X-Git-Url: https://hackdaworld.org/gitweb/?a=blobdiff_plain;f=posic%2Fthesis%2Fbasics.tex;h=1dbe1c0de4000f5bfee5a3eb7c17ab41629cffc3;hb=71d8ad22f7c8fd77cc86c966aa209d5cd4c14ac0;hp=c25e8587a8d18d929a28a15b8bb8b51cbe549229;hpb=19c81fd5d9039d184c432861bdb995503d1959c1;p=lectures%2Flatex.git

diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex
index c25e858..1dbe1c0 100644
--- a/posic/thesis/basics.tex
+++ b/posic/thesis/basics.tex
@@ -34,6 +34,7 @@ The forces ${\bf F}_i$ are obtained from the potential energy $U(\{{\bf r}\})$:
 \begin{equation}
 {\bf F}_i = - \nabla_{{\bf r}_i} U({\{\bf r}\}) \, \textrm{.}
 \end{equation}
+
 Given the initial conditions ${\bf r}_i(t_0)$ and $\dot{{\bf r}}_i(t_0)$ the equations can be integrated by a certain integration algorithm.
 The solution of these equations provides the complete information of a system evolving in time.
 
@@ -107,8 +108,72 @@ F_i^j = - \frac{\partial}{\partial x_i} U^{LJ}(r) \, \textrm{.}
 \end{equation}
 
 \subsubsection{The Stillinger Weber potential}
-\subsubsection{The Stillinger Weber potential}
-\subsubsection{The Stillinger Weber potential}
+
+The Stillinger Weber potential \cite{stillinger_weber} \ldots
+
+\begin{equation}
+U = \sum_{i,j} U_2({\bf r}_i,{\bf r}_j) + \sum_{i,j,k} U_3({\bf r}_i,{\bf r}_j,{\bf r}_k)
+\end{equation}
+
+\begin{equation}
+U_2(r_{ij}) = \left\{
+  \begin{array}{ll}
+    \epsilon A \Big( B (r_{ij} / \sigma)^{-p} - 1\Big) \exp \Big[ (r_{ij} / \sigma - 1)^{-1}  \Big]  & r_{ij} < a \sigma \\
+    0 & r_{ij} \ge a \sigma
+  \end{array} \right.
+\end{equation}
+
+\begin{equation}
+U_3({\bf r}_i,{\bf r}_j,{\bf r}_k) = 
+\epsilon \Big[ h(r_{ij},r_{ik},\theta_{jik}) + h(r_{ji},r_{jk},\theta_{ijk}) + h(r_{ki},r_{kj},\theta_{ikj}) \Big]
+\end{equation}
+
+\begin{equation}
+h(r_{ij},r_{ik},\theta_{jik}) =
+\lambda \exp \Big[ \gamma (r_{ij}/\sigma -a)^{-1} + \gamma (r_{ik}/\sigma - a)^{-1} \Big] \Big( \cos \theta_{jik} + \frac{1}{3} \Big)^2
+\end{equation}
+
+\subsubsection{The Tersoff potential}
+
+Tersoff proposed an empirical interatomic potential for covalent systems.
+The Tersoff potential explicitly incorporates the dependence of bond order on local envirenments, permitting an improved description of covalent materials.
+Tersoff applied the potential to silicon \cite{tersoff_silicon1,tersoff_silicon2,tersoff_silicon3}, carbon \cite{tersoff_carbon} and also to multicomponent systems like $SiC$ \cite{tersoff_multi}.
+
+The basic idea is that, in real systems, the bond order depends upon the local environment.
+An atom with many neighbours forms weaker bonds than an atom with few neighbours.
+
+
+
+\begin{equation}
+V_{ij} = f_C(r_{ij}) [ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) ]
+\end{equation}
+
+The total energy is then given by
+\begin{equation}
+E = \frac{1}{2} \sum_{i \ne j} V_{ij} \, \textrm{.}
+\end{equation}
+
+
+\begin{equation}
+f_R(r_{ij}) = A_{ij} \exp (- \lambda_{ij} r_{ij} ) \\
+\end{equation}
+
+\begin{equation}
+f_A(r_{ij}) = -B_{ij} \exp (- \mu_{ij} r_{ij} ) \\
+\end{equation}
+
+The function $f_C$ is the potential cutoff function designed to have a smooth transition between $R_{ij}$ and $S_{ij}$.
+\begin{equation}
+f_C(r_{ij}) = \left\{
+  \begin{array}{ll}
+    1 & r_{ij} < R_{ij} \\
+    \frac{1}{2} + \frac{1}{2} \cos [ \pi (r_{ij} - R_{ij})/(S_{ij} - R_{ij}) ] & R_{ij} < r_{ij} < S_{ij} \\
+    0 & r_{ij} > S_{ij}
+  \end{array} \right.
+\end{equation}
+
+
+\subsubsection{The Brenner potential}
 
 \subsection{Statistical ensembles}
 \label{subsection:statistical_ensembles}