A variation-perturbation (VP) treatment of inelastic and rearrangement scattering processes is presented. We show that the first-order perturbation functions determine the scattering parameters through third order and develop a variation principle for the approximate calculation of these functions. It is seen how adjustable parameters in the zeroth-order potentials can be optimized separately for each scattering event by the Hulthèn variation condition, which yields a first-order function that is normalizable in the exit channel. We also propose that terms coupling different channels be included in the initial approximation. A comparison of the VP method with the usual distorted-wave Born series reveals certain practical advantages of the former, despite the formal similarity. Some of these advantages are illustrated in a sample calculation on a system characterized by a model inelastic potential.

The complete electron-impact spectra, including the elastic scattering, for He have been obtained at the scattering angles 1°, 1.5°, 2°, 3°, 4°, 5°, 7°, and 10° using 25-keV incident electrons. The data cover the momentum-transfer region from 0.8 to 7.5 in atomic units (a.u.). Measured intensities were converted approximately to relative generalized oscillator strengths for different values of energy transfer, and placed on an absolute scale by use of the Bethe sum rule. Although sum-rule normalization using the energy-loss spectrum at a fixed angle is not rigorous, it is justified in the large-angle high-energy limit. Comparison of the results with the theory proved that use of sum-rule normalization was justified at angles as small as 1°. The first-Born theory was shown to be in agreement with experiment over the complete data range. In the case of the binary-encounter theory, good agreement with experiment was obtained from momentum-transfer values greater than 4 a.u. Below 4 a.u. substantial deviations were observed, especially in regions removed from the maximum of the spectral distribution. Sums of the generalized optical-oscillator strength, the x-ray incoherent-scattering factor $S\left(K\right)\mathrm{}$,, the elastic differential, and total scattering cross sections were all computed from the experimental data. The experimental values of $S\left(K\right)\mathrm{}$ and the elastic differential cross section agreed with accurate theoretical calculations to an average accuracy of 2%. Compton profiles obtained at 5°, 7°, and 10° were found to be in excellent agreement with previous electron-impact, x-ray, and $\gamma$-ray measurements. The unresolved autoionizing double excitations ($2s2p$, $2p^{ 2}$, etc.) at an energy loss of 60 eV were observed at both 1° and 2.7°. The $2{}_{}{}^1{}_{}{}^{}P\leftarrow 1{}_{}{}^1{}_{}{}^{}S$ transition at 21.2 eV was also a prominent feature of the Bethe surface and could still be observed at angles as large as 4°.

Approximate analytic expression are given for the limitation due to backward stimulated light scattering on the diameter of filaments in self-focusing liquids. The calculation is carried out for both $Q$-switched and picosecond laser pulses. The effect of transient gain is considered. Raman scattering is found to be the dominant stimulated process. The results are compared with available experimental data.

The quantum-theoretic method developed in one of our earlier papers is applied to the interaction of an electron with a multicavity system. This analysis is extended to the two-cavity oscillator, linear accelerator (linac), and frequency mixer. It is found that the experimental observations done so far in the above systems do not depend on the quantum state of the cavity field. A comparison of the oscillator theory and the linac given here indicates that the weak-field limit should be taken as the correct classical limit according to the generalized correspondence principle. Finally, a possible refinement of linac design, as a result of the present calculation, is pointed out.

A measurement is presented of the liquid-structure factor of $p-n$-heptyloxyazoxybenzene in the nematic phase using monochromatic Cu $K\alpha$ radiation. This material has a smectic $C$ phase at lower temperature and exhibits strong pretransition scattering in the nematic phase. Near the phase transition the liquid-structure factor peaks at a tilt angle of 30° between the plane normal and the nematic director. By using a simple Landau-like theory to describe the phenomenon, one can map out the free energy of a smectic $C$ density wave as a function of wavelength, tilt angle, and temperature.

Formulas for the stopping power of a nonrelativistic ionized gas for electrons, protons, and other structureless charged particles are derived. The primary velocities considered extend from values comparable to those of the plasma electrons up to values in the relativistic range. New results are obtained for the lower primary velocities, in that full account is taken of the effect of the initial velocity distribution of the plasma electrons on binary collisions as well as on collective excitations. Formulas for the case of non-negligible damping of plasma oscillations are given and comparisons are made with the impact-parameter method and with particle kinetic theories.

A quantum lattice-gas model of adsorbed systems is studied to determine the importance of quantum-mechanical processes upon order-disorder transition temperatures. The phase diagram of systems interacting with nearest-neighbor attractive and repulsive potentials is obtained within the Bethe-Peierls approximation. It is found that the transition temperature depends, in order of increasing importance, upon particle statistics, quantum tunneling, and the nearest-neighbor interaction strength. The results are applied to systems of helium adsorbed on graphite.

A three-dimensional theory for the resonant interaction of electromagnetic waves with a gas of two-level atoms is formulated in terms of macroscopic variables. The theory is utilized to find the steady-state attenuation of a plane wave in the presence of another plane wave running in the opposite direction with different amplitude. Contributions are included from the reflection of the oppositely running wave by an induced standing-wave inhomogeneity in the population inversion of the medium. The resulting attenuation and reflection coefficients are expressed as velocity integrals of continued fractions. Correspondence is made with existing gas-laser theories, yielding the formulation of a high-intensity ring-laser theory. Analytic approximations for the coefficients are presented for the Doppler-limit cases of both waves weak, one wave weak, and negligible reflection (rate-equation approximation). More-general cases have been calculated numerically. The attenuation coefficients exhibit a Lamb-dip feature. The relative depth of the dip increases rapidly with power at low saturation levels, slowly at high saturation, and is greater in the attenuation of the weaker wave. The width of the dip is nonlinearly power broadened. The shape of the dip is very nearly Lorentzian, except for one special case at high power in which the line splits. The propagation equations for the two waves are integrated over long absorption paths. A large resulting attenuation increases the relative size of the dip while decreasing the power broadening.

From an absolute wavelength measurement of the $2s {}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}-2p {}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}$ transition in ${}_{}{}^7{}_{}{}^{}\mathrm{Li}$ II we deduce a Lamb shift of 1.274±0.015 ${\mathrm{cm}}^{-1\mathrm{}}$, compared with a theoretical value of 0.99±0.04 ${\mathrm{cm}}^{-1\mathrm{}}$ for the $2s {}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}$ level. We suggest that the difference is due to a Lamb shift of the $2p {}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}$ level and show that other measurements in isoelectronic two-electron spectra are in accord with this prediction.

The structure factor $S\left(Q\right)\mathrm{}$ for liquid aluminum was measured at 703 and 1029 °C in the region $0<Q<\mathrm{}15\mathrm{}$ ${\mathrm{\AA }}^{-1\mathrm{}}$ using neutron-diffraction techniques. A description of the experimental apparatus and procedure is presented, as well as the method of reducing the neutron intensity to the structure factor. The results are compared with other simple liquid metals, and the electrical resistivity is calculated using the Ziman formulation. Excellent agreement is obtained between calculated and experimentally observed resistivity, both in absolute value and in temperature dependence. This agreement lends credence to both the measured structure factor and to the Ziman formulation for resistivity.

In McMillan's first-order calculation, the transport coefficients obtained are a factor of 2 too large. Here we do the calculation by including the multiple-scattering contribution. By carefully picking the cutoff, we get a good agreement with experiment.

The electron temperatures and the electron energy distribution function in a helium afterglow, for pressures of 10-80 Torr and electron densities of ${10}^{11}$-${10}^{13}$ ${\mathrm{cm}}^{-3\mathrm{}}$, are calculated by a technique which readily allows the determination of the relaxation processes controlling the energy input into the electron gas. Comparisons are made to existing experimental data. The energy-distribution-function evaluations show that careful consideration of this parameter must be made, for it can be extremely non-Maxwellian.

The time dependence of the densities of ${\mathrm{He}}^{+}$, $\mathrm{He}_2^{}{}_{}{}^{+}$, ${\mathrm{H}}^{+}$,$\mathrm{H}_2^{}{}_{}{}^{+}$,$\mathrm{H}_3^{}{}_{}{}^{+}$, ${\mathrm{HeH}}^{+}$, and ${\mathrm{He}}_2$${\mathrm{H}}^{+}$ ions was measured in the afterglow period of plasmas produced in helium containing 0.01, 0.02, and 0.1% hydrogen for total gas pressures varying from 1 to 10 Torr. The rate constant for the ionization of ${\mathrm{H}}_2$ by $\mathrm{He}\left(2\mathrm{} {}_{}{}^3{}_{}{}^{}S\right)$ was found to be 5.2×${10}^{-11\mathrm{}}$ cm ${\sec }^{-1\mathrm{}}$. The studies resulted in the observation, for the first time, of the production of $\mathrm{H}_2^{}{}_{}{}^{+}$ by mutual collisions between metastable hydrogen molecules. The radiative lifetime of these molecules was measured to be 2.7 ± 0.2 msec. The occurence of several other collision processes was also established. The mobility of $\mathrm{H}_3^{}{}_{}{}^{+}$ in helium was determined to be ${\mu }_0=40\mathrm{}\pm 0.5$ ${\mathrm{cm}}^2$ ${\mathrm{}\left(\mathrm{V}\sec \right)\mathrm{}}^{-1\mathrm{}}$.

The nuclear-charge-expansion method is used to calculate the radiative probabilities of the dipole transitions $2s^{a}2p^b-2s^{a-1\mathrm{}}2p^{b+1\mathrm{}}$ of ionic systems. An extensive comparison with more-elaborate calculations and with experimental data shows that after a few stages of ionization the method gives results of uniformly high accuracy.

A computer simulation of recent self-induced-transparency experiments in ruby demonstrates the general appropriateness of a previously derived two-level model for a wide range of attenuator temperatures. The only significant deviation is a sharper rise in the experimental delay curve. The temperature dependence of the homogeneous phase relaxation $T_2^{\prime }$ was introduced by including a "direct" phonon transition process in our program. The parameters for best fit provide ${T^{\prime }}_2\simeq 50$ nsec for $T\simeq 0$ °K, which emphasizes the importance of phase-memory effects in low-temperature atomic-coherence experiments in ruby.

An exact renormalization equation is derived by making an infinitesimal change in the cutoff in momentum space. From this equation the expansion for critical exponents around dimensionality 4 and the limit $n=\infty$ of the $n$-vector model are calculated. We obtain agreement with the results of Wilson and Fisher, and with the spherical model.

The electron-ion recombination coefficients for $\mathrm{H}_3^{}{}_{}{}^{+}$ and $\mathrm{H}_5^{}{}_{}{}^{+}$ ions have been determined by means of a microwave afterglow/mass-spectrometer apparatus. Measurements of electron density decays in helium-hydrogen mixtures are correlated with the decay of mass-identified ion currents to the wall of the microwave cavity. At low partial pressures of hydrogen in the mixture, the ion $\mathrm{H}_3^{}{}_{}{}^{+}$ dominates the ion composition and the ion wall current "tracks" the electron density decay curves. From recombination controlled electron density decay curves, the values $\alpha \left({{\mathrm{H}}_3}^{+}\right)=(2.9\mathrm{}\pm 0.3),(2.3\pm 0.3),\mathrm{and} (2.0\mathrm{}\pm 0.2)\times {10}^{-7\mathrm{}}$ ${\mathrm{cm}}^3$/sec are obtained at 205, 300, and 450 K, respectively. At higher partial pressures of hydrogen and low temperatures, where $\mathrm{H}_5^{}{}_{}{}^{+}$ is the dominant ion, the value $\alpha \left({{\mathrm{H}}_5}^{+}\right)=(3.6\mathrm{}\pm 1.0)\times {10}^{-6\mathrm{}}$ ${\mathrm{cm}}^3$/sec is obtained at 205 K. The implications of these results concerning ionization levels in the atmospheres of the outer planets and in the intersteller medium are discussed.

Recombination coefficients of electrons with HC${\mathrm{O}}^{+}$ ions have been determined with a microwave afterglow/mass-spectrometer apparatus. Afterglow measurements of electron density decays in neon-hydrogen-carbon-monoxide mixtures are correlated with the decay of mass-identified ion currents to the wall of the microwave cavity. At the appropriate partial pressures of hydrogen and carbon monoxide in the mixture, the ion HC${\mathrm{O}}^{+}$ dominates the ion composition and its wall current approximately "tracks" the electron density decay curve. From recombination controlled electron density decay curves, the values $\alpha \left({\mathrm{HCO}}^{+}\right)=(3.3\mathrm{}\pm 0.5) \mathrm{and} (2.0\mathrm{}\pm 0.3)\times {10}^{-7\mathrm{}}$ ${\mathrm{cm}}^3$/sec are obtained at 205 and 300 K, respectively. The implications of these results for models of polyatomic-molecule formation in dense interstellar clouds are briefly discussed.

The statistical dynamics of a classical random variable that satisfies a nonlinear equation of motion is recast in terms of closed self-consistent equations in which only the observable correlations at pairs of points and the exact response to infinitesimal disturbances appear. The self-consistent equations are developed by introducing a second field that does not commute with the random variable. Techniques used in the study of the interacting quantum fields can then be employed, and systematic approximations can be obtained. It is also possible to carry out a "charge normalization" eliminating the nonlinear coupling in favor of a dimensionless parameter which measures the deviation from Gaussian behavior. No assumptions of spatial or time homogeneity or of small deviation from equilibrium enter. It is shown that previously inferred renormalization schemes for homogeneous systems were incomplete or erroneous. The application of the method to classical microscopic systems, where it leads from first principles to a coupled-mode description is briefly indicated.

The dipole-dipole dispersion force between carbon and hydrogen and between oxygen and hydrogen was calculated by many-body perturbation theory. Unperturbed orbitals of carbon and oxygen were calculated in a $V^{N-1\mathrm{}}$ potential. Many-body diagrams were evaluated in second and third order. Estimates of higher-order diagrams were made by geometric summation. The values obtained for $C_6$ for C-H were 14.4 a.u. in the $\Sigma$ state and 14.8 a.u. in the $\Pi$ state. For O-H a value of 9.13 a.u. was obtained for the $\Sigma$ state and a value of 9.22 a.u. was obtained for the $\Pi$ state.