We adapt and modify the eigenfunction method of computing the power-law spectrum of particles accelerated at a relativistic shock front via the first-order Fermi process [ 6 ] to apply to shocks of arbitrarily high Lorentz factor .
The power-law index of accelerated particles undergoing isotropic small-angle scattering at an ultrarelativistic , unmagnetized shock is found to be s = 4.23 \pm 0.2 ( where s = d \ln f / d \ln p , with f the Lorentz-invariant phase-space density and p the momentum ) , in agreement with the results of Monte-Carlo simulations .
We present results for shocks in plasmas with different equations of state and for Lorentz factors ranging from 5 to infinity .