Acceleration gradient of a plasma wakefield accelerator

The phase velocity of the wakefield waves is identical to the electron beam velocity. A theoretical analysis indicates that the acceleration gradient of the wakefield accelerator normalized by the wave breaking amplitude is K₀(ξ)/K₁(ξ), where K₀(ξ) and K₁(ξ) are the modified Bessel functions of the second kind of order zero and one, respectively and ξ is the beam parameter representing the beam intensity. It is also shown that the beam density must be considerably higher than the diffuse plasma density for the large radial velocity of plasma electrons that are required for a high acceleration gradient