Arbitrary Angular Dependence Research Articles

Analytical Study of the Propagation Modes Into a Perturbed Border Waveguide

This article analytically approaches the problem of a propagating wave in a waveguide with a border characterized by an arbitrary angular dependence. We find a relationship between the Fourier coefficients of the function describing the border of the waveguide and the propagation constant β of the propagating wave. The proposed solution to the problem provides an analytical tool to obtain information about the shape of a waveguide. The analysis of the propagating wave reveals that, from a theoretical point of view, it is possible to partially, or even completely, reconstruct the border of the waveguide.

Electrostatic field of angular-dependent surface electrodes

We present an analytic strategy to find the electric field generated by surface electrode SE with angular dependent potential. This system is a planar region $\mathcal{A}$ kept at a fixed but non-uniform electric potential $V(\phi)$ with an arbitrary angular dependence. We show that the generated electric field is due to the contribution of two fields: one that depends on the circulation on the contour of the planar region ---in a Biot-Savart-Like (BSL) term---, and another one that accounts for the angular variations of the potential in $\mathcal{A}$. This approach can be used to find exact solutions of the BSL electric field for circular or polygonal contours of the planar region with periodic distributions of the electric potential. Analytic results are validated with numerical computations and the Finite Element Method. Keywords: Surface-electrode, Biot-Savart law, electrostatic problems, exactly solvable models.

Hořava gravity versus thermodynamics: The black hole case

Under broad assumptions breaking of Lorentz invariance in gravitational theories leads to tension with unitarity because it allows for processes that apparently violate the second law of thermodynamics. The crucial ingredient of this argument is the existence of black hole solutions with the interior shielded from infinity by a causal horizon. We study how the paradox can be resolved in the healthy extension of Horava gravity. To this aim we analyze classical solutions describing large black holes in this theory with the emphasis on their causal structure. The notion of causality is subtle in this theory due to the presence of instantaneous interactions. Despite this fact, we find that within exact spherical symmetry a black hole solution contains a space-time region causally disconnected from infinity by a surface of finite area -- the `universal horizon'. We then consider small perturbations of arbitrary angular dependence in the black hole background. We argue that aspherical perturbations destabilize the universal horizon and, at non-linear level, turn it into a finite-area singularity. The causal structure of the region outside the singularity is trivial. If the higher-derivative terms in the equations of motion smear the singularity while preserving the trivial causal structure of the solutions, the thermodynamics paradox would be obviated. As a byproduct of our analysis we prove that the black holes do not have any non-standard long-range hair. We also comment on the relation with Einstein-aether theory, where the solutions with universal horizon appear to be stable.

Collapse and stable self-trapping for Bose-Einstein condensates with1/rb-type attractive interatomic interaction potential

We consider dynamics of Bose-Einstein condensates with long-range attractive interaction proportional to $1/r^b$ and arbitrary angular dependence. It is shown exactly that collapse of Bose-Einstein condensate without contact interactions is possible only for $b\ge 2$. Case $b=2$ is critical and requires number of particles to exceed critical value to allow collapse. Critical collapse in that case is strong one trapping into collapsing region a finite number of particles. Case $b>2$ is supercritical with expected weak collapse which traps rapidly decreasing number of particles during approach to collapse. For $b<2$ singularity at $r=0$ is not strong enough to allow collapse but attractive $1/r^b$ interaction admits stable self-trapping even in absence of external trapping potential.

Small-angle impurity scattering and the spin Hall conductivity in two-dimensional semiconductor systems

An arbitrarily small concentration of impurities can affect the spin Hall conductivity in a two-dimensional semiconductor system. We develop a Boltzmann-like equation that can be used for impurity scattering with arbitrary angular dependence, and for arbitrary angular dependence of the spin-orbit field b(k) around the Fermi surface. For a model applicable to a 2D hole system in GaAs, if the impurity scattering is not isotropic, we find that the spin Hall conductivity depends on the derivative of b with respect to the energy and on deviations from a parabolic band structure, as well as on the angular dependence of the scattering. In principle, the resulting spin Hall conductivity can be larger or smaller than the ``intrinsic value'', and can have opposite sign. In the limit of small angle scattering, in a model appropriate for small hole concentrations, where the band is parabolic and b ~ k^3, the spin Hall conductivity has opposite sign from the intrinsic value, and has larger magnitude. Our analysis assumes that the spin-orbit splitting $b$ and the transport scattering rate tau^{-1} are both small compared to the Fermi energy, but the method is valid for for arbitrary value of b*tau.

Oscillating Dependence of the Acoustic Attenuation Coefficient in Thin Metal Layers

Dimensional and frequency dependences of the energy absorption coefficient Γ of a longitudinal sound wave in a plane parallel metal layer are theoretically investigated for arbitrary relationship between the layer thickness d and the electron free path l. Exact and asymptotic (for thick (d >> l) and thin (d << l) metal layers) formulas are derived for the coefficient Γ for an arbitrary angular dependence of the probability of specular reflection of charge carriers from the sample surface q(ϕ). A square-root dependence of the acoustic absorption coefficient on the thin film thickness is predicted. If the direction of sound wave propagation is perpendicular to the thin metal layer boundary, the acoustic absorption coefficient becomes an oscillating function of the layer thickness; its amplitude decreases with increasing kd, where k is the acoustic wave number.

Degeneracies and scaling relations in general power-law models for gravitational lenses

The time delay in gravitational lenses can be used to derive the Hubble constant in a relatively simple way. The results of this method are less dependent on astrophysical assumptions than in many other methods. The most important uncertainty is related to the mass model used. We discuss a family of models with a separable radial power-law and an arbitrary angular dependence for the potential psi = r^beta * F(theta). Isothermal potentials are a special case of these models with beta=1. An additional external shear is used to take into account perturbations from other galaxies. Using a simple linear formalism for quadruple lenses, we can derive H0 as a function of the observables and the shear. If the latter is fixed, the result depends on the assumed power-law exponent according to H0 proportional to (2-beta)/beta. The effect of external shear is quantified by introducing a `critical shear' gamma_c as a measure for the amount of shear that changes the result significantly. The analysis shows, that in the general case H0 and gamma_c do not depend on the position of the lens galaxy. We discuss these results and compare with numerical models for a number of real lens systems.

Universal weight functions for elastically anisotropic, angularly inhomogeneous media with notches or cracks

Two-dimensional weight function theory is extended in this paper so as to compute stress intensity factors for inhomogeneous anisotropic elastic solids with notches and cracks. Elasticity constants have an arbitrary angular dependence around the notch (crack) tip. Particular crack and inhomogeneity geometries are analysed including interface cracks in anisotropic bimaterials. Specimens of finite size and arbitrary geometry, with both forces and dislocations present, are considered. Weight functions for forces and dislocations are treated equally by means of a six-dimensional real form representation. Following the Bueckner concepts of regular and fundamental fields, it is suggested that the fundamental field asymptote should be derived explicitly close to the tip, and then the reciprocity theorem for the evaluation of the stress intensity factors should be applied.

Higher order weight functions in fracture mechanics of inhomogeneous anisotropic solids

A two-dimensional theory of higher-order weight functions is developed for inhomogeneous anisotropic elastic solids with notches or cracks. Elasticity constants have an arbitrary angular dependence around the notch (crack) tip. Specimens of finite size and arbitrary geometry, subjected to both surface displacements and tractions, are considered. The theory allows the computation of a complete set of expansion coefficients for the stress field into a series over the infinite notched elastic plane eigenfunctions. The approach used is based on the Bueckner concept of regular and fundamental fields, to which the reciprocity theorem is applied to obtain the expansion coefficients.

Theory of weak supercrystallinity in melts of two-component copolymers of complicated chemical structure. Fluctuation effects

The influence of fluctuation effects on supercrystallization (formation of domain structure) in melts of two-component copolymers of complicated chemical structure was studied by a refined variation method, generalizing the results of the Brazovskii theory of weak crystallization to the case of arbitrary angular dependence of node functions. By means of numerical calculations, the phase diagrams of transitions into the supercrystalline state were constructed and analysed in a space of parameters describing the chemical structure and temperature of the copolymer, for the following types of copolymeric systems: asymmetrical triblock and trigraft copolymers, star copolymers (A n ) k (B m ) k , as well as polyblock and polygraft copolymers. By taking account of the fluctuation effects, qualitative agreement is obtained with the results of Fredrickson and Helfand on changes in the structure of the transition sequence previously found by the authors using the mean field approximation.

The Lorentz model for neutrinos: Exact solution

The initial value problem for the linear Boltzmann equation describing the propagation of neutrinos in a medium composed of nuclei is rigorously solved. The structure of the equation corresponds to the Lorentz model with an angle-dependent scattering cross-section. A method yielding the rigorous solution in the case of an arbitrary angular dependence of the cross-section is described.

Calculation of efficiency and cross section of cylindrical scintillators in radiation fields of arbitrary angular dependence

In a previous paper 1) the authors have developed a computational method for the accurate numerical calculation of efficiency and cross section of cylindrical scintillators in axisymmetrical radiation fields with direction of symmetry parallel to the scintillator axis. This method is generalized here to radiation fields of arbitrary angular dependence. Sample results are presented for 3″ × 3″ and 5″ × 5″ NaI scintillators in fields of multiply scattered γ-radiation with oblique direction of symmetry (initial energies E 0 = 1 MeV). The angular distributions of this radiation are approximated by Gaussian exponential functions broadening with decreasing scattering energy as in extended scattering media. The angles δ between the central directions of these radiation fields and the axis of the scintillator were taken to be 22.5°, 45°, 67.5° and 90°. The results are compared with efficiency and cross section in Gaussian distribution fields with δ = 0° and in isotropic radiation fields from the article mentioned above.

Molecular four-centre integrals for tensor interelectronic interactions

A new method is presented for evaluating four-centre electronic interaction integrals for molecular systems when the electrons are coupled by two-body tensor interactions. The wave functions considered are Slater orbitals with arbitrary angular dependence.