Vlasov formalism of the laser driven ion channel x-ray laser

An intense short pulse laser, undergoing relativistic self-focusing in a plasma, pushes the low energy electrons radially outwards to create an ion channel, while it accelerates, directly by itself or by the wakefield driven plasma wave, the energetic ones axially to multi-MeV energies. When the energy spread of the energetic electrons is small, they form a beam that executes betatron oscillations in the channel and drives x-ray radiation unstable via betatron resonance, omega-kv(z) = omega(b), producing coherent x-rays of frequency omega = 2 gamma(2 omega)(0)(b), with growth rate approximate to omega p(n(ob)/n(o))(1/3)/2 gamma(5/6)(0), where k = omega/c, omega(b) = omega(p)/(2(gamma 0))(1/2) is the betatron frequency, omega(p) is the plasma frequency, gamma(0) is the Lorentz factor of the beam, v(z) is its axial velocity and n(ob)/n(o) is the ratio of beam density to plasma density. The fractional energy spread in excess of 2 Gamma m/3 omega(b) reduces the growth rate, where Gamma m = (omega(2)(pb)omega(p))(1/3)/2 gamma(5/6)(0) and omega(pb) = (4 pi n(0b)e(2)/m)(1/2).