Finite-size modifications of the magnetic properties of clusters

The spin-wave spectrum of Heisenberg spin clusters of various structures (bcc, fcc, and disordered) ranging in size between 9 and 749 spins is calculated by a self-consistent diagonalization of the equation of motion of ${\mathit{S}}^{+}$ in real space. The spin-wave spectrum of the clusters is strongly modified relative to the bulk, and the consequent neutron-scattering cross section exhibits discretely spaced wave-vector-broadened eigenstates. The implications of the finite size on thermodynamic properties, like the temperature dependence of the magnetization and the critical temperature, are also elucidated. We find the temperature dependence of the cluster magnetization to be well described by an effective power law, ${\mathit{M}}_{\mathrm{mean}}$∝1-${\mathit{BT}}^{\mathrm{\alpha }}$, with a size-dependent, but structure-independent, exponent larger than the bulk value. The critical temperature of the clusters is calculated from the spin-wave spectrum by a method based on the correlation theory and the spherical approximation generalized to the case of finite systems. A size-dependent reduction of the critical temperature by up to 50% for the smallest clusters is found. The trends found for the model clusters are extrapolated to the size regime of nanoscale particles.