Analytical expressions for bulk moduli and frequencies of volumetrical vibrations of fullerenes C$_{20}$ and C$_{60}$

In the present paper simple analytical expressions connecting bulk moduli for fullerenes C$_{20}$ and C$_{60}$ with stiffness of interatomic bond and geometrical characteristics of the fullerenes are derived. Ambiguities related to definition of the bulk modulus are discussed. Nonlinear volumetrical deformation of the fullerenes is considered. Pressure-volume dependence for the fullerenes under volumetrical compression are derived. Simple analytical model for volumetrical vibrations of the fullerenes is proposed. The expression connecting frequencies of volumetrical vibrations for fullerenes C$_{20}$ and C$_{60}$ with parameters of interatomic interactions are obtained.