Low-temperature properties of fcc Al from modified Debye theory

The value of the lattice parameter of fcc Al at low temperatures, a(T), is shown to be accurately determined by the Debye theory of lattice vibrations and first-principles total energy band calculations. The Debye theory uses the well-known generalization to direction-dependent elastic mode frequencies and is further modified by introducing a parameter βz, which gives the fraction of the full Debye zero-point energy that is possessed by the actual dispersive mode frequencies. The calculation of a(T) is simplified by minimizing the Gibbs free energy of the vibrating lattice rather than the usual procedure of minimizing the total energy at constant volume. The parameter βz is shown to affect the value of a(T), but comparison of a(T) with experiment is not able to evaluate βz because of the inherent inaccuracy of the band calculations (the GGA value is 0.7% high, and the LSDA value is 1.2% low for a(T)). By using experimental values of a(T) and elastic constants cij(T),ij = 11, 12, 44 and modified Debye theory, βz is evaluated as 0.48 without band calculations. From βz the rigid lattice of a is shown to be 4.01 Å, midway between the GGA and LSDA values.