Schematic diagram showing the relationship between solidlike and liquidlike clusters and the dissociation threshold that results from the loss of a single atom from the cluster. At the cluster is solidlike but dissociation occurs from the liquidlike cluster. is the energy difference between the solidlike cluster and solidlike products, and is the energy difference between liquidlike cluster and liquidlike products. is the latent heat for melting of the cluster at its melting temperature (where the number of solidlike clusters equals the number of liquidlike). and are the energy differences between the liquidlike and solidlike states at for the atom and atom clusters, respectively.
Plot of the dissociation energies of the liquidlike and solidlike clusters obtained from the analysis described in the text. The unfilled green circles show the values (for dissociation from the liquidlike cluster to give liquidlike products) and the filled black circles show the values (for dissociation from the solidlike cluster to give solidlike products). The values are obtained directly from analysis of the measured dissociation thresholds, the values result from adding to the values the contribution from the latent heats of the reactants and products.
The upper panel shows the cohesive energies for solidlike clusters obtained from the analysis described in the text. The filled black points are the values determined from the experimental measurements. The unfilled red points show cohesive energies determined with density-functional theory for the lowest energy structures that were found for each cluster size in the calculations. The lower panel shows the latent heat per cluster plotted against cluster size (from Ref. 19).
A selection of cluster structures. The different rows, from top to bottom, show examples of distorted decahedral fragments, perfect fcc, fcc with stacking faults (SFs), and disordered polytetrahedral isomers, respectively. The number of atoms is shown in the left side of each structure. For two of the disordered structures, we also show the first coordination shell of the innermost atom.
(a) A perfect 54-atom decahedron. The thick dashed line shows the atoms that must be removed in order to obtain a 35-atom decahedral fragment, which exposes two energetically unfavorable (100) facets, shown with arrows in (b). Adding one more atom (yellow or light gray) to the bottom row of atoms and displacing this row removes the (100) facets and results in the perfect 36-atom cluster shown in (c). Further distortion creates an additional fivefold axis as demonstrated in the rotated view. This cluster exposes only (111) compact facets. (d) shows how the 44-atom cluster can also be considered an undistorted fragment of the same 54-atom parent decahedron. This cluster does not preserve the fivefold axis and belongs therefore to the SF family.
Structural indicators for clusters from the calculations. (a) shows the average bond length, (b) its root-mean-squared deviation, (c) the number of atoms at (100) facets and (d) the asphericity shape parameter, for the global minimum structures found in this work. For , there is one disordered isomer nearly degenerate with the ordered ground state, so we quote the structural indicators for both isomers, joined with a dotted line.
Vertical electron affinities (VEAs) of clusters, calculated for each size as the total energy difference between the cation and the neutral cluster at the optimal geometry of the cation.
The left panel shows a comparison of dissociation and cohesive energies for neutral sodium clusters with 135–154 atoms. Note that the dissociation energy of is substantially lower than the cohesive energy of . The right panel shows a blowup of the cohesive energies. A significant drop only occurs between 147 and 148 atoms.
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