Classical Heisenberg model of magnetic molecular rings 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, C-N(n,T), and the zero-held magnetic susceptibility, chi(N)(T), are known for the classical Heisenberg ring, their evaluation involves summing infinite series of modified spherical Bessel functions. By contrast, the formula C-N(n,T)=(u(n)+u(N-n))/(1+u(N)), where u(K)=cothK-K-1 is the Langevin function and K=JS(S+1)/(k(B)T) is the nearest-neighbor dimensionless coupling constant, provides an excellent approximation if N greater than or equal to 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, C-N(n,T)=C-N(N-n,T), By way of illustration, we show that, for T>50 K, 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 "ferric wheel" 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. (C) 1998 American Institute of Physics.