  We study the X-ray emission in a sample of galaxy clusters using the BeppoSAX
PDS instrument in the 20 -- 80 keV energy band. The non-thermal hard X-ray
cluster emission (HXR) is detected at a 2 sigma level in 50% of the
non-significantly AGN-contaminated clusters: A2142, A2199, A2256, A3376, Coma,
Ophiuchus and Virgo. The data are consistent with a scenario whereby relaxed
clusters have no hard X-ray component of non-thermal origin, whereas merger
clusters do, with a 20-80 keV luminosity of 10^(43-44) erg/s. The co-added
spectrum of the above clusters indicates a power-law spectrum for the HXR with
a photon index of 2.8+0.3-0.4 in the 12-115 keV band, and we find indication
that it has extended distribution. These indications argue against significant
contamination from obscured AGN, which have harder spectra and centrally
concentrated distribution. These results are supportive of the assumption of
the merger shock acceleration of electrons in clusters. Assuming that the
Cosmic Microwave Background photons experience Inverse Compton scattering from
the merger-accelerated relativistic electrons, and thus produce the observed
HXR, the measured hard X-ray slope corresponds to a differential momentum
spectra of the relativistic electrons with a slope of mu = 3.8-5.0. The
observed slope of HXR is also consistent with that predicted by the non-thermal
bremsstrahlung, which thus cannot be ruled by the fit to the current data, even
though this model requires an extreme, untenable cluster energetics. Assuming
centrally concentrated distribution of HXR, the data requires a harder slope
for the HXR spectrum, which is consistent with secondary electron models, but
this model yields a worse fit to the PDS data and thus seems to be disfavored
over the primary electron Inverse Compton model.
