Rietveld Structure Refinement and Cation Distribution of Cr 3 + Substituted Nanocrystalline Ni-Zn Ferrites

Ferrite nanoparticles of Ni0.7Zn0.3CrxFe2−xO4 were prepared by a sol-gel autocombustion method. The prepared samples were shown to have a cubic spinel structure by applying the full pattern fitting of the Rietveld method. The unit cell dimension, discrepancy factor, and interatomic distance have been determined. As the Cr3+ content x increases, the unit cell dimensions and crystallite size are decreased. The IR spectra show two absorption bands in the wave number range of 400 to 600 cm−1.


Introduction
Ni-Zn ferrites are one of the most versatile soft magnetic materials.Recently, the technological application of these materials has been studied extensively, primarily due to their applicability in many electronic devices owing to their high permeability at high frequency, remarkably high-electrical resistivity, low-eddy current loss, and reasonable cost [1][2][3].The Ni-Zn ferrite is a well-known mixed inverse spinel, whose unitary cell is represented by the formula (Zn x Fe 1−x ) [Ni 1−x Fe 1+x ]O 4 [4].The addition of impurities induces changes in the defect structure and texture of the crystal [5], creating significant modifications in the magnetic and electrical properties of these materials.Nickel chromate is a normal spinel while nickel ferrites are inverse spinels.To determine unit cell dimension imperfections and their nature to find the reason for the phase transformation of the synthesized material, X-ray characterization techniques based on structure and microstructure refinement are usually preferred [6,7].Rietveld analysis based on structure and microstructure refinement [8,9] has been adopted in the present study for precise determination of several microstructural parameters.The aim of the present work is to study the structure refinement by the Rietveld method and the study of cation distribution of chromium substituted Ni-Zn ferrites prepared by a sol-gel method.

Experimental
The ferrite powders with a generic formula Ni 0.7 Zn 0.3 Cr x Fe 2−x O 4 (where x = 0.0-0.5, in the step of 0.1) were synthesized by the sol-gel auto-combustion method using the AR grade citric acid ), and iron nitrate (Fe(NO 3 ) 3 •9H 2 O).The as-prepared powders of all the samples were sintered at 600 • C for 4 h to get the final product.Details of the sol-gel combustion technique procedure have been reported in our previous publications [10,11].The samples were X-ray examined by Phillips X-ray diffractometer (Model 3710) with Cu − K α radiation (λ = 1.5405Å).The data were processed to analyze all of the samples using the computer Program FullProf.2k(Version 4.30-Apr, 2008-ILL JRC) in the Rietveld method for structure refinement.Program refinement of the first samples was started with the space group Fd3m, origin at −3 m, O in 32e − , A site in 8f, and B site in 16c.In the first step the global parameters, such as 2θ-zero and background, were refined.In the next step, the structural parameters, such as lattice parameter, atomic coordinates, and site occupancy, were refined.IR spectra were recorded in the range 350-800 cm −1 at room temperature by an IR spectrometer (Bruker).

Results and Discussion
The XRD patterns confirm the single phase cubic spinel structure, and XRD refinement was continuous until convergence was reached with a goodness factor very close to 1.The values of the discrepancy factor (R wp ), expected values (R exp ), and Bragg value (R Bragg ) with the goodness of fit index (x 2 ) are listed in Table 1, our Rietveld refined values are in good agreement with the literature report for other ferrite systems [12,13].Figure 1 represents the typical Rietveld refined Xray pattern for sample x = 0.1.The lattice constant calculated by the Rietveld method and the values are listed in Table 1.
The lattice constant initially increases and then begins to decrease, the initial increase of the lattice constant from x = 0.0 to 0.2 may be due to the fact that the substitution of Cr 3+ up to x = 0.2 does not affect the lattice.The decrease in the lattice constant above x > 0.2 is related to the difference in ionic radii of Fe 3+ and Cr 3+ .In the present ferrite system, Fe 3+ ions (0.67 Å) ions are replaced by the relatively small Cr 3+ ions (0.64 Å).The system Ni 0.7 Zn 0.3 Cr x Fe 2−x O 4 under investigation is neither completely normal nor completely inverse.We have previously reported similar behavior of lattice constant with Cr 3+ content x [14].The average crystallite size (t) was determined using the line broadening of the most intense (311) diffraction peak using the Debye-Scherrer formula [15].The values of the crystallite size are given in Table 1.The crystallite size decreases from 35 nm to 20 nm with increasing Cr content.The cation distribution in spinel ferrite can be obtained from an analysis of the X-ray diffraction pattern.In the present work, the Rietveld refinement method [8] and Bertaut method [16] are used to determine the cation distribution.The best information on cation distribution is achieved by comparing the experimental and calculated intensity ratios for reflections whose intensities (i) are nearly independent of the oxygen parameter, (ii) vary with the cation distribution in opposite ways, and (iii) do not significantly differ.The final results of cation distribution obtained from the Rietveld refinement and the analysis of Xray diffraction and are given in Tables 2 and 3, respectively.In these tables, the fraction of Fe 3+ ions in either site is listed.The results demonstrate that Ni 2+ ions occupy B sites, whereas Zn 2+ ions occupy tetrahedral A sites.Cr 3+ preferentially replaces Fe 3+ from octahedral sites because of favorable crystal-field effects (Cr 3+ 6/5Δ 0 , Cr 3+ 0Δ 0 ) [17].The data in Tables 2 and 3 shows that Cr 3+ ions predominately occupy the octahedral sites, which is consistent with the preference for large octahedral-site energy.With increasing Cr 3+ content, the fraction Cr 3+ ions in octahedral sites increases, whereas the fraction of Fe 3+ ions in octahedral sites decreases linearly.
Using the values of a, the radius of oxygen ion R O = 1.32 Å, and radius of tetrahedral A site (r A ) in the following expression, the oxygen positional parameter u can be calculated [18] Table 3 shows the increasing value of the oxygen positional parameter from 0.3888 to 0.3893 Å.In most oxide spinels, the oxygen ions are larger than the metallic ions.In spinel-like structures, the oxygen positional parameter has a value near 0.375 Å, for which the arrangement of O 2− ions is equal, exactly a cubic closed packing, but in an actual spinel lattice, this ideal pattern is slightly deformed.u has a value of 0.37 Å when the origin is chosen on the tetrahedral sites.However, the structure is a centric and the structure factor calculation is less direct [19].Our value of u is larger than the ideal value (u = 0.375 Å), which may probably be due to many reasons, including the history of the samples and experimental or measurement errors, for example, the precision of the observed X-ray intensity and the theoretical data used for the scattering model of the system.In most spinels, u > 0.375 is obtained of a small displacement of the anions due to the expansion of the tetrahedral interstices.In the present work, u > 0.375 may be due to an anion displacement from the ideal position [20].
The lattice disturbance is confirmed by the data for the lattice constant and the oxygen positional parameter.
The infrared spectra can give some additional information on valance state and the different vibrational modes of the crystal lattice.The band position obtained from these IR spectra is given in Table 4.The IR spectra of the series Ni 0.7 Zn 0.3 Cr x Fe 2−x O 4 are shown in Figure 2. The high-frequency band v 1 is in the range of 575-89 cm −1 and the low-frequency band v 2 is in the range of 400-420 cm −1 .The absorption bands observed within this range are an indication of the formation of the single phase spinel structure.Vibrational bands v 1 and v 2 are assigned to the intrinsic vibration of tetrahedral and octahedral sites [21].A small band v 3 is observed at 493 cm −1 and 501 cm −1 for x = 0.4 and 0.5, respectively.Similar IR spectra have been reported in the literature for a ceramically prepared Cr substituted ferrite system [22].It can be noticed from IR spectra that tetrahedral complex (v 1 ) have more intense absorption than octahedral complex (v 2 ).This is a consequence of the first selection rule: transitions between d orbitals in a complex having a center of symmetry are forbidden.As a result, absorption bands for octahedral complex are weak as compared to tetrahedral complex, the lack of centre of symmetry makes transition between d orbitals more allowed [23].The change in band position is attributed to the change in Fe 3+ -O 2 distance for the tetrahedral and octahedral complex.The small change in frequency of the bands v 1 and v 2 is due to the distribution of Cr 3+ ions, which replace the Fe 3+ ion only at the octahedral B site only, thus making no significant change in size of the octahedral site.Further, the decrease in the Fe B 3+ -O 2 2− intermolecular distance increases the metal-oxygen vibrational energies, which arises from the decrease in the number of Fe 3+ -O 2 2− complexes caused by the increase of the number of Cr 3+ -O 2 2− complexes [24] and the formation of Me 3+ O 2 2− at A and B sites (Me = Ni 2 + ).The force constant is the second derivative of the potential energy with respect to the site radius with the other independent parameters kept constant.The force constant for tetrahedral site (K t ) and octahedral site (K o ) were calculated using the method suggested by Waldron [21].According to Waldron, the force constants K t , and K o for respective sites are given by where M 1 and M 2 are molecular weight of cations on A and B sites, respectively.The bond lengths R A and R B have been calculated using the formula given by Gorter [25].The  molecular weights of the tetrahedral M 1 and octahedral M 2 sites have been calculated using the cation distribution data in Table 3.The values of R A , R B the force constants K t and K o are listed in Table 4.The force constants K t and K o are found to increase, whereas R B initially increases and then decreases with Cr content.It is an established fact in IR studies that the force constant is inversely proportional to the bond length [26].

Conclusions
In conclusion, the substitution of Cr 3+ has induced significant changes in the structural properties of Ni-Zn ferrite.Experimental results revealed that the lattice constant and cell volume decrease with increasing Cr 3+ content in Ni-Zn ferrite.The cation distribution suggests that Cr 3+ and Zn 2+ both have a strong preference towards the octahedral B site and that Ni 2+ also occupies the B site, whereas Fe 3+ occupies both the A and B sites.The IR spectra confirm the formation of the spinel structure and give the distribution of ions between A and B sites.

Table 3 :
Cation distribution and oxygen positional parameter (u) of Ni 0.7 Zn 0.3 Fe 2−x Cr x O 4 .

Table 4 :
Band positions (v 1 and v 2 ), force constants (K o and K t ), and bond lengths (R A and R B ) of Ni 0.7 Zn 0.3 Fe 2−x Cr x O 4 .