diff --git a/lepaper/arrayscat.lyx b/lepaper/arrayscat.lyx index 9861623..e71697a 100644 --- a/lepaper/arrayscat.lyx +++ b/lepaper/arrayscat.lyx @@ -790,13 +790,22 @@ noprefix "false" \end_layout \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. Make it more explanatory and human-readable. \end_layout -\begin_layout Itemize -Check whether everything written is correct also for non-symmorphic space - groups. +\end_inset + + \end_layout \begin_layout Standard diff --git a/lepaper/p6m_kpoint.png b/lepaper/p6m_kpoint.png index 125f46a..4147b9e 100644 Binary files a/lepaper/p6m_kpoint.png and b/lepaper/p6m_kpoint.png differ diff --git a/lepaper/symmetries.lyx b/lepaper/symmetries.lyx index de25f9c..cfc2c8d 100644 --- a/lepaper/symmetries.lyx +++ b/lepaper/symmetries.lyx @@ -318,14 +318,14 @@ noprefix "false" and the properties of the gradient operator under coordinate transforms, vector spherical harmonics -\begin_inset Formula $\vsh 2lm,\vsh 3lm$ +\begin_inset Formula $\vsh2lm,\vsh3lm$ \end_inset transform in the same way, \begin_inset Formula \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\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\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\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_inset @@ -337,8 +337,8 @@ 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), +\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\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_inset @@ -349,14 +349,14 @@ status open \end_inset but the remaining set -\begin_inset Formula $\vsh 1lm$ +\begin_inset Formula $\vsh1lm$ \end_inset transforms differently due to their pseudovector nature stemming from the cross product in their definition: \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 @@ -411,8 +411,8 @@ noprefix "false" : \begin_inset Formula \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\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\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\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_inset @@ -1328,8 +1328,8 @@ horizontal the same unit cell, e.g. \begin_inset Formula \begin{align*} -\outcoeffp{\vect0A} & \overset{\sigma_{xz}}{\longmapsto}\tilde{J}\left(\sigma_{xz}\right)\outcoeffp{\vect0E},\\ -\outcoeff_{\vect0C} & \overset{\sigma_{xz}}{\longmapsto}\tilde{J}\left(\sigma_{xz}\right)\outcoeffp{\vect0C}, +\outcoeffp{\vect 0A} & \overset{\sigma_{xz}}{\longmapsto}\tilde{J}\left(\sigma_{xz}\right)\outcoeffp{\vect 0E},\\ +\outcoeff_{\vect 0C} & \overset{\sigma_{xz}}{\longmapsto}\tilde{J}\left(\sigma_{xz}\right)\outcoeffp{\vect 0C}, \end{align*} \end_inset @@ -1374,8 +1374,8 @@ vertical , \begin_inset Formula \begin{align*} -\outcoeffp{\vect0A} & \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}, +\outcoeffp{\vect 0A} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\left(0,1\right)E},\\ +\outcoeff_{\vect 0C} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\left(1,0\right)C}, \end{align*} \end_inset @@ -1385,22 +1385,22 @@ but we want \end_inset 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 : 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 , -\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 so \begin_inset Formula \begin{align*} -\outcoeffp{\vect0A} & \overset{\sigma_{yz}}{\longmapsto}-\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\vect0E},\\ -\outcoeff_{\vect0C} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\vect0C}. +\outcoeffp{\vect 0A} & \overset{\sigma_{yz}}{\longmapsto}-\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\vect 0E},\\ +\outcoeff_{\vect 0C} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\vect 0C}. \end{align*} \end_inset @@ -1439,19 +1439,19 @@ the original rotation, as an example we have \begin_inset Formula \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},\\ -\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_{\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}, +\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_{\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_{\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_inset 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 , -\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 @@ -1463,12 +1463,13 @@ placement document alignment document wide false sideways false -status collapsed +status open \begin_layout Plain Layout \align center \begin_inset Graphics filename p6m_mpoint.png + lyxscale 20 width 100col% \end_inset @@ -1477,8 +1478,10 @@ status collapsed \end_layout \begin_layout Plain Layout +\align center \begin_inset Graphics filename p6m_kpoint.png + lyxscale 20 width 100col% \end_inset @@ -1490,6 +1493,20 @@ status collapsed \begin_inset Caption Standard \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 LatexCommand label name "Phase factor illustration"