The diffusion of high-energy electrons (≥ 10 keV) in the presence of an electric field is discussed. Solutions are obtained by extending the Bethe age theory to include effects of the applied electric field. Since we restrict ourselves to the $P_1$ approximation, the results are only valid for diffuse beams, i.e., beams whose distribution function can be adequately described by a current and number density. The solution predicts a range enhancement in the presence of an accelerating electric field. A simple exponential decay is obtained when the electric field is large enough so that the electrons gain as much energy in the electric field as they lose by inelastic collisions. The validity and accuracy of the solutions and predictions are discussed.

A simple modification of the Bethe approximation for high-energy electron-atom collisions is considered for the purpose of analyzing the medium-energy scattering by highly stripped ionic targets. The effect of amplitude and phase distortions of the projectile-electron wave function on the excitation and ionization cross sections is approximately taken into account using the Elwert factor and the eikonal procedure, respectively. To facilitate the cross-section calculation further, the momentum-transfer-dependent part in the excitation matrix element is factored out in terms of an average transfer. The energy dependence of the various modification factors is studied in detail, showing that the distortion effect is large for ionic targets and decays only slowly with the scattering energy, as $E^{-\frac{1}{2}}$. For a neutral target, however, the distortion effects roughly cancel with each other, thus resulting in the effectiveness of the unmodified Bethe cross section at much lower energies than one usually expects. The improved formalism is applied to estimate the total cross sections for the direct excitation and ionization of outer-shell electrons of the ionic target. The results are drastically changed from the unmodified Bethe approximation, and are generally in fair agreement with experimental data for all energies except in the threshold region.

The Glauber approximation is known to give poor results for elastic electron-atom scattering at large scattering angles at intermediate energies. A simple method based on the two-potential Glauber-Born approximation is proposed to improve the behavior at large scattering angles and is applied to $e$-H elastic scattering. The main feature of the present method is the simplicity of calculations; it is hardly any more difficult than the ordinary Glauber approximation.

We present experimental data on relative excitation cross sections from the collisions ${\mathrm{Li}}^{+}$-He, -Ne, -${\mathrm{C}}_6$${\mathrm{H}}_6$; ${\mathrm{Ne}}^{+}$-, ${\mathrm{Na}}^{+}$-, ${\mathrm{Mg}}^{+}$-, ${\mathrm{Mg}}^{2+}$-He; and ${\mathrm{Mg}}^{+}$-${\mathrm{C}}_6$${\mathrm{H}}_6$ at 10-150 keV. The ion-atom data show a preferential excitation of $d$ levels for fixed principal quantum number, whereas no preferential excitation exists in the ion-benzene collisions. The data are compared with various model predictions. The preferential $d$-level population seems to be a result of the symmetry properties of the transient quasimolecule formed during the collision for projectile velocities of the order of one atomic unit of velocity. It is concluded that neither the Born approximation, the Brinkman-Kramers approach, a statistical population viewpoint, or a velocity-matching viewpoint explains the data satisfyingly well in our energy regime. For $v\gg v_0$ (1 a.u.), $s>p>d>\mathrm{}\cdots$.

Cross sections for the $1{}_{}{}^1{}_{}{}^{}S\to 4{}_{}{}^1{}_{}{}^{}S$ excitation of He proton and $\alpha$-particle impact are calculated with incident energy ranging from 10 to 5000 keV using the two-state distortion approximation and a corresponding second-Born approximation. Two sets of wave functions are used. The results are compared with the existing first-Born calculations and the experimental data. The generalized oscillator strengths for the transition are also presented. The nonorthogonality of the wave functions has a very marked effect on the cross sections for almost the entire energy range considered. For proton impact, use of a properly orthogonal set of wave functions reduces the first-Born cross-section values obtained from an improperly orthogonalized set of wave functions by a factor of almost 2 at high energies, in agreement with the experimental values. The second-Born and distortion approximations give better agreement at lower energies also and the latter correctly predicts the functional dependence of cross sections on energy down to an incident energy of 20 keV. The necessity of using properly orthogonal sets of wave functions in the calculation of scattering processes involving high atomic excitations is thereby stressed.

The perturbation theory expression for the HD molecule vibration-rotation dipole-transition probabilities is cast into the form of an instantaneous dipole-moment function $D\left(R\right)\mathrm{}$. Two large components of $D\left(R\right)\mathrm{}$ can be obtained from accurately known ground-electronic-state expectation values, whereas the rest is computed by a sum-over-states procedure similar to the one we used previously for the dynamic polarizability of ${\mathrm{H}}_2$. From this $D\left(R\right)\mathrm{}$ we compute dipole-transition matrix elements for the 0-0 through 6-0 vibrational bands, for all $P$ and $R$ branch lines with $J_i\le 3$. Our results agree well with the recent calculations of Wolniewicz [Can. J. Phys. 53, 1207 (1975); 54, 672 (1976)]. They also agree well with experiment for the 1-0 through 5-0 bands, but, like the results of Wolniewicz, are a factor of approximately $\surd 2$ larger than the 0-0 band results of M. Trefler and H. P. Gush [Phys. Rev. Lett. 20, 703 (1968)].

Using a cw dye laser beam normally incident on a glass-Na-vapor interface, fluorescence excitation spectra of the $D_1$ line are measured. By observing the fluorescence transmitted into the glass above the angle of total reflection, only evanescent light emitted within about a wavelength distance from the interface is observed. The measured spectra show a cutoff on the high-frequency side which is direct evidence for the transient behavior of the induced polarization of resonantly excited vapor atoms which have collided at the solid interface.

Radiative lifetimes (and $f$ values) have been measured for the $n=3\mathrm{}$ levels of the sodiumlike ion, ${\mathrm{Cu}}^{18+}$, using the beam-foil method. Strong cascade tails attributable to the decay of long-lived Rydberg or "yrast" levels were observed.

Absolute values for the differential cross section $\frac{d\sigma }{d\Omega d\omega }$ of Cu $K\alpha$ x rays inelastically scattered from the series of elements Ni-Ge are presented for the region 2.5-8 keV. These data display the resonance x-ray. Raman effect discovered by Sparks, and are also in very good qualitative agreement with a model H-atom calculation by Gavrila and Tugulea which predicts that at still lower energies the scattering cross section diverges.

We present an analytic calculation of the differential cross section $\frac{d^2\sigma }{d{\Omega }_{{\overrightarrow{\mathrm{k}}}_f}d{\omega }_f}$ using relativistic wave functions for the initial ($s$ wave only) and final electron states. Comparisons are made with other theoretical calculations. Experimental results are given as a test of the theoretical calculation: energy spectra for 662-keV $\gamma$ rays scattered incoherently through 90° and 135° by Ge bound electrons, and for 60-keV photons scattered by Cu and Mo $K$-shell electrons through 50°. Good agreement has been obtained between experimental and theoretical results.

Determinations of the mean lives of the 27887- and 16900-${\mathrm{cm}}^{-1\mathrm{}}$ levels in Ui have been made, each at a single vapor density, using electronic excitation and a method of delayed coincidence. The lifetime values were obtained by optically detecting the decays of the 3584.9- and 5915.4-Å resonance transitions, respectively. Using branching ratios obtained from known relative $f$ values, the averages of the measured lifetimes were corrected for imprisonment of the resonance radiation giving the following results: ${\tau }_0^{27887}=10.9\mathrm{}\pm 0.8$ ns and ${\tau }_0^{16900}=205\mathrm{}\pm 20$ ns. These mean lives were then used to place a system of relative $\mathrm{gf}$ values on two independent absolute scales, which were found to be in very close agreement. The lifetimes and $\mathrm{gf}$ values are presented in comparison with the experimental results of others.

We study numerically the solution to Schrödinger's equation for the radiative interaction of fictitious $N$-level model atomic and molecular systems with monochromatic radiation fields. The numerical method of solution allows us to study multiphoton transitions in $N$-level systems unhampered by the assumptions usually adopted, such as weak fields or large detunings of intermediate states, in perturbative and quasiperturbative analytic studies. We point out regularities and near regularities in the population dynamics that appear to be qualitatively different from those encountered in two- and three-level systems.

We show that a class of multiphoton-absorption problems has exact solutions. In these problems an $N$-level atom or molecule absorbs energy from near-resonance monochromatic laser fields, and one asks for the time-dependent populations of the energy levels of the absorber. We find that the equal-Rabi and the harmonic-Rabi $N$-level systems studied numerically in the preceding paper are associated with Chebyshev and Hermite polynomials. Using the exact solutions we find exact expressions for the time averages of occupation probabilities in terms of sums of squares of polynomials. In addition we find the first approximate analytic expression for the long-period population oscillations observed in numerical solutions; and we show that a steady-state population inversion between the highest and lowest levels may be maintained by multiphoton pumping.

The basic concept of classical, stochastic electrodynamics is the assumption that there exists a classical, fluctuating, electromagnetic ground-state radiation, even at absolute zero of temperature, with energy density $\left(\frac{1}{2}\right)\hslash \omega$ per mode. This ground-state radiation is incorporated into the semiclassical radiation theory in which particle systems are quantized and radiation is a classical field. It is shown that the ground-state radiation removes the difficulties associated with spontaneous emission, and that all radiation processes can be treated by one single perturbation scheme which closely resembles that of quantum electrodynamics. The transition probabilities for various first- and second-order radiation processes and the Lamb shift for energy levels are calculated in agreement with the results of quantum electrodynamics. Small deviations from the latter are found only in two processes which are probably inobservable in principle. Processes in which ground-state radiation is spontaneously absorbed by an atom have to be excluded by postulate. However, it is suggested that radiation-reaction effects of the type introduced by Crisp and Jaynes could provide a mechanism for the exclusion of spontaneous absorption.

The magnetic quadrupole transition probabilities for the $1s^{2}2s2p$ ${}_{}{}^3{}_{}{}^{}P_2^{}{}_{}{}^{}-1\mathrm{}{s}^{2}2s^2$ ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ and $1s^{2}2s2p$ ${}_{}{}^3{}_{}{}^{}P_2^{}{}_{}{}^{}-1\mathrm{}{s}^{2}2p^2$ ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ transitions of elements of the beryllium isoelectronic sequence are calculated using eigenfunctions obtained by the $Z$-expansion method. Matrix elements are presented for $Z=6-30\mathrm{}$ and transition probabilities are obtained for all elements with $Z\ge 6$ for which excitation energies are available. Good agreement is obtained with other calculations when common values for the transition energies are used.

A theory is presented to calculate vectorial time correlation functions for a classical system of free asymmetric rotors. The Euler equations of a freely rotating rigid body are solved first, and Eulerian angles between the laboratory-fixed and the molecule-fixed coordinate systems are deduced. The results are expressible in terms of the Jacobi theta functions. Further calculation implies the construction of the $Q$ matrix. Its general properties are examined with the help of group theory. The final integration over the components of the angular momentum vector is in part analytical and in part numerical. The calculation is simplified by considering the properties of the Jacobi zeta function. General properties of free asymmetric rotor correlation functions and of their Fourier transforms are discussed.

A method is proposed for solving the equations of perturbation theory appropriate for multiphoton excitation or ionization of complex atoms, and detailed calculations are carried out for two-photon ionization of He. The method consists of writing the wave function for order $N$ as a product of $N\mathrm{th}$- and zeroth-order wave functions for the electron absorbing the photon ("optical" electron) and the remaining electrons ("passive" electrons), respectively, with symmetrization of the spatial wave function appropriate for an eigenfunction of spin. Projection onto the space of the passive electrons then generates an integro-differential equation for the optical electron. The Coulomb Green's function is used to convert this equation into an integral equation, which is solved numerically using a modified version of the noniterative method of solution used extensively in scattering theory. Results obtained with the Hartree-Fock zeroth-order model show good agreement with the results of coupled Hartree-Fock perturbation theory. Results are also obtained ignoring exchange, and the error is observed to be about a factor of 2 over a wide range of frequencies.

We report on computer-generated solutions to the coupled matter-field equations which provide the first evidence for distortionless optical pulse propagation in an absorbing medium of three-level atoms. The associated phenomenon represents the three-level equivalent of self-induced transparency in two-level atoms. Results have application to the propagation of short, coherent optical pulses in a medium in which the atoms have two distinct transition frequencies in near resonance with the pulse carrier frequency. We relate in particular to the case of pulse propagation in a magnetically perturbed sodium vapor where the pulse carrier frequency is tuned near the frequencies of the well-known sodium $D$ lines. Here we obtain particular solutions corresponding to three and five peak pulses with "areas" of approximately $6\pi$ and $10\pi$, respectively. The solutions are strongly supported by a number of analytical results. In particular, simple algebraic relations are found which give the populations of the levels in terms of the field variables.

We report observations of the rotation of the plane of polarization of photon echoes by a weak axial magnetic field. The echoes are generated on the $6S_{\frac{1}{2}}$ to $7P_{\frac{1}{2}}$ transition in cesium (4593 Å) by a sequence of two collinear laser pulses polarized in the same direction. When a magnetic field is applied along the propagation axis the polarization vector of the echo is rotated from that of the exciting pulses. We measure the intensity of the photon echo signal polarized orthogonal to the excitation pulse polarization as a function of magnetic field and pulse separation. The theoretical analysis decribes cesium atoms as a nondegenerate multilevel system, consisting of ground and excited Zeeman-split hyperfine-structure levels. The theory presented predicts the observed dependence on magnetic field strength, pulse separation, and excited- and ground-state $g$ factors. A novel method of observing photon echoes is described which uses the polarization rotation effect to replace electro-optical shutters often employed in photon echo experiments.

This paper presents an analysis of the penetration of an intense electromagnetic beam in an inhomogeneous plasma. It is seen that on account of the electron density gradient created by an intense transversely inhomogeneous beam a region is created in the overdense plasma where the beam can propagate. Typically a C${\mathrm{O}}_2$ laser beam of width $r_0=100\mathrm{}$ μm, power $P=15\mathrm{}$ GW, and $\omega =1.7\mathrm{}\times {10}^{14}$ ${\sec }^{-1\mathrm{}}$ can penetrate to a region where $\frac{{\omega }_p^2}{{\omega }^2}=20\mathrm{}$ in a 500-eV inhomogeneous plasma characterized by $\frac{{\omega }_p^2}{{\omega }^2}=\left(\frac{{\omega }_{p0\mathrm{}}^2}{{\omega }^2}\right)\mathrm{}(1+Bz)\mathrm{}$.