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 ...
