Soft-mode transitions of alkaline-earth 122 pnictides

$A$-122 pnictides ($A$=Ca, Sr, Ba) exhibit three pressure-driven transitions: a first order enthalpic transition at $P_H$ from the striped AFM orthorhombic (OR) to a tetragonal (T) or a collapsed tetragonal (cT) phase; a transition at $P_M > P_H$ from the metastable AFM OR to a T or cT phase; a Lifshitz transition at $P_L$ that causes T to collapse to a cT phase. Transitions at $P_H$ and $P_L$ were previously examined through total energy and band structure calculations \footnote{M. Widom and K. Quader, Phys. Rev. B 88, 045117 (2013)}\footnote{K. Quader and M. Widom, Phys. Rev. B 90, 144512 (2014)}. Here we address the transition at $P_M$, beyond which the metastable AFM OR state ceases to exist. We show this transition occurs through a loss of elastic stability caused by softening of a shear mode associated with stretching along the c-axis. Simultaneously, magnetism and orthorhombicity approach limiting values with an approximately square-root singularity \footnote{M. Widom and K. Quader, arXiv:1508.07932}. Together these suggest a strong magneto-elastic coupling that may be relevant to a further understanding of the $A$-122-pnictides under pressure.