Cluster approximation forq-dependent correlations in magnetic and ferroelectric systems

Abstract

A self-consistent cluster approximation is developed for the wave-vector ($\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}$)-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 $\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}$-dependent susceptibilities with clusters of size $N$ is found to be ${x}_{\ensuremath{\nu}}^{\ensuremath{-}1}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})={C}^{\ensuremath{-}1}T[{M}_{\ensuremath{\nu}}^{\ensuremath{-}1}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})\ensuremath{-}(1\ensuremath{-}C)]$, $\ensuremath{\nu}=1,2,\dots{},N$, where ${M}_{\ensuremath{\nu}}^{\ensuremath{-}1}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ are the eigenvalues of the Fourier transform of ${({M}^{\ensuremath{-}1})}_{\mathrm{ij}}$ where ${M}_{\mathrm{ij}}$ 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 $\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}$ 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 K${\mathrm{D}}_{2}$P${\mathrm{O}}_{4}$. We also investigate the ice-rule limit of our results. In that case a singularity of the type ${\ensuremath{\chi}}^{\ensuremath{-}1}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})\ensuremath{\simeq}{\ensuremath{\chi}}^{\ensuremath{-}1}(0)+B(T)\frac{q_{z}^{}{}_{}{}^{2}}{({2q}_{\ensuremath{\perp}}^{2}+{q}_{z}^{2})}$ for $q\ensuremath{\rightarrow}0$ is found, similar to that generated by long-range dipolar forces.