Cluster approximation for q-dependent correlations in magnetic and ferroelectric systems

A self-consistent cluster approximation is developed for the wave-vector (q→)-dependent spin-spin correlation in Ising models describing magnetic and ferroelectric systems. The method is particularly suitable for describing systems with competing short-range interactions. The selfconsistent approximation for the q→-dependent susceptibilities with clusters of size N is found to be xν-1(q→)=C-1T[Mν-1(q→)-(1-C)], ν=1,2,...,N, where Mν-1(q→) are the eigenvalues of the Fourier transform of (M-1)ij where Mij is the pair-correlation matrix of spins within the cluster calculated by the exact Hamiltonian of the cluster. The constant C is the ratio of the number of nearest neighbors inside the cluster to the total number of nearest neighbors. The method is applied to calculate scattering intensities in potassium-dihydrogen-phosphate-type hydrogen-bonded ferroelectrics. We find a strong anisotropy in the q→ dependence of the intensity, exhibiting a strong suppression of fluctuations along the easy (z) axis. The results are found to be in good agreement with neutron scattering data in KD2PO4. We also investigate the ice-rule limit of our results. In that case a singularity of the type χ-1(q→)χ-1(0)+B(T)qz2(2q2+qz2) for q→0 is found, similar to that generated by long-range dipolar forces.