Improvement of Structural, Elastic, and Magnetic Properties of Vanadium-Doped Lithium Ferrite

Abstract

The influence of vanadium substitution on the structure, elastic, mechanical, and magnetic behavior of lithium ferrite (Li_0.5+xV_xFe_2.5−2xO₄; x = 0.00–0.2) was systematically studied. X-ray diffraction (XRD) was used to investigate the crystal structure, and infrared spectroscopy (IR) was used to determine the cation distribution between the two ferrite sublattices, in addition to the elastic and mechanical behavior of Li_0.5+xV_xFe_2.5−2xO₄ ferrites. X-ray analysis revealed a monotonic decrease in lattice parameter from 8.344 Å to 8.320 Å with increasing V⁵⁺ content, confirming lattice contraction and stronger metal–oxygen bonding. Despite a moderate increase in porosity (from 6.9% to 8.9%), the elastic constants C₁₁ and C₁₂ increased, indicating improved stiffness and reduced compressibility. The derived Young’s, bulk, and rigidity moduli rose with the doping of V⁵⁺. Correspondingly, the longitudinal, shear, and mean velocities (V_l, V_s, and V_m) increased. The Debye temperature also showed a linear rise from 705 K to 723 K with V⁵⁺ doping, directly reflecting enhanced lattice stiffness and phonon frequency. Furthermore, both the saturation magnetization (M_S) and the initial permeability (μ_i) increased up to V⁵⁺ concentration x = 0.1 and then decreased. Curie temperature (T_C) decreased with increasing V⁵⁺ concentration, while both the saturation magnetization (M_S) and the initial permeability (μ_i) increased up to V⁵⁺ concentration x = 0.1 and then decreased, while the coercivity (H_C) showed the reverse trend. These results confirm that V⁵⁺ incorporation significantly enhances the Li ferrite, improving its elastic strength, lattice energy, thermal stability, and magnetically controlling properties and making them suitable for a variety of daily uses such as magneto-elastic sensors, high-frequency devices, and applications requiring mechanically robust ferrite materials.

1. Introduction

Ferrites have recently gained significant attention due to their versatile applications in modern technologies such as lithium-ion batteries, communication systems, information storage, microwave devices, biosensors, magnetic fluids, hyperthermia treatment, water purification, and renewable energy systems [1,2,3,4].

Spinel ferrites, with the general formula AB₂O₄, exhibit three structural types: normal, inverse, and mixed spinel ferrite. In normal spinel, divalent cations occupy the tetrahedral (A) sites while Fe³⁺ ions occupy the octahedral [B] sites. In inverse spinel, Fe³⁺ ions occupy the (A) sites, and both Fe³⁺ and divalent cations share the [B] sites. Mixed spinel ferrite exhibits a random distribution of cations over both sublattices. The magnetic and electrical properties of ferrites—saturation magnetization, permeability, Curie temperature, and coercivity—are strongly linked to cation distribution, particle size, crystal structure, and synthesis conditions [5,6,7]. Applications in materials science, telecommunication, biomedicine, and data storage all depend on an understanding of these phenomena. Magnetic characteristics may be optimized for applications by adjusting particle size and morphology. The inverse spinel crystal structure of lithium ferrite, which possesses soft ferromagnetic characteristics, has the A-site for Fe³⁺ ions and the B-site for Li⁺ and the remaining Fe³⁺ ions [8]. Lithium ferrite has many features that make it eligible for participating vigorously in the electronic and technological industries, such as microwave devices and rechargeable Li-ion batteries, as well as circulators, filters, adjustable electronic modulators, and memory cores [9]. Additionally, there is a great focus on doped Li-ferrites for their possible applications in Li batteries as cathode materials [10]. The high values of magnetization, permeability, and Curie temperature are some of the Li-ferrite advantages. Many authors have studied the structural and magnetic properties of Li-ferrite [1,2,4,11]. Furthermore, the modulations of Li-ferrite magnetic properties via doping with different ions (substitution and addition) were studied. For example, Li-Cd ferrite [12], Li-Ni ferrite [13], Li-Co ferrite [13], Li-Ge ferrite [14], and Li-Zn-Ti ferrite [13] were all investigated. On the other hand, pentavalent vanadium was shown to have an interesting effect on the different properties of ferrites. Both magnetic and crystallographic properties of vanadium-substituted Li-Co-Ti ferrite were investigated [15]. Increasing the vanadium content led to an increase in the saturation magnetization and then to some decrease, whereas the coercivity decreased. The effect of V⁵⁺ substitution on the dielectric and magnetic properties of Li-Zn-Ti ferrite showed that the density, the magnetization, and the Curie temperature all decreased with increasing vanadium content. The dielectric behavior increased with the increase in temperature while showing dispersion with the frequency [16,17]. Magnetic and electric studies of only one sample of very small content (only 0.02) of vanadium ions substituted Li-ferrite showed that the lattice parameter, Curie temperature, and the dc electrical resistivity decrease while the initial permeability and the magnetization increase [18]. Moreover, the effect of V₂O₅ additions on the ordering of Li-Zn ferrite was investigated by T. Iimura [19], where the superlattice disappeared by increasing (V⁵⁺) content. Incorporating V₂O₅ was observed to alter the distribution of cations within the spinel structure. V⁵⁺ ions likely integrate into the lattice, impacting the arrangement of Li, Zn, and Fe ions, which in turn modified magnetic properties, like saturation magnetization and Curie temperature. The research showed that minor quantities of V₂O₅ might boost elastic and magnetic performance by refining the cation distribution [19].

From the previous literature survey, it is observed that the previous studies seem insufficient to understand the detailed role of the vanadium ion in changing the elastic magnetic properties of ferrites, especially lithium ferrite. In addition, the linking of these changes with the effect of the vanadium ion on the crystal structure is not clearly discussed in these studies. For these reasons, this work deals, for the first time, with the effect of V⁵⁺ ion as a substitution on the structure, physical, elastic, and magnetic properties of Li-ferrite, aiming to improve its mechanical and magnetic properties. Elastic parameters such as the elastic constants (C₁₁ and C₁₂), bulk modulus (B), Young’s modulus (Y), rigidity modulus (R), and Debye temperature are critical for understanding lattice dynamics, interatomic bonding, and mechanical stability. Since these parameters are directly linked to interatomic force constants and sound velocities, their variation with V⁵⁺ content provides deeper insight into how substitution influences lattice stiffness and energy. Moreover, relatively high concentrations of vanadium were incorporated into the ferrite lattice to arrive at a complete discussion and understanding of its role in the changes of these properties. Lastly, the magneto-mechanical characteristics of Li_0.5+xV_xFe_2.5−2xO₄ ferrite open the door for several applications, such as magneto-elastic sensors and high-frequency devices, which make several everyday uses possible.

2. Materials and Methods

2.1. Preparation of Samples

Samples of the chemical formula Li_0.5+xV_xFe_2.5−2xO₄ (x = 0.0 to 0.2, step 0.05) were synthesized using the usual solid-state synthesis. Oxides (purity = 99.99%, Sigma Aldrich, Carlsbad, CA, USA) of Fe₂O₃ and V₂O₅, together with Li₂CO₃, were mixed according to their molecular masses. Each sample’s mixture was ground for 24 h to a fine powder using a machine type (Retsch RM100, Düsseldorf, Germany), presintered at 800 °C for 6 h in air, ground again, and pressed under 700 MPa at room temperature into toroidal forms. At last, all samples were sintered at 1050 °C for 3 h in air and then slowly cooled to room temperature. Figure 1 illustrates a schematic diagram for solid-state synthesis for Li_0.5+xV_xFe_2.5−2xO₄ synthesis. The reaction is as follows:

$$\left(0.25+0.5x\right){\mathrm{L}\mathrm{i}}_2\mathrm{C}{\mathrm{O}}_3+0.5x{\mathrm{V}}_2{\mathrm{O}}_5+\left(1.25-x\right){\mathrm{F}\mathrm{e}}_2{\mathrm{O}}_3\to {\mathrm{L}\mathrm{i}}_{0.5+\mathrm{x}}{\mathrm{V}}_{\mathrm{x}}{\mathrm{F}\mathrm{e}}_{2.5-2\mathrm{x}}{\mathrm{O}}_4+(0.25+0.5x)\mathrm{C}{\mathrm{O}}_2$$

2.2. Measurements and Calculations

An X’Pert Graphics diffractometer (PANalytical, Almelo, The Netherlands) was used to obtain X-ray diffractograms (CuK_α, 40 kV, 30 mA, and λ = 1.54056 Å). Step-scan mode was used to measure diffraction intensities in the angular range of 20–80° (step size 2θ = 0.02°; counting time 2 s). Theoretical X-ray density (${\mathrm{\rho }}_{\mathrm{x}}$) of the samples was calculated according to the formula ${\mathrm{\rho }}_{\mathrm{x}}={\displaystyle \frac{8\mathrm{m}}{{\mathrm{N}}_{\mathrm{A}}{a}^3}}$, where $\mathrm{m}$ is the molecular mass, ${\mathrm{N}}_{\mathrm{A}}$ is Avogadro’s number, and $a$ is the lattice parameter [20]. Archimedes’ principle was used to measure the density of samples ($\mathrm{\rho }$). The porosity percentage P% was calculated according to the relation $\mathrm{P}=100[1-({\displaystyle \frac{\mathrm{\rho }}{{\mathrm{\rho }}_{\mathrm{x}}}}\left)\right]$ [21]. In the wavenumber range of 400–4000 cm⁻¹, Fourier transform infrared spectroscopy (FTIR) was carried out at room temperature in transmission mode using a Spectrometer JASCO, 6300, Hachioji, Japan. Magnetization (M) was measured using a room temperature vibrating sample magnetometer (VSM) (model Lake Shore no. 7410, Westerville, OH, USA). A toroid of each sample was used as a transformer core for measuring the initial permeability µ_i as a function of temperature at a constant frequency f = 10 kHz. The value of µ_i was calculated using Poltinnikov’s formula [22], ${\mathrm{V}}_{\mathrm{s}}={\displaystyle \frac{{\mathrm{\mu }}_{\mathrm{o}}{\mathrm{n}}_{\mathrm{p}}{\mathrm{n}}_{\mathrm{s}}{\mathrm{I}}_{\mathrm{p}}\mathrm{A}\mathrm{\omega }}{\mathrm{L}}}{\mathrm{\mu }}_{\mathrm{i}}$, where V_s is the induced voltage in the secondary coil, n_p and n_s are the number of turns of the primary and secondary coils, respectively, I_p is the current in the primary coil, A is the cross-sectional area of the sample, ω is the angular frequency, and L is the average path of the magnetic flux. To ensure reproducibility, all samples were synthesized under identical conditions using the same batch of precursors, and each characterization (XRD, FTIR, and VSM) was repeated at least three times, yielding deviations within ±3%. These controls support the reliability of the reported data and trends.

3. Results

3.1. Structure and Physical Properties

X-ray diffraction patterns for all samples are illustrated in Figure 2a. It is noted that they all have the cubic spinel structure, and the Miller indices of all diffraction peaks are indexed using the ordered Li_0.5Fe_2.5O₄ JCPDS card number 75-0407. Moreover, the 1:3 octahedral ordering (superstructure planes) or (long-range order) reflections for the cubic primitive Li-ferrite also appear, and they are observed to vanish gradually on increasing V⁵⁺ ion concentration. A similar effect of the addition of V₂O₅ to Li-Zn [19] and Mn-Zn [23] ferrite was previously reported.

However, Figure 3 shows that minority extra phases of very small relative intensities (<5% maximum) of α-Fe₂O₃, V₂O₅ [24], and FeVO₄ for two samples (x = 0 and 0.2) were detected with increasing V⁵⁺ ion concentration. Moreover, Figure 2b illustrates the shift of the positions of the crystallographic planes for all Li_0.5+xV_xFe_2.5−2xO₄ samples. One can observe that all reflections are shifted to higher 2θ values, which implies an increase in the inter-planar spacing according to Bragg’s law. This must lead the lattice parameter to decrease due to V⁵⁺ substitution. To make sure of that, the relation $a_{exp}= {\mathrm{d}}_{\mathrm{h}\mathrm{k}\mathrm{l}}\sqrt{{\mathrm{h}}^2+{\mathrm{k}}^2+{\mathrm{l}}^2}$ was used to calculate the experimental lattice parameter ($a_{exp}$) of the Li_0.5+xV_xFe_2.5−2xO₄ phase, where ${\mathrm{d}}_{\mathrm{h}\mathrm{k}\mathrm{l}}$ is the inter-planer spacing and h, k, and l are the Miller indices of each plane. These values of ($a_{exp}$) were plotted versus the Nelson–Riley function $\mathrm{F}(\mathrm{\theta })$, where $\mathrm{F}\left(\mathrm{\theta }\right)={\displaystyle \frac{1}{2}}\left[{\displaystyle \frac{{\mathrm{c}\mathrm{o}\mathrm{s}}^2\mathrm{\theta }}{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\theta }}}+{\displaystyle \frac{{\mathrm{c}\mathrm{o}\mathrm{s}}^2\mathrm{\theta }}{\mathrm{\theta }}}\right]$. Straight lines were obtained, and the extrapolation of them to $\mathrm{F}(\mathrm{\theta })=0$ gives the accurate values of ($a_{exp}$) [25]. Further, both lattice parameters and densities for all investigated samples are listed in

Table 1. Structural parameters obtained from X-ray diffraction of Li_0.5+xV_xFe_2.5−2xO₄ ferrites.

	Lattice Parameters (Å) [±0.002]		Porosity [±0.01]	X-Ray Density (g cm−3) [±0.001]
x	${\mathit{a}}_{\mathit{e}\mathit{x}\mathit{p}}$	${\mathit{a}}_{\mathit{t}\mathit{h}}$	P(%)	${\mathbf{\rho }}_{\mathbf{x}}$
0	8.340	8.090	5.31	4.742
0.05	8.331	8.086	7.62	4.695
0.1	8.329	8.082	8.34	4.637
0.15	8.328	8.078	8.61	4.577
0.2	8.325	8.074	8.83	4.521

. Figure 4a shows the lattice parameter ($a_{exp}$) variation with V⁵⁺ concentration (x). The value of the lattice parameter for the unsubstituted sample Li_0.5Fe_2.5O₄ (8.340 Å) compares well with that recorded before (8.332 Å [18]). It is clear from Figure 4a that the lattice parameter decreases with increasing V⁵⁺ content. Such a decrease could be understood, noting that the average value of V⁵⁺ and Li¹⁺ ions radii is less than the radius of the replaced Fe³⁺ ion. Furthermore, the X-ray detected oxide phases (α-Fe₂O₃, V₂O₅, and FeVO₄) reside at the grain boundaries and may cause a pressure on the unit cell and in turn share in the decrease in the lattice parameter. For more confirmation, the theoretical values of the lattice parameter ($a_{th}$) were calculated using the following cation distribution between the two sublattices (A, tetrahedral, and B, octahedral) of the spinel structure [18].

$${\left({\mathrm{V}}_{\mathrm{x}}^{5+}{\mathrm{F}\mathrm{e}}_{1-\mathrm{x}}^{3+}\right)}_{\mathrm{A}}{\left[{\mathrm{L}\mathrm{i}}_{0.5+\mathrm{x}}^{1+}{\mathrm{F}\mathrm{e}}_{1.5-\mathrm{x}}^{3+}\right]}_{\mathrm{B}}{\mathrm{O}}_4^{2-}$$

The Li¹⁺ ion has a strong preference to occupy the B-site [2] and the previously reported preference of V⁵⁺ ion to occupy the A-site [18]. This distribution itself will soon be assured by an analysis of the IR spectra of our samples. According to this cation distribution, the theoretical values of the lattice parameter ($a_{th}$) were calculated by Equation (1) as follows:

$$a_{th}={\displaystyle \frac{8}{3\sqrt{3}}}[\left({\mathrm{r}}_{\mathrm{A}}+{\mathrm{r}}_{\mathrm{O}}\right)+\sqrt{3}\left({\mathrm{r}}_{\mathrm{B}}+{\mathrm{r}}_{\mathrm{O}}\right)]$$

(1)

where r_A, r_B, and r_O are the radii of the A-site, B-site, and oxygen ion, respectively [8]. The fact that the values of all ionic radii depend on the coordination number was considered in the substitution in Equation (1). A schematic for the cubic and polyhedron structures of Li_0.5+xV_xFe_2.5−2xO₄ ferrite is shown in Figure 5 [25]. Both values (theoretical and experimental) are plotted in Figure 4b, and it is clear that they have a decreasing behavior. Moreover, it can be noticed that $a_{exp}$ > $a_{th}$ for all V⁵⁺ concentrations. This could be attributed to the formation of Fe²⁺ ions (of larger radius) according to the oxidation process and the metal–metal charge transfer transition. On the other hand, Figure 4a illustrates that the porosity P% increases with increasing V⁵⁺ concentration. Similar results were previously reported for vanadium-substituted Li-ferrite [18]. Such an increase in porosity is expected because the density decreases for Li_0.5+xV_xFe_2.5−2xO₄ as heavier Fe³⁺ ions (molecular mass = 55.847 g/mole) are replaced by lighter Li¹⁺ and V⁵⁺ ions (average molecular mass = 28.941 g/mole). This result may also be explained by noting the competition between the two types of porosities, P_intra (intragranular) and P_inter (intergranular) [25]. P_intra is related to the ionic radii in the lattice, while P_inter is directly proportional to the grain size. For Li_0.5+xV_xFe_2.5−2xO₄, the replacement of Fe³⁺ ions by Li¹⁺ and V⁵⁺ ions of a smaller average radius causes a decrease in P_intra values. Moreover, an increase in V⁵⁺ content in ferrites leads the grain size to increase [15] and hence increases P_inter. This, in turn, supports that the increase in the intergranular porosity is the dominant factor in increasing the total porosity P%.

3.2. Infrared Spectroscopic Analysis

In fact, one of the most important sources for collecting information on positions of different ions in the crystal unit cell is what is provided by infrared (IR) spectra [26]. In view of this, the IR transmission spectra of all studied ferrite samples were obtained to help in the determination of the cation distribution of Li_0.5+xV_xFe_2.5−2xO₄ ferrite. Figure 6 illustrates the IR spectra of all Li_0.5+xV_xFe_2.5−2xO₄ samples, and

Table 2. Variation of wave number (cm⁻¹) of the IR observed bands, force constants (N/m), and Poisson’s ratio with V⁵⁺ concentration for all investigated ferrite samples.

	Tetrahedral Bands	Octahedral Bands		Lattice Vibration	F × 105		σ
x	ν1	ν2	ν3	ν4	Ft	Fo
0	589–540	460–392	335	235–259–281	2.56	1.23	0.3006
0.05	595–543	460–392	340	236–259–281	2.61	1.22	0.2982
0.1	605–548	460–391	346	236–259–280	2.69	1.22	0.2955
0.15	610–554	460–391	349	235–260–282	2.73	1.20	0.2949
0.2	615–562	460–390	352	236–259–279	2.78	1.19	0.2941

has the wavenumber positions of their bands. Each IR spectrum is found to have mainly four bands. These bands demonstrate the four fundamental vibrations for spinel ferrites (namely, ν₁, ν₂, ν₃, and ν₄) [27]. ν₁ refers to tetrahedral complexes, ν₂ and ν₃ refer to octahedral complexes, and ν₄ refers to the lattice vibrations [13]. It is valuable to note here that ν₃ is usually assigned to Fe²⁺- O complexes, which indicates that Fe²⁺ ions exist in all our ferrite samples, as concluded in the previous discussion of the lattice parameter behavior. Moreover, the band positions of the Li_0.5Fe_2.5O₄ sample agree well with those reported earlier [28]. On the other hand, it is clear that other bands appear within the wavenumber range of 690–760 cm⁻¹. These bands are mainly associated with the stretching vibrations of metal–oxygen (Fe³⁺–O²⁻) bonds within the octahedral positions of the spinel lattice and tend to disappear by increasing the concentration of V⁵⁺ ions (i.e., their intensity decreases). Such bands are due to the long-range order (superstructure), and they ensure the role of V⁵⁺ ion substitution in the transformation from ordered to disordered Li-ferrite [13]. Moreover, the presence of Fe²⁺ ions in Li_0.5+xV_xFe_2.5−2xO₄ ferrites has been demonstrated to result in either this band splitting or the formation of shoulders on the absorption bands. This phenomenon is attributed to the Jahn–Teller distortion induced by Fe²⁺, which creates local distortions in the crystal field potential, consequently leading to splitting the band to 690 cm⁻¹ and 760 cm⁻¹ [29]. This enforces our previous interpretation of these superstructure planes in X-ray diffractograms.

Furthermore, Figure 7 shows the variations of ν₁ and ν₂ for all Li_0.5+xV_xFe_2.5−2xO₄ ferrite samples. It is obvious that ν₁ increases with increasing V⁵⁺ concentration while ν₂ is nearly constant. Such an observation implies that V⁵⁺ ions occupy the tetrahedral sites only. The behavior of ν is generally governed by the force constant (F) and the mass of the ion (m) forming the bond through the relation $\mathrm{\nu } \mathrm{\alpha }\sqrt{{\displaystyle \frac{\mathrm{F}}{\mathrm{m}}}}$ [30]. Increasing V⁵⁺ concentration, the force constant (F) increases as it is inversely proportional to the lattice parameter (which was found to decrease with increasing x. Conversely, the mass (m) decreases because the mass of V⁵⁺ (50.942 g/mole) is lighter than that of Fe³⁺ (55.847 g/mole). It can be concluded that both the increase in (F) and the decrease in (m) lead to an increase in ν₁, which is found experimentally. Thus, the increase in ν₁ supports that V⁵⁺ ions replace Fe³⁺ ions in the A-site. The preference of V⁵⁺ ions to enter the A-site matches well with previous studies [31]. From the above discussion, and according to the fact that Li¹⁺ ions have a strong preference to occupy the B-site [12], one can propose the cation distribution to be as follows:

$${\left({\mathrm{V}}_{\mathrm{x}}^{5+}{\mathrm{F}\mathrm{e}}_{1-\mathrm{x}}^{3+}\right)}_{\mathrm{A}}{\left[{\mathrm{L}\mathrm{i}}_{0.5+\mathrm{x}}^{1+}{\mathrm{F}\mathrm{e}}_{1.5-\mathrm{x}}^{3+}\right]}_{\mathrm{B}}{\mathrm{O}}_4^{2-}$$

For tetrahedral and octahedral sites, respectively, the force constants (F_t) and (F_o) could be computed using the following relation [32]:

$${\mathrm{F}}_{\mathrm{t}}=7.62\times {\mathrm{M}}_1\times {\upsilon }_t\times {10}^{-7}$$

(2)

$${\mathrm{F}}_{\mathrm{o}}=5.31\times {\mathrm{M}}_2\times {\upsilon }_o\times {10}^{-7}$$

(3)

where M₁ and M₂ are the molecular weight of both tetra- and octahedral sites, respectively, and ${\upsilon }_t$ and ${\upsilon }_o$ are the corresponding centre frequencies of both sites. Table 2 provides a summary of the force values. While F_o decreases, the force constant F_t increases as V⁵⁺ content increases. This variance can be explained by the difference in ionic radii between Fe³⁺ and V⁵⁺ions, as well as their occupancy at the A- and B-sites. Like most thermal quantities, the Debye temperature (θ_IR) connects the elastic characteristics to thermodynamic ones. So, it is one of the most crucial characteristics. As the molecular structure of ferrites may be inferred from the IR spectra, (θ_IR) could be determined using the following relation, which is where the greatest lattice vibrations occur [32]:

$${\mathrm{\theta }}_{\mathrm{I}\mathrm{R}}={\displaystyle \frac{\mathrm{h}\mathrm{c}{\upsilon }_{av}}{2\mathrm{\pi }{\mathrm{k}}_{\mathrm{B}}}}$$

(4)

where h, c, k_B, and ${\upsilon }_{av}$ are Planck’s constant, the velocity of light, Boltzmann’s constant, and the average value of the wavenumbers.

Figure 8 illustrates the variation of Debye temperature with V⁵⁺ content. One can see that, with an increase in V⁵⁺ concentration, the Debye temperature became higher. The rise in the wavenumber of IR bands may be the cause of this change. According to the specific heat theory, the rising trend of the Debye temperature may be explained by the fact that electrons absorb some of the heat, which causes the value of θ_IR to rise. It is important to note that higher V⁵⁺ content results in a higher rate, demonstrating once more the strong correlation between V⁵⁺ doping and the change in (θ_IR). Additionally, these variations in the Debye temperature imply that electrons in n-type semiconductors, rather than other processes, are responsible for the conduction in these samples [29].

3.3. Elastic Properties

Elastic parameters (as constants and moduli) that characterize a solid’s capacity to stretch or compress in various directions can be categorized into two terms: elastic constant and/or elastic moduli. Although there are generally thirty-six elastic moduli, only three moduli are present in spinel ferrites [32]. Elastic moduli and Debye temperature (θ_E) were determined using the IR and structural data details of the investigated ferrite as lattice parameters, X-ray density, porosity, and average force constant values. The following formula was used to calculate the average force constant (F_ave.):

$${\mathrm{F}}_{\mathrm{a}\mathrm{v}\mathrm{e}.}={\displaystyle \frac{{\mathrm{F}}_{\mathrm{t}}+{\mathrm{F}}_{\mathrm{o}}}{2}}$$

(5)

The stiffness constants (C₁₁) and (C₁₂) were calculated using the following relations [32,33]:

$${\mathrm{C}}_{11}={\displaystyle \frac{{\mathrm{F}}_{\mathrm{a}\mathrm{v}\mathrm{e}.}}{a}}$$

(6)

$${\mathrm{C}}_{12}={\displaystyle \frac{{\mathrm{C}}_{11}\times \sigma }{(1-\sigma )}}$$

(7)

where $\left(a\right)$ and (σ) are the lattice parameter and Poisson’s ratio, respectively, which depend on the porosity according to the following relation [33]:

$$\sigma =0.324 (1-1.0143 P)$$

(8)

It was found that Poisson’s ratio values, which are shown in Table 2, fall between 0.3006 and 0.2941, and this indicates that the samples are elastically stable and the bonds are still partly ionic (Fe–O, Li–O) but have some covalent contribution in line with the isotropic elasticity hypothesis [32,34]. The slight decrease from 0.3006 to 0.2941 as V⁵⁺ content increases means that the material becomes slightly more rigid and less compressible, and the lattice becomes slightly stiffer and less ductile.

Moreover, Figure 9 shows that the stiffness constants (C₁₁ and C₁₂) increased as the concentration of V⁵⁺ increased. The force constant and the tightness of the atom bonding really affect the stiffness constant values. Accordingly, it can be proposed that the emerging bond between V⁵⁺ and Fe³⁺ bonds in the current spinel system (K increases $a$ decreases) is the reason why the stiffness constants rose with the V⁵⁺ concentration. Additionally, using these two stiffness constants, the elastic moduli, such as Young’s modulus (E), rigidity modulus (R), and bulk modulus (B), were calculated using the following relations [33].

Young’s modulus

$$\left(\mathrm{E}\right)={\displaystyle \frac{({\mathrm{C}}_{11}-{\mathrm{C}}_{12})({\mathrm{C}}_{11}+2{\mathrm{C}}_{12})}{({\mathrm{C}}_{11}+{\mathrm{C}}_{12})}}$$

(9)

Rigidity modulus

$$\left(\mathrm{R}\right)={\displaystyle \frac{\mathrm{E}}{2(\mathrm{\sigma }+1)}}$$

(10)

Bulk modulus

$$\left(\mathrm{B}\right)={\displaystyle \frac{1}{3}}({\mathrm{C}}_{11}+2{\mathrm{C}}_{12})$$

(11)

Figure 10 illustrates how the elastic moduli increase as the concentration of V⁵⁺ ions increases because the interatomic linkages between the various atoms eventually gain their effectiveness. Based on Wooster’s model [33,35], the behaviour of elastic moduli is associated with the weakening of interatomic bonding between various atoms in the current spinel system. Moreover, the increase in elastic moduli with V⁵⁺ doping in Li ferrite arises from lattice contraction and enhanced metal–oxygen bond strength due to the smaller, higher-valence, and more covalently bonded V⁵⁺ ions. Furthermore, as V⁵⁺ concentration rises, the repulsion between electrons may increase because V⁵⁺ ions replace Fe³⁺ ions in the outermost orbital configuration in the Li_0.5+xV_xFe_2.5−2xO₄ ferrite system. This, in turn, causes the elastic moduli to further go up.

Moreover, using the following formulas proposed by Waldron [33], the longitudinal and shear elastic wave velocities ($V_l$) and ($V_s$), respectively, were calculated as follows:

$$V_l= {\left(\frac{C_{11}}{{\mathrm{\rho }}_{\mathrm{x}}}\right)}^{\frac{1}{2}}$$

(12)

$$V_{s }={\left(\frac{R}{{\mathrm{\rho }}_{\mathrm{x}}}\right)}^{\frac{1}{2}}$$

(13)

Then, ($V_l$) and ($V_s$) were then used to compute the mean wave velocity ($V_m$). Figure 11 depicts the variation of $V_l$, $V_s$, and $V_m$ against V⁵⁺ content. The decreasing trend, as shown in Figure 11, of $V_l$, $V_s$, and $V_m$ increased with V⁵⁺ substitution in Li_0.5+xV_xFe_2.5−2xO₄. This could be due to (i) the lattice contraction, in which the substitution of a smaller radius of V⁵⁺ decreases the lattice constant, and the shorter cation–oxygen distances mean stiffer bonds with higher force constants, leading to large elastic moduli and, consequently, high sound velocities. (ii) V⁵⁺ has a higher positive charge and stronger electrostatic attraction to O²⁻. So, both effects increase bond strength and interatomic force constants, which directly raise elastic wave velocities. Further, the partial replacement of Fe³⁺ with V⁵⁺ enhances the cation–oxygen–cation bridge stiffness (B–O–B), which reduces internal lattice vibrations and increases the resistance to deformation.

Additionally, using Anderson’s formula [32,36], the Debye temperature from elastic parameters (${\theta }_E$) was determined using the mean wave velocity ($V_m$)

$${\theta }_E={\displaystyle \frac{h}{k_B}} {\left[{\displaystyle \frac{3\mathrm{Q}{N}_A{\mathrm{\rho }}_{\mathrm{x}}}{4\pi \mathrm{m}}}\right]}^{1/3}{V}_m$$

(14)

where Q and m are the number of atoms in the unit formula and molecular weight, respectively.

The Debye temperature (${\theta }_{IR}$) values derived from IR data are clearly comparable to those derived from elastic constant data (${\theta }_E$), as shown in Figure 8. However, (${\theta }_{IR}$) is smaller than (${\theta }_E$), and the existence of low-frequency vibrational bands in the infrared spectrum is the primary cause of this discrepancy [37]. The increased rigidity of our studied Li_0.5+xV_xFe_2.5−2xO₄ ferrite may be the cause of the rise in Debye temperature with increasing V⁵⁺ concentration (Figure 8). Both the fluctuation of the lattice constant, which is primarily influenced by the concentration of V⁵⁺, and the variation of elastic constants and Debye temperature often have an inverse connection. Once more, this demonstrates the close relationship between the elastic characteristics of the Li_0.5+xV_xFe_2.5−2xO₄ ferrite system and its V⁵⁺ concentration.

However, it is interesting to note that, as Figure 12 illustrates, the average sound velocity ($V_m$) rises linearly with an increase in (${\theta }_E$). This behaviour suggests a clear relationship between an acoustic parameter ($V_m$) and a thermodynamic one (${\theta }_E$), and this rise reflects enhanced interatomic force constants, stronger V–O and Fe–O bonds, and reduced lattice compressibility. The lattice becomes more rigid, phonons propagate faster, and the material exhibits greater elastic stability and higher lattice energy, confirming that vanadium doping effectively stiffens the crystal network at the atomic level. Other reports have noted the same difference for other mixed ferrites [20,29,32,35,38].

3.4. Magnetization

Figure 13 illustrates the hysteresis curves of Li_0.5+xV_xFe_2.5−2xO₄ ferrite samples as a function of the applied magnetic field (H). It can be seen that by increasing the magnetic field, the magnetization of all samples increases and tends to saturation at high field values. Moreover, Figure 13 shows the variation of the saturation magnetization (M_S) with V⁵⁺ concentration (x). Moreover,

Table 3. Magnetic parameters of Li_0.5+xV_xFe_2.5−2xO₄ ferrites.

x	Ms (emu/g) [±0.1]	Hc (G) [±0.05]	Br (emu/g) [±0.1]	M (g) [±0.1]	Ms (μB) [exp.]	Mag. Mom. (μB) [calc.]	ΘYK (°)	Initial Permeability $({\mathbf{\mu }}_{\mathbf{i}}$)	Tc (K) [±2]
0	41.42	211.36	8.9	207.08	1.54	2.5	29.31	18.02	889
0.05	48.72	156.54	12	204.39	1.78	2.5	25.75	31.61	882
0.1	50.64	132.68	13	201.70	1.83	2.5	25.27	37.21	875
0.15	46.75	145.69	13	199.01	1.67	2.5	28.71	12.79	869
0.2	44.94	163.51	13	196.32	1.58	2.5	30.86	5.01	865

observed that M_S increases with V⁵⁺ concentration to reach its maximum value at x = 0.1 and then decreases for higher V⁵⁺ concentrations. Considering the proposed cation distribution, which was concluded from the IR spectra, the value of the magnetic moment per formula unit, which represents the total magnetization, is given by the following:

$${\mathrm{M}}_{\mathrm{S}}\left(\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}\right)={\mathrm{M}}_{\mathrm{B}}-{\mathrm{M}}_{\mathrm{A}}=2.5{\mathrm{\mu }}_{\mathrm{B}}$$

(15)

where M_A, M_B, and μ_B are the magnetic moments of ions in the A- and B-sites and the Bohr magneton, respectively. As is known, according to Neel theory, A-B interaction is the most important factor that affects magnetization. In fact, other factors combine to create this interaction. This means that M_S must have the same value for all samples, which was not found experimentally, as is clear in Table 3. But, according to the non-collinear model, the experimental magnetic moment per formula unit is given by the following equation [30]:

$${\mathrm{M}}_{\mathrm{S}}\left(\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{l}\right)={\mathrm{M}}_{\mathrm{B}}\cos {\mathrm{\theta }}_{\mathrm{Y}\mathrm{K}}-{\mathrm{M}}_{\mathrm{A}}$$

(16)

where θ_YK is the Yafet–Kittel canting angle between the moments in the B-sites. It is noted that increasing V⁵⁺ concentration leads the concentration of Fe³⁺ in the B-site to decrease. Thus, the B-B interaction begins to decrease, and in turn, θ_YK decreases, as shown in Table 3. According to the last Equation (16), such a decrease in θ_YK leads to an increase in M_S, which explains its behavior for 0 ≤ x ≤ 0.1. On the other hand, the decrease in M_S for V⁵⁺ concentration x > 0.1 may be due to the continuous decrease in Fe³⁺ ions with a higher magnetic moment (5 μ_B) in the B-site and the formation of Fe²⁺ ions with a lower magnetic moment (4 μ_B), which reduces the magnetization of these sites more than expected. The large decrease in the long-range ordering may also contribute to decreasing M_S in this range of V⁵⁺ concentration [39]. Furthermore, the initial increase in M_s with low V⁵⁺ substitution in Li ferrite (0 ≤ x ≤ 0.1) can be attributed to cation redistribution between the A- and B-sites, where limited V⁵⁺ incorporation drives Fe³⁺ migration from the A- to B-sites, strengthening the net B-sublattice magnetization. Such an oxygen vacancy-induced reduction in magnetization has been similarly reported for doped Li-ferrites and other spinel systems [11,40], where oxygen deficiency leads to a partial disruption of magnetic ordering. However, further V⁵⁺ addition introduces a large number of nonmagnetic ions and oxygen vacancies for charge compensation, which disrupts Fe–O–Fe superexchange pathways and induces Fe³⁺ → Fe²⁺ conversion. These effects generate lattice strain, spin canting, and magnetic dilution, collectively leading to the subsequent decline in M_s at higher doping levels. Meanwhile, the decrease in M_s at higher V⁵⁺ content could be due to the formation of nonmagnetic secondary phases (α-Fe₂O₃, FeVO₄), which dilute the ferrimagnetic spinel network and disrupt Fe–O–Fe superexchange interactions. These structural inhomogeneities induce spin canting and magnetic dilution, reducing the overall magnetization.

3.4.1. Initial Permeability

Figure 14 shows the change in the initial permeability (µ_i) as a function of temperature for Li_0.5+xV_xFe_2.5−2xO₄ ferrite samples, and the values are recorded in Table 3. It can be seen that µ_i increases with increasing temperature, and it begins to decrease faster near the Curie temperature T_C. The Globus relation (Equation (17)) [41] is usually used to explain such a behavior as follows:

$${\mathrm{\mu }}_{\mathrm{i}}\mathrm{\alpha }{\displaystyle \frac{{\mathrm{M}}_{\mathrm{S}}^2\mathrm{D}}{{\mathrm{k}}_1^{0.5}}}$$

(17)

where (D) is the average grain size and (k₁) is the anisotropy constant. According to this relation, the behavior of µ_i is controlled by the competition of both M_S and k₁. As the decrease in k₁ with temperature is much faster than M_S, µ_i tends to increase with increasing temperature. For temperatures higher than T_C, the random orientations of the magnetic moments in the paramagnetic state are established, and the magnetization vanishes, which leads μ_i to decrease with an increase in temperature.

Furthermore, the behavior of µ_i, at room temperature, with V⁵⁺ concentration (x) is shown in Figure 15. The composition dependence of initial permeability of ferrites could be explained by taking the Globus relation (Equation (17)) into consideration and the fact that the porosity has an inverse relation with μ_i, as it hinders the domain wall motion [25]. It is clear that µ_i has a similar behavior to (x) as that of M_S (plotted in the same Figure 15). This observation indicates that M_S is the dominant factor in the behavior of μ_i in the whole range of V⁵⁺ concentration for Li_0.5+xV_xFe_2.5−2xO₄ ferrite. Moreover, for x ≤ 0.1, the increase in µ_i is supported by the decrease in k₁ (due to the decrease in the iron concentration) and the probable increase in the grain size (due to the increase in V⁵⁺ concentration [15,42]). On the other hand, for x > 0.1, the decrease in µ_i is supported by the increase in porosity and the formation of extra phases (detected by X-ray).

3.4.2. Curie Temperature (T_C)

The thermal stability of the magnetic ordering in ferrites is measured by the Curie temperature (T_C). Figure 16 and Table 3 show the values of the temperature at which the initial permeability tends to zero for all our ferrite samples. These values were taken as Curie temperatures (T_C) and plotted as a function of V⁵⁺ concentration in Figure 16. A close result was reported for the T_C of lithium ferrite (T_C $\approx$ 889 K) [43]. It can be noticed that T_C decreases continuously with increasing V⁵⁺ concentration. As is known, T_C depends mainly on the A-B interaction that is affected by the following factors: first, the magnitude of the magnetic moments in A- and B-sites (in Li_0.5+xV_xFe_2.5−2xO₄, the only moments are those of Fe³⁺ ions); second, the relative concentration of the moments on both sites (N_A/N_B); and third, the distance between the A- and B-sites, which is proportional to the variation of the lattice parameter. According to these factors, one can discuss the decrease in T_C for Li_0.5+xV_xFe_2.5−2xO₄ ferrite as follows. According to the previous cation distribution, (N_A/N_B) is found to decrease extremely with increasing (x) (as shown in Figure 16), which clearly explains the decrease in T_C in the whole range of V⁵⁺ concentration. Moreover, the formation of Fe²⁺ ions (of smaller magnetic moment 4 μ_B) and the detected minority extra phases of α-Fe₂O₃, V₂O₅, and FeVO₄ in the whole range of V⁵⁺ concentration give further contribution in decreasing T_C. Furthermore, the decrease in M_S for x > 0.1 leads the A-B interaction to decrease and enhances the decrease in T_C. It is to be noted here that the decrease in the lattice parameter (Figure 5) has the effect of increasing T_C. So, it could be deduced that the previous factors are dominant against the effect of the decrease in the lattice parameter in the behavior of T_C. Furthermore, high Curie temperatures of lithium ferrites are suitable for high-temperature applications where thermal stability is essential, such as inductors, transformers, and microwave devices [1,2]. Introducing vanadium ions allows the tuning of Tc of lithium ferrite, promoting the creation of ferrites with unique magnetic characteristics adapted to different operating conditions.

4. Conclusions

Replacing vanadium in lithium ferrite Li_0.5+xV_xFe_2.5−2xO₄ leads to a marked improvement in its thermodynamic, elastic, and magnetic properties. The reduced lattice parameter, resulting from the smaller radius of the V⁵⁺ ion, promotes V–O and Fe–O bonding while simultaneously increasing porosity. Cation distribution obtained from IR analysis shows the preference of V⁵⁺ ions to occupy the tetrahedral sites. Furthermore, both the X-ray and the IR spectra confirmed the existence of the long-range order (superstructure) and ensured the role of V⁵⁺ ion substitution in the transformation from ordered to disordered Li-ferrite. The corresponding increase in C₁₁ and C₁₂ and the elastic moduli indicate improved lattice rigidity and mechanical resilience. The increased mean sound velocity and Debye temperature with doping confirm that the intermolecular force constants and lattice vibration energy are significantly increased, despite the slight rise in porosity. Moreover, doping lithium ferrite with vanadium leads to enhancing the saturation magnetization, Curie temperature, and the permeability. The Curie temperature decreased with increasing V⁵⁺ concentration, allowing for tuning of the T_C of Li_0.5+xV_xFe_2.5−2xO₄ ferrites. On the other hand, saturation magnetization and initial permeability were enhanced up to a V⁵⁺ concentration of x = 0.1. These results demonstrate that V⁵⁺ substitution effectively enhances the mechanical and magnetic properties of Li_0.5+xV_xFe_2.5−2xO₄ ferrite, making it a promising candidate for magneto-elastic sensors, high-frequency devices, and applications requiring mechanically robust ferrite materials.