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Review of the finite multiple scattering group action part. 

Former-commit-id: ad6e51190c52daa35522daa7654937f642161f60
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Marek Nečada 2019-08-05 18:32:09 +03:00
parent 306cb1bef8
commit 055ad7a1be
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5
lepaper/arrayscat.lyx
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@ -399,6 +399,11 @@ status open
\end_inset
\begin_inset FormulaMacro
\newcommand{\groupop}[1]{\hat{P}_{#1}}
\end_inset
\end_layout
\begin_layout Standard

208
lepaper/symmetries.lyx
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@ -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
@ -261,25 +290,64 @@ where
denotes the elements of the
\emph on
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}
@ -642,7 +718,55 @@ 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
.
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 \\