Solid state reaction technique is assumed to synthesis Ti⁴⁺ substituted Ni-Cu-Zn ferrites of stoichiometric compositions Ni_0.4Cu_0.2Zn_0.4Fe_2−xTi_xO₄ (x = 0.0, 0.02, 0.05, 0.07, 0.10) in the present study. Ni₂O₃ (99.99%), CuO (99.99%), ZnO, TiO₂, Fe₂O₃ are used as raw materials for the preparation of Ni-Cu-Zn ferrites. The constituent components in required stoichiometric proportions are weighted and the weighted powders are mixed mechanically by milling to decrease the particle size of the powders to the micron range. For this preparation, mortar and pestle are used for milling operation and each sample was milled for 6 hours. The resulting powder is calcined at 800˚C for 4 hours and grounded into fine powder by milling for 2 hours. Grounded powder is then used in X-ray diffractometer for XRD for phase identification, Scanning Electron Microscope (SEM) for micro-structural investigation and Vibrating Sample Magnetometer (VSM) for magnetic properties measurement. For the investigation of dielectric properties, the powders are then pressed into disc and toroid shaped sample at a pressure of 10 - 15 KN/m². These disc and toroid are sintered at 1200˚C for 4 hours.

XRD was performed using PHILIPS PW 3040 X-ray diffractometer with Cu-Kα radiation in the range 5˚ to 70˚. Lattice parameter (a₀), theoretical density (ρ_th), porosity (P), bulk density (ρ_B) were calculated for all the samples using the following Equations (1)-(5) and were tabulated in Table 1 [16].

d h k l 2 = a 2 h 2 + k 2 + l 2 (For cubic spinel) (1)

F ( θ ) = 1 2 ( cos 2 θ / sin θ ) + ( cos 2 θ / θ ) (2)

ρ t h = 8 M N A a 0 3 g/cm³ (3)

P ( % ) = ( ρ x − ρ B ) ρ x × 100 % (4)

ρ B = m π r 2 t (5)

where h, k and l represent the crystal plane’s Miller indices, N_A denotes the number of Avogadro, M means molecular weight, a₀ is the lattice constant of spinel structure, m equal mass, r indicates radius and t signifies the pellet or ring thickness [16].

The grain structures of the sample were visualized under Field Emission Scanning Electron Microscope (FESEM). Measurement of the magnetic hysteresis loop and the magnetic properties of the materials were performed using a Vibrating Sample Magnetometer (VSM) within a range of 10 kOe (1 Tesla). For dielectric tests, etched, well-polished pellets were used. For good electrical contact, silver paste had been used on both sides of the samples. Wayne Kerr Impedance Analyzer is used to carry dielectric measurements at room temperature in the frequency ranged 1 kHz to 100 kHz. The variation of dielectric properties (dielectric constant, dielectric loss) and magnetic properties (magnetic loss, relative quality factor, resistivity) for all the samples were measured and calculated using the following Equations (6)-(9) [2] [3].

Dielectric constant, ε ′ = C C 0 (6)

Dielectric loss, tan δ E = ε ″ ε ′ (7)

RQF, RQF = ε ′ tan δ E (8)

Resistivity, ρ = R A l (9)

where, C is the capacitance of the samples and C₀ is the capacitance of the capacitor without the dielectric materials and C 0 = ε 0 A / l , ϵ 0 is the permittivity of the free space and A = πr² is the area of the cross section of the pellet shaped sample, R is the resistance of the sample, l is the thickness of the pellets and A is the surface area.

The measurement of complex permeability of the ring shaped samples was performed at room temperature in the frequency range from 1 KHz to 100 MHz using the following formulas,

Real part, μ ′ i = L s L 0 (10)

Imaginary part,

μ ″ i = μ ′ i tan δ M (11)

where, L_S is the inductance of the sample and L o = μ 0 N 2 S π d ¯ and that is derived from geometric of the samples, L₀ is the inductance of the winding coil except the sample, N represents the number of turns of the coil (N = 5), S indicates the cross sectional area of the ring shaped samples as given below,

S = d × h (12)

And

d = d 2 − d 1 2 (13)

where, d₁ is the inner diameter and d₂ is the outer diameter, h represents the height of the ring. Again, d ¯ indicates the mean diameter of the toroidal sample and can be written as,

d ¯ = d 1 + d 2 2 (14)

Then Equation (10) can be written as

μ ′ i = π L S d ¯ μ 0 N 2 ( d 2 − d 1 2 ) h (15)

X-ray diffraction pattern of Ni_0.4Cu_0.2Zn_0.4Fe_2−xTi_xO₄ (x = 0.0, 0.02, 0.05, 0.07, 0.10) ferrite is shown in Figure 1. In XRD patter, from the reflected peaks it is clear that the samples are of crystalline cubic spine structure. Having differences in the ionic radius of Ti⁴⁺ (0.68 Å) with that of Fe³⁺ (0.64 Å), there is an increasing trend of lattice parameter a₀ of the lattice with increase in Ti⁴⁺ concentration. The increase in lattice parameter could be ascribed to Ti⁴⁺ ions can be located at the boundaries of grain [16]. That’s why it hinders the growth of grain and increases the rinsing of grains and the tips to increase the lattice parameter.

Figure 2 shows that lattice parameter increases with the Ti content except x = 0.02. It has been observed by Mali et al. that the rise of the crystallite size with the substitution of Ti⁴⁺ is attributed to the substituting of Ti⁴⁺ (0.68 Å) at Fe³⁺ (0.64 Å) site [17].

Figure 3 illustrates the variation of bulk and theoretical densities of Ni_0.4Cu_0.2Zn_0.4Fe_2−xTi_xO₄ (x = 0.0, 0.02, 0.05, 0.07, 0.10). From this figure it is observed that theoretical density of these samples reduces almost linearly with the concentration of Ti except for x = 0.02. Theoretical density decreases because the atomic mass of Ti (47.867 g/mol) is less than that of Fe (55.845 g/mol). And for x = 0.02, the theoretical density rises because in this case the lattice constant (a₀) decreases. From Figure 3, it has also been noted that bulk density of these samples rises linearly with the concentration of Ti except for x = 0.02 and x = 0.05. It is obvious that theoretical density is higher than bulk density, this variation is explained in the way that when the samples were sintered, they may contain crack and microscopic pores and lattice vacancies on the atomic scale [18]. But theoretical density which is precisely measured from lattice parameter and unit cell volume of the sample is free from these defects.

Figure 4 represents the variant of the porosity and bulk density ( ρ B ) with the concentration of Ti. The variant of ρ B with the concentration of Ti might be understood from the microstructure. The bulk density decreases initially for x = 0.02 and x = 0.05 and for these two samples the microstructure contains intergranular porosity. Intra-granular porosity leading to poor mechanical and magnetic properties is practically impossible to eliminate [19]. When the growth rate of the grain is very high, the pores may be left behind the rapidly moving grain boundary and trapped inside the grain. The increase of bulk density for x = 0.07 and x = 0.10 is due to the compact microstructure of the samples. It is evident from Figure 4, that the ρ B and porosity show the opposite behavior to each other. The structural parameters of Ni_0.4Cu_0.2Zn_0.4Fe_2−xTi_xO₄ ferrites have been presented in Table 1.

Scanning Electron Microscopy of Ni_0.4Cu_0.2Zn_0.4Fe_2−xTi_xO₄ (x = 0.0, 0.02, 0.05, 0.07, 0.10) compositions with 5000× magnification are shown in Figure 5. Micro structural images of Ni-Cu-Zn-Ti ferrite samples are taken for morphological study, so that an insight of grain structures can be seen. The average grain size (D) of the samples is estimated using a linear intercept technique [20]. Value of D reduces with the increase in concentration of Ti except for x = 0.02 and decreases from 18.89 μm to 10.85 μm (for x = 0.02 the grain size is 19.76 μm). Pores at grain boundary becomes very less when the grain size is smaller. The addition of Ti content might have suppressed the growth of grain boundary. Hence, the D reduces with the increase in concentration of Ti. Decrease in D could also be due to a decrease in oxygen vacancy [21].

Hysteresis loops (M-H) of different ferrite samples are shown in Figure 6. Saturation magnetizations (M_s), Coercivity (H_c), Anisotropy constant (K) and Remanent magnetization (M_r) of Ni_0.4Cu_0.2Zn_0.4Fe_2−xTi_xO₄ are measured and tabulated in Table 2. From Table 2 we see that the samples have low H_c and high M_s that enable them to be used as soft magnetic materials. From Table 2, it’s also noted that, though the (M_s) is decreased with the concentration of Ti but coercivity (H_c) is increased except for x = 0.02 i.e.; there is an inverse relationship between M_s and H_c as shown in Figure 7.

It could be concluded that value of coercivity rises with the rising concentration of Ti except for x = 0.02 due to the enough defects or porosity in the sample. Moreover, as can be seen clearly in Figure 8 that anisotropy K reduces with increase in Ti concentration due to decreasing grain size which is evident from SEM images.

Figure 9(a), Figure 9(b) and Figure 9(c) illustrate the frequency dependency of initial permeability (μ_i), relative quality factor (RQF), magnetic loss tangent (tanδ) for Ni_0.4Cu_0.2Zn_0.4Fe_2−xTi_xO₄ under work. From Figure 9(a), it is seen that the permeability increases with Ti substitution of x = 0.02 and x = 0.05 and it reduces for x = 0.07 and x = 0.10. The μ i of a ferrite depends on average grain size, intra-granular porosity, domain walls bulging and reversible domain walls displacement etc. [22].

Figure 9(b) illustrates the frequency dependency of magnetic loss (tanδ). The tanδ value declines with frequency, approaching the minimum value, and then becomes almost static. The reason behind the reduction of tanδ with rising frequency may be above a certain frequency of applied external electric field; the motion of the domain wall may not match the external electric field. Figure 9(c) shows the variation in relative quality factor (RQF) with frequency. It is revealed that RQF rises with the increase of frequency and after a certain frequency RQF slightly decreases with frequency increase. In fact RQF has the maximum value at that frequency level where the magnetic loss tangent has the minimum value.

Figure 10(a) shows the variation of dielectric constants with frequency range 1 kHz to100 MHz at room temperature. From that figure, it is evident that the dielectric constant decreases with increasing frequency at higher frequencies easily reaching constant value. The variance shows the dispersion due to interfacial polarization of the form Maxwell-Wagner [23] [24] that is confirmed by the phenomenological theory of Koop [25]. The Koop’s phenomenological theory explains the higher dielectric constant values observed at lower frequencies. It may happen owing to the majority of the class of similar Fe²⁺ ions, oxygen vacancy, interfacial disruption pile-ups and grain boundary imperfection, etc. Due to the fact that any species paying for polarizability is bound to show the sheathing behind the applied field at higher frequencies, the dielectric constant is reduced with frequency.

Figure 10(b) indicates the frequency dependence of the dielectric loss tangent (tanδ) at room temperature. Dielectric loss decreases as the frequency increases. The tanδ decreases significantly with increasing frequency for each ferrite composition and becomes flat. If the frequency of the externally applied ac electric field at neighboring octahedral sites is much lower than the rate of hopping electrons between Fe²⁺ and Fe³⁺ ions, the electrons will obey the field and the loss is therefore maximum. The rate of hopping of the electron exchange between Fe²⁺ and Fe³⁺ ions at higher frequencies of the applied electric field cannot match the applied field beyond certain critical frequency [26] and the loss is the least. It is seen that with increasing Ti⁴⁺ content at lower concentration the tanδ decreases

and again increases with higher concentration this may be due to the presence of impurity in the samples although no unreacted component is seen in XRD phase identification. The values of dielectric loss may be subject to content ratio, availability of Fe²⁺ and Fe³⁺ content and mechanical stability of the composition and preparation technique.