We present the supersymmetric formulation of quaternionic quantum mechanics. We show that most of the results of supersymmetric quantum mechanics over the complex numbers go through, but unlike complex quantum mechanics, there is no general expression for the ground state in terms of the superpotential.

We consider the transition rate β(n,l,T,Z) for a hydrogenlike state characterized by quantum numbers n and l, and nuclear charge Z, induced by a Planck blackbody distribution at temperature T. We expand the Planck distribution in a power series in |${\mathrm{E}}_{\mathrm{n}}$-${\mathrm{E}}_{\mathrm{n}\prime }$|/${\mathrm{k}}_{\mathrm{B}}$T, where n′ defines one of the set of states representing a transition and where the energies ${\mathrm{E}}_{\mathrm{n}}$ and ${\mathrm{E}}_{\mathrm{n}\prime }$ are hydrogenic. We take into account transitions to both bound and continuum states by using sum rules (some not previously reported) for the completeness of hydrogenlike states. The method is not applicable to states of low l and is not accurate for low T. However, for moderate (and high) values of l and T, we obtain analytic upper and lower bounds on β which provide rather accurate estimates of β. The method is quick and convenient as opposed to the straightforward numerical calculation which requires, for contributions of transitions to the continuum, double integration of hypergeometric functions over the Planck distribution. A semiclassical approach is also discussed. Some illustrative numerical results are provided.

The logarthmic derivative and the second derivative of the spherically averaged charge density ρ(r), denoted as ρ’(r)/ρ(r) and ρ’’(r), respectively, are numerically studied in a Hartree-Fock framework for all ground-state atoms from hydrogen (Z=1) through uranium (Z=92). It is observed that (i) the logarithmic derivative of ρ(r) always attains its absolute minimum at the nucleus for Z=1–92, which extends a previous result for Z≤54, (ii) the second derivative of ρ(r) presents pairs of local minima and maxima, the number of which never decreases with increasing nuclear charge, and (iii) the occurrence of new local maxima and minima in ρ’’(r) always corresponds to the addition of an electron in a new subshell. The regularity in the behavior of the local characteristics of ρ’’(r) not only suggests an alternative way of studying the atomic shell structure by means of the one-particle density, but also provides further evidence of the hierarchical arrangement of the atomic charge density.

The great majority of measured values of the Coster-Kronig probability ${\mathit{f}}_{23}$ derive from x-ray coincidence experiments using radionuclide sources or from synchrotron-radiation excitation of thin foils. Very little attention has been paid in published work to the role of line-shape effects and satellites in the x-ray spectra that are obtained. It is shown that this neglect can influence ${\mathit{f}}_{23}$ values derived from such measurements. This may explain in part the well-established trends for measured ${\mathit{f}}_{23}$ values to fall below the theoretical predictions for elements with atomic number Z<90 and to fall above when Z>92. In elemental analysis techniques based upon x-ray emission spectroscopy, where a database of atomic parameters such as Coster-Kronig probabilities is used, the same approximations are made in treating the spectra. Until a more sophisticated approach to spectrum reduction is developed, it may be preferable in this context to use ‘‘nominal’’ Coster-Kronig probabilities derived from experiment in preference to theoretical values.

The A ${}_{}{}^2{}_{}{}^{}\mathrm{\Sigma }_{1/2}^{+}{}_{}{}^{}$ state of ${\mathrm{HBr}}^{+}$ has been studied experimentally using He i photoelectron spectroscopy and theoretically employing an ab initio multireference configuration-interaction method. Vibrational structure has been observed throughout the band up to the dissociation limit. The vibrational lines corresponding to v=3–9 are much broadened due to strong interaction with repulsive states as found in earlier studies. In recordings with a resolution of about 4–5 meV, rotational structure has been observed for the v=0, 1, 2, 11, 12, and 13 vibrational states. The spectra have been fitted using calculated rotational constants. It is found that the ΔN=±1 branches dominate the photoelectron bands but also the ΔN=±3 transitions are significant. For the lowest states the T branch (ΔN=3) forms a well-separated band.

We consider 51 odd-parity configurations of Pb i utilizing all data available from absorption and emission spectroscopy measurements, photoionization experiments, and ejected-electron spectroscopy. These include detailed results of our earlier laser spectroscopic studies of the 6${\mathit{s}}^2$6pns, 6${\mathit{s}}^2$6pnd, and 6${\mathit{s}}^2$6png Rydberg levels [P. Buch, J. Nellessen, and W. Ertmer, Phys. Scr. 38, 664 (1988)]. The analysis has been performed with the aid of a linked-parameter method of level-fitting calculations in a large multiconfiguration basis. New identifications of the 6s6${\mathit{p}}^3$ levels are proposed and reasons for their assignment are presented. On the basis of our level-fitting calculations precise assignments of 242 odd-parity electronic energy levels are given. Eigenvectors obtained in these calculations can be used to calculate the values of other physical observables with high precision. In addition, we have also made appropriate pseudorelativistic Hartree-Fock ab initio calculations of level structure, oscillator strengths, and autoionization widths for procedural guidance and comparison purposes.

The dual-laser-produced plasma (DLP) photoabsorption technique has been used to study 2p→3s excitations in the isoelectronic species Mg, ${\mathrm{Al}}^{+}$, and ${\mathrm{Si}}^{2+}$ prepared in the excited configuration 2${\mathit{p}}^6$3s3p. The autoionizing upper states belong to the 2${\mathit{p}}^5$3${\mathit{s}}^2$3p even-parity configuration. The versatility of the technique is demonstrated through a careful combination of space- and time-resolved photoabsorption scans. Plasma conditions optimized for the observation of the inaccessible parity regime were successfully reproduced along the isoelectronic sequence of interest. All the observed transitions were interpreted with the help of multiconfigurational atomic structure calculations. In the case of magnesium, the photoabsorption data are compared with the ejected-electron spectra excited by low-energy electron impact of Pejcev et al. [J. Phys. B 10, 2389 (1977)].

Isotope shifts and hyperfine structures of four transitions in W i have been measured as well as isotope shifts in Lu i by means of atomic-beam laser spectroscopy. Magnetic-dipole hyperfine constants A of ${}_{}{}^{183}{}_{}{}^{}\mathrm{W}$ are determined for the atomic states ${}_{}{}^7{}_{}{}^{}$${\mathit{D}}_1$, ${}_{}{}^7{}_{}{}^{}$${\mathit{F}}_1$, and ${}_{}{}^7{}_{}{}^{}$${\mathit{F}}_3$ of the 5${\mathit{d}}^4$6s6p configuration. The nuclear parameters λ and δ〈${\mathit{r}}^2$〉 are derived for W stable isotopes. J dependences of the isotope shifts are observed for the ${}_{}{}^2{}_{}{}^{}$D term of the 5d6${\mathit{s}}^2$ configuration in Lu i and the ${}_{}{}^5{}_{}{}^{}$D term of the 5${\mathit{d}}^4$6${\mathit{s}}^2$ configuration in W i. Parameters ${\mathit{z}}_{5\mathit{d}}$ of the crossed-second-order effects are derived for the Lu i 5d6${\mathit{s}}^2$ and W i 5${\mathit{d}}^4$6${\mathit{s}}^2$ configurations, and ${\mathit{z}}_{5\mathit{d}}$/λ is found to be 411(26) MHz/${\mathrm{fm}}^2$ for W. A systematic behavior of the normalized parameter ${\mathit{z}}_{\mathit{n}\mathit{l}}$/λ${\mathrm{\zeta }}_{\mathit{n}\mathit{l}}$ is discussed.

Angular correlation parameters (${\mathit{A}}_{22}$) between the K${\mathrm{\alpha }}_1$ and ${\mathit{L}}_3$ x rays were measured for gadolinium in the electron-capture decay of the isotope ${}_{}{}^{157}{}_{}{}^{}\mathrm{Tb}$. The results are ${\mathit{A}}_{22}$(K${\mathrm{\alpha }}_{1\mathrm{-}}$Ll)=0.272±0.042, ${\mathit{A}}_{22}$(K${\mathrm{\alpha }}_{1\mathrm{-}}$Lα)=0.032±0.015, ${\mathit{A}}_{22}$(K${\mathrm{\alpha }}_{1\mathrm{-}}$L${\mathrm{\beta }}_{2,15}$)=0.061±0.018. These values are in very good agreement with the theoretical calculations of Scofield [Lawrence Livermore Laboratory Report No. UCRL 51231, 1972 (unpublished)]; however, they differ from the earlier experimental data for the K${\mathrm{\alpha }}_{1\mathrm{-}}$Ll cascade in the 60<Z<70 atomic-number region.

Pump-and-probe methods of laser spectroscopy, used together with a jet-expansion molecular beam, were used to record A$0^{+}$(5 ${}_{}{}^3{}_{}{}^{}$${\mathit{P}}_1$)←X$0^{+}$(5 ${}_{}{}^1{}_{}{}^{}$${\mathit{S}}_0$) and D1(5 ${}_{}{}^1{}_{}{}^{}$${\mathit{P}}_1$)←X$0^{+}$(5 ${}_{}{}^1{}_{}{}^{}$${\mathit{S}}_0$) excitation spectra of CdNe and CdAr van der Waals molecules. The spectra were subjected to vibrational analysis and the assignments of the components were made with the aid of a computer-simulated spectrum based on Franck-Condon factors calculated assuming the Morse potential as representative of the lower and upper electronic states. The analysis yielded spectroscopic constants for the X$0^{+}$, A$0^{+}$, and D1 states of CdNe and CdAr. Considerations of the linear dependence of the dissociation energy ${\mathit{D}}_{\mathit{e}}$ of CdRG molecules on rare-gas (RG) polarizability led to the development of an empirical calculation of ${\mathit{D}}_{\mathit{e}}$ and bond lengths for various states in metal–rare-gas van der Waals complexes; the calculations also yielded polarizabilities of some ground-state and excited metal atoms.

Theoretical treatment of resonance radiation trapping under steady-state conditions, particularly of quenching experiments, is presented. Applying operator manipulations, Holstein’s equation of radiation trapping [Phys. Rev. 72, 1212 (1947)] is solved by a perturbation expansion, which can be physically interpreted in terms of multiple scatterings. The resulting expression shows that in a steady-state quenching experiment radiation trapping invalidates the relation of the Stern-Volmer type except when the amount of the buffer gas is small. In the limit of low buffer-gas pressure, a simple expression is given for the effective lifetime in the Stern-Volmer relation: the average time of the detected resonant photons remaining within the enclosure. While it is true that this suggested approach is perturbative in nature, the asymptotic behavior of the high-order expansion coefficients, however, enables us to reduce the summation over infinitely many alternatives to some leading terms. Such a reduction procedure helps curtail the computation time and improve the precision. Finally, some numerical illustrations of our suggested ideas are presented, which shows this approach is a good tool to treat the steady-state trapping effect.

A generalized coupling operator method is employed to construct a single Fock operator for matrix Dirac-Fock self-consistent-field calculations on general open-shell multiplet states of atoms. The approach handles any number of open electronic shells. The matrix Dirac-Fock procedure is implemented with analytic basis sets of Gaussian-type functions for expansion of the large and small components of the Dirac four-spinors. Open-shell Dirac-Fock calculations in the single-Fock-operator formalism are performed on the ground and low-lying excited states of Li, B, Na, Al, K, Ga, and In.

I derive an approximate estimate for the number N of states in a vibrational band of a diatomic molecule. The number N may be used to redefine and improve by a factor π a well-known estimate of Born and Oppenheimer [Ann. Phys. (Leipzig) 84, 457 (1927)]. The formulas for N are compared to data for ${\mathrm{H}}_2$ and a few other molecules.

Narrow-band extreme-ultraviolet (xuv) radiation in the range 73.5–75.5 nm has been produced in Xe by generating the third harmonic of light from a frequency-doubled, pulsed blue-dye laser. In a one-xuv-photon plus one-uv-photon ionization process both 2${\mathit{p}}^{6\mathrm{-}}$2${\mathit{p}}^5$3s[3/2${]}_1$ and 2${\mathit{p}}^{6\mathrm{-}}$2${\mathit{p}}^5$3s’[1/2${]}_1$ transitions in ${}_{}{}^{20}{}_{}{}^{}\mathrm{Ne}$ and ${}_{}{}^{22}{}_{}{}^{}\mathrm{Ne}$ were observed. Frequencies were calibrated using a tellurium absorption spectrum of the fundamental blue light. Frequencies of 134 459.33(5) ${\mathrm{cm}}^{\mathrm{-}1}$ for the 2${\mathit{p}}^{6\mathrm{-}}$2${\mathit{p}}^5$3s[3/2${]}_1$ and 135 888.73(5) ${\mathrm{cm}}^{\mathrm{-}1}$ for the 2${\mathit{p}}^6$-2${\mathit{p}}^5$3s’[1/2${]}_1$ transition in ${}_{}{}^{20}{}_{}{}^{}\mathrm{Ne}$ are obtained. In both transitions the isotope shift ${(}^{20}$Ne${\mathrm{-}}^{22}$Ne) was measured. The values are -0.014(3) ${\mathrm{cm}}^{\mathrm{-}1}$ and +0.003(3) ${\mathrm{cm}}^{\mathrm{-}1}$, respectively. Combined with existing values for the 3s[3/2${]}_1$ and 3s’[1/2${]}_1$ level shifts, a value for the ground-state specific mass shift of 9.54(15) GHz is deduced, in good agreement with a theoretical value of 10.1 GHz.

In the work formalism for the determination of electronic structure, the exchange-correlation energy and (local) potential of the electrons both arise via Coulomb’s law from the same source, viz., the quantum-mechanical Fermi-Coulomb hole charge. The potential is the work ${\mathit{W}}_{\mathrm{xc}}$(r) done to move an electron in the field of its Fermi-Coulomb hole and the energy is the interaction energy between the electronic and hole charge densities. For nonsymmetrical electronic density systems for which the curl of the field may not vanish, a local effective exchange-correlation potential ${\mathit{W}}_{\mathrm{xc}}^{\mathrm{eff}}$(r) is determined from the irrotational component of the field, the solenoidal component being neglected. In this paper we investigate this approximation further to better understand its accuracy for nonspherical density systems by application to a degenerate ground state of the carbon atom within the Pauli-correlated approximation. The results of the non-self-consistent calculations indicate that the solenoidal component of the force field due to the Fermi hole is negligible and two orders of magnitude smaller than the irrotational component. Therefore, essentially all the effects of Pauli correlation are accounted for by the latter and the effective exchange potential ${\mathit{W}}_{\mathit{x}}^{\mathrm{eff}}$(r) is thereby an accurate representation of the local exchange potential in the atom. Further, the solenoidal component of the field vanishes at the nucleus, in the classically forbidden region and along certain axes of symmetry. Thus the work ${\mathit{W}}_{\mathit{x}}$(r) in the field of the Fermi hole for such nonspherical atoms is path independent over substantial regions of configuration space. Finally, we discuss the structure of the local many-body potential of nonspherical-density atoms when both Pauli and Coulomb correlations are present.

A model with all radial degrees of freedom frozen is used to simulate the ${}_{}{}^{2\mathit{S}+1\mathrm{}}{}_{}{}^{}$${\mathit{S}}^{\mathit{e}}$ intrashell states of four-valence-electron atoms. By inspecting the two-body densities, the latitudinal distribution functions, and by inspecting directly the eigenfunctions in appropriate subspaces, the features of geometric structure and of internal motion have been obtained. The effect of the spatial permutation symmetry and the parity have been emphasized. All even-parity states under consideration prefer the coplanar axial-symmetric structure. Two modes of motion are found, namely, the triple two-body collision mode and the rectangle-square-rectangle mode (or planar double two-body collision mode). It is found that the basic modes in quantum states are associated with periodic solutions of classical mechanics.

The large error of the local-density approximation for the transverse photon-electron or Breit interaction is analyzed, transforming the Breit interaction into a sum of terms suitable to be handled in a nonrelativistic formalism: the local approximation for each of these terms is built up and an alternative expression for the relativistic exchange energy is proposed. Calculations using this expression are made for atoms ranging throughout the periodic table and the results show a sharp improvement when compared to those obtained by the previous approximation.

The dynamic dipole polarizabilities at real and imaginary frequencies have been determined for the helium atom in its lowest singlet and triplet 2P states, using our time-dependent gauge-invariant method. Contributions of the 2S states to static polarizabilities are strong and negative, particularly in the case of the singlet 2 ${}_{}{}^1{}_{}{}^{}$P state, for which the static scalar polarizability ${\mathrm{\alpha }}_0$ is negative. The tensor polarizability ${\mathrm{\alpha }}_2$ corresponding to the differential Stark shift between the m Zeeman sublevels was also calculated and compared to previous theoretical and experimental results for the singlet 2 ${}_{}{}^1{}_{}{}^{}$P state. An evaluation of the ${\mathit{C}}_6$ dispersion coefficients for the 2 ${}_{}{}^1{}_{}{}^{}$P–2 ${}_{}{}^1{}_{}{}^{}$P and 2 ${}_{}{}^3{}_{}{}^{}$P–2 ${}_{}{}^3{}_{}{}^{}$P systems is also given, derived from the dynamic polarizabilities at imaginary pulsations ω.

An algorithm is formulated for the construction of many-particle permutational symmetry adapted functions in hyperspherical coordinates. A recursive procedure is proposed, introducing hyperspherical coefficients of fractional parentage (hscfps). These coefficients are the eigenvectors of the transposition class sum of the symmetric group in an appropriate basis. Only the matrix element of the transposition of the last two particles has to be calculated in each step. This matrix element is obtained by using the hscfps calculated in the preceding step as well as the Raynal-Revai and the T coefficients. The results are applicable to the study of the atomic, molecular, and nuclear few-body problem.

Measurements were made of the absorption spectrum of hydrogen atoms to levels near the ionization threshold in a strong magnetic field. Taking advantage of a classical scaling law, we varied the photon energy and the magnetic-field strength simultaneously, and measured absorption versus ${\mathit{B}}^{\mathrm{-}1/3}$ at fixed scaled energy, ɛ=E/(B/${\mathit{B}}_{\mathrm{o}}$${)}^{2/3}$. The absorption rate has sinusoidal fluctuations which are correlated with closed classical orbits of the electron. Fourier transformation of this signal gives peaks, which we interpret as ‘‘recurrence strength,’’ as a function of the classical action of the closed orbit. Closed-orbit theory gives formulas for these recurrence strengths. We find that the formulas are in good agreement with the measurements. As the scaled energy is increased, observed recurrences proliferate, consistent with a change from orderly to chaotic motion of the electron. Bifurcation theory provides organizing principles for understanding this proliferation and for interpreting the data. New ‘‘exotic’’ orbits suddenly appear through saddle-node bifurcations. The ‘‘main sequence’’ of orbits is produced from an orbit parallel to B through a sequence of pitchfork and period-doubling bifurcations. Other recurrences are created by period-tripling and higher-order bifurcations of existing orbits. These bifurcations can have ‘‘generic’’ structure, or sometimes the structures are modified by symmetries of the system. Focusing effects associated with these bifurcations cause some recurrences to be particularly strong.