The entropy of formation of anion Frenkel defects in fluorites: a quasiharmonic calculation for calcium fluoride

Abstract

A Green function formalism, as first used by Govindarajan et al. (1979) for Schottky defects in alkali halides, has been reformulated on a more rigorous basis and then applied to the calculation of the entropy of formation of an interstitial and of an anion vacancy in the alkaline earth fluoride, CaF2. The relevant lattice Green functions are calculated using a shell model. The force constant changes are obtained from the coordinates of the relaxed ions which are computed using the Hades program due to Norgett (1974) and the potential parameters of Catlow, Norgett and Ross (1977). This calculation of the entropy of formation of a Frenkel defect in CaF2 is a quasiharmonic one at the temperatures 300, 1000 and 1500K. The calculated values are significantly lower than experimental results. The principal reason for this discrepancy is that the calculations refer to the entropy of formation at constant volume (sv) whereas the experiments measure the entropy of formation at constant pressure (sp). A calculation of sp-sv is presented and it is demonstrated that this term is an important component of sp.

Spontaneous Frenkel pair formation in zirconium carbide

“…The values of S − S reported in Table III are positive which agrees in sign with Frenkel pair entropies reported for other materials. [21][22][23][24] The unbound defects have larger vibrational entropies than bound defects, which is also consistent with reports for other materials, e.g. the Frenkel pairs in In 2 O 3 reported by Walsh et al.…”

Formation entropies of intrinsic point defects in cubic In2O3 from first-principles density functional theory calculations

“…We have calculated the relaxation volumes for each individual defect by iteratively scaling the volume until the residual pressure on the cell was less than 1 kBar. In addition, we applied eqn (8) and calculated the relaxation volume from the hydrostatic stress obtained from the constant volume calculation described before. Both methods yielded the same results within the numerical accuracy of the relaxation scheme.…”

“…So far, only formation entropies at a fairly high temperature of 1000 K were discussed and little has been said on the temperature dependence of S f . Also, in the literature, this aspect has rarely been considered, 8 since most of the calculations rely on the high temperature limit of eqn (2) for simplicity. Since we have evaluated the full temperature dependent phonon entropy, the present study is not limited to the high temperature region and we are also able to access the temperature dependence.…”

“…This is because the dynamical matrix has to be calculated for the ideal as well as the defective material at different cell sizes. 6 In the past, calculations of defect formation entropies were therefore mainly carried out by means of analytic interatomic potentials [6][7][8][9][10][11] and, to a lesser extent, by first-principles DFT calculations. 4,5,12 For treating elastic contributions to the defect formation entropy, different approaches like the embedded cluster method and the Greens function method as well as the supercell method were devised and tested.…”