Light scattering from Brownian particles suspended in a viscous fluid was used to measure the macroscopic velocity of the fluid confined within a finite lateral system whose top and bottom were kept at different temperatures. The results indicated that a finite heat conduction of the side wall causes a macroscopic fluid motion even much below the critical Rayleigh number for a finite system, and that the subcritical flow could suppress critical fluctuations which are expected for an infinite system. The measured value of the macroscopic fluid motion agreed qualitatively as well as quantitatively for $-{10}^{-2\mathrm{}}\lesssim \epsilon =\frac{(\Delta T-\Delta T_c)}{\Delta T_c}\lesssim -0.5\mathrm{}$ with the theoretical prediction of Daniels and with that of Hall and Walton. Required experimental conditions necessary to observe critical fluctuations are discussed.

The influence of dissipative effects on the nonlinear hydrodynamic equations of various types of liquid crystals like nematics, cholesterics, smectics $A$, $C$, is investigated. An experiment for uniaxial nematics is proposed which allows the detection of one group of the novel terms presented. It is shown that the viscous nonlinearities in the stress tensor of uniaxial nematics discussed by Moritz and Franklin do not exist.

The x-ray photoelectron shake-up spectra on the Ar $2p$ and Kr $3d$ core levels have been obtained to high statistics. The shake-up energies and intensities on the Ne $1s$, Ar $2p$, Kr $3d$, and Xe $3d$ levels have been calculated using Hartree-Fock-Slater $X\alpha$ calculations. These calculations generally show good agreement with experiment and confirm the assignments. The shake-up energies are in good agreement with the optical-transition energies for the equivalent core alkali-metal ions. The relaxation energies from shake-up have been calculated from the experimental spectra, and are shown to be much smaller than the relaxation energies from shake-off.

To every solution of the time-independent Dirac equation for the electron in an arbitrary magnetic field there belongs a second solution. A characteristic twofold degeneracy, suggested by this circumstance, is shown to be excluded for the ground state at the energy $E=m{c}^2$,, with the further distinction of this state that the current density vanishes here at all points in space. The conditions under which the excited states exhibit such a degeneracy are discussed and in the case of its occurrence are shown to allow the construction of a particularly symmetrical pair of mutually orthogonal solutions.

The accuracy of an energy-density functional that contains kinetic-energy terms up to fourth order in the gradient expansion and exchange-energy terms up to second order, is tested by constructing a trial charge density in the same way as is done in a minimal-basis-set-Hartree-Fock-type calculation. It is found that while the total energies for He, Ne, Ar, and Kr, obtained by optimization of the parameters through the statistical functional differ slightly from those determined by the Hartree-Fock method. The electron densities are poorly described, especially in Ar and Kr. The results for the molecules C${\mathrm{H}}_4$, N$\mathrm{H}_4^{}{}_{}{}^{+}$, B$\mathrm{H}_4^{}{}_{}{}^{-}$, Si${\mathrm{H}}_4$, and Ge${\mathrm{H}}_4$ within the one-center approximation are also reported and compared with similar Hartree-Fock calculations to test the accuracy of the model to predict equilibrium distances. The same type of density is used in connection with an energy-density functional whose kinetic-energy part is given by a partitioned-density model recently proposed by Bader, which leads to similar results.

Hitherto known formulas for the translation of exponential-type functions (ETF's) from one center to another (i.e., addition theorems) encounter serious difficulties of one kind or another in practical applications. In contrast, the recently derived new addition theorem of $\Lambda$ functions appears to be free of many of those difficulties. The $\Lambda$ functions are a special class of ETF's defined by a product of an exponential, a Laguerre function, and a regular solid spherical harmonic: ${\Lambda }_{\mathrm{nl}}^m(r,\theta ,\phi )=2^{\frac{3}{2}}N(n,l)\mathrm{}{e}^{-r}{L}_{n-l-1\mathrm{}}^{\mathrm{}(2l+2)\mathrm{}}\mathrm{}\left(2r\right)\mathrm{}{\mathrm{}\left(2r\right)\mathrm{}}^l{Y}_l^m(\theta ,\phi ).\mathrm{}$

We have examined the new one-range addition theorem for its applicability to quantum-mechanical multicenter problems from the computational point of view and found it very useful. Test calculations on molecular multicenter one-electron integrals are reported. Under term-by-term integration, accuracy of ${10}^{-6\mathrm{}}$ could be achieved for all these integrals with an acceptable number of expansion terms.

Within the framework of a partial-wave expansion, the Schrödinger equation for the asymptotic form of the wave function for two electrons in a Coulomb field is reduced to a set of first-order coupled differential equations in one variable ($\frac{r_{<}}{r_{>}}$). The solution of these equations is found both for the monopole interaction only, and the monopole and dipole interactions. The monopole result greatly simplifies an expression derived in an earlier paper. These results can be applied to the calculation of ionization processes for any atom or ion.

The retarded van der Waals force between a pair of spheres with finite electrical conductivity is calculated in various limits.

This paper constructs a general method for obtaining series expansions of the logarithm of the configuration function $F$. For an $N$-body quantum system with Hamiltonian $H$ and inverse temperature $\beta$, the configuration function is the ratio of the exact to the free coordinate-space heat kernels: $F=\frac{〈\overrightarrow{\mathrm{x}}\left|e^{-\beta H}\right|{\overrightarrow{\mathrm{x}}}^{\prime }〉}{〈\overrightarrow{\mathrm{x}}\left|e^{-\beta H_0}\right|{\overrightarrow{\mathrm{x}}}^{\prime }〉}$. Expansion of $\ln F$ in Planck's constant $h$ leads to the semiclassical expansion of Wigner and Kirkwood, whereas the series in the variable $\beta$ provides the high-temperature expansion. By the introduction of an appropriate linked-graph method, it is shown how to obtain explicit formulas for the coefficient functions that enter either of these two series. Further, it is established that the same results can be derived by using the Feynman-Kac path-integral description of the partition function.

Measured cross sections for single ionization of ${\mathrm{Mg}}^{+}$, ${\mathrm{Al}}^{2+}$, and ${\mathrm{Si}}^{3+}$ by electron impact are reported. Crossed beams of electrons and ions have been employed to study the absolute cross sections as a function of collision energy in detail. Near threshold the cross sections for ${\mathrm{Mg}}^{+}$ and ${\mathrm{Al}}^{2+}$ are roughly 70% of the predicted direct-ionization cross sections, while ${\mathrm{Si}}^{3+}$ is in reasonable agreement with the predictions. Contributions to the total cross section by indirect processes, principally inner-shell-excitation autoionization, are specifically identified in each case and compared with theoretical results. These comparisons demonstrate specific failures of the predictions which rely on addition of excitation cross sections to the direct-ionization cross section.

The contributions of excitation autoionization to electron-impact ionization in the Nalike ions ${\mathrm{Mg}}^{+}$, ${\mathrm{Al}}^{2+}$, and ${\mathrm{Si}}^{3+}$ are calculated using the distorted-wave approximation with exchange. Inner-shell excitations of the type $2p^{6}3s\to 2p^{5}3snl \mathrm{}(n=3,4;l=0,1,2\mathrm{})\mathrm{}$ are included. The calculations indicate that the largest contribution should be due to the $2p\to 3p$ excitations in all three elements, although the relative importance of this transition decreases with ionic charge. Comparisons of these results with recent experimental measurements indicate that the distorted-wave approximation overestimates the magnitude of the monopole-dominated $2p\to \mathrm{np}$ transitions. Possible reasons for this discrepancy between experiment and theory are discussed, and additional calculations and experiments are suggested.

Measurements of the enhanced Lyman-$\alpha$ radiation following beam-foil excitation of $\mathrm{H}_2^{}{}_{}{}^{+}$ and $\mathrm{H}_3^{}{}_{}{}^{+}$ molecular ions are presented for incident ion energies of 0.2-1.75 MeV/amu. The enhancement is with respect to radiation observed with the use of incident protons of the same velocity. This enhancement is small (factors of 2-10) at large foil dwell times (greater than 1 fs) and large (factors of 50) at small dwell times (less than 1 fs). A phenomenological model which fits the data for both dwell-time regions, and at the same time fits other results of the transmission of $\mathrm{H}_2^{}{}_{}{}^{+}$ and of the production of ${\mathrm{H}}^0$ atoms from incident $\mathrm{H}_2^{}{}_{}{}^{+}$ ions, is presented.

Quantum-mechanical calculations of the probability of $K$-shell ionization in the $\alpha$ decay of Po and in $\alpha +\mathrm{Pb}$ collisions are made. The initial and final electronic wave functions are approximated by Dirac hydrogenic wave functions centered on the Pb nucleus. Also, we include the contribution from the part of the trajectory where the $\alpha$ particle tunnels through the Coulomb barrier. The contribution from tunneling is small except at small $\alpha$-particle velocities.

The structures of product angular distributions for the state-to-state rearrangement collision of an atom — diatomic-molecule system are examined as a function of the orbital (relative) angular momentum and incident kinetic energy. It is found that a coordinated study of the kernel- and radial-overlap distribution is useful for understanding the predicted structures of the angular distributions of the scattered product. The cause of both backward and sideways scattering are explainable from such study. Broadening of the angular distributions in the case of backward scattering is found to occur due to the increased number of partial waves which overlap with the kernel distribution. In addition, we find that the shift of the peak position or the change from the backward to sideways scattering at increased kinetic energies is due to the enhanced radial overlap between the kernel function and partial waves of larger orbital angular momentum.

An experiment is described in which 25-eV electrons are allowed to collide with sodium atoms in selected $\mathrm{nd}$ Rydberg states with $36\le n\le 50$. The excited collision products are analyzed by use of selective field ionization. Collisions are found to result in state changing, and cross sections for this process are estimated.

Explicit plane-wave Born-approximation (PWBA) calculations are presented for electron ionization of ions of carbon, nitrogen, oxygen, and fluorine. Calculations of innershell excitation followed by autoionization are included. For ions of carbon, nitrogen, and oxygen, agreement with Coulomb-Born calculations is better than 15% except near threshold; agreement with experiment is better than 25% except near threshold, and except for the Be-like ions ${\mathrm{N}}^{3+}$ and ${\mathrm{O}}^{4+}$. For fluorine ions, explicit PWBA calculations are presented. These cross sections are as much as 40% higher than results obtained via a scaling procedure. A reexamination of the scaling mechanics indicates an inherent 15% error at high incident electron energy. An attempt is made to correct the error near the cross-section maximum by using an integral over the optical limit of the generalized oscillator strength and its momentum-transfer derivative. The correction is only partially successful because of significant changes in the shape of the generalized oscillator strength with increasing degree of ionization, near squared momentum transfer equal to ionization energy, and secondary electron energy much less than ionization energy.

A theoretical treatment of ionization in atomic collisions is developed for quantum-mechanical and semiclassical relative motions. It is based on a molecular-wave-function expansion containing both discrete and continuum spectrum states. The continuum is discretized by further expansion, which leads to normalizable "electronic state packets." A procedure is described to calculate scattering amplitudes that avoids having to orthogonalize continuum states to all the (usually unknown) discrete ones.

The general-reaction theory of Feshbach is applied, within the framework of the distorted-wave approximation, to the calculation of excitation-autoionization resonances in the electron-impact ionization of ${\mathrm{Ga}}^{+}$. Although the spectrum of autoionizing levels for ${\mathrm{Ga}}^{+}$ is quite complex, we focus our attention on the important $3d^{10}4s^2\to 3d^{9}4s^{2}4p$ inner-shell excitations. For excitation of the $3d^{9}4s^{2}4p {}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}$ autoionizing level we make a general-reaction-theory calculation for the dominant partial-wave cross section and compute a typical resonance profile in the ejected-electron differential cross section. We find that the quantum-mechanical interference between the direct and indirect processes has a small effect on the total ionization cross section. Employing an independent-processes approximation we calculate excitation-autoionization contributions to all twelve levels of the $3d^{9}4s^{2}4p$ configuration. Using the results of our calculations and their comparison with a recent crossed-beam experiment by Rogers et al., we discuss the accuracy of the distorted-wave method and the effects of configuration interaction on energy levels and excitation cross sections.

The total and the differential cross section for the elastic scattering of electrons on argon is calculated for energies below the ionization threshold. The phase shifts were evaluated initially in the frozen-core Hartree-Fock approximation. The correlation corrections to them are found by solving the integral equation for the reducible self-energy part of the one-particle Green's function; this procedure is equivalent to solving the Schrödinger equation with the optical potential. The optical potential, being equal to the self-energy part of the one-particle Green's function, is found using many-body perturbation theory. Presented are the results of calculations for $s,p,d, \mathrm{and} f$ phase shifts and for the differential and the total cross sections. Comparison is made with other calculations and experimental data, and a good agreement is obtained.

An extension of the adiabatic-nuclei approximation appropriate for electron collisions with polar molecules is discussed. The method will find most useful application, but is not restricted, to molecules with large permanent dipole moments. Treatment of molecules with small or negligible dipole moments but significant quadrupole moments and/or dipole polarizabilities is also within its purview. The essence of the method consists of extracting the effects of the long-range interactions from the usual adiabatic-nuclei expressions, and reintroducing them in the laboratory frame in a self-consistent manner. The first Born approximation is the simplest, but not the only possible, vehicle for this approach. The method is closely related to the angular frame-transformation method. Illustrative applications are presented.