  In this third paper of a series, we discuss the physics of the population of
accelerated particles in the precursor of an unmagnetized, relativistic
collisionless pair shock. In particular, we provide a theoretical estimate of
their scattering length $l_{scatt}(p)$ in the self-generated electromagnetic
turbulence, as well as an estimate of their distribution function. We obtain
$l_{scatt}(p) \simeq (\gamma_p /\epsilon_B)(p/\gamma_{\infty} mc)^2
(c/\omega_p)$, with p the particle momentum in the rest frame of the shock
front, $\epsilon_B$ the strength parameter of the microturbulence, $\gamma_p$
the Lorentz factor of the background plasma relative to the shock front and
$\gamma_{\infty}$ its asymptotic value outside the precursor. We compare this
scattering length to large-scale PIC simulations and find good agreement for
the various dependencies.

High Energy Astrophysical Phenomena;Plasma Physics