We show that Compton scattering by electrons of the hot intergalactic gas in galaxy clusters should lead to peculiar distortions of the cosmic background X-ray and soft gamma-ray radiation - an increase in its brightness at E<60-100 keV and a drop at higher energies. The distortions allow the most important cluster parameters to be measured. The spectral shape of the distortions and its dependence on the gas temperature, optical depth, and surface density distribution law have been studied using Monte Carlo computations and confirmed by analytical estimations. In the cluster frame the maximum of the background decrease due to the recoil effect occurs at ~500-600 keV. The photoionization of H- and He-like iron and nickel ions leads to additional distortions in the background spectrum - a strong absorption line with the threshold at ~9 keV (and also to an absorption jump at ~2 keV for cold clusters). The absorption of intrinsic thermal radiation from the cluster gas by these ions also leads to such lines. In nearby (z<1) clusters the line at ~2 keV is noticeably enhanced by absorption in the colder (~10^6 K) plasma of their peripheral (~3 Mpc) regions; moreover, the absorption line at ~1.3 keV splits off from it. The redshift of distant clusters shifts the absorption lines in the background spectrum (at ~2, ~9, and ~500 keV) to lower energies. Thus, in contrast to the microwave background scattering effect, this effect depends on the cluster redshift z, but in a very peculiar way. When observing clusters at z>1, the effect allows one to determine how the X-ray background evolved and how it was gathered with z. To detect the effect, the accuracy of measurements should reach ~0.1%. We consider the most promising clusters for observing the effect and discuss the techniques whereby the influence of the thermal gas radiation hindering the detection of background distortions should be minimal.
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