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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 temperatures 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 possessing 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} Molecular Dynamics simulations of freezing of single component metalic nanoclusters have shown a tendency for icosohedral structure formation particularly at the surface.\cite{Gafner:2004bg,PhysRevLett.89.275502,Ascencio:2000qy,Chen:2004ec} |
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Frank first proposed icosahedral arrangement of atoms as a model for |
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structure supercooled atomic liquids.\cite{19521106} The ability to |
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cool simple liquid metals well below their equilibrium melting |
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temperatures was attributed to this local icosahedral ordering. Frank |
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further showed that a 13-atom icosahedral cluster has a 8.4\% higher |
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binding energy the either a face centered cubic ({\sc fcc}) or |
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hexagonal close-packed ({\sc hcp}) crystal structure. Icosahedra also |
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have six fivefold symmetry axes that cannot be extended indefinitely |
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in three dimensions making them incommensurate with long-range |
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translational order. This does not preclude icosahedral clusters from |
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possessing long-range {\it orientational} order. The ``frustrated'' |
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packing of these icosahedral structures into dense clusters has been |
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proposed as a model for glass formation.\cite{19871127} The size of |
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the icosahedral clusters is thought to increase until frustration |
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prevents any further growth.\cite{HOARE:1976fk} Molecular dynamics |
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simulations of a two-component Lennard-Jones glass showed that |
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clusters of face-sharing icosahedra are distributed throughout the |
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material.\cite{PhysRevLett.60.2295} Simulations of freezing of single |
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component metalic nanoclusters have shown a tendency for icosohedral |
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structure formation particularly at the surfaces of these |
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clusters.\cite{Gafner:2004bg,PhysRevLett.89.275502,Ascencio:2000qy,Chen:2004ec} |
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Experimentally, the splitting (or shoulder) on the second peak of the |
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X-ray structure factor in binary metallic glasses has been attributed |
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to the formation of tetrahedra that share faces of adjoining |
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icosahedra.\cite{Waal:1995lr} |
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Various structural probes have been used to characterize structural order in systems including common neighbor analysis, voronoi-analysis and orientational bond-order parameters.\cite{HoneycuttJ.Dana_j100303a014,Iwamatsu:2007lr,hsu:4974,nose:1803} Experimentally, the splitting (or shoulder) on the second peak of the X-ray structure factor in binary metal glasses has been attributed to the formation of face-sharing tetrahedra.\cite{Waal:1995lr} These tetraherda form structural units that are linked by sharing of an icosohedron creating face sharing icosohedron linked by tetrahedral structures. |
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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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Various structural probes have been used to characterize structural |
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order in systems including: common neighbor analysis, voronoi-analysis |
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and orientational bond-order |
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parameters.\cite{HoneycuttJ.Dana_j100303a014,Iwamatsu:2007lr,hsu:4974,nose:1803} |
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One method that has been used extensively for determining local and |
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extended orientational symmetry in condensed phases is the |
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bond-orientational analysis formulated by Steinhart |
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et.al.\cite{Steinhardt:1983mo} In this model, a set of spherical |
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harmonics is associated with each of the near neighbors of a central |
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atom. Neighbors (or ``bonds'') are defined as having a distance from |
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the central atom that is within the first peak in the radial |
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distribution function. The spherical harmonic between a central atom |
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$i$ and a neighboring atom $j$ is |
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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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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 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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where, $\{ Y_{lm}(\theta,\phi)\}$ are spherical harmonics, and |
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$\theta(\vec{r})$ and $\phi(\vec{r})$ are the polar and azimuthal |
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angles made by the bond vector $\vec{r}$ with respect to a reference |
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coordinate system. We chose for simplicity the origin as defined by |
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the coordinates for our nanoparticle. (Only even-$l$ spherical |
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harmonics are considered since permutation of a pair of identical |
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particles should not affect the bond-order parameter.) The local |
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environment surrounding atom $i$ can be defined by |
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the average over all neighbors, $N_b(i)$, surrounding that 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}(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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We can further define a global average orientational-bond order over |
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all $\bar{q}_{lm}$ by calculating the average of $\bar{q}_{lm}(i)$ |
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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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\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 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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The $\bar{Q}_{lm}$ contained in Equation(\ref{eq:sys_avg_bo}) is not |
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necessarily invariant with respect to rotation of the arbitrary reference |
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coordinate system. |
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Second- and third-order rotationally invariant combinations, $Q_l$ and |
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$W_l$, can be taken by summing over $m$ values 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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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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where |
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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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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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The factor in parentheses in Eq. \ref{eq:third_inv} is the Wigner-3$j$ |
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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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\caption{Values of bond orientational order parameters for |
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simple structures cooresponding to $l=4$ and $l=6$ spherical harmonic |
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functions.\cite{wolde:9932} $Q_4$ and $\hat{W}_4$ have values for {\it |
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individual} icosahedral clusters, but these values are not invariant |
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under rotations of the reference coordinate systems. Similar behavior |
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is observed in the bond-orientational order parameters for individual |
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liquid-like structures.} |
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\begin{center} |
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\begin{tabular}{ccccc} |
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\hline |
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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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Icosahedral & - & 0.663 & - & -0.170\\ |
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(liquid) & 0 & 0 & 0 & 0\\ |
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(liquid) & - & - & - & -\\ |
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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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For ideal face-centered cubic ({\sc fcc}), body-centered cubic ({\sc |
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bcc}) and simple cubic ({\sc sc}) as well as hexagonally close-packed |
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({\sc hcp}) structures, these rotationally invariant bond order |
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parameters have fixed values independent of the choice of coordinate |
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reference frames. For ideal icosahedral structures, the $l=6$ |
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invariants, $Q_6$ and $\hat{W}_6$ are also independent of the |
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coordinate system. $Q_4$ and $\hat{W}_4$ will have non-vanishing |
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values for {\it individual} icosahedral clusters, but these values are |
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not invariant under rotations of the reference coordinate systems. |
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Similar behavior is observed in the bond-orientational order |
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parameters for individual liquid-like structures. |
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|
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Additionally, both $Q_6$ and $\hat{W}_6$ have extreme values for the |
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icosahedral clusters. This makes the $l=6$ bond-orientational order |
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parameters particularly useful in identifying the extent of local |
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icosahedral ordering in condensed phases. For example, a local |
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structure which exhibits $\hat{W}_6$ values near -0.17 is easily |
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identified as an icosahedral cluster and cannot be mistaken for |
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distorted cubic or liquid-like structures. |
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|
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|
