  In hybrid improper ferroelectric systems, polarization arises from the onset
of successive nonpolar lattice modes. In this work, measurements and modeling
were performed to determine the spatial symmetries of the phases involved in
the transitions to these modes. Structural and optical measurements reveal that
the tilt and rotation distortions of the MnO6 or TiO6 polyhedra relative to the
high symmetry phases driving ferroelectricity in the hybrid improper Ca3X2O7
system (X=Mn and Ti) condense at different temperatures. The tilt angle
vanishes abruptly at T$_T$ ~ 400 K for Ca3Mn2O7 (and continuously for X=Ti) and
the rotation mode amplitude is suppressed at much higher temperatures T$_R$
~1060 K. Moreover, Raman measurements in Ca3Mn2O7 under isotropic pressure
reveal that the polyhedral tilts can be suppressed by very low pressures
(between 1.4 and2.3 GPa) indicating their softness. These results indicate that
the Ca3Mn2O7 system provides a new platform for strain engineering of
ferroelectric properties in film based systems with substrate induced strain.
