The magnetic quadrupole polarizability tensor determines the quadrupole moment induced in an atomic system by a nonuniform external magnetic field. From the analogy to the corresponding electric field case, this quantity is defined and simplified for a spherically symmetric system in an axially symmetric external field. Expressions are presented for the magnetic vector potential corresponding to the first three terms of a Taylor-series expansion for the magnetic field vector, and the Hamiltonian operator is obtained for a many-electron atom in an axially symmetric magnetic field with first-order gradients. Calculations are presented for the magnetic quadrupole polarizability for closed-shell atoms and ions with two to eighteen electrons. These calculations employ the fully coupled Hartree-Fock variation-perturbation procedure. The magnetic quadrupole polarizability shows a rapid decrease with atomic number within each isoelectronic series. Positive values are obtained for all atoms and ions except ${\mathrm{Li}}^{-}$, Be, and ${\mathrm{Na}}^{-}$.

A precision redetermination of the fine-structure interval of the ground state of positronium, $\Delta v$, has been made in an experiment similar to earlier experiments. The frequency difference between the $M=\pm 1$ and the $M=0\mathrm{}$ Zeeman levels of orthopositronium was measured in a static magnetic field of about 7900 G. The dependence of $\Delta \nu$ on gas pressure (argon) was observed for the first time. The result for $\Delta \nu$ is $\Delta {\nu }_{\exp }=(2.034\mathrm{} 03\pm 0.00012)\times {10}^5$ Mc/sec, where the error quoted is one standard deviation and represents an improvement in accuracy by a factor of 4 compared to earlier work. This value is in reasonable agreement with the theoretical value of $\Delta {\nu }_{\mathrm{theor}}=2.034\mathrm{}27\times {10}^5$ Mc/sec, but a calculation of the ${\alpha }^4-\mathrm{R}\mathrm{y}$ term in the theory is needed. The result for the fractional fine-structure pressure shift is $\left(\frac{1}{\Delta \nu }\right){\left(\frac{\partial \Delta \nu }{\partial P}\right)}_{300{}_{}{}^{\circ }{}_{}{}^{}\mathrm{K}}=\frac{(-0.93\mathrm{}\pm 0.18)\times {10}^{-7\mathrm{}}}{\mathrm{Torr}}$ of Ar, in which a one-standard-deviation statistical error is quoted. This value has the same sign but is 20 times larger than the fractional hfs pressure shift for hydrogen in argon. The experiment also yields the first direct measurement of the parapositronium annihilation rate ${\lambda }_{p\mathrm{expt}}=0.799\mathrm{}\pm 0.011\times {10}^{10}$ ${\sec }^{-1\mathrm{}}$, where a one-standard-deviation error is given. This value is in excellent agreement with the theoretical value of ${\lambda }_{p\mathrm{theor}}=0.798\mathrm{}\times {10}^{10}$ ${\sec }^{-1\mathrm{}}$.

The relative Auger intensities for the $K-\mathrm{LM}$ group with respect to the $K-L_1{L}_1$ line are presented as a function of the atomic number. The theoretical values are determined by the relativistic Hartree-Fock-Slater atomic model. The available experimental data are in excellent agreement with theoretical relative intensities for $Z\ge 55$. It is concluded that the relativistic effects with a realistic atomic model are as important as effects of the intermediate coupling scheme for medium $Z$ values in the $K-\mathrm{LM}$ group. Additional measurements of the $K-\mathrm{LM}$ relative intensities for intermediate atomic numbers are needed to pinpoint the relative contributions of the relativistic model and the intermediate coupling scheme. Theoretical results are also presented for $\Sigma K-\mathrm{LM} \Sigma K-\mathrm{LL}$ versus the atomic number.

The matrix elements of a c.m. transformed molecular Hamiltonian for a wave function constructed from a symmetrized product of one-particle orthonormal spin functions are given in terms of the integrals over the coordinates of the particles. These matrix elements are necessary to study the structure of deuterated molecules such as ${\mathrm{D}}_2$O, N${\mathrm{D}}_3$, C${\mathrm{D}}_4$, etc. The integrals are the same as those found for antisymmetric wave functions with the exception of the two-particle integrals of the form $\int {g_i}^{*}\mathrm{}\left(1\right)\mathrm{}{g_i}^{*}\mathrm{}\left(2\right)\mathrm{} {\left(r_{12}\right)}^{-1\mathrm{}}{g}_j\mathrm{}\left(1\right)\mathrm{}{g}_j\mathrm{}\left(2\right)\mathrm{} \mathrm{dV}$, when two bosons occupy the same spin function. There are, however, significant differences in the matrix elements.

In this paper the slope at $q^2=0\mathrm{}$ of the Dirac form factor for the electron vertex in fourth order is calculated. The result is ${\kappa }^2\partial \frac{F_1^{\mathrm{}\left(4\right)\mathrm{}}}{\partial q^2{|}_{q^2=0\mathrm{}}}={\left(\frac{\alpha }{\pi }\right)}^2\left(0.107\mathrm{}648\right)$. This result affects the energy levels of hydrogenic atoms and can be represented as an addition to the interaction energy density of the electron with the external field in the form $-ie\left[\overline{\Psi }\mathrm{}\right(x),{\gamma }_{\mu }\Psi \mathrm{}(x\left)\right]\left(\frac{{\square }^2}{{\kappa }^2}\right)A_{\mu }^e\mathrm{}\left(x\right)\mathrm{}\times \left[\right(\frac{{\alpha }^2}{4{\pi }^2}\left)\right(0.215\mathrm{}296\left)\right]$, where $\kappa$ is the mass of the electron. This term contributes to the level shift in a hydrogenic atom an energy of ${\delta }_{\mathrm{lo}}\left[\frac{4{\left(z\alpha \right)}^4\kappa c^2}{n^3}\right]{\left(\frac{\alpha }{\pi }\right)}^2\left(0.107\mathrm{}648\right)$. This calculation reduces the uncertainty in the theoretical result by about 50% to ± 0.1 MHz, which is about the size of the experimental error.

The Heitler-London curves for the interaction of two hydrogen atoms are calculated as a function of the internuclear distance and velocity. Electron-exchange terms are found to dominate the interaction energy at laboratory energy 1 keV, to be quite significant at 25 keV, and to be negligible at 400 keV. The exchange energy is attractive at 25 keV and below, but at 100 keV it is repulsive. Hence, in going from 25 to 100 keV, the single and triplet curves go from attractive and repulsive to repulsive and attractive, respectively, suggesting that an oscillatory behavior of the cross section versus energy is possible. These results have important implications for the calculation of atom-atom excitation cross sections in this energy range.

Drift velocities and reactions of negative ions have been investigated in oxygen at temperatures between 310 and 370 °K at values of $\frac{E}{N}$ (the ratio of electric field to neutral-gas number density) between 2 and 400 × ${10}^{-17\mathrm{}}$ V ${\mathrm{cm}}^2$, and at pressures ($P_0$) between 0.5 and 4 Torr. Electrons were produced from a hot filament and reacted with the neutral-gas molecules to produce ions. The ions were then gated into a drift cell through a double grid, withdrawn through an aperture, and their mass and intensity were determined as a function of drift time by a quadrupole mass analyzer and a gated particle multiplier. Comparison of experimental results with numerical calculations allowed the determination of ion drift times and reaction rates. The rate constant for the reaction ${\mathrm{O}}^{-}+{\mathrm{O}}_2+{\mathrm{O}}_2\to {\mathrm{O}}_3^{-}+{\mathrm{O}}_2$ was found to be 9 × ${10}^{-31\mathrm{}}$ ${\mathrm{cm}}^6$/sec at low $\frac{E}{N} (<\mathrm{}30\mathrm{}\times {10}^{-17\mathrm{}} \mathrm{V} {\mathrm{cm}}^2)$. At higher $\frac{E}{N} (>\mathrm{}70\mathrm{}\times {10}^{-17\mathrm{}} \mathrm{V} {\mathrm{cm}}^2)$ the reaction ${\mathrm{O}}^{-}+{\mathrm{O}}_2\to {\mathrm{O}}_2^{-}+\mathrm{O}$ was observed with a rate that increased sharply with $\frac{E}{N}$ to a value of 5 × ${10}^{-11\mathrm{}}$ ${\mathrm{cm}}^3$/sec at $\frac{E}{N}=300\mathrm{}\times {10}^{-17\mathrm{}}$ V ${\mathrm{cm}}^2$.

Elastic differential cross sections for proton-helium scattering are calculated in the semi-classical approximation for two assumed interaction potentials. Both potentials are of the form $V(r;A,B,C)=\left(\frac{2}{r}\right)e^{\frac{-r}{A}}[1+\frac{r}{A}+\frac{1}{2}{r}^2(\frac{1}{A^2}-\frac{U}{B}\left)\right]-U{[1+\frac{r}{B}+{\left(\frac{r}{C}\right)}^2+\frac{2U{r}^4}{\alpha }]}^{-1\mathrm{}},$ where $U$ is the difference between the ground-state energies of He and ${\mathrm{Li}}^{+}$ (4.373 11 hartree) and $\alpha$ is the polarizability of He (1.3835 ${\mathrm{bohr}}^3$). The first potential $V_M\equiv V(r;0.423,0.483,0.441\mathrm{})\mathrm{}$ fits the He-${\mathrm{H}}^{+}$ ground-state energies which Michels has calculated. The second, $V_W\equiv V(r;0.442,0.505,0.451\mathrm{})\mathrm{}$, is similar to $V_M$ except that its minimum is decreased by 10% to agree with the value obtained by Wolniewicz. The cross sections for these two potentials are shown for protons incident at energies $T$ of 7, 19, 58, and 116 eV in the laboratory frame and for scattering angles, at each energy, out to the rainbow angle ${\theta }_R$. ${\theta }_R$ is given in center-of-mass coordinates by the expression ${\theta }_{R}T\cong 0.1$ rad hartree. As the collision energy decreases, the cross sections develop oscillatory structure not present in the classical cross sections. This structure and the rainbow angle are sensitive to the choice of potential, which suggests that measurements of ${\mathrm{H}}^{+}$-He cross sections may be used to test the suitability of, e.g., the Born-Oppenheimer potential for scattering phenomena. It is also suggested that many-body calculations of these cross sections would allow, by comparison with the present results, an evaluation of the potential scattering model.

In a previous publication, an $n$-body potential was used to transform the one-channel particle-bound-state scattering problem into a discrete-eigenvalue problem. A basic relationship was established in one particular case and used to study electron-hydrogen-atom scattering. A general relationship is now established, subject to some WKB estimates, providing the basis of a general theory of particle-bound-state scattering. The theory is extended to the $n$-channel case, both direct and exchange scattering being treated. The theory provides an explicit test of convergence of calculations based on it. An approximation designed to simplify computational labor is discussed.

Expressions correlating the transition operators to Faddeev's $T^{\mathrm{}\left(i\right)\mathrm{}}$'s are given for a three-body scattering problem. Using the results of Ball, Chen, and Wong, a revised partial-wave analysis for various transition processes in a three-particle Coulomb system is suggested. Some of the apparent contradictions between the off-shell two-particle partial-wave $T$ matrix derived by Ball et al. and the full two-particle Coulomb $T$ matrix derived by Nutt are resolved. In an Appendix, it is shown that $T^{\mathrm{}\left(i\right)\mathrm{}}$ is related to the transition operators. This establishes the equivalence between the limiting procedure of Ball et al. and the conventional procedure for obtaining the transition amplitudes.

We discuss electrostatic instabilities in an infinite uniform homogeneous plasma imbedded in a uniform magnetic field that are driven by the anisotropy of the velocity distribution function of only one of the species that comprise the plasma. Specifically, we discuss three instabilities that arise when the temperature of the driving species perpendicular to the applied magnetic field is different from its temperature parallel to the magnetic field, and the velocity distribution of the driving species perpendicular to the field is (a) a Maxwellian, (b) a losscone type, and (c) a $\delta$ function. Results are presented for two cases when the driving species are (a) the ions and (b) the electrons. Results show that these instabilities appear only when the plasma frequency of the driving species becomes comparable to or larger than its cyclotron frequency.

Thermomagnetic gas torque experiments on dilute polyatomic gases indicate that a circumferential flow of gas exists in the annular space between inner and outer cylinders when a temperature difference and an axial magnetic field coexist. Experiments on a hollow pendulum show that the torques on the inner and outer cylinders are in opposite directions.

A computer-linked spectroscopic system with a high-sensitivity-resolution product has been used to measure the time-dependent net rate of production of neutral helium molecules resulting from the recombination of $\mathrm{He}_2^{}{}_{}{}^{+}$ with electrons in a pulsed helium afterglow at a neutral pressure of 44.6 Torr. The relative recombination-rate cofficient of $\mathrm{He}_2^{}{}_{}{}^{+}$ as a function of afterglow time has been obtained by dividing these data by the unscaled product of the electron and molecular ion density as determined from levels in Saha equilibrium with the free electrons. Simultaneous measurement of the electron density with a 35-Gc/sec microwave interferometer has given the unscaled rate coefficient as a continuous function of electron density over the range ${10}^{10}$-${10}^{12}$ ${\mathrm{cm}}^{-3\mathrm{}}$ without the need of a priori assumptions about the functional form. Scale has been established through integration of the rate equation for the loss of ions. Empirically, the rate coefficient can be represented as $\alpha =2.8\mathrm{}\times {10}^{-11\mathrm{}}{\mathrm{}\left[e\right]\mathrm{}}^{0.185}$ ${\mathrm{cm}}^3$/sec, where $e$ is the electron density in ${\mathrm{cm}}^{-3\mathrm{}}$.

The mathematical dilemma of the direct determination of the reduced density matrix for an $N$-particle system in a stationary state is examined. An energy eigenvalue equation pertaining to the statistical Hamiltonian of a subsystem which determines the reduced density matrix is formulated. It is shown that the construction of this equation represents a classical, semi-quantum-mechanical attitude. The dilemma, in fact, arises from this attitude.

The solution to the problem of determining the transition radiation emitted when a charged particle moves uniformly in a periodically stratified cold, isotropic plasma is obtained. The electromagnetic field may be expressed in terms of an infinite number of normal modes. The conditions under which some of these modes are radiative are discussed, and expressions for the field components and the energy spectral density are obtained. It is found that radiation may be emitted for arbitrarily small particle velocities and that the strongest emission takes place in a frequency band just above the average plasma frequency. Numerical results are presented to illustrate the behavior of the spectra of the various radiative modes as the frequency and plasma parameters are varied.

A single-particle-propagator (i.e., Green's function) theory of evaporation from quantum liquids is developed and applied to the energy distribution of ${\mathrm{He}}^3$ atoms evaporating from ${\mathrm{He}}^3$-${\mathrm{He}}^4$ mixtures. The resulting theoretical energy distribution contains no unknown parameters. We predict a distribution described in terms of an effective temperature which is somewhat higher than the temperature of the liquid. However, the effect is not so pronounced as the temperature shift observed by King and Johnston in the pure-${\mathrm{He}}^4$ evaporation experiment. We suggest that experiments of this type might be used to measure the Bose condensate fraction of superfluid ${\mathrm{He}}^4$.

The radiation spectrum from an anisotropic turbulent plasma is found to have a definite relationship to the turbulence spectrum. The intensity of such a radiation is estimated to be significantly greater than that expected from a scattering experiment.

The extent to which the renormalization of critical-point behavior should be visible experimentally is investigated on the basis of detailed numerical calculations for a three-dimensional soluble model (a mobile-electron Ising ferromagnet). If a dilution parameter $x$ is defined such that the change in critical temperature from the "pure" or unrenormalized system is $\left|T_c\mathrm{}\right(x)-T_c^0|=\frac{x{T}_c}{f}$, where $f=0.6-0.9\mathrm{}$, then we conclude that the effective exponents ${\beta }_{\mathrm{fit}}\mathrm{}\left(x\right)\mathrm{}$ and ${\gamma }_{\mathrm{fit}}\mathrm{}\left(x\right)\mathrm{}$ which will be observed experimentally, vary roughly as ${\beta }_{\mathrm{fit}}\approx \beta +x\Delta {\beta }_X$ and ${\gamma }_{\mathrm{fit}}\simeq \gamma +\frac{1}{5}x\times (1+2\mathrm{}{x}^2)\Delta {\gamma }_X$. Here $\beta$ and $\gamma$ are the ideal exponents for the order parameter and total fluctuation or susceptibility of the pure system, while $\Delta {\beta }_X={\beta }_X-\beta$ and $\Delta {\gamma }_X={\gamma }_X-\gamma$, in which ${\beta }_X=\frac{\beta }{(1-{\alpha }^{\prime })}$ and ${\gamma }_X=\frac{\gamma }{(1-\alpha )}$ are the fully renormalized exponents, while $\alpha$ and ${\alpha }^{\prime }$ (assumed positive) describe the divergence of the specific heats of the pure system. [Theoretically the true limiting asymptotic behavior at the transition is described by $\beta \mathrm{}\left(x\right)={\beta }_X$ and $\gamma \mathrm{}\left(x\right)={\gamma }_X$ for all $x>\mathrm{}0\mathrm{}$.] The renormalized specific heats are found to be sensitive to $x$ but their true renormalized behavior is not evident until $x\gtrsim 0.3$. Various techniques of data analysis such as logarithmic, semilogarithmic, Heller-Benedek, and Kouvel-Fisher plots have been tested.

To investigate the diffusive motion of a test particle in a classical many-body system, the projection-operator formalism of Zwanzig has been applied to the calculation of the velocity autocorrelation function. The memory kernel in this formalism is investigated by using a perturbation expansion in a parameter characteristic of the time scale of correlations in the system. General studies of the asymptotic time behavior of the correlation function are given, as well as several explicit calculations for modeled interparticle potentials.

The quantum-statistical theory of a nonrelativistic fully ionized gas in thermal equilibrium is developed using the well-known linked-cluster expansion of the grand potential. A systematic analysis of the self-energy structures leads to a master-graph formulation of quantum statistics. This provides a much simpler derivation and improved version, with important differences, of the earlier work of Mohling and Grandy. In particular, the analysis of the photon self-energy structures is now entirely different. The present formulation includes the technique of adding and subtracting in the Hamiltonian a single-particle operator, and the relation of this technique to the $\Lambda$-transformation method of Mohling, RamaRao, and Shea is indicated. As an application of the general theory, the lowest-order calculations of the photon self-energy and photon momentum distribution are presented. The result for the photon momentum distribution is different from that of Hwang and Grandy, and the reason for the difference is given. Finally, explicit connections between the master-graph line factors and Green's functions are outlined, and the consequences of such a connection are indicated.