Classical Heisenberg model of magnetic molecular ring clusters: Accurate approximants for correlation functions and susceptibility

We show that the measured magnetic susceptibility of molecular ring clusters can be accurately reproduced, for all but low temperatures T, by a classical Heisenberg model of N identical spins S on a ring that interact with isotropic nearest-neighbor interactions. While exact expressions for the two-spin correlation function, CN(n,T), and the zero-field magnetic susceptibility, χN(T), are known for the classical Heisenberg ring, their evaluation involves summing infinite series of modified spherical Bessel functions. By contrast, the formula CN(n,T)=(un+uN-n)/(1+uN), where u(K)=cothK-K-1 is the Langevin function and K=JS(S+1)/(kBT) is the nearest-neighbor dimensionless coupling constant, provides an excellent approximation if N≥6 for the regime [K]<3. This choice of approximant combines the expected exponential decay of correlations for increasing yet small values of n, with the cyclic boundary condition for a finite ring, CN(n,T)=CN(N-n,T). By way of illustration, we show that, for T>50K, the associated approximant for the susceptibility derived from the approximate correlation function is virtually indistinguishable from both the exact theoretical susceptibility and the experimental data for the open-quotes ferric wheelclose quotes molecular cluster ([Fe(OCH3)2(O2CCH2Cl)]10), which contains N=10 interacting Fe3+ ions, each of spin S=5/2, that are symmetrically positioned in a nearly planar ring. copyright 1998 American Institute of Physics