Rates are calculated for the decay of metastable $2s_{\frac{1}{2}}$ ions to the ground state by the simultaneous emission of two photons. The calculation includes all relativistic and retardation effects, and all combinations of photon multipoles which make significant contributions up to $Z=100\mathrm{}$. Summations over intermediate states are performed by constructing a finite-basis-set representation of the Dirac Green's function. The estimated accuracy of the results is ± 10 ppm for all $Z$ up to 100. The decay rates are about $20 {\left(\alpha Z\right)}^2$% larger than an earlier calculation by Johnson owing to the inclusion of higher-order retardation effects. The general question of gauge invariance in two-photon transitions is discussed.

A simple approximate theory is developed for high-$J$ spin-rotational levels and very-high-resolution spectra of octahedral $X{Y}_6$ ($X$ spin-zero, $Y$ spin-1/2) molecules. The structure and theory of S${\mathrm{F}}_6$ spectra is reintroduced and the properties of rotational energy level and spectral clusters are explained in terms of angular-momentum uncertainty relations and axis tunneling. An analogy between axis tunneling and nuclear-spin tunneling is used to provide a simple quasiparticle picture of hyperfine effects in strong (case-2) rotational clusters. Analogous types of clusters within clusters are discussed using this theoretical model. The possibility emerges for new and very sensitive type of spectra which we label superhyperfine structure. This structure has some remarkable similarities with nuclear magnetic resonance (NMR) spectra, and it might be an even more revealing indicator of internal molecular dynamics than NMR. Model Hamiltonians are represented in all the elementary types of cluster bases and some examples are solved using tableau techniques. Level correlations between case-1 and case-2 clusters are sketched. Nomograms for visualizing laser-saturation-absorption spectra are introduced, and examples of case-1 to case-1 and case-2 to case-1 transitions are given.

Theoretical parameters $a_t$ for the self-consistent-field—$\Xi a(\mathrm{S}\mathrm{C}\mathrm{F}-\Xi a)$ method are derived by considering the Fermi hole, modified from the form used to generate the theoretical $\alpha$ parameters in the $X\alpha$ method, because of including explicit self-interaction in the Hamiltonian, which scales the exchange potential. The $a_t$ for all elements in the Periodic Table are given, to compare with previous compilations. For $Z>\mathrm{}30\mathrm{}$ they are in error due to relativistic effects becoming important. The results from the $\mathrm{S}\mathrm{C}\mathrm{F}-\Xi a$ method with $a_t$ have been compared with those previously calculated for ${\mathrm{Cu}}^{+}$, ${\mathrm{Mn}}^{2+}$, Ne, and Ar for the one-electron energies -${\epsilon }_i$, the total energies $E$, the Fermi contact terms $\chi$, and the core-spin densities.

The $2{}_{}{}^3{}_{}{}^{}P_0^{}{}_{}{}^{}-2{}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}$ fine-structure interval ${\nu }_{01}$ has been measured using the optical-microwave atomic beam magnetic resonance technique. The result, a major improvement upon previous measurements, is ${\nu }_{01}=29\mathrm{}616.864\pm 0.036$ MHz (1.2 ppm). This compares well with the latest theoretical value ${\nu }_{01}=29\mathrm{}616.914\pm 0.043$ MHz (1.4 ppm), and provides an important test of the theory of fine structure of the two-electron atom. Alternatively, our experiment can be regarded as an independent measurement of the fine-structure constant with the result ${\alpha }^{-1\mathrm{}}=137.036\mathrm{}08\pm 0.00013$ (0.94 ppm), which is consistent with the more accurate currently accepted value ${\alpha }^{-1\mathrm{}}=137.035\mathrm{}963\pm 0.000015$ (0.11 ppm).

The $2{}_{}{}^3{}_{}{}^{}P_0^{}{}_{}{}^{}-2{}_{}{}^3{}_{}{}^{}P_2^{}{}_{}{}^{}$ fine-structure interval ${\nu }_{02}$ has been measured using the optical-microwave atomic beam magnetic resonance technique with the result ${\nu }_{02}=31\mathrm{}908.040$ (20) MHz (0.6 ppm). This is in excellent agreement with the sum of previous separate measurements of the $2{}_{}{}^3{}_{}{}^{}P_0^{}{}_{}{}^{}-2{}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}$ and $2{}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}-2{}_{}{}^3{}_{}{}^{}P_2^{}{}_{}{}^{}$ intervals, which gives ${\nu }_{02}=31\mathrm{}908.060$ (36) MHz, and establishes internal consistency for all the measurements. The most accurate comparison with theory is for ${\nu }_{01}$, and we obtain the most accurate experimental value for this by subtracting the earlier measurement of ${\nu }_{12}$ from the present result for ${\nu }_{02}$. The result is ${\nu }_{01}=29\mathrm{}616.844\pm 0.021$ MHz (0.7 ppm) (experiment) compared to ${\nu }_{01}=29\mathrm{}616.914\pm 0.043$ MHz (1.5 ppm) (theory). Alternatively our experiment can be regarded as an independent measurement of the fine-structure constant with the result ${\alpha }^{-1\mathrm{}}=137.036\mathrm{}12$ (11) (0.8 ppm) which is consistent with the more accurate currently accepted value ${\alpha }^{-1\mathrm{}}=137.035\mathrm{}963$ (15) (0.11 ppm).

Optical-double-resonance measurements using ionization detection have been carried out in the $3S_{\frac{1}{2}}-3P_{\frac{1}{2}}-4D$ atomic-sodium system. Asymmetries observed in production of $4D$ atoms from the two components of the Stark-split $3P_{\frac{1}{2}}$ state are found to be controlled by the far, very weak wings of the 17-MHz full-width-at-half-maximum laser line which is used to drive the $3S_{\frac{1}{2}}-3P_{\frac{1}{2}}$ transition at detunings in the range 0-70 GHz. Suppression of the wings with a Fabry-Perot filter causes a pronounced reversal of the asymmetry.

The $L$-shell x-ray production process in cadmium for proton bombardment over the energy range 120 - 420 keV was studied using thin Cd targets (∼ 30 μ g/${\mathrm{cm}}^2$). The x rays were observed with a high-purity Ge low-energy-photon-spectroscopy detector. The total x-ray cross sections were determined at 20-keV intervals in the bombarding energy range. Experimental cross sections were compared to the predictions of the nonrelativistic plane-wave Born approximation and the perturbed stationary-state theory with relativistic and Coulomb deflection corrections (CPSSR). The total $L$-shell x-ray cross sections are in reasonable agreement with the CPSSR theory. The complex $L$ x-ray spectrum was resolved into several components and the relative x-ray cross sections of these components were determined.

The intensity of the resonant third-harmonic generation in ${\mathrm{I}}_2$ vapor is found to decrease with increasing strength of an external magnetic field even when the exciting light is polarized parallel to the magnetic field. This effect is explained with a theory based on the interference between different molecules. The gyromagnetic ratio of the excited state is determined in this work. This technique may also be applied to study the mechanism of the various processes of saturation in resonant wave mixing.

Relativistic Dirac-Fock calculations are presented for the photoelectron angular distribution asymmetry parameter $\beta$ corresponding to $4d$-subshell photoionization in Cd and for the linear polarization $P_L$ of the subsequent fluorescence radiation, for photon energies less than 185 eV. Near threshold, our results for $\beta$ lie lower than previous relativistic Dirac-Slater and nonrelativistic many-body-perturbation-theory (MBPT) calculations. The agreement with very recent measurements by Heinzmann and Schönhense is very good. Our calculations show that $P_L({}_{}{}^2{}_{}{}^{}D_{\frac{5}{2}}^{}{}_{}{}^{}\to {}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{})=-0.059\mathrm{} \mathrm{and} -0.058\mathrm{}$ at incident photon energies $h\nu =40.2\mathrm{} \mathrm{and} 45.6$ eV, respectively. These results lie above the nonrelativistic independent-electron theory absolute maximum of -0.061 for $P_L$, but the agreement with recent measurements lying above this maximum is still not very good. Except for this peaking of $P_L({}_{}{}^2{}_{}{}^{}D_{\frac{5}{2}}^{}{}_{}{}^{}\to {}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{})$ above the nonrelativistic theory maximum for $40.2\le h\nu \le 45.6$ eV, our results at other energies are in general agreement with the nonrelativistic MBPT calculations of Carter and Kelly. We also find that $P_L({}_{}{}^2{}_{}{}^{}D_{\frac{3}{2}}^{}{}_{}{}^{}\to {}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{})=+0.134$ at $h\nu =21.2\mathrm{}$ eV, in excellent agreement with the experimental value of + 0.12±0.04 of Caldwell and Zare.

A Dirac-Hartree-Fock potential has been used to evaluate all zeroth- and first-order effects on the parity-nonconserving optical rotation for the 876-nm and 648-nm transitions in Bi. For the 876-nm transition, the zeroth-order result is -9.6×${10}^{-8\mathrm{}}$ for the ratio $R=\frac{\mathrm{Im}\mathrm{}\left(E1\mathrm{}\right)\mathrm{}}{M1\mathrm{}}$, and the result after addition of the first-order effect is -7.0×${10}^{-8\mathrm{}}$. The ratios for the 648-nm transition are -12.8×${10}^{-8\mathrm{}}$, respectively. To obtain more accurate results, the parity-nonconserving parts of the orbitals were then obtained in a self-consistent procedure by including the weak parity-nonconserving interaction in the one-particle Hamiltonian of the Hartree-Fock equations. Similarly, the electric-dipole functions were modified by taking into account the shielding of the external electric field due to a polarization of the core electrons. Re-evaluation of the first-order diagrams using these functions changes the ratio to $R=-8.3\mathrm{}\times {10}^{-8\mathrm{}}$ for the 876-nm transition and to $R=-11.1\mathrm{}\times {10}^{-8\mathrm{}}$ for the 648-nm transition.

We present a theory of spin polarization of electrons ejected from atoms by two-photon resonant three-photon ionization. Our treatment includes saturation effects and pays proper attention to the finite rise time of the laser pulse and the non-Lorentzian wings of the laser spectrum. Numerical results for the cesium atoms indicate that due to saturation effects the spin polarization is generally less than its value predicted by perturbation theory. The different saturation behavior of the ionization channels causes a saturation dip to develop in the degree of spin polarization.

The photoionization cross section of the metastable $3p {}_{}{}^3{}_{}{}^{}P_{}^o{}_{}{}^{}$ term of neutral Mg was measured at $\lambda =248.6\mathrm{}$ nm, near threshold. The Mg vapor was produced in a pulsed, hollow cathode, discharge tube. The density of metastable Mg atoms was determined by using the hook method. The cross section was determined by a photoelectric measurement of the absorption of light by the Mg vapor. The result is (46±12) Mb. This value is larger than that derived from quantum-defect-theory calculations and from a previous measurement. Use of the larger value in models of the solar chromosphere improves the agreement with the observed solar spectrum.

A new formulation of the moderation and capture of negative mesons by atoms has been developed. The scattering is described by a complex potential with a real part given by a diabatic interaction potential and an imaginary part given by the ionization width of the diabatic state embedded in the electronic continuum. Results are presented for collisions of ${\mu }^{-}$, ${\pi }^{-}$, $K^{-}$, and $p^{-}$ with the hydrogen atom. The differential energy-loss cross sections and stopping powers obtained are large. The energy spectrum of stopped mesons is determined and most captures are found to occur at collision energies below or near the ionization potential. The principal-quantum-number distributions of the initially formed mesic atoms peak close to the orbital giving optimal overlap with the displaced electronic orbital. The angular-momentum distributions are not too different from statistical, except that they tend to cut off at $l$ significantly smaller than $n-1\mathrm{}$ in the case of large $n$. Comparison to a very recent experiment is made in a note added.

The analyticity properties and the asymptotic behavior of the quadratic Zeeman energy shifts $\delta E$ for an atom are analyzed, leading to once-subtracted dispersion relations for the energy shifts. In conjunction with the WKB expression for $\mathrm{Im}\delta E$ in the weak-field limit and perturbation expressions for the susceptibility and the strong-field limit of $\mathrm{Im}\delta E$, these dispersion relations allow us to obtain (i) the large-$n$ behavior of the ratios of the expansion coefficients of the perturbation series, and (ii) the energy shifts for the symmetric ground state of He and ${\mathrm{Li}}^{+}$.

An exact solution of the two-dimensional rigid rotator in a static uniform electric field of arbitrary strength is given. The eigenvalue spectrum is found to consist of the characteristic values of the even-index Mathieu functions of $\pi$ periodicity. At low field strengths $\mathcal{E}$ the double degeneracy of the rotational states $\pm m$ is broken in order ${\mathcal{E}}^{2m}$. The broken degeneracy persists for arbitrary field strength. Results are compared with perturbation theory.

The two-dimensional rigid rotator in a static uniform magnetic field of arbitrary strength is solved exactly. The eigenvalue spectrum is found to consist of the characteristic values of the Mathieu functions of both even and odd index and of both $\pi$ and $2\pi$ periodicities. For low field strengths $B$ the double degeneracy of the rotatinal states $\pm m$ is broken in order $B^m$. For high field strengths the system exhibits a quasi-Landau spectrum with a regenerated double degeneracy.

Dispersive optical bistability with fluctuations is analyzed in the mean-field approximation. It is shown that the phase shift of the transmitted field in the purely dispersive case exhibits hysteresis and bistable behavior in a manner similar to that of the transmitted intensity. The Fokker-Planck equation describing the time evolution of the quasiprobability distribution for the transmitted-field amplitude is obtained. The time-dependent moments of the transmitted light are calculated with the use of a Monte Carlo integration scheme.

It is shown that the paraxial Gaussian beam becomes the complex-source-point spherical wave when all-order corrections are made according to the method of Lax, Louisell, and McKnight. Apparent contradictions between previously published first-order corrections are also discussed.

How much information about a laser field can be reconstructed is considered if multiphoton transitions have been measured but if the laser-flux distribution in the experiment is unknown. It is shown how this reconstruction is done in detail if the relative strengths of the photon replicas of some atomic or molecular transitions are given. The resulting information about the laser field turns out to be the power moments of a weight function which indicates the relative weight with which a particular laser-flux value has contributed to the observed multiphoton transitions. Some analytically soluble examples are presented for the connection between the laser-flux distribution in space and time, the resulting moments of the weight function, and the weight function itself.

The probability distribution $p\left(n\right)\mathrm{}$ for the detection of $n$ photons emitted by a single-mode dye laser in a short time at various excitations is measured, together with the distribution of time intervals between pairs of detected photons. The first measurement yields the probability density $\mathcal{p}\mathrm{}\left(I\right)\mathrm{}$ and the moments of the light intensity $I$, and the second one the two-time intensity-correlation function. Measurements are also performed at three different wavelengths. $\mathcal{p}\mathrm{}\left(I\right)\mathrm{}$ is found to have a two-component structure, and the relative intensity fluctuations $\frac{〈{\left(\Delta I\right)}^2〉}{{〈I〉}^2}$ grow, apparently without limit, as the excitation tends to zero, so that no thermal state is reached. A possible explanation in terms of pumping fluctuations is discussed. The correlation function has the form of a sum of three exponential functions, and the amplitudes and decay constants of the three components are derived. The effective decay-time constant exhibits a thirteenth-power-law dependence on frequency of the laser light.