Fractional phase transition in medium size metal clusters and some remarks on magic numbers in gravitationally and weakly interacting clusters

TL;DR

This paper introduces a fractional rotation group model to predict magic numbers in metal clusters and suggests a phase transition in cluster size, with potential implications for understanding fundamental forces.

Contribution

It develops a fractional extension of the rotation group to model cluster magic numbers and links fractional derivatives to phase transitions and fundamental interactions.

Findings

Predicts magic numbers in metal clusters using fractional rotation groups.

Identifies a second order phase transition in cluster size around N=200-300.

Connects fractional derivatives to fundamental forces in nature.

Abstract

Based on the Riemann- and Caputo definition of the fractional derivative we use the fractional extensions of the standard rotation group SO(3) to construct a higher dimensional representation of a fractional rotation group with mixed derivative types. An analytic extended symmetric rotor model is derived, which correctly predicts the sequence of magic numbers in metal clusters. It is demonstrated, that experimental data may be described assuming a sudden change in the fractional derivative parameter $\alpha$ which is interpreted as a second order phase transition in the region of cluster size with $200 \leq N \leq 300$. Furthermore it is demonstrated, that the four different realizations of higher dimensional fractional rotation groups may successfully be connected to the four fundamental interaction types realized in nature and may be therefore used for a prediction of magic numbers and binding energies of clusters with gravitational force and weak force respectively bound constituents. The results presented lead to the conclusion, that mixed fractional derivative operators might play a key role for a successful unified theoretical description of all four fundamental forces realized in nature.