| 2 |
|
|
| 3 |
|
One method of analysis that has been used extensively for determining local and extended orientational symmetry of a central atom with its surrounding neighbors is that of Bond-orientational analysis as formulated by Steinhart et.al.\cite{Steinhardt:1983mo} |
| 4 |
|
In this model of bond-orientational analysis, a set of spherical harmonics is associated with its near neighbors as defined by the first minimum in the radial distribution function forming an association that Steinhart et.al termed a "bond". More formally, this set of numbers is defined as |
| 5 |
– |
|
| 5 |
|
\begin{equation} |
| 6 |
|
Y_{lm}\left(\vec{r}_{ij}\right)=Y_{lm}\left(\theta(\vec{r}_{ij}),\phi(\vec{r}_{ij})\right) |
| 7 |
|
\label{eq:spharm} |
| 8 |
|
\end{equation} |
| 9 |
< |
where, $\{ Y_{lm}(\theta,\phi)\}$ are spherical harmonic functions, and $\theta(\vec{r})$ and $\phi(\vec{r})$ are the polar angles made by the bond with respect to a reference coordinate system. We chose for simplicity the origin as defined by the coordinates for our nanoparticle. The local environment surrounding any given atom in our system can be defined by the average over all bonds surrounding that central atom |
| 11 |
< |
|
| 9 |
> |
where, $\{ Y_{lm}(\theta,\phi)\}$ are spherical harmonic functions, and $\theta(\vec{r})$ and $\phi(\vec{r})$ are the polar angles made by the bond with respect to a reference coordinate system. We chose for simplicity the origin as defined by the coordinates for our nanoparticle. The local environment surrounding any given atom in a system can be defined by the average over all bonds, $N_b(i)$, surrounding that central atom |
| 10 |
|
\begin{equation} |
| 11 |
|
\bar{q}_{lm}(i) = \frac{1}{N_{b}(i)}\sum_{j=1}^{N_b(i)} Y_{lm}(\vec{r}_{ij}). |
| 12 |
|
\label{eq:local_avg_bo} |
| 13 |
|
\end{equation} |
| 14 |
< |
|
| 17 |
< |
We can further define a system average orientational-bond order over all $\bar{q}_{lm}$ |
| 14 |
> |
We can further define a global average orientational-bond order over all $\bar{q}_{lm}$ by calculating the average of $\bar{q}_{lm}(i)$ over all $N$ particles |
| 15 |
|
\begin{equation} |
| 16 |
|
\bar{Q}_{lm} = \frac{\sum^{N}_{i=1}N_{b}(i)\bar{q}_{lm}(i)}{\sum^{N}_{i=1}N_{b}(i)} |
| 17 |
|
\label{eq:sys_avg_bo} |
| 18 |
|
\end{equation} |
| 19 |
< |
The $\bar{Q}_{lm}$ contained in Equation(\ref{eq:sys_avg_bo}) is not necessarily invariant with respect to rotation to the reference coordinate system. To solve this |
| 20 |
< |
|
| 19 |
> |
The $\bar{Q}_{lm}$ contained in Equation(\ref{eq:sys_avg_bo}) is not necessarily invariant with respect to rotation to the reference coordinate system. This issue can be addressed by constructing rotationally invariant combinations |
| 20 |
> |
\begin{equation} |
| 21 |
> |
Q_{l} = \left( \frac{4\pi}{2 l+1} \sum^{l}_{m=-l} \left| \bar{Q}_{lm} \right|^2\right)^{1/2} |
| 22 |
> |
\label{eq:sec_ord_inv} |
| 23 |
> |
\end{equation} |
| 24 |
> |
and |
| 25 |
> |
\begin{equation} |
| 26 |
> |
\hat{W}_{l} = \frac{W_{l}}{\left( \sum^{l}_{m=-l} \left| \bar{Q}_{lm} \right|^2\right)^{3/2}} |
| 27 |
> |
\label{eq:third_ord_inv} |
| 28 |
> |
\end{equation} |
| 29 |
> |
\begin{equation} |
| 30 |
> |
W_{l} = \sum_{\substack{m_1,m_2,m_3 \\m_1 + m_2 + m_3 = 0}} \left( \begin{array}{ccc} l & l & l \\ m_1 & m_2 & m_3 \end{array}\right) \\ \times \bar{Q}_{lm_1}\bar{Q}_{lm_2}\bar{Q}_{lm_3} |
| 31 |
> |
\label{eq:third_inv} |
| 32 |
> |
\end{equation} |
| 33 |
> |
where $Q_l$ and $W_l$ are the second and third order invariant combinations of $\bar{Q}_{lm}$. |
