The analytical method developed by the authors for the determination of the expectation value of single-particle operators $W={\Sigma }_i{W(\overrightarrow{\mathrm{r}}}_i)$ correct to second order is employed in both the first and second decoupling approximations to obtain analytic expressions for the coherent x-ray scattering factor $F\left(\overrightarrow{\mathrm{k}}\right)$ for the ground state of the helium isoelectronic sequence valid for all values of momentum transfer. The trial wave function ${\psi }_{0T}$ employed is an energy-minimized Hartree product of hydrogenic states. For helium the results for the form factor in the second decoupling approximation are found to be superior to those of the first, and are within 1.2% of the highly accurate values calculated using a 120-parameter configuration-interaction wave function and have an accuracy equivalent to that of an analytical Hartree-Fock treatment. This error is further reduced as the atomic number is increased. In order to demonstrate the internal self-consistency of the technique, we prove that the expectation value of any single-particle operator as obtained by direct use of the analytical method is the same as the expectation value obtained employing the form factor provided that the latter has also been calculated by the analytical method employing the same trial wave function. Finally, we extend our calculations to the infinite-momentum-transfer range and study our results in this limit by employing a cusp condition for the exact ground-state wave function of two-electron atomic systems written in terms of the logarithmic derivative of the electron density at the origin. We observe that in the infinite-momentum-transfer limit our results for helium are in error by 0.57% and that this error is further diminished for each successive element of the isoelectronic sequence. In addition, we note that the cusp condition is exactly satisfied via the formalism of the second decoupling approximation and is independent of the variational parameter employed.

X-ray scattering intensities from unoriented samples of cholesteryl nonanoate and myristate are reported for several temperatures in the smectic $A$, cholesteric, and isotropic liquid phases. The measured Bragg-scattering intensities from the smectic planes are used to test a recent theoretical model of the smectic $A$ phase. Strong pretransition scattering (short-range-order or order-parameter fluctuations) are observed in the cholesteric phase and a Landau theory is constructed to describe this effect.

Spontaneous-radiation processes associated with a number of multiatom systems are studied. Green's-function techniques are used in conjunction with a model of $N$ two-level atoms interacting with a quantized radiation field to investigate the assumption of the independence of the spontaneous-radiation properties of a given atom from the states of the other atoms in the system. It is shown that the natural linewidth of an excited atom in the presence of a deexcited atom is different from that of an isolated excited atom. For interatomic separations smaller than a critical separation, the two-atom system is best considered as a collective unit with regard to its spontaneous-radiation properties. The influence of these "radiative cooperative effects" is studied also for the systems of two initially excited atoms. Also the influence of radiative cooperative effects in the scattering of a photon by a system of two deexcited atoms is studied. It is shown that the frequency distribution of the scattered radiation exhibits a double-peak structure for interatomic separations smaller than $\frac{3}{4}\lambda$. The techniques of Green's functions are then applied to the study of radiative cooperative effects in several many-atom systems. It is shown that within the context of the model certain information is obtained exactly via the Green's-function techniques as applied to the manyatom problems of a single excited atom in the presence of $N\gg 1$ deexcited atoms and the scattering of a photon by a system of $N\gg 1$ deexcited atoms. The necessity of treating manyatom systems from a "collective" point of view is easily seen from our results.

The Brueckner-Goldstone many-body perturbation theory is applied to study the electronic structure of the atomic nitrogen in its ground ${}_{}{}^4{}_{}{}^{}S$ state. A complete orthonormal set of the $V^{N-1\mathrm{}}$ single-particle states with angular symmetry up to $l=5\mathrm{}$ is explicitly used to calculate the correlation-energy diagrams. The contribution from higher $l(>5\mathrm{})\mathrm{}$ is estimated by a hydrogenic approximation of the continuum wave functions. Our final correlation energy is -0.1895 ± 0.003 a.u., as compared with -0.1886 ± 0.0094 a.u. from the semiempirical estimate made by Veillard and Clementi. The sum of the exchange-core-polarization and pair-correlation-energy diagrams give a total of -0.207 28 a.u., which is 110% of the semiempirical estimate. This contains a contribution of -0.006 10 a.u. (3.2%) from $l>\mathrm{}3\mathrm{}$ states and -0.058 25 a.u. (31%) from the excitations into the valence $2p$ orbitals, the latter arising from the open-shell nature of the nitrogen atom. The overestimate (10%) of the correlation energy is remedied by the pair-pair correlation of 0.011 37 a.u. (6%) and the many-electron (three and four) effect of 0.006 42 a.u. (3.4%). A comparison with earlier configuration-interaction results is also made.

The linked-cluster many-body perturbation theory has been applied to the calculation of the correlation energies of the molecular systems ${\mathrm{H}}_2$O, N${\mathrm{H}}_3$, and C${\mathrm{H}}_4$. The Hartree-Fock and correlation energies are obtained separately by combining the contributions from the pertinent diagrams. The calculated Hartree-Fock energies are in satisfactory agreement with the best previous theoretical results. The correlation energies (in a. u.) that we have obtained are -0.338, -0.317, and -0.312 for ${\mathrm{H}}_2$O, N${\mathrm{H}}_3$, and C${\mathrm{H}}_4$, respectively. These latter energies are utilized for two purposes. One is to combine them with the total Hartree-Fock energies to obtain the total energies to compare with experiment. Secondly, they are subtracted from the "experimental" total energies to obtain reference Hartree-Fock energies with which to compare the energies from theoretical one-electron calculations.

A trapped-electron technique is used to determine the integral cross sections for production of secondary electrons in the energy range from 0 to 1 eV resulting from electron-impact ionization of He, Ne, and Ar. For incident electron energies above 150 eV the slow-electron production cross section decreases approximately as the total-ionization cross section. A simple Born calculation for the case of atomic hydrogen also shows this general behavior. The measured absolute cross sections compare well with recently measured relative angular distributions and with Born and Born-exchange calculations.

The electron-impact excitation cross sections of the $a{}_{}{}^1{}_{}{}^{}\Pi _g^{}{}_{}{}^{}$, $c^{\prime }{}_{}{}^1{}_{}{}^{}\Sigma _u^{+}{}_{}{}^{}$, $a^{\prime \prime }{}_{}{}^1{}_{}{}^{}\Sigma _g^{+}{}_{}{}^{}$, $w{}_{}{}^1{}_{}{}^{}\Delta _u^{}{}_{}{}^{}$, $b^{\prime }{}_{}{}^1{}_{}{}^{}\Sigma _u^{+}{}_{}{}^{}$, $b{}_{}{}^1{}_{}{}^{}\Pi _u^{}{}_{}{}^{}$, $A{}_{}{}^3{}_{}{}^{}\Sigma _u^{+}{}_{}{}^{}$, $B{}_{}{}^3{}_{}{}^{}\Pi _g^{}{}_{}{}^{}$, $C{}_{}{}^3{}_{}{}^{}\Pi _u^{}{}_{}{}^{}$, $D{}_{}{}^3{}_{}{}^{}\Sigma _u^{+}{}_{}{}^{}$, $W{}_{}{}^3{}_{}{}^{}\Delta _u^{}{}_{}{}^{}$, and $E{}_{}{}^3{}_{}{}^{}\Sigma _g^{+}{}_{}{}^{}$ states of ${\mathrm{N}}_2$ have been calculated over the range of 0-2000 eV for the singlet states and 0-40 eV for the triplet states by means of the Born approximation with Ochkur's and Rudge's scheme for treating the electron-exchange-scattering amplitude. The Franck-Condon-factor approximation was used to obtain the excitation cross sections to each vibrational level of the electronic states. The computation of the scattering-amplitude integrals was greatly facilitated by expressing the molecular wave functions in terms of atomic Gaussian-type orbitals. Four sets of self-consistent-field molecular wave functions have been employed for the calculations in order to test the sensitivity of the calculated cross sections to the choice of the wave functions. With the exception of the $c^{\prime }{}_{}{}^1{}_{}{}^{}\Sigma _u^{+}{}_{}{}^{}$ state, the cross sections based on three of the four sets vary by typically about 15%. As a test of the Born cross sections for singlet-singlet excitation, comparison between the theoretical and the available experimental values shows 25% agreement for the $a{}_{}{}^1{}_{}{}^{}\Pi _g^{}{}_{}{}^{}$ state at 900 eV and 50% for the $a^{\prime \prime }{}_{}{}^1{}_{}{}^{}\Sigma _g^{+}{}_{}{}^{}$ state at 80 eV. In the case of triplet excitation, the theoretical cross sections of the $C{}_{}{}^3{}_{}{}^{}\Pi _u^{}{}_{}{}^{}$ state agree very well with the experimental data, but for the $A{}_{}{}^3{}_{}{}^{}\Sigma _u^{+}{}_{}{}^{}$ and $B{}_{}{}^3{}_{}{}^{}\Pi _g^{}{}_{}{}^{}$ states the discrepancy is generally as large as a factor of 2.

A new cascade-decoupling technique has been used to measure hyperfine intervals (in MHz) of the following excited $S$ states of alkali-metal atoms: ${\mathrm{K}}^{39}\left(6\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\right)46\mathrm{}\pm 2$; ${\mathrm{K}}^{41}\left(6\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\right)26\mathrm{}\pm 2$; ${\mathrm{Rb}}^{85}\left(7\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\right)270\mathrm{}\pm 10$; ${\mathrm{Rb}}^{85}\left(8\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\right)135\mathrm{}\pm 15$; ${\mathrm{Rb}}^{87}\left(7\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\right)565\mathrm{}\pm 60$; ${\mathrm{Rb}}^{87}\left(8\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\right)290\mathrm{}\pm 20$; ${\mathrm{Cs}}^{133}\left(8\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\right)680\mathrm{}\pm 120$; and ${\mathrm{Cs}}^{133}\left(9\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\right)405\mathrm{}\pm 30$.

Intensities and energies of gold $L\alpha$, $L\beta$, and $L\gamma$ composite x-ray peaks have been measured with a Si(Li) detector as a function of proton bombarding energies from 0.5-3.0 MeV. The ratios of $\frac{L\alpha }{L\beta }$ and $\frac{L\alpha }{L\gamma }$ peaks reach a maximum value at $E_p\approx 1$ MeV. An upward shift in the energy of the $L\gamma$ peak of ≈ 100 eV was observed at $E_p=0.5\mathrm{}$ MeV. Both effects (shifts and ratios) are due, at least in part, to the energy dependence of the $L$-subshell hole-production cross sections. The experimental results for the intensity ratios are in better agreement with theory if relativistic $L$-shell wave functions are used in the plane-wave Born-approximation calculations for $L$-subshell hole production.

The projection operators onto various subsets of states of a quantum-mechanical system are constructed in a semiclassical approximation based on the Wigner transformation of statistical mechanics. As illustrations, explicit operator expressions are derived for the cases of central Coulomb potential, one-dimensional harmonic oscillator, and radial Coulombic states of specified angular momenta. The accuracy of these operators is then examined in some detail in terms of the overlap integrals and dipole transition probabilities. The semiclassical approximation is found to be effective in the energy regions away from the classical turning points. Extensions of the approach to partially projected Green's functions and other related moments are discussed and their applications to scattering problems pointed out.

A new measurement of the Lamb shift in the $n=3\mathrm{}$ state of atomic hydrogen is reported. The experiment was performed in zero magnetic field with a fast hydrogen beam. The radio-frequency spectroscopy measurements were made with both a single oscillating field and two separated oscillating fields. The final result for the Lamb shift was $\mathcal{S}(\mathrm{H}, n=3)=(314.819\mathrm{}\pm 0.048)$ MHz. In a further experiment, the separated-oscillating-field technique was used to narrow the resonance so that the linewidth was less than the natural linewidth.

The hyperfine-structure widths of $1snl-1sn{l}^{\prime }$ transitions in heliumlike ions are estimated for $n=2, 3 \mathrm{and} Z$ up to 30. Relevant parameters such as spin-orbit and electrostatic parameters are also calculated on the basis of screened hydrogenic functions. It is found that the hyperfine-structure widths are greater than 10 Å at accessible visible wavelengths in some cases. Also, absolute wavelength shifts due to Lamb and nuclear-volume effects are of the order of several angstroms.

Spin-free 1 matrices $\gamma \left(\overrightarrow{\mathrm{r}}\right|{\overrightarrow{\mathrm{r}}}^{\prime })\text{'}\mathrm{s}$ are obtained in algebraic form from simple Hylleraas-type (HY) wave functions for the ${}_{}{}^1{}_{}{}^{}S$ ground states of He and several He-like ions. The algebraic $\gamma \left(\overrightarrow{\mathrm{r}}\right|{\overrightarrow{\mathrm{r}}}^{\prime })\text{'}\mathrm{s}$ are then Fourier transformed to momentum space in order to obtain radial momentum distributions $I_0\mathrm{}\left(p\right)\text{'}\mathrm{s}$. The analytical method developed here allows $I_0\mathrm{}\left(p\right)\text{'}\mathrm{s}$. to be calculated in a direct manner from HY wave functions. Although the present emphasis is on the method rather than on the "goodness" of individual $I_0\mathrm{}\left(p\right)\text{'}\mathrm{s}$, results for the ${}_{}{}^1{}_{}{}^{}S$ He and ${\mathrm{Li}}^{+}$ ground states are obtained from six-term HY functions which incorporate 99.1 and 98.0%, respectively, of the correlation energy.

Variational equations are derived for the optimum orbitals for constructing a multiconfiguration expansion of the wave function for a many-electron system. Both the case where all Slater determinants are constructed from a common set of orbitals and the case where each determinant is constructed from an independent set of orbitals are considered. An equation with a single orbital operator is obtained in the former case; a separate operator for each determinant in the latter. The latter case corresponds to a physical picture in which the orbitals fluctuate as the system makes virtual transitions from one Slater determinant to another. The invariance properties of the wave function and of the variational equations with respect to linear transformations of the orbitals are analyzed, and various procedures are given for obtaining equations which select a unique set of orbitals. A prescription of Adams can be used in either case to localize the orbitals at a selected physical subsystem (an atomic shell, an atom, or a group of atoms). Another prescription can be used with a common set of orbitals to obtain an equation for natural spin orbitals.

Using the beam-foil technique we have observed radiative transitions between doubly excited states of He i. From previous calculations, we have been able to classify many of these transitions. Some of the lifetimes of the upper states were obtained from decay-time measurements. There is good agreement between theory and experiment for both wavelengths and radiative lifetimes. A transition between doubly excited states in Li ii has also been observed, and we compare experiment and theory for higher members of the He i isoelectronic sequence.

We apply laser photodetachment and photoelectron spectrometry for the first time to the study of molecular negative ions. We describe in detail the study of the nitric oxide in N${\mathrm{O}}^{-}$; a following paper reports results for ${\mathrm{O}}_2^{-}$. We use the N${\mathrm{O}}^{-}$ results to develop and illustrate in detail the principles and applications of the technique. A mass-selected N${\mathrm{O}}^{-}$ beam (680 eV) is crossed with a linearly polarized monochromatic (4880-Å) argon-ion laser beam, and electrons photodetached into a $\frac{4\pi }{2000}$-sr solid angle perpendicular to the crossed beams are energy analyzed using a hemispherical electrostatic monochromator. The data yield a set of vertical detachment energies between vibrational states of N${\mathrm{O}}^{-}$ and NO, and relative intensities for these transitions. Angular distributions about the polarization direction are studied by rotating the laser polarization while maintaining the mutually perpendicular ion-beam-laser-beam-electron-collection geometry. For each transition we measure the anisotropy parameter $\beta$, corresponding to the form [$1+\beta P_2\left(cos\theta \right)$] for the angular distribution. Several arguments, including data on N${\mathrm{O}}^{18-}$ photodetachment, are used to identify the initial and final vibrational states. Molecular rotational effects, and effects associated with the spin-orbit splitting of the final $\mathrm{NO}\left(X{}_{}{}^2{}_{}{}^{}\Pi \right)$ state are identified and included in the analysis. A Franck-Condon-factor analysis of the observed relative cross sections, parametrized by trial values of the N${\mathrm{O}}^{-}$ molecular constants, is used to determine that for N${\mathrm{O}}^{-}$ ${\omega }_e^{\prime \prime }=1470\mathrm{}\pm 200$ ${\mathrm{cm}}^{-1\mathrm{}}$, ${\mathcal{r}}_e^{\prime \prime }=1.258\mathrm{}\pm 0.010$ Å, and $B_e^{\prime \prime }=1.427\mathrm{}\pm 0.02$ ${\mathrm{cm}}^{-1\mathrm{}}$. Using these constants, the measured vertical detachment energy is reduced by rotational and spin-orbit effects to the electron affinity $E_A\left(\mathrm{NO}\right)={24}_{-5\mathrm{}}^{+10\mathrm{}}$ meV.

A beam of ${\mathrm{O}}_2^{-}$ ions, extracted from a glow discharge in ${\mathrm{N}}_2$O, is crossed with the linearly polarized intracavity photon beam of an argon-ion laser (4880 Å). Electrons photodetached at right angles to the crossed beams are energy filtered by a hemispherical analyzer. The electron energy spectra are characteristic of photodetachment from the $v^{\prime }=0\mathrm{}$ state of ${\mathrm{O}}_2^{-}$ to the $X{}_{}{}^3{}_{}{}^{}\Sigma _g^{-}{}_{}{}^{}$ and $a{}_{}{}^1{}_{}{}^{}\Delta _g^{}{}_{}{}^{}$ states of ${\mathrm{O}}_2$. Vibrational state analysis is facilitated by the use of isotopes. The electron affinity obtained is 0.440 ± 0.008 eV. Additionally, we have measured the relative transition probabilities as a function of final vibrational state and the angular distributions of the outgoing electrons. The relative intensities, corrected by the angular distributions, determine through Franck-Condon-factor analysis the internuclear distance for the negative ion. We find ${\mathcal{r}}_e^{\prime \prime }=1.341\mathrm{}\pm 0.010$ Å and therefore $B_e^{\prime \prime }=1.17\mathrm{}\pm 0.02$ ${\mathrm{cm}}^{-1\mathrm{}}$.

Corrections to the interaction integrals of valence electrons, which are due to the interaction of configurations for which a single electron is excited from the core, are calculated using a perturbed-function approach. Numerical results are given for ${\mathrm{Pr}}^{3+}$, ${\mathrm{Dy}}^{2+}$, ${\mathrm{Er}}^{3+}$, and ${\mathrm{Pu}}^{3+}$.

It is well known that the Glauber approximation for scattering amplitudes is obtained by applying the eikonal approximation to the Fourier transform of the transition operator. The eikonal approximation can also be applied to the coupled equations of scattering obtained by the expansion of the state function in terms of a suitable set of functions. The scattering amplitude can thus be obtained by solving the set of eikonal coupled equations. The latter approach is analyzed for a special class of channel-coupling potentials. The first-order approximation to the derived eikonal coupled equations is the eikonal Born approximation. Numerical illustrations in this approximation are given for the $1s-2s$ and $1s-2p$ excitations of hydrogen atoms by electron and positron impact. The results are compared with those obtained in the Glauber eikonal approximation and with experimental measurements.

High-energy inner-shell ionization processes generate a spectrum of lower-energy shake-up lines and shake-off band satellites of a main electron energy peak, corresponding to excitations and multiple ionizations of the daughter ion. Such satellite final states give reduced screening for the ejected inner electron, and consequently the observed rise in binding energy. Experimentally we here verify the conventional view that the effective binding energy corresponding to the main peak is independent of transition energy in such fast ionizations; four transitions in ${}_{95}{}^{241}{}_{}{}^{}\mathrm{Am}$, one only 7 keV above the $K$ binding energy of 125 keV, gave ($L_1-K$) internal-conversion electron energy differences of ∼101 keV, constant within 5 eV, over an electron energy range from 7 keV ($\frac{v}{c}=0.14\mathrm{}$, nearly adiabatic) to 450 keV ($\frac{v}{c}=0.85\mathrm{}$). This implied invariance of the $K$ mainline binding energy within 5 eV is contrasted to a calculated rearrangement energy value of 88 eV from a Dirac-Hartree-Fock (DHF) "frozen-orbital" eigenvalue minus the full atom-ion DHF "adiabatic" energy difference. Further, the main peak binding energy should be the threshold, adiabatic value; comparison of our ($L_2-K$) electron energy difference, 102.031±0.005 keV, with the $K{\alpha }_2$ x-ray energy, 102.033±0.010 keV, verifies this to 12 eV. The displacement of the centroid of the satellite spectrum from the main peak, i.e., the mean increase in binding energy averaged over this spectrum as the transition energy increases infinitely above threshold, should equal the rearrangement energy of the ion. Based on $\beta$-decay shake-off intensity predictions for all shells of Carlson, Nestor, Tucker, and Malik, and on our experimental results for $K$ and $L$ shells at lower $Z$, we calculate a centroid shift of 89±10 eV for Am.