Sets of energy-loss spectra for ${\mathrm{Li}}^{+}$ incident on He that were taken at a series of scattering angles were used to calculate differential cross sections. Differential cross sections are given for elastic scattering and for excitation of the He ($n=2\mathrm{}$) states. The differential cross section is also given for excitation of states corresponding to a 60-eV energy loss. These inelastic differential cross sections exhibit maxima at scattering angles greater than zero. The velocity dependence of the cross sections can be compensated by displaying the elastic data in $\rho$-vs-$\tau$ plots and the inelastic data in $\rho$-vs-$E^{\frac{3}{2}}\theta$ plots. When displayed in terms of these variables, the data are strikingly similar to those reported for ${\mathrm{Li}}^{+}$ + He collisions at much lower impact energies.

Absolute and relative cross sections were obtained for the Penning-ionization (PI) reaction ${\mathrm{Ne}}^{*}+\mathrm{Ar}\to \mathrm{Ne}+{\mathrm{Ar}}^{+}+e$ and the associative-ionization (AI) reaction ${\mathrm{Ne}}^{*}+\mathrm{Ar}\to \mathrm{Ne}{\mathrm{Ar}}^{+}+e$ by a merging-beams technique over a range of interaction energies $W$ from 0.01 to 600 eV. The Ne* represents a composite of $\mathrm{Ne}\left(3s{}_{}{}^3{}_{}{}^{}P_2^{}{}_{}{}^{}\right)$ and $\mathrm{Ne}\left(3s{}_{}{}^3{}_{}{}^{}P_0^{}{}_{}{}^{}\right)$. The PI cross section $Q_{{\mathrm{Ar}}^{+}}$ rises with decreasing $W$ except for a small dip near 0.05 eV and exhibits a rather weak dependence on $W$. For a change of almost five orders of magnitude in $W$, $Q_{{\mathrm{Ar}}^{+}}$ only changes by about a factor of 7. The AI cross section $Q_{\mathrm{Ne}{\mathrm{Ar}}^{+}}$ monotonically increases with decreasing $W$ and shows a large dependence on $W$. The $Q_{\mathrm{Ne}{\mathrm{Ar}}^{+}}$ at $W=0.01\mathrm{}$ eV is about 70 times larger than that at 0.5 eV. The total cross section $Q_T=Q_{{\mathrm{Ar}}^{+}}+Q_{\mathrm{Ne}{\mathrm{Ar}}^{+}}$ is compared with theoretical and other experimental values. At low $W$ our absolute $Q_T$ are considerably larger than those of Tang, Marcus, and Muschlitz although the relative $Q_T$ for both experiments are in fair agreement.

The change in the $K\alpha$ x-ray spectra of neon is studied as a function of projectiles of ${\mathrm{C}}^{+3,+4,+5,+6\mathrm{}}$, ${\mathrm{N}}^{+4,+6,+7\mathrm{}}$, ${\mathrm{O}}^{+4,+5,+6,+7,+8\mathrm{}}$, and ${\mathrm{F}}^{+9\mathrm{}}$ having an energy between 1 and 2 MeV/amu. The spectra are observed using a curved crystal spectrometer. Combined with previous measurements data are obtained for projectiles from carbon to argon. X rays are observed from different ionization states of Ne designated by $K{L}^n$ for $n=1-7\mathrm{}$. An 81% correction between the $K{L}^5$ peak and $K{L}^6$ peak is derived. Semiempirical fluorescence yields are applied to the relative x-ray intensities to obtain relative multiple ionization cross sections. These cross sections are fitted by a binomial distribution as predicted by Coulomb ionization. All cases give good fits except for the ${\mathrm{F}}^{+9\mathrm{}}$ induced spectra. A possible explanation is that a different excitation mechanism may exist for this near-symmetric collision case.

Results of previous structure calculations by the authors on the ground and excited states of ${\mathrm{Ne}}_2$ and ${\mathrm{Ne}}_2^{+}$ are used to calculate total, diffusion, and excitation - (or charge-) transfer cross sections for collisions of ${\mathrm{Ne}}^{*}(3s, {}_{}{}^3{}_{}{}^{}P_{2,1,0}^{}{}_{}{}^{} \mathrm{and} {}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{})$ and ${\mathrm{Ne}}^{+}\left({}_{}{}^2{}_{}{}^{}P_{\frac{3}{2},\frac{1}{2}}^{}{}_{}{}^{}\right)$ with the ground-state neon atom. The calculations have been carried out both including and excluding spin-orbit coupling, to demonstrate its effect. The influences of curve crossings (and adiabatically avoided crossings), potential maxima, and long-range forces on the magnitude and shape of the cross sections are emphasized. It is shown that the inference of excited-state potential curves from experimental scattering data can be a much more difficult task than is commonly assumed. The calculated cross sections and related quantities are in generally good agreement with the available experimental measurements.

$K$-shell x-ray and ionization cross sections are reported for protons on Ag, Cd, Sn, Sb, Te, Ba, and La over the ion energy range of 0.6-2.0 MeV. The data are compared with the predictions of the nonrelativistic plane-wave Born approximation (PWBA), the nonrelativistic PWBA with corrections for binding energy and/or Coulomb deflection, binary-exchange approximation (BEA), and the constrained BEA predictions. In general the nonrelativistic PWBA with binding-energy corrections gives the best over-all agreement with the measurements of proton-induced x-ray processes for the $K$ shells of the elements studied in this work. The data further suggest the need for relativistic PWBA treatment of the interactions in the $K$ shell for the range of binding energies represented by the elements investigated in this work.

The electron-temperature dependence of the recombination of mass-identified N${\mathrm{O}}^{+}$ ions with electrons is studied by means of a microwave-afterglow/mass-spectrometer apparatus employing microwave heating of the electrons. Under conditions where the N${\mathrm{O}}^{+}$-ion wall current satisfactorily tracks the volume electron density decay, the electron decay data indicate that the recombination coefficient, $\alpha \left(\mathrm{N}{\mathrm{O}}^{+}\right)$, decreases as $T_e^{-(0.37\mathrm{}\pm 0.03)}$ over the range $380\le T_e\le 5470$ K with $T_{+}=T_n=380\mathrm{}$ K. The values of $\alpha \left(\mathrm{N}{\mathrm{O}}^{+}\right)$ obtained at $T_e=T_{+}=T_n=300\mathrm{} \mathrm{and} 380$ K are (${4.3}^7$±0.3)×${10}^{-7\mathrm{}}$ and (${3.5}^0$±0.3)×${10}^{-7\mathrm{}}$ ${\mathrm{cm}}^3$/sec, respectively. These values apply to N${\mathrm{O}}^{+}$ ions in their ground electronic and probably the $v=1\mathrm{} or 0$ vibration state, and are in good agreement with previous afterglow measurements. Our $T_e^{-0.37\mathrm{}}$ variation agrees qualitatively with theoretical predictions of a $\sim T_e^{-0.5\mathrm{}}$ variation, but is not in agreement with the $\sim T_e^{-0.83\mathrm{}}$ variation obtained for N${\mathrm{O}}^{+}$ ($v=0\mathrm{}$) ions using ion-storage techniques. We are unable to account for the discrepancies in the energy dependences obtained in the two measurements.

Absolute cross sections for the destruction of hydrogen metastable atoms [$\mathrm{H}\mathrm{}\left(2s\right)\mathrm{}$] have been measured for impact on He, Ar, ${\mathrm{H}}_2$, and ${\mathrm{N}}_2$ in the energy range from 20 to 120 keV. Cross sections have also been measured for the collisional ionization of $\mathrm{H}\mathrm{}\left(2s\right)\mathrm{}$ by impact on ${\mathrm{H}}_2$ in this same energy range. The analysis of the collisional destruction of $\mathrm{H}\mathrm{}\left(2s\right)\mathrm{}$ includes electron capture into unbound states of the negative ion with subsequent loss of the excited state as one of the quenching mechanisms in this energy range. Collisional ionization (direct electron loss) of the metastable projectile accounts for most of the remaining destruction. Within experimental error, collisional ionization and electron capture can entirely account for the destruction by impact on He. Additional quenching mechanisms are required for impact on the other targets. It is found that about 70% of the noncapture destruction is accounted for by collisional ionization in these cases. The results for destruction by impact on the molecular gases, ${\mathrm{N}}_2$ and ${\mathrm{H}}_2$, appear to be adequately explained by the inclusion of dipole-quadrupole electric-field quenching as the third mechanism.

A unified method for calculating both resonance and continuum contributions to photoionization cross sections by bound-state techniques is developed. The stationary scattering wave function for the atomic system is expanded in a subspace of discrete functions which is large enough to be nearly complete for the region near the nucleus involved in the photoionization process. Some of the functions in the subspace are selected by root-stabilization techniques to be accurate representations of the resonances while others represent continuum functions. The expression for the cross section is derived from the Fano treatment of autoionizing states. The accuracy of the method is demonstrated by the calculation of cross sections for photoionization of the ground state $1{}_{}{}^1{}_{}{}^{}S$ of the hydrogen negative ion and the helium atom, and for the metastable $2{}_{}{}^1{}_{}{}^{}S$ state of helium at energies in the continuum and through the resonance $2{}_{}{}^1{}_{}{}^{}S-\left(2s2p\right)\mathrm{}{}_{}{}^1{}_{}{}^{}P$ transition.

We have measured target $K$ x-ray production cross sections for lithium projectiles incident on thin aluminum, scandium, copper, and tin targets at energies from 4 to 34 MeV. The Sn $L$ x-ray cross sections for those energies have also been obtained. In addition, similar measurements for protons incident at velocities matching those of the lithium projectiles have been made. The absolute magnitudes of the lithium ionization cross sections as deduced from the above results are well predicted by the binary-encounter approximation. Substantial deviations downward occur in the Sc and Cu cases for projectile velocities below one-half of the $L$-shell electron orbital velocities. These decreases in the cross sections, which are also evident in the data when scaled by the corresponding proton values, are examined in light of a previous treatment of increased binding effects.

Single- and double-electron-transfer cross sections have been measured for a ${}_{}{}^4{}_{}{}^{}\mathrm{He}_{}^{++}{}_{}{}^{}$ beam incident on thin He, Ar, ${\mathrm{H}}_2$, and ${\mathrm{N}}_2$ gas targets over the energy range 15-125 keV. Comparison with previous ${}_{}{}^3{}_{}{}^{}\mathrm{He}_{}^{++}{}_{}{}^{}$ experimental results is made at the same incident particle velocity. For ${\mathrm{He}}^{++}$ collisions with He, we find good agreement with the theory for double electron transfer, but conclude that the single-electron-transfer cross section is far from being understood. Interpretation of the data in terms of simple theoretical ideas indicates that capture into excited states of the ${\mathrm{He}}^{+}$ ion can be the dominant single-electron-transfer process in some keV-energy ${\mathrm{He}}^{++}$ atomic collisions.

Cross sections for single-electron transfer forming the metastable state ${}_{}{}^4{}_{}{}^{}\mathrm{He}_{}^{+}{}_{}{}^{}$ ${}_{}{}^4{}_{}{}^{}\mathrm{He}_{}^{+}{}_{}{}^{}\mathrm{}\left(2s\right)\mathrm{}$ have been measured for a ${}_{}{}^4{}_{}{}^{}\mathrm{He}_{}^{++}{}_{}{}^{}$ beam incident on thin He, Ar, ${\mathrm{H}}_2$, and ${\mathrm{N}}_2$ gas targets. The ${}_{}{}^4{}_{}{}^{}\mathrm{He}_{}^{++}{}_{}{}^{}$ energy range was 7-70 keV. The experiments utilized detection of the $2s$ state via a delayed time coincidence between a field-induced Lyman-$\alpha$ photon and the ion itself. While the measured cross sections are believed accurate to ±35%, they are three times larger than the recent cross sections of Shah and Gilbody using incident ${}_{}{}^3{}_{}{}^{}\mathrm{He}_{}^{++}{}_{}{}^{}$ ions. The present time-coincidence technique directly measured the fraction of the total electron-transfer cross section resulting in metastable production with results ranging between 5% and 17%.

A novel theory is developed to cope with the difficulty of the multiple-scattering problem in a random medium (RM). The theory is given for a one-dimensional homogeneous RM which is represented by a strictly stationary random process. Some possible forms of the stochastic solution are determined by a group-theoretic consideration based on the shift-invariance property of the homogeneous RM. The form of the solution has some analogy with Floquet's solution for a periodic medium. It is shown that there are two kinds of solutions in the one-dimensional RM: a travelling-wave mode and a cutoff mode. The former exists only when the power spectrum of the medium becomes zero at nearly double the wave number. Otherwise the wave is in the cutoff mode which is almost a standing wave whose envelope increases or decreases exponentially with distance. For a Gaussian RM with small fluctuations, an approximate stochastic solution given in the possible form is obtained in terms of multiple Wiener integrals with respect to the Brownian-motion process. The average and variance are calculated for the phase and amplitude of the wave in terms of the power spectrum of the RM. The law of large numbers is shown to hold concerning the fluctuations of the phase and amplitude. The average value of the wave and the transmission coefficient of a medium with finite thickness are also studied using the stochastic solution.

The initial-value problem of the one-dimensional wave propagation in a homogeneous random medium is treated by means of the "Laplace transform," based again on a group-theoretic consideration introduced in the preceding paper. We first define the "Fourier transform" of a random process regarded as a function on the translation group associated with the homogeneity. The inverse "Fourier transform" then gives a general representation of a nonstationary random process generated by a stationary process. Similarly, we define the "Laplace transform" of a random process vanishing on the negative coordinate axis as well as the "Laplace transform" of its derivatives. The one-dimensional wave equation together with the random initial values can be directly treated by means of the "Laplace-transform" technique and is solved approximately in two Gaussian cases where the random media are represented by the well-known O-U (Ornstein-Uhlenbeck) process and by the $Z_0$ process having zero spectrum at the origin. Various statistical parameters associated with the solution can be calculated from the stochastic solution by the averaging procedure. It is shown that the behavior of the average wave is quite different between the two cases and that the result is in agreement with that of the preceding paper. The average of the absolute square of the wave is also calculated using the stochastic solution, and its range of validity is discussed by comparing with the previous results.

The averaged Lagrangian method of Whitham for treating nonlinear, dispersive wave propagation is used to formulate the problem of optical pulse propagation in a resonant medium. For the particular case of self-steepening, it is shown that this formulation takes the same mathematical form as Light-hill's treatment of the propagation of wave trains on deep water. Certain aspects of the solution are discussed and compared with alternative treatments of self-steepening. Some comments are given concerning the applicability of the averaged Lagrangian method to other problems of coherent, nonlinear optics.

We consider the evaluation of multitime-correlation functions for quantum statistical systems characterized by dynamical variables of the angular-momentum type (such as, for example, paramagnetic or nuclear spins or collections of two-level atoms), and show how the atomic coherent-state representation of Arecchi et al. can be used to express the multitime averages in terms of phase-space integrals. Our calculation suggests a convenient classical-quantum correspondence rule for angular-momentum degrees of freedom.

We investigate properties of the cooperative emission of radiation from $N$ identical two-level atoms by using, for the collective atomic operators, the Holstein-Primakoff transformation to boson operators. In the basis of the Glauber states, a weight function is associated with the atomic density operator. The equation of motion for the weight function is derived from the master equation and approximate solutions are given in the two limit cases of superradiance and diffusion. The moments of the radiation intensity are calculated and are found to be in close agreement with previous works.

A soft-x-ray laser utilizing electron pickup by a stripped nucleus is proposed. A particular example using electron exchange between hydrogen and an $\alpha$ particle is analyzed and is shown to produce inversion between the ${\mathrm{He}}_{2p}^{+}$ and ${\mathrm{He}}_{1s}^{+}$ levels giving stimulated emission at 304 Å. In this scheme the $\alpha$ particles are swept along the target at the speed of light. The cross sections involved in the electron pickup in a hydrogen target and subsequent collisional deexcitation are analyzed to ascertain the optimum target thickness for population inversion. Also the effects of atomic collisions on the stimulated-emission cross section are estimated (Doppler broadening). The dynamics of the beam including focusing and space-charge effects are discussed. The stimulated-emission cross sections for a gas and a solid thin-film target are calculated. For a 30-mA $\alpha$-particle current and a 10-cm lasing length, we estimate gains of 1.1 and 50, respectively.

A theorem is proved which gives the necessary and sufficient conditions on the $p$-particle reduced-density operator restricted to a functional subspace that ensures the separability of the wave function of a fermion system. This theorem is the theoretical basis of a general method of analyzing atomic and molecular wave functions, a method which is able to reveal the existence of somehow chemically discernible subsystems of electrons.

Working within the adiabatic approximation in the gauge where the vector potential is zero at the nucleus, Ramsey's diamagnetic part of the shielding constant was calculated using an 87-term James-Coolidge-type wave function. The paramagnetic part of the shielding was determined by means of spin-rotation interaction and isotope-shift data. Results for the shielding constant in isotopes of the hydrogen molecule are reported for a range of temperatures between 0 and 500 K. For ${\mathrm{H}}_2$ at 295 K and zero pressure, $\sigma =26.366\mathrm{}\pm 0.070$ ppm was obtained, with the error meant to take at least partly into account presently uncalculated nonadiabatic and relativistic effects. This value is in agreement with $\sigma =26.23\mathrm{}\pm 0.18$ ppm from experiment.

A variational formalism for the determination of the spinless single-particle density matrix $\gamma (\overrightarrow{\mathrm{r}}, {\overrightarrow{\mathrm{r}}}^{\prime })$ for an interacting many-particle system, correct to second order, is developed. The method is based on variational principles for determining single-particle expectation values, correct to second order, without requiring highly accurate wave functions in order to represent the interacting system. The expectation value of the operator $X=Y+iZ$, where $Y=\left(\frac{1}{2}\right){\Sigma }_j\left[W\right({\overrightarrow{\mathrm{r}}}_i\left)T_j\right(\overrightarrow{\mathrm{a}})+W({\overrightarrow{\mathrm{r}}}_j^{\prime }\left)T_j\right(-\overrightarrow{\mathrm{a}}\left)\right]$, $Z=-\left(\frac{i}{2}\right){\Sigma }_j\left[W\right({\overrightarrow{\mathrm{r}}}_j\left)T_j\right(\overrightarrow{\mathrm{a}})-W({\overrightarrow{\mathrm{r}}}_j^{\prime }\left)T_j\right(-\overrightarrow{\mathrm{a}}\left)\right]$, $W\left({\overrightarrow{\mathrm{r}}}_j\right)=\delta ({\overrightarrow{\mathrm{r}}}_j-\overrightarrow{\mathrm{r}})$, $T_j\left(\overrightarrow{\mathrm{a}}\right)$ is a translation operator, and $\overrightarrow{\mathrm{a}}={\overrightarrow{\mathrm{r}}}^{\prime }-\overrightarrow{\mathrm{r}}$, is the reduced single-particle density matrix. For a given choice of a Slater-determinant-type trial wave function it is shown that the density matrix may be obtained from the charge density of the system where the latter too is determined via the same formalism to second order as the expectation value of the operator $W$ employing the same system trial wave function. The applicability of the technique is then demonstrated by showing that for the model hydrogen-atom problem it leads to highly accurate results for the momentum density and the Compton profile in the impulse approximation, and that the virial theorem is closely satisfied, even when the calculations of the various properties employing an approximate trial wave function alone are substantially in error.