Improving the text on symmetries, WIP figs

Former-commit-id: 49e6eabb7188f98308c1b6d1661dbd0b89a79a71

lepaper/finite.lyx

@@ -306,8 +306,8 @@ outgoing
 , respectively, defined as follows:
 \begin_inset Formula 
 \begin{align}
-\vswfrtlm 1lm\left(\kappa\vect r\right) & =j_{l}\left(\kappa r\right)\vsh 1lm\left(\uvec r\right),\nonumber \\
+\vswfrtlm1lm\left(\kappa\vect r\right) & =j_{l}\left(\kappa r\right)\vsh1lm\left(\uvec r\right),\nonumber \\
-\vswfrtlm 2lm\left(\kappa\vect r\right) & =\frac{1}{\kappa r}\frac{\ud\left(\kappa rj_{l}\left(\kappa r\right)\right)}{\ud\left(\kappa r\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{j_{l}\left(\kappa r\right)}{\kappa r}\vsh 3lm\left(\uvec r\right),\label{eq:VSWF regular}
+\vswfrtlm2lm\left(\kappa\vect r\right) & =\frac{1}{\kappa r}\frac{\ud\left(\kappa rj_{l}\left(\kappa r\right)\right)}{\ud\left(\kappa r\right)}\vsh2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{j_{l}\left(\kappa r\right)}{\kappa r}\vsh3lm\left(\uvec r\right),\label{eq:VSWF regular}
 \end{align}
 
 \end_inset
@@ -315,8 +315,8 @@ outgoing
 
 \begin_inset Formula 
 \begin{align}
-\vswfouttlm 1lm\left(\kappa\vect r\right) & =h_{l}^{\left(1\right)}\left(\kappa r\right)\vsh 1lm\left(\uvec r\right),\nonumber \\
+\vswfouttlm1lm\left(\kappa\vect r\right) & =h_{l}^{\left(1\right)}\left(\kappa r\right)\vsh1lm\left(\uvec r\right),\nonumber \\
-\vswfouttlm 2lm\left(\kappa\vect r\right) & =\frac{1}{kr}\frac{\ud\left(\kappa rh_{l}^{\left(1\right)}\left(\kappa r\right)\right)}{\ud\left(\kappa r\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{h_{l}^{\left(1\right)}\left(\kappa r\right)}{\kappa r}\vsh 3lm\left(\uvec r\right),\label{eq:VSWF outgoing}\\
+\vswfouttlm2lm\left(\kappa\vect r\right) & =\frac{1}{kr}\frac{\ud\left(\kappa rh_{l}^{\left(1\right)}\left(\kappa r\right)\right)}{\ud\left(\kappa r\right)}\vsh2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{h_{l}^{\left(1\right)}\left(\kappa r\right)}{\kappa r}\vsh3lm\left(\uvec r\right),\label{eq:VSWF outgoing}\\
  & \tau=1,2;\quad l=1,2,3,\dots;\quad m=-l,-l+1,\dots,+l,\nonumber 
 \end{align}
 
@@ -387,9 +387,9 @@ vector spherical harmonics
 
 \begin_inset Formula 
 \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,\nonumber \\
+\vsh1lm\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),\nonumber \\
+\vsh2lm\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).\label{eq:vector spherical harmonics definition}
+\vsh3lm\left(\uvec r\right) & =\uvec r\ush lm\left(\uvec r\right).\label{eq:vector spherical harmonics definition}
 \end{align}
 
 \end_inset
@@ -408,7 +408,7 @@ electric dipolar
 \end_inset
 
  waves 
-\begin_inset Formula $\vswfrtlm 21m$
+\begin_inset Formula $\vswfrtlm21m$
 \end_inset
 
 , they vanish.
@@ -605,7 +605,7 @@ noprefix "false"
 \end_inset
 
  inside a ball 
-\begin_inset Formula $\openball{R^{>}}{\vect 0}$
+\begin_inset Formula $\openball{R^{>}}{\vect0}$
 \end_inset
 
  with radius 
@@ -615,7 +615,7 @@ noprefix "false"
  and center in the origin, were it filled with homogeneous isotropic medium;
  however, if the equation is not guaranteed to hold inside a smaller ball
  
-\begin_inset Formula $\closedball{R^{<}}{\vect 0}$
+\begin_inset Formula $\closedball{R^{<}}{\vect0}$
 \end_inset
 
  around the origin (typically due to presence of a scatterer), one has to
@@ -624,7 +624,7 @@ noprefix "false"
 \end_inset
 
  to have a complete basis of the solutions in the volume 
-\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}=\openball{R^{>}}{\vect 0}\setminus\closedball{R^{<}}{\vect 0}$
+\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect0}=\openball{R^{>}}{\vect0}\setminus\closedball{R^{<}}{\vect0}$
 \end_inset
 
 .
@@ -1541,6 +1541,34 @@ where
 
 matrices as the off-diagonal blocks, whereas the diagonal blocks are set
  to zeros.
+ 
+\end_layout
+
+\begin_layout Standard
+We note that eq.
+ 
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "eq:Multiple-scattering problem block form"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+ with zero right-hand side describes the normal modes of the system; the
+ methods mentioned later in Section 
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "sec:Infinite"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+ for solving the band structure of a periodic system can be used as well
+ for finding the resonant frequencies of a finite system.
 \end_layout
 
 \begin_layout Standard

lepaper/symmetries.lyx

@@ -92,9 +92,8 @@ name "sec:Symmetries"
 
 \begin_layout Standard
 If the system has nontrivial point group symmetries, group theory gives
- additional understanding of the system properties, and can be used to reduce
+ additional understanding of the system properties, and can be used to substanti
- the computational costs.
+ally reduce the computational costs.
- 
 \end_layout
 
 \begin_layout Standard
@@ -175,8 +174,21 @@ noprefix "false"
 \end_inset
 
  matrix, not the linear algebra.
- However, the lattice modes can be searched for in each irrep separately,
+ However, decomposition of the lattice mode problem 
- and the irrep dimension gives a priori information about mode degeneracy.
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "eq:lattice mode equation"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+ into the irreducible representations of the corresponding little co-groups
+ of the system's space group is nevertheless a useful tool in the mode analysis:
+ among other things, it enables separation of the lattice modes (which can
+ then be searched for each irrep separately), and the irrep dimension gives
+ a priori information about mode degeneracy.
 \end_layout
 
 \begin_layout Subsection
@@ -193,8 +205,8 @@ TODO Zkontrolovat všechny vzorečky zde!!!
 
 \end_inset
 
-In order to use the point group symmetries, we first need to know how they
+In order to make use of the point group symmetries, we first need to know
- affect our basis functions, i.e.
+ how they affect our basis functions, i.e.
  the VSWFs.
 \end_layout
 
@@ -1170,8 +1182,7 @@ Periodic systems
 \end_layout
 
 \begin_layout Standard
-For periodic systems, we can in similar manner also block-diagonalise the
+Also for periodic systems, 
- 
 \begin_inset Formula $M\left(\omega,\vect k\right)=\left(I-T\left(\omega\right)W\left(\omega,\vect k\right)\right)$
 \end_inset
 
@@ -1196,7 +1207,7 @@ noprefix "false"
 
 \end_inset
 
-.
+ can be block-diagonalised in a similar manner.
  Hovewer, in this case, 
 \begin_inset Formula $W\left(\omega,\vect k\right)$
 \end_inset
@@ -1218,25 +1229,29 @@ s happens unless
 
  lies somewhere in the high-symmetry parts of the Brillouin zone.
  However, the high-symmetry points are usually the ones of the highest physical
- interest, for it is where the band edges 
+ interest, for it is where the band edges are typically located.
+ The subsection does not aim for an exhaustive treatment of the topic of
+ space groups in physics (which can be found elsewhere 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "dresselhaus_group_2008,bradley_mathematical_1972"
+literal "false"
+
+\end_inset
+
+), here we rather demonstrate how the group action matrices are generated
+ on a specific example of a symmorphic space group.
+ 
 \begin_inset Note Note
 status open
 
 \begin_layout Plain Layout
-or 
+better formulation
-\begin_inset Quotes eld
-\end_inset
-
-dirac points
-\begin_inset Quotes erd
-\end_inset
-
-
 \end_layout
 
 \end_inset
 
- are typically located.
+
 \end_layout
 
 \begin_layout Standard
@@ -1468,6 +1483,36 @@ because in this case, the Bloch condition gives
 
 \end_layout
 
+\begin_layout Standard
+Having the group action matrices, we can construct the projectors and decompose
+ the system into irreducible representations of the corresponding point
+ groups analogously to the finite case 
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "eq:SAB unitary transformation operator"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+.
+ This procedure can be repeated for any system with a symmorphic space group
+ symmetry, where the translation and point group operations are essentially
+ separable.
+ For systems with non-symmorphic space group symmetries (i.e.
+ those with glide reflection planes or screw rotation axes) a more refined
+ approach is required 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "bradley_mathematical_1972,dresselhaus_group_2008"
+literal "false"
+
+\end_inset
+
+.
+\end_layout
+
 \begin_layout Standard
 \begin_inset Float figure
 placement document
@@ -1537,20 +1582,6 @@ name "Phase factor illustration"
 
 \end_layout
 
-\begin_layout Standard
-More rigorous analysis can be found e.g.
- in 
-\begin_inset CommandInset citation
-LatexCommand cite
-after "chapters 10–11"
-key "dresselhaus_group_2008"
-literal "true"
-
-\end_inset
-
-.
-\end_layout
-
 \begin_layout Standard
 \begin_inset Note Note
 status open