N = 5 cluster has no transition structure
If we consider before and after snapshots of an N = 5 cluster going through a transition, we observe that a outer sphere has moved into the available vacancy, and the vacancy is now at the coordinate position the outer sphere de-occupied. However, careful observation of the transition shows that this is not accomplished by the jumping of a single outer sphere. Rather, the cluster enters a metastable set of states, the non-entropically preferred continuum, and then collapses into a nearby square pyramid state. We demonstrate this in the figure. At its densest, the state of the N=5 cluster can be specified by exactly two numbers, the smallest two angles between outer spheres on the equator of the cluster. We plot (θ1, θ2, θ3) the three angles between the outer spheres on the equator. This vector is restricted to a planar triangle. When the cluster is in a square pyramid (SP) state, the state of the system is near a corner of the triangle. Illustrations of the three SP states are shown. The rest of the triangle represents the phase space of all the states that are not a square pyramid. The triangular bipyramid (TBP) is in the center of the phase space. In the figure below, we show the path of the cluster through state space during three transitions. There is no single point that the paths pass through. Rather the cluster enters the metastable middle of state space and then collapses to a nearby corner of state space.
Below is the triangular bipyramid structure that corresponds to the center of the triangle above.
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