A unified hydrodynamic theory is presented that is appropriate for crystals; smectic, cholesteric, and nematic liquid crystals; glasses; and normal fluids. In the theory, the increased spatial degeneracy as the system progresses from crystalline and mesomorphic phases to the isotropic fluid phase is marked by successive reductions in the number of firstorder elastic constants and in the number of transport coefficients. Distinction between local lattice dilations and local mass changes, and recognition of processes like vacancy diffusion that this difference makes possible, are crucial for understanding the connection between theories in different phases. Formulas are derived that give the number of hydrodynamic modes and the frequencies, lifetimes, and intensities of these modes in all of the above systems. In the nematic and cholesteric phases, the results agree with some found previously. In more complex systems, they are new. An attempt is made to explain the differences between the present hydrodynamic theory and other phenomenological proposals.

A Fokker-Planck equation for the joint probability density of the orientation and angular velocity of a body of general shape is derived by use of a rotational Langevin equation. Equations governing the seperate distributions of orientation and angular velocity are deduced from the equation for the joint probability density. For the special case of a spherical body, two expressions for the orientation distribution are calculated, one valid for small values of the frictional constant occurring in the rotational Langevin equation, and the other valid for large values of the frictional constant. The latter expression includes previously presented results of rotational-diffusion theory and Steele's modification of rotational-diffusion theory, and the calculation provides conditions of validity for these theories. Expressions are calculated for time-correlation functions of spherical tensors, such as spherical harmonics, which involve functions of the orientation of a body.

We calculate the two-photon exchange (retarded van der Waals) potential between neutral spinless systems, including the effects of higher partial waves in the atom-photon scattering amplitude. This is equivalent to including higher multipoles in the interaction of the charges in the atoms. We show that this potential can be expressed as an infinite series of terms, with coefficients that can, in principle, be measured in atom-photon scattering. The behavior of the various terms, at small separation and large separation of the system, is discussed. We show that the leading term in the contribution of each multipole has the property that it has one more power of $R^{-1\mathrm{}}$ in its large-$R$ behavior than in its small-$R$ behavior.

A model is proposed to explain the superfluid-film formation in dilute liquid mixtures of ${\mathrm{He}}^4$ in ${\mathrm{He}}^3$. It is shown that the difference of mass between the isotopes increases the ${\mathrm{He}}^4$ adsorption near the walls. Previous and new measurements are in good agreement with the model.

In an earlier paper, the problem of the optically pumped gas laser was treated within the formalism of a modified Boltzmann equation and the equation of radiative transfer. The laser cavity was tuned precisely to the center of the atomic lasing frequency and only one cavity mode was excited above threshold. These restrictions are now removed and the effects on the laser population inversion, gain, and power output are presented. In the limit that Doppler broadening dominates collisional broadening, it is found that the power output for the single-mode operation increases as the laser cavity is detuned, until an optimum occurs when the cavity detuning is equal to the collisional linewidth. In the limit that collisional broadening dominates Doppler effects, power output is independent of detuning, until the detuning approaches the collisional linewidth. For multimode operation, it is shown that power output increases approximately linearly with the number of modes in oscillation, provided that the Doppler width is much larger than the collisional width and mode interaction is not occurring. When collisional broadening dominates, no increase in total power output is achieved by allowing more than one mode to go into oscillation. Most of these results are well known, for low levels of lasing intensity, through the semiclassical analysis of Lamb. However, the formulation here provides a more quantitative evaluation of the relative roles of inhomogeneous and homogeneous broadening for all collisional linewidths and high lasing intensities.

The low-temperature behavior of the third-quantum-cluster coefficient is investigated using the multiple-scattering form of the binary-collision expansion. For hard spheres and Boltzmann statistics we find $b_3=2\mathrm{}{\left(\frac{a}{\lambda }\right)}^2-\frac{4}{3}\sqrt{2}\pi {\left(\frac{a}{\lambda }\right)}^3-\frac{16}{3}(4\mathrm{}\pi -3\mathrm{}\sqrt{3}){\left(\frac{a}{\lambda }\right)}^4\ln \left(\frac{a}{\lambda }\right)+O\left({\left(\frac{a}{\lambda }\right)}^4\right)$ where $a$ is the sphere diameter and $\lambda$ is the thermal wavelength. The first two terms were obtained some time ago by Lee and Yang and by Pais and Uhlenbeck. The occurrence of a term of the form ${\lambda }^{-4\mathrm{}}\ln \lambda$ was predicted recently by Adhikari and Amado. The expansion is also given for Bose-Einstein and Fermi-Dirac statistics, and for the case of an intermolecular potential without bound states. The limitations of such low-temperature expansions are discussed.

The possibility of extending theories of strictly one-dimensional systems to linear systems is examined. Shockley's theorem is proved for the former case and shown not to be valid for the latter. On the subject of localized states, it is shown that only a limited generalization is possible, but that this may be sufficient for calculations on real disordered polymer chains.

The various proposals made for the evaluation of the second-order matrix elements of the electron-collision operator of partially degenerate lines are compared numerically for the He i 4471-Å line emitted from a thermal plasma. No significant discrepancy is found. An interpretation of this result is given.

Using only the analytic structure of the amplitude for atom-atom scattering as derived by Feinberg and Sucher, a direct derivation of the threshold behavior of the partial-wave amplitudes is given. The behavior is, as expected, characteristic of scattering from an $R^{-7\mathrm{}}$ potential.

Extensive numerical results are presented concerning the energies of highly excited circular and nearly circular orbits of some heliumlike exotic atoms. The atoms, together with the ranges of principal quantum numbers considered, are the $\alpha {\pi }^{-}{e}^{-}$ with $13\le n\le 20$, the $\alpha K^{-}{e}^{-}$ with $23\le n\le 36$, the $\alpha \overline{p}{e}^{-}$ with $31\le n\le 47$, and the $\alpha {\Sigma }^{-}{e}^{-}$ with $34\le n\le 52$.

Previously reported experiments by Pusey, Bak, and Goldburg showed an asymmetry in the light scattering in the high- ($h$) and low- ($l$) density phases of critical mixtures of phenol and water at temperatures below the critical temperature $T_c$. Those experiments have been reanalyzed to evaluate the exponents $\tau$ and $\theta$ in the asymptotic expressions $I_{h\left(l\right)\mathrm{}}\propto ({\mathrm{}\left|t\right|\mathrm{}}^{-\gamma }\pm B^{\prime }{\mathrm{}\left|t\right|\mathrm{}}^{-\theta })$ and ${\Gamma }_{h\left(l\right)\mathrm{}}\propto ({\mathrm{}\left|t\right|\mathrm{}}^{{\nu }^{*}}\mp B{\left|\tau \right|}^{\tau })$, where $I$ and $\Gamma$ are, respectively, the intensity and spectral width of the scattered light and $t=\frac{(T-T_c)}{T_c}$. The experimental results are discussed in terms of recent ideas concerning expected nonsymmetric behavior in pure fluids and binary mixtures. The value of $\tau$ the experiments yield can be accounted for in terms of simple linear dependence of the viscosity on concentration.

Positions of the lowest odd-parity ${}_{}{}^{1,3}{}_{}{}^{}D$ states in He have been calculated variationally. The trial wave function of Hylleraas type with four nonlinear parameters has been used. The results for the lowest ${}_{}{}^{1,3}{}_{}{}^{}D$ states obtained by using 112 terms in the wave function are lower than the previous results. The mass-polarization correction has been also calculated. The transition $2{}_{}{}^3{}_{}{}^{}P_{}^e{}_{}{}^{}\to 3{}_{}{}^3{}_{}{}^{}D_{}^0{}_{}{}^{}$ is of wavelength 3014.16 Å, which agrees well with the previous calculation and the experimental result. The ${}_{}{}^{1,3}{}_{}{}^{}D_{}^0{}_{}{}^{}$ states of ${\mathrm{H}}^{-}$ were also investigated, but no states of odd parity were found in this calculation.

Single-quantum annihilation of positrons in an atomic field is considered. Numerical calculations of the differential and total cross sections and the various polarization correlation functions are presented for the $K$ and $L$ shells of a range of elements from $Z=47\mathrm{}$ to $Z=92\mathrm{}$. The positron wave function is described by a partial-wave expansion in angular momentum eigenstates, and the interaction of the positron and bound electron with the radiation field is treated in lowest-order perturbation theory. Numerical programs are constructed for the solution of the radial part of the positron wave function and for the partial-wave phase shifts and normalization factors in ȧn arbitrary non-Coulomb central potential, and for the evaluation of the differential and total cross sections. The effects of screening are included by using the bound-state wave functions and central potentials predicted by the relativistic Hartree-Fock-Slater atomic model. Screening corrections to the Coulomb $K$-shell total cross sections are found to be sizable for large atomic numbers and low positron energy, and the ratio of $L$- to $K$-shell total cross sections is found to be significant for heavy atoms. The angular distributions for this atomic potential exhibit the sharp forward peak predicted in previous work assuming a purely Coulombic potential.

The atomic shielding or antishielding factor $R$ for the quadrupole hyperfine structure has been obtained for seven atomic ground states ranging from F $2p^5$ to Br $4p^5$. The values of $R$ were determined by means of the perturbed wave functions $v_1^{\prime }(nl\to l^{\prime })$, as obtained by solving the inhomogeneous Schrödinger equation for each type of excitation of the core electrons by the nuclear quadrupole moment $Q$. The resulting values of $R$ have been listed, together with those for four atomic states which had been previously investigated. For the atomic ground states in this region of the Periodic Table, $R$ is generally positive (shielding), and of the order of +0.1, except for Al $3p$ and Ga $4p$, for which $R$ is negative, because of the antishielding provided by the $2p\to p$ and $3p\to p$ perturbations, respectively. The resulting correction factors $C=\frac{1}{\mathrm{}(1-R)\mathrm{}}$ have been applied to the quadrupole moments of 11 nuclear isotopes. We have also obtained the ionic antishielding factor ${\gamma }_{\infty }$ for the ${\mathrm{Sc}}^{3+}$ ion, ${\gamma }_{\infty }\left({\mathrm{Sc}}^{3+}\right)\cong -11.2\mathrm{}$.

Term splittings, ionization potentials, and electron affinities of states in the lowest configurations of atoms with $Z$ from 5 to 10 have been computed. Correlation-energy terms are obtained from variational solution of one- and two-particle Bethe-Goldstone equations, formulated in terms of configurational virtual excitations. This procedure differs from previous calculations (which used orbital virtual excitations) in that $\mathrm{LS}$ eigenfunctions are obtained at each stage of the hierarchy of computations. The present procedure makes it possible to compute correlation energies for multideterminantal $\mathrm{LS}$ states. Computed results are in reasonable agreement with experiment, but are less satisfactory than results obtained previously with orbital virtual excitations, including three-particle terms.

The atomic-beam magnetic-resonance technique has been used to measure the $g$-factor ratio of the ground state of nitrogen to the ground state of potassium. The result is $\frac{g_J({\mathrm{N}}^{14}, {}_{}{}^4{}_{}{}^{}S_{\frac{3}{2}}^{}{}_{}{}^{})}{g_J({\mathrm{K}}^{39}, {}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{})}=0.999\mathrm{}9196\left(20\right)\mathrm{}$, where the quoted error represents the 90% confidence level. Combining this result with that of other researchers, we find the absolute $g_J$ factor for nitrogen to be $g_J({\mathrm{N}}^{14}, {}_{}{}^4{}_{}{}^{}S_{\frac{3}{2}}^{}{}_{}{}^{})=2.002\mathrm{}134\left(5\right)\mathrm{}$. This value is in much better agreement with the result of a recent calculation carried out according to the theory of Kambe and Van Vleck than is the earlier measured value, which was obtained by electron paramagnetic resonance. The discrepancy between experiment and theory is reduced from 13.5 ± 2 to 4 ± 2.5 ppm. The atomic nitrogen was generated in an electrodeless discharge and detected with a new mass-spectrometer universal detector; this latter instrument is described in some detail.

The mean radiative lifetimes of several excited levels of O ii have been measured using ${\mathrm{O}}_2^{+}$ and ${\mathrm{O}}^{+}$ ion beams in the 20-30-keV energy range. The levels were populated by directing the beams through a differentially pumped collision cell containing gaseous helium at pressures of a few mTorr. The decays in the intensities of selected electronic transitions, arising from the levels of interest, were obtained as a function of the distance $x$ downstream from the exit aperture by using a ½-m grating monochromator equipped with a photomultiplier detector. These decay curves were fitted to an expression of the form $I=A{e}^{-\frac{x}{v{\tau }_k}}+\Sigma j^{}{B}_j{e}^{-\frac{x}{v{\tau }_j}}+C{e}^{-k^{\prime }x}$, where $A$, $B_j$, and $C$ are constants, $v$ is the speed of the ions, ${\tau }_k$ is the lifetime of the level $k$ of interest, ${\tau }_j$ is the lifetime of a level $j$ which cascades into level $k$, and $k^{\prime }$ is a decay constant associated with the drop in gas pressure downstream from the exit aperture of the collision chamber arising from gas streaming through the aperture. The intensity data and the computed fits were plotted either by hand or by a plotter connected to a computer. The number of exponentials in a given fit was adjusted until a good fit was obtained, as determined by visual inspection in conjunction with a ${\chi }^2$ test. Lifetimes (in nsec) obtained in this way are as follows: $\tau \left(3\mathrm{}{p}^4{D}_{\frac{7}{2},\frac{1}{2}}^0\right)=15.0\mathrm{}\pm 0.8$, $\tau \left(3\mathrm{}{p}^4{D}_{\frac{5}{2}}^0\right)=14.9\mathrm{}\pm 0.4$, $\tau \left(3\mathrm{}{p}^4{D}_{\frac{3}{2}}^0\right)=15.6\mathrm{}\pm 0.5$, $\tau \left(3\mathrm{}{p}^4{P}_{\frac{5}{2},\frac{3}{2}}^0\right)=8.5\mathrm{}\pm 0.6$, $\tau \left(3\mathrm{}{p}^2{D}^0\right)=11.6\mathrm{}\pm 0.3$, $\tau \left(3\mathrm{}{p}^2{P}^0\right)=9.2\mathrm{}\pm 0.1$, $\tau \left(3\mathrm{}{p}^{\prime }{}_{}{}^2{}_{}{}^{}F_{}^0{}_{}{}^{}\right)=12.1\mathrm{}\pm 0.3$, and $\tau \left(3\mathrm{}{p}^{\prime }{}_{}{}^2{}_{}{}^{}D_{}^0{}_{}{}^{}\right)=9.2\mathrm{}\pm 0.4$. These results are believed to be good to 10% or better. In addition to these results, measurements were also made on several other levels with lifetimes in the 6-nsec range. The reported values for the latter levels are upper limits to the actual lifetimes and are otherwise unreliable because of the possibility of significant errors arising from gas streaming. An extensive discussion of the treatment of possible errors from gas streaming, cascading, and blending of spectral lines is given.

It is found that a nonspherical Hartree-Fock potential may occur even for atomic systems generally thought of as having a complete shell structure in their ground state. Some conditions, under which a perturbation treatment of correlation effects starting from deformed Hartree-Fock states may be performed, are discussed.

Pressure broadening by neutral atoms is treated in a time-dependent formalism making use of generalized cumulants due to Kubo. A thermal-equilibrium initial density matrix is assumed, unlike in previous theories by Baranger and Fano who neglect initial correlations between atoms; it is pointed out that the wings of the spectrum depend in an essential manner on these initial correlations. The treatment centers on a time-evolution operator $U\left(t\right)={\left({\mathrm{Tr}}_B\rho \right)}^{-1\mathrm{}}\times {\mathrm{Tr}}_B\rho e^{\mathrm{it}{H}^{\mathrm{x}}}$ operating in the Liouville space of the radiating atom and governing its motion under the influence of the perturbing gas or bath {${\mathrm{Tr}}_B$ is the trace over bath coordinates, $\rho \sim e^{-\beta H}$ is the density matrix, and $H^{\mathrm{x}}$ is the quantum-mechanical Liouvillian: $H^{\mathrm{x}}\left( \right)=[H, ( \left)\right]$; $H$ is the total Hamiltonian for the system radiator plus bath}; $U\left(t\right)\mathrm{}$ is written as $U\left(t\right)=T_{\to }\exp \left[i\int 0^{t}d{t}^{\prime }L\left(t^{\prime }\right)\right]$, $L\left(t\right)=H_s^{\mathrm{x}}+R\left(t\right)\mathrm{}$ ($T_{\to }$ is a time-ordering operator), where the effect of the bath is contained in the time-dependent non-Hermitian perturbation $R\left(t\right)\mathrm{}$ added to the Liouvillian $H_s^{\mathrm{x}}$ of the unperturbed radiator. The operator $R\left(t\right)\mathrm{}$ is expanded in powers of a "reduced" density equal to the perturbing-gas density multiplied by the ratio of the fugacities corresponding to mutually interacting and noninteracting perturbing atoms, respectively; the terms of the expansion are expressed by means of generalized cumulants, and describe interactions of the radiator with clusters of perturbers. By setting $R\left(t\right)=\overline{R}+\tilde{R}\mathrm{}\left(t\right)\mathrm{}$, where $\overline{R}\equiv {\lim }_{T\to \infty }\left[\right(\frac{1}{T}\left)\int 0^T\mathrm{dtR}\left(t\right)\mathrm{}\right]$, the spectrum is written as the sum of its impact approximation, determined by $\overline{R}$, plus a correction expanded in powers of $\tilde{R}\mathrm{}\left(t\right)\mathrm{}$, which to first order in the gas density equals the one-perturber spectrum minus its singularities at the resonance frequencies.