Sintering process is necessary step in the preparation of ferrite by standard ceramic method, and the sintering temperature plays a key role in crystalline type and particle size of ferrite [13] [14] . Figure 1 shows the XRD patterns of the prepared ferrite samples indicating that the samples in a single phase cubic spinel structure, which indicates that the crystallographic planes of the characteristic peaks of cubic spinel structure that perfectly matched with the theoretical data of Franklinite spinel structure (ICDD card No. 74-2402). Figure 2 shows that the positions of peaks were shifted to a higher value of 2θ with substitution of Zn²⁺ ions, indicating that the lattice parameter for the cubic Mn_1−xZn_xFe₂O₄ samples decreased with Zn-content (x). The crystallite size (t) of the prepared samples was calculated using Scherrer’s Equation [15] ,

where β is the peak width at half maximum, λ is wavelength and θ corresponds to the peak position. The most intense peak at (220), (311) and (400) are used for calculation. The calculated crystallite size (t) as a function of Zn content (x) shows that the substitution the Mn²⁺ ions by Zn²⁺ ions results in decrease in crystallite size from 115 nm to 90 nm as shown in Table 1.

The experimental lattice parameter (a_exp) of the prepared samples was calculated and plotted as a function of Zn content (x) which decreases with increasing Zn content (x) in accordance to gradual replacement of bigger Mn²⁺ ions of higher ionic radius (0.75 Å) with smaller Zn²⁺ ions of ionic radius (0.68 Å) as shown in Figure 3 [16] -[18] . The correlation between the ionic radii and the theoretical lattice constant (a_th) is calculated using the equation [19] :

where R_o is the radius of the oxygen ion (1.38 Å), r_A and r_B are the ionic radii of tetrahedral A- and octahedral B- site, respectively. In order to calculate r_A and r_B it is necessary to know the cation distribution for a given system. In general, Zn²⁺ ions have higher preference for A-sites while Mn²⁺ and Fe³⁺ ions are distributed between the tetrahedral A- and octahedral B-sites, so the formula of the cation distribution can be written as:

where the brackets ( ) and [ ] denote to A- and B-sites respectively. The ionic radius of each site was calculated according to the following equations:

where, and are the ionic radii of Zn²⁺, Mn²⁺ and Fe³⁺ ions respectively. The theoretical and experimental values of lattice parameter (a_th and a_exp) are plotted against Zn content (x) in Figure 3. From Figure 3, one can see that (a_th and a_exp) decreased slightly with increasing Zn content (x). This decrease may be due to substitutional occupancy since the ionic radii of Zn²⁺, Mn²⁺ and Fe³⁺ ions are 0.68, 0.75 and 0.64 Å respectively. Figure 3 also shows that the experimental lattice parameter is greater than theoretical lattice parameter; this can be attributed to the formation of Fe²⁺ ions during the sintering process, which have an ionic radius greater than Fe³⁺.

The X-ray density (D_x) was calculated using the relation [20] :

where 8 is the number of molecules in a unit cell of spinel lattice, M is the molecular weight of the sample, a_exp is the experimental lattice parameter of the ferrite and N is the Avogadro’s number. The theoretical density D_x depends on the lattice constant and molecular weight of the sample. The theoretical density D_x as a function of Zn concentration is shown in Figure 4. It can be seen from the figure that the X-ray density increases with the increase in Zn-content (x), where it is inversely proportional to the lattice constant, which decreases with increasing Zn content. In addition, the difference in atomic weight between Mn (54.938) and Zn (65.37) contributes to this increase in D_x. Pores trapped inside the grains must be explained in terms of rapid grain growth, with increasing the Zn²⁺ ions concentration. It is clear from Figure 5 that, as Zn content (x) increases the porosity increases this can be explained as follow. It is well known that the porosity of ceramic samples results from two sources namely intragranular (P_intra) and intergranular porosity (P_inter), so that the total porosity (P%) could be written as the sum of two types

The percentage porosity (P%) was calculated from the relation

where D_app is the apparent density, it was found that the apparent density D_app values are less than those of X-ray density D_x as shown in Table 1, such behavior was expected due to the presence of pores formed during sintering process, therefore the replacements of Mn²⁺ ions by Zn²⁺ ions caused a relative decrease in apparent density D_app of the sample and a corresponding increase in porosity P%. In addition the decrease of density and the

increase of porosity with increasing Zn content (x) are due to the increase of oxygen vacancies which plays a predominant role in accelerating densification; i.e. the decrease in oxygen ion (anion) diffusion would retard the densification. The presence of Zn²⁺ ions reduces the population of Fe³⁺ ions in B-sites resulting in the decrease of density as well as the increase in porosity [21] .

Figure 6 shows the IR absorption spectra for Mn_1−xZn_xFe₂O₄ (0.0 ≤ x ≤ 0.5), from which it can be seen that there are five bands characterizing spinel ferrites in the range 200 - 1000 cm⁻¹ for the observed samples [22] . As seen in Table 2 the positions of five bands are recorded, the five bands can be classified into two groups; two high-frequency bands and three low-frequency bands. In the high-frequency bands (υ₁) is in range of 547 - 555 cm⁻¹ and is related to intrinsic vibrations of tetrahedral group, the second band (υ₂) is in the range of 407 - 430 cm⁻¹ and is related to intrinsic vibrations of octahedral group. Band (υ₁) (near 600 cm⁻¹) arises due to tetrahedral complexes (the stretching vibration of tetrahedral metal-oxygen band) and band (υ₂) (near 400 cm⁻¹) is due to octahedral complexes (metal-oxygen vibration in octahedral sites). The difference in position of two strong bands (υ₁) and (υ₂) could be related to difference in Fe³⁺-O²⁻ distance for A-sites (0.189 nm) and that of the B-sites (0.199 nm). The three low-frequency bands, (υ₃) is in the range 380 - 387 cm⁻¹ and is related to the divalent octahedral metal-oxygen ion complexes. The fourth band (υ₄) and fifth band (υ₅) in range 215 - 284 cm⁻¹ were observed and can be assigned to the divalent tetrahedral vibrations (lattice vibration). The presence of shoulder (υ’) could be ascribed to the linear combination of the bands (υ₂) at 407 cm⁻¹ and (υ₃) at 380 cm⁻¹ (407 + 380) = 787 cm⁻¹ at x = 0.1, (υ₂) at 423 cm⁻¹ and (υ₃) at 380 cm⁻¹ (423 + 380) = 803 cm⁻¹ at x = 0.4 [23] . The data in Table 2 confirm these results.

The electric properties of the ferrite to spinel system Mn_1−xZn_xFe₂O₄, was studied for (0.0 ≤ x ≤ 0.5 in steps of 0.1), over temperature range from room temperature to 650 K and represented in Figure 7. Figure 7 shows the variation of dc resistivity, expressed as (ln ρ) with absolute temperature, expressed as 1000/T. The obtained results could be described according to relation,

where E is the activation energy, K is the Boltzmann constant, ρ_o is the temperature independent constant. The temperature dependence of resistivity exhibits two breaks and distinct regions (I, II and III). Such a break was associated with a change in the slope which is attributed to the change of magnetic order and lowering the generation of charge carrier [24] confirmed this discussion and suggested that the change in the slope can be either

linked with magnetic ordering or with a conduction mechanism. The first region is attributed to extrinsic conduction mechanism (impurities); it extends from room temperature up to the first transition temperature (T₁) ≈ 305:355 K for all studied samples. The second transition temperature (T_c)_elec. is always attributed to the magnetic phase transition from ferrimagnetic to paramagnetic state. The activation energies for the conduction process were calculated from slopes of each line according to Equation (8). The activation energies (E_ferri.) for region II and (E_para.) for region III and the transition temperature (T_c)_elec. between ferrimagnetic and paramagnetic region are given in Table 3 which are ranged from 396 to 515 K for all samples and agrees with the published data for spinel ferrites. Figure 8 shows the relation between (lnρ) and 1000/T for sample (x = 0.2) i.e., Mn_0.8Zn_0.2Fe₂O₄, as an example of studied samples to illustrate the transition temperatures. The change in

activation energies between region II and region III is attributed to magnetic transition from order to disorder state. It is observed that (E_para.) is greater than (E_ferri.) for all investigated samples as shown Table 3. The increase of the activation energy from order to disorder state was attributed to the volume expansion above T_c. The lower activation energy in the ferrimagnetic region is attributed to the magnetic spin disordering due to decrease in the concentration of current carriers [25] [26] , while the change in activation energy is attributed to the change in conduction mechanism (polaron hopping). This decrease of resistivity in ferrite is due to the exponential increase of drift mobility of charge carriers with temperature.

The temperature dependence of Seebeck voltage coefficient for the investigated samples for Mn_1−xZn_xFe₂O₄ with 0.0 ≤ x ≤ 0.5 is shown in Figure 9. From the figure it is clear that, the investigated samples with negative sign indicates the semiconducting n- type behavior of the samples which is predominantly due to hopping of electron from Fe²⁺ to Fe³⁺ ions (Fe²⁺ ↔ Fe³⁺ + e⁻) [27] [28] . It is clear that the number of charge carriers is nearly constant. In other words, increasing Zn²⁺ ion substitution leads to the migration of more Fe³⁺ from tetrahedral site to octahedral one, hence increasing hopping electron between Fe²⁺ ↔ Fe³⁺. This means that the variation in the conductivity is due to thermally activated mobility and not due to thermal activated creation of charge carriers.