid.

A method for obtaining analytical expressions for $2N\pi$ pulses in an unbroadened resonant medium is derived by a systematic application of the Bäcklund transformation. The resulting large-area pulses display the expected property of pulse breakup into separate $2\pi$ hyperbolic secant pulses.

This is a note concerned with a previous publication by R. Klingbeil, V. G. Kaveeshwar, and R. P. Hurst [Phys. Rev. A 4, 1760 (1971)] giving specific results for neon.

An estimate of ${\alpha }^4{R}_{\infty }$ contributions to positronium and muonium hfs is made in somewhat greater detail than heretofore. Comparison with the most recent experimental data suggests that such additional corrections could well improve agreement between theory and experiment. Indeed, the one immediately calculable ${\alpha }^4{R}_{\infty }$ contribution almost totally accounts for present discrepancies.

It is pointed out, with specific examples, that the physically relevant liquid-vapor transition can have a singular coexistence-curve diameter, even in lattice-gas models described by Hamiltonians with particle-hole symmetry.

The many-body perturbation theory of Brueckner and Goldstone is used to calculate the correlation energy of the neutral argon atom. The result is -0.685 a. u. Three-body effects among electrons in the outer shells are examined in detail, and their contribution to the correlation energy is 0.0553 a. u. Estimates are given for four-body and higher-order terms.

The spin-orbit interaction for $p$ electrons is shown to occur in the $K$ shell for all atoms and to vary with atomic number $Z$ according to $\Delta \mu \propto H{\alpha }^2\frac{(0, Z)}{Z}$, where $\Delta \mu$ is the difference in quantum defect between the fine-structure levels evaluated at the series limit, $\alpha (0, Z)$ is the amplitude function for a nonrelativistic zero-energy $p$ wave at $r=0\mathrm{}$, and $H$ is a relativistic correction factor. The quantity $\Delta \mu$ is evaluated and found to vary as $Z^{2.33}$, which is in remarkable agreement with experiment. In addition to this gross behavior, the variation of ${\alpha }^2\frac{(0, Z)}{Z}$ exhibits a structure tied to the Periodic Table. This structure derives from the influence of the valence region of the atomic potential, which affects the spin-orbit interaction through the normalization factor $\alpha (0, Z)$. Neither inner screening effects, embodied in Casimir's $\sigma \mathrm{}\left(Z\right)\mathrm{}$ parameter, nor deviations of the atomic potential from $Z{r}^{-1\mathrm{}}$ in the interaction region appear very relevant to this problem.

Effect of correlation between the two participating electrons is investigated for $\mathrm{KLL}$ Auger transitions in the region $12<~Z<~21$ using a simple ansatz for the correlated two-electron wave function. For simplicity, the screened-hydrogenic model is used for these wave functions in the limit of no correlation. Our results for relative intensities are compared with the uncorrelated hydrogenic intensities and with the results of earlier hydrogenic calculations. An enhancement of these intensities is observed in the right direction.