Showing papers in "Physical Review in 1964"

Inhomogeneous Electron Gas

Abstract: This paper deals with the ground state of an interacting electron gas in an external potential $v(\mathrm{r})$. It is proved that there exists a universal functional of the density, $F[n(\mathrm{r})]$, independent of $v(\mathrm{r})$, such that the expression $E\ensuremath{\equiv}\ensuremath{\int}v(\mathrm{r})n(\mathrm{r})d\mathrm{r}+F[n(\mathrm{r})]$ has as its minimum value the correct ground-state energy associated with $v(\mathrm{r})$. The functional $F[n(\mathrm{r})]$ is then discussed for two situations: (1) $n(\mathrm{r})={n}_{0}+\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{n}(\mathrm{r})$, $\frac{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{n}}{{n}_{0}}\ensuremath{\ll}1$, and (2) $n(\mathrm{r})=\ensuremath{\phi}(\frac{\mathrm{r}}{{r}_{0}})$ with $\ensuremath{\phi}$ arbitrary and ${r}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}$. In both cases $F$ can be expressed entirely in terms of the correlation energy and linear and higher order electronic polarizabilities of a uniform electron gas. This approach also sheds some light on generalized Thomas-Fermi methods and their limitations. Some new extensions of these methods are presented.

Superconductivity in a Strong Spin-Exchange Field

Abstract: A strong exchange field, such as produced by ferromagnetically aligned impurities in a metal, will tend to polarize the conduction electron spins. If the metal is a superconductor, this will happen only if the spin-exchange field is sufficiently strong compared to the energy gap. When the field is strong enough to break many electron pairs, the self-consistent gap equation is modified and a new type of depaired superconducting ground state occurs. In the idealization of a spatially uniform exchange field with no scattering, it is found that the depaired state has a spatially dependent complex Gorkov field, corresponding to a nonzero pairing momentum in the BCS model. The presence of the "normal" electrons from the broken pairs reduces the total current to zero, gives the depaired state some spin polarization, and results in almost normal Sommerfeld specific heat and single-electron tunneling characteristics. The nonzero value of the pairing momentum also gives rise to an unusual anisotropic electrodynamic behavior of the superconductor, as well as to a degenerate ground state and low-lying collective excitations, in accordance with Goldstone's theorem. The effects of scattering in an actual superconducting ferromagnetic alloy have not been studied and may interfere with experimental investigation of the theoretical results found in this paper for the idealized model.

Correlations in the Motion of Atoms in Liquid Argon

Abstract: A system of 864 particles interacting with a Lennard-Jones potential and obeying classical equations of motion has been studied on a digital computer (CDC 3600) to simulate molecular dynamics in liquid argon at 94.4\ifmmode^\circ\else\textdegree\fi{}K and a density of 1.374 g ${\mathrm{cm}}^{\ensuremath{-}3}$. The pair-correlation function and the constant of self-diffusion are found to agree well with experiment; the latter is 15% lower than the experimental value. The spectrum of the velocity autocorrelation function shows a broad maximum in the frequency region $\ensuremath{\omega}=0.25(\frac{{k}_{B}T}{\ensuremath{\hbar}})$. The shape of the Van Hove function ${G}_{s}(r, t)$ attains a maximum departure from a Gaussian at about $t=3.0\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}12}$ sec and becomes a Gaussian again at about ${10}^{\ensuremath{-}11}$ sec. The Van Hove function ${G}_{d}(r, t)$ has been compared with the convolution approximation of Vineyard, showing that this approximation gives a too rapid decay of ${G}_{d}(r, t)$ with time. A delayed-convolution approximation has been suggested which gives a better fit with ${G}_{d}(r, t)$; this delayed convolution makes ${G}_{d}(r, t)$ decay as ${t}^{4}$ at short times and as $t$ at long times.

Gravitational Radiation and the Motion of Two Point Masses

Abstract: The expansion of the field equations of general relativity in powers of the gravitational coupling constant yield conservation laws of energy, momentum, and angular momentum. From these laws, the loss of energy, momentum and angular momentum of a system due to the radiation of gravitational waves is found. Two techniques, radiation reaction and flux across a large sphere, are used in this calculation and are shown to be in agreement over a time average. These results are then applied to the system of two point masses moving in elliptical orbits around each other. The secular decays of the semi-major axis and eccentricity are found as functions of time and are integrated to specify the decay by gravitational radiation of such systems as functions of their initial conditions. For completeness, the secular change in the perihelion of the orbit for two arbitrary masses is found by a method which is in the spirit of the above calculations. The case of gravitational radiation when the bodies are relativistic is then considered, and an equation for the radiation similar to that of electromagnetic radiation is found. Also a proof is given that, regardless of coordinate systems or conditions, the energy of a system must decrease as a result of the radiation of gravitational waves, providing the potentials are inversely proportional to the distance from the source for large distances.

Linear Magnetic Chains with Anisotropic Coupling

Abstract: Linear chains (and rings) of $S=\frac{1}{2}$ spins with the anisotropic (Ising-Heisenberg) Hamiltonian $\mathcal{H}=\ensuremath{-}2J\ensuremath{\Sigma}\stackrel{N}{i=1}{{{S}_{i}}^{z}{{S}_{i+1}}^{z}+\ensuremath{\gamma}({{S}_{i}}^{x}{{S}_{i+1}}^{x}+{{S}_{i}}^{y}{{S}_{i+1}}^{y})}\ensuremath{-}g\ensuremath{\beta}\ensuremath{\Sigma}\stackrel{N}{i=1}\mathrm{H}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{i}$ have been studied by exact machine calculations for $N=2 \mathrm{to} 11$, $\ensuremath{\gamma}=0 \mathrm{to} 1$ and for ferro- and antiferro-magnetic coupling. The results reveal the dependence on finite size and anisotropy of the spectrum and dispersion laws, of the energy, entropy, and specific heat, of the magnetization and susceptibilities, and of the pair correlations. The limiting $N\ensuremath{\rightarrow}\ensuremath{\infty}$ behavior is accurately indicated, for all $\ensuremath{\gamma}$, in the region $\frac{\mathrm{kT}}{|J|}g~0.5$ which includes the maxima in the specific heat and susceptibility. The behavior of thermal and magnetic properties of infinite chains at lower temperatures is estimated by extrapolation. For infinite antiferromagnetic chains the ground-state degeneracy, the anisotropy gap, and the magnetization, perpendicular susceptibility, and pair correlations at $T=0$ are similarly studied. Estimates of the long-range order suggest that it vanishes only at the Heisenberg limit $\ensuremath{\gamma}=1$ and confirm the accuracy of Walker's perturbation series in $\ensuremath{\gamma}$.

Thermal Conductivity of Silicon and Germanium from 3°K to the Melting Point

Abstract: The thermal conductivity $K$ of single crystals of silicon has been measured from 3 to 1580\ifmmode^\circ\else\textdegree\fi{}K and of single crystals of germanium from 3 to 1190\ifmmode^\circ\else\textdegree\fi{}K. These measurements have been made using a steady-state, radial heat flow apparatus for $Tg300\ifmmode^\circ\else\textdegree\fi{}$K and a steady-state, longitudinal flow apparatus for $Tl300\ifmmode^\circ\else\textdegree\fi{}$K to give absolute $K$ values. This radial flow technique eliminates thermal radiation losses at high temperatures. The accuracy of both the low-temperature apparatus and the high-temperature apparatus is approximately \ifmmode\pm\else\textpm\fi{}5%. Some special experimental techniques in using the high-temperature apparatus are briefly considered. At all temperatures the major contribution to $K$ in Si and Ge is produced by phonons. The phonon thermal conductivity has been calculated from a combination of the relaxation times for boundary, isotope, three-phonon, and four-phonon scattering, and was found to agree with the experimental measurements. Above 700\ifmmode^\circ\else\textdegree\fi{}K for Ge and 1000\ifmmode^\circ\else\textdegree\fi{}K for Si an electronic contribution to $K$ occurs, which agrees quite well with the theoretical estimates. At the respective melting points of Si and Ge, electrons and holes are responsible for 40% of the total $K$ and phonons are responsible for 60%. The measured electronic $K$ yields values for the thermal band gap at the melting point of 0.6\ifmmode\pm\else\textpm\fi{}0.1 eV for Si and 0.26\ifmmode\pm\else\textpm\fi{}0.08 eV for Ge.

Relativistic Equations for Adiabatic, Spherically Symmetric Gravitational Collapse

Abstract: Relativistic gravitational collapse equations assuming spherical symmetry, adiabatic flow and pressure gradient forces

Theory of an Optical Maser

Abstract: A theoretical model for the behavior of an optical maser is presented in which the electromagnetic field is treated classically, and the active medium is made up of thermally moving atoms which acquire nonlinear electric dipole moments under the action of the field according to the laws of quantum mechanics. The corresponding macroscopic electric polarization of the medium acts as a source for an electromagnetic field. The self-consistency requirement that a quasistationary field should be sustained by the induced polarization leads to equations which determine the amplitudes and frequencies of multimode oscillation as functions of the various parameters characterizing the maser. Among the results obtained are: threshold conditions, single-mode output as a function of cavity tuning, frequency pulling and pushing, mode competition phenomena including frequency locking, production of combination tones, and population pulsations. A more approximate discussion of maser action using rate equations is also given in which the concept of "hole burning" plays a role.

Degenerate Systems and Mass Singularities

Abstract: For a system with degenerate energies, the power series expansions of the $S$-matrix elements may become singular. An elementary theorem in quantum mechanics is proved which shows that under certain general conditions such singularities do not appear in the power series expansions of the transition probabilities, provided these are averaged over an appropriate ensemble of degenerate states. Application of this theorem leads to the cancellations of mass singularities and infrared divergences in quantum electrodynamics. The question of whether a charged particle can have zero mass is studied.

Possibility of synthesizing an organic superconductor

Abstract: London’s idea that superconductivity might occur in organic macromolecules is examined in the light of the BCS theory of superconductivity. It is shown that the criterion for the occurrance of such a state can be met in certain organic polymers. A particular example is considered in detail. From a realistic estimation of the matrix elements and density of states in this polymer it is concluded that superconductivity should occur even at temperatures well above room temperature. The physical reason for this remarkable high transition temperature is discussed. It is shown further that the superconducting state of these polymers should be distinguished by certain unique chemical properties which could have considerable biological significance.

Einstein Relations Connecting Broadband Emission and Absorption Spectra

Abstract: Relations described by Einstein connecting the rates of spontaneous emission, stimulated emission, and absorption of radiation by an atomic system in free space are generalized to apply to broadband spectra of quantized systems dilutely distributed in a dielectric medium. Although the gross features of the broadband emission and absorption spectra can be qualitatively different (especially at low temperatures), the various spectra are connected at any specific frequency by simple expressions. For a two-level system imbedded in a medium at temperature $T$, typical equations connecting stimulated-emission and absorption cross sections $\ensuremath{\sigma}(\ensuremath{\omega})$ to the rate $f(\ensuremath{\omega})$ of spontaneous emission of photons per unit solid angle per unit frequency interval are: ${\ensuremath{\sigma}}_{e}(\ensuremath{\omega})={\ensuremath{\sigma}}_{a}(\ensuremath{\omega})\mathrm{exp}[\frac{\ensuremath{\hbar}(\ensuremath{\mu}\ensuremath{-}\ensuremath{\omega})}{\mathrm{kT}}]=f(\ensuremath{\omega}){(\frac{2\ensuremath{\pi}c}{\ensuremath{\omega}n})}^{2}$, where $\ensuremath{\hbar}\ensuremath{\mu}$ is a temperature-dependent excitation potential and $n$ is the index of refraction of the host material.

Exact Solution of Generalized Percus-Yevick Equation for a Mixture of Hard Spheres

Abstract: The Percus-Yevick approximate equation for the radial distribution function of a fluid is generalized to an $m$-component mixture. This approximation which can be formulated by the method of functional Taylor expansion, consists in setting $\mathrm{exp}[\ensuremath{-}\ensuremath{\beta}{\ensuremath{\phi}}_{\mathrm{ij}}(r)]{C}_{\mathrm{ij}}(r)$ equal to ${g}_{\mathrm{ij}}(r)[{e}^{\ensuremath{-}\ensuremath{\beta}{\ensuremath{\phi}}_{\mathrm{ij}}(r)}\ensuremath{-}1]$, where ${C}_{\mathrm{ij}}$, ${g}_{\mathrm{ij}}$, and ${\ensuremath{\phi}}_{\mathrm{ij}}$ are the direct correlation function, the radial distribution function and the binary potential between a molecule of species $i$ and $a$ molecule of species $j$. The resulting equation for ${C}_{\mathrm{ij}}$ and ${g}_{\mathrm{ij}}$ is solved exactly for a mixture of hard spheres of diameters ${R}_{i}$. The equation of state obtained from ${C}_{\mathrm{ij}}(r)$ via a generalized Ornstein-Zernike compressibility relation has the form $\frac{p}{\mathrm{kT}}={[\ensuremath{\Sigma}{\ensuremath{\rho}}_{i}][1+\ensuremath{\xi}+{\ensuremath{\xi}}^{2}]\ensuremath{-}\frac{18}{\ensuremath{\pi}}\ensuremath{\Sigma}{ilj}^{}{\ensuremath{\eta}}_{i}{\ensuremath{\eta}}_{j}{({R}_{i}\ensuremath{-}{R}_{j})}^{2}\ifmmode\times\else\texttimes\fi{}[{R}_{i}+{R}_{j}+{R}_{i}{R}_{j}(\ensuremath{\Sigma}{\ensuremath{\eta}}_{l}R_{l}^{}{}_{}{}^{2})]}{(1\ensuremath{-}\ensuremath{\xi})}^{\ensuremath{-}3}$, where ${\ensuremath{\eta}}_{i}=\frac{\ensuremath{\pi}}{6}$ times the density of the $i\mathrm{th}$ component and $\ensuremath{\xi}=\ensuremath{\Sigma}{\ensuremath{\eta}}_{l}R_{l}^{}{}_{}{}^{3}$. This equation yields correctly the virial expansion of the pressure up to and including the third power in the densities and is in very good agreement with the available machine computations for a binary mixture. For a one-component system our solution for $C(r)$ and $g(r)$ reduces to that found previously by Wertheim and Thiele and the equation of state becomes identical with that found on the basis of different approximations by Reiss, Frisch, and Lebowitz.

Detailed Magnetic Behavior of Nickel Near its Curie Point

Abstract: The temperature dependence of the initial susceptibility ${\ensuremath{\chi}}_{0}$ of nickel above the Curie point ${T}_{c}$ and the field dependence of its magnetization at ${T}_{c}$ are deduced from the data of Weiss and Forrer and found to be at variance with the simple molecular-field model. Instead, the experimental ${{\ensuremath{\chi}}_{0}}^{\ensuremath{-}1}$-versus-$T$ curve just above ${T}_{c}$ is shown to follow the simple relation ${{\ensuremath{\chi}}_{0}}^{\ensuremath{-}1}=A{(T\ensuremath{-}{T}_{c}]}^{\ensuremath{\gamma}}$, with $\ensuremath{\gamma}=1.35\ifmmode\pm\else\textpm\fi{}0.02$, in excellent agreement with the $\frac{4}{3}$-power relation recently predicted from the exact series for the Heisenberg model. From the coefficient $A$, it is deduced that ${\ensuremath{\mu}}_{0}$, the average atomic moment, is 0.642 ${\mathrm{\ensuremath{\mu}}}_{\mathrm{B}}$ and that the individual electron moments are in a state corresponding to $S=\frac{1}{2}$. At higher temperatures, the ${{\ensuremath{\chi}}_{0}}^{\ensuremath{-}1}$-versus-$T$ curve deviates from the Heisenberg-model predictions, possibly because of a gradual rise in ${\ensuremath{\mu}}_{0}$ with increasing temperature. Up to the highest field $H$ of measurement, the magnetization at ${T}_{c}$ is shown to vary as ${H}^{\ensuremath{\epsilon}}$ with $\ensuremath{\epsilon}=0.237$, which is consistent with the exponent values for an analogous empirical relationship between the density and pressure of several different gases at their critical points.

Time Symmetry in the Quantum Process of Measurement

Abstract: We examine the assertion that the "reduction of the wave packet," implicit in the quantum theory of measurement introduces into the foundations of quantum physics a time-asymmetric element, which in turn leads to irreversibility. We argue that this time assymmetry is actually related to the manner in which statistical ensembles are constructed. If we construct an ensemble time symmetrically by using both initial and final states of the system to delimit the sample, then the resulting probability distribution turns out to be time symmetric as well. The conventional expressions for prediction as well as those for "retrodiction" may be recovered from the time-symmetric expressions formally by separating the final (or the initial) selection procedure from the measurements under consideration by sequences of "coherence destroying" manipulations. We can proceed from this situation, which resembles prediction, to true prediction (which does not involve any postselection) by adding to the time-symmetric theory a postulate which asserts that ensembles with unambiguous probability distributions may be constructed on the basis of preselection only. If, as we believe, the validity of this postulate and the falsity of its time reverse result from the macroscopic irreversibility of our universe as a whole, then the basic laws of quantum physics, including those referring to measurements, are as completely time symmetric as the laws of classical physics. As a by-product of our analysis, we also find that during the time interval between two noncommuting observations, we may assign to a system the quantum state corresponding to the observation that follows with as much justification as we assign, ordinarily, the state corresponding to the preceding measurement.

Third-Order Elastic Constants and the Velocity of Small Amplitude Elastic Waves in Homogeneously Stressed Media

Abstract: Third-order elastic constants can be determined from the velocity of small amplitude sound waves in statically stressed media. For this purpose exact expressions are derived for the sound velocity and for a natural velocity and their stress derivatives, evaluated at zero stress, in terms of second- and third-order elastic constants. The formulas apply to arbitrary crystal symmetry and to arbitrary stress systems depending on a single scalar variable. Special formulas for hydrostatic pressure and uniaxial stress are listed for the cubic point groups $O$, ${O}_{h}$, ${T}_{d}$, and for isotropic materials. Attention is given to the proper variation of propagation direction with static stress in order to maintain propagation normal to a given crystal face as in ultrasonic experiments, and to the proper separation of isothermal and isentropic coefficients in the results. The simplest and most convenient from of the results employs the natural velocity (natural unstressed length at the same temperature divided by the transit time), which is computed directly from experimental data without correcting the path length for the effect of stress.

Photons and Gravitons in S-Matrix Theory: Derivation of Charge Conservation and Equality of Gravitational and Inertial Mass

Abstract: We give a purely S-matrix-theoretic proof of the conservation of charge (defined by the strength of soft photon interactions) and the equality of gravitational and inertial mass. Our only assumptions are the Lorentz invariance and pole structure of the S matrix, and the zero mass and spins 1 and 2 of the photon and graviton. We also prove that Lorentz invariance alone requires the S matrix for emission of a massless particle of arbitrary integer spin to satisfy a “mass-shell gauge invariance” condition, and we explain why there are no macroscopic fields corresponding to particles of spin 3 or higher.

Theory of Phonon-Terminated Optical Masers

Abstract: A simple dielectric theory is used to describe the operating properties of phonon-terminated masers of the type reported by Johnson, Dietz, and Guggenheim. Basic to this model is a broadband gain characteristic which describes the frequency-dependent gain of the active maser material as a function of the populations of metastable electronic levels and of the temperature or temperatures describing lattice vibrations. The power levels required to produce phonon saturation are estimated to be extremely high (typically, \ensuremath{\sim}${10}^{10}$ W/${\mathrm{cm}}^{2}$ power output). Because phonon saturation does not ordinarily occur, a single-lattice temperature is generally sufficient. In that case, details of the electron-phonon coupling are unimportant, and the gain can be related by detailed balance to fluorescence and absorption spectra. Effects of phonon saturation are briefly discussed in the event that they might pertain to exceptional systems and because they give insight into the principles of operation of these masers.

Photoemission Studies of Copper and Silver: Theory

Abstract: Theoretical expressions are derived for the quantum yield and for the energy distribution of photoelectrons assuming bulk photoemission from a solid. The effects of electrons which escape without inelastic scattering after optical excitation, and of those electrons which escape after one inelastic-scattering event, are considered. The expressions relate optical transition probabilities, optical constants, and mean free paths for inelastic scattering in a solid to quantities which can be measured in photoemission experiments. Examples of photoemission data are interpreted to show how the contribution of once-scattered electrons can be separated from the contribution of those electrons which have not suffered an inelastic-scattering event before escaping. The contribution to photoemission of those electrons which have not been scattered is analyzed to show the way in which direct and nondirect optical transitions can be identified and the way in which the density of states in a solid can be determined. The contribution of once-scattered electrons to photoemission is analyzed to show the way in which the nature and strength of inelastic-scattering mechanisms can be determined. The effects of electron-electron scattering, scattering by plasmon creation, and the Auger process are described, and methods of obtaining mean free paths and other scattering parameters are suggested.

Spontaneous and Stimulated Recombination Radiation in Semiconductors

Abstract: Spectral line shapes of the radiation produced by band-to-band recombination of excess carriers in semi-conductors are calculated under the assumption that the momentum matrix element is the same for all initial and final states, i.e., that there is no momentum selection rule. The peak of the stimulated radiation falls at a lower photon energy than does the peak of the spontaneous radiation, except when $T=0$\ifmmode^\circ\else\textdegree\fi{}K. Some numerical results are given for simple parabolic bands, specifically for the case of electron injection into $p$-type GaAs, and are used to deduce the temperature dependence of the forward current which is necessary to maintain a fixed gain in the active region of a diode. The result is closely related to the temperature dependence of the threshold current in an injection laser, and gives reasonable agreement with experiment. The effect of a conduction band tail is briefly considered.

Theory of the insulating state

Abstract: In this paper a new and more comprehensive characterization of the insulating state of matter is developed. This characterization includes the conventional insulators with energy gap as well as systems discussed by Mott which, in band theory, would be metals. The essential property is this: Every low-lying wave function $\ensuremath{\Phi}$ of an insulating ring breaks up into a sum of functions, $\ensuremath{\Phi}={{\ensuremath{\Sigma}}_{\ensuremath{-}\ensuremath{\infty}}}^{\ensuremath{\infty}}{\ensuremath{\Phi}}_{M}$, which are localized in disconnected regions of the many-particle configuration space and have essentially vanishing overlap. This property is the analog of localization for a single particle and leads directly to the electrical properties characteristic of insulators. An Appendix deals with a soluble model exhibiting a transition between an insulating and a conducting state.

Feynman Rules for Any Spin

Abstract: The explicit Feynman rules are given for massive particles of any spin j, in both a 2j+1-component and a 2(2j+l)-component formalism. The propagators involve matrices which transform like symmetric traceless tensors of rank 2j; they are the natural generalizations of the 2×2 four-vector σ µ and 4×4 four-vector γ µ for \(j\frac{1}{2}\). Our calculation uses field theory, but only as a convenient instrument for the construction of a Lorentz-invariant S matrix. This approach is also used to prove the spin-statistics theorem, crossing symmetry, and to discuss T, C, and P.

Electric-Field Gradient Tensor in Ferrous Compounds

Abstract: The electric-field gradient tensor at the iron nucleus in ferrous (${\mathrm{Fe}}^{2+}$) compounds is investigated. One sees that under the combined action of axial and rhombic crystalline fields and the spin-orbit interaction, the ferrous-ion $(^{5}D,3{d}^{6}){d}_{\ensuremath{\epsilon}}$ states produce a large, temperature-dependent, contribution to the electric-field gradient tensor. It is found that this direct contribution is diminished by that from the lattice itself (the second-order axial and rhombic components of the crystalline field), as well as Sternheimer polarization effects and covalency. The results of this investigation are then applied to M\"ossbauer results in FeSi${\mathrm{F}}_{6}$\ifmmode\cdot\else\textperiodcentered\fi{}6${\mathrm{H}}_{2}$O to obtain an estimate of the electric quadrupole moment of ${\mathrm{Fe}}^{57m}(0.29\ifmmode\pm\else\textpm\fi{}0.02\mathrm{b})$, which is in agreement with that from ferric (${\mathrm{Fe}}^{3+}$) studies. Finally, estimates also based upon M\"ossbauer measurements, are made of the ${d}_{\ensuremath{\epsilon}}$ energy splittings in the ferrous compounds, FeSi${\mathrm{F}}_{6}$\ifmmode\cdot\else\textperiodcentered\fi{}6${\mathrm{H}}_{2}$O, FeS${\mathrm{O}}_{4}$\ifmmode\cdot\else\textperiodcentered\fi{}7${\mathrm{H}}_{2}$O, Fe${\mathrm{C}}_{2}$${\mathrm{O}}_{4}$\ifmmode\cdot\else\textperiodcentered\fi{}2${\mathrm{H}}_{2}$O, Fe${(\mathrm{N}{\mathrm{H}}_{4}\mathrm{S}{\mathrm{O}}_{4})}_{2}$\ifmmode\cdot\else\textperiodcentered\fi{}6${\mathrm{H}}_{2}$O, FeS${\mathrm{O}}_{4}$, Fe${\mathrm{Cl}}_{2}$\ifmmode\cdot\else\textperiodcentered\fi{}4${\mathrm{H}}_{2}$O, and Fe${\mathrm{F}}_{2}$.

Interstitials and Vacancies in α Iron

Abstract: Tile migration energies and atomic configurations for mono- and di-interstitials and mono- and di-vacancies in α-iron have been calculated using a classical model. About 530 atoms surrounding the defect were treated as individual particles, each with three degrees of freedom, while the remainder of the crystal was treated as an elastic continuum with atoms imbedded in it. A two-body central force was devised which matched the elastic moduli, was sharply repulsive at close separation, and which went to zero midway between the second and third neighboring atoms. Configurations were found by choosing a starting configuration roughly approximating the situation under consideration and successively adjusting the value of each variable occurring in the energy equation so that the magnitude of the generalized force associated with it was zero until equilibrium was reached. Tile energy calculations include changes in bond energy in the discrete region, energy in the elastic field, and work done against cohesive forces, but neglect changes due to the redistribution of electrons. Calculated activation energies for motion of mono- and di-interstitials and mono- and di-vacancies were 0.33 eV, 0.18 eV, 0.68 eV and 0.66 eV respectively, and binding energies of di-interstitials and di-vacancies were 1.08 eV and 0.20 eVmore » respectively. The stable interstitial was a split configuration in which two atoms were symmetrically split in a (110) direction about a vacant normal lattice site, and the stable di-interstitial consisted of two parallel split interstitials at nearest neighbor lattice sites.with their axes perpendicular to the line joining their centers. In the vacancy configuration an atom was missing from a normal lattice site, and the di-vacancy consisted of two vacancies at second nearest neighbor lattice sites. (auth)« less

Lattice Dynamics and Phase Transitions of Strontium Titanate

Abstract: The crystal dynamics of strontium titanate has been studied both experimentally and theoretically. The frequency versus wave-vector dispersion curves for some of the normal modes propagating along the [0,0,1] direction have been measured by neutron spectrometry at 90 and 296\ifmmode^\circ\else\textdegree\fi{}K. The experiments were performed at the Chalk River Laboratories of Atomic Energy of Canada Ltd., using a triple-axis crystal spectrometer. The temperature dependence of the transverse optic mode of the lowest frequency has been found to be in agreement with the temperature dependence of the dielectric constant, as predicted by Cochran. The experimental results have been used to obtain the parameters of several models, more than one of which gives reasonable agreement with the experimental results. It is suggested that the anomalous behavior of the elastic properties and the phase transition at 110\ifmmode^\circ\else\textdegree\fi{}K are associated with an accidental degeneracy of two branches of the dispersion curves; the longitudinal acoustic branch and the transverse optic branch of lowest frequency. The origin of the temperature dependence of this transverse optic mode and the relevance of lattice dynamics to the phase transitions in other perovskites are discussed.

Theory of Thermal Transport Coefficients

Abstract: A simple proof of the usual correlation-function expressions for the thermal transport coefficients in a resistive medium is given. This proof only requires the assumption that the phenomenological equations in the usual form exist. It is a "mechanical" derivation in the same sense that Kubo's derivation of the expression for the electrical conductivity is. That is, a purely Hamiltonian formalism with external fields is used, and one never has to make any statements about the nature or existence of a local equilibrium distribution function, or how fluctuations regress. For completeness the analogous formulas for the viscosity coefficients and the heat conductivity of a simple fluid are given.

Defects in Irradiated Silicon: Electron Paramagnetic Resonance and Electron-Nuclear Double Resonance of the Si-E Center

Abstract: The Si-$E$ center is one of the dominant defects produced by electron irradiation in phosphorus-doped vacuum floating zone silicon. It introduces an acceptor level at $\ensuremath{\sim}({E}_{c}\ensuremath{-}0.4)$ eV and gives rise to an electron paramagnetic resonance when this level does not contain an electron. As a result of electron paramagnetic resonance (EPR) and electron-nuclear double resonance (ENDOR) studies described in this paper, we conclude that the defect is a lattice vacancy trapped next to a substitutional phosphorus atom, with EPR arising from the neutral charge state. The observed hyperfine interactions with ${\mathrm{P}}^{31}$ and neighboring ${\mathrm{Si}}^{29}$ nuclei, as well as the observed $g$-tensor anisotropy, are discussed in terms of a simple linear combination of atomic orbitals (LCAO) molecular orbital treatment. In addition to the anisotropy associated with the phosphorus-vacancy direction in the lattice, an additional distortion of the defect occurs which is identified in the LCAO treatment as a manifestation of the Jahn-Teller effect. Thermally activated reorientation from one Jahn-Teller distortion to another causes motional broadening and narrowing effects upon the EPR spectrum in the temperature region 60-150\ifmmode^\circ\else\textdegree\fi{}K. The motion is also studied by stress-induced alignment at lower temperatures and the activation energy for this process is determined to be \ensuremath{\sim}0.06 eV. Alignment of the phosphorus-vacancy direction in the lattice is also achieved by stressing at elevated temperatures. The activation energy for this motion is 0.93\ifmmode\pm\else\textpm\fi{}0.05 eV. The magnitude and sense of the alignment in both kinds of stress experiments are consistent with the microscopic model of the defect. The role of the phosphorus-vacancy interaction in the diffusion of phosphorus in unirradiated silicon is discussed. Using the published value for the diffusion activation energy for phosphorus in silicon, we estimate the appropriate value for silicon self-diffusion to be 3.94\ifmmode\pm\else\textpm\fi{}0.33 eV and the formation energy for the lattice vacancy in silicon to be 3.6\ifmmode\pm\else\textpm\fi{}0.5 eV. These are quantities for which no direct experimental values are available. Also included is an appendix which gives estimates of ${|{\ensuremath{\psi}}_{3s}(0)|}^{2}$ and $〈{{r}_{3p}}^{\ensuremath{-}3}〉$ for the $3p$ atoms aluminum through chlorine.

Thermodynamic Definition of Higher Order Elastic Coefficients

Abstract: General thermodynamic definitions of the higher order elastic coefficients of thermoelastic media are presented in tensor and engineering notation. They are natural generalizations of the customary definitions of second-order coefficients, they retain the usual conventions relating tensor and engineering stresses and strains, and they simplify thermodynamic calculations.

Magnetic Translation Group

Abstract: In this paper a group-theoretical approach to the problem of a Bloch electron in a magnetic field is given. A magnetic translation group is defined and its properties, in particular its connection with the usual translation group, are established.

Ionization Rates of Holes and Electrons in Silicon

Abstract: The ionization rates of charge carriers in silicon have been measured and fit to the recent theoretical calculations of Baraff; in contrast, none of the existing published data could be fit to these theoretical curves The study has been made using microplasma-free junctions of demonstrably high, uniform local multiplication A new and considerably simplified approach to the problem of extracting the ionization rates from the multiplication data has been used By employing much more precise control of the electron and hole currents used to initiate the multiplication process, the hole ionization rate at electric fields less than 300 kV/cm is found to be more than an order of magnitude smaller than any previously published measurements Hole and electron ionization rates have been measured in the same junction and consequently in the identical scattering environment The threshold energy is determined to be ${E}_{g}\ensuremath{\le}{E}_{i}\ensuremath{\le}15{E}_{g}$, and the mean free path for scattering of high-energy electrons is $50 \AA{}\ensuremath{\le}{\ensuremath{\lambda}}_{e}\ensuremath{\le}70 \AA{}$ and for energetic holes $30 \AA{}\ensuremath{\le}{\ensuremath{\lambda}}_{h}\ensuremath{\le}45 \AA{}$ Measurement of ionization rates at various temperatures substantiates the assumption that the energy-loss mechanism is the emission of optical phonons In addition, significant differences of the electrical breakdown characteristics of microplasma-free junctions are discussed as well as their preparation