  Several models for type Ia-like supernovae events rely on the production of a
self-sustained detonation powered by nuclear reactions. In the absence of
hydrogen, the fuel that powers these detonations typically consists of either
pure helium (He) or a mixture of carbon and oxygen (C/O). Studies that
systematically determine the conditions required to initiate detonations in C/O
material exist, but until now no analogous investigation of He matter has been
conducted. We perform one-dimensional reactive hydrodynamical simulations at a
variety of initial density and temperature combinations and find critical
length scales for the initiation of He detonations that range between 1 -
$10^{10}$ cm. A simple estimate of the length scales over which the total
consumption of fuel will occur for steady-state detonations is provided by the
Chapman-Jouguet (CJ) formalism. Our initiation lengths are consistently smaller
than the corresponding CJ length scales by a factor of $\Sim 100$, providing
opportunities for thermonuclear explosions in a wider range of low-mass white
dwarfs (WDs) than previously thought possible. We find that virialized WDs with
as little mass as 0.24 $M_\odot$ can be detonated, and that even less massive
WDs can be detonated if a sizable fraction of their mass is raised to a higher
adiabat. That the initiation length is exceeded by the CJ length implies that
certain systems may not reach nuclear statistical equilibrium within the time
it takes a detonation to traverse the object. In support of this hypothesis, we
demonstrate that incomplete burning will occur in the majority of He WD
detonations and that $^{40}$Ca, $^{44}$Ti, or $^{48}$Cr, rather than $^{56}$Ni,
is the predominant burning product for many of these events. We anticipate that
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