Nefunkční Xu A-koefficienty

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Marek Nečada 2015-12-03 03:49:29 +02:00
parent 0a9e2e5815
commit 86622c8b89
1 changed files with 258 additions and 11 deletions

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@ -190,7 +190,7 @@ Expressions for VSWF in Xu
\begin_inset CommandInset citation \begin_inset CommandInset citation
LatexCommand cite LatexCommand cite
after "(2)" after "(2)"
key "xu_calculation_1996" key "xu_electromagnetic_1995"
\end_inset \end_inset
@ -199,13 +199,13 @@ key "xu_calculation_1996"
\begin_layout Standard \begin_layout Standard
\begin_inset Formula \begin_inset Formula
\begin{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},\\ \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}\\ \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)\\ & & +\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},\\ & & \phantom{+}\times\frac{1}{kr}\frac{\ud\left(rz_{n}^{(J)}(kr)\right)}{\ud r}e^{im\phi},\nonumber \\
& = & ... & = & ...\nonumber
\end{eqnarray*} \end{eqnarray}
\end_inset \end_inset
@ -242,7 +242,7 @@ Expansions for the scattered fields are
\begin_inset CommandInset citation \begin_inset CommandInset citation
LatexCommand cite LatexCommand cite
after "(4)" after "(4)"
key "xu_calculation_1996" key "xu_electromagnetic_1995"
\end_inset \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? are still consistent, are they not?
\end_layout \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 \begin_layout Subsubsection
Mie scattering Mie scattering
\end_layout \end_layout
@ -348,7 +365,7 @@ For the exact form of the coefficients following from the boundary conditions
\begin_inset CommandInset citation \begin_inset CommandInset citation
LatexCommand cite LatexCommand cite
after "(1213)" after "(1213)"
key "xu_calculation_1996" key "xu_electromagnetic_1995"
\end_inset \end_inset
@ -358,7 +375,7 @@ key "xu_calculation_1996"
\begin_inset CommandInset citation \begin_inset CommandInset citation
LatexCommand cite LatexCommand cite
after "(1415)" after "(1415)"
key "xu_calculation_1996" key "xu_electromagnetic_1995"
\end_inset \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). (which is not surprising and probably follows from the Wigner-Eckart theorem).
\end_layout \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 \begin_layout Subsection
T-Matrix resummation (multiple scatterers) T-Matrix resummation (multiple scatterers)
\end_layout \end_layout