**3.1 XRD analysis**

**Figure 2** displays the XRD Rietveld Refinement corresponding to samples of CoErxFe2-xO4 with values of x between 0.00 to 0.030 (x = incremented by 0.005).

**Figure 2.** *XRD Rietveld refinement pattern of Er-substituted CoFe2O4.*

*Crystal Chemistry, Rietveld Analysis, Structural and Electrical Properties of Cobalt… DOI: http://dx.doi.org/10.5772/intechopen.98864*

It is observed that the peaks analogous to diffraction planes [111], [320], [311], [400], [511] and [440] match with usual data (JCPDS card no. 022–1086) confirming FCC cubic spinel structure for samples investigated [12, 13]. **Figure 3** shows shift in XRD peaks towards left hand side with increasing concentration of Er+3 ions in CoFe2O4 particles in concurrence with 'a' value. **Table 1** lists different parameters of XRD calculated for CoErxFe2-xO4 nanoparticles. The values of 'a' were calculated from the equation given [14].

$$\mathbf{a} = \mathbf{d} \ast \left(\mathbf{h}^2 + \mathbf{k}^2 + \mathbf{l}^2\right)^{1/2} \tag{1}$$

where cell constant is given by 'a', inter planer spacing calculated from Bragg's equation (2 dsin *θ* = *nλ*) is denoted by 'd' and miller indices are done by '*h,k,l'*.

It was reported that, low concentration RE (rare earth) doping in spinel ferrite experience phase separation and grain boundary diffusion giving rise to precipitation of additional crystalline phases like hematite (a-Fe2O3), metal monoxides and orthoferrites (REFeO3) [15–17]. Hence in case of rare earth doped ferrites, Er+3doped CFO having no impurity phase (*x* ≤ *0.010)* is exceptional and is because of auto-combustion. Induced effect due to substitution of erbium on the structure reflects two main observations given by decrease in size of crystal and increase in lattice constant both on small scale. The value of lattice constant slightly enhanced between 8.361 Å to 8.398 Å for *x* = 0.000 to *x* = 0.030 as per Law of Vegard [18]. Scherrer formula was used to calculate the crystallite size given by [19]:

$$\mathbf{L} = \frac{\mathbf{0}.9 \ast \boldsymbol{\lambda}}{\beta \mathrm{Cos}\theta} \tag{2}$$

**Figure 3.** *XRD pattern of Er-substituted CoFe2O4 and shifting of peaks.*



*Structural parameters of the prepared Co-Er nano ferrite sample.* *Crystal Chemistry, Rietveld Analysis, Structural and Electrical Properties of Cobalt… DOI: http://dx.doi.org/10.5772/intechopen.98864*

where 'λ' = wavelength of x ray,'β' = peak width at half maximum height and constant 'K' = 0.9. The data related to intense peak (311) was used in estimating size (L). The results indicated reduction in size of crystallite from 20.84nmto14.40 nm (for x = 0.0 to 0.030). Further, the high intense peak (311) shifts towards the lower angle with increasing values of *x* (**Figure 3**). **Table 1** lists the physical parameters obtained from XRD which indicated increase in lattice constant of Co-Fe-Er spinel lattice which might be due to replacement of 8 smallCo2+and Fe3+ ions with big Er3+ ions. Huge difference in radii of these three ions induce strain during formation of lattice and diffusion processes. Requirement of more energy in absorbing RE3+ ions with more radii while replacing Fe3+to form RE-O bond decreases crystallization energy and leads to particles of small size. Earlier literature reported similar results on RE-ion substituted cobalt ferrite [20–23]. From **Table 2**, EDAX confirmed the effect of incorporating Er3+ into CFO and stoichiometric amount of O, Fe, Co, and Er atoms. Therefore, XRD results are liable for expansion of unit cell due to larger Er3+ ion doping in CFO. Calculation of X-ray density (*Dx*) was done using [24]:

$$\mathbf{d}\_{\mathbf{x}} = \frac{\mathbf{8} \ast M}{\mathrm{Na} \mathbf{3}} \tag{3}$$

Here

*'M' =* compositionmolecular weight.

'*N' =* Avogadro's number.

'*a*' = lattice constant.

X-ray density value is found to increase from 5.3344gm/cm3 to 5.3392gm/cm3 (*x* = 0.00 to *x* = 0.030) with increasing Er3+ content. The bulk density increased from 3.2113to 3.2141 (x = 0.00 to x = 0.030). At the same time, CoFe2�xErxO4 ceramics having more Er content (x = 0.015) exhibited lower ErFeO3 orthoferrite amount along with primary spinel ferrite phase. Cobaltferrite in inverse spinel form has tetrahedral site occupied by half of Fe+3 while the remaining half of Fe+3 and Co�<sup>2</sup> occupy octahedral sites [25]. Any change in site occupation of Fe+3 and Co�<sup>2</sup> might be because of preparation technique and affect cell constant. Bulk densities were found from the relation [26].

$$\mathbf{d} = \frac{m}{\pi r 2h} \tag{4}$$

where pellets mass, thickness and radius are given by '*m', 'h' and 'r'*. Bulk densities exhibit inhomogeneous behavior due to pallets variable thickness and mass. The values of porosity in percentage were found using the relation.


**Table 2.**

*Summarizes different bond lengths of A, B sites due to Er+3 ion doping in spinel lattice.*

*Ferrites - Synthesis and Applications*

$$\mathbf{P}\mathfrak{W} = (\mathbf{1}\mathbf{-d}/\mathbf{d}\_{\mathbf{x}}) \times \mathbf{100} \tag{5}$$

Here d and dx are apparent and experimental densities. Surface area was calculated by using the Eq. (16).

$$\mathbf{S} = \frac{\mathbf{6000}}{D \ast d} \tag{6}$$

Here, S = area of surface, D = crystallite size, d = Bulk density. Strain was calculated by using the following Equation [27].

$$\text{Strain}\left(\boldsymbol{\varepsilon}\right) = \mathbf{1}/D^{\gamma}\mathbf{2}\tag{7}$$

Dislocation Density calculated by using following equation

$$\text{Dislocation density} \left( \delta \right) = \mathbf{15} \epsilon / aD \tag{8}$$

Here *ε* is strain, a is lattice constant, D is crystallite size. Packing factor is calculated by using following equation

$$\mathbf{P} = \frac{Lnm}{d} \tag{9}$$

Here L is crystallite size, d is inter planner spacing.

Cationic distributions that depend on factors like synthesis, total energy and thermal history are useful in understanding spinel ferrites behavior (electric and magnetic). Cationic calculations play important role in this regard. Average ionic radii of A, B sites were calculated from Stanley's equations:

$$\mathbf{r}\_{\mathbf{A}} = [\mathbf{u} - \mathbf{1}/4]\mathbf{a} \times (\mathbf{3})^{0.5} - \mathbf{R}\_{\mathbf{o}} \tag{10}$$

$$\mathbf{r\_B} = (\mathbf{5/8} - \mathbf{u})\mathbf{a} - \mathbf{R\_o} \tag{11}$$

Here Ro is the radius of the oxygen ion (1.35 Å), 'u' is the oxygen parameter whose ideal value is 0.375Å and experimental value is 0.383Å.

Bonding lengths and hopping lengths are calculated by using following formulas [28].

Bonding lengths: Hoping lengths:

$$d\_{A-A} = \frac{d}{4} \times \left(\mathfrak{Z}\right)^{0.5} \tag{12}$$

$$d\_{B-B} = \frac{a}{4} \times \left(2\right)^{0.5} a \tag{13}$$

$$d\_{A-B} = \frac{a}{8} \times \left(\mathbf{11}\right)^{0.5} \tag{14}$$

$$d\_{A-OA} = a\left(u - \frac{1}{4}\right) \* \left(\mathfrak{d}\right)^{0.5} \tag{15}$$

$$d\_{B-OB} = a \left\{ 2 \left( u - \frac{3}{8} \right)^2 + \left( \frac{5}{8} - u \right)^2 \right\}^{0.5} \tag{16}$$

$$L\_A = \frac{\sqrt{3}}{4} \times \mathfrak{a} \tag{17}$$

$$L\_B = \frac{\sqrt{2}}{4} \times a \tag{18}$$

*Crystal Chemistry, Rietveld Analysis, Structural and Electrical Properties of Cobalt… DOI: http://dx.doi.org/10.5772/intechopen.98864*

The difference in 'u' value in comparison with its ideal value on substituting Er+3 ions has been explained with rA values. Increasing rA values increase 'u'showing distortion in CoFe2O4 spinel lattice. Calculated values of ionic radii for B-sites are slightly higher than A-site because more Er+3 ions reside at B-site than A-site. Hopping length is the gap between magnetic ions at A, B sites. The hopping lengths between magnetic ions at A, B sites are denoted by LA and LB whose values reduce with addition of Er+3 content and is consistent with variation in lattice constant on adding Er+3 ions. The determined values from the formulas (10), (11), (17) and (18) are listed in **Table 1**.

By using the relations below structural parameters associated with A, B sites are calculated. Magnetic interactions and their strengths among AA, BB and AB sites mainly depend on bond length and bond angle existing between positive and negative ions. Increase in bond angle increases magnetic interaction strength while it reduces with increasing bond length as the strength has direct relation with bond angle and inverse relation with bond length. **Table 3** summarizes different bond lengths of A, B sites (dA-A, dB-B, dA-B, dA-OA, dB-OB) which depict an increase in bond lengths of tetrahedral and octahedral sites which is due to Er+3 ion doping in spinel lattice which might be due to larger Er+3 ions replacing smaller Fe+3 ions.

#### **3.2 EDAX analysis**

**Figure 4** displays the EDAX spectrums that analyzed elemental and atomic percentages of CoFe2-xErxO4 nanoparticles for x = 0.0, 0.005, 0.010, 0.015, 0.020, 0.025 and 0.030. It confirmed the presence of Co, O, Fe and Er. Er peak confirms Erbium substitution in the Fe2-x lattice.

**Table 2** summarizes atomic percentages of individual in CoFe2-xErxO4 nanoparticles. EDAX confirmed the effect of incorporating Er3+ into CFO and stoichiometric amount of O, Fe, Co, and Er atoms.

#### **3.3 Field emission scanning Electron microscopy (FE-SEM)**

**Figure 5** shows studies on surface morphology of ferrite powders with the help of FE-SEM. The nature of ferrite particle in the samples is uniform indicating fine form of agglomeration and grain growth. Agglomerate formation specifies strong magnetic nature of erbium doped ferrites. These studies also confirm microstructure changes on doping Er+3. A close look at these microstructures indicate improvement in microstructure and spherical shaped grains in all samples. Apart from this Erbium doping increases percentage of porosity in small range between 39.8001to 39.8018 illustrating individual grains and grain boundaries are separated.


#### **Table 3.**

*summarizes atomic percentages of CoFe2-xErx04 nanoparticles for x = 0.0, 0.005, 0.010, 0.015, 0.020, 0.025 and 0.030.*

**Figure 4.** *Displays the EDAX spectrums that of CoFe2-xErxO4 nanoparticles.*

### **3.4 Atomic force microscopy (AFM)**

AFM was used to characterize the surface roughness of CoErxFe2-xO4 nano ferrite samples of the synthesized nanoparticles. The three-dimensional arrangement of the spherical nanoparticles and diameter are shown in **Figure 6**. The surface roughness increased when the coercivity increases, but in this work all the parameters crystallite size, saturation magnetization, remanent magnetization, coercivity decreased with the increasing of Er dopant from x = 0.00 to 0.030 in the cobalt ferrite. In view of the above, the largest surface roughness is observed for x = 0.0 sample and the lowest surface roughness is obtained for Er (x = 0.030) doped samples. This indicates that the surface activity of x = 0.0 ferrite has higher values compared to the range x = 0.005–0.030 ferrite samples. The largest surface roughness is observed for x = 0.0 sample that is, it behaves like hard ferrite and the *Crystal Chemistry, Rietveld Analysis, Structural and Electrical Properties of Cobalt… DOI: http://dx.doi.org/10.5772/intechopen.98864*

**Figure 5.** *Displays FE-SEM images of CoFe2-xErxO4 nanoparticles.*

lowest surface roughness is obtained for Er (x = 0.030) doped samples. That is, it behaves like soft ferrite, hereby the ferrite is transformed from hard ferrite to soft ferrite due to the doping of Er content.
