  We report on experimental spectra of photons radiated by 50 GeV positrons
crossing silicon single crystals of thicknesses 1.1 mm, 2.0 mm, 4.2 mm, and 6.2
mm at sufficiently small angles to the (110) planes that their motion
effectively is governed by the continuum crystal potential. The experiment
covers a new regime of interaction where each positron emits several hard
photons, whose recoil are not negligible and which are formed on lengths where
the variation of the crystal field cannot be ignored. As a result neither the
single-photon semiclassical theory of Baier et al. nor the conventional cascade
approach to multiple hard photon emissions (quantum radiation reaction) based
on the local constant field approximation are able to reproduce the
experimental results. After developing a theoretical scheme which incorporates
the essential physical features of the experiments, i.e., multiple emissions,
photon recoil and background field variation within the radiation formation
length, we show that it provides results in convincing agreement with the data.
