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even more so
author
hackbard
<hackbard@hackdaworld.org>
Tue, 19 Jun 2012 09:46:45 +0000
(11:46 +0200)
committer
hackbard
<hackbard@hackdaworld.org>
Tue, 19 Jun 2012 09:46:45 +0000
(11:46 +0200)
physics_compact/solid.tex
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diff --git
a/physics_compact/solid.tex
b/physics_compact/solid.tex
index
507de12
..
7fa936f
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}
\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}
E^{\text{SO,KB}}_l\braket{\chi_{lm}}{\vec{r''}}
\text{ .}
\label{eq:solid:so_me}
@@
-310,14
+310,26
@@
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''}}\\
&=&
\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'}
\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} \,
&&\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}
+\frac{2l+1}{4\pi}\\
+&=&
+-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\nonumber\\
\end{eqnarray}
\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}
Finally, to evaluate $V^{\text{SO}}_l\ket{\Psi}$, the integral \ldots
\begin{equation}
\end{equation}
Finally, to evaluate $V^{\text{SO}}_l\ket{\Psi}$, the integral \ldots
\begin{equation}
\end{equation}
+