I think it is because of less strain in the larger crystalite size. The smaller ones (nano scale) might have more strain in their lattice, so the lattice parameter could be affected.

Well, thanks Mr. Mohammad Khodaei for your answer. This is helpful.

I cannot see any straightforward correlation. Ni2+ (0.078nm) and Zn2+ (0.083nm) have different radii so that a change of the lattice parameter should not surprize. However, with increasing Ni the lattice parameter should actually decrease and not increase. Of course, it is a very simple approximation bt commonly it works not that bad. The same is related to the crystallite size. Why the higher Ni concentration should automatically decrease the crystal size? Could you say anything about you x? About how much you are talking? How did you synthesize your spinels?

The change of unit cell parameter consistents with the values of the ionic radii of cations in tetrahedral positions of the spinel structure. Parameter of NixZn(1-x)Fe2O4 ferrite monotonically decreases with increasing Ni content as a(Ni,Zn)Fe2O4=a(NiFe2O4)*x+a(ZnFe2O4)*(1-x) in accordance with Vegard rule.

As to the change of grain size, among other reasons, this may be due to the nature and concentration of the adsorption centers on the surface of the ferrites in the process of their formation. So, for example, there is naturally change of the acid-base properties of the surface depending on the nature of the cation. For NixZn(1-x)Fe2O4 ferrite, increasing Zn concentration leads to the increase in the proportion of strong acid centers on the surface. And with increasing Ni content, the contribution of acid sites of weak and medium strength increases. There may be other reasons for the differences in the conditions of formation of crystallites.

Alexander, his measurement obviously showed that with increasing Ni content the lattice parameters increase, not decrease as you (and I) proposed. Any idea, why? Or he simply concluded from a shift to higher Bragg angles that the lattice parameters increases (what is actually inverse)....

To Gert Nolze: unfortunately, he does not give further explanation.

Perhaps, he estimates the crystallite size incorrect too, and all is on the contrary. It may be advisable to test this alternative way, at least by microscopy.

Regarding lattice parameter:

Nickel zinc ferrite has the cubic (fcc) spinel-type structure, space group Fd3m, where cations occupy one eight of the tetrahedral holes (or "sites") and half of the octahedral holes. However, cation distribution among those tetrahedral and octahedral sites depends on the relative coordination preference (tetrahedral or octahedral) of the cation with the oxygen anions. Zn(2+) is known to have a strong tetrahedral preference while Ni(2+) has a strong octaherdal preference, and Fe(3+) does not have much of a preference for any of the sites. Hence, when the concentration of Ni is increased those added Ni(2+) cations should be expected to displace Fe(3+) cations from octahedral sites to the tetrahedral sites, made available because of the smaller Zn concentration. In other words, that renders Vegard's rule of little use, since that rule refers to isostructural substitution at the same lattice site. At this point, one could be tempted to conclude that, since the Ni(2+) ion is larger than Fe(3+), we have the increasing lattice prarameter (when Ni concentration is increased) explained. But, the Fe(3+) ions (displced form their octahedral sites) go now to occupy vacant  tetrahedral sites, and Fe(3+) is smaller than Zn(2+)! To put things worse, in the spinel-type structure, the size of the tetrahedral hole can be changed (till some extent) to accommodate cations of different size, by slightly diplacing the nearby anions (which would be reflected in the anion position parameter).. But that would take us a bit to far, I think.

So far as the question made by our colleague goes, I would say that, in principle, there is no reason why the lattice parameter  should not increase (at least slightly) when Ni concentration of the nickel zinc ferrite samples is increased.

Regarding crystalite size:

Assuming that sample preparation was always carried out in the same conditions (temperature and time), the reason for having smaller crystallite size when increasing Ni content is likely to be a smaller diffusion coefficient of the Ni(2+) cation, as compared to that of Zn(2+).

I agree with Carlos Otero Areán. He's right. I got excited and did not realize that the Zn spinel is normal, and Ni spinel is converted. Therefore, the crystal-chemical formula of NiZn spinel has the form: Zn(1-x)Fex[NixFe(2-x)]O4, and substitutions of Ni and Zn occur in different sublattices. The increase of the Ni content in the octahedra and the Fe content in the tetrahedra occur synchronically (proportional to x). I not undertake to assert with full confidence, nevertheless, but taking into account the ratio of ionic radii within each sublattice, we can assume (or expect) that increasing the Ni content in the octahedra should not lead to a stronger increase of the lattice parameter than the decrease of the parameter by increasing the Fe content and reduction in the number of large Zn ions in the tetrahedra.

Carlos Otero Areán correctly explained the situation as is usually found in bulk crystals of Ni-Zn ferrites. Despite the complication with various preferences of bivalent Ni and Zn ions for tetrahedral and octahedral sites in the spinel lattice, the "naive" Vegard's  rule is surprisingly obeyed. The point of the original question is that I Putu Tedy Indrayana found opposite behaviour of the lattice constant with the Ni content than in numerous published works.

One of the reasons may be the following: the cation distribution in nanoparticles was proved to be different form the bulk crystals and depends on the particles size. In particular the Zn ferrite with all Zn in tetrahedral (A) sites is antiferromagnetic with rather low ordering temperature of the order of 10 K. But nanoparticles are found to be ferrimagnetic with relatively high magnetic moment and also structural studies show that in nanoparticles Fe may partly enter the (A) sites.

The funny result with the lattice constant may thus be connected to the change of cation preferences for (A) and [B] spinel sites with changing the particle size down to the nano scale.

And a remark en passant: the change of the "cationic radius" will first deform the relevant oxygen tetra- or octa-hedron and the change of the lattice parameter will be a mediated (secondary) effect.