  The $^{11}$B$(p,\alpha)2\alpha$ fusion reaction is particularly attractive
for energy production purposes because of its aneutronic character and the
absence of radioactive species among reactants and products. Its exploitation
in the thermonuclear regime, however, appears to be prohibitive due to the low
reactivity of the $^{11}$B fuel at temperatures up to 100 keV. A fusion chain
sustained by elastic collisions between the alpha particles and fuel ions, this
way scattered to suprathermal energies, has been proposed as a possible route
to overcome this limitation. Based on a simple model, this work investigates
the reproduction process in an infinite, non-degenerate $^{11}$B plasma, in a
wide range of densities and temperatures which are of interest for laser-driven
experiments ($10^{24} \lesssim n_e \lesssim 10^{28} {\rm cm}^{-3}$, $T_e
\lesssim 100$ keV, $T_i \sim$ 1 keV). In particular, cross section data for the
$\alpha$-$p$ scattering which include the nuclear interaction have been used.
The multiplication factor, $k_\infty$, increases markedly with electron
temperature and less significantly with plasma density. However, even at the
highest temperature and density considered, and despite a more than twofold
increase by the inclusion of the nuclear scattering, $k_\infty$ turns out to be
of the order of $10^{-2}$ only. In general, values of $k_\infty$ very close to
1 are needed in a confined scheme to enhance the suprathermal-to-thermonuclear
energy yield by factors of up to $10^3$-$10^4$.
