Multiple Scattering in Solids by Antonios Gonis
By Antonios Gonis
The origins of a number of scattering idea (MST) should be traced again to Lord Rayleigh's booklet of a paper treating resistivity of an ar ray of spheres, which seemed greater than a century in the past. At its most simple, MST offers a method for fixing a linear partial differential equa tion outlined over a quarter of area via dividing house into nonoverlapping subregions, fixing the differential equation for every of those subregions individually after which assembling those partial recommendations right into a worldwide phys ical resolution that's gentle and non-stop over the whole area. This strategy has given upward thrust to a wide and transforming into checklist of functions either in classical and quantum physics. almost immediately, the strategy is being utilized to the research of membranes and colloids, to acoustics, to electromagnetics, and to the answer of the quantum-mechanical wave equation. it really is with this latter software, specifically, with the answer of the SchrOdinger and the Dirac equations, that this ebook is basically involved. we'll additionally display that it presents a handy method for fixing the Poisson equation in good fabrics. those differential equations are vital in sleek calculations of the digital constitution of solids. the applying of MST to calculate the digital constitution of reliable ma terials, which originated with Korringa's well-known paper of 1947, supplied an effective strategy for fixing the one-electron Schrodinger equation.
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Additional resources for Multiple Scattering in Solids
117) In view of the second inequality, the Hankel function in the last equation can be further expanded to yield the result, L' x = [~GL'LII(-ro - h(ro) b) Jh ll (r)V(r)1/JL(r)d 3 rj , ~k LJL'( -b) [L GL'LII(-ro-b)iLIILj. 118) L" The quantities G LL' (R) are the so-called real-space structure constants and are the elements of the free-particle propagator in the angular-momentum representation. Explicit expressions for the structure constants are given in Appendix D. Evidently, the last equation yields an expression for the wave function at a point inside the moon region explicitly in terms of the cell t-matrix.
81) SLL' = -k / ds· [Jv(r)'\7'I/JL(r) - 'l/JLCr)'\7Jv(r)]. 84) for points outside the circumscribing sphere. 55). 80) (with T replacing S). Note that 'ljJL will no longer behave like h near the origin, but like C-IIJ) . Similarly, if the phase-function matrix 5.. 59). It should be noted, however, that for nonspherical potentials, S is likely to be singular or nearly singular depending on the potential and on the maximum number of angular momentum values used to represent the matrix, S LV. The discussion just given was based on the choice of the Neumann functions as the irregular solutions of the free-space Schrodinger equation (Helmholtz equation).
80) (with T replacing S). Note that 'ljJL will no longer behave like h near the origin, but like C-IIJ) . Similarly, if the phase-function matrix 5.. 59). It should be noted, however, that for nonspherical potentials, S is likely to be singular or nearly singular depending on the potential and on the maximum number of angular momentum values used to represent the matrix, S LV. The discussion just given was based on the choice of the Neumann functions as the irregular solutions of the free-space Schrodinger equation (Helmholtz equation).