#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\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