Angular correlations between $K$ and $L$ x rays in Nd and Tb were measured. The Nd and Tb x rays were obtained from radioactive sources of ${}_{}{}^{145}{}_{}{}^{}\mathrm{Pm}$ and ${}_{}{}^{159}{}_{}{}^{}\mathrm{Dy}$, respectively. The $K$ and $L$ x rays were detected with cooled Ge(Li) and Si(Li) detectors, respectively. The following values were obtained for the coefficient $A_{22}$ in the angular-correlation function $W\left(\theta \right)=1+A_{22}{P}_2\left(cos\theta \right)$: for the $K\alpha -L{\alpha }_1+L{\alpha }_2$ cascade (2.7 ± 0.3) × ${10}^{-2\mathrm{}}$ in Nd and (2.6 ± 0.5) × ${10}^{-2\mathrm{}}$ in Tb; for the $K\alpha -L\beta$ cascade (0.5 ± 0.4) × ${10}^{-2\mathrm{}}$ in Nd and (-0.1 ± 0.5) × ${10}^{-2\mathrm{}}$ in Tb; for the $K\alpha -Ll$ cascade (15 ± 3) × ${10}^{-2\mathrm{}}$ in Nd and (14 ± 5) × ${10}^{-2\mathrm{}}$ in Tb. The agreement of these results with theory is good only for the $K\alpha -L{\alpha }_1+L{\alpha }_2$ cascade.

The stabilization method for positions and widths of resonances has been applied up to now only to model problems or to problems for which definitive calculations are not possible. The purpose of this paper is to examine the method for autoionization states in He and ${\mathrm{H}}^{-}$, where accurate calculations are available. It is concluded that reasonably accurate results can be obtained by this method.

The linked-cluster many-body perturbation theory is used to calculate the Sternheimer antishielding factor ${\gamma }_{\infty }$ for the ${\mathrm{Fe}}^{+3\mathrm{}}$ ion. The net result for ${\gamma }_{\infty }$ comes out as -9.03 and is composed of -8.69 and -0.34, respectively, for zero and higher orders in the electron-electron interaction. The zero-order result is in good agreement with earlier work and derives a major contribution close to 85% from the $3p\to p$ radial excitations. The first-order effect represents the role of self-consistency, while the second-order one includes, in addition, the influence of correlation. Reasons for the smallness of the consistency and correlation contributions will be discussed.

An accurate analytic approximate solution of the hydrogen-Stark-effect Schrödinger equation is obtained. In contrast to previous non-numerical solutions, this solution (which is piecewise continuous, and easily calculable) is accurate for applied electric fields of arbitrary strength—including weak fields ($F\lesssim 0.1F_{\mathrm{ion}}$; the Stark-shift regime), strong fields ($F\lesssim F_{\mathrm{ion}}$), and ultrastrong fields ($F\gtrsim {10}^2{F}_{\mathrm{ion}}$), where $F_{\mathrm{ion}}=3\mathrm{}\times {10}^8$ V / cm is the classical ionization field of hydrogen. The ultrastrong-field regime is experimentally accessible in solid-state physics where, for hydrogenic excitons, $F_{\mathrm{ion}}$ may be as small as ${10}^2$ V / cm.

The effect of radiative corrections and of retardation in electron-electron interactions is discussed for a many-electron atom in an external homogeneous magnetic field. The results indicate that for $S$ states, corrections of relative order ${\alpha }^3{\mu }_{B}H$ are obtained by inclusion of Pauli interactions added to the Breit Hamiltonian of a many-electron atom coupled minimally to the external magnetic field. For states with nonzero orbital angular momentum additional order ${\alpha }^3{\mu }_{B}H$ corrections may be present. These arise from the Lamb-shift-type radiative correction combined with retardation corrections to the Breit potential.

We have used perturbation theory to calculate the shift due to an electric field E in the frequency of the hfs transition $(F=I+\frac{1}{2},0)\leftrightarrow (F=I-\frac{1}{2},0)$ for the ground states of the Na and K atoms. The frequency shift $\delta f=-k\times {10}^{-6\mathrm{}}{E}^2\frac{\mathrm{Hz}}{{\left(\frac{\mathrm{V}}{\mathrm{cm}}\right)}^2}$, and we calculate $k$ to be 0.111 and 0.061 for Na and K, respectively. Electron correlations are not included. The corresponding experimental values measured by Mowat are 0.124 ± 0.004 and 0.071 ± 0.002.

It is shown that the "accelerated" or "multistep" ladder-operator procedure, when introducing pseudo-key matrix elements, leads to closed-form expressions for the discrete-continuous hydrogenic $r^k$ matrix elements.

The rotational spectrum of ${\mathrm{T}}_2$O in the ground vibrational state has been measured with high-resolution microwave techniques in the 80-640-GHz region. Forty-six previously unreported transitions have been measured and assigned. Analysis of these results and the six previously observed lines yielded the following rotation and distortion parameters (in MHz) of the Watson formulation: $\mathcal{A}=338\mathrm{}810.923\pm 0.076$, $\mathcal{B}=145\mathrm{}665.417\pm 0.044$, $\mathcal{C}=100\mathrm{}259.415\pm 0.044$, ${\Delta }_J=4.145597\mathrm{}\pm 0.0014$, ${\Delta }_{\mathrm{JK}}=-22.03898\mathrm{}\pm 0.008$, ${\Delta }_K=144.13766\mathrm{}\pm 0.006$, ${\delta }_J=1.609823\mathrm{}\pm 0.0008$, and ${\delta }_K=5.44092\mathrm{}\pm 0.01$. The usual distortion-free rotational constants derived from these are (in MHz) $A^{\prime }=338\mathrm{}808.0$, $B^{\prime }=145\mathrm{}631.9$, and $C^{\prime }=100\mathrm{}291.9$. The effective ground-state structures obtained for ${\mathrm{T}}_2$O are 0.9623 Å for the bond length and 104.6° for the bond angle. The observed inertial defect $\Delta =0.0772\mathrm{}$ amu ${\mathrm{\AA }}^2$ agrees well with the theoretically derived value of 0.0753 amu ${\mathrm{\AA }}^2$.

The hyperfine structure of several of the excited $S$ states of the stable isotopes of potassium, rubidium, and cesium has been measured by the method of cascade radio-frequency spectroscopy. We find, for the magnetic dipole coupling constant $A$, in MHz, for ${\mathrm{K}}^{39}$: $A\left(5\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\right)=55.50\mathrm{}\left(60\right)\mathrm{}$, $A\left(6\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\right)=21.81\mathrm{}\left(18\right)\mathrm{}$; for ${\mathrm{K}}^{41}$: $A\left(5\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\right)=30.75\mathrm{}\left(75\right)\mathrm{}$, $A\left(6\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\right)=12.03\mathrm{}\left(40\right)\mathrm{}$; for ${\mathrm{Rb}}^{85}$: $A\left(6\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\right)=239.3\mathrm{}\left(1.2\right)\mathrm{}$, $A\left(7\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\right)=94.00\mathrm{}\left(64\right)\mathrm{}$, $A\left(8\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\right)=45.5\mathrm{}\left(2.0\right)\mathrm{}$; for ${\mathrm{Rb}}^{87}$: $A\left(6\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\right)=809.1\mathrm{}\left(5.0\right)\mathrm{}$, $A\left(7\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\right)=318.1\mathrm{}\left(3.2\right)\mathrm{}$, $A\left(8\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\right)=158.0\mathrm{}\left(3.0\right)\mathrm{}$; for ${\mathrm{Cs}}^{133}$: $A\left(7\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\right)=546.3\mathrm{}\left(3.0\right)\mathrm{}$, $A\left(8\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\right)=218.9\mathrm{}\left(1.6\right)\mathrm{}$, and $A\left(9\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\right)=109.5\mathrm{}\left(2.0\right)\mathrm{}$.

The hyperfine structure of the $4{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}$ state of ${\mathrm{Na}}^{23}$ has been measured by the method of cascade radio-frequency spectroscopy. We find, for the magnetic dipole coupling constant, $A\left(4\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\right)=202\mathrm{}\pm 3$ MHz.

In an earlier paper, published by the authors elsewhere, it was shown that for 661.6-keV $\gamma$ rays the measurements of $\gamma$-ray attenuation coefficients would greatly improve if one uses the counting sequence of Conner et al. together with a new criterion $\mu t<\mathrm{}1\mathrm{}$, where $\mu$ is the $\gamma$-ray attenuation coefficient and $t$ is the thickness of the sample. In this paper the authors report results which establish the above criterion for 84-keV $\gamma$ rays also. Using the above method the total $\gamma$-ray attenuation coefficients have been measured in carbon, aluminum, copper, tin, and lead at $\gamma$-ray energies of 84, 129, 145.41, 279.12, 322, and 411.8 keV.

A new method is given for deriving the angular factors of energy matrices in $l^n$ atomic configurations, and the need for coefficients of fractional parentage, Racah coefficients, algebraic formulas, and chain calculations is avoided. Instead, matrix elements are obtained directly by a comparatively simple digital counting procedure.

Using basis functions with complex coordinates, it is possible to construct discrete approximations to the Fredholm determinant directly at real energies using only square-integrable functions. For the case of purely elastic scattering, the procedure is equivalent to an analytic continuation of the coordinate dependence of the Hamiltonian. It is shown that improved convergence is obtained by an application of the "dispersion-correction" method. The method is generalized to allow calculation of the "substituted" Fredholm determinants needed to construct the $S$ matrix for many-channel potential-scattering problems. This generalization is not equivalent to a simple continuation of the coordinate dependence of the many-channel Hamiltonian. Results of calculations on several model problems are presented.

This paper deals with the apparent breakdown of gauge invariance in atomic-level-width formulas. Specifically, it is shown that in the analysis of the measurement od the Lamb shift the use of the $\frac{\overrightarrow{\mathrm{p}}·\overrightarrow{\mathrm{A}}}{\left(mc\right)}$ and the $\overrightarrow{\mathrm{E}}·\overrightarrow{\mathrm{r}}$ forms of the interaction yields almost identical results for the shape of the resonance curve and equal positions for the line center. This is achieved by adding the contribution of all intermediate states to the basic three-level resonance formula. While the contribution of these intermediate states in the $\overrightarrow{\mathrm{E}}·\overrightarrow{\mathrm{r}}$ language is negligible, it is nevertheless significant when the $\overrightarrow{\mathrm{p}}·\overrightarrow{\mathrm{A}}\left(mc\right)$ form of the interaction is used. Although this particular example would indicate that the use of the $\overrightarrow{\mathrm{E}}·\overrightarrow{\mathrm{r}}$ form of the interaction is computationally simpler, it is shown that no general statement to this effect is permissible. A simple counter example is presented within the context of elastic photon scattering off atomic targets.

A calculation of higher-order relativistic contributions to the Zeeman effect in the $2{}_{}{}^3{}_{}{}^{}P$, $2{}_{}{}^3{}_{}{}^{}S$, $3{}_{}{}^3{}_{}{}^{}P$, $4{}_{}{}^3{}_{}{}^{}P$, and $5{}_{}{}^3{}_{}{}^{}P$ states of helium is given. The contributions to the $g$ factors $g_S^{\prime }$, $g_L^{\prime }$, and $g_X$, have been calculated to order ${\alpha }^2$ and $\frac{m}{M}$ where $m$ is the electron mass and $M$ is the helium mass. Relativistic contributions are obtained from the matrix elements of a reduced Breit Hamiltonian with a series of numerical wave functions using a Hylleraas basis. The comparison of the theoretical values with the experimental determinations of the $g$ factors is given for the $2{}_{}{}^3{}_{}{}^{}P$ and $2{}_{}{}^3{}_{}{}^{}S$ states. The value of $g_S^{\prime }$ calculated for $2{}_{}{}^3{}_{}{}^{}P$ helium is within two standard deviations of the experimental error of the measurement of Lewis et al. A recent precise measurement of the $g$ factor in $2{}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}$ helium by Brossel et al. is three standard deviations of the experimental error from the value calculated here and from the result of Perl and Hughes. These higher-order contributions are necessary for the determination of the $2{}_{}{}^3{}_{}{}^{}P$ fine-structure intervals from the Zeeman effect, and hence to test the theory of fine structure for helium and to determine precisely the fine-structure constant $\alpha$ from the helium fine structure. The relativistic contributions to $g_S^{\prime }$ have been calculated for the helium isoelectronic series from Li II through Ne IX for the $2{}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}$, $3{}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}$, $4{}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}$, and $5{}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}$ states.

By using the method proposed in a previous paper the differential and integrated capture cross sections for the processes $\mathrm{p}+\mathrm{H}\mathrm{}\left(1s\right)\mathrm{}\to \mathrm{H}\left(nl\right)+p$ are computed as a function of incident kinetic energy, for energies of 25 keV and higher, and compared with experiment. The results are compared with the Jackson and Schiff scaling rules and are used to compute the total charge-exchange cross section for proton—hydrogen-atom collisions.

Differential and integrated cross sections for proton—hydrogen-atom charge-exchange collisions are calculated for capture into highly excited states using a method which takes the nonorthogonality of initial and final states into account. The differential cross sections show dramatic features that persist from charge exchange into the lower quantum states. The integrated-cross-section results agree well with experiments over the range of validity of the method (incident proton energies above 20 keV) and are a considerable improvement over previous calculations, which are an order of magnitude larger than experiment. The postulated scaling rules of Jackson and Schiff and of Omidvar are compared with the calculations, and the former is in better agreement.

Cross-section measurements of the dissociation modes $\mathrm{H}_3^{}{}_{}{}^{+}$ → (3H and H + ${\mathrm{H}}_2$), H + $\mathrm{H}_2^{}{}_{}{}^{+}$, ${\mathrm{H}}^{+}$ + 2H, ${\mathrm{H}}^{+}$ + ${\mathrm{H}}_2$, H + 2${\mathrm{H}}^{+}$, ${\mathrm{H}}^{+}$ + $\mathrm{H}_2^{}{}_{}{}^{+}$, and 3${\mathrm{H}}^{+}$ are reported for $\mathrm{H}_3^{}{}_{}{}^{+}$ energies between 400 and 1800 keV in targets of ${\mathrm{H}}_2$, ${\mathrm{N}}_2$, and Li. For the ${\mathrm{H}}_2$ target, comparisons are made with available previous measurements of H, ${\mathrm{H}}^{+}$, ${\mathrm{H}}_2$, and $\mathrm{H}_2^{}{}_{}{}^{+}$ production cross sections at lower energies. Cross-section measurements for 270- to 1200-keV ${\mathrm{H}}_2$ → $\mathrm{H}_2^{}{}_{}{}^{+}$, H + ${\mathrm{H}}^{+}$, and 2${\mathrm{H}}^{+}$ are also reported.

Theoretical $K$-shell Auger groups rates, $K\alpha$ and $K\beta \mathrm{x}$-ray rates, and fluorescence yields are presented for 47 defect electronic configurations of argon. Energy shifts from the normal x rays and the Auger-electron energies are also given for these defect configurations. These theoretical results can be conveniently used in the analysis of experiments involving heavy-ion—argon collisions.

The vibrational cross section to $v=8,9\mathrm{}$, and 10 by electron impact on ${\mathrm{N}}_2$ and CO has been measured in the energy range 2-4 eV. In this energy range vibrational excitation proceeds predominantly via a compound state, and decay into high vibrational states can occur. At the first peak the ratio of cross sections for $v=8:9:10\mathrm{}$ is 1: 0.31: 0.11 in ${\mathrm{N}}_2$, and 1: 0.51: 0.25 in CO. These ratios are largely determined by the quantum-mechanical penetration through the centrifugal barrier.