# Effect of Nd3+ Substitution on the Structural, Magnetic, Magnetocaloric, and Dielectric Performance of M-Type Hexaferrites

Amal Homri, Jose Ordonez-Miranda, Kawther Laajimi, Malek Gassoumi, M.K.M. Ali

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Amal Homri, Jose Ordonez-Miranda, Kawther Laajimi, Malek Gassoumi, M.K.M. Ali. Effect of Nd3+ Substitution on the Structural, Magnetic, Magnetocaloric, and Dielectric Performance of M-Type Hexaferrites. Ceramics International, In press, 10.1016/j.ceramint.2026.02.243. hal-05520948

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Effect of Nd³⁺ Substitution on the Structural, Magnetic, Magnetocaloric, and Dielectric Performance of M-Type Hexaferrites

Amal Homri<sup>a*</sup>, Jose Ordonez-Miranda<sup>b</sup>, Kawther Laajimi<sup>c</sup>, Malek Gassoumi<sup>a</sup>, M. K. M. Ali<sup>d</sup>

<sup>a</sup>Laboratory of Condensed Matter and Nanoscience, Faculty of Sciences Monastir, University of Monastir, LR11ES40, 5000 Monastir, Tunisia

<sup>b</sup>Sorbonne Université, CNRS, Institut de Nanosciences (INSP), 4 place Jussieu, F-75005 Paris, France

<sup>c</sup>CRISMAT-ENSICAEN and IUT-Caen, University of Caen Normandy, Campus 2, 6, Bd. M. Juin, Caen 14050, France

<sup>d</sup>Physics Department, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 13318, Saudi Arabia
**Abstract**

Nd³⁺-doped Ba–Ca M-type hexaferrites Ba₀.₅Ca₀.₅NdₓFe₁₂-xO₁₉ (x = 0.0, 0.1), hereafter referred to as BCNFO, were synthesized via a sol–gel auto-combustion route to investigate the influence of Nd incorporation on their structural, magnetic, magnetocaloric, and dielectric properties.

Rietveld-refined XRD patterns combined with TEM observations confirm the formation of a single-phase magnetoplumbite structure with well-defined hexagonal grains and a slight lattice expansion attributed to the larger ionic radius of Nd³⁺. Raman spectra further support the structural stability and reveal phonon mode shifts associated with local distortions induced by doping.

Magnetic characterization shows that Nd³⁺ substitution weakens the Fe–O–Fe superexchange network, leading to a decrease in saturation magnetization (98.87 → 86.62 emu.g⁻¹) and Curie temperature (729 → 722 K), while maintaining high magnetic anisotropy. Although the maximum magnetic entropy change slightly decreases, the relative cooling power notably increases (up to 158 J.kg⁻¹), owing to the broadening of the magnetic transition and improved heat-transfer capability. Dielectric analyses reveal enhanced permittivity and pronounced Maxwell–Wagner relaxation, linked to strengthened interfacial polarization and increased Fe²⁺/Fe³⁺ electron hopping.

The combined enhancement in magnetocaloric efficiency, dielectric behavior, and magnetic hardness highlights Nd-doped Ba-Ca hexaferrites as promising multifunctional materials for magnetic refrigeration, high-frequency electronics, and microwave device applications.

**Keywords:** M-type hexaferrite; Nd substitution; magnetic anisotropy; magnetocaloric effect; dielectric relaxation; multifunctional materials.
# 1. Introduction

The growing demand for sustainable energy and high-performance electronic technologies has driven extensive research into multifunctional oxide materials with coupled magnetic, dielectric, and magnetocaloric properties [1–3]. Among these, ferrites stand out due to their chemical stability, low cost, and structural versatility, making them suitable for applications in permanent magnets, data storage, sensors, and microwave devices [1–3]. Their physical properties are largely governed by the interplay between crystal structure, cation distribution, and superexchange interactions between Fe–O–Fe networks [4,5].

M-type hexaferrites (BaFe₁₂O₁₉) are particularly attractive owing to their large magnetocrystalline anisotropy, high Curie temperature, and excellent chemical stability [6–8]. These materials have demonstrated remarkable multifunctionality combining magnetic hardness, significant magnetocaloric effects (MCE), and favorable dielectric performance, which makes them promising candidates for emerging spintronic, microwave, and energy-efficient refrigeration systems [9–12].

Several studies have shown that rare-earth (RE³⁺) substitution (e.g., Dy³⁺, Ce³⁺, Sm³⁺, Tb³⁺, and Nd³⁺) is an effective strategy to enhance and diversify the functional properties of M-type hexaferrites [12–16]. Owing to their large ionic radii and strong spin–orbit coupling associated with partially filled 4f orbitals, rare-earth ions can induce lattice distortions, modify cation distribution, and alter Fe–O–Fe superexchange interactions. These effects effectively tune magnetic anisotropy, coercivity, dielectric response, and magnetocaloric performance, often resulting in improved relative cooling power and enhanced electromagnetic characteristics [17, 18]. Consequently, RE-doped M-type hexaferrites have attracted considerable attention for permanent magnet, microwave absorption, and magnetocaloric refrigeration applications.

In particular, Ba- and Ba–Ca-based M-type hexaferrites are highly sensitive to rare-earth incorporation, where even low substitution levels can induce pronounced changes in magnetic and dielectric behavior without compromising phase stability [19,20].

Despite these advances, systematic investigations focusing on low-level Nd³⁺ substitution in Ba–Ca M-type hexaferrites remain limited, especially concerning the coupled magnetic, magnetocaloric, and dielectric properties. Most reported works explore wide substitution ranges, which often promote secondary phase formation and obscure the intrinsic structure–property relationships. Therefore, a targeted study at low Nd³⁺ content is essential to clarify the intrinsic role of rare-earth ions in tailoring multifunctional behavior while preserving the magnetoplumbite structure.
In this context, the present work investigates Ba0.5Ca0.5NdₓFe12-ₓO19 (x = 0.0, 0.1) compounds synthesized via a sol-gel auto-combustion route. A comprehensive experimental approach combining XRD, SEM, TEM, Raman spectroscopy, magnetic, magnetocaloric, dielectric, and impedance measurements was employed to elucidate the correlations between structure, magnetic exchange interactions, magnetocaloric response, and dielectric relaxation induced by Nd³⁺ substitution. The results demonstrate that even a moderate Nd³⁺ substitution level (x = 0.1) leads to significant modifications in superexchange interactions, enhanced dielectric response, and improved relative cooling power, while preserving the single-phase magnetoplumbite structure.

It should be emphasized that this study deliberately focuses on a low Nd³⁺ substitution level in order to probe the intrinsic effects of rare-earth incorporation without triggering secondary phase formation or severe structural disorder. Unlike many previous reports that explore wide substitution ranges, the restricted doping regime adopted here enables a clear and unambiguous interpretation of the structure-property relationships.

By limiting the substitution range, the present work provides new insight into how subtle Nd-induced lattice perturbations govern the coupled magnetic, magnetocaloric, and dielectric behaviors in Ba-Ca M-type hexaferrites. This approach highlights the technological relevance of low-level rare-earth substitution for the design of multifunctional ceramic materials suitable for magnetic refrigeration and high-frequency electronic applications.

## 2. Materials and Methods

### 2.1 Chemicals

The samples were prepared from barium nitrate (Ba(NO₃)₂), calcium nitrate (Ca(NO₃)₂), iron nitrate (Fe(NO₃)₃.9H₂O), neodymium nitrate (Nd(NO₃)₃.6H₂O), citric acid (C₆H₈O₇), and ammonia, all obtained from Sigma-Aldrich.

### 2.2 Synthesis of Ba0.5Ca0.5NdₓFe12-ₓO19 (x = 0.0, 0.1) Nanoparticles

The Ba0.5Ca0.5NdₓFe12-ₓO19 (x = 0.0, 0.1) nanoparticles were prepared via the sol-gel auto-combustion method. Stoichiometric amounts of (Ba(NO₃)₂).4H₂O, citric acid (C₆H₈O₇.H₂O), aqueous ammonia (NH₄OH), Co(NO₃)₂.6H₂O, and Fe(NO₃)₃.9H₂O were dissolved separately in deionized water to form 0.4 M solutions. The pH was adjusted to ~7 using a 2 M ammonia solution. The mixtures were then heated and stirred at 70 °C for 4 hours to form a fluffy gel, which was subsequently dried in an oven at 150 °C. The dried gel was ground with an agate
mortar and pestle, transferred to crucibles, and sintered at 900 °C for 9 hours under an inert atmosphere. The sintered material was finally ground into a fine powder and stored in glass vials.

### 2.2.1 Characterizations

Structural and microstructural analyses were carried out to verify phase formation and investigate morphology. XRD patterns were recorded using a PANalytical X'Pert PRO diffractometer with Cu-Kα radiation (λ = 0.154056 nm), scanning from 20° to 80° in 2θ with 0.02° increments and 1 s acquisition per step. The surface and microstructure of the samples were characterized using scanning electron microscopy (SEM), transmission electron microscopy (TEM), and high-resolution TEM (HR-TEM) at the Institute of Mineralogy, Physics of Materials, and Cosmochemistry (IMPMC), Sorbonne University. Grain size distributions were evaluated with ImageJ by measuring over 150 grains per sample to ensure statistical reliability. Elemental composition was analyzed via energy-dispersive X-ray spectroscopy (EDX) integrated with SEM. Magnetic properties were investigated using a Quantum Design Physical Property Measurement System (QD-PPMS) in the 300-900 K temperature range. Isothermal magnetization (M-H) curves and zero-field-cooled/field-cooled (ZFC/FC) measurements were performed under magnetic fields up to 10 T to probe both magnetic behavior and magnetocaloric characteristics. Disks of the BCNFO compounds (pellets with 8 mm diameter and 2 mm thickness) were prepared for electrical measurements. Aluminum electrodes were subsequently deposited on both surfaces of each pellet. The measurements were then performed over a wide range of frequencies and temperatures using an Agilent 4294A impedance analyzer and a cold plate within a liquid nitrogen-cooled cryostat.

## 3. Results and Discussions

### 3.1 XRD Analysis of Ba<sub>0.5</sub>Ca<sub>0.5</sub>Nd<sub>x</sub>Fe<sub>12-x</sub>O<sub>19</sub> (x = 0.0, 0.1) Nanoparticles

Figure 1(a) presents the X-ray diffraction (XRD) patterns of neodymium-substituted M-type hexaferrites with the formula Ba<sub>0.5</sub>Ca<sub>0.5</sub>Nd<sub>x</sub>Fe<sub>12-x</sub>O<sub>19</sub> (x = 0.0, 0.1). All samples crystallize in a hexagonal lattice belonging to the P63/mmc space group, and the observed diffraction peaks are consistent with JCPDS card no. 43-0002 [20,21]. No extra peaks were observed. The main diffraction peaks corresponded to the (006), (110), (008), (114), (205), (206), (1011), (209), (2010), (218), (0014), and (0016) planes.

The XRD results confirm that Nd<sup>3+</sup> substitution up to x = 0.1 preserves the single-phase magnetoplumbite structure without detectable secondary phases, indicating good solubility of Nd<sup>3+</sup> ions within the Ba-Ca M-type hexaferrite lattice at low substitution levels. This
observation supports the deliberate choice of a limited Nd³⁺ content, since higher rare-earth concentrations in hexaferrite systems are frequently reported to promote impurity phase formation and increased lattice distortion. Therefore, restricting the Nd³⁺ substitution to x = 0.1 ensures phase stability while enabling a clear investigation of the intrinsic effects of Nd incorporation on the magnetic, magnetocaloric, and dielectric properties.

### 3.1.2 Rietveld analysis

Figure 1(b–c) displays the Rietveld refinement results for the BCNFO samples. The refinements were performed using HighScore software [22]. The refinement parameters included background, lattice constants, scale factor, isotropic thermal factors, profile half-width parameters (u, v, w), atomic positions and occupancies, together with the agreement factors R<sub>wp</sub>, R<sub>p</sub>, and the goodness-of-fit (GOF) [23,24]. The refined structural parameters are summarized in Table 1. The GOF values are close to unity and remain below 2 for all samples, confirming the good quality of the refinements.

In order to gain insight into the site occupancy preference of Nd³⁺ ions, the refined atomic positions listed in Table 1 were carefully examined. It is observed that the most pronounced variations upon Nd substitution occur at the Fe/Nd mixed crystallographic sites, particularly the 12k and 4f sites, whereas the Ba/Ca site remains essentially unchanged. Notably, the refined x and z coordinates of the 12k site exhibit a clear shift when x = 0.1, suggesting that Nd³⁺ ions are likely to preferentially substitute Fe³⁺ at the octahedral 12k site.

This preferential occupation is consistent with crystal-chemical considerations and previous reports on rare-earth-substituted M-type hexaferrites, where large trivalent ions tend to occupy the most spacious octahedral sites (especially 12k) due to steric constraints [25]. Moreover, the 12k site plays a key role in determining the magnetocrystalline anisotropy of M-type hexaferrites.

The larger ionic radius of Nd³⁺ compared to Fe³⁺ leads to a slight lattice expansion upon substitution. Accordingly, both lattice parameters a and c increase from 5.8436 Å and 23.1454 Å for x = 0.0 to 5.9337 Å and 23.4265 Å for x = 0.1, respectively, indicating that the magnetoplumbite structure accommodates the larger ions without altering its hexagonal symmetry.

The Scherrer formula was employed to calculate the crystallite size of the prepared nanoparticles [26]:

$$D = \frac{k\lambda}{\beta\cos\theta} \qquad (1)$$
In this formula, k denotes the Scherrer constant (0.9), and β represents the full width at half maximum (FWHM) of the diffraction peak and λ is the X-ray wavelength, which is 1.54 Å [27]. The crystallite size was estimated from the most intense diffraction peak of each sample, yielding values in the range of 38.07–43.90 nm. In M-type hexaferrites, crystallite size can be effectively controlled by synthesis parameters such as calcination temperature and dwell time. A crystallite size below approximately 50 nm is generally considered a favorable structural condition for high-density magnetic recording, as it contributes to the reduction of magnetic noise [27]. However, crystallite size alone is not sufficient to ensure data storage performance. Other magnetic parameters, including saturation magnetization, coercivity, magnetocrystalline anisotropy, and thermal stability, also play a crucial role.

In this context, the BCNFO samples investigated in this work exhibit crystallite sizes below 50 nm, which represents a supportive structural feature when combined with their magnetic properties, rather than a decisive criterion on its own [27].

The unit cell volume was calculated using the following relation [28] :

$$V_{\text{cell}} = 0.8666a^2c, \qquad (2)$$

The cell volume was calculated to lie between 686.81 to 695.09 Å³ to and which is consistent with the value reported in the JCPDS card.

### 3.1.2.1 Relative density and porosity

Using the experimentally determined bulk density (d<sub>B</sub>), the X-ray density (d<sub>x</sub>) and the relative density and apparent porosity (P) of the Ba<sub>0.5</sub>Ca<sub>0.5</sub>Nd<sub>x</sub>Fe<sub>12-x</sub>O<sub>19</sub> (x = 0.0, 0.1) ceramics were evaluated using the following relations :

The bulk density (d<sub>B</sub>) of all samples was measured by the Archimedes principle according to [29] :

$$d_B = \left( \frac{w_{\text{air}}}{\text{Weightloss in the liquid}} \right) \times D_t \qquad (3)$$

where w<sub>air</sub> is the weight of the sample in air and D<sub>t</sub> is the density of toluene (0.857 g/cm³). The theoretical X-ray density (d<sub>x</sub>) was calculated using the crystallographic relation [29,30] :

$$d_x = \left( \frac{2M}{N_A V} \right) \qquad (4)$$

where M is the molecular weight, N<sub>A</sub> is Avogadro's number and V is the unit cell volume. The apparent porosity was calculated using [30] :
$$ \%P = \left(1 - \frac{d_B}{d_x}\right) \times 100 \qquad (5) $$

The bulk density, theoretical density and porosity values are summarized in Table 1. The theoretical densities were found to be 5.139 g/cm³ for x = 0.0 and 5.120 g/cm³ for x = 0.1, while the corresponding experimental bulk densities are 4.653 g/cm³ and 4.831 g/cm³, leading to porosity values of 9.46% and 5.65%, respectively. These values indicate a good densification of the ceramics synthesized by the sol-gel auto-combustion method and sintered at 900 °C, and are consistent with literature reports on M-type hexaferrites.

It is well established that density and porosity strongly influence the physical properties of ferrite ceramics, particularly their dielectric and magnetic behavior. For the present BCNFO hexaferrite samples, the influence of these microstructural parameters has already been discussed in detail in the preceding sections.

The Williamson-Hall (W-H) method was used to further assess crystallite size and lattice strain, attributing XRD peak broadening to combined size and strain contributions. The (W-H) is given by [31]:

$$ \beta_{hkl} = \beta_{\text{size}} + \beta_{\text{strain}} \qquad (6) $$

where $\beta_{\text{strain}}$ represents the broadening due to lattice strain. Accordingly, Equation (4) can be rewritten as [31]:

$$ \beta_{hkl} \cos \theta = \frac{k\lambda}{D} + 4 \varepsilon \sin \theta \qquad (7) $$

Where $\varepsilon$ is the strain.

The Williamson-Hall plots shown in Fig. 2(a–b) represent $\beta_{hkl} \cos(\theta)$ on the Y-axis and $4\varepsilon \sin(\theta)$ on the X-axis. The crystallite size ($D_{WH}$) is determined from the Y-axis intercept after performing a linear fit, while the strain within the samples is derived from the slope of the plot. For the sample with x = 0.0, the average crystallite size was measured at 67 nm, whereas in the substituted sample, this size slightly decreased, to 43 nm for x = 0.1. It is noted that crystallite sizes obtained via Scherrer's method are smaller than those calculated by the Williamson-Hall approach, which is attributed to the latter's correction for lattice strain ($\varepsilon$) in the analysis [32,33].
## 3.2. Transmission electron microscopy

TEM images (Figs. 3(a–b)) show that the samples exhibit a well-defined hexagonal morphology. The average grain size was estimated from the TEM images using ImageJ software. For each composition, more than 100 particles were randomly measured from different regions of the micrographs to ensure a reliable particle size distribution. The calculated particle sizes range from approximately 38 to 58 nm, which lie between the crystallite sizes obtained from the Scherrer and Williamson-Hall methods.

The crystallite sizes estimated from the Scherrer and Williamson-Hall methods differ from the particle sizes observed by TEM due to the fundamental differences between these techniques. XRD-based methods provide the size of coherently diffracting domains, whereas TEM reflects the actual particle size, which may consist of one or several crystallites and may also include agglomeration effects. The Williamson-Hall method, by considering both size and lattice strain contributions, offers a more realistic estimation of the intrinsic crystallite size in strained systems, while TEM observations confirm the nanometric nature of the synthesized particles.

## 3.3. Raman Spectroscopy

Raman spectroscopy was employed to confirm the structural results derived from the XRD analysis of the BCNFO samples at 300 K, as illustrated in Fig. 4(a). To improve the reliability of Raman mode assignments, the experimental spectra were further analyzed using peak fitting and deconvolution based on a multi-Lorentzian function after baseline correction.

Figures 4(b) and 4(c) display the deconvoluted Raman spectra for $x = 0.0$ and $x = 0.1$, respectively, where the experimental data, fitted curves, and individual vibrational modes are clearly resolved. A detailed examination of the spectra reveals the presence of twelve active Raman modes in BCNFO samples. According to group theory analysis, the 189 optical phonon modes in a single unit cell of barium hexaferrite can be described by $D_{6h}$ symmetry [34,35]. For M-type hexaferrites with a hexagonal structure (space group P63/mmc) at 300 K, 42 Raman-active modes are theoretically expected, including $11A_1g$, 14 $E_1g$, and 17 $E_2g$ modes. The Raman mode positions were extracted from Lorentzian peak fitting of the experimental spectra shown in Fig. 4(b) and (c). The symmetry assignments and corresponding site occupancies of the observed peaks are summarized in Table 2 and are consistent with previously reported Raman studies on BaM hexaferrites [35,36]. As shown in Fig. 4(b), taking the $x = 0.0$ sample as a representative example, the most intense Raman band observed at $686 \text{ cm}^{-1}$ corresponds to the $A_1g$-symmetric vibrations of the bi-pyramidal (2b) sites. Additional modes
located at approximately 612, 523, 466, 407, 306, 284 cm⁻¹ originate from vibrations of the 4f₂, 12k, 2a, 12k, (2a + 12k), and octahedral sites, respectively. The Raman mode shifts and slight broadening associated with the 12k and 2b sites upon Nd substitution, as revealed by the fitting analysis, indicate that these sites are particularly sensitive to Nd incorporation. Raman-active modes at 174 and 208 cm⁻¹, exhibiting E₁g symmetry, are attributed to vibrations of the entire spinel block [36]. The mode observed near 332 cm⁻¹ arises from O–Fe–O bending vibrations and exhibits E₂g symmetry [36]. The incorporation of Nd³⁺ ions into the BaM lattice induces slight shifts in Raman peak positions which reveals in Fig. 4(c). This behavior is mainly attributed to the substitution of Fe³⁺ ions (0.64 Å) by larger Nd³⁺ ions (0.80 Å), leading to local lattice distortion without altering the hexagonal crystal structure, in agreement with XRD results.

### 3.4. Magnetic properties

BCNFO hexaferrites (x = 0.0, 0.1) were characterized at 300 K, 600 K, 700 K, and 800 K using a Vibrating Sample Magnetometer (VSM) with magnetic fields up to ±10 T to obtain their hysteresis loops.

The resulting magnetization curves are presented in Fig. 5(a-b). Doping with Nd significantly influence the shape and characteristics of the M–H hysteresis loops. In general, undoped M-type hexaferrites (x = 0.0), which exhibit uniaxial magnetic anisotropy, tend to display broader hysteresis loops. BCNFO samples display clear ferromagnetic behavior. The Ms decreases exponentially with increasing temperature from 300 K to 800 K. With increasing thermal fluctuations, the magnetic moments of blocked particles gain enough energy to overcome the magnetic anisotropy barrier. As anticipated, with a rise in temperature, a weaker magnetic field is required to reverse the direction of magnetization. Consequently, the Ms is higher at 300 K than that at 800 K for our samples. With decreasing temperature, the reduction in magnetic Gibbs free energy causes the magnetic spins to lock in the direction of the applied field [37-39].

In a similar way, as the T rises, the thermal energy causes greater fluctuations in the magnetic moment.

Both coercivity and saturation magnetization are seen to decline with rising temperature, as represented by the following equation [40]:

$$ \frac{dH_c}{dT} = \alpha (2 \frac{dK_1}{dT} - H_a \frac{dM_s}{dT}) M_s, \qquad (8) $$
$\alpha$ represent the orientation coefficient. The value of $dH_C/dT$ is strongly dependent on the parameters $dK_1/dT$ and $dM_S/dT$.

Above 700 K, the ferromagnetic M-H loops transition to paramagnetic behavior. The wide, smooth hysteresis loops confirm the materials' hard ferromagnetic character. Furthermore, the Law of Approach to Saturation (LAS) was employed to determine $M_S$, the magnetocrystalline anisotropy constant ($K_{eff}$), the anisotropy field ($H_a$) and the maximum magnetic energy product (BHmax), using the following equation [40,41]:

$$M = M_S \left[ 1 - \frac{A}{H} - \frac{B}{H^2} \right] + \chi H, \qquad (9)$$

In this framework, the constants A and B reflect contributions from the magnetocrystalline anisotropy, intrinsic material characteristics including structural inhomogeneities and the saturation magnetization of the sample.

The magnetization behavior in the high magnetic field region is influenced by the strength of the applied field, while at elevated temperatures, the susceptibility $\chi$ becomes significant. Consequently, the term $A/H$ loses its relevance, leading to a simplified form of the LAS equation as follows [42]:

$$M = M_S \left[ 1 - \frac{B}{H^2} \right]. \qquad (10)$$

The following equations [42,43] were used to calculate the magnetocrystalline anisotropy constant ($K_{eff}$) and the anisotropy field ($H_a$):

$$K_{eff} = \sqrt{\frac{15}{2} M_S B}. \qquad (11)$$

$$H_a = \frac{2k_{eff}}{M_S}. \qquad (12)$$

Figure 6 shows the LAS fitting results for all samples, where the experimental data are represented by open circles and the fitted curves by solid lines. The corresponding magnetic parameters, including remanent magnetization ($M_r$), coercivity ($H_C$), and saturation magnetization ($M_S$), are summarized in Table 3.

It is clearly observed from the LAS analysis and Table 3 that Nd doping leads to a gradual decrease in $M_S$, $M_r$, and $H_C$.

The saturation magnetization ($M_S$) decreases from 98.87 to 86.62 emu.g$^{-1}$ as $x$ increases from 0.0 to 0.1. To explain this decrease, it is well established in the literature that the saturation
magnetization of M-type hexaferrites is governed by the number of uncompensated electron spins in the magnetic sublattices. According to the classical Gorter model [44,45], the twelve Fe³⁺ ions are distributed among five crystallographic sites: three octahedral sites (2a, 12k, and 4f₂), one tetrahedral site (4f₁), and one trigonal bipyramidal site (2b). The Fe³⁺ ions occupying the 12k, 2a, and 2b sites have their spins aligned parallel (spin-up), whereas those located at the 4f₁ and 4f₂ sites possess antiparallel (spin-down) orientations [46-48]. Consequently, the magnetic behavior strongly depends on the preferential occupation of dopant ions among these sublattices. Substitution at spin-up sites generally leads to a reduction in magnetization, whereas substitution at spin-down sites may result in an enhancement of the net magnetic moment [49,50].

Although Nd³⁺ substitution is often assumed to occur at Fe³⁺ sites in M-type hexaferrites, it should be emphasized that Nd³⁺ has a relatively large ionic radius and may also partially occupy the alkaline-earth (Ba²⁺/Ca²⁺) sites [51]. Substitution at these sites requires charge compensation mechanisms, such as the formation of oxygen vacancies and/or partial reduction of Fe³⁺ to Fe²⁺. Such defect-related effects have been widely reported in rare-earth-substituted hexaferrites and can significantly influence magnetic properties. Therefore, the observed reduction in magnetization in the present samples is likely associated with a combined effect of disturbed sublattice balance, defect formation, and possible mixed-site occupation of Nd³⁺ ions. In contrast, the LAS analysis reveals an enhancement of the effective magnetocrystalline anisotropy constant (Kₐ₇₅) from 3.85 × 10⁵ to 4.02 × 10⁵ J·m⁻³ and of the anisotropy field (Hₐ) from 14.7 to 17.5 kOe as x increases from 0.0 to 0.1, indicating that Nd incorporation strengthens the intrinsic magnetocrystalline anisotropy of the BCNFO system.

This behavior can be attributed to the larger ionic radius of Nd³⁺, which induces lattice distortion and local strain, as evidenced by structural analysis. These distortions modify the crystal-field environment at the anisotropy-sensitive 2b and 12k sites, which play a dominant role in the uniaxial anisotropy of M-type hexaferrites.

Notably, the refined x and z coordinates of the 12k site exhibit a clear shift when x = 0.1, suggesting that Nd³⁺ ions are likely to preferentially substitute Fe³⁺ at the octahedral 12k site. Therefore, the preferential incorporation of Nd³⁺ at this site provides a structural explanation for the experimentally observed enhancement of the magnetocrystalline anisotropy constant. The Raman mode shifts associated with vibrations of the 12k and 2b sites further support the sensitivity of these sites to Nd incorporation.
In addition, the strong spin-orbit coupling associated with the localized Nd³⁺ 4f electrons contributes to the enhancement of anisotropy energy through indirect 4f-3d interactions with neighboring Fe³⁺ ions.

The magnetic moment per formula unit (nB, in µB) was calculated using the relation reported in [52]:

$$n_B = \frac{M * M_S}{5585} \qquad (13)$$

where M is the molecular weight of the sample and Ms is the saturation magnetization.

As shown in Table 3, the magnetic moment per formula unit decreases with increasing Nd content. This reduction follows the decrease in Ms and can be attributed to the disturbance of the magnetic sublattice balance, weakening of Fe³⁺-O-Fe³⁺ superexchange interactions, and the possible replacement or dilution of Fe³⁺ ions by Nd³⁺ ions, whose magnetic contribution is weakly coupled with temperature.

### 3.4.1 Zero-Field-Cooled and Field-Cooled Magnetization Behavior

To investigate the influence of Nd doping on the magnetic behavior of Ba₀.₅Ca₀.₅NdₓFe₁₂-xO₁₉, samples with x = 0.0 and 0.1 were examined. The magnetization as a function of temperature (M-T) was measured between 300 and 900 K under an applied magnetic field of 0.05 T, using both zero-field-cooled (ZFC) and field-cooled (FC) protocols, as shown in Figures 7(a-b).

The results reveal that doping with Nd markedly influences the magnetization behavior. Specifically, both FC and ZFC magnetization values show a gradual decline with higher x values compared to the undoped sample (x = 0.0). This finding is consistent with the M-H results. Moreover, the absence of a blocking temperature within the 300-900 K range for all samples confirms their characteristic ferromagnetic behavior, as further supported by the M-H curve. With increasing temperature, the FC magnetization gradually decreases. Similar trends have been reported in [53,54]. Moreover, both samples undergo a paramagnetic-to-ferromagnetic (PM-FM) transition upon cooling, which takes place at the Curie temperature (Tc).

The Tc was determined from the magnetization-temperature (M-T) curves measured under a constant applied magnetic field (0.05 T). Specifically, Tc was identified as the inflection point of the M(T) curve, corresponding to the minimum of the derivative dM/dT, as illustrated in Fig. 8. This method provides a reliable and reproducible determination of the PM-FM transition temperature in M-type hexaferrites and is widely adopted in the literature [55].
A noticeable decline in the Tc is observed, dropping from 729 K for the undoped sample (x = 0.0) to 722 K for the Nd-doped one (x = 0.1), suggesting that the incorporation of Nd ions weakens the magnetic exchange interactions in M-type barium hexaferrites. These Tc values show excellent concordance with previously studies [56,57], confirming the reliability of the present results.

To examine how the magnetic spins are organized and behave within the high-temperature paramagnetic (PM) phase, the inverse magnetic susceptibility $\chi^{-1}(T) = H/M$ was analyzed based on the magnetization-temperature (M-T) data, as presented in Figure 9. In this temperature range, the magnetic susceptibility follows the Curie-Weiss (CW) law [58]:

$$ \chi = \frac{C}{T - \theta_{CW}}. \qquad (14) $$

Here, $\theta_{CW}$ represents the paramagnetic Curie temperature and $C = \frac{N_A \mu_B^2}{3k_B} \mu_{eff}^2$ is the Curie constant.

The obtained Curie-Weiss temperatures ($\theta_{CW}$) were found to be slightly lower than the corresponding Curie temperatures (Tc) for both samples ($\theta_{CW} = 700$ K and Tc = 729 K for x = 0.0; $\theta_{CW} = 690$ K and Tc = 722 K for x = 0.1). This difference ($\theta_{CW} < T_c$) suggests the presence of short-range spin fluctuations above Tc and indicates that the magnetic interactions in the paramagnetic region deviate slightly from the ideal Curie-Weiss behavior. Such a behavior is commonly observed in hexaferrites, where competing superexchange interactions between different magnetic sublattices (e.g., Fe$^{3+}$ in 12k, 2a, and 4f sites) and partial substitution of Fe$^{3+}$ by Nd$^{3+}$ ions lead to a weakening of the overall exchange coupling. The reduction in $\theta_{CW}$ and The observed decrease in Tc with doping by Nd ions provides additional evidence that substituting highly magnetic Fe$^{3+}$ ions Fe$^{3+}$ 5 ≈ µB (spin pur S = 5/2) with the less magnetic Nd$^{3+}$ ≈ 3.27 µB ions weakens the overall magnetic interactions.

The samples' isothermal magnetization (M-H) behavior was investigated over the temperature range of 600-900 K, with an applied magnetic field varying from -10 T to +10 T as illustrated in Figures 10(a-b). For all samples, the magnetization rises sharply at low fields (below 2 T) and then gradually saturates at higher fields. At 600 K and 10 T, the magnetization reaches an almost steady, quasi-saturated state. The corresponding Arrott plots (M$^2$ versus µ$_0$H/M), shown in Figures 11(a-b) for BCNFO (x = 0.0 and 0.1), the high-field region exhibits a positive slope, indicating that the samples undergo a second-order magnetic phase transition [59,60].
### 3.4.2 Magnetocaloric effect properties MCE

To explore the applicability of Ba0.5Ca0.5NdₓFe12-xO19, (x = 0.0 and 0.1) hexaferrites in magnetic refrigeration, the temperature-dependent magnetic entropy change (–ΔS<sub>M</sub>(T, H)) was computed from the isothermal magnetization data using Maxwell’s equations [61,62] :

$$ \Delta S_M(T, \mu_0 H) = S_M(T, \mu_0 H) - S_M(T, 0) = \int_0^{\mu_0 H_{\max}} \left( \frac{\partial S}{\partial \mu_0 H} \right)_T \mu_0 dH \quad (15) $$

The variation of magnetic entropy change (ΔS<sub>M</sub>) was determined by increasing the magnetic field from zero to 10 T.

$$ \left(\frac{\partial M}{\partial T}\right)_{\mu_0 H} = \left(\frac{\partial S}{\partial \mu_0 H}\right)_T \quad (16) $$

The peaks observed in the samples coincide with those obtained experimentally near the transition temperature, $\left(\frac{\partial M}{\partial T}\right)_{(\mu_0 H)}$ as determined from the M(T) curves under an applied magnetic field (H). The magnetic entropy change was calculated using the following relation [63] :

$$ \Delta S_M \left( \frac{T_1 + T_2}{2} \right) = \left( \frac{1}{T_1 - T_2} \right) \left[ \int_0^{\mu_0 H_{\max}} M(T_2, \mu_0 H) \mu_0 dH - \int_0^{\mu_0 H_{\max}} M(T_1, \mu_0 H) \mu_0 dH \right] \quad (17) $$

Here, T₁ and T₂ represent the temperatures of the hot and cold reservoirs, respectively.

Figure 12(a–b) shows the temperature dependence of –ΔS<sub>M</sub> under different applied magnetic fields, indicating that –ΔS<sub>M</sub> increases with increasing external magnetic field strength. The reduction in the maximum magnetic entropy change from 1.59 to 1.44 J kg<sup>-1</sup>.K<sup>-1</sup> upon Nd substitution is attributed to the weakening of magnetic exchange interactions caused by Nd incorporation. Substituting Fe<sup>3+</sup> ions (with a magnetic moment of 5 µB) by Nd<sup>3+</sup> ions (with a lower effective magnetic contribution) weakens the Fe–O–Fe superexchange interactions, reducing the overall magnetization response to the applied magnetic field. Consequently, the field-induced change in magnetic entropy decreases with increasing Nd content.

The Relative Cooling Power (RCP) is a key parameter for evaluating magnetic refrigerant materials, as it quantifies the amount of heat transferred between the cold and hot reservoirs during an ideal refrigeration cycle and is defined as [64] :

$$ RCP = -\Delta S_M^{\max} \times \delta T_{FWHM} \quad (18) $$

Here, |ΔS<sub>M</sub>|<sup>max</sup> denotes the peak value of the magnetic entropy change, while δT<sub>FWHM</sub> corresponds to the full width at half maximum.
Figure 13 illustrates the variation of RCP with applied magnetic field (μ₀H). An increase in the applied magnetic field leads to higher RCP values for x = 0.1, which can be attributed to the weakening of spin coupling under strong magnetic fields [65]. Moreover, the RCP shows a marked increase with Nd content, mainly due to the broadening of the entropy peak, even although |Δ*S<sub>M</sub>|<sup>max</sup> does not increase significantly. A broader transition enhances magnetic refrigeration efficiency by enabling heat exchange over a wider temperature range. Additionally, Nd substitution effectively reduces magnetic anisotropy, facilitating easier spin reorientation under an applied magnetic field and thereby enhancing the magnetocaloric performance.

The observed evolution of the magnetocaloric response is therefore a direct consequence of the Nd-induced magnetic modifications discussed in the previous section. The weakening of Fe-O-Fe superexchange interactions combined with the effective reduction of magnetic anisotropy leads to a broader paramagnetic-ferromagnetic transition. Although this slightly reduces |Δ*S<sub>M</sub>|<sup>max</sup>, it significantly enhances the relative cooling power by widening the operational temperature range. This clear magnetic-magnetocaloric correlation highlights the intrinsic coupling between structural distortion, magnetic interactions, and thermal entropy change in the BCNFO system.

To facilitate a comprehensive comparison of the magnetocaloric performance, Table 4 summarizes the magnetic entropy change (Δ*S<sub>M</sub>*) and relative cooling power (RCP) of the Ba<sub>0.5</sub>Ca<sub>0.5</sub>Fe<sub>12-x</sub>Nd<sub>x</sub>O<sub>19</sub> (x = 0.0 and 0.1) samples in comparison with previously reported Ferrite- based magnetocaloric materials [66-70]. Although the maximum Δ*S<sub>M</sub>* values obtained in the present study are relatively modest compared to those of giant magnetocaloric intermetallic compounds [71], the Nd-doped sample exhibits a clear enhancement in RCP relative to the undoped composition and several previously reported ferrite systems. This improvement is mainly attributed to the broadened operating temperature span δT<sub>FWHM</sub> associated with the second-order paramagnetic-ferromagnetic transition. It is worth noting that the comparison involves different applied magnetic fields (ΔH), which directly influence the absolute values of Δ*S<sub>M</sub>* and RCP. Nevertheless, the obtained RCP values confirm the competitive magnetocaloric performance of the present materials within the ferrite family. Furthermore, compared to benchmark rare-earth-based magnetocaloric materials, Nd-substituted M-type hexaferrites offer several practical advantages, including low material cost, high chemical and thermal stability, corrosion resistance, and environmental friendliness. These features, combined with their magnetocaloric response near and above room temperature,
highlight the potential of BCNFO hexaferrites for magnetic refrigeration applications in medium- and high-temperature regimes.

The change in heat capacity, ΔCp, induced by an applied magnetic field from 1 to 10 T is another key parameter for evaluating magnetic refrigeration and can be determined using [71]:

$$ \Delta C_p(T, \mu_0 H) = C_p(T, \mu_0 H) - C_p(T, 0) = T \frac{\partial \Delta S_M(T, \mu_0 H)}{\partial T} \quad (19) $$

Figures 14(a–b) show that ΔCp undergoes a sharp change from negative to positive around Tc. The heating or cooling performance of a magnetic refrigerator is strongly affected by the sign of ΔCp. Furthermore, ΔCp generally increases as the external magnetic field becomes stronger. These findings are consistent with recent reports [70,71].

### 3.5 Frequency- and temperature-dependent dielectric studies

Hexaferrites are semiconducting materials characterized by a relatively wide band gap, which originates from the nature of the Fe³⁺–O bonds. Chemical substitution can significantly modify the electrical properties of hexaferrite solid solutions. In the case of heterovalent substitution, such as the partial replacement of Fe³⁺ by Nd³⁺, charge compensation mechanisms may lead to variations in the Fe oxidation state, thereby influencing the polarization and dielectric behavior of the material.

In Nd-doped hexaferrites, the substitution of Fe³⁺ by Nd³⁺ can induce charge compensation through the formation of Fe²⁺ ions. The coexistence of Fe²⁺ and Fe³⁺ ions promotes electron hopping between neighboring sites, thus enhancing the electrical conductivity of the material.

In this context, the dielectric response of Ba₀.₅Ca₀.₅NdₓFe₁₂-xO₁₉ (x=0.0, 0.1, BCNFO) was investigated in the temperature range of 300–700 K and over a frequency range from 10 Hz to 10⁶ Hz. A central aspect of this study is the analysis of the complex permittivity, which characterizes the response of the material to an applied electromagnetic field.

The dielectric constant (ε') of ferrite materials is affected by several structural and microstructural factors and was calculated using the following equation [72, 73]:

$$ \epsilon' = \frac{C_p d}{\epsilon_0 A} \quad (20) $$

Here, C is the capacitance of the pellet (in farads), d is its thickness (in meters), A is the cross-sectional area of the pellet’s flat surface, and ε₀ denotes the permittivity of free space.

The dielectric loss tangent (tan δ) is determined using the relation tan δ = 1/(2πfRₚ Cₚ) where δ is the loss angle, f denotes the frequency, Rₚ is the equivalent parallel resistance, and Cₚ is the equivalent parallel capacitance.
Figure 15(a–b) illustrates the frequency dependence of the dielectric constant (ε') for BCNFO samples. It is observed that ε' decreases as frequency increases and becomes almost constant at higher frequencies, reflecting the typical dielectric behavior of ferrite materials [72, 73].

This behavior can also be interpreted using Koops' theory [74], which relies on the Maxwell-Wagner model of interfacial polarization [75, 76]. As per this theory, ferrite materials consist of well-conducting grains that are separated by thin layers of poorly conducting grain boundaries. At low frequencies, the dielectric response is mainly governed by the grain boundaries, whereas at higher frequencies, the contribution from the bulk grains becomes predominant [76,77]. When an AC electric field is applied, electron hopping occurs between Fe²⁺ and Fe³⁺ ions. However, as these electrons encounter the insulating grain boundaries, their movement is hindered, leading to the formation of space charge polarization. Space charge polarization can also arise from oxygen vacancies formed during the sintering of the pellets [76]. Additionally, the presence of free charge carriers at the grain boundaries contributes to this polarization. At low frequencies, the induced dipole moments or electron hopping processes remain well synchronized with the applied alternating electric field. Therefore, a higher value of ε' is observed at low frequencies. As the frequency increases, electrons oscillate more rapidly, facing greater resistance to their motion within the material. This reduces the accumulation of charge carriers at the grain boundaries, leading to a decline in permittivity (dielectric constant). Consequently, the pronounced dielectric dispersion at low frequencies which tends to stabilize at higher frequencies indicates the presence of induced electric dipole moments under the applied electric field. It was found that in BCNFO samples, Nd doping (x = 0.1) leads to a significant increase in the dielectric constant compared to x = 0.0. This effect can be attributed to the substitution of Nd³⁺ ions at the pseudotetrahedral (2b) and tetrahedral (4f₁) sites. The incorporation of Nd³⁺ at these positions may cause displacement of Fe³⁺ ions from tetrahedral to octahedral sites. This migration of Fe³⁺ ions enhances electron hopping, leading to an increase in interfacial polarization.

The increase in ε' with Nd³⁺ ions can be explained in terms of ionic polarizability. Since the ionic polarizability of Nd³⁺ is higher than that of Fe³⁺, substituting Nd³⁺ for Fe³⁺ leads to an increase in ε' up to a certain concentration.

As reported, materials with high permittivity are crucial for reducing the size of microwave devices [77]. In this study, Ba₀.₅Ca₀.₅Fe₁₂O₁₉ compound substituted with Nd³⁺ ions exhibit higher dielectric constants than the undoped Ba₀.₅Ca₀.₅Fe₁₂O₁₉. Therefore, these Nd³⁺-doped
hexaferrites can be considered promising candidates for the miniaturization of microwave devices.

Dielectric loss refers to the energy dissipated by a material due to the motion of ions under an external electric field. Structural defects, impurities, and other imperfections arising from the synthesis method and calcination temperature also contribute to this loss [76,77]. According to Koops' theory, grain boundaries provide higher resistance at low frequencies, which increases the energy required for electron hopping and results in higher dielectric loss. At higher frequencies, the resistance of the grains is low, allowing ions to move more freely between sites, and consequently, the dielectric loss factor decreases [77,78].

In addition to intrinsic charge transport mechanisms, microstructural parameters such as density and porosity can also influence the dielectric response of ferrite ceramics. As reported in Table 1, the porosity decreases from 9.46% for x = 0.0 to 5.65% for x = 0.1, indicating improved densification upon Nd substitution. Lower porosity reduces the presence of air gaps and weakly connected grain boundaries, which are known to act as sources of interfacial polarization and dielectric loss. Consequently, the relatively low porosity of the present samples suggests that the observed dielectric dispersion and dielectric loss are mainly governed by intrinsic mechanisms, such as Maxwell-Wagner interfacial polarization and Fe²⁺/Fe³⁺ electron hopping, rather than by extrinsic microstructural effects. The slightly lower porosity of the Nd-doped sample may also contribute marginally to the improved dielectric response at low frequencies. Overall, the close density values between the compositions confirm that the evolution of the dielectric properties is primarily related to Nd-induced structural modifications rather than to differences in densification.

The dielectric loss behavior of Ba₀.₅Ca₀.₅Fe₁₂-xNdₓO₁₉ (x = 0.0 and 0.1) samples was analyzed using the imaginary part of the complex permittivity (ε''), which represents dielectric energy dissipation, and the loss tangent (tan δ), related through the expression [77–79].

$$ \tan \delta = \frac{\varepsilon''}{\varepsilon'} \qquad (21) $$

Figures 16(a–b) present the frequency-dependent variation of ε'' at different temperatures. At low frequencies, ε'' exhibits high values due to space-charge polarization and charge carrier accumulation at grain boundaries, resulting in increased energy dissipation. As the frequency increases, ε'' decreases significantly, indicating that electron hopping between Fe²⁺ and Fe³⁺ ions cannot follow the rapidly oscillating electric field, leading to reduced dielectric losses. The temperature dependence of dielectric loss reveals a clear frequency–temperature correlation. At
low frequencies, ε'' increases with temperature due to thermally activated charge carrier hopping. This behavior can be described using Debye's relaxation model [80,81] :

$$ \mathrm{Tan}\delta = \frac{\omega}{1 + \omega^2\tau^2} $$

where ω is the angular frequency and τ is the relaxation time. With increasing temperature, the relaxation time decreases, leading to enhanced dielectric loss. However, at higher frequencies, ε'' and tan δ remain relatively low and weakly temperature-dependent, indicating dielectric behavior under operational conditions. Such low dielectric loss at high frequencies is desirable for high-frequency and microwave device applications.

The dielectric behavior observed in the present samples is consistent with recent studies on substituted hexaferrites. For instance, investigations on Ba0.5Ca0.5DyxFe12-xO19 systems [82] have shown that rare-earth substitution significantly influences dielectric relaxation by modifying charge carrier dynamics and microstructural characteristics. Similarly, impedance spectroscopy and dielectric analyses of Cr3+ and Al3+ co-substituted SrFe12-2xCrxAlxO19 (x = 0.0 and 0.4) [83] revealed enhanced electrical conductivity, reduced dielectric loss tangent, and improved polarization mechanisms. These findings strongly support our interpretation that the dielectric response of BCNFO samples is closely linked to Nd-induced structural modifications and localized charge effects.

## 3.6 Impedance spectroscopy

The electrical behavior of Ba0.5Ca0.5DyxFe12-xO19 (x = 0.0 and 0.1) was studied using complex impedance spectroscopy (CIS). To obtain a comprehensive understanding of the material, it is essential to convert the dielectric and electrical data into various formalisms and analyze them accordingly. Figure 17(a-b) The variations of the real part of the impedance (Z') as a function of frequency and temperature, measured for the x = 0 and x = 0.1 samples, exhibit a typical polycrystalline ferrite behavior governed by the distinct contributions of grains and grain boundaries, in agreement with the Maxwell-Wagner and Koops models. At low frequencies, Z' reaches high values and decreases sharply with increasing frequency; beyond a transition frequency (~10²–10³ Hz in our measurements), the curves at different temperatures converge toward a nearly zero value.

This behavior can be explained as follows: at low frequencies, free charges (electrons involved in Fe2+↔Fe3+ hopping and thermal carriers) have enough time to accumulate at the poorly conducting grain boundaries, leading to strong space-charge polarization and thus high
impedance [84,85]. As the frequency increases, the induced dipoles and carriers cannot follow
the alternating field accumulation at the boundaries decreases, and the response becomes
dominated by the conductivity of the grain cores, resulting in the drop of Z' and stabilization at
high frequencies. The temperature dependence confirms a thermally activated conduction:
increasing T reduces Z' at all frequencies, indicating enhanced carrier mobility and/or a higher
number of free carriers (reduction of the resistance of grains and grain boundaries). Moreover,
the dispersion region shifts to higher frequencies as temperature rises, showing that the
relaxation processes are accelerated by heat, as expected for electronic hopping mechanisms.
Comparison between x = 0 and x = 0.1: the introduction of Nd³⁺ (x = 0.1) notably increases the
Z' values at low frequencies compared to the undoped sample. This suggests that Nd doping
enhances the effective resistance, likely by disturbing the Fe²⁺-Fe³⁺ hopping paths (reducing
the number of active sites/carriers) or by modifying the microstructure (more resistive grain
boundaries, increased defects/oxide variations), which promotes stronger space-charge
polarization. In other words, Nd tends to reinforce the contribution of grain boundaries to the
overall impedance. In summary, the Z'(f, T) measurements confirm that the dielectric/electrical
response of Ba₀.5Ca₀.5NdₓFe₁₂-xO₁₉ is controlled by space-charge polarization at grain
boundaries at low frequency and by grain conductivity at high frequency; the incorporation of
Nd increases the resistive contribution of grain boundaries and slows down the electronic
hopping processes, as evidenced by higher Z' values and enhanced dispersion for x = 0.1.

**4. Conclusion**

A simple and cost-effective sol–gel auto-combustion route followed by sintering at 900 °C was successfully employed to synthesize Ba₀.₅Ca₀.₅NdₓFe₁₂-ₓO₁₉ (x = 0.0, 0.1) M-type hexaferrite nanoparticles. X-ray diffraction results confirmed the formation of a single-phase magnetoplumbite hexagonal structure for all compositions. The average crystallite size, estimated using the Debye–Scherrer method, ranged from 38.07 to 43.90 nm, indicating nanoscale dimensions suitable for magnetic applications.

Magnetic measurements revealed that Nd substitution (x = 0.1) leads to a reduction in both
saturation magnetization and coercivity, which is attributed to the modification of Fe–O–Fe
superexchange interactions and microstructural effects. The combination of reduced coercivity
and nanosized crystallites suggests that these materials are promising for magnetic recording
media.

Dielectric studies showed that the dielectric constant decreases with increasing frequency,
consistent with Maxwell–Wagner interfacial polarization and space-charge effects. Impedance
analysis further demonstrated that grain boundaries play a dominant role in governing the electrical response of the samples.

Overall, the results indicate that Nd-doped Ba-Ca M-type hexaferrites exhibit a balanced combination of structural stability, tunable magnetic behavior, and favorable dielectric properties, making them attractive candidates for multifunctional ceramic applications.

## Acknowledgments

This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-DDRSP2602).
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**Table 1:** Unit cell parameters, cation coordinates and standard reliability factors and bulk density ($d_B$), x-ray density ($d_x$), porosity (%)(P) for the solid solutions of Ba$_{0.5}$Ca$_{0.5}$Nd$_x$Fe$_{12-x}$O$_{19}$ $x = 0$ and 0.1 hexaferrites at 300 K in frame of SG P6$_3$/mmc (No 194) obtained by Rietveld method in Fullprof compute program.

**Table 2.** Raman characteristics with attributed vibrational modes of Ba$_{0.5}$Ca$_{0.5}$Nd$_x$Fe$_{12-x}$O$_{19}$ ($x = 0.0, 0.1$).

**Table 3:** Data of $M_s$, $K_{eff}$ and $H_c$ values for Ba$_{0.5}$Ca$_{0.5}$Nd$_x$Fe$_{12-x}$O$_{19}$ ($x = 0.0, 0.1$)

**Table 4:** Comparison between MCE properties of Ba$_{0.5}$Ca$_{0.5}$Nd$_x$Fe$_{12-x}$O$_{19}$ ($x = 0.0, 0.1$) samples compared with other ferrite materials.
<table><caption>Table 1</caption><thead><tr><th rowspan="2">No</th><th rowspan="2"></th><th rowspan="2"></th><th colspan="2">x</th></tr><tr><th>0</th><th>0.1</th></tr></thead><tbody><tr><td>1.</td><td rowspan="3">unit cell parameters</td><td>a, (Å)</td><td>5.8436(4)</td><td>5.9337(8)</td></tr><tr><td>2.</td><td>c, (Å)</td><td>23.1454(5)</td><td>23.4265(7)</td></tr><tr><td>3.</td><td>V, (Å<sup>3</sup>)</td><td>686.81(7)</td><td>695.09(1)</td></tr><tr><td>4.</td><td rowspan="15">cation coordinates</td><td>Ba/Ca, x</td><td>0.66671(2)</td><td>0.66671(3)</td></tr><tr><td>5.</td><td>Fe<sub>3</sub>/Nd<sup>3+</sup> (4f<sub>iv</sub>), z</td><td>0.0255(6)</td><td>0.0267(1)</td></tr><tr><td>6.</td><td>Fe<sub>4</sub>/Nd<sup>3+</sup>(4f<sub>v1</sub>), z</td><td>0.1909(7)</td><td>0.1885(4)</td></tr><tr><td>7.</td><td>Fe<sub>5</sub>/Nd<sup>3+</sup> (12k), x</td><td>0.1659(4)</td><td>0.1678(5)</td></tr><tr><td>8.</td><td>Fe<sub>5</sub>/Nd<sup>3+</sup> (12k), z</td><td>0.6083(7)</td><td>0.6082(9)</td></tr><tr><td>9.</td><td>O<sub>1</sub> (4e), z</td><td>0.1358(12)</td><td>0.1552(17)</td></tr><tr><td>10.</td><td>O<sub>2</sub> (4f), z</td><td>0.5555(7)</td><td>0.5574(8)</td></tr><tr><td>11.</td><td>O<sub>3</sub> (6h), x</td><td>0.1744(9)</td><td>0.1822(3)</td></tr><tr><td>12.</td><td>O<sub>4</sub> (12k), x</td><td>0.1537(1)</td><td>0.1573(7)</td></tr><tr><td>13.</td><td>O<sub>4</sub> (12k), z</td><td>0.0532(6)</td><td>0.05495(4)</td></tr><tr><td>14.</td><td>O<sub>5</sub> (12k), x</td><td>0.5163(10)</td><td>0.5077(7)</td></tr><tr><td>15.</td><td>O<sub>5</sub> (12k), z</td><td>0.1257(4)</td><td>0.1542(3)</td></tr><tr><td>16.</td><td rowspan="4">reliability factors</td><td>R<sub>wp</sub>, %</td><td>8.3(3)</td><td>7.9(5)</td></tr><tr><td>17.</td><td>R<sub>p</sub>, %</td><td>10.04(2)</td><td>9.37(2)</td></tr><tr><td>18.</td><td>R<sub>f</sub>, %</td><td>8.28(3)</td><td>7.67(1)</td></tr><tr><td>19.</td><td>χ<sup>2</sup></td><td>1.65(4)</td><td>1.42(3)</td></tr><tr><td>20.</td><td rowspan="3">bulk density<br>x-ray density and<br>porosity</td><td>(d<sub>B</sub>) (g/cm<sup>3</sup>)</td><td>4.653(5)</td><td>4.831(6)</td></tr><tr><td>21.</td><td>(d<sub>x</sub>) (g/cm<sup>3</sup>)</td><td>5.139(2)</td><td>5.120(2)</td></tr><tr><td>22.</td><td>(P)%</td><td>9.46(1)</td><td>5.65(4)</td></tr></tbody></table>
<table><caption>Table 2</caption><thead><tr><th>Raman active mode (cm<sup>-1</sup>)</th><th>Symmetry</th><th>Mode assignment</th><th>References</th></tr></thead><tbody><tr><td>x = 0.0</td><td>174</td><td>E<sub>1g</sub> Whole spinel block</td><td>[35,36]</td></tr><tr><td>x = 0.1</td><td>178</td><td>E<sub>1g</sub> Whole spinel block</td><td>[35,36]</td></tr><tr><td>x = 0.0</td><td>206</td><td>E<sub>1g</sub> Dominant octahedral (12k) site</td><td>[35,36]</td></tr><tr><td>x = 0.1</td><td>208</td><td>E<sub>1g</sub> Dominant octahedral (12k) site</td><td>[35,36]</td></tr><tr><td>x = 0.0</td><td>283</td><td>E<sub>1g</sub> Dominant octahedral (12k) site</td><td>[35,36]</td></tr><tr><td>x = 0.1</td><td>284</td><td>E<sub>1g</sub> Dominant octahedral (12k) site</td><td>[35,36]</td></tr><tr><td>x = 0.0</td><td>306</td><td>E<sub>1g</sub> octahedral (12k) site</td><td>[35,36]</td></tr><tr><td>x = 0.1</td><td>308</td><td>E<sub>1g</sub> octahedral (12k) site</td><td>[35,36]</td></tr><tr><td>x = 0.0</td><td>332</td><td>E<sub>2g</sub> octahedral (12k) site</td><td>[35,36]</td></tr><tr><td>x = 0.1</td><td>334</td><td>E<sub>2g</sub> octahedral (12k) site</td><td>[35,36]</td></tr><tr><td>x = 0.0</td><td>407</td><td>A<sub>1g</sub> Dominant octahedral (12k) site</td><td>[35,36]</td></tr><tr><td>x = 0.1</td><td>408</td><td>A<sub>1g</sub> Dominant octahedral (12k) site</td><td>[35,36]</td></tr><tr><td>x = 0.0</td><td>466</td><td>A<sub>1g</sub> octahedral (2a) site</td><td>[35,36]</td></tr><tr><td>x = 0.1</td><td>467</td><td>A<sub>1g</sub> octahedral (2a) site</td><td>[35,36]</td></tr><tr><td>x = 0.0</td><td>523</td><td>E<sub>1g</sub> Dominant octahedral (12k) site</td><td>[35,36]</td></tr><tr><td>x = 0.1</td><td>524</td><td>E<sub>1g</sub> Dominant octahedral (12k) site</td><td>[35,36]</td></tr><tr><td>x = 0.0</td><td>612</td><td>A<sub>1g</sub> octahedral (4f2) site</td><td>[35,36]</td></tr><tr><td>x = 0.1</td><td>613</td><td>A<sub>1g</sub> octahedral (4f2) site</td><td>[35,36]</td></tr><tr><td>x = 0.0</td><td>685</td><td>A<sub>1g</sub> Bipyramidal (2b) site</td><td>[35,36]</td></tr><tr><td>x = 0.1</td><td>686</td><td>A<sub>1g</sub> Bipyramidal (2b) site</td><td>[35,36]</td></tr></tbody></table>
<table><caption>Table 3</caption><thead><tr><th>x</th><th>M<sub>s</sub>(emu/g)</th><th>M<sub>r</sub>(emu/g)</th><th>M<sub>r</sub>/M<sub>s</sub></th><th>H<sub>c</sub>(k Oe)</th><th>K<sub>eff</sub> (J/m<sup>3</sup>10<sup>5</sup>)</th><th>H<sub>a</sub> (k Oe)</th></tr></thead><tbody><tr><td>0</td><td>98.87</td><td>24.7</td><td>0.24</td><td>5.71</td><td>3.85</td><td>14.7</td></tr><tr><td>0.1</td><td>86.62</td><td>21.69</td><td>0.25</td><td>4.37</td><td>4.02</td><td>17.5</td></tr></tbody></table>

<table><caption>Table 4</caption><thead><tr><th>Sample name</th><th>ΔH (T)</th><th>ΔS<sub>M</sub><sup>max</sup><br>(J/kg.K)</th><th>RCP<br>(J/kg)</th><th>References</th></tr></thead><tbody><tr><td>Ba<sub>0.5</sub>Ca<sub>0.5</sub>Fe<sub>12</sub>O<sub>19</sub></td><td>10</td><td>1.64</td><td>137</td><td>Present work</td></tr><tr><td>Ba<sub>0.5</sub>Ca<sub>0.5</sub>Nd<sub>0.1</sub>Fe<sub>11.9</sub>O<sub>19</sub></td><td>10</td><td>1.43</td><td>156</td><td>Present work</td></tr><tr><td>Zn<sub>0.2</sub>Ni<sub>0.4</sub>Cu<sub>0.4</sub>Fe<sub>2</sub>O<sub>4</sub></td><td>5</td><td>1.61</td><td>233</td><td>[66]</td></tr><tr><td>Mg<sub>0.6</sub>Cu<sub>0.4</sub>Fe<sub>2</sub>O<sub>4</sub></td><td>5</td><td>1.09</td><td>136</td><td>[67]</td></tr><tr><td>BaFe<sub>11.3</sub>Gd<sub>0.7</sub>O<sub>19</sub></td><td>4.4</td><td>1.36</td><td>107</td><td>[68]</td></tr><tr><td>Ba<sub>0.5</sub>Sr<sub>1.5</sub>Zn<sub>2</sub>(Fe<sub>0.92</sub>Al<sub>0.08</sub>)<sub>12</sub>O<sub>22</sub></td><td>7</td><td>0.8</td><td>187</td><td>[69]</td></tr><tr><td>Sr<sub>2</sub>Co<sub>2</sub>Fe<sub>28</sub>O<sub>46</sub></td><td>5</td><td>1.1</td><td>-</td><td>[70]</td></tr></tbody></table>
Figures list

**Fig.1:** (a) XRD pattern of Ba<sub>0.5</sub>Ca<sub>0.5</sub>Nd<sub>x</sub>Fe<sub>12-x</sub>O<sub>19</sub>, (x=0.0 and 0.1) samples at room temperature;
(b–d) Rietveld refinement of the XRD diagrams of the samples determined using the FullProf
program.

**Fig.2 : (a-b)** The variation of βcosθ versus 4sinθ of of Ba<sub>0.5</sub>Ca<sub>0.5</sub>Nd<sub>x</sub>Fe<sub>12-x</sub>O<sub>19</sub>, (x=0.0 and 0.1) hexaferrites.

**Fig.3:**(a–b) TEM images of Ba<sub>0.5</sub>Ca<sub>0.5</sub>Nd<sub>x</sub>Fe<sub>12-x</sub>O<sub>19</sub>, (x=0.0 and 0.1) M type hexaferrites : a) x = 0.0, b) x = 0.1.

**Fig.4 :** (a-c) Experimental and deconvoluted Raman spectra of of Ba<sub>0.5</sub>Ca<sub>0.5</sub>Nd<sub>x</sub>Fe<sub>12-x</sub>O<sub>19</sub>,
samples: (a) comparison of experimental Raman spectra for x = 0.0 and x = 0.1; (b)
Lorentzian peak fitting and individual vibrational mode contributions for x = 0.0; (c)
Lorentzian peak fitting and individual vibrational mode contributions for x = 0.1.

**Fig.5(a-b):** Hysteresis loops of Ba<sub>0.5</sub>Ca<sub>0.5</sub>Nd<sub>x</sub>Fe<sub>12-x</sub>O<sub>19</sub>, (x=0.0 and 0.1) hexaferrites type M at high temperature 300 K to 800 K.

**Fig.6:** The Law of Approach to Saturation fitting of virgin M-H curves for (x=0.0 and x=0.1).

**Fig.7:** (a–b) Shows temperature-dependences of ZFC and FC magnetization for Ba<sub>0.5</sub>Ca<sub>0.5</sub>Nd<sub>x</sub>Fe<sub>12-x</sub>O<sub>19</sub>, (x=0.0 and 0.1) compounds under a magnetic field of 0.05 T.

**Fig.8:** transition temperature, $T_c$ obtained from the minima of the $dM^2/dT^2$ curve.

**Fig.9:** inverse of susceptibility as a function of temperatures for Ba<sub>0.5</sub>Ca<sub>0.5</sub>Nd<sub>x</sub>Fe<sub>12-x</sub>O<sub>19</sub>, (x=0.0 and 0.1) samples determined in a field of 0.05 T.

**Fig.10:(a-b)** The Isothermal magnetization for all samples at different temperatures.

**Fig.11:** (a–b) Arrott plots for Ba<sub>0.5</sub>Ca<sub>0.5</sub>Nd<sub>x</sub>Fe<sub>12-x</sub>O<sub>19</sub>, (x=0.0 and 0.1) samples at different temperatures above and below T<sub>c</sub>.

**Fig.12:** (a–b) Temperature dependence of magnetic entropy change at magnetic field, H=1-10T.

**Fig.13:** The absolute value of RCP for the samples versus applied field for Ba<sub>0.5</sub>Ca<sub>0.5</sub>Nd<sub>x</sub>Fe<sub>12-x</sub>O<sub>19</sub>, (x=0.0 and 0.1).

**Fig.14:** (a–b) The variation of -ΔC<sub>P</sub> around (T<sub>C</sub>) for all compounds under different magnetic fields.

**Fig. 15. (a–b)** Behavior of dielectric constant as function of temperature for Ba<sub>0.5</sub>Ca<sub>0.5</sub>Nd<sub>x</sub>Fe<sub>12-x</sub>O<sub>19</sub>, (x=0.0 and 0.1) at high frequency measured at different temperatures in the range of 300 K–700 K

**Fig. 16 : (a-b)** Dielectric loss ε'' as a function of frequency for Ba<sub>0.5</sub>Ca<sub>0.5</sub>Nd<sub>x</sub>Fe<sub>12-x</sub>O<sub>19</sub>, (x=0.0 and 0.1).
**Fig. 17. (a–b) Frequency variation of reel part of impedance (Z') of Ba0.5Ca0.5NdₓFe12-ₓO19(x=0.0 and 0.1).**

<figure><img src="image_2.png" /></figure>
<figure><img src="image_3.png" /><figcaption>Fig.1(a)</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Fig.1(b-c)</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig.2(a-b)</figcaption></figure>

<figure><img src="image_6.png" /></figure>
<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>
<figure><img src="image_9.png" /><figcaption>Fig.4(a-c)</figcaption></figure>
<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>
<figure><img src="image_12.png" /><figcaption>Fig.5(a-b)</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Fig.6(a-b)</figcaption></figure>
<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>
<figure><img src="image_16.png" /><figcaption>Fig.8(a-b)</figcaption></figure>

<figure><img src="image_17.png" /></figure>
<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /></figure>
<figure><img src="image_20.png" /><figcaption>Fig.10(a-b)</figcaption></figure>

<figure><img src="image_21.png" /></figure>
<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /></figure>
<figure><img src="image_24.png" /><figcaption>Fig.12</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>Fig.13(a-b)</figcaption></figure>

<figure><img src="image_26.png" /></figure>
<figure><img src="image_27.png" /><figcaption>Fig. 14(a-b)</figcaption></figure>

<figure><img src="image_28.png" /></figure>
<figure><img src="image_29.png" /><figcaption>Fig.15(a-b)</figcaption></figure>

<figure><img src="image_30.png" /></figure>
<figure><img src="image_31.png" /><figcaption>Fig.16(a-b)</figcaption></figure>

<figure><img src="image_32.png" /></figure>
<figure><img src="image_33.png" /><figcaption>Fig.17(a-b)</figcaption></figure>

<figure><img src="image_34.png" /></figure>
**Declaration of interests**

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.