Critical behavior of energy-energy, strain-strain, higher-harmonics, and similar correlation functions

The structure factor associated with general biquadratic correlation functions is calculated for an n-component order parameter using ε-expansion techniques in d = 4 - εdimensions. The results apply to energy-energy and strain-strain correlations as well as to correlations of higher harmonics in density wave systems. We find the correlations of these secondary order parameters to be characterized by a correlation length ξ̂=ξ̂₀[(T-T_C)/T_C]^-v, with the same n-dependent exponent v as for the correlation length characterizing fluctuations of the primary order parameter, which is denoted by ξ. The amplitude ratio X̂ = (ξ̂₀/ξ₀)² is universal, and we obtain X_T= γ_T/6γ+O(ε³) for quadratic order parameters transforming like a traceless spin tensor in n-component space (with γ_T characterizing the divergence of the corresponding susceptibility) and X_E= α/6γ+O(ε³) for energy-energy correlations, where α and γ denote the usual specific heat and susceptibility critical exponents, respectively. The universal amplitude ratio for the second harmonic in density wave systems is given by _T with n=2 and takes the value X₂= ε/20-ε²/100+0(ε³), thus being very small. This naturally explains previously puzzling experimental results for the critical behavior of the second harmonic structure factor at the nematic-smectic-A₂ transition of a thermotropic liquid crystal. Applications to sound attenuation in liquids or solids close to critical transitions and to colloidal interactions in near-critical binary mixtures are briefly discussed.