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The response of a magnetic calorimeter with a sapphire crystal serving as an X-ray absorber has been studied as a function of temperature. Several different Au films were used to connect thermally the magnetic sensor to the absorber. The amplitude and time dependence of the signal resulting from the absorption of an X-ray were fit using an idealized model for the calorimeter. The values of the various parameters resulting from a fit of the data are internally consistent and provide a physical understanding of the processes determining the performance of the calorimeter. The fraction of the energy of the X-ray that is captured by the film without having first been down-converted to thermal phonons in the sapphire is found to depend on both the area and the thickness of the film. The rate at which the energy is transferred between thermal phonons in the sapphire and the electrons in the film is determined by the electron/phonon interaction in the gold. Also, an additional heat capacity was observed to be present in the sapphire, which, for want of a better means of characterization, is ascribed to the tunneling systems. The magnitude of this additional heat capacity and its thermal coupling to the lattice has been studied. [Copyright &y& Elsevier]