pre3 (equation enumeration)

[lectures/latex.git] / nlsop / nlsop_emrs_2004.tex

diff --git a/nlsop/nlsop_emrs_2004.tex b/nlsop/nlsop_emrs_2004.tex
index a7191d0..4df76ab 100644 (file)
--- a/nlsop/nlsop_emrs_2004.tex
+++ b/nlsop/nlsop_emrs_2004.tex
@@ -11,6 +11,8 @@
 \usepackage[dvips]{graphicx}
 \graphicspath{{./img/}}
 
+\usepackage[lmargin=3cm,rmargin=2cm,tmargin=2cm,bmargin=2cm,noheadfoot]{geometry}
+
 %\usepackage{./graphs}
 
 \title{Modelling of a selforganization process leading to periodic arrays of nanometric amorphous precipitates by ion irradiation}
@@ -87,12 +89,15 @@ 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/re-crystallization process, the carbon incorporation and finally the carbon diffusion.
+The simulation algorithm consists of three parts, the amorphization/recrystallization process, the carbon incorporation and finally the carbon diffusion.
 
 For the amorphization/recrystallization process random values are computed to specify the volume which is hit by an impinging carbon ion. Two random numbers $x,y \in [0,1]$ are generated and mapped to the coordinates $k,l$ using a uniform probability distribution, $p(x)dx=dx \textrm{, } p(y)dy=dy$. A random number $z$ corresponding to the $m$ coordinate is distributed according to the linear approximated nuclear stopping power, $p(z)dz=(s z+s_0)dz$, where $s$ and $s_0$ are simulation parameters describing the nuclear energy loss. After calculating the local probability of amorphization $p_{c \rightarrow a}(k,l,m)$ of the selected volume another random number determines depending on the current status whether the volume turns amorphous, recrystallizes or remains unchanged. This step is looped for the average hits per ion in the simulation window as extracted from TRIM \cite{9} collision data.