  Ferropericlase (Fp), (Mg$_\mathrm{1-x}$Fe$_\mathrm{x}$)O, is the second most
abundant phase in the Earths lower mantle. At relevant pressure-temperature
conditions, iron in Fp undergoes a high spin (HS), S=2, to low spin (LS), S=0,
state change. The nature of this phenomenon is quite well understood now, but
there are still basic questions regarding the structural stability and the
existence of soft phonon modes during this iron state change. General theories
exist to explain the volume reduction, the significant thermo-elastic
anomalies, and the broad nature of this HS-LS crossover. These theories make
extensive use of the quasi-harmonic approximation (QHA). Therefore, dynamical
and structural stability is essential to their validity. Here, we investigate
the vibrational spectrum of Fp throughout this spin-crossover using $\textit{ab
initio}$ DFT+Usc calculations. We address vibrational modes associated with
isolated and (2nd) nearest neighbor iron ions undergoing the HS-LS state
change. As expected, acoustic modes of this solid solution are resilient while
optical modes are the most affected. We show that there are no soft phonon
modes across this HS-LS crossover, and Fp is dynamically stable at all relevant
pressures.
