Wave Vector Space Research Articles

Calculation of the condensate fraction in liquid helium-4

We adduce the results of the condensate fraction calculation for liquid helium-4. The method is derived from the first principles and involves minimum assumptions. The only experimental quantity we need in our calculations is the static structure factor which is easily measurable. We use the approximation in which expressions contain one summation in the wave vector space (or one integration it the radius vector space) in order to demonstrate the validity of our method. The computed values of the condensate fraction lie within 8.8 ÷ 14% depending on the model potential for the short-range interaction. This result agrees with recent experimental measurements and numerical estimations.

Scattering T-matrix theory in wave-vector space for surface-enhanced Raman scattering in clusters of nanoscale spherical metal particles

Very large enhancements up to 14 orders of magnitude in the Raman cross section from a molecule adsorbed on a single cluster of a few nanoscale metal particles has been reported recently. The enhancement is believed mainly due to the enhanced electric field because of the excitation of the localized surface plasmon modes. We have developed a Green's function theory using scattering matrix approach in the wave-vector space to solve the Maxwell equations for the enhanced field near a spherical metal particle cluster. The advantage of working in the wave-vector space is that one does not need to use complicated translational addition theorem required in the real space as used in earlier calculations. Therefore our theory can be easily extended to any shape or size of the cluster. We consider clusters of two, three, and four spherical particles forming a linear chain, triangle, and square and calculate their localized surface modes. These modes have much more localized field near the cluster compared to those of single metal sphere and are redshifted. We find the enhancement in the Raman cross section can reach up to 10 orders of magnitude due to the resonant excitation of these modes for silver particle clusters and is in a broad frequency range. We also find new results that chainlike clusters of three or more particles have very sharp resonant features that give a dramatic increase in the enhancement near the resonance. The results for gold particle clusters are also presented.

Characteristics of field‐aligned density depletion irregularities in the auroral ionosphere that duct Z‐ and X‐mode waves

The small‐scale and two‐point nature of the Observations of Electric‐field Distributions in the Ionospheric Plasma—A Unique Strategy C (OEDIPUS‐C, OC) dual‐payload propagation experiment in the auroral ionosphere in 1995 has permitted improved measurements of the parameters of magnetic field‐aligned density irregularities. Comparatively strong and dispersed pulses were observed at frequencies f just above the electron plasma frequency fp when the electron gyrofrequency fc was less than fp. The waves are interpreted as quasielectrostatic Z‐mode propagation with dispersion surfaces close to those of the Langmuir solutions in wave vector space, albeit at somewhat lower refractive indices of about 50. If mission length surveys of the Z‐wave intensities are aligned with histories of fp at the payload and of the strength of X‐ and fast Z‐mode ionospheric reflection echoes, a strong positive correlation is found at momentary relative depletions of the ambient density. These observations are taken as evidence of ducting in the field‐aligned depletions. The spectra of these strong Z‐mode transmissions are similar to those of slow Z ducted spectra observed at similar f, fp, and fc values in the OEDIPUS‐A experiment in 1989. The magnitudes of the density depletions are found to lie in the range 7–21% and to have cross‐field dimensions of a few kilometers. The present duct dimensions are of the same order as the previous findings from ionospheric X‐mode electromagnetic echoes on OC, but the depletions are up to 10 times deeper. Measurements of ducting irregularities can lead to insights into their formation. This will be important for our understanding of the interfaces of the ionospheric or magnetospheric topologies where irregularity formation is an important link in the large‐scale flow of energy.

Stratified quantization approach to dissipative quantum systems: Derivation of the Hamiltonian and kinetic equations for reduced density matrices

A technique for describing dissipative quantum systems that utilizes a fundamental Hamiltonian, which is composed of intrinsic operators of the system, is presented. The specific system considered is a capacitor (or free particle) that is coupled to a resistor (or dissipative medium). The microscopic mechanisms that lead to dissipation are represented by the standard Hamiltonian. Now dissipation is really a collective phenomenon of entities that comprise a macroscopic or mesoscopic object. Hence operators that describe the collective features of the dissipative medium are utilized to construct the Hamiltonian that represents the coupled resistor and capacitor. Quantization of the spatial gauge function is introduced. The magnetic energy part of the coupled Hamiltonian is written in terms of that quantized gauge function and the current density of the dissipative medium. A detailed derivation of the kinetic equation that represents the capacitor or free particle is presented. The partial spectral densities and functions related to spectral densities, which enter the kinetic equations as coefficients of commutators, are evaluated. Explicit expressions for the nonMarkoffian contribution in terms of products of spectral densities and related functions are given. The influence of all two-time correlation functions are considered. Also stated is a remainder term that is a product of partial spectral densities and which is due to higher order terms in the correlation density matrix. The Markoffian part of the kinetic equation is compared with the Master equation that is obtained using the standard generator in the axiomatic approach. A detailed derivation of the Master equation that represents the dissipative medium is also presented. The dynamical equation for the resistor depends on the spatial wavevector, and the influence of the free particle on the diagonal elements (in wavevector space) is stated.

Spacecraft observations of solar wind turbulence: an overview

Spacecraft measurements in the solar wind offer the opportunity to study magnetohydrodynamic (MHD) turbulence in a collisionless plasma in great detail. We review some of the key results of the study of this medium: the presence of large amplitude Alfvén waves propagating predominantly away from the Sun; the existence of an active turbulent cascade; and the presence of intermittency similar to that in neutral fluids. We also discuss the presence of anisotropy in wavevector space relative to the local magnetic field direction. Some models suggest that MHD turbulence can evolve to a state with power predominantly in wavevectors either parallel to the magnetic field (‘slab’ fluctuations) or approximately perpendicular to it (‘2D’). We review the existing evidence for such anisotropy, which has important consequences for the transport of energetic particles. Finally, we present the first results of a new analysis which provides the most accurate measurements to date of the wave-vector anisotropy of wavevector power in solar wind MHD turbulence.

Anisotropic electrostatic turbulence and zonal flow generation

In this paper we analyse the running and asymptotic diffusion coefficients of a plasma in the case of zonal flow generation by an anisotropic stochastic electrostatic potential. Both the weak and relatively strong turbulence regimes were analysed. The analysis of the diffusion coefficients in wave vector space provides an illustration of the fragmentation of drift wave structures in the radial direction and the generation of long-wavelength structures in the poloidal direction that are identified as zonal flows. We have shown that the fragmentation of drift wave structures is strongly influenced by the anisotropy parameter, the electrostatic Kubo number and by the initial values of the wave vector.

The k‐filtering technique applied to wave electric and magnetic field measurements from the Cluster satellites

The k‐filtering technique is a method to characterize stationary fluctuations in space plasmas in terms of the wave energy distribution in the frequency and wave vector space. It has the ability to distinguish between wave modes of the same frequency in the spacecraft frame of reference, but with different wave vectors. This method is based on simultaneous multi‐point measurements of the wave field components, where a filter bank is used to enhance the spatial resolution. We have for the first time combined electric field data from the Electric Field and Wave (EFW) instrument and magnetic field data from the Spatio‐Temporal Analysis of Field Fluctuation (STAFF) instrument on the four Cluster spacecraft in order to determine the wave energy distribution. The k‐filtering technique has previously been performed with only the magnetic field measurements. The reason to include the electric field measurements is that it is important to include as much data as possible in order to get the best possible estimation of the wave energy distribution. Another reason is that it also enables comparisons between the electric and the magnetic part of the wave energy distribution to make it possible to differentiate the observed waves according to their polarization. The k‐filtering method has been extended in order to allow for two measured components of the electric field, and also for the possibility that the electric field measurements from one or more satellites cannot be used for k‐filtering. The technique has been applied on satellite data from the magnetosheath and the foreshock, and from these examples it is clear that k‐filtering using both electric and magnetic field measurements is a good tool for characterizing the waves that are observed in space.

Ferromagnetism of (ScCa)Co 2 Laves phase compound synthesized under high pressure

The appearance of ferromagnetism in (Sc 0.85Ca 0.15)Co 2 Laves phase synthesized under high pressure of 6 GPa was confirmed. A magnetically homogeneous state was achieved by applying pressure of 6 GPa in 1373 K for 48 h. This is the first study that confirmed a stable ferromagnetic state in exchange-enhanced Pauli paramagnet ScCo 2-based compound system. The quantities of pressure P and magnetization M dependence of the Curie temperature T C, dln T C/d P and (dln T C/d P)/(dln M/d P) = dln T C/dln M for (Sc 0.85Ca 0.15)Co 2 are smaller than those for Lu(Co 1 − x Al x ) 2 compounds, and accordingly the magnetic moment of (Sc 0.85Ca 0.15)Co 2 is stiffer against pressure. The ratio of the spin wave stiffness constant D s to T C, D s/ T C, in (Sc 0.85Ca 0.15)Co 2 is smaller than that of Lu(Co 1 − x Al x ) 2, implying that the spin wave in (Sc 0.85Ca 0.15)Co 2 spread over wider wavevector space.

Transport equations for waves in randomly perturbed Hamiltonian systems, with application to Rossby waves

Transport equations for linear waves in randomly perturbed media are derived for a general class of Hamiltonian systems. These equations govern the evolution in position–wavevector space of the energy density of the waves; they describe the modulations caused by large-scale inhomogeneities of the medium, and the scattering due to weak random perturbations with spatial scales comparable to the wavelengths. Particular attention is paid to the conservative properties of the transport equations inherited from the Hamiltonian structure of the systems considered. The general results are used to derive transport equations for Rossby waves propagating in a two-layer model with random topography.

Role of nonlinear toroidal coupling in electron temperature gradient turbulence

Global gyrokinetic particle simulation and nonlinear gyrokinetic theory find that electron temperature gradient (ETG) instability saturates via nonlinear toroidal coupling, which is a nonlocal interaction in the wave vector space that transfers energy successively from unstable modes to damped modes preferentially with lower toroidal mode numbers. The electrostatic ETG turbulence is dominated by nonlinearly generated radial streamers. The length of the streamers scales with the device size, which is longer than the distance between mode rational surfaces and electron radial excursions. Both fluctuation intensity and transport level at saturation are independent of the streamer length, and are much smaller than the mixing length estimates.

Intermittency in MHD turbulence and coronal nanoflares modelling

Abstract. High resolution numerical simulations, solar wind data analysis, and measurements at the edges of laboratory plasma devices have allowed for a huge progress in our understanding of MHD turbulence. The high resolution of solar wind measurements has allowed to characterize the intermittency observed at small scales. We are now able to set up a consistent and convincing view of the main properties of MHD turbulence, which in turn constitutes an extremely efficient tool in understanding the behaviour of turbulent plasmas, like those in solar corona, where in situ observations are not available. Using this knowledge a model to describe injection, due to foot-point motions, storage and dissipation of MHD turbulence in coronal loops, is built where we assume strong longitudinal magnetic field, low beta and high aspect ratio, which allows us to use the set of reduced MHD equations (RMHD). The model is based on a shell technique in the wave vector space orthogonal to the strong magnetic field, while the dependence on the longitudinal coordinate is preserved. Numerical simulations show that injected energy is efficiently stored in the loop where a significant level of magnetic and velocity fluctuations is obtained. Nonlinear interactions give rise to an energy cascade towards smaller scales where energy is dissipated in an intermittent fashion. Due to the strong longitudinal magnetic field, dissipative structures propagate along the loop, with the typical speed of the Alfvén waves. The statistical analysis on the intermittent dissipative events compares well with all observed properties of nanoflare emission statistics. Moreover the recent observations of non thermal velocity measurements during flare occurrence are well described by the numerical results of the simulation model. All these results naturally emerge from the model dynamical evolution without any need of an ad-hoc hypothesis.

The Effect of the Localization in a Quantum Well on the Lifetime of the States of Shallow Impurity Centers

The results of theoretical and experimental studies into the effect of the localization in a quantum well on the lifetime of the states of shallow impurity centers are reported. The transitions between excited impurity states are induced by an interaction with acoustic phonons whose wave vectors exceed the scale of localization of the wave function in the wave-vector space and, therefore, are determined by the asymptotic behavior of the wave function of impurity states at large wave vectors. It is shown that the localization in a quantum well results in a slower decay of the wave functions of the impurity states in the wave vector space and can result in an exponential decrease in the lifetime of impurity states along with the width of the well. The theoretical results are in good agreement with the experimental data on the lifetime of acceptor states in GaAs/AlAs:Be structures.

Level repulsions of bulk acoustic waves in composite materials

Level repulsions of bulk acoustic waves in two-dimensional composite materials and phononic crystals are analyzed in this paper. Detail dispersion curves and displacement fields (polarizations) of bulk acoustic modes in the $\mathbf{k}$ (wave vector) space around the region of level repulsion are calculated and discussed. We show that some cross points of the modes that appear in the dispersion curves in the literature are not true intersections. Those cross points due to the level repulsions in the dispersion curves imply mode anticrossing. The displacement fields of the two modes around the cross points interchange with each other abruptly. On the other hand, the other cross points appearing in the dispersion curves are true intersections which implies that the couplings between the modes are vanishingly small. The continuous displacements of the two wave modes around the cross points provide the continuous dispersion curves. The polarizations of the different modes could be used as the criterion of real or apparent cross points in the dispersion curves.

Photonic band gap enhancement in frequency-dependent dielectrics

We illustrate a general technique for evaluating photonic band structures in periodic d -dimensional microstructures in which the dielectric constant epsilon (omega) exhibits rapid variations with frequency omega . This technique involves the evaluation of generalized electromagnetic dispersion surfaces omega ( k--> ,epsilon) in a (d+1) -dimensional space consisting of the physical d -dimensional space of wave vectors k--> and an additional dimension defined by the continuous, independent, variable epsilon . The physical band structure for the photonic crystal is obtained by evaluating the intersection of the generalized dispersion surfaces with the "cutting surface" defined by the function epsilon (omega) . We apply this method to evaluate the band structure of both two- and three-dimensional (3D) periodic microstructures. We consider metallic photonic crystals with free carriers described by a simple Drude conductivity and verify the occurrence of electromagnetic pass bands below the plasma frequency of the bulk metal. We also evaluate the shift of the photonic band structure caused by free carrier injection into semiconductor-based photonic crystals. We apply our method to two models in which epsilon (omega) describes a resonant radiation-matter interaction. In the first model, we consider the addition of independent, resonant oscillators to a photonic crystal with an otherwise frequency-independent dielectric constant. We demonstrate that for an inhomogeneously broadened distribution of resonators impregnated within an inverse opal structure, the full 3D photonic band gap (PBG) can be considerably enhanced. In the second model, we consider a coupled resonant oscillator mode in a photonic crystal. When this mode is an optical phonon, there can be a synergetic interplay between the polaritonic resonance and the geometrical scattering resonances of the structured dielectric, leading to PBG enhancement. A similar effect may arise when resonant atoms that are coupled radiatively through resonance dipole-dipole interaction are placed in a photonic crystal.

Rashba effect on the plasma oscillations in a coupled bilayer of electrons and holes

We consider the effect of spin-orbit (SO) coupling on the plasma oscillations for a coupled system of two-dimensional electron or heavy hole (HH) gases. The Rashba effect lifts the degeneracy of the energy spectrum and produces a linear term for electrons but a cubic term for HH states in wave vector space. The SO coupling gives rise to two plasmon branches for a single layer due to transitions within a subband and between subbands. The interlayer coupling splits each of these branches. We present numerical results for the frequency of the collective excitations as a function of wave vector and the Rashba parameter. The application to optical characterization studies is discussed.

Unified particle approach to Wigner-Boltzmann transport in small semiconductor devices

Small semiconductor devices can be separated into regions where the electron transport has classical character, neighboring with regions where the transport requires a quantum description. The classical transport picture is associated with Boltzmann-like particles that evolve in the phase-space defined by the wave vector and real space coordinates. The evolution consists of consecutive processes of drift over Newton trajectories and scattering by phonons. In the quantum regions, a convenient description of the transport is given by the Wigner-function formalism. The latter retains most of the basic classical notions, particularly, the concepts for phase-space and distribution function, which provide the physical averages. In this work we show that the analogy between classical and Wigner transport pictures can be even closer. A particle model is associated with the Wigner-quantum transport. Particles are associated with a sign and thus become positive and negative. The sign is the only property of the particles related to the quantum information. All other aspects of their behavior resemble Boltzmann-like particles. The sign is taken into account in the evaluation of the physical averages. The sign has a physical meaning because positive and negative particles that meet in the phase space annihilate one another. The Wigner and Boltzmann transport pictures are explained in a unified way by the processes drift, scattering, generation, and recombination of positive and negative particles. The model ensures a seamless transition between the classical and quantum regions. A stochastic method is derived and applied to simulation of resonant-tunneling diodes. Our analysis shows that the method is useful if the physical quantities do not vary over several orders of magnitude inside a device.

Spin dependent momentum density and Fermi surface of ferromagnetic Ni obtained by positron annihilation experiments

AbstractThe spin‐dependent momentum density and Fermi surface of ferromagnetic Ni have been obtained through positron annihilation experiments. The measurements were carried out through 2D angular correlation of annihilation radiation (ACAR) using longitudinally polarized positrons. The magnetic field direction was reversed in order to study the effect of the spin‐dependent positron–electron momentum space density on the Fermi surface of Ni. The results showed that ferromagnetic Ni had different Fermi surfaces for the majority‐spin and minority‐spin states. The differences due to the spin‐states were studied in the momentum space and in the wave vector space. In general, the experimental results showed good agreement with previous theoretical calculations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Neutron scattering study of PbMg $_{\mathsf{1/3}}$Ta $_{\mathsf{2/3}}$O$_{\mathsf 3}$ and BaMg $_{\mathsf{1/3}}$Ta $_{\mathsf{2/3}}$O$_{\mathsf 3}$ complex perovskites

Neutron scattering investigations were carried out in PbMg1/3Ta2/3O3 and BaMg1/3Ta2/3O3 complex perovskites. The crystal structure of neither compound shows any phase transition in the temperature range 1.5-730 K. Whereas the temperature dependence of the lattice parameter of BaMg1/3Ta2/3O3 follows the classical expectations, the lattice parameter of relaxor ferroelectric PbMg1/3Ta2/3O3 exhibits anomalies. One of these anomalies is observed in the same temperature range as the peak in the dielectric susceptibility. Below $T\sim180$ K PbMg1/3Ta2/3O3 exhibits very small thermal expansion, similar to invar alloys. We find that in PbMg1/3Ta2/3O3, lead ions are displaced from the ideal positions in the perovskite structure at all temperatures. Consequently short-range order is present. This induces strong diffuse scattering with an anisotropic shape in wavevector space. The temperature dependences of the diffuse neutron scattering intensity and of the amplitude of the lead displacements are similar.

Invariant formulation of the strain dependence of the critical temperature Tc of Nb 3Sn in a three term approximation

It has been shown previously in a full detailed analysis that the strain dependence of the critical temperature may be obtained from a general strain invariant formulation of T c in strong superconductivity. A physical model was presented in which the phonon frequency spectrum is represented through generalized elastic stiffness coefficients that include strain dependence. The primary purpose of the present work is to achieve a simplification of the analysis in order to facilitate calculation and reveal the essential physical content. The formulation in wave vector space of the equations for T c in strong superconductivity is reviewed. The method of simplification employs a succession of approximations to the effective elastic constants that enter the relation between phonon frequency and wave number. It is found that the effective elastic constants in the crystal symmetry directions may be grouped into sets having similar form, and this form includes terms in common among the sets and difference terms. The difference terms are found to be in the nature of gradients and may be eliminated to good approximation. The common terms include the strain dependence in a form identified as a deformation strain parameter. The analysis treats spherical (hydrostatic) and deformation strain dependence under longitudinal and transverse applied strain for wire and tape conductor. The analysis is applicable over a full range of applied strain, including small strains often described by a power law strain dependence, and larger strains often described by a deviatoric strain approach. A comparison is provided between the results of the full detailed analysis and the results of the approximate treatment showing the degree of agreement in the various applied strain orientations.

Supercontinuum generation from intense laser–plasma interactions

The parametric instabilities of intense laser–plasma interaction are investigated by one-dimensional (1D) dispersion relation and particle-in-cell (PIC) simulations. We show the evolution of laser spectrum from discrete to continuous with increase of the laser intensity and plasma density. An extremely broad, supercontinuum (SC) radiation decays slowly in frequency is generated from an intense subpicosecond laser pulse propagating in plasmas. The continuous spectrum is produced owing to the merging of stimulated Raman scattering instability with relativistic self-modulation instability in their wave-vector space when the laser intensity and plasma density are high enough. This results in strong nonlinear coupling of these instabilities. The extents of SC spectra can be enlarged if the unstable regions get broaden by selecting suitable parameter windows for the laser intensity and the plasma density. The effects of laser pulse polarizations, pulse duration and propagating distance on SC generation are also discussed. It is found that continuum spectra from circularly polarized laser pulses are similar to these from linearly polarized pulses.