From: hackbard Date: Tue, 11 Jul 2006 15:48:49 +0000 (+0000) Subject: more theory X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=9761bf24f47daee3d2c82f870321bddbab0e32f4;p=lectures%2Flatex.git more theory --- diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex index c25e858..8cf717b 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 +108,65 @@ 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} + +Ther Tersoff potential \cite{tersoff1} \ldots + +\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} diff --git a/posic/thesis/literature.tex b/posic/thesis/literature.tex index e478cad..002da01 100644 --- a/posic/thesis/literature.tex +++ b/posic/thesis/literature.tex @@ -6,15 +6,20 @@ Oeuvres Compl\`etes de Laplace, volume VII. Paris, 1820, Gauthier-Villars. \bibitem{alder1} - B. J. Alder, T.E. Wainwright. + B. J. Alder, T. E. Wainwright. J. Chem. Phys. 27 (1957) 1208. \bibitem{alder2} - B. J. Alder, T.E. Wainwright. + B. J. Alder, T. E. Wainwright. J. Chem. Phys. 31 (1959) 459. + \bibitem{stillinger_weber} + F. H. Stillinger, T. A. Weber. + Phys. Rev. B 31 (1985) 5262. + \bibitem{tersoff1} + J. Tersoff. + Phys. Rev. B 39 (1989) 5566. \bibitem{example} \selectlanguage{german} F. Zirkelbach, M. H"aberlen, J. K. N. Lindner, B. Stritzker. \selectlanguage{english} - {\em Modelling of a selforganization process leading to periodic arrays of nanometric amorphous precipitates by ion irradiation.} Comp. Mater. Sci. 33 (2005) 310. \end{thebibliography}