Spinel oxides have been the subject of great interest because of their wide applications in the high frequency devices [1] -[4] . Spinel ferrite plays an important role in a variety of technical fields such as telecommunication, AC and DC motors, switched mode power supplies, power distribution transformers, communication systems, digital memories, multilayer chip inductors, video and audio applications, radar and satellite communications and in many other technological applications like photoelectrical devices and gas sensors, magneto-mechanical stress and torque sensors, microwave devices and biomedical applications [5] -[7] .

The general chemical formula of spinel ferrite is AB₂O₄, where A is a divalent metal ion (cation) and B is a trivalent metal ion (cation). The oxygen ions form a face centered cubic (fcc) lattice and the cations occupy the interstitial positions. The A cations are located in tetrahedral sites, and the B cations are in octahedral sites. The cubic structure permits to substitute cations selectively in the octahedral or tetrahedral sites or in both the sites. This helps to modify electrical properties and magnetic ordering in these systems. Extensive work has been done by various workers to upgrade the properties of ferrites by substituting different types and amount of impurities [8] -[16] .

The physical properties of the spinel ferrites such as the electrical, magnetic and elastic properties are governed by the type of magnetic ions residing on the tetrahedral (A) site and octahedral (B) site of the spinel lattice and the relative strength of the inter- and intra-sublattice interactions [8] [17] . Neutron diffraction, in many cases, can give an unambiguous picture of the cation distribution since in neutron scattering, the adjacent elements in the periodic table, unlike X-ray scattering, are distinguishable [16] [18] [19] . The marked differences in nuclear scattering amplitude of the cations and interaction of their magnetic moments with the neutron magnetic moment make possible the accurate determination of cation distribution and magnetic structure in the spinel system [8] [16] [17] [19] .

Cobalt ferrite (CoFe₂O₄) is a well-known hard magnetic material with remarkably high electrical resistivity, large permeability at high frequency and cost effectiveness. In the present study, Co ions have been substituted by divalent Ni ions in order to determine the cation distribution, and other crystallographic and magnetic properties in the spinel system Ni_xCo_1−xFe₂O₄ using neutron diffraction technique.

In the spinel systems, the magnetic unit cell is the same as the crystallographic unit cell having identical symmetry relations. Hence the nuclear and magnetic contribution to the Bragg reflections occurs at the same angular position. Thus the Bragg diffraction peaks in the neutron diffraction data contains both nuclear and magnetic contributions. Due to the strong form dependence of neutrons the magnetic contributions rapidly fall off with scattering angle. Therefore, in order to avoid any ambiguities arising from the magnetic contribution to the Bragg intensities, higher angle neutron diffraction data (50˚ ≤ 2θ ≤ 110˚), where the magnetic contribution is negligible, were profile analyzed using the computer program FullProf [20] in the Reitveld method [21] for the determination of cation distributions and other crystallographic parameters for all the samples. Figure 2 shows once such representative fitted neutron diffraction pattern confined to higher scattering angle (50˚ ≤ 2θ ≤ 110˚) for composition x = 0.2, showing an excellent agreement between the observed and calculated profiles in the nuclear phase. The space group F d-3m was used in the data analysis for generating the calculated profile because the symmetry of the crystal structure of spinel system belongs to this space group. Rietveld agreement factors (R-factors), cation distribution and crystallographic parameters obtained from the refinement of the neutron diffraction data have been summarized in Table 1.

The total neutron diffraction data (10˚ ≤ 2θ ≤ 110˚) containing both nuclear and magnetic contributions were analyzed using the program FullProf in two different phases, the nuclear phase and the magnetic phase, using an additional space group F-1 for the magnetic phase. The occupation number of cations on both the A and B sites as determined from the higher angle neutron diffraction data were kept constant and all the other parameters were varied freely in this part of refinement. The symmetry operators for the magnetic atoms were supplied as external input to the program. Figure 3 shows a plot of the Reitveld refinement result of the total neutron diffraction data containing both nuclear and magnetic contributions for the composition x = 0.8. There is excellent agreement between the observed and calculated profiles in the nuclear and magnetic phases.

The neutron diffraction patterns at room temperature for different compositions in the 2q range 15˚ - 70˚ are shown in Figure 4. The intensity change in the Bragg peaks with increasing x, as evident from Figure 4, is a manifestation of the substitution effect in the system. A small fluctuation in the Bragg intensities is present but the position of the Bragg peaks does not show any significant change with increasing x. Thus, the replacement of Co²⁺ ions by Ni²⁺ ions in these spinel oxides does not show any remarkable change in the lattice dimension. The spinel system under study consists of three different types of cations (Ni²⁺, Co²⁺ and Fe³⁺) which are distributed over the A and B sites. The coherent scattering lengths of Ni, Co and Fe for neutrons are 10.30, 2.49 and 9.45 fm, respectively. The significant difference in the coherent scattering lengths is advantageous for the correct determination of cation occupancies. In the refinement process, initially Ni was kept fixed on the B sites, because it shows such site preferences [18] [19] . Then, occupancies of Co and Fe on the two sites were varied within the stoichiometric limit. Finally, the occupancy of Ni was varied separately with Co and Fe one by one.

From the refinement, it is observed that for x ≤ 0.6 Ni ions invariably occupy the octahedral (16d) site only. For x ≥ 0.8, some Ni ions enter into the tetrahedral (8a) site and their occupation in this site gradually increases with increasing x. On the other hand, Co ions are distributed over both 8a and 16d sites for the composition with x ≤ 0.6, while all the Co ions enter into the 16d site for x ≥ 0.8. However, the occupation of Co gradually decreases in both the sites for all values of x. The Fe ions are distributed over both 8a (A) and 16d (B) sites for the whole compositional range investigated. The occupation of Fe increases in A site and decreases in B site for x ≤ 0.6, while it decreases in A site and increases in B site for x ≥ 0.8. The variation of the occupation factors of individual cations as a function of composition over the tetrahedral and octahedral sites are presented in Figure 5.

The lattice constants decreases and the oxygen position parameters increases with increasing Ni content (x) in the system. The dependence of the lattice parameter (a) and the oxygen position parameter (u) on Ni content (x) is shown in Figure 6. It is quite evident from the figure that the lattice parameter exhibits a linear dependence on x obeying the Vegard’s law [22] . The decrease in the lattice parameter with Ni concentration could be attributed to the slightly smaller ionic radius of Ni²⁺ (0.69 Ǻ) which has been increasingly substituted for Co²⁺ (0.745 Ǻ) in the system. The oxygen position parameter u lies within the normal limit of 0.25 - 0.27 for the spinel oxides

showing linear concentration dependence (Figure 6).

The tetrahedral and octahedral cation-anion bond distances d_A-O and d_B-O have been calculated using the Piox data [23] [24] : d_A-O(Ni²⁺-O) = 1.970 Å, d_B-O(Ni²⁺-O) = 2.088 Å; d_A-O(Co²⁺-O) = 1.967 Å, d_B-O(Co²⁺-O) = 2.131 Å and d_A-O(Fe³⁺-O) = 1.855 Å, d_B-O(Fe³⁺-O) = 2.020 Å. Since there are more than one kind of ions present in each site the weighted mean values of bond distances have been calculated using the cation distribution obtained from the Rietveld refinement of our neutron diffraction data listed in Table 1. The mean values of A-O and B-O bond distances have been obtained from the output of the Rietveld refinement of experimental neutron diffraction data. The calculated bond distances from Piox data are compared with those obtained from the refinement of experimental data in Table 2.

The deviation in the A-O bond lies in the range 0.26% - 2.15% while that in the B-O bond ranges from 1.47% - 1.70%. It is quite obvious from the table that the values of experimental A-O and B-O bond distances decrease with increasing x. As before, this may be explained as an effect of the substitution of slightly smaller Ni ions for Co ions in the system. However, the calculated bond distance on the tetrahedral site decreases upto x ≥ 0.6 and then increases depending on the cations occupied in this site, while on the octahedral site it decreases with increasing x for the whole compositional range investigated. This result indicates that the Piox data may not be suitable for describing the cation-anion distance in the present system where many different types of cations are present at the same site with various dimensions and concentrations.

In the spinel structure, the relation between lattice parameter and bond distances is expressed as [25] :

where d_A-O and d_B-O are the average cation-anion bond lengths of A and B site respectively. Using the above relation, the lattice parameters are also calculated and compared with those obtained from Rietveld refinement of neutron diffraction data in Table 1. The maximum deviation found between the experimental and calculated lattice parameters is less than 0.1%.

In spinel oxides, the magnetic moments are mainly from the parallel uncompensated electron spins of the individual ions and according to Néel’s two sublattice model, the spin alignments in the two sublattices are antiparallel. The net magnetic moments per formula unit, n_B is expressed as: n_B = M_B-M_A, where M_A and M_B are tetrahedral and octahedral sublattice magnetic moments in μ_B, respectively. In spinels, the magnetic and crystallographic unit cells and the associated symmetry relations are the same. Therefore, both the nuclear and magnetic scattering contributions to the Bragg peaks occur at the same scattering angles giving rise to an enhanced intensity. In order to take the magnetic contribution to the Bragg diffraction peaks into account, the neutron diffraction data in the entire angular range for each member of the series were refined with an additional phase for the magnetic part (Figure 3). The sublattices and net magnetization obtained from the two-phase refinement of the room temperature neutron diffraction data have been plotted in Figure 7. Significant magnetic moments have been deduced from the refinement of neutron diffraction data from all the compositions indicating that the system possess ferrimagnetic ordering at room temperature. From the figure it is observed that the B site moments decrease slightly with increasing x, where as the A site moments vary within the error limit and remain almost

the same throughout the compositional range investigated. The net moments decrease slightly with increasing x in the system.

The ferrite system Ni_xCo_1−xFe₂O₄ (x = 0.0, 0.2, 0.4, 0.6, 0.8 and 1.0) synthesized in the conventional solid state sintering technique in air has been characterized to be polycrystalline single phase cubic spinel belonging to the space group F d-3m. The final sintering temperature, as determined by trial and error method, is found to be 1300˚C in air. The neutron diffraction measurements have given a clear picture of the distribution of cations for each member of the series having three different types of cations of which some of them are very close in atomic number. The occupation numbers of different cations over the A and B sites have been determined precisely since this technique is capable of differentiating them due to the large variation of the scattering length of these atoms for neutron. For all the compositions, Fe ions are distributed over both A and B sites. Ni ions invariably occupy the octahedral (16d) site only for x ≤ 0.6, while Co ions are distributed in both A and B sites for this compositional range. For x ≥ 0.8, some Ni ions enter into the tetrahedral (A) site, while all the Co ions enter into the octahedral (B) site. The knowledge of the cation distribution will be helpful in determining the magnetic and other properties of the system. The deduced sublattice and net moments reveal the system exhibited ferrimagnetic ordering at room temperature for the whole range of composition studied. In order to know the detailed magnetic behavior of the system, temperature dependent neutron diffraction studies will be performed and reported later on.

One of the author (Sadia Khanam) are grateful to the Reactor and Neutron Physics Division (RNPD), Institute of Nuclear Science & Technology (INST) and Reactor Operation and Maintenance Unit (ROMU), Atomic Energy Research Establishment (AERE), Bangladesh Atomic Energy Commission (BAEC), Savar, Dhaka for the support in order to carry out this research work.