L-shaped Region Research Articles

On the variational approach to defining splines on L-shaped regions

On the variational approach to defining splines on L-shaped regions

The Exact Dimensions of a Family of Rectangular Coaxial Lines with Given Impedance

The potential problem associated with a capacitor bounded by concentric rectangles may be solved by conformably mapping a certain L-shaped region onto the upper half plane. In certain cases, the integral, by which the Schwarz-Christoffel transformation may be expressed, can be evaluated in terms of two elliptic integrals of the first kind. Then the odd- and even-mode impedance of the related transmission line is readily found.

On Direct Methods for Solving Poisson’s Equations

Some efficient and accurate direct methods are developed for solving certain elliptic partial difference equations over a rectangle with Dirichlet, Neumann or periodic boundary conditions. Generalizations to higher dimensions and to L-shaped regions are included.

Quantum Mechanical Model of Three-Body Rearrangement Scattering. I. Formal Solution of the Helmholtz Equation on a Square

Solutions of the Helmholtz equation (∂2 / ∂x2 + ∂2 / ∂y2 + k2 + 1)Ψ = 0 describing a simple scattering process in an L-shaped region are studied. To obtain the correct expansion for ψ(x, y) on the square corner of the L shape when k = 1, it is found necessary to replace terms sinkx siny and sinx sinky by sinx siny and (x cosx siny–y cosy sinx). Terms similar to the latter have been neglected by previous authors and spurious resonances found in consequence.

Classical Model for the Study of Isotope Effects in Energy Exchange and Particle Exchange Reactions

An idealized model of a colinear collision between a diatomic molecule AB and an atom C is studied. The three-atom potential-energy surface is an L-shaped region consisting of two flat-bottomed troughs with vertical walls; and the rectangular corner region of the L is at a higher uniform potential. The dependence of the classical energy exchange probability on the relative masses of the three particles is investigated systematically for this model. All possible collisions are considered in which the initial distribution of energy between the vibrational and relative translational components is fixed. Formulas are derived which express the average fraction of vibrational energy after collision as a function of the fraction of vibrational energy before collision. In the reaction AB+C→A+BC, the dependence of the classical particle-exchange probability on the relative mass of Particles A and C is determined in the limiting case where the mass of Particle B is infinite. In addition to the relative mass parameter, the particle exchange probability depends upon the initial vibrational and relative translational energies and the height of the potential barrier in the corner region VII. A thermal average probability for particle exchange is determined which depends upon two parameters: the relative mass of A and C and VII/kT. This dependence on the relative mass, an isotope effect, is a product of classical mechanics.