From: Frank Zirkelbach Date: Mon, 18 Jun 2012 12:15:16 +0000 (+0200) Subject: more on so X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=fa167b99a2c520549296e61b92d56bbdd44d3849;p=lectures%2Flatex.git more on so --- diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex index e4c2362..c4cf869 100644 --- a/physics_compact/solid.tex +++ b/physics_compact/solid.tex @@ -169,12 +169,34 @@ HSC potential \ldots \subsubsection{Fully separable form of the pseudopotential} +KB transformation \ldots +\subsection{Spin-orbit interaction} -\subsection{Spin orbit interaction} - - -\subsubsection{Perturbative treatment} - -\subsubsection{Non-perturbative method} - +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. +This is advantageous since \ldots +With the solutions of the all-electron Dirac equations, the new pseudopotential reads +\begin{equation} +V(r)=\sum_{l,m}\left[ +\ket{l+\frac{1}{2},m+{\frac{1}{2}}}V_{l,l+\frac{1}{2}}(r) +\bra{l+\frac{1}{2},m+{\frac{1}{2}}} + +\ket{l-\frac{1}{2},m-{\frac{1}{2}}}V_{l,l-\frac{1}{2}}(r) +\bra{l-\frac{1}{2},m-{\frac{1}{2}}} +\right] \text{ .} +\end{equation} +By defining an averaged potential weighted by the different $j$ degeneracies of the $\ket{l\pm\frac{1}{2}}$ states +\begin{equation} +\bar{V}_l(r)=\frac{1}{2l+1}\left( +l V_{l,l-\frac{1}{2}}(r)+(l+1)V_{l,l+\frac{1}{2}}(r)\right) +\end{equation} +and a potential describing the difference in the potential with respect to the spin +\begin{equation} +V^{\text{SO}}_l(r)=\frac{2}{2l+1}\left( +V_{l,l+\frac{1}{2}}(r)-V_{l,l-\frac{1}{2}}(r)\right) +\end{equation} +the total potential can be expressed as +\begin{equation} +V(r)=\sum_l \ket{l}\left[\bar{V}_l(r)+V^{\text{SO}}_l(r)LS\right]\bra{l} +\text{ ,} +\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.