Nonplanar Electron - Acoustic Shock Waves with Superthermal Hot Electrons

Nonplanar electron-acoustic shock waves having superthermal hot electrons are investigated with two temperature electrons model in unmagnetized plasma. Using reductive perturbation method, Korteweg-de Vries-Burgers (KdVB) equation is obtained in the cylindrical/spherical coordinates. Dissipation effect is introduced in the model by means of kinematic viscosity term. On the basis of the solutions of KdVB equation, variation of shock waves features (amplitude, velocity and width) with different plasma parameters are analysed. KdV-Burgers equation always leads to monotonic solitons and no oscillatory peak may appear. The combined effect of particle density (α), superthermal parameter (κ), electron temperature ratio (??) and kinetic viscosity (η₀) is numerically studied, and it is observed that these parameters significantly change the properties of the shock waves in nonplanar geometry especially in spherical coordinates. Results could be helpful to analyse the soliton features in laboratory as well as in the space environments.     

Exact Solutions of the Gardner Equation and their Applications to the Different Physical Plasmas

Traveling wave solution of the Gardner equation is studied analytically by using the two dependent (G^′/G,1/G)-expansion and (1/G^′)-expansion methods and direct integration. The exact solutions of the Gardner equations are obtained. Our analytic solutions are applied to the unmagnetized four-component and dusty plasma systems consisting of hot protons and electrons to investigate dynamical features of the solitons and shock waves produced in these systems. A wide variety of parameters of the plasma is used, and the basic features of the Gardner solitons that are beyond the existing study in literature are found. It is observed that the analytic solutions from (G^′/G,1/G)-expansion and (1/G^′)-expansion methods only produce shock waves but the solitary waves are found from the analytic solutions derived from the direct integration. It is also noted that the superhot electrons and relative mass density of the electrons significantly effect the soliton’s amplitude, width, and position. We have also numerically proved that the combination of every value of nomalized density μ₁ or temperature ratio σ₁ with the other sets of plasma parameters creates a region where the solutions have similar physical properties. The time-dependent behavior of the soliton is also studied, and a periodic motion of soliton along the phase variable η is found during the evolution. The investigations and the limits presented in this study may be helpful for studying and understanding the nonlinear properties of the solitary and shock waves seen in various physical and astrophysical plasma systems.     

Notes on number density fluctuations and the scattering cross section for electromagnetic waves scattered from a thermal nonequilibrium plasma

The scattering cross section and the Doppler spectrum for electromagnetic waves scattered by the electron density fluctuations of a plasma, where the mean kinetic temperature of the electronsT _e may differ from that of the ionsT _i , has been obtained among others byFejer, Buneman, Renau, Camnitz andFlood, andSalpeter. These authors use different methods of approach to calculate the autocovariance of the electron number-density fluctuations (from the mean) and then obtain the scattering cross section. Because of the differing results, the methods, concepts, and derivations of the scattering cross section are carefully examined in this paper. It is shown that the short-time dynamical considerations incorporated in the formulation of the statistical theory of the electron number-density fluctuations of the plasma as used by several authors (for instanceFejer, Buneman, Salpeter,) leads to results of limited validity. In addition, a fundamental error in calculating the electron density fluctuations leads these latter authors to an incorrect scattering cross section. The theory of scattering of electromagnetic waves from a plasma, where the electrons arenot in thermal equilibrium with the ions but statistical equilibrium exists, is developed in a general way. The covariance of the number-density fluctuations from the mean of the charged species of the plasma and the scattering cross sectionσ(q) are obtained. In particular it is shown that for a wavelength λ much greater than the effective Debye lengthd, the backscattering cross section increases and approaches complete incoherent scattering asT _e /T _i increases. This result is explained by noting that in the case of thermal equilibrium, the predicted value of the back-scattering cross section is smaller than that of the backscattering cross section from completely uncorrelated electron density fluctuations because the electrostatic interaction between the charged particles of the plasma, which is a function ofT _e andT _i , introduces a certain amount of organization in otherwise completely uncorrelated electron density fluctuations. When the mean temperature of the electrons increases relative to that of the ions, the organization introduced in the fluctuations diminishes because of the increasing thermal agitation of the electrons relative to that of the ions, and the backscattering process approaches that of incoherent backscattering (Thomson-type scattering). The spectrum function of incoherent scattering of electromagnetic waves from a nonequilibrium plasma is obtained and some cases of current interest are plotted.     

Kinetic Theory and Lax Equations for Shock Clustering and Burgers Turbulence

We study shock statistics in the scalar conservation law ? _t u+? _x f(u)=0, x∈?, t>0, with a convex flux f and spatially random initial data. We show that the Markov property (in x) is preserved for a large class of random initial data (Markov processes with downward jumps and derivatives of Lévy processes with downward jumps). The kinetics of shock clustering is then described completely by an evolution equation for the generator of the Markov process u(x,t), x∈?. We present four distinct derivations for this evolution equation, and show that it takes the form of a Lax pair. The Lax equation admits a spectral parameter as in Manakov (Funct. Anal. Appl. 10:328–329, 1976), and has remarkable exact solutions for Burgers equation (f(u)=u ²/2). This suggests the kinetic equations of shock clustering are completely integrable.     

Compression of whistler waves in a plasma with a nonstationary magnetic field

We present the results of our experiments in which the propagation of whistler waves in a plasma with a nonstationary magnetic-field perturbation (B=B₀+δB(t), δB/B₀ ≤ 5%) was investigated. The parametric and dispersive phenomena in a variable magnetic field were studied on the unique Krot plasma bench (the plasma column was 4 m in length and 1.5 m in diameter). A periodic field perturbation is shown to lead to an amplitude-frequency modulation of the whistler wave and to fragmentation of the signal into separate frequency-modulated wavepackets followed by their compression. The formation and compression of pulses is attributable to strong whistler group-velocity dispersion near the electron cyclotron frequency (ω ≤ ω_H). The results can be used to interpret the spectral shapes of the signals received from the Earth’s magnetosphere and ionosphere in the electron and ion whistler frequency ranges.     

Propagation of electromagnetic waves in hot magnetoactive plasma

The paper derives the general form of the tensor of dielectric permittivity? _ij(ω,k), Eq. (15), of non-relativistic hot magnetoactive collisionless plasma taking into consideration the influence of spatial dispersion. The general form of the tensor? _ij(ω,k) is used to express the tensorε _ij(ω,k) in the region of weak and strong spatial dispersion and in some special cases. A general dispersion equation (30) is derived and an analysis is made of the waves propagating in hot magnetoactive plasma. The expressions derived are used to investigate the damping of a right-handed circularly polarized wave propagating in hot magnetoactive plasma in the direction of the magnetic field.     

On the theory of the resonant interaction of electrons with a high-frequency electric field in one-dimensional two-barrier nanostructures with symmetric barriers of finite height and width

The theory of the interaction of electrons with a high-frequency electric field in one-dimensional two-barrier nanostructures with symmetric barriers of finite height and widths was developed. An exact solution to the Schrödinger equation was found for electrons in this nanostructure in the absence of high-frequency electric field. An analytical expression for the direct current I ₀ induced in this structure by an incident electron flux with energy ε differing slightly from the resonant level energy ε _r (|ε ? ε _r | << ε _r ) was derived. In the small-signal approximation, the active (field-phased) component I _c of the alternating electric current was calculated. At ε > ε _r , the current I _c is negative in the entire frequency range, which suggests the possibility of ac electric field amplification and generation in the two-barrier resonant-tunneling structure with the barriers of finite height and width. Within the applicability of the theory (?ω << ε _r ), the frequency at which amplification and generation of the ac electric field are possible reaches ω ? 10¹³ s ^?1; the power transferred by electrons to the field is ～1 W/cm².     

On resonance and nonresonance interactions between electrons and spatially periodic clusters

The effects of resonance and nonresonance interactions between electrons and spherical structures with spatial periodicity in the radial direction (clusters) were studied. It was shown analytically and by numerical calculations that the δ _l phase shift of the wave function, which arises in resonance electron scattering by such a periodic structure of a fairly large radius r₀, was not small even at a small ratio between the U₀ amplitude of the periodic potential and scattered electron energy E(ε₀=U₀/E?1) and equaled |δ _l |=π/4 (modulo π). This phase shift corresponded to the limiting case of a large Born parameter for the cluster, ξ₀=r₀U₀/?v?1, where v is the characteristic velocity of the electron. The effect of nonresonance electron scattering by a periodic potential whose spatial period was incommensurate with the Brillouin wavelength of the scattered electron was considered analytically. The effect of nonresonance scattering was shown to be of a higher order in the ε₀?1 parameter than resonance scattering. The cross section of electron scattering by a cluster was calculated, which allowed the conductivity of a medium containing clusters to be estimated.     

Control of the perturbation-wave-vector range of modulation instability of dispersive electromagnetic waves in the nonlocal Josephson electrodynamics of a thin superconducting film

Modulation instability of dispersive electromagnetic waves propagating through a Josephson junction in a thin superconducting film is investigated in the framework of the nonlocal Josephson electrodynamics. A dispersion relation is found for the time increment of small perturbations of the amplitude. For dispersive waves, it is first established that spatial nonlocality suppresses the modulation instability in the range of perturbation wave vectors 0≤Q≤Q_B1(k), i.e., in the long-wavelength range of experimental interest. The modulation instability range Q_B1(k)<Q<Q_B2(k, A, L) can be controlled (which is a unique possibility) by varying a dispersion parameter, namely, the wave vector k [or the frequency ω(k)] of linear-approximation waves. In the wave-vector ranges 0≤Q≤Q_B1(k) and Q≤Q_B2(k, A, L), waves are shown to be stable.     

Collision between two ortho-positronium (Ps) atoms: A four-body Coulomb problem

The elastic collision between two ortho-positronium (e.g. S = 1) atoms is studied using an ab-initio static exchange model (SEM) in the centre of mass (CM) frame by considering the system as a four-body Coulomb problem where all the Coulomb interaction terms in the direct and exchange channels are treated exactly. A coupled channel methodology in momentum space is used to solve Lippman–Schwinger equation following the integral approach. A new SEM code is developed in which the Born–Oppenheimer (BO) scattering amplitude acts as input to derive the SEM amplitude adapting the partial wave analysis. The s-, p- and d-wave elastic phase shifts and the corresponding partial cross-sections for the spin alignment S = 0, i.e., singlet (+) and S = 2, i.e., triplet (?) states are studied. An augmented Born approximation is used to include the contribution of higher partial waves more accurately to determine the total/integrated elastic cross-section (σ), the quenching cross-section (σ_q) and ortho-to-para conversion ratio (σ/σ_q). The effective range theory is used to determine the scattering lengths and effective ranges in the s-wave elastic scattering. The theory includes the non-adiabatic short-range effects due to exchange.     

Amplitude of Aharonov-Bohm oscillations in a small semiconductor ring interferometer in the tunneling regime

The amplitude g_AB of Aharonov-Bohm oscillations in a small semiconductor ring interferometer is studied as a function of the average conductance G_AV. Experimentally, it is found that, in the tunneling regime, the relative amplitude g_AB/G_AV of h/e oscillations is constant in the rings under investigation and smaller than unity. The small value of g_AB/G_AV in ring interferometers in the tunneling regime at low temperatures is explained by the difference in the amplitudes of the interfering electron waves.     

Electron–phonon heat exchange in quasi-two-dimensional nanolayers

We study the heat power P transferred between electrons and phonons in thin metallic films deposited on free-standing dielectric membranes. The temperature range is typically below 1 K, such that the wavelengths of the excited phonon modes in the system is large enough so that the picture of a quasi-two-dimensional phonon gas is applicable. Moreover, due to the quantization of the components of the electron wavevectors perpendicular to the metal film’s surface, the electrons spectrum forms also quasi two-dimensional sub-bands, as in a quantum well (QW). We describe in detail the contribution to the electron–phonon energy exchange of different electron scattering channels, as well as of different types of phonon modes. We find that heat flux oscillates strongly with thickness of the film d while having a much smoother variation with temperature (T _e for the electrons temperature and T _ph for the phonons temperature), so that one obtains a ridge-like landscape in the two coordinates, (d, T _e ) or (d, T _ph ), with crests and valleys aligned roughly parallel to the temperature axis. For the valley regions we find P ∝ T _e ^3.5 – T _ph ^3.5 . From valley to crest, P increases by more than one order of magnitude and on the crests P cannot be represented by a simple power law. The strong dependence of P on d is indicative of the formation of the QW state and can be useful in controlling the heat transfer between electrons and crystal lattice in nano-electronic devices. Nevertheless, due to the small value of the Fermi wavelength in metals, the surface imperfections of the metallic films can reduce the magnitude of the oscillations of P vs. d, so this effect might be easier to observe experimentally in doped semiconductors.     

Surface electromagnetic waves at a superconductor-dielectric interface

This paper reports on the results of investigating the frequency dependence, the structure, and the polarization-and energy-related characteristics of surface electromagnetic waves propagating along a superconductor-dielectric interface. An expression for the complex permittivity of a superconductor is derived in the approximation of a two-component plasma containing “normal” and “superconducting” electrons. Basic relations are obtained in the general case at temperatures T≤T _c (where T _c is the critical temperature) and in the limiting case at T?T _c, when the contribution from normal electrons to the permittivity of the superconductor can be disregarded.     

Thermodynamic properties of rhodium at high temperature and pressure by using mean field potential approach

The thermophysical properties of rhodium are studied up to melting temperature by incorporating anharmonic effects due to lattice ions and thermally excited electrons. In order to account anharmonic effects due to lattice vibrations, we have employed mean field potential (MFP) approach and for thermally excited electrons Mermin functional. The local form of the pseudopotential with only one effective adjustable parameter r_c is used to construct MFP and hence vibrational free energy due to ions – F_ion. We have studied equation of state at 300 K and further, to access the applicability of present conjunction scheme, we have also estimated shock-Hugoniot and temperature along principle Hugoniot. We have carried out the study of temperature variation of several thermophysical properties like thermal expansion (β_P), enthalpy (E_H), specific heats at constant pressure and volume (C_P and C_V), specific heats due to lattice ions and thermally excited electrons (C_V^ion and C_V^el, isothermal and adiabatic bulk moduli (B_T and B_s) and thermodynamic Gruneisen parameter (γ_th) in order to examine the inclusion of anharmonic effects in the present study. The computed results are compared with available experimental results measured by using different methods and previously obtained theoretical results using different theoretical philosophy. Our computed results are in good agreement with experimental findings and for some physical quantities better or comparable with other theoretical results. We conclude that local form of the pseudopotential used accounts s-p-d hybridization properly and found to be transferable at extreme environment without changing the values of the parameter. Thus, even the behavior of transition metals having complexity in electronic structure can be well understood with local pseudopotential without any modification in the potential at extreme environment. Looking to the success of present scheme (MFP + pseudopotential) we would like to extend it further for the study of liquid state properties as well as thermophysical properties of d and f block metals.     

Analysis of the angular distribution of Auger electrons in a Xe atom

The angular distribution of Auger electrons, and the numerically calculated anisotropy parameters of the angular distribution α for (M₃ → N₂N₃), (M₃ → N₃N₃), (M₄ → N₁N₃), (M₄ → N₄N₅), (M₄ → N₅N₅), and (M_4,5 → O_{2, 3}O_{2, 3}) transitions in a Xe atom are given. The matrix elements are calculated by using the nonrelativistic Hartree-Fock method in LS coupling and the relativistic Hartree-Fock-Dirac method in both jj coupling (the single-configuration approximation) and intermediate coupling (the multiconfiguration approximation).     

A comparison on absorption coefficients for secondary electron emission obtained from two different formulas

In this work, the absorption coefficients for secondary electron emißsion, α and β, that appeared respectively in the two different formulas, \(\delta (E_p ) = k\int_0^\infty {\left( {\frac{{dE}}{{dz}}} \right)E_p \exp ( - \alpha z)dz} \) and \(\delta (E_p ) = k\int_0^\infty {\left( {\frac{{dE}}{{dz}}} \right)E_p \exp ( - \alpha z)dz} \), were derived with a standard deviation rate analysis method based on a Monte Carlo simulated secondary electron yield, δ(E_p). Both the energy dissipation in depth for primary electrons, \(\left( {dE/dz} \right)E_p \), and the depth distribution for the number of secondary electrons including cascade electrons, n(z, E_p), were obtained by the same Monte Carlo method, in which the discrete inelastic scattering model was employed. The calculation results for Cu and Mg show that the n(z, E_p)-curve differs significantly from the \(\left( {dE/dz} \right)E_p \)-curve, and thus as well as a from b, for varied incidence angles (0°–80°) and low-energy primary electrons (up to 3 keV). The absorption coefficient β-values derived from a realistic depth distribution of cascade secondary electrons, n(z, E_p), then describe more accurately the nature of attenuation behavior of secondary electrons than a-values that associated with the approximate formula.     

Effect of positron density and temperature on the electron acoustic waves in a magnetized dissipative plasma

Zakharov–Kuznetsov–Burgers (ZKB) equation is derived for electron acoustic shock waves in magnetized e–p–i plasma. In the present model, magnetized plasma containing two electron population with kappa distributed positrons has been considered. The propagation characteristics of three dimensional electron acoustic (EA) shock waves have been studied under the influence of magnetic field. Our present plasma model supports the negative potential shocks. Combined action of dissipation (η), superthermality (κ), concentration of positrons (β), temperature ratio of cold electrons to positrons (σ), and magnetic field (ω_c) significantly modify the properties of EA shock waves. The width and amplitude of the shock structures are modified by various physical parameters. It is found that shock wave width decreases with increase in β, η₀, and ω_c whereas it becomes wider for κ and σ. Further, potential of the shock wave decreases as one departs away from superthermal distribution.     

Alfvén-drift turbulence suppression as triggering mechanism for LH transition

The Alfvén drift turbulence suppression at the plasma edge is suggested as a triggering mechanism for the L to H transition. The stability theory of Alfvén drift-waves shows that with increasing plasma pressure the Alfvén waves get coupled to electron drift waves and as a consequence the unstable long wavelength perturbations (most important for transport) are suppressed. The instability can be characterised by two significant parameters, i.e. the normalised plasma beta, β _n , and the normalised collision frequency, v _n . The resulting turbulent transport coefficient is suppressed when the normalised beta is greater than a critical value, i.e. β _n >1+v _n ^2/3 , which depends on the normalised collision frequency v _n . The transport coefficients change their dependence on plasma parameters at this threshold. Therefore, the possible scenario for the development of the H-mode could be associated with the stabilisation of the electron fluctuation at the plasma edge. The Alfvén drift-wave model predicts the experimental trend of a roughly linear dependence of threshold temperature on magnetic field, with a weak dependence on density at high densities and a strong dependence on density at lower densities.     

Electron gas induced in SrTiO<Subscript>3</Subscript>

This mini-review is dedicated to the 85th birthday of Prof. L.V. Keldysh, from whom we have learned so much. In this paper, we study the potential and electron density depth profiles in surface accumulation layers in crystals with a large and nonlinear dielectric response such as SrTiO₃ (STO) in the cases of planar, spherical, and cylindrical geometries. The electron gas can be created by applying an induction D₀ to the STO surface. We describe the lattice dielectric response of STO using the Landau–Ginzburg free energy expansion and employ the Thomas–Fermi (TF) approximation for the electron gas. For the planar geometry, we arrive at the electron density profile n(x) ∝ (x + d)^–12/7, where d ∝ D₀^–12/7. We extend our results to overlapping electron gases in GTO/STO/GTO heterojunctions and electron gases created by spill-out from NSTO (heavily n-type doped STO) layers into STO. Generalization of our approach to a spherical donor cluster creating a big TF atom with electrons in STO brings us to the problem of supercharged nuclei. It is known that for an atom with a nuclear charge Ze where Z > 170, electrons collapse onto the nucleus, resulting in a net charge Zn < Z. Here, instead of relativistic physics, the collapse is caused by the nonlinear dielectric response. Electrons collapse into the charged spherical donor cluster with radius R when its total charge number Z exceeds the critical value Z_c ≈ R/a, where a is the lattice constant. The net charge eZ_n grows with Z until Z exceeds Z* ≈ (R/a)^9/7. After this point, the charge number of the compact core Z_n remains ≈ Z*, with the rest Z* electrons forming a sparse TF atom with it. We extend our studies of collapse to the case of long cylindrical clusters as well.     

Equilibrium and stability of the charged particles’ fluxes drifting crosswise to the strong magnetic field at the expense of the Hall charge separation

The current equilibrium is investigated, where the generation of the Hall electric field on the magnetic Debye radius r _B = B ₀/(4πen _e) is considered by the drifting of the relativistic electrons crosswise to the strong magnetic field. In this case, the electron propagation is possible at the distance d that is essentially larger than the electron radius of the backward reflection in the magnetic field r ₀ ? m _e v _z c/(eB ₀). The instability of the joint drift motion of ions and electrons is investigated for the frequency oscillation w much higher than the ion cyclotron frequency w _Bi and by 4π n _i m _i c ² ? B ₀ ² and (k · B ₀) = 0. It is shown that the resonance effects by the ion beam’s plasma frequency w ? kv ₀ = w _pi leads to the generation of the nonpotential perturbations with the characteristic increment Imw ～ 10^?1 ÷ 10^{? 2} w _pi. Estimates show that the instability, associated with the propagation of the high-energy ion beam through the strong magnetic field, can essentially be like the edge-localized mode in tokamaks.