basically finished tersoff now

diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex
index 2493076..5aa6fe8 100644 (file)
--- a/posic/thesis/basics.tex
+++ b/posic/thesis/basics.tex
@@ -11,7 +11,7 @@ Although classical potentials are often most successful and at the same time com
 Thus, in order to obtain more accurate results quantum-mechanical calculations from first principles based on density functional theory (DFT) were performed.
 The Vienna {\em ab initio} simulation package ({\textsc vasp}) \cite{kresse96} is used for this purpose.
 The relevant basics of DFT are described in section \ref{section:dft} while an overview of utilities mainly used to create input or parse output data of {\textsc vasp} is given in appendix \ref{app:code}.
-The gain in accuracy achieved by this method, however, is accompanied by an increase in computational effort constraining the system to be described to be much smaller in size.
+The gain in accuracy achieved by this method, however, is accompanied by an increase in computational effort constraining the simulated system to be much smaller in size.
 Thus, investigations based on DFT are restricted to single defects or combinations of two defects in a rather small Si supercell, their structural relaxation as well as some selected diffusion processes.
 Next to the structure, defects can be characterized by the defect formation energy, a scalar indicating the costs necessary for the formation of the defect, which is explained in section \ref{section:basics:defects}.
 The method used to investigate migration pathways to identify the prevalent diffusion mechanism is introduced in section \ref{section:basics:migration} and modifications to the {\textsc vasp} code implementing this method are presented in appendix \ref{app:patch_vasp}.
@@ -113,9 +113,6 @@ Since the forces for the new positions are required to update the velocity the d
 \subsection{Interaction potentials for silicon and carbon}
 \label{subsection:interact_pot}
 
-% todo
-% rewrite!
-
 The potential energy of $N$ interacting atoms can be written in the form
 \begin{equation}
 U(\{{\bf r}\}) = \sum_i U_1({\bf r}_i) + \sum_i \sum_{j>i} U_2({\bf r}_i,{\bf r}_j) + \sum_i \sum_{j>i} \sum_{k>j>i} U_3({\bf r}_i,{\bf r}_j,{\bf r}_k) \ldots
@@ -137,12 +134,14 @@ Tersoff proposed an empirical interatomic potential for covalent systems \cite{t
 The Tersoff potential explicitly incorporates the dependence of bond order on local environments, permitting an improved description of covalent materials.
 Due to the covalent character Tersoff restricted the interaction to nearest neighbor atoms accompanied by an increases in computational efficiency for the evaluation of forces and energy based on the short-range potential.
 Tersoff applied the potential to silicon \cite{tersoff_si1,tersoff_si2,tersoff_si3}, carbon \cite{tersoff_c} and also to multicomponent systems like silicon carbide \cite{tersoff_m}.
-...
-The basic idea is that, in real systems, the bond order depends upon the local environment.
-An atom with many neighbors forms weaker bonds than an atom with few neighbors since .
-
-Here comes an explanation, energy per bond monotonically decreasing with the amount of bonds and so on and so on \ldots
-
+The basic idea is that, in real systems, the bond order, i.e. the strength of the bond, depends upon the local environment \cite{abell85}.
+Atoms with many neighbors form weaker bonds than atoms with only a few neighbors.
+Although the bond strength intricately depends on geometry the focus on coordination, i.e. the number of neighbors forming bonds, is well motivated qualitatively from basic chemistry since for every additional formed bond the amount of electron pairs per bond and, thus, the strength of the bonds is decreased.
+If the energy per bond decreases rapidly enough with increasing coordination the most stable structure will be the dimer.
+In the other extreme, if the dependence is weak, the material system will end up in a close-packed structure in order to maximize the number of bonds and likewise minimize the cohesive energy.
+This suggests the bond order to be a monotonously decreasing function with respect to coordination and the equilibrium coordination being determined by the balance of bond strength and number of bonds.
+
+Tersoff incorporated the concept of bond order based on pseudopotential theory \cite{abell85} in a three-body potential formalism.
 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{ ,} \\