From af10a7cb7b83bf31dfbb61af01ec87b718f3fd0d Mon Sep 17 00:00:00 2001
From: hackbard <hackbard@sage.physik.uni-augsburg.de>
Date: Wed, 11 Nov 2009 18:46:09 +0100
Subject: [PATCH] next: start with si self-ints

---
 posic/thesis/defects.tex | 33 +++++++++++++++++++++++++--------
 1 file changed, 25 insertions(+), 8 deletions(-)

diff --git a/posic/thesis/defects.tex b/posic/thesis/defects.tex
index e763151..89bd382 100644
--- a/posic/thesis/defects.tex
+++ b/posic/thesis/defects.tex
@@ -15,17 +15,25 @@ The ions are relaxed by a conjugate gradient method.
 The cell volume and shape is allowed to change using the pressure control algorithm of Parinello and Rahman \cite{}.
 Periodic boundary conditions in each direction are applied.
 
-\begin{figure}
+\begin{figure}[h]
 \begin{center}
-\includegraphics[width=10cm]{unit_cell_e.eps}
+\includegraphics[width=9cm]{unit_cell_e.eps}
 \end{center}
-\caption{Insertion positions for the tetrahedral ({\color{red}$\bullet$}), hexagonal  ({\color{green}$\bullet$}), \hkl<1 0 0> dumbbell ({\color{yellow}$\bullet$}) and \hkl<1 1 0> dumbbell ({\color{magenta}$\bullet$}) interstitial configurations.}
+\caption{Insertion positions for the tetrahedral ({\color{red}$\bullet$}), hexagonal  ({\color{green}$\bullet$}), \hkl<1 0 0> dumbbell ({\color{yellow}$\bullet$}), \hkl<1 1 0> dumbbell ({\color{magenta}$\bullet$}) and bond-centered ({\color{cyan}$\bullet$}) interstitial configuration.}
 \label{fig:defects:ins_pos}
 \end{figure}
 
-The interstitial atom positions are displayed in Fig. \ref{fig:defects:ins_pos}.
-In seperated simulation runs the silicon or carbon atom is inserted at the tetrahedral $(0,0,0)$ ({\color{red}$\bullet$}), the hexagonal $(-1/8,-1/8,1/8)$ ({\color{green}$\bullet$}), the nearly \hkl<1 0 0> dumbbell $(-1/4,-1/4,-1/8)$ ({\color{yellow}$\bullet$}) and the nearly \hkl<1 1 0> dumbbell $(-1/8,-1/8,-1/4)$ ({\color{magenta}$\bullet$}) interstitial position.
-For the dumbbell configurations the nearest silicon atom is displaced by $(0,0,-1/8)$ and $(-1/8,-1/8,0)$ respectively of the unit cell length to avoid to high forces.
+The interstitial atom positions are displayed in figure \ref{fig:defects:ins_pos}.
+In seperated simulation runs the silicon or carbon atom is inserted at the
+\begin{itemize}
+ \item tetrahedral, $\vec{p}=(0,0,0)$, ({\color{red}$\bullet$})
+ \item hexagonal, $\vec{p}=(-1/8,-1/8,1/8)$, ({\color{green}$\bullet$})
+ \item nearly \hkl<1 0 0> dumbbell, $\vec{p}=(-1/4,-1/4,-1/8)$, ({\color{yellow}$\bullet$})
+ \item nearly \hkl<1 1 0> dumbbell, $\vec{p}=(-1/8,-1/8,-1/4)$, ({\color{magenta}$\bullet$})
+ \item bond-centered, $\vec{p}=(-1/8,-1/8,-3/8)$, ({\color{cyan}$\bullet$})
+\end{itemize}
+interstitial position.
+For the dumbbell configurations the nearest silicon atom is displaced by $(0,0,-1/8)$ and $(-1/8,-1/8,0)$ respectively of the unit cell length to avoid too high forces.
 A vacancy or a substitutional atom is realized by removing one silicon atom and switching the type of one silicon atom respectively.
 
 From an energetic point of view the free energy of formation $E_{\text{f}}$ is suitable for the characterization of defect structures.
@@ -33,13 +41,22 @@ For defect configurations consisting of a single atom species the formation ener
 \begin{equation}
 E_{\text{f}}=\left(E_{\text{coh}}^{\text{defect}}
                   -E_{\text{coh}}^{\text{defect-free}}\right)N
+\label{eq:defects:ef1}
 \end{equation}
 where $N$ and $E_{\text{coh}}^{\text{defect}}$ are the number of atoms and the cohesive energy per atom in the defect configuration and $E_{\text{coh}}^{\text{defect-free}}$ is the cohesive energy per atom of the defect-free structure.
-Evtl Paper mit Ef rauskramen lenen schreiben ...
-Defects consisting of two or more atom species ...
+The formation energy of defects consisting of two or more atom species is defined as
+\begin{equation}
+E_{\text{f}}=E-N_1\mu_1-N_2\mu_2 - \ldots
+\label{eq:defects:ef2}
+\end{equation}
+where $E$ is the free energy of the interstitial system and $N_i$ and $\mu_i$ are the amount of atoms and the chemical potential of species $i$.
+The chemical potential is determined by the cohesive energy of the structure of the specific type in equilibrium at zero Kelvin.
+For a defect configuration of a single species equation \ref{eq:defects:ef2} is equivalent to equation \ref{eq:defects:ef1}.
 
 \section{Silicon self-interstitials}
 
+
+
 \section{Carbon related point defects}
 
 \section[Migration of the carbon \hkl<1 0 0> interstitial]{\boldmath Migration of the carbon \hkl<1 0 0> interstitial}
-- 
2.20.1