From 055ad7a1be69190c0d93f2769b46773250b74792 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Mon, 5 Aug 2019 18:32:09 +0300 Subject: [PATCH] Review of the finite multiple scattering group action part. Former-commit-id: ad6e51190c52daa35522daa7654937f642161f60 --- lepaper/arrayscat.lyx | 5 + lepaper/symmetries.lyx | 212 ++++++++++++++++++++++++++++++----------- 2 files changed, 161 insertions(+), 56 deletions(-) diff --git a/lepaper/arrayscat.lyx b/lepaper/arrayscat.lyx index dae1cde..d2c3f23 100644 --- a/lepaper/arrayscat.lyx +++ b/lepaper/arrayscat.lyx @@ -399,6 +399,11 @@ status open \end_inset +\begin_inset FormulaMacro +\newcommand{\groupop}[1]{\hat{P}_{#1}} +\end_inset + + \end_layout \begin_layout Standard diff --git a/lepaper/symmetries.lyx b/lepaper/symmetries.lyx index 540de85..c7f11f2 100644 --- a/lepaper/symmetries.lyx +++ b/lepaper/symmetries.lyx @@ -203,8 +203,8 @@ Let \begin_inset Formula $g$ \end_inset - be a member of orthogonal group -\begin_inset Formula $O(3)$ + be a member of the orthogonal group +\begin_inset Formula $\mathrm{O}(3)$ \end_inset , i.e. @@ -225,16 +225,45 @@ Let \end_inset +With +\begin_inset Formula $\groupop g$ +\end_inset + + we shall denote the action of +\begin_inset Formula $g$ +\end_inset + + on a field in real space. + For a scalar field +\begin_inset Formula $w$ +\end_inset + + we have +\begin_inset Formula $\left(\groupop gw\right)\left(\vect r\right)=w\left(R_{g}^{-1}\vect r\right)$ +\end_inset + +, whereas for a vector field +\begin_inset Formula $\vect w$ +\end_inset + +, +\begin_inset Formula $\left(\groupop g\vect w\right)\left(\vect r\right)=R_{g}\vect w\left(R_{g}^{-1}\vect r\right)$ +\end_inset + +. +\end_layout + +\begin_layout Standard Spherical harmonics \begin_inset Formula $\ush lm$ \end_inset -, being a basis the +, being a basis of the \begin_inset Formula $l$ \end_inset -dimensional representation of -\begin_inset Formula $O(3)$ +\begin_inset Formula $\mathrm{O}(3)$ \end_inset , transform as @@ -249,7 +278,7 @@ literal "false" \begin_inset Formula \[ -\ush lm\left(R_{g}\uvec r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\ush l{m'}\left(\uvec r\right) +\left(\groupop g\ush lm\right)\left(\uvec r\right)=\ush lm\left(R_{g}^{-1}\uvec r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\ush l{m'}\left(\uvec r\right) \] \end_inset @@ -260,26 +289,65 @@ where denotes the elements of the \emph on -Wigner matrix +Wigner matrix +\begin_inset Note Note +status open + +\begin_layout Plain Layout +TODO explicit formulation +\end_layout + +\end_inset + + \emph default representing the operation \begin_inset Formula $g$ \end_inset . - By their definition, vector spherical harmonics + From their definitions +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:vector spherical harmonics definition" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + and the properties of the gradient operator under coordinate transforms, + vector spherical harmonics \begin_inset Formula $\vsh 2lm,\vsh 3lm$ \end_inset transform in the same way, \begin_inset Formula \begin{align*} -\vsh 2lm\left(R_{g}\uvec r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh 2l{m'}\left(\uvec r\right),\\ -\vsh 3lm\left(R_{g}\uvec r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh 3l{m'}\left(\uvec r\right), +\left(\groupop g\vsh 2lm\right)\left(\uvec r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh 2l{m'}\left(\uvec r\right),\\ +\left(\groupop g\vsh 3lm\right)\left(\uvec r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh 3l{m'}\left(\uvec r\right), \end{align*} \end_inset + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula +\begin{align*} +\left(\groupop g\vsh 2lm\right)\left(\uvec r\right) & =R_{g}\vsh 2lm\left(R_{g}^{-1}\uvec r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh 2l{m'}\left(\uvec r\right),\\ +\left(\groupop g\vsh 3lm\right)\left(\uvec r\right) & =R_{g}\vsh 2lm\left(R_{g}^{-1}\uvec r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh 3l{m'}\left(\uvec r\right), +\end{align*} + +\end_inset + + +\end_layout + +\end_inset + but the remaining set \begin_inset Formula $\vsh 1lm$ \end_inset @@ -288,7 +356,7 @@ but the remaining set cross product in their definition: \begin_inset Formula \[ -\vsh 3lm\left(R_{g}\uvec r\right)=\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vsh 3l{m'}\left(\uvec r\right), +\left(\groupop g\vsh 1lm\right)\left(\uvec r\right)=\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vsh 1l{m'}\left(\uvec r\right), \] \end_inset @@ -301,12 +369,20 @@ where \begin_inset Formula $g$ \end_inset - is a proper rotation, but for spatial inversion operation + is a proper rotation, +\begin_inset Formula $g\in\mathrm{SO(3)}$ +\end_inset + +, but for spatial inversion operation \begin_inset Formula $i:\vect r\mapsto-\vect r$ \end_inset we have -\begin_inset Formula $\widetilde{D_{m,m'}^{l}}\left(i\right)=\left(-1\right)^{l+m}D_{m,m'}^{l}\left(i\right)$ +\begin_inset Formula $D_{m,m'}^{l}\left(i\right)=\left(-1\right)^{l}$ +\end_inset + + but +\begin_inset Formula $\widetilde{D_{m,m'}^{l}}\left(i\right)=\left(-1\right)^{l+1}$ \end_inset . @@ -335,8 +411,8 @@ noprefix "false" : \begin_inset Formula \begin{align*} -\vswfouttlm 1lm\left(R_{g}\vect r\right) & =\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vswfouttlm 1l{m'}\left(\vect r\right),\\ -\vswfouttlm 2lm\left(R_{g}\vect r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vswfouttlm 2l{m'}\left(\vect r\right), +\left(\groupop g\vswfouttlm 1lm\right)\left(\vect r\right) & =\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vswfouttlm 1l{m'}\left(\vect r\right),\\ +\left(\groupop g\vswfouttlm 2lm\right)\left(\vect r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vswfouttlm 2l{m'}\left(\vect r\right), \end{align*} \end_inset @@ -360,7 +436,7 @@ TODO víc obdivu. \begin_inset Formula $D_{m,m'}^{\tau l}$ \end_inset - that describes the transformation of both types ( + that describes the transformation of both ( \begin_inset Quotes eld \end_inset @@ -376,10 +452,10 @@ electric \begin_inset Quotes erd \end_inset -) of waves at once: +) types of waves at once: \begin_inset Formula \[ -\vswfouttlm{\tau}lm\left(R_{g}\vect r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(\vect r\right). +\groupop g\vswfouttlm{\tau}lm\left(\vect r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(\vect r\right). \] \end_inset @@ -397,7 +473,7 @@ noprefix "false" of the electric field around origin in a rotated/reflected system, \begin_inset Formula \[ -\vect E\left(\omega,R_{g}\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\vect r\right)+\outcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\vect r\right)\right), +\left(\groupop g\vect E\right)\left(\omega,\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\vect r\right)+\outcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\vect r\right)\right), \] \end_inset @@ -505,10 +581,10 @@ noprefix "false" \end_inset -, we have +, we have (CHECK THIS CAREFULLY AND EXPLAIN) \begin_inset Formula \begin{multline} -\vect E\left(\omega,R_{g}\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m',\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-R_{g}\vect r_{p}\right)\right)\right.+\\ +\left(\groupop g\vect E\right)\left(\omega,\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m',\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-R_{g}\vect r_{p}\right)\right)\right.+\\ +\left.\outcoeffptlm p{\tau}lmD_{m',\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-R_{g}\vect r_{p}\right)\right)\right).\label{eq:rotated E field expansion around outside origin} \end{multline} @@ -642,8 +718,56 @@ With these transformation properties in hand, we can proceed to the effects \begin_inset Formula $p\in\mathcal{P}$ \end_inset +; their positions transform as +\begin_inset Formula $\vect r_{\pi_{g}p}=R_{g}\vect r_{p}$ +\end_inset + +, +\begin_inset Formula $\vect r_{\pi_{g}^{-1}p}=R_{g}^{-1}\vect r_{p}$ +\end_inset + . - For a given particle + In the symmetric multiple-scattering problem, transforming the whole field + according to +\begin_inset Formula $g$ +\end_inset + +, in terms of field expansion around a particle originally labelled as +\begin_inset Formula $p$ +\end_inset + + +\begin_inset Formula +\begin{align*} +\left(\groupop g\vect E\right)\left(\omega,\vect r\right) & =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-R_{g}\vect r_{p}\right)\right)\right.+\\ + & \quad+\left.\outcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-R_{g}\vect r_{p}\right)\right)\right)\\ + & =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{\pi_{g}p}\right)\right)\right.\\ + & \quad+\left.\outcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{\pi_{g}p}\right)\right)\right)\\ + & =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm{\pi_{g}^{-1}q}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{q}\right)\right)\right.\\ + & \quad+\left.\outcoeffptlm{\pi_{g}^{-1}q}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{q}\right)\right)\right) +\end{align*} + +\end_inset + +In the last step, we relabeled +\begin_inset Formula $q=\pi_{g}p$ +\end_inset + +. + This means that the field expansion coefficients +\begin_inset Formula $\rcoeffp p,\outcoeffp p$ +\end_inset + + transform as +\begin_inset Formula +\begin{align} +\rcoeffptlm p{\tau}lm & \overset{g}{\longmapsto}\rcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right),\nonumber \\ +\outcoeffptlm p{\tau}lm & \overset{g}{\longmapsto}\outcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right).\label{eq:excitation coefficient under symmetry operation} +\end{align} + +\end_inset + +For a given particle \begin_inset Formula $p$ \end_inset @@ -683,43 +807,19 @@ noprefix "false" \end_inset . - The importance of the particle orbits stems from the following: in the - multiple-scattering problem, outside of the scatterers -\begin_inset Note Note -status open - -\begin_layout Plain Layout -< FIXME -\end_layout + The importance of the particle orbits stems from fact that the expansion + coefficients belonging to particles in different orbits are not related + together under the group action in +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:excitation coefficient under symmetry operation" +plural "false" +caps "false" +noprefix "false" \end_inset - one has -\begin_inset Formula -\begin{align} -\vect E\left(\omega,R_{g}\vect r\right) & =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{\pi_{g}(p)}\right)\right)\right.+\label{eq:rotated E field expansion around outside origin-1}\\ - & \quad+\left.\outcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{p}\right)\right)\right)\\ - & =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{p}\right)\right)\right.+\\ - & \quad+\left.\outcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{p}\right)\right)\right). -\end{align} - -\end_inset - -This means that the field expansion coefficients -\begin_inset Formula $\rcoeffp p,\outcoeffp p$ -\end_inset - - transform as -\begin_inset Formula -\begin{align} -\rcoeffptlm p{\tau}lm & \overset{g}{\longmapsto}\rcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right),\nonumber \\ -\outcoeffptlm p{\tau}lm & \overset{g}{\longmapsto}\outcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right).\label{eq:excitation coefficient under symmetry operation} -\end{align} - -\end_inset - -Obviously, the expansion coefficients belonging to particles in different - orbits do not mix together. +. As before, we introduce a short-hand pairwise matrix notation for \begin_inset CommandInset ref LatexCommand eqref @@ -730,7 +830,7 @@ noprefix "false" \end_inset - (TODO avoid notation clash here in a more consistent and readable way! + (TODO avoid notation clash here in a more consistent and readable way!) \begin_inset Formula \begin{align} \rcoeffp p & \overset{g}{\longmapsto}\tilde{J}\left(g\right)\rcoeffp{\pi_{g}^{-1}(p)},\nonumber \\