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\section{Analysis} |
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Frank first proposed icosahedral arrangement of atoms as a model for structure supercooled atomic liquids.\cite{19521106} The ability to cool simple liquid metals well below their equilibrium melting tempatures was attributed to this icosahedral local ordering. Frank further showed that a 13-atom icosahedral cluster has a 8.4\% higher binding energy the either a face center cubic or hexagonal close packed crystal structure. Icosahedra also have six fivefold symmetry axes that cannot be extended indefinitely in three dimensions making them incommensurate with long-range positional crystallographic order. This does not preclude icosahedral clusters from posessing long-range orientational order. The "frustrated" packing of these icosahedral structures into dense clusters has been proposed as a model for glass formation.\cite{19871127} The size of the icosahedral clusters increase until frustration prevents any further growth near the glass .\cite{HOARE:1976fk} Molecular Dynamics calculations of a Lennard-Jones binary glass shows that a two component glass has clusters of face-sharing icosahedra that are distributed throughout the material.\cite{PhysRevLett.60.2295} |
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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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where, $\{ Y_{lm}(\theta,\phi)\}$ are spherical harmonics, and $\theta(\vec{r})$ and $\phi(\vec{r})$ are the polar and azimuthal angles made by the bond vector $\vec{r}$ with respect to a reference coordinate system. We chose for simplicity the origin as defined by the coordinates for our nanoparticle. (Only even-$l$ spherical harmonics are considered since permutation of a pair of identical particles should not affect the bond-order parameter.) 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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\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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The $\bar{Q}_{lm}$ contained in Equation(\ref{eq:sys_avg_bo}) is not necessarily invariant with respect to rotation of 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} |
