In this paper (arXiv:2311.08435) the one-loop quantum corrections to the internal energy of lattices due to the quantum fluctuations of the scalar field of phonons are studied. The band spectrum of the lattice is characterised in terms of the scattering data, allowing to compute the total Helmholtz free energy and the entropy at finite non zero temperature. Some examples of three dimensional periodic potentials built from the repetition of the same punctual or compact supported potential are addressed: the generalised Dirac comb and the Pöschl-Teller comb, respectively.