Volume 102A, number 3 PHYSICS LETTERS 7 May 1984 PARAMETRIC BEAM INSTABILITY OF RELATIVISTIC CHARGED PARTICLES IN A CRYSTAL V.G. BARYSHEVSKY and I.D. FERANCHUK Department of Physics, Byelorussian State University, Minsk-80, USSR Received 14 September 1983 Revised manuscript received 27 February 1984 A new mechanism of the instability of an electron (positron) beam under conditions of the parametric X-rays is discov- ered. Equations, which define the amplitude increment of the instability and the starting current, are obtained. It is shown that a considerable part of the particle energy transforms into coherent X-ray energy due to this parametric instability. Known schemes of coherent X-ray and gamma-ray sources are basically connected with the creation of the conditions, when resonant transitions of MSss- bauer nuclei or the internal electron shell of heavy atoms have an inverse population. It was shown in refs. [1,2] that these conditions were realized for a relatively small threshold current (j ~ 10 2 A/cm2), when parametric X-rays [2,3] from the modulated electron beam were used. But in this scheme a solu- tion of the very complicated problem of X-ray modu- lation of the ultrarelativistic particle beam is re- quired. An essentially different mechanism of the coher- ent X-ray generation is considered in the present paper. It is shown that parametric X-rays lead to lon- gitudinal self-modulation of the relativistic chan- nelled particle beam and as a result the parametric beam instability arises. In this process the longitudi- nal energy of the particles efficiently transforms into the energy of the coherent monochromatic X-rays. The mechanism considered is in many aspects analo- gous to the processes which are used in the powerful generators of superhigh-frequency radiation and free- electron lasers [4-7]. The interaction of the channelled particle beam with a radiation field and the crystal is described by the following self-consistent system of the Maxwell equations and the particle's equations of motion k 2 ~ ~(k, 60) - k~(k. ~) _ 602 ~ ea#(k, kr ' 60) 6#(kr, 60) 7" = 4rri60]a(k , 60), m dt)i/dt - 7-1F(ri) (1) = --(e/~/) [~(ri, t) + uiX B(ri, t) - ui(• i. ~)], (2) /i=c=l, e 2=1/137, k r =k+x, 7 =E/m, where r i and o i are the coordinate and velocity of the ith particle of the beam; E(t) is the total energy of the particle at the time t; e~#(k, kr; 60) are the Fou- rier components of the dielectric constant of the crys- tal [2] ; * is the reciprocal lattice vector of the crys- tal, ~ (r, t) is the electric radiation field: ~(r, t) = f dk d60 £(k, 60) exp(ik.r - i60t), and j(r, t) is the current density which may be writ- ten as j(r,t) = e ~. oi6 [r-ri(t)]. l (3) Let us consider the simplest case of planar chan- nelling, when the interplane potential is periodic along the X-axis with period d (see fig. 1). Then the 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) 141