A study is made of the distance dependence of carrier phase and average frequency of an electromagnetic pulse propagating through a quantum two-level system. Phenomenological differential equations are introduced to describe the distance rate of change of first- and second-moment deviations of the pulse carrier frequency from its original input frequency. In equilibrium the equations predict characteristics of pulse propagation in steady state. For both absorber and amplifier cases, computer plots are presented of the distance dependence of pulse shape, carrier phase, Fourier amplitude, and pulse energy for selected types of homogeneous and inhomogeneous line broadening and for specific input-pulse conditions. Off-resonance formation of a single $2\pi$ hyperbolic secant pulse of self-induced transparency may evolve from input-pulse areas less than $\pi$, as well as slightly greater than $\pi$, accompanied by frequency modulation and pulse breakup. The amplification of pulsed carrier radiation is accompanied by frequency pulling toward resonance. With and without phase modulation, stability conditions for stable pulse propagation are obtained. Either frequency pulling toward or frequency pushing away from resonance is dominant for an absorber, according to the dominance, respectively, of homogeneous or inhomogeneous line broadening. In the slowly-varying-pulse-envelope approximation, a steady-state solution is found for a symmetrically-phase-modulated pulse propagating in an amplifier with scattering losses. The mean carrier frequency for the pulse is displaced off resonance by one linewidth above the resonance transition frequency of a homogeneously broadened system.

The ordered state of a $d$-dimensional isotropic system with an $n$-vector ($n\ge 2$) order parameter is considered. By the imposition of suitable boundary conditions it is shown how to define explicitly a helicity modulus $\Upsilon \mathrm{}\left(T\right)\mathrm{}$ which measures the free-energy increment associated with "twisting" the direction of the order parameter. For a Bose system the superfluid density is seen to be ${\rho }_s\mathrm{}\left(T\right)={\left(\frac{m}{\hslash }\right)}^2\Upsilon \mathrm{}\left(T\right)\mathrm{}$. A critical exponent $v$ is defined by $\Upsilon \mathrm{}\left(T\right)\mathrm{}\sim {|T-T_c|}^v$ as $T\to T_c$; for an ideal Bose gas and spherical model ($n\to \infty$), $v=1\mathrm{}$ is an exact result for all $d>\mathrm{}2\mathrm{}$. The difficulties of defining a correlation length in the ordered phase are discussed. A full scaling theory of the correlations avoids these problems and may be linked to a phenomenological hydrodynamic approach, to clarify and rederive Josephson's relation $v=2\mathrm{}\beta -\eta \nu =2-\alpha -2\mathrm{}\nu$. This reduces to $v=(d-2\mathrm{})\mathrm{}\nu$ (used by some authors with $d=3\mathrm{}$), only if one accepts $d$-dependent, "hyperscaling" relations such as $d\nu =2-\alpha$; however, both these latter relations fail for the ideal Bose gas when $d>\mathrm{}4\mathrm{}$. An alternative derivation of the formula $v=2-\alpha -2\mathrm{}\nu$ is based on the scaling theory for systems with a large but finite dimension.

The specific heat at constant density of an ideal-Bose-fluid film of thickness $L=l{\rho }^{-\frac{1}{3}}$, but infinite lateral extent, is calculated analytically to order $l^{-2\mathrm{}}$. Both hard-wall and periodic boundary conditions are considered. Good agreement with the numerical calculations of Goble and Trainor for the total specific heat under hard-wall conditions is obtained down to $l\simeq 10$. In the critical region, the large-$l$ behavior accords with the scaling theory of finite-size effects. The appropriate scaling functions and the surface specific heat are explicitly calculated.

The $e-\mathrm{R}\mathrm{b}$ and $e-\mathrm{N}\mathrm{a}$ spin-exchange cross sections have been measured in a spin-exchange optical-pumping experiment as a function of temperature over the range 300-660 °K for Rb and 400-725 °K for Na. Within experimental error, the cross sections are essentially constant as a function of temperature. Measurements at 293 °K give ${\sigma }_{\mathrm{SE}}\left(\mathrm{Rb}\right)=(7.0\mathrm{}\pm 2.0)\times {10}^{-14\mathrm{}}$ ${\mathrm{cm}}^2$ and measurements at 403 °K give ${\sigma }_{\mathrm{SE}}\left(\mathrm{Na}\right)=(3.2\mathrm{}\pm 1.0)\times {10}^{-14\mathrm{}}$ ${\mathrm{cm}}^2$.

An approximate expression for the critical screening parameters associated with the attractive generalized screened Coulomb potential (GSCP) is obtained by utilizing the variational principle and the virial theorem. Subject to certain conditions, this expression is shown to represent a lower bound to the true critical screening parameters associated with the GSCP. Specific results are reported for the bound states $n$, $l=n-1\mathrm{}$ of the Debye-Hückel potential and for the ground state and first excited state of the exponential-cosine screened Coulomb potential.

A previous paper by Starace showed that the length formula for the electric-dipole matrix element is the only consistent one for calculations that solve for the exact eigenfunctions of an approximate Hamiltonian containing a nonlocal potential. It is emphasized here that disagreement between such a length-formula calculation and experiment indicates inadequacy of the approximate Hamiltonian and not that the velocity formula may be preferable.

The $M_{2,3}$ x-ray emission spectrum of gaseous krypton excited by direct electron bombardment has been recorded with a scanning single-flat crystal spectrometer. The most prominent spectral features are two peaks at 187 and 203 eV, with the 187-eV peak having an extended low-energy tail. This low-energy tail is attributed to double-electron single-vacancy transitions owing to the strong mixing of the final-state configuration $4s4p^6{}_{}{}^2{}_{}{}^{}S$ with the even levels $4s^{2}4p^4\mathrm{ns}{}_{}{}^2{}_{}{}^{}S \mathrm{and} 4s^{2}4p^4\mathrm{nd}{}_{}{}^2{}_{}{}^{}S$. The 203-eV peak is identified as probably the single-electron double-vacancy $M_{2,3}{M}_{2,3}\to M_{2,3}{N}_1$ multiplet complex.

Dynamic multipole polarizabilities of a number of simple atomic closed-shell systems in their ground states have been calculated within the coupled time-dependent Hartree-Fock scheme. The systems here investigated include He, ${\mathrm{Li}}^{+}$, ${\mathrm{Be}}^{2+}$, ${\mathrm{B}}^{3+}$, ${\mathrm{C}}^{4+}$, Be, ${\mathrm{B}}^{+}$, and ${\mathrm{C}}^{2+}$. In order to acquire some feeling about the obtained results, the dipole polarizabilities have been evaluated either by using a scalar- or a vector-potential representation for the oscillating external field: these two choices give rise to the well-known polarizability expressions in terms of "length" and "velocity" oscillator strengths, respectively. Furthermore, a unitary transformation of the Hartree-Fock Hamiltonian is devised, which allows one to use a perturbation operator corresponding to that involved in the "acceleration" oscillator-strength expression. Estimates of transition energies to low-lying excited states have been also obtained from the frequency values at which the various polarizabilities display discontinuities.

Ambiguities arising in the construction of potentials from the scattering amplitude at a given energy are studied. The functional space of all possible potentials is explored in general by means of the JWKB approximation and in particular by the generalized Born approximation. Typical kinds of ambiguities are exhibited. Certain of them ("window" ambiguities) arise from the fact that wave functions are unable to test analytical details over distances much smaller than a wavelength. Others are owing to the parts of the potential that are located between turning points and therefore not "visible" in the classical limit. These parts can be deformed slightly without perturbing the phase shifts (continuous ambiguity) and besides can also be modified drastically so as to include more nodes of the wave function. This strong modification yields classes of equivalent potentials which are layered near the origin. As a first approximation, it is very remarkable that the layering does not depend on any detail of the shape of the potential.

Within the framework of time-dependent perturbation theory and the Born—Oppenheimer approximation, general expressions are derived for the cross sections of multiphoton processes induced by radiation acting on a homopolar diatomic molecule. A Morse-type model of potential-energy curves of excited electronic states is proposed which gives a good representation of true curves. The procedure used to evaluate the sums over intermediate states is described. The method is based on the use of the internuclear potential Green's functions obtained in closed form. Analytic expressions are derived for one-photon and two-photon cross sections. The results for higher-order processes are presented in a form which is convenient for numerical computation. As an illustration, the cross section for one-photon dissociation of $\mathrm{H}_2^{}{}_{}{}^{+}$ from the $v=0\mathrm{}$ vibrational level is calculated as a function of wavelength. A comparison of cross sections calculated using the Morse-type model of potential curves with those using accurate wave functions of $\mathrm{H}_2^{}{}_{}{}^{+}$ shows a good precision of the model.

Within the framework of the Born—Oppenheimer approximation questions are considered involving application of time-dependent perturbation theory to diatomic heteronuclear molecules. A potential-energy model of the form $V_i\mathrm{}\left(r\right)=\frac{A_i}{{(r-r_i)}^2}-\frac{B_i}{(r-r_i)}+Q_i$ is proposed which gives a good representation of the true curves of the ground and excited states of a molecule. The relation ${\omega }_e{x}_e=6\mathrm{}\frac{D^2}{m}=\frac{3{\omega }_e^{\frac{4}{3}}}{2m^{\frac{1}{3}}}$ has been obtained between the dissociation energies into free ions $D$, vibrational frequencies ${\omega }_e$, and vibrational anharmonicity parameters ${\omega }_e{x}_e$ of the ground state of polar molecules. Analytical expressions are derived for vibration-rotation spectra, wave functions, and radial matrix elements considered in the first-order perturbation theory. An explicit expression obtained for the Green's function of the model enables a closed form to be given for matrix elements appearing in the higher-order perturbation theory involved in calculation of multiphoton molecular transitions.

Within the framework of the Born—Oppenheimer approximation and the theory of multiphoton molecular transitions, a method is proposed for calculating matrix elements of multiphoton processes in a heteropolar diatomic molecule. The method is based on the application of the explicit expressions of the Green's functions of the internuclear potentials. Analytical expressions are derived for the two-photon transitions. Results for higher-order processes are presented in a form which is convenient for numerical computation.

A calculation of higher-order relativistic contributions to the combined Zeeman and motional Stark effects in positronium is presented. These contributions are necessary for the determination of the fine-structure interval in the ground state from the Zeeman effect and may be important in future experiments on the first excited states of positronium. The contributions to the $g_J$ factors have been calculated to order ${\alpha }^2$ for all the $S$ states and for the $2P$ states. The energy levels and the higher-order corrections to the motional Stark effect in the first excited state are also presented. Relativistic contributions are obtained from the matrix elements of a Hamiltonian containing the Breit interaction, a Pauli Hamiltonian, and the virtual annihilation interaction. For all $nS$ states the relativistic contributions to the Zeeman effect may be accounted for by the replacement of $g_e$ by $g_e(1-\frac{5{\alpha }^2}{24n^2}-\frac{T}{2m{c}^2})$, where $T$ is the kinetic energy of the atom and $g_e$ is the gyromagnetic ratio of the free electron.

The stability conditions which ensure that the Brueckner (maximum-overlap) solution maximizes locally the overlap with the exact wave function are derived. These stability conditions are then specified for the closed-shell electronic systems with a spin-independent Hamiltonian yielding the singlet and nonsinglet ("triplet") stability conditions in complete analogy with the Hartree-Fock stability problem. The stability conditions are then applied to the $\pi$-electronic model of benzene. Critical coupling constants are determined for both singlet and nonsinglet instabilities and an example of space-symmetry-broken Brueckner solution is given. The main characteristics of the pure singlet, broken-symmetry Hartree-Fock, and Brueckner solutions are briefly outlined.

X-ray rates, Auger group rates, and $K$-shell fluorescence yields are presented for variously ionized states of neon. The nonrelativistic Hartree-Fock-Slater atomic model, with the exchange approximation of Herman-Van Dyke-Ortenburger, is used. The x-ray and Auger transition energies were obtained by the adiabatic method. The theoretical results presented here are applicable to the analysis of defect configurations which may be produced in a heavy-ion-neon collision. It is shown that the commonly used statistical scaling procedure to obtain the $K$-shell fluorescence yield for defect configuration leads to significant errors for neon.

Transition dipole moments have been calculated for lines in the ${\nu }_3$, ${\nu }_4$, $2{\nu }_3$, and $3{\nu }_3$ vibration-rotation bands of ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$${\mathrm{H}}_4$. The values of ${\mu }^2$ are 0.76 ± 0.10, 0.69 ± 0.04, (1.86 ± 0.07) × ${10}^{-3\mathrm{}}$, and (2.96 ± 0.16) × ${10}^{-6\mathrm{}}$ ${\mathrm{D}}^2$, respectively. These results have been applied to predict absolute line intensities. In particular, for the $P7F$ line at 2947.912 ${\mathrm{cm}}^{-1\mathrm{}}$, the calculated intensity is (1.34 ± 0.17) ${\mathrm{cm}}^{-2\mathrm{}}$ ${\mathrm{atm}}^{-1\mathrm{}}$ at 300° K. Furthermore, ultraweak absorption features observed in the $3{\nu }_3$ region may be accounted for on the basis of transitions with $J\approx 30$.

We describe a systematic procedure for the construction of variational principles for the variational estimation of an extremely wide class of functionals $F({\phi }^{†},\phi )$. The quantities $\phi$ are well defined but not known exactly because the equations defining the set $\phi$ are not solvable exactly. The procedure is illustrated for $F$, the off-diagonal matrix element ${\phi }_1^{†}\mathrm{W}{\phi }_2$ of an arbitrary Hermitian linear operator $W$, for ${\phi }_1$ and ${\phi }_2$ nondegenerate normalized bound-state eigenfunctions of the same Hamiltonian with a specified relative phase.

A beam containing argon metastable atoms is crossed by the output beam of a pulsed tunable uv laser. The resulting ion production is measured as a function of wavelength within the range $2945<\mathrm{}\lambda <3075\mathrm{}$ Å and displays several sharp resonances. These arise from excitation of $\mathrm{Ar}\left({}_{}{}^3{}_{}{}^{}P_0^{}{}_{}{}^{}\right)$ metastable atoms to high-lying $3p {}_{}{}^5{}_{}{}^{}\left({}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}\right)n{p}^{\prime }$ levels which subsequently undergo autoionization. The term values of these levels are measured to an accuracy of ± 6 ${\mathrm{cm}}^{-1\mathrm{}}$ and are in excellent agreement with values obtained from a quantum-defect extrapolation of the known lower terms of this series. An unambiguous assignment of principal quantum numbers to the observed levels is therefore possible. The lifetimes against autoionization of these levels are longer than 2 ± ${10}^{-12\mathrm{}}$ sec.

Exact relativistic Coulomb wave functions, in the form of partial-wave expansions, are used to obtain analytic formulas for the differential cross sections and energy spectrum for electron-positron pair production by photons. Radiative corrections, which are believed to be small (of the order of 1%), are neglected. The errors introduced by the finite nuclear size and recoil effects are quite negligible for the energy region considered here. Apart from these approximations, the present formulas give the exact cross sections for the unscreened atomic field. Numerical results are obtained for positron energy spectra for a large number of photon energies and atomic numbers in the ranges $2m_e{c}^2<k\le 10m_e{c}^2$ and $1\le Z\le 100$. Total cross sections are obtained by numerical integration of the spectra. A representative choice of spectra is given. The approximately 500 total cross sections calculated are given in tables and diagrams. By one- or two-way interpolation in these data the total cross section for any photon energy and nuclear charge within the given limits may be obtained with an accuracy of 1% or better. For heavy elements large Coulomb corrections to the Bethe-Heitler Born-approximation results are found. The positron energy spectra obtained are asymmetric, favoring positrons of high energy. The total cross section $\sigma$ is smaller than the Bethe-Heitler total cross section ${\sigma }_B$ for photon energies very close to the threshold (owing to the repulsion of the positron), while $\sigma$ is generally larger than ${\sigma }_B$ in the energy region $2.2m_e{c}^2\lesssim k\lesssim 10m_e{c}^2$.

In a recent paper, the atomic quadrupole shielding factor $R_{\Theta }$ has been calculated for the excited states $n{p}^5{n}^{\prime }s (n^{\prime }=n+1)$ of the rare gases, in connection with an experiment of Sandars and Stewart. The previous calculations included only the first-order term $R_{\Theta ,1\mathrm{}}$ in the perturbation ${v_1}^{\prime }(n^{\prime }s\to d)$ of the excited $n^{\prime }s$ electron. The values of $R_{\Theta ,1\mathrm{}}$ were found to be in good agreement with experiment. In the present paper, the terms of second order in ${v_1}^{\prime }$ have been evaluated. Their inclusion improves appreciably the agreement of $R_{\Theta ,\mathrm{calc}}$ with the experimental values of Sandars and Stewart, in particular for the cases of Kr and Xe.