3.1. Analysis of structure and composition
Fig. 1 shows the X-ray diffraction (XRD) patterns of Ni1-x (Zn0.6 Mg0.2Cu0.2) x Fe2O4 (x= 0.0, 0.3, 0.5 and 0.7) samples. It can be seen that the prominent hkl planes (220), (311), (222), (400), (422), (511), and (440) were identified. The patterns were confirmed the formation cubic spinel structure of samples due to the well matching with the standard XRD patterns (JCPDS PDF cars No. 00-008-0234). Similar results have been revealed in previous work [13, 14]. The average crystallite size (D) was calculated by using the Scherer-Debye equation :
From Table 1 which shows XRD parameters of ferrites can be seen that the average crystallite size raised from 6 nm to 11 nm with increasing the doping ratio of magnesium, copper, and zinc. The increase in the average crystallite size may be due to the liquid phase in the calcination process. With the addition of Zn2+, Mg2+, and Cu2+ ions, the contact area of solid reaction increases, and the growth of the crystallite size accelerates . Also, the lattice constant found to be increased little from 8.331 Å to 8.372 Å with increasing the x content (0.0 to 0.7). The lattice constant is dependent on the difference between the substitution ions and host ions radii and the cationic distribution among the interstitial tetrahedral (A) and octahedral (B) sites . The increase in lattice constant is due to larger ionic radii of Cu2+ (0.73 Å), Mg2+ (0.72 Å) and Zn2+ (0.82 Å) than nickel (0.69 Å) . This agrees with other works observed byWhere “a” is the lattice constant, d is inter planer spacing, (hkl) are Miller indices of respective peaks.
Roy et al.  and Thorat et al. . The variations in lattice constant and crystallite size as x contents are shown in Fig. 2.
Fig. 3 indicates the FE-SEM micrographs for all the samples. The substitution of Cu, Mg, and Zn ions into nickel ferrites has influenced on their size and structure. The average particle size increases with the addition of Cu, Mg, and Zn ions from 21 nm to 26 nm, as listed in Table 1. The spherical nature of the particles and the agglomeration of grains is shown in the FE-SEM images. The variation in particle size as x concentration are shown in Fig. 4. The increase in lattice volume has an important role in grain boundary diffusion . With increasing lattice volume, the diffusion path, and the rate of the cation interdiffusion increase. Furthermore, the growth of grain is related to the grain’s boundary mobility . Recrystallization and grains growth lead to the movement of grains boundaries and the variations in permeability, density, and resistivity . So, the grain boundary diffusion was increased by increasing Cu2+, Mg2+, and Zn2+ substitutions.
Using EDS, the distribution of elements in selected samples with x= 0.3, 0.5 were analyzed. The atomic percentage of elements are given in Fig. 5. The spectra confirmed that the final compositions of ferrites were the same as those of compositions without any extra impurity elements.
3.2. Analysis of magnetic performance
Fig. 6 shows the initial permeability (µˊ) as a function of frequency for Ni1-x (Zn06Mg0.2Cu0.2) x Fe2O4 (x=0.0, 0.3, 0.5 and 0.7) ferrites under the frequency range of 10 kHz-10 MHz. As depicted clearly, µˊincreases with the increasing zinc, magnesium, and copper and reaches the maximum value µˊ= 76 at x= 0.7. The increase in initial permeability has been attributed to grain size. The variations of the initial Permeability as a function of x concentration and particle size in the frequency range of 10 kHz- 10 MHz are shown in Fig. 7 and Fig. 8. The observed variation in initial permeability can be due to the basis of changes that took place in grain size as evident from FE-SEM micrographs. The variations in initial permeability are parallel to the changes observed in the particle size. The initial permeability correlates by two different magnetizing mechanisms of ferrites; the movement of domain wall and the spin rotation . The spin rotation is smaller than domain walls at the low-frequency region, the domain wall movement is reversible because of the presence of a weak magnetic field [21, 23].
As can see in Fig. 8, the increase in particle size causes an increase in domain wall contribution, and the increasing initial permeability . It is a well-known fact that the increase in grain size reduces the number of grain boundaries in a sample and causes an increase in the initial permeability . The higher µˊobtained for the composition x = 0.7 at the frequency of 10 kHz is contributed by the domain wall oscillations in the bigger grains of this composition. Furthermore, the µˊremained steady over a wide range of frequency 10 kHz -1 MHz, and then decreases rapidly greater than 1 MHz applied frequency, as indicated in Fig. 6. The stability of µˊ from 10 kHz to 1 MHz is due to the domain wall motion. The frequency of 1 MHz is named the zone of utility , which is equal to the external magnetic field frequency and a desirable characteristic for various applications of ferrites in high-frequency, such as broadband pulse transformers and wideband read-write heads for video recording . After the resonance frequency (1 MHz) the µˊ decreases due to the absorption of magnetic energy by spin moments . The required energy for the displacement of the domain wall is lower than that required for domain rotation. The constancy in permeability throughout the frequency range studied from 10 kHz to1 MHz indicates the compositional stability and quality of the samples. The present investigation of Cu2+, Mg2+, and Zn2+ substituted Ni ferrites reveal that the ferrites with the high initial permeability are an excellent choice as magnetic cores.
The characteristics of µˊhave been reported by Roy et al.  for Ni-Zn-Mg-Cu ferrites prepared by the auto combustion method and Thorat et al.  for Ni-Zn-Mg-Cu ferrites prepared by citrate assisted sol-gel method, which agrees with our reported values. The results are listed in Table 2.
The initial permeability is influenced by compositions, impurity contents, preparation methods, grain size, saturation magnetization, magnetostriction [17, 22]. The µˊvalues reported by Roy et al.  and Thorat et al.  are higher than those obtained for the Ni1-x (Zn0.6 Mg0.2Cu0.2) x Fe2O4 in the present investigation. It may be due to the different preparation methods and the higher calcination and sintering temperature of samples than those ferrite samples in the present investigation. The density and grain size of the ferrites increase with increasing sintering temperature, and they would affect magnetic properties directly . Higher crystallite and particle sizes, and higher temperature used for calcination and sintering samples cause to increase in µˊ. The magnetization mechanism of soft magnetic materials is domain wall motion, which generates high initial permeability (µ). Although pores and grain boundary would obstruct the movement of the domain wall, the fewer amounts of pores and grain boundary could be obtained at higher sintering temperature and lead to easy movement of domain wall and high initial permeability . Furthermore, the values of µˊreported by Roy et al.  indicated that the increase in permeability is attributed to the increase in particle size.
Fig. 9 illustrates the magnetization (M) as a function of field (H) curves for the Ni1-x (Zn0.6 Mg0.2Cu0.2) x Fe2O4 (x=0.0, 0.3, 0.5 and 0.7) ferrite at room temperature and maximum magnetic field of 9 kOe. The magnetization is not saturated until 9 kOe in all hysteresis curves. From the plotted M-H curves, the maximum magnetization (M), and coercivity (Hc) were measured as reported in Table 1. With increasing Cu2+, Mg2+, and Zn2+ ions in the nickel ferrite the magnetization increases. The increase in magnetization can be due to the variations of exchange interactions between A and B sites, an increase in the crystalline nature, and a narrow particle size distribution . The magnetization is calculated about 57.94 emu/g, 68.41 emu/g, and 71.37 emu/g for samples with x= 0.3, 0.5, and 0.7 respectively, which are higher than that of the unsubstituted nickel ferrite sample (32.5 emu/g). Moreover, the M value of ferrite with x=0.7 is higher than that of the substituted nickel ferrite samples. The magnetic properties are influenced by the composition and different cation distribution as described by the Neel model . According to Neel's theory, the net magnetic moment can be defined as following :
Where MA and MB are the magnetic moments of A and B sublattices .
The replacement of cations substituted in the spinel ferrites leads to weak or strong interaction among magnetic ions . It is reported in previous research that the Zn2+ and Mg2+ ions have a strong preference for occupancy the A sites; the Ni2+ and Cu2+ ions have a strong preference towards B sites, while the Fe3+ ions distribute over both A and B sites [28, 29]. The magnetic moment of Zn2+, Cu2+, Ni2+, Fe+3 and Mg2+ ions is 0, 1.3, 2.3, 5 and 0 µB, respectively [15, 19]. With the increasing of Cu2+, Mg2+, and Zn2+ ions on nickel ferrite, the copper ions occupy B sites, due to their high preference towards B sites, and a smaller concentration of copper may appear at A sites. The diamagnetic Zn2+ and Mg2+ ions occupy the A sites, where cause the movement of Fe3+ ions from A sites to B sites. Therefore the increase in magnetization is due to the higher magnetic moment of B than A sublattices . The variations of magnetization and coercivity of ferrite samples as a function of x are shown in Fig. 10.
The magnetic characteristics are dependent on the particle size, anisotropy, density, cationic stoichiometry, random canting of particle spins, and surface effects [29, 31]. The increase of M can be explained with the particle size trend. As reported in the structural analysis of this series published elsewhere, the particle size has increased with an increase in the amount of doped copper, zinc, and magnesium. The variation of magnetization and coercivity dependence of average particle size is shown in Fig. 11. As can be seen the magnetization increases by the enhancement of particle size. As the particle size increases, the number of magnetic domains increases, and the movement of the domain wall facilitates, which leads to the enhancement of magnetization . This behavior can also be justified according to the core-shell model . In this model, each particle consists of a magnetically ordered core and a spin glass surface with a constant thickness without net magnetization . The disordering of the surface layer spins may be due to the broken super-exchange bonds and unlike local symmetry for those atoms near the surface layer . In smaller particles, the surface to volume ratio of particles is higher, which leads to a decrease in magnetization . With an increase in the average particle size, due to the diminution in the surface to volume ratio, the magnetization increases.
Coercivity (Hc) gives the value of the applied field, at which the induced magnetization is zero. The values of Hc for ferrite samples are listed in Table 1. As can be seen from Fig. 10 the coercivity decreases with Cu2+, Mg2+, and Zn2+ substitution. The Hc is dependent on the particle size, grain boundaries, anisotropy, and precipitates . The increase in particle size of ferrites decreases the surface area to volume ratio and surface anisotropy of the crystal. The coercivity property of the samples originates from a multi-domain structure. In the multi-domain region, the increase in particle size decreases the coercivity . Referring to the variations of M and µˊas a function of average particle size, it is understood that the dependence of magnetization to the initial permeability is direct. Therefore, the increase in particle size leads to an increase in domain wall contribution and an increase in initial permeability .