The first-order Fermi acceleration process at an ultra-relativistic shock wave is expected to create a particle spectrum with the unique asymptotic spectral index \sigma _ { \gamma \gg 1 } \approx 2.2 .
Below , we discuss this result and differences in its various derivations , which – explicitly or implicitly – always require highly turbulent conditions downstream of the shock .
In the presence of medium amplitude turbulence the generated particle spectrum can be much steeper than the above asymptotic one .
We also note problems with application of the pitch angle diffusion model for particle transport near the ultra-relativistic shocks .