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Thermodynamics of spin crossover in ferropericlase: an improved LDA+$U_{sc}$ calculation

We present LDA+$U_{sc}$ calculations of high-spin (HS) and low-spin (LS) states in ferropericlase (fp) with an iron concentration of 18.75$\%$. The Hubbard parameter $U$ is determined self-consistently with structures optimized at arbitrary pressures. We confirm a strong dependence of $U$ on the pressure and spin state. Static calculations confirm that the antiferromagnetic configuration is more stable than the ferromagnetic one in the HS state, consistent with low-temperature measurements. Phonon calculations guarantee the dynamical stability of HS and LS states throughout the pressure range of the Earth mantle. Compression curves for HS and LS states agree well with experiments. Using a non-ideal mixing model for the HS to LS states solid solution, we obtain a crossover starting at $\sim$45 GPa at room temperature and considerably broader than previous results. The spin-crossover phase diagram is calculated, including vibrational, magnetic, electronic, and non-ideal HS-LS entropic contributions. Our results suggest the mixed-spin state predominates in fp in most of the lower mantle.