Showing papers in "Physical Review A in 1977"

Hydrodynamic fluctuations at the convective instability

Abstract: The effects of thermal fluctuations on the convective instability are considered. It is shown that the Langevin equations for hydrodynamic fluctuations are equivalent, near the instability, to a model for the crystallization of a fluid in equilibrium. Unlike the usual models, however, the free energy of the present system does not possess terms cubic in the order parameter, and therefore the system undergoes a second-order transition in mean-field theory. The effects of fluctuations on such a model were recently discussed by Brazovskii, who found a first-order transition in three dimensions. A similar argument also leads to a discontinuous transition for the convective model, which behaves two dimensionally for sufficiently large lateral dimensions. The magnitude of the jump is unobservably small, however, because of the weakness of the thermal fluctuations being considered. The relation of the present analysis to the work of Graham and Pleiner is discussed.

Large-distance and long-time properties of a randomly stirred fluid

Abstract: Dynamic renormalization-group methods are used to study the large-distance, long-time behavior of velocity correlations generated by the Navier-Stokes equations for a randomly stirred, incompressible fluid. Different models are defined, corresponding to a variety of Gaussian random forces. One of the models describes a fluid near thermal equilibrium, and gives rise to the usual long-time tail phenomena. Apart from simplifying the derivation of the latter, our methods clearly establish their universality, their connection with Galilean invariance, and their analytic form in two dimensions, $\ensuremath{\sim}\frac{{(logt)}^{\ensuremath{-}\frac{1}{2}}}{t}$. Nontrivial behavior results when the model is formally continued below two dimensions. Although the physical interpretation of the Navier-Stokes equations below $d=2$ is unclear, the results apply to a forced Burger's equation in one dimension. A large class of models produces a spectral function $E(k)$ which behaves as ${k}^{2}$ in three dimensions, as expected on the basis of equipartition. However, nonlinear effects (which become significant below four dimensions) control the infrared properties of models which force the Navier-Stokes equations at zero wave number.

Relation between the transport coefficients and the internal entropy of simple systems

Abstract: The transport coefficients of hard-sphere, soft-sphere, Lennard-Jones, and one-component-plasma systems are expressed in terms of the corresponding internal entropies, to reveal some universal characteristics. The transport of all the inverse $n\mathrm{th}$-power repulsive potentials appears to be related, and can be numerically estimated by the hard-sphere model, although the hard-sphere transport is different. The entropy dependence of the coefficients of self-diffusion and (to a lesser extent) shear viscosity, for systems with purely repulsive potentials, is only slightly affected by the addition of an attractive potential.

Charge-transfer and impact-ionization cross sections for fully and partially stripped positive ions colliding with atomic hydrogen

Abstract: Classical trajectory calculations have been used to predict the charge-transfer and impact-ionization cross sections for collisions of ${A}^{+q}$+H in the velocity range of (2-7) \ifmmode\times\else\texttimes\fi{} ${10}^{8}$ cm/sec. The calculations employ a three-dimensional Monte Carlo approach that uses a classical description of the H atom and the Coulomb forces among all particles to obtain the cross sections. The positive ions studied include fully stripped ${A}^{+q}$, with $q=1\ensuremath{-}8, 10, 14, 18, 26, \mathrm{and} 36$, and partially stripped ${\mathrm{B}}^{+q}$, ${\mathrm{C}}^{+q}$, ${\mathrm{N}}^{+q}$, and ${\mathrm{O}}^{+q}$, with $q\ensuremath{\ge}3$. The total electron-loss cross sections (sum of charge exchange and impact ionization) vary only slightly with velocity for the higher charge states and reach a value of 2 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}14}$ ${\mathrm{cm}}^{2}$ for ${\mathrm{Kr}}^{+36}$ + H collisions. The importance of impact ionization relative to charge transfer is found to decrease with increasing $q$ in the velocity range studied. Transition probabilities versus impact parameters are presented for the ${\mathrm{H}}^{+}$ + H and ${\mathrm{Ar}}^{+18}$ + H reactions. The calculations are in reasonable agreement with experimentally measured cross sections for the ${\mathrm{H}}^{+}$, ${\mathrm{C}}^{+q}$, ${\mathrm{N}}^{+q}$, and ${\mathrm{O}}^{q+}$+H systems.

Wigner function as the expectation value of a parity operator

Abstract: It is pointed out that the Wigner function $f(r, p)$ is $\frac{2}{h}$ times the expectation value of the parity operator that performs reflections about the phase-space point $r$, $p$. Thus $f(r, p)$ is proportional to the overlap of the wave function $\ensuremath{\psi}$ with its mirror image about $r$, $p$; this is clearly a measure of how much $\ensuremath{\psi}$ is centered about $r$, $p$, and the Wigner distribution function now appears physically more meaningful and natural than it did previously.

Partial photoionization cross sections of N 2 and CO using synchrotron radiation

Abstract: The photon energy dependence of the partial photoionization cross sections and branching ratios for CO and ${\mathrm{N}}_{2}$ have been measured using the continuum radiation produced by an electron storage ring. These measurements were conducted over the photon energy range of 14 to 50 eV. The partial photoionization cross sections in the energy region of the photoionization continuum are in good agreement with recent calculations where both the theory and experiment exhibit scattering resonances for specific states of the ion. The continuously variable photon energy source was utilized to examine the effect of autoionization on the partial photoionization cross section, showing in several cases quite different structure depending upon the final state of the ion.

Compact and accurate integral-transform wave functions. I. The 1 S 1 state of the helium-like ions from H − through Mg 1 0 +

Abstract: Accurate and compact integral-transform wave functions are constructed for the $1^{1}S$ state of the helium-like ions from ${\mathrm{H}}^{\ensuremath{-}}$ through ${\mathrm{Mg}}^{10+}$. The variational ansatz is of the form $\ensuremath{\Psi}({r}_{1}, {r}_{2}, {r}_{12})={(4\ensuremath{\pi})}^{\ensuremath{-}1}{\ensuremath{\Sigma}}_{k=1}^{N}{c}_{k}(1+{P}_{12})\mathrm{exp}(\ensuremath{-}{\ensuremath{\alpha}}_{k}{r}_{1}\ensuremath{-}{\ensuremath{\beta}}_{k}{r}_{2}\ensuremath{-}{\ensuremath{\gamma}}_{k}{r}_{12})$ where the ${c}_{k}$ are found by solving the secular equation and the exponents ${\ensuremath{\alpha}}_{k}$, ${\ensuremath{\beta}}_{k}$, and ${\ensuremath{\gamma}}_{k}$ are chosen to be the abscissas of Monte Carlo and number-theoretic quadrature formulas for a variationally optimized parallelotope in $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}\ensuremath{-}\ensuremath{\gamma}$ space. A 66-term function of this type for the helium atom yields an energy of -2.903 724363 a.u. as compared with the 1078-term function of Pekeris which yields an energy of -2.903 724376 a.u. In order to test the accuracy of the wave functions a number of properties including $〈{r}^{n}〉$ and $〈{r}_{12}^{n}〉$ with $n=\ensuremath{-}2, \ensuremath{-}1, 1, \dots{}, 4$, $〈{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{1}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{2}〉$, $〈{cos\ensuremath{\theta}}_{12}〉$, $〈\ensuremath{\delta}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{1})〉$, and $〈\ensuremath{\delta}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{12})〉$ are computed and compared with the best available results. The electric dipole polarizability is computed from a simple formula due to Thorhallsson, Fisk, and Fraga. Comments on the limiting accuracy of this formula are made. Electron-nuclear and electron-electron cusp condition tests are made for the functions. Detailed convergence studies are presented for ${\mathrm{H}}^{\ensuremath{-}}$ and He in the form of a sequence of functions with increasing $N$. The functions are found to be rather accurate and more compact than any other functions available in the literature with the exception of those containing logarithmic terms. Possible refinements to the basis set used are discussed.

Electron impact excitation of the electronic states of N 2 . I. Differential cross sections at incident energies from 10 to 50 eV

Abstract: Normalized differential cross sections (DCS's) for the electron impact excitation of the lowest three singlet (${a}^{\ensuremath{'}}, a, \ensuremath{\omega}$), and lowest five triplet ($A, B, W, {\mathrm{B}}^{\ensuremath{'}}, C$) valence electronic states of ${\mathrm{N}}_{2}$, and of the two ($3s{\ensuremath{\sigma}}_{g}$) Rydberg states ($E, {a}^{\ensuremath{'}\ensuremath{'}}$) have been determined at seven incident electron energies ranging from 10 to 50 eV. These DCS's were obtained for the scattering-angle range 5\ifmmode^\circ\else\textdegree\fi{} to 138\ifmmode^\circ\else\textdegree\fi{} by analyzing electron energy-loss spectra in ${\mathrm{N}}_{2}$ at a number of fixed scattering angles within that range. These data, which are the first that cover such a large incident energy and scattering-angle range for such a wide variety of final target states, show that each of the DCS's for the ten final target states falls into one of four classes. Since the initial molecular target state is a singlet, and none of the final states studied here is dipole allowed, these four classes are determined according to whether the final target state is triplet, singlet, parity-unfavored, or Rydberg in character. The DCS's for all the final target states change rapidly in shape as the incident electron energy is varied. Theoretical DCS's obtained from first-order perturbation models (Born-Ochkur and Ochkur-Rudge) agree poorly with the measured DCS's.

Effect of rapid spin exchange on the magnetic-resonance spectrum of alkali vapors

Abstract: It was recently discovered that when there are sufficiently rapid spin-exchange collisions in an alkali-metal vapor, the rf magnetic-resonance frequency of the ground-state atoms becomes a constant fractional value (e.g., $\frac{4}{11}$ for Cs) of the Larmor frequency, and the linewidth becomes inversely proportional to the spin-exchange rate. These properties are in complete contrast to those in the slow spin-exchange limit. We present a theoretical analysis of these properties, and we predict some additional new effects for rapid spin exchange. Four theories are presented; an intuitive-vector model, a perturbation treatment, numerical solutions of complex non-Hermitian matrices, and a statistical analysis of the effect of rapid transfers between two Zeeman multiplets. These theories all provide, to varying degrees, consistent quantitative answers to the problem of how spin exchange affects magnetic resonance in an alkali vapor.

Radiative decays of the n = 2 states of He-like ions

Abstract: The relativistic random-phase approximation (RRPA) is applied to study radiative transitions from $n=2$ states along the He isoelectronic sequence. The strengths of various decay modes and the energy splittings of the $n=2$ multiplets are investigated. At low $Z$ the present results agree with earlier nonrelativistic studies, whereas, at high $Z$ our results provide new information about oscillator strengths, branching ratios, and multiplet structure for the $n=2$ states.

Solitons under perturbations

Abstract: A direct perturbation method is developed in order to study the interactions of $N$ solitons with imperfections and with each other in the presence of imperfections. The leading-order effects are obtained directly from the $N$-soliton waveform without invoking methods from inverse-scattering theory. The method is based upon a Green's function and a "two time" procedure from classical perturbation theory. An example of a single soliton is developed in detail. Finally, this perturbation method is compared with other approaches in the literature.

Superbroadening in H2O and D2O by self-focused picosecond pulses from a YAlG:Nd laser

Abstract: The dielectric-breakdown intensity threshold, the critical power for self-focusing, and the power threshold for the production of spectral superbroadening have been measured in H2O and D2O. For bandwidth-limited pulses of 30 psec duration at 1.06 microns and of 21 psec duration at 0.53 micron, the superbroadening in water always required power levels sufficient for catastrophic self-focusing and intensities equal to the electric breakdown threshold.

Measurement of the spectrum of resonance fluorescence from a two-level atom in an intense monochromatic field

Abstract: We have made measurements of the spectrum of resonance fluorescence emitted by sodium atoms in an intense, uniform, monochromatic field. The transition used in the experiment was between the $3^{2}S_{\frac{1}{2}}$ ($F=2$, ${m}_{F}=2$) ground state and the $3^{2}P_{\frac{3}{2}}$ (${F}^{\ensuremath{'}}=3$, ${m}_{F}^{\ensuremath{'}}=3$) excited state. A separate light beam, circularly polarized and resonant with this transition, was used to prepare the sodium atoms in the $F=2$, ${m}_{F}=2$ magnetic sublevel. This pumping scheme was used to obtain the spectrum for both on- and off-resonance excitations. The measurements are in good agreement with Mollow's calculation. In addition, we describe conditions under which we observed asymmetric spectra similar to those of earlier experiments.

Independent electron approximation for atomic scattering by heavy particles

Abstract: Scattering from a multielectron target by a heavy particle is approximated in terms of amplitudes for scattering from individual target electrons. By treating the motion of the projectile classically and ignoring correlations, the wave function for the system is expressed as a product of single-electron wave functions. The probability amplitude for scattering into specific states is then a product of single-electron scattering amplitudes. In this approximation, cross sections for excitation and ionization involving many electrons are expressed in terms of a binomial distribution of single-electron probabilities. The standard connection of this amplitude for multiple excitation and ionization, ${A}^{\mathrm{if}}(B)$, to the corresponding scattering amplitude, $f(\ensuremath{\theta})$, is given, and the validity of this approximation is discussed.

Collisional redistribution and saturation of near-resonance scattered light

Abstract: This paper reports on our studies of near-resonant scattering of laser light in a collisional environment. A pulsed dye laser with a peak power of 55 MW/${\mathrm{cm}}^{2}$ was tuned near the 460.73-nm resonance line of strontium and the side emission was observed from an oven containing both strontium vapor and argon buffer gas. The emission was composed of three spectral components: Rayleigh scattering at the frequency of the laser ${\ensuremath{\omega}}_{L}$, fluorescence at the resonance frequency ${\ensuremath{\omega}}_{0}$ of strontium, and a third component at $2{\ensuremath{\omega}}_{L}\ensuremath{-}{\ensuremath{\omega}}_{0}$. These three components have been studied as a function of the frequency and intensity of the laser and also as a function of the argon buffer gas pressure. While the Rayleigh emission was found to vary as ${\ensuremath{\Delta}}^{\ensuremath{-}2}$ ($\ensuremath{\Delta}={\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{L}$), the fluorescence component, which was produced by Sr-Ar collisions, was found to be asymmetric with the sign of $\ensuremath{\Delta}$ as predicted by line-broadening theory. By measuring the ratio of the intensities of the fluorescence and Rayleigh components, we were able to measure directly the collisional redistribution function, important in the study of radiative transfer in stellar and planetary atmospheres. At high laser intensities all three components were found to saturate. The results were compared with the theoretical predictions of Mollow's steady-state theory. Theoretical fits for the high-intensity results were obtained when the collisional cross sections were taken to be considerably smaller than in our low-intensity measurements. We believe the discrepancy lies in the use of a steady-state theory for a transient experiment. Effects of radiative trapping and spatial averaging are also discussed.

Stopping power of Al, Cu, Ag, and Au for MeV hydrogen, helium, and lithium ions. Z 3 1 and Z 4 1 proportional deviations from the Bethe formula

Abstract: Stopping power of Al, Cu, Ag, and Au for 0.8-7.2-MeV/amu hydrogen, helium, and lithium ions has been measured by the calorimetric-compensation method to an accuracy of 0.5%. The data agree with most other published results and confirm the validity of earlier measurements by Andersen and co-workers. Higher-order ${Z}_{1}$ deviations from the Bethe formula have been examined. The experimental results agree with Lindhard's calculation of the ${Z}_{1}^{3}$ correction combined with a ${Z}_{1}^{4}$ term, which is slightly larger than Bloch's value. The experimental higher-order ${Z}_{1}$ contributions have a significant influence on the evaluation of empirical shell corrections, and the data are used to extract more realistic values. The deduced shell corrections show good agreement with Bonderup's calculations.

Cross sections for ionization of metastable rare-gas atoms (Ne*, Ar*, Kr*, Xe*) and of metastable N/sub 2/*,CO* molecules by electron impact. [6 to 250 eV, Born and binary encounter approximations]

Abstract: Cross sections for the collisional ionization of the metastable atoms Ne*, Ar*, Kr*, and Xe* by electrons with impact energy E in the range 6 < E < 250 eV are determined in the Born and binary-encounter approximations. For low energies epsilon of ejection, the s-d and s-f bound-free transitions dominate the associated form factor, while transitions to continuum states with progressively higher angular momentum gain importance with increasing epsilon. While up to nine partial waves are normally sufficient for convergence of the bound-free form factor at a given energy epsilon and momentum change K of the ejected electron, as many as 30 are required for those K in the vicinity of the Bethe ridge at large epsilon. These properties are the origin of the overall closeness obtained between the Born and binary-encounter cross sections. Also inner-shell ionization, as described by the binary-encounter treatment, becomes increasingly important as the target atom becomes more complex. Cross sections for ionization of metastable N/sub 2/* and CO* are also determined. Good agreement with available measurements (for Ne* and Ar*) is obtained. (AIP)

Thermodynamics in finite time. I. The step-Carnot cycle

Abstract: The object of this paper is the beginning of a formulation of a method to find bounds to process functions, such as work and heat, for processes occurring in finite time. A general variational statement of the problem is given. Then model problems are solved, all but one of which are based on the "step-Carnot" cycle. This is similar to the reversible Carnot cycle but with the external pressure varying in finite steps. Such a system only needs to go through a finite number of equilibrium states during its cycle. The problems are the maximization of effectiveness of the step-Carnot cycle, the maximization of efficiency of the same cycle, the determination of optimal period for a step-Carnot cycle whose contact with the external reservoirs has finite heat conductance, and the determination of the maximum power and the rate at which maximum power is obtained, for a continuous Carnot cycle with finite heat conductance between system and thermostats.

Optical model for electron scattering from inert gases

Abstract: A local, complex potential is used to compute differential cross sections, spin polarizations, and total nonelastic cross sections for electrons incident on inert gases at energies ranging from 20 to 3000 eV. Excellent comparison with experimental data is achieved in most cases. In addition to Hartree-Fock wave functions, the input to the model involves two parameters. The strength $W$ of the imaginary potential is determined by fitting the total nonelastic cross section. The static polarizability $\ensuremath{\alpha}$ is well known from independent determinations.

Structure and thermodynamics of liquid metals and alloys

Abstract: Ab initio calculations of the structure and the thermodynamic properties of simple liquid metals and alloys are presented. The basic ingredients of the theory are as follows: (a) An orthogonalized-plane-wave-based first-principles pseudopotential is used to describe the interatomic forces. The pseudopotential is optimized specifically for binary systems. The $X\ensuremath{\alpha}$ method is used to construct the electron-ion potential. (b) A system of hard spheres is used as a reference system for describing the liquid structure. Effective hard-sphere diameters are determined by a variational method based on the Gibbs-Bogolyubov inequality. The method of this paper yields encouraging results for the structure factors, the excess volume, and the enthalpy and entropy of formation for alloys with a nearly random distribution of the components. Difficulties arise where nonrandomness has to be expected. The theory provides a basis for a microscopic understanding of the thermodynamics of alloys of simple metals in terms of their electronic structure.

Statistical mechanics of dense ionized matter. VII. Equation of state and phase separation of ionic mixtures in a uniform background

Abstract: By solving the hypernetted chain (HNC) equation for the pair distribution functions, and by Monte Carlo simulations, we have calculated the equation of state and the pair structure of dense binary mixtures of classical point ions in a rigid, uniform background of opposite charge. All of our results indicate that the excess internal and free energies of mixing are negligible compared to the energies of the mixture or the pure phases, for ionic-charge ratios $\frac{{Z}_{2}}{{Z}_{1}}=2 \mathrm{and} 3$, in the strong coupling (high density or low temperature) regime. This feature allows us to write down a simple equation of state for such mixtures, which is used to determine the phase diagram of pressure-ionized H-He and H-Li mixtures in a rigid background of degenerate electrons. We then treat the polarization of the electron gas by the ionic-charge distribution by perturbation theory and include quantum corrections to the free energy of the ions. Both effects do not drastically modify the phase diagrams. The applicability of our results in astrophysical situations is discussed.

Analytic bound-state solutions in a relativistic two-body formalism with applications in muonium and positronium

Abstract: This work advocates the use in atomic physics of a new relativistic two-body formalism equal in rigor to the Bethe-Salpeter formalism and clearly superior to it in several respects. Outstanding among these is the existence of a Coulomb-like kernel for which the exact analytic solutions of the bound-state equations are known. These solutions are derived and applied in a calculation of the O (..cap alpha../sup 6/m ln..cap alpha../sup -1/) contributions to hfs in muonium and positronium. Three previously unknown contributions are found. Theory and experiment are compared.

Resonance fluorescence with excitation of finite bandwidth

Abstract: The problem of resonance fluorescence from a two-level atom that is driven near resonance by a laser beam of finite bandwidth is tackled via the Heisenberg equations of motion, by a simple generalization of our previous treatment. It is found that the two-time intensity correlation function is again factorizable into a product of the mean light intensities at the two times, each of which is slightly modified under broad-band excitation from its value under monochromatic excitation. The spectrum of the fluorescence is affected rather more drastically, and, in general, becomes asymmetric under broad-band excitation.

Statistical atomic models with piecewise exponentially decaying electron densities

Abstract: If it is assumed that the electron density of an atom in its ground state is piecewise exponentially decreasing as a function of the distance from the nucleus, then it is shown that much improved values for the energy, and electron densities exhibiting shell structure, are obtained from optimization of the density functionals of Thomas-Fermi, Thomas-Fermi-Dirac, and Thomas-Fermi-Dirac with inhomogeneity correction. An inhomogeneity correction which is one-ninth of the original Weizsacker correction is favored. Numerical results are presented for all first-row atoms and selected second-row atoms, and comparisons are made with results of other methods.

Continuum orbitals, complex scaling problem, and the extended virial theorem

Abstract: A complex scale transformation of the time-independent Schr\"odinger equation leads to a symmetric eigenvalue problem containing both bound states and resonance (complex) eigenvalues as solutions. An extended virial theorem is stated, and its necessary fulfillment is pointed out. The latter, in conjunction with a symmetric stationary principle, allows for determination of resonance (complex) eigenvalues by means of elementary matrix manipulations. Application to the Stark effect in the hydrogen atom shows agreement with previous calculations based on numerical integration.

Time and spectral resolution in resonance scattering and resonance fluorescence

Abstract: The theory of resonance Raman effect, resonantly enhanced two-photon absorption, and resonance fluorescence---valid also for strong fields---is treated in a unified and simplified way. Dressed states, Bloch equations, and perturbation theory are used to calculate line positions, and integrated intensities both for steady state and transient excitation. The case of adiabatic following is solved explicitly; it occurs when an off-resonance incident pulse is turned on and off slowly. It leads to the identification of Raman scattering, two-photon absorption, and Rayleigh scattering with an adiabatic process while fluorescence and consecutive two-photon absorption with nonadiabaticity. Time-dependent spectra are defined in a rigorous way. Our formulas agree in the various limits with those that appear in the literature.

Low-energy electron-molecule scattering: Application of coupled-channel theory to e-CO/sub 2/ collisions. [0. 07 to 10. 0 eV]

Abstract: A theoretical coupled-channels investigation of $e$-C${\mathrm{O}}_{2}$ scattering is reported for incident electron energies from 0.07 to 10.0 eV. The fixed-nuclei approximation is made with the molecule in the ground $X^{1}\ensuremath{\Sigma}_{\mathrm{g}}^{+}$ state and the nuclei frozen at their equilibrium positions. The $e$-C${\mathrm{O}}_{2}$ interaction potential consists of an ab initio electrostatic Hartree potential, an approximate local exchange potential, and a semiempirical polarization potential. The coupled-channel equations are formulated in a body-fixed reference frame using single-center coordinates and are solved by means of an integral-equations algorithm. Convergence of the highly anisotropic interaction potential and of the expansion of the scattering function are discussed. The asymptotic decoupling approximation and the Born approximation are also studied and found to be unsatisfactory methods for computing quantitatively accurate cross sections for low-energy $e$-C${\mathrm{O}}_{2}$ collisions. Converged coupled-channel total integrated, momentum-transfer and differential cross sections are presented, and the former are compared with experimental results, with special attention given to low scattering energies (\ensuremath{\lesssim}0.1 eV).

Electron capture from K shells by fully stripped ions

Abstract: Cross sections for electron capture from inner atomic shells by fully stripped ions, of velocities high compared to the electron velocities in the inner-shell orbits, are calculated in the second Born approximation. The theory of Drisko for electron capture by protons from hydrogen is generalized to projectiles and targets of arbitrary atomic numbers ${Z}_{1}$ and ${Z}_{2}$, respectively. For ions of low velocity, the effects of binding and Coulomb deflection are accounted for in a manner similar to that of Brandt and his co-workers in the theory for direct ionization to the continuum of the target atom. The results are in good agreement with experimental capture cross sections, whereas the Oppenheimer-Brinkman-Kramers approximation overestimates such cross sections. The contribution of electron capture to inner-shell ionization cross sections increases with increasing $\frac{{Z}_{1}}{{Z}_{2}}$ and brings them into agreement with experiment.