Stacking Principle and Magic Sizes of Transition Metal Nanoclusters Based on Generalized Wulff Construction

Nanoclusters with extra stability at certain cluster sizes are known as magic clusters with exotic properties. The classic Wulff construction principle, which stipulates that the preferred structure of a cluster should minimize its total surface energy, is often invoked in determining the cluster magicity, resulting in close-shelled Mackay icosahedronal clusters with odd-numbered magic sizes of 13, 55, 147, etc. Here we use transition metal clusters around size 55 as prototypical examples to demonstrate that, in the nanometer regime, the classic Wulff construction principle needs to be generalized to primarily emphasize the edge atom effect instead of the surface energy. Specifically, our detailed calculations show that nanoclusters with much shorter total edge lengths but substantially enlarged total surface areas are energetically much more stable. As a consequence, a large majority of the nanoclusters within the $3d$-, $4d$-, and $5d$-transition metal series are found to be fcc or hcp crystal fragments with much lower edge energies, and the widely perceived magic size of 55 is shifted to its nearby even numbers.