Structural characteristics, cation distribution, and elastic properties of Cr3+ substituted stoichiometric and non-stoichiometric cobalt ferrites

Structural, elastic and cation distribution properties have been investigated on stoichiometric and non-stoichiometric cobalt ferrites. Crystal structure, formation of spinel type ferrite, chemical bonding, cation distribution, and thermal properties of two series of Cr3+ substituted stoichiometric and non-stoichiometric various cobalt ferrites with general formula Co1−xCrxFe2O4 (S1), and Co1+xCrxFe2−xO4 (S2) were reported. Samples are synthesized by the solid-state reaction technique via planetary ball milling. X-ray diffraction (XRD) analysis confirms the formation of a single phase cubic spinel structure with the space group Fd3̄m. Rietveld refinement results show that Cr occupies both the tetrahedral (A-site) and octahedral sites (B-site). The experimental lattice parameters show increasing trends for both the series with increase of Cr content. The cation–anion vacancies, chemical bonding, and the displacement of oxygen have been evaluated to understand the effect of Cr substitution and how the non-stoichiometry affects the physical and chemical properties of the material. The crystallite size is found to be the decreasing value with an increase of Cr concentration for both series of samples. Specific vibrational modes from the FTIR spectra suggest a gradual change of inversion of the ferrite lattice with the increase of Cr concentration which is also evident from Rietveld refinement data. The elastic properties analysis reveals that the synthesized samples for both series are ductile in nature. The non-stoichiometric structure with excess Co2+ may pave a new way to realize the lowering of Curie temperature of ferrite that is expected to improve the magnetocaloric properties.


Introduction
Ferrites are ferrimagnetic materials usually used in high density recording media and satellite communication as well as microwave devices in electronic industries due to their ease of fabrication, low cost, moderate saturation magnetization, mechanical strength, and chemical stability. 1 In the last decade, the synthesis of spinel ferrites emerged extensively due to their diversied use in technological applications. 2,3 In particular, cobalt ferrite (COF) has got enormous importance both in technological as well as biomedical applications. [3][4][5][6][7][8][9][10][11][12][13] Intensive studies on COF have been carried out with varying doping and/or substituting the divalent or trivalent cations and synthesizing them by different methods. 3,14,15 In order to conrm the crystallinity and the phase of the synthesized materials, Rietveld renement has been signicantly used on the X-ray diffraction (XRD) data of the samples. 1,[9][10][11]16 Amri et al. 1 have calculated the theoretical lattice parameters along with hopping lengths, oxygen positional parameters, cationcation and cation-anion bond lengths of Ni-Zn-Al ferrite. Kumar et al. 14 have reported on the estimation of the lattice parameters by using Nelson-Riley (N-R) function, intrinsic strain and crystallite size by using Williamson-Hall (W-H) method. It has also been studied that the crystallite size and intrinsic strain can be determined from the XRD peak broadening analysis by using several ways such as W-H method, size strain plot (SSP), Scherer method, and the modied Scherer method (MS). 14,17,18 However, it is reported that SSP and MS methods are found more signicant compared to that of W-H and Scherer methods, because both SSP and MS can be observed at a lower angle where the accuracy is found to be very high. 18 On the other hand, the formation of ferrite and identication of the chemical bonds can be analyzed from the Fourier transform infrared (FTIR) spectra. 1,[19][20][21] Several techniques for the measurement mechanical properties have been used to calculate the elastic and mechanical properties. FTIR is one of the suitable techniques found to calculate the elastic and thermal properties. The elastic moduli, Debye 2. Experimental procedure

Synthesis technique
The selected two series of stoichiometric and nonstoichiometric COF denoted as S1 and S2 have been synthesized by the standard solid-state reaction method. The required amount of Co 2 O 3 (98.0%), Cr 2 O 3 (99.9%), and Fe 2 O 3 (96.0%) are weighed in an analytical balance and hand mixed in a mortar pestle for 2 h. The mixed powders are milled in a planetary ball mill (MSK-SFM-1) for 12 h. The milled powders are calcined at 800 C for 6 h for a complete solid-state reaction through diffusion of particles. Then the powders of each composition are pressed into pellets by applying a hydraulic pressure of 16 000 psi. Finally, the pellets are sintered at 1200 C for 6 h in a furnace and rcrushed in to ne powders.

Characterizations
The X-ray diffraction (XRD) data of the synthesized samples are taken using a Rigaku Smart Lab X-ray diffractometer with Cu-Ka radiation (l ¼ 1.5406Å) with the scanning angle 2q within the range of 10 to 80 . The Rietveld renement was carried out using the Full-Prof soware integrated in Match-3 soware. Room temperature Fourier transform infrared (FTIR) spectroscopy (Spectrometer model-IR Prestige-21) in the region 350-4000 cm À1 was used to nd out the functional groups and vibrational structure of the synthesized samples. The parameters related to structural, elastic, and thermal properties are calculated using the XRD data and FTIR spectra.

XRD analysis
The X-ray diffraction patterns along with Rietveld rened data have been shown in Fig. 1 29 No impurity phase has been observed in the pattern for the parent COF which signies the phase purity of COF. The XRD patterns for the samples of the stoichiometric series S1 and non-stoichiometric series S2 shown in Fig. 2 and 3, reveal the formation of spinel structure (COD No. 910063). The peak positions of (311) plane for both the series S1 and S2 have been shied towards the lower 2q angle with an increase of Cr content as shown in Fig. 2(b) and 3(b), which imply an increase of the lattice parameters as a result of Cr substitution.
3.1.1 Estimation of cation distribution. The cation distribution of the samples at the tetrahedral and octahedral sites of spinel structure has been determined by comparing the observed X-ray intensities from the specic planes of (220), (440), (400), and (422). The ratios of observed and calculated intensities viz. I 220 /I 400 , I 220 /I 440 , I 422 /I 440, and I 400 /I 440 are considered to evaluate the cation distribution. From the calculation of structure factor, it is reported that (220) and (440) are sensitive to cation distribution at the tetrahedral site, while (400) and (422) are sensitive to cation distribution at the octahedral site. 30 The general formulae for cation distribution at the tetrahedral site and octahedral site for samples of the series S1, and S2, are as follows: where a,b, and g are the cationic parameters for the tetrahedral site and a + b + g ¼ 1. For COF x ¼ b ¼ 0. The cation distribution has been calculated by the above formulae and tabulated in Table 1. The value of the inversion parameter as determined by calculating the ratio between Fe 3+ at the tetrahedral site and Fe 3+ at octahedral site Fe tetra /Fe octa has also been tabulated in Table 1. Using the exact cation distribution the ionic radius for the tetrahedral site (r tet ), and octahedral site (r oct ), have been calculated by the following relations: 16 r tet ¼ ar Co + br Cr + gr Fe for both the series, where r Co , r Cr , and r Fe are the ionic radii of Co 2+ , Cr 3+ , and Fe 3+ , respectively. The calculated ionic radii for the tetrahedral site and octahedral site are tabulated in Table 1. Fig. 4 represents the Energy Dispersive (EDS) spectra for all samples. The elemental analysis are performed from these spectra. From the data of EDS spectra, it is conrmed that chemical composition of the synthesized samples are close to the respective nominal composition.
3.1.2 Lattice parameter estimation. To determine true value of lattice parameter (a true ) from XRD data, Nelson-Riley (N-R) extrapolating function (eqn (A2)) has been used. 17 The lattice parameter for each Bragg position has been calculated using eqn (A3). The N-R function vs. a hkl graph has been plotted for each sample which is shown in Fig. 5 for both the series S1 and S2, respectively. From Fig. 5 the true value of lattice parameters for each sample have been determined by extrapolating q ¼ 0 . The extracted value of a true for each sample have been illustrated in Fig. 6. The lattice parameters calculated from the XRD data by using Match-3 soware are denoted by a exp , and these values for the both series of samples have been illustrated in Fig. 6. The lattice parameters calculated using the eqn (A1) are denoted by a th , and these values also have been illustrated in Fig. 6. It is observed that a exp for the sample of both series follows the similar increasing trend of the a th and a true with the Cr content which implies that a exp is reliable. This increase in lattice parameter for S1 series may be caused due to the larger ratio of Cr 2+ /Co 2+ at tetrahedral site than the Cr 3+ / Co 2+ ratio at octahedral site. The increasing trends of lattice parameter for S2 are due to the excess Co 2+ with an ionic radius of 0.75Å which is larger than that of Cr 3+ (0.63Å). 31 Since unit cell volume is directly proportional to the lattice constant, hence it follows a similar trend of the lattice constant.
3.1.3 Density and porosity estimation. The bulk density d B , crystal density d x , and porosity (P) for both series of samples have been calculated using eqn (A4), (A5), and (A6), respectively and have been illustrated as a function of Cr concentration in Fig. 7. The density extracted from the Rietveld rened data is denoted as d reit is also presented in Fig. 7. These parameters have been listed in Table 2. It is seen that the d x , and d B show a similar decreasing trend with the increase in Cr content which agrees with the Rietveld rened density d reit . The decreasing trend of the density is due to the molecular weight loss of the investigated samples. Furthermore, it is seen that the d x and d B follow the increasing trend with the Cr content that is also agrees with the d reit for the samples of the series S2. This increasing nature of density may have been caused by the molecular weight gain due to the non-stoichiometry of samples of the series S2. The porosity is found to increase with the addition of Cr content for the samples of both series.
3.1.4 Oxygen positional parameter, interionic distance and bond angle estimation. When tetrahedral interstices are occupied by divalent ion then the expansion in tetrahedral site is relatively larger than octahedral site due to the difference of ionic radii of cations. This expansion can be explained by oxygen positional parameter, u, which gives quantitative measure of the displacement of oxygen ions. This displacement takes place whenever there is a difference in the radii of substituted and replaced ions. The value of u can be calculated from the a th using the eqn (A7).
The calculated u values for both series are listed in Table 4. As seen in that table, u value is almost invariant with Cr content for both series found to be almost equal to that of the parent COF. In a cubic spinel structure the ideal value of u is equal to 3/ 8 ¼ 0.375. The calculated value of u is slightly larger than that of ideal value, which may be due to the anion displacement from its ideal position. From the analysis of u and d (¼ u À 0.375), it is conrmed that lattice is slightly distorted for all the samples of both series. 33 The inter-ionic distance between magnetic ions is named as hopping length in tetrahedral site (L A ) and octahedral site (L B ) which gives information about the strength of spin interaction of ions. The hopping lengths have been calculated by using Stanley's equations (eqn (A8)). 1 The calculated values of L A and L B have been tabulated in Table 3 for both the series. Hopping lengths decrease for S1 series, and increase for S2 series with the increasing Cr content.
The tetrahedral and octahedral bond lengths (d AL ) and (d BL ), tetrahedral edge length (d AE ), and shared octahedral edge lengths (d BE ) and un-shared lengths (d BEU ) have been calculated using eqn (A9)-(A13). The calculated d AL , d BL , d AE , d BE, and d BEU values for both the series have been listed in Table 3. From this Table, it is seen that the values of d AL , d AE have increased while the values of d BL , d BE, and d BEU have reduced with an increase in Cr content for both the series S1, and S2.
Magnetic interaction strength for cubic spinel and spinellike ferrite depends on the cation-cation (Me-Me) bond length and cation-anion (Me-O) bond length and bond angle that have been calculated using the eqn (A14)-(A27). All the calculated bond lengths and bond angles have been presented in Table 3. As seen in Table 3, there is decreasing and increasing trend in Me-Me for the S1 and S2 series, respectively. The Me-O distance shows an increasing trend for both the series. The bond angles show a decreasing trend with the increasing Cr content for both series. The decreasing trend of Me-Me and the bond angles may be due to the smaller ionic radius of Cr 3+ than that of the Co 2+ for the samples of the series S1. But in the case of the samples of series S2, this decreasing in Me-Me and bond angles is most likely due to the combined effects of smaller ionic radius of Cr 3+ and excess of Co 2+ ions therein. This behavior of bond length and bond angles for the samples of the series S1, and S2 imply the lattice expansion with the increase in Cr content, which is found to be in agreement with the lattice volume. To conrm the presence of cation or anion vacancies, estimation of ionic packing coefficient is required. The ionic packing coefficient of tetrahedral site P tet and octahedral site P oct have been calculated by the eqn (A28) and (A29), respectively. The values for P tet and P oct have been presented in Table 4 for both the series. From Table 4, it is observed that the values of P tet and P oct are less than one, which suggests the existence of cation and anion vacancies in the parent COF. 34 For series S1, the values of P tet and P oct increase for increasing Cr content and the values are close to 1, which indicates the reduction of ion (cation and anion) vacancies. On the other hand, P tet and P oct have been found to be decrease with increasing Cr 3+ content which indicates enhancement of ion (cation and anion) vacancies.
The degree of ionic packing coefficient can be evaluated by calculating the fulllment coefficient (a) of the unit cell using eqn (A30). The vacancy parameter b is dened as normalized values of ions at the nodal point of the spinel structure which is calculated using eqn (A31). The values of a (tabulated in Table  4) show close to 0.58 for all the samples which conrm that our synthesized samples exhibit inverse spinel structure as explained in the earlier literature. 34 The values of b listed in Table 4 show a low vacancy parameter which implies that the missing ions are lesser for parent COF. For series S1 the values of b shows the decreasing trend up to x ¼ 0.375 that implies reducing the missing ions due to Cr substitution. But at higher Cr content, it shows the negative value, which are most likely due to the excess of ions. The excess of ions dominates due to higher Cr content in series S1. But in series S2, b values increase strongly with an increase of Cr content that dominates the enhancement of metallic behavior from semiconducting behavior due to the higher values of Co 2+ ions. 35 The tolerance factor (T) is another property to get an idea about the impurities of these type of materials. Hence the T have been calculated using eqn (A32) that have been tabulated in Table 4 for both the series. It is observed that T is slightly   higher than one which indicates that the synthesized sample is slightly distorted from the inverse spinel structure. In both the series, the values of T are found to decrease with the increase of Cr content indicating the reduction of distortion from the central atom resulting the improvement of inverse spinel structure.

Crystallite size estimation.
To evaluate crystallite size, analysis of X-ray proles is the most effective and easiest way. To calculate crystallite size and strain of the powdered samples various methods have been used such as: Scherrer, Modied Scherrer, Size strain plot, and Williamson-Hall plot. In every case, XRD data have generally been used because of X-ray line broadening comes out mainly from three factors: (i) instrumental effect, (ii) crystallite size, and (iii) local lattice strain. To exclude the instrumental broadening, a standard silicon X-ray powder diffraction data is recorded under the same condition and eliminated from the observed peak width. The full width at half maximum (FWHM) of all peak positions from the XRD has been estimated using a nonlinear combined curve tting function that includes Gaussian and Lorentzian functions. The FWHM data has been calculated by using the eqn (A33). In addition, the instrumental broadening (b i ) is removed by using the eqn (A34). The values for b i are 0.092 , 0.099 , 0.117 , 0.186 , 0.21 , and 0.208 for the (220), (311), (400), (511), and (440) peaks, respectively. The average crystallite size (D) have been calculated by using observed FWHM of the most intense peak (311) with the help of Scherrer equation (eqn (A35)). The calculated values of D using this method are listed in Table 5.  Table. Scherrer and modied Scherrer methods can provide the only information about the crystallite size but not the information about the intrinsic strain of the lattice. As such, Williamson-Hall method has been utilized to calculate both D and Here, b hkl is the total broadening due to strain and size in a particular peak having the (hkl) value which is written exchange of b tot . Fig. 9 show the b hkl cos q vs. 4 sin q graph corresponding to each diffraction peak for the samples of series S1, and S2, respectively. In every case for all the samples, the slope and y-intercept have been noted from the linear tted curve, which gives the 3 and D of the investigated samples. The 3 and D for all the samples have been listed in Table 5 in Appendix B.
Williamson-Hall method describes isotropic peak broadening due to the combination of size and strain-induced effect as a function of a 2q.To better evaluation of 3 and D another model known as "size-strain plot (SSP)" has been used. In this method, higher angle reections are less important than lower angle reections. In the SSP method, the following relation has been considered: 17 Here, d hkl is lattice spacing for different (hkl) planes. The (d hklb hkl cos q) 2 are plotted as a function of d hkl 2 b hkl cos q and their linear tted curve have been drawn using the origin soware as shown in Fig. 10. The slope and y-intercept have been extracted for all the samples that provides the values of D and 3 of both series and presented in Table 5. In addition, the D and 3 values extracted by the Match-3 soware from the Rietveld rened XRD data have also been listed in Table 5. The D values calculated by using all the methods is found to be decreasing trend with the increasing Cr content for both series as shown in Fig. 11. This decreasing trend is most likely due to the peak broadening. Fig. 12 illustrate the FTIR spectrum in the wavenumber range of 350-3000 cm À1 for the samples of both series at room temperature. From the FTIR spectra, two distinct absorption bands at  554 cm À1 (n 1 ) and 372 cm À1 (n 2 ) are observed for the parent COF. The symmetrical stretching vibrations of metal-oxygen at tetrahedral and octahedral sites have been observed at the higher band n 1, and lower band n 2 , respectively. For all samples, it is seen that both n 1, and n 2 are increased with the increase of Cr content as shown in the inset of Fig. 12. The values n 1, and n 2 have been listed in Table 6. A clear absorption band has been noticed at around 3000 cm À1 (n 3 ). This may be attributed to the H-O-H stretching vibrations due to the effect of moisture during fabrication of the studied samples. 3.2.1. Calculation of force constant. The wavenumbers n 1 and n 2 of the infrared active phonon mode is directly connected to the force constant. The force constant k t and k o at the tetrahedral and octahedral site of cubic spinel structure has been calculated by the Waldron relation (eqn (A39)-(A41)). The average force constant k av ¼ (k t + k o )/2 have shown in Fig. 13 for the all samples of both series. From the Fig. 13, it is seen that the average force constant k av increases for the samples of both series which is as usually related to the Me-Me bond distances and bond angles.

Elastic properties analysis.
Ferrites demonstrate important elastic properties and thermal behavior due to their interatomic and interionic forces. Although the elastic properties of such materials are determined by applying external stress, according to the analysis of various research, a technique based on the structural and FTIR data parameter related to elastic properties along with thermal properties has been calculated for exploring the correlations with the other properties. 1,[20][21][22] According to Hook's law, the stress h i , strain 3 ij and stiffness C ij are correlated on the basis of the stress-strain approach. 40 The stiffness C ij are used to calculate the elastic constants. For cubic symmetry, only three stiffness C 11 , C 12 , and C 44 are considered to be dominant, where C 11 represents the elasticity in length and C 12 and C 44 represents the elasticity in shape. The stiffness constant C 11 and C 12 are calculated using eqn (A43) and (A44), and the values are tabulated in Table 6.
The bulk modulus, B, rigidity modulus, G, Young's modulus, E, Poisson's ratio, s, longitudinal wave velocity, v l , transverse wave velocity, v t , and the mean velocity, v m , have also been calculated for all samples of both series using the eqn (A46)-  (A51). The calculated values of all these elastic constants with Cr content for both the series have been graphically presented in Fig. 14. The values of E, B, and G are found to increase with increasing Cr content. The measured elastic moduli however do not provide enough information on the mechanical properties of the investigated samples due to the presence of porosity. Hence, to improve the elastic nature of the materials, the corrected zero porosity elastic moduli have been calculated by using Hasselman and Fularth's formula (eqn (A51)-(A54)). The corrected zero porosity elastic moduli (E 0 , B 0 , and G 0 ) have been listed in Table 6 which shows the larger value than the measured E, B, and G. From Table 6, it is seen that the values of Ductility and brittleness behavior of a material can be estimated by the Pugh's ratio. 42 The Pugh's ratios (B/G) are illustrated in Fig. 15 for the samples of both series. According to Frabtsevich et al. 43 the brittleness and ductility nature can be conrmed from the value of s. The calculated values of s are also illustrated in Fig. 15. It is observed that both the Pugh's ratio and s are higher than that of their respective critical values 1.75 and 0.26, respectively. This reveals the ductile nature of the synthesized samples. 42,43 The ductility decrease with the increase of Cr content for both the series which may be due to the substitution of brittle Cr with ductile Co and Fe for the sample of both series. In addition, s for all the samples are in the range of 0.27-0.30 which lies in between À1 to 0.5 which implies that the investigated samples are of isotropic elastic in nature.
3.2.3 Thermodynamic properties. The Debye temperature, q D , is a signicant parameter to know about the thermodynamic properties of a solid that originates from the maximum lattice vibration of the atoms. According to Anderson's formula the q D can be calculated using the eqn (A55). Thermal conductivity is one of the thermodynamic properties of a material that indicates the ability to conduct heat. The minimum value of thermal conductivity, K min , have been calculated by the eqn (A56). The calculated q D and K min have been presented in Fig. 16 as a function of Cr content for all samples of both series. It is observed that q D and K min increases with the increase of Cr content for S1 series. However, for S2 series both q D and K min shows the maximum value for x ¼ 0.125, and beyond this value of Cr content they decrease.

Discussion
The analysis of structural, mechanical, and thermal properties of the samples of two series S1, and S2, show almost similar behavior in most of their properties. For the stoichiometric series S1, the XRD pattern shows a single phase cubic structure for all Cr contents. For a lower value of Cr 3+ substitution with Co 2+ , cation-anion vacancies dominated. However, for higher values of Cr 3+ , negative values of vacancy parameter have been found. This negative vacancy parameter dominates the excess   Cr 3+ replaced with the Co 2+ and these excess ions are not too high, so no impurity peaks have been observed in the XRD patterns. A slight distortion of inverse spinel structure has been conrmed from the tolerance factor for all samples of both series. The dominance of Cr 3+ with extra Co 2+ ions due to lowering the packing fraction demonstrates strongly increasing trend of vacancy parameter with increase of Cr 3+ and leading to transform semiconducting to metallic behavior in S2 series. However, a single phase cubic structure is observed for the samples of S2 series without any impurity peak in the XRD patterns. The observed anion displacement for the samples of two series are dominated by the deviation of oxygen positional parameters from their standard value of 0.375. Expansion of lattice volume for the series S1 has been observed due to the larger ratio of Cr 2+ /Co 2+ at tetrahedral site than the Cr 3+ /Co 2+ ratio at octahedral site. In S2 series, Fe 3+ has been replaced by Cr 3+ with a simultaneous addition of Co 2+ , resulting an increase in lattice volume. In series S1, both the d x and d B show a decreasing trend, which may be attributed to the increase of Cr 3+ with a lower density of 7.14 g cm À3 than that of Co 2+ (8.9 g cm À3 ). 45 Although pelletization pressure and sintering temperature for all samples were the same, there is an increasing trend of ductility with an increase of Cr content. At higher sintering temperature (>1100 C) the ductility of Cr/Crbased alloys and composite is reported to improve. 46 In series S2, Cr metal having lower density replaced the Fe (7.87 g cm À3 ) with slightly higher density. 45 However, addition of same amount of Co metal with high density, resulted in the increasing trend of d B , and d x with a increase in porosity. However, the ductility is found to decrease possibly due to excess Co 2+ . In addition, an increase of excess Co with lower shear modulus presumably plays a prominent role in lowering the ductile nature for the sample series S2. The decreasing trend of ductility as a function of Cr content are observed due to the increase of the Cr 3+ /Fe 2+ ratio. In series S1, the q D shows an increasing trend due to the greater q D of Cr than that of Co. 47 In series S2, the q D increases at lower Cr content (¼0.125) but at a higher Cr content (>0.125), q D decreases due to the increase of Co/Cr ratio, and decrease of Fe/Cr ratio.

Conclusions
Two series of Cr 3+ substituted cobalt ferrite with stoichiometric (S1 series) and non-stoichiometric (S2 series) ratios have been synthesized by the standard solid-state reaction technique. XRD patterns for all samples of both series indicate a single phase cubic spinel structure with a space group of Fd 3m. FTIR spectra of the synthesized samples also conrm the spinel structure. The cation distribution calculated from the extracted Rietveld rened data conrms the mixed spinel structure of all samples of both series. In both the stoichiometric and non-stoichiometric series, an enhancement of lattice parameter have been found. From the analysis of elastic properties, it is found that all the samples for both series are ductile in nature. For the stoichiometric series Cr 3+ together with Co 2+ is assumed to have inuenced the structural and mechanical properties. In addition, nonstoichiometric compositions provide an opportunity to tailor Co 2+ /Cr 3+ and Fe 3+ /Cr 3+ in the inverse spinel ferrite which causes the defect in structure, and modication of mechanical behavior.

Appendix-A
Equations belonging to Section 3: Equation for theoretical lattice parameter: 1 where R O (¼1.32Å) is the ionic radius of oxygen. The N-R function is as follows: 17 The lattice parameter for each Bragg position: 30 Here, a hkl , d hkl are the lattice constant and interplanar spacing respectively. Equations for bulk density d B and X-ray density d x