Effect of Cd2+ Substitution on Structural–Magnetic and Dielectric Properties of Ni–Cu–Zn Spinel Ferrite Nanomaterials by Sol–Gel

CdxNi0.5−xCu0.2Zn0.3Fe2O4 (0 ≤ x ≤ 0.50) ferrite with a spinel structure was prepared using the sol–gel self-propagation method. The effects of Cd2+ doping on the structure, morphology, dielectric, and magnetic properties of Ni–Cu–Zn ferrite were examined using XRD, SEM, EDX, FTIR, MPMS, and dielectric tests. The cubic spinel structure was verified by XRD and FTIR analyses. The crystallite size and particle size information of the samples were obtained with XRD and SEM analysis. The sample particle size belonged to a class of nanoscale materials with a particle size range of 1–100 nm. The minor difference between the grain size and particle size indicated that the sample nanoparticles were composed of numerous microcrystals. The EDX spectra indicated that the samples contained all stoichiometric elements. MPMS was used to measure the hysteresis lines of the samples. According to the hysteresis line, the saturation magnetization intensity (Ms), coercivity (Hc), and magnetic moment (μB) of the sample increased and then decreased with the increase in cadmium concentration. The magnetization strength (Ms) is between 4–67 emu/g, and the coercivity (Hc) is between 9–46 Oe. The curves of the real part of the dielectric constant (ε′), the imaginary part of the dielectric constant (ε″), and the loss factor (tanδ) with frequency were measured in the frequency range 100 Hz–100 kHz by means of an impedance analyzer. The complex modulus spectrum was analyzed to understand the dynamics of the conduction process.

Ni-Cu-Zn spinel ferrite nanomaterials have a wide range of applications in both high-frequency and magnetic memory devices. The most common material for producing multilayer chip inductors (MLCIs) is Ni-Cu-Zn spinel ferrite [18,19]. MLCIs are made of alternating layers of silver electrodes and spinel ferrites. A key strategy for reducing the number of MLCI layers and increasing efficiency is to further reduce the size of the electronics. By modifying the metal ion ratio and/or replacing alternative metal ions, the Ni-Cu-Zn spinel ferrite produced has high resistivity, low coercivity, and a low dielectric constant [19]. Juan Carlos Aphesteguy et al. [20] prepared Ni 0.35 Cu 0.15 Zn 0.5 Fe 2 O 4 magnetic nanoparticles using the sol-gel combustion method. Mössbauer spectroscopy results indicated the presence of a paramagnetic doublet state in the samples, and the coercivity field of Ni 0.35 Cu 0.15 Zn 0.5 Fe 2 O 4 magnetic nanoparticles decreased with Fe 2+ substitution at low temperature (260 Oe). Qing Lin et al. [21] prepared Ni 0.6 Cu 0.2 Zn 0.2 Ce x Fe 2−x O 4 (0 ≤ x ≤ 0.85) using the sol-gel method. XRD spectral analysis indicated that the average grain size of the samples decreased with increasing Ce 3+ substitution, and the lattice parameters varied with the x content. The Mössbuaer room-temperature spectra confirm that the [Fe 3+ -O 2− -Fe 3+ ] superexchange interactions are reduced as a result of Ce 2+ substitution. Substitution of divalent or trivalent metal ions such as Ni-Cu-Zn-Fe-V [22], Ni-Cu-Zn-Fe-Gd [23], and Ni-Cu-Zn-Fe-La-Y [19] for Fe 3+ in Ni-Cu-Zn spinel ferrites has been frequently reported to modify the structural, electrical, and magnetic characteristics of these materials. However, studies on the replacement of Ni/Cu/Zn ions in Ni-Cu-Zn spinel ferrite by divalent or trivalent transition metal ions to obtain quaternary materials are relatively limited. The doping of cadmium ions significantly impacts the structural, magnetic, and electrical properties of the material [1,10] due to the large radius of cadmium ions and their preference for a tetrahedral environment. Therefore, utilizing isolator-doped Ni-Cu-Zn spinel ferrite will be of significant interest.
In this study, we innovatively doped Ni-Cu-Zn spinel ferrite with divalent excess metal particles (Cd 2+ ) and investigated the influence of Cd 2+ doping on the lattice structure of Ni-Cu-Zn spinel prepared with the sol-gel method. Furthermore, we discussed the effects of Cd 2+ doping on the morphology, magnetic properties, and dielectric properties of Ni-Cu-Zn spinel ferrite. Figure 1 illustrates the XRD patterns of the spinel ferrite system Cd x Ni 0.5−x Cu 0.2 Zn 0.3 Fe 2 O 4 (0 ≤ x ≤ 0.50, in steps of 0.10). The Miller index (hkl) of the spinel phase is marked in the figure, corresponding to the (220), (311), (222), (400), (422), (511), and (400) crystal planes. The XRD patterns were analyzed using Jade 5 software, revealing the formation of a cubic spinite structure with space group Fd-3m, which was in accordance with the JCPDS standard card number 74-0748. Moreover, we observed a second phase apart from the sample (x = 0). The second phase was a peak of the hematite hexagonal phase α-Fe 2 O 3 (JCPDS # 86-0550), with space group R-3c and lattice parameters a = b = 5.03 Å and c = 13.73 Å. M. H. Nasr et al. used Ni-doped Fe-Cd spinel nanoferrite [8], which also produced the secondary phase -Fe 2 O 3 as the Ni doping increased. Vivek Verma et al. [9] synthesized L i0.35 Cd 0.3 Fe 2.35 O 4 lithium-cadmium ferrite with secondary-phase α-Fe 2 O 3 generation using a hybridized citrate gel precursor technique in a different pH medium. Munish Gupta et al. [14] prepared cadmium ferrite (CdLi x Fe (2−x) O (4−x) ); 0 ≤ x ≤ 0.50) using ammonium hydroxide as a co-precipitant, and also observed the presence of Fe 2 O 3 . This result shows that the solid solution limit of Cd-doped Ni-Cu-Zn spinel ferrite is less than x = 0.1. XRD patterns can be derived for the lattice constant (a), grain size (D), and X-ray density (ρ x ).
CdxNi0.5−xCu0.2Zn0.3Fe2O4 (0 ≤ x ≤ 0.50, in steps of 0.10). The Miller index (hkl) of the spinel phase is marked in the figure, corresponding to the (220), (311), (222), (400), (422), (511), and (400) crystal planes. The XRD patterns were analyzed using Jade 5 software, revealing the formation of a cubic spinite structure with space group Fd-3m, which was in accordance with the JCPDS standard card number 74-0748. Moreover, we observed a second phase apart from the sample (x = 0). The second phase was a peak of the hematite hexagonal phase α-Fe2O3 (JCPDS # 86-0550), with space group R-3c and lattice parameters a = b = 5.03 Å and c = 13.73 Å. M. H. Nasr et al. used Ni-doped Fe-Cd spinel nanoferrite [8], which also produced the secondary phase -Fe2O3 as the Ni doping increased. Vivek Verma et al. [9] synthesized Li0.35Cd0.3Fe2.35O4 lithium-cadmium ferrite with secondary-phase α-Fe2O3 generation using a hybridized citrate gel precursor technique in a different pH medium. Munish Gupta et al. [14] prepared cadmium ferrite (CdLixFe(2−x)O(4−x)); 0 ≤ x ≤ 0.50) using ammonium hydroxide as a co-precipitant, and also observed the presence of Fe2O3. This result shows that the solid solution limit of Cd-doped Ni-Cu-Zn spinel ferrite is less than x = 0.1. XRD patterns can be derived for the lattice constant (a), grain size (D), and Xray density (ρx).   The inset image in Figure 1 illustrates how the main peak of spinel ferrite (311) changes to a lower angle with the increase in cadmium doping.
This phenomenon can be explained by Bragg's law and formula in the following ways: where d is the crystal plane spacing, λ is the X-ray wavelength, and θ is the Bragg angle. The doping of cadmium ions into Ni-Cu-Zn spinel ferrite increases the ferrite crystal plane spacing (d) because cadmium has a higher ionic radius (0.97 Å) than nickel (0.69 Å). This eventually results in a shift from the Bragg angle to a smaller angle. Additionally, Figure 2 shows a positive correlation between increasing Cd content and the change in both the X-ray density (ρ x ) and the cell parameters of the samples. The X-ray density can be estimated using the lattice constant and Equation (2).
where M is the molecular weight of the sample, N is the Avogadro constant (6.023 × 10 23 mol −1 ), a is the lattice constant, and Z is the number of molecules in each cell. The equation shows that the ρ x value is inversely related to a 3 . However, because the ρ x value is positively related to M, and the sample's molecular weight increases with the increase in Cd (Cd molecular weight is larger than Ni ions), the overall ρ x value and the cell parameters of the sample are positively correlated. As shown in Figure 3, Sample grain size and lattice strain can be estimated by linearly fitting the Williamson-Hall formula with 4sin(θ) as the horizontal coordinate and βcos(θ) as the vertical coordinate. The slope of the plot represents the strain (ε), while the intercept on the y-axis determines the grain size.
changes to a lower angle with the increase in cadmium doping.
This phenomenon can be explained by Bragg's law and formula in the following ways: where d is the crystal plane spacing, λ is the X-ray wavelength, and θ is the Bragg angle. The doping of cadmium ions into Ni-Cu-Zn spinel ferrite increases the ferrite crystal plane spacing (d) because cadmium has a higher ionic radius (0.97 Å) than nickel (0.69 Å). This eventually results in a shift from the Bragg angle to a smaller angle. Additionally, Figure 2 shows a positive correlation between increasing Cd content and the change in both the X-ray density (ρx) and the cell parameters of the samples. The X-ray density can be estimated using the lattice constant and Equation (2).
where M is the molecular weight of the sample, N is the Avogadro constant (6.023 × 10 23 mol −1 ), a is the lattice constant, and Z is the number of molecules in each cell. The equation shows that the ρx value is inversely related to a 3 . However, because the ρx value is positively related to M, and the sample's molecular weight increases with the increase in Cd (Cd molecular weight is larger than Ni ions), the overall ρx value and the cell parameters of the sample are positively correlated. As shown in Figure 3, Sample grain size and lattice strain can be estimated by linearly fitting the Williamson-Hall formula with 4sin(θ) as the horizontal coordinate and βcos(θ) as the vertical coordinate. The slope of the plot represents the strain (ε), while the intercept on the y-axis determines the grain size.  In Figure 3, a linear fit to the Williamson-Hall equation may be used to estimate the grain size and lattice strain. Table 1 presents the structural features of Cd 2+ -doped Ni-Cu-Zn spinel ferrite nanomaterials. In this study, the Williamson-Hall formula is used to determine the induced strain brought on by lattice defects and distortions in the nanocrystals.
The grain size and lattice strain of the Cd x Ni 0.5−x Cu 0.2 Zn 0.3 Fe 2 O 4 sample were estimated using the deformation model (UDM) given in Equation (3) [8]. The Williamson-Hall equation for determining the average grain size and lattice strain is as follows: where D is the grain size, λ is the X-ray wavelength, K is a constant (0.9), and β is the half-peak width.

XRD Rietveld Refinement
The XRD of the spinel ferrite system Cd x Ni 0.5−x Cu 0.2 Zn 0.3 Fe 2 O 4 (0 ≤ x ≤ 0.50, in steps of 0.10) was subjected to Rietveld refinement in order to further investigate the crystal structure of the samples, as shown in Figure 4.The refinement results indicate that samples have an Fd-3m space group with increasing lattice constants and an increasing trend in the volume fraction of α-Fe 2 O 3 . The lattice constants of the samples obtained after the Rietveld refinement are very similar to those obtained by the XRD fit. The ionic radii of the doped ions can account for the increase in lattice constant with increased Cd concentration. Moreover, the significant difference in ionic radii between Cd 2+ (0.97 Å) and Ni 2+ (0.69 Å) may be the primary cause of this phenomenon. The R-factor, refinement parameters, and goodness-of-fit are reported in Table 2. Harpreet Kaur et al. [7] used Zn 2+ and Cd 2+ -doped cobalt ferrite, and the experimental results found that the lattice constant of the sample becomes larger after doping. The explanation is that the ionic radii of Zn 2+ (0.74 Å) and Cd 2+ (0.97 Å) are larger than that of Co 2+ (0.70 Å), which is in agreement with experimental results of the present work. M. H. Nasr et al. [8] employed Ni 2+ -doped cadmium ferrite and found that the lattice constant keeps getting smaller with increasing Ni content. This experimental result is corroborated by our experimental results.     Table 3 shows  The distance between the A-site cations (hopping length L A ) and the distance between the B-site cations (hopping length L B ) was estimated using the following equations: It is clear from Equations (4) and (5) that the A-position hopping length (L A ) and the B-position hopping length (L B ) are directly related to the lattice constants. The hopping lengths L A and L B of A-and B-positions keep increasing with the increase in Cd incorporation. This can be explained by the cell expansion due to the increase in site radius, caused by the entry of larger Cd 2+ ions into the tetrahedral lattice and smaller ions into the octahedral lattice. The experimental findings are consistent with the cell and bond length variation patterns obtained by M.H. Nasr et al. [8] using Ni 2+ ion-doped Cd ferrite. The following equations can be used to determine the radius of the A-site cation (r A ) and the radius of the B-site cation (r B ):

SEM Analysis
SEM tests were conducted on representative samples where x = 0.1 in order to examine the surface morphology of spinel-structured ferrite. Figure 5 indicates that the surface morphology of the sample where x = 0.1 is inhomogeneous, spherical, and highly agglomerated. The agglomeration phenomenon of the sample particles is a natural consequence of the sol-gel self-propagation method of synthesis [6]. The particle distribution of the samples was plotted. The mean particle size was determined by fitting the histogram of the particle distribution to a normal distribution function. The average grain size obtained statistically in SEM is approximately 85.23 nm, which is larger than the grain size estimated by the Williamson-Hall formula. This indicates that each nanoparticle in the synthesized sample consists of numerous microcrystals. Harpreet Kaur. et al. [7] prepared Cd-doped Co ferrite samples using the sol-gel method. In addition, the experimental results indicated that the particle size (D SEM ) obtained by scanning electron microscopy statistics of the samples was 58.83 nm, and the microcrystal size (D W-H ) estimated by the Williamson-Hall formula was 47.79 nm. This result is consistent with the results of the present work.

4
It is clear from Equations (4) and (5) that the A-position hopping length (LA) and the B-position hopping length (LB) are directly related to the lattice constants. The hopping lengths LA and LB of A-and B-positions keep increasing with the increase in Cd incorporation. This can be explained by the cell expansion due to the increase in site radius, caused by the entry of larger Cd 2+ ions into the tetrahedral lattice and smaller ions into the octahedral lattice. The experimental findings are consistent with the cell and bond length variation patterns obtained by M.H. Nasr et al. [8] using Ni 2+ ion-doped Cd ferrite. The following equations can be used to determine the radius of the A-site cation (rA) and the radius of the B-site cation (rB):

SEM Analysis
SEM tests were conducted on representative samples where x = 0.1 in order to examine the surface morphology of spinel-structured ferrite. Figure 5 indicates that the surface morphology of the sample where x = 0.1 is inhomogeneous, spherical, and highly agglomerated. The agglomeration phenomenon of the sample particles is a natural consequence of the solgel self-propagation method of synthesis [6]. The particle distribution of the samples was plotted. The mean particle size was determined by fitting the histogram of the particle distribution to a normal distribution function. The average grain size obtained statistically in SEM is approximately 85.23 nm, which is larger than the grain size estimated by the Williamson-Hall formula. This indicates that each nanoparticle in the synthesized sample consists of numerous microcrystals. Harpreet Kaur. et al. [7] prepared Cd-doped Co ferrite samples using the sol-gel method. In addition, the experimental results indicated that the particle size (DSEM) obtained by scanning electron microscopy statistics of the samples was 58.83 nm, and the microcrystal size (DW-H) estimated by the Williamson-Hall formula was 47.79 nm. This result is consistent with the results of the present work.  Energy dispersive X-ray spectroscopy (EDX) analysis was conducted on a representative sample where x = 0.1 in order to determine the elemental composition of the synthesized samples. Figure 6 indicates the elemental analysis obtained from the EDX analysis of Cd x Ni 0.5−x Cu 0.2 Zn 0.3 Fe 2 O 4 nanoparticles. The energy spectrum lines associated with Cd, Ni, Cu, Zn, Fe, and O can be found in the spectra. The values calculated from the spectra for the sample's elemental composition in atomic and weight percentages are recorded in the embedded table in Figure 6. The results demonstrate that the Cd x Ni 0.5−x Cu 0.2 Zn 0.3 Fe 2 O 4 sample indicated only the emission peaks of its constituent elements, and no additional impurities were found. sized samples. Figure 6 indicates the elemental analysis obtained from the EDX analysis of CdxNi0.5−xCu0.2Zn0.3Fe2O4 nanoparticles. The energy spectrum lines associated with Cd, Ni, Cu, Zn, Fe, and O can be found in the spectra. The values calculated from the spectra for the sample's elemental composition in atomic and weight percentages are recorded in the embedded table in Figure 6. The results demonstrate that the CdxNi0.5−xCu0.2Zn0.3Fe2O4 sample indicated only the emission peaks of its constituent elements, and no additional impurities were found. Figure 6. EDA X-ray spectra of a CdxNi0.5−xCu0.2Zn0.3Fe2O4 sample. Figure 7 indicates the FTIR spectra of the spinel ferrite system CdxNi0.5−xCu0.2Zn0.3Fe2O4 (0 ≤ x ≤ 0.50, in steps of 0.10). There are two distinct bands between 467 and 580 cm −1 ; the low-frequency absorption band υ2 located approximately 460 m −1 , attributed to stretching vibrations of bonds between tetrahedral (A-site) metal oxides; and the high-frequency absorption band υ1 located near 580 cm −1 , attributed to stretching vibrations of bonds between octahedral (B-site) metal oxides [17,24]. These bands prove the presence of ferrite with spinel structure in the sample [7]. The force constants KT and KO were also calculated from the frequency band data with the following equations [16]:   [17,24]. These bands prove the presence of ferrite with spinel structure in the sample [7]. sized samples. Figure 6 indicates the elemental analysis obtained from the EDX analysis of CdxNi0.5−xCu0.2Zn0.3Fe2O4 nanoparticles. The energy spectrum lines associated with Cd, Ni, Cu, Zn, Fe, and O can be found in the spectra. The values calculated from the spectra for the sample's elemental composition in atomic and weight percentages are recorded in the embedded table in Figure 6. The results demonstrate that the CdxNi0.5−xCu0.2Zn0.3Fe2O4 sample indicated only the emission peaks of its constituent elements, and no additional impurities were found. Figure 6. EDA X-ray spectra of a CdxNi0.5−xCu0.2Zn0.3Fe2O4 sample. Figure 7 indicates the FTIR spectra of the spinel ferrite system CdxNi0.5−xCu0.2Zn0.3Fe2O4 (0 ≤ x ≤ 0.50, in steps of 0.10). There are two distinct bands between 467 and 580 cm −1 ; the low-frequency absorption band υ2 located approximately 460 m −1 , attributed to stretching vibrations of bonds between tetrahedral (A-site) metal oxides; and the high-frequency absorption band υ1 located near 580 cm −1 , attributed to stretching vibrations of bonds between octahedral (B-site) metal oxides [17,24]. These bands prove the presence of ferrite with spinel structure in the sample [7]. The force constants KT and KO were also calculated from the frequency band data with the following equations [16]: The force constants K T and K O were also calculated from the frequency band data with the following equations [16]:

FTIR Analysis
where M is the molecular weight and υ 1 and υ 2 are the bands. The frequency bands are inversely related to the bond length. Since υ 1 is greater than υ 2 , the dimensions of the octahedron (B-site) are larger than those of the tetrahedron (A-site) [3,4]. As shown in Table 4, the range of υ 1 band decreases as the Cd 2+ concentration increases, while the υ 2 band remains constant. This is because the ionic radius of Cd 2+ is greater than that of Ni 2+ . Moreover, the doping of Cd 2+ into the tetrahedra increases the Fe-O bond length of the tetrahedra (A-site). On the other hand, the υ 2 band remains unchanged because nonmagnetic divalent Cd 2+ ions prefer to occupy tetrahedral sites. M.H. Nasr et al. synthesized nanocrystalline spinel ferrites with the general formula Cd 1−x Ni x Fe 2 O 4 (0 ≤ x ≤ 1.0) using a flash auto-combustion method. With increasing amounts of nickel ions replacing cadmium ions, the υ 1 and K T values increase. This result is corroborated by the results of the present work [8]. In addition to these two bands, Figure 7 also displays the bands of -NO 3 ions [9], carboxyl groups [9,11], and hydroxyl groups [4,9,10] appearing at 1390.06, 1630.06, and 3444.35 cm −1 , respectively.

Dielectric Analysis
where M is the molecular weight and υ1 and υ2 are the bands. The frequency bands are inversely related to the bond length. Since υ1 is greater than υ2, the dimensions of the octahedron (B-site) are larger than those of the tetrahedron (A-site) [3,4]. As shown in Table  4, the range of υ1 band decreases as the Cd 2+ concentration increases, while the υ2 band remains constant. This is because the ionic radius of Cd 2+ is greater than that of Ni 2+ . Moreover, the doping of Cd 2+ into the tetrahedra increases the Fe-O bond length of the tetrahedra (A-site). On the other hand, the υ2 band remains unchanged because non-magnetic divalent Cd 2+ ions prefer to occupy tetrahedral sites. M.H. Nasr et al. synthesized nanocrystalline spinel ferrites with the general formula Cd1−xNixFe2O4 (0 ≤ x ≤ 1.0) using a flash auto-combustion method. With increasing amounts of nickel ions replacing cadmium ions, the υ1 and KT values increase. This result is corroborated by the results of the present work [8]. In addition to these two bands, Figure 7 also displays the bands of -NO 3 ions [9], carboxyl groups [9,11], and hydroxyl groups [4,9,10] appearing at 1390.06, 1630.06, and 3444.35 cm −1 , respectively.  Figure 8 shows the variation curves of the dielectric constant's real part of the dielectric constant (ε′), and the imaginary part of the dielectric constant (ε″) with frequency for the spinel ferrite system CdxNi0.5−xCu0.2Zn0.3Fe2O4 (0.00 ≤ x ≤ 0.50 with a step size of 0.10) in the frequency range of 100 Hz to 100 KHz.  The real part of the dielectric constant is the relative dielectric constant under an electrostatic field, which reflects a material's ability to store charge. The dielectric constant's imaginary part represents the energy dissipation that occurs when a dipole in a dielectric overcomes the interference of mechanical collisions under the influence of an electric field, and is oriented back and forth in different directions.

Dielectric Analysis
In dielectric materials, the dielectric loss can be expressed by the loss factor, which is determined by the following equation: where ε is the dielectric constant's real part, ε is the dielectric constant's imaginary part, and δ is the loss angle. It can be observed in Figure 9 that the dielectric constant real part (ε ) decreases as the frequency increases, and eventually remains constant at high frequencies. The dielectric constant of ferrite depends on the electron transfer mechanism. The electron hopping between Fe 3+ and Fe 2+ ions is the primary cause of the ferrite conduction mechanism. Moreover, the hopping electrons have a direct impact on the polarization. The Maxwell-Wagner theory of interfacial polarization provides a detailed explanation for the variation of the dielectric constant with frequency [4,16,17]. According to this polarization theory, grain boundaries with high resistance have a greater impact on the dielectric constant at low frequencies. Moreover, highly conductive grains have a greater impact on the sample at high frequencies. During the annealing treatment of ferrite materials, the irregular distribution of oxygen ions in grains and grain boundaries results in ion and electron displacement polarization at high frequencies, and interfacial polarization at low frequencies. Due to the long response time of interfacial polarization, there is not enough time for response as the frequency increases, resulting in a decrease in the material's dielectric constant. The dielectric constant eventually maintains a constant value at high frequencies, mainly due to electron displacement polarization. The same results were obtained by M. Moazzam. Hossen et al. in a study using La-doped Ni-Cu-Cd ferrite [4].
trostatic field, which reflects a material's ability to store charge. The dielectric constant's imaginary part represents the energy dissipation that occurs when a dipole in a dielectric overcomes the interference of mechanical collisions under the influence of an electric field, and is oriented back and forth in different directions. In dielectric materials, the dielectric loss can be expressed by the loss factor, which is determined by the following equation: tan δ = ε″ ε′ (10) where ε′ is the dielectric constant's real part, ε″ is the dielectric constant's imaginary part, and δ is the loss angle. It can be observed in Figure 9 that the dielectric constant real part (ε′) decreases as the frequency increases, and eventually remains constant at high frequencies. The dielectric constant of ferrite depends on the electron transfer mechanism. The electron hopping between Fe 3+ and Fe 2+ ions is the primary cause of the ferrite conduction mechanism. Moreover, the hopping electrons have a direct impact on the polarization. The Maxwell-Wagner theory of interfacial polarization provides a detailed explanation for the variation of the dielectric constant with frequency [4,16,17]. According to this polarization theory, grain boundaries with high resistance have a greater impact on the dielectric constant at low frequencies. Moreover, highly conductive grains have a greater impact on the sample at high frequencies. During the annealing treatment of ferrite materials, the irregular distribution of oxygen ions in grains and grain boundaries results in ion and electron displacement polarization at high frequencies, and interfacial polarization at low frequencies. Due to the long response time of interfacial polarization, there is not enough time for response as the frequency increases, resulting in a decrease in the material's dielectric constant. The dielectric constant eventually maintains a constant value at high frequencies, mainly due to electron displacement polarization. The same results were obtained by M. Moazzam. Hossen et al. in a study using La-doped Ni-Cu-Cd ferrite [4]. The loss factor (tanization theory) and highly resistive grain boundaries have a greater impact on the dielectric constant at low frequencies. The loss factor is high at low frequencies and becomes constant at high frequencies. This is in line with Koop's theoretical model, which suggests that crystal plane polarization is generated at lower frequencies and that crystal plane polarization primarily contributes to the loss factor at lower frequencies. Due to the long response time of crystal plane polarization, electron transfer and ion exchange at high frequencies cannot be responded to, resulting in a decreasing loss factor with increasing frequency. Ruqayya Zakir., et al. obtained the same results using Ce-doped Co-Mg-Cd ferrites [5]. The loss factor (tanization theory) and highly resistive grain boundaries have a greater impact on the dielectric constant at low frequencies. The loss factor is high at low frequencies and becomes constant at high frequencies. This is in line with Koop's theoretical model, which suggests that crystal plane polarization is generated at lower frequencies and that crystal plane polarization primarily contributes to the loss factor at lower frequencies. Due to the long response time of crystal plane polarization, electron transfer and ion exchange at high frequencies cannot be responded to, resulting in a decreasing loss factor with increasing frequency. Ruqayya Zakir., et al. obtained the same results using Ce-doped Co-Mg-Cd ferrites [5].
The real and imaginary parts of the dielectric modulus of the spinel ferrite system Cd x Ni 0.5−x Cu 0.2 Zn 0.3 Fe 2 O 4 (0 ≤ x ≤ 0.50, in steps of 0.10) are calculated. Figure 10a,b indicate the variation curves of the dielectric modulus real part and the dielectric modulus imaginary part with the external field frequency in the frequency range from 100 Hz to 100 KHz. When the frequency is below 10 3 Hz, the M value is close to zero, which is known as the low-frequency region. The M value increases continuously with the increase in frequency and reaches the highest point of saturation in the highfrequency region when the frequency is greater than 10 3 Hz. The spatial variation of the charge carriers (release of space-charge polarization) in the short-range region causes the saturation maxima of M in the high-frequency region during the conductivity process. Moreover, the minimum values result from the spatial variation of charge carriers in the long-range region during the conductivity process [4,10]. Figure 10b illustrates that the curve of M with frequency has a unique peak (except where x = 0.0). The peak changes to higher frequencies as the cadmium content increases. These peaks provide crucial information regarding the type of carrier migration within the system. The peak divides the frequency into two regions: low and high frequencies. In the low-frequency region, the carriers undergo long-range displacement, whereas in the high-frequency region, they experience short-range movement.
M = ε ε + ε (12) Figure 10a,b indicate the variation curves of the dielectric modulus real part and the dielectric modulus imaginary part with the external field frequency in the frequency range from 100 Hz to 100 KHz. When the frequency is below 10 3 Hz, the M′ value is close to zero, which is known as the low-frequency region. The M′ value increases continuously with the increase in frequency and reaches the highest point of saturation in the high-frequency region when the frequency is greater than 10 3 Hz. The spatial variation of the charge carriers (release of space-charge polarization) in the short-range region causes the saturation maxima of M′ in the high-frequency region during the conductivity process. Moreover, the minimum values result from the spatial variation of charge carriers in the long-range region during the conductivity process [4,10]. Figure 10b illustrates that the curve of M″ with frequency has a unique peak (except where x = 0.0). The peak changes to higher frequencies as the cadmium content increases. These peaks provide crucial information regarding the type of carrier migration within the system. The peak divides the frequency into two regions: low and high frequencies. In the low-frequency region, the carriers undergo long-range displacement, whereas in the high-frequency region, they experience short-range movement. The sample's Nyquist plot is shown in Figure 11. Evenly curving arcs can be observed in the figure (except for where x = 0.0). This indicates that grain boundaries occupy a larger volume and that grain boundaries dominate the conductive behavior [16]. By analyzing the complex modulus, it is possible to study materials with the same resistance but different capacitances. M.M. Hossen et al. [4,10] obtained dielectric experimental results using La and Mn -doped Ni-Cu-Cd ferrites, which are in agreement with our experimental results. The sample's Nyquist plot is shown in Figure 11. Evenly curving arcs can be observed in the figure (except for where x = 0.0). This indicates that grain boundaries occupy a larger volume and that grain boundaries dominate the conductive behavior [16]. By analyzing the complex modulus, it is possible to study materials with the same resistance but different capacitances. M.M. Hossen et al. [4,10] obtained dielectric experimental results using La and Mn -doped Ni-Cu-Cd ferrites, which are in agreement with our experimental results.  Figure 12 depicts the measured hysteresis lines of the spinel ferrite system CdxNi0.5−xCu0.2Zn0.3Fe2O4 (0 ≤ x ≤ 0.50, in steps of 0.10) at room temperature. The sample    Figure 12 depicts the measured hysteresis lines of the spinel ferrite system CdxNi0.5−xCu0.2Zn0.3Fe2O4 (0 ≤ x ≤ 0.50, in steps of 0.10) at room temperature. The sample hysteresis line is S-shaped when x = 0.0-0.4, indicating that the sample is ferromagnetic. The hysteresis line of the x = 0.5 sample is close to a straight line, indicating that the sample is weakly ferromagnetic and close to being paramagnetic.  Table 5. Evidently, the saturation magnetization intensity of the CdxNi0.5−xCu0.2Zn0.3Fe2O4 sample increases initially and then decreases with increasing Cd content. The saturation magnetization intensity of ferrite is dependent on the distribution of magnetic cations in the A and B sites, the sintering temperature, grain size, chemical composition, density, etc. The saturation magnetization strength of ferrite depends  Table 5. Evidently, the saturation magnetization intensity of the Cd x Ni 0.5−x Cu 0.2 Zn 0.3 Fe 2 O 4 sample increases initially and then decreases with increasing Cd content. The saturation magnetization intensity of ferrite is dependent on the distribution of magnetic cations in the A and B sites, the sintering temperature, grain size, chemical composition, density, etc. The saturation magnetization strength of ferrite depends strongly on the exchange interaction between the A-and B-site cations [10,17]. The intensity of the exchange interactions in a ferrite system is proportional to the distance between the cations. Therefore, ion-doped ferrite with different particle sizes can alter the ferrite's magnetization strength. The magnetic variation of ferrite can be explained by Neel theory. Based on Neel theory, the M s of spinel-type ferrite can be expressed by the following equation:

Magnetic Properties
where M A and M B are the magnetization intensities of the A-and B-site ions, respectively. The increase and then decrease in magnetization intensity of the sample with increasing Cd 2+ content may be a result of the tendency of non-magnetic Cd 2+ ions to occupy the A-site (M A decreases), causing the Fe 3+ ions at the A-site to migrate to the B-site after being compressed (M B increases). This, based on Neel's theory, increases the sample's magnetization intensity. The subsequent decrease in the samples' magnetization intensity may be a result of the production of the paramagnetic material α-Fe 2 O 3 in the sample. The magnetic moment for each formula unit in the Bohr magnetron is obtained using the following equation [8]: where M s is the saturation magnetization intensity and M W is the sample's molecular weight. In a Bohr magnetron, the trend of the magnetic moment for each formula unit is the same as the saturation magnetization intensity. The same phenomenon was discovered by Kaur, Harpreet, et al. using non-magnetic ions in Cd-doped CoFe 2 O 4 [25]. The coercivity of the samples decreases as the Cd 2+ increases, as can be observed from the lower right inset of Figure 12. The variation of coercivity is determined by various factors, such as particle size, crystallinity, crystal structure, morphology, strain, and anisotropy [24,26]. Based on the Stoner Wolfarth findings, values for the theoretical squareness (M r /M s ) below 0.5 indicate that the nanoparticles are uniaxially anisotropic, single-domain materials [3,6,24]. The squareness of the samples (M r /M s ) falls in the range of 0.0120-0.1672, indicating that the material is anisotropic and single-domain. Table 5 illustrates the various anisotropies [3,4,10], where M s is the saturation magnetization intensity and H c is the coercivity [8]. The Cd x Ni 0.5−x Cu 0.2 Zn 0.3 Fe 2 O 4 sample has excellent hysteresis lines and low coercivity, indicating that the sample is easily demagnetized and has the potential to be an excellent electromagnet material [11,14].

Experimental Section
The sol-gel method was used to prepare cadmium-doped Ni-Cu-Zn ferrite ( Figure 13). The general formulation was Cd x Ni 0. 5 . All metal nitrates were weighed based on the stoichiometric ratio and dissolved in distilled water, and then the solution was stirred using a magnetic stirrer and stirred evenly to ensure complete dissolution. Citric acid was added at a ratio of 1:1 (nitrate to citric acid), and the solution continued to be stirred at a constant rate. To create a pH = 7 precursor solution, ammonia was added dropwise at room temperature. The precursor solution was heated in a water bath at 80 • C while being constantly stirred. Eventually, the solution became a wet gel, which was then baked in a drying oven at 100 • C for 24 h until completely dry. Finally, the soft powder-like material was obtained by self-propagation. The same method was used to prepare each sample. The samples were then crushed and calcined for 5 h at 700 • C at a rate of 10 • C per minute to create extremely crystalline samples. XRD analysis was performed on an X'Pert3 powder-type multifunctional X-ray diffractometer. Cu Kα was utilized as the radiation source (λ = 0.15406 nm), and the scanning range was 20-80 • . The crystal structure and lattice parameters of the samples were determined using the Fullprof and Rietveld methods. The infrared spectra of the samples were measured using a NICOLET 6700 Fourier transform infrared spectrometer with a scan range of 450-4000 Hz. The indicated morphology and microstructure of the samples were examined using a TASKEN MIRALMS scanning electron microscope (SEM) with an inbuilt Oxford Xplore 30 energy spectrometer (EDS). The hysteresis lines of the samples were measured using the Magnetic Property Measurement System (MPMS3) at room temperature, with a maximum applied magnetic field of 5 kOe. Frequency-dependent dielectric spectra were obtained using a WayneKer impedance analyzer (model 6500P) in the range of 100 Hz to 100 kHz at room temperature.
ning range was 20-80°. The crystal structure and lattice parameters of the samples were determined using the Fullprof and Rietveld methods. The infrared spectra of the samples were measured using a NICOLET 6700 Fourier transform infrared spectrometer with a scan range of 450-4000 Hz. The indicated morphology and microstructure of the samples were examined using a TASKEN MIRALMS scanning electron microscope (SEM) with an inbuilt Oxford Xplore 30 energy spectrometer (EDS). The hysteresis lines of the samples were measured using the Magnetic Property Measurement System (MPMS3) at room temperature, with a maximum applied magnetic field of 5 kOe. Frequency-dependent dielectric spectra were obtained using a WayneKer impedance analyzer (model 6500P) in the range of 100 Hz to 100 kHz at room temperature.

Conclusions
Using the sol-gel self-propagation method, Cd x Ni 0.5−x Cu 0.2 Zn 0.3 Fe 2 O 4 (0 ≤ x ≤ 0.50) spinel-type ferrite was prepared using the sol-gel self-propagation technique in this study. In addition, XRD analysis indicated that the samples formed a space group of Fd-3m cubic spinel structure ferrite, while the second phase α-Fe 2 O 3 appeared in the samples with Cd doping, indicated that the solid solution limit of cadmium-doped Ni-Cu-Zn ferrite was less than x = 0.1. The lattice constant of the samples increased from 8.3794 Å to 8.5397 Å with the continuous increase of the Cd content, and the increase of lattice constants The increase in lattice constant is due to the larger ionic radius of Cd (0.97 Å) than that of Ni (0.69 Å). FTIR analysis determined two distinct bands between 467 and 580 cm −1 , corresponding to the stretching vibrations of the bonds between tetrahedral and octahedral metal oxides. This results in the band υ1 decreasing from 580.50 cm −1 to 556.23 cm −1 , and the tetrahedral force constant decreasing from 3.1880 × 10 2 N/m to 3.0921 × 10 2 N/m caused by the preference of Cd 2+ ions to occupy the tetrahedral sites. The decrease in the sample dielectric property parameters with increasing frequency can be explained in detail by Maxwell-Wagner interface polarization theory, where the high-frequency dielectric constant remains constant at low values resulting from the major role of electron displacement polarization. Magnetic analysis shows that the samples are ferromagnetic when x = 0.0-0.4, and weakly ferromagnetic and nearly paramagnetic when x = 0.5. The saturation magnetization intensity of the samples increases with the increase of Cd content to 67.2958 emu/g (x = 0.1) and then decreases rapidly to 4.4228 emu/g (x = 0.5). The hysteresis return lines of Cd x Ni 0.5−x Cu 0.2 Zn 0.3 Fe 2 O 4 (0.0 ≤ x ≤ 0.4) samples are good, with low coercivity and squareness. These nanoparticles are easy to demagnetize and may be suitable for magnetic recording and soft magnetic ferrite applications, including biomedical applications (e.g., thermal therapy).