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 frequency range x = 3.5 - 4.5 .
Therefore , while the amplitude of the SZ _ { kin } Â produces a systematic bias in the position of the crossover frequency X _ { 0 } , it does not affect significantly the spectral slope of the SZE .
We therefore propose using 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 .