Normally the Debye–Waller factor vanishes in 1D and 2D at finite temperature, and in 1D at zero temperature as well. The situation can be different for cases with long-range coupling between atoms, which can result in non-linear phonon dispersion of the form ω(k)∼kα at long wavelength, with α<1.
(i) Show that the Debye–Waller factor does not vanish for a 2D lattice at finite T, as long as 0<α<1. Show that the same is true for a 1D lattice at T=0.
(ii) Find the range of α within which the Debye–Waller factor does not vanish for the 1D lattice at finite T.
(iii) Now consider a more exotic phonon dispersion at long wavelength of the form ω(k)∼k∣lnk∣γ. Find the conditions γ must satisfy in order for the Debye–Waller factor not to vanish for a 2D lattice at finite T, or a 1D lattice at T=0, respectively.