From 15013e6764d118844536ce9917bb90e19b6ecbda Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= <marek.necada@aalto.fi>
Date: Tue, 15 Aug 2017 16:15:20 +0300
Subject: [PATCH] [ewald] dudom

Former-commit-id: 911eb634ab3d609e8d008a66a4e5d7de2353c954
---
 ewald.lyx | 217 ++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 217 insertions(+)
 create mode 100644 ewald.lyx

diff --git a/ewald.lyx b/ewald.lyx
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@@ -0,0 +1,217 @@
+#LyX 2.1 created this file. For more info see http://www.lyx.org/
+\lyxformat 474
+\begin_document
+\begin_header
+\textclass article
+\begin_preamble
+\usepackage{unicode-math}
+
+% Toto je trik, jimž se z fontspec získá familyname pro následující
+\ExplSyntaxOn
+\DeclareExpandableDocumentCommand{\getfamilyname}{m}
+ {
+  \use:c { g__fontspec_ \cs_to_str:N #1 _family }
+ }
+\ExplSyntaxOff
+
+% definujeme novou rodinu, jež se volá pomocí \MyCyr pro běžné použití, avšak pro účely \DeclareSymbolFont je nutno získat název pomocí getfamilyname definovaného výše
+\newfontfamily\MyCyr{CMU Serif}
+
+\DeclareSymbolFont{cyritletters}{EU1}{\getfamilyname\MyCyr}{m}{it}
+\newcommand{\makecyrmathletter}[1]{%
+  \begingroup\lccode`a=#1\lowercase{\endgroup
+  \Umathcode`a}="0 \csname symcyritletters\endcsname\space #1
+}
+\count255="409
+\loop\ifnum\count255<"44F
+  \advance\count255 by 1
+  \makecyrmathletter{\count255}
+\repeat
+
+\renewcommand{\lyxmathsym}[1]{#1}
+\end_preamble
+\use_default_options true
+\maintain_unincluded_children false
+\language english
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman TeX Gyre Pagella
+\font_sans default
+\font_typewriter default
+\font_math default
+\font_default_family default
+\use_non_tex_fonts true
+\font_sc false
+\font_osf true
+\font_sf_scale 100
+\font_tt_scale 100
+\graphics default
+\default_output_format pdf4
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref true
+\pdf_title "Accelerating lattice mode calculations with T-matrix method"
+\pdf_author "Marek Nečada"
+\pdf_bookmarks true
+\pdf_bookmarksnumbered false
+\pdf_bookmarksopen false
+\pdf_bookmarksopenlevel 1
+\pdf_breaklinks false
+\pdf_pdfborder false
+\pdf_colorlinks false
+\pdf_backref false
+\pdf_pdfusetitle true
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\quotes_language english
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Title
+\begin_inset FormulaMacro
+\newcommand{\vect}[1]{\mathbf{#1}}
+\end_inset
+
+Accelerating lattice mode calculations with 
+\begin_inset Formula $T$
+\end_inset
+
+-matrix method
+\end_layout
+
+\begin_layout Author
+Marek Nečada
+\end_layout
+
+\begin_layout Section
+Formulation of the problem
+\end_layout
+
+\begin_layout Standard
+Assume a system of compact EM scatterers in otherwise homogeneous and isotropic
+ medium, and assume that the system, i.e.
+ both the medium and the scatterers, have linear response.
+ A scattering problem in such system can be written as
+\begin_inset Formula 
+\[
+A_{α}=T_{α}P_{α}=T_{α}(\sum_{β}S_{α\leftarrowβ}A_{β}+P_{0α})
+\]
+
+\end_inset
+
+where 
+\begin_inset Formula $T_{α}$
+\end_inset
+
+ is the 
+\begin_inset Formula $T$
+\end_inset
+
+-matrix for scatterer α, 
+\begin_inset Formula $A_{α}$
+\end_inset
+
+ is its vector of the scattered wave expansion coefficient (the multipole
+ indices are not explicitely indicated here) and 
+\begin_inset Formula $P_{α}$
+\end_inset
+
+ is the local expansion of the incoming sources.
+ 
+\begin_inset Formula $S_{α\leftarrowβ}$
+\end_inset
+
+ is ...
+ and ...
+ is ...
+\end_layout
+
+\begin_layout Standard
+...
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\sum_{β}(\delta_{αβ}-T_{α}S_{α\leftarrowβ})A_{β}=T_{α}P_{0α}.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Now suppose that the scatterers constitute an infinite lattice
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{\vect aα}S_{\vect aα\leftarrow\vect bβ})A_{\vect bβ}=T_{\vect aα}P_{0\vect aα}.
+\]
+
+\end_inset
+
+Due to the periodicity, we can write 
+\begin_inset Formula $S_{\vect aα\leftarrow\vect bβ}=S_{α\leftarrowβ}(\vect b-\vect a)$
+\end_inset
+
+.
+ In order to find lattice modes, we search for solutions with zero RHS
+\begin_inset Formula 
+\[
+\sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{\vect aα}S_{\vect aα\leftarrow\vect bβ})A_{\vect bβ}=0
+\]
+
+\end_inset
+
+and we assume periodic solution 
+\begin_inset Formula $A_{\vect b\alpha}(\vect k)=A_{\vect a\alpha}e^{i\vect k\cdot\vect r_{\vect b-\vect a}}$
+\end_inset
+
+.
+\end_layout
+
+\end_body
+\end_document