Showing papers in "Physical Review in 1962"

Optical absorption intensities of rare-earth ions

Abstract: Electric dipole transitions within the $4f$ shell of a rare-earth ion are permitted if the surroundings of the ion are such that its nucleus is not situated at a center of inversion. An expression is found for the oscillator strength of a transition between two states of the ground configuration $4{f}^{N}$, on the assumption that the levels of each excited configuration of the type $4{f}^{N}{n}^{\ensuremath{'}}d$ or $4{f}^{N}{n}^{\ensuremath{'}}g$ extend over an energy range small as compared to the energy of the configuration above the ground configuration. On summing over all transitions between the components of the ground level ${\ensuremath{\psi}}_{J}$ and those of an excited level ${{\ensuremath{\psi}}^{\ensuremath{'}}}_{{J}^{\ensuremath{'}}}$, both of $4{f}^{N}$, the oscillator strength $P$ corresponding to the transition ${\ensuremath{\psi}}_{J}\ensuremath{\rightarrow}{{\ensuremath{\psi}}^{\ensuremath{'}}}_{{J}^{\ensuremath{'}}}$ of frequency $\ensuremath{ u}$ is found to be given by $P=\ensuremath{\Sigma}{T}_{\ensuremath{\lambda}}\ensuremath{ u}{({\ensuremath{\psi}}_{J}\ensuremath{\parallel}{U}^{(\ensuremath{\lambda})}\ensuremath{\parallel}{{\ensuremath{\psi}}^{\ensuremath{'}}}_{{J}^{\ensuremath{'}}})}^{2},$ where ${\mathrm{U}}^{(\ensuremath{\lambda})}$ is a tensor operator of rank $\ensuremath{\lambda}$, and the sum runs over the three values 2, 4, and 6 of $\ensuremath{\lambda}$. Transitions that also involve changes in the vibrational modes of the complex comprising a rare-earth ion and its surroundings, provide a contribution to $P$ of precisely similar form. It is shown that sets of parameters ${T}_{\ensuremath{\lambda}}$ can be chosen to give a good fit with the experimental data on aqueous solutions of Nd${\mathrm{Cl}}_{3}$ and Er${\mathrm{Cl}}_{3}$. A calculation on the basis of a model, in which the first hydration layer of the rare-earth ion does not possess a center of symmetry, leads to parameters ${T}_{\ensuremath{\lambda}}$ that are smaller than those observed for ${\mathrm{Nd}}^{3+}$ and ${\mathrm{Er}}^{3+}$ by factors of 2 and 8, respectively. Reasons for the discrepancies are discussed.

Interactions between light waves in a nonlinear dielectric

Abstract: The induced nonlinear electric dipole and higher moments in an atomic system, irradiated simultaneously by two or three light waves, are calculated by quantum-mechanical perturbation theory. Terms quadratic and cubic in the field amplitudes are included. An important permutation symmetry relation for the nonlinear polarizability is derived and its frequency dependence is discussed. The nonlinear microscopic properties are related to an effective macroscopic nonlinear polarization, which may be incorporated into Maxwell's equations for an infinite, homogeneous, anisotropic, nonlinear, dielectric medium. Energy and power relationships are derived for the nonlinear dielectric which correspond to the Manley-Rowe relations in the theory of parametric amplifiers. Explicit solutions are obtained for the coupled amplitude equations, which describe the interaction between a plane light wave and its second harmonic or the interaction between three plane electromagnetic waves, which satisfy the energy relationship ${\ensuremath{\omega}}_{3}={\ensuremath{\omega}}_{1}+{\ensuremath{\omega}}_{2}$, and the approximate momentum relationship ${\mathrm{k}}_{3}={\mathrm{k}}_{1}+{\mathrm{k}}_{2}+\ensuremath{\Delta}\mathrm{k}$. Third-harmonic generation and interaction between more waves is mentioned. Applications of the theory to the dc and microwave Kerr effect, light modulation, harmonic generation, and parametric conversion are discussed.

Nonlinear Dielectric Polarization in Optical Media

Abstract: The physical mechanisms which can produce second-order dielectric polarization are discussed on the basis of a simple extension of the theory of dispersion in ionic crystals. Four distinct mechanisms are described, three of which are related to the anharmonicity, second-order moment, and Raman scattering of the lattice. These mechanisms are strongly frequency dependent, since they involve ionic motions with resonant frequencies lower than the light frequency. The other mechanism is related to electronic processes of higher frequency than the light, and, therefore, is essentially flat in the range of the frequencies of optical masers. Since this range lies an order of magnitude higher than the ionic resonances, the fourth mechanism may be the dominant one. On the other hand, a consideration of the linear electro-optic effect shows that the lattice is strongly involved in this effect, and, therefore, may be very much less linear than the electrons. It is shown that the question of the mechanism involved in the second harmonic generation of light from strong laser beams may be settled by experiments which test the symmetry of the effect. The electronic mechanism is subject to further symmetry requirements beyond those for piezoelectric coefficients. In many cases, this would greatly reduce the number of independent constants describing the effect. In particular, for quartz and KDP there would be a single constant.

Wave-Number-Dependent Dielectric Function of Semiconductors

Abstract: Expressions for the wave-number-dependent dielectric function are derived for various models of a semiconductor The calculation is carried out for the diagonal part of the dielectric function at zero frequency It is found that calculations based on plane wave models (such as the free electron model) give poor results for small values of the wave number due to neglect of both Bragg reflections and Umklapp processes We use instead an isotropic version of the nearly free electron model in which dielectric function depends on only one parameter ${E}_{g}$ representing an average energy gap that can be determined from optical data It is noted that for small wave numbers Umklapp processes give the major contribution to the dielectricfunction, where-as for large wave numbers normal processes dominate The dielectric function is evaluated numerically for a value of ${E}_{g}$ appropriate to Si

Nuclear Double Resonance in the Rotating Frame

Abstract: A double nuclear resonance spectroscopy method is introduced which depends upon effects of magnetic dipole-dipole coupling between two different nuclear species. In solids a minimum detectability of the order of ${10}^{14}$ to ${10}^{16}$ nuclear Bohr magnetons/cc of a rare $b$ nuclear species is predicted, to be measured in terms of the change in a strong signal displayed by an abundant $a$ nuclear species. The $a$ magnetization is first oriented by a strong radio-frequency field in the frame of reference rotating at its Larmor frequency. The $b$ nuclear resonance is obtained simultaneously with a second radio-frequency field; and with the condition that the $a$ and $b$ spins have the same Larmor frequencies in their respective rotating frames, a cross relaxation will occur between the two spin systems. The cross-relaxation interaction, which lasts for the order of a long spin-lattice relaxation time of the $a$ magnetization, is arranged to produce a cumulative demagnetization of the $a$ system when maximum sensitivity is desired. Final observation of the reduced $a$ magnetization indicates the nuclear resonance of the $b$ system. The concepts of uniform spin temperature, when it is valid, and of nonuniform spin temperature where spin diffusion is important, are applied. The density matrix method formulates the double resonance interaction rate in second order. Preliminary tests of the double resonance effect are carried out with a nuclear quadrupole system.

Symmetries of baryons and mesons

Abstract: The system of strongly interacting particles is discussed, with electromagnetism, weak interactions, and gravitation considered as perturbations. The electric current jα, the weak current Jα, and the gravitational tensor θαβ are all well-defined operators, with finite matrix elements obeying dispersion relations. To the extent that the dispersion relations for matrix elements of these operators between the vacuum and other states are highly convergent and dominated by contributions from intermediate one-meson states, we have relations like the Goldberger-Treiman formula and universality principles like that of Sakurai according to which the ρ meson is coupled approximately to the isotopic spin. Homogeneous linear dispersion relations, even without subtractions, do not suffice to fix the scale of these matrix elements; in particular, for the nonconserved currents, the renormalization factors cannot be calculated, and the universality of strength of the weak interactions is undefined. More information than just the dispersion relations must be supplied, for example, by field-theoretic models; we consider, in fact, the equal-time commutation relations of the various parts of j4 and J4. These nonlinear relations define an algebraic system (or a group) that underlies the structure of baryons and mesons. It is suggested that the group is in fact U(3)×U(3), exemplified by the symmetrical Sakata model. The Hamiltonian density θ44 is not completely invariant under the group; the noninvariant part transforms according to a particular representation of the group; it is possible that this information also is given correctly by the symmetrical Sakata model. Various exact relations among form factors follow from the algebraic structure. In addition, it may be worthwhile to consider the approximate situation in which the strangeness-changing vector currents are conserved and the Hamiltonian is invariant under U(3); we refer to this limiting case as "unitary symmetry." In the limit, the baryons and mesons form degenerate supermultiplets, which break up into isotopic multiplets when the symmetry-breaking term in the Hamiltonian is "turned on." The mesons are expected to form unitary singlets and octets; each octet breaks up into a triplet, a singlet, and a pair of strange doublets. The known pseudoscalar and vector mesons fit this pattern if there exists also an isotopic singlet pseudoscalar meson χ0. If we consider unitary symmetry in the abstract rather than in connection with a field theory, then we find, as an attractive alternative to the Sakata model, the scheme of Ne'eman and Gell-Mann, which we call the "eightfold way"; the baryons N, Λ, Σ, and Ξ form an octet, like the vector and pseudoscalar meson octets, in the limit of unitary symmetry. Although the violations of unitary symmetry must be quite large, there is some hope of relating certain violations to others. As an example of the methods advocated, we present a rough calculation of the rate of K+→μ++ν in terms of that of π+→μ++ν.

Gauge Invariance and Mass. II

Abstract: The possibility that a vector gauge field can imply a nonzero mass particle is illustrated by the exact solution of a one-dimensional model.

Self-Consistent Approximations in Many-Body Systems

Abstract: The criteria for maintenance of the macroscopic conservation laws of number, momentum, and energy by approximate two-particle correlation functions in manybody systems are investigated. The methods of generating such approximations are the same as in a previous paper. However, the derivations of the conservation laws given here clarify both why the approximation method works and the connection between the macroscopic conservation laws and those at the vertices. Conserving nonequilibrium approximations are based on self-consistent approximations to the one-particle Green's function. The same condition that ensures that the nonequilibrium theory be conserving also ensures that the equilibrium approximation has the following properties. The several common methods for determining the partition function from the one-particle Green's function all lead to the same result. When applied to a zero-temperature normal fermion system, the approxi-mation procedure maintains the Hugenholtz-Van Hove theorem. Consequently, the self-consistent version of Brueckner's nuclear matter theory obeys this theorem. (auth)

Light waves at the boundary of nonlinear media

Abstract: Solutions to Maxwell's equations in nonlinear dielectrics are presented which satisfy the boundary conditions at a plane interface between a linear and nonlinear medium. Harmonic waves emanate from the boundary. Generalizations of the well-known laws of reflection and refraction give the direction of the boundary harmonic waves. Their intensity and polarization conditions are described by generalizations of the Fresnel formulas. The equivalent Brewster angle for harmonic waves is derived. The various conditions for total reflection and transmission of boundary harmonics are discussed. The solution of the nonlinear plane parallel slab is presented which describes the harmonic generation in experimental situations. An integral equation formulation for wave propagation in nonlinear media is sketched. Implications of the nonlinear boundary theory for experimental systems and devices are pointed out.

Asymptotic symmetries in gravitational theory

Abstract: It is pointed out that the definition of the inhomogeneous Lorentz group as a symmetry group breaks down in the presence of gravitational fields even when the dynamical effects of gravitational forces are completely negligible. An attempt is made to rederive the Lorentz group as an "asymptotic symmetry group" which leaves invariant the form of the boundary conditions appropriate for asymptotically flat gravitational fields. By analyzing recent work of Bondi and others on gravitational radiation it is shown that, with apparently reasonable boundary conditions, one obtains not the Lorentz group but a larger group. The name "generalized Bondi-Metzner group" ("GBM group") is suggested for this larger group.It is shown that the GBM group contains an Abelian normal subgroup whose factor group is isomorphic to the homogeneous orthochronous Lorentz group; that the GBM group contains precisely one Abelian four-dimensional normal subgroup, which can be identified with the group of rigid translations; that the GBM group contains an infinite number of different subgroups isomorphic to the inhomogeneous orthochronous Lorentz group; that the infinitesimal GBM group algebra permits at least one nontrivial representation, which is directly analogous to the rest-mass-zero and spin-zero representation of the Lorentz group; that in any representation of the infinitesimal GBM group algebra there is a "rest mass" operator which commutes with all the other operations; and that no similar "spin" operator appears to exist. It is argued that the GBM group is so similar to the inhomogeneous Lorentz group that the former may be compatible as a symmetry group with present microphysics.Two applications are given: Certain quantum commutation relations covariant under GBM transformations are presented; and a denumerably infinite set of integral invariants, for classical asymptotically flat gravitational fields, are derived. The four simplest integral invariants constitute the total energy momentum radiated to infinity by gravitational waves.

Optical Properties of Ag and Cu

Abstract: Experimental data for the optical constants of Ag and Cu extending to 25 eV are discussed in terms of three fundamental physical processes: (1) free-electron effects, (2) interband transitions, and (3) collective oscillations. Dispersion theory is used to obtain an accurate estimate of the average optical mass characterizing the free-electron behavior over the entire energy range below the onset of interband transitions. The values are ${m}_{a}=1.03\ifmmode\pm\else\textpm\fi{}0.06$ for Ag and 1.42\ifmmode\pm\else\textpm\fi{}0.05 for Cu. The interband transitions to 11 eV are identified tentatively using Segall's band calculations. Plasma resonances involving both the conduction band and $d$ electrons are identified and described physically.

Optical Absorption of Gallium Arsenide Between 0.6 and 2.75 eV

Abstract: The optical absorption coefficient of high-resistivity gallium arsenide has been measured over the range of photon energy 0.6 to 2.75 eV, at temperatures from 10 to 294\ifmmode^\circ\else\textdegree\fi{}K. The main absorption edge shows a sharp peak due to the formation of excitons. The energy gap and exciton binding energy are deduced from the shape of the absorption curve above the edge. Their values at 21\ifmmode^\circ\else\textdegree\fi{}K are 1.521 and 0.0034 eV, respectively. Absorption from the split-off valence band is observed, the spin-orbit splitting being 0.35 eV at the center of the zone. The exciton line shows unexplained structure on the low-energy side. Application of a stress splits the exciton line by 12 eV per unit [111] shear, and shifts it by -10 eV per unit dilation. Absorption due to the ionization of deep-lying impurity levels is observed, with thresholds at 0.70, 0.49, and 0.27 eV from the main absorption edge.

Quantum Theory of the Dielectric Constant in Real Solids

Abstract: The quantum theory of the frequency- and wave-number-dependent dielectric constant in solids is extended in order to study the full dielectric constant tensor and to include local-field effects. Within the framework of the band theory, an explicit expression for the dielectric constant tensor, neglecting local-field effects, is derived. In addition to components which are the ordinary longitudinal and transverse dielectric constants, there are components which couple transverse and longitudinal electromagnetic disturbances. A formalism for calculating the local-field corrections to the dielectric constant is developed in detail for the case of the longitudinal dielectric constant of a cubic insulating solid. In the coarsest (dipole) approximation, the theory gives a Lorenz-Lorentz formula modified by self-polarization corrections arising from the polarization of the charge in a unit cell by its own field.

Theoretical Investigation of Some Magnetic and Spectroscopic Properties of Rare-Earth Ions

Abstract: Investigations of some magnetic and spectroscopic properties of rare-earth ions based on approximate Hartree-Fock calculations are reported. First, a set of conventional, nonrelativistic Hartree-Fock wave functions were obtained for ${\mathrm{Ce}}^{3+}$, ${\mathrm{Pr}}^{3+}$, ${\mathrm{Nd}}^{3+}$, ${\mathrm{Sm}}^{3+}$, ${\mathrm{Eu}}^{2+}$, ${\mathrm{Gd}}^{3+}$, ${\mathrm{Dy}}^{3+}$, ${\mathrm{Er}}^{3+}$, and ${\mathrm{Yb}}^{3+}$; second, calculations for ${\mathrm{Ce}}^{3+}$ were carried out in which spin-orbit coupling was directly included in the conventional Hartree-Fock equations in order to obtain some estimate of wave-function dependence on $J$ and the resulting effects on experimental quantities. These results are then used to discuss spin-orbit splittings, hyperfine interactions, and the determination of nuclear magnetic moments, the Slater ${F}^{k}(4f, 4f)$ integrals, and the crystal-field parameters, ${{V}_{n}}^{m}={{A}_{n}}^{m}〈{r}^{n}〉$, all of which depend fairly critically on the precise form of the $4f$ wave functions. Comparisons are made with experiment and with the result of previous theoretical investigations which relied on either Hartree or modified hydrogenic wave functions or on semiempirical parametrizations. The usual spin-orbit formula, $〈\frac{{r}^{\ensuremath{-}1}\mathrm{dV}}{\mathrm{dr}}〉$, is found not to give agreement with experiment; the reasons for this are discussed, and some evidence is described which indicates the importance of including spin-orbit exchange terms between the $4f$ electrons and the core. The implications of this result for efforts to relate $〈{r}^{\ensuremath{-}3}〉$ integrals to experimentally observed spin-orbit coupling parameters are discussed, as is the relation (and use) of $〈{r}^{\ensuremath{-}3}〉$ integrals to the determination of nuclear magnetic moments. Our $〈{r}^{\ensuremath{-}3}〉$ values agree very closely (i.e., to within 5%) with Bleaney's parametrized values, and, hence, so do our estimates for the hyperfine interactions. A sampling of estimated rare-earth nuclear magnetic moments, based on the conventional Hartree-Fock $〈{r}^{\ensuremath{-}3}〉$ data, is given; comparison with previous estimates are made; and several causes of the uncertainty in these and all other estimates are discussed. The spectroscopic properties of these ions in a crystalline field are interpreted on the basis of the simple crystal-field theory. The $〈{r}^{n}〉$ integrals are found to be in good agreement for $n=2, 4, \mathrm{and} 6$ with the Elliott and Stevens parametrization formula, but the assumption of the constancy with $Z$ of the ${{A}_{n}}^{m}$ is not valid, as is shown by analysis of the available trichloride and ethyl-sulfate data. Systematic discrepancies between experimental and theoretical ${F}^{k}(4f, 4f)$ have been found which are similar to but greater than what has been previously observed for smaller ions. Finally, the role of spin polarization and aspherical distortions (of the closed shells and the $4f$ electrons) is indicated, particularly from the "unrestricted" Hartree-Fock point of view, and an estimate of the field due to polarization of the core electrons is given for all the ions. Results for smaller ions and their implications for the interpretation of observed rare-earth magnetic and spectroscopic properties are sketched.

Mach's Principle and Invariance under Transformation of Units

Abstract: A gravitational theory compatible with Mach's principle was published recently by Brans and Dicke. It is characterized by a gravitational field of the Jordan type, tensor plus scalar field. It is shown here that a coordinate-dependent transformation of the units of measure can be used to throw the theory into a form for which the gravitational field appears in the conventional form, as a metric tensor, such that the Einstein field equation is satisfied. The scalar field appears then as a "matter field" in the theory. The invariance of physical laws under coordinate-dependent transformations of units is discussed.

Gauge Invariance and Mass

Abstract: It is argued that the gauge invariance of a vector field does not necessarily imply zero mass for an associated particle if the current vector coupling is sufficiently strong This situation may permit a deeper understanding of nucleonic charge conservation as a manifestation of a gauge invariance, without the obvious conflict with experience that a massless particle entails

Calculation of the superconducting state parameters with retarded electron- phonon interaction

Abstract: The energy gap and other parameters of the superconducting state are calculated from the Bardeen-Cooper-Schrieffer theory in Gor'kov-Eliashberg form, using a realistic retarded electron-electron interaction via phonons and including the Coulomb repulsion. The solution is facilitated by observing that only the local phonon interaction, mediated entirely by short-wavelength phonons, is important, and that a good approximation for the phonon spectrum is therefore an Einstein model rather than Debye model. The resulting equation is solved by an approximate iteration procedure. The results are similar to earlier gap equations but the derivation gives a precise meaning to the interaction and cutoff parameters of earlier theories. The numerical results are in good order-of-magnitude agreement with the observed transition temperatures but lead to an isotope effect at least 15% less than the accepted -\textonehalf{} exponent (${T}_{c}$ proportional to ${M}^{\ensuremath{-}\frac{1}{2}}$). Also, the present theory predicts that all metals should be superconductors, although those not observed to do so would have remarkably low transition temperatures.

Magnetic Disorder as a First-Order Phase Transformation

Abstract: The exchange interaction that gives rise to ordered magnetic states depends upon interatomic spacing. If the lattice is deformable, then a spontaneous distortion of the lattice will occur in the ordered state. We have calculated, in the molecular field approximation, the properties of a system in which the exchange energy dependence is given by ${T}_{c}={T}_{0}[1+\frac{\ensuremath{\beta}(v\ensuremath{-}{v}_{0})}{{v}_{0}}]$. ${T}_{c}$ is the Curie temperature appropriate to a lattice volume $v$ while ${v}_{0}$ is the equilibrium volume in the absence of magnetic interactions. The course of the magnetization with temperature of such a system depends upon the steepness $\ensuremath{\beta}$ of the exchange interaction dependence on interatomic distance, the compressibility $K$, and ${T}_{0}$. The behavior may be the usual second-order transition to paramagnetism, but it can in fact become a first-order transition with the properties usually associated thereto, e.g., latent heat and discontinuous density change. In the absence of an externally applied pressure, the transition will be of the first order if $\ensuremath{\eta}\ensuremath{\equiv}\frac{40NkK{T}_{0}{\ensuremath{\beta}}^{2}{[j(j+1)]}^{2}}{[{(2j+1)}^{4}\ensuremath{-}1]}g1$. In this inequality, $N$ is the number per unit volume of magnetic ions of angular momentum $j\ensuremath{\hbar}$ while $k$ is the Boltzmann constant.We have reviewed the experimental evidence on the nature of the first-order magnetic transition in MnAs. We find that this evidence indicates the transition to be one from ferromagnetism to paramagnetism rather than ferromagnetism to antiferromagnetism as has been generally assumed. Application of the theory noted above gives $\ensuremath{\eta}=2$ for this transition. In addition, we derive a value for the volume strain sensitivity, $\ensuremath{\beta}=19$ and infer the compressibility to be 2.2\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}12}$ ${\mathrm{cm}}^{2}$/d.

Spin Density Waves in an Electron Gas

Abstract: It is shown rigorously that the paramagnetic state of an electron gas is never the Hartree-Fock ground state, even in the high-density---or weak-interaction---limit. The paramagnetic state is always unstable with respect to formation of a static spin density wave. The instability occurs for spin-density waves having a wave vector $Q\ensuremath{\approx}2{k}_{F}$, the diameter of the Fermi sphere. It follows that the (Hartree-Fock) spin susceptibility of the paramagnetic state is not a monotonic decreasing function with increasing $Q$, but rather a function with a singularity near $Q=2{k}_{F}$. Rather convincing experimental evidence that the antiferromagnetic ground state of chromium is a large-amplitude spin density wave state is summarized. A number of consequences of such states are discussed, including the problem of detecting them by neutron diffraction.

Perturbation Theory for the Fock-Dirac Density Matrix

Abstract: In Hartree-Fock theory and its various generalizations, it is customary to solve an eigenvalue problem involving an effective one-body Hamiltonian. The eigenvectors determine the Fock-Dirac density matrix, which also appears in the effective Hamiltonian, and solution proceeds iteratively until self-consistency is achieved.An alternative (necessary and sufficient) condition for a solution is that the density matrix ($\ensuremath{\rho}$) is idempotent and commutes with the Hamiltonian (h). The change in $\ensuremath{\rho}$, accompanying a change $\ensuremath{\Delta}$ in h, can then be expressed as a perturbation series. Formulas for the perturbation, to all orders, are obtained in terms of the unperturbed Hamiltonian and density matrix. It is also shown that the whole perturbation may be obtained directly, without separating the orders, and that the approach is related to earlier steepest-descent methods.

Scattering of neutrons by an anharmonic crystal

Abstract: The one-phonon differential scattering cross section for the coherent scattering of thermal neutrons by an anharmonic Bravais crystal is obtained correct to the lowest nonvanishing order in the anharmonic force constants. Cubic and quartic anharmonic terms are retained in the crystal's Hamiltonian. It is found that the $\ensuremath{\delta}$-function peaks in the energy distribution of the scattered neutrons for a fixed momentum transfer (which occur at the unperturbed phonon energies) in the harmonic approximation are broadened and their positions are shifted in an anharmonic crystal. Some numerical results for the magnitudes of the phonon widths and shifts are obtained for a simple model of a face-centered cubic crystal.

Theory of Transient Space-Charge-Limited Currents in Solids in the Presence of Trapping

Abstract: The problem of transient space-charge-limited currents in insulating and conducting crystals is treated mathematically. With a number of simplifying assumptions, solutions are derived for the time-dependent current and space-charge distribution following the onset of injection, the latter taking place via an ohmic contact under an applied voltage-pulse. Exact analytical solutions are given for the two limiting cases of no trapping and fast trapping. For flow in an insulator under slow trapping, approximate expressions are derived which are valid or trapping-times larger than twice the transit time. For shorter trapping times the equations of flow are solved numerically and the solutions presented in graphical form.

Statistical Mechanics of the Anisotropic Linear Heisenberg Model

Abstract: The anisotropic Hamiltonian, $H=\ensuremath{-}\frac{1}{2}\ensuremath{\Sigma}({J}_{x}{{\ensuremath{\sigma}}_{l}}^{x}{{\ensuremath{\sigma}}_{l+1}}^{x}+{J}_{y}{{\ensuremath{\sigma}}_{l}}^{y}{{\ensuremath{\sigma}}_{l+1}}^{y}+{J}_{z}{{\ensuremath{\sigma}}_{l}}^{z}{{\ensuremath{\sigma}}_{l+1}}^{z})\ensuremath{-}m\mathcal{H}\ensuremath{\Sigma}{{\ensuremath{\sigma}}_{l}}^{z},$ of the linear spin array in the Heisenberg model of magnetism is examined. The eigenstate and the partition function for the case ${J}_{z}=0$ are obtained exactly for a finite system and for an infinite system with the aid of annihilation and creation operators, and the free energy $F$ of the latter is given by $\ensuremath{-}\frac{F}{\mathrm{NkT}}=(\frac{1}{\ensuremath{\pi}})\ensuremath{\int}{0}^{\ensuremath{\pi}}\mathrm{ln}{2cosh{[{{K}_{x}}^{2}+{{K}_{y}}^{2}+2{K}_{x}{K}_{y}cos2\ensuremath{\omega}\ensuremath{-}2C({K}_{x}+{K}_{y})cos\ensuremath{\omega}+{C}^{2}]}^{\frac{1}{2}}}d\ensuremath{\omega},$ where ${K}_{x}=\frac{{J}_{x}}{2kT}$, ${K}_{y}=\frac{{J}_{y}}{2kT}$, $C=\frac{m\mathcal{H}}{\mathrm{kT}}$. The case ${J}_{x}={J}_{y}={J}_{z}=J$ is discussed with the aid of a high-temperature expansion and of analysis of small systems. Specific heats and susceptibilities in special cases: (i) ${J}_{x}={J}_{y}=J$, ${J}_{z}=0$, (ii) ${J}_{x}=J$, ${J}_{y}={J}_{z}=0$, (${\mathrm{iii}}_{\mathrm{f}}$) ${J}_{x}={J}_{y}=0$, ${J}_{z}=Jg0$, (${\mathrm{iii}}_{\mathrm{a}}$) ${J}_{x}={J}_{y}=0$, ${J}_{z}=Jl0$, (${\mathrm{iv}}_{\mathrm{f}}$) ${J}_{x}={J}_{y}={J}_{z}=Jg0$, (${\mathrm{iv}}_{\mathrm{a}}$) ${J}_{x}={J}_{y}={J}_{z}=Jl0$ are compared and it is shown that (i), (${\mathrm{iii}}_{\mathrm{a}}$), and (${\mathrm{iv}}_{\mathrm{a}}$) have the characteristic features of the observed parallel susceptibility of an antiferromagnetic substance, (ii) those of perpendicular susceptibility, and (${\mathrm{iii}}_{\mathrm{f}}$) and (${\mathrm{iv}}_{\mathrm{f}}$) those of paramagnetic susceptibility, even though they have no singularities. The distribution of the zeros of the partition function is also discussed.

Spin-Wave Spectrum of the Antiferromagnetic Linear Chain

Abstract: The methods of Bethe and Hulth\'en are used to build spin-wave states for the antiferromagnetic linear chain. These states, of spin 1 and translational quantum number $k$, are eigenstates of the Hamiltonian $H=\ensuremath{\Sigma}{j}^{}{\mathrm{S}}_{j}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{j+1}$ with periodic boundary conditions. For an infinite chain, their spectrum is ${\ensuremath{\epsilon}}_{k}=(\frac{\ensuremath{\pi}}{2})|sink|$, whereas Anderson's spin-wave theory gives ${\ensuremath{\epsilon}}_{k}=|sink|$. For finite chains it has been verified by numerical computation that these states are the lowest states of given $k$, but no rigorous proof has been given for an infinite chain.

Spectral Diffusion Decay in Spin Resonance Experiments

Abstract: In spin resonance experiments random flipping by ${T}_{1}$ or ${T}_{2}$ processes of nearby, nonresonant spins introduces fluctuations into the precessional frequency of the observed spins. These fluctuations may be described by means of a stochastic model, and for wide classes of both Markoffian and non-Markoffian distributions we make predictions for the line shape, for the free induction decay, and for various spin-echo signals. If the homogeneous broadening of the line is due to a dipolar interaction term, then we find that the conditional distribution for the precessional frequency has the shape of a Lorentzian with a cutoff on the wings, rather than a Gaussian shape as commonly assumed. The causes and consequences of Lorentzian diffusion are analyzed in detail for samples in which ${T}_{1}$ processes control the source of local frequency fluctuations and for samples in which ${T}_{2}$ processes dominate. Recent two- and three-pulse spin-echo experiments of Mims et al. dramatically confirm the predictions of Lorentzian diffusion for electron paramagnetic resonances in samples with temperature-dependent diffusion, as well as with temperature-independent diffusion. "Instantaneous" diffusion caused by the action of the applied pulses is predicted by our model and explains features of Mims' data. The generality of our principal results still permits the outcome of various resonance experiments to be predicted, even when a simple dipolar interaction is no longer an adequate model.

Photoionization from Outer Atomic Subshells. A Model Study

Abstract: Calculations of photoionization cross sections are reported which emphasize the spectral range (from threshold to \ensuremath{\sim}150 eV above it) where the bulk of the optical oscillator strength is distributed and where the cross sections are large but experimental evidence is scarce. We have used as a model the light absorption by a single electron moving in a potential similar to the Hartree-Fock potential appropriate to the outer subshell of each atom. Data are reported for the rare gases He, Ne, Ar, and Kr, for Na, and for the closed-shell ions ${\mathrm{Cu}}^{+}$ and ${\mathrm{Ag}}^{+}$. Sum rules are used to analyze the oscillator strength spectral distribution and to attempt extrapolations to still higher energies. The results suggest a classification of atomic subshells into two types with fundamentally different spectral distributions of oscillator strength. One type consists of the subshells $1s$, $2p$, $3d$, $4f$, with nodeless radial wave functions, the other type includes all remaining subshells. The present calculations are regarded as a first approximation to be improved upon by taking into account configuration interaction.

Phase Transition in Elastic Disks

Abstract: The study of a two-dimensional system consisting of 870 hard-disk particles in the phase-transition region has shown that the isotherm has a van der Waals-like loop. The density change across the transition is about 4% and the corresponding entropy change is small.

Electric Dipole Ground-State Transition Width Strength Function and 7-Mev Photon Interactions

Abstract: Formulas are given which describe some significant effects that a Porter-Thomas distribution of ground-state transition widths would have on the interpretation of the nuclear interaction of photons which reach closely spaced, but separated energy levels. When these formulas are used to reinterpret existing data, the parameters implied by photon interaction become consistent with those resulting from neutron capture data.These compatible parameters are further shown to be consistent with a crude generalized extrapolation of the giant dipole resonance. At energies near 7 Mev, the average photon absorption cross section can be written approximately as $〈{\ensuremath{\sigma}}_{a}〉=5.2$ mb ${(\frac{E}{7}\mathrm{Mev})}^{3}$ ${(\frac{A}{100})}^{\frac{8}{3}}$. This extrapolation also implies a ground-state transition width strength function which does not have the ${E}^{2}{A}^{\frac{2}{3}}$ dependence usually used because of single-particle model predictions. Near 7 Mev, $\frac{〈{\ensuremath{\Gamma}}_{0}〉}{D}=2.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$ ${(\frac{E}{7} \mathrm{Mev})}^{5}$ ${(\frac{A}{100})}^{\frac{8}{3}}$; below 3 Mev, $\frac{〈{\ensuremath{\Gamma}}_{0}〉}{D}=6.7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9} {(\frac{E}{1}\mathrm{Mev})}^{4} {(\frac{A}{100})}^{\frac{7}{3}}$. These estimates, while subject to refinements, are in better accord with experiments than are the more popular single-particle estimates.

Diffusion of Hot and Cold Electrons in Semiconductor Barriers

Abstract: The electron current in a semiconductor at uniform lattice temperature ${T}_{0}$, with a nonuniform electric field distribution (e.g., a barrier layer), consists of terms arising from conduction, diffusion, and thermal diffusion. The first two terms involve the mobility and diffusion coefficient which are functions of the electron temperature $T$ or, more generally, depend on certain averages over the nonequilibrium, field-dependent electron energy distribution function. The third term is due to the electron temperature gradient and is analogous to conventional thermal diffusion of a gas in a temperature gradient. In conventional theory, which neglects electron heating or cooling, the mobility and diffusion coefficient are material constants and thermal diffusion is absent. Contrary to the case of uniform fields, $T$ is not a unique function of the local field; it also depends on the current and can only be determined by a simultaneous solution of the equations for current flow and conservation of energy with boundary conditions for a particular structure. As an example, a one carrier metal-semiconductor contact rectifer has been analyzed in detail including a discussion of the Peltier effect. In the barrier region $T$ is greater than ${T}_{0}$ (i.e., hot electrons) for a reverse bias but less than ${T}_{0}$ (i.e., cold electrons) for a forward bias. Computer solutions have been obtained for a Schottky barrier and electron scattering due to acoustic phonons only.

Green function theory of ferromagnetism

Abstract: A theory of ferromagnetism for general spin, approximately valid through the entire temperature range, is given. At low temperatures the magnetization agrees with the Dyson results, having no term in ${T}^{3}$ and having a term in ${T}^{4}$ equal to that found by Dyson in first Born approximation; terms arising from the approximations of the theory first appear in order ${T}^{\frac{3(2S+1)}{2}}$, so that a spurious ${T}^{3}$ term does appear for $S=\frac{1}{2}$, but for no other spin. Curie temperatures are within a few percent of the Brown and Luttinger estimates for spins greater than unity, and agree within 1% of the Domb and Sykes estimate of the large-spin limit. The susceptibility at high temperatures agrees with the Opechowski expansion to terms in $\frac{1}{{T}^{2}}$. The quasiparticle energies are renormalized by the energy at low temperature and by the magnetization at higher temperature. The Green function is decoupled by a physical criterion involving self-consistency of the decoupling at all temperatures. The Green function method is extended to higher spin by a technique of parametrizing the Green function and explicitly finding the functional dependence on this parameter by solution of an auxiliary differential equation.