X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=physics_compact%2Fsolid.tex;h=306d23523eecaf24bcfa1e653be492f7b9196177;hp=ee0cc2a0adef0bd3e8fa124d4dcbdfe45fc06b02;hb=f4666008c79f572cbe7cbfa5f9a7e306bfa1637c;hpb=93808b285afe6d16ac131af43108b975a0cc9042

diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex
index ee0cc2a..306d235 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 +198,56 @@ 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)
+\end{equation}
+and corresponding eigenvalue
+\begin{eqnarray}
+j(j+1)-l(l+1)-s(s+1)&=&
+(l\pm\frac{1}{2})(l\pm\frac{1}{2}+1)-l^2-l-\frac{3}{4} \nonumber\\
+&=&
+l^2\pm\frac{l}{2}+l\pm\frac{l}{2}+\frac{1}{4}\pm\frac{1}{2}-l^2-l-\frac{3}{4}
+\nonumber\\
+&=&\pm(l+\frac{1}{2})-\frac{1}{2}=\left\{\begin{array}{rl}
+l & \text{for } j=l+\frac{1}{2}\\
+-(l+1) & \text{for } j=l-\frac{1}{2}
+\end{array}\right.
+\text{ ,}
+\end{eqnarray}
+which, if used in equation~\eqref{eq:solid:so_bs2}, gives the same (diagonal) matrix elements
+\begin{eqnarray}
+\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) \nonumber\\
+&=&\bar{V}_l(\vec{r})+\frac{1}{2}V^{\text{SO}}_l(\vec{r})
+\left\{\begin{array}{rl}
+l & \text{for } j=l+\frac{1}{2}\\
+-(l+1) & \text{for } j=l-\frac{1}{2}
+\end{array}\right. \nonumber\\
+&=&\frac{1}{2l+1}\left(lV_{l,l-\frac{1}{2}}(\vec{r})+
+                       (l+1)V_{l,l+\frac{1}{2}}(\vec{r})\right)+\nonumber\\
+&&+\frac{1}{2l+1}\left\{\begin{array}{rl}
+l\left(V_{l,l+\frac{1}{2}}(\vec{r})-V_{l,l-\frac{1}{2}}(\vec{r})\right) &
+ \text{for } j=l+\frac{1}{2}\\
+-(l+1)\left(V_{l,l+\frac{1}{2}}(\vec{r})-V_{l,l-\frac{1}{2}}(\vec{r})\right) &
+ \text{for } j=l-\frac{1}{2}
+\end{array}\right.
+\end{eqnarray} 
+as equation~\eqref{eq:solid:so_bs1}
+\begin{equation}
+\text{ .}
+\end{equation}
+
+\end{proof}
 
 
 \subsubsection{Excursus: Real space representation within an iterative treatment}