  The effects of magnetism on the Bain transformation of $\alpha$-phase FeNi
systems are investigated by using the full potential linearized augmented plane
wave (FLAPW) method based on the generalized gradient approximation (GGA). We
found that Ni impurity in bcc Fe increases the lattice constant in
ferromagnetic (FM) states, but not in the nonmagnetic (NM) states. The shear
modulus $G$ and Young's modulus $E$ of bcc Fe are also increased by raising the
concentration of nickel. All the compositions considered show high shear
anisotropy and the ratio of the bulk to shear modulus is greater than 1.75
implying ductility. The mean sound velocities in the $[100]$ directions are
greater than in the $[110]$ directions. The Bain transformation, which is a
component of martensitic transformation, has also been studied to reveal that
Ni$_{x}$Fe$_{1-x}$ alloys are elastically unstable in the NM states, but not so
in the FM states. The electronic structures explain these results in terms of
the density of states at the Fermi level. It is evident that magnetism cannot
be neglected when dealing with the Bain transformation in iron and its alloys.
