Cylindrical particle T-matrix notes.
Former-commit-id: c670cd97009e4035059f78f252c2f86ac9c37da6
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Marek Nečada
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\begin_layout Title
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T-matrix of an axially symmetric particle
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So we have
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after "(9.12)"
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key "kristensson_scattering_2016"
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literal "false"
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:
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\begin_inset Formula
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\begin{align*}
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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,\\
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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,
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\end{align*}
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where
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\begin_inset Formula $S_{s}$
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is the scatterer surface and
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\begin_inset Formula $\uvec{\nu}$
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\end_inset
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is the outwards pointing unit normal to it; then
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\begin_inset Formula
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\[
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T_{nn'}=-\sum_{n''}R_{nn''}Q_{n''n}^{-1}.
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\]
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\begin_inset Formula $z$
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axis and parametrise the integral in terms of polar angle
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\begin_inset Formula $\theta$
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.
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Let
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\begin_inset Formula $\beta$
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\end_inset
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be the angle between the surface normal
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\begin_inset Formula $\uvec{\nu}$
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\end_inset
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and the coordinate radial direction
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\begin_inset Formula $\uvec r$
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\end_inset
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.
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The infinitesimal surface area element is then
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\begin_inset Formula
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\[
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\ud S\left(\theta\right)=\frac{\left(r\left(\theta\right)\right)^{2}\sin\theta}{\cos\beta\left(\theta\right)}\ud\theta\,\ud\phi
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\]
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\end_inset
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and the surface normal in local coordinates
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\begin_inset Formula
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\[
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\uvec{\nu}\left(\theta\right)=\uvec r\cos\beta\left(\theta\right)+\uvec{\theta}\sin\beta\left(\theta\right),
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\]
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\end_inset
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which also sets a convention for the sign of
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\begin_inset Formula $\beta$
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\end_inset
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.
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\begin_layout Standard
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For fully axially symmetric particles the integrals vanish for
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\begin_inset Formula $m\ne-m'$
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\end_inset
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due to the
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\begin_inset Formula $e^{i\left(m+m'\right)}$
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asimuthal factor in the integrand.
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width 50text%
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Parametrisation of cylindrical particle surface.
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For cylindrical particle of radius
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\begin_inset Formula $R$
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\end_inset
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and height
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\begin_inset Formula $h$
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\end_inset
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, we can divide the parametrisation into three intervals
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\begin_inset Formula $\left(0,\theta_{1}\right),\left(\theta_{1},\theta_{2}\right),\left(\theta_{2},\pi\right)$
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\end_inset
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where
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\begin_inset Formula $\theta_{1}=\tan^{-1}\left(2R/h\right),\theta_{2}=\pi-\tan^{-1}\left(2R/h\right)$
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:
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In the first section,
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\begin_inset Formula $0<\theta<\theta_{1}$
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\end_inset
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,
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\begin_inset Formula
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\begin{align*}
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r & =\frac{h}{2\cos\theta},\\
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\beta & =-\theta.
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\end{align*}
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In the second section,
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\begin_inset Formula $\theta_{1}<\theta<\theta_{2}$
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,
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\begin_inset Formula
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\begin{align*}
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r & =\frac{R}{\cos\left(\theta-\pi/2\right)}=\frac{R}{\sin\theta},\\
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\beta & =-\theta+\pi/2.
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In the third section,
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\begin_inset Formula $\theta_{2}<\theta<\pi$
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,
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\begin_inset Formula
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\begin{align*}
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r & =\frac{h}{2\cos\left(\theta-\pi\right)}=-\frac{h}{2\cos\theta},\\
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\beta & =-\theta+\pi.
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\end{align*}
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