Showing papers on "Dispersion relation published in 1968"

Low-Energy Theorem for Graviton Scattering

Abstract: A low-energy theorem for the scattering of gravitons from spin-0 particles is derived. We use the dispersion-theoretic method, recently utilized by Abarbanel and Goldberger to derive low-energy theorems for the Compton scattering of photons, to write unsubtracted dispersion relations for physical helicity amplitudes. The scattering amplitude at fixed angle is shown to be given by the Born approximation up to fourth-order terms in the graviton energy.

Theory of hydromagnetic propagation in the ionospheric waveguide

Abstract: In this paper, we consider the propagation of low-frequency (Pc 1) hydromagnetic waves in the ionospheric duct, which results from the minimum in the Alfven speed near the F2 ionization peak. We consider an inhomogeneous (in the vertical direction) waveguide in the presence of a uniform static magnetic field and treat the case of horizontal propagation in the plane of the magnetic meridian. Propagation in the waveguide is governed by two coupled equations for the horizontal components of the electric field. For nighttime conditions, where the ionized region of interest starts at sufficiently high altitude that the collision frequency of ions with neutrals is small compared with the ion cyclotron frequency, the two components become uncoupled, and only the isotropic, or fast wave, need be considered. For daytime conditions, however, coupling between the fast and slow waves must be taken into account; to simplify the analysis, the static field is assumed uniform in this case. To obtain analytic solution of the waveguide equations, it is necessary to have analytic approximations to the height dependence of the relevant ionospheric parameters; such approximations are constructed to fit some representative tabulated ionospheric profiles for a variety of conditions of time of day and sunspot activity. Solution of the waveguide equations subject to the appropriate boundary conditions results in a rather complicated dispersion relation between complex (horizontal) wave number and angular frequency. The solutions of the dispersion relation prescribe the allowed bands of propagation in the guide. It is shown that each band, including the lowest, has a low-frequency cutoff, and that the existence of a low-frequency cutoff is a consequence of the boundary conditions and not of attenuation, as assumed in some theories. Numerical solutions of the dispersion equation are obtained for the two lowest bands for each of the conditions considered. From the dependence of the real part of the wave number on frequency, the cutoff frequencies and the phase and group velocities are determined, while the imaginary part provides the attenuation length. For nighttime conditions, attenuation is not large, and propagation can take place over distances of thousands of km, whereas daytime propagation is restricted by attenuation to distances of the order of hundreds of km. For nighttime minimum conditions, the calculated waveguide cutoff is about 0.4 cps, and the group velocity for the lowest band is about 720 km/sec, both in reasonable agreement with experiment. Finally, it is pointed out that there is an effective high-frequency ‘cutoff’ (in any band) as far as ground-level signals are concerned due to an exponential decrease of transmission coefficient with frequency at high frequencies.

The onset of superconductivity in the time dependent Ginzburg-Landau theory

Abstract: The fluctuations of the local order parameter above the superconducting transition temperature give rise to singularities in the electrical conductivity and the diamagnetic susceptibility. Using the time dependent Ginzburg-Landau equation the fluctuation of the current density is calculated. By means of the fluctuation-dissipation theorem and a dispersion relation the electromagnetic response function is then determined for small frequencies and wave-numbers. The dynamical conductivity for bulk material, thin films, and thin wires shows an increasing peak at zero frequency the width of which decreases as the transition temperature is approached. This structure should be observable in microwave experiments.

Transition of the Dispersion Roots from Beam‐Type to Landau‐Type Solutions

Abstract: The roots of the dispersion relation describing the linear interaction between a plasma and a relatively low density electron beam are investigated. The dispersion curves exhibit a nontrivial change in topology at a particular value of the appropriately scaled thermal velocity of the beam.

Spin Waves in 3d Metals

Abstract: A review is given of recent measurements of spin‐wave dispersion relations of the 3d metals by the Brookhaven neutron diffraction group using the diffraction technique and triple‐axis spectrometry. The parameters D and β in the relation ħω = Dq2(1 − βq2) have been determined at 295°K for Fe, Co, Ni, and some of their alloys. These values are compared with those obtained by thin‐film resonance and small‐angle scattering.The most extensive measurements were carried out on Fe using a triple‐axis spectrometer. The dispersion relation was measured along the three principal symmetry directions for wavevectors up to q/qmax = 0.4. The stiffness constant D as well as the linewidth of selected spin waves were studied for the temperature range between 77°K and the Curie temperature, 1042°K. Well‐defined magnons were observed up to a reduced temperature T/TC = 0.995, but not above TC.

Adjusting Poles and Zeros of Dielectric Dispersion to Fit Reststrahlen of Pr Cl 3 and La Cl 3

Abstract: The dielectric dispersion function of a medium can be defined by the locations of the poles and zeros of that function in the complex-frequency plane. In this form the dispersion function is not restricted by special characteristics of any specific physical model from which dispersion might be derived. The locations of these poles and zeros are subject to several more or less fundamental physical restrictions which are described in this paper. The connections between the locations of the poles and zeros, the frequencies and damping constants of certain optical modes, and the Lyddane-Sachs-Teller relation are discussed. Adjustments of locations of poles and zeros, consistent with the physical restrictions, were performed to obtain least-squares fits of reststrahlen data from Pr${\mathrm{Cl}}_{3}$ and La${\mathrm{Cl}}_{3}$ using a small number of poles and zeros. In this way, approximate dispersion functions for the two dielectric tensor components of each crystal were obtained at room temperature and at lower temperatures.

Lattice Dynamics of Niobium-Molybdenum Alloys

Abstract: The frequency-versus-wave-vector dispersion relations for the normal modes of vibration of a series of alloys of the transition metals niobium and molybdenum have been measured at 296\ifmmode^\circ\else\textdegree\fi{}K, and previous measurements on the pure metals have been extended, using coherent, one-phonon scattering of thermal neutrons. The phonon dispersion relations are very different for the two pure metals, suggesting that the electronic structure, acting through the electron-phonon interaction, plays a significant role in the determination of the dynamics of these materials. The observed neutron groups corresponding to the phonons in the alloys are not significantly broader than in the pure metals. The dependence of the dispersion curves on alloy composition is found to be complicated, both the general level of frequencies and the shape of the curves changing significantly. Fourier analysis indicates that the interatomic forces in the metals are oscillatory and of long range. Suspected Kohn anomalies are observed on several branches of the dispersion curves. With the assumption of a rigid-band model, the positions of several of these anomalies correlate with the calculated electron band structure for tungsten. The dimensions of the Fermi surface obtained from this correlation are in agreement with other Fermi-surface information.

Coherence and band structure of inertial motion in the sea

Abstract: It is now well established by observation that a peak in the spectrum of horizontal motion should be anticipated everywhere in the ocean near the local inertia frequency, 2ω sine (latitude). The theory of wave motion in a weakly stratified, rotating ocean of constant depth explains this observation either by the existence of a frequency condensation point in wave-number space or, alternatively, by the vanishing of the meridional group velocity. This explanation is independent of a specific generating mechanism, such as tidal forcing. The details of the wave structure and dispersion relation are readily obtained when, as seems both likely and desirable, it is permissible to ignore the discrete normal-mode-producing effects of distant lateral boundaries. This theory predicts a spectral peak slightly above the inertia frequency, and this displacement depends on the zonal and vertical wave numbers. The peak frequency in the North Atlantic measurements by Fofonoff and Webster implies vertical modes of O(10) and a zonal wave number of O (several hundred cycles per earth circumference). When these numbers are applied to a simple coherence model, assuming phase independence between different wave numbers, one can account for the observed lack of coherence between stations separated in depth or longitude. This theory also defines a latitudinal scale; for vertical wave number 10 this is, typically, of O(25 km), which is in qualitative agreement with Hendershott's observations in the eastern North Pacific. The present theoretical model is appropriate for random distributed sources. The observations, however, indicate a higher degree of intermittency than is implied by this model. We conclude that both random distributed sources and intermittent discrete sources must be taken into account for a satisfactory description of the phenomena.

Anisotropic Temperature Plasma Instabilities

Abstract: An anisotropic plasma exhibits a transverse instability due to the mutual attraction of the microscopic current elements. A detailed study is made of this instability in an electron‐ion plasma without any external magnetic field. The system is characterized by a temperature tensor and described by a quasi‐Maxwellian distribution function. The dielectric tensor for such a system is calculated and then the dispersion relation is considered. It is found that there are two unstable modes: (i) a purely transverse mode and (ii) a coupled or transverse‐longitudinal mode. The latter separates into two branches: (a) the first is an essentially unstable quasitransverse mode, (b) the second is the quasi‐longitudinal ion acoustic mode driven into instability through its interaction with the quasitransverse mode. Growth rate, k and angular dependence, domain of instability, and polarization are analyzed in detail.

Magnetic excitations in uranium dioxide.

Abstract: The spin-wave dispersion relations have been measured in antiferromagnetic uranium dioxide by inelastic neutron-scattering techniques. The triple-axis crystal spectrometer at the C5 facility of NRU was used throughout in its constant-Q mode of operation. The dispersion relations were obtained for spin waves propagating along the main symmetry directions at 9\ifmmode^\circ\else\textdegree\fi{}K, and less complete measurements were made at higher temperatures both above and below the N\'eel temperature. The theory of spin waves in U${\mathrm{O}}_{2}$ is developed and various models are used in attempts to deduce the exchange and anisotropy parameters from the experimental results. None of the models are completely satisfactory, because of the difficulties arising from the multidomain character of the specimen, and from the strong interaction between the magnons and the phonons. A theory of this interaction is also developed which gives quite reasonable agreement with experiment.

Low-Energy Theorems, Dispersion Relations, and Superconvergence Sum Rules for Compton Scattering

Abstract: A derivation of low-energy theorems for Compton scattering from spin-0 and spin-½ targets is given within the framework of dispersion theory. We work exclusively with physical helicity amplitudes and utilize the zeros of these amplitudes forced by angular momentum conservation to write unsubtracted dispersion relations. The conventional requirement of gauge invariance is replaced in our work by Lorentz in-variance together with the knowledge that the photon is a massless spin-1 particle. From the dispersion relations we extract a number of sum rules of the superconvergence type, one example of which reduces the Drell-Hearn result in the forward direction.

Theories of Elastic Continua and Crystal Lattice Theories

Abstract: This paper is concerned with the relations between discrete and continuum theories of the elastic behavior of perfect crystals. The point of view adopted is that the validity of any extension of the classical theory of elasticity, intended to accommodate effects of the atomic structure of crystalline solids, can be tested by comparison with an appropriate lattice theory. The particular test to be applied is how well and to how short wave lengths the dispersion relation for plane waves, deduced from the continuum theory, reproduces that for the lattice.

Structure of generalized ion bernstein modes from the full electromagnetic dispersion relation.

Abstract: The structure of modes for which k┴B0 and ω≪Ω_ (ion waves) has been studied qualitatively in the two limits κR+ ≪ 1 and κR+ ≫ 1, where R+ = (κT+/MΩ+)½ is the mean thermal Larmor radius, without the usual electrostatic approximation. Asymptotic forms of the dielectric tensor elements eij are developed in these two limits. The modes having appreciable k × E can be called ‘generalized’ Bernstein modes. The approximation which yields the familar electrostatic Bernstein modes is exx = 0. This approximation is shown to be valid only for large κR+ and low β. However, for small and moderate values of κR+ the generalized Bernstein modes partake of a ‘mixed’ electromagnetic- electrostatic character. In particular, for κR+ ≲ 0(1) (but ω/κ < c) the electrostatic Bernstein modes are incorrect approximations. The warm plasma electromagnetic theory is discussed with reference to cold plasma theory for a low β plasma, and it is shown that: (1) the lower hybrid frequency is only an approximate resonance in the warm plasma; (2) electromagnetic cut-offs occur at all harmonics of the gyrofrequency as k → 0; (3) electrostatic resonances occur at all harmonics of the gyrofrequency as k→∞; (4) propagation can occur in warm plasmas at frequencies above the lower hybrid.

Dispersion of electron plasma waves

Abstract: The propagation of electron plasma waves along a cylindrical plasma column in a strong magnetic field has been studied both theoretically and experimentally. The dispersion equation for such waves has been solved numerically where the plasma density profile is assumed to be either uniform or parabolic and the electron energy distribution Maxwellian. Dispersion curves have been computed for the m = 0 azimuthal mode and compared with experimental points obtained from a thermally ionized plasma in a strong magnetic field over the range of electron density 107-1010 cm-3.

Two-Pole Approximation for the Plasma Dispersion Function

Abstract: A simple algebraic approximation to the Hilbert transform of the Gaussian, valid over most of the complex plane, is given.

Magnetic Interactions in Tb and Tb‐10% Ho from Inelastic Neutron Scattering

Abstract: The magnon dispersion relations and lifetimes have been measured in Tb and a Tb‐10% Ho alloy by inelastic neutron scattering, in regions of both ferromagnetic and spiral ordering In the ferromagnetic phase, the magnon energy is generally finite at zero wavevector and rises quadratically at low q The magnon energies scale approximately with the magnetization In the spiral phase the magnon energy rises linearly from zero at low q The Fourier‐transformed exchange parameter J(q) has pronounced peaks in the c direction, which are ascribed to transitions between states close to the Fermi surface These peaks are less pronounced in the ferromagnetic phase The primary mechanism limiting the magnon lifetimes appears to be interaction with the conduction electrons In the alloy, the lifetime for magnons propagating in the c direction in the ferromagnetic phase falls abruptly at about q = 035 A−1, and this may be due to the exchange splitting of the conduction‐electron energy bands The dispersion curve for ma

Magneto--gravity waves and the heating of the solar corona.

Abstract: It is generally believed that the heating of the solar corona is caused by waves originating in the photosphere and propagating into the corona where their energy is dissipated. The medium through which these waves propagate is in general permeated by magnetic fields complicating the behaviour of this propagation considerably. We have therefore analysed the wave motions in a plasma permeated by constant magnetic and gravitational fields. In general, three waves modes were found, which we called the + mode, −mode, and the Alfven mode. Each mode was found to be strongly coupled to each of the three kinds of motion; acoustic, gravity, and hydromagnetic. However, the Alfven mode was found to be separable from the dispersion relation, and therefore independent of compressibility and gravity. The local dispersion relation is derived and expressed in nondimensional form independent of the constants that describe a particular atmosphere. From the dispersion relation one can show that rising waves propagate either with a constant or a growing wave amplitude depending on the magnitudes and directions of the gravitational field, magnetic field, and the wave vector. The variation of the density with height is taken into account by a generalized W.K.B. method. Equations are found which give the height at which wave reflection occurs, giving the upper bound for possible wave propagation.

Dispersion of Surface Elastic Waves Produced by a Conducting Grating on a Piezoelectric Crystal

Abstract: The dispersion relation for surface elastic waves propagating under a thin metallic grating on the basal plane of a hexagonal piezoelectric crystal is investigated theoretically. Purely reactive loads are assumed connected between adjacent electrodes of the grating. The effect of mass loading due to the grating is neglected. The presence of the grating imposes periodic boundary conditions on the piezoelectric fields associated with the surface wave. The resulting boundary‐condition problem is solved to obtain the dispersion relation for surface elastic waves. For a material with a large electromechanical coupling coefficient an inductance connected between adjacent electrodes can cause a large reduction in the phase velocity of the surface wave. (For example, in PZT‐4, this reduction is about 10%.)

Light Scattering from a Multiply Relaxing Fluid

Abstract: Formulas are derived for the roots of the dispersion equation for temporally damped hydrodynamic waves in a viscous, thermally conducting liquid having also a frequency‐dependent viscosity with one or more relaxation times Values of the roots are then computed for CCl4 at 25° for practically the entire range of wavenumbers, under the assumption that the extra viscosity has a relaxation time of 543 × 10−11 sec Next, the values of the roots corresponding to k ≈ 2 × 105cm−1 (for 90° scattering of He–Ne laser light) are used to calculate both the unbroadened and the instrumentally broadened Rayleigh–Brillouin spectrum A similar calculation under the assumption that CCl4 is doubly relaxing shows radically different velocity dispersion curves according to the relaxation time τ2 assigned to the longitudinal viscosity For τ2 = 10−13sec, viscous overdamping reduces the sound velocity to zero for a narrow range of wavenumbers above about k = 12 × 107cm−1 For τ2 = 10−12sec, the velocity is an increasing funct

Propagation of acoustic waves through a system undergoing phase transformations

Abstract: The propagation of a sound wave through a system in which phase transformations are possible is examined. Dispersion and attenuation of the wave are found if the wave frequency is comparable to the reaction rate. Sharp changes in the phase velocity and relative energy loss per cycle are expected when phase boundaries are crossed. The attenuation due to a phase transformation can be sufficiently large to contribute to seismic losses in the mantle.

Kelvin‐Helmholtz Instability for a Collisionless Plasma Model

Abstract: The dispersion relation of the Kelvin‐Helmholtz instability due to a shear in ion fluid velocity parallel to the magnetic field is calculated from the Vlasov equation to account for effects of ion Landau damping and finite Larmor radius. Due to the relatively short wavelength parallel to the magnetic field, resonant particle effects are strong and the dispersion relation is considerably modified from that obtained by D'Angelo with a fluid treatment.

Excitons and Plasmons in Insulators

Abstract: The relation between excitons and “plasmons” in insulators is studied, and it is shown that both excitons and “plasmons” may be obtained in the singlet longitudinal states. When the energy of interband transition is nearly equal to the free electron plasmon energy \(\hbar\omega_{p}\), the energy and the angular dispersion of “plasmon” loss peaks are closely related to the structure of the pair band. In this case the “plasmon” solution of the dispersion relation is expected on the higher energy side of M 3 -(or M 2 -) van Hove singularity provided the mean value of imaginary part of the dielectric constant exceeds some critical value below this energy and is small above this energy. Recent experimental data of energy loss spectra on alkali-halides are analyzed making use of the optical data and the results of band structure calculations.

The behaviour of the π0π0 → π0π0 S-wave below threshold

Abstract: The properties of the π0π0 → π0π0S-wave amplitude in the interval 0≤s≤4 are investigated using crossing symmetry and positivity properties due to unitarity. It is shown that the above-mentioned amplitude has a unique minimum in this region and that it is located betweens=1.29 ands=1.7(*).

Time‐Harmonic Fields in Source‐Free Bianisotropic Media

Abstract: A bianisotropic medium is defined as one in which the field vectors D and H depend on both E and B, but may not be parallel to either. A moving medium appears bianisotropic to the laboratory observer, even if it is isotropic in its rest frame. It is shown that, for time‐harmonic fields, a bianisotropic medium can be viewed as one which is electrically and magnetically anisotropic with properly defined dyadic operators. In particular, a moving medium can be characterized by a permittivity tensor and a permeability tensor for plane‐wave propagation. This characterization simplifies the solution of electromagnetic problems in moving media. As illustrated, the dispersion relations in a moving uniaxially anisotropic medium are obtained, and the problem of a plane wave normally incident upon a moving uniaxial medium is solved.