Several calculational procedures for generating approximate solutions to the time-dependent Schrödinger equation of the impact-parameter model are discussed in this work. It is emphasized that in these calculational procedures a trial wave function is selected from a definite class of functions the class of functions which have the asymptotic form of the exact wave function and then some property satisfied by the exact wave function is used to fix the trial wave function. The customary set of coupled equations can be obtained by requiring that the Schrödinger equation be satisfied in the subspace defined by the basis set used to express the trial wave function. Recently, it has been popular to show that these coupled equations follow by making certain functionals stationary. It is shown, however, that these functionals are not stationary about the exact solution for the variations represented by available trial wave functions which, as they must, reflect our ignorance of the exact nature of the true wave function. In this work, a functional related to the norm of the exact wave function is given, and this functional is shown to be stationary about the exact wave function for the variations represented by available trial wave functions. A calculational procedure based upon the stationary property of this functional is given by which variational parameters in a trial wave function can be determined. If these parameters are taken to be time-varying nuclear charges in the basis functions, one obtains the equations given by Cheshire. Finally, the question of convergence of the two-centered traveling hydrogenic orbital expansion is considered. We conclude that the two-centered expansion does not provide a scheme by which a sequence of approximate wave functions can be generated that converges to the exact wave function. However, it appears from the comparison of the approximate cross sections obtained in two-, four-, eight-, and 14-state calculations that the cross sections for the $1s$ exchange reaction and possibly for the $2s$ transitions may be tending to limiting values.

The boundary condition (at $t\to \infty$) previously shown to be associated with the customary time-dependent Schrödinger equation of the impact-parameter method places a restriction upon the approximate or trial wave functions which are to be used in the usual coupled-state calculations. For the trial wave function to be physically acceptable, it must belong to the class of functions which have the asymptotic form of the exact wave function. This simply means that one must obtain from the coupled-state calculation, approximate transition amplitudes for which the absolute squares are time independent as $t\to \infty$. It is noted that the trial wave function in the six-, eight-, and nine-state Sturmian calculations are not physically acceptable. The trial wave function in the pseudostate calculation of Cheshire, Gallaher, and Taylor is acceptable; however, if one attempted to include an additional pseudostate with the same symmetry as one already in the set, then this trial wave function would not be physically acceptable.

A stationary functional and two variational principles are given in this work by which approximate transition amplitudes for the charge-exchange and electronic excitation processes occurring in proton — hydrogen-atom scattering can be calculated. The calculated transition amplitudes will be "second-order accurate." It is demonstrated how the stationary property of the functional can be used for a general variation to determine optimal approximate wave functions. The equations for these wave functions should be no more difficult to solve than the usual set of coupled equations, and the solution will automatically satisfy the unitary condition and the condition imposed by the time-reversal symmetry of the Hamiltonian. An upper bound on the second-order error term is derived which only depends upon the interaction and the trial wave function. The upper bound approaches zero as the error between the exact and trial wave functions vanishes, and thus the magnitude of the upper bound is a measure of the quality of a trial wave function. In addition, this upper bound can be used in conjunction with the variational principles to obtain upper and lower bounds which bracket the absolute values of the exact transition amplitudes; thus, one can in principle obtain exact values for the transition amplitudes to any desired accuracy. One of the variational principles offers the advantage that the trial wave function need not satisfy any asymptotic conditions. With this variational principle, one can therefore circumvent the difficulties associated with the use of Sturmian or pseudostate basis functions.

Electron excitation functions of some thirty states of the $3p^{5}ns$, $3p^{5}np$,, and $3p^{5}nd$ configurations of the argon atom have been measured by the optical method. In order to evaluate the level cross sections from the optical excitation cross sections, we have determined the branching ratios experimentally by measuring in an argon discharge tube the relative intensities of the emission lines originating from the level of interest. This procedure gives more accurate level cross sections than does the conventional approach of using theoretical transition probabilities to form the branching ratios, since for an atom as complex as argon the wave functions of the excited states calculated by a semiempirical treatment of the fine structure are not always of sufficient accuracy to give reliable transition probabilites. Analysis of the shape of the excitation functions is facilitated by expressing the wave functions of the excited states of argon in terms of the $LS$ eigenfunctions, and the special features of the experimental data of excitation functions can be readily explained by generalizing the results of helium. Analogous to the case of neon, one can show theoretically that within a configuration $3p^{5}nl$,, the states with odd values of $J+l$ in general have larger excitation cross sections than the ones with even values at incident energies well above the threshold, and that in general $Q(J=0\mathrm{})>Q(J=2\mathrm{})\mathrm{}$, $Q(J=1\mathrm{})>Q(J=3\mathrm{})\mathrm{}$ for the $np$ states, and $Q(J=1\mathrm{})>Q(J=3\mathrm{})\mathrm{}$, $Q(J=2\mathrm{})>Q(J=0,4\mathrm{})\mathrm{}$ for $nd$. The experimental results of the excitation cross sections are found to be in good agreement with these rules.

Measurements of nonresonant charge transfer between atomic ions and atoms have been made in a drift-tube—mass-spectrometer apparatus as a function of $\frac{E}{p}$ (drift field/gas pressure) at 295 K. At $\frac{E}{p}=0\mathrm{}$ the charge-transfer coefficients are: $k({\mathrm{He}}^{+},\mathrm{Hg})=(1.6\mathrm{}\pm 0.3)\times {10}^{-9\mathrm{}}$ ${\mathrm{cm}}^3$/sec, $k({\mathrm{Ar}}^{+},\mathrm{Hg})=(1.52\mathrm{}\pm 0.3)\times {10}^{-11\mathrm{}}$, $k({\mathrm{Kr}}^{+},\mathrm{Hg})=(8.5\mathrm{}\pm 2)\times {10}^{-12\mathrm{}}$. For ${\mathrm{Ne}}^{+}$ + Hg and ${\mathrm{Xe}}^{+}$ + Hg, $k$ is ≤ ${10}^{-12\mathrm{}}$, while for ${\mathrm{He}}^{+}$ + Ne and ${\mathrm{He}}^{+}$ + Ar, $k$ is ≤ ${10}^{-13\mathrm{}}$ ${\mathrm{cm}}^3$/sec. Little dependence of $k({\mathrm{He}}^{+},\mathrm{Hg})$ and $k({\mathrm{Ar}}^{+},\mathrm{Hg})$ on ion energy is found up to mean energies of $\sim 0.25$ eV. Comparison of the measured rates with available theories suggests an inability to predict the moderately slow reaction rates.

The Fano-Lichten model is known to be in good accord with available inner-shell excitation data on gaseous targets, whereas there exist several examples of discrepancies between the model and results of solid-target experiments. As will be discussed in detail in this paper, these apparent discrepancies may be explained by so-called "solid effects." An equation taking due account of energy-loss straggling, x-ray generation in recoil collisions, and targets absorption of emitted x rays is used to relate the measured x-ray yield to an x-ray production cross section averaged over the equilibrium distribution of projectile charge states in the solid. Experimental cross sections are reported for the Al $K$ x-ray production in Al targets bombarded with N, Al, and Ar ions, and for the Ne $K$ x-ray production of Ne projectiles incident on Mg, Al, Si, and P targets. The neglect of solid effects in published x-ray data is commented upon.

By using a monochromator-analyzer combination, electron-excitation functions of the lowest-lying electronic states of krypton have been obtained at a scattering angle of 45 ° in the near-threshold region (9-14 eV). All excitation functions display a complex resonant structure. Additional data have been obtained on elastic scattering at 45 ° to provide further information on the resonances. In addition to the $4p^{5}5s^2 J=\frac{3}{2},\frac{1}{2}$ resonance doublet, a number of narrow resonances appear below about 12 eV in both the excitation data and elastic scattering which correlate in energy with recent measurements on transmission by Sanche and Schulz. Broad resonant structure is found in a number of the excitation curves and the excitation process in the near-threshold region appears to be dominated by resonant effects. An analysis of the elastic-scattering data on the $4p^{5}5s^2$ doublet is presented and resonance parameters obtained compared with theory. A comparison of the narrow resonant structure data with optical absorption data on Rb I obtained by Beutler indicates that the narrow resonances in the 10.5-12-eV region are probably due to levels of the $4p^{5}5s4d$ configuration.

Thin gas targets of argon were bombarded with C, N, O, and F projectiles of various incident charge states at energies from 1 to 2 MeV/amu. Argon $K$ x rays were observed with a Si(Li) detector, and x-ray-production cross sections were obtained. For a given-energy projectile, the cross sections decrease approximately exponentially as the number of electrons bound to the incident projectile increases from 0 to 3. For three or more electrons, the measured cross sections change very little. Hartree-Fock calculations of fluorescence yields as a function of target configuration were used to obtain argon $K$-shell vacancy-production cross sections. Target configurations were chosen by comparing the experimental argon $K\alpha$ and $K\beta$ x-ray energies and $\frac{K\beta }{K\alpha }$ relative intensities with the values obtained by Hartree-Fock calculations. For projectiles with more than one electron, the vacancy-production cross sections scale as the square of projectile nuclear charge as expected for Coulomb ionization, but the absolute magnitudes of the cross sections do not agree with Coulomb-ionization results. For projectiles with 0 to 1 electron, however, Coulomb ionization alone cannot explain the observed cross sections. Charge exchange from the argon $K$ shell to projectile bound states has been proposed as a significant vacancy-producing mechanism for collisions involving highly charged projectiles. Although calculations of the charge-exchange cross sections are in reasonable relative agreement with the data, additional vacancy-producing mechanisms cannot be ignored.

The optical pumping of atoms by a multimode gas laser, in the presence of a magnetic field, is studied theoretically. The atoms are described by their density matrix, which is expanded on an irreducible tensorial set. The atomic relaxation is assumed to be isotropic. In order to avoid the usual perturbation theory, we use the so-called "Broad-Line approximation" (BLA). BLA is valid when the width of the "Bennett hole" created by one laser mode in the velocity distribution of the excited atoms is comparable to the spacing between modes; under these conditions, the atomic response does not depend on the velocity and the multimode laser irradiation is equivalent to a broad-line excitation. For the atomic system, BLA leads to a set of coupled equations which is valid at arbitrary intensities of the laser field and which depends on this field through a unique parameter, $\gamma$. This parameter can be understood as a laser-induced transition probability. In this article, we analyze the nonlinear effects which can be deduced from this theory, for any value of the angular momenta involved in the laser transition: Hanle-effect broadening, alignment coupling, saturation resonances on the populations are shown. The exact calculations, valid at arbitrary laser intensities for $J=1\mathrm{}\leftrightarrow J=0\mathrm{}$, $J=1\mathrm{}\leftrightarrow J=1\mathrm{}$, and $J=1\mathrm{}\leftrightarrow J=2\mathrm{}$ transitions, and the corresponding experimental results will be presented in forthcoming papers.

The time dependence of ${\mathrm{Xe}}^{+}$ ion density was measured during the decay period of plasmas produced in xenon for reduced gas pressures varying from 0.10 to 4.0 Torr and a gas temperature of 300 °K. The reaction rate constant for the termolecular ion-neutral association reaction ${\mathrm{Xe}}^{+}$ + 2Xe → $\mathrm{Xe}_2^{}{}_{}{}^{+}$ + Xe was found to be $k=(2.0\mathrm{}\pm 0.2)\times {10}^{-31\mathrm{}}$ ${\mathrm{cm}}^6$ ${\sec }^{-1\mathrm{}}$. The value of the mobility of ${\mathrm{Xe}}^{+}$ in xenon was confirmed to be ${\mu }_0\left({\mathrm{Xe}}^{+}\right)=0.62\mathrm{}\pm 0.03$ ${\mathrm{cm}}^2$${\mathrm{V}}^{-1\mathrm{}}$${\sec }^{-1\mathrm{}}$. At gas pressures smaller than about 1.0 Torr the phenomenon of diffusion cooling was observed.

The zero-field line crossing of the mercury isotope ${}_{}{}^{204}{}_{}{}^{}\mathrm{Hg}$ was studied in forward-scattered light, with the scattering atoms both in a foreign gas and in vacuum. With sufficient foreign-gas pressure, a Lorentzian shape was obtained, in contrast to the Gaussian shape found for atoms in vacuum. An approximate theoretical treatment gives a field dependence of the forward-scattered intensity in satisfactory agreement with the measurements. Line crossings of Zeeman components of different isotopes, contained in separate cells, are also discussed.

The radiative lifetimes and alignment depolarization cross sections due to collisions with ground-state atoms of the $\left(4\mathrm{}{p}^{5}5p\right){}_{}{}^1{}_{}{}^{}{D}^{\prime }_2^{}{}_{}{}^{}$ and ${}_{}{}^3{}_{}{}^{}D_3^{}{}_{}{}^{}$ levels in Kr have been measured by the Hanle-effect method. Our results are $\tau \left({}_{}{}^1{}_{}{}^{}D_2^{\prime }{}_{}{}^{}\right)=3.02\mathrm{}\left(31\right)\mathrm{}\times {10}^{-8\mathrm{}}$ sec, $\tau \left({}_{}{}^3{}_{}{}^{}D_3^{}{}_{}{}^{}\right)=2.81\mathrm{}\left(25\right)\mathrm{}\times {10}^{-8\mathrm{}}$ sec, ${\overline{\sigma }}_2\left({}_{}{}^1{}_{}{}^{}D_2^{\prime }{}_{}{}^{}\right)=4.49\mathrm{}\left(32\right)\mathrm{}\times {10}^{-14\mathrm{}}$ ${\mathrm{cm}}^2$, and ${\overline{\sigma }}_2\left({}_{}{}^3{}_{}{}^{}D_3^{}{}_{}{}^{}\right)=4.79\mathrm{}\left(27\right)\mathrm{}\times {10}^{-14\mathrm{}}$ ${\mathrm{cm}}^2$. The cross sections were measured at $T=292\mathrm{}\left(6\right)\mathrm{}$ °K.

$\frac{K\alpha }{K\beta }$ x-ray yield ratios have been determined for eleven elements, Ti, V, Fe, Ni, Cu, Ge, Rb, Zr, Ag, Sn, and Sb, when bombarded with protons of 1.00, 2.25, and 3.00 MeV. Excitation functions of $\frac{K\alpha }{K\beta }$ ratios for Ni, Ge, and Ag were also measured from 1.00 to 3.70 MeV in 100-keV steps, and these ratios were found to be constant with proton energy. For the ten elements Ce, Sm, Dy, Tm, W, Au, Pb, Bi, Th, and U, the yield ratios $\frac{L\alpha }{L\beta }$, $\frac{L\alpha }{L\gamma }$, and $\frac{L\alpha }{Ll}$ were determined for proton bombarding energies of 1.00, 2.25, and 3.00 MeV. Excitation functions for these same ratios were measured for Sm, W, and Th for proton energies between 1.00 and 3.70 MeV in 100-keV steps. While the $\frac{L\alpha }{L\beta }$ and $\frac{L\alpha }{L\gamma }$ ratios varied significantly with proton energy, the $\frac{L\alpha }{Ll}$ ratio was found to be constant as a function of proton energy. Additionally, the energy of the centroid of the peak corresponding to the $L\alpha ,L\beta$ and $L\gamma$ x-ray transitions was measured as a function of proton energy for Sm, W, and Th. The $L\beta$ and $L\gamma$ centroid energies changed appreciably with proton energy, while the $L\alpha$ centroid energy remained constant. The experimental yield ratios and centroid energies are compared to plane-wave-Born- and binary-encounter-approximation predictions.

A formalism, previously presented, for describing atom-atom scattering in the adiabatic representation to higher orders in $\frac{m}{M}$ is generalized to describe the problem of different isotopes and different atoms. This is accomplished by a sequence of nonlinear coordinate transformations which introduce a "switching function" which is defined only in the asymptotic limit of separated atoms. It is found that the effective potential for elastic scattering contains energy-dependent terms in order $\frac{m}{M}$ and that it also depends upon the switching function in this order so that the potential is not unique. It is also found that in this order the definition of adiabatic states is not a trivial problem. A variational choice is made for the switching function and explicit calculations are presented for the He-He interaction. The coupled equations describing transitions are then derived and it is shown that terms are omitted from the usual coupling operator which are of the same order of magnitude as the one usually kept. This leads to a modification of the usual Landau-Zener transition formula.

In this paper we analyze the energy eigenvalues of the Dicke Hamiltonian which describes the interaction of $N$ two-level atoms with an electromagnetic field. Using as a starting point the classification of the collective states proposed by Tavis and Cummings, we first derive approximate analytic expressions for the energy eigenvalues which are appropriate for special values of the collective parameters. We next evaluate a first-order correction which allows the application of our results to a wider class of collective states. An accurate description of the energy spectrum for arbitrary values of the collective parameters is provided using a method first introduced by Scharf for the case of resonant interaction, suitably generalized to account for possible frequency detuning between the radiation and the atomic resonant frequency. The limit of validity of the approximation is analyzed and the results compared with the numerical diagonalization of the Dicke Hamiltonian. With the exception of asymptotically small ranges of values of the parameters, the analytic expression for the density of energy eigenvalues derived in this paper compares very well with the results of the numerical diagonalization.

The problem of the characterization of the fluctuations in statistical systems brought about by nonequilibrium states owing to external perturbations is considered. The time evolution of the second-order parameters of nonequilibrium fluctuations in paramagnetic systems is approached in detail. A connection is established between the temporal derivative of the dispersion of the magnetic-moment vector $\overrightarrow{\mathrm{M}}$ and the expectation values of the components of this vector. This connection is regarded as a particular case of fluctuation-dissipation relations, which are functional relations between the different order parameters of the fluctuations. It is shown that, in the limit of the approximations in which the phenomenological equations of Bloch type are valid, the time evolution of the dispersion of the magnetic moment is directly conditioned only by the internal relaxation processes and not by the external magnetic fields.

The properties of Fermi gases are investigated in the near-classical limit. In this region of temperature and density the diffraction effects due to the uncertainty principle are neglected, but degeneracy effects due to the Pauli exclusion principle are included exactly. The basic approach involves the evaluation of terms in the Bloch-de Dominicis quantum-perturbation expansion of the grand canonical partition function. In the near-classical limit the perturbation terms reduce to the classical results, modified by statistics factors in the form of Fermi integrals. Using these statistics factors, a modified form of the Mayer cluster expansion that includes degeneracy is obtained. Also a new form of the Boltzmann factor including degeneracy is derived. As an illustrative example these results are applied to the calculation of the near-classical limit of the free energy for plasmas.

The damping coefficient (imaginary part of the frequency) for ripplons in superfluid ${\mathrm{He}}^4$ at zero temperature is found to be ${\Gamma }_q=0.162\mathrm{} \hslash \frac{q^5}{{\rho }_0}$ owing to a process where one ripplon decays into two.

A physical picture of the smectic $C$ phase is proposed in which the molecules rotate freely about their long axes in the smectic $A$ phase and this rotation is frozen out in the smectic $C$ phase. This picture is in contrast with the accepted view that the long axis of the molecules is normal to the smectic planes in the smectic $A$ phase and tilted in the smectic $C$ phase. A microscopic theory of this phase transition is constructed using the dipole-dipole interaction of the permanent molecular dipole moments. Three ordered phases are found, one with the physical properties of the smectic $C$ phase (tilted director, optically biaxial, second-order smectic-$A$-smectic-$C$ phase transition). The second-ordered phase is a two-dimensional ferroelectric and the third, low-temperature ordered phase is both tilted and ferroelectric.

The ground-state wave function of a finite drop of liquid ${\mathrm{He}}^4$ is taken as a product of two-body and one-body factors. The surface energy is calculated, taking into account the variation of the two-particle-correlation function in the surface region, and is minimized by a density profile which falls off outside the liquid with a scale length 0.7 Å and approaches the interior density with scale length 2 Å. The calculated surface tension is 10% higher than the observed value.