Showing papers on "Dispersion relation published in 1971"

Phonon Dispersion Relations in Ge at 80 °K

Abstract: Phonon dispersion relations in Ge have been measured at 80 \ifmmode^\circ\else\textdegree\fi{}K for all principal symmetry directions and on some lines at the boundary of the Brillouin zone. The measurements were performed by neutron inelastic scattering using a three-axis crystal spectrometer. In order to achieve maximum accuracy, care was taken to reduce resolution widths as far as possible. In general, the estimated uncertainties of the measured phonon frequencies range from 0.3% to 0.5% for optical and from 0.3% to 1% for acoustic phonons. Frequencies of 238 phonons are tabulated and phonon linewidths are given for the symmetry directions.

Crystal Dynamics and Electronic Specific Heats of Palladium and Copper

Abstract: Using inelastic neutron scattering, the frequency – wave vector dispersion relations for the lattice vibrations in a single crystal of palladium have been determined at 120, 296, 673, and 853 °K. A...

Electrostatic oscillations in cold inhomogeneous plasma. i. differential equation approach.

Abstract: Small amplitude electrostatic oscillations in a cold plasma with continuously varying density have been investigated. The problem is the same as that treated by Barston (1964) but instead of his normal-mode analysis we employ the Laplace transform approach to solve the corresponding initial-value problem. We construct the Green function of the differential equation of the problem to show that there are branch-point singularities on the real axis of the complex frequency-plane, which correspond to the singularities of the Barston eigenmodes and which, asymptotically, give rise to non-collective oscillations with position-dependent frequency and damping proportional to negative powers of time. In addition we find an infinity of new singularities (simple poles) of the analytic continuation of the Green function into the lower half of the complex frequency-plane whose position is independent of the spatial co-ordinate so that they represent collective, exponentially damped modes of plasma oscillations. Thus, although there may be no discrete spectrum, in a more general sense a dispersion relation does exist but must be interpreted in the same way as in the case of Landau damping of hot plasma oscillations.

Surface Plasmon in a Semi-Infinite Free-Electron Gas

Abstract: When electron-lifetime effects, electron-hole pair excitations, or both are included in the description of an electron gas, the frequency associated with the surface plasmon is a complex quantity, the imaginary part providing a measure of the damping of the plasmon. The surface-plasmon dispersion relation then involves the specification of this complex frequency as a function of the wave vector parallel to the surface. A general theory is developed for such a surface-plasmon dispersion relation in a semi-infinite free-electron gas bounded by a surface that scatters the electrons specularly. The properties of the electron gas enter through the nonlocal transverse and longitudinal dielectric functions ${\ensuremath{\epsilon}}_{t}(q,\ensuremath{\omega})$ and ${\ensuremath{\epsilon}}_{l}(q,\ensuremath{\omega})$, both of which include a finite electron lifetime here. The results obtained using local and hydrodynamic approximations for the dielectric functions are presented briefly, and the self-consistent-field approximation is discussed in detail. The calculations are done both with and without retardation.

Theory of triggered VLF emissions

Abstract: A theory is presented to explain the origin of triggered discrete VLF emissions that is more complete than earlier theories, in that it is not restricted to the discussion of kinematical relations but evaluates the dynamics of the problem. Resonant electrons are phase correlated with the wave magnetic field by a finite length whistler train moving in the opposite direction. The time for phase correlation is of the order of the period of oscillation of a particle in the effective ‘potential well’ of the wave. It is recognized that the wave acceleration due to the inhomogeneous magnetic field of the earth must be small enough for the particle to stay trapped in the potential well. The phase-correlated electrons are subject to an instability in the form of an emitted whistler with a growth rate γ/ω ∼ (n/no)2/5(υ⊥/c)2/5 (ωp/Ω)2/5, where (n/no) is the fractional density of the resonant particles, υ⊥ is their mean transverse velocity, and ωp and Ω are local cold plasma and gyrofrequencies. The emitted frequency varies according to ω = k(ω)υ∥ − Ω, where υ∥ is the zero order longitudinal velocity of the resonant electron, and the wave vector k is a function of frequency ω through the whistler dispersion relation. The theory is in good agreement with observation.

Optically Excited Longitudinal Plasmons in Potassium

Abstract: Oscillatory structure in the absorption of thin potassium films has been observed for energies just above the plasmon energy in a wavelength-modulated photoemission experiment. This structure is shown to be due to resonant excitation of longitudinal plasmons. Observation of up to ten resonances allows the evaluation of the dispersion relation, which shows a quadratic behavior in agreement with a theoretical formula based on the Boltzmann equation.

Ion Acoustic Waves in a Multi‐Ion Plasma

Abstract: An exact treatment of the multispecies ion acoustic dispersion relation is given for an argon/helium plasma. Phase velocity and damping are obtained as a function of ion‐electron temperature ratio and relative densities of the two species. There are two important modes in the plasma, with quite different phase velocities, which are referred to as principal heavy ion mode and principal light ion mode. Which of these is dominant depends on the relative densities of the two components, but, in general, the light ion mode becomes important for surprisingly small light ion contamination. Approximate analytic expressions are derived from damping rates and phase velocities and their domains of validity are investigated. Relevance of the results for the investigation of collisionless shocks is discussed.

Thermalization in the earth's bow shock.

Abstract: Thermalization processes in the earth's bow shock are examined with particular emphasis on ion heating. A two-dimensional analysis of the ion-ion streaming instability using the complete electromagnetic dispersion relation with magnetized electrons is presented. For Ti/Te≲0.2 the peak growth rates are approximately a tenth of the total ion plasma frequency. As the ions are heated above Ti/Te≈0.2, the growth rates drop sharply to a fraction of (m/M)1/2Ω e, where Ωe is the electron gyrofrequency. For Ti/Te≳0.6 the growth rates are comparable to the ion gyrofrequency. We conclude that ion gyroeffects play a very important role in the ion thermalization process and are the source of the observed secondary peak in the ion velocity distribution. The ion-ion streaming instability provides only fine-scale diffusion in ion velocity space. Electron heating to βe≈1 is shown to be consistent with a c/ωpe length scale for changes in the magnetic field and the operation of the electron-ion drift instability in the leading edge of the earth's bow shock.

Structure of the Electromagnetic Field in a Spatially Dispersive Medium

Abstract: The exact mode expansion is derived for the electromagnetic field in a spatially dispersive model dielectric occupying the volume 0<∼z<∼d. The dispersion relations for the transverse as well as the longitudinal waves are deduced and the nature of the modes is briefly discussed.

Linearized dispersion relation and green's function for discrete-charge-transfer devices with incomplete transfer

Abstract: A small-signal (linearized) theory of discrete-charge-transfer-device performance is presented for the case of incomplete charge transfer. Specifically, the dispersion relation is derived which relates the charge-transfer efficiencies presently characterizing these discrete (in space and time) devices to the usual measures of device or transmission-line performance based on the attenuation, dispersion, phase velocity, etc., of sine waves. In a more general sense this emphasizes the applicability of conventional signal theory to these new devices. The impulse solution or Green's function is then shourn to be the equivalent of a bivariate distribution in probability theory. More generally the utility of (deterministically interpreted) probability theory is emphasized by showing the equivalence of a general small-signal theory to a random-walk process.

Verification of Theory on Weak Turbulence Relating to the Sequence of Diffuse Plasma Resonances in Space

Abstract: An interpretation of the sequence of diffuse plasma resonances observed by space probes (Alouette 2 and ISIS-1 satellites) is developed in terms of wave-particle nonlinear interaction in a weakly turbulent plasma including the electrostatic electron cyclotron harmonic wave instability. The longest time duration of the center frequency of the diffuse plasma resonance is found to coincide with the most favorable condition for the electrostatic electron cyclotron harmonic wave instability which is obtained by solving the dispersion equation obtained for a linear approximation of the kinetic wave equation for the warm magnetoactive plasma. The electrostatic field due to the transmission of the intense rf pulse produces plasma turbulence involving nonlinear wave-wave interaction and temperature anisotropy which leads to instability. This instability supplies energy to the turbulence. The process can be thought of as a feedback system.

Velocity shear and low-frequency plasma instabilities

Abstract: The effects of transverse velocity shear on the low‐frequency stability of a plasma are examined theoretically for a low‐β resistive plasma in a uniform magnetic field. Cylindrical geometry is used and the velocity shear is introduced by a nonuniform E × B rotation. Both numerical and analytic methods are used. The principal analytic result is a dispersion relation for instabilities caused by a thin velocity shear layer. This dispersion relation describes the Q machine edge oscillation, which is identified either as a Kelvin‐Helmholtz instability or as a velocity‐shear analog of the resistive drift wave, depending on the parallel wavelength. The numerical results show that properties of instabilities observed in several experiments agree reasonably well with theory. The effect of velocity shear on the drift instability is to make it into either a local or nonlocal type of normal mode.

Theoretical Considerations on Bragg-case Diffraction of X-rays at a Small Glancing Angle

Abstract: X-ray diffraction phenomenon, for which the glancing angle of the incident beam is so small that the effect of specular reflection can not be neglected, is studied theoretically. Two wave fields, one of which is usually neglected, are considered as the solution of the dispersion equation, and the boundary conditions of the waves are considered for the gradient of amplitude normal to the surface besides the amplitude. Numerical calculations are made for (220) reflection of CuKα1 from a germanium single crystal. The diffracted beam is stronger in intensity and broader in half-value width than that where specular reflection is not considered.

An Improved Numerical Technique for Solving Multi-Dimensional Miscible Displacement Equations

Abstract: Analog difference equations of high order accuracy describing stable miscible displacement are presented. The high order difference scheme eliminates almost all the numerical smearing, which is the result of the normally used approximations, and leaves only the effect of the physical dispersion in the solution. In a one-dimensional system, the technique involves an addition of a negative dispersion term to the continuity equation. The negative dispersion term which is called the numerical dispersion coefficient, depends upon the flow velocity, the time-step size, and the block size. The procedure is extended to multi-dimensional systems. As a check, comparisons of the computed results with analytical solutions in one and 2- dimensional systems are made. The error function solution in a one-dimensional miscible system and a 5-spot fractional flow curve computed from the potentiometric model are considered as the analytical solutions in the 2 cases.

Experimental Verification of Optically Excited Longitudinal Plasmons

Abstract: Anomalous optical absorption is observed in thin Ag films just above the bulk plasma frequency. The result is interpreted as excitation of longitudinal plasmons, an effect which is normally neglected when discussing optical properties. The dispersion relation of the plasma wave is obtained from the data.

Normal Modes of Vibrations in CuCl

Abstract: The frequency-wave-vector dispersion relation $\ensuremath{ u}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ for the normal vibrations of a CuCl single crystal at room temperature has been measured for the [100]-, [110]-, and [111]-symmetric directions using inelastic neutron scattering.

Time-harmonic waves traveling obliquely in a periodically laminated medium.

Abstract: : The dispersion relation is presented for time-harmonic waves propagating in an arbitrary direction in a periodically laminated medium. The analysis is based on two-dimensional equations of elasticity. Limiting phase velocities are presented for infinite wavelength for any angle of propagation in the form of a fourth-order determinant that illustrates the influence of an arbitrary angle. For the cases when the propagation is along or across the layers, the determinant reduces to two determinants of second order that yield the limiting phase velocities directly. Numerical results are presented to indicate the dependence of dispersion upon the angle of propagation. Also, a comparison with an approximate continuum theory is included; agreement is satisfactory for those angles where the dispersion is the strongest. (Author)

Growth rates and stability limits for beam--plasma interaction.

Abstract: A treatment is given of the one‐dimensional problem of beam‐plasma interaction, including the effects of plasma and beam thermal velocities and momentum transfer collisions in the plasma. The conditions under which the Landau damping due to the plasma is negligible are first investigated. Neglecting this damping, a weak‐beam approximation to the dispersion relation is derived, and conditions for its validity are established. Expressions are derived for the maximum temporal and spatial growth rates in the limits of cold and hot beams, and few and many collisions. The growth rates given by the weak‐beam approximation are compared with those calculated from the full dispersion relation. The transition from a cold to a hot beam is discussed in terms of the topology of the roots of the dispersion relation, and also in terms of the location of the phase velocities of the most unstable waves relative to the bump in the distribution.

Concept of Group Velocity in Resonant Pulse Propagation

Abstract: It is shown that the prediction of Garrett and McCumber that a light pulse's velocity may exceed the speed of light in a resonantly absorbing medium is due to asymmetric absorption of energy from the light pulse. More energy is absorbed from the trailing half of the pulse than from the front half, causing the center of gravity of the pulse to move at a velocity greater than the phase velocity of light. It is also shown that in certain cases a pulse's maximum will propagate at the group velocity even when the pulse as a whole is distorted by dispersion.

Wave Propagation in a Fluid‐Filled Tube

Abstract: A circular cylindrical viscoelastic solid tube filled with a compressible viscous fluid is considered. The outer surface of the tube is assumed to be constrained so that it cannot move freely. Axially symmetric wave solutions are obtained for the linearized equations governing the motion of the fluid and the solid. The solutions lead to a complicated transcendental dispersion equation relating the wave frequency and the propagation constant. This equation is simplified by taking the fluid to be inviscid and the tube to be thin. Then it is studied by both analytical and numerical means. Formulas and graphs are given for the propagation constant, the phase velocity, the group velocity, etc., as functions of the frequency for various sets of parameter values. There are infinitely many modes of propagation, two of which are “tube modes” and the rest of which are acoustic modes. The case of a viscous fluid and a tube of any thickness are to be treated in a subsequent paper.

Perturbation Solutions of the Dispersion Equation in Porous Mediums

Abstract: The solution of the dispersion equation in porous mediums is difficult because of the dependence of the dispersion tensor on the velocity. Approximate solutions of dispersion in nonuniform two-dimensional flows are sought by perturbation expansions. An inner boundary layer solution for the transition zone between two fluids in steady and unsteady conditions is derived. Two examples of dispersion, radial flow and coastal aquifers, are solved approximately.

Propagation Characteristics of Periodic Arrays of Dielectric Slabs

Abstract: Propagation along periodic arrays of dielectric slabs in the direction transverse to that of periodicity is studied as a function of the prescribed phase delay per period for two polarizations. Classification of modes is achieved with the help of "stability diagrams." In contrast with previous work, the rigorous dispersion relation and exact mode functions are considered. Calculated dispersion curves and closed form mode functions serve to illustrate the guiding properties of the structure and are, in turn, explained in terms of stability diagrams and equivalent networks.

Relativistic kinetic theory of the large- amplitude transverse Alfven wave

Abstract: Relativistic kinetic theory of large amplitude transverse Alfven wave, discussing propagation in collisionless plasma