A systematic investigation was conducted on the effects of spin-dependent interactions on the energy-level schemes of the $3d^N$ configurations ($N=2, 3, \dots , 8$) in the third spectra of the iron group. The spin-dependent interactions considered were the spin—other-orbit, the spin-spin, and the effective electrostatic-spin-orbit interactions. The appropriate radial integrals $M^k(k=0, 2)$ and $Q^k(k=2, 4)$ were considered as adjustable parameters and evaluated by means of diagonalization-least-squares procedures. The introduction of the spin-dependent interactions greatly improved the fit between calculated and observed multiplet splittings in all investigated configurations. Reliable and consistent values of the appropriate parameters were obtained, which agree with available theoretical predictions. In particular, the $M^k$ values are in excellent agreement with corresponding values obtained by the Hartree-Fock method. The variation of the various parameters with $k$ and their dependence on the effective nuclear charge were determined.

A high-resolution and high-sensitivity electron transmission spectrometer is used to study sharp structures in the total scattering cross section of ${\mathrm{H}}_2$, ${\mathrm{D}}_2$, ${\mathrm{O}}_2$, NO, ${\mathrm{N}}_2$, and CO. Many new resonances are found, some of which form bands which consist of progressions of negative-ion vibronic states. The existence of bands makes the identification of temporary negative-ion states easier, since the spacing and the magnitudes of the structure can be compared with the appropriate parameters of various positive-ion cores. The most prominent features observed consist of two Rydberg electrons attached to a particular positive-ion core.

Linearization of the differences between the lowest levels of the $f^{q}p{s}^2$, $f^{q}d{s}^2$, and $f^{q+1\mathrm{}}{s}^2$ electron configurations as a function of $q$ is demonstrated for the lanthanide and actinide series. A linear extrapolation of these differences estimates the energy of the lowest level of the $4f^{14}6p6s^2$ electron configuration of Lu to be 3.5 × ${10}^3$ ${\mathrm{cm}}^{-1\mathrm{}}$ higher than the measured value. This deviation from linearity is attributed to expected discontinuities at the $q=0\mathrm{} \mathrm{and} 14$ ends of the $f^{q}d{s}^2$ series. In the actinide series the corresponding linear extrapolation predicts the lowest level of the $5f^{14}7p7s^2$ electron configuration of Lr to be (2.3 ± 3) × ${10}^3$ ${\mathrm{cm}}^{-1\mathrm{}}$ above the expected $5f^{14}6d7s^2$ ground state, after correction for a corresponding 3.3 × ${10}^3$ ${\mathrm{cm}}^{-1\mathrm{}}$ deviation expected at the end of the actinide series.

The reflection, refraction, and associated production of photoelectrons by ultrasoft x rays (10-100 Å) can be important bases for the determination of material constants such as the linear x-ray absorption coefficients and the electron mean free paths. These may then be used to establish directly the photoionization cross sections and the electron-collision cross sections which account for the dominant energy-absorbing processes within solids for this energy region. Because the effective sample depths for these interactions are typically less than 100 Å, they constitute an important practical basis for surface characterization. By applying the exact theory for the reflection-refraction of a plane electromagnetic wave at an absorbing dielectric interface to the shorter-wavelength region (<10 Å), it can be shown that the conventional approximate theory of x-ray reflection is adequate. However, the more exact theory must be applied in the region of longer x-ray wavelengths (>50 Å). Although the derivations of the exact theory are tedious, the results can be expressed in relatively simple form as functions of two material constants $\alpha$ and $\gamma$, which are identifiable as the unit decrements to a complex dielectric constant, of the grazing-incidence angle, and of a parameter which is a function of this grazing angle and which becomes the angle of refraction for small angles of incidence. X-ray absorption coefficients and electron mean-free-path values have been determined from x-ray reflection and refraction and photoelectron excitation data. These values have been shown to agree reasonably well with such material constants as determined by transmission measurements through thin samples.

The bound-state hydrogen-atom radial functions are used to construct a basis for a unitary representation of the algebra $O(3, 2)$. With the use of the $O\left(3\right)\mathrm{}$ subalgebra of this "radial algebra" matrix elements of $r^s$ between states of different energy are shown to be completely expressible in terms of the matrix elements of $r^{-2\mathrm{}}$. Matrix elements corresponding to the radial matrix elements of the dipole operator are shown to be proportional to the overlap of the radial functions. The continuum radial functions of constant energy are shown to provide a basis for a unitary representation of $O(2, 1)$.

Using recently calculated values of radiative, Coster-Kronig, and Auger transition rates, and the hypothesis that the satellite structure arises from doubly ionized atoms (with $L$ and $N$ holes), and an assumed incident x-ray spectrum, we calculate the relative intensities of the Ag $L$-series x-ray spectrum. The calculations are in best agreement with Parratt's measurements when Ag $L_1-L_3{M}_{4,5}$ Coster-Kronig transitions are forbidden. Comparison of the calculations with the satellite components leads to unsatisfactory agreement. Reasons for this are discussed. Modified values of $f_{12}$, $f_{13}$, and ${\omega }_1$ for the Ag and Sn $L$ shells are reported.

The electron-impact-broadened profiles of the resonance lines of singly ionized calcium ($\lambda \sim 3934$ Å) and magnesium ($\lambda \sim 2802$ Å) have been measured using an electromagnetically driven shock tube and a rapid-scanning Fabry-Perot spectrometer. For $N_e\simeq {10}^{17}$ ${\mathrm{cm}}^{-3\mathrm{}}$ and $T\simeq 19000°$K, we found the Lorentizian half-width of the ${\mathrm{Ca}}^{+}$ line to be 0.086 Å ± 10% and of the ${\mathrm{Mg}}^{+}$ line to be 0.044 Å ± 10%. Using the quantum-mechanical theory of Barnes and Peach and our semiclassical calculation for the calcium lines, we found that the temperature dependence of the theoretical curves is close to that measured, although both theories predict actual values which are somewhat large. Using the quantum-mechanical theory of Bely and Griem and our semiclassical calculation for the magnesium lines, once again we found that the shape of both curves agrees quite well with the experimental points, although the quantum-mechanical predictions are somewhat too small. The probable sources of error are the use of the classical assumption in the semiclassical calculations and the use of an incomplete set of perturbing levels in the quantum-mechanical calculations.

The bremsstrahlung energy distribution from a thick tungsten target is calculated from the Sommerfeld and Born-approximation thin-target formulas, taking into account electron energy losses, electron backscatter losses, and photon attenuation in the target. Over-all agreement with measurement is 20% in the 12-100-keV energy range, increasing to 50% at 300 keV. Semiempirical expressions are developed that give 20% over-all agreement with measurement in the 12-300-keV energy range.

A complete discussion is given of our precision measurement of the hyperfine-structure interval $\Delta \nu$ of the ground state of muonium at a strong magnetic field of 5000 G. Particular emphasis is placed on refinements to the theory of the resonance line shape and on a detailed discussion of the sources of error and of the capability of our method. The final result for $\Delta \nu$ is $\Delta \nu =4463.24\mathrm{}\pm 0.12$ MHz (27 ppm), in which the 1-standard-deviation error is caused principally by the statistical counting error and magnetic field inhomogeneity. Allowance is included for a quadratic as well as a linear dependence of $\Delta \nu$ on pressure. This result is in good agreement with other, more precise, experimental values and with the current theoretical value $\Delta \nu =4463.326\mathrm{}\pm 0.019$ MHz (4.2 ppm).

The ground-state hyperfine interval $\Delta \nu$ of muonium has been redetermined in argon by measuring the (1, 1) ↔ (1, 0) Zeeman frequency ${\nu }_1$ at that external field value where $\frac{\partial {\nu }_1}{\partial B}=0\mathrm{}$. The measurements were performed with good statistics at much lower pressures than heretofore, greatly reducing uncertainties in extrapolating to zero pressure. Our value, $\Delta \nu \mathrm{}\left(0\right)=4463.313\mathrm{}\mathrm{}\left(18\right)\mathrm{}$, disagrees with the linearly extrapolated values found in higher-pressure Ar and Kr by Crane et al. However, a new interpretation, namely, a pressure extrapolation based on a quadratic function with the linear coefficient fixed at the "atomic" value (i.e., assumption of no isotope dependence in the linear pressure shift), results in consistency between all present (data published before April, 1971) determinations of $\Delta \nu$.

The irreducible-tensor theory is extended to the double group O. A complex basis is used because the representations $E^{\prime }$, $E^{\prime \prime }$, and $U^{\prime }$ are necessarily complex, and a set of phase factors are presented. The presence of repeated representations in the reduction of direct products of representations implies that more than one set of $V$ coefficients may be associated with a given set of representations $\mathrm{abc}$. For the group O, representations are repeated a maximum of two times in any direct product and the corresponding sets of $V$ coefficients are labeled $V_1$ and $V_2$. This requires a modification in the standard definition of $W$ so that the subscripts on the $V$ coefficients are included in the definition. This in turn calls for a rederivation of the useful matrix elements of double-tensor operators and several of the more important formulas are developed with particular emphasis on matrix elements in a spin-orbit basis. Complete tables of all $V$ and $W$ coefficients for O are also given.

A symmetry-adapted pair-correlation approximation has been used to calculate the electron affinity of the oxygen atom. The calculated electron affinity is 1.47 eV which is in reasonable agreement with experimental values. The present results add to the evidence that the energy difference between similar systems can be calculated with reasonable accuracy using symmetry-adapted pair energies.

The $\frac{L{\gamma }_2}{L{\beta }_3}$, $\frac{L{\gamma }_3}{L{\beta }_3}$, $\frac{L\eta }{L{\beta }_1}$, $\frac{L{\beta }_5}{L{\alpha }_1}$, $\frac{L{\beta }_6}{L{\alpha }_1}$, and $\frac{\mathrm{Ll}}{L{\alpha }_1}$ radiative-transition probability ratios have been measured for 13 elements ranging in atomic number from ${}_{62}{}^{}{}_{}{}^{}\mathrm{Sm}$ to ${}_{92}{}^{}{}_{}{}^{}\mathrm{U}$. This has been accomplished by bombarding the elements with an electron beam of constant energy and flux, and counting the x-ray photons emitted by a given electronic transition. The corrected measured ratios are compared with the results calculated on the basis of the relativistic-Hartree-Fock-Slater (RHFS) potential and those calculated on the basis of the relativistic-screened-Coulomb (RSC) potential. In general, values calculated on the basis of RHFS potential are favored except for the $\frac{\mathrm{Ll}}{L{\alpha }_1}$ ratio where the measured values and those obtained from a RSC-potential calculations are in excellent agreement and are as much as 40% lower than the values obtained assuming a RHFS potential. The $\frac{L{\gamma }_2}{L{\beta }_3}$ and $\frac{L{\gamma }_3}{L{\beta }_3}$ values obtained with a RSC potential are as much as an order of magnitude greater than experimental values at high $Z$, and are not included in the graphs. The estimated error of the experimental results is from 10 to 15%. Other experimental values are also included for comparison.

The lowest-order configuration-space Hamiltonian (CSH) for a heavy atom is constructed from quantum electrodynamics by a variational procedure. A variational potential function $\Omega$ is introduced which in effect allows some freedom in the choice of the definition of the difference between electrons and positrons. The optimization of $\Omega$ results in a nonlinear equation from which it is shown that $\Omega$ is probably not small for relativistic electrons. The procedure results in a CSH which contains a new two-body interaction which is relativistic in origin and which is apparently of the same order of magnitude as the Breit interaction when acting between relativistic electrons.

We propose a new type of expansion for obtaining accurate wave functions for ${}_{}{}^{1,3}{}_{}{}^{}S$ states of two-electron atoms, expressing $\Psi$ as a sum of "configurations" of the form $(1\mathrm{}\pm P_{12})\exp (-{\alpha }_i{r}_1-{\beta }_i{r}_2-{\gamma }_i{r}_{12})$. Matrix elements of both H and ${\mathrm{H}}^2$ can be easily evaluated for these configurations. Results of an illustrative calculation of the $2{}_{}{}^{1,3}{}_{}{}^{}S$ states of ${\mathrm{H}}^{-}$, which proves them to be unbound, are given.

A new expression of the differential cross section of bremsstrahlung in nuclear fields is given. Higher-order Coulomb correction terms are included, but the effect of the atomic electron screening, the radiative corrections, and the nucleus size are neglected. Validity of the formula obtained here is discussed. It is expected to give good accuracy in the energy region from a few MeV to about 50 MeV for high values of the atomic number of the target, as long as the screening effects are negligible. Some remarks concerning computation of the cross section are included.