A simple classical model is used to estimate the deviations of the ${\mathrm{He}}^3$ quasiparticle energy in ${\mathrm{He}}^4$ at $T=0\mathrm{}$ from the parabolic form $\epsilon \mathrm{}\left(p\right)=-E_0+\frac{p^2}{2m}$. The result is in reasonable agreement with a recent experimental determination of this effect.

The atomic antishielding factor $R$ for the quadrupole hyperfine structure has been evaluated for the lowest three excited $\mathrm{np}$ states for each of the alkali atoms, i.e., for a total of fifteen atomic states. The values of $R$ were determined by means of the perturbed wave functions $v_1^{\prime }(nl\to l^{\prime })$, as obtained by solving the inhomogeneous Schrödinger equation for each type of excitation of the core electrons by the nuclear quadrupole moment $Q$. In order to obtain reliable values of $R$, we have obtained accurate valence wave functions $v\left(\mathrm{np}\right)\mathrm{}$ for the $\mathrm{np}$ excited states from the requirement that the calculated energy eigenvalues $E_0\mathrm{}\left(\mathrm{np}\right)\mathrm{}$ should reproduce the observed energy levels. The wave function $v\left(n_{0}p\right)$ for the lowest valence $\mathrm{np}$ state ($n=n_0$, e.g., $n_0=3\mathrm{}$ for Na) was checked for each alkali atom by calculating the oscillator strength $f_{n_{0}s\to n_{0}p}$ for the lowest $n_{0}s\to \mathrm{np}$ transition. The calculated values of $f_{n_{0}s\to n_{0}p}$ were found to be in good agreement with experiment. As an example of the values of $R\left(\mathrm{np}\right)\mathrm{}$ obtained, we have found for Rb: $R\left(5p\right)=-0.257\mathrm{}$; $R\left(6p\right)=-0.207\mathrm{}$; and $R\left(7p\right)=-0.190\mathrm{}$. The resulting correction factors $\mathcal{C}=\frac{1}{\mathrm{}(1-R)\mathrm{}}$ are $\mathcal{C}\mathrm{}\left(5p\right)=0.796\mathrm{}$, $\mathcal{C}\mathrm{}\left(6p\right)=0.829\mathrm{}$, and $\mathcal{C}\mathrm{}\left(7p\right)=0.840\mathrm{}$. At the end of the paper we give the corrected values of $Q$ for twelve alkali isotopes, for which experimental data are available, namely, for the following nuclei: ${\mathrm{Na}}^{23}$, ${\mathrm{K}}^{39}$, ${\mathrm{K}}^{40}$, ${\mathrm{K}}^{41}$, ${\mathrm{Rb}}^{85}$, ${\mathrm{Rb}}^{87}$, ${\mathrm{Cs}}^{131}$, ${\mathrm{Cs}}^{132}$, ${\mathrm{Cs}}^{133}$, ${\mathrm{Cs}}^{134}$, ${\mathrm{Cs}}^{135}$, and ${\mathrm{Cs}}^{137}$. We have also given a brief summary of the present experimental evidence supporting the existence of the quadrupole shielding and antishielding effects for atomic states.

Angular correlations between $K$ and $L$ x rays in Ta and Tl were measured. The $K$ x rays were detected with a NaI(Tl) crystal, and the $L$ x rays with a Ge(Li) spectrometer. The angular-correlation functions for the $K-L\alpha$ and $K-L\beta$ cascades in Ta were found to be $W\left(\theta \right)=1+(2.48\mathrm{}\pm 0.41)\times {10}^{-2\mathrm{}}{P}_2\left(cos\theta \right)$ and $W\left(\theta \right)=1-(0.12\mathrm{}\pm 0.46)\times {10}^{-2\mathrm{}}{P}_2\left(cos\theta \right)$, respectively. The result for the $K-L\alpha$ cascade is in fair agreement with theory, while the result for the $K-L\beta$ cascade deviates significantly from the theoretical prediction. The angular-correlation functions for the $K-L\alpha$, $K-L\beta$, and $K-\mathrm{Ll}$ cascades in Tl were found to be $W\left(\theta \right)=1+(3.63\mathrm{}\pm 0.32)\times {10}^{-2\mathrm{}}{P}_2\left(cos\theta \right)$, $W\left(\theta \right)=1+(1.31\mathrm{}\pm 0.36)\times {10}^{-2\mathrm{}}{P}_2\left(cos\theta \right)$, and $W\left(\theta \right)=1+(21.8\mathrm{}\pm 2.0)\times {10}^{-2\mathrm{}}{P}_2\left(cos\theta \right)$. These results are in relatively good agreement with theory when the $M2\mathrm{}$ admixtures in the x-ray transitions are taken into account. Better agreement with theory would be obtained for slightly higher $M2\mathrm{}$ admixtures in the $L$ x-ray transitions than theoretically calculated on the basis of an independent-particle model.

A precision comparison is made between the Stark-broadened profiles of ${\mathrm{H}}_{\beta }$, ${\mathrm{H}}_{\gamma }$, and ${\mathrm{H}}_{\delta }$ measured by Hill and Gerardo and the theoretical profiles calculated by Kepple and Griem. In addition, the effects of inelastic collisions between perturbing electrons and the radiating atom are investigated. It is found that the inclusion of the inelastic collisions improves the qualitative agreement between theory and experiment near the line center, but that the Gaunt factors used to estimate the broadening strength of the inelastic collisions overestimate the effect. In terms of half-widths, the best comparison to experiment was obtained using theoretical profiles which omitted inelastic collisions. The electron densities determined from these profiles are 2, 3, and 6% less than the interferometeric values for ${\mathrm{H}}_{\beta }$, ${\mathrm{H}}_{\gamma }$, and ${\mathrm{H}}_{\delta }$, respectively.

The justification of the use of unprojected many-electron wave functions for solids and large molecules is examined from the point of view of an extended Hartree-Fock method in the one-$\lambda$ approximation presented in previous papers. It is shown that, in the limit of a large number of electrons, the Hartree-Fock operators are the same for the projected and unprojected case. It is further shown that the expectation value of purely spatial operators for the projected and unprojected functions are equal in this limit. Graphs of the differences of the relevant quantities as a function of number of electrons are given.

X-ray line emission originating from "optical" (dipole and intercombination) transitions in heliumlike copper, iron, and titanium and hydrogenic iron and titanium ions has been observed in a concentrated dense plasma formed in a linear low-pressure discharge. In addition, emission from inner-shell x-ray transitions in Fe XIX-XXIV and Ti XV-XX ions has been identified. The measured wavelengths are compared with theoretical predictions and other observations. Continuum emission at wavelengths shorter than the Lyman-series limit for hydrogenic species has been used for electron temperature and density determination and as evidence supporting a thermal-electron collisional ionization model. Plasma dynamics studies indicate the existence of an axially propagating radial pinch, collapsing near the anode to a minute point.

A full discussion is given of the microwave-spectroscopy resonance method of measuring the hyperfine-structure interval $\Delta \nu$ of the ground state of muonium, including the theoretical value for $\Delta v$, the transition frequencies, and the resonance line shape. A complete description is also given of our experimental method of measuring $\Delta \nu$ at strong magnetic field. The initial experimental results gave $\Delta \nu =4461.3\mathrm{}\pm 2.0$ MHz, in good agreement with the theoretical value $\Delta \nu =4463.282\mathrm{}\pm 0.062$ MHz. The first observations of muonium chemistry are described.

Precision measurements of the energy differences ${\mathcal{S}}_4(4\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}-4\mathrm{}{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{})$ and $\Delta E_4-{\mathcal{S}}_4(4\mathrm{}{}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}-4\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{})$ in ${\left({\mathrm{He}}^4\right)}^{+}$ are reported. The experimental results, ${\mathcal{S}}_4=1768\mathrm{}\pm 5$ MHz and $\Delta E_4-{\mathcal{S}}_4=20179.7\mathrm{}\pm 1.2$ MHz, agree with the values predicted by quantum electrodynamic theory, ${\mathcal{S}}_4=1768.34\mathrm{}\pm 0.51$ MHz and $\Delta E_4-{\mathcal{S}}_4=20180.78\mathrm{}\pm 0.56$ MHz. An electron gun excites the states of interest in a section of waveguide containing helium gas. A magnetic field applied perpendicular to the waveguide axis is used to scan a microwave resonance between suitable Zeeman levels for a fixed frequency of oscillating electric field in the waveguide. Any induced electric dipole transitions $4{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\to 4{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}$ or $4{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\to 4{}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}$ reduce the intensity of 1215-Å radiation which is emitted in the natural decay of $4S$ to $2P$. This light is directed onto an ultraviolet-detecting phototube whose output is measured by a lock-in detector for which the synchronous signal is provided by square-wave amplitude modulation of the microwave field. The resonances obtained by varying the magnetic field are fitted to a theoretical line-shape formula by a computerized least-squares program. The resulting best-fit parameters include either ${\mathcal{S}}_4$ or $\Delta E_4-{\mathcal{S}}_4$, depending on the transition studied. Consideration is given to the dependence of the resonance center on the gun current, helium pressure, and microwave power level.

A high-resolution Ge(Li) x-ray spectrometer was used for measuring relative radiative decay rates of vacancies in the $K$ shell of the following atoms: Au, Hg, Tl, Pb, Bi, Rn, Ra, Th, and U. In the most favorable cases, the x rays following the filling of a $K$ vacancy by $L_{\mathrm{II}}$, $L_{\mathrm{III}}$, $M_{\mathrm{II}}$, $M_{\mathrm{III}}$, $M_{\mathrm{IV}}$, $M_{\mathrm{V}}$, $N_{\mathrm{II}}$, $N_{\mathrm{III}}$, $N_{\mathrm{IV}-\mathrm{V}}$, and $O_{\mathrm{II}-\mathrm{III}}$ electrons were observed and measured. The results were compared with recent relativistic calculations carried out by Scofield.

It has recently been established that the radiative lifetime of the metastable $2{}_{}{}^3{}_{}{}^{}S$ state of helium and the heliumlike ions is determined by single-photon magnetic dipole ($M1\mathrm{}$) transitions to the ground state, rather than the two-photon process proposed by Breit and Teller. The theory of $nl-n^{\prime }l$ $M1\mathrm{}$ transitions with $n\ne n^{\prime }$ is developed in the Pauli approximation and extended to two-electron systems. Terms arising from relativistic energy corrections and finite-wavelength effects are included. The results for hydrogenic systems are shown to be identical to those obtained in the relativistic four-component Dirac formulation. The coefficients in the $Z^{-1\mathrm{}}$ perturbation expansion of the $1s2s{}_{}{}^3{}_{}{}^{}S-1s^2{}_{}{}^1{}_{}{}^{}S$ $M1\mathrm{}$ transition integral are evaluated through ninth order and used to calculate the $M1\mathrm{}$ emission probabilities from the $2{}_{}{}^3{}_{}{}^{}S$ state of the two-electron ions up to Fe XXV. The emission probability for neutral helium is 1.27 × ${10}^{-4\mathrm{}}$ ${\sec }^{-1\mathrm{}}$. The results are compared with recent solar coronal observations by Gabriel and Jordan, and with a measurement of the $2{}_{}{}^3{}_{}{}^{}S$ state lifetime in Ar XVII by Schmieder and Marrus.

The lifetimes of the gallium $5{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}$ state, the indium $6{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}$ state, and the thallium $7{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}$ state were measured using the zero-field level-crossing (Hanle-effect) technique. The life-times obtained were (6.8±0.3) × ${10}^{-9\mathrm{}}$, (7.0±0.3) × ${10}^{-9\mathrm{}}$, and (7.45±0.2) × ${10}^{-9\mathrm{}}$ sec, respectively. Anomalous contributions to the level-crossing signals, from the wavelength dependence of the exciting light intensity and self-absorption by the fluorescing atomic beam, were investigated in detail.

The optical potential of Bell and Squires is used to formulate the problem of the elastic scattering of positrons from atoms. An application is made to the scattering of $S$-wave positrons from hydrogen below the threshold for positronium formation. The single-particle states of the electron are chosen to be the hydrogenic solutions, while the positron states are obtained by using a single-particle potential which includes the Hartree potential plus a model polarization potential. Various models are investigated and a choice is made which effectively maximizes the calculated phase shifts through second order in the perturbation expansion of the optical potential. Phase shifts thus obtained yield approximately 67-80% of the difference between the static Hartree results and the variational results of Schwartz. Higher-order effects are evaluated using the techniques of Kelly. Intermediate states of multipole $l\le 3$ are included.

Collisional ionization rates for C IV, N V, and O VI as well as for O V and Ne VII are deduced from the time history of spectral lines emitted by these ions in a hot plasma. The plasma (in the electron temperature range 100-260 eV) is produced in a 15-kJ theta-pinch device, and it is analyzed using the light-scattering technique. The results are consistent with theoretical calculations for O VI and with a semiempirical formula based on theoretical calculations for hydrogenic ions but smaller by factors 1.5-2.

Double-differential cross sections (ddcs), relative to ejection energy and angle, have been calculated for electrons emitted from helium by proton bombardment for all excited states of the resulting helium ion through the third hydrogenic shell. Comparable calculations were made by using both a 12-term correlated wave function and a product of one-electron wave functions for the initial state of the helium atom. Results are presented for an incident proton energy of 300 keV with various ejection energies and angles. The effect of the correlated wave function was significant for cases involving excitation to the $P$ states. The contribution from these states represented an appreciable percentage of the total ddcs over all states for low and intermediate ejection energies, which was not indicated by the corresponding results from the uncorrelated wave function.

The shift, broadening, and line shape of the absorption line Ca 4227 in the presence of Ar and He were observed under a wide range of pressures. The relative densities (rd) employed extended from 1 to about 245 (15 450 psi at 638 °C) for Ar and to 58 (2563 psi at 638 °C) for He. Both Ar and He produce a red shift, -0.019 ${\mathrm{cm}}^{-1\mathrm{}}$/rd for He and -0.023 ${\mathrm{cm}}^{-1\mathrm{}}$/rd for Ar. The dependence of the half-width on rd appears to adhere to linearity less than does the dependence of the shift on the rd. The broadening at low rd (<10) is 0.64 ${\mathrm{cm}}^{-1\mathrm{}}$/rd for He and 0.74 ${\mathrm{cm}}^{-1\mathrm{}}$/rd for Ar. Some typical intensity profiles are given showing also the presence of a very extended blue wing due to He and a broad violet satellite due to Ar.

The first Born approximation, with the prior interaction, is used to calculate the total cross section for the process ${\mathrm{H}}^{+}+\mathrm{He}{\mathrm{}\left(1s\right)\mathrm{}}^2\to \mathrm{H}\mathrm{}\left(1s\right)+{\mathrm{He}}^{++}+e^{-}$. Comparison of calculated with existing experimental values is made to the extent that it is possible. Fair agreement is obtained for an impact energy of 50 keV; however, more refined calculations are needed for complete clarification.

Ionization growth rates of hydrogen, in the range of $\frac{E}{p}$ between 100 and 10 000 V/cm Torr are presented. They have been deduced from microwave measurements of electron densities in a ring-shaped discharge, produced by a uniform magnetic field linearly increasing in time. The experimental conditions are described and the meaning of the obtained data is discussed as well as compared with the more usual results in a straight uniform electric field. At $\frac{E}{p}<~1000$, the obtained values of $\frac{\beta }{p}$, where $\beta$ is the ionization growth rate, increase with $\frac{E}{p}$ and are in good agreement both with theory and earlier experiments. At $\frac{E}{p}>2000\mathrm{}$, on the contrary, $\frac{\beta }{p}$ decreases with increasing $\frac{E}{p}$ as suggested by theory, but not yet measured experimentally. The maximum of $\frac{\beta }{p}\simeq 1.3\times {10}^9$ sec ${\mathrm{Torr}}^{-1\mathrm{}}$ corresponds to $\frac{E}{p}\simeq 2000$.

Calculations are made of the minimum dipole moments necessary to bind an electron to a nonstationary finite electric dipole in a number of rotationally excited states. The critical moment for a given dipolar system is found to depend on the dipole length, the moment of inertia, and the rotational quantum state of the finite dipole. The properties of the dipolar system are discussed as to their implications for electron scattering by polar molecules.

A variation-perturbation treatment of potential scattering by a central field is presented. The first-order correction $X_l^{\mathrm{}\left(1\right)\mathrm{}}$, to the lth partial wave, which gives the phase shift ${\eta }_l$, through third order, may be determined from a variation principle or as the solution of an inhomogeneous differential equation. Optimization of the initial approximation by means of the Hulthén condition yields a vanishing ${\eta }_l^{\mathrm{}\left(1\right)\mathrm{}}$ and a bound $X_l^{\mathrm{}\left(1\right)\mathrm{}}$. Encouraging results are obtained in trial calculations on the exponential and Yukawa potentials treated as perturbed spherical holes. The likelihood of extending this approach to include the most powerful techniques of bound-state perturbation theory is anticipated.