  It is shown that strong 0+2 -> 0+1 E0 transitions provide a clear signature
of phase transitional behavior in finite nuclei. Calculations using the IBA
show that these transition strengths exhibit a dramatic and robust increase in
spherical-deformed shape transition regions, that this rise matches well the
existing data, that the predictions of these E0 transitions remain large in
deformed nuclei, and that these properties are intrinsic to the way that
collectivity and deformation develop through the phase transitional region in
the model, arising from the specific d-boson coherence in the wave functions,
and that they do not necessarily require the explicit mixing of normal and
intruder configurations from different IBA spaces.
