A quantum-mechanical study is made of reactive scattering in the (H, ${\mathrm{H}}_2$) system. The problem is formulated in terms of a form of the distorted-wave Born approximation (DWBA) suitable for collisions in which all particles have finite mass. For certain incident energies, differential and total cross sections, as well as other attributes of the reactive collisions, (e.g., the reaction configuration) are determined. Two limiting models in the DWBA formulation are compared; in one, the molecule is unperturbed by the incoming atom, and in the other, the molecule adiabatically follows the incoming atom. For thermal incident energies and the semiempirical interaction potential employed, the adiabatic model seems to be more appropriate. Since the DWBA method is too complicated for a general study of the (H, ${\mathrm{H}}_2$) reaction, a much simpler approximation method, the "linear model," is developed. This model is very different in concept from treatments in which the three atoms are constrained to move on a line throughout the collision. The present model includes the full three-dimensional aspect of the collision, and it is only the evaluation of the transition matrix element itself that is simplified. It is found that the linear model, when appropriately normalized, gives results in good agreement with that of the DWBA method. By application of this model, the energy dependence, rotational-state dependence, and other properties of the total and differential reaction cross sections are determined. These results of the quantum-mechanical treatment are compared with the classical calculation for the same potential surface. The most important result is that, in agreement with the classical treatment, the differential cross sections are strongly backward peaked at low energies and shift toward the forward direction as the energy increases. Finally, the implications of the present calculations for a theory of chemical kinetics are discussed.

The charge-exchange process has been studied at much higher energies than previously examined. Equilibrium charge distributions have been measured in foils of Mylar, Al, Ni, and Ta for protons of 155 and 600 MeV, and deuterons of 80 MeV. Estimates have been made for the loss cross sections ${\sigma }_l$, and the resulting capture cross sections ${\sigma }_c$ are compared with theoretical expectations. Up to 155 MeV, ${\sigma }_c$ can be reasonably well accounted for by the nonradiative capture process. Above this energy there are significant departures from this simple behavior which are consistent with the predicted onset of radiative capture. However, because of possible relativistic effects, such an interpretation is not certain. The present data suggest several interesting areas for future theoretical and experimental study.

The ionization extrapolated ranges of ${\mathrm{H}}^{+}$ and ${\mathrm{He}}^{+}$ in Ar, ${\mathrm{N}}_2$, ${\mathrm{O}}_2$, and air have been measured from 200 keV to 2.0 MeV with an accuracy of ± 1%. A magnetically analyzed beam of ions was brought through a differential pumping system into a gas stopping cell. A single-grid ionization chamber, movable parallel to the beam axis, was used to obtain the data necessary to plot a Bragg curve. The observed range-energy relations are compared with previous data where possible.

Electron-impact excitation cross sections for the resonant transition ($2s-2p$) in the lithium atom have been calculated under the Born approximation for incident energies up to 60 eV. The polarization of the target atom caused by the incident charged particle is considered by modifying the target wave function. The exchange has also been included via the Ochkur approximation. The cross sections obtained here are compared with the other theoretical calculations and the experimental data.

Energies of the $1s$, $2s$, $3s$, and $4s$ states are obtained by a perturbation calculation and by a one-parameter variational calculation for a two-particle system interacting through a static screened Coulomb potential. The variational results are practically identical with those recently obtained by Rogers, Graboske, and Harwood by the numerical integration of the wave equation, except in the region very close to the critical screening.

A new variational principle for multichannel scattering is derived which combines certain simplifying features of both Kohn and minimum principles. The rigorous bound property of the latter is relaxed slightly to gain simplifications and efficiency, especially for collisions involving particle rearrangements. The approach is essentially equivalent to treating variationally the open-channel part of the wave function, rather than solving for it exactly as required by the minimum principle. However, variations of parameters for the open and closed channels are carried out separately using a double functional.

Two approximations to Seaton's unitarization scheme are described which make it possible to impose the conservation-of-particle-flux requirements on an approximate calculation of inelastic scattering without making calculations of elastic scattering in the two channels which the transition connects. One approximation is based on the black-sphere model of Feshbach and Weisskopf. Another approximation corrects only for back coupling and can be equivalent to a method of Levine. The methods are illustrated by applying them to the Born approximation for the $1s-2s$ excitation of the hydrogen atom by electron impact.

Upper and lower bounds for three-body scattering are studied using Hilbert-space results previously obtained. The evaluation of the bounds entails using a set of basis functions to approximate the exact wave function and requires a lower bound to an associated eigenvalue problem. Our Hilbert-space results yield the lower bound without additional calculations when the set of basis functions can adequately approximate the exact wave function. Our bounds do not require knowledge of any bound state and indeed show how a trial bound-state wave function should be determined to minimize the error in a particle bound-state scattering process. Both one-channel and two-channel processes are treated.

Within the framework of time-dependent perturbation theory, general expressions are derived for the cross section of multiphoton processes induced by radiation acting on a hydrogen atom. Bound-bound transitions involving emission of a single quantum and bound-free transitions are calculated for a wide range of photon energies. In the case of resonance, the transition probability is also discussed. The procedure used to evaluate the sums over intermediate states is described. The numerical computation allows one to compare the results obtained for various competing processes.

Some of the key approximations involved in the application of the eikonal method to atomic physics are analyzed, and the method is related to the close-coupling approach. The importance of an accurate treatment of kinematics is stressed, and a Monte Carlo integration technique is used to facilitate the application of eikonal equations without simplifying kinematical assumptions. Results are presented on the excitation of the $2{}_{}{}^1{}_{}{}^{}P$ state of helium by electron impact, on the excitation of the $2p$ state of atomic hydrogen by electron impact, and on the excitation of the $2s$ state of hydrogen by electron and positron (or proton) impact. Detailed comparison with experiment is made wherever possible.

Excitation effects due to rotational motion of the internuclear line during an atomic collision are considered in the two-state approximation. These effects are governed by a pair of first-order differential equations which couple adiabatic levels of different angular momentum (e.g., $\sigma$ to $\pi$ transitions), which actually do cross. The equations are cast in such a form that the solution for rotational excitation may be expressed in terms of the well-known Landau-Zener solution. It is found, however, that application of these results to a real collision suffers from defects which are worse for the case of rotational excitation than for ordinary Landau-Zenner transitions. The coupled differential equations are then solved numerically to document the shortcomings of the Landau-Zener approach to a real collision and to present cross-section results which are free from these defects.

The channel operators ${\tau }_{\gamma \alpha }$ describing scattering from configuration $\alpha$ to configuration $\gamma$ are utilized in considering a model for three-body rearrangement scattering. The ${\tau }_{\gamma \alpha }$ are those defined by $V_{\alpha }+V_{\gamma }{(E-H+i\epsilon )}^{-1\mathrm{}}{V}_{\alpha }$. Two well-known forms of integral equations for the ${\tau }_{\gamma \alpha }$ obtained from this expression are explicitly solved for the model potential surface. Using these explicit solutions, results are examined in the limit that no dissociative continuum is present. It is found that the integral equations in which the ${\tau }_{\gamma \alpha }$ are not explicitly coupled do not yield the correct results for this limiting case. Integral equations explicity coupling the ${\tau }_{\gamma \alpha }$ give limiting results in agreement with those obtained by more-common boundary-matching techniques. These results indicate that the major effects of the dissociative continuum may be accounted for by considering the coupled equations for the ${\tau }_{\gamma \alpha }$ (at least so long as one is well below threshold for the production of three free particles).

A study has been made of the polarized-orbital method (POM) for electron-atom scattering. Within this framework, an exact calculation has been carried out for the electron-helium-atom case. Our study confirms the importance of the second-order distortion-exchange terms, which provide a nonadiabatic effect. A study is made of the much-criticized step function, and its inclusion within the POM framework is found to be not crucial. An attempt is made to clear up some of the present confusion in the literature by distinguishing between the POM and related methods.

The results of some recent molecular-beam magnetic resonance experiments on HD and ${\mathrm{D}}_2$ are reported. A discussion is given of the shifts and distortions of the separated oscillatory field resonance pattern produced by inhomogeneities in the external magnetic field and by the Bloch-Siegert effect. The sign of the electron-coupled spin-spin interaction constant in HD was remeasured to be positive. Its magnitude was determined to be 47(7) Hz. The spectrum of the $J=1\mathrm{}$ state of ${\mathrm{D}}_2$ was observed for external fields less than 160 G, and the spin-rotation and second-rank tensor interaction constants were determined to be $c_d=+8.768\mathrm{}\left(3\right)\mathrm{}$ and $d=+25.2414\mathrm{}\left(14\right)\mathrm{}$ kHz. Using the calculated value $d_M^{\prime }=+2.737\mathrm{}\left(1\right)\mathrm{}$ kHz for the magnetic spin-spin interaction constant, the value of the electric quadrupole interaction constant was determined from the observed value $d$ to be $\frac{\mathrm{eqQ}}{h}=+225.044\mathrm{}\left(24\right)\mathrm{}$ kHz in the first rotational state of ${\mathrm{D}}_2$. Hyperfine transitions were observed for the first time in the second rotational state of ${\mathrm{D}}_2$. The hyperfine constants were determined from the transitions in the limit of zero external magnetic field to be $c_d=+8.723\mathrm{}\left(20\right)\mathrm{}$, $d_M^{\prime }=2.725\mathrm{}\left(14\right)\mathrm{}$, and $\frac{\mathrm{eqQ}}{h}=+223.38\mathrm{}\left(18\right)\mathrm{}$ kHz in the second rotational state. The effects of isotopic substitution and nuclear motion on the hyperfine constants in molecular hydrogen are discussed.

The cross sections for the dissociation modes ${}_{}{}^4{}_{}{}^{}\mathrm{He}_{}^1{}_{}{}^{}$ ${\mathrm{H}}^{+}$ → ${\mathrm{He}}^0$ + ${\mathrm{H}}^0$, ${\mathrm{He}}^0$ + ${\mathrm{H}}^{+}$, ${\mathrm{He}}^{+}$ + ${\mathrm{H}}^0$, ${\mathrm{He}}^{+}$ + ${\mathrm{H}}^{+}$, ${\mathrm{He}}^{++}$ + ${\mathrm{H}}^0$, or ${\mathrm{He}}^{++}$ + ${\mathrm{H}}^{+}$ are reported for the He${\mathrm{H}}^{+}$ energy range of 0.5-1 MeV in the target gases ${\mathrm{H}}_2$, He, ${\mathrm{N}}_2$, and Ne. Comparisons are made with available previous measurements.

In the energy range 7-40 MeV, electron-capture and electron-loss cross sections have been determined for incident oxygen ions of charges +2 to +8 passing through argon, nitrogen, and helium. Thin-target conditions have been used in all cases. The analysis technique used to extract the cross sections allows relative cross sections to be obtained without consideration of the measurement of the gas target thickness. Multiple-transfer cross sections can thereby be determined reliably in the presence of cross sections three orders of magnitude greater. Double, triple, and quadruple transfers were observed. The electron-loss cross sections appear to pass through maxima when the velocity of the ion is in the vicinity of the velocity of the electron to be lost. Single-electron-capture cross sections have magnitudes as large as ${10}^{-17\mathrm{}}$ ${\mathrm{cm}}^2$ at 40 MeV in argon, and depend on velocity approximately as $V^{-n}$, where $n$ lies between 3 and 6 for the heavy gases. The importance of capture from inner shells of the heavy gases is inferred by comparison with helium, in which $n$ is 8 or 9 and the cross sections are relatively small. The possibility of capture into the $K$ shell of the oxygen ion is suggested by the systematic differences observed in capture by +7 and +8 oxygen ions from capture by lower-charged ions.

Experimental data on electron-capture cross sections are analyzed in order to obtain the cross sections for removal of $n$ given atomic electrons, where $n=1, 2, or 3$. Probabilities of capture averaged over certain regions of impact parameter are obtained from ratios of these cross sections. Maxima are found in the capture probabilities for +8 oxygen ions in helium, and for +8 and +7 ions in nitrogen and argon. No maxima are found for ions of lower charge. The large cross sections at high velocities in nitrogen and argon are attributed to capture from the $K$ shell of nitrogen and the $L$ shell of argon, which contain electrons with momenta great enough to satisfy conservation requirements. The larger cross sections for +8 ions and the maxima in capture probability are attributed to exoergic capture from atomic energy levels into the deeply bound $K$ shell of the oxygen +8 ion. A criterion based on an atomic-orbital theory is given for the velocity at which maxima due to exoergic capture should occur, and this is in rough agreement with the observed velocities. Electron-capture cross sections from atoms with tightly bound shells can generally be expected to be large for heavy ions even at velocities greater than ${10}^9$ cm/sec.

An application of the Faddeev-Watson multiple-scattering expansion for rearrangement collision to an arbitrary three-body Coulomb system is presented. The importance of the Coulomb cuts on the energy shell is emphasized. A demonstration of how the second-order Born term and the truncated second-order multiple-scattering term considered by Carpenter and Tuan get canceled out in the second-order multiple-scattering term is given. It is then clear that considerable problems remain in the determination of the high-energy behavior of the cross section for rearrangement collisions. A number of errors which have recently appeared in the literature concerning the treatment of the branch-point singularities and the intermediate two-body bound- and antibound-state poles in the off-shell two-body Coulomb $T$ matrix are rectified. Detailed expressions for the three-body rearrangement-collision amplitude in the first-order multiple-scattering approximation are derived and expressed as a sum of three contributions coming from (a) the bound-state poles in the initial and final wave functions, (b) the intermediate two-body antibound-state poles, and (c) the branch-point singularities in the off-shell two-body Coulomb $T$ matrix.

The high-energy limit of the differential scattering cross section, according to the exact Born approximation can have either an $E^{-2\mathrm{}}$ or an $E^{-6\mathrm{}}$ energy dependence, depending on the scattering angle and masses of the system. When the target is made of equal-mass particles having charges which are equal but opposite in sign, the first-order Born approximantion predicts a zero for the elastic scattering amplitude. This anomalous behavior does not appear in the first-order Faddeev-Watson multiple-scattering approximation. An application of the first-order Faddeev-Watson multiple-scattering approximation to a three-body Coulomb system with arbitrary masses is considered. The three-body scattering amplitude is derived and expressed in terms of an integral consisting of contributions coming from (a) the two-body bound-state poles in the initial and final wave functions, (b) the branch-point singularities in the off-shell two-body Coulomb $T$ matrices, and (c) the intermediate two-body bound- and antibound-state poles.

Explicit relations between the on-shell and off-shell three-body scattering and rearrangement-collision amplitudes are given. It is shown that the off-shell approach to the determination of the on-shell amplitudes is equivalent to the conventional method of taking on-shell matrix elements. As an example, the bare-potential term of the Faddeev-Watson multiplescattering expansion for rearrangement collisions is evaluated using the two alternative approaches.