From 93808b285afe6d16ac131af43108b975a0cc9042 Mon Sep 17 00:00:00 2001 From: hackbard Date: Wed, 20 Jun 2012 19:20:33 +0200 Subject: [PATCH] after ice SO stuff --- physics_compact/math.tex | 2 +- physics_compact/phys_comp.tex | 3 ++- physics_compact/solid.tex | 47 ++++++++++++++++++++++------------- 3 files changed, 33 insertions(+), 19 deletions(-) diff --git a/physics_compact/math.tex b/physics_compact/math.tex index b649597..bd2a888 100644 --- a/physics_compact/math.tex +++ b/physics_compact/math.tex @@ -21,7 +21,7 @@ which satisfies the properties of an inner product (see \ref{math_app:product}) \begin{equation} ||\vec{a}||=\sqrt{(\vec{a},\vec{a})} \end{equation} -that just corresponds to the length of vector \vec{a}. +that just corresponds to the length of vector $\vec{a}$. Evaluating the scalar product $(\vec{a},\vec{b})$ by the sum representation of \eqref{eq:vec_sum} leads to \begin{equation} (\vec{a},\vec{b})=(\sum_i\vec{e}_ia_i,\sum_j\vec{e}_jb_j)= diff --git a/physics_compact/phys_comp.tex b/physics_compact/phys_comp.tex index b4d5e66..5974619 100644 --- a/physics_compact/phys_comp.tex +++ b/physics_compact/phys_comp.tex @@ -9,6 +9,7 @@ \usepackage[T1]{fontenc} \usepackage{amsmath} \usepackage{amssymb} +\usepackage{bm} \usepackage{ae} \usepackage{aecompl} \usepackage[dvips]{graphicx} @@ -108,7 +109,7 @@ \newcommand{\RM}[1]{\MakeUppercase{\romannumeral #1{}}} % vectors are simply represented by bold font characters -\renewcommand{\vec}[1]{{\bf #1{}}} +\renewcommand{\vec}[1]{{\bm #1{}}} % % theorem environment diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex index 01f545c..ee0cc2a 100644 --- a/physics_compact/solid.tex +++ b/physics_compact/solid.tex @@ -203,7 +203,7 @@ V(\vec{r})=\sum_l where the first term correpsonds to the mass velocity and Darwin relativistic corrections and the latter is associated with the spin-orbit (SO) coupling. -\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 +272,8 @@ 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 +299,25 @@ 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 +336,28 @@ 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 +367,4 @@ 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} -- 2.20.1