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%!TEX root = /Users/charles/Desktop/nanoglass/nanoglass.tex |
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\section{Analysis} |
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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} |
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In this model, 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 |
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\begin{equation} |
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Y_{lm}\left(\vec{r}_{ij}\right)=Y_{lm}\left(\theta(\vec{r}_{ij}),\phi(\vec{r}_{ij})\right) |
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\label{eq:spharm} |
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\end{equation} |
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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 |
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\begin{equation} |
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\bar{q}_{lm}(i) = \frac{1}{N_{b}(i)}\sum_{j=1}^{N_b(i)} Y_{lm}(\vec{r}_{ij}). |
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\label{eq:local_avg_bo} |
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\end{equation} |
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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 |
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\begin{equation} |
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\bar{Q}_{lm} = \frac{\sum^{N}_{i=1}N_{b}(i)\bar{q}_{lm}(i)}{\sum^{N}_{i=1}N_{b}(i)} |
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\label{eq:sys_avg_bo} |
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\end{equation} |
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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 can be addressed by constructing the following rotationally invariant combinations $Q_l$ and $W_l$ as the second and third order invariant combinations of $\bar{Q}_{lm}$. |
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\begin{equation} |
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Q_{l} = \left( \frac{4\pi}{2 l+1} \sum^{l}_{m=-l} \left| \bar{Q}_{lm} \right|^2\right)^{1/2} |
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\label{eq:sec_ord_inv} |
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\end{equation} |
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and |
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\begin{equation} |
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\hat{W}_{l} = \frac{W_{l}}{\left( \sum^{l}_{m=-l} \left| \bar{Q}_{lm} \right|^2\right)^{3/2}} |
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\label{eq:third_ord_inv} |
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\end{equation} |
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\begin{equation} |
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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} |
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\label{eq:third_inv} |
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\end{equation} |
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where the term in parentheses is Wigner-3$j$ symbol. |
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\begin{table} |
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\caption{Calculated values of bond orientational order parameters for simple structures cooresponding to $l=4$ and $l=6$ spherical harmonic functions.\cite{wolde:9932}} |
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\begin{center} |
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\begin{tabular}{ccccc} |
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\hline |
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\hline |
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& $Q_4$ & $Q_6$ & $\hat{W}_4$ & $\hat{W}_6$\\ |
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fcc & 0.191 & 0.575 & -0.159 & -0.013\\ |
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hcp & 0.097 & 0.485 & 0.134 & -0.012\\ |
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bcc & 0.036 & 0.511 & 0.159 & 0.013\\ |
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sc & 0.764 & 0.354 & 0.159 & 0.013\\ |
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Icosahedral & 0 & 0.663 & 0 & -0.170\\ |
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(liquid) & 0 & 0 & 0 & 0\\ |
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\hline |
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\end{tabular} |
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\end{center} |
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\label{table:bopval} |
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\end{table} |
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