From 1b9983243e764a394f2ba2d7d1aa5babece75218 Mon Sep 17 00:00:00 2001
From: hackbard <hackbard>
Date: Thu, 14 Jun 2007 07:38:59 +0000
Subject: [PATCH] tersoff pot, albe still missing ...

---
 posic/talks/md_simulation_von_silizium.tex | 97 ++++++++++++++++++----
 1 file changed, 83 insertions(+), 14 deletions(-)

diff --git a/posic/talks/md_simulation_von_silizium.tex b/posic/talks/md_simulation_von_silizium.tex
index b78dd11..6590157 100644
--- a/posic/talks/md_simulation_von_silizium.tex
+++ b/posic/talks/md_simulation_von_silizium.tex
@@ -565,29 +565,98 @@ Problemstellung: Finden der Nachbarn f"ur Wechselwirkung
 \end{itemize}
 \end{slide}
 
+%\begin{slide}
+%{\large\bf
+% Thermodynamische Gr"o"sen
+%}
+%\begin{itemize}
+% \item W"armekapazit"at
+% \item Struktur Werte
+% \item Diffusion
+%\end{itemize}
+%\end{slide}
+
 \begin{slide}
 {\large\bf
- Thermodynamische Gr"o"sen
+ Idee des Tersoff Potentials
 }
+       \begin{picture}(350,10)
+       \end{picture}
 \begin{itemize}
- \item W"armekapazit"at
- \item Struktur Werte
- \item Diffusion
+ \item Potential f"ur kovalente Bindungen\\
+       ($Si$: $sp^3$-Hybridisierung, 4 "au"sere Elektronen,
+        4 gerichtete Bindungen, Winkel: $109,47 ^{\circ}$)\\
+       $\Rightarrow$ Bindungsenergie von 3 Atomen $i,j,k$
+       abh"angig von $r_{ij},r_{ik},r_{jk}$ {\color{red} und}
+       $\theta_{ijk},\theta_{ikj},\theta_{kij}$
+ \item {\em\color{blue} bond order} Potential
+       im Gegensatz zu {\em explicit angular}\\
+       \[
+       \pot = \pot_R(r_{ij}) + {\color{blue} b_{ijk}} \pot_A(r_ij)
+       \]
+       \begin{picture}(350,10)
+       \end{picture}
+       \begin{itemize}
+        \item $b_{ijk}$: umgebungsabh"angiger Term
+        \item $b_{ijk}=const.$ $\Rightarrow$ Paarpotential
+        \item Schw"achung der Paarbindung je mehr Nachbarn vorhanden\\
+              qualitative Motivation: Anzahl der Elektronenpaare pro Bindung
+        \item St"arke der Bindung monoton fallend mit Koordinationszahl\\
+              steiler Abfall $\Rightarrow$ Dimer\\
+              schwacher Abfall $\Rightarrow$ maximale Koordinationszahl
+              (hcp-Struktur)
+        \item Pseudopotentialtheorie:
+              \[
+               b_{ijk} \sim Z^{-\delta}
+              \]
+              \begin{center}
+              {\scriptsize Abell et al. Phys. Rev. B 31 (1985) 6184.}
+              \end{center}
+       \end{itemize}
 \end{itemize}
 \end{slide}
 
 \begin{slide}
 {\large\bf
- Tersoff
-}
+ Form des Tersoff Potentials:
+}\\
+Gesamtenergie:
+\[
+E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
+\pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right]
+\]
+Repulsiver und attraktiver Beitrag:
+\begin{eqnarray}
+f_R(r_{ij}) &=& A_{ij} \exp(-\lambda_{ij} r_{ij}) \nonumber \\
+f_A(r_{ij}) &=& - B_{ij} \exp(-\mu_{ij} r_{ij}) \nonumber
+\end{eqnarray}
+Cut-Off Funktion:
+\[
+f_C(r_{ij})=\left\{\begin{array}{ll}
+ 1, & r_{ij} < R_{ij} \\
+ \frac{1}{2} +
+ \frac{1}{2} \cos \Big[ \pi (r_{ij} - R_{ij})/(S_{ij} - R_{ij}) \Big],
+ & R_{ij} < r_{ij} < S_{ij} \\
+ 0, & r_{ij} > S_{ij}
+\end{array} \right.
+\]
+{\em bond order} Term:
+\begin{eqnarray}
+b_{ij} &=& \chi_{ij} (1 + \beta_i^{n_i} \zeta^{n_i}_{ij})^{-1/2n_i}
+\nonumber \\
+\zeta_{ij} &=& \sum_{k \ne i,j} f_C (r_{ik}) \omega_{ik} g(\theta_{ijk})
+\nonumber \\
+g(\theta_{ijk})&=&1+c_i^2/d_i^2 - c_i^2/[d_i^2 + (h_i - \cos \theta_{ijk})^2]
+\nonumber
+\end{eqnarray}
 \end{slide}
 
-\begin{slide}
-{\large\bf
- EAM
-}
-
-\end{slide}
+%\begin{slide}
+%{\large\bf
+% EAM
+%}
+%
+%\end{slide}
 
 \begin{slide}
 {\large\bf
@@ -609,10 +678,10 @@ Gear Predictor Corrector & ${\color{red} \times}$ & GEAR-5 & $\bullet\bullet$ \\
 {\bf Potential} & & & \\
 Harmonischer Oszillator & ${\color{green} \surd}$ & & - \\
 Lennard-Jones &$ {\color{green} \surd}$ & & - \\
-Tersoff/Albe & ${\color{green} \surd\surd}$ & & - \\
+Tersoff & ${\color{green} \surd}$ & & - \\
+Albe & ${\color{green} \surd}$ & & - \\
 Tersoff/Albe (inkl. $\lambda^3$) & ${\color{red} \times\times}$ &
                                  &  $\bullet\bullet\bullet$ \\
-EAM & ${\color{red} \times}$ & & $\bullet\bullet$ \\
 {\bf Ensembles} & & & \\
 {\em temperature scaling} & ${\color{green} \surd}$ & & - \\
 {\em pressure scaling} & ${\color{green} \surd}$ & & - \\
-- 
2.20.1