The agreement between the fragments internal and kinetic temperatures with the breakup temperature is investigated using a Statistical Multifragmentation Model which makes no a priori as- sumption on the relationship between them. We thus examine the conditions for obtaining such agreement and find that, in the framework of our model, this holds only in a relatively narrow range of excitation energy. The role played by the qualitative shape of the fragments state densities is also examined. Our results suggest that the internal temperature of the light fragments may be affected by this quantity, whose behavior may lead to constant internal temperatures over a wide excitation energy range. It thus suggests that the nuclear thermometry may provide valuable information on the nuclear state density.
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