Replace with correct drawing
Former-commit-id: 7a3ad9060c7adcb2a5ad5b0e52f4b1eb6ccf0e33
This commit is contained in:
Marek Nečada
parent
9aff527bd9
commit
9657dc613e
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@ -790,13 +790,22 @@ noprefix "false"
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\end_layout
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\begin_layout Itemize
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Check whether everything written is correct also for non-symmorphic space
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groups.
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\end_layout
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\begin_layout Itemize
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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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The text about symmetries is pretty dense.
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Make it more explanatory and human-readable.
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\end_layout
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\begin_layout Itemize
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Check whether everything written is correct also for non-symmorphic space
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groups.
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\end_inset
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\end_layout
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\begin_layout Standard
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@ -318,14 +318,14 @@ noprefix "false"
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and the properties of the gradient operator under coordinate transforms,
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vector spherical harmonics
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\begin_inset Formula $\vsh 2lm,\vsh 3lm$
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\begin_inset Formula $\vsh2lm,\vsh3lm$
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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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\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),\\
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\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),
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\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),\\
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\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),
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\end{align*}
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\end_inset
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@ -337,8 +337,8 @@ status open
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\begin_layout Plain Layout
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\begin_inset Formula
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\begin{align*}
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\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),\\
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\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),
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\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),\\
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\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),
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\end{align*}
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\end_inset
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@ -349,14 +349,14 @@ status open
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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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\begin_inset Formula $\vsh1lm$
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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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\[
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\left(\groupop g\vsh 1lm\right)\left(\uvec r\right)=\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vsh 1l{m'}\left(\uvec r\right),
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\left(\groupop g\vsh1lm\right)\left(\uvec r\right)=\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vsh1l{m'}\left(\uvec r\right),
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\]
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\end_inset
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@ -411,8 +411,8 @@ noprefix "false"
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:
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\begin_inset Formula
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\begin{align*}
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\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),\\
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\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),
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\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),\\
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\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),
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\end{align*}
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\end_inset
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@ -1328,8 +1328,8 @@ horizontal
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the same unit cell, e.g.
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\begin_inset Formula
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\begin{align*}
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\outcoeffp{\vect0A} & \overset{\sigma_{xz}}{\longmapsto}\tilde{J}\left(\sigma_{xz}\right)\outcoeffp{\vect0E},\\
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\outcoeff_{\vect0C} & \overset{\sigma_{xz}}{\longmapsto}\tilde{J}\left(\sigma_{xz}\right)\outcoeffp{\vect0C},
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\outcoeffp{\vect 0A} & \overset{\sigma_{xz}}{\longmapsto}\tilde{J}\left(\sigma_{xz}\right)\outcoeffp{\vect 0E},\\
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\outcoeff_{\vect 0C} & \overset{\sigma_{xz}}{\longmapsto}\tilde{J}\left(\sigma_{xz}\right)\outcoeffp{\vect 0C},
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\end{align*}
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\end_inset
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@ -1374,8 +1374,8 @@ vertical
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,
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\begin_inset Formula
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\begin{align*}
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\outcoeffp{\vect0A} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\left(0,1\right)E},\\
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\outcoeff_{\vect0C} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\left(1,0\right)C},
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\outcoeffp{\vect 0A} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\left(0,1\right)E},\\
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\outcoeff_{\vect 0C} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\left(1,0\right)C},
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\end{align*}
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\end_inset
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@ -1385,22 +1385,22 @@ but we want
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\end_inset
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to operate only inside one unit cell, so we use the Bloch condition
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\begin_inset Formula $\outcoeffp{\vect n,\alpha}=\outcoeffp{\vect0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}$
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\begin_inset Formula $\outcoeffp{\vect n,\alpha}=\outcoeffp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}$
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\end_inset
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: in this case, we have
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\begin_inset Formula $\outcoeffp{\left(0,1\right)\alpha}=\outcoeffp{\vect0\alpha}e^{i\vect M_{1}\cdot\vect a_{2}}=\outcoeffp{\vect0\alpha}e^{i0}=\outcoeffp{\vect0\alpha}$
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\begin_inset Formula $\outcoeffp{\left(0,1\right)\alpha}=\outcoeffp{\vect 0\alpha}e^{i\vect M_{1}\cdot\vect a_{2}}=\outcoeffp{\vect 0\alpha}e^{i0}=\outcoeffp{\vect 0\alpha}$
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\end_inset
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,
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\begin_inset Formula $\outcoeffp{\left(1,0\right)\alpha}=e^{i\vect M_{1}\cdot\vect a_{2}}\outcoeffp{\vect0\alpha}=e^{i\pi}\outcoeffp{\vect0\alpha}=-\outcoeffp{\vect0\alpha},$
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\begin_inset Formula $\outcoeffp{\left(1,0\right)\alpha}=e^{i\vect M_{1}\cdot\vect a_{2}}\outcoeffp{\vect 0\alpha}=e^{i\pi}\outcoeffp{\vect 0\alpha}=-\outcoeffp{\vect 0\alpha},$
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\end_inset
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so
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\begin_inset Formula
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\begin{align*}
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\outcoeffp{\vect0A} & \overset{\sigma_{yz}}{\longmapsto}-\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\vect0E},\\
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\outcoeff_{\vect0C} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\vect0C}.
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\outcoeffp{\vect 0A} & \overset{\sigma_{yz}}{\longmapsto}-\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\vect 0E},\\
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\outcoeff_{\vect 0C} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\vect 0C}.
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\end{align*}
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\end_inset
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@ -1439,19 +1439,19 @@ the original
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rotation, as an example we have
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\begin_inset Formula
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\begin{align*}
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\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},\\
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\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},\\
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\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},
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\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},\\
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\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},\\
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\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},
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\end{align*}
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\end_inset
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because in this case, the Bloch condition gives
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\begin_inset Formula $\outcoeffp{\left(0,-1\right)\alpha}=\outcoeffp{\vect0\alpha}e^{i\vect K\cdot\left(-\vect a_{2}\right)}=\outcoeffp{\vect0\alpha}e^{-4\pi i/3}=\outcoeffp{\vect0\alpha}e^{2\pi i/3}=\outcoeffp{\vect0\alpha}$
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\begin_inset Formula $\outcoeffp{\left(0,-1\right)\alpha}=\outcoeffp{\vect 0\alpha}e^{i\vect K\cdot\left(-\vect a_{2}\right)}=\outcoeffp{\vect 0\alpha}e^{-4\pi i/3}=\outcoeffp{\vect 0\alpha}e^{2\pi i/3}=\outcoeffp{\vect 0\alpha}$
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\end_inset
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,
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\begin_inset Formula $\outcoeffp{\left(1,-1\right)\alpha}=\outcoeffp{\vect0\alpha}e^{i\vect K\cdot\left(\vect a_{1}-\vect a_{2}\right)}=e^{-2\pi i/3}\outcoeffp{\vect0\alpha}.$
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\begin_inset Formula $\outcoeffp{\left(1,-1\right)\alpha}=\outcoeffp{\vect 0\alpha}e^{i\vect K\cdot\left(\vect a_{1}-\vect a_{2}\right)}=e^{-2\pi i/3}\outcoeffp{\vect 0\alpha}.$
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\end_inset
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@ -1463,12 +1463,13 @@ placement document
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alignment document
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wide false
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sideways false
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status collapsed
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status open
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\begin_layout Plain Layout
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\align center
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\begin_inset Graphics
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filename p6m_mpoint.png
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lyxscale 20
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width 100col%
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\end_inset
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@ -1477,8 +1478,10 @@ status collapsed
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\end_layout
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\begin_layout Plain Layout
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\align center
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\begin_inset Graphics
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filename p6m_kpoint.png
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lyxscale 20
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width 100col%
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\end_inset
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@ -1490,6 +1493,20 @@ status collapsed
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\begin_inset Caption Standard
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\begin_layout Plain Layout
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Representing symmetry action on electromagnetic Bloch waves in a lattice
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with
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\begin_inset Formula $p6m$
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\end_inset
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wallpaper group symmetry for
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\begin_inset Formula $M$
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\end_inset
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(top) and
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\begin_inset Formula $K$
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\end_inset
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(bottom) points.
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\begin_inset CommandInset label
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LatexCommand label
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name "Phase factor illustration"
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