From 8d1d7674ef08669827ef927728256566d173a1b3 Mon Sep 17 00:00:00 2001 From: Frank Zirkelbach Date: Wed, 20 Jun 2012 16:08:50 +0200 Subject: [PATCH] ce --- physics_compact/solid.tex | 138 ++++++++++++++++++++------------------ 1 file changed, 74 insertions(+), 64 deletions(-) diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex index e8c61e1..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(\vec{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} @@ -261,70 +263,78 @@ 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\\ -&=& -\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'}) -\end{eqnarray} -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''}}= +\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_mY^*_{lm}(\Omega_{r''})Y_{lm}(\Omega_{r'}) \text{ ,} -\end{equation} -which can be rewritten as -\begin{equation} -\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''}\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''} -\frac{2l+1}{4\pi}P_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right) -\end{equation} -using the vector addition theorem +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_{r''})Y_{lm}(\Omega_{r'}) +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 potential for angular momentum $l$ can be calculated as +In total, the matrix elements of the SO potential can be calculated by \begin{eqnarray} -\bra{\vec{r'}}V^{\text{KB,SO}}\ket{\vec{r''}}&=& -\bra{\vec{r'}}(r\times p)\ket{\chi_{lm}}E^{\text{SO,KB}}_l -\braket{\chi_{lm}}{\vec{r''}}\\ -&=& --i\hbar(\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}''}=\\ +=-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)\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 +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)\times\nonumber\\ -&&\left(\frac{\vec{r'}\times\vec{r''}}{r'r''}\right) +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} + {\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} +\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}) -- 2.20.1