X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=physics_compact%2Fsolid.tex;h=01f545c984d26622bff40ebf8c614ee38efd8b18;hp=33fe8a4c970900514004a73f54600c7b26875566;hb=8d1d7674ef08669827ef927728256566d173a1b3;hpb=283813d62a10b6805e95f65eee8e2de5000d8d94

diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex
index 33fe8a4..01f545c 100644
--- a/physics_compact/solid.tex
+++ b/physics_compact/solid.tex
@@ -207,49 +207,51 @@ where the first term correpsonds to the mass velocity and Darwin relativistic co
 
 In the following, the spin-orbit part is evaluated in real space.
 Since spin is treated in another subspace, it can be treated separately.
-The matrix elements of the orbital angular momentum part of the potential in KB form for orbital angular momentum $l$ read
+The matrix elements of the orbital angular momentum part of the potential in KB form read
 \begin{equation}
-\bra{\vec{r'}}(r\times p)\ket{\chi_{lm}}E^{\text{SO,KB}}_l\braket{\chi_{lm}}{\vec{r''}}
+\sum_{lm}
+\bra{\vec{r}'}(r\times p)\ket{\chi_{lm}}E^{\text{SO,KB}}_l
+\braket{\chi_{lm}}{\vec{r}''}
 \text{ .}
 \end{equation}
 With
 \begin{eqnarray}
-\bra{\vec{r'}}p\ket{\chi_{lm}} & = & -i\hbar\nabla_{\vec{r'}} \braket{\vec{r'}}{\chi_{lm}}
-=-i\hbar\nabla_{\vec{r'}}\,\chi_{lm}(\vec{r'}) \\
-r\ket{\vec{r'}} & = & r'\ket{\vec{r'}}
+\bra{\vec{r}'}r\ket{\chi_{lm}} & = & \vec{r}'\braket{\vec{r}'}{\chi_{lm}}\\
+\bra{\vec{r}'}p\ket{\chi_{lm}} & = & -i\hbar\nabla_{\vec{r}'}
+\braket{\vec{r}'}{\chi_{lm}}
 \end{eqnarray}
 we get
 \begin{equation}
--i\hbar(r'\times \nabla_{\vec{r'}})\braket{\vec{r'}}{\chi_{lm}}
-E^{\text{SO,KB}}_l\braket{\chi_{lm}}{\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}''}
 \text{ .}
 \label{eq:solid:so_me}
 \end{equation}
 To further evaluate this expression, the KB projectors
 \begin{equation}
-\chi_{lm}=\frac{\ket{\delta V_l^{\text{SO}}\Phi_{lm}}}
+\ket{\chi_{lm}}=\frac{\ket{\delta V_l^{\text{SO}}\Phi_{lm}}}
 {\braket{\delta V_l^{\text{SO}}\Phi_{lm}}
         {\Phi_{lm}\delta V_l^{\text{SO}}}^{1/2}}
 \end{equation}
-must be known in real space (with respect to $\vec{r'}$).
+must be known in real space (with respect to $\vec{r}'$).
 \begin{equation}
-\braket{\vec{r'}}{\chi_{lm}}=
-\frac{\braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}}{
+\braket{\vec{r}'}{\chi_{lm}}=
+\frac{\braket{\vec{r}'}{\delta V_l^{\text{SO}}\Phi_{lm}}}{
 \braket{\delta V_l^{\text{SO}}\Phi_{lm}}{\Phi_{lm}\delta V_l^{\text{SO}}}
 ^{1/2}}
 \end{equation}
 and
 \begin{equation}
-\braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}=
-\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}Y_{lm}(\Omega_{r'})
+\braket{\vec{r}'}{\delta V_l^{\text{SO}}\Phi_{lm}}=
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}Y_{lm}(\Omega_{\vec{r}'})
 \text{ .}
 \label{eq:solid:so_r1}
 \end{equation}
 In this expression, only the spherical harmonics are complex functions.
-Thus, the complex conjugate with respect to $\vec{r''}$ is given by
+Thus, the complex conjugate with respect to $\vec{r}''$ is given by
 \begin{equation}
-\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r''}}=
-\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}Y^*_{lm}(\Omega_{r''})
+\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r}''}=
+\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}Y^*_{lm}(\Omega_{\vec{r}''})
 \text{ .}
 \label{eq:solid:so_r2}
 \end{equation}
@@ -261,29 +263,95 @@ Using the orthonormality property
 of the spherical harmonics, the squared norm of the $\chi_{lm}$ reduces to
 \begin{eqnarray}
 \braket{\delta V_l^{\text{SO}}\Phi_{lm}}{\Phi_{lm}\delta V_l^{\text{SO}}} &=&
-\int \braket{\delta V_l^{\text{SO}}\Phi_{lm}}{\vec{r'}}
-\braket{\vec{r'}}{\Phi_{lm}\delta V_l^{\text{SO}}} d\vec{r'}\\
+\int \braket{\delta V_l^{\text{SO}}\Phi_{lm}}{\vec{r}'}
+\braket{\vec{r}'}{\Phi_{lm}\delta V_l^{\text{SO}}} d\vec{r}'\\
 &=&\int 
-{\delta V_l^{\text{SO}}}^2(r')\frac{u_l^2(r')}{r'^2}Y^*_{lm}(\Omega_{r'})
-Y_{lm}(\Omega_{r'})
-r'^2 dr' d\Omega_{r'} \\
+{\delta V_l^{\text{SO}}}^2(r')\frac{u_l^2(r')}{r'^2}Y^*_{lm}(\Omega_{\vec{r}'})
+Y_{lm}(\Omega_{\vec{r}'})
+r'^2 dr' d\Omega_{\vec{r}'} \\
 &=&\int_{r'}
 {\delta V_l^{\text{SO}}}^2(r') u_l^2(r') dr'
-\int_{\Omega_{r'}}Y^*_{lm}(\Omega_{r'})Y_{lm}(\Omega_{r'}) d\Omega_{r'}\\
+\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{ .}
 \end{eqnarray}
-To obtain a final expression for the matrix elements \eqref{eq:solid:so_me}, the product of \eqref{eq:solid:so_r1} and \eqref{eq:solid:so_r2} must be further evaluated.
+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}
-\braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}
-\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r''}}&=&
-\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}Y_{lm}(\Omega_{r'})
-\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}Y^*_{lm}(\Omega_{r''})\nonumber\\
-&=&
+\sum_{lm}
+\braket{\vec{r}'}{\delta V_l^{\text{SO}}\Phi_{lm}}
+\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r}''}&=&\sum_{lm}
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}Y_{lm}(\Omega_{\vec{r}'})
+\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}Y^*_{lm}(\Omega_{\vec{r}''})\nonumber\\
+&=&\sum_l
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
+\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}\sum_m
+Y^*_{lm}(\Omega_{\vec{r}''})Y_{lm}(\Omega_{\vec{r}'})\nonumber\\
+&=&\sum_l
 \delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
 \delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}
-Y^*_{lm}(\Omega_{r''})Y_{lm}(\Omega_{r'})
+P_l\left(\frac{\vec{r}'\vec{r}''}{r'r''}\right)\frac{2l+1}{4\pi}\nonumber\\
+\end{eqnarray}
+due to the vector addition theorem
+\begin{equation}
+P_l\left(\frac{\vec{r}'\vec{r}''}{r'r''}\right)=
+\frac{4\pi}{2l+1}\sum_mY^*_{lm}(\Omega_{\vec{r}''})Y_{lm}(\Omega_{\vec{r}'})
+\text{ .}
+\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}'})
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
+P_l\left(\frac{\vec{r}'\vec{r}''}{r'r''}\right)\cdot\nonumber
+\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{ ,}
+\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}.
+\begin{eqnarray}
+\left(\vec{r}\times\nabla_{\vec{r}}\right)f(r)&=&
+\left(\begin{array}{l}
+r_y\frac{\partial}{\partial r_z}f(r)-r_z\frac{\partial}{\partial r_y}f(r)\\
+r_z\frac{\partial}{\partial r_x}f(r)-r_x\frac{\partial}{\partial r_z}f(r)\\
+r_x\frac{\partial}{\partial r_y}f(r)-r_y\frac{\partial}{\partial r_x}f(r)
+\end{array}\right)
+\label{eq:solid:rxp1}
 \end{eqnarray}
-All megnetic states $m=-l,-l+1,\ldots,l-1,l$ contribute to the potential for angular momentum $l$.
+\begin{eqnarray}
+r_i\frac{\partial}{\partial r_j}f(r)-r_j\frac{\partial}{\partial r_i}f(r)&=&
+r_if'(r)\frac{\partial}{\partial r_j}(r_x^2+r_y^2+r_z^2)^{1/2}-
+r_jf'(r)\frac{\partial}{\partial r_i}(r_x^2+r_y^2+r_z^2)^{1/2}\nonumber\\
+&=&
+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
+\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}
+