  Ferropericlase (fp), (Mg$_{1-x}$Fe$_x$)O, the second most abundant mineral in
the Earth's lower mantle, is expected to be an essential component of
super-Earths' mantles. Here we present an ab initio investigation of the
structure and magnetic ground state of fp up to $\sim$ 3 TPa with iron
concentrations (x$_{Fe}$) varying from 0.03 to 0.12. Calculations were
performed using LDA+U$_{sc}$ and PBE exchange-correlation functionals to
elucidate the pressure range for which the Hubbard U (U$_{sc}$) is required.
Similar to the end-members FeO and MgO, fp also undergoes a B1 to B2 phase
transition that should be essential for modeling the structure and dynamics of
super-Earths' mantles. This structural transition involves a simultaneous
change in magnetic state from a low spin (LS) B1 phase with iron total spin S=0
to an intermediate spin (IS) B2 phase with S=1. This is a rare form of
pressure/strain-induced magnetism produced by local cation coordination
changes. Phonon calculations confirm the dynamical stability of the iron B2-IS
state. Free energy calculations are then carried out including vibrational
effects and electronic and magnetic entropy contributions. The phase diagram is
then obtained for low concentration fp using a quasi-ideal solid solution
model. For x$_{Fe}$ > 0.12 this approach is no longer valid. At ultra-high
pressures, there is an IS to LS spin state change in Fe in the B2 phase, but
the transition pressure depends sensitively on thermal electronic excitations
and on x$_{Fe}$.
