%0 Journal Article %1 Fisher1967Theory %A Fisher, Michael E. %A Burford, Robert J. %D 1967 %I American Physical Society %J Physical Review Online Archive (Prola) %K ising\_model correlations critical-points %N 2 %P 583--622 %R 10.1103/physrev.156.583 %T Theory of Critical-Point Scattering and Correlations. I. The Ising Model %U http://dx.doi.org/10.1103/physrev.156.583 %V 156 %X The theory of the correlations and critical scattering of two- and three-dimensional nearest-neighbor Ising models is discussed critically. A distinction is drawn between κ(T), the true inverse range of exponential decay of the correlations, and κ1(T), the effective range determined from the low-angle scattering intensity. Ten to eleven terms of appropriate high-temperature series exapnsions for κ and κ1 are determined for the square and simple cubic lattices, and shorter series are given for the triangular, fcc, and bcc lattices. For the former lattices, the complete correlation expansions are obtained to the same order. It is shown that κ and κ1 vary as (T-T0)ν when T→Tc, with ν=1 for dimensionality d=2, but ν=0.6430±0.0025≃9/14 for d=3. The asymptotic decay of correlation at T=Tc is found to be 1/rd-2+η, where η is related to the exponent γ of the divergence of the susceptibility by (2-η)ν=γ, Numerical values are η=1/4 for d=2 and η=0.056±0.008≃1/18 for d=3. The relative scattering intensity χ̂ as a function of wave number k is given to high accuracy for all T>\~Tc by χ̂(k,T)≃(a/r1)2-η(κ1a)2+φ2a2K2(k)η/2/(κ1a)2+ψa2K2(k), where (i) a is the lattice spacing, (ii) a2K2=2d1-q-1Σexp(ik·r)≃(ka)2, the sum being over the q nearest-neighbor lattice sites, (iii) r1(T) is a slowly-varying decreasing function near Tc, (iv) ψ=1+1/2ηφ2, and (v) φ(T) is slowly varying with a magnitude at Tc of 0.03 for d=2 and of 0.06 to 0.09 for d=3. Explicit formulas are given for κ1, r1, and φ as functions of T. The correlations and the scattering are isotropic near Tc. The critical scattering isotherm is curved for low k according to χ̂-1∼k2-η and it intersects the isotherms for T>Tc. Correspondingly, χ̂(k,T) exhibits a maximum for fixed k, at a temperature above Tc; for d=2 the maxima are very well marked, but for d=3 they are smaller and occur closer to Tc. The theory is compared favorably with recent neutron-scattering experiments on pure beta-brass.

@article{Fisher1967Theory, abstract = {{The theory of the correlations and critical scattering of two- and three-dimensional nearest-neighbor Ising models is discussed critically. A distinction is drawn between κ(T), the true inverse range of exponential decay of the correlations, and κ1(T), the effective range determined from the low-angle scattering intensity. Ten to eleven terms of appropriate high-temperature series exapnsions for κ and κ1 are determined for the square and simple cubic lattices, and shorter series are given for the triangular, fcc, and bcc lattices. For the former lattices, the complete correlation expansions are obtained to the same order. It is shown that κ and κ1 vary as (T-T0)ν when T→Tc, with ν=1 for dimensionality d=2, but ν=0.6430±0.0025≃9/14 for d=3. The asymptotic decay of correlation at T=Tc is found to be 1/rd-2+η, where η is related to the exponent γ of the divergence of the susceptibility by (2-η)ν=γ, Numerical values are η=1/4 for d=2 and η=0.056±0.008≃1/18 for d=3. The relative scattering intensity χ̂ as a function of wave number k is given to high accuracy for all T>\~{}Tc by χ̂(k,T)≃(a/r1)2-η[(κ1a)2+φ2a2K2(k)]η/2/[(κ1a)2+ψa2K2(k)], where (i) a is the lattice spacing, (ii) a2K2=2d[1-q-1Σexp(ik·r)]≃(ka)2, the sum being over the q nearest-neighbor lattice sites, (iii) r1(T) is a slowly-varying decreasing function near Tc, (iv) ψ=1+1/2ηφ2, and (v) φ(T) is slowly varying with a magnitude at Tc of 0.03 for d=2 and of 0.06 to 0.09 for d=3. Explicit formulas are given for κ1, r1, and φ as functions of T. The correlations and the scattering are isotropic near Tc. The critical scattering isotherm is curved for low k according to χ̂-1∼k2-η and it intersects the isotherms for T>Tc. Correspondingly, χ̂(k,T) exhibits a maximum for fixed k, at a temperature above Tc; for d=2 the maxima are very well marked, but for d=3 they are smaller and occur closer to Tc. The theory is compared favorably with recent neutron-scattering experiments on pure beta-brass.}}, added-at = {2019-06-10T14:53:09.000+0200}, author = {Fisher, Michael E. and Burford, Robert J.}, biburl = {https://www.bibsonomy.org/bibtex/2a098cc6528908c66a9a35efb9c7de8b6/nonancourt}, citeulike-article-id = {7560081}, citeulike-linkout-0 = {http://dx.doi.org/10.1103/physrev.156.583}, citeulike-linkout-1 = {http://link.aps.org/abstract/PR/v156/i2/p583}, citeulike-linkout-2 = {http://link.aps.org/pdf/PR/v156/i2/p583}, doi = {10.1103/physrev.156.583}, interhash = {09e03f7b9e32bbcfd7364f2ac89d8bd5}, intrahash = {a098cc6528908c66a9a35efb9c7de8b6}, journal = {Physical Review Online Archive (Prola)}, keywords = {ising\_model correlations critical-points}, month = apr, number = 2, pages = {583--622}, posted-at = {2012-08-22 11:38:47}, priority = {2}, publisher = {American Physical Society}, timestamp = {2019-08-23T10:59:34.000+0200}, title = {{Theory of Critical-Point Scattering and Correlations. I. The Ising Model}}, url = {http://dx.doi.org/10.1103/physrev.156.583}, volume = 156, year = 1967 }