  The effects of Ni$^{2+}$ substitution on the structure (lattice parameters,
cations distribution, average cations radii, cation-cation/anion bond lengths
and angles) and the soft magnetic properties of Mg$_{1-x}$Ni$_x$Fe$_2$O$_4$ ($0
\le x \le 1.0$) nanocrystals have been studied by x-ray diffraction with
Rietveld refinement, transmittance electron microscopy, Mossbauer spectroscopy
and magnetization measurements. The mostly inverse spinel structure was found
in MgFe$_2$O$_4$ nanoparticles and the inversion factor is further increased by
Ni-substitution up to a complete inversion in NiFe$_2$O$_4$. The synthesized
ferrites with a small particle size (20 - 30 nm) exhibit soft magnetic
properties of a single-domain behavior with critical domain size about 30 - 40
nm. An observed monotonic increase in magnetization and drastic decrease in
coercivity at 5 K by Ni-substitution result in a lowest anisotropy constant Ku
(2.8 kJ/m3) for x = 0.5. We discuss the evolution of magnetic properties by
Ni-substitution in a correlation with a subsequent cations-redistribution
between sites of the two sublattices and corresponding structural changes in
the inter-ionic distances.
