Isosceles Right Triangle Research Articles

Fröhlich electron–phonon interaction Hamiltonian and potential distribution of a polar optical phonon mode in wurtzite nitride triangular nanowires

Polar optical phonon modes of wurtzite triangular nanowires (NWs) with three different cross sections, including the hemi-equilateral triangle (HET), the isosceles right triangle (IRT), and the equilateral triangle (ET), are deduced and analyzed using the dielectric continuum model. The exact and analytical phonon states of exactly confined (EC) modes in nitride NWs with HET, IRT, and ET cross sections are derived. The characteristic frequency of EC phonon modes in the triangular nitride NW systems is specified. Fröhlich electron–phonon interaction Hamiltonians in wurtzite NWs with three types of triangular cross sections are obtained. It is found from the numerical results that, among the three types of GaN NWs, the electron–phonon coupling of EC modes in NWs with an HET cross section is the weakest one, that in NWs with an ET cross section is the strongest one, and that in NWs with an IRT cross section is in the middle. The electrostatic potentials of EC modes in HET NWs are neither symmetric nor antisymmetric. The potential functions of EC modes in the ET NW structures have one (three) symmetric axis (axes) as the quantum numbers p and q take fractions (integers). The potential functions of EC modes in IRT NWs behave either symmetrically or anti-symmetrically, which are closely dependent on the parities of the quantum numbers p and q. With the increase of order-number of EC modes, the electron–phonon coupling becomes weaker and weaker. This reveals that cross-sectional morphology of quantum structures has an important influence on the symmetries of phonon modes and electron–phonon coupling strengths in low-dimensional quantum systems.

Generalised formulation for electromagnetic scattering from finite arbitrarily shaped grooves in a perfect conducting plane

This study presents an analytical-based solution of electromagnetic plane wave scattering from a finite number of general-shaped grooves in a perfectly conducting plane. The formulation, for the near as well as far-field, is based on Fourier-integral representation of the scattered field and stair-case approximation of the grooves' fields. An efficient formulation of the scattering matrix based on hierarchical and modular formulations, which combines the boundary conditions between fields in groove layers and upper half space is demonstrated. The scattering matrix construction consists of two levels of matrices. The first level of matrices is the individual grooves scattering matrices and the coupling matrices between grooves. The second level of the matrices contains all the geometrical and physical parameters of the grooves as well as the incident field. The near and far-field results of finite rectangular and isosceles right triangle (IRT) grooves are in excellent agreement with a solution from high frequency simulation software, a finite-element simulator and previously published results. The proposed method is able to optimise finite asymmetric gratings for optical and terahertz applications.

Three Familiar Formulas for π via Geometry

(1986). Three Familiar Formulas for π via Geometry. The College Mathematics Journal: Vol. 17, No. 4, pp. 339-339.

A Theorem on the Tangram

(1942). A Theorem on the Tangram. The American Mathematical Monthly: Vol. 49, No. 9, pp. 596-599.