  The method of calculating the free energy and thermodynamic characteristics
of the classical n-vector three-dimensional (3D) magnetic model at the
microscopic level without any adjustable parameters is proposed. Mathematical
description is perfomed using the collective variables (CV) method in the
framework of the $\rho^4$ model approximation. The exponentially decreasing
function of the distance between the particles situated at the N sites of a
simple cubic lattice is used as the interaction potential. Explicit and
rigorous analytical expressions for entropy,internal energy, specific heat near
the phase transition point as functions of the temperature are obtained. The
dependence of the amplitudes of the thermodynamic characteristics of the system
for $T>T_c$ and $T<T_c$ on the microscopic parameters of the interaction
potential are studied for the cases $n=1,2,3$ and $n\to\infty$. The obtained
results provide the basis for accurate analysis of the critical behaviour in
three dimensions including the nonuniversal characteristics of the system.
