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In this paper we discuss the self-stabilization of the so-called fast ion instability of an electron beam in the case when coherent oscillations of bunches can be described in the linear approximation. Such a possibility occurs due to an increase in the number of ions along the electron beam and a proportional increase in the betatron tune shifts of electrons in subsequent bunches. The tune variation along the beam results in the BNS damping of the coherent oscillations of electron bunches

Natural BNS damping of the fast ion instability