| 271 |
|
As a useful set of correlation functions to describe |
| 272 |
|
position-orientation correlation, rotation invariants were first |
| 273 |
|
applied in a spherical symmetric system to study x-ray and light |
| 274 |
< |
scatting\cite{Blum1971}. Latterly, expansion of the orientation pair |
| 274 |
> |
scatting\cite{Blum1972}. Latterly, expansion of the orientation pair |
| 275 |
|
correlation in terms of rotation invariant for molecules of |
| 276 |
|
arbitrary shape was introduce by Stone\cite{Stone1978} and adopted |
| 277 |
< |
by other researchers in liquid crystal studies\cite{Berardi2000}. |
| 277 |
> |
by other researchers in liquid crystal studies\cite{Berardi2003}. |
| 278 |
|
|
| 279 |
|
\begin{eqnarray} |
| 280 |
< |
S_{22}^{220} (r) & = & \frac{1}{{4\sqrt 5 }}\left\langle {\delta (r |
| 281 |
< |
- r_{ij} )((\hat x_i \cdot \hat x_j )^2 - (\hat x_i \cdot \hat |
| 282 |
< |
y_j )^2 - (\hat y_i \cdot \hat x_j )^2 + (\hat y_i \cdot \hat |
| 283 |
< |
y_j |
| 284 |
< |
)^2 ) \right.\\ |
| 285 |
< |
& & \left.- 2(\hat x_i \cdot \hat y_j )(\hat y_i \cdot \hat x_j ) - |
| 286 |
< |
2(\hat x_i \cdot \hat x_j )(\hat y_i \cdot \hat y_j ))} |
| 287 |
< |
\right\rangle |
| 280 |
> |
S_{22}^{220} (r) & = & \frac{1}{{4\sqrt 5 }} \left< \delta (r - |
| 281 |
> |
r_{ij} )((\hat x_i \cdot \hat x_j )^2 - (\hat x_i \cdot \hat y_j |
| 282 |
> |
)^2 - (\hat y_i \cdot \hat x_j )^2 + (\hat y_i \cdot \hat y_j |
| 283 |
> |
)^2 ) \right. \\ |
| 284 |
> |
& & \left. - 2(\hat x_i \cdot \hat y_j )(\hat y_i \cdot \hat x_j ) - |
| 285 |
> |
2(\hat x_i \cdot \hat x_j )(\hat y_i \cdot \hat y_j )) \right> |
| 286 |
|
\end{eqnarray} |
| 287 |
|
|
| 288 |
|
\begin{equation} |
