Surface spin fluctuations in the paramagnetic phase

We have analyzed the static and dynamic properties of the surface spin fluctuations (SSF) within a Ginzburg-Landau-type theory for the paramagnetic phase. We have limited ourselves to only the quadratic terms in the fluctuations from mean-field value and have derived the generalized-susceptibility, contribution to the specific heat due to the surface and the friction coefficient $\eta$ that determines the viscous part of the Brownian motion of a spin of mass $M$ at the surface. We find that the specific heat due to SSF is divergent as ${(T-T_c)}^{-\frac{1}{2}}$ when the surface contribution to the free energy is positive and ${(T-T_s)}^{-1\mathrm{}}$ when the surface contribution is negative. $T_s$ is the ordering temperature for surface spins such that $T_s>T_c$. Away from $T_c$, the time scale of relaxation to equilibrium is governed by the bulk relaxation time $\tau$, but at $T_c$, when $\tau$ is divergent, for positive surface energy the spins relax with a $t^{-\frac{3}{2}}$ law. For motion perpendicular to the surface ${\eta }_{\mathrm{zz}}$ has a cusp at $T_c$ for a positive surface-energy term and diverges as ${(T-T_s)}^{-1\mathrm{}}$ when it is negative. In contrast, for surface diffusion ${\eta }_{11}$ is finite at $T_c$ and decays linearly for positive surface energy. It has a logarithmic divergence at $T_s$ for negative surface energy.