X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=physics_compact%2Fsolid.tex;h=306d23523eecaf24bcfa1e653be492f7b9196177;hp=01f545c984d26622bff40ebf8c614ee38efd8b18;hb=f4666008c79f572cbe7cbfa5f9a7e306bfa1637c;hpb=8d1d7674ef08669827ef927728256566d173a1b3

diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex
index 01f545c..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}
@@ -197,13 +198,59 @@ 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}
+
+\subsubsection{Excursus: Real space representation within an iterative treatment}
 
 In the following, the spin-orbit part is evaluated in real space.
 Since spin is treated in another subspace, it can be treated separately.
@@ -272,7 +319,8 @@ r'^2 dr' d\Omega_{\vec{r}'} \\
 &=&\int_{r'}
 {\delta V_l^{\text{SO}}}^2(r') u_l^2(r') dr'
 \int_{\Omega_{\vec{r}'}}Y^*_{lm}(\Omega_{\vec{r}'})Y_{lm}(\Omega_{\vec{r}'}) d\Omega_{\vec{r}'}\\
-&=&\int_{r'}{\delta V_l^{\text{SO}}}^2(r') u_l^2(r') dr' \text{ .}
+&=&\int_{r'}{\delta V_l^{\text{SO}}}^2(r') u_l^2(r') dr' \text{ .}\\
+&=&\braket{\delta V_l^{\text{SO}}u_l}{u_l\delta V_l^{\text{SO}}}
 \end{eqnarray}
 To obtain a final expression for the matrix elements \eqref{eq:solid:so_me}, the sum of the products of \eqref{eq:solid:so_r1} and \eqref{eq:solid:so_r2} must be further evaluated.
 \begin{eqnarray}
@@ -298,25 +346,25 @@ P_l\left(\frac{\vec{r}'\vec{r}''}{r'r''}\right)=
 \end{equation}
 In total, the matrix elements of the SO potential can be calculated by
 \begin{eqnarray}
--i\hbar\sum_{lm}(\vec{r}'\times \nabla_{\vec{r}'})\braket{\vec{r}'}{\chi_{lm}}
-E^{\text{SO,KB}}_l\braket{\chi_{lm}}{\vec{r}''}=\\
-=-i\hbar\sum_l(\vec{r}'\times \nabla_{\vec{r}'})
+&&-i\hbar\sum_{lm}(\vec{r}'\times \nabla_{\vec{r}'})\braket{\vec{r}'}{\chi_{lm}}
+E^{\text{SO,KB}}_l\braket{\chi_{lm}}{\vec{r}''}=\nonumber\\
+&=&-i\hbar\sum_l(\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)\cdot\nonumber
+P_l\left(\frac{\vec{r}'\vec{r}''}{r'r''}\right)\cdot
 \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} \cdot
 \frac{2l+1}{4\pi}\nonumber\\
-=
+&=&
 -i\hbar\sum_l
 \delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
 P'_l\left(\frac{\vec{r}'\vec{r}''}{r'r''}\right)\cdot
 \left(\frac{\vec{r}'\times\vec{r}''}{r'r''}\right)\cdot
 \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}\text{ ,}
+\frac{2l+1}{4\pi}\text{ ,}\nonumber\\
 \label{eq:solid:so_fin}
 \end{eqnarray}
-where the derivatives of functions that depend on the absolute value of $\vec{r}'$ do not contribute due to the cross product as can be seen from equations \eqref{eq:solid:rxp1} and \eqref{eq:solid:rxp2}.
+since derivatives of functions that depend on the absolute value of $\vec{r}'$ do not contribute due to the cross product as is illustrated below (equations \eqref{eq:solid:rxp1} and \eqref{eq:solid:rxp2}).
 \begin{eqnarray}
 \left(\vec{r}\times\nabla_{\vec{r}}\right)f(r)&=&
 \left(\begin{array}{l}
@@ -335,13 +383,28 @@ r_if'(r)\frac{1}{2r}2r_j-r_jf'(r)\frac{1}{2r}2r_i=0
 \label{eq:solid:rxp2}
 \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
+If these projectors are considered to be centered around atom positions $\vec{\tau}_{\alpha n}$ of atoms $n$ of species $\alpha$, the variable $\vec{r}'$ in the previous equations is changed to $\vec{r}'_{\alpha n}=\vec{r}'-\vec{\tau}_{\alpha n}$, which implies
+\begin{eqnarray}
+r'&\rightarrow&r_{\alpha n}=|\vec{r}'-\vec{\tau}_{\alpha n}|\\
+\Omega_{\vec{r}'}&\rightarrow&\Omega_{\vec{r'}-\vec{\tau}_{\alpha n}}\\
+\delta V_l(r')&\rightarrow&\delta V_l(|\vec{r}'-\vec{\tau}_{\alpha n}|)\\
+u_l(r')&\rightarrow&u_l(|\vec{r}'-\vec{\tau}_{\alpha n}|)\\
+Y_{lm}(\Omega_{\vec{r}'})&\rightarrow&
+Y_{lm}(\Omega_{\vec{r}'-\vec{\tau}_{\alpha n}})
+\text{ .}
+\end{eqnarray}
+Within an iterative treatment on a real space grid consisting of $n_{\text{g}}$ grid points, the sum
 \begin{equation}
+\sum_{\vec{r}''_{\alpha n}}
+\sum_{lm}-i\hbar(\vec{r}'_{\alpha n}\times \nabla_{\vec{r}'_{\alpha n}})
+\braket{\vec{r}'_{\alpha n}}{\chi^{\text{SO}}_{lm}}
+E^{\text{SO,KB}}_l\braket{\chi^{\text{SO}}_{lm}}{\vec{r}''_{\alpha n}}
+\braket{\vec{r}''_{\alpha n}}{\Psi}
+\qquad\forall\,\bra{\vec{r}'_{\alpha n}}
 \end{equation}
+to obtain all elements $\bra{\vec{r}'_{\alpha n}}$, involves $n_{\text{g}}^2$ evaluations of equation~\eqref{eq:solid:so_fin} for eeach atom, if the projectors are short-ranged, i.e.\  $\delta V_l=0$ outside a certain cut-off radius.
+Thus, this method scales linearly with the number of atoms.
+
 The $E_l^{\text{SO,KB}}$ are given by
 \begin{equation}
 E_l^{\text{SO,KB}}=
@@ -351,7 +414,4 @@ E_l^{\text{SO,KB}}=
      {\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}