The effect of the earth's gravity on the total irradiance of light scattered from a simple fluid near its critical point is investigated. It is found that gravity modifies the results of the usual theories for the total scattering (turbidity) qualitatively in two ways. First, the total scattering at the critical point (in the Born approximation) does not diverge. Second, the turbidity does not achieve its maximum value at the critical temperature, but rather above it. For example, the turbidity of xenon with a sample height of 1 cm reaches its maximum value at about ${10}^{-2\mathrm{}}$ K° above its critical temperature. By comparing these results with those of the modified Ornstein-Zernike theory (critical exponent $\eta \ne 0$) for a uniform fluid, an apparent gravity-induced $\eta$ is defined. This is a temperature- and sample-size-dependent quantity which, when substituted for $\eta$ in the modified Ornstein-Zernike theory for a uniform fluid, produces the same turbidity as the gravity-modified classical Ornstein-Zernike theory for a nonuniform fluid. It is surprising that the apparent $\eta$ increases rapidly, until it is of order unity, within a very short temperature range near the critical point, as the critical temperature is approached from above. It is more surprising that the sudden change in the apparent $\eta$ is quantitatively commensurate, through the scaling law $\gamma =\nu (2-\eta )$, with the rapid change in the exponent $\gamma$ observed by a number of experimenters.

We have found a new instability for a Maxwellian plasma immersed in a uniform magnetic field with frequency near the lower-hybrid frequency. For zero perpendicular temperature and equal ion and electron temperatures, the mode is unstable for $\beta =\frac{4\pi \mathrm{nT}}{B^2}>0.549\mathrm{}$ with maximum growth for almost perpendicular propagation. This instability is in addition to the usual "hose" instability for parallel propagation and the two modes with zero real part (the ion mode of Forslund and our recent electron mode), both of which grow fastest for almost perpendicular propagation.

Asymptotic expansions for the thermodynamic functions of an ideal spin-½ Fermi gas in a weak magnetic field are obtained about the point $z=1\mathrm{}$, where $z$ is the fugacity. We consider paramagnetism, diamagnetism, and paramagnetism plus diamagnetism together. To relate to liquid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ films and bulk fluid, we study the spin paramagnetism in two and three dimensions.

Jahnke, Meyer, and Rice have reported anomalous maxima in the low-field mobility of excess electrons in fluid Ar near the liquid-gas critical temperature. Lekner has proposed a zero-scattering-length model to explain the maxima. We propose an alternative explanation which depends on structural changes induced in a compressible fluid during a mobility experiment rather than some dramatic change in the electron-atom scattering length with density. The basic idea is that inhomogeneities of density higher than that of the normal fluid may be established and sustained in the compressible fluid through the attractive electron-Ar interaction provided by a steady-state flux of excess electrons. Once established, these dense regions may provide contiguous channels of relatively high conductivity since condensed phase diffraction effects in dense, incompressible Ar more effectively reduce the excess electron scattering. Furthermore, excess electrons are preferably channeled into the denser regions since they provide a more attractive volume-averaged effective electron-Ar interaction (i.e., optical potential). Numerical calculations of the mobility are performed employing a simple two-phase model of the electronically perturbed fluid. The lower-density phase is regarded as the normal fluid with the usual equilibrium density at a given temperature and pressure. The volume fraction of the denser phase is assumed to grow as ${\xi }^3$, where $\xi$ is the Ornstein-Zernike long-range correlation length. Estimates of the density calculated from linear-response theory indicate the denser regions are solid-like except at the highest temperatures where the mobility was measured. Electron propagation is favored in the denser regions by an attractive optical potential which is assumed to be proportional to the density difference of the two phases. The mobilities calculated from this model are in reasonably good agreement with experiment in the critical region; the magnitudes and location of the mobility maxima are well predicted. The volume fraction of the denser domains required for the model to yield reasonable experimental agreement does not exceed ${10}^{-7\mathrm{}}$.

Properties of ${\mathrm{H}}_2$ are investigated using an analytic anisotropic potential which has been deduced from recent atomic orbital and perturbation calculations. The low-pressure solid results are based upon a spherical average of the anisotropic potential. The calculated ground-state energy is $E_0=-88.76\mathrm{}\pm 2$ K. The pressure-volume curve agrees with experiment to within 10% over the range $9\le V\le 22.65$ ${\mathrm{cm}}^3$/mole ${\mathrm{H}}_2$. The high-pressure solid properties are calculated using the anisotropic potential for particular frozen orientations, as well as the spherically averaged potential. The structures investigated are the $P_{a}3$ and $\frac{P{4}_2}{\mathrm{mnm}}$ orientations. The $\frac{P{4}_2}{\mathrm{mnm}}$ orientation yields energies and pressures 10-20% lower than either the spherical average or the $P_{a}3$ arrangement. Agreement with experimental shock-wave data is tolerable. The metal-insulator phase-transition pressure is predicted to be between 1.61 × ${10}^6$ and 3.76 × ${10}^6$ atm, depending on the metallic equation of state used. Second virial coefficients $B\left(T\right)\mathrm{}$ are calculated for ${\mathrm{H}}_2$ and ${\mathrm{D}}_2$ over the range $60 \mathrm{K}\le T\le 523 \mathrm{K}$, using a formalism which fully accounts for the potential anisotropies and the discrete rotational spectrum. The results are in excellent agreement with experiment except at high temperatures, where the discrepancy is nearly 10%. A comparison of the results with those obtained using the spherically averaged potential indicates that the effect of anisotropies on $B\left(T\right)\mathrm{}$ is small. This coupled with the results from solid calculations implies that anisotropies are generally not very important except at extremely high pressures. The difference in $B\left(T\right)\mathrm{}$ between ortho and para ${\mathrm{H}}_2$ and ${\mathrm{D}}_2$ is also calculated.

A modified version of the Van der Waals theory of fluid interfaces is presented. The modified theory is shown to retain the qualitative simplicity of the original theory while yielding a much more quantitative description of fluid properties.

Cross sections for $K$-shell ionization have been determined from relative x-ray yields normalized to Auger-electron-production data at low proton energies. The asymptotic form of the plane-wave Born approximation reproduces the data at high energies, but the binary encounter approximation underestimates the cross sections in this region by a factor of 2.

We discuss a priori calculations of the three-body, nonadditive, inert-atom interaction energies using an electron-gas approximation recently introduced by Gordon and Kim. The results are compared with other theoretical estimates and shown to considerably underestimate crystal cohesive energies; inadequacies and refinements of the approach are suggested.

The effect of neglecting ionization in the calculation of numerical bounds on transition probabilities is examined for proton-hydrogen-atom scattering, and it is found to be significant. This implies that numerical bounds on transition probabilities must necessarily be of limited accuracy, since it does not appear to be possible to take ionization even partially into account without the bounds becoming infinite.

It is demonstrated that a good share of the error Shakeshaft attributes to the failure to account for ionization in customary impact-parameter calculations for proton-hydrogen-atom scattering amplitudes really results from the inadequacy of the traveling hydrogenic basis set to account for the dynamic polarization of the hydrogen atom by the moving proton. The lower limit for the first-order bound can be reduced by using hydrogenlike basis functions that allow for this polarization. Bounds on the cross sections obtained by using the bound ${\Delta }_1$ need not be infinite. The inclusion of time-dependent adjustable parameters in the basis functions provides a method for modifying the projection of the deviation vector or error term in the Schrödinger equation in the continuum. The exploratory work of Storm and Rapp appears to offer hope that reasonably accurate bounds on at least the $1s$ charge exchange amplitudes and cross sections can be obtained by employing only square-integrable basis functions that contain time-dependent variable parameters. However, if it is necessary to account for the flux in the ionization channels, it is shown that an account could be made without the bound becoming infinite.

As an addendum to the recent work of Stillinger and Herrick, we have constructed a separable nonlocal potential which generates a bound state in the continuum.

This note draws attention to a small correction to the calculated value of the polarizability of the hydrogen molecule, arising from the use of the Born-Oppenheimer approximation. After including the correction, the discrepancy between theory and experiment is identical in the two cases of ${\mathrm{H}}_2$ and ${\mathrm{D}}_2$.

Nonrelativistic wave functions and energy values for the ground state of helium are calculated accurately by using the conventional Rayleigh-Ritz technique. However, the trial wave functions used are more general than those used by Hylleraas and others. They contain interelectronic separation coordinates in the exponential function beside the variational parameters. Groups of three- and four-parameter wave functions are obtained with comparatively smaller energies. Also, improved wave functions and energies have been obtained by solving determinants of orders 7, 13, 22, 34, 50, 70, 95, and 125, using a simple model. The last determinant yields an energy value of -2.903 724 371 a.u., as compared with -2.903 724 370 a.u. obtained by Pekeris by solving a determinant of order 715. The first determinant (ten parameters) yields an energy value of -2.903 425 858 a.u., as compared with the recent value of -2.9022 a.u. computed by Tweed with the use of a 41-parameter wave function.

Using the ab initio energy-minimization procedure of Bass, Green, and Wood, we determine two potential parameters, $\xi$ and $\eta$, characterizing the independent-particle-model potential of Green, Sellin, and Zachor (GSZ) for atoms and positive ions with $36<Z\le 54$. This extends earlier modified-Hartree-Fock (MHF) calculations of Szydlik and Green and of Green, Garvey, and Jackman. We find that both of the parameters in question display, to a good approximation, a linear dependence on the degree of ionization $Z-N$ for fixed numbers of electrons $N$. The slopes and $y$ intercepts associated with the linear dependence of $\xi$ display marked shell-like behavior, while those associated with $\eta$ vary rather smoothly with $N$. Our determinations of total energies are usually within 50 ppm of earlier Hartree-Fock calculations for those cases in which such calculations exist. Using the entire collection of energies and GSZ minimization parameters now available, we reexamine a modified version of the Thomas-Fermi statistical model (MTF) due to Green, Sellin, and Darewych. We show that this model is capable of yielding the linear $Z-N$ dependence of the GSZ parameters which we found empirically in the MHF work. By numerical adjustment of the coefficients of our MTF model, we obtain energies of stable atoms and ions, as well as GSZ potential parameters which are in good agreement with the MHF calculations.

The contact hyperfine structure constant for the ${}_{}{}^4{}_{}{}^{}S$ state of the nitrogen atom is computed variationally by a multiconfiguration self-consistent-field method. The present calculations take account of radial-correlation contributions to spin density, but do not take account of any significant part of angular correlations. It is found that the calculated value of 0.0977 for the spin density is in excellent agreement with the experimental value of 0.0972. A mechanism called "core radial polarization," for providing contributions mainly to the spin density at the nucleus, is proposed. It is noted that the mechanism provides a main contribution for the ${}_{}{}^4{}_{}{}^{}S$ state of nitrogen as well as for the ${}_{}{}^2{}_{}{}^{}P$ state of lithium. It is also claimed that the proper description of radial correlations should be a prerequisite for a good spin density.

Diagrammatic quasidegenerate Rayleigh-Schrödinger perturbation theory is used for the construction of an effective Hamiltonian, defined completely in a finite-dimensional model space. This effective Hamiltonian, containing one-, two-, three-, etc. particle terms, may serve as a useful tool for constructing an ab initio theory of quantum semiempirical effective Hamiltonians, widely used in quantum molecular and atomic physics, and also in some modified form in solid-state physics.

The influence of electron-electron interaction effects on nuclear quadrupole antishielding factors ${\gamma }_{\infty }$ is studied for the negative halide ions ${\mathrm{F}}^{-}$, ${\mathrm{Cl}}^{-}$, and ${\mathrm{Br}}^{-}$. Using a procedure developed earlier which utilizes the nuclear-quadrupole-moment-perturbed and external-point-charge-perturbed wave functions for the ions, obtained by the differential equation procedure for the zero-order antishielding factor ${\left({\gamma }_{\infty }\right)}_0$, we have computed the consistency contributions ${\left({\gamma }_{\infty }\right)}_1$ to be +2.28, -1.50, and -7.16 in ${\mathrm{F}}^{-}$, ${\mathrm{Cl}}^{-}$, and ${\mathrm{Br}}^{-}$, respectively. The resulting values of ${\gamma }_{\infty }$ correct to first order in electron-electron interaction are then -19.29, -56.58, and -140.83. The appearance of negative signs for ${\left({\gamma }_{\infty }\right)}_1$ in ${\mathrm{Cl}}^{-}$ and ${\mathrm{Br}}^{-}$ is seen to be due to large negative interactions between $p$ electrons belonging to different shells. This intershell effect is not present in ${\mathrm{F}}^{-}$ ($1s^{2}2s^{2}2p^6$). The contributions to ${\left({\gamma }_{\infty }\right)}_1$ for all three ions are analyzed in detail, and a comparison is made of ${\left({\gamma }_{\infty }\right)}_1$ for ${\mathrm{F}}^{-}$ and the isoelectronic positive ion ${\mathrm{Na}}^{+}$. An explanation is proposed for the comparable values of the ratio $\frac{{\left({\gamma }_{\infty }\right)}_1}{{\left({\gamma }_{\infty }\right)}_0}$ found for these two ions, even though the ${\mathrm{F}}^{-}$ ion is substantially more deformable with a larger value of ${\left({\gamma }_{\infty }\right)}_0$. From the results of the present investigations, as well as earlier ones in related positive ions, it is felt that the consistency contribution ${\left({\gamma }_{\infty }\right)}_1$ should, in general, be less than 15% of ${\left({\gamma }_{\infty }\right)}_0$ for most ions.

A formalism is discussed whereby many-electron systems may be described by a great number of product functions of different configurations in a manageable way. Calculations are made to find triply excited energy levels of three-electron atomic systems, using products of hydrogenic functions with a $Z$ equal to that of the nucleus. Results for the lowest ${}_{}{}^2{}_{}{}^{}P_{}^o{}_{}{}^{}$ and ${}_{}{}^2{}_{}{}^{}S_{}^e{}_{}{}^{}$ levels in the Li isoelectronic sequence, $Z=1\mathrm{} \mathrm{to} 10$, are presented. The results of calculations for the energies of ${}_{}{}^2{}_{}{}^{}D_{}^e{}_{}{}^{}$ triply excited states in ${\mathrm{He}}^{-}$ are also reported. The calculated energies are compared with available experimental and other theoretical values. Finally, the lowest ${}_{}{}^2{}_{}{}^{}S_{}^e{}_{}{}^{}$ and two lowest ${}_{}{}^2{}_{}{}^{}P_{}^o{}_{}{}^{}$ energy levels in the isoelectronic sequence are fitted by a $\frac{1}{Z}$ perturbation expansion. It is seen that this expansion can be used to estimate the energy levels of such states for higher nuclear charges.

A new method for determining atomic shielding factors to use in hydrogenlike wave functions which gives "exact" Compton profiles in close agreement with experiment was recently developed by the present authors. These shielding factors $Z^{*}\mathrm{}\left(0\right)\mathrm{}$ are determined by imposing the requirement for each atomic orbital that the "impulse" hydrogenic Compton-profile value at its center, $q=0\mathrm{}$, match the Hartree-Fock Compton-profile value. We now compare the use of $Z^{*}\mathrm{}\left(0\right)\mathrm{}$ hydrogenic values in the calculation of the moments $〈r〉$, $〈r^2〉$, and $〈r^4〉$ with values of these moments obtained using Slater, Clementi, Burns, and binding-energy-matched $Z^{*}\text{'}\mathrm{s}$ used in hydrogenic wave functions for $K$-, $L$-, and $M$-shell orbitals for rare-gas atoms up through xenon. Relative accuracy is determined by comparison to Hartree-Fock results for these moments. It is found that for $1s$, $2p$, and $3d$ states, our $Z^{*}\mathrm{}\left(0\right)\mathrm{}$ values give the most consistently accurate results. A table of recommended $Z^{*}$ values is given for all atoms and states considered.

The low-lying excited states of ${\mathrm{He}}_2$ have been examined using spatially projected generalized-valence-bond wave functions. Typically the excited-state potential curves are repulsive at large $R$ (internuclear distance) but exhibit an attractive well at small $R$. We find that the repulsion at large $R$ results from an unfavorable exchange interaction of a Rydberg orbital mostly on one He with the two core orbitals on the other He. This arises from the restrictions on the molecular wave function implicit in the Pauli principle and is similar in nature to the repulsive pair-pair interactions between ground state He atoms. For $R<\mathrm{}3\mathrm{}{a}_0$ the Rydberg orbital is large compared with $R$ and the shape of the potential curve is determined by core-core interactions. For the $A{}_{}{}^1{}_{}{}^{}{\Sigma }_u_{}^{+}{}_{}{}^{}$ state, the calculated maximum in the potential curve is 0.0607 eV at 3.09 Å, in excellent agreement with experimentally derived values. Results are presented for several other excited states.