Enhanced magnetic permeability in Ni$_{1-x}$(Zn$_{0.6}$Mg$_{0.2}$Cu$_{0.2})_x$ Fe$_2$O$_4$ synthesized by auto combustion method

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Enhanced magnetic permeability in Ni

_12x

(Zn

_0.6

Mg

_0.2

Cu

_0.2

)

_x

Fe

₂

O

₄

synthesized by auto combustion method

S SHAFIEE¹, A ARAB^1,* and N RIAHI-NOURI²

1Faculty of Applied Science, Malek Ashtar University of Technology, Shahin Shahr 83, Iran

2Non-Metallic Materials Research Group, Niroo Research Institute, Tehran 13, Iran

*Author for correspondence (aa.arab1344@gmail.com) MS received 30 September 2020; accepted 2 January 2021

Abstract. Ni1-x(Zn0.6Mg0.2Cu0.2)xFe2O4(wherex= 0.0, 0.3, 0.5 and 0.7) was prepared by the auto combustion method and was characterized by X-ray diffraction (XRD). From XRD analysis, the average crystallite size and lattice constant were determined in the range of 6–11 nm and 8.331–8.372 A˚ , respectively. The structure of single-phase cubic in samples was also confirmed because the main peaks of the patterns were related to the position of the peaks in the mentioned standard (JCPDS card no. 00-008-0234). The average size of the particle was calculated in the range of 21–26 nm from the field emission scanning electron microscopy (FESEM). The atomic percentage of elements was obtained from the energy dispersive spectroscopy (EDS). From room-temperature hysteresis loops, the magnetization was evaluated in a magnetic field of 9 kOe. It was observed that the magnetization was not saturated and was increased with the increment inxvalues.

According to the core–shell model, the magnetic behaviour of samples were discussed. The results illustrated that the magnetization of the sample withx= 0.7, due to the larger particle size than that of other samples, was the highest. In addition, the increase in magnetization was also discussed based on Neel’s theory and the site’s occupation of the substituted cations on the sub-lattices. Also, the variation of coercivity was investigated. The coercivity, due to the multi- domain structure of samples, was decreased with doping Zn, Mg and Cu. Among all the samples, the sample withx= 0.7 indicated the lowest coercivity and the sample with x = 0 showed the highest coercivity. Based on the domain wall movement, the variations of initial permeability as a function of frequency in the frequency range of 10 kHz–10 MHz were discussed. The results indicated that the initial permeability of the sample withx= 0.7, similar to its magnetization, was the highest. The constancy ofl⁰in the frequency range of 10 kHz–1 MHz was due to the motion of the domain wall, which was indicated by the compositional stability and quality of the samples. Finally, the results obtained from the initial permeability in the present work were compared to the values obtained from the previous researchers. Generally, it was observed that the sample with x = 0.7, due to the maximum magnetization, highest initial permeability and lowest coercivity could be suitable for high-frequency applications.

Keywords. ZnMgCu spinel ferrite; magnetic properties; core–shell; multi-domain; initial permeability; high frequency.

1. Introduction

Nowadays, spinel ferrites have shown excellent magnetic features in the high-frequency [1] and have been used in the core of power transformers, noise filters and choke coils [1].

The structure and morphology of spinel ferrites were characterized by the lattice of oxygen ions. In the spinel structure, 4 and 6 oxygen ions were placed in the tetrahedral (A) and octahedral (B) sites, respectively. The radii of metal ions are smaller than that of oxygen ions [2,3]. The magnetic behaviours can be influenced by the super-exchange interactions like A–B, A–A and B–B, which are mediated by the intermediate O^2-ions and the distribution of cations [4]. Moreover, the preparation process of ferrites and compositions are influenced by the structural features [5].

The increment in the initial permeability is due to the

recrystallization and growth of grains, which causes the movement of grain boundaries [6]. For developing the magnetic properties, the spinel ferrites were substituted by divalent cations like chromium, copper, manganese and zinc [4].

Various methods have been used to prepare the spinel ferrites. Among all techniques, auto combustion, due to the high temperature of materials sintering, homogeneous distribution of particle size, low-time consumed, is often used for the preparation of ferrites [7,8].

Different researchers reported the magnetic properties of spinel ferrites based on the site’s occupation by cations [9,10]. The investigations showed that the Zn^2?ions preferred to occupy the A sites [9], whereas Ni^2?ions preferred to occupy the B sites. Hence, the addition of Ni^2?ions on Zn ferrite causes to occupy B sites by Ni^2? ions, which https://doi.org/10.1007/s12034-021-02429-y

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results in the migration of some Fe^3?ions from B site to A site [10].

Many researchers studied the properties of Ni ferrite by varying the compositions [6,11,12]. From the literature, it was observed that the incorporation of new elements into the Ni ferrite, like Mg^2?, Cu^2?and Zn^2?can increase the magnetization and initial permeability. Besides, it has been found from the work of different researchers that the Mg^2?

ion-substituted ferrite system shows enhancement in properties, such as improved magnetization and resistivity, reduction in power loss and also good thermal stability [13–15]. Besides, Thoratet al[11] studied the effect of Ni substitution on the magnetization and initial permeability of Mg_0.25-xNi_xCu_0.25Zn_0.5Fe₂O₄; the results reported the maximum value of initial permeability (l⁰ = 2620) at the sample with x = 0.05. Gangaswamyet al [6] investigated the magnetic behaviour of Ni_0.65-xMg_xZn_0.35Fe₂O₄. The results showed that the saturation magnetization increased by doping of Mg. Roy and Bera [12] studied the electro- magnetic properties of Ni_0.25-xMg_xCu_0.2Zn_0.55Fe₂O₄ prepared by the auto combustion method; the results indicated the high permeability of Ni0.07Mg0.18Cu0.2Zn0.55Fe2O4.

Hence, the work in this paper is focussed on tailoring this composition by substituting various concentrations of Mg^2?, Cu^2?and Zn^2?ions in the place of Ni^2?ion to study the effect of magnesium on magnetic properties and initial permeability. Therefore, the spinel ferrites of Ni_1-x(Zn_0.6 Mg_0.2Cu_0.2)_xFe₂O₄ (where x = 0.0, 0.3, 0.5 and 0.7) are prepared by the auto combustion method. It is observed that the magnetic properties of these ferrites are influenced by the particle size. The reduction of particle size led to a variety of novel phenomena like the decrease in magnetization and initial permeability and the enhancement in coercivity. Moreover, the application of samples in the high-frequency was discussed and the results were compared to the values reported in other works.

2. Experimental

2.1 Preparation

To prepare Ni_1-x(Zn_0.6Mg_0.2Cu_0.2)_xFe₂O₄(x= 0.0, 0.3, 0.5 and 0.7) from the auto combustion method, the stoichio- metric amounts of Zn(NO₃)₂6H2O, Cu(NO₃)₂3H2O, Mg(NO₃)₂6H2O, Ni(NO₃)₂6H2O and Fe(NO₃)₂9H2O (Merck, with purity C98%) were dissolved in deionized water. Then, the NH₅C₂O₂solution was added to the mixture. The mixture was continuously stirred at 80–200°C for 2 h. The gel obtained in this process was completely dried at 250°C for 2 h in air atmosphere. Finally, the powders were calcined at 600°C for 4 h.

The powders using polyvinyl alcohol (PVA) were gran- ulated and were pressed into toroidal shape (with a diameter of about 4 mm and a thickness of about 3 mm) at a pressure of 3 ton cm^-2. To sum up, the toroidal ferrites were

sintered at 900°C for 8 h and were wound by copper wire at room temperature.

2.2 Characterization

The powders were analysed by XRD (X’Pert Pro MPD model) and the wavelength of CuKaradiation is about 1.54 A˚ . The cubic structure of the samples was confirmed by the XRD pattern. The morphology, shape and size of the particle were carried out by FESEM (Hitachi S-4160 model).

The percentage of elements was measured from the energy- dispersive X-ray spectroscopy (EDS). Magnetic measure- ments were obtained using the alternating gradient force magnetometer (AGFM) in a magnetic field of 9 kOe. The initial permeability of toroidal ferrites was determined by the LCR meter (Hioki IM3536 model) in the frequency range from 10 kHz to 10 MHz.

3. Results and discussion 3.1 Structural properties

Figure 1 illustrates the XRD patterns of Ni_1-x(Zn_0.6Mg_0.2 Cu_0.2)_xFe₂O₄ (x = 0.0, 0.3, 0.5 and 0.7) samples. The diffractograms of the samples prepared with the auto combustion method revealed the reflections of (220), (311), (222), (400), (422), (511) and (440). Comparison peaks position with the mentioned standard (JCPDS PDF card no.

00-008-0234) confirmed the single-phase cubic structure of samples. Similar results have also been explained in different studies [16,17]. The average crystallite size was calculated using the Scherrer–Debye formula [18]:

D¼ 0:9k

bcosh; ð1Þ

where D is the average crystallite size, h the Bragg’s diffraction angle,kthe incident beam wavelength andbthe FWHM peaks [18].

Figure 1. Indexed XRD patterns of Ni1-x(Zn0.6Mg0.2Cu0.2)xFe2

O4withxequal to: (a) 0.0, (b) 0.3, (c) 0.5 and (d) 0.7.

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The lattice constant was obtained from the below equation [19]:

a¼d h²þk²þl²¹₂

; ð2Þ

whereais the lattice constant,dthe interplanar spacing, (h k l) are Miller indices of respective peaks [19].

The variations of average crystallite size and lattice constant concerning x values are shown in figure 2. As seen, the average crystallite size and lattice constant are increased with increasing xconcentration and were calculated in the range of 6–11 nm and 8.331–8.372 A˚ , respectively, as shown in table1. From the XRD patterns, it is observed that the sample with x = 0, due to the smaller size of crystallite than that of other samples, had a broader peak.

The enhancement of average crystallite size can be due to the existence of liquid phase in the calcination process;

increased contact area of solid reaction and the acceleration of the growth in crystallite sizes [20,21]. The lattice constant can be influenced by the difference between the substituted ions and host ions radii and the distribution of cations in the A and B sites [21]. Also, the variations of

lattice constant are caused by the different distances between magnetic ions, the changes in the super-exchange interactions and the magnetic behaviours of ferrites [20–22].

Generally, the increment of lattice constant is explained based on ionic radii. The ionic radii of Zn^2?(0.82 A˚ ), Mg^2?

(0.72 A˚ ) and Cu^2? (0.73 A˚ ) are larger than that of Ni^2?

(0.69 A˚ ) [22]. Therefore, with the replacement of Ni^2?by Cu^2?in B sites and the occupancy of A sites by Zn^2?and Mg^2?, the unit cell increases, which leads to the increase in lattice constant. The results obtained in this work agree with those of reported works by Thoratet al [11] and Roy and Bera [12]. Hence, among all samples, the sample withx= 0.7 has the largest average crystallite size and the lattice constant.

To investigate the morphology and compositional analysis of the synthesized composites, FESEM micrographs with EDS analysis have been recorded at various magnifications.

Figure3indicates the FESEM graphs for all the samples.

From FESEM images, the shape of the crystallite was analysed. The FESEM images show that the crystallites are narrowly distributed and are nearly spherical in shape, thus confirming the structure. Also, the particles are agglomer- ated because of the interaction between magnetic particles [23].

The variations in average particle size concerningxcon- tent is shown in figure4. As seen, with the addition of Cu, Mg and Zn ions, the average particle size increased in the range of 21–26 nm, as reported in table1.

The increase in average particle size is attributed to the increase in lattice constant. Furthermore, the particle size determined by FESEM is larger than that of the crystallite size estimated from Scherrer’s formula [24,25]. This is indicative of the fact that each particle is approximately formed by aggregation of crystallites [26].

Figure 5 presents the distribution of elements in the selected samples withx= 0.3 and 0.5 using EDS data. The

8.31 8.32 8.33 8.34 8.35 8.36 8.37 8.38

0 2 4 6 8 10 12

0 0.3 0.5 0.7

Average crystallite size (nm)

Composition (x)

Lattice constant (A)

Figure 2. Variations in the lattice parameter and average crystallite size concerning x values for Ni_1-x(Zn_0.6 Mg0.2Cu0.2)xFe2O4.

Table 1. The contents ofD,a,d,l⁰,MandH_cof Ni_1-x(Zn_0.6Mg_0.2Cu_0.2)_xFe₂O₄for differentxvalues.

Parameter x= 0 x= 0.3 x= 0.5 x= 0.7

Average particle size (d), nm 6 8 10 11

Standard deviation (SD) 0.81 2.38 1.52 0.81

Standard error (SE) 0.36 1.06 0.68 0.36

Lattice constant (a), A˚ 8.331 8.357 8.369 8.372

Average particle size (d), nm 21 23 25 26

Standard deviation (SD) 2.86 2.40 1.77 1.79

Standard error (SE) 1.28 1.07 0.79 0.80

Maximum magnetization (M), emu g^-1 32.50 57.94 68.41 71.37

Coercivity field (Hc), Oe 198 131 96 92

Initial permeability (l⁰),H/M 100 kHz–1 MHz

32 58 71 76

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atomic percentage of elements was measured, as listed in table2. EDS micrographs corresponding to FESEM images show the clear weight percentage of all the elements, which are also in line with the formation of the composite phase of samples.

3.2 Magnetic properties

Figure 6 exhibited the variations in M–H curves for Ni_1-x(Zn_0.6Mg_0.2Cu_0.2)_xFe₂O₄at the room temperature and at a magnetic field of 9 kOe. It is observed that the saturation magnetization is not attained for all the samples even at the maximum applied field of 9 kOe. The maximum magnetization (M) and coercivity (H_c) for samples derived from the magnetization plots are given in table1.

As seen in table1, the magnetization, due to the growth of grains [27], is increased by the increment in x values.

Hence, the magnetization of the sample with x = 0.7 is observed to be 71.37 (emu g^-1), higher than that reported for others. Generally, the magnetization is influenced by the stoichiometry of cations, spin-canting of the particle, effects of the surface, size of crystallite, exchange interactions between cations and disordered layer [4,28]. The variations of magnetization as a function ofxvalues are indicated in figure 7. The increment of M with the addition of Cu^2?, Mg^2?and Zn^2?ions can be due to the increase in the size of the particle, which is discussed based on the core–shell model. According to this model, each particle consisted of a Figure 3. Morphology of Ni_1-x(Zn_0.6Mg_0.2Cu_0.2)_xFe₂O₄withxequal to: (a) 0.0, (b) 0.3, (c) 0.5 and (d) 0.7.

15 17 19 21 23 25 27 29

0 0.3 0.5 0.7

Average particle size (nm)

Composition (x)

Figure 4. Variations in the average particle size concerning xvalues for Ni1-x(Zn0.6Mg0.2Cu0.2)xFe2O4.

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magnetic core and a surface layer with the disorder spins [24]. The disordering spins of the surface layer are inde- pendent of the broken super-exchange bonds and unlike isochronism of atoms near the surface layer [24]. The enhancement of the surface to volume ratio in the small particle causes an increase in the dead layer and a decrease in magnetization [24]. Thus, theMof the sample withx= 0 due to the smallest particle size, is the lowest.

The increment in M with doping of Cu^2?, Mg^2? and Zn^2?ions can be explained based on Neel’s theory and the occupation of the site by substituted cations on sublattices [16,24]. According to this theory, there exists three kinds of

exchange interactions between the unpaired electrons in the tetrahedral (A) and octahedral (B) sites. The interaction between the various magnetic ions located at A site (A–A interaction), B site (B–B interaction), and A site with those at B site (A–B interaction). The A–B interactions are the strongest and in which the alignment of all the magnetic spins at A site in one direction and those at B site in the opposite direction [24]. The super-exchange interaction between magnetic cations is dependent on the different condensations and magnetic moments of Zn^2?, Ni^2?, Cu^2?

and Mg^2?ions and the spin-canting of cations at A and B sites [16,24]. The magnetic moments of cations in A and B

Figure 5. EDX spectrum of Ni1-x(Zn0.6Mg0.2Cu0.2)xFe2O4withxequal to: (a) 0.3 and (b) 0.5.

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sublattices are oppositely aligned in the spinel structure.

The magnetic moment of Zn^2?, Ni^2?, Cu^2?, Mg^2?and Fe^3?

ions is 0, 2.3, 1, 0 and 5 lB, respectively [18,22]. The net magnetic moments result from the difference between the magnetic moments of B and A sub-lattices, which was theoretically expressed by the below equation [29–32]:

M ¼ jMBMAj; ð3Þ

where M_A andM_B are the magnetic moments of A and B sites, respectively [31].

The net magnetic moment of Ni_1-x(Zn_0.6Mg_0.2Cu_0.2)_x Fe₂O₄ is dependent on the distribution of Zn^2?, Cu^2?and Mg^2? ions in A and B sites. Generally, in the spinel structure, Ni^2? (2.3 lB) and Cu^2? (1 lB) ions tend to occupy B sites, while Zn^2? (0lB) and Mg^2?(0 lB) ions prefer to occupy A sites. Fe^3? (5 lB) ions also have a preference to occupancy of both the sites [33–35]. The increase in magnetization with increasing x values is attributed to the Ni^2?replacement with Cu^2?ions in B sites, which caused the migration of some Fe^3?from B sites to A sites and the decrease in the magnetization of B sites [35].

Also, the occupancy of A sites with Mg^2?and Zn^2?ions is caused by the movement of some Fe^3? from A sites to B sites and the decrease in the magnetization of A sites [36].

Thus, the net magnetization, M_B - M_A, increases with increasingxcontents.

Furthermore, the increase in the lattice constant of samples leads to the expansion of unit cell volume, which increased the inter-atomic distance between ions and affected the magnetic properties.

The variations of the coercivity (Hc) concerning the xcontents are exhibited in figure7. TheH_cis dependent on the size of the particle, boundary of grain, anisotropy and precipitates [37]. Also, it directly varies with the porosity and inversely with grain sizes [19]. The replacement of Fe^3?with Cu^2?ions at B sites leads to the variations of effective anisotropy constant and coercivity [27,38].

The reduction ofH_cby the addition of Cu^2?, Mg^2?and Zn^2? ions can be due to the multi-domain structure of samples [39]. So, among all samples, the sample withx= 0.7, due to the largest particle size, has the lowest coercivity [39].

The low value of coercivity indicates the soft nature, easy to magnetize and demagnetize of ferrites, and it suggests that it can be used in high-density data storage devices [40].

Figure 8 shows the initial permeability (l⁰) for Ni_1-x (Zn_0.6Mg_0.2Cu_0.2)_xFe₂O₄as a function of frequency within the frequency range of 10 kHz–10 MHz. The initial permeability can be influenced by the compositions, impurity contents and preparation techniques, size of the grain, saturation magnetization and magnetostriction [20,41]. The initial permeability is impressed by two magnetic functions like the domain wall movement and the spin rotation [41].

The spin rotation often used in the low-frequency region [42]. The domain wall movement, due to the existence of Table 2. Weight percentage of the elements for the samples withx= 0.3 and 0.5.

xvalue Parameter O Mg Fe Ni Cu Zn Al Au

0.3 Elements (weight %) 17.49 3.42 55.49 11.9 3.51 8.19 — —

0.5 16.32 1.48 48.22 9.98 2.98 6.65 3.66 10.76

0.3 Atomic (%) 41.87 5.39 38.06 7.76 2.12 4.8 — —

0.5 41.6 2.48 35.21 6.93 1.86 4.15 5.54 2.23

Figure 6. Room temperature hysteresis loops for Ni_1-x(Zn_0.6 Mg0.2Cu0.2)xFe2O4 withx equal to: (a) 0.0, (b) 0.3, (c) 0.5 and (d) 0.7.

0 50 100 150 200 250

15 25 35 45 55 65 75

0 0.3 0.5 0.7

Magnetization (emu/g)

Composition (x)

M Hc

Coercivity (Oe)

Figure 7. Variations in magnetization and coercivity of Ni1-x(Zn0.6Mg0.2Cu0.2)xFe2O4as a function ofxvalues.

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weak magnetic field, is reversible [42]. Moreover, the increase in grain size causes a decrease in the number of grain boundaries and the increment in initial permeability [43]. Thus, with increasing particle size, the contribution of the domain wall increases, which leads to the initial permeability [30].

The variations in the initial permeability as a function of xvalues are shown in figure9. The initial permeability of the sample with x = 0.7 (76 H/M) similar to its magnetization, is a maximum value. In addition, thel⁰has remained constant in the frequency range of 10 kHz–1 MHz, and then, it is fallen to 10 MHz. The constancy ofl⁰ is due to the domain wall motion, which indicated the compositional stability and quality of the samples. The frequency of 1 MHz, due to its equality with the external magnetic field frequency, is named as the utility zone [35], which is revealed as a desirable characteristic of ferrite applications in high-frequency [35] like broadband pulse transformers and wideband read–write heads for video recording [35].

The reduction ofl⁰ after the resonance frequency (1 MHz) can be attributed to the magnetic energy absorption by spin moments [30]. Furthermore, the energy required for dis- placement of the domain wall is lower than that of domain rotation.

To sum up, the substitution of Cu^2?, Mg^2?and Zn^2?ions on the Ni ferrite reveals that the sample with high magnetization and initial permeability and low coercivity can be an excellent choice for use in magnetic cores.

Although previous studies have been reported the initial permeability of ZnMgCuNi-ferrites [6,11,12], few studies are compared to the initial permeability results to other works.

Comparison of initial permeability obtained in this work with previous results in researches [11,12], as listed in table 3, shows that the l⁰ values can be influenced by the preparation method, temperature in calcination and sintering processes, particle size, density of cations and magnetization. Thus, the higher initial permeability in the previous works than that of samples in the present work is due to the higher temperature used in the process of calcination and sintering, the larger size of particles and different cations density. Besides, the results reported by Roy and Bera [12]

show the dependence of magnetization on the initial permeability is direct [12]. Thus, the increase in particle size causes the contribution of the domain wall and the increase in the initial permeability [30]. Also, the increase in density and grain size with an increment in sintering temperature can affect the magnetic properties [44].

0 10 20 30 40 50 60 70 80

10K 50K 100K 500K 1M 5M 10M

x=0.7 x=0.5 x=0.3 x=0

Frequency (Hz)

Initial permeability (H/m)

Figure 8. Variations in initial permeability concerning frequency for ferrite samples withxequal to 0.0, 0.3, 0.5 and 0.7.

0 10 20 30 40 50 60 70 80

0 0.3 0.5 0.7

10 kHz 50 kHz 100 kHz 500 kHz 1 MHz 5 MHz 10 MHz

Composition (x)

Initial permeability (H/m)

Figure 9. Variations in initial permeability concerningxvalues for Ni_1-x(Zn_0.6Mg_0.2Cu_0.2)_xFe₂O₄.

Table 3. Contents ofD,a,d,l⁰reported by Thoratet al[11] and Roy and Bera [12] for different ferrite samples.

Samples

Average crystallite size (D), nm

Lattice constant (a), A˚

Initial permeability (l⁰),H/M

100 kHz–1 MHz Method of synthesis

Ni0.18Zn0.55Mg0.7Cu0.2Fe2O4 124 8.3990 680 Auto combustion

Ni0.12Zn0.55Mg0.13Cu0.2Fe2O4 121.9 8.3991 940

Ni0.7Zn0.55Mg0.18Cu0.2Fe2O4 127.6 8.4097 2420

Ni0.5Zn0.5Mg0.2Cu0.25Fe2O4 48 8.392 2620 Citrate-assisted sol–gel

Ni_0.1Zn_0.5Mg_0.15Cu_0.25Fe₂O₄ 51 8.400 1930

Ni0.15Zn0.5Mg0.1Cu0.25Fe2O4 47 8.405 1569

Ni_0.2Zn_0.5Mg_0.5Cu_0.25Fe₂O₄ 49 8.407 522

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4. Conclusion

The effect of Cu^2?, Mg^2? and Zn^2? substitution on the structural and magnetic properties, and initial permeability of Ni_1-x(Zn_0.6Mg_0.2Cu_0.2)_xFe₂O₄samples prepared by the auto combustion has been investigated. The XRD results confirmed the formation of a single-phase fcc spinel structure in all the samples. By the enhancement ofxvalues, the lattice constant and average crystallite sizes were increased. From FESEM images, the average particle size was calculated in the range of 21–26 nm. The hysteresis studies showed that the magnetization increased by doping Mg, Cu and Zn ions on Ni ferrite. Analysis of the magnetic features of samples based on the core–shell model showed that the reduction of particle size resulted in the enhancement of the surface to volume ratio, which led to a decrease in magnetization.

Among all the samples, the sample with x= 0.7, due to its largest particle size, indicated the highest magnetization.

Besides, the increase in magnetization was discussed according to Neel’s theory. The substitution of Ni^2?cation by Mg^2?, Zn^2?and Cu^2?ions increased the total magnetic moment. The reduction of coercivity with increasingxvalues showed the multi-domain structure of samples. So, the coercivity of the sample withx= 0.7 was the lowest.

The increased initial permeability with increasingxcontents was due to the growth of grains. The constancy of initial permeability in the studied frequency ranges from 10 kHz to 1 MHz, indicated the compositional stability and the quality of samples. The reduction of initial permeability from 1 to 10 MHz illustrated the external frequency in the magnetic field. Thus, the Cu^2?, Mg^2?and Zn^2?doping on Ni ferrites indicated that the sample with x= 0.7 with the highest magnetization, highest initial permeability and lowest coercivity is the excellent choice for a magnetical core. Comparison of the initial permeability in the present work with the results obtained in the previous works indicated that the lower values ofl⁰in the present study than those reported in previous studies are due to the lower temperature of calcination, sintering and smaller particle size. Also, this behaviour can be due to the different preparation methods and cations density. The spinel ferrites in the present work indicated a considerable performance in magnetic characteristics and high-frequency permeability.

To sum up, the auto combustion method can be suitable for obtaining fine nanoparticles, and Cu^2?, Mg^2?and Zn^2?substitution on nickel ferrite makes them soft magnetic materials.

Furthermore, the sample with x= 0.7, due to the maximum magnetization, maximum initial permeability and minimum coercivity can be used for high-frequency applications.

Acknowledgements

This study was completed at Malek Ashtar University of technology and was supported by Niroo Research Institute.

We are grateful to the institute for their support.

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