We present a method of calculating elastic photon-hydrogen scattering at resonance fluorescence, where the result of Gavrila is invalid. The effects of the unitarity-violating direct nonresonant and crossed terms are also calculated and shown to be small, as expected.

A more extensive analysis of the effects of different assumptions about initial state populations is added to a previous paper on the coherent excitation of hydrogen fine structure by foils. It is found that the previously postulated alignment of angular momenta perpendicular to the beam direction is a sufficient but not necessary assumption; an alternative assumption that the broad $l$-state distribution of the collision products has a maximum for $d$ states also matches the experimental traces adequately. A method of obtaining initial state populations in a manner which suppresses the corrections required for cascade effects is discussed. The method avoids the errors of extrapolating decay-curve fits.

This comment suggests mathematical modifications to the solution of the Woo-Whealton model which facilitate the calculations.

The atomic-beam method has been used to study the Stark effect in the resonance lines of indium ($6s{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\to 5p{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}$, 4102 Å) and thallium ($7s{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\to 6p{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}$, 3776 Å). The difference between the atomic polarizability of the ${}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}$ state and that of the ${}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}$ state [$\Delta \alpha \mathrm{}\left(\mathrm{sp}\right)\mathrm{}$] has been determined and compared with calculations in the Coulomb approximation. For indium, we find $\Delta \alpha \mathrm{}\left(\mathrm{sp}\right)=138\mathrm{}\left(11\right)\mathrm{}\times {10}^{-24\mathrm{}}$ ${\mathrm{cm}}^{-3\mathrm{}}$ and for thallium $\Delta \alpha \mathrm{}\left(\mathrm{sp}\right)=115\mathrm{}\left(12\right)\mathrm{}\times {10}^{-24\mathrm{}}$ ${\mathrm{cm}}^3$.

We propose fractional integrals (Euler transforms) of conventional quantum-mechanical functions as special integral-transform trial functions. Calculations on the first four members of the He isoelectronic sequence with three different-order Euler transforms of both Slater and Gaussian $1s$ orbitals are reported. A pilot calculation for ${\mathrm{H}}_2$ with an Euler transform of an LCGTO (linear combination of Gaussian-type orbitals) molecular orbital gives a total energy $E=-1.10903\mathrm{}$ a.u. at $R=1.4\mathrm{}$ a.u. (exact value is -1.17444 a.u.). A generalization of the shape function used in this work is given and the general applicability of Euler transforms discussed.

Spin-exchange optical pumping has been used to measure the hyperfine pressure shift of deuterium in helium as a function of temperature. A new technique for optical pumping at low temperatures has been combined with a previously reported technique for optical pumping at high temperatures to make measurements between -135 and 400 °C. The experimental results are compared with existing theoretical calculations.

Cross sections for pair production by 2.615-MeV $\gamma$ rays incident on Al, Cu, Cd, and Pb have been found by subtracting tabulated values of the photoelectric cross section from measured values of total cross sections. The latter were determined to be 129.8, 137.0, 147.6, and 181.9 mb per electron, respectively, for the above elements. The statistical uncertainty is ±0.2%. Subtraction of photoelectric cross sections gives 0.0642 (±5%), 0.343 (±2.3%), 0.991 (±1.4%), and 3.55 (±0.8%) b per atom for pair production. The present result in Pb is 38.9% higher than the Bethe-Heitler Born approximation result and is about 14% higher than Jaeger-Hulme non-Born theoretical results but only 6% higher than the recent, exact non-Born calculation (neglecting screening) by Øverbø, Mork, and Olsen. Differences between our values for Al, Cu, and Cd and the calculated values of these authors are even smaller, namely, 0.5% for Al and Cd, and 3% for Cu. It is pointed out that pair-production cross sections can be found from Born-approximation values by multiplication by $a+bZ$ for $0<Z<\mathrm{}50\mathrm{}$, with $a=0.97\mathrm{}\pm 0.01$ and $b=4\mathrm{}\times {10}^{-3\mathrm{}}$, and by a factor $1+\alpha Z^2$ for $Z\ge 50$ with $\alpha =(5.8\mathrm{}\pm 0.1)\times {10}^{-5\mathrm{}}$.

Two-point Padé approximants of the dynamic polarizabilities are used to calculate the leading term of the long-range (van der Waals) force between atoms. The bounding properties of these approximants are established explicitly. A successive-approximation procedure is introduced. It keeps each step simple and makes the higher-order approximation expressible in closed forms and readily applicable. The results of hydrogen and noble gases show that, with the known moments, the present method gives the tightest upper and lower bounds yet established.

$K_{\alpha }$ and $K_{\beta }$ x rays produced in the bombardment of Cu with protons of 6- to 10-MeV incident energy and with ${\mathrm{O}}^{4+}$ ions of 15- to 19-MeV incident energy have been measured using a 4-mm-thick cooled Si(Li) detector with an instrumental resolution of 280 eV. The $\frac{K_{\alpha }}{K_{\beta }}$ transition ratios are observed to be constant over the energy range studied with a value of 7.8±0.1 for proton excitation and 7.0±0.1 for the oxygen excitation, compared to a previous experimental value of 7.33 for electron excitation and a calculated value of 8.2. The cross section for x-ray production at 90° to the beam direction for 7-MeV protons on Cu is 23.9±3.6 b/sr and for 15-MeV oxygen on Cu is 0.88±0.27 b/sr. The $K_{\beta }$ peak produced by oxygen bombardments is observed to shift upwards in energy by 97±15 eV relative to the $K_{\beta }$ peak produced in proton bombardments.

Using an effective interaction or reaction matrix obtained from a modified Brueckner theory, various low-temperature properties are calculated or estimated for dilute solutions of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ in liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$. The system is regarded as a low-density Fermi liquid with ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ quasiparticles created by ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ atoms in superfluid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$. The single-particle energy spectrum is given by an effective mass, and an effective interaction between the ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ quasiparticles is derived. The calculations are done for two different two-body potentials: an Yntema-Schneider potential given by Brueckner and Gammel, and a Frost-Musulin potential given by Bruch and McGee. The Landau $f$ function is estimated from the reaction matrix, and the coefficients of the expansion of the Landau $f$ function in terms of Legendre polynomials are calculated. The estimated values are in reasonably good agreement with experimental results. The exclusion-principle sum rule is also roughly satisfied by the lowest-order coefficients. Low-temperature properties, such as the compressibility, the quasiparticle effective mass or specific-heat ratio, and the magnetic susceptibility, are estimated; results are in fair agreement with experimental values. The various properties are also given as functions of the ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ concentration in the solution, and the maximum solubility of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ in liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ is estimated in good agreement with the experimental value. Also, transport coefficients - i. e., viscosity, thermal conductivity and spin diffusion - are estimated after identification of the reaction matrix with the scattering amplitude in the formulas developed by Abrikosov and Khalatnikov and by Hone. The agreement with experimental results is surprisingly good, considering that the results depend very much on the value chosen for the effective mass.

Using restricted Hartree-Fock (RHF) wave functions and two previously defined types of configuration interaction (CI) wave functions, the polarization wave function and the first-order wave function, we have made ab initio calculations of the hyperfine structure (hfs) of: ${}_{}{}^2{}_{}{}^{}P {\mathrm{C}}^{+}$; ${}_{}{}^3{}_{}{}^{}P$ and ${}_{}{}^1{}_{}{}^{}D {\mathrm{N}}^{+}$; ${}_{}{}^4{}_{}{}^{}S$, ${}_{}{}^2{}_{}{}^{}D$, and ${}_{}{}^2{}_{}{}^{}P {\mathrm{O}}^{+}$; ${}_{}{}^3{}_{}{}^{}P$ and ${}_{}{}^1{}_{}{}^{}D {\mathrm{F}}^{+}$; ${}_{}{}^2{}_{}{}^{}P {\mathrm{Ne}}^{+}$; ${}_{}{}^3{}_{}{}^{}P$ and ${}_{}{}^1{}_{}{}^{}D {\mathrm{B}}^{-}$; ${}_{}{}^4{}_{}{}^{}S$, ${}_{}{}^2{}_{}{}^{}D$, and ${}_{}{}^2{}_{}{}^{}P {\mathrm{C}}^{-}$; ${}_{}{}^3{}_{}{}^{}P$ and ${}_{}{}^1{}_{}{}^{}D {\mathrm{N}}^{-}$; ${}_{}{}^2{}_{}{}^{}P {\mathrm{O}}^{-}$. For the polarization wave functions, the calculated parameters $〈{{\mathcal{r}}_l}^{-3\mathrm{}}〉$, $〈{{\mathcal{r}}_s}^{-3\mathrm{}}〉$, and $〈{{\mathcal{r}}_q}^{-3\mathrm{}}〉$ are expected to be within 2% of the exact values, and the calculated spin densities are expected to be between 55% and 80% of the exact values. All calculated spin densities except that for ${}_{}{}^3{}_{}{}^{}P {\mathrm{B}}^{-}$ are positive, and the polarization spin density is usually slightly larger than the first-order value. For all positive ions and neutral atoms, the relationship $〈{{\mathcal{r}}_q}^{-3\mathrm{}}〉<〈{{\mathcal{r}}_l}^{-3\mathrm{}}〉<{〈{\mathcal{r}}^{-3\mathrm{}}〉}_{\mathrm{RHF}}<〈{{\mathcal{r}}_s}^{-3\mathrm{}}〉$ holds. Calculated diamagnetic susceptibilities and nuclear magnetic shielding constants are reported for both the ions and the corresponding neutral atoms, using the present CI wave functions. Finally, we tabulate the angular integration coefficients needed to compute $〈{{\mathcal{r}}_s}^{-3\mathrm{}}〉$ and $〈{{\mathcal{r}}_q}^{-3\mathrm{}}〉$, and discuss a procedure used in this work to compact the form of a CI wave function.

A theory for the structure of one-positron many-electron atoms and molecules in bound states is presented. The purpose of the theory is to permit the accurate calculation of positron lifetimes, angular correlation for two-photon annihilation, and accurate positron binding energies, but not necessarily quantities which depend upon purely electronic correlation. We therefore use a modified orbital approximation for the electrons, the modification consisting of the introduction of explicit electron-positron correlation into each electronic orbital. The kinetic energies of the electrons and positron are treated on the same footing, and the Born-Oppenheimer approximation is applied to the nuclei. No spin-dependent or relativistic terms are included in the Hamiltonian. In this paper we treat only those systems for which the electrons constitute a closed shell, and we do not consider the spatial symmetry of the atom or molecule.

Equilibrium charge fractions have been measured for aluminum ions at energies from 0.4 to 4.0 MeV passing through nitrogen gas. Charge states from 0 to $7^{+}$ were observed. The mean charge increased from 1.47 at 0.4 MeV to 3.84 at 4.0 MeV.

Differential measurements are made for close encounters of protons with helium atoms, measuring the charge state after collision of both particles in coincidence. Incident proton energies range from 5 to 170 keV and the scattering angle is 10°. The probabilities versus energies of three reactions are studied: electron transfer $P_{01}$, scattering without change of charge of either particle $P_{10}$, and ionization of the target (so that both particles are singly ionized after the collision) $P_{11}$. Here $P_{01}$ is found to exhibit a damped resonant structure plotted versus energy and $P_{10}$ shows a complementary oscillatory structure. The ionization probability $P_{11}$ increases smoothly with energy.

Independent heat reservoirs are postulated to interact with the atoms in a harmonic system as a substitute for real nonharmonic forces. The temperature of each reservoir is determined by the condition that it exchange no energy with the system in the steady state. An explicit solution for the covariance matrix is obtained for the case of the linear chain. The thermal conductivity is finite and inversely proportional to the coupling to the reservoirs.

The polarizations of Lyman-$\alpha$ radiation produced (i) by electron impact on atomic and molecular hydrogen and (ii) by electric field quenching of metastable $2S$ hydrogen atoms have been measured. The polarization analyzer was a LiF crystal set at the Brewster angle, with reflected radiation being detected by an iodine-vapor photon counter. Generally good agreement between the measured polarization fractions and recent theoretical predictions was found. The radiation produced by quenching metastable atoms in an electric field was found to be polarized with a polarization fraction of -0.30±0.02, which compares with the theoretical value -0.323.

The cross section for excitation of atomic hydrogen to the $2{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}$ state by electron impact has been remeasured in a modulated-crossed-beam experiment in the energy range extending from threshold to 1000 eV. Absolute values for the $2S$ cross section were obtained by measuring the ratio of the $2S$ to $2P$ cross sections and using previous measurements of the $2P$ cross section, which were normalized to the Born approximation at high energies. The atoms excited to the $2P$ and $2S$ states were detected by observing the Lyman-$\alpha$ photons produced by radiative decay and by quenching in an electrostatic field, respectively. In order to determine the total cross section, it was necessary to consider the polarization of the Lyman-$\alpha$ radiation emitted in each method of detection. The remeasured cross section is found to be in agreement with the Born approximation above 200 eV. The measured ratios are in reasonable agreement with the ratios that would be predicted by the close-coupling theory of Burke et al. The peak value of the cross-section curve is $0.168\pm 0.020\pi {a_0}^2$, and occurs at 11.6±0.2 eV.