diff --git a/notes/hexlattice_kpoint_projections.lyx b/notes/hexlattice_kpoint_projections.lyx new file mode 100644 index 0000000..cf39545 --- /dev/null +++ b/notes/hexlattice_kpoint_projections.lyx @@ -0,0 +1,355 @@ +#LyX 2.1 created this file. For more info see http://www.lyx.org/ +\lyxformat 474 +\begin_document +\begin_header +\textclass article +\use_default_options true +\maintain_unincluded_children false +\language finnish +\language_package default +\inputencoding auto +\fontencoding global +\font_roman TeX Gyre Pagella +\font_sans default +\font_typewriter default +\font_math auto +\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 "Sähköpajan päiväkirja" +\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 swedish +\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 +Symmetry-adapted basis functions for honeycomb lattice at +\begin_inset Formula $K$ +\end_inset + +-point +\end_layout + +\begin_layout Section +Generation theorem +\end_layout + +\begin_layout Standard +Let +\begin_inset Formula $\mathbf{G}$ +\end_inset + + be a group and +\begin_inset Formula $\Gamma^{i}\left\{ R\to\mathbf{D}^{i}\left(R\right)\right\} $ +\end_inset + + some +\begin_inset Formula $d_{i}$ +\end_inset + +-dimensional rep of +\begin_inset Formula $\mathbf{G}$ +\end_inset + +. + Let the group ring (corresponding to the given rep indexed by +\begin_inset Formula $i$ +\end_inset + +) elements be defined as [Bradley&Cracknell (2.2.2)] +\begin_inset Formula +\[ +W_{ts}^{i}=\frac{d_{i}}{\left|\mathbf{G}\right|}\sum_{R\in\mathbf{G}}\mathbf{D}^{i}\left(R\right)_{ts}^{*}R. +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +From [Bradley&Cracknell, theorem 2.2.1]: +\end_layout + +\begin_layout Standard +If +\begin_inset Formula $\phi$ +\end_inset + + is an arbitrary function of +\begin_inset Formula $V$ +\end_inset + + (a linear space in which the realisation of the group operation act) such + that +\begin_inset Formula $W_{ss}^{i}\phi\ne0$ +\end_inset + + ( +\begin_inset Formula $s$ +\end_inset + + is fixed and is a number in the range 1 to +\begin_inset Formula $d_{i}$ +\end_inset + +; +\begin_inset Formula $i$ +\end_inset + + is idx of the rep) then the funs +\begin_inset Formula $W_{ts}^{i}\phi=\phi_{t}^{i}$ +\end_inset + +, +\begin_inset Formula $t=1$ +\end_inset + + to +\begin_inset Formula $d_{i}$ +\end_inset + +, form a basis for the rep +\begin_inset Formula $\Gamma^{i}$ +\end_inset + +. +\end_layout + +\begin_layout Section +Particle-centered transformations +\end_layout + +\begin_layout Standard +Now let's see what are the point group actions on SVWF in the origin [Schulz]: +\end_layout + +\begin_layout Standard +\begin_inset Tabular + + + + + + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $Z$ +\end_inset + +-axis rotation by +\begin_inset Formula $2\pi/N$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $C_{N}M_{l}^{m}=e^{\pm?i2\pi m/N}M_{l}^{m}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $C_{N}N_{l}^{m}=e^{\pm?i2\pi m/N}N_{l}^{m}$ +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +Horizontal (xy) reflection +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\sigma_{h}M_{l}^{m}=\left(-1\right)^{m+l+1}M_{l}^{m}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\sigma_{h}N_{l}^{m}=\left(-1\right)^{m+l}N_{l}^{m}$ +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +Vertical (yz) reflection +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\sigma_{yz}M_{l}^{m}=-M_{l}^{-m}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\sigma_{yz}N_{l}^{m}=N_{l}^{-m}$ +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +Vertical (xz) reflection +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\sigma_{xz}M_{l}^{m}=\left(-1\right)^{m+1}M_{l}^{-m}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\sigma_{xz}N_{l}^{m}=\left(-1\right)^{m}N_{l}^{-m}$ +\end_inset + + +\end_layout + +\end_inset + + + + +\end_inset + + +\end_layout + +\begin_layout Section +Transformations in a lattice +\end_layout + +\begin_layout Standard + +\end_layout + +\end_body +\end_document