more so

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

diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex
index 507de12..e8c61e1 100644 (file)
--- a/physics_compact/solid.tex
+++ b/physics_compact/solid.tex
@@ -220,7 +220,7 @@ r\ket{\vec{r'}} & = & r'\ket{\vec{r'}}
 \end{eqnarray}
 we get
 \begin{equation}
--i\hbar(r'\times \nabla_{\vec{r'}})\braket{\vec{r'}}{\chi_{lm}}
+-i\hbar(\vec{r'}\times \nabla_{\vec{r'}})\braket{\vec{r'}}{\chi_{lm}}
 E^{\text{SO,KB}}_l\braket{\chi_{lm}}{\vec{r''}}
 \text{ .}
 \label{eq:solid:so_me}
@@ -283,7 +283,7 @@ To obtain a final expression for the matrix elements \eqref{eq:solid:so_me}, the
 \delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}
 Y^*_{lm}(\Omega_{r''})Y_{lm}(\Omega_{r'})
 \end{eqnarray}
-and if all megnetic states $m=-l,-l+1,\ldots,l-1,l$ that contribute to the potential for angular momentum $l$ are considered
+and if all states with magnetic quantum numbers $m=-l,-l+1,\ldots,l-1,l$ that contribute to the potential for angular momentum $l$ are considered
 \begin{equation}
 \braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}
 \braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r''}}=
@@ -310,14 +310,38 @@ In total, the matrix elements of the potential for angular momentum $l$ can be c
 \bra{\vec{r'}}(r\times p)\ket{\chi_{lm}}E^{\text{SO,KB}}_l
 \braket{\chi_{lm}}{\vec{r''}}\\
 &=&
--i\hbar(r'\times \nabla_{\vec{r'}})
+-i\hbar(\vec{r'}\times \nabla_{\vec{r'}})
 \delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
-P_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right)\times\\
+P_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right)\times\nonumber\\
 &&\times\frac{E^{\text{SO,KB}}_l\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}}
              {\int_{r}{\delta V_l^{\text{SO}}}^2(r) u_l^2(r) dr} \,
+\frac{2l+1}{4\pi}\\
+&=&
+-i\hbar
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
+P'_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right)\times\nonumber\\
+&&\left(\frac{\vec{r'}\times\vec{r''}}{r'r''}\right)
+\frac{E^{\text{SO,KB}}_l\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}}
+        {\int_{r}{\delta V_l^{\text{SO}}}^2(r) u_l^2(r) dr} \,
 \frac{2l+1}{4\pi}
 \end{eqnarray}
-
+If the potential at position $\vec{r}$ is considered a sum of atomic potentials $v_{\alpha}(\vec{r}-\vec{\tau}_{\alpha n})$ (atom $n$ of species $\alpha$)
+\begin{equation}
+V(\vec{r})=\sum_{\alpha}\sum_n v_{\alpha}(\vec{r}-\vec{\tau}_{\alpha n})
+\end{equation}
+and the SO projectors are likewise centered on atoms, the SO potential contribution reads
+\begin{equation}
+\end{equation}
+The $E_l^{\text{SO,KB}}$ are given by
+\begin{equation}
+E_l^{\text{SO,KB}}=
+\frac{\braket{\delta V_lu_l}{u_l\delta V_l}}
+     {\bra{u_l}\delta V_l\ket{u_l}}=
+\frac{\int_{r}\delta V^2_l(r)u^2_l(r)}r^2dr
+     {\int_{r'}\int_{r''}\braket{u_l}{r'}\bra{r'}\delta V_l
+\ket{r''}\braket{r''}{u_l}}=
+\end{equation}
 Finally, to evaluate $V^{\text{SO}}_l\ket{\Psi}$, the integral \ldots
 \begin{equation}
 \end{equation}
+