Temperature-dependent atomic B factor: An ab initio calculation

The Debye-Waller factor explains the temperature dependence of the intensities of X-ray or neutron diffraction peaks. It is defined in terms of the B matrix whose elements B ?? are mean-square atomic displacements in different directions. These quantities, introduced in several contexts, account for the effects of temperature and quantum fluctuations on the lattice dynamics. This paper presents an implementation of the B factor (8? B ??) in the thermo-pw software, a driver of Quantum ESPRESSO routines that provides several thermodynamic properties of materials. The B factor can be calculated from the ab initio phonon frequencies and displacements or can be estimated, although less accurately, from the elastic constants, using the Debye model. The B factors are computed for a few elemental crystals: silicon, ruthenium, magnesium and cadmium; the harmonic approximation at fixed geometry is compared with the quasi-harmonic approximation where the B factors are calculated accounting for thermal expansion. The results are compared with the available experimental data.