Cylindrical particle T-matrix notes.

Former-commit-id: c670cd97009e4035059f78f252c2f86ac9c37da6

notes/cylinderT.lyx

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+#LyX 2.4 created this file. For more info see https://www.lyx.org/
+\lyxformat 584
+\begin_document
+\begin_header
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+
+\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\vect r\right)+\wfkcreg_{\overline{n'}}\left(k\vect r\right)\times\wfkcreg_{n}\left(k\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\vect r\right)+\wfkcout_{\overline{n'}}\left(k\vect r\right)\times\wfkcreg_{n}\left(k\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 and 
+\begin_inset Formula $\uvec{\nu}$
+\end_inset
+
+ is the outwards pointing unit normal to it; then
+\begin_inset Formula 
+\[
+T_{nn'}=-\sum_{n''}R_{nn''}Q_{n''n}^{-1}.
+\]
+
+\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.
+\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
+\begin_inset CommandInset bibtex
+LatexCommand bibtex
+btprint "btPrintCited"
+bibfiles "Electrodynamics"
+options "plain"
+encoding "default"
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document

notes/makrot.lyx

@@ -0,0 +1,144 @@
+#LyX 2.4 created this file. For more info see https://www.lyx.org/
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+\newcommand{\ud}{\mathrm{d}}
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+
+\begin_inset FormulaMacro
+\newcommand{\abs}[1]{\left|#1\right|}
+\end_inset
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+ 
+\begin_inset FormulaMacro
+\newcommand{\vect}[1]{\mathbf{#1}}
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+
+\begin_inset FormulaMacro
+\newcommand{\uvec}[1]{\hat{\mathbf{#1}}}
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+
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+
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+\newcommand{\wfkcout}{\vect u}
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+
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+\newcommand{\wfkcreg}{\vect v}
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+
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+\newcommand{\wckcreg}{a}
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+
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+\newcommand{\wckcout}{f}
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+
+
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+
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