Spherical waves expansion

Andersen, O.K. (1971). Comments on the KKR wavefunctions extension of the spherical wave expansion beyond the muffin tins. In Computational Methods in Band Theory, eds. P.M. Marcus, J.F. Janak, A.R. Williams (Plenum, New York), 178-182. 

Plane waves are often used when two conditions are satisfied 1) many (but not necessarily all) atomic layers of the surface have a two-dimensional periodicity, and 2) either a plane-wave incident electron beam is present or angle-resolved electron detection is applied. Computation based on the plane-wave expansion are often much more efficient than those based on the spherical-wave expansion. This explains their frequent use even in problems that do not involve strict two-dimensional periodicity, as with disordered overlayers on an otherwise periodic substrate. 

This Gookl equally be done by making a spherical wave expansion for V, as proposed by Koide but his method for calculating is practical only for atoms. 

Spherical harmonic functions are important in many problems in Chemistry and Physics. Spherical harmonic functions are central in discussions of rotation, motion in a central potential, multipole expansion, cluster bonding, spherical wave expansions and many more topics. The calculation of symmetrized powers of representations give a way of obtaining the... 

Spectral representations of the fields (plane, cylindrical, spherical wave expansions Hermite-Gaussian and Laguerre-Gaussian beams prolate spheroidal harmonics)... 

In this section, we characterize the polarization state of vector plane waves and derive vector spherical wave expansions for the incident field. The first topic is relevant in the analysis of the scattered field, while the second one plays an important role in the derivation of the transition matrix. 

The vector spherical waves expansion of the incident field reads as... 

Considering the scattered field representation (2.5), we replace the surface fields by their approximations and use the vector spherical wave expansion of the dyad gl on a sphere enclosing D to obtain... 

Passing to spherical coordinates, using the spherical wave expansion of the plane waves exp (jp-r) and assuming spherical symmetry for H, i.e., H p) = H p), we obtain... 

Thus, the derivation of the vector spherical wave expansion of the electromagnetic field scattered by an arbitrary infinite surface involves the following... 

Computation of the vector spherical wave expansion of the scattered field... 

The spherical wave expansion of the plane wave exp(jfe r) is of significant importance in electromagnetic scattering and is given by [215]... 

Inserting the spherical wave expansion of the quasi-plane wave Q k,/3, a, ro) into (B.61) yields... 

As in the scalar case, integral and series representations for the translation coefficients can be obtained by using the integral representations for the vector spherical wave functions. First we consider the case of regular vector spherical wave functions. Using the integral representation (B.26), the relation r = 0 + T" ) and the vector spherical wave expansion... 

Inserting the spherical wave expansion of the plane wave expQfe Tq) and integrating over a, we derive the following series representation for the translation coefficients ... 

Restricting r to lies inside a sphere enclosed in S2 and using the vector spherical wave expansion of the dyad gl, we see that the first set of closure relations gives 5 = 0 in D fi. Analogously, but restricting r to lies outside a sphere enclosing Si, we deduce that the second set of closure relations yields = 0 in )<,. Theorem 1 can now be used to conclude. ... 

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