From 36cc152166100e84935b9cfc296ba2095c02968f Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Thu, 1 Aug 2019 04:38:51 +0300 Subject: [PATCH] Point group transformation of VSWFs, t-matrices. Former-commit-id: 07695e5d1e8969a72fa9068d85ca359b4ebf4512 --- lepaper/arrayscat.lyx | 17 ++- lepaper/finite.lyx | 55 +++++--- lepaper/symmetries.lyx | 314 ++++++++++++++++++++++++++++++++++++++++- 3 files changed, 363 insertions(+), 23 deletions(-) diff --git a/lepaper/arrayscat.lyx b/lepaper/arrayscat.lyx index 7eeb2cf..94cbed1 100644 --- a/lepaper/arrayscat.lyx +++ b/lepaper/arrayscat.lyx @@ -1,5 +1,5 @@ #LyX 2.4 created this file. For more info see https://www.lyx.org/ -\lyxformat 583 +\lyxformat 584 \begin_document \begin_header \save_transient_properties true @@ -734,6 +734,21 @@ Concrete comparison with other methods. Fix and unify notation (mainly indices) in infinite lattices section. \end_layout +\begin_layout Itemize +Carefully check the transformation directions in sec. + +\begin_inset CommandInset ref +LatexCommand eqref +reference "sec:Symmetries" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + +\end_layout + \begin_layout Standard \begin_inset CommandInset include LatexCommand include diff --git a/lepaper/finite.lyx b/lepaper/finite.lyx index b2fcf41..104377f 100644 --- a/lepaper/finite.lyx +++ b/lepaper/finite.lyx @@ -1,5 +1,5 @@ #LyX 2.4 created this file. For more info see https://www.lyx.org/ -\lyxformat 583 +\lyxformat 584 \begin_document \begin_header \save_transient_properties true @@ -289,20 +289,20 @@ outgoing , respectively, defined as follows: \begin_inset Formula -\begin{align*} -\vswfrtlm 1lm\left(k\vect r\right) & =j_{l}\left(kr\right)\vsh 1lm\left(\uvec r\right),\\ -\vswfrtlm 2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krj_{l}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{j_{l}\left(kr\right)}{kr}\vsh 3lm\left(\uvec r\right), -\end{align*} +\begin{align} +\vswfrtlm 1lm\left(k\vect r\right) & =j_{l}\left(kr\right)\vsh 1lm\left(\uvec r\right),\nonumber \\ +\vswfrtlm 2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krj_{l}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{j_{l}\left(kr\right)}{kr}\vsh 3lm\left(\uvec r\right),\label{eq:VSWF regular} +\end{align} \end_inset \begin_inset Formula -\begin{align*} -\vswfouttlm 1lm\left(k\vect r\right) & =h_{l}^{\left(1\right)}\left(kr\right)\vsh 1lm\left(\uvec r\right),\\ -\vswfouttlm 2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krh_{l}^{\left(1\right)}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{h_{l}^{\left(1\right)}\left(kr\right)}{kr}\vsh 3lm\left(\uvec r\right),\\ - & \tau=1,2;\quad l=1,2,3,\dots;\quad m=-l,-l+1,\dots,+l, -\end{align*} +\begin{align} +\vswfouttlm 1lm\left(k\vect r\right) & =h_{l}^{\left(1\right)}\left(kr\right)\vsh 1lm\left(\uvec r\right),\nonumber \\ +\vswfouttlm 2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krh_{l}^{\left(1\right)}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{h_{l}^{\left(1\right)}\left(kr\right)}{kr}\vsh 3lm\left(\uvec r\right),\label{eq:VSWF outgoing}\\ + & \tau=1,2;\quad l=1,2,3,\dots;\quad m=-l,-l+1,\dots,+l,\nonumber +\end{align} \end_inset @@ -325,11 +325,11 @@ vector spherical harmonics \emph default \begin_inset Formula -\begin{align*} -\vsh 1lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\times\left(\vect r\ush lm\left(\uvec r\right)\right)=\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\ush lm\left(\uvec r\right)\times\vect r,\\ -\vsh 2lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}r\nabla\ush lm\left(\uvec r\right),\\ -\vsh 3lm\left(\uvec r\right) & =\uvec r\ush lm\left(\uvec r\right). -\end{align*} +\begin{align} +\vsh 1lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\times\left(\vect r\ush lm\left(\uvec r\right)\right)=\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\ush lm\left(\uvec r\right)\times\vect r,\nonumber \\ +\vsh 2lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}r\nabla\ush lm\left(\uvec r\right),\nonumber \\ +\vsh 3lm\left(\uvec r\right) & =\uvec r\ush lm\left(\uvec r\right).\label{eq:vector spherical harmonics definition} +\end{align} \end_inset @@ -517,7 +517,7 @@ doplnit frekvence a polohy \begin_inset Formula \begin{equation} -\vect E\left(\omega,\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\left(\rcoefftlm{\tau}lm\vswfrtlm{\tau}lm+\outcoefftlm{\tau}lm\vswfouttlm{\tau}lm\right).\label{eq:E field expansion} +\vect E\left(\omega,\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\left(\rcoefftlm{\tau}lm\vswfrtlm{\tau}lm\left(k\vect r\right)+\outcoefftlm{\tau}lm\vswfouttlm{\tau}lm\left(k\vect r\right)\right).\label{eq:E field expansion} \end{equation} \end_inset @@ -603,14 +603,27 @@ noprefix "false" \end_inset are the effective induced electric ( -\begin_inset Formula $\tau=1$ -\end_inset - -) and magnetic ( \begin_inset Formula $\tau=2$ \end_inset -) multipole polarisation amplitudes of the scatterer. +) and magnetic ( +\begin_inset Formula $\tau=1$ +\end_inset + +) multipole polarisation amplitudes of the scatterer, and this is why we + sometimes refer to the corresponding VSWFs as the electric and magnetic + VSWFs. + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +TODO mention the pseudovector character of magnetic VSWFs. +\end_layout + +\end_inset + + \end_layout \begin_layout Standard diff --git a/lepaper/symmetries.lyx b/lepaper/symmetries.lyx index 27527c1..aac8e5e 100644 --- a/lepaper/symmetries.lyx +++ b/lepaper/symmetries.lyx @@ -1,5 +1,5 @@ #LyX 2.4 created this file. For more info see https://www.lyx.org/ -\lyxformat 583 +\lyxformat 584 \begin_document \begin_header \save_transient_properties true @@ -183,6 +183,318 @@ noprefix "false" Finite systems \end_layout +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout +TODO Zkontrolovat všechny vzorečky zde!!! +\end_layout + +\end_inset + +In order to use the point group symmetries, we first need to know how they + affect our basis functions, i.e. + the VSWFs. +\end_layout + +\begin_layout Standard +Let +\begin_inset Formula $g$ +\end_inset + + be a member of orthogonal group +\begin_inset Formula $O(3)$ +\end_inset + +, i.e. + a 3D point rotation or reflection operation that transforms vectors in + +\begin_inset Formula $\reals^{3}$ +\end_inset + + with an orthogonal matrix +\begin_inset Formula $R_{g}$ +\end_inset + +: +\begin_inset Formula +\[ +\vect r\mapsto R_{g}\vect r. +\] + +\end_inset + +Spherical harmonics +\begin_inset Formula $\ush lm$ +\end_inset + +, being a basis the +\begin_inset Formula $l$ +\end_inset + +-dimensional representation of +\begin_inset Formula $O(3)$ +\end_inset + +, transform as +\begin_inset CommandInset citation +LatexCommand cite +after "???" +key "dresselhaus_group_2008" +literal "false" + +\end_inset + + +\begin_inset Formula +\[ +\ush lm\left(R_{g}\uvec r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\ush l{m'}\left(\uvec r\right) +\] + +\end_inset + +where +\begin_inset Formula $D_{m,m'}^{l}\left(g\right)$ +\end_inset + + denotes the elements of the +\emph on +Wigner matrix +\emph default + representing the operation +\begin_inset Formula $g$ +\end_inset + +. + By their definition, 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), +\end{align*} + +\end_inset + +but the remaining set +\begin_inset Formula $\vsh 1lm$ +\end_inset + + transforms differently due to their pseudovector nature stemming from the + cross product in their definition: +\begin_inset Formula +\[ +\vsh 3lm\left(R_{g}\uvec r\right)=\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vsh 3l{m'}\left(\uvec r\right), +\] + +\end_inset + +where +\begin_inset Formula $\widetilde{D_{m,m'}^{l}}\left(g\right)=D_{m,m'}^{l}\left(g\right)$ +\end_inset + + if +\begin_inset Formula $g$ +\end_inset + + is a proper rotation, but for spatial inversion operation +\begin_inset Formula $i:\vect r\mapsto-\vect r$ +\end_inset + + we have +\begin_inset Formula $\widetilde{D_{m,m'}^{l}}\left(i\right)=\left(-1\right)^{l+m}D_{m,m'}^{l}\left(i\right)$ +\end_inset + +. + The transformation behaviour of vector spherical harmonics directly propagates + to the spherical vector waves, cf. + +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:VSWF regular" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:VSWF outgoing" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +: +\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), +\end{align*} + +\end_inset + +(and analogously for the regular waves +\begin_inset Formula $\vswfrtlm{\tau}lm$ +\end_inset + +). + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +TODO víc obdivu. +\end_layout + +\end_inset + + For convenience, we introduce the symbol +\begin_inset Formula $D_{m,m'}^{\tau l}$ +\end_inset + + that describes the transformation of both types ( +\begin_inset Quotes eld +\end_inset + +magnetic +\begin_inset Quotes erd +\end_inset + + and +\begin_inset Quotes eld +\end_inset + +electric +\begin_inset Quotes erd +\end_inset + +) of waves at once: +\begin_inset Formula +\[ +\vswfouttlm{\tau}lm\left(R_{g}\vect r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(\vect r\right). +\] + +\end_inset + +Using these, we can express the VSWF expansion +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:E field expansion" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + of the electric field around origin in a rotated/reflected system, +\begin_inset Formula +\[ +\vect E\left(\omega,R_{g}\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfrtlm{\tau}lm\left(k\vect r\right)+D_{m,m'}^{\tau l}\left(g\right)\outcoefftlm{\tau}lm\vswfouttlm{\tau}lm\left(k\vect r\right)\right), +\] + +\end_inset + +which, together with the +\begin_inset Formula $T$ +\end_inset + +-matrix definition, +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:T-matrix definition" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + can be used to obtain a +\begin_inset Formula $T$ +\end_inset + +-matrix of a rotated or mirror-reflected particle. + Let +\begin_inset Formula $T$ +\end_inset + + be the +\begin_inset Formula $T$ +\end_inset + +-matrix of an original particle; the +\begin_inset Formula $T$ +\end_inset + +-matrix of a particle physically transformed by operation +\begin_inset Formula $g\in O(3)$ +\end_inset + + is then +\begin_inset Note Note +status open + +\begin_layout Plain Layout +check sides +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +T'_{\tau lm;\tau'l'm'}=\sum_{\mu=-l}^{l}\sum_{\mu'=-l'}^{l'}\left(D_{\mu,m}^{\tau l}\left(g\right)\right)^{*}T_{\tau l\mu;\tau'l'm'}D_{m',\mu'}^{\tau l}\left(g\right).\label{eq:T-matrix of a transformed particle} +\end{equation} + +\end_inset + +If the particle is symmetric (so that +\begin_inset Formula $g$ +\end_inset + + produces a particle indistinguishable from the original one), the +\begin_inset Formula $T$ +\end_inset + +-matrix must remain invariant under the transformation +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:T-matrix of a transformed particle" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, +\begin_inset Formula $T'_{\tau lm;\tau'l'm'}=T{}_{\tau lm;\tau'l'm'}$ +\end_inset + +. + Explicit forms of these invariance properties for the most imporant point + group symmetries can be found in +\begin_inset CommandInset citation +LatexCommand cite +key "schulz_point-group_1999" +literal "false" + +\end_inset + +. +\end_layout + +\begin_layout Standard +With these point group transformation properties in hand, we can proceed + to rotating (or mirror-reflecting) the whole many-particle system. +\end_layout + \begin_layout Subsection Periodic systems \end_layout