Diffuse radio sources are not the only indication of non-thermal activity in the ICM. The prospects for the X-ray detection of inverse-Compton emission originating from radio emitting electrons and photons of the microwave background were presented about 30 years ago [108, 109]. Significant progress in the search of non-thermal emission in the hard X-ray band (> 20 keV, HXR) has been recently made owing to the improved sensitivity and wide spectral capabilities of the BeppoSAX and the Rossi X-ray Timing Explorer (RXTE) satellites (see the review by Fusco-Femiano et al. ).
Evidence for the presence of HXR radiation in excess to the thermal emission has been obtained in the spectrum of Coma [111, 112, 113, 114]. In Fig. 5, we report the combined hard X-ray spectrum of the Coma cluster, obtained using BeppoSAX data of two independent observations of 90 ksec and 300 ksec . The non-thermal excess with respect to the thermal emission is at a confidence level of ~ 4.8 and has a flux of (1.5 ± 0.5) × 10-11 erg cm-2 s-1 in the 20-80 keV energy band (assuming a photon index X = 2.0). In the framework of the IC model (see Sec. 3.4) the combination of the radio and non-thermal X-ray fluxes allows an estimate of the volume-averaged intracluster magnetic field of ~ 0.2 µG.
Figure 5. Coma cluster -- PDS data. The continuous line represents the best fit with a thermal component at the average cluster gas temperature of 8.1 keV. The errors bars are quoted at the 1 level 114.
In addition to Coma, HXR non-thermal emission has been detected in A2256 [115, 116]. The flux in the 20 - 80 keV energy range is ~ 9 × 10-12 erg cm-2 s-1. A magnetic field of ~ 0.05 µG is derived for the northern cluster region, where the radio relic is located, while a higher field value, ~ 0.5 µG, could be present at the cluster center, in the region of the radio halo. A HXR detection at low confidence level is obtained in A754 [117, 118], where however a radio galaxy with BL Lac characteristics could be responsible for the emission.
The detection in A2199, which is a cooling core cluster with no extended diffuse radio emission, is controversial [110, 119]. A marginal detection has been obtained in A119 , a merging cluster without a radio halo, but the presence of several point sources in the field of view makes the IC interpretation unlikely.
For the clusters A3667  and A2163  only upper limits to the non-thermal X-ray emission have been derived. A possible detection in A2319 with RXTE  leads to a magnetic field of 0.1 - 0.3 µG. Localized IC emission associated with the radio relic and with merger shocks in A85 has been claimed from ROSAT observations . The derived magnetic field is ~ 1 µG.
The value of the magnetic field derived in the Coma cluster by the IC HRX emission is quite consistent with that obtained by the radio halo Coma C under equipartition conditions (Sec. 4.1), but it is much lower than that derived from RM measurements (see next section). Therefore, alternative interpretations to the IC model for the non-thermal radiation detected in the Coma cluster have been proposed. A suggested mechanism is the non-thermal bremsstrahlung from supra-thermal electrons formed through the current acceleration of the thermal gas [123, 124, 125, 126]. However, Petrosian  pointed out that due to the low efficiency of the bremsstrahlung mechanism, these models would require an unrealistically high energy input.
We will show in Sec. 9 that the disagreement between IC and RM magnetic field measurements can be at least partially alleviated. Future studies of non-thermal X-ray emission in clusters will be possible with the ASTRO-E2 satellite.