First, the appropriate mole of Mg(NO₃)₂·4H₂O, Cd(NO₃)₂.4H₂O, Cu(NO₃)₂·3H₂O, Zn(NO₃)₂·4H₂O, Fe(NO₃)₃·9H₂O according to Table 1, with citric acid (1:1 mole ratio), was completely dissolved with a minimum of water and its pH value noted as 2 (Gholizadeh, 2017b; Gholizadeh and Malekzadeh, 2017; Gholizadeh, 2019). Then, the resulted solution was put in an oven for 24 h at 80°C and kept until complete drying. After that, the dried materials were powdered and finally calcined at 500°C for 3 h.

X-ray diffraction patterns (XRD) of the powders were recorded using Bruker’s Advanced-8D at room temperature (RT) in the range of 2θ = 20–80°. XRD device was equipped with a Cu-Kα (λ = 1.5404 Å) radiation source and a nickel filter (35 mA, 30 kV). Fourier-transform infrared (FTIR) spectra of the samples were measured with Perkin-Elmer device at RT. The crystal structure and lattice parameters were determined by a Rietveld method using the Fullprof program. Halder-Wagner (H-W) method was employed for calculating the crystallite size (D) and microstrain (ε) of the samples (Gholizadeh et al., 2016): ( β h k l ∗ d h k l ∗ ) 2 = ( ε 2 ) 2 + ( β h k l ∗ d h k l ∗ 2 ) ( 1 D ) (1) where, β h k l ⁡ cos ⁡ θ λ = β h k l ∗ and 2 ⁡ sin ⁡ θ λ = d h k l ∗ . The surface images of the samples were recorded by the field-emission scanning electron microscope (FE-SEM, MlRA3 model), and the values of average particle sizes ​​were calculated from the FESEM images by digimizer image analysis software. Also, the atomic and weight percentage of the constituent elements of the compound was calculated by the X-Ray energy dispersive (EDX) spectroscope equipped inside the FE-SEM microscope. To study the optical properties of the samples, the absorption spectra of these nanoparticles were obtained by UV-Vis spectroscopy (UNICO model 4,802) in the UV-Vis region. The energy gap of the samples was obtained using Tauc’s method according to Ref. (Gholizadeh and Tajabor, 2010). The magnetic hysteresis loop of the samples was measured at RT using a vibrating sample magnetometer (VSM, 7,400 Lake Shore) and in a variable magnetic field up to 2T. The powders were compressed into pallets with diameter 14 mm and sintered at 500 °C for 1 h. Then pellets were coated by Ag in order to investigate the electrical behaviour. The dielectric measurements were performed in the frequency range (100 Hz-10 MHz) using a precision LCR meter (GW Instek 8110G).

XRD pattern of Cd _x Mg_0.3-x Cu_0.2Zn_0.5Fe₂O₄ nanoparticles are shown in Figure 1. The Miller indices (hkl) of the spinel phase are labeled on the diffraction peaks in Figure 1, which corresponds to the (220), (311), (222), (400), (422), (511), (440) planes. Structural analysis of samples x = 0.00, 0.05, 0.10, 0.15, 0.20, using X’pert package indicates the formation of a pure cubic spinel structure with Fd 3 ¯ m space group, which is in agreement with JCPDS standard cards numbered 47-2,399 (Beyranvand and Gholizadeh, 2020; Gholizadeh and Beyranvand, 2020). However, this analysis for samples with x = 0.25, 0.30 indicates the presence of CdO impurity phase in addition to spinel phase diffraction peaks as shown in Figure 1. For more detailed structural studies, Rietveld refinement of X-ray diffraction patterns using Fullprof software for the samples was performed to calculate the lattice parameter of the unit cell.

The lattice parameter values (a) of all the samples are given in Table 2, and the change in the a with the Cd content is drawn in Figure 2. The a shows a decreasing trend by increasing the amount of cadmium from x = 0.00 to x = 0.15, and then the substitution goes through an increasing trend. The ionic radii of the atoms participating in the A-site (CN: 4) for Mg²⁺, Cd²⁺, Zn²⁺, Cu²⁺, Zn²⁺, Fe²⁺, Fe³⁺, O^2- are: 0.49, 0.84, 0.62, 0.60, 0.63, 0.49, 1.38 Å, respectively, and in the B site (CN: 6) are: 0.72, 0.95, 0.73, 0.74, 0.77, 0.64, 1.40 Å, respectively (William, 1944; Birajdar et al., 2002). It is observed that the lattice parameter decreases with cadmium content obeying Vegard’s law for x up to 0.15 (Bachhav et al., 2011). However, as reported by the structural analysis, there is the traces of CdO impurity phase in addition to spinel phase diffraction peaks for x = 0.25, 0.30. This result shows that the solid solubility limit in our samples is for x = 0.15. Also, decrease in the lattice parameter a results in the decrease of the bond lengths with Cd content. It can be concluded that there is increase of ionic-covalent character of the spinel with Cd content (Yue et al., 2001; Bachhav et al., 2011).

The linear fitting curves of ( β h k l d h k l ∗ ) 2 versus ( β h k l ∗ d h k l ∗ 2 ) in the H-W method are plotted for all samples in Figure 3 (Gholizadeh and Jafari, 2017; Gholizadeh and Beyranvand, 2020). In the H-W method, the crystallite size values (D _H-W) and micro-strain (ε) were obtained using the slope inverse and the y-intercept of the linearly fitted data, respectively as given in Table 2. These results demonstrated that the lattice parameter shows dependence on size of particle, decreasing for smaller particles, increasing strain-surface effects (Bachhav et al., 2011).

The theoretical density (ρ _XRD) of Cd _x Mg_0.30-x Cu_0.20Zn₀.₅₀Fe₂O₄ ferrites was obtained from the following equation (Gholizadeh, 2018): ρ X R D = 8 M N A α 3 (2)

In the above relation, M, and a are the molar mass, and the lattice parameter of the samples, respectively. An increase in the theoretical density with the Cd substitution shows that the increasing trend of the molar mass overcomes the non-uniform change of the a. Also, the experimental density (ρ _exp) of the studied ferrites was obtained by the following equation (Gholizadeh and Jafari, 2017): A × d = V → ρ = m V (3)

In the above relation, A is the surface of the pellet, d is the thickness of the pellet, m is the weight of the pellet, V is also the pellet’s volume. Using the theoretical and experimental densities of all samples, the values of porosity percentage (P) were calculated with the following relation (William, 1944): P = ρ X R D − ρ exp ρ X R D × 100 (4)

The dislocation density (δ) contains information on the improvement of the crystal structure and was calculated according to δ = 1 D 2 (ZeidAbo Zeida et al., 2019). The values of dislocation density, theoretical density, experimental density, and porosity are given in Table 2. As can be seen from Table 2, there is approximately equal and low porosity in the studied ferrites. Also, one can notice the dependence of density on the Cd content. The density increases with increasing Cd content, indicating improved densification by incorporating Cd into the ferrite samples (Yue et al., 2001). Further, X-ray density of all the ferrite samples is higher than the corresponding bulk density, due to the existence of the voids and pores. The decrease in the dislocation density (δ) values and, consequently, the lattice strain, is caused by an increase in the Cd content (ZeidAbo Zeida et al., 2019). There is a decrease in the size of crystallites and an increase in the dislocation density of the samples. In this case, the resulting defects and dislocations result in distortion and deformation of the crystal lattice with the formation of disordered regions in the structure (ZeidAbo Zeida et al., 2019).

The FE-SEM images of the samples with x = 0.00, 0.05, 0.10, 0.15 are shown in Figure 4. As can be seen, the surface morphology of the samples consists of ordered grains composed of smaller nanoparticles with an unequal size that are distributed throughout the samples with small porosity. A particle size distribution diagram of the samples was drawn and the mean particle size (D _FESEM) was obtained using fitting the particle size distribution histogram with the lognormal distribution function (Mahmoudi and Gholizadeh, 2018). P ( d ) = 1 d σ d 2 π exp ( − 1 2 σ d 2 ln ( d D F E S E M ) ) (5)

The D_FESEM values calculated are given in Table 2 and are plotted in Figure 2. The particle size obtained by this method is slightly larger than the D _H-W obtained by the Halder-Wagner method. This is because the grains consist of crystallite size. As shown in Figure 2, the variation of the D_FE-SEM is compatible with the values of the a, and D_H-W.

The X-ray Energy-dispersive spectroscopy (EDX) attached to the FESEM microscope was used for elemental analysis of the samples. Figure 5 shows the elemental analysis spectra obtained from the EDX analysis for Cd _x Mg_0.30-x Cu_0.20Zn_0.50Fe₂O₄ nanoparticles. In the desired spectrum, energy lines related to Mg, Cd, Cu, Zn, Fe, O can be seen. The values of atomic percentage and weight percentage calculated from the spectrum for the constituent elements of the FESEM samples are recorded as a table in the inset of Figure 5. The data in these tables show a good agreement with the formula composition of each sample.

The FTIR spectra of all samples are shown in Figure 6 in the wavelength range of 400–900 cm⁻¹. Two strongly absorption bands are observed around 450 and 560 cm⁻¹, which are familiar to those that observed in the spinel ferrites and are mainly related to the iron-oxygen vibrations. The frequency band υ_t at around 560 cm⁻¹ related to the oxygen-iron vibration at the A site, the frequency band υ_o around 450 cm⁻¹ refers to the oxygen-iron vibration at the B site, so that the appearance of these two vibration modes confirms the formation of the spinel structure of the samples (Gholizadeh and Jafari, 2017). However, the position and width of the bands are affected by factors such as grain size, sintering conditions, and preparation method (Gholizadeh and Jafari, 2017).

As shown in Figure 6, the value of υ_t at the A site for the samples with x = 0.00, 0.05, 0.10, 0.15, 0.20 is almost constant, but υ_o at the B site for these samples shows a decreasing trend. However, the value of υ_t in the samples with x = 0.25, 0.30 shows a decreasing trend, and υ_o for these samples shows an increasing trend. The difference in the band’s position can be interpreted in terms of the mass and radius of the participating atoms at the two sites. Here, the change in the Fe³⁺-O^2- bond length at the A and B sites explains the changes observed in the a. Since the ionic radius of Cd²⁺ is greater than other divalent cations participated at the A and B sites, it can mainly occupy the B site, which explains the variation of the υ₂.

Figure 7 shows the magnetization curves of Cd _x Mg_0.30-x Cu_0.20Zn_0.50Fe₂O₄ ferries versus applied magnetic field. The magnetic hysteresis confirms the prepared ferrite material is magnetically ordered. The magnetic parameters of the studied ferrites can be obtained from these curves.

The According to law of approach to saturation (LAS), the magnetization of ferromagnetic nanoparticles at high magnetic fields follows the following formula (Gholizadeh and Jafari, 2017; Beyranvand and Gholizadeh, 2020; Gholizadeh and Beyranvand, 2020). M ( H ) = M s [ 1 − ( b H 2 ) − ( c H 3 ) ] + χ H (6) where,   M s is the contribution of ferromagnetic saturation magnetization. The LAS fitting of the magnetization curves with this formula is shown in Figure 8 for all the samples.

In general, the total M _s of the ferrites depends on the distribution of magnetic cations in the A and B sites, sintering temperature, grain size, chemical composition, density, etc. (Gholizadeh and Jafari, 2017; Gholizadeh, 2018; Shamgani and Gholizadeh, 2019; Beyranvand and Gholizadeh, 2020; Gholizadeh and Beyranvand, 2020). Magnetization of ferrites is strongly dependent on the exchange interactions between cations present at A and B sites. These exchange interactions are A–B, B–B and A–A interactions, either positive or negative. The parallel arrangement of cations is represented by positive and negative sign denotes anti-parallel arrangement of cations. These interactions depend on magnitude, sign, the extent and form of overlap of the 3d electron wave functions of the neighboring ions (Hakim et al., 2015; Vara Prasad, 2015; THORAT et al., 2018). The strength of the interaction is proportional to the distance between ions and the cations present in the ferrite system. Therefore, when the system is altered with larger/smaller cations it may influence the magnetization. Magnetization of ferrites also can be explained on Neel’s theory of ferrimagnetism (two sub lattice model) (Hakim et al., 2015; Vara Prasad, 2015; THORAT et al., 2018). According to the Néel theory, the A-B negative exchange interaction dominates over the A-A and B-B positive exchange interactions. The magnetic moments in each sublattice are oriented in parallel and relative to each are antiparallel. According to Neil’s theory, the M _s of the spinel ferrites can be expressed by (Gholizadeh and Jafari, 2017; Beyranvand and Gholizadeh, 2020; Gholizadeh and Beyranvand, 2020): M s = | M B − M A | (7) where, M _A and M _B are the M _s of ions at the A- and B-sites, respectively. Two factor are determining the decrease of M _s with Cd content. First factor for a gradual decrease in the magnetization of the samples with the increasing the Cd²⁺ content is originated from the presence of non-magnetic Cd²⁺ ions at the B site which induces a pressure on the Fe³⁺ (5 μ_B) ions at the B site and their transfer to the A site. This reduces the magnetization of the samples in accordance with the Néel relation. Second factor in decrease of M _s with Cd content is the decrease of grain size. Since, magnetization caused by domain wall movement requires less energy than that requires by domain rotation, it is easy for the domain wall movement to magnetize or demagnetize the samples with larger grain size (Hakim et al., 2015). Samples with larger grain are expected to have high M _s. However, a high decrease in the M _s of the samples with x = 0.25, 0.30 can also be attributed to the CdO impurity phase. The CdO is the II-VI group semiconductor that shows the paramagnetism nature at RT. The coexistence of the CdO phase and spinel ferrite can potentially explain the decrease of M _s (Beyranvand and Gholizadeh, 2020).

Investigation of optical properties of Mg_0.30-xCd_xCu_0.2Zn_0.5Fe₂O₄ nanoparticles is possible by using the absorption spectrum of the samples in the visible-ultraviolet region. The direct energy gap of all the studied samples was obtained by extrapolating the linear part of the diagram of (αhʋ)² versus (hʋ) as shown in Figure 9. The error in bandgap value, measured from Tauc plot method is ∼0.02 eV. The energy gap of the samples with increasing the Cd substitution up to x = 0.20 shows an increasing trend and then decreases with the substitution. The obtained values ​​of the bandgap are in agreement with the values reported in other articles for ferrites (Ajmal and Maqsood, 2008; Banerjee et al., 2007). The energy gap of the samples may shift due to structural changes in the crystal lattice parameters listed in Table 2 (Andrade et al., 2017). It can be interpreted in terms of the changes in the electronic structure of the samples which is originated from the changes in the Fe³⁺-O^2-- Fe³⁺ bond angle and the Fe³⁺-O^2- bond length. Also, Figure 9 depicts that the energy band gap increases as the crystallite size decreases with the increase of Cdsubstitution for x up to 0.20 (see Table 2). Such behavior may be explained on the bases of Brass’s effective mass model (Lin et al., 2005; Jauhar et al., 2011; Agrawal et al., 2016), where the measured band gap E g ∗ can be expressed as a function of particle size as: E g ∗ = E g b u l k + h 2 8 e r 2 ( 1 m e + 1 m h ) − 1.8   e 2 4 π ε o   ε   r (8) where E g b u l k is the bulk energy gap, r is the particle size, m _e is the effective mass of electrons, m _h is the effective mass of holes, ε is the relative permittivity, ε ₀ is the permittivity of free space, h is Planck constant and e is electron charge. According to this model, the energy band gap increases as the particle size decrease.

The measured values of capacitance (C) and dielectric loss tangent (tan δ) of the pellets were used for obtaining the real ( ε r ' ) and imaginary ( ε r ' ' ) dielectric constant by the following relations (Beyranvand and Gholizadeh, 2020; Gholizadeh and Beyranvand, 2020); ε r '   =   C t ε 0 A (9) ε r ' ' =   ε r '   tan ⁡ δ (10) where t, and A are the thickness, and the surface area of the pellets. The ε_o is the permittivity of free space.

Changes in the real and imaginary component of the dielectric constant are shown in Figure 10A,B as a function of frequency for Mg_0.30-x Cd _x Cu_0.2Zn_0.5Fe₂O₄ (x = 0.00, 0.10, 0.20, 0.30) ferrites. The ε′ has larger values at low frequencies and then, the values of the dielectric constants decrease uniformly and become very low in the high frequency region which can be mainly interpreted with a Maxwell’s theory of the interfacial polarization (Beyranvand and Gholizadeh, 2020). Based on the proposed mechanism for this type of polarization, the dielectric structure of the ferrites is generally divided into two layers; Layers of high conductivity grains and grain boundaries that act as insulators. The higher accumulation of electrons at the grain boundary than within the grain results in a more significant bilayer structure and interfacial polarization. At low frequencies, electrons due to transfer and electron exchange between ions can travel more than one atomic distance and pass through the grain and accumulate at the grain boundary. But, the further increase in the frequency causes to change the direction of the electron’s motion before reaching the boundaries, resulting in fewer electrons reaching the grain boundary. Hence, the weaker bilayer structure and lower interfacial polarization are created which results in a decrease of dielectric constant (Amini and Gholizadeh, 2020). As shown in Figures 10A,B, the maximum value for the dielectric constant (real and imaginary part) are observed in sample x = 0.20 and the lowest value in sample x = 0.00. The higher value of the real dielectric constant of sample x = 0.20 may be related to its smaller grain size so that at low frequencies, electrons due to easier transfer and electron exchange between Fe³⁺and M²⁺ ions can pass through the grain and accumulate more at the grain boundary. However, a decrease in the real dielectric constant of sample x = 0.25 can also be attributed to the presence of CdO impurity phase.

Figure 10B shows the imaginary part of dielectric constant which means the loss of energy in dielectric materials. As it can be seen, at low frequencies the loss is very high, and as the frequency increases, the loss decreases steadily. As the frequency increases, the amount of loss reaches a fixed value. High amounts of loss at low frequency can be attributed to the accumulation of electrons in high-resistive grain boundaries. Therefore, much energy is needed to electron transfer between Fe³⁺ and M²⁺ ions. The higher the amount of energy consumed for electron exchange, the greater the energy loss. On the other hand, due to the obvious role of the grain in the electron exchange and the lower resistivity of the grains at higher frequencies, the amount of energy required for electron transfer is reduced and consequently the amount of dielectric loss is reduced (Amini and Gholizadeh, 2020; Beyranvand and Gholizadeh, 2020; Gholizadeh and Beyranvand, 2020). The strong dispersion in the low frequency region and the high value of the measured dielectric constant of sample x = 0.20 being large is due to the combined influence of space charge effects arising from species like Fe²⁺ ions, oxygen vacancies, and grain boundary defects.

In the Mg_0.30Cu_0.2Zn_0.5Fe₂O₄ ferrites, the Zn²⁺ ions prefer to occupy the tetrahedral sites, while most of the Mg²⁺ and Cu²⁺ions occupy the octahedral sites. The Fe³⁺ ions are distributed among the tetrahedral and octahedral sites (Shamgani and Gholizadeh, 2019). However, the FTIR results of the samples Mg_0.30Cu_0.2Zn_0.5Fe₂O₄ suggest that Cu²⁺ ions prefer to occupy the octahedral sites. Thus the cation distribution for the Mg_0.30-x Cd _x Cu_0.2Zn_0.5Fe₂O₄ ferrites can be written as (Zn_0.5Fe_0.5)_A (Cd _x Mg_0.3-x Cu_0.2Fe_1.5)_B.

By using molecular weights of cations M_A and M_B on tetrahedral and octahedral sites, the values of force constants K_t and K_o for A- and B-sites, respectively, which suggested by Waldron, were calculated as follow (WALDRON, 1955): K t ( N m ) = 7.62   M A   υ t 2   10 − 7 (11) K o ( N m ) = 10.62 ( M B / 2 )   υ o 2   10 − 7 (12)

The stiffness constant C ₁₁ can be obtained from K= (K _t + K _o )/2 = a C ₁₁ (Gholizadeh, 2017a), where K and a are the average force constant and lattice parameter. In cubic crystals, the other elastic constants of the studied ferrites have been calculated using following equations (WALDRON, 1955; Mohammed et al., 2012; Gholizadeh, 2017a; Dubey and Lahiri, 2021): Velocity   of   longitudinal   wave: V L = C 11 / ρ (13) Velocity   of   shear wave: V S = C 44 / ρ = V L / 3 (14) Longitudinal   Modulus: L = ρ ( V l ) 2 (15) Rigidity   Modulus: G = ρ ( V s ) 2 (16) Poisson's   Ratio: σ = ( L − 2 G ) / ( 2 ( L − G ) ) (17) Stiffness   constant: C 12 = σ C 11 / ( 1 − σ ) (18) Mean   wave   velocity: 3 V m 3 = 1 V L 3 + 2 V S 3 (19) Debye   temperature: θ D = h V m k B 3 ρ q N A 4 π M 3 (20) Young's   modulus:E = ( C 11 + 2 C 12 ) ( C 11 − C 12 ) ( C 11 + C 12 ) (21) Bulk   modulus:B = ( C 11 + 2 C 12 ) 3 (22) where h, k_B, M, q and ρ are the Plank constant, the Boltzmann constant, the molecular weight, the number of atoms in the unit formula and density, respectively. The calculated values for the elastic constants are tabulated in Table 3.

It is observed from Table 3 that the values of the K, C ₁₁, C ₄₄, C₁₂, L, G, E, and B increase with the increase of Cd²⁺ substitution. It results in increase of the strength of the interatomic interactions between various atoms. This behavior can be attributed to the substitution of heavier Cd²⁺ for lighter Mg²⁺ ions which need higher energy to vibration. The value of σ = 0.2354 for the samples lie in the range from -0.9 to 0.5 obtained from the theory of isotropic elasticity (Mohammed et al., 2012; Gholizadeh, 2017a; Dubey and Lahiri, 2021).

Looking at Table 3, we can observe that the velocity of shear waves (V _S ) is less than longitudinal waves (V _L ). This is due to when a wave travels through a material, it makes the particle vibrate. Due to transfer of energy, the vibrating particles collide with other particles, which results in other particles to vibrate. In case of shear waves, vibrates, and hence it requires a larger energy to make the neighboring particle vibrate (Gholizadeh, 2017a). Since, the energy required to vibrate the particle perpendicular to the direction of wave propagation in a material is smaller than parallel vibration; hence the velocity of the shear waves is less than that of longitudinal waves. The values of V _m and θ D given in Table 3 for Cd-substituted samples are lower than the pure sample due to the decrease in the wave number of IR bands.