Showing papers on "Normal mode published in 1970"

Uniqueness in the Inversion of Inaccurate Gross Earth Data

Abstract: A gross Earth datum is a single measurable number describing some property of the whole Earth, such as mass, moment of inertia, or the frequency of oscillation of some identified elastic-gravitational normal mode. We suppose that a finite set G of gross Earth data has been measured, that the measurements are inaccurate, and that the variance matrix of the errors of measurement can be estimated. We show that some such sets G of measurements determine the structure of the Earth within certain limits of error except for fine-scale detail. That is, from some setsG it is possible to compute localized averages of the Earth structure at various depths. These localized averages will be slightly in error, and their errors will be larger as their resolving lengths are shortened. We show how to determine whether a given set G of measured gross Earth data permits such a construction of localized averages, and, if so, how to find the shortest length scale over which G gives a local average structure at a particular depth if the variance of the error in computing that local average from G is to be less than a specified amount. We apply the general theory to the linear problem of finding the depth variation of a frequency-independent local elastic dissipation ( Q ) from the observed damping rates of a finite number of normal modes. We also apply the theory to the nonlinear problem of finding density against depth from the total mass, moment and normal-mode frequencies, in case the compressional and shear velocities are known.

Raman-Scattering Selection-Rule Breaking and the Density of States in Amorphous Materials

Abstract: We present a calculation of the dielectric correlation function in glasses showing how the assumption of short correlation length for normal modes breaks the momentum selection rules and leads to expressions for the first-order Raman-scattering intensity in terms of the density-of-states functions and known frequency-dependent amplitudes

Atomic vibrations in vitreous silica

Abstract: Frequency spectra and normal modes of vibration have been computed for vitreous silica. They have been calculated from atomic arrangements in physical models based on the random network theory. The positions of bands in the computed spectra agree well with observed features in the experimental infra-red and Raman spectra of the glass. Detailed analysis of the normal modes indicates that the bands at 1050, 750 and 400 cm–1 are associated with bond-stretching, bending and rocking motions, respectively, of the oxygen atoms. Atomic vibrations in the glass are, on the whole, less extended in space than the plane wave-like modes which prevail in perfect crystals. The spatial localization tends to be greatest at high frequencies and near band edges. If non-bridging oxygen atoms are present in the structure, the frequenncy spectrum exhibits an additional band of very intense localization, associated with bond-stretching vibrations of the non-bridging atoms.

Infrared and microwave magnetoplasma effects in semiconductors

Abstract: The interaction of electromagnetic waves with free-carrier plasmas in semiconductors and semimetals is analysed, with particular emphasis on microwave and infrared effects arising in the presence of external magnetic fields. The general frequency- and field-dependent dielectric tensor is initially developed for a single isotropic band via the Boltzmann equation in the local approximation. Solution of the electromagnetic wave equation in the tensor medium yields two normal modes, whose complex propagation constants and polarizations are determined by the microscopic medium parameters as well as the angle θ between wave vector q and applied magnetic field B0. We first consider wave propagation in the lossless (collisionless) limit, and concentrate on the properties of normal modes in the Faraday (q parallel B0) and Voigt (q perpendicular B0) geometries. Essential features of the wave interaction with the medium are conveniently set forth in `contour maps' which display zeroes and infinities of the lossless dielectric constants, indicating resonances, dielectric anomalies and black-out regions for each mode of propagation over a wide range of frequencies and fields. Losses are then explicitly incorporated into the dielectric tensor, and their effect discussed in some detail. A number of experimental examples associated with the single band model are presented, including free-carrier absorption, cyclotron resonance, Faraday and Voigt effects, helicon waves, magnetoplasma reflection and small-particle effects. The presentation is subsequently generalized to more complicated systems. The dielectric tensor is developed for a multiple-carrier plasma and for anisotropic bands. A variety of new effects is seen to arise, including hybrid resonances, tilted-orbit resonances and Alfven wave propagation. `Contour maps' of appropriate lossless dielectric constants are again found convenient in discussing these new features. Effects of spatial dispersion (nonlocal phenomena) and of orbital quantization on the high-frequency response of a free-carrier system are then considered. The contribution of the polar lattice, which gives rise to a vast array of coupled magnetoplasma-phonon modes, is finally examined. A further elaboration of certain topics arising in this review is presented in an appendix, including a detailed discussion of electron dynamics, the `polariton' formalism, fundamentals of Kramers-Kronig analysis and magneto-optics of birefringent media.

Role of Landau damping in crossed-field electron beams and inviscid shear flow

Abstract: Under certain circumstances the problem of the stability of low‐density crossed‐field electron beams is precisely analogous to the problem of the stability of incompressible inviscid shear flows The electron density, the drift velocity, and the potential correspond to the fluid vorticity, velocity, and the stream function, respectively Neutrally stable normal modes of oscillation that are found when certain suitable step‐function unperturbed electron density (vorticity) profiles are assumed seem to disappear when small gradients are present In the context of a Laplace transform solution of the initial value problem, the vanishing normal modes can be found by analytic continuation of the solution onto another sheet of the cut complex ω plane The “recovered” modes are found to be damped by a fluid‐wave resonant interaction which closely resembles the Landau damping of plasma oscillations in warm plasmas by particle‐wave resonance An interesting feature of the neutral waves is that they have a negative energy and so are destabilized by the removal of energy from the system (for example, by slightly dissipative walls) The sign and magnitude of the Landau damping term is evaluated for a simple case and compared (for the electronic case) with the growth provoked by slightly lossy walls

Wave‐driven inertial oscillations

Abstract: In a nonrotating system, the shear Reynolds stresses exerted by surface or internal gravity waves vanish on account of the exact quadrature between the horizontal and vertical orbital velocities. It is shown that a rotation of the system induces small in‐phase perturbations, resulting in a mean Reynolds stress which can generate low frequency currents. If both the wave field and the ocean are homogeneous with respect to the horizontal coordinates, the low‐frequency response is an undamped inertial oscillation. If either the wave field or the ocean are weakly inhomogeneous, the oscillation disperses in the vertical and horizontal directions due to phase‐mixing of modes with closely neighboring frequencies. Other effects which produce small frequency shifts also contribute to phase‐mixing, for example the horizontal component of the Coriolis vector and nonlinear interactions with geo‐strophic currents. The analysis is based on operator representations which avoid normal mode decomposition and yield...

Localization of normal modes in vitreous silica, germania and beryllium fluoride

Abstract: The authors previously (1968) presented vibrational spectra for vitreous silica, germania and beryllium fluoride, calculated from atomic arrangements in physical models based on the random network theory. The present work describes the localization properties of the normal modes of vibration. It is shown that the vibrations are, on the whole, less extended in space than the plane wave-like modes which characterize regular crystals. Localization tends to be greatest at high frequencies and, especially, near band edges. If nonbridging oxygen atoms are present in the structure, the spectra of vitreous silica and germania exhibit an additional band of very intense localization; atomic motions associated with modes in this band are effectively restricted to no more than a few atoms.

Quantum electrodynamics of spinless particles between conducting plates

Abstract: The energy of any system containing charged particles arises partly from its coupling to the quantized electromagnetic field; it changes on inserting the system between conducting plates, because these alter the normal modes of the field relative to free space. We calculate to order e2 such energy shifts for a free electron and for a hydrogen atom, neglecting spin, but allowing in full for electrostatic and retardation corrections as well as for changes in the Lamb shift proper. The potential between a particle and a single plate follows as a limiting case. An appendix details the distinction between classical and quantum effects.

Ferromagnetic Resonance in Thin Films. I. Theory of Normal-Mode Frequencies

Abstract: A theory is developed for the frequencies of the general ferromagnetic normal modes of a sample of arbitrary shape and size with both exchange and demagnetization energies included. The frequencies of the modes of rectangular and circular films are calculated by casting the linearized equation of motion of the magnetization into the form of an eigenvalue equation, which is solved by a variational method. The results explain the experiments of Dillon, Besser, Sparks et al., Freedman and Brundle, and Voltmer in detail qualitatively and typically to within \ensuremath{\sim}5-10% quantitatively for the mode spacings, with possible exceptions for the first few low-order modes in some samples, for which several contributions to the line spacings are difficult to estimate accurately. Pinning the surface spins has little effect on the frequencies and intensities of magnetostatic modes (with negligible exchange energy). The theory has implications concerning the main-resonance position in finite films, and together with experiments, further verifies Portis's modespacing theory. A simple physical explanation of the results is given, and the relation of the results for for finite films to those for infinite films is given.

Crystal Dynamics and the Ferroelectric Phase Transition of Sodium Nitrite

Abstract: The crystal dynamics of sodium nitrite has been studied by coherent neutron inelastic scattering techniques. The normal modes of vibration propagating along the three principal symmetry axes in the crystal at 296°K have been measured. The results have been used to find the parameters of rigid ion models for NaNO 2 in which the NO 2 ion was treated as a group. Although none of the models are entirely satisfactory they do give a fair description of the results. The temperature dependence of the frequencies of some of the normal modes was measured and none were found to be very temperature dependent near the ferroelectric transition. Ferroelectric critical scattering was observed which was quasi-elastic showing that the ferroelectric fluctuations change slowly compared with the frequencies of almost all the normal modes of vibration. A model is deduced for the ferroelectric fluctuations incorporating the motion of the atoms but it is found that the motion of the atoms in these fluctuations cannot be describe...

Measurements of mode attenuation coefficients in shallow water

Abstract: An experiment was conducted in the Gulf of Mexico to evaluate a technique for measuring modal attenuations. Use of short pulses allowed individual normal modes to be observed and then identified by comparing observed pressure distributions and arrival times with those predicted by a normal mode model. Variable depth sources for 400, 750, and 1500 Hz were mounted on a fixed platform. A vertical string of hydrophones was suspended from a ship, which anchored at several ranges. The measurements were made under downward refracting conditions in 30 m of water with a level sand bottom. For the first two modes at 400 Hz, vertical pressure distributions, relative arrival times and time spreading due to dispersion were in good agreement with theory. Higher modes and modes at 750 Hz and 1500 Hz were not identified. By comparing peak amplitudes at different ranges, a mode attenuation coefficient of 1.3 dB/km was determined for the first mode at 400 Hz.

Young's and Shear Moduli of Unidirectional Composites by a Resonant Beam Method

Abstract: The resonant frequencies of unidirectional graphite epoxy com posite beams were found to deviate markedly from classical beam theory predictions at higher modes of vibration. Timoshenko beam theory was used to account quantitatively for the dependence of ex perimental resonant frequencies on mode of vibration, length/thickness ratio, and the ratio of Young's modulus/shear modulus of the com posite beam. By using reasonable values for the longitudinal-trans verse shear moduli, the longitudinal Young's modulus of anisotropic composite beams under vibration agreed well with values determined by static tests and became independent of mode of vibration and L/t. The vibrating beam test method was employed for the deter mination of E11, E22, and G12.

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.

Visualization of Plane-Strain Vibration Modes of a Long Cylinder Capable of Producing Sound Radiation

Abstract: This paper points out that a class of plane‐strain vibration modes in an isotropic solid cylinder has properties suitable for interaction with a sound field in a fluid medium surrounding the cylinder. A resonance‐type theory of such modes is presented. Experimental verification of this theory is obtained by observing stress birefringence patterns of resonating glass cylinders. The circumferential waves arising from these modes are likely to play a part in the scattering phenomena occurring in the medium surrounding the cylinder.

Rare‐Gas Scattering from LiF: Correlation with Lattice Properties. II

Abstract: The scattering of thermal energy atomic beams of Ne, Ar, Kr, and Xe from the (001) face of LiF has been studied. Two dominant peaks in the scattering distributions in the plane of incidence are observed: one peak remains fixed with respect to the crystal whereas the position of the other displays a strong dependence on angle of incidence, incident energy, and azimuthal orientation of the target. The former peak is associated with the normal modes of the lattice, in particular with dipole–induced dipole attractive forces produced by longitudinal optical modes in next‐nearest‐neighbor unlike ion directions. The latter peak is discussed in terms of the surface properties of the crystal, e.g., interaction potential and elastic properties, the latter acting on the transfer of tangential momentum. For Ne, additional peaks occur which arise from diffraction effects. The present results provide additional insight into the phenomenon of preferential scattering reported earlier and strongly emphasize the inadequacies of existing theoretical descriptions of thermal energy atomic and molecular (TEAM) scattering from surfaces, especially when applied to the surfaces of ionic crystals.

Physical Interpretation of the WKB or Eikonal Approximation for Waves and Vibrations in Inhomogeneous Beams and Plates

Abstract: The WKB or eikonal approximation is derived for flexural waves of constant frequency in inhomogeneous Euler‐Bernoulli beams and plates and for waves in inhomogeneous Timoshenko beams. The derivation is based on the assumption of negligible partial reflection of waves and holds when the relative variation of parameters characterizing the medium is small over distances comparable to one wavelength. The approximate relations governing the variation of wave amplitude from point to point are derived from the requirement that no energy be lost from the wave. The extension of this technique to include evanescent waves as well as propagating waves is also discussed. As an application of the method, some simple results are given for the modal frequencies and normal modes of a cantilevered inhomogeneous Euler‐Bernoulli beam. A general development of the eikonal method applicable to a wide class of mechanical systems is also given, which establishes the consistency of the energy‐conservation technique with the results obtained by taking the eikonal approximation as the zeroth‐order approximation to a series.

Waveforms in rotating Couette flow

Abstract: An experimental survey has been made of the different kinds of coherent waveforms which occur in the finite amplitude region of unstable circular Couette flow. Gap ratios ranging from quite wide to very narrow and Taylor numbers up to the onset of turbulence have been investigated. Throughout this range all the waveforms can be assigned to five basic types: regular Taylor cells, spirals, a subharmonic asymmetric mode and two other types of rotationally asymmetric modes. Sketches of the basic types of cells and a description of how they are modified in various regions of the modal diagram are presented. It is shown how the normal modes of the system may be combined to form the basic modes. A comparison is made between the present data and previous reports of the higher modes; the theoretical work on this problem is also related to the observed results.

Studies of the Dynamical Jahn-Teller Effect on Static Magnetic Susceptibility

Abstract: Static magnetic susceptibilities of simplified model vibronic systems where non-Kramers paramagnetic ions in triplet ground state interact either with localized tetragonal modes of vibration with sharp frequency spectrum at ω (localized model), or with tetragonal modes composed of the superposition of phonons with continuous frequency spectrum (Debye model) are calculated in a rigorous way. The susceptibility is simply expressed in terms of a temperature dependent reduction factor γ L ( T ) for the localized model, or γ D ( T ) for the Debye model, which diminishes the effective Curie constant and makes the susceptibility deviate from the Curie-Weiss law. The theory explains the characteristic features of the susceptibilities of dilute solid solutions of UO 2 in ThO 2 measured by Slowinski and Elliott and by Comly, for example, the convex curvature of reciprocal susceptibility vs. temperature plot. The values of parameters determined to fit the observed susceptibilities are reasonable in orders of magnitude.

The Free Oscillations of Fluid on a Hemisphere Bounded by Meridians of Longitude

Abstract: A precise calculation is presented of the normal modes of oscillation of an ocean of uniform depth which is bounded by two meridians of longitude separated by an angle of 180°. The calculation takes full account of the horizontal divergence of the motion, and so is applicable to both barotropic and baroclinic modes of oscillation.At small values of the parameter (defined fully in § 1) the calculation yields both the familiar gravity waves and also the nondivergent planetary waves computed in an earlier paper (Longuet-Higgins 1966). At large, positive values of e , corresponding to baroclinic waves, new types of oscillation appear in which the flux of energy is concentrated near the equator, the circuit being completed by Kelvin waves along the meridianal boundaries. The calculated frequencies are compared with asymptotic expressions derived from a recent beta-plane analysis by D. W. Moore. Solutions are also found corresponding to negative values of e . These must be included in a complete calculation of the response of the ocean to external forces. At small values of e these solutions resemble the planetary waves. At large (negative) values of e they represent almost-inertial motions concentrated near the poles, having a phase-velocity towards the east and an amplitude modulated so as to vanish at the boundaries. The calculations are relevant to the real ocean in so far as they show the kinds of oscillation that might be expected in any ocean basin including any section of the equator (or including a pole). They also indicate the degree of accuracy to be expected in computing the frequencies of the normal modes by beta-plane methods.

Dynamics of the rutile structure. I. Space group representations and the normal mode analysis

Abstract: Multiplier group approach has been used to analyse the normal vibrations of the rutile structure. The irreducible representations at various critical points in the Brillouin zone of the crystal are listed. The basis vectors for the stud of the lattice vibrations have been derived using the projection operator technique and are tabulated. The general form of the atomic displacements at the critical points has been discussed.

Exact Natural Frequencies for Cross-Ply Laminates

Abstract: Vibration of a simply supported cross-ply laminated plate infinite in length is analyzed. Existence of plane strain conditions allow an exact solution of the field equations. Curves of variation of exact natural frequencies versus wavelength for several cases of a two-ply laminated plate are calculated. The first two mode shapes are plotted for various wavelengths and compared to classical plate theory. Flex ural vibration appears to be well approximated by classical theory up to moderate wavelengths. Extensional vibration is approximated mar ginally only for very long wavelengths.

Transport of Resonance Radiation in an Infinite Cylinder

Abstract: The Holstein-Biberman integrodifferential equation for the transport of resonance radiation has been solved for a gas contained in a long cylindrical container The solution involves the following assumptions: (i) The excited atoms are initially distributed uniformly along the axis of the cylinder; (ii) the pressure and temperature of the gas are such that the absorption coefficient has either a pure Doppler-broadened or a pure pressure-broadened profile; (iii) the radius of the cylinder corresponds to many optical depths at the frequency corresponding to the maximum of the absorption coefficient It is also shown that steady-state solutions corresponding to a line source along the axis of the cylinder can be calculated by the same procedure The geometry and the initial conditions considered here are of particular interest because the situation can be simulated experimentally Hurst and Thonnard have pointed out that an approximate line of excited atoms can be produced by sending a well-collimated pulse of protons or electrons down the axis of a long cylinder containing the gas of interest The present paper also considers the decay of resonance radiation from a steady state built up by a line source The latter situation is even easier to arrange experimentally All experimental work to date has concerned itself with situations where the initial distribution of excited atoms is not known, and theory and experiment can only be compared at times which are large enough so that the system has decayed to its lowest eigenmode