Showing papers on "Dispersion relation published in 1970"

Propagation of a Gaussian Light Pulse through an Anomalous Dispersion Medium

Abstract: The propagation of a Gaussian light pulse through a medium having a positive or negative absorption line is examined. Analytical approximations are obtained for the case where the spectral width of the pulse is much smaller than that of the line. It is shown that the pulse remains substantially Gaussian and unchanged in width for many exponential absorption depths, and that the locus of instants of maximum amplitude follows the classical expression for the group velocity, even if this is greater than the velocity of light, or negative. Numerical calculations have been used to examine what happens beyond the limit of usefulness of the analytical approximations.

Influence of the Electron Charge Distribution on Surface-Plasmon Dispersion

Abstract: The effects of the spatial variation of the electron density on the surface-plasmon dispersion relation are investigated. We show that measurements of that relation are a useful probe of the electron density in the surface region. Previous calculations on homogeneous materials have predicted a linear or quadratic dependence of the frequency on momentum parallel to the surface. We find that the usual surface-plasmon resonance frequency at first decreases with increasing momentum and then increases with further increases in momentum. This behavior agrees with the experimentally observed dispersion. Additional higher-frequency surface modes, similar to those observed in laboratory plasmas, are identified.

Cyclotron harmonic wave propagation and instabilities: I. Perpendicular propagation

Abstract: Parts I and II of this paper present a comprehensive picture of longitudinal wave propagation in a warm homogeneous magnetoplasma. Part I discusses computed dispersion characteristics for propagation perpendicular to the static magnetic field. For a ring electron velocity distribution it is found that mode coupling and absolute instability can occur. Similar effects are predicted for a spherical shell distribution. The Maxwellian distribution gives rise to stable propagation of undamped waves, and attenuating standing waves. A mixture of ring and Maxwellian distributions can give absolute instability with stronger growth and lower instability thresholds than for the ring distribution alone. Propagation oblique to the static magnetic field will be dealt with in Part II.

The Dispersion of Surface Waves in Multilayered Anisotropic Media

Abstract: Summary The matrix procedure, developed by W.'T. Thompson and N. A. Haskell, to obtain the phase velocity dispersion relations for Rayleigh and Love waves in multilayered isotropic media, is extended to obtain the dispersion relation for generalized surface waves in multilayered media, where the layers may be either isotropic or anisotropic. The procedure is extended to anisotropic layers by describing the anisotropy by the fourth-rank tensor of elastic coefficients, and by reorientating the tensor so that the direction of wave propagation is along the x-axis.

Lattice Vibrations in Crystals with Deformable Molecules; a Calculation for Naphthalene

Abstract: A calculation has been done to show the effect of molecular distortion on the external modes of the molecular crystal naphthalene. This calculation also yields the dispersion relation for the internal modes caused by the crystalline forces. The method used is quite general. The force field for the isolated molecule comes from spectroscopic frequency fitting, and to this is added the effect of crystalline forces using a Buckingham atom–atom potential. The shift and splitting of all the modes calculated at zero wave vector are given, and also the dispersion relation for the lower modes for wave vectors parallel to the crystal screw‐diad axis. A suggestion is made concerning the analysis of the results of neutron coherent inelastic scattering, which until now has assumed molecular rigidity.

Longitudinal waves in a perpendicular collisionless plasma shock: I. Cold ions

Abstract: This paper considers electrostatic waves in a Vlasov plasma of unmagnetized ions and magnetized, Maxwellian electrons. The linear dispersion relation is derived for waves in a perpendicular shock such that the most important sources of instability are the E × B and ∇B electron drifts. For the case of cold ions, propagation perpendicular to the applied magnetic field, and the E × B drift alone, a numerical analysis of frequency vs. wave-number is presented. The effects of the ∇B drift are also considered, and it is shown that the maximum growth rate can be larger than the maximum growth rate for the zero magnetic field ion acoustic instabifity under comparable conditions.

Surface Waves on a Plasma Half‐Space

Abstract: The effect of density gradients and a finite temperature on the dispersion relation for surface waves on a plasma half‐space has been investigated analytically. The full set of Maxwell's equations is used to obtain the dispersion of surface waves on a warm homogeneous plasma, thus complementing earlier work on the electrostatic mode. The full surface‐wave dispersion relation is then derived for a cold plasma with arbitrary but weak density profile in the WKB limit. Finally, the dispersion of electrostatic surface modes on a cold plasma with a linear density profile of arbitrary strength is obtained. It is shown that when the density variation over a wavelength is very large, a new type of damped surface wave with a frequency higher than the surface plasma frequency is possible.

Alfven wave damping from finite gyroradius coupling to the ion acoustic mode

Abstract: The collisionless damping of shear Alfven waves in the limit of low frequency and small but finite Larmor radius is discussed. Because the averaging of the wave electric field over the Larmor circle (the so‐called finite cyclotron radius effect) creates a small difference in the transverse velocities of the ions and electrons, a longitudinal electric field appears in the Alfven wave which is responsible for the transfer of the wave energy into the thermal motion of the “resonant” particles Vz = ω/kz. Numerical solutions (in the complex ω plane) of the dispersion relation for two wave modes have been obtained: the shear Alfven mode and the least damped Fried and Gould ion acoustic mode. The normalized damping rate (Im ω)/(Re ω) has been computed for both waves, and it is shown that the damping factor for the Alfven wave is maximal when the Alfven velocity is equal to the real part of the phase velocity of the ion acoustic wave. Moreover, there exists a particular direction of propagation with respect to th...

Forward Compton Scattering

Abstract: An analysis of the recently measured photon-proton total cross sections is performed. Smooth fits to the cross sections are obtained and used to calculate, by means of the forward dispersion relation, the real part of the spin-averaged forward amplitude. The resulting predictions for the real part are given. At high energies, the fits to the present total cross-section data, together with the calculated real part, suggest the presence in the high-energy behavoir of an extra real constant in addition to what one would have predicted from Regge theory and the high-energy behavior of the imaginary part. This extra real constant, which is consistent in sign and magnitude with the Thomson limit, −αMN, could correspond to a fixed pole at J=0 in Regge-pole language. Possible ways to test the forward dispersion relation are discussed.

Inelastic neutron scattering investigation of spin waves and magnetic interactions in α-Fe2O3

Abstract: Spin waves in antiferromagnetic α-Fe2O3 have been studied at temperatures of 240 and 290° K by means of inelastic neutron scattering. We report here the first determination of the acoustical branch throughout the entire Brillouin zone and the first observation of the optical branch. The latter shows little dispersion throughout the zone, and has an energy of 1125° K (97 meV) at the zone centre. At the zone boundaries a gap of 20 to 70° K exists between the two branches. Heisenberg interaction parameters Jm defined through a Hamiltonian were obtained through fitting of the data to theoretical expressions for the dispersion relations. The following values were obtained for the first five nearest neighbours: J1 = 6.0 ± 1.6° K, J2 = 1.6 ± 0.6° K, J3 = −29.7 ± ± 2.0° K, J4 = −23.2 ± 1.0° K and J5 = −1.0 ± 1.0° K. Interactions to farther neighbours were found to be weak. Neutron intensity data were partly invoked in obtaining the interaction parameters, as two sets of parameters could fit the energy data almost equally well, but they predicted different relative intensities for the two branches in certain regions of the reciprocal space. All spin waves of energy larger than 80° K, including the entire optical branch, were found to be unaffected by the Morin spin-flip transition at 261° K temperature. The spin-wave data were used to calculate the sublattice magnetization, the Neel and the Curie-Weiss temperatures, the perpendicular susceptibility at low temperatures and the density of spin-wave state spectrum. [Russian Text Ignored]

X-ray determination of phonon dispersion in vanadium.

Abstract: Phonon frequencies along principal symmetry directions were measured in vanadium by means of thermal diffuse scattering of x rays. A polishing and annealing procedure was adopted which eliminated extra scattering associated with impurities. The dispersion curves have initial slopes in agreement with the measured elastic constants. We found no evidence of any irregularity at low frequencies in any of the dispersion curves and in the calculated frequency spectrum. A low-frequency peak in the frequency spectrum has been reported in two inelastic neutron-scattering experiments and was attributed to a Kohn effect. A Born-von K\'arm\'an model with seven-neighbor forces gave an adequate fit to the dispersion curves within experimental uncertainty. The derived interatomic force constants are not consistent with an axially symmetric model. The calculated frequency spectrum is in generally good agreement with the inelastic neutron determinations, but indicates the presence of more detailed structure. The high-energy tail observed in many experiments is not confirmed, nor does the $\frac{g(\ensuremath{ u})}{{\ensuremath{ u}}^{2}}$ curve exhibit any hump in the low-frequency region.

Vlasov Equilibria and Stability of an Electron Gas

Abstract: Self‐consistent Vlasov equilibria with space charge are constructed for a pure electron gas radially confined in a uniform external magnetic field. A sufficient condition is obtained for the stability of these nonuniform equilibria to electrostatic perturbations. For the case in which the electron density is constant in the column interior, the dispersion relations for small‐amplitude interior perturbations, electrostatic and electromagnetic, are shown to be similar in structure to the corresponding results for a neutral plasma. Algorithms are given for obtaining electron gas stability information from the neutral plasma literature.

Spin-Wave Dispersion Relations in Gadolinium

Abstract: Spin-wave dispersion relations have been measured in high-symmetry directions for metallic Gd. Analysis shows that at least five interplanar constants are required for a satisfactory fit to the data. The energy gap at $q=0$ is unmeasurably small. In the $c$ direction the measured dispersion curve gives directly the Fourier-transformed exchange interaction $J(0)\ensuremath{-}J(q)$. This exhibits no other extreme value except that at the origin.

Electrostatic Plasma Instabilities Excited by a High‐Frequency Electric Field

Abstract: The electrostatic plasma waves excited by a uniform, alternating electric field of arbitrary intensity are studied on the basis of the Vlasov equation; their dispersion relation, which involves the determinant of either of two infinite matrices, is derived. For ω0 ≫ ωpi (ω0 being the applied frequency and ωpi the ion plasma frequency) the waves may be classified in two groups, each satisfying a simple condition; this allows writing the dispersion relation in closed form. Both groups coalesce (resonance) if (a) ω0 ≈ ωpe/r (r any integer) and (b) the wavenumber k is small. A nonoscillatory instability is found; its distinction from the DuBois‐Goldman instability and its physical origin are discussed. Conditions for its excitation (in particular, upper limits to ω0,k, and k·vE, vE being the field‐induced electron velocity), and simple equations for the growth rate are given off‐resonance and at ω0 ≈ ωpi. The dependence of both threshold and maximum growth rate on various parameters is discussed, and the results are compared with those of Silin and Nishikawa. The threshold at ω0 ≈ ωpi/r,r ≠ 1, is studied.

Is a subtraction really necessary in the isospin even π N dispersion relation

Abstract: It is shown that the assumption of anunsubtracted fixed-t dispersion relation for the isospin evenπN amplitudeA + is compatible with all data. The prediction for the subtraction function agrees reasonably well with independent determinations. The assumption has the consequence that the ratio of thes-channel helicity flip and non flip amplitudes vanishes in the high energy limit.

Thermal conductivity and low frequency waves in collisional plasmas.

Abstract: Thermal fluctuations and heat flow terms are retained in the two‐fluid equations which are used to derive a localized dispersion relation for low‐frequency electrostatic waves in a nonuniform collisional plasma. A slab plasma immersed in a uniform magnetic field is assumed. The Nyquist analysis shows two unstable waves: the collisional drift instability and a new overstability of the entropy wave. The collisional drift wave is most easily excited for parallel wavenumbers given by k‖2 = 0.15 α2(me/mi)1/2 where α ≡ |≇ In n| (all quantities in cgs units). The stability condition at this k| is B/k⊥ < 104 (min/κTe|α|)1/2 G‐cm. These are in markedly improved agreement with the experimental data. Heat transport reduces the maximum growth rate, especially at high density. A small equilibrium temperature gradient tends to stabilize the drift waves in the large magnetic field region. The entropywave instability propagates along ion drift direction and is found in the range 0.4 ≤ d¯ ≤ 2 with d¯ ≈ 0.15 (k⊥κ2Ti/Ωi2mi)...

Lagrangian methods in plasma dynamics. I. General theory of the method of the averaged Lagrangian

Abstract: Whitham's averaged Lagrangian method is used to yield a relativistically covariant formalism for wave packets in a weakly inhomogeneous (and time dependent) medium. Provided the physics of the medium can be based on a Lagrangian density, a procedure of expansion and averaging is available which gives separate Lagrangians for the dynamics of the background and of the waves. The Euler—Lagrange equations then give immediately the equations of motion for the background, including non-linear reaction of the waves, and the dispersion relation, equations for ray tracing, conservation of wave action and non-linear coupling coefficients, for the waves. Many of these results can be interpreted in an illuminating way by considering the corresponding expansion of the canonical four-dimensional stress tensor.

Optical constants of graphite

Abstract: The high-frequency optical constants of graphite are computed from electron energy loss results by using a dispersion relation for anisotropic materials The values of ɛ⊥(ω) are in reasonable agreement with those obtained by Taft and Philipp from optical measurements The values of ɛ‖(ω) connect with those previously obtained at lower frequency from reflectivity measurements, and show a structure with two absorption peaks at 112 eV and 16 eV and two plasmons at 14 eV and 19 eV The results are explained in terms of interband transitions, using a two-dimensional band structure The experimental peaks in the energy loss function are shown to be due both to plasmons and to a different kind of excitation To understand the peak position in energy as a function of the direction of the momentum transfer, a noncrossing rule due to the interaction between these two excitations is required

Turbulent Plasmas in a Magnetic Field—A Statistical Theory

Abstract: A statistical theory of plasma turbulence in a magnetic field is described. This theory makes use of averaging operators to formally solve the Vlasov equation and the stochastic acceleration problem. Sets of turbulence equations are thereby derived for determining the ensemble average of the oneparticle distribution function and the electric field spectrum. The average distribution function is seen to satisfy a diffusion equation to all orders in the perturbation—even in the presence of a magnetic field. The mentioned equations are used to derive a nonlinear dispersion relation for waves in turbulent plasmas in a magnetic field. This dispersion relation is a nonlinear generalization of Bernstein's linear dispersion relation for electrostatic waves. The nonlinearity manifests itself as a damping factor proportional to mean square deviations from mean particle trajectories. A simplifying feature of the present work is the use of cumulant expansions, which also avoid certain heuristic arguments. A previous result, given by Dupree, is shown to be a special case of the present result.

Lattice dynamics of argon at 4 degrees K

Abstract: Neutron scattering techniques have been used to measure the phonon dispersion relations in a single crystal of 36Ar at 4 degrees K Data have been taken along the (110) and the (111) crystallographic directions The analysis is expressed in terms of a two-neighbour force-constant model with five disposable parameters; one linear combination of three of these parameters is not determined well by the data It is shown that the measurements are not inconsistent with a Mie-Lennard-Jones potential where the parameters are determined entirely from other macroscopic measurements on argon However, there is a discrepancy with recent measurements of the elastic constants

Temperature Effects on High‐Frequency Beam Plasma Interaction

Abstract: The dispersion equation corresponding to the interaction of an electron beam with a Maxwellian plasma has been solved in quasistatic and infinite geometry approximations. Two representative beams are considered: a parallel beam with isotropic velocity spread and a helical beam with different longitudinal and transverse temperatures. The aim of the work is to predict the frequency, growth rate, and wavenumbers of the most unstable waves as functions of the basic beam and the plasma parameters. The results of the calculations are presented in graphs suitable for comparison with the experiments. The effect of plasma and beam temperatures on the spectrum of the instabilities is discussed in detail. It is shown that the plasma temperature reduces the growth rate of the instabilities at cyclotron harmonic frequencies while a velocity spread in the helical beam suppresses the instabilities everywhere, except in the vicinity of the cyclotron harmonics.

Lattice Dynamics of Terbium

Abstract: The frequency-wave-vector dispersion relation for the normal modes of vibration of terbium at room temperature has been measured by means of slow-neutron inelastic scattering techniques. The triple-axis spectrometer at the Oak Ridge high flux isotope reactor was used, mostly in the constant-$Q$ mode of operation. Phonon frequencies for wave vectors along the principal symmetry directions have been determined and, in addition, measurements of phonon frequencies along the boundaries of the Brillouin zone and along a more general direction are reported. The data have been fitted with a Born-von K\'arm\'an force model which includes interactions out to the eighth nearest neighbor. The interactions have been assumed to be general (tensor) out to the fourth neighbor and axially symmetric beyond. The model has been used to calculate a frequency distribution function $g(\ensuremath{ u})$ and related quantities such as the lattice specific heat and Debye temperature.