  The three dimensional mean spherical model on a hypercubic lattice with a
film geometry $L\times \infty ^2$ under periodic boundary conditions is
considered in the presence of an external magnetic field $H$. The universal
Casimir amplitude $\Delta $ and the Binder's cumulant ratio $B$ are calculated
exactly and found to be $\Delta =-2\zeta (3)/(5\pi)\approx -0.153051$ and
$B=2\pi /(\sqrt{5}\ln ^3[(1+\sqrt{5})/2]).$ A discussion on the relations
between the finite temperature $C$-function, usually defined for quantum
systems, and the excess free energy (due to the finite-size contributions to
the free energy of the system) scaling function is presented. It is
demonstrated that the $C$-function of the model equals 4/5 at the bulk critical
temperature $T_c$. It is analytically shown that the excess free energy is a
monotonically increasing function of the temperature $T$ and of the magnetic
field $|H|$ in the vicinity of $T_c.$ This property is supposed to hold for any
classical $d$-dimensional $O(n),n>2,$ model with a film geometry under periodic
boundary conditions when $d\leq 3$. An analytical evidence is also presented to
confirm that the Casimir force in the system is negative both below and in the
vicinity of the bulk critical temperature $T_c.$

Statistical Mechanics;High Energy Physics - Theory