It is shown that the decay in photon-echo amplitude with interval $T$ between pulses provides a new method for the determination of the small-angle dependence of the forward-elastic-scattering cross section $\sigma (V,\theta )$, and that the experiment of Schmidt, Berman, and Brewer yields an analysis of $\sigma (V,\theta )$ for C${\mathrm{H}}_3$F collisions. The result is in agreement with the $\frac{C}{r^3}$ interaction between molecules with permanent electric dipole moments.

The use of high-resolution x-ray measurements in determining the electronic vacancy configuration of an energetic ion inside a solid has been applied to 4.0- and 5.0-MeV Ar projectiles incident on a KCl target. The Ar $K{\alpha }_{1,2}$ line is found to be shifted to higher energies by 6.1 ± 0.6 and 8.1 ± 0.6 eV, respectively, at these bombarding energies. These shifts correspond to five and six vacancies in the Ar $3s-3p$ subshells. The Cl $K{\alpha }_{1,2}$ line is used as a reference line and thin-target x-ray cross-section measurements are used to show that Cl and Ar are excited in the same manner.

Resonances in $e^{-}-{\mathrm{N}}_2$ scattering have been studied from about 11 to 16 eV by simultaneous measurement of the transmitted electron spectrum, and both the prompt and delayed emission functions for the $C {}_{}{}^3{}_{}{}^{}\Pi _u^{}{}_{}{}^{}\to B {}_{}{}^3{}_{}{}^{}\Pi _g^{}{}_{}{}^{}\mathrm{}(0,0)\mathrm{}$ transition. In this energy domain 28 structures have been observed, 25 of which have been observed in both the electron transmission spectrum and the total optical excitation function. The delayed emission in this energy range is attributed to the decay of the various states of $\mathrm{N}_2^{}{}_{}{}^{-}$ to the $E {}_{}{}^3{}_{}{}^{}\Sigma _g^{{}^{+}}{}_{}{}^{}$ state of ${\mathrm{N}}_2$ which is collisionally de-excited to the $C {}_{}{}^3{}_{}{}^{}\Pi _u^{}{}_{}{}^{}$ state of ${\mathrm{N}}_2$. Negative-ion resonances which decay to both the $E {}_{}{}^3{}_{}{}^{}\Sigma _g^{+}{}_{}{}^{}$ and $C {}_{}{}^3{}_{}{}^{}\Pi _u^{}{}_{}{}^{}$ contribute strongly to the 3371 emission in this energy domain.

The stopping powers of six gaseous compounds ${\mathrm{C}}_2$${\mathrm{H}}_2$${\mathrm{F}}_2$, ${\mathrm{C}}_2$${\mathrm{H}}_4$${\mathrm{F}}_2$, ${\mathrm{C}}_3$${\mathrm{H}}_8$, ${\mathrm{C}}_4$${\mathrm{H}}_6$, ${\mathrm{C}}_4$${\mathrm{H}}_{10}$, and ${\mathrm{C}}_3$${\mathrm{H}}_4$ (allene) for $\alpha$ particles of energies 0.3 to 2.0 MeV have been measured in a differentially-pumped gas-cell system with probable random errors ranging from 0.7% to 1.3%. An ambiguity in the interpretation of previous stopping-power measurements in hydrocarbon and halogen-carbon compounds has been resolved, and it is shown that all the previous data, as well as the present data involving straight-chain single-bond compounds, can be fitted by an atomic stopping power for carbon identical to the vapor-deposited solid-carbon stopping power. A rule has been obtained to predict the molecular stopping power of aliphatic saturated hydrocarbons. It is found that due consideration must be given to molecular structure when predicting molecular stopping powers from atomic stopping powers for double-bonded compounds.

Cross sections for electron capture into each excited state of hydrogen up to $4f$ by proton impact on hydrogen and on helium in the energy range of 15-1000 keV have been calculated by means of the Born approximation (Bates's version). In addition we have used the method of close coupling to calculate the $2s$, $3s$, $2p$, $3p$, and $3d$ cross sections for the proton-helium process in the energy range of 30-300 keV. Evaluation of the two-center integrals occurring in the Born amplitude was facilitated by expanding the atomic wave functions in terms of Gaussian-type orbitals. At high energies the relative cross sections of the $\mathrm{nl}$ states (obtained from the Born approximation) for a given $l$ may be related to the importance of close encounters. For the close-coupling calculations we have included as many as eleven states in the multistate expansion of the total wave function. The effects of the intermediate states on the cross sections of the various excited states have been examined. The convergence of multistate cross sections to the two-state cross sections and to the Born cross sections is found to be slow, particularly for $p$ and $d$ states. Comparison of the 11-state close-coupling cross sections of the proton-helium process with experiment shows substantial agreement for the $2s$ and $2p$ states, and, at lower energies, for the $3s$, $3p$, and $3d$ states. At higher energies, however, differences between theory and experiment for the $3s$, $3p$, and $3d$ states somewhat exceed estimated experimental errors.

A semiclassical model for vibrational excitation in molecular collisions is proposed, in which three-dimensional classical trajectory calculations are used to evaluate the quantum vibrational transition probabilities. Using an accurate analytic fit to 138 $\mathrm{ab}$ initio points in the important region of the $\mathrm{H}_3^{}{}_{}{}^{+}$ potential surface and a new Monte Carlo interpolation method for averaging over initial conditions, we apply the model to vibrational excitation in collisions of ${\mathrm{H}}^{+}$ with ${\mathrm{H}}_2$, HD, and ${\mathrm{D}}_2$. The calculations reveal the necessity of correcting our previously reported experimental vibrational transition probabilities for rotational contributions. Once this is done, the experimental data and the model calculations are in very good agreement. The theoretical results support our previous conclusion that vibrational excitation in this system is caused primarily by dilution of the molecular bond by the passing proton. The observed maximum in the experimental inelastic differential cross sections at approximately half the rainbow angle is shown to be associated with the second classical rainbow, which results from the anisotropy of the potential.

A systematic investigation of the interference of energetically neighboring quasimolecular terms giving rise to oscillatory structure in the total cross sections for diabatic processes has been performed by optical studies of selected spectral lines produced in ion-atom collisions. All ions ranging from ${\mathrm{Li}}^{+}$ to ${\mathrm{Al}}^{+}$ have been used as projectiles with energies varying from 0.1 to 15 keV using Ne as the target. Interference between direct-excitation and charge-exchange collision channels leading to regularly spaced oscillations which are 180° out of phase have been seen in the ${\mathrm{N}}^{+}$-Ne, ${\mathrm{O}}^{+}$-Ne, ${\mathrm{Na}}^{+}$-Ne, and ${\mathrm{Mg}}^{+}$-Ne systems. The behavior of the oscillations present in the total emission cross sections for the $3p$ levels in Ne cannot be accounted for by the theoretical predictions of Ankundinov et al. Strong polarization effects have been seen in the optical radiation from the $3p^{\prime }{\left[\frac{3}{2}\right]}_1$ level in Ne in all systems and for the $3p {}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}$ level in Mg in the ${\mathrm{Mg}}^{+}$-Ne system. The oscillations in the Ne excitation function are mainly due to the polarization component of the radiation perpendicular to the beam direction for the ${\mathrm{N}}^{+}$-Ne, ${\mathrm{O}}^{+}$-Ne, and ${\mathrm{Na}}^{+}$-Ne systems, whereas the parallel component also plays a dominant role in the ${\mathrm{Mg}}^{+}$-Ne system. The molecular-orbital (MO) model is used to account for the polarization data.

A method of variable reduction of the dimensionality of the coupled equations for inelastic scattering is presented. The method is based upon a projection operator $P$ with restricted ranges of values for the orbital angular momentum and coupled rotational angular momenta of the two molecules. For rotational states restricted by $0\le j_1\le j_1^{*}$ and $0\le j_2\le j_2^{*}$ and total angular momentum large, the coupled equations have dimensionality $2(j_1^{*}+1) (j_2^{*}+1)-1\mathrm{}\le N\le {(j_1^{*}+1)}^2{(j_2^{*}+1)}^2$, where $N$ is controlled by the choice of $P$. This is in contrast to conventional partitioning techniques which utilize further restrictions on the important molecular rotational states. In both cases the well-known parity conservation causes blocking of the equations into two sets, each of which is essentially half the size of the original set. The dynamics in the $P$ subspace and its complementary $Q$ subspace are decoupled by various approximations on the equation of motion of $Q{\psi }_{\mathrm{scat}}$. Information about scattering into the $Q$ subspace is retained within these approximations and is reintroduced at the end of the computation with little additional labor. An expression for the error in the calculation of the resultant approximate reactance operators is derived and its implications discussed. The general formal equations are then applied to the case of scattering of two rigid rotors, although the inclusion of vibrational modes would in no way affect the procedures outlined. Various possible choices for $P$ are presented, and additional constraints on $P$ for the case of two indistinguishable molecules are discussed. A method of solution for the $P{\psi }_{\mathrm{scat}}$ equation is suggested, and it is shown to lead to the possibility of substantial savings in computational labor.

The angular distributions of electrons emitted in the decay of the autodetaching $\left(2\mathrm{}{s}^2\right){}_{}{}^1{}_{}{}^{}S$, $\left(2s2p\right){}_{}{}^3{}_{}{}^{}P$, $\left(2\mathrm{}{p}^2\right){}_{}{}^1{}_{}{}^{}D$, and $\left(2s2p\right){}_{}{}^1{}_{}{}^{}P$ states of ${\mathrm{H}}^{-}$ have been measured from 10° to 150° for collisions of ${\mathrm{H}}^{-}$ on He, Ar, and ${\mathrm{H}}_2$ between 100 eV and 10 keV. The intensity of the $\left(2\mathrm{}{s}^2\right){}_{}{}^1{}_{}{}^{}S$ line, in the "rest" reference frame of ${\mathrm{H}}^{-}$, was isotropic for high collision energies. For low-energy collisions the distribution has a component which is asymmetric about 90° and peaked at forward angles. The intensity of the $\left(2s2p\right){}_{}{}^3{}_{}{}^{}P$ line, observed only in high-energy collisions, was approximately isotropic. The intensity of the unresolved $\left(2\mathrm{}{p}^2\right){}_{}{}^1{}_{}{}^{}D$ and $\left(2s2p\right){}_{}{}^1{}_{}{}^{}P$ lines, for collisions above 1 keV, was symmetric about 90° exhibiting a ($1+C {cos}^2\theta$) behavior with angle. For lower collision energies, the angular distribution of the unresolved $\left(2\mathrm{}{p}^2\right) {}_{}{}^1{}_{}{}^{}D-\left(2s2p\right) {}_{}{}^1{}_{}{}^{}P$ lines became asymmetric. The results are interpreted in terms of the orientation of the excited ${\mathrm{H}}^{-}$ state. We discuss the effect of interference between discrete and continuum states and the effect of molecular distortion due to the target atom on the measured angular distribution.

In this paper, a systematic study of x-ray production in ion-atom collisions is presented for electron promotion by both $3d\sigma$ and $4f\sigma$ molecular orbitals. $L$ x-ray production cross sections and high-resolution $L$ x-ray spectral measurements are presented for ${\mathrm{P}}^{+}$, ${\mathrm{S}}^{+}$, ${\mathrm{Cl}}^{+}$, ${\mathrm{Ar}}^{+}$, and ${\mathrm{K}}^{+}$ projectiles incident on Ar and C${\mathrm{H}}_4$ gas targets for projectile energies between 15 and 200 keV. Average values of the fluorescence yield and total $L$-shell ionization cross sections are determined. A dramatic $Z$ dependence observed in the x-ray production cross section is attributed to changes in fluorescence yield. Theoretical calculations of fluorescence yield are found not to agree with experimentally determined values. Finally, inner-shell vacancy production by $3d\sigma$ electron promotion is found to be much less efficient than $4f\sigma$ electron promotion for the collision systems studied in this paper.

The $2p$ fluorescence yield of Ar in the presence of zero to six $3p$ holes has been calculated by statistically averaging the fluorescence yields of initial states that consist of individual multiplet configurations, formed by coupling the $2p$ vacancy to the partially filled $3p$ shell. The $L_{2,3}$ fluorescence yields for the ${\mathrm{}\left(2p\right)\mathrm{}}^{-1\mathrm{}}{\mathrm{}\left(3p\right)\mathrm{}}^{-n}$ configurations of Ar are found to be 1.48, 17.97, 24.83, 37.84, 79.61, 112.16, and 121.48 × ${10}^{-4\mathrm{}}$ for $n=0,1,2,3,4,5, \mathrm{and} 6$, respectively. Results agree reasonably well with experimental fluorescence yields deduced from ion-atom collision measurements.

A method is described for computing scattering amplitudes for a cylindrically symmetric potential using square-integrable basis functions. The addition of variational corrections using the Kato formula is also outlined. Results for a two-center Gaussian potential are presented.

In a previous study of bound-state matrix elements of a Hermitian operator $W$, it was possible to obtain at most an upper (or lower) stationary bound. The possibility arose only for the diagonal matrix element case, and only for $W$ nonpositive (or nonnegative). In the present treatment, both upper and lower stationary bounds are obtained, for diagonal and off-diagonal matrix elements, and, though some restrictions on $W$ remain, the requirement that $W$ be of well-defined sign can be dropped. The derivation also improves upon that given previously in that the possibility of any difficulty with near singularities in the equation defining the trial auxiliary (or Lagrange) function is unambiguously avoided. As an example, the method is applied to the problem of the zero-energy scattering of positrons by atoms or ions, and an expression is derived which provides a rigorous stationary upper bound on the scattering length; the target ground-state wave function need not be known exactly. Crude but rigorous numerical results are obtained quite simply in the Born approximation.

An integral form of the close-coupling approximation has been applied to the scattering of electrons by hydrogen atoms with the retention of only the $1s$ state, the effect of exchange being included. Results for the differential and the total cross sections have been obtained from 1 to 500 eV. The calculated results have been compared with the experimental findings and other theoretical results and are found to be fairly reliable throughout the entire energy range. The present results differ from Born values by about 4% even at 500 eV.

A multichannel eikonal treatment of atomic collisions is presented and applied to the excitation of atomic helium by electrons with incident energy $E$ in the range $50\le E\le 500$ eV. Two different sets of orthogonal wave functions are employed. A four-channel description yields differential- and total- excitation cross sections in satisfactory agreement with experiment.

We have studied current transients of pre-breakdown electron avalanches in uniform fields in pure oxygen, and in mixtures of oxygen and hydrogen. The goal of this work is to determine an effective first Townsend coefficient $\alpha$, which accounts for instantaneous ionization processes by electron impact. Following the reasoning of Price, Lucas, and Moruzzi, the mixtures were chosen to remove the primary negative ion ${\mathrm{O}}^{-}$ by a rapid associative detachment reaction with ${\mathrm{H}}_2$, thus effectively making oxygen behave as a nonattaching gas. This simplifies the analysis of the transients. A simple extrapolation to zero hydrogen pressure is used to determine $\frac{\alpha }{p}$, the ratio of the ionization coefficient to pure oxygen pressure. The values of $\frac{\alpha }{p}$ thus obtained are in agreement with pulse measurements in pure oxygen and supplement the pure oxygen data at lower field-strength-to-pressure ratios (between 30 to 40 V/cm Torr) and at higher pressures (16 to 32 Torr). Because of the observed presence of a detachment reaction in the mixture, the role of associative detachment in making oxygen behave as a nonattaching gas is reexamined.

We have used the acoustic-beam probing technique of Gordon and Cohen to measure the speed and attenuation of 36-MHz sound in sulfur hexafluoride near its critical point. The measurements were made along the critical isochore. We have also analyzed all existing sound-speed and attenuation data on the critical isochore in the temperature range $0.1\le T-T_c\le 10.0$ °C. We find that it is possible to model all available data with a frequency-dependent viscosity, involving a slightly modified form of the frequency dependence calculated by Kawasaki using the mode-mode coupling approach of Kadanoff and Swift, together with a single relaxation to account for energy exchange with the vibrational states. We also find that a single-relaxation model involving no critical contribution is not adequate to account for the data.

X-ray absorption and small-angle scattering have been used to study the equilibrium equation of state of argon in the neighborhood of the liquid-vapor critical point. The form of the liquid-vapor coexistence curve was determined from the absorption measurements. The quantity ${\left(\frac{\rho }{{\rho }_c}\right)}^2{P}_c{K}_T$, where $\rho$ is the number of molecules per unit volume, ${\rho }_c$ is the value of $\rho$ at the critical point, $P_c$ is the critical pressure, and $K_T$ is the isothermal compressibility, was obtained by extrapolating the scattering curves to obtain the zero-angle scattered intensity. The long-range correlation length $\xi$, which characterizes the extent of the long-range density fluctuations which occur when a fluid is near its critical point, was computed from the angular dependence of the scattered intensity. In the analysis of the scattering data, the consistency of the results obtained by different methods of analysis was used to determine the value of the critical exponent $\eta$, which is a measure of the deviation of the scattered intensity predicted by the Ornstein-Zernike theory. This analysis indicated that $\eta =0.10\mathrm{}\pm 0.05$. The temperature dependence of ${\left(\frac{\rho }{{\rho }_c}\right)}^2{P}_c{K}_T$ and $\xi$ was studied, both at the critical density above the critical temperature $T_c$ and also in the liquid and vapor phases on the coexistence curve below $T_c$. Along the coexistence curve, the temperature dependence of ${\left(\frac{\rho }{{\rho }_c}\right)}^2{P}_c{K}_T$ has the liquid-vapor asymmetry predicted by several authors. The deviation of ${\left(\frac{\rho }{{\rho }_c}\right)}^2{P}_c{K}_T$ from the form given by the scaling laws was found to be much greater in the liquid phase than in the vapor. The angular dependence of the scattered intensity was different above and below $T_c$, with the Fisher-Burford equation giving a good description of the scattering at the critical density above $T_c$ and with the Tarko-Fisher equation applying on the coexistence curve below $T_c$. Several critical exponents were evaluated, including $\gamma$ and ${\gamma }^{\prime }$, which give the temperature dependence of ${\left(\frac{\rho }{{\rho }_c}\right)}^2{P}_c{K}_T$ above $T_c$ at the critical density and below $T_c$ on the coexistence curve, respectively; $\nu$ and ${\nu }^{\prime }$, which describe the temperature dependence of $\xi$ at the critical density above $T_c$ and on the coexistence curve below $T_c$, respectively; and $\beta$, which specifies the temperature variation of the densities of the two coexisting phases below $T_c$. The short-range correlation length $R={\left(kT\rho K_T\right)}^{-\frac{1}{(2-\eta )}}\xi$, where $k$ is Boltzmann's constant, was calculated from the x-ray data, both above and below $T_c$.

The velocity field of an ideal fluid in the presence of a moving spherical boundary and a vortex line of arbitrary configuration is found, and the hydrodynamic force acting on the sphere determined. Using this formalism, one may compute the motion of the sphere and the vortex line in the general case where the sphere has inertia and is subject to externally applied forces. The resulting algorithm can be utilized to study the complicated hydrodynamic interactions between charge carriers and quantized vortices in superfluid helium. As simple examples of the application of the hydrodynamic formalism, the force exerted on a moving sphere by an instantaneously rectilinear vortex is evaluated, and the interaction of a vortex ring with a fixed sphere is considered.

A new feature of optical free induction decay (FID) is observed in a coherently prepared sample of ${}_{}{}^{13}{}_{}{}^{}\mathrm{CH}$${}_3{}^{}{}_{}{}^{}\mathrm{F}$ by the method of Stark switching. The preparative phase is due to the resonant steady-state excitation by a cw laser beam, and is followed by FID upon sudden application of a dc Stark field that switches the molecular sample out of resonance. The emission is not observed as a simple decay but instead appears as a train of sharp pulses regularly spaced in time as a result of a repetitive interference. This situation arises because an entire set of infrared transitions within the Stark split manifold are initially prepared, in contrast to our previous study of a nondegenerate transition. The emission, which beats with the laser, produces a heterodyne beat spectrum, consisting of a set of regularly spaced frequencies, that is the Fourier transform of the slowly decaying pulse progression observed. We thus demonstrate what is the optical analog of the well-known NMR method of high-resolution pulse Fourier spectroscopy. The detailed behavior of the pulse train agrees well with an FID theory that assumes the transitions to be uncoupled. The subtle behavior of FID near the time origin is explored also by approximate analytic expressions that reveal either a near zero amplitude or sizable amplitude depending on the degree of saturation in the preparative stage. The experimental technique discussed offers an attractive way for obtaining high-resolution optical spectra without Doppler broadening, and for generating an optical pulse train whose time scale can be compressed by simply increasing the Stark field.