The Fullerene-like Structure of Superheavy Element Z=120 (Greinerium)-a tribute to Walter Greiner

Abstract: We review the basic ideas and some theoretical models behind the concept of fullerene-like structures made of $\alpha$-particles. The possibility of such a peculiar nuclear shape developing in a double magic superheavy nucleus with $Z$=120 was mentioned for the first time in the literature by Walter Greiner and then constantly defended by him. We provide estimates of the energy of such metastable states within the liquid-drop model. In the second part of this paper we discuss a simple model rooted in the nontopological soliton model consisting of a complex scalar field describing a finite system of $\alpha$ bosons with non-linear self-interactions coupled to the electromagnetic field. We demonstrate that this model predicts density depletion in the central region of a soliton-like structure for a range of model parameters.