Nefunkční Xu A-koefficienty
Former-commit-id: 28c0f6aa29732d4830f8b7de6a1050a98b08087f
This commit is contained in:
Marek Nečada
parent
0a9e2e5815
commit
86622c8b89
269
worknotes.lyx
269
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
|
||||
|
|
|
|||
Loading…
Reference in New Issue