Improving the text on symmetries, WIP figs
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Marek Nečada
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@ -306,8 +306,8 @@ 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(\kappa\vect r\right) & =j_{l}\left(\kappa r\right)\vsh 1lm\left(\uvec r\right),\nonumber \\
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\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}
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\vswfrtlm1lm\left(\kappa\vect r\right) & =j_{l}\left(\kappa r\right)\vsh1lm\left(\uvec r\right),\nonumber \\
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\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}
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\end{align}
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\end_inset
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@ -315,8 +315,8 @@ outgoing
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\begin_inset Formula
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\begin{align}
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\vswfouttlm 1lm\left(\kappa\vect r\right) & =h_{l}^{\left(1\right)}\left(\kappa r\right)\vsh 1lm\left(\uvec r\right),\nonumber \\
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\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}\\
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\vswfouttlm1lm\left(\kappa\vect r\right) & =h_{l}^{\left(1\right)}\left(\kappa r\right)\vsh1lm\left(\uvec r\right),\nonumber \\
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\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}\\
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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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@ -387,9 +387,9 @@ vector spherical harmonics
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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,\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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\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 \\
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\vsh2lm\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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\vsh3lm\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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@ -408,7 +408,7 @@ electric dipolar
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\end_inset
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waves
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\begin_inset Formula $\vswfrtlm 21m$
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\begin_inset Formula $\vswfrtlm21m$
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\end_inset
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, they vanish.
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@ -605,7 +605,7 @@ noprefix "false"
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\end_inset
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inside a ball
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\begin_inset Formula $\openball{R^{>}}{\vect 0}$
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\begin_inset Formula $\openball{R^{>}}{\vect0}$
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\end_inset
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with radius
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@ -615,7 +615,7 @@ noprefix "false"
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and center in the origin, were it filled with homogeneous isotropic medium;
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however, if the equation is not guaranteed to hold inside a smaller ball
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\begin_inset Formula $\closedball{R^{<}}{\vect 0}$
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\begin_inset Formula $\closedball{R^{<}}{\vect0}$
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\end_inset
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around the origin (typically due to presence of a scatterer), one has to
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@ -624,7 +624,7 @@ noprefix "false"
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\end_inset
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to have a complete basis of the solutions in the volume
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\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}=\openball{R^{>}}{\vect 0}\setminus\closedball{R^{<}}{\vect 0}$
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\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect0}=\openball{R^{>}}{\vect0}\setminus\closedball{R^{<}}{\vect0}$
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\end_inset
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.
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@ -1541,6 +1541,34 @@ where
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matrices as the off-diagonal blocks, whereas the diagonal blocks are set
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to zeros.
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\end_layout
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\begin_layout Standard
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We note that eq.
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Multiple-scattering problem block form"
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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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with zero right-hand side describes the normal modes of the system; the
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methods mentioned later in Section
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "sec:Infinite"
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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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for solving the band structure of a periodic system can be used as well
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for finding the resonant frequencies of a finite system.
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\end_layout
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\begin_layout Standard
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@ -92,9 +92,8 @@ name "sec:Symmetries"
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\begin_layout Standard
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If the system has nontrivial point group symmetries, group theory gives
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additional understanding of the system properties, and can be used to reduce
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the computational costs.
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additional understanding of the system properties, and can be used to substanti
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ally reduce the computational costs.
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\end_layout
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\begin_layout Standard
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@ -175,8 +174,21 @@ noprefix "false"
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\end_inset
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matrix, not the linear algebra.
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However, the lattice modes can be searched for in each irrep separately,
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and the irrep dimension gives a priori information about mode degeneracy.
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However, decomposition of the lattice mode problem
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:lattice mode equation"
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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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into the irreducible representations of the corresponding little co-groups
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of the system's space group is nevertheless a useful tool in the mode analysis:
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among other things, it enables separation of the lattice modes (which can
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then be searched for each irrep separately), and the irrep dimension gives
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a priori information about mode degeneracy.
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\end_layout
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\begin_layout Subsection
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@ -193,8 +205,8 @@ TODO Zkontrolovat všechny vzorečky zde!!!
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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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In order to make use of the point group symmetries, we first need to know
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how they affect our basis functions, i.e.
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the VSWFs.
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\end_layout
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@ -1170,8 +1182,7 @@ Periodic systems
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\end_layout
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\begin_layout Standard
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For periodic systems, we can in similar manner also block-diagonalise the
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Also for periodic systems,
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\begin_inset Formula $M\left(\omega,\vect k\right)=\left(I-T\left(\omega\right)W\left(\omega,\vect k\right)\right)$
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\end_inset
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@ -1196,7 +1207,7 @@ noprefix "false"
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\end_inset
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.
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can be block-diagonalised in a similar manner.
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Hovewer, in this case,
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\begin_inset Formula $W\left(\omega,\vect k\right)$
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\end_inset
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@ -1218,25 +1229,29 @@ s happens unless
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lies somewhere in the high-symmetry parts of the Brillouin zone.
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However, the high-symmetry points are usually the ones of the highest physical
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interest, for it is where the band edges
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interest, for it is where the band edges are typically located.
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The subsection does not aim for an exhaustive treatment of the topic of
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space groups in physics (which can be found elsewhere
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\begin_inset CommandInset citation
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LatexCommand cite
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key "dresselhaus_group_2008,bradley_mathematical_1972"
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literal "false"
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\end_inset
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), here we rather demonstrate how the group action matrices are generated
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on a specific example of a symmorphic space group.
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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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or
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\begin_inset Quotes eld
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\end_inset
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dirac points
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\begin_inset Quotes erd
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\end_inset
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better formulation
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\end_layout
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\end_inset
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are typically located.
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\end_layout
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\begin_layout Standard
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@ -1468,6 +1483,36 @@ because in this case, the Bloch condition gives
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\end_layout
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\begin_layout Standard
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Having the group action matrices, we can construct the projectors and decompose
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the system into irreducible representations of the corresponding point
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groups analogously to the finite case
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:SAB unitary transformation operator"
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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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This procedure can be repeated for any system with a symmorphic space group
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symmetry, where the translation and point group operations are essentially
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separable.
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For systems with non-symmorphic space group symmetries (i.e.
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those with glide reflection planes or screw rotation axes) a more refined
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approach is required
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\begin_inset CommandInset citation
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LatexCommand cite
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key "bradley_mathematical_1972,dresselhaus_group_2008"
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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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\begin_inset Float figure
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placement document
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@ -1537,20 +1582,6 @@ name "Phase factor illustration"
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\end_layout
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\begin_layout Standard
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More rigorous analysis can be found e.g.
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in
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\begin_inset CommandInset citation
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LatexCommand cite
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after "chapters 10–11"
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key "dresselhaus_group_2008"
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literal "true"
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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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\begin_inset Note Note
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status open
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