more tersoff

diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex
index 1dbe1c0de4000f5bfee5a3eb7c17ab41629cffc3..e2958aec4a6eec739c22f4ffa86d390fe227a1e0 100644 (file)
--- a/posic/thesis/basics.tex
+++ b/posic/thesis/basics.tex
@@ -94,7 +94,7 @@ Chosing $12$ as the exponent of the repulsive term it is just the square of the
 The constants $\epsilon$ and $\sigma$ are usually determined by fitting to experimental data.
 $\epsilon$ accounts to the depth of the potential well, where $\sigma$ is regarded as the radius of the particle, also known as the van der Waals radius.
 
-Writing down the derivation of the Lennard-Jones potential in respect to $x_i$ (the $i$th component of the distance vector $\vec{r}$)
+Writing down the derivation of the Lennard-Jones potential in respect to $x_i$ (the $i$th component of the distance vector ${\bf r}$)
 \begin{equation}
 \frac{\partial}{\partial x_i} U^{LJ}(r) = 4 \epsilon x_i \Big( -12 \frac{\sigma^{12}}{r^{14}} + 6 \frac{\sigma^6}{r^8} \Big)
 \label{eq:lj-d}
@@ -138,39 +138,62 @@ h(r_{ij},r_{ik},\theta_{jik}) =
 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}
-
-
+The interatomic potential is taken to have the form
+\begin{eqnarray}
+E & = & \sum_i E_i = \frac{1}{2} \sum_{i \ne j} V_{ij} \textrm{ ,} \\
+V_{ij} & = & f_C(r_{ij}) [ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) ] \textrm{ .}
+\end{eqnarray}
+$E$ is the total energy of the system, constituted either by the sum over the site energies $E_i$ or by the bond energies $V_{ij}$.
+The indices $i$ and $j$ correspond to the atoms of the system with $r_{ij}$ being the distance from atom $i$ to atom $j$.
+
+The functions $f_R$ and $f_A$ represent a repulsive and an attractive pair potential.
+The repulsive part is due to the orthogonalization energy of overlapped atomic wave functions.
+The attractive part is associated with the bonding.
+\begin{eqnarray}
+f_R(r_{ij}) & = & A_{ij} \exp (- \lambda_{ij} r_{ij} ) \\
+f_A(r_{ij}) & = & -B_{ij} \exp (- \mu_{ij} r_{ij} )
+\end{eqnarray}
+The function $f_C$ is the potential cutoff function to limit the range of the potential.
+It is designed to have a smooth transition of the potential at distances $R_{ij}$ and $S_{ij}$.
 \begin{equation}
-f_R(r_{ij}) = A_{ij} \exp (- \lambda_{ij} r_{ij} ) \\
+f_C(r_{ij}) = \left\{
+  \begin{array}{ll}
+    1, & r_{ij} < R_{ij} \\
+    \frac{1}{2} + \frac{1}{2} \cos \Big[ \pi (r_{ij} - R_{ij})/(S_{ij} - R_{ij}) \Big], & R_{ij} < r_{ij} < S_{ij} \\
+    0, & r_{ij} > S_{ij}
+  \end{array} \right.
 \end{equation}
 
+The function $b_{ij}$ represents a measure of the bond order, monotonically decreasing with the coordination of atoms $i$ and $j$.
+It is of the form:
+\begin{eqnarray}
+b_{ij} & = & \chi_{ij} (1 + \beta_i^{n_i} \zeta^{n_i}_{ij})^{-1/2n_i} \\
+\zeta_{ij} & = & \sum_{k \ne i,j} f_C (r_{ik}) \omega_{ik} g(\theta_{ijk}) \\
+g(\theta_{ijk}) & = & 1 + c_i^2/d_i^2 - c_i^2/[d_i^2 + (h_i - \cos \theta_{ijk})^2]
+\end{eqnarray}
+where $\theta_{ijk}$ is the bond angle between bonds $ij$ and $ik$.
+This is illustrated in Figure \ref{img:tersoff_angle}.
+
+\printimg{!h}{width=8cm}{tersoff_angle.eps}{Angle between bonds of atoms $i,j$ and $i,k$.}{img:tersoff_angle}
+
+In order to calculate the forces the derivation of the potential with respect to $x^i_n$ (the $n$th component of the position vector of atom $i$ $\equiv$ ${\bf r}_i$) has to be known.
+This is gradually done in the following.
+The cutoff function $f_C$ derivated with repect to $x^i_n$ is
 \begin{equation}
-f_A(r_{ij}) = -B_{ij} \exp (- \mu_{ij} r_{ij} ) \\
+\partial_{x^i_n} f_C(r_{ij}) =
+\frac{1}{2} \sin \Big( \pi (r_{ij} - R_{ij}) / (S_{ij} - R_{ij}) \Big) \frac{\pi x^i_n}{(S_{ij} - R_{ij}) r_{ij}} \textrm { .}
 \end{equation}
+The angle $\theta_{ijk}$ can be expressed by the atom distances with the law of cosines:
+\begin{eqnarray}
+\theta_{ijk} & = & \arccos \Big( (r_{ij}^2 + r_{ik}^2 - r_{jk}^2)/(2 r_{ij} r_{ik}) \Big) \\
+\partial_{x^i_n} \theta_{ijk} & = & 
+\frac{-1}{\sqrt{1 - ((r_{ik}^2+r_{ij}^2-r_{jk}^2)/2r_{ik}r_{ij})^2}}
+\Big( \frac{4 r_{ik}r_{ij} (2 x^i_n - x^k_n - x^j_n) + 2(x^j_n - x^i_n)\frac{r_{ik}}{r_{ij}} + 2(x^k_n - x^i_n)\frac{r_{ij}}{r_{ik}} }{4 r^2_{ik} r^2_{ij}}\Big)
+\end{eqnarray}
 
-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}