From: Frank Zirkelbach Date: Thu, 14 Feb 2013 13:19:57 +0000 (+0100) Subject: added so from scratch doc :) X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=cb8c7d0935379dbb747f3cef38ca532f0bbf4f26;p=lectures%2Flatex.git added so from scratch doc :) --- diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex index ee0cc2a..5b79972 100644 --- a/physics_compact/solid.tex +++ b/physics_compact/solid.tex @@ -173,7 +173,7 @@ KB transformation \ldots \subsection{Spin-orbit interaction} -Relativistic effects can be incorporated in the normconserving pseudopotential method up to but not including order $\alpha^2$ with $\alpha$ being the fine structure constant. +Relativistic effects can be incorporated in the normconserving pseudopotential method up to but not including terms of order $\alpha^2$ \cite{kleinman80,bachelet82} with $\alpha$ being the fine structure constant. This is advantageous since \ldots With the solutions of the all-electron Dirac equations, the new pseudopotential reads \begin{equation} @@ -183,6 +183,7 @@ V(\vec{r})=\sum_{l,m}\left[ \ket{l-\frac{1}{2},m-{\frac{1}{2}}}V_{l,l-\frac{1}{2}}(\vec{r}) \bra{l-\frac{1}{2},m-{\frac{1}{2}}} \right] \text{ .} +\label{eq:solid:so_bs1} \end{equation} By defining an averaged potential weighted by the different $j$ degeneracies of the $\ket{l\pm\frac{1}{2}}$ states \begin{equation} @@ -197,10 +198,36 @@ V_{l,l+\frac{1}{2}}(\vec{r})-V_{l,l-\frac{1}{2}}(\vec{r})\right) the total potential can be expressed as \begin{equation} V(\vec{r})=\sum_l -\ket{l}\left[\bar{V}_l(\vec{r})+V^{\text{SO}}_l(\vec{r})LS\right]\bra{l} +\ket{l,m}\left[\bar{V}_l(\vec{r})+V^{\text{SO}}_l(\vec{r})LS\right]\bra{l,m} \text{ ,} +\label{eq:solid:so_bs2} \end{equation} where the first term correpsonds to the mass velocity and Darwin relativistic corrections and the latter is associated with the spin-orbit (SO) coupling. +\begin{proof} +This can be shown by rewriting the $LS$ operator +\begin{equation} +J=L+S \Leftrightarrow J^2=L^2+S^2+2LS \Leftrightarrow +LS=\frac{1}{2}\left(J^2-L^2-S^2\right) +\text{ ,} +\end{equation} +which, if used in equation~\eqref{eq:solid:so_bs2}, gives the same (diagonal) matrix elements +\begin{align} +\bra{l\pm\frac{1}{2},m\pm\frac{1}{2}}V(\vec{r}) +\ket{l\pm\frac{1}{2},m\pm\frac{1}{2}}&= +\bar{V}_l(\vec{r})+V^{\text{SO}}_l(\vec{r}) +\frac{1}{2}\left(l(l+1)-j(j+1)-\frac{3}{4}\right) \\ +&= \bar{V}_l(\vec{r})+\frac{1}{2}V^{\text{SO}}_l(\vec{r}) +\left\{\begin{array}{rl} +-\left(l+\frac{3}{2}\right) & \text{ for } j=l+\frac{1}{2}\\ +\left(l-\frac{1}{2}\right) & \text{ for } j=l-\frac{1}{2} +\end{array}\right. +\end{align} +as equation~\eqref{eq:solid:so_bs1} +\begin{equation} +\text{ .} +\end{equation} + +\end{proof} \subsubsection{Excursus: Real space representation within an iterative treatment}