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We study the energetics of galaxy cluster atmospheres by analyzing the SZeffect spectra around the crossover frequency. We calculated analytically theexpressions of both the crossover frequency and the spectral slope of the SZEaround the crossover frequency in various cases: a thermal electron population;a power-law, non-thermal electron population; and a population of electronsexperiencing a stochastic acceleration. We find that the value of the crossoverfrequency X_0 of the SZE depends significantly on the cluster peculiar velocityV_r (which determines the amplitude of the kinematic SZE), while the value ofthe slope of the SZE does not depend on the kinematic SZE spectrum in theoptimal 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 amplitudeof the kinematic SZE produces a systematic bias in the position of thecrossover frequency, it does not affect significantly the spectral slope of theSZE. We therefore propose to use measurements of the spectral slope of the SZEto obtain unbiased information about the specific properties of variouselectron distributions in galaxy clusters as well as in other cosmic structuresin which a SZE can be produced.

Studying the leptonic structure of galaxy cluster atmospheres from the spectral properties of the SZ effect