  We investigate the spatial behaviour of the total comptonization parameter
$y_{tot}$ evaluated for a galaxy cluster containing two population of
electrons: the thermal population, with energy around some KeV and whose trace
is evident in the X-ray emission of the ICM (Intra-Cluster Medium), and the
relativistic population, which give rise to the radio halo emission found in
several clusters of galaxies. We present the first results obtained from our
analysis showing that there are remarkable features in such spatial trend,
which might throw a new light in understanding the cluster internal processes.
