Effective real-space correlations of crystal lattice vibrations

Effective phonon fields coupled to a temperature-dependent permeability tensor are introduced to model empirical thermodynamic functions of crystal lattices and the temperature variation of Debye-Waller factors inferred by X-ray, gamma-ray or neutron diffraction. The permeabilities generate a varying Debye temperature and a temperature-dependent spectral cutoff in the partition function as well as an effective temperature-dependent oscillator mass, to be calculated from diffraction and heat capacity data. The zero-point internal energy of the phonon field is extracted from low-temperature Debye-Waller B-factor measurements. The varying spectral cutoff, Debye temperature and oscillator mass determine the temperature evolution of real-space correlation functions. Closed integral representations are derived for the effective two-point function correlating isotropic lattice vibrations in monatomic cubic crystals as well as for the reduced four-point function correlating fluctuations around the mean-squared atomic displacement. The correlations are long-range with power-law tails and become oscillatory at low temperature. The formalism is illustrated with the correlation functions of copper.