  We present evidence that relativistic shocks propagating in unmagnetized
plasmas can self-consistently accelerate particles. We use long-term
two-dimensional particle-in-cell simulations to study the well-developed shock
structure in unmagnetized pair plasma. The particle spectrum downstream of such
a shock consists of two components: a relativistic Maxwellian, with
characteristic temperature set by the upstream kinetic energy of the flow, and
a high-energy tail, extending to energies >100 times that of the thermal peak.
This tail is best fitted as a power law in energy with index -2.4+-0.1,
modified by an exponential cutoff. The cutoff moves to higher energies with
time of the simulation, leaving a larger power law range. The number of
particles in the tail is ~1% of the downstream population, and they carry ~10%
of the kinetic energy in the downstream. Upon investigation of the trajectories
of particles in the tail, we find that the energy gains occur as particles
bounce between the upstream and downstream regions in the magnetic fields
generated by the Weibel instability. We compare this mechanism to the first
order Fermi acceleration, and set a lower limit on the efficiency of shock
acceleration process.
