From b80e7607f8ee61eb7f50350322563e7bfa1bd3e3 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Fri, 5 Jun 2020 10:59:41 +0300 Subject: [PATCH] Simplify tr.op. expression. Scattered field in periodic system. Former-commit-id: 0b93f3bdf78a3a3d6a9a2cf628012f80790daebd --- lepaper/arrayscat.lyx | 5 ++ lepaper/finite.lyx | 63 +++++++++++++++----- lepaper/infinite.lyx | 133 ++++++++++++++++++++++++++++++++++++++++-- 3 files changed, 181 insertions(+), 20 deletions(-) diff --git a/lepaper/arrayscat.lyx b/lepaper/arrayscat.lyx index cca8ce3..3e19445 100644 --- a/lepaper/arrayscat.lyx +++ b/lepaper/arrayscat.lyx @@ -395,6 +395,11 @@ status open \end_inset +\begin_inset FormulaMacro +\newcommand{\tropcoeff}{C} +\end_inset + + \begin_inset FormulaMacro \newcommand{\truncated}[2]{\left[#1\right]_{l\le#2}} \end_inset diff --git a/lepaper/finite.lyx b/lepaper/finite.lyx index 22db654..59a6ca5 100644 --- a/lepaper/finite.lyx +++ b/lepaper/finite.lyx @@ -1919,10 +1919,27 @@ outside. \begin_layout Standard In our convention, the regular translation operator elements can be expressed explicitly as +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout \begin_inset Formula \begin{align} \tropr_{\tau lm;\tau l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|}^{l+l'}C_{lm;l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)j_{\lambda}\left(d\right),\nonumber \\ -\tropr_{\tau lm;\tau'l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}D_{lm;l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)j_{\lambda}\left(d\right),\quad\tau'\ne\tau,\label{eq:translation operator} +\tropr_{\tau lm;\tau'l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}D_{lm;l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)j_{\lambda}\left(d\right),\quad\tau'\ne\tau,\label{eq:translation operator-1} +\end{align} + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{align} +\tropr_{\tau lm;\tau'l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|+\left|\tau-\tau'\right|}^{l+l'}\tropcoeff_{\tau lm;\tau'l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)j_{\lambda}\left(d\right),\label{eq:translation operator} \end{align} \end_inset @@ -1936,10 +1953,27 @@ and analogously the elements of the singular operator \end_inset ) in the radial part instead of the regular bessel functions, +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout \begin_inset Formula \begin{align} \trops_{\tau lm;\tau l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|}^{l+l'}C_{lm;l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)h_{\lambda}^{(1)}\left(d\right),\nonumber \\ -\trops_{\tau lm;\tau'l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}D_{lm;l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)h_{\lambda}^{(1)}\left(d\right),\quad\tau'\ne\tau,\label{eq:translation operator singular} +\trops_{\tau lm;\tau'l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}D_{lm;l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)h_{\lambda}^{(1)}\left(d\right),\quad\tau'\ne\tau,\label{eq:translation operator singular-1} +\end{align} + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{align} +\trops_{\tau lm;\tau'l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|+\left|\tau-\tau'\right|}^{l+l'}\tropcoeff_{\tau lm;\tau'l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)h_{\lambda}^{(1)}\left(d\right),\label{eq:translation operator singular} \end{align} \end_inset @@ -2135,15 +2169,26 @@ m & -m' & m'-m \end_inset +\begin_inset Formula +\[ +\tropcoeff_{\tau lm;\tau'l'm'}^{\lambda}=\begin{cases} +A_{lm;l'm'}^{\lambda} & \tau=\tau',\\ +B_{lm;l'm'}^{\lambda} & \tau\ne\tau', +\end{cases} +\] + +\end_inset + + \begin_inset Formula \begin{multline} -C_{lm;l'm'}^{\lambda}=\left(-1\right)^{\frac{l'-l+\lambda}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\right)}}\times\\ +A_{lm;l'm'}^{\lambda}=\left(-1\right)^{\frac{l'-l+\lambda}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\right)}}\times\\ \times\begin{pmatrix}l & l' & \lambda\\ 0 & 0 & 0 \end{pmatrix}\begin{pmatrix}l & l' & \lambda\\ m & -m' & m'-m \end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right),\\ -D_{lm;l'm'}^{\lambda}=-i\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\right)}}\times\\ +B_{lm;l'm'}^{\lambda}=-i\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\right)}}\times\\ \times\begin{pmatrix}l & l' & \lambda-1\\ 0 & 0 & 0 \end{pmatrix}\begin{pmatrix}l & l' & \lambda\\ @@ -2222,16 +2267,6 @@ todo different notation for the complex conjugation without transposition??? , which has to be taken into consideration when evaluating quantities such as absorption or scattering cross sections. Similarly, the full regular operators can be composed -\begin_inset Note Note -status open - -\begin_layout Plain Layout -better wording -\end_layout - -\end_inset - - \begin_inset Formula \begin{equation} \troprp ac=\troprp ab\troprp bc\label{eq:regular translation composition} diff --git a/lepaper/infinite.lyx b/lepaper/infinite.lyx index 3c6dcae..25d2195 100644 --- a/lepaper/infinite.lyx +++ b/lepaper/infinite.lyx @@ -140,6 +140,13 @@ Topology anoyne? \begin_layout Subsection Formulation of the problem +\begin_inset CommandInset label +LatexCommand label +name "subsec:Quasiperiodic scattering problem" + +\end_inset + + \end_layout \begin_layout Standard @@ -1061,6 +1068,10 @@ W_{\alpha\beta}(\vect k)\equiv\sum_{\vect m\in\ints^{d}}\left(1-\delta_{\alpha\b \end_inset +\begin_inset Note Note +status open + +\begin_layout Plain Layout \begin_inset Formula \begin{align*} W_{\alpha,\tau lm;\beta,\tau l'm'}(\vect k) & =\sum_{\lambda=\left|l-l'\right|}^{l+l'}C_{lm;l'm'}^{\lambda}\sigma_{\lambda,m-m'}\left(\vect k,\vect r_{\beta}-\vect r_{\alpha}\right),\\ @@ -1070,6 +1081,19 @@ W_{\alpha,\tau lm;\beta,\tau'l'm'}(\vect k) & =\sum_{\lambda=\left|l-l'\right|+1 \end_inset +\end_layout + +\end_inset + + +\begin_inset Formula +\[ +W_{\alpha,\tau lm;\beta,\tau'l'm'}(\vect k)=\sum_{\lambda=\left|l-l'\right|+\left|\tau-\tau'\right|}^{l+l'}\tropcoeff_{\tau lm;\tau'l'm'}^{\lambda}\sigma_{\lambda,m-m'}\left(\vect k,\vect r_{\beta}-\vect r_{\alpha}\right),\quad\tau'\ne\tau, +\] + +\end_inset + + \begin_inset Note Note status open @@ -1913,7 +1937,8 @@ Whatabout different geometries? \end_inset - However, at greater wavelengths + However, at larger wavelengths, TODO BLA BLA BLA which is detrimental for + accuracy in floating point arithmetics. \end_layout \begin_layout Standard @@ -2015,14 +2040,14 @@ noprefix "false" Fortunately, these can be obtained easily from the expressions for the translation operator: \begin_inset Formula -\begin{align*} -\vswfrtlm{\tau}lm\left(\kappa\vect r\right) & =\sum_{m'=-1}^{1}\tropr_{\tau lm;21m'}\left(\kappa\vect r\right)\vswfrtlm 21{m'}\left(0\right),\\ -\vswfouttlm{\tau}lm\left(\kappa\vect r\right) & =\sum_{m'=-1}^{1}\trops_{\tau lm;21m'}\left(\kappa\vect r\right)\vswfrtlm 21{m'}\left(0\right), -\end{align*} +\begin{align} +\vswfrtlm{\tau}lm\left(\kappa\vect r\right) & =\sum_{m'=-1}^{1}\tropr_{\tau lm;21m'}\left(\kappa\vect r\right)\vswfrtlm 21{m'}\left(0\right),\nonumber \\ +\vswfouttlm{\tau}lm\left(\kappa\vect r\right) & =\sum_{m'=-1}^{1}\trops_{\tau lm;21m'}\left(\kappa\vect r\right)\vswfrtlm 21{m'}\left(0\right),\label{eq:VSWFs expressed as translated dipole waves} +\end{align} \end_inset -where we used eqs. +which follows from eqs. \begin_inset CommandInset ref LatexCommand eqref @@ -2048,6 +2073,102 @@ noprefix "false" \end_inset vanish at origin. + For the quasiperiodic scattering problem formulated in section +\begin_inset CommandInset ref +LatexCommand ref +reference "subsec:Quasiperiodic scattering problem" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, the total electric field scattered from all the particles at point +\begin_inset Formula $\vect r$ +\end_inset + + located outside all the particles' circumscribing sphere reads, using eqs. + +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:translation operator singular" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:sigma lattice sums" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:scalar spherical wavefunctions" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula +\begin{align} +\trops_{\tau lm;\tau'l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|+\left|\tau-\tau'\right|}^{l+l'}\tropcoeff_{\tau lm;\tau'l'm'}^{\lambda}\psi_{\lambda,m-m'}\left(\vect d\right),\label{eq:translation operator singular-1} +\end{align} + +\end_inset + + +\begin_inset Formula +\begin{equation} +\sigma_{l,m}\left(\vect k,\vect s\right)=\sum_{\vect n\in\ints^{d}}\left(1-\delta_{\vect{R_{n}},\vect s}\right)e^{i\vect{\vect k}\cdot\left(\vect R_{\vect n}-\vect s\right)}\sswfoutlm lm\left(\kappa\left(\vect{R_{n}}-\vect s\right)\right),\label{eq:sigma lattice sums} +\end{equation} + +\end_inset + + +\begin_inset Formula +\begin{align*} +\vect E_{\mathrm{scat}}\left(\vect r\right) & =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect n,\alpha}{\tau}lm\vect u_{\tau lm}\left(\kappa\left(\vect r-\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\\ + & =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lme^{i\vect k\cdot\vect R_{\vect n}}\sum_{m'=-1}^{1}\trops_{\tau lm;21m'}\left(\kappa\left(\vect r-\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\vswfrtlm 21{m'}\left(0\right)\\ + & =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lme^{-i\vect k\cdot\vect R_{\vect n}}\sum_{m'=-1}^{1}\trops_{\tau lm;21m'}\left(\kappa\left(\vect r+\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\vswfrtlm 21{m'}\left(0\right),\text{FIXME signs}\\ + & =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lme^{-i\vect k\cdot\vect R_{\vect n}}\sum_{m'=-1}^{1}\sum_{\lambda=\left|l-1\right|+\left|\tau-2\right|}^{l+1}\tropcoeff_{\tau lm;21m'}^{\lambda}\psi_{\lambda,m-m'}\left(\kappa\left(\vect r+\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\vswfrtlm 21{m'}\left(0\right)\\ + & =\sum_{\alpha\in\mathcal{P}_{1}}e^{-i\vect k\cdot\left(\vect r-\vect r_{\alpha}\right)}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lm\sum_{m'=-1}^{1}\sum_{\lambda=\left|l-1\right|+\left|\tau-2\right|}^{l+1}\tropcoeff_{\tau lm;21m'}^{\lambda}\sigma_{\lambda,m-m'}\left(\vect k,\vect r-\vect r_{\alpha}\right) +\end{align*} + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +TODO fix signs and exponential phase factors +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{align*} +\vect E_{\mathrm{scat}}\left(\vect r\right) & =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect n,\alpha}{\tau}lm\vect u_{\tau lm}\left(\kappa\left(\vect r-\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\\ + & =\sum_{\alpha\in\mathcal{P}_{1}}e^{-i\vect k\cdot\left(\vect r-\vect r_{\alpha}\right)}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lm\sum_{m'=-1}^{1}\sum_{\lambda=\left|l-1\right|+\left|\tau-2\right|}^{l+1}\tropcoeff_{\tau lm;21m'}^{\lambda}\sigma_{\lambda,m-m'}\left(\vect k,\vect r-\vect r_{\alpha}\right). +\end{align*} + +\end_inset + + \end_layout \end_body