Heat capacity and entropy of nonstoichiometric magnetite ${\mathrm{Fe}}_{3\left(1\mathrm{-}\mathrm{\delta }\right)}$${\mathrm{O}}_4$: The thermodynamic nature of the Verwey transition

The heat capacity of ${\mathrm{Fe}}_{3\left(1\mathrm{-}\mathrm{\delta }\right)}$ ${\mathrm{O}}_4$, was systematically measured for nine samples in the composition range 0<δ<0.012 and for temperatures between 5 and 200 K. The character of the heat-capacity anomaly at the Verwey transition changed at δ=${\mathrm{\delta }}_{\mathit{c}}$≊0.004: for δ<${\mathrm{\delta }}_{\mathit{c}}$ or δ>${\mathrm{\delta }}_{\mathit{c}}$ the transformation was of first or second order, respectively. The transition disappears altogether for δ>3${\mathrm{\delta }}_{\mathit{c}}$≃0.012. From these measurements the entropy of the transition and the temperature dependence of the entropy have been calculated. The entropy change at the transition is R ln2 for δ=0 and decreases steadily with increasing δ. The implications of these findings are discussed in terms of a mean-field model that includes the Coulomb interactions among electrons located on neighboring octahedral sites. The charge carriers resonate between the two equivalent octahedral positions above the Verwey transition, and freeze out well below the transition. The model predicts a molar entropy change of R ln2 for discontinuous transitions and 2R ln2 for continuous transitions.