 First-order Fermi acceleration

In first-order Fermi acceleration, converging scattering media produce a mean fractional energy gain linear in their relative speed divided by the particle speed. At a nonrelativistic shock wave, an almost isotropic ultrarelativistic population gains $2(u_1-u_2)/(3c)$ per crossing and $4(u_1-u_2)/(3c)$ per complete return cycle. The distinction between a crossing and a cycle must also be respected when assigning an escape probability.