  With a view to the design of hard magnets without rare earths we explore the
possibility of large magnetocrystalline anisotropy energies in Heusler
compounds that are unstable with respect to a tetragonal distortion. We
consider the Heusler compounds Fe$_2$YZ with Y = (Ni, Co, Pt), and Co$_2$YZ
with Y = (Ni, Fe, Pt) where, in both cases, Z = (Al, Ga, Ge, In, Sn). We find
that for the Co$_2$NiZ, Co$_2$PtZ, and Fe$_2$PtZ families the cubic phase is
always, at $T=0$, unstable with respect to a tetragonal distortion, while, in
contrast, for the Fe$_2$NiZ and Fe$_2$CoZ families this is the case for only 2
compounds -- Fe$_2$CoGe and Fe$_2$CoSn. For all compounds in which a tetragonal
distortion occurs we calculate the MAE finding remarkably large values for the
Pt containing Heuslers, but also large values for a number of the other
compounds (e.g. Co$_2$NiGa has an MAE of -2.11~MJ/m$^3$). The tendency to a
tetragonal distortion we find to be strongly correlated with a high density of
states at the Fermi level in the cubic phase. As a corollary to this fact we
observe that upon doping compounds for which the cubic structure is stable such
that the Fermi level enters a region of high DOS, a tetragonal distortion is
induced and a correspondingly large value of the MAE is then observed.
