High-resolution Ge(Li) and Si(Li) x-ray spectrometers and a wall-less anticoincidence multiwire proportional counter were used for coincidence measurements between various $K$ and $L$ x rays and $M$ x rays of Np (Z=93) and Cm ($Z=96\mathrm{}$) following $\alpha$ decay of ${\mathrm{Am}}^{241}$ and ${\mathrm{Cf}}^{249}$, respectively. The effect of multiple vacancies in the $M$ shell was taken into account in the determination of the following mean $M$-subshell fluorescence yields: ${\nu }_1^M=0.065\mathrm{}\pm 0.014$, ${\nu }_2^M=0.080\mathrm{}\pm 0.029$, ${\nu }_4^M=0.062\mathrm{}\pm 0.005$, ${\nu }_{4,5}^M=0.065\mathrm{}\pm 0.012\approx {\omega }_5^M$ at $Z=93\mathrm{}$; and ${\nu }_1^M=0.081\mathrm{}\pm 0.016$, ${\nu }_2^M=0.068\mathrm{}\pm 0.023$, ${\nu }_3^M=0.062\mathrm{}\pm 0.019$, ${\nu }_4^M=0.080\mathrm{}\pm 0.006$, and ${\nu }_{4,5}^M=0.075\mathrm{}\pm 0.012\approx {\omega }_5^M$ at $Z=96\mathrm{}$. The close agreement of the various mean $M$-subshell fluorescence yields with that of the $M_{4,5}$ subshells for each element indicates that Coster-Kronig transitions are very strong in the $M$ shell. Including earlier measurements of mean $M$-shell fluorescence yields at lower $Z$, corrected for multiple vacancies, and accepting Bhalla's theoretical calculations of radiative $M_{4,5}$-subshell widths, one concludes that the nonradiative widths of the $M_{4,5}$ subshells are essentially constant from $Z=76\mathrm{} \mathrm{to} 96$. Comparison of fluorescence yields of the $K$ shell and the $L_3$ and $M_{4,5}$ subshells at the same energy of the principal radiative transition (i. e., at different values of $Z$ indicates that the ratio of radiative to Auger widths decreases from the $1s_{\frac{1}{2}}$ to $2p_{\frac{3}{2}}$ state, but remains constant from $2p_{\frac{3}{2}}$ to $3d_{\frac{5}{2}}$. The resolution of the Si(Li) detector was sufficient to separate the radiative transitions from fillings of the $M_{1,2}$- and $M_{3,4,5}$-subshell groups, so that from $K$ and $L$ x-ray coincidences with the $M_{1,2}$ group of x rays, it was possible to determine the following quantities: ${\omega }_1^M+f_{12}^M{\omega }_2^M=\left({2.0}_{-2.0\mathrm{}}^{+3.1\mathrm{}}\right)\times {10}^{-3\mathrm{}} \mathrm{and} \left({7.5}_{-7.5\mathrm{}}^{+8.9\mathrm{}}\right)\times {10}^{-3\mathrm{}}$ at $Z=93\mathrm{} \mathrm{and} 96$, respectively, and ${\omega }_2^M=\left({4.6}_{-4.6\mathrm{}}^{+5.1\mathrm{}}\right)\times {10}^{-3\mathrm{}}$ at $Z=96\mathrm{}$. Thus, about 94% of $M_2$-subshell vacancies undergo Coster-Kronig shifts to higher subshells before filling from $N$ and higher major shells occurs. Observation of the radiative $L_1-L_3$ transition was made for the first time in the high-$Z$ region, and its intensity is in reasonable agreement with extrapolated theoretical predictions (6-7% of x rays filling $L_1$-subshell vacancies). The radiative fraction ${\omega }_{13}^L$ in the Coster-Kronig yield $f_{13}^L$ is found to be about 2% in the region of $Z=93\mathrm{}-96$, and ${\omega }_{13}^L=0.011\mathrm{}\pm 0.005 \mathrm{and} 0.009\pm 0.005$ at $Z=93\mathrm{} \mathrm{and} 96$, respectively. The absolute emission rate of $M$ x rays from ${\mathrm{Am}}^{241}$ was measured with multiwire and single-wire proportional counters, and values are found of (6.35 ± 0.60) × ${10}^{-2\mathrm{}}$ $M$ x rays per $\alpha$ decay and 0.470 ± 0.045 $M$ x rays per $L_{\alpha }$ x ray. These intensities were used in determinations of efficiency at 3.25 keV of the semiconductor x-ray detectors.

The ground-state energies of the He atom and the ${\mathrm{Li}}^{+}$ ion have been calculated using the Faddeev formalism. For the off-shell two-body collision amplitude $t$ matrix), Sturmian-function (SF) as well as Coulomb-function representations have been employed. For the He atom, calculations have been performed retaining $1s$, $1s$ and $2s$, and $1s$, $2s$, and $3s$ states in the SF representation for the two-body $t$ matrix, whereas only the $1s$ term has been retained in the mixed-mode (MM) representation. For the ${\mathrm{Li}}^{+}$ ion, computations have been done after retaining terms up to $2s$ in the SF representation and only the $1s$ term in the MM representation. The results obtained by retaining only the $1s$ term are in marked disagreement with the experimental values in both representations. It is noticed that the results for the MM representation are almost as much below the experimental values as those for the SF representation are above. The results for the MM representation are slightly better than those for the SF representation. This is seen more clearly in the case of the ${\mathrm{Li}}^{+}$ ion. The results in the SF representation are found to be in good agreement with the experimental findings after inclusion of the $2s$ term.

A gas-driven shock tube was used to measure the absolute strengths of 21 P I lines and 126 P II lines ($3300\AA <\lambda <6900\mathrm{}\AA$). Accuracy for prominent, isolated neutral and ionic lines is estimated to be 28-40% and 18-30%, respectively. The data and the corresponding theoretical predictions are examined for conformity with the sum rules.

As a test of accurate valence wave functions which have been previously obtained for the excited $\mathrm{np}$ states of the alkali atoms, we have calculated the fine-structure splittings $\Delta \nu$ using these wave functions. The resulting values of ${\Delta \nu }_{\mathrm{theor}}$ are generally in good agreement with the corresponding experimental values ${\Delta \nu }_{\mathrm{expt}}$. Calculations have also been carried out for the atomic quadrupole shielding factor $R$ for the ground states of boron ($\mathrm{B} 2p$) and aluminum ($\mathrm{Al} 3p$), and the ionic antishielding factor ${\gamma }_{\infty }$ for the ions ${\mathrm{B}}^{+}$, ${\mathrm{Al}}^{+}$, and ${\mathrm{Al}}^{3+}$. The value of $R\left(\mathrm{B} 2p\right)=+0.048$ is in good agreement with the results of calculations using spin-polarization wave functions. The result for $R\left(\mathrm{Al} 3p\right)=-0.063$ leads to a corrected value of the nuclear quadrupole moment $Q\left({\mathrm{Al}}^{27}\right)=0.140\mathrm{}\pm 0.002$ b, using the atomic-beam result of Lew and Wessel. The ionic antishielding factors ${\gamma }_{\infty }$ have the following calculated values: ${\gamma }_{\infty }\left({\mathrm{B}}^{+}\right)=+0.773\mathrm{}$, ${\gamma }_{\infty }\left({\mathrm{Al}}^{+}\right)=-1.68\mathrm{}$, and ${\gamma }_{\infty }\left({\mathrm{Al}}^{3+}\right)\cong -2.4\mathrm{}$.

The $\frac{L{\beta }_3}{L{\beta }_4}$, $\frac{L{\gamma }_1}{L{\beta }_1}$, $\frac{L{\alpha }_2}{L{\alpha }_1}$, $\frac{L{\beta }_{2,15}}{L{\alpha }_1}$ and the $\frac{L{\beta }_{2,15}}{L{\alpha }_2}$ transition probabilities have been measured for some 27 elements ranging in atomic number from ${}_{57}{}^{}{}_{}{}^{}\mathrm{La}$ to ${}_{92}{}^{}{}_{}{}^{}\mathrm{U}$. Other $L$ transitions have also been measured for a smaller number of elements in the same range of atomic numbers. This has been accomplished by bombarding these elements with a constant energy electron beam and measuring their x-ray emission spectrum. The measured ratios are compared with the most recent calculation based on a relativistic Hartree-Fock-Slater (RHFS) potential, and screened Coulomb potential. In general, the form of the dependence of these ratios on atomic number predicted by calculations based on a RHFS potential is favored and the agreement between theory and experiment ranges from excellent for the $\frac{L{\alpha }_2}{L{\alpha }_1}$ ratio to a discrepancy of about 22% in the $\frac{L{\beta }_{2,15}}{L{\alpha }_1}$ ratio for elements of large atomic number.

The probability of simultaneous emission of an x-ray photon and an electron is considered. It is shown that this process can occur as a consequence of the change in the average potential acting on the electrons when the x-ray hole is moving from an inner to an outer shell (shake-off) and as a consequence of interaction between single- and double-hole configurations in the final state. Relative shake-off probabilities of $K\to L^2$, $K\to M^2$, and $L_{2,3}\to M_1{M}_{2,3}$ transitions are calculated for ${\mathrm{F}}^{-}$, Ne, ${\mathrm{Na}}^{+}$, ${\mathrm{Cl}}^{-}$, Ar, and ${\mathrm{K}}^{+}$ from Bagus's self-consistent-field wave functions of single-hole x-ray states. The relative probability of $K\to L^2$ and $K\to M^2$ transitions is estimated for the rest of the atoms in the range $9<~Z<~22$. The estimated $K\to L^2$ and $K\to M^2$ probabilities are of the same order of magnitude as the observed probabilities. However, for low $Z$, they overestimate the observed probabilities, which may be because the shake-off calculations do not account properly for the interaction between single- and double-hole configurations in the final state.

Electron shake-off has been investigated in the decay of ${}_{}{}^{204}{}_{}{}^{}\mathrm{Tl}$ in a measurement of the relative intensities of the Hg and Pb $K$ and $L$ x rays. Values of (1.0 ± 0.1) × ${10}^{-2\mathrm{}}$% and (20.0 ± 2.0) × ${10}^{-2\mathrm{}}$% have been obtained for the $K$-shell and total $L$-shell electron shake-off probabilities, respectively. Approximations of individual $L$-subshell shake-off probabilities resulted in values of (2.0 ± 0.6) × ${10}^{-2\mathrm{}}$%, (9.3 ± 0.6) × ${10}^{-2\mathrm{}}$%, and (9.0 ± 0.6) × ${10}^{-2\mathrm{}}$% for the $L_{\mathrm{I}}$, $L_{\mathrm{II}}$, and $L_{\mathrm{III}}$ subshells, respectively. These values, in some cases, differ by as much as a factor of 3 from theoretical predictions.

Measured values of the relative intensities of several $K$-fluorescent x rays are reported for 28 elements ranging from calcium through plutonium. Solid-state Si(Li) and Ge(Li) spectrometers and a Bragg-diffraction spectrometer were used in making the measurements. Fluorescence was excited in thin samples by higher-energy x rays. Corrections were made to the measured relative intensities for sample absorption, air absorption, and spectrometer efficiency. The accuracy of most of the measurements is ± 2 to 5%. The results are compared with those of other experimenters and with the Scofield calculations.

High-resolution measurements of ${\mathrm{S}}^{-}$ photodetachment near threshold are interpreted on the basis of an intermediate $\left[p^4\right({}_{}{}^3{}_{}{}^{}P)+e(l=0\mathrm{}\left)\right]{}_{}{}^2{}_{}{}^{}P$ complex which dissociates into $\mathrm{jj}$-coupled constituents. The $e-\mathrm{S}$ interaction is represented by doublet and quartet scattering lengths $a_d$ and $a_q$. Relative cross sections are calculated in terms of electron energies, and of $a_d$ and $a_q$. Thereby we generalize to a multichannel process the scattering-length theory of cross sections near $s$-wave thresholds. Many features of the experimental data are accounted for. Predictions are made on the spin-polarization effects of circularly polarized light.

We consider the lowest-order nonlinear contributions to the electric dipole moment induced in closed-shell atomic systems by intense electromagnetic fields. These contributions are comprised of two terms: (i) an intensity-dependent refractive-index coefficient ${\chi }_{\mathrm{zzzz}}(-\omega ;\omega ,\omega ,-\omega )$, and (ii) a third-harmonic coefficient ${\chi }_{\mathrm{zzzz}}(-3\mathrm{}\omega ;\omega ,\omega ,\omega )$. The problem is formulated within the framework of time-dependent Hartree-Fock perturbation theory. The expressions for ${\chi }_{\mathrm{zzzz}}(-\omega ;\omega ,\omega ,-\omega )$ and ${\chi }_{\mathrm{zzzz}}(-3\mathrm{}\omega ;\omega ,\omega ,\omega )$ contain third- and lower-order frequency-dependent wave functions. It is found possible to eliminate the third-order terms from the expressions for the $\chi \text{'}\mathrm{s}$. A variational method for solving the required second-order integrodifferential equations is proposed. Numerical results for helium are obtained. A zero-frequency Hartree-Fock hyperpolarizability for helium of 35.8 a. u. is obtained which agrees reasonably well with the previous Hartree-Fock calculations of Sitter and Hurst. The Hartree-Fock wave functions give static hyperpolarizability results which are about 17% smaller than more accurate calculations. A "static second-order function" approximation for calculating the $\chi \text{'}\mathrm{s}$ is developed and is shown to be useful for obtaining the $\chi \text{'}\mathrm{s}$ at low frequencies with a substantial saving in computing effort.

The three independent coefficients in the magnetic-dipole hyperfine Hamiltonian cannot be determined by measurements of the hfs constants of the two atomic states $3d4s^2{}_{}{}^2{}_{}{}^{}D_{\frac{3}{2},\frac{5}{2}}^{}{}_{}{}^{}$. However, the interaction between these states at a high magnetic field shifts the resonance frequencies for transitions between magnetic substates of either state, and the necessary additional information for evaluating the coefficients may be obtained by observing these frequency shifts. Such measurements, together with more accurate low-field determinations of the hfs constants, lead to ratios of the coefficients in the dipole hfs Hamiltonian in agreement with a recent theoretical prediction. More precise values for the electron $g$ factors are also reported.

The $\alpha$-parametrized Hartree-Fock-Slater ($X\alpha$) equations are solved in the framework of $Z^{-1\mathrm{}}$ expansion theory for the two- and four-electron ions in their ground-state configurations ($1s^2$) and ($1s^{2}2s^2$), respectively. The first two terms in the $Z^{-1\mathrm{}}$ expansion of $\alpha$ are given when $\alpha$ is determined by various criteria.

Spontaneous emission from two systems, namely, $N$ identical harmonic oscillators and $N$ identical two-level atoms, is studied without the use of the rotating-wave approximation. Certain new features of spontaneous emission, for instance, the dependence of the radiation rate on the initial dipole moment phase, are discussed.

This paper constructs a special perturbation theory which directly describes the alteration of decaying states by an external perturbation. The complex energies of the decaying states appear in this theory. When lifetime effects are taken into account, we find that the criteria for quantum degeneracy are slightly strengthened and sharpened. We also discuss the smooth merger of the properties of decaying states and those of true eigenstates in the case of long lifetimes.

A general theory of spontaneous emission was developed in Papers I and II of this series. In the present paper, the mastre equation describing spontaneous emission from $N$ identical two-level atoms is investigated further. It is shown that the equation of motion for the $p(<N)\mathrm{}$-atom density matrix is coupled to the ($p+1\mathrm{}$)-atom density matrix. We first consider the initial excitation of the system to state $|{\theta }_0,{\varphi }_0〉$ (${\theta }_0<\pi$) and solve the hierarchy of equations so obtained in the "uncorrelated approximation." It is found that the presence of other atoms is equivalent to an external field which is determined from a self-consistent analysis. It is also found that in this approximation the state of the atom at time $t$ can be described by a single parameter $\theta \mathrm{}\left(t\right)\mathrm{}$ ($\theta <\pi$), with ${sin}^2\frac{1}{2}\theta$ giving the probability that the atom is to be found in the excited state. Next, some improvements over the uncorrelated approximation are given. This is done by decoupling the equation of motion for higher-order mean values. It is found that the fluctuation $〈(S_i^Z-〈S_i^Z〉)(S_j^Z-〈S_j^Z〉)〉$ for any pair of atoms is of the order of $\frac{1}{N}$. The case when ${\theta }_0=\pi$ is considered separately and we obtain an expression for the radiation rate by making a "Hartree-Fock" type of approximation on the two-particle mean values. In this case, the behavior of the radiation rate is found to differ markedly from the "sech" behavior. It is then shown that the entire dynamics of two-level atoms emitting spontaneously can be described by a set of $2N$ coupled first-order equations which clearly exhibit the type of nonlinearity (which is the analog of the van der Pol type of nonlinearity) for this problem and provide a better understanding of spontaneous emission. Finally in Sec. V, the theory is extended to include the effects of inhomogeneous broadening, and the functional dependence of the radiation rate on the atomic-line-shape factor is obtained.

We present the details of a calculation of the $m{\alpha }^6\ln {\alpha }^{-1\mathrm{}}$ contributions to the triplet-singlet splitting of the positronium ground state. Our result $\Delta \nu \left({\alpha }^6\ln {\alpha }^{-1\mathrm{}}\right)=\frac{3}{4}{\alpha }^4{R}_{\infty }\ln {\alpha }^{-1\mathrm{}}=34\mathrm{}$ MHz places the theoretical value for the transition frequency $\nu$ approximately 1 standard deviation above the most recent experimental value. The calculation is performed using a perturbation theory based on the Bethe-Salpeter equation and on a wave function obtained by a single iteration from the nonrelativistic Pauli wave function. After a study of all relevant terms, we conclude that the only diagrams which contribute to our order are those involving the exchange of one or two virtual transverse photons. These contributions are explicitly evaluated, and we show that other possible contributions actually vanish.

A general computational procedure is described for locating electron-atom scattering resonances and for computing accurate values of the resonance parameters (eigenchannel vector, energy, width, and background eigenphase). The procedure makes use of the anomaly-free multichannel variational method and of the hierarchical continuum Bethe-Goldstone formalism published previously. Eigenvalues of the electronic Hamiltonian in a variational Hilbert space are located by an accurate procedure that takes advantage of analytic properties of the matrices constructed in a variational phase shift calculation. Without computing the Hilbert space eigenvector, effective bound-free matrix elements for a resonance are computed from residues of matrix elements that have poles at the Hilbert space eigenvalues. Each eigenvalue associated with a resonance defines an initial approximation to the resonance parameters, which are then refined by an iterative process that uses variational calculations of the eigen-phases. The proposed method combines the ability of stabilization methods to locate narrow resonances with an accurate computation of phase shifts in the resonant region. Calculations of electron-hydrogen resonances are given as examples of the method.

The method of extrapolation from complex energies is used to calculated $s$-wave electron-hydrogen-atom elastic scattering from the ground state at energies in the region ranging from 1.21 to 2.25 Ry.

Total cross sections have been measured for electron transfer from ${\mathrm{O}}^{-}$ to ${\mathrm{Na}}^{+}$, resulting in optically allowed transitions to $\mathrm{Na}\left({}_{}{}^2{}_{}{}^{}S\right)$ ground state. A merged-beam technique was used to study the collisions in the energy range 0.1-7 eV. Structure in the curves of cross sections vs relative energy indicates that the excitation of oxygen plays an important role in the charge-transfer process. Cross sections for an endoergic process, giving $\mathrm{Na}\mathrm{}\left(4p\right)\mathrm{}$, are of order ${10}^{-14\mathrm{}}$ ${\mathrm{cm}}^2$; cross sections for the exoergic processes leading to production of $\mathrm{Na}\mathrm{}\left(3d\right)\mathrm{}$ and $\mathrm{Na}\mathrm{}\left(3p\right)\mathrm{}$ are of order ${10}^{-13\mathrm{}}$ ${\mathrm{cm}}^2$. Production of $\mathrm{Na}\mathrm{}\left(3d\right)\mathrm{}$ is nearly a resonant process; production of $\mathrm{Na}\mathrm{}\left(3p\right)\mathrm{}$ is apparently due to a cascade from $\mathrm{Na}\mathrm{}\left(3d\right)\mathrm{}$.

A discussion of relativistic-pair-production cross sections is given for incident-photon energies in the range 1.073-2.615 MeV. Cross sections are obtained numerically for a screened potential, using an exact partial-wave formulation. Comparisons are made with previous authors' results. For low photon energies the effect of atomic-electron screening on pari-production cross sections is significant. The screening increases the cross section, since the atomic electrons decrease the nuclear Coulomb repulsion of the positron. Comparisons of screened with unscreened results show that angular distributions are essentially unaffected in shape by the screening; cross sections are simply renormalized by replacing ${\left|N_{+}{N}_{-}\right|}^2$ in the point-Coulomb field by ${\left|N_{+}{N}_{-}\right|}^2$ in the screened field, where $N_{\pm }$ are the normalizations of the positron and electron wave functions at the origin.