On the nature of two superconducting transitions in the specific heat of PrOs$_4$Sb$_{12}$: Effects of crystal grinding
Specific heat, dc- and ac-magnetic susceptibility are reported for a large single crystal of PrOs$_4$Sb$_{12}$ and, after grinding, its powder. The room temperature effective paramagnetic moment of the crystal was consistent with the Pr$^{3+}$ ionic configuration and full occupancy of the Pr-sublattice. The crystal showed two distinct anomalies in the specific heat and an overall discontinuity in $C/T$ of more than 1000 mJ/K$^2$mol. The upper transition (at $T_{c1}$) was rounded, in an agreement with previous reports. The anomaly at $T_{c2}$ was very sharp, consistent with a good quality of the crystal. We observed a shoulder in $\chi$' and two peaks in $\chi$'' below $T_{c1}$. However, there were no signatures in $\chi$' of the lower temperature transition. PrOs$_4$Sb$_{12}$ is extremely sensitive to grinding, which suppresses the upper superconducting transition in both the specific heat and magnetic susceptibility. $\Delta C/T_{c}$ was reduced to 140 mJ/K$^2$ mol in the powdered sample. Existing data on ground, polished, and sliced crystals suggests the existence of a length scale of order 100 $\mu$, characterizing the higher temperature superconducting phase.