Fine Structures of the Permittivity and Characteristic Losses of Electrons in Fluorite

Experimental-computational spectra of the permittivity and characteristic losses –Im^–1 for energies in the range 5–21 eV at a temperature of 4.2 K and theoretical spectra of and –Im^–1 of a fluorite crystal are resolved into elementary transition bands. The parameters of transition bands (energies of their maxima E _i, band halfwidths H _i and areas S _i, and oscillator forces f _i) are determined. A correlation of the spectral bands of and –Im^–1is established, and their specific features are elucidated.     

Band-gap structure of the spectrum of periodic Maxwell operators

We investigate the band-gap structure of some second-order differential operators associated with the propagation of waves in periodic two-component media. Particularly, the operator associated with the Maxwell equations with position-dependent dielectric constant (x),xR ³, is considered. The medium is assumed to consist of two components: the background, where (x) = _b , and the embedded component composed of periodically positioned disjoint cubes, where (x) = _a . We show that the spectrum of the relevant operator has gaps provided some reasonable conditions are imposed on the parameters of the medium. Particularly, we show that one can open up at least one gap in the spectrum at any preassigned point provided that the size of cubesL, the distancel=L betwen them, and the contrast = _b / _a are chosen in such a way thatL ^–2, and quantities ^-1-3/2 and ² are small enough. If these conditions are satisfied, the spectrum is located in a vicinity of widthw(^3/2)^-1 of the set {² L ^-2 k ²:kZ³}. This means, in particular, that any finite number of gaps between the elements of this discrete set can be opened simultaneously, and the corresponding bands of the spectrum can be made arbitrarily narrow. The method developed shows that if the embedded component consists of periodically positioned balls or other domains which cannot pack the space without overlapping, one should expect pseudogaps rather than real gaps in the spectrum.     

Dielectric constant characterization of large-area substrates in millimeter wave band

A possibility of the dielectric constant measurement for substrates with permittivity=+i without an essential restriction on their area has been shown experimentally. The method uses frequency measurement of quasioptical dielectric resonator (QDR) with two slots oriented along the QDR radius with a dielectric substrate in one of them. Taking QDR of teflon in 8mm waveband as an example it is found that measurable values of can ran up 15 _q , where _q is the QDR material permittivity. Absolute error of the measurements is determined by an accuracy with which the permittivity of calibrated (standard) samples is known. The relative measuring error is determined by the accuracy of the QDR frequency measurement and can be quite a small. As an example the method is demonstrated forLaAlO ₃ single crystals.     

Reflection and Absorption Spectra of Fluorite in the UV Spectral Region

We have calculated the R(E) and ₁(E) spectra from the theoretical ₂(E) spectra of five models in the region 8–27 eV and the ₂(E) and ₁(E) spectra from the experimental R(E) spectrum in the region 6–35 eV. The results are compared with the known theoretical ₂(E) spectra of five models. The basic features of all of the R(E), ₂(E), and ₁(E) spectra have been revealed. It is established that the experimental R(E) spectrum and the ₂(E) and ₁(E) spectra calculated with the use of experimental data are in good agreement with the results of theoretical calculations for the models of ₂(E). On the basis of the known theoretical calculations of the fluorite zones, a scheme of the nature of the principal maxima of R(E) and ₂(E) is suggested.     

Correlation tensors of the blackbody radiation in rectangular cavities

Electromagnetic equilibrium fluctuations in finite cavities filled with a dissipative medium (dielectric function ()=+i) and bounded by walls of infinite conductivity are considered. Expanding the fields in terms of a complete and orthonormal set of functions and solving the Maxwell equations the response of the EM field to external forces (polarization and magnetization) is obtained. With the aid of the fluctuation dissipation theorem and the linear response functions the 2nd order correlation tensors of the EM field are derived.For rectangular cavities explicit considerations are made. In the case of transparent media (=0) the spectral energy density of the EM radiation is calculated.     

Unmixing of binary alloys by a vacancy mechanism of diffusion: a computer simulation

The initial stages of phase separation are studied for a model binary alloy (AB) with pairwise interactions _AA , _AB , _BB between nearest neighbors, assuming that there is no direct interchange of neighboring atoms possible, but only an indirect one mediated by vacancies (V) occurring in the system at a concentrationc _v and which are strictly conserved, as are the concentrationsc _A andc _B of the two species.A-atoms may jump to vacant sites with jump rate _A , B-atoms with jump rate _B (in the absence of interactions). Particular attention is paid to the question to what extent nonuniform distribution of vacancies affects the unmixing kinetics. Our study focuses on the special case _A = _B on a square lattice, considering three different choices of interactions with the same = _AB – ( _AA + _BB )/2: (i) _AB =, _AA = _BB = 0; (ii) _AA = 0, _AA = _BB ; = ; (iii) _AB = _BB = 0, _AA = –2. We obtain both the time evolution of the structure factorS(k,t) following a quench from infinite temperature to the considered temperature, and the timedependence of the mean cluster size and the various neighborhood probabilities of a vacancy. While in case (i) forc _V 0.16 the distribution of vacancies in the system stays nearly random, in case (ii) the vacancies cluster in theA-B interfacial region, and in case (iii) they get nearly completely expelled from theA-rich regions. While phase separation proceeds in case (i) only slightly faster than in case (ii), a significant slowing down of the relaxation is observed for case (iii), which shows up in a strong reduction of the effective exponents describing the growth.     

s-Polarized nonlinear surface polaritons

An exact solution of Maxwell's equations is found, corresponding to ans-polarized nonlinear surface polarition, at the planar interface between two dielectric media, one of which is optically unaxial and is characterized by a diagonal dielectric tensor whose elements depend on the amplitude of the electric field according to ₁₁=₂₂=₀()+a() (|E ₁|²+|E ₂|²), ₃₃=(). Such modes have no counterpart in the corresponding linear system.     

P-Polarized nonlinear surface polaritons in layered structures

We found an exact solution of Maxwell's equations, which describes the propagation ofp-polarized nonlinear surface polaritons and ofp-polarized nonlinear guided wave polaritons in two cases:i) in a film of a surface active material placed on a substrate described by a diagonal dielectric tensor whose elements depend on the amplitude of the electric field according to _{1 1}=_{2 2}= + (|E ₁|² + |E ₂|²), _{3 3}=, andii) in a film described by the same dielectric tensor (optically uniaxial nonlinear crystal) placed on a substrate with dielectric constant ₃ (optically linear medium). The power carried in the surface waves has also been exactly calculated.     

Frequency Dependence of the Components of Dielectric Permeability and Characteristic Losses of Fluorite

The ₁(E), –Im^–1, and Re^–1 spectra of the fluorite crystal are calculated on the basis of the experimental (10–35 eV) and theoretical spectra ₂(E) (10–27 and 8–20 eV). They were employed to decompose the ₂(E) and –Im^–1 spectra into elementary components. The most intense transverse and longitudinal components of transitions and their parameters have been determined. The correlation between two types of components of transitions and their distinguishing features have been established.     

On the cauchy problem for the discrete Boltzmann equation with initial values inL 1 + (ℝ)

We prove a global existence theorem for a discrete velocity model of the Boltzmann equation when the initial values _i (x) have finite entropy and, for some constant>0, (1+|x|) _i (x)L ₁ ⁺ ().     

Structure of Electron Transitions of Corundum

The integral spectra of the dielectric permeability of corundum within the region 8–40 eV in the polarizations E C and E C are decomposed into 18 Lorentz elementary oscillators. Their principal parameters, namely, the energy E _i , halfwidth H _i , band area S _i , and height of the maximum _2max, are calculated and the main features are established. An analysis of the wellknown results obtained by two models of the reproduction of the ₂ spectra with the aid of a large number of fitting parameters and theoretical calculations of the corundum bands is carried out.     

Photonic pseudogaps for periodic dielectric structures

We consider the problems of existence and structure of gaps (pseudogaps) in the spectra associated with Maxwell equations and equations that govern the propagation of acoustic waves in periodic two-component media. The dielectric constant is assumed to be real and positive, and the value of = _b on the background is supposed to be essentially larger than the value of = _a on the embedded component. We prove the existence of pseudogaps in the spectra of the relevant operators. In particular, we give an accurate treatment of the term pseudogap. We also show that if the contrast _b / _a approaches infinity, then the bands of the spectrum shrink to a discrete set which can be identified with the set of eigenvalues of a Neumann-type boundary value problem and thus can be effectively calculated.     

Optical Spectra of Residual Porous Silicon

The spectra of absorption (), reflection (R), ₂, and E ²₂ of residual silicon (r-Si) were calculated using the R spectra of porous silicon in the range from 0 to 20 eV and with ₂(E) in the range 2.5–5.0 eV of porous silicon specimens with P = 0.57, 0.66, and 0.77. The ₂ spectra of r-Si were decomposed into elemental components. We calculated their main parameters: the energies of maxima E _i and halfwidths H _i of bands, their areas S _i , and heights I _i , and oscillator strengths f _i . The two-phase Bruggeman model of effective dielectric function and Kramers–Kronig analysis were applied in the calculations. The essential differences between the optical spectra of the residual and cubic silicon were established. They are at least partially attributed to the quantum dimensional effects. The data obtained are compared with the known theoretical spectra of silicon clusters.     

Averaged electron Green function and CDW pinning in one-dimensional random potential

The averaged retarded electron Green functionG ⁺(,k) in 1d disordered metal is calculated using the Berezinsky diagram technique. Using the Gorkov's theory it is shown, that the substitution of inG ⁺ (,k) by the square of the external frequency atk=0 gives the dependence of Fröhlich conductivity _F(). This dependence describes the impurity pinning of CDW in 1d disordered metals. The good agreement of this dependence with experimental data Zeller et al. about _F() in quasi-1d conductor KCP is found     

On the Hamiltonian interpolation of near-to-the identity symplectic mappings with application to symplectic integration algorithms

We reconsider the problem of the Hamiltonian interpolation of symplectic mappings. Following Moser's scheme, we prove that for any mapping , analytic and -close to the identity, there exists an analytic autonomous Hamiltonian system, H such that its time-one mapping _H differs from by a quantity exponentially small in 1/. This result is applied, in particular, to the problem of numerical integration of Hamiltonian systems by symplectic algorithms; it turns out that, when using an analytic symplectic algorithm of orders to integrate a Hamiltonian systemK, one actually follows exactly, namely within the computer roundoff error, the trajectories of the interpolating Hamiltonian H, or equivalently of the rescaled Hamiltonian K=^-1H, which differs fromK, but turns out to be ⁵ close to it. Special attention is devoted to numerical integration for scattering problems.     

Relationship between low frequency dielectric dispersion and ferroelectric phase transition in pulsed-laser-deposited Bi<Subscript>4-x</Subscript>La<Subscript>x</Subscript>Ti<Subscript>3</Subscript>O<Subscript>12</Subscript> thin film

The frequency and temperature dependence of the complex dielectric constant of Bi_4-xLa_xTi₃O₁₂ (BLT, x=0.9) ferroelectric thin film was studied in the frequency range of 10^-110⁶ Hz and the temperature range of 298673 K. A low frequency dielectric dispersion (LFDD) was found. A model was proposed to account for this observed phenomena. The complex dielectric constant data obtained in the measured frequency and temperature ranges have been found to fit very well to the dielectric dispersion relation: ^*=+i/₀+[B(i)^n-1]/₀. The knee in the log of the electrical conductivity versus the reciprocal temperature curve occurs at T_c. The activation energies associated with charge conduction are E_a,II=0.73 eV below T_c and E_a,I=0.95 eV above T_c. The occurrence of an anomaly in both the n and parameters near T_c indicates a coupling between charge carries and phonons. PACS 77.55.+f; 77.80.-e; 77.22.Jp     

Random shearing direction models for isotropic turbulent diffusion

Recently, a rigorous renormalization theory for various scalar statistics has been developed for special modes of random advection diffusion involving random shear layer velocity fields with long-range spatiotemporal correlations. New random shearing direction models for isotropic turbulent diffusion are introduced here. In these models the velocity field has the spatial second-order statistics of an arbitrary prescribed stationary incompressible isotropic random field including long-range spatial correlations with infrared divergence, but the temporal correlations have finite range. The explicit theory of renormalization for the mean and second-order statistics is developed here. With the spectral parameter, for –<<4 and measuring the strength of the infrared divergence of the spatial spectrum, the scalar mean statistics rigorously exhibit a phase transition from mean-field behavior for <2 to anomalous behavior for with 2<<4 as conjectured earlier by Avellaneda and the author. The universal inertial range renormalization for the second-order scalar statistics exhibits a phase transition from a covariance with a Gaussian functional form for with <2 to an explicit family with a non-Gaussian covariance for with 2<<4. These non-Gaussian distributions have tails that are broader than Gaussian as varies with 2<<4 and behave for large values like exp(–C _c |x|^4–), withC _c an explicit constant. Also, here the attractive general principle is formulated and proved that every steady, stationary, zero-mean, isotropic, incompressible Gaussian random velocity field is well approximated by a suitable superposition of random shear layers.     

TM-polarized nonlinear waves guided by asymmetric dielectric layered structures

We found the field structure, exact dispersion relations and power flow ofp-polarized nonlinear guided and surface waves travelling along a three-component layered structure consisting of a film of thicknessd with dielectric constant _b bounded at the negativez-side by a linear medium with dielectric constant _a and at the positivez-side by a nonlinear uniaxial substrate characterized by the diagonal dielectric tensor ₁₁ = ₂₂ = + (|E ₁|² + |E ₂|²), ₃₃ = , <0 (self-defocusing medium),E ₁ andE ₂ being the components of the electric field in thex andy-direction, respectively. It is shown that for sufficiently smalld/ (: wavelength) the nonlinear wave may exist only at power flows exceeding some certain minimum values. For sufficiently larged/ to some values of the power flow there correspond two distinct values of the propagation constant. In this case with increasing of the power flow the number of waveguide modes is decreasing and for higher-order modes the film-waveguide exhibits an optical-power limiter from the above behaviour.     

The Schrödinger equation and canonical perturbation theory

LetT ₀(, )+V be the Schrödinger operator corresponding to the classical HamiltonianH ₀()+V, whereH ₀() is thed-dimensional harmonic oscillator with non-resonant frequencies =(₁, ... , _d ) and the potentialV(q ₁, ... ,q _d) is an entire function of order (d+1)^–1. We prove that the algorithm of classical, canonical perturbation theory can be applied to the Schrödinger equation in the Bargmann representation. As a consequence, each term of the Rayleigh-Schrödinger series near any eigenvalue ofT ₀(, ) admits a convergent expansion in powers of of initial point the corresponding term of the classical Birkhoff expansion. Moreover ifV is an even polynomial, the above result and the KAM theorem show that all eigenvalues _n (, ) ofT ₀+V such thatn coincides with a KAM torus are given, up to order , by a quantization formula which reduces to the Bohr-Sommerfeld one up to first order terms in .     

Power law scaling of the top Lyapunov exponent of a Product of Random Matrices

A sequence of i.i.d. matrix-valued random variables with probabilityp and with probability 1–p is considered. Leta() = a ₀ + O(), c() = c ₀ + O() lim ₀ b() = Oa ₀,c ₀, >0, andb()>0 for all >0. It is shown show that the top Lyapunov exponent of the matrix productX _n X _n-1...X ₁, = lim_n (1/n) n X _n X _n-1...X ₁ satisfies a power law with an exponent 1/2. That is, lim ₀(ln /ln ) = 1/2.