Showing papers on "Dispersion relation published in 1978"
IBM1
TL;DR: In this paper, the phase shift method was used to solve the migration of zero-offset seismic records with laterally invariant velocities, and the migration process was solved very accurately by the phase-shift method.
Abstract: Accurate methods for the solution of the migration of zero-offset seismic records have been developed. The numerical operations are defined in the frequency domain. The source and recorder positions are lowered by means of a phase shift, or a rotation of the phase angle of the Fourier coefficients. For applications with laterally invariant velocities, the equations governing the migration process are solved very accurately by the phase-shift method. The partial differential equations considered include the 15 degree equation, as well as higher order approximations to the exact migration process. The most accurate migration is accomplished by using the asymptotic equation, whose dispersion relation is the same as that of the full wave equation for downward propagating waves. These equations, however, do not account for the reflection and transmission effects, multiples, or evanescent waves. For comparable accuracy, the present approach to migration is expected to be computationally more efficient than finite-difference methods in general.
885 citations
TL;DR: In this article, a new technique is developed to determine the dispersion relation and the propagational speeds of waves in dispersive solids, which can be applied to measurements of acoustic or electromagnetic wave speeds in other dispersive media.
Abstract: A new technique is developed to determine the dispersion relation and the propagational speeds of waves in dispersive solids. The phase spectrum of a broadband pulse is linearly related to the dispersion relation of the dispersive medium. The method is simpler than the continuous‐wave phase comparison technique. Application is made to measure the phase and group velocities of waves in fiber‐reinforced composite materials and in thin wires. This technique is expected to be applicable to measurements of acoustic or electromagnetic wave speeds in other dispersive media.
476 citations
TL;DR: The dispersion relations for molecular orientation fluctuations in a freely suspended ferroelectric smectic-C$ liquid-crystal film have been determined by inelastic light scattering.
Abstract: The dispersion relations for molecular-orientation fluctuations in a freely suspended ferroelectric smectic-$C$ liquid-crystal film have been determined by inelastic light scattering The smectic-$C$ film geometry appears to be ideal for the experimental study of the space-time behavior of a two-dimensional orientational field
241 citations
TL;DR: The wave plasma experiment onboard satellite S3-3 has detected electrostatic hydrogen cyclotron waves at altitudes near 1R/sub E/ in the Earth's polar magnetosphere as discussed by the authors.
Abstract: The wave plasma experiment onboard satellite S3-3 has detected electrostatic hydrogen cyclotron waves at altitudes near 1R/sub E/ in the Earth's polar magnetosphere. The observed wave properties are in excellent agreement with the theory theoretical dispersion relation as well as the linear instability theory for current driven ion cyclotron waves and are used to determine that the plasma is more than 90%hydrogen, that the electron temperature is about 3.5 electron volts and that T/sub e//T/sub i/ is in the range of 1 or larger. The strongest wave events are associated with large (> or =120mV/m) Dc electric fields described in an earlier paper (Mozer et al., 1977) as paired electrostatic shocks. Other wave events, which are not as clearly related to shock structures, usually are imbedded in or specifically correlate with fluxes of electrons from .074 keV to 5.04 keV and turbulent Dc electric fields. These results are consistent with a lower threshold for ion cyclotron waves than for shocks or double layers.
201 citations
IBM1
TL;DR: Using angle-resolved photoemission and synchrotron radiation, the authors determined the energy-versus-momentum valence-band dispersion relations for a Ni(111) crystal.
Abstract: Using angle-resolved photoemission and synchrotron radiation, we have determined the energy-versus-momentum valence-band dispersion relations for a Ni(111) crystal. The temperature-dependent ferromagnetic exchange splitting has been directly observed. Both the $d$-band width (\ensuremath{\sim}3.4 eV at $L$) and exchange splitting (0.31 eV) are much smaller than theoretical estimates (\ensuremath{\sim}4.5 eV wide at $L$ with \ensuremath{\sim}0.7-eV splitting, respectively, at 293 K).
200 citations
TL;DR: In this article, a scattering matrix approach, that involves only the transition matrix of a single obstacle, is proposed for studying the multiple scattering of elastic waves in a medium (matrix) containing identical, long, parallel, randomly distributed cylinders of arbitrary cross section.
Abstract: A scattering matrix approach, that involves only the transition matrix of a single obstacle, is proposed for studying the multiple scattering of elastic waves in a medium (matrix) containing identical, long, parallel, randomly distributed cylinders of arbitrary cross section The elastic properties of the cylinders are assumed to be different from those of the matrix A statistical approach in conjunction with Lax’s ’’quasicrystalline’’ approximation is employed to obtain equations for the average amplitudes of the scattered and exciting fields which may then be solved to yield the dispersion relations of the composite medium Dynamic elastic properties of the composite medium containing circular and elliptical cylinders are found in the Rayleigh or low‐frequency limit Numerical results displaying phase velocity and damping effect of the composite medium are presented for a wide range of frequencies
174 citations
TL;DR: In this article, it was shown that the eigenmodes of the collisionless drift wave in slab geometry are stable and that the electron dynamics of the drift wave can be treated in an incomplete manner.
Abstract: It is shown that the eigenmodes of the collisionless drift wave in slab geometry are stable. Previous studies yielding instability (the "universal" instability) were based upon an incomplete treatment of the electron dynamics; i.e., the principal part of the plasma dispersion function was ignored.
161 citations
TL;DR: In this paper, the dispersion relation of magnetostatic surface waves propagating in a ferromagnetic film of finite dimensions is derived and a consideration of the finite width of the sample is introduced.
Abstract: The dispersion relation of magnetostatic surface waves propagating in a ferromagnetic film of finite dimensions is derived. Specifically, a consideration of the finite width of the sample is introduced to the previous treatments of magnetostatic surface‐wave propagation. The inclusion of a finite width produces a multiplicity of propagating modes, which can be excited individually or collectively, depending on experimental conditions. In addition, the finite width allows the excitation of magnetostatic volume waves, which exist over a narrow frequency range just below the range of magnetostatic surface waves. All aspects of the theoretical analysis are supported by measurements made on magnetostatic surface‐wave propagation at frequencies around 1 and 9 GHz in yttrium iron garnet samples of different widths.
138 citations
TL;DR: In this article, the role of finite electron Larmor radius effects on wave absorption was discussed and a numerical analysis of the relevant dispersion relation in the weakly relativistic and nonrelativistic limit was presented.
Abstract: Wave absorption for frequencies near the electron cyclotron frequency for the ordinary and extraordinary modes is investigated. The role of finite electron Larmor radius effects is discussed. A numerical analysis of the relevant dispersion relation in the weakly relativistic and nonrelativistic limit is presented. The density dependence (1/ne) of the damping of the extraordinary wave for not too small densities and the strong absorption of the ordinary wave for quasi‐perpendicular propagation in a hot plasma (Te≳1 keV) are investigated numerically for several angles of incidence.
112 citations
TL;DR: In this paper, a perturbation formulation of the equations of linear piezoelectricity is obtained using a Green's function approach, which is applied in the calculation of the attenuation of surface waves due to the finite electrical conductivity of thin metal films plated on the surface and air loading.
Abstract: A perturbation formulation of the equations of linear piezoelectricity is obtained using a Green’s function approach. Although the resulting equation for the first perturbation of the eigenvalue strictly holds for real perturbations of real eigenvalues only, it is formally extended to the case of purely imaginary perturbations of real eigenvalues. The extended equation is applied in the calculation of the attenuation of surface waves due to the finite electrical conductivity of thin metal films plated on the surface and air loading. The influence of the viscosity of the air is included in the air‐loading analysis, and the calculated attenuation increases accordingly. Since the metal films are thin compared with a wavelength, an approximate thin‐plate conductivity equation is employed in the determination of the attenuation due to the electrical conductivity of the films. The resulting attenuation is obtained over a very large range of values of sheet conductivity. This is accomplished by using the equatio...
99 citations
TL;DR: In this paper, a new interpretation of a nonlinear wind-wave system is proposed, where the spectral components of the wind wave system are not considered a random collection of free waves, each obeying the usual dispersion relation, but are effectively non-dispersive bound-wave components of a single dominant wave, travelling at the speed of the dominant wave.
Abstract: A new interpretation of a nonlinear wind-wave system is proposed. It is proposed that, for steady wind blowing in one direction, a nonlinear wind-wave system can be completely characterized, to a good first approximation, by a single nonlinear wave train having a carrier frequency equal to that of the dominant frequency in the wind-wave spectrum. In this model, the spectral components of the wind-wave system are not considered a random collection of free waves, each obeying the usual dispersion relation, but are effectively non-dispersive bound-wave components of a single dominant wave, travelling at the speed of the dominant wave. To first order, the nonlinear wind-wave system is considered to be a coherent bound-wave system which propagates energy only at the group velocity of the dominant wave and is governed by nonlinear self-interactions of the type found in amplitude-modulated wave trains. The role of short free waves in the system is discussed. Results of laboratory experiments performed by the authors and by Ramamonjiarisoa & Coantic (1976) are found to provide evidence supporting the applicability of such a model to wind waves under virtually all laboratory conditions. Preliminary consideration is given to possible application of the model to oceanic wind waves and conditions are identified for which the model would be most likely to apply.
TL;DR: In this paper, the energy and momenta of the particles involved in a two-photon Raman scattering process were calculated self-consistently from the direction of the exciting photons, their energy, and direction of scattered light.
Abstract: Starting from Hopfield's theory of the excitonic polariton, the energies and momenta of the particles involved in a two-photon Raman scattering process may be calculated self-consistently from the direction of the exciting photons, their energy, and the direction of the scattered light. Attributing to the calculated momenta the measured energies of the emitted photons, one obtains the dispersion relation of the excitonic polaritons with a high accuracy in a semiexperimental way.
TL;DR: In this article, the authors presented numerical calculations of unstable discrete spiral modes for gas disks based on some recent developments of the asymptotic theory of density waves, and derived a local dispersion relation for these spiarl waves.
Abstract: Numerical calculations of unstable discrete spiral modes for gas disks are presented. These calculations are based on some recent developments of the asymptotic theory of density waves. In addition to Toomre's stability index Q, another local parameter, which depends primarily on the surface density and on the amount of differential rotation, is found to be essential for the growth of spiral modes and waves. A local dispersion relation for these spiarl waves is derived, and examined in the light of previous studies.
TL;DR: In this paper, the kinetic equation for electrons including Coulomb collisions and scattering by ion sound and related spectra is reduced to a system of equations for the energy distribution and the anisotropic part.
Abstract: The kinetic equation for electrons including Coulomb collisions and scattering by ion sound and related spectra is reduced to a system of equations for the energy distribution and the anisotropic part. The energy distribution is obtained for the cases where Coulomb collisions, runaway or turbulent heating dominates. Resistivity, heating rate, and the dispersion relation are significantly modified for the self‐consistent non‐Maxwellian distribution. Applications to turbulent heating by ion sound are made.
TL;DR: In this article, the results of a generalized polarization potential calculation of the particle and spin density fluctuation excitation spectra of 3He in the low-temperature limit are described and compared with the recent neutron scattering experiments of Skold and Pelizzari.
Abstract: The results of a generalized polarization potential calculation of the particle and spin density fluctuation excitation spectra of 3He in the low-temperature limit are described and compared with the recent neutron scattering experiments of Skold and Pelizzari. The extent to which the range and shape of the effective repulsive interaction between particles of parallel and antiparallel spin may differ is explored, and it is shown that changes of only a few percent have experimentally observable consequences. Good agreement with experiment is found for the dispersion of zero sound; the wave vector dependence of the enhancement of the low-frequency part of the spin density fluctuation excitations, and the coherent and incoherent static structure factors, are examined and compared with experiment for identical and slightly different repulsive parallel and antiparallel spin interactions. Simple qualitative arguments are presented which suggest that for wave vectors 0.4⩽q⩽ 1.2»-1, the zero-sound mode will be little affected as the temperature is raised to 1.2K, while the low-frequency spin fluctuation excitation spectrum undergoes considerable broadening, again in accord with experiment. Changes in the zero-sound dispersion relation with pressure are shown to be sensitive to the range of the repulsive part of the effective quasiparticle interaction, and it is suggested that experiments at 20 atm will help determine the physical origin of the comparatively large range (∼3 ») required to explain the experimental results for 3He at SVP.
TL;DR: In this paper, the physics, spectra and damping rates of high and low-frequency surface waves in semi-infinite and cylindrical plasmas are studied.
Abstract: The physics, spectra and damping rates of high- and low-frequency surface waves in semi-infinite and cylindrical plasmas are studied. The boundary conditions (both macroscopic and microscopic) imposed on the plasma fluid/particles are discussed. The solution of the Vlasov-Maxwell set of equations for semi-infinite and cylindrical plasmas assuming mirror reflection of plasma particles at the boundary is given. The dispersion relations are derived and analysed. The dispersion properties obtained from the kinetic model are compared with those derived from the fluid model.
TL;DR: In this article, the authors studied the hydromagnetic wave propagation characteristics in a mixture of cold and hot plasma in the presence of an inhomogeneous magnetic field and found that the storm-associated magnetic field oscillations with periods of 100-600 s might be caused by the coupling of Alfven waves and the unstable drift mirror waves under certain conditions.
Abstract: The article studies the hydromagnetic wave propagation characteristics in a mixture of cold and hot plasma in the presence of an inhomogeneous magnetic field. Electron and ion distribution functions with a temperature anisotropy and a density gradient are used to obtain the dispersion equation by solving the Vlasov equation and Maxwell equations. From the solutions of the dispersion equation we find that the Alfven waves can couple to unstable drift mirror waves under certain conditions. The polarization of the coupled waves is studied for varying parameters of temperature anisotropy and the cold to hot density ratio. From detailed comparison of the theoretical results with the low-frequency wave properties observed in the magnetosphere we propose that the storm-associated magnetic field oscillations with periods of 100-600 s might be caused by the coupling of Alfven waves and the unstable drift mirror waves.
TL;DR: In this article, a density depression is mainly created by the effect of the antenna nearzone field rather than by the wave radiation pressure, and the intense localized field near the antenna gives rise to electron heating which leads to the density trough.
Abstract: Whistler waves launched from an antenna damp away for small incident power. With increasing power, undamped nondiverging waves (the ducted waves) are observed, together with a field‐aligned density trough and electron heating. However, the density trough is found not only in the wave propagation regime (ω/ωc<1), but also in the wave evanescence regime (ω/ωc≳1). This implies that the density depression is mainly created by the effect of the antenna near‐zone field rather than by the wave radiation pressure. The intense localized field near the antenna gives rise to electron heating which leads to the density trough. The ducting of antenna‐launched whistler waves has been explained as a filamentation instability in terms of nonlinear wave‐plasma interactions.
TL;DR: In this paper, the two-fluid equations describing transient nonequilibrium liquid-vapour flow have been used to derive a general dispersion relation for acoustic waves for both dispersed and separated flow regimes.
TL;DR: In this paper, the energy levels of two simple examples of screened Coulomb potentials have been analyzed using nonperturbative methods, and the analysis indicates that the energy level as a function of the perturbation parameter has a branch cut along the negative real axis; starting from the origin.
Abstract: The energy levels of two simple examples of screened Coulomb potentials have been analyzed using nonperturbative methods. The analysis indicates that the energy levels as a function of the perturbation parameter ) have a branch cut along the negative real axis; starting from the origin. Furthermore, there are singularities on the second sheet, along ~X~e '' for ~X ~0. As a consequence of these singularities, the energy levels have an asymptotic series in X, which means that orie cannot use a power series in X to describe the energy levels to an arbitrary accuracy, The approximate but nonperturbative expression for the energy levels, which has been obtained by using dispersion relations, predicts energy levels which are in good agreement with those obtained from variational calculations.
TL;DR: In this article, the drift-tearing instability for m ≥ 2 is re-examined, by using two-fluid equations in a cylindrical tokamak plasma.
Abstract: The drift-tearing instability for m ≥ 2 is re-examined, by using two-fluid equations in a cylindrical tokamak plasma. Strong deviations from the previously given analytic dispersion relation are found in the most interesting regime, ω* > γT, owing to a non-local structure of the drift mode.
TL;DR: In this article, room temperature measurements of phonon dispersion relations in ferroelectric lithium niobate have been carried out by inelastic neutron scattering, and most of the measurements were made along the c* direction with two acoustic branches measured along an a* direction.
Abstract: Room temperature measurements of the phonon dispersion relations in ferroelectric lithium niobate have been carried out by inelastic neutron scattering. Most of the measurements were made along the c* direction with two acoustic branches measured along an a* direction. The q to 0 slopes of the acoustic branches are found to be in good agreement with the ultrasonic sound velocity measurements. Results on the phonon frequencies are compared with the optical data. Measurements carried out at elevated temperatures up to 900K failed to detect any softening of the 'ferroelectric mode' (the lowest-frequency optic Gamma 1 mode), thereby suggesting that the ferroelectric phase transition in lithium niobate is not associated with a soft phonon-mode instability. The possibility that the phase transition is of the order-disorder type, rather than the displacive type, is discussed, and, in this respect, similarities with the isomorphous lithium tantalate are pointed out.
TL;DR: In this paper, the evolution and effects of current-driven instabilities in isothermal, inhomogeneous plasmas are investigated in both theory and experiment, and anomalous effects resulting from waveparticle interactions, i.e., enhanced resistivity, ion heating, and electron viscosity, are compared with predictions based on quasilinear calculations.
Abstract: The evolution and effects of current‐driven instabilities in isothermal, inhomogeneous plasmas are investigated in both theory and experiment. Sucessive destabilizations of four different instabilities, low‐frequency drift wave, ion‐cyclotron drift wave, high‐frequency (continuous‐spectrum ion‐cyclotron) drift wave, and high‐frequency electron wave, are observed in Q‐device plasmas with increased current, and explained by a theory based on fluid and kinetic equations. Anomalous effects resulting from wave‐particle interactions, i.e., enhanced resistivity, ion heating, and electron viscosity, are compared with predictions based on quasi‐linear calculations. Analogous to ion sound causing important anomalies in the transport coefficients for plasmas with Te/Ti≫1, high‐frequency, continuous‐spectrum drift waves determine anomalous plasma behavior in inhomogeneous plasmas with Te/Ti≃1.
TL;DR: In this paper, small amplitude, axially symmetric waves in a thin-walled viscoelastic tube containing a viscous compressible fluid are considered and asymptotic formulae for the propagation constant k are obtained for both low and high frequencies and for various ranges of the parameters characterizing the tube and the fluid.
Abstract: Small amplitude, axially symmetric waves in a thin-walled viscoelastic tube containing a viscous compressible fluid are considered. Previous authors have found two modes of propagation for such waves but have studied them only in the low frequency, long wavelength limit. We show that there are infinitely many modes and study them at all frequencies. The appropriate dispersion equation was derived previously (Rubinow & Keller 1971) and analysed for an inviscid fluid. Now it is analysed for a viscous fluid. Asymptotic formulae for the propagation constant k are obtained for both low and high frequencies and for various ranges of the parameters characterizing the tube and the fluid. Special attention is paid to the case of a rigid tube and to parameter values that characterize the flow of blood in mammalian arteries. In addition, numerical results are obtained which complement the asymptotic formulae. Graphs of the velocity c vs. the frequency ω are presented for various modes and for various ranges of the parameters. Transmission-line equations and formulae for the impedance and compliance of the fluid-tube system are obtained, together with asymptotic and numerical results.
TL;DR: In this paper, a formalism for simultaneous inversion of surface wave phase velocity and attenuation to determine shear wave velocity and Q−1 in the earth is developed, which is mathematically more complete than the approach of correcting phase velocity data for the intrinsic dispersion due to anelasticity and inverting the corrected velocity data alone.
Abstract: A formalism is developed for simultaneous inversion of surface wave phase velocity and attenuation to determine shear wave velocity and Q−1 in the earth. A simultaneous inversion takes full account of the dependence of surface wave velocity and attenuation on both the elastic and the dissipative part of an earth structure and permits inclusion of the physical relationship between anelasticity and intrinsic dispersion that arises from linearity. The procedure is mathematically more complete than the approach of correcting phase velocity data for the intrinsic dispersion due to anelasticity and inverting the corrected velocity data alone, and it gives different results. The proposed formalism, including resolution analysis, weighted least squares inversion, and extremal inversion, is applied to Love waves in western North America. Various intrinsic dispersion-attenuation relations are tested, including Q independent of frequency, Q varying as a power of frequency, and Q specified by a sum of relaxation mechanisms. The results of the inversions confirm the coincidence of the low-velocity and low Q zones beneath western North America for frequencies in the surface wave band. Compared with previous inversion of Q−1 data alone, the simultaneous inversion results in improved depth resolution of Qs−1 and the elimination of an apparent incompatibility of low-attenuation data at 20- to 25-s periods. The Love wave data do not discriminate among the various dispersion-attenuation relations, though a constant Q leads to the removal of the requirement for a low-velocity zone at frequencies above 1 Hz. The predicted intrinsic dispersion within the low-velocity zone varies from 1% to 10% between 0.01 and 1 Hz for the various models; broadband measurements of body wave dispersion offer the greatest promise for choosing among the models.
TL;DR: In this article, a dual-frequency L-band radar was used to detect small-scale water waves induced by larger-scale 2 to 18 m gravity waves by using a coherent, dualfrequency radar technique.
Abstract: The modulation of small-scale centimeter (and later decimeter) water waves induced by larger-scale 2 to 18 m gravity waves has been studied by using a coherent, dual-frequency radar technique. Experiments using a prototype CW X-band radar operating at 9.3 GHz have been performed. Experience with the X-band system led to the subsequent development of a pulsed dual-frequency L-band radar operating at 1.2 GHz. The gravity wave modulation manifests itself as a narrow, Doppler-shifted, resonance peak in the product power spectrum of the backscattered returns. The dispersion relation (for both deep and shallow water) of the modulation pattern matches that of gravity waves. Modulation amplitude spectra have been experimentally obtained which, after sufficient averaging, closely resemble directional gravity wave spectra simultaneously obtained from capacitance wave probe and Sea Photo Analysis measurements. Temporal stationarity of the large-scale gravity wave structure may only be assumed for finite data acquisition times of, perhaps, the order of one hour, or less. Experiments with the X-band system, however, have shown that high resolution, sufficiently averaged, densely sampled spectra require measurement periods much longer than one hour. To reduce the overall data acquisition time, multiplexing of the individual spectral samples has been employed. The amount of spectral averaging required was substantially reduced by developing and using a dual-frequency L-band radar (Bragg resonant with 12 cm short-gravity waves) which proved to be much less sensitive to wind-induced fluctuations in the small-scale wave return. A method is also presented for enhancing the amplitude of the resonance peak through selective Doppler filtering. This technique was subsequently used to facilitate detection of small tidal currents flowing in the Chesapeake Bay.
TL;DR: In this article, a three-pass plus two-pass tandem Fabry-Perot interferometer was used to analyze the backscattered light from the polycrystalline or single crystal samples.
Abstract: Light scattering measurements from thermal acoustic magnons have been performed for the first time in the metals iron and nickel. A three-pass-plus-two-pass tandem Fabry-Perot interferometer was used to analyze the backscattered light from the polycrystalline or single crystal samples. The magnetic field was parallel to the sample surface and perpendicular to the plane of incidence. Two modes can be observed: bulk magnons and surface magnons. The dependence of the frequency on the applied field for iron can be described by a magnetostatic dispersion relation for an insulating ferromagnet. In nickel the measured frequencies are considerably higher than predicted. Some of the problems arising from the metallic character of the samples are discussed.
TL;DR: In this paper, Taylor's analysis is extended to a shear flow in which the velocity vector rotates with depth but is horizontally homogeneous, and a tensor coefficient represents the effect of the vertical shear.
Abstract: In 1953, G. I. Taylor showed that longitudinal dispersion of matter in a pipe can be represented by a one-dimensional dispersion equation and derived the magnitude of a coefficient to represent the effect of the shear flow. Taylor's analysis is extended to a shear flow in which the velocity vector rotates with depth but is horizontally homogeneous. We show that horizontal dispersion can be represented by a two-dimensional equation in which a tensor coefficient represents the effect of the vertical shear. The terms in the tensor are derived for an approximation to the mean flow on the Middle Atlantic Bight of the U.S. Atlantic continental shelf.
TL;DR: In this paper, the properties of Bernstein modes were investigated through computer simulations using two-dimensional and two-and-one-half-dimensional (i.e., two spatial and three velocity coordinates) electrostatic models with fixed magnetic field.
Abstract: The properties of Bernstein modes were investigated through computer simulations using two‐dimensional and two‐and‐one‐half‐dimensional (i.e., two spatial and three velocity coordinates) electrostatic models with fixed magnetic field. The measured discrete spectrum was found to agree with the linear dispersion relation for these modes. The quasi‐periodic phenomenon of early phase‐mixing damping and later recurrence, predicted by Baldwin and Rowlands, was observed. For large wavenumber k⊥, the initial damping rate is the same as that for Landau damping in an unmagnetized plasma; for small k⊥, however, it is much stronger. The recurrence peaks slowly damp in time at a rate proportional to k2⊥D, where D is the measured cross‐field particle diffusion coefficient which is dominated by convective transport. Finally, splitting of the main spectral peaks and the appearance of subpeaks at half‐integral multiples of the cyclotron frequency are observed and may be explained by nonlinear mode coupling.
TL;DR: Inelastic neutron scattering has been used to measure the phonon energy dispersion relation along the (001), (110) and (111) directions of a single crystal of lead fluoride, PbF2, at 10K as mentioned in this paper.
Abstract: Inelastic neutron scattering has been used to measure the phonon energy dispersion relation along the (001), (110) and (111) directions of a single crystal of lead fluoride, PbF2, at 10K. Shell model parameters have been successfully fitted to the data, and those giving the best fit have been used to calculate the density of phonon states and values for the elastic constants and the Debye temperature. The results are compared with data obtained by other methods.