Showing papers in "Physical Review in 1965"

Self-Consistent Equations Including Exchange and Correlation Effects

Abstract: From a theory of Hohenberg and Kohn, approximation methods for treating an inhomogeneous system of interacting electrons are developed. These methods are exact for systems of slowly varying or high density. For the ground state, they lead to self-consistent equations analogous to the Hartree and Hartree-Fock equations, respectively. In these equations the exchange and correlation portions of the chemical potential of a uniform electron gas appear as additional effective potentials. (The exchange portion of our effective potential differs from that due to Slater by a factor of $\frac{2}{3}$.) Electronic systems at finite temperatures and in magnetic fields are also treated by similar methods. An appendix deals with a further correction for systems with short-wavelength density oscillations.

New method for calculating the one-particle green's function with application to the electron-gas problem

Abstract: A set of successively more accurate self-consistent equations for the one-electron Green's function have been derived. They correspond to an expansion in a screened potential rather than the bare Coulomb potential. The first equation is adequate for many purposes. Each equation follows from the demand that a corresponding expression for the total energy be stationary with respect to variations in the Green's function. The main information to be obtained, besides the total energy, is one-particle-like excitation spectra, i.e., spectra characterized by the quantum numbers of a single particle. This includes the low-excitation spectra in metals as well as configurations in atoms, molecules, and solids with one electron outside or one electron missing from a closed-shell structure. In the latter cases we obtain an approximate description by a modified Hartree-Fock equation involving a "Coulomb hole" and a static screened potential in the exchange term. As an example, spectra of some atoms are discussed. To investigate the convergence of successive approximations for the Green's function, extensive calculations have been made for the electron gas at a range of metallic densities. The results are expressed in terms of quasiparticle energies E(k) and quasiparticle interactions f(k, k′). The very first approximation gives a good value for the magnitude of E(k). To estimate the derivative of E(k) we need both the first- and the second-order terms. The derivative, and thus the specific heat, is found to differ from the free-particle value by only a few percent. Our correction to the specific heat keeps the same sign down to the lowest alkali-metal densities, and is smaller than those obtained recently by Silverstein and by Rice. Our results for the paramagnetic susceptibility are unreliable in the alkali-metal-density region owing to poor convergence of the expansion for f. Besides the proof of a modified Luttinger-Ward-Klein variational principle and a related self-consistency idea, there is not much new in principle in this paper. The emphasis is on the development of a numerically manageable approximation scheme. (Less)

Thermal Properties of the Inhomogeneous Electron Gas

Abstract: A variational property of the ground-state energy of an electron gas in an external potential $v(\mathrm{r})$, derived by Hohenberg and Kohn, is extended to nonzero temperatures. It is first shown that in the grand canonical ensemble at a given temperature and chemical potential, no two $v(\mathrm{r})$ lead to the same equilibrium density. This fact enables one to define a functional of the density $F[n(\mathrm{r})]$ independent of $v(\mathrm{r})$, such that the quantity $\ensuremath{\Omega}=\ensuremath{\int}v(\mathrm{r})n(\mathrm{r})d\mathrm{r}+F[n(\mathrm{r})]$ is at a minimum and equal to the grand potential when $n(\mathrm{r})$ is the equilibrium density in the grand ensemble in the presence of $v(\mathrm{r})$.

Solution of the Schrödinger Equation with a Hamiltonian Periodic in Time

Abstract: The interaction of a quantum system with an oscillating field is studied in a formalism which replaces the semiclassical time-dependent Hamiltonian with a time-independent Hamiltonian represented by an infinite matrix. The formalism is developed as a mathematical equivalent to the semiclassical treatment, and interpreted as a classical approximation to the quantum treatment of the field. Combined with a perturbation theory for two nearly degenerate states, the formalism provides a convenient method for determining resonance transition probabilities including frequency shifts and multiple quantum transitions. The theory is illustrated by a detailed study of the simple case of a two-state system excited by a strong oscillating field.

Infrared photons and gravitons

Abstract: It is shown that the infrared divergences arising in the quantum theory of gravitation can be removed by the familiar methods used in quantum electrodynamics. An additional divergence appears when infrared photons or gravitons are emitted from noninfrared external lines of zero mass, but it is proved that for infrared gravitons this divergence cancels in the sum of all such diagrams. (The cancellation does not occur in massless electrodynamics.) The formula derived for graviton bremsstrahlung is then used to estimate the gravitational radiation emitted during thermal collisions in the sun, and we find this to be a stronger source of gravitational radiation (though still very weak) than classical sources such as planetary motion. We also verify the conjecture of Dalitz that divergences in the Coulomb-scattering Born series may be summed to an innocuous phase factor, and we show how this result may be extended to processes involving arbitrary numbers of relativistic or nonrelativistic particles with arbitrary spin.

Study of Optical Effects Due to an Induced Polarization Third Order in the Electric Field Strength

Abstract: This paper presents the results of a series of experiments in which a giant pulsed ruby laser is used to study several different nonlinear optical effects arising from an induced optical polarization third order in the electric field strength. The various phenomena studied are special cases of either frequency mixing or intensity-dependent changes in the complex refractive index, including Raman laser action at a focus. A wide range of crystalline and isotropic materials was studied. The theory for these effects is extended to cover resonant interactions. The experimental results are interpreted in terms of simplified models, and quantitative values for the nonlinear polarizability coefficients are given. The rather large experimental uncertainties in these coefficients are discussed.

Theory of the Motion of Vortices in Superconductors

Abstract: The theory of the motion of vortex lines in the mixed state of type II superconductors is derived on the basis of a local model that is a generalization of the London theory It is believed the model simulates reasonably well the behavior of relatively pure superconductors ($lg{\ensuremath{\xi}}_{0}$), giving a vortex line with a normal core It is found that if the force on a line is produced by a uniform transport current ${\mathrm{J}}_{T}$, electric fields generated by the motion drive the current through the core, so that the total current flow is ${\mathrm{J}}_{T}+{\mathrm{J}}_{0}(\mathrm{r}\ensuremath{-}{\mathrm{v}}_{L}t)$, where ${\mathbf{J}}_{0}$(r) is the circulation of a stationary vortex and ${\mathrm{v}}_{L}$ is the velocity of the line In part ${\mathrm{J}}_{T}$ represents superfluid flow and in part normal flow Expressions derived for the viscosity and flow resistivity are nearly identical with empirical laws of Kim and co-workers The Hall angle expected in the mixed state is the same as in the normal state for a magnetic field equal to that in the core

Theory of Surface States

Abstract: The properties of metal-to-semiconductor junctions and of free semiconductor surfaces are usually explained on the basis of surface states. The theory of the metal contacts is discussed critically, because strictly speaking localized surface states cannot exist in such junctions. However, it is shown that virtual or resonance surface states can exist which behave for practical purposes in the same way. They are really the tails of the metal wave functions rather than separate states. In the past, the length of this tail has often been ignored. Some estimates of its length are made and its consequences pointed out. A semiquantitative discussion is given of various recent data, including the effect of an oxide layer on barrier height, the variation of barrier height with the metal, the work function of a free surface at high doping, and the effect of a cesium layer on the work function.

Classical Theory of Atomic Collisions. I. Theory of Inelastic Collisions

Abstract: In this paper, a classical theory of inelastic atomic collisions is evolved on the basis of the relations for binary collisions as well as for the Coulomb collisions derived in the laboratory system of coordinates. Built up as an approximation based on the binary collisions, i.e., the independent pair interactions of the individual elements of the colliding systems, the theory, with its immense simplicity, not only permits a clear qualitative interpretation of the atomic collisions, but also describes well their quantitive aspect. In terms of that theory, a majority of basic inelastic processes accompanying the atomic collisions are analyzed. In particular, calculations are made for the following: (i) ionization of atoms and molecules by light particles (electrons), as well as by heavy particles (protons, deuterons), including inner-shell ionization and double ionization; (ii) excitation of single and triplet lines (excitation with exchange and without exchange); (iii) capture of electrons in orbit; (iv) slowing down of heavy charged particles (with consideration of the capture process as well as of the interaction with the Coulomb field of the nucleus); (v) inelastic scattering of electrons on atoms and molecules. According to the theory developed, the "diffraction" of elementary particles on atomic systems is explained on the basis of corpuscular mechanics; it is shown that the discrete energy states of the scatterer electrons---and the anisotropy in the space orientation of their velocities in the case of crystals---are responsible for the main features of the diffraction pattern. Having at our disposal a simple theory without any arbitrary parameters except those describing the target system, we find it a useful tool for the investigation of atomic structures.

Optical Absorption and Photoconductivity in the Band Edge of β − Ga 2 O 3

Abstract: Optical absorption and photoconductivity have been observed in the ultraviolet in single crystals of nominally pure $\ensuremath{\beta}\ensuremath{-}{\mathrm{Ga}}_{2}{\mathrm{O}}_{3}$. At room temperature a steep absorption edge, characteristic of a band-to-band transition, is observed at 2700 \AA{}. The edge is shifted approximately 100 \AA{} toward shorter wavelengths when the temperature is reduced to 77\ifmmode^\circ\else\textdegree\fi{}K. Photoconductivity begins coincident with the absorption edge at 77\ifmmode^\circ\else\textdegree\fi{}K, but could not be detected at room temperature. A model is proposed in which the absorption arises as a result of excitation of an electron from the oxygen $2p$ band to the gallium $4s$ band. Calculations using this model and the Born-Haber cycle are in good agreement with the observed band gap of 4.7 eV. It is suggested that the much smaller band gap of $\ensuremath{\beta}\ensuremath{-}{\mathrm{Ga}}_{2}{\mathrm{O}}_{3}$ as compared with sapphire is due to the reduced coordination number of the ions involved in the transition.

Optical Properties of Graphite

Abstract: The complex dielectric constant $\ensuremath{\epsilon}(\ensuremath{\omega})={\ensuremath{\epsilon}}_{1}+i{\ensuremath{\epsilon}}_{2}$ and associated functions are derived by application of the Kramers-Kronig relation to reflectance data for graphite obtained in the energy range to 26 eV. It is possible to divide the optical properties into two spectral regions. In the range 0 to 9 eV, intra- and interband transitions involve mainly the $\ensuremath{\pi}$ bands. At higher energies, a broad absorption peak near 15 eV is associated with interband transitions involving the 3 $\ensuremath{\sigma}$ electrons per atom. This viewpoint is strongly supported by evaluation of the sum rules for ${n}_{\mathrm{eff}}$. Plasma resonances which produce peaks in the energy-loss function $\ensuremath{-}\mathrm{Im}{\ensuremath{\epsilon}}^{\ensuremath{-}1}$ at 7 and 25 eV are identified and described physically. At low energies, structure in the reflectance curve near 0.8 eV is attributed to the onset of transitions between the ${E}_{2}$ and ${E}_{3}$ bands at the point $K$. This yields a value for ${\ensuremath{\gamma}}_{1}$ of \ensuremath{\approx}0.4 eV.

Correlation of Electrons in a Narrow s Band

Abstract: The ground-state wave function for the electrons in a narrow $s$ band is investigated for arbitrary density of electrons and arbitrary strength of interaction. An approximation is proposed which limits all the calculations to counting certain types of configurations and attaching the proper weights. The expectation values of the one-particle and two-particle density matrix are computed for the ferromagnetic and for the non-ferromagnetic case. The ground-state energy is obtained under the assumption that only the intra-atomic Coulomb interaction is of importance. Ferromagnetism is found to occur if the density of states is large at the band edges rather than in the center, and if the intra-atomic Coulomb repulsion is sufficiently strong. The relation of this approximation to certain exact results for one-dimensional models is discussed.

Kinetics of Radiative Recombination at Randomly Distributed Donors and Acceptors

Abstract: The kinetics of the recombination of holes trapped on acceptors with electrons trapped on donors have been treated theoretically and the results compared with the observed low-temperature fluorescent decay of GaP doped with sulphur donors and silicon acceptors. The assumptions are made that the distribution of donors and acceptors is random and that the recombination rate depends exponentially on the donor and acceptor separation. The theoretical problem can be exactly solved when either the donor or acceptor is in excess, and an approximate solution is given for the case of exact compensation. The decay of the total pair emission has been measured over many factors of 10 of intensity and time and can be satisfactorily accounted for by the theory using only two adjustable parameters. If the decay occurs from a system in which all the donors and acceptors are not initially neutral, different decay curves are obtained which can be explained by assuming that the capture cross section of a pair for a hole or electron depends approximately on the square of the internuclear pair separation. The spectra have also been measured after flash excitation, and they are found to change in shape and position as a function of time. These changes can be quantitatively explained in terms of the simple theory, provided residual broadening effects from phonon interactions and other sources are included. The effects of a magnetic field at low temperature on the decay kinetics have been predicted and observed and allow the determination of certain $g$ values. At high-impurity concentrations there are deviations from simple behavior, evidently caused by complications in the chemical doping of the crystals. Experimental evidence for pair effects in the edge emission of CdS is also briefly reported. It is likely that similar decay phenomena observed in other seniconductors and phosphors can be explained on a similar basis without invoking arbitrary trap distributions. It is only necessary to assume a random distribution of donors and acceptors throughout the crystal.

Line Profiles in the Far-uv Absorption Spectra of the Rare Gases

Abstract: The line profiles of a dozen series of widely different appearance, observed recently in the rare-gas optical absorption spectra between 20 and 100 eV, are analyzed by adapting and extending previously developed theory. Theory characterizes each profile by its linewidth, $q$ index, and correlation coefficient $\ensuremath{\rho}$ and relates these parameters to matrix elements of the energy and of the electric dipole moment. Because the profiles result from interference effects, the signs of these matrix elements are highly relevant. Crude estimates of such, hitherto hardly accessible, matrix elements are obtained from experimental evidence and from atomic theory, and are intercompared and found to be in general agreement. Similarities and differences among the line profiles are thereby tentatively explained. The connection of Rydberg series and absorption edges is discussed and the occurrence of downward jumps at some edges is explained. This exploratory investigation demonstrates how one can treat experimental data, and it surveys the kind and volume of information on the dynamics of highly and multiply excited atoms made accessible by recent experimental techniques.

Dynamical Jahn-Teller Effect in Paramagnetic Resonance Spectra: Orbital Reduction Factors and Partial Quenching of Spin-Orbit Interaction

Abstract: It is shown that the dynamical Jahn-Teller effect in a complex having orbital degeneracy may partially quench spin-orbit interaction, the orbital parts of the Zeeman and hyperfine interactions, and other orbital operators governing response to perturbations such as strain or applied electric fields. Such dynamical quenching thus decreases the value of orbital reduction factors usually attributed in paramagnetic resonance studies to covalent bonding, without necessarily causing anisotropy in the spectrum of an individual complex. The dynamical Jahn-Teller effect may also substantially enhance various second-order effects. Such dynamic effects thus may make important changes in the parameters of the spin Hamiltonian without changing its symmetry. It is shown that the dynamical Jahn-Teller effect accounts qualitatively for unusual features in the spectra of interstitial transition-metal ions ${\mathrm{Cr}}^{0}$, ${\mathrm{Mn}}^{+}$, ${\mathrm{Mn}}^{0}$, and ${\mathrm{Fe}}^{+}$ in silicon and that it is probably of importance equal to or greater than that of covalent bonding in the interpretation of the spectrum of ${\mathrm{Fe}}^{2+}$ in MgO and CaO. A mathematical analysis of the dynamical effects is given for an orbital triplet state in interaction with a doublet or triplet vibrational mode, and some results are given also when the coupling is with the phonon continuum.

Theory of Stimulated Brillouin and Raman Scattering

Abstract: Stimulated Raman and Brillouin scattering can be described as the interaction of several light waves with optical and acoustic-phonon waves, respectively. The coupling parameters can be derived both classically and quantum mechanically. A prototype solution is given for the coupling of infinite plane Stokes and anti-Stokes waves satisfying appropriate boundary conditions on a plane-parallel Raman cell, in which the laser intensity is assumed a constant parameter. Saturation effects, generation of higher order Raman radiation, and the effect of mode structure in the laser beam are treated in a more approximate and qualitative fashion. The theory can explain at least qualitatively most of the experimental findings, including the directional properties of the Raman radiation. Ideal experiments for clarifying the mechanism of the Raman effect are suggested.

Consistency Conditions on the Strong Interactions Implied by a Partially Conserved Axial-Vector Current

Abstract: Consequences of the partially conserved axial-vector current (PCAC) hypothesis are explored. A set of simple rules is derived which relate the matrix element for any strong interaction process with the matrix element for the corresponding process in which an additional zero-mass, zero-energy pion is emitted or absorbed. A generalization to include lowest order electromagnetic processes is given. A theorem is stated and proved which shows how divergence equations of the form ${\ensuremath{\partial}}_{\ensuremath{\lambda}}{J}_{\ensuremath{\lambda}}=D$ are modified when a minimal electromagnetic interaction is switched on.

Optical Properties and Band Structure of SrTi O 3 and BaTi O 3

Abstract: The normal-incidence reflectance spectra of SrTi${\mathrm{O}}_{3}$ and BaTi${\mathrm{O}}_{3}$ and the optical constants derived by the Kramers-Kronig method are reported. These spectra are very similar to those of rutile. The strongest optical transitions ($A$) occur at about 4.4 eV: They are responsible for the dispersion observed below the fundamental absorption edge (3.2 eV). A tentative interpretation of these spectra in terms of the energy band structure of the materials is offered. The $A$ peak seems due to transitions between $2p$ levels of oxygen and $3s$ levels of titanium at the $X$ point of the Brillouin zone. The splitting of the $A$ peak is assigned to the splitting of the $2p$ levels of the oxygen in the cubic field. Transitions between $2p$ oxygen and $4s$ titanium levels, between $2s$ oxygen and $3d$ titanium levels, and between $2s$ oxygen and $5s$ titanium or $6s$ barium levels are also observed. Plasma oscillations are seen around 21 eV.

Photons and gravitons in perturbation theory: Derivation of maxwell's and einstein's equations

Abstract: The $S$ matrix for photon and graviton processes is studied in perturbation theory, under the restriction that the only creation and annihilation operators for massless particles of spin $j$ allowed in the interaction are those for the physical states with helicity $\ifmmode\pm\else\textpm\fi{}j$. The most general covariant fields that can be constructed from such operators cannot represent real photon and graviton interactions, because they give amplitudes for emission or absorption of massless particles which vanish as ${p}^{j}$ for momentum $p\ensuremath{\rightarrow}0$. In order to obtain long-range forces it is necessary to introduce noncovariant "potentials" in the interaction, and the Lorentz invariance of the $S$ matrix requires that these potentials be coupled to conserved tensor currents, and also that there appear in the interaction direct current-current couplings, like the Coulomb interaction. We then find that the potentials for $j=1$ and $j=2$ must inevitably satisfy Maxwell's and Einstein's equations in the Heisenberg representation. We also show that although the existence of magnetic monopoles is consistent with parity and time-reversal invariance [provided that $P$ and $T$ are defined to take a monopole into its antiparticle], it is nevertheless impossible to construct a Lorentz-invariant $S$ matrix for magnetic monopoles and charges in perturbation theory.

Nuclear-Magnetic-Resonance Line Narrowing by a Rotating rf Field

Abstract: A nuclear-magnetic-resonance method is explored, which effectively attenuates the dipolar interaction in solids. The experimental technique corresponds to the observation of a free-induction decay in a frame of reference rotating with the frequency of an applied rf field. When the amplitude ${H}_{1}$ of this field is much greater than the local field in the solid, and when its frequency is appropriately chosen, the secular part of the dipolar interaction is removed. As a result the rotary saturation line is extremely narrowed. At smaller values of ${H}_{1}$, nonsecular terms in the dipolar interaction come into play and contribute to line broadening. These nonsecular effects are investigated both theoretically and experimentally. All the measurements were made in single crystals of calcium fluoride. The calculation of the nonsecular contribution to the line width utilizes the unitary transformation method of Jordhal and Pryce. Theory and experiment are in good agreement.

Three Triplet Model with Double SU(3) Symmetry

Abstract: With a view to avoiding some of the kinematical and dynamical difficulties involved in the single-triplet quark model, a model for the low-lying baryons and mesons based on three triplets with integral charges is proposed, somewhat similar to the two-triplet model introduced earlier by one of us (Y. N.). It is shown that in a $U(3)$ scheme of triplets with integral charges, one is naturally led to three triplets located symmetrically about the origin of ${I}_{3}\ensuremath{-}Y$ diagram under the constraint that the Nishijima-Gell-Mann relation remains intact. A double $\mathrm{SU}(3)$ symmetry scheme is proposed in which the large mass splittings between different representations are ascribed to one of the $\mathrm{SU}(3)$, while the other $\mathrm{SU}(3)$ is the usual one for the mass splittings within a representation of the first $\mathrm{SU}(3)$.

Defects in Irradiated Silicon: Electron Paramagnetic Resonance of the Divacancy

Abstract: Two electron paramagnetic resonance spectra produced in silicon by 1.5-MeV electron irradiation are described. Labeled Si-G6 and Si-G7, they are identified as arising from the singly positive and singly negative charged states of the divacancy, respectively. The observed hyperfine interactions with neighboring ${\mathrm{Si}}^{29}$ nuclei and $g$ tensors are discussed in terms of a simple molecular-orbital treatment of the defect by the method of linear combination of atomic orbitals. In addition to the anisotropy associated with the vacancy-vacancy direction in the lattice, an additional distortion occurs which is identified as a manifestation of the Jahn-Teller effect. Thermally activated reorientation from one Jahn-Teller distortion direction to another causes motional broadening and narrowing effects upon both spectra in the temperature region 40-110\ifmmode^\circ\else\textdegree\fi{}K. The motion is also studied by stress-induced alignment at lower temperatures, and the activation energy for the process is found to be \ensuremath{\sim}0.06 eV for each charge state. Alignment of the vacancy-vacancy axis direction in the lattice is also achieved by stressing at elevated temperatures. The activation energy for this reorientation process is \ensuremath{\sim}1.3 eV. The magnitude and sense of the alignment in both kinds of stress experiments are consistent with the microscopic model of the defect. It is pointed out that when the divacancy reorients its vacancy-vacancy axis, it is also diffusing through the lattice. The 1.3 eV is therefore also its activation energy for diffusion. Analysis of higher temperature annealing studies allow a lower limit estimate for the binding energy of the two vacancies as $\ensuremath{\gtrsim}1.6$ eV. The electrical level structure is deduced and it is concluded that the divacancy introduces one donor and two acceptor levels in the forbidden gap.

Optical Modes of Vibration in an Ionic Crystal Slab

Abstract: The normal modes of vibration in an ionic crystal of finite thickness are found both by using lattice dynamics and by using electrodynamics, and neglecting retardation If the wavelength is much larger than the lattice parameter, both methods give coupled integral equations involving the ionic displacements and the normal-mode frequencies There are two classes of normal modes; those with an oscillatory spatial dependence and frequencies equal to ${\ensuremath{ u}}_{\mathrm{TO}}$ and ${\ensuremath{ u}}_{\mathrm{LO}}$, the usual transverse optical (TO) and longitudinal optical (LO) frequencies at $k\ensuremath{\approx}0$ in an infinite crystal, and those with an exponential dependence on distance across the slab and frequencies between ${\ensuremath{ u}}_{\mathrm{TO}}$ and ${\ensuremath{ u}}_{\mathrm{LO}}$ A qualitative connection between the normal modes and optical absorption in a slab is presented

Quantum Density Oscillations in an Inhomogeneous Electron Gas

Abstract: An infinite system of electrons in its ground state, subject to a very slowly varying external potential, has slowly varying properties, and can be described by a gradient expansion theory. However, when, in addition, either (1) there is a spatially rapidly varying perturbing potential, or (2) there are regions in which the electron density drops to zero (e.g., electrons in an oscillator potential), the density and other properties of the system exhibit additional spatial oscillations, which we call quantum oscillations. An example are the Friedel oscillations in metals. In the present paper we develop a general theory of these oscillations for one-dimensional, noninteracting electrons. Illustrations are worked out which show that when the quantum oscillations are superposed on the smooth "semiclassical" results, one can obtain very accurate approximations to the exact densities.

Self-Consistent-Field Wave Functions for Hole States of Some Ne-Like and Ar-Like Ions

Abstract: Analytic self-consistent-field (SCF) wave functions were computed for the ground states of the closed-shell atomic systems ${\mathrm{F}}^{\ensuremath{-}}$, Ne, and ${\mathrm{Na}}^{+}$, and ${\mathrm{Cl}}^{\ensuremath{-}}$, Ar, and ${\mathrm{K}}^{+}$; and for those ground and excited states of the open-shell systems which are obtained by removing a single electron from any one of the occupied shells of these closed-shell systems. Details of the calculation of the functions are presented, with emphasis on a justification of the procedures used for the calculations for excited states. A high accuracy is obtained; the calculations for the closed-shell systems give the most accurate analytic SCF wave functions which have yet been reported. Ionization potentials are calculated and compared with experimental values. Computed ionization potentials for the removal of a $2s$ electron from ${\mathrm{Cl}}^{\ensuremath{-}}$, Ar, and ${\mathrm{K}}^{+}$, for which no direct experimental data are available, are estimated to be accurate to 1%. It is found that the removal of an electron from the outermost $s$ shell increases the correlation energy, in contradiction to the predictions of a recently proposed semiempirical scheme for estimating the correlation energy. For example, the magnitude of the correlation energy of the lowest $^{2}S$ state of ${\mathrm{Ar}}^{+}$ is \ensuremath{\sim}4 eV greater than the magnitude of the correlation energy of neutral Ar. The effect of the nonzero off-diagonal Lagrangian multipliers is considered and found to be important for the inner shell hole states.

Ground State of Liquid He 4

Abstract: The properties of the ground state of liquid ${\mathrm{He}}^{4}$ are studied using a variational wave function of the form ${\ensuremath{\Pi}}_{ilj}f({r}_{\mathrm{ij}})$. The Lennard-Jones 12-6 potential is used with parameters determined from the gas data by deBoer and Michiels. The configuration space integrals are performed by a Monte Carlo technique for 32 and 108 atoms in a cube with periodic boundary conditions. With $f(r)=\mathrm{exp}[\ensuremath{-}{(\frac{2.6 \mathrm{\AA{}}}{r})}^{5}]$, the ground-state energy is found to be -0.78 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}15}$ ergs/atom, which is 20% above the experimental value. The liquid structure factor and the two-particle correlation function are in reasonably good agreement with the x-ray and neutron scattering experiments.

Two-Particle Collisions. I. General Relations for Collisions in the Laboratory System

Abstract: The relations between the energies and angles for two-particle collisions are given for the case of conservative central forces. They have been derived in the laboratory system of coordinates, since this system is of prime importance for the interpretation of the experimental results. The dependence on the law of interaction enters the obtained relations through the function for the c.m.-system scattering angle. Some remarks on the statistics of encounters are given, and the cross sections for collisions in which the colliding particles experience given changes in energy and direction are derived. These cross sections can be used to determine the cross sections describing various physical processes. On the basis of the cross sections derived for the scattering problem, one can suggest that the "diffraction" pattern of scattered particles is due to quantized states of the scattering particles, rather than to the law of interaction.

Electrical and Optical Properties of Epitaxial Films of PbS, PbSe, PbTe, and SnTe

Abstract: Single-crystal films of PbS, PbTe, PbSe, and SnTe have been grown on heated alkali-halide substrates. The temperature dependence of the mobility, Hall coefficient, and resistivity between 77\ifmmode^\circ\else\textdegree\fi{}K and 300\ifmmode^\circ\else\textdegree\fi{}K and the dependence of the magnetoresistance upon sample orientation and magnetic field strength at 77\ifmmode^\circ\else\textdegree\fi{}K have been studied. Analysis of the refractive indices, measured interferometrically in the 2.0- to 15.0-\ensuremath{\mu} region, has yielded optical dielectric constants and the direct energy gaps as functions of temperature. These studies indicate that the single-crystal films have electrical and optical properties comparable to those found in bulk material. Discussions of film formation and strain phenomena are presented and compared with the experimental results. Some of the limitations of these materials are discussed with particular emphasis on the role of structure of the films on the electrical properties.

Flux-flow resistance in type-ii superconductors

Abstract: Electrical resistance of type-II superconductors measured by a dc method is related to the motion of Abrikosov vortices. Flux lines in the mixed state are driven into a viscous-flow state by increasing the Lorentz force J\ifmmode\times\else\texttimes\fi{}H beyond the depinning threshold; the voltage observed under this condition is linear in current and the slope $\frac{\ensuremath{\Delta}V}{\ensuremath{\Delta}I}$ is interpreted to represent the flow rate. At low temperatures, the flow resistivity ${\ensuremath{\rho}}_{f}$ so measured follows an empirical relation $\frac{{\ensuremath{\rho}}_{f}}{{\ensuremath{\rho}}_{n}}=\frac{H}{{H}_{c2}}$, where ${\ensuremath{\rho}}_{n}$ is the normal-state resistivity and ${H}_{c2}$ the upper critical field predicted by the Ginsburg-Landau-Abrikosov-Gor'kov (GLAG) theory. In high-field superconductors the GLAG upper critical fields are not accessible for conventional measurements since the paramagnetic spin alignment quenches superconductivity at much lower fields, but the above empirical rule still holds and makes it possible to determine the GLAG upper critical fields from the flow-resistivity measurements. The empirical relation also suggests that the frictional drag on a moving flux line arises from normal currents flowing in the core of the vortex line. The observed damping is, however, much larger than can be accounted for by eddy currents induced by the moving magnetic field of the vortex line. Two new mechanisms, one proposed by Tinkham and another by Stephen and Bardeen, appear to correct this difficulty and they are discussed in the text. Also reported is a new type of resistance-minimum phenomenon: if the temperature $T$ is increased from zero at a fixed field $H$, the resistance $R(T)$ of a type-II specimen decreases initially and goes through a minimum before it reaches the normal resistance at ${T}_{c}(H)$. This behavior is qualitatively accounted for by including the dissipation outside of the vortex core.

Magnetostriction, Forced Magnetostriction, and Anomalous Thermal Expansion in Ferromagnets

Abstract: In previous papers we have formulated the theory of magnetostriction arising from single-ion crystal-field effects, for cubic crystals, and we have shown that the theory accounts extremely well for the temperature dependence of the magnetostriction in yttrium iron garnet. We here summarize that theory, extend it to arbitrary crystal symmetry, augment it by the inclusion of two-ion interactions, analyze the dependence on magnetic field strength (the "forced magnetostriction"), and discuss the totally symmetric component (which exhibits itself as an anomalous thermal expansion). The first part of this paper is concerned with the above matters, and the analysis is based entirely on symmetry considerations. It culminates in expressions relating the macroscopic magnetostriction coefficients to the product of microscopic magnetoelastic coupling constants and certain spin correlation functions. Only three such correlation functions appear, and all temperature and field dependence enters through these correlation functions. In the second part of the paper, we evaluate these correlation functions by various approximate theories: molecular field theory, a cluster theory, and the random-phase approximation. Applications to Dy and to EuS are cited, and a detailed application to Gd is given. In Gd the sign inversion of a particular magnetostriction coefficient, and its full temperature dependence, are accurately accounted for theoretically. The field dependence of the forced magnetostriction of Gd is also discussed, with special reference to the observed persistence of a pseudo-linearity slightly above the Curie temperature.