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Expressing VSWF in terms of SSWF

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Marek Nečada 2022-03-29 14:32:06 +03:00
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33
notes/VSWF_from_SSWF.md Normal file
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VSWF expansions in terms of SSWF
================================
From
\cite necada_multiple-scattering_2021, eq. (2.19)
\f[
\wfkcout_{\tau lm}\left(\kappa (\vect r - \vect r_1) \right) =
\sum_{\tau'l'm'} \tropSr{\kappa(\vect r_2 - \vect r_1)}_{\tau l m;\tau'l'm} \wfkcreg_{\tau'l'm'}(\vect r -\vect r_2),
\qquad |\vect r -\vect r_2| < |\vect r_1 - \vect r_2|,
\f]
setting \f$ \vect r = \vect r_2\f$ and considering that
\f$ \wfkcreg_{\tau'l'm'}(\vect 0) \ne \vect 0 \f$ only for electric dipole waves (\f$ \tau = \mathrm{E}, l=1 \f$),
we have
\f[
\wfkcout_{\tau lm}\left(\kappa (\vect r - \vect r_1) \right) =
\sum_{m'} \tropSr{\kappa(\vect r - \vect r_1)}_{\tau l m;\mathrm{E}1m} \wfkcreg_{\mathrm{E}1m'}(\vect 0),
\qquad \vect r \ne \vect r_1 .
\f]
Combining this with
\cite necada_multiple-scattering_2021, eq. (2.25)
\f[
\tropSr{\vect d}_{\tau l m; \tau' l' m'} = \sum_{\lambda =|l-l'|+|\tau-\tau'|}^{l+l'}
C^{\lambda}_{\tau l m;\tau' l'm'} \underbrace{ \spharm{\lambda}{m-m'}(\uvec d) h_\lambda^{(1)}(d)}_{\sswfout_\lambda^{m-m'}(\vect d)},
\f]
we get
\f[
\wfkcout_{\tau lm}(\vect d) = \sum_{m'=-1}^1 \wfkcreg_{\mathrm{E}1m'}(\vect 0)
\sum_{\lambda=l-1+|\tau-\tau'|}^{l+1}
C^\lambda_{\tau l m;\mathrm{E}1m'} \sswfout_\lambda^{m-m'}(\vect d).
\f]
Note that the VSWF components in this expression are given in global "cartesian" basis,
*not* the local orthonormal basis derived from spherical coordinates.
(This is mostly desirable, because in lattices we need to work with flat coordinates anyway.)

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notes/mathjax_newcommands.js
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@ -31,8 +31,15 @@ MathJax.Hub.Config({
spharm: ["{{Y_{\\mathrm{#1}}}_{#2}^{#3}}", 3, ""], // Spherical harmonics
spharmR: ["{{Y_{\\mathrm{#1}}}_{\\mathrm{#1}{#2}{#3}}", 4, ""], // Spherical harmonics
csphase: "\\mathsf{C_{CS}}", // Condon-Shortley phase
tropSrr: ["{{S^\\mathrm{#1}}\\pr{{#2} \\leftarrow {#3}}}", 3, ""], // Translation operator singular
tropRrr: ["{{R^\\mathrm{#1}}\\pr{{#2} \\leftarrow {#3}}}", 3, ""], // Translation operator regular
tropS: "{\\mathcal{S}}", // Translation operator singular
tropR: "{\\mathcal{R}}", // Translation operator regular
tropSr: ["{{\\mathcal{S}^\\mathrm{#1}}\\pr{{#2}}}", 2, ""], // Translation operator singular
tropRr: ["{{\\mathcal{R}^\\mathrm{#1}}\\pr{{#2}}}", 2, ""], // Translation operator regular
tropSrr: ["{{\\mathcal{S}^\\mathrm{#1}}\\pr{{#2} \\leftarrow {#3}}}", 3, ""], // Translation operator singular
tropRrr: ["{{\\mathcal{R}^\\mathrm{#1}}\\pr{{#2} \\leftarrow {#3}}}", 3, ""], // Translation operator regular
sswfout: "{\\psi}", // outgoing SSWF
sswfreg: "{\\phi}", // regular SSWF
// Kristensson's VSWFs, complex version (2014 notes)
wfkc: "{\\vect{y}}", // any wave