added so from scratch doc :)

[lectures/latex.git] / physics_compact / solid.tex

diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex
index ee0cc2a..5b79972 100644 (file)
--- 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}