From 2cd85054a17c8d327445289a5bcaaf0d9fe8b11a Mon Sep 17 00:00:00 2001
From: hackbard <hackbard>
Date: Fri, 30 Apr 2004 12:20:12 +0000
Subject: [PATCH] pre3 (equation enumeration)

---
 nlsop/nlsop_emrs_2004.tex | 9 ++++++---
 1 file changed, 6 insertions(+), 3 deletions(-)

diff --git a/nlsop/nlsop_emrs_2004.tex b/nlsop/nlsop_emrs_2004.tex
index 4d1ccf9..4df76ab 100644
--- a/nlsop/nlsop_emrs_2004.tex
+++ b/nlsop/nlsop_emrs_2004.tex
@@ -89,10 +89,13 @@ Before discussing the implementation some assumptions and approximations have to
 The target is devided into $64 \times 64 \times 100$ cells with a side length of $3 \, nm$. Each of it has a state (crystalline/amorphous) and keeps the local carbon concentration. The cell is addressed by a position vector $\vec r = (k,l,m)$, where $k$, $l$, $m$ are integers.
 
 The probability of amorphization is assumed to be proportional to the nuclear stopping power. A local probability of amorphization at any point in the target is composed of three contributions, the ballistic amorphization, a carbon-induced and a stress-induced amorphization. The ballistic amorphization is proportional to the nuclear stopping power as mentioned before. The carbon-induced amorphization is a linear function of the local carbon concentration. The stress-induced amorphization is proportional to the compressive stress originating from the amorphous volumes in the vicinity, the stress amplitude decreasing with the square of distance $d=|\vec r - \vec{r'}|$. Thus the probability of a crystalline volume getting amorphous can be calculated as
-\[
+\begin{equation}
  p_{c \rightarrow a}(\vec r) = p_{b} + p_{c} \, c_{carbon}(\vec r) + \sum_{amorphous \,  neighbours} \frac{p_{s} \, c_{carbon}(\vec{r'})}{d^2}
-\]
-with $p_{b}$, $p_{c}$ and $p_{s}$ being simulation parameters to weight the three different mechanisms of amorphization. The probability $p_{a \rightarrow c}$ of an amorphous volume to turn crystalline should behave contrary to $p_{c \rightarrow a}$ and is thus assumed as $p_{a \rightarrow c} = 1 - p_{c \rightarrow a}$.
+\end{equation}
+with $p_{b}$, $p_{c}$ and $p_{s}$ being simulation parameters to weight the three different mechanisms of amorphization. The probability $p_{a \rightarrow c}$ of an amorphous volume to turn crystalline should behave contrary to $p_{c \rightarrow a}$ and is thus assumed as:
+\begin{equation}
+ p_{a \rightarrow c} = 1 - p_{c \rightarrow a}
+\end{equation}
 
 The simulation algorithm consists of three parts, the amorphization/recrystallization process, the carbon incorporation and finally the carbon diffusion.
 
-- 
2.20.1