The role of electron scattering from registration detector in a MAC-E type spectrometer

There is a proposal to search for a sterile neutrino in a few keV mass range by the "Troitsk nu-mass" facility. In order to estimate sterile neutrino mixing one needs to make precision spectrum measurements well below the endpoint using the existing electrostatic spectrometer with a magnetic adiabatic collimation, or MAC-E filter. The expected signature will be a kink in the electron energy spectrum in tritium beta-decay. In this article we consider the systematic effect of electron backscattering on the detector used in the spectrometer. For this purpose we provide a set of Monte-Carlo simulation results of electron backscattering on a silicon detector with a thin golden window with realistic electric and magnetic fields in the spectrometer. We have found that the probability of such an effect reaches up to 20-30\%. The scattered electron could be reflected backwards to the detector by electrostatic field or by magnetic mirror. There is also a few percent probability to escape from the spectrometer through its entrance. A time delay between the scattering moment on the detector and the return of the reflected electron can reach a couple of microseconds in the Troitsk spectrometer. Such estimations are critical for the planning upgrades of the detector and the registration electronics.