Point group transformation of VSWFs, t-matrices.

Former-commit-id: 07695e5d1e8969a72fa9068d85ca359b4ebf4512
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Marek Nečada 2019-08-01 04:38:51 +03:00
parent 91e2ae9e4d
commit 36cc152166
3 changed files with 363 additions and 23 deletions

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#LyX 2.4 created this file. For more info see https://www.lyx.org/ #LyX 2.4 created this file. For more info see https://www.lyx.org/
\lyxformat 583 \lyxformat 584
\begin_document \begin_document
\begin_header \begin_header
\save_transient_properties true \save_transient_properties true
@ -734,6 +734,21 @@ Concrete comparison with other methods.
Fix and unify notation (mainly indices) in infinite lattices section. Fix and unify notation (mainly indices) in infinite lattices section.
\end_layout \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_layout Standard
\begin_inset CommandInset include \begin_inset CommandInset include
LatexCommand include LatexCommand include

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#LyX 2.4 created this file. For more info see https://www.lyx.org/ #LyX 2.4 created this file. For more info see https://www.lyx.org/
\lyxformat 583 \lyxformat 584
\begin_document \begin_document
\begin_header \begin_header
\save_transient_properties true \save_transient_properties true
@ -289,20 +289,20 @@ outgoing
, respectively, defined as follows: , respectively, defined as follows:
\begin_inset Formula \begin_inset Formula
\begin{align*} \begin{align}
\vswfrtlm 1lm\left(k\vect r\right) & =j_{l}\left(kr\right)\vsh 1lm\left(\uvec r\right),\\ \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), \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{align}
\end_inset \end_inset
\begin_inset Formula \begin_inset Formula
\begin{align*} \begin{align}
\vswfouttlm 1lm\left(k\vect r\right) & =h_{l}^{\left(1\right)}\left(kr\right)\vsh 1lm\left(\uvec r\right),\\ \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),\\ \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, & \tau=1,2;\quad l=1,2,3,\dots;\quad m=-l,-l+1,\dots,+l,\nonumber
\end{align*} \end{align}
\end_inset \end_inset
@ -325,11 +325,11 @@ vector spherical harmonics
\emph default \emph default
\begin_inset Formula \begin_inset Formula
\begin{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,\\ \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),\\ \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). \vsh 3lm\left(\uvec r\right) & =\uvec r\ush lm\left(\uvec r\right).\label{eq:vector spherical harmonics definition}
\end{align*} \end{align}
\end_inset \end_inset
@ -517,7 +517,7 @@ doplnit frekvence a polohy
\begin_inset Formula \begin_inset Formula
\begin{equation} \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{equation}
\end_inset \end_inset
@ -603,14 +603,27 @@ noprefix "false"
\end_inset \end_inset
are the effective induced electric ( are the effective induced electric (
\begin_inset Formula $\tau=1$
\end_inset
) and magnetic (
\begin_inset Formula $\tau=2$ \begin_inset Formula $\tau=2$
\end_inset \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 \end_layout
\begin_layout Standard \begin_layout Standard

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#LyX 2.4 created this file. For more info see https://www.lyx.org/ #LyX 2.4 created this file. For more info see https://www.lyx.org/
\lyxformat 583 \lyxformat 584
\begin_document \begin_document
\begin_header \begin_header
\save_transient_properties true \save_transient_properties true
@ -183,6 +183,318 @@ noprefix "false"
Finite systems Finite systems
\end_layout \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 \begin_layout Subsection
Periodic systems Periodic systems
\end_layout \end_layout