421 lines
10 KiB
Plaintext
421 lines
10 KiB
Plaintext
#LyX 2.4 created this file. For more info see https://www.lyx.org/
|
|
\lyxformat 584
|
|
\begin_document
|
|
\begin_header
|
|
\save_transient_properties true
|
|
\origin unavailable
|
|
\textclass article
|
|
\use_default_options true
|
|
\maintain_unincluded_children false
|
|
\language english
|
|
\language_package default
|
|
\inputencoding utf8
|
|
\fontencoding auto
|
|
\font_roman "default" "default"
|
|
\font_sans "default" "default"
|
|
\font_typewriter "default" "default"
|
|
\font_math "auto" "auto"
|
|
\font_default_family default
|
|
\use_non_tex_fonts false
|
|
\font_sc false
|
|
\font_roman_osf false
|
|
\font_sans_osf false
|
|
\font_typewriter_osf false
|
|
\font_sf_scale 100 100
|
|
\font_tt_scale 100 100
|
|
\use_microtype false
|
|
\use_dash_ligatures true
|
|
\graphics default
|
|
\default_output_format default
|
|
\output_sync 0
|
|
\bibtex_command default
|
|
\index_command default
|
|
\float_placement class
|
|
\float_alignment class
|
|
\paperfontsize default
|
|
\spacing single
|
|
\use_hyperref false
|
|
\papersize default
|
|
\use_geometry false
|
|
\use_package amsmath 1
|
|
\use_package amssymb 1
|
|
\use_package cancel 1
|
|
\use_package esint 1
|
|
\use_package mathdots 1
|
|
\use_package mathtools 1
|
|
\use_package mhchem 1
|
|
\use_package stackrel 1
|
|
\use_package stmaryrd 1
|
|
\use_package undertilde 1
|
|
\cite_engine basic
|
|
\cite_engine_type default
|
|
\biblio_style plain
|
|
\use_bibtopic false
|
|
\use_indices false
|
|
\paperorientation portrait
|
|
\suppress_date false
|
|
\justification true
|
|
\use_refstyle 1
|
|
\use_minted 0
|
|
\use_lineno 0
|
|
\index Index
|
|
\shortcut idx
|
|
\color #008000
|
|
\end_index
|
|
\secnumdepth 3
|
|
\tocdepth 3
|
|
\paragraph_separation indent
|
|
\paragraph_indentation default
|
|
\is_math_indent 0
|
|
\math_numbering_side default
|
|
\quotes_style english
|
|
\dynamic_quotes 0
|
|
\papercolumns 1
|
|
\papersides 1
|
|
\paperpagestyle default
|
|
\tablestyle default
|
|
\tracking_changes false
|
|
\output_changes false
|
|
\html_math_output 0
|
|
\html_css_as_file 0
|
|
\html_be_strict false
|
|
\end_header
|
|
|
|
\begin_body
|
|
|
|
\begin_layout Standard
|
|
\begin_inset CommandInset include
|
|
LatexCommand input
|
|
filename "makrot.lyx"
|
|
literal "false"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Title
|
|
T-matrix of an axially symmetric particle
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
So we have
|
|
\begin_inset CommandInset citation
|
|
LatexCommand cite
|
|
after "(9.12)"
|
|
key "kristensson_scattering_2016"
|
|
literal "false"
|
|
|
|
\end_inset
|
|
|
|
:
|
|
\begin_inset Formula
|
|
\begin{align*}
|
|
R_{nn'} & =ik^{2}\iint_{S_{s}}\left(\frac{\eta}{\eta_{1}}\wfkcreg_{n}\left(k\vect r\right)\times\wfkcreg_{\overline{n'}}\left(k_{1}\vect r\right)+\wfkcreg_{\overline{n}}\left(k\vect r\right)\times\wfkcreg_{n'}\left(k_{1}\vect r\right)\right)\cdot\uvec{\nu}\,\ud S,\\
|
|
Q_{nn'} & =ik^{2}\iint_{S_{s}}\left(\frac{\eta}{\eta_{1}}\wfkcout_{n}\left(k\vect r\right)\times\wfkcreg_{\overline{n'}}\left(k_{1}\vect r\right)+\wfkcout_{\overline{n}}\left(k\vect r\right)\times\wfkcreg_{n'}\left(k_{1}\vect r\right)\right)\cdot\uvec{\nu}\,\ud S,
|
|
\end{align*}
|
|
|
|
\end_inset
|
|
|
|
where
|
|
\begin_inset Formula $S_{s}$
|
|
\end_inset
|
|
|
|
is the scatterer surface,
|
|
\begin_inset Formula $\uvec{\nu}$
|
|
\end_inset
|
|
|
|
is the outwards pointing unit normal to it, and the subscript
|
|
\begin_inset Formula $_{1}$
|
|
\end_inset
|
|
|
|
refers to the particle inside; then
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
T_{nn'}=-\sum_{n''}R_{nn''}Q_{n''n}^{-1}.\label{eq:T matrix from R and Q}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Let us consider the case with full rotational symmetry around the
|
|
\begin_inset Formula $z$
|
|
\end_inset
|
|
|
|
axis and parametrise the integral in terms of polar angle
|
|
\begin_inset Formula $\theta$
|
|
\end_inset
|
|
|
|
.
|
|
Let
|
|
\begin_inset Formula $\beta$
|
|
\end_inset
|
|
|
|
be the angle between the surface normal
|
|
\begin_inset Formula $\uvec{\nu}$
|
|
\end_inset
|
|
|
|
and the coordinate radial direction
|
|
\begin_inset Formula $\uvec r$
|
|
\end_inset
|
|
|
|
.
|
|
The infinitesimal surface area element is then
|
|
\begin_inset Formula
|
|
\[
|
|
\ud S\left(\theta\right)=\frac{\left(r\left(\theta\right)\right)^{2}\sin\theta}{\cos\beta\left(\theta\right)}\ud\theta\,\ud\phi
|
|
\]
|
|
|
|
\end_inset
|
|
|
|
and the surface normal in local coordinates
|
|
\begin_inset Formula
|
|
\[
|
|
\uvec{\nu}\left(\theta\right)=\uvec r\cos\beta\left(\theta\right)+\uvec{\theta}\sin\beta\left(\theta\right),
|
|
\]
|
|
|
|
\end_inset
|
|
|
|
which also sets a convention for the sign of
|
|
\begin_inset Formula $\beta$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
For fully axially symmetric particles the integrals vanish for
|
|
\begin_inset Formula $m\ne-m'$
|
|
\end_inset
|
|
|
|
due to the
|
|
\begin_inset Formula $e^{i\left(m+m'\right)}$
|
|
\end_inset
|
|
|
|
asimuthal factor in the integrand.
|
|
One then has
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
T_{nn'}=-\sum_{n''}R'_{nn''}Q'_{n''n}^{-1}\label{eq:T-matrix from reduced R and Q}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
where
|
|
\begin_inset Formula
|
|
\begin{align*}
|
|
R'_{nn'} & =\int_{0}^{\pi}\left(\frac{\eta}{\eta_{1}}\wfkcreg_{n}\left(k\vect r\right)\times\wfkcreg_{\overline{n'}}\left(k_{1}\vect r\right)+\wfkcreg_{\overline{n}}\left(k\vect r\right)\times\wfkcreg_{n'}\left(k_{1}\vect r\right)\right)\cdot\left(\uvec r\cos\beta\left(\theta\right)+\uvec{\theta}\sin\beta\left(\theta\right)\right)\frac{\left(r\left(\theta\right)\right)^{2}\sin\theta}{\cos\beta\left(\theta\right)}\ud\theta,\\
|
|
Q'_{nn'} & =\int_{0}^{\pi}\left(\frac{\eta}{\eta_{1}}\wfkcreg_{n}\left(k\vect r\right)\times\wfkcreg_{\overline{n'}}\left(k_{1}\vect r\right)+\wfkcreg_{\overline{n}}\left(k\vect r\right)\times\wfkcreg_{n'}\left(k_{1}\vect r\right)\right)\cdot\left(\uvec r\cos\beta\left(\theta\right)+\uvec{\theta}\sin\beta\left(\theta\right)\right)\frac{\left(r\left(\theta\right)\right)^{2}\sin\theta}{\cos\beta\left(\theta\right)}\ud\theta
|
|
\end{align*}
|
|
|
|
\end_inset
|
|
|
|
where
|
|
\begin_inset Formula $\vect r=\vect r\left(\theta\right)=\left(r\left(\theta\right),\theta,0\right)$
|
|
\end_inset
|
|
|
|
.
|
|
Matrices
|
|
\begin_inset Formula $Q',R'$
|
|
\end_inset
|
|
|
|
differ from the original
|
|
\begin_inset Formula $R,Q$
|
|
\end_inset
|
|
|
|
matrices in
|
|
\begin_inset CommandInset ref
|
|
LatexCommand eqref
|
|
reference "eq:T matrix from R and Q"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
by a factor of
|
|
\begin_inset Formula $2\pi ik^{2}$
|
|
\end_inset
|
|
|
|
, but this cancels out in the matrix product.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float figure
|
|
placement document
|
|
alignment document
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\align center
|
|
\begin_inset Graphics
|
|
filename cylinder.png
|
|
lyxscale 30
|
|
width 50text%
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
Parametrisation of cylindrical particle surface.
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
For cylindrical particle of radius
|
|
\begin_inset Formula $R$
|
|
\end_inset
|
|
|
|
and height
|
|
\begin_inset Formula $h$
|
|
\end_inset
|
|
|
|
, we can divide the parametrisation into three intervals
|
|
\begin_inset Formula $\left(0,\theta_{1}\right),\left(\theta_{1},\theta_{2}\right),\left(\theta_{2},\pi\right)$
|
|
\end_inset
|
|
|
|
where
|
|
\begin_inset Formula $\theta_{1}=\tan^{-1}\left(2R/h\right),\theta_{2}=\pi-\tan^{-1}\left(2R/h\right)$
|
|
\end_inset
|
|
|
|
:
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
In the first section,
|
|
\begin_inset Formula $0<\theta<\theta_{1}$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula
|
|
\begin{align*}
|
|
r & =\frac{h}{2\cos\theta},\\
|
|
\beta & =-\theta.
|
|
\end{align*}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
In the second section,
|
|
\begin_inset Formula $\theta_{1}<\theta<\theta_{2}$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula
|
|
\begin{align*}
|
|
r & =\frac{R}{\cos\left(\theta-\pi/2\right)}=\frac{R}{\sin\theta},\\
|
|
\beta & =-\theta+\pi/2.
|
|
\end{align*}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
In the third section,
|
|
\begin_inset Formula $\theta_{2}<\theta<\pi$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula
|
|
\begin{align*}
|
|
r & =\frac{h}{2\cos\left(\theta-\pi\right)}=-\frac{h}{2\cos\theta},\\
|
|
\beta & =-\theta+\pi.
|
|
\end{align*}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Let's write VSWFs in terms of the power-normalised
|
|
\begin_inset Formula $p,\pi,\tau$
|
|
\end_inset
|
|
|
|
funs:
|
|
\begin_inset Formula
|
|
\begin{align*}
|
|
\vsh_{1lm} & =\left(\uvec{\theta}\pi_{lm}-\uvec{\phi}\tau_{lm}\right)e^{im\phi}\\
|
|
\vsh_{2lm} & =\left(\uvec{\theta}\tau_{lm}+\uvec{\phi}\pi_{lm}\right)e^{im\phi}\\
|
|
\vsh_{3lm} & =\sqrt{l\left(l+1\right)}p_{lm}e^{im\theta}
|
|
\end{align*}
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula
|
|
\begin{align*}
|
|
\vect y_{\kappa1lm} & =\underbrace{h_{l}^{\kappa}e^{im\phi}}_{c_{\kappa lm}^{1}}\left(\uvec{\theta}\pi_{lm}-\uvec{\phi}\tau_{lm}\right)\\
|
|
\vect y_{\kappa2lm} & =\frac{1}{kr}e^{im\phi}\left(\frac{\ud\left(krh_{l}^{\kappa}\right)}{\ud\left(kr\right)}\left(\uvec{\theta}\tau_{lm}+\uvec{\phi}\pi_{lm}\right)+h_{l}^{\kappa}l\left(l+1\right)\uvec rp_{lm}\right)\\
|
|
& =c_{\kappa lm}^{2}\left(\uvec{\theta}\tau_{lm}+\uvec{\phi}\pi_{lm}\right)+c_{\kappa lm}^{3}\uvec rp_{lm}
|
|
\end{align*}
|
|
|
|
\end_inset
|
|
|
|
The triple products than are (reminder:
|
|
\begin_inset Formula $\uvec{\nu}\left(\theta\right)=\uvec r\cos\beta\left(\theta\right)+\uvec{\theta}\sin\beta\left(\theta\right))$
|
|
\end_inset
|
|
|
|
:
|
|
\begin_inset Formula
|
|
\begin{align*}
|
|
\left(\vect y_{\kappa1lm}\times\vect v_{1l'm'}\right)\cdot\uvec{\nu} & =\cos\beta c_{\kappa lm}^{1}c_{\mathrm{R}l'm'}^{1}\left(-\pi_{lm}\tau_{l'm'}+\tau_{lm}\pi_{l'm'}\right)\\
|
|
\left(\vect y_{\kappa1lm}\times\vect v_{2l'm'}\right)\cdot\uvec{\nu} & =\cos\beta c_{\kappa lm}^{1}c_{\mathrm{R}l'm'}^{2}\left(\pi_{lm}\pi_{l'm'}+\tau_{lm}\tau_{l'm'}\right)\\
|
|
& +\sin\beta c_{\kappa lm}^{1}c_{\mathrm{R}l'm'}^{3}\left(-\tau_{lm}p_{l'm'}\right)\\
|
|
\left(\vect y_{\kappa2lm}\times\vect v_{1l'm'}\right)\cdot\uvec{\nu} & =\cos\beta c_{\kappa lm}^{2}c_{\mathrm{R}l'm'}^{1}\left(-\pi_{lm}\pi_{l'm'}-\tau_{lm}\tau_{l'm'}\right)\\
|
|
& +\sin\beta c_{\kappa lm}^{3}c_{\mathrm{R}l'm'}^{1}\left(p_{lm}\tau_{l'm'}\right)\\
|
|
\left(\vect y_{\kappa2lm}\times\vect v_{2l'm'}\right)\cdot\uvec{\nu} & =\cos\beta c_{\kappa lm}^{2}c_{\mathrm{R}l'm'}^{2}\left(\tau_{lm}\pi_{l'm'}-\pi_{lm}\tau_{l'm'}\right)\\
|
|
& -\sin\beta c_{\kappa lm}^{3}c_{\mathrm{R}l'm'}^{2}p_{lm}\pi_{l'm'}\\
|
|
& +\sin\beta c_{\kappa lm}^{2}c_{\mathrm{R}l'm'}^{3}\pi_{lm}p_{l'm'}
|
|
\end{align*}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset CommandInset bibtex
|
|
LatexCommand bibtex
|
|
btprint "btPrintCited"
|
|
bibfiles "Electrodynamics"
|
|
options "plain"
|
|
encoding "default"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_body
|
|
\end_document
|