From 86622c8b89422673a7154595163217c37debde5c Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Thu, 3 Dec 2015 03:49:29 +0200 Subject: [PATCH] =?UTF-8?q?Nefunk=C4=8Dn=C3=AD=20Xu=20A-koefficienty?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Former-commit-id: 28c0f6aa29732d4830f8b7de6a1050a98b08087f --- worknotes.lyx | 269 +++++++++++++++++++++++++++++++++++++++++++++++--- 1 file changed, 258 insertions(+), 11 deletions(-) diff --git a/worknotes.lyx b/worknotes.lyx index 0ad9319..3ab50f1 100644 --- a/worknotes.lyx +++ b/worknotes.lyx @@ -190,7 +190,7 @@ Expressions for VSWF in Xu \begin_inset CommandInset citation LatexCommand cite after "(2)" -key "xu_calculation_1996" +key "xu_electromagnetic_1995" \end_inset @@ -199,13 +199,13 @@ key "xu_calculation_1996" \begin_layout Standard \begin_inset Formula -\begin{eqnarray*} -\vect M_{mn}^{(J)} & = & \left(i\uvec{\theta}\pi_{mn}(\cos\theta)-\uvec{\phi}\tau_{mn}(\cos\theta)\right)z_{n}^{(J)}(kr)e^{im\phi},\\ -\vect N_{mn}^{(J)} & = & \uvec rn(n+1)P_{n}^{m}(\cos\theta)\frac{z_{n}^{(J)}(kr)}{kr}e^{im\phi}\\ - & & +\left(\uvec{\theta}\tau_{mn}(\cos\theta)+i\uvec{\phi}\pi_{mn}(\cos\theta)\right)\\ - & & \phantom{+}\times\frac{1}{kr}\frac{\ud\left(rz_{n}^{(J)}(kr)\right)}{\ud r}e^{im\phi},\\ - & = & ... -\end{eqnarray*} +\begin{eqnarray} +\vect M_{mn}^{(J)} & = & \left(i\uvec{\theta}\pi_{mn}(\cos\theta)-\uvec{\phi}\tau_{mn}(\cos\theta)\right)z_{n}^{(J)}(kr)e^{im\phi},\nonumber \\ +\vect N_{mn}^{(J)} & = & \uvec rn(n+1)P_{n}^{m}(\cos\theta)\frac{z_{n}^{(J)}(kr)}{kr}e^{im\phi}\label{eq:vswf}\\ + & & +\left(\uvec{\theta}\tau_{mn}(\cos\theta)+i\uvec{\phi}\pi_{mn}(\cos\theta)\right)\nonumber \\ + & & \phantom{+}\times\frac{1}{kr}\frac{\ud\left(rz_{n}^{(J)}(kr)\right)}{\ud r}e^{im\phi},\nonumber \\ + & = & ...\nonumber +\end{eqnarray} \end_inset @@ -242,7 +242,7 @@ Expansions for the scattered fields are \begin_inset CommandInset citation LatexCommand cite after "(4)" -key "xu_calculation_1996" +key "xu_electromagnetic_1995" \end_inset @@ -337,6 +337,23 @@ It should be possible to just take it away and the abovementioned expansions are still consistent, are they not? \end_layout +\begin_layout Standard +In +\begin_inset CommandInset citation +LatexCommand cite +after "sec. 4A" +key "xu_electromagnetic_1995" + +\end_inset + +, there are formulae for translation of the plane wave between VSWF with + different origins. +\end_layout + +\begin_layout Standard +o +\end_layout + \begin_layout Subsubsection Mie scattering \end_layout @@ -348,7 +365,7 @@ For the exact form of the coefficients following from the boundary conditions \begin_inset CommandInset citation LatexCommand cite after "(12–13)" -key "xu_calculation_1996" +key "xu_electromagnetic_1995" \end_inset @@ -358,7 +375,7 @@ key "xu_calculation_1996" \begin_inset CommandInset citation LatexCommand cite after "(14–15)" -key "xu_calculation_1996" +key "xu_electromagnetic_1995" \end_inset @@ -382,6 +399,236 @@ in other words, the Mie coefficients do not depend on (which is not surprising and probably follows from the Wigner-Eckart theorem). \end_layout +\begin_layout Subsubsection +Translation coefficients +\end_layout + +\begin_layout Standard +A quite detailed study can be found in +\begin_inset CommandInset citation +LatexCommand cite +key "xu_calculation_1996" + +\end_inset + +, I have not read the recenter one +\begin_inset CommandInset citation +LatexCommand cite +key "xu_efficient_1998" + +\end_inset + + which deals with efficient evaluation of Wigner 3jm symbols and Gaunt coefficie +nts. + +\end_layout + +\begin_layout Standard +With the VSWF as in +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:vswf" + +\end_inset + + and translation relations in the form +\begin_inset CommandInset citation +LatexCommand cite +after "(38,39)" +key "xu_calculation_1996" + +\end_inset + + +\begin_inset Formula +\begin{eqnarray*} +\vect M_{\mu\nu}^{(J)l} & = & \sum_{n=1}^{\infty}\sum_{m=-n}^{n}\left[A_{mn}^{\mu\nu}\vect M_{mn}^{(1)j}+B_{mn}^{\mu\nu}\vect N_{mn}^{(1)j}\right],\quad r\le d_{lj},\\ +\vect N_{\mu\nu}^{(J)l} & = & \sum_{n=1}^{\infty}\sum_{m=-n}^{n}\left[B_{mn}^{\mu\nu}\vect M_{mn}^{(1)j}+A_{mn}^{\mu\nu}\vect N_{mn}^{(1)j}\right],\quad r\le d_{lj},\\ +\vect M_{\mu\nu}^{(J)l} & = & \sum_{n=1}^{\infty}\sum_{m=-n}^{n}\left[A_{mn}^{\mu\nu}\vect M_{mn}^{(J)j}+B_{mn}^{\mu\nu}\vect N_{mn}^{(J)j}\right],\quad r\ge d_{lj},\\ +\vect N_{\mu\nu}^{(J)l} & = & \sum_{n=1}^{\infty}\sum_{m=-n}^{n}\left[B_{mn}^{\mu\nu}\vect M_{mn}^{(J)j}+A_{mn}^{\mu\nu}\vect N_{mn}^{(J)j}\right],\quad r\ge d_{lj}, +\end{eqnarray*} + +\end_inset + +the translation coefficients (which should in fact be also labeled with + their origin indices +\begin_inset Formula $l,j$ +\end_inset + +) are +\begin_inset CommandInset citation +LatexCommand cite +after "(82,83)" +key "xu_calculation_1996" + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{multline*} +A_{mn}^{\mu\nu}=\\ +\frac{(-1)^{m}i^{\nu+n}(n+2)_{n-1}\left(\nu+2\right)_{\nu+1}(n+\nu+m-\mu)!}{4n(n+\nu+1)_{n+\nu}(n-m)!(\nu+m)!}\\ +\times e^{i(\mu-m)\phi_{lj}}\sum_{q=0}^{q_{\mathrm{max}}}(-1)^{q}\left[n(n+1)+\nu(\nu+1)-p(p+1)\right]\\ +\times\tilde{a}_{1q}\begin{pmatrix}z_{p}^{(J)}(kd_{lj})\\ +j_{p}(kd_{lj}) +\end{pmatrix}P_{p}^{\mu-m}(\cos\theta_{lj}),\qquad\begin{pmatrix}r\le d_{lj}\\ +r\ge d_{lj} +\end{pmatrix}; +\end{multline*} + +\end_inset + + +\begin_inset Formula +\begin{multline*} +B_{mn}^{\mu\nu}=\\ +\frac{(-1)^{m}i^{\nu+n+1}(n+2)_{n+1}\left(\nu+2\right)_{\nu+1}(n+\nu+m-\mu+1)!}{4n(n+1)(n+m+1)(n+\nu+2)_{n+\nu+1}(n-m)!(\nu+m)!}\\ +\times e^{i(\mu-m)\phi_{lj}}\sum_{q=0}^{Q_{\mathrm{max}}}(-1)^{q}\Big\{2(n+1)(\nu-\mu)\tilde{a}_{2q}-\\ +-\left[p(p+3)-\nu(\nu+1)-n(n+3)-2\mu(n+1)\right]\tilde{a}_{3q}\Big\}\\ +\times\begin{pmatrix}z_{p+1}^{(J)}(kd_{lj})\\ +j_{p+1}(kd_{lj}) +\end{pmatrix}P_{p+1}^{\mu-m}(\cos\theta_{lj}),\qquad\begin{pmatrix}r\le d_{lj}\\ +r\ge d_{lj} +\end{pmatrix}; +\end{multline*} + +\end_inset + +where +\begin_inset CommandInset citation +LatexCommand cite +after "(79,80)" +key "xu_calculation_1996" + +\end_inset + + +\begin_inset Formula +\begin{eqnarray*} +\tilde{a}_{1q} & = & a(-m,n,\mu,\nu,n+\nu-2q)/a(-m,n,\mu,\nu,n+\nu),\\ +\tilde{a}_{2q} & = & a(-m-1,n+1,\mu+1,\nu,n+\nu+1-2q)/\\ + & & /a(-m-1,n+1,\mu+1,\nu,n+\nu+1),\\ +\tilde{a}_{3q} & = & a(-m,n+1,\mu,\nu,n+\nu+1-2q)/\\ + & & /a(-m,n+1,\mu,\nu,\mu+\nu+1), +\end{eqnarray*} + +\end_inset + + +\begin_inset Formula +\begin{eqnarray*} +p & = & n+\nu-2q\\ +q_{\max} & = & \min\left(n,\nu,\frac{n+\nu-\left|m-\mu\right|}{2}\right),\\ +Q_{\max} & = & \min\left(n+1,\nu,\frac{n+\nu+1-\left|m-\mu\right|}{2}\right), +\end{eqnarray*} + +\end_inset + +with the Pochhammer symbol / falling factorial (hope it is the +\emph on +falling +\emph default + one; Xu does not explain the notation anywhere) +\begin_inset Formula +\[ +(x)_{n}=x\left(x-1\right)\left(x-2\right)\dots\left(x-n+1\right)=\frac{x!}{\left(x-n\right)!} +\] + +\end_inset + +(in contrast to the rising factorial +\begin_inset Formula +\[ +x^{(n)}=x(x+1)(x+2)\dots(x+n-1)=\frac{(x+n-1)!}{(x-1)!}, +\] + +\end_inset + + their mutual relation should then be +\begin_inset Formula $(x)_{n}=(x-n+1)^{(n)}$ +\end_inset + +). +\end_layout + +\begin_layout Standard +The implementation should be checked with +\begin_inset CommandInset citation +LatexCommand cite +after "Table II" +key "xu_calculation_1996" + +\end_inset + + +\end_layout + +\begin_layout Subsubsection +Equations for the scattering problem +\end_layout + +\begin_layout Standard +The linear system for the scattering problem reads +\begin_inset CommandInset citation +LatexCommand cite +after "(30)" +key "xu_electromagnetic_1995" + +\end_inset + + +\begin_inset Formula +\begin{eqnarray*} +a_{mn}^{j} & = & a_{n}^{j}\left\{ p_{mn}^{j,j}-\sum_{l\neq j}^{(1,L)}\sum_{\nu=1}^{\infty}\sum_{\mu=-\nu}^{\nu}\left[a_{\mu\nu}^{l}A_{mn}^{\mu\nu;lj}+b_{\mu\nu}^{l}B_{mn}^{\mu\nu;lj}\right]\right\} \\ +b_{mn}^{j} & = & b_{n}^{j}\left\{ q_{mn}^{j,j}-\sum_{l\neq j}^{(1,L)}\sum_{\nu=1}^{\infty}\sum_{\mu=-\nu}^{\nu}\left[a_{\mu\nu}^{l}B_{mn}^{\mu\nu;lj}+b_{\mu\nu}^{l}A_{mn}^{\mu\nu;lj}\right]\right\} +\end{eqnarray*} + +\end_inset + +where +\begin_inset Formula $p_{mn}^{j,j},q_{mn}^{j,j}$ +\end_inset + + are the expansion coefficients of the initial incident waves in the +\begin_inset Formula $j$ +\end_inset + +-th particle's coordinate system +\begin_inset CommandInset citation +LatexCommand cite +after "sec. 4A" +key "xu_electromagnetic_1995" + +\end_inset + +. + +\emph on +TODO expressions for +\begin_inset Formula $p_{mn}^{j,j},q_{mn}^{j,j}$ +\end_inset + + in the case of dipole initial wave. +\end_layout + +\begin_layout Subsubsection +Solving the linear system +\end_layout + +\begin_layout Standard +\begin_inset CommandInset citation +LatexCommand cite +after "sec. 5" +key "xu_electromagnetic_1995" + +\end_inset + + +\end_layout + \begin_layout Subsection T-Matrix resummation (multiple scatterers) \end_layout