It is observed that there is an appreciable increase in the lattice parameter with Gd concentration indicating lattice expansion which may be attributed to larger ionic radius of Gd3+

It is observed that there is an appreciable increase in the lattice parameter with Gd concentration indicating lattice expansion which may be attributed to larger ionic radius of Gd3+ (0.938A?) compared to that of Fe3+ (0.67 A?)20,21. Gd ions enter the octahedral site B at the expense of Fe3+ ions which results in internal stress that makes the lattice distorted and produces lattice expansion 22.
The crystallite size is found to decrease with the gadolinium content. Various researchers have reported similar trends in ferrite nanoparticles doped with rare earth ions 23,24. The Gd3+ ions have larger ionic radius than Fe3+ and requires more activation energy for it to enter octahedral site as the bond energy of Gd-O is higher when compared to that of Fe-O. Thus the higher energy requirement impedes grain growth as it is difficult to complete crystallization and growth of grains 25. The X-ray density increases with Gd content as it basically depends upon the molecular weight of the sample. However, there is an exception for the sample with x=0.02 which has the maximum value of lattice constant. The low porosity values in the present study indicate that the substituted cations have completely dissolved and entered the spinel lattice.
The cation distribution for the studied samples are proposed based on the reports available in the literature on the preferred site occupancy of the substituted transition metal cations. Smaller ions should get located at the sites with smaller available space in the lattice. It is noted that the available space at the (A) sites is smaller than that at the B sites in the spinel ferrites. Ni2+ has a strong octahedral site preference whereas Zn2+ occupies only tetrahedral site 26,27. Gd3+ possesses a large affinity to occupy the octahedral site due to its larger ionic radius 27. On the other hand, the Mn2+ ions get distributed over tetrahedral (80%) and octahedral site (20%) 12.
The proposed cation distribution is thus
(Mn0.16 Zn0.2Fe0.64 )A Mn0.04Ni0.6 Gdx Fe1. 36-xO4.B O4

Theoretical lattice constants for each composition have been calculated using the above proposed cationic distribution and the following relation for the theoretical lattice constant (ath)

a_th=(8/3?3) r_A+r_O+?3(r_B+r_O)

where rO is the radius of oxygen ion O2-(1.38 Å); rA and rB are the ionic radii of tetrahedral (A) and octahedral (B) sites, respectively. The radii rA and rB for the studied spinel systems have been calculated on the basis of the cation distribution using the given relations 28
rA = 0.16RMn2+ +0.2RZn2+ + 0.64RFe3+
rB = (1/2)0.6RNi2+ + 0.04RMn2+ +xRGd3+ +(1.36-x)RFe3+

where RMn2+ , RZn2+ and RFe3+ are the ionic radii of Mn2+ ion (0.83 Å), Zn2+ ion (0.74 Å) and Fe3+ ion (0.49 Å) at the A-site respectively. RNi2+, RMn2+ , RGd3+ and RFe3+ are the ionic radii of Ni2+ ion (0.69 Å), Mn2+ ion (0.91 Å), Gd 3+ ion (0.938 Å) and Fe3+ ion (0.645Å) at the B-site respectively. The value of rA was obtained as 0.594 Å which is independent of Gd concentration as the probability of existence of Gd 3+ ions at A-site is negligible 27 and the rB values are presented in Table 1.
The calculated lattice constants are found to be in close agreement with the experimentally observed lattice constant values.