The relativistic fireball model assumes that the event which gives
rise to a gamma-ray burst (GRB) deposits an energy into a small
baryonic mass *M*, such that .Typically is assumed to be of order 10^{2} to 10^{3}. After an
initial acceleration phase, such a fireball reaches a "coasting" stage
with Lorentz factor . The fireball drives a blast wave
of Lorentz factor into the surrounding
medium. The energy expended to shock the surrounding medium eventually
decelerates the fireball; in the adiabatic case, the Lorentz factor of
the blast wave then decreases with radius as (Rees & Mészáros 1992).

While the GRB emission itself may come from internal shocks
(Rees & Mészáros 1994),
the afterglow emission is generally
thought to be due to electrons accelerated by the blast wave in its
deceleration phase (e.g.
Mészáros & Rees 1997;
Vietri 1997).
Fireball models of GRB afterglows (e.g.
Sari et al. 1998)
usually assume that these accelerated
electrons have a power-law spectrum of unspecified index *p*.
Waxman (1995)
and
Vietri (1995)
also
suggested that ultra-high-energy cosmic rays (UHECRs), with energies
eV, might be accelerated in these relativistic
fireballs. In particular,
Vietri (1995)
argued that in
Fermi-type acceleration at an ultra-relativistic shock, a particle
could have an energy gain per shock crossing
cycle.

Motivated by these considerations, we examine acceleration of the Fermi type at an ultra-relativistic shock in more detail, considering first the spectral index of the accelerated particles, and then the maximum energy attainable by this process.

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