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Replace with correct drawing 

Former-commit-id: 7a3ad9060c7adcb2a5ad5b0e52f4b1eb6ccf0e33
This commit is contained in:
Marek Nečada 2019-08-07 07:19:38 +03:00
parent 9aff527bd9
commit 9657dc613e
3 changed files with 53 additions and 27 deletions
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15
lepaper/arrayscat.lyx
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@ -790,13 +790,22 @@ noprefix "false"
\end_layout \end_layout
\begin_layout Itemize \begin_layout Itemize
Check whether everything written is correct also for non-symmorphic space
groups.
\end_layout
\begin_layout Itemize
\begin_inset Note Note
status open
\begin_layout Plain Layout
The text about symmetries is pretty dense. The text about symmetries is pretty dense.
Make it more explanatory and human-readable. Make it more explanatory and human-readable.
\end_layout \end_layout
\begin_layout Itemize \end_inset
Check whether everything written is correct also for non-symmorphic space
groups.
\end_layout \end_layout
\begin_layout Standard \begin_layout Standard

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lepaper/p6m_kpoint.png
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65
lepaper/symmetries.lyx
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@ -318,14 +318,14 @@ noprefix "false"
and the properties of the gradient operator under coordinate transforms, and the properties of the gradient operator under coordinate transforms,
vector spherical harmonics vector spherical harmonics
\begin_inset Formula $\vsh 2lm,\vsh 3lm$ \begin_inset Formula $\vsh2lm,\vsh3lm$
\end_inset \end_inset
transform in the same way, transform in the same way,
\begin_inset Formula \begin_inset Formula
\begin{align*} \begin{align*}
\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\vsh2lm\right)\left(\uvec r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh2l{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), \left(\groupop g\vsh3lm\right)\left(\uvec r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh3l{m'}\left(\uvec r\right),
\end{align*} \end{align*}
\end_inset \end_inset
@ -337,8 +337,8 @@ status open
\begin_layout Plain Layout \begin_layout Plain Layout
\begin_inset Formula \begin_inset Formula
\begin{align*} \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\vsh2lm\right)\left(\uvec r\right) & =R_{g}\vsh2lm\left(R_{g}^{-1}\uvec r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh2l{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), \left(\groupop g\vsh3lm\right)\left(\uvec r\right) & =R_{g}\vsh2lm\left(R_{g}^{-1}\uvec r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh3l{m'}\left(\uvec r\right),
\end{align*} \end{align*}
\end_inset \end_inset
@ -349,14 +349,14 @@ status open
\end_inset \end_inset
but the remaining set but the remaining set
\begin_inset Formula $\vsh 1lm$ \begin_inset Formula $\vsh1lm$
\end_inset \end_inset
transforms differently due to their pseudovector nature stemming from the transforms differently due to their pseudovector nature stemming from the
cross product in their definition: cross product in their definition:
\begin_inset Formula \begin_inset Formula
\[ \[
\left(\groupop g\vsh 1lm\right)\left(\uvec r\right)=\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vsh 1l{m'}\left(\uvec r\right), \left(\groupop g\vsh1lm\right)\left(\uvec r\right)=\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vsh1l{m'}\left(\uvec r\right),
\] \]
\end_inset \end_inset
@ -411,8 +411,8 @@ noprefix "false"
: :
\begin_inset Formula \begin_inset Formula
\begin{align*} \begin{align*}
\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\vswfouttlm1lm\right)\left(\vect r\right) & =\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vswfouttlm1l{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), \left(\groupop g\vswfouttlm2lm\right)\left(\vect r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vswfouttlm2l{m'}\left(\vect r\right),
\end{align*} \end{align*}
\end_inset \end_inset
@ -1328,8 +1328,8 @@ horizontal
the same unit cell, e.g. the same unit cell, e.g.
\begin_inset Formula \begin_inset Formula
\begin{align*} \begin{align*}
\outcoeffp{\vect0A} & \overset{\sigma_{xz}}{\longmapsto}\tilde{J}\left(\sigma_{xz}\right)\outcoeffp{\vect0E},\\ \outcoeffp{\vect 0A} & \overset{\sigma_{xz}}{\longmapsto}\tilde{J}\left(\sigma_{xz}\right)\outcoeffp{\vect 0E},\\
\outcoeff_{\vect0C} & \overset{\sigma_{xz}}{\longmapsto}\tilde{J}\left(\sigma_{xz}\right)\outcoeffp{\vect0C}, \outcoeff_{\vect 0C} & \overset{\sigma_{xz}}{\longmapsto}\tilde{J}\left(\sigma_{xz}\right)\outcoeffp{\vect 0C},
\end{align*} \end{align*}
\end_inset \end_inset
@ -1374,8 +1374,8 @@ vertical
, ,
\begin_inset Formula \begin_inset Formula
\begin{align*} \begin{align*}
\outcoeffp{\vect0A} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\left(0,1\right)E},\\ \outcoeffp{\vect 0A} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\left(0,1\right)E},\\
\outcoeff_{\vect0C} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\left(1,0\right)C}, \outcoeff_{\vect 0C} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\left(1,0\right)C},
\end{align*} \end{align*}
\end_inset \end_inset
@ -1385,22 +1385,22 @@ but we want
\end_inset \end_inset
to operate only inside one unit cell, so we use the Bloch condition to operate only inside one unit cell, so we use the Bloch condition
\begin_inset Formula $\outcoeffp{\vect n,\alpha}=\outcoeffp{\vect0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}$ \begin_inset Formula $\outcoeffp{\vect n,\alpha}=\outcoeffp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}$
\end_inset \end_inset
: in this case, we have : in this case, we have
\begin_inset Formula $\outcoeffp{\left(0,1\right)\alpha}=\outcoeffp{\vect0\alpha}e^{i\vect M_{1}\cdot\vect a_{2}}=\outcoeffp{\vect0\alpha}e^{i0}=\outcoeffp{\vect0\alpha}$ \begin_inset Formula $\outcoeffp{\left(0,1\right)\alpha}=\outcoeffp{\vect 0\alpha}e^{i\vect M_{1}\cdot\vect a_{2}}=\outcoeffp{\vect 0\alpha}e^{i0}=\outcoeffp{\vect 0\alpha}$
\end_inset \end_inset
, ,
\begin_inset Formula $\outcoeffp{\left(1,0\right)\alpha}=e^{i\vect M_{1}\cdot\vect a_{2}}\outcoeffp{\vect0\alpha}=e^{i\pi}\outcoeffp{\vect0\alpha}=-\outcoeffp{\vect0\alpha},$ \begin_inset Formula $\outcoeffp{\left(1,0\right)\alpha}=e^{i\vect M_{1}\cdot\vect a_{2}}\outcoeffp{\vect 0\alpha}=e^{i\pi}\outcoeffp{\vect 0\alpha}=-\outcoeffp{\vect 0\alpha},$
\end_inset \end_inset
so so
\begin_inset Formula \begin_inset Formula
\begin{align*} \begin{align*}
\outcoeffp{\vect0A} & \overset{\sigma_{yz}}{\longmapsto}-\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\vect0E},\\ \outcoeffp{\vect 0A} & \overset{\sigma_{yz}}{\longmapsto}-\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\vect 0E},\\
\outcoeff_{\vect0C} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\vect0C}. \outcoeff_{\vect 0C} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\vect 0C}.
\end{align*} \end{align*}
\end_inset \end_inset
@ -1439,19 +1439,19 @@ the original
rotation, as an example we have rotation, as an example we have
\begin_inset Formula \begin_inset Formula
\begin{align*} \begin{align*}
\outcoeffp{\vect0A} & \overset{C_{3}}{\longmapsto}\tilde{J}\left(C_{3}\right)\outcoeffp{\left(0,-1\right)E}=e^{2\pi i/3}\tilde{J}\left(C_{3}\right)\outcoeffp{\vect0E},\\ \outcoeffp{\vect 0A} & \overset{C_{3}}{\longmapsto}\tilde{J}\left(C_{3}\right)\outcoeffp{\left(0,-1\right)E}=e^{2\pi i/3}\tilde{J}\left(C_{3}\right)\outcoeffp{\vect 0E},\\
\outcoeff_{\vect0C} & \overset{C_{3}}{\longmapsto}\tilde{J}\left(C_{3}\right)\outcoeffp{\left(1,-1\right)A}=e^{-2\pi i/3}\tilde{J}\left(C_{3}\right)\outcoeffp{\vect0A},\\ \outcoeff_{\vect 0C} & \overset{C_{3}}{\longmapsto}\tilde{J}\left(C_{3}\right)\outcoeffp{\left(1,-1\right)A}=e^{-2\pi i/3}\tilde{J}\left(C_{3}\right)\outcoeffp{\vect 0A},\\
\outcoeff_{\vect0B} & \overset{C_{3}}{\longmapsto}\tilde{J}\left(C_{3}\right)\outcoeffp{\left(1,-1\right)B}=e^{-2\pi i/3}\tilde{J}\left(C_{3}\right)\outcoeffp{\vect0B}, \outcoeff_{\vect 0B} & \overset{C_{3}}{\longmapsto}\tilde{J}\left(C_{3}\right)\outcoeffp{\left(1,-1\right)B}=e^{-2\pi i/3}\tilde{J}\left(C_{3}\right)\outcoeffp{\vect 0B},
\end{align*} \end{align*}
\end_inset \end_inset
because in this case, the Bloch condition gives because in this case, the Bloch condition gives
\begin_inset Formula $\outcoeffp{\left(0,-1\right)\alpha}=\outcoeffp{\vect0\alpha}e^{i\vect K\cdot\left(-\vect a_{2}\right)}=\outcoeffp{\vect0\alpha}e^{-4\pi i/3}=\outcoeffp{\vect0\alpha}e^{2\pi i/3}=\outcoeffp{\vect0\alpha}$ \begin_inset Formula $\outcoeffp{\left(0,-1\right)\alpha}=\outcoeffp{\vect 0\alpha}e^{i\vect K\cdot\left(-\vect a_{2}\right)}=\outcoeffp{\vect 0\alpha}e^{-4\pi i/3}=\outcoeffp{\vect 0\alpha}e^{2\pi i/3}=\outcoeffp{\vect 0\alpha}$
\end_inset \end_inset
, ,
\begin_inset Formula $\outcoeffp{\left(1,-1\right)\alpha}=\outcoeffp{\vect0\alpha}e^{i\vect K\cdot\left(\vect a_{1}-\vect a_{2}\right)}=e^{-2\pi i/3}\outcoeffp{\vect0\alpha}.$ \begin_inset Formula $\outcoeffp{\left(1,-1\right)\alpha}=\outcoeffp{\vect 0\alpha}e^{i\vect K\cdot\left(\vect a_{1}-\vect a_{2}\right)}=e^{-2\pi i/3}\outcoeffp{\vect 0\alpha}.$
\end_inset \end_inset
@ -1463,12 +1463,13 @@ placement document
alignment document alignment document
wide false wide false
sideways false sideways false
status collapsed status open
\begin_layout Plain Layout \begin_layout Plain Layout
\align center \align center
\begin_inset Graphics \begin_inset Graphics
filename p6m_mpoint.png filename p6m_mpoint.png
lyxscale 20
width 100col% width 100col%
\end_inset \end_inset
@ -1477,8 +1478,10 @@ status collapsed
\end_layout \end_layout
\begin_layout Plain Layout \begin_layout Plain Layout
\align center
\begin_inset Graphics \begin_inset Graphics
filename p6m_kpoint.png filename p6m_kpoint.png
lyxscale 20
width 100col% width 100col%
\end_inset \end_inset
@ -1490,6 +1493,20 @@ status collapsed
\begin_inset Caption Standard \begin_inset Caption Standard
\begin_layout Plain Layout \begin_layout Plain Layout
Representing symmetry action on electromagnetic Bloch waves in a lattice
with
\begin_inset Formula $p6m$
\end_inset
wallpaper group symmetry for
\begin_inset Formula $M$
\end_inset
(top) and
\begin_inset Formula $K$
\end_inset
(bottom) points.
\begin_inset CommandInset label \begin_inset CommandInset label
LatexCommand label LatexCommand label
name "Phase factor illustration" name "Phase factor illustration"