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A theory for Raman backscattering, in the rest frame of
an electron beam with ion-channel guiding, is presented when the
electrons are making small amplitude betatron oscillations. A
relativistic theory of this problem is also presented when the
electrons are moving through relativistic and helical
trajectories. The relativistic theory is used to find the growth
rate of a free-electron laser in Raman regime. The wiggler
effects, on the linear dispersion relations of the space charge
wave and radiation, are included in the analysis. The growth rate
and free-space wavelength of the radiation are studied numerically
at different operating regimes corresponding to small and large
ion-channel densities. An improved dispersion relation for FEL
without any guiding is derived
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