diff --git a/lepaper/figs/hex/pokus.svg b/lepaper/figs/hex/pokus.svg index 303c3aa..5372eed 100644 --- a/lepaper/figs/hex/pokus.svg +++ b/lepaper/figs/hex/pokus.svg @@ -14,13 +14,83 @@ height="1056" width="816" xml:space="preserve" - inkscape:version="0.92.4 (5da689c313, 2019-01-14)" + inkscape:version="0.91 r13725" version="1.1" - id="svg4457">image/svg+xml \ No newline at end of file + d="m -191.73685,426.22846 310.48774,0" + style="fill:#ff00ff;fill-rule:evenodd;stroke:#ff00ff;stroke-width:0.59999996;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:2.39999987, 1.19999994;stroke-dashoffset:0;stroke-opacity:1" + sodipodi:nodetypes="cc" />Phase factors \ No newline at end of file diff --git a/lepaper/finite.lyx b/lepaper/finite.lyx index 59a6ca5..e0c3e90 100644 --- a/lepaper/finite.lyx +++ b/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 \\ -\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} +\vswfrtlm1lm\left(\kappa\vect r\right) & =j_{l}\left(\kappa r\right)\vsh1lm\left(\uvec r\right),\nonumber \\ +\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 \\ -\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}\\ +\vswfouttlm1lm\left(\kappa\vect r\right) & =h_{l}^{\left(1\right)}\left(\kappa r\right)\vsh1lm\left(\uvec r\right),\nonumber \\ +\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 \\ -\vsh 2lm\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} +\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 \\ +\vsh2lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}r\nabla\ush lm\left(\uvec r\right),\nonumber \\ +\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 diff --git a/lepaper/symmetries.lyx b/lepaper/symmetries.lyx index fb3e723..d4fa818 100644 --- a/lepaper/symmetries.lyx +++ b/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 - the computational costs. - + additional understanding of the system properties, and can be used to substanti +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, - and the irrep dimension gives a priori information about mode degeneracy. + However, decomposition of the lattice mode problem +\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 - affect our basis functions, i.e. +In order to make use of the point group symmetries, we first need to know + 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 -\begin_inset Quotes eld -\end_inset - -dirac points -\begin_inset Quotes erd -\end_inset - - +better formulation \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