Point group transformation of VSWFs, t-matrices.
Former-commit-id: 07695e5d1e8969a72fa9068d85ca359b4ebf4512
This commit is contained in:
Marek Nečada
parent
91e2ae9e4d
commit
36cc152166
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#LyX 2.4 created this file. For more info see https://www.lyx.org/
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\lyxformat 583
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\lyxformat 584
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\begin_document
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\begin_header
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\save_transient_properties true
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@ -734,6 +734,21 @@ Concrete comparison with other methods.
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Fix and unify notation (mainly indices) in infinite lattices section.
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\end_layout
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\begin_layout Itemize
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Carefully check the transformation directions in sec.
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "sec:Symmetries"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset CommandInset include
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LatexCommand include
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@ -1,5 +1,5 @@
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#LyX 2.4 created this file. For more info see https://www.lyx.org/
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\lyxformat 583
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\lyxformat 584
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\begin_document
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\begin_header
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\save_transient_properties true
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@ -289,20 +289,20 @@ outgoing
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, respectively, defined as follows:
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\begin_inset Formula
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\begin{align*}
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\vswfrtlm 1lm\left(k\vect r\right) & =j_{l}\left(kr\right)\vsh 1lm\left(\uvec r\right),\\
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\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),
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\end{align*}
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\begin{align}
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\vswfrtlm 1lm\left(k\vect r\right) & =j_{l}\left(kr\right)\vsh 1lm\left(\uvec r\right),\nonumber \\
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\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}
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\end{align}
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\end_inset
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\begin_inset Formula
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\begin{align*}
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\vswfouttlm 1lm\left(k\vect r\right) & =h_{l}^{\left(1\right)}\left(kr\right)\vsh 1lm\left(\uvec r\right),\\
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\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),\\
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& \tau=1,2;\quad l=1,2,3,\dots;\quad m=-l,-l+1,\dots,+l,
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\end{align*}
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\begin{align}
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\vswfouttlm 1lm\left(k\vect r\right) & =h_{l}^{\left(1\right)}\left(kr\right)\vsh 1lm\left(\uvec r\right),\nonumber \\
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\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}\\
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& \tau=1,2;\quad l=1,2,3,\dots;\quad m=-l,-l+1,\dots,+l,\nonumber
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\end{align}
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\end_inset
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@ -325,11 +325,11 @@ vector spherical harmonics
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\emph default
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\begin_inset Formula
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\begin{align*}
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\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,\\
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\vsh 2lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}r\nabla\ush lm\left(\uvec r\right),\\
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\vsh 3lm\left(\uvec r\right) & =\uvec r\ush lm\left(\uvec r\right).
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\end{align*}
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\begin{align}
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\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 \\
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\vsh 2lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}r\nabla\ush lm\left(\uvec r\right),\nonumber \\
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\vsh 3lm\left(\uvec r\right) & =\uvec r\ush lm\left(\uvec r\right).\label{eq:vector spherical harmonics definition}
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\end{align}
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\end_inset
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@ -517,7 +517,7 @@ doplnit frekvence a polohy
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\begin_inset Formula
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\begin{equation}
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\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}
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\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}
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\end{equation}
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\end_inset
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@ -603,14 +603,27 @@ noprefix "false"
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\end_inset
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are the effective induced electric (
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\begin_inset Formula $\tau=1$
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\end_inset
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) and magnetic (
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\begin_inset Formula $\tau=2$
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\end_inset
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) multipole polarisation amplitudes of the scatterer.
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) and magnetic (
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\begin_inset Formula $\tau=1$
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\end_inset
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) multipole polarisation amplitudes of the scatterer, and this is why we
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sometimes refer to the corresponding VSWFs as the electric and magnetic
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VSWFs.
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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TODO mention the pseudovector character of magnetic VSWFs.
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Standard
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@ -1,5 +1,5 @@
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#LyX 2.4 created this file. For more info see https://www.lyx.org/
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\lyxformat 583
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\lyxformat 584
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\begin_document
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\begin_header
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\save_transient_properties true
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@ -183,6 +183,318 @@ noprefix "false"
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Finite systems
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\end_layout
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\begin_layout Standard
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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TODO Zkontrolovat všechny vzorečky zde!!!
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\end_layout
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\end_inset
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In order to use the point group symmetries, we first need to know how they
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affect our basis functions, i.e.
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the VSWFs.
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\end_layout
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\begin_layout Standard
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Let
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\begin_inset Formula $g$
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\end_inset
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be a member of orthogonal group
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\begin_inset Formula $O(3)$
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\end_inset
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, i.e.
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a 3D point rotation or reflection operation that transforms vectors in
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\begin_inset Formula $\reals^{3}$
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\end_inset
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with an orthogonal matrix
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\begin_inset Formula $R_{g}$
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\end_inset
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:
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\begin_inset Formula
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\[
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\vect r\mapsto R_{g}\vect r.
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\]
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\end_inset
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Spherical harmonics
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\begin_inset Formula $\ush lm$
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\end_inset
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, being a basis the
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\begin_inset Formula $l$
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\end_inset
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-dimensional representation of
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\begin_inset Formula $O(3)$
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\end_inset
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, transform as
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\begin_inset CommandInset citation
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LatexCommand cite
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after "???"
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key "dresselhaus_group_2008"
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literal "false"
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\end_inset
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\begin_inset Formula
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\[
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\ush lm\left(R_{g}\uvec r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\ush l{m'}\left(\uvec r\right)
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\]
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\end_inset
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where
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\begin_inset Formula $D_{m,m'}^{l}\left(g\right)$
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\end_inset
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denotes the elements of the
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\emph on
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Wigner matrix
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\emph default
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representing the operation
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\begin_inset Formula $g$
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\end_inset
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.
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By their definition, vector spherical harmonics
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\begin_inset Formula $\vsh 2lm,\vsh 3lm$
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\end_inset
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transform in the same way,
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\begin_inset Formula
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\begin{align*}
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\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),\\
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\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),
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\end{align*}
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\end_inset
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but the remaining set
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\begin_inset Formula $\vsh 1lm$
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\end_inset
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transforms differently due to their pseudovector nature stemming from the
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cross product in their definition:
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\begin_inset Formula
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\[
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\vsh 3lm\left(R_{g}\uvec r\right)=\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vsh 3l{m'}\left(\uvec r\right),
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\]
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\end_inset
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where
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\begin_inset Formula $\widetilde{D_{m,m'}^{l}}\left(g\right)=D_{m,m'}^{l}\left(g\right)$
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\end_inset
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if
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\begin_inset Formula $g$
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\end_inset
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is a proper rotation, but for spatial inversion operation
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\begin_inset Formula $i:\vect r\mapsto-\vect r$
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\end_inset
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we have
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\begin_inset Formula $\widetilde{D_{m,m'}^{l}}\left(i\right)=\left(-1\right)^{l+m}D_{m,m'}^{l}\left(i\right)$
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\end_inset
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.
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The transformation behaviour of vector spherical harmonics directly propagates
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to the spherical vector waves, cf.
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:VSWF regular"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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,
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:VSWF outgoing"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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:
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\begin_inset Formula
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\begin{align*}
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\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),\\
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\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),
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\end{align*}
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\end_inset
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(and analogously for the regular waves
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\begin_inset Formula $\vswfrtlm{\tau}lm$
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\end_inset
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).
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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TODO víc obdivu.
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\end_layout
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\end_inset
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For convenience, we introduce the symbol
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\begin_inset Formula $D_{m,m'}^{\tau l}$
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\end_inset
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that describes the transformation of both types (
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\begin_inset Quotes eld
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\end_inset
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magnetic
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\begin_inset Quotes erd
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\end_inset
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and
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\begin_inset Quotes eld
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\end_inset
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electric
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\begin_inset Quotes erd
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\end_inset
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) of waves at once:
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\begin_inset Formula
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\[
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\vswfouttlm{\tau}lm\left(R_{g}\vect r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(\vect r\right).
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\]
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\end_inset
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Using these, we can express the VSWF expansion
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:E field expansion"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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of the electric field around origin in a rotated/reflected system,
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\begin_inset Formula
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\[
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\vect E\left(\omega,R_{g}\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfrtlm{\tau}lm\left(k\vect r\right)+D_{m,m'}^{\tau l}\left(g\right)\outcoefftlm{\tau}lm\vswfouttlm{\tau}lm\left(k\vect r\right)\right),
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\]
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\end_inset
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which, together with the
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\begin_inset Formula $T$
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\end_inset
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-matrix definition,
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:T-matrix definition"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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can be used to obtain a
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\begin_inset Formula $T$
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\end_inset
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-matrix of a rotated or mirror-reflected particle.
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Let
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\begin_inset Formula $T$
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\end_inset
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be the
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\begin_inset Formula $T$
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\end_inset
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-matrix of an original particle; the
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\begin_inset Formula $T$
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\end_inset
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-matrix of a particle physically transformed by operation
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\begin_inset Formula $g\in O(3)$
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\end_inset
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is then
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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check sides
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\end_layout
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\end_inset
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\begin_inset Formula
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\begin{equation}
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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}
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\end{equation}
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\end_inset
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If the particle is symmetric (so that
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\begin_inset Formula $g$
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\end_inset
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produces a particle indistinguishable from the original one), the
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\begin_inset Formula $T$
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\end_inset
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-matrix must remain invariant under the transformation
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:T-matrix of a transformed particle"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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,
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\begin_inset Formula $T'_{\tau lm;\tau'l'm'}=T{}_{\tau lm;\tau'l'm'}$
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\end_inset
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.
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Explicit forms of these invariance properties for the most imporant point
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group symmetries can be found in
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\begin_inset CommandInset citation
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LatexCommand cite
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key "schulz_point-group_1999"
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literal "false"
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\end_inset
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.
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\end_layout
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\begin_layout Standard
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With these point group transformation properties in hand, we can proceed
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to rotating (or mirror-reflecting) the whole many-particle system.
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\end_layout
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\begin_layout Subsection
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Periodic systems
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\end_layout
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