The $4{}_{}{}^3{}_{}{}^{}P$ state of helium has been excited and aligned by electron impact in a strong magnetic field. Transitions between Zeeman levels of the fine-structure states are induced by microwave frequency radiation and are detected through the resulting change in polarization of 3188-Å fluorescence. The derived values of the fine-structure intervals are $4{}_{}{}^3{}_{}{}^{}P_0^{}{}_{}{}^{}-4{}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}=3306.6\mathrm{}\pm 1.0$ MHz and $4{}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}-4{}_{}{}^3{}_{}{}^{}P_2^{}{}_{}{}^{}=270.7\mathrm{}\pm 0.8$ MHz. The diamagnetic Zeeman interaction produces an observable effect, and the magnitude of its anisotropic part has been measured. The atomic radius and quadrupole moment are derived from this measurement and agree within experimental error with the values for a hydrogenic $4p$ orbital.

The problem of the screening of many-electron atoms (ions) is considered. Using the scaled Thomas-Fermi (STF) method of Stewart and Rotenberg for approximating the core potential of an atom (ion) with nuclear charge $Z$ and net core charge $Z^{*}$, a screened scaled Thomas-Fermi potential (SSTF) is presented: $V\left(r\right)=-\left(\frac{Z{e}^2}{r}\right)U_{\mathrm{SSTF}}$, where $\{U_{\mathrm{SSTF}},=U(r)-Ar, 0\le r\le a\}\{=BU(r)\mathrm{}{e}^{\frac{\mathrm{}(a-r)\mathrm{}}{D}}, r\ge a.\mathrm{}\}$ Here $D$ is the screening radius; $A$ and $B$ are constants; and $U\left(r\right)\mathrm{}$ is given by $\left\{U\right(r)=U_{\mathrm{STF}}\mathrm{}(r)\mathrm{},=\varphi (\frac{r}{\alpha \mu })+\frac{\mathrm{qr}}{r_c}, r\le r_c\}\{=q, r\ge r_c\}$ where $\varphi \left(\frac{r}{\alpha \mu }\right)$ is the well-known Thomas-Fermi function, $r_c$ is the STF core radius with $\mu =\frac{1}{4}{\left(\frac{9{\pi }^2}{2Z}\right)}^{\frac{1}{3}}=0.885\mathrm{}34Z^{\frac{-1\mathrm{}}{3}}$, $\alpha$ is the adjustable scaling factor, and $q=\frac{Z^{*}}{Z}$. The constants $B$ and $A$ are given by $B=\frac{D\left[a{U}^{\prime }\mathrm{}\right(a)-U(a\left)\right]}{\left\{D\right[a{U}^{\prime }\mathrm{}\left(a\right)-U\left(a\right)]-aU(a\left)\right\}}$ and $A=(1-B)\mathrm{}\frac{U\left(a\right)\mathrm{}}{a}$, where $U^{\prime }\mathrm{}\left(a\right)=\frac{\partial U}{\partial r}$ at $r=a$. Eigenvalues of the Schrödinger equation with the SSTF potential are given for the $3d$, $4s$, $4p$, $4d$, $4f$, and $5s$ orbitals of FeI and FeVIII. Comparisons between the ions and corresponding hydrogenic orbitals show that the variations of the SSTF eigen-values and the limiting screening radii are generally very different, with different level crossings and different relative energies. It is concluded that in order to obtain a correlation between a limiting screening radius and the observed disappearance of lines from a many-electron atom (ion), SSTF solutions are needed for the ion of interest. It is also concluded that until an accurate external screening function is obtained and applied to the screening of Hartree-Fock isolated-atom (-ion) solutions, SSTF solutions will be useful for the very important astrophysical problems of calculating equations of state and opacities for high-$Z$ matter at stellar densities and temperatures.

A theory for the spectral wings of collision-induced scattered light is presented. It differs from previous theories in that realistic potentials and polarizabilities are employed. The line shape at relatively low frequencies is determined largely by the effects of distant collisions. At high frequencies it is determined primarily by the repulsive part of the interatomic potential. At intermediate frequencies it is sensitive to both the attractive part of the potential and the short-range part of the polarizability anisotropy. A tecnique in which the analytic behavior of the trajectories in the complex time plane plays a key role is employed.

Individual-term electron transition rates for initial $L$-shell holes are presented in $j-j$ coupling. Comparison with experimental-term intensities are made for platinum, tellurium, uranium, argon, and krypton. For the latter two elements agreement is poor. Comparison with experiment indicates some prominent peaks in the uranium $L_2$ Coster-Kronig spectrum are improperly identified.

The low portion of an extensive level structure derived from analysis of the optical spectrum is reported. Positions, $J$ values, and $g_J$ factors are given for 98 levels, including all 91 levels expected below 10 000 ${\mathrm{cm}}^{-1\mathrm{}}$. A previous report on this analysis showed the ground level to be $4f5d6s^2 {}_{}{}^1{}_{}{}^{}G_4^{\circ }{}_{}{}^{}$ and gave the lowest levels of $4f5d^{2}6s$. Comparison of the observed odd-parity levels with calculations by Goldschmidt and Salomon shows that all 86 of the odd levels tabulated here belong to these configurations. All but seven of these odd levels are assigned $\mathrm{LS}$ names, although the calculations show that many of them have low $\mathrm{LS}$ purities, and a few have strong mixtures of the two configurations. The much simpler system of even levels below 10 000 ${\mathrm{cm}}^{-1\mathrm{}}$ includes only the six levels of $4f^{2}6s^2 {}_{}{}^3{}_{}{}^{}H$ and ${}_{}{}^3{}_{}{}^{}F$, beginning with ${}_{}{}^3{}_{}{}^{}H_4^{}{}_{}{}^{}$ at 4762.718 ${\mathrm{cm}}^{-1\mathrm{}}$ above the ground-state level. The table of even levels also includes $4f^{2}6s^2 {}_{}{}^1{}_{}{}^{}G_4^{}{}_{}{}^{}$ and the lowest two levels of each of the lowest two terms of $4f^{2}5d6s$, ${}_{}{}^5{}_{}{}^{}I_{4,5}^{}{}_{}{}^{}$, and ${}_{}{}^5{}_{}{}^{}K_{5,6}^{}{}_{}{}^{}$.

Using the Thomas-Fermi model of the atom, we calculate the damping of collective oscillations due to the excitation of bound electrons to free states. We find that in some cases the decay time is much longer than the period of the oscillation. Our results indicate that weakly damped plasma oscillations can exist in atoms.

Electron densities are calculated from approximate solutions to the Thomas-Fermi-Dirac equation for homonuclear diatomic molecules. The accuracy of expectation values calculated from these densities is assessed. In general, one obtains fair agreement with self-consistent-field and experimental results, but this is insufficient when a property is a difference between electronic and nuclear contributions. An important example is the Hellmann-Feynman force on a nucleus, the net force necessarily being repulsive, as for closed-shell atoms. Forc and energy results for such a situation are compared with experiment. Finally, it is shown that a modified theory, previously applied to atoms, gives improvement in expectation values depending on the electron density near the nuclei.

The expression for the annihilation rate of positrons in scattering from atoms at an energy above the positronium (Ps) formation threshold is examined. A model describing the physical effects of annihilation is constructed and it is shown that no unexpected terms appear in the annihilation rate even when almost degenerate states of Ps are open channels. A previous calculation of positron-helium scattering is used to calculate the annihilation rate just above the Ps formation threshold where two competing effects complicate the expression. We conclude that the annihilation rate above threshold comes almost completely from the formation of real Ps.

The level shift $\mathcal{S}({}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}-{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{})$ and the fine-structure interval $\Delta E-\mathcal{S}({}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}-{}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{})$ have been measured in the $n=3\mathrm{}$ level of the helium ion ${\mathrm{He}}^{+}$ with a microwave-optical technique. Electron bombardment of helium gas produces excited ions. The 1640-ÅA light from the $n=3\mathrm{} \mathrm{to} 2$ decay undergoes a decrease when a microwave field induces an $S$-to-$P$ transition. Extensive data were taken on three transitions as a function of microwave power, helium pressure, and bombarding current in a search for systematic effects. The major effect was a slight shift of the resonance curves with microwave power due to transitions in higher excited states. The high-frequency transitions $\alpha c$ and $\beta d$ yield $\Delta E-\mathcal{S}=47\mathrm{} 844.05\pm 0.48$ MHz. Using the Taylor, Parker, and Langenberg value for the fine-structure constant to calculate $\Delta E$, we obtain an indirect measurement of $\mathcal{S}=4183.73\mathrm{}\pm 0.51$ MHz. Measurements on $\alpha e$ yield a direct value of $\mathcal{S}=4183.17\mathrm{}\pm 0.54$ MHz, which, combined with the indirect value, yields $\mathcal{S}=4183.45\mathrm{}\pm 0.52$ MHz, in agreement with the theoretical value of 4184.4 ± 0.5 MHz of Applequist and Brodsky. (All uncertainties are expressed as 68% confidence values.)

Rotational transitions of a number of isotopic species of the hydrogen halides have been measured in the 1.0- to 0.38-mm wavelength region of the spectrum. These transitions have been measured with a submillimeter-wave spectrometer which employs a klystron-driven crystal harmonic generator and a 1.6°K InSb photoconducting detector. The following results have been obtained: for H ${}_{}{}^{35}{}_{}{}^{}\mathrm{Cl}$, $B_0=312\mathrm{} 989.297\pm 0.020$ Mc/sec, $D_0=15.836\mathrm{}$ Mc/sec, $r_e=1.274\mathrm{} 5991$ Å; for H ${}_{}{}^{37}{}_{}{}^{}\mathrm{Cl}$, $B_0=312\mathrm{} 519.121\pm 0.020$ Mc/sec, $D_0=15.788\mathrm{}$ Mc/sec, $r_e=1.274\mathrm{} 5990$ Å; for D ${}_{}{}^{35}{}_{}{}^{}\mathrm{Cl}$, $B_0=161\mathrm{} 656.238\pm 0.014$ Mc/sec, $D_0=4.196\mathrm{}\pm 0.003$ Mc/sec, $r_e=1.274\mathrm{} 5990$ Å; for D ${}_{}{}^{37}{}_{}{}^{}\mathrm{Cl}$, $B_0=161\mathrm{} 183.122\pm 0.016$ Mc/sec, $D_0=4.162\mathrm{}\pm 0.003$ Mc/sec, $r_e=1.274\mathrm{} 5988$ Å; for H ${}_{}{}^{127}{}_{}{}^{}\mathrm{I}$, $B_0=192\mathrm{} 657.577\pm 0.019$ Mc/sec, $D_0=6.203\mathrm{}\pm 0.003$ Mc/sec, $r_e=1.609\mathrm{} 018$ Å; for D ${}_{}{}^{127}{}_{}{}^{}\mathrm{I}$, $B_0=97\mathrm{} 537.092\pm 0.009$ Mc/sec, $D_0=1.578\mathrm{}\pm 0.001$ Mc/sec, $r_e=1.609\mathrm{} 018$ Å; for D ${}_{}{}^{79}{}_{}{}^{}\mathrm{Br}$, $B_0=127\mathrm{} 357.639\pm 0.012$ Mc/sec, $D_0=2.6529\mathrm{}\pm 0.0014$ Mc/sec, $r_e=1.414\mathrm{} 4698$ Å; for D ${}_{}{}^{81}{}_{}{}^{}\mathrm{Br}$, $B_0=127\mathrm{} 279.757\pm 0.017$ Mc/sec, $D_0=2.6479\mathrm{}\pm 0.0020$ Mc/sec, $r_e=1.414\mathrm{} 4698$ Å; for D ${}_{}{}^{19}{}_{}{}^{}\mathrm{F}$, $B_0=325\mathrm{} 584.98\pm 0.300$ Mc/sec, $D_0=17.64\mathrm{}$ Mc/sec, $r_e=0.916\mathrm{} 914$ Å.

The one-electron reduced density matrix (one-matrix) is reported for the Weiss $\mathrm{Li}\left({}_{}{}^2{}_{}{}^{}P\right)$ configuration interaction (CI) wave function and for the $\mathrm{B}\left({}_{}{}^2{}_{}{}^{}P\right)$ polarization, first-order, and 187-term CI functions of Schaefer, Harris, and Klemm. The one-matrix of the $\mathrm{Li}\left({}_{}{}^2{}_{}{}^{}P\right)$ function demonstrated none of the symmetry mixing expected of a ${}_{}{}^2{}_{}{}^{}P$ state due to the "${}_{}{}^2{}_{}{}^{}S$-type" restrictions imposed on the wave function. Both the 187-term and first-order wave functions for $\mathrm{B}\left({}_{}{}^2{}_{}{}^{}P\right)$ do not fully describe the hyperfine structure because the 187-term function lacks an adequate basis and the first-order function systematically excludes some important radial-type correlation terms in the $L$ shell. By including a few such terms, this function could represent an effective systematic approach to hyperfine-structure determinations. The polarization function excludes too many important terms to be reliable. These studies indicate that the polarization to the $1s$ and $2s$ orbitals decreased with the inclusion of more correlation in the wave function and that the inclusion of the $2s-2p$ near-degeneracy effect did materially influence the forms of the dominant natural spin orbitals. Pertinent suggestions are presented for constructing improved CI wave functions for $\mathrm{B}\left({}_{}{}^2{}_{}{}^{}P\right)$.

New identities are derived which enable the construction of closed formulas for the matrix elements of the electrostatic interaction between (two-particle) configurations comprising an electron and a hole. Relations between these matrix elements and those corresponding to two-electron configurations are obtained. Formulas are given in which the matrix elements of the electrostatic interaction between three-particle configurations are expressed as linear combinations of matrix elements between two-particle configurations.

The effect of a static electromagnetic field on the $2P-\mathrm{nS}$ and $2S-\mathrm{nP}$ He I isolated lines is systematically investigated. Exact expressions are given for the shift of the allowed component, and the behavior of the corresponding transition probability under static perturbation is considered in detail. The usual hydrogenic approximations for the Stark constants are discussed with reference to the most recent values for the oscillator strengths available. They are found to be consistent, expecially for the $2{}_{}{}^1{}_{}{}^{}P-n{}_{}{}^1{}_{}{}^{}S$ and $2{}_{}{}^1{}_{}{}^{}S-n{}_{}{}^1{}_{}{}^{}P$ lines with $n\ge 3$, when accurate data are used for the unperturbed excitation energies. The shifts determined with either nondegenerate or degenerate perturbation theory are critically compared, and the greatest difference is found for the magnetic sublevels in the presence of crossed fields. The fourth-order Stark effect is also considered. Calculated shifts are compared with the existing experimental data, and attention is also given to the methods used in plasma spectroscopy for the evaluation of the static Stark effect.

An energy sum rule for fermion levels split by $n$-body exchange forces is derived and expressed in terms of Young tableau "hook lengths." The Franzini-Radicati nuclear mass formula is rederived and generalized to include the effect of $n$-body operators. The form of a mass splitting by a three-body term is computed in detail.

The multiple diffusion theory of Barrat has been extended by expanding the density matrix in irreducible tensors to calculate level-crossing signals arising from observation of the back-scattered light. In the framework of an approximation equivalent to the coherent scattering of radiative-transfer theory, we first calculate the different Hanle-effect signals, using, in particular, the results of Chandrasekhar which give light intensities in zero magnetic field. We then roughly evaluate the effect of the redistribution in frequency of the radiation and the variation of absorption with frequency. The narrowing predicted for the Hanle-effect curves is much less important than for lateral diffusion.

We present experimental verification of a certain number of theoretical predictions given in the preceding paper. The narrowing of the different Hanle-effect signals by multiple diffusion is clearly weaker if observed using the light scattered in the backward direction than if observed using the laterally scattered light. This narrowing can become very small if the vapor is illuminated with light whose frequency is in the wings of the Doppler absorption width of the vapor. The values of the ratios of the widths and intensities of the different Hanle-effect signals that were measured are in good agreement with the values predicted by theory.

The Auger rates, x-ray rates, and fluorescence yields are presented for elements $Z=4\mathrm{}-54$ when a vacancy is in the $K$ shell. Numerical calculations have been performed using the Hartree-Fock-Slater approach with Kohn-Sham and Gaspar exchange. Comparisons of different exchange approximations on the Auger and x-ray total transition rates and the fluorescence yields are reported. The differences are found to be ≅ 10-20% at low $Z$ and to decrease with increasing $Z$.

A measurement of the low-velocity moderation of ${\Sigma }^{-}$ hyperons in hydrogen has been performed in a bubble-chamber experiment. The technique employed determines very accurately the ${\Sigma }^{-}$ velocity at decay, $\Delta \beta =\pm 0.004$, by using ${\Sigma }^{-}\to {\pi }^{-}+n$ decays accompanied by neutron scatters with visible proton recoils. The results of the study, based on 23 200 ${\Sigma }^{-}$ leading to 151 events with recoils in the region ${\beta }_{{\Sigma }^{-}}<0.10\mathrm{}$, indicate that the ${\Sigma }^{-}$ moderation time for $\beta >0.02\mathrm{}$ agrees with the predictions of energy-loss theory using the first-order Born approximation and with experimental energy-loss data for protons. However, for the region $0.01<\mathrm{}\beta <0.02\mathrm{}$, the moderation-time data for ${\Sigma }^{-}$ (and ${\pi }^{-}$) in hydrogen no longer agree with calculations based on the Born approximation or with experimental data for protons. The results for hydrogen are discussed and interpreted as being related to an "excess-range" effect previously observed for negative particles in emulsion.

We have searched for radiative transitions from doubly excited levels in He I, using the beam-foil method. The spectra showed the $1s2p{}_{}{}^3{}_{}{}^{}P-2p^2{}_{}{}^3{}_{}{}^{}P$ transition at 320. 4 Å and some weaker lines, which are also interpreted as transitions from the doubly excited system. Radiative lifetimes were measured for three lines. Our value for the $2p^2{}_{}{}^3{}_{}{}^{}P$ level, 0.080 ± 0.007 nsec, is in agreement with the recent theoretical value of 0.0803 nsec. The possibilities of observing transitions from doubly excited levels in the He I isoelectronic sequence are briefly discussed.

The Roothaan method has been used to calculate analytic self-consistent-field functions for the excited $\mathrm{}\left(3d\right)\mathrm{}\left(4s\right)\mathrm{}$ configuration for the first-row transition-metal ions which normally have a ${\mathrm{}\left(3d\right)\mathrm{}}^2$ ground configuration. Wave functions are calculated for both ${}_{}{}^1{}_{}{}^{}D$ and ${}_{}{}^3{}_{}{}^{}D$ multiplets arising from the excited configuration. The separations of these two multiplet levels from the ground ${\mathrm{}\left(3d\right)\mathrm{}}^2 {}_{}{}^3{}_{}{}^{}F$ level deviate from experiment by 3-8%.