# Showing papers in "Physical Review A in 1971"

••

TL;DR: In this paper, a spin-1 Ising model is introduced and solved in the mean-field approximation, and the phase diagram is qualitatively similar to that observed experimentally and phase separation appears as a consequence of the superfluid ordering.

Abstract: A spin-1 Ising model, which simulates the thermodynamic behavior of ${\mathrm{He}}^{3}$-${\mathrm{He}}^{4}$ mixtures along the $\ensuremath{\lambda}$ line and near the critical mixing point, is introduced and solved in the mean-field approximation. For reasonable values of the parameters of the model the phase diagram is qualitatively similar to that observed experimentally and the phase separation appears as a consequence of the superfluid ordering. Changing the parameters produces many different types of phase diagram, including as features $\ensuremath{\lambda}$ lines, critical points, tricritical points, and triple points. Certain thermodynamic features which differ from the ${\mathrm{He}}^{3}$-${\mathrm{He}}^{4}$ experiments may be artifacts of the mean-field theory.

1,094 citations

••

TL;DR: In this paper, a system of either fermions or bosons interacting in one dimension by a two-body potential with periodic boundary conditions was investigated, and expressions for the one-particle density matrix at zero temperature and particular (nontrivial) values of the coupling constant $g, as a determinant of order $N\ifmmode\times\else\texttimes\fi{}N$ were presented.

Abstract: We investigate exactly a system of either fermions or bosons interacting in one dimension by a two-body potential $V(r)=\frac{g}{{r}^{2}}$ with periodic boundary conditions. In addition to rederiving known results for correlation functions and thermodynamics in the thermodynamic limit, we present expressions for the one-particle density matrix at zero temperature and particular (nontrivial) values of the coupling constant $g$, as a determinant of order $N\ifmmode\times\else\texttimes\fi{}N$. These concise expressions allow a discussion of the momentum distribution in the thermodynamic limit. In particular, for a case of repulsive bosons, the determinant is evaluated explicitly, exhibiting a weak (logarithmic) singularity at zero momentum, and vanishing outside of a "Fermi" surface.

993 citations

••

Bell Labs

^{1}TL;DR: In this article, the Maier-Saupe model with an orientational order parameter is extended to the smectic $A$ phase by introducing a new order parameter, the amplitude of a density wave in the direction of the nematic preferred axis.

Abstract: The Maier-Saupe model of the nematic phase with an orientational order parameter is extended to the smectic $A$ phase by introducing a new order parameter, the amplitude of a density wave in the direction of the nematic preferred axis. Self-consistent equations for the two order parameters are derived from an anisotropic model interaction and are solved numerically. We calculate the order parameters, the entropy, and the specific heat as a function of temperature for several values of dimensionless interaction strength $\ensuremath{\alpha}$ for the smectic $A$ phase. The transition temperatures plotted versus $\ensuremath{\alpha}$ provide a theoretical phase diagram which resembles experimental plots of transition temperature versus alkyl chain length for homologous series of compounds. The model qualitatively reproduces chemical trends in transition entropies. Experiments are suggested to measure the order parameters in the smectic $A$ phase.

873 citations

••

TL;DR: In this article, the spin correlation functions of the one-dimensional $XY$ model were studied in the presence of a constant magnetic field and it was shown that the asymptotic behavior of these correlation functions depends strongly on the various parameters of the Hamiltonian.

Abstract: The spin-correlation functions of the one-dimensional $XY$ model are studied in the presence of a constant magnetic field. We find that the asymptotic behavior of these correlation functions depends strongly on the various parameters of the Hamiltonian.

679 citations

••

TL;DR: In this paper, a generalized cluster expansion is derived relating the thermodynamic and structural properties of such systems to those of the hard-sphere fluid, and the first (lowestorder) approximation generated by the expansion equates the free energy and $y(r)$ for the fluid to the respective functions appropriate to a system of hard spheres with diameter $d.

Abstract: We consider the equilibrium statistical mechanics of classical fluids in which the potential energy is decomposable into repulsive pair interactions. A generalized cluster expansion is derived relating the thermodynamic and structural properties of such systems to those of the hard-sphere fluid. The expansion is ordered by a softness parameter $\ensuremath{\xi}$ which is essentially the range of intermolecular distances in which the difference between the Mayer $f$ functions for the repulsive potential and an appropriate reference hard-sphere potential is nonzero. The first (lowest-order) approximation generated by the expansion equates the free energy and $y(r)$ for the fluid to the respective functions appropriate to a system of hard spheres with diameter $d$. Here $y(r)=g(r) {e}^{+\ensuremath{\beta}u(r)}$, where $g(r)$ and $u(r)$ denote the radial distribution function and repulsive pair potential, respectively. A prescription is given for choosing a temperature- and density-dependent diameter $d$ in the reference hard-sphere fluid so that the first approximation for the free energy contains errors of order ${\ensuremath{\xi}}^{4}$ only, and the corrections to the first approximation for $g(r)$ are of order ${\ensuremath{\xi}}^{2}$. The method is used to calculate the properties of a fluid whose intermolecular potential varies as ${r}^{\ensuremath{-}12}$. The repulsive potential that produces the repulsive forces in the Lennard-Jones potential is also studied. Since the properties of the hard-sphere fluid are known from the results of computer calculations and conveniently summarized by analytic equations, the application of the first approximation is numerically very simple. With this approximation, the results obtained for both model systems agree closely with those obtained by Monte Carlo calculations.

524 citations

••

TL;DR: In this article, the energy levels and other properties of the states of the helium isoelectronic sequence up to F VIII were investigated, including the contributions from the mass-polarization correction and the relativistic effects of order ε = 2.

Abstract: Calculations have been made of the energy levels and other properties of the states $n^{1}S$, $n^{3}S$, $n^{1}P$, and $n^{3}P$, $n=2 \mathrm{to} 5$, for atoms belonging to the helium isoelectronic sequence up to $Z=10$, and also for the higher excited $S$ states of helium. The theoretical term values, including the contributions from the mass-polarization correction and the relativistic effects of order ${\ensuremath{\alpha}}^{2}$ are listed. A detailed comparison with the experimentally determined energy differences between $S$ and $P$ states for He I up to F VIII shows a satisfactory agreement in almost every case, provided that we include an estimate of the Lamb-shift correction to the $S$-state energy level when considering transitions involving the $1^{1}S$, $2^{1}S$, or $2^{3}S$ states.

365 citations

••

TL;DR: In this article, it was shown that large quantum fluctuations occur only if the initial total occupancy of the excited state differs from the total number of atoms at most by a number of order unity.

Abstract: We discuss the solution of the "superradiance master equation" derived in a preceding paper. During the first few photon transient times the cooperative atomic decay goes through a non-adiabatic oscillatory regime. For later times the decay takes place monotonically in time with the electromagnetic field following it adiabatically. The emitted light pulse has different statistical properties for an incoherently and a coherently prepared "superradiant" atomic initial state. The former case is characterized by large quantum fluctuations and strong atom-atom and atom-field correlations. In the latter case quantum fluctuations are small and the system behaves essentially classically. By also solving for a class of coherently prepared intermediate initial states we show that large quantum fluctuations occur only if the initial total occupancy of the excited state differs from the total number of atoms at most by a number of order unity.

362 citations

••

TL;DR: In this article, a set of intermediate-frequency long-wavelength waves were found to be unstable for any temperature ratio, and growth rates were on the order of the ion plasma (or lower hybrid) frequency, depending on the density and field strength.

Abstract: Electrons drifting relative to ions across a magnetic field are found to drive a set of intermediate-frequency long-wavelength waves (${\ensuremath{\omega}}_{\mathrm{ci}}l\ensuremath{\omega}l{\ensuremath{\omega}}_{\mathrm{ce}}$) unstable for any temperature ratio $\frac{{T}_{e}}{{T}_{i}}$ One instability is caused by the coupling of a drift wave in a nonuniform plasma to either the ion plasma oscillation or a lower hybrid oscillation, depending on density and field strength. Another instability is a form of the two-stream instability, shown here to exist even for drift speeds less than electron thermal speeds. Growth rates are on the order of the ion plasma (or lower hybrid) frequency, depending on the density and field strength.

359 citations

••

TL;DR: By the superposition of Bethe's wave functions, using the Lieb's solution for the system of identical bosons interacting in one dimension via a $\ensuremath{delta}$-function potential, this paper constructed the wave function of the corresponding system enclosed in a box by imposing the boundary condition that the wave functions must vanish at the two ends of an interval.

Abstract: By the superposition of Bethe's wave functions, using the Lieb's solution for the system of identical bosons interacting in one dimension via a $\ensuremath{\delta}$-function potential, we construct the wave function of the corresponding system enclosed in a box by imposing the boundary condition that the wave function must vanish at the two ends of an interval. Coupled equations for the energy levels are derived, and approximately solved in the thermodynamic limit in order to calculate the boundary energy of this Bose gas in its ground state. The method of superposition is also applied to the analogous problem of the Heisenberg-Ising chain (not the ring).

333 citations

••

TL;DR: In this article, a detailed error analysis is presented showing that the over-all root-mean-square error in $a(q)$ never exceeds 2.5% for any value of the momentum transfer.

Abstract: Highly accurate x-ray diffraction measurements are presented for the static structure factor $a(q)$ for liquid Na (at 100 and 200 \ifmmode^\circ\else\textdegree\fi{}C) and liquid K (at 65 and 135 \ifmmode^\circ\else\textdegree\fi{}C). A detailed error analysis is presented showing that the over-all root-mean-square error in $a(q)$ never exceeds 2.5% for any value of the momentum transfer $q$ and the relative root-mean-square error in $a(q)$ between different temperatures is always less than 1.5%. We discuss and demonstrate the reliability of the tabulated values for the atomic form factor and the Compton-scattering correction. A brief discussion is included of the relative merits of x-ray vs neutron diffraction for obtaining the static structure factor.

244 citations

••

TL;DR: In this paper, a single set of theoretical parameters which pertain to collisions of the $e+{\mathrm{Xe+}+}$ system with $J=1$ and characterize the five close-coupling eigenchannels of this system were obtained by fitting the experimental data.

Abstract: Experiments on the uv photoabsorption of Xe have determined level positions, line intensities, Land\'e $g$ factors, intensity profiles in the auto-ionization region, and the branching ratio of photoelectron groups. This paper expresses all these data in terms of a single set of theoretical parameters which pertain to collisions of the $e+{\mathrm{Xe}}^{+}$ system with $J=1$ and characterize the five close-coupling eigenchannels of this system. Values of the parameters, obtained by fitting the experimental data, provide (1) values of the zero-energy scattering eigenphases, (2) evidence that the orbital angular momentum of the free electron ($l=0 or 2$) is a good quantum number to within \ensuremath{\sim} 2%, and (3) evidence that the $\mathrm{LS}$ coupling classification of the $e+{\mathrm{Xe}}^{+}$ complex holds approximately, but only for $l=2$. The whole analysis correlated diverse experimental data into a unified pattern and can be extended to other values of $J$ and $l$ to other rare gases and to spectra of other elements.

••

TL;DR: In this paper, the Schrodinger equation for two electrons in a Coulomb field is studied in the critical region where both electrons have near-zero kinetic energies, and the main feature of this problem is that the mutual screening between the two electrons determines and is determined by the partition of the available energy between them.

Abstract: The Schr\"odinger equation for two electrons in a Coulomb field is studied in the critical region where both electrons have near-zero kinetic energies. The main feature of this problem is that the mutual screening between the two electrons determines and is determined by the partition of the available energy between them. This energy-dependent screening can be taken into account to yield a complex potential in the radial variable $R={({r}_{1}^{2}+{r}_{2}^{2})}^{\frac{1}{2}}$ of the six-dimensional configuration space of the two electrons. Solutions of this equation are obtained and are shown to correspond to the classical orbits given in an early paper by Wannier. A possible way is indicated of using these wave functions to establish the Wannier threshold law which, for ionization of neutral atoms, is $\ensuremath{\sigma}\ensuremath{\propto}{E}^{1.127}$. Finally, the interplay between the total energy and the Coulomb potential is discussed both for this problem and for the case of one electron in the field of a nucleus.

••

TL;DR: In this article, a model of a helium monolayer constrained to motion in two dimensions is considered, and the ground-state energy per particle and condensate depletion are found to be in the low-density limit.

Abstract: As a model of a helium monolayer a system of hard-core bosons of mass $m$ and diameter $a$ constrained to motion in two dimensions is considered at absolute zero. In the low-density limit, the ground-state energy per particle and condensate depletion are found to be $\frac{E}{N}=\ensuremath{-}\frac{2\ensuremath{\pi}{\ensuremath{\hbar}}^{2}n}{m\mathrm{ln}n{a}^{2}}$ and ${n}_{0}=n(1+\frac{1}{\mathrm{ln}n{a}^{2}})$, where $n$ is the areal density of the system. The expansion parameter $\ensuremath{-}\frac{1}{\mathrm{ln}n{a}^{2}}$ is approximately equal to unity for real helium monolayers. The variation of the above results with temperature is discussed for a system of finite size.

••

TL;DR: In this paper, a stochastic-type theory related to similar theories of infrared and NMR spectra was proposed to explain the Raman spectra of inert solutions of diatomic molecules.

Abstract: A theory is proposed to explain the Raman spectra of inert solutions of diatomic molecules; this theory can also be considered as an approximate theory of Raman spectra of pure liquids The theory is a stochastic-type theory related to similar theories of infrared and NMR spectra The vapor-solution band shifts are shown to depend on the difference between the solvent-solute interaction energies of the two vibrational states involved in the transition The analysis of the band profiles is made separately for the isotropic and anisotropic components, respectively, of a Raman spectrum The former spectrum is produced by vibrational relaxation mechanisms alone; the bands are all asymmetric although, in certain cases, the asymmetric perturbation is small enough to be neglected The latter spectrum is produced by both vibrational and reorientational relaxation mechanisms The theory predicts the existence of a continuous sequence of band forms comprising, among others, the profile with an $O\ensuremath{-}Q\ensuremath{-}S\ensuremath{-}$ type structure, the Lorentzian profile, the Gaussian profile, the Voigt profile, and several sorts of asymmetric profiles The resulting Raman spectrum appears as a superposition, with appropriate coefficients, of an isotropic and anisotropic spectrum A procedure is indicated permitting a separate study of vibrational and reorientational relaxation effects

••

TL;DR: In this paper, it has been shown that the radiative lifetime of the metastable two-electron system is determined by a single-photon magnetic dipole transition to the ground state, rather than the twophoton process proposed by Breit and Teller.

Abstract: It has recently been established that the radiative lifetime of the metastable $2^{3}S$ state of helium and the heliumlike ions is determined by single-photon magnetic dipole ($M1$) transitions to the ground state, rather than the two-photon process proposed by Breit and Teller. The theory of $nl\ensuremath{-}{n}^{\ensuremath{'}}l$ $M1$ transitions with $n\ensuremath{
e}{n}^{\ensuremath{'}}$ is developed in the Pauli approximation and extended to two-electron systems. Terms arising from relativistic energy corrections and finite-wavelength effects are included. The results for hydrogenic systems are shown to be identical to those obtained in the relativistic four-component Dirac formulation. The coefficients in the ${Z}^{\ensuremath{-}1}$ perturbation expansion of the $1s2s^{3}S\ensuremath{-}1{s}^{2}^{1}S$ $M1$ transition integral are evaluated through ninth order and used to calculate the $M1$ emission probabilities from the $2^{3}S$ state of the two-electron ions up to Fe XXV. The emission probability for neutral helium is 1.27 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}4}$ ${\mathrm{sec}}^{\ensuremath{-}1}$. The results are compared with recent solar coronal observations by Gabriel and Jordan, and with a measurement of the $2^{3}S$ state lifetime in Ar XVII by Schmieder and Marrus.

••

TL;DR: In this article, the generalization of the Breit interaction with rearrangement and its generalizations are reviewed with applications to heavy atoms in mind, and numerical results for the energy shifts of atomic ground states are presented for selected atoms in the range $Z = 2$ to $Z=102$.

Abstract: The Breit interaction is reviewed with applications to heavy atoms in mind. Generalizations of the Breit interaction which avoid expansion in powers of the electron velocities are discussed. Two-particle matrix elements of the Breit interaction and its generalizations are given in a form convenient for numerical applications. Expressions are derived for evaluating configuration-averaged atomic energy shifts for the Breit interaction and its generalizations. Numerical results for the energy shifts of atomic ground states are presented for selected atoms in the range $Z=2$ to $Z=102$; interpolated values of the energy shifts are given graphically for all atoms in the range considered. A breakdown of the interelectron contributions to the Breit energy shift is given for Ne and for $K$ electrons in Hg. "Frozenorbital" calculations of Breit corrections to electron binding energies in Hg are given. The binding of $K$ electrons in W, Hg, Pb, and Rn including the generalized Breit interaction with rearrangement are determined; when considered together with Lamb shift and correlation effects, these calculations reduce the discrepancy between theoretical and experimental $K$ binding energies to about 0.1 Ry.

••

••

Duke University

^{1}TL;DR: In this article, the rotational transitions of a number of isotopic species of the hydrogen halides have been measured in the 1.0-to 0.38-mm wavelength region of the spectrum.

Abstract: Rotational transitions of a number of isotopic species of the hydrogen halides have been measured in the 1.0- to 0.38-mm wavelength region of the spectrum. These transitions have been measured with a submillimeter-wave spectrometer which employs a klystron-driven crystal harmonic generator and a 1.6\ifmmode^\circ\else\textdegree\fi{}K InSb photoconducting detector. The following results have been obtained: for H $^{35}\mathrm{Cl}$, ${B}_{0}=312 989.297\ifmmode\pm\else\textpm\fi{}0.020$ Mc/sec, ${D}_{0}=15.836$ Mc/sec, ${r}_{e}=1.274 5991$ \AA{}; for H $^{37}\mathrm{Cl}$, ${B}_{0}=312 519.121\ifmmode\pm\else\textpm\fi{}0.020$ Mc/sec, ${D}_{0}=15.788$ Mc/sec, ${r}_{e}=1.274 5990$ \AA{}; for D $^{35}\mathrm{Cl}$, ${B}_{0}=161 656.238\ifmmode\pm\else\textpm\fi{}0.014$ Mc/sec, ${D}_{0}=4.196\ifmmode\pm\else\textpm\fi{}0.003$ Mc/sec, ${r}_{e}=1.274 5990$ \AA{}; for D $^{37}\mathrm{Cl}$, ${B}_{0}=161 183.122\ifmmode\pm\else\textpm\fi{}0.016$ Mc/sec, ${D}_{0}=4.162\ifmmode\pm\else\textpm\fi{}0.003$ Mc/sec, ${r}_{e}=1.274 5988$ \AA{}; for H $^{127}\mathrm{I}$, ${B}_{0}=192 657.577\ifmmode\pm\else\textpm\fi{}0.019$ Mc/sec, ${D}_{0}=6.203\ifmmode\pm\else\textpm\fi{}0.003$ Mc/sec, ${r}_{e}=1.609 018$ \AA{}; for D $^{127}\mathrm{I}$, ${B}_{0}=97 537.092\ifmmode\pm\else\textpm\fi{}0.009$ Mc/sec, ${D}_{0}=1.578\ifmmode\pm\else\textpm\fi{}0.001$ Mc/sec, ${r}_{e}=1.609 018$ \AA{}; for D $^{79}\mathrm{Br}$, ${B}_{0}=127 357.639\ifmmode\pm\else\textpm\fi{}0.012$ Mc/sec, ${D}_{0}=2.6529\ifmmode\pm\else\textpm\fi{}0.0014$ Mc/sec, ${r}_{e}=1.414 4698$ \AA{}; for D $^{81}\mathrm{Br}$, ${B}_{0}=127 279.757\ifmmode\pm\else\textpm\fi{}0.017$ Mc/sec, ${D}_{0}=2.6479\ifmmode\pm\else\textpm\fi{}0.0020$ Mc/sec, ${r}_{e}=1.414 4698$ \AA{}; for D $^{19}\mathrm{F}$, ${B}_{0}=325 584.98\ifmmode\pm\else\textpm\fi{}0.300$ Mc/sec, ${D}_{0}=17.64$ Mc/sec, ${r}_{e}=0.916 914$ \AA{}.

••

TL;DR: In this article, the length formula for electric dipole transitions is given by the Hartree-Fock, configuration-interaction, and related approximations, which involve the diagonalization of an approximate, but nonlocal, Hamiltonian.

Abstract: The matrix element for electric dipole transitions is correctly given by the length formula in the Hartree-Fock, configuration-interaction, and related approximations, which involve the diagonalization of an approximate, but nonlocal, Hamiltonian.

••

TL;DR: The theory of Brownian motion of a quantum oscillator is described by a model Hamiltonian which is taken to be the one describing the interaction between this oscillator and a reservoir as discussed by the authors.

Abstract: The theory of Brownian motion of a quantum oscillator is developed. The Brownian motion is described by a model Hamiltonian which is taken to be the one describing the interaction between this oscillator and a reservoir. Use is made of the master equation recently derived by the author, to obtain the equation of motion for the various reduced phase-space distribution functions that are obtained by mapping the density operator onto $c$-number functions. The equations of motion for the reduced phase-space distribution functions are found to be of the Fokker-Planck type. On transforming the Fokker-Planck equation to real variables, it is found to have the same form as the Fokker-Planck equation obtained by Wang and Uhlenbeck to describe the Brownian motion of a classical oscillator. The Fokker-Planck equation is solved for the conditional probability (Green's function) which is found to be in the form of a two-dimensional Gaussian distribution. This solution is then used to obtain various time-dependent quantum statistical properties of the oscillator. Next, the entropy for a quantum oscillator undergoing Brownian motion is calculated and we show that this system approaches equilibrium as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. Finally we show that in the weak-coupling limit the Fokker-Planck equation reduces to the one obtained by making the usual rotating-wave approximation.

••

TL;DR: In this paper, the asymptotic time behavior of the velocity autocorrelation function and of the kinetic parts of the correlation functions for the shear viscosity and the heat conductivity is derived on the basis of a local equilibrium assumption and the linearized Navier-Stokes equations.

Abstract: The asymptotic time behavior ($\ensuremath{\sim}c{t}^{\ensuremath{-}\frac{d}{2}}$, where $d$ is the dimensionality of the system) of the velocity autocorrelation function and of the kinetic parts of the correlation functions for the shear viscosity and the heat conductivity is derived on the basis of a local equilibrium assumption and the linearized Navier-Stokes equations. The coefficients $c$ are expressed in terms of the transport coefficients and thermodynamic quantities. The physical mechanism responsible for the long-time tail is indicated, and the connections between the present work and investigations based on molecular dynamics and on kinetic theory are discussed.

••

TL;DR: In this paper, the theory of measurements in which photons are detected in delayed coincidence with a scattered particle is developed in a form specifically applicable to atomic collisions, and equations which relate the measured coincidence rate to excitation amplitudes are obtained.

Abstract: The theory of measurements in which photons are detected in delayed coincidence with a scattered particle is developed in a form specifically applicable to atomic collisions. Equations are obtained which relate the measured coincidence rate to excitation amplitudes. These equations incorporate the polarization of the radiation, the fine and hyperfine structure of the atomic levels, the coherence of the radiation, and the time dependence of the radiation intensity. A semiclassical model for certain transitions is introduced to illustrate new features of coincidence measurements. The $\mathrm{Ly}\ensuremath{-}\ensuremath{\alpha}$ transitions in hydrogen and $^{1}P\ensuremath{\rightarrow}^{1}S$ transitions in He are treated in detail to illustrate the general theory.

••

TL;DR: In this article, the authors presented a statistical model of the atom and measurements of energy dissipation by protons at energies above 100 keV, which were based on a refinement of a procedure suggested by Lindhard and Scharff.

Abstract: Calculations and measurements of energy dissipation by protons at energies above \ensuremath{\sim} 100 keV are presented. The calculations, which make use of a statistical model of the atom, are based on a refinement of a procedure suggested by Lindhard and Scharff. The theoretical section of the present paper is concerned with energy straggling, as stopping powers were dealt with in an earlier publication. Measurements of stopping power and energy straggling for 100-500-keV protons have been made in various gases, viz. hydrogen, helium, air, neon, argon, and krypton. The stopping-power data are in good agreement with theory and earlier experimental work. For the heavy gases, the experimental straggling values are seen to be an increasing function of energy, as expected from theory. In a more quantitative comparison, however, some discrepancy between theory and experiment is observed.

••

TL;DR: In this article, the authors compute the correlation functions for the one-dimensional model in the presence of a magnetic field at a constant T = 0, where T is the number of points in the model.

Abstract: We compute the correlation functions $〈{S}_{0}^{x}(t){S}_{R}^{x}(0)〉$ and $〈{S}_{0}^{y}(t){S}_{R}^{y}(0)〉$ at $T=0$ for the one-dimensional $\mathrm{XY}$ model in the presence of a magnetic field.

••

TL;DR: In this article, the effective charge for the continuum-electron Coulomb wave function is taken to be the geometric mean of the effective charges appropriate for the state from which the electron originates and the next higher state.

Abstract: Radiationless transition probabilities to the atomic $1s$ shell are calculated for all transitions that contribute measurably to the Auger effect. Screened nonrelativistic hydrogenic wave functions are used. The effective charge for the continuum-electron Coulomb wave function is taken to be the geometric mean of the effective charges appropriate for the state from which the electron originates and the next-higher state. The results are combined with Scofield's radiative transition probabilities to derive theoretical $K$-shell fluorescence yields. Agreement with a selected set of most reliable ${\ensuremath{\omega}}_{K}$ measurements is very good in the range from $Z=10 \mathrm{to} 55$, and total $K$-level widths agree very well with measured values.

••

TL;DR: In this paper, a wave function for the ground state of the beryllium atom was constructed from a basis set consisting of $s$ and $p$ Slater-type orbitals and powers of interelectronic coordinates.

Abstract: A method is proposed for the accurate determination of atomic wave functions and energies by the explicit introduction of interelectronic coordinates into a configuration-interaction wave function. This is accomplished by choosing the configurations in the wave function to be antisymmetrized projected products of one-electron functions with powers of interelectronic coordinates. A 107-configuration wave function for the ground state of the beryllium atom was constructed from a basis set consisting of $s$ and $p$ Slater-type orbitals and powers of interelectronic coordinates: ${r}_{\mathrm{ij}}^{v} (v.=0,1,2)$. The energy obtained from this wave function ($E=\ensuremath{-}14.66654$ a.u.) is an upper bound to the "exact" nonrelativistic energy of this state and it is believed to be within 0.0002a. u. of the "exact" value. The advantages that the present method offers for extending accurate Hylleraas-method calculations to atomic systems with $Ng3$ are discussed.

••

TL;DR: In this article, the classical van der Waals model of fluids is modified by a more accurate equation of state for hard spheres, which is used to predict the critical constants of metals, as well as the equation of states, cohesive energy, and coexistence curves near the critical point.

Abstract: The classical van der Waals model of fluids is modified by a more accurate equation of state for hard spheres. The hard-sphere diameter and the van der Waals constant $a$ are obtained from experimental data. The model is used to predict the critical constants of metals, as well as the equation of state, cohesive energy, and coexistence curves near the critical point. The model is also applied to rare gases and ionic salts. The semiquantitative predictions of the model are at least as accurate as those of other theories. Of the three critical constants, the critical temperature is most accurately predicted, being within 11% of experiment. More information about the interatomic potential in metals is needed before the theory can be substantially improved.

••

••

TL;DR: In this article, a unitary transformation of the complete Breit Hamiltonian is made in order to simplify further the calculation of small corrections to the electron and nuclear $g$ factors.

Abstract: The interaction of hydrogenic atoms with a weak constant magnetic field is discussed in detail. The Breit Hamiltonian, minimally coupled to the external magnetic field, is treated in several different ways. First, approximate eigenfunctions are obtained in the nonrelativistic nucleus approximation. These wave functions are used to treat perturbatively the residual terms dependent on the magnetic field, and to identify the magnetic moment of the bound electron in the ground state. The corrections previously given by us, of relative order ${(Z\ensuremath{\alpha})}^{2}$, $\frac{{(Z\ensuremath{\alpha})}^{2}m}{M}$, $\ensuremath{\alpha}{(Z\ensuremath{\alpha})}^{2}$, and $\frac{\ensuremath{\alpha}{(Z\ensuremath{\alpha})}^{2}m}{M}$, are confirmed including lowest-order radiative corrections. Next, a unitary transformation of the complete Breit Hamiltonian is made in order to simplify further the calculation of small corrections to the electron and nuclear $g$ factors. The physical origin of this unitary transformation, which is similar to a gauge transformation, is discussed extensively, and it is shown that the transformed Hamiltonian for a neutral system commutes with $\stackrel{\ensuremath{\rightarrow}}{\mathrm{P}}$, the momentum conjugate to the center-of-mass position $\stackrel{\ensuremath{\rightarrow}}{\mathrm{X}}$. This new Hamiltonian, which treats the electron and the nucleus on equal footing, is then transformed by means of the Chraplyvy-Barker-Glover reduction. The electron and the nuclear $g$ factors are calculated, this time including terms of relative order $\frac{{(Z\ensuremath{\alpha})}^{2}{m}^{2}}{{M}^{2}}$ and $\frac{\ensuremath{\alpha}{(Z\ensuremath{\alpha})}^{2}{m}^{2}}{{M}^{2}}$. These computations yield the magnetic moments for hydrogenic atoms in their ground state. The theoretical results are summarized and compared with recent experiments.