  We study the energetics of galaxy cluster atmospheres by analyzing the SZ
effect spectra around the crossover frequency. We calculated analytically the
expressions of both the crossover frequency and the spectral slope of the SZE
around the crossover frequency in various cases: a thermal electron population;
a power-law, non-thermal electron population; and a population of electrons
experiencing a stochastic acceleration. We find that the value of the crossover
frequency X_0 of the SZE depends significantly on the cluster peculiar velocity
V_r (which determines the amplitude of the kinematic SZE), while the value of
the slope of the SZE does not depend on the kinematic SZE spectrum in the
optimal frequency range around the crossover frequency of the thermal SZE, i.e.
in the (a-dimensional) frequency range x = 3.5 - 4.5. Thus, while the amplitude
of the kinematic SZE produces a systematic bias in the position of the
crossover frequency, it does not affect significantly the spectral slope of the
SZE. We therefore propose to use measurements of the spectral slope of the SZE
to obtain unbiased information about the specific properties of various
electron distributions in galaxy clusters as well as in other cosmic structures
in which a SZE can be produced.
