Magnetoelectric effects theory by heisenberg method based on permutation group symmetry of nanoparticles

The Heisenberg theory of ferromagnetism is widened to include external electric field action. The material relations are derived by means of differentiation of logarithm of partition function with respect to the magnetic and electric fields. The mean energy coefficients as the exchange integrals combinations are expressed via characters of irreducible representations of corresponding permutation groups by the Heitler method. The thermodynamic equations of state for polarization and magnetization, as functions of the electric and magnetic fields, are derived and illustrated by figures. The magnetization and hysteresis curves in magnetization — magnetic field components plane are built. The theory is applied to nanoparticles, the particle partition function is modeled as the product of the surface and bulk parts. The statistical sum is constructed having explicit expressions for the mean energy in terms of exchange integrals and number of closest neighbors for surface and bulk atoms. The relative contribution of the surface and bulk terms is evaluated.