--- trunk/tengDissertation/LiquidCrystal.tex	2006/06/20 13:46:10	2870
+++ trunk/tengDissertation/LiquidCrystal.tex	2006/06/23 21:33:52	2882
@@ -212,8 +212,8 @@ every banana shaped molecule has been represented thre
 
 A series of molecular dynamics simulations were perform to study the
 phase behavior of banana shaped liquid crystals. In each simulation,
-every banana shaped molecule has been represented three GB particles
-which is characterized by $\mu = 1,~ \nu = 2,
+every banana shaped molecule has been represented by three GB
+particles which is characterized by $\mu = 1,~ \nu = 2,
 ~\epsilon_{e}/\epsilon_{s} = 1/5$ and $\sigma_{e}/\sigma_{s} = 3$.
 All of the simulations begin with same equilibrated isotropic
 configuration where 1024 molecules without dipoles were confined in
@@ -222,6 +222,8 @@ To investigate the phase structure of the model liquid
 barostat of 50~ps were used to equilibrate the system to desired
 temperature and pressure.
 
+\subsection{Order Parameters}
+
 To investigate the phase structure of the model liquid crystal, we
 calculated various order parameters and correlation functions.
 Particulary, the $P_2$ order parameter allows us to estimate average
@@ -254,26 +256,44 @@ where $X$, $Y$ and $Z$ are axis of the director frame.
 \end{equation}
 where $X$, $Y$ and $Z$ are axis of the director frame.
 
+\subsection{Structure Properties}
 
-The density correlation along the director is
-\begin{equation}g(z) =< \delta (z-z_{ij})>_{ij} / \pi R^{2} \rho
+It is more important to show the density correlation along the
+director
+\begin{equation}
+g(z) =< \delta (z-z_{ij})>_{ij} / \pi R^{2} \rho
 \end{equation},
-where $z_{ij} = r_{ij} cos \beta_{r_{ij}}$ was measured in the
-director frame and $R$ is the radius of the cylindrical sampling
-region.
+where $z_{ij} = r_{ij} \dot Z$ was measured in the director frame
+and $R$ is the radius of the cylindrical sampling region.
+
+\subsection{Rotational Invariants}
+
+As a useful set of correlation functions to describe
+position-orientation correlation, rotation invariants were first
+applied in a spherical symmetric system to study x-ray and light
+scatting\cite{Blum1971}. Latterly, expansion of the orientation pair
+correlation in terms of rotation invariant for molecules of
+arbitrary shape was introduce by Stone\cite{Stone1978} and adopted
+by other researchers in liquid crystal studies\cite{Berardi2000}.
 
+\begin{eqnarray}
+S_{22}^{220} (r) & = & \frac{1}{{4\sqrt 5 }} \left< \delta (r -
+r_{ij} )((\hat x_i  \cdot \hat x_j )^2  - (\hat x_i  \cdot \hat y_j
+)^2  - (\hat y_i  \cdot \hat x_j )^2  + (\hat y_i  \cdot \hat y_j
+)^2 ) \right. \\
+ & & \left. - 2(\hat x_i  \cdot \hat y_j )(\hat y_i \cdot \hat x_j ) -
+2(\hat x_i  \cdot \hat x_j )(\hat y_i  \cdot \hat y_j )) \right>
+\end{eqnarray}
 
+\begin{equation}
+S_{00}^{221} (r) =  - \frac{{\sqrt 3 }}{{\sqrt {10} }}\left\langle
+{\delta (r - r_{ij} )((\hat z_i  \cdot \hat z_j )(\hat z_i  \cdot
+\hat z_j  \times \hat r_{ij} ))} \right\rangle
+\end{equation}
+
 \section{Results and Conclusion}
 \label{sec:results and conclusion}
 
 To investigate the molecular organization behavior due to different
 dipolar orientation and position with respect to the center of the
 molecule,
-
-
-
-\section{\label{liquidCrystalSection:methods}Methods}
-
-\section{\label{liquidCrystalSection:resultDiscussion}Results and Discussion}
-
-\section{Conclusion}
