Gerassimos C. Kokkorakis

Local electric field at the emitting surface of a carbon nanotube

We present a method for the calculation of the local electric field at the surface of a nanoscopic emitting structure. The method is here applied to carbon nanotubes (NT) where symmetry makes the application of the method easier. The NT is simulated as a cylindrical array of touching spheres, each sphere representing an atom of the tube. The electrostatic potential is written as a linear combination of the potentials produced by each of the spheres. We calculate the local electric field and the corresponding enhancement factor γ for both open and closed nanotubes. For a closed NT we find for γ a simple polynomial expression in terms of the ratio of the height h of the tube to its radius R, which for h/R<40 reduces to a frequently quoted formula of γ. For an open single-wall NT we find that γ is three times greater than that of a single-wall NT of the same h/R. As the thickness of the wall increases this difference diminishes. From these results one may deduce a possible explanation as to why in some experiments a closed NT emits more current than a corresponding open one while in other experiments the opposite holds true.

Enhancement factor of open thick-wall carbon nanotubes

We have calculated the electric field around and on the surface of an open thick-wall carbon nanotube (CNT) of height h, external radius R, and wall thickness w. To accomplish that we simulate the CNT as a vertical array of touching toroids, each of external radius R and cross section radius w/2, and then we express the problem in toroidal coordinates. From our calculations we obtain the enhancement factor γ as a function of h, R, and w. By fitting to our numerical results we obtain an empirical but simple formula for γ, which extrapolates to that of a closed CNT in the limiting case of w=R.

Self focusing of field emitted electrons at an ellipsoidal tip

In models of field emission the needle is usually terminated by a hemispherical cap. Here we choose to terminate it with a hemiellipsoidal cap and use a three-dimensional Wentzel–Kramers–Brillouin method for the computations. This has two important consequences: as the ellipsoid becomes more elongated, (a) the effective emission area is decreased and (b) the quantum mechanically computed electron paths converge toward the needle axis. Both mechanisms produce a self-focusing of the field emitted electrons.

Electromagnetic Scattering by an Inhomogeneous Gyroelectric Sphere Using Volume Integral Equation and Orthogonal Dini-Type Basis Functions

The electromagnetic scattering by an inhomogeneous gyroelectric sphere is studied in this work. The solution is constructed using the well-known electric field volume integral equation (EFVIE). The kernel that incorporates the Greens function is expanded in spherical vector wave functions, and the unknown electric field and electric displacement is expanded in spherical vector wave functions using fully orthogonal Dini-type basis functions. The permittivity tensor of the spherical object supports a continuously varying radial inhomogeneity, while its permeability is that of free space. This approach allows a fully analytical computation of the unknown matrix coefficients in special cases of homogeneous permittivity tensors. The validation of our method is performed by comparisons with published results in the literature. New numerical results are given for various continuously varying permittivity tensors.

Acoustic scattering by a circular cylinder parallel with another of small radius

The scattering of a plane acoustic wave by an infinite penetrable or impenetrable circular cylinder, parallel with another one, also penetrable or impenetrable, of acoustically small radius, is considered. The method of separation of variables, in conjunction with translational addition theorems for cylindrical wave functions, is used. Analytical expressions are obtained for the scattered pressure field and the various scattering cross sections, for normal incidence. Numerical results are given for penetrable and impenetrable cylinders.

Field induced in inhomogeneous spheres by external sources. I. The scalar case

The evaluation of acoustic or electromagnetic fields induced in the interior of inhomogeneous penetrable bodies by external sources is based on well-known volume integral equations; this is particularly true for bodies of arbitrary shape and/or composition, for which separation of variables fails. In this paper the investigation focuses on acoustic (scalar fields) in inhomogeneous spheres of arbitrary composition, i.e., with r-, θ- or even φ-dependent medium parameters. The volume integral equation is solved by a hybrid (analytical–numerical) method, which takes advantage of the orthogonal properties of spherical harmonics, and, in particular, of the so-called Dini’s expansions of the radial functions, whose convergence is optimized. The numerical part comes at the end; it involves the evaluation of certain definite integrals and the matrix inversion for the expansion coefficients of the solution. The scalar case treated here serves as a steppingstone for the solution of the more difficult electromagnetic problem.

Acoustic eigenfrequencies in a spheroidal cavity with a concentric penetrable sphere

The acoustic eigenfrequencies fnsm in a spheroidal cavity containing a concentric penetrable sphere are determined analytically, for both Dirichlet and Neumann conditions in the spheroidal boundary. Two different methods are used for the evaluation. In the first, the pressure field is expressed in terms of both spherical and spheroidal wave functions, connected with one another by well-known expansion formulas. In the second, a shape perturbation method, this field is expressed in terms of spherical wave functions only, while the equation of the spheroidal boundary is given in spherical coordinates. The analytical determination of the eigenfrequencies is possible when the solution is specialized to small values of h=d/(2R2), (h≪1), with d the interfocal distance of the spheroidal boundary and 2R2 the length of its rotation axis. In this case exact, closed-form expressions are obtained for the expansion coefficients gnsm(2) and gnsm(4) in the resulting relation fnsm(h)=fns(0)[1+h2gnsm(2)+h4gnsm(4)+O(h6)]. ...

Acoustic field induced in spheres with inhomogeneous density by external sources

Acoustic or electromagnetic fields induced in the interior of inhomogeneous penetrable bodies by external sources can be evaluated via well-known volume integral equations. For bodies of arbitrary shape and/or composition, for which separation of variables fails, a direct attack for the solution of these integral equations is the only available approach. In a previous paper by the same authors the scalar (acoustic) field in inhomogeneous spheres of arbitrary compressibility, but with constant density, was considered. In the present one the direct hybrid (analytical-numerical) method applied to the much simpler integral equation for spheres with constant density is generalized to densities that vary with r, theta, or even psi. This extension is by no means trivial, owing to the appearance of the derivatives of both the density and the unknown function in the volume integral, a fact necessitating a more subtle and accuracy-sensitive approach. Again, the spherical shape allows use of the orthogonal spherical harmonics and of Dinis expansions of a general type for the radial functions. The convergence of the latter, shown to be superior to other possible sets of orthogonal expansions, can be further optimized by the proper selection of a crucial parameter in their eigenvalue equation.

Resonant Frequencies in an Electromagnetic Cylindrical/Spherical Cavity with an Internal Off-Axis Small Dielectric Sphere

ABSTRACT Analytical expressions for the resonant frequencies in an electromagnetic cylindrical/spherical cavity with an off-axis inner electrically small dielectric sphere are derived, for both magnetic and electric modes. The walls of the cylindrical/spherical cavity are perfectly conducting. Cylindrical/spherical vector wave functions and related addition theorems, as well as expansion of cylindrical wave functions in terms of spherical ones, are used. Our results are useful in problems connected with resonant cavities containing small inhomogeneities, as well as in cases that we want to determine the permittivities or magnetic permeabilities of various materials, by measuring the resonant frequency shifts caused by the introduction of small samples of them, inside cavities. Graphical results for some of the lower order modes are given, for various values of the parameters and for both kinds of modes.

Electromagnetic Scattering by a General Rotationally Symmetric Inhomogeneous Anisotropic Sphere

In this work, the electromagnetic scattering in general rotationally symmetric inhomogeneous anisotropic spheres is investigated. A rigorous full-wave solution is presented based on the volume integral equation. The solution is achieved by expanding the unknown fields in fully orthogonal Dini-type spherical vector wave functions. This formulation allows the study of general rotationally symmetric gyrotropic spheres with varying permittivity and permeability tensors. We investigate the validity of the developed method, which is next applied on the computation of the scattering cross sections of inhomogeneous rotationally symmetric gyroelectric and gyromagnetic spheres.