  We study Ni80Fe20-based permalloys with the relativistic spin-polarized
Korringa-Kohn-Rostoker electronic structure method. Treating the compositional
disorder with the coherent potential approximation, we investigate how the
magnetocrystalline anisotropy, K, and magnetostriction, lambda, of Ni-rich
Ni-Fe alloys vary with the addition of small amounts of non-magnetic transition
metals, Cu and Mo. From our calculations we follow the trends in K and lambda
and find the compositions of Ni-Fe-Cu and Ni-Fe-Mo where both are near zero.
These high permeability compositions of Ni-Fe-Cu and Ni-Fe-Mo match well with
those discovered experimentally. We monitor the connection of the magnetic
anisotropy with the number of minority spin electrons, Nmin. By raising Nmin
via artificially increasing the band-filling of Ni80Fe20, we are able to
reproduce the key features that underpin the magnetic softening we find in the
ternary alloys. The effect of band-filling on the dependence of
magnetocrystalline anisotropy on atomic short-range order in Ni80Fe20 is also
studied. Our calculations, based on a static concentration wave theory,
indicate that the susceptibility of the high permeability of the Ni-Fe-Cu and
Ni-Fe-Mo alloys to their annealing conditions is also strongly dependent on the
alloys' compositions. An ideal soft magnet appears from these calculations.
