Large reversible magnetocaloric effect in GdPO4
nano-spherical particles at ultralow temperatures

Meryam Lassri, Said El Ouahbi, Imane Abdelrhafore, Lhaj El Hachemi
Omari, Abdelhadi El Hachmi, Hassan Hamouda, Benaissa Harradi, Yousra
Ounza, Gianluca Viscusi, Claire Minaud, et al.

► To cite this version:

Meryam Lassri, Said El Ouahbi, Imane Abdelrhafore, Lhaj El Hachemi Omari, Abdelhadi El Hachmi, et al.. Large reversible magnetocaloric effect in GdPO4 nano-spherical particles at ultralow temperatures. Journal of Magnetism and Magnetic Materials, 2025, Journal of Magnetism and Magnetic Materials, 629, pp.173291. 10.1016/j.jmmm.2025.173291. hal-05236233

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# Large reversible magnetocaloric effect in GdPO₄ nano-spherical particles at ultralow temperatures

Meryam Lassri<sup>a,b</sup>, Said El Ouahbi<sup>a</sup>, Imane Abdelrhafore<sup>a,b</sup>, Lhaj El Hachemi Omari<sup>a,*</sup>, Abdelhadi El Hachmi<sup>c</sup>, Hassan Hamouda<sup>a</sup>, Benaissa Harradi<sup>a</sup>, Yousra Ounza<sup>a</sup>, Gianluca Viscusi<sup>d,*</sup>, Claire Minaud<sup>e</sup>, Charafeddine Jama<sup>f,g</sup>, Hassan Lassri<sup>a</sup>

## ARTICLE INFO

### Keywords:

Magnetocaloric effect

Gadolinium phosphate

Magnetic refrigeration

Ultralow temperature

Magnetic entropy change

## ABSTRACT

This study reports the synthesis of gadolinium phosphate (GdPO₄) with a monoclinic crystal structure (P2₁/n) using a high-temperature solid-state reaction method. Rietveld refinement of the X-ray diffraction (XRD) data revealed lattice constants of *a* = 0.66497(5) nm, *b* = 0.68457(5) nm, and *c* = 0.63327(4) nm. The magnetic and magnetocaloric properties of GdPO₄ were investigated in detail. The ferromagnetic (FM) to paramagnetic (PM) phase transition temperature was determined to be around 1.8 K. The transition was strongly supported by Arrott plots and a universal phenomenological curve describing the magnetocaloric effect (MCE). GdPO₄ exhibits significant magnetic entropy changes (−ΔS<sub>M</sub>). The maximum values of −ΔS<sub>M</sub><sup>max</sup> are 20.43 J/(kg·K) and 44.79 J/(kg·K) for a magnetic field change (μ<sub>0</sub>ΔH) of 2 T and 9 T, respectively. Our analysis indicates that the GdPO₄ nano-spherical particles exhibit short-range magnetic ordering, characterized by the coexistence of ferromagnetic (FM) and antiferromagnetic (AFM) interactions.

## 1. Introduction

Magnetic refrigeration offers several advantages over traditional gas-compression cooling, including higher energy efficiency and reduced environmental impact due to the absence of ozone-depleting gases [1–3]. Magnetic cooling technology is attracting interest for its potential to provide low-noise, low-vibration, and eco-friendly cooling solutions. This technology holds promise for various applications, from fundamental research to cooling superconducting magnets in medical imaging and hydrogen liquefaction [4]. To exploit these benefits, researchers are actively searching for new magnetic materials with a large MCE [5–7]. The latter describes the change in a material’s isothermal magnetic entropy (ΔS<sub>M</sub>) or adiabatic temperature (ΔT<sub>ad</sub>) under a varying magnetic field [8–10]. Based on the MCE of materials, adiabatic demagnetization refrigeration has gained significant interest for its potential applications in quantum technologies, space science, and achieving ultra-low temperatures [4,11–14]. Therefore, developing new magnetic

refrigerants with a large MCE across a broad temperature range, particularly at sub-kelvin temperatures, is highly desirable. Compared to other rare earth elements, gadolinium-based materials are of particular interest due to the large magnetic moment of the Gd<sup>3+</sup> ion (S = 7/2) and minimal magnetic anisotropy [15–17]. Encouraged by reports of promising values of (−ΔS<sub>M</sub>) in materials like Gd<sub>2</sub>(SO<sub>4</sub>)<sub>3</sub>·8H<sub>2</sub>O [18], researchers are exploring their potential as refrigerants based on the adiabatic demagnetization principle. Previous studies have shown promising MCE values in this material, indicating its potential use in refrigerant manufacturing to achieve ultra-low temperatures below 1 K. Also, Gd(OH)CO<sub>3</sub> [19] and GdF<sub>3</sub> [20] possess large −ΔS<sub>M</sub> of 33.5 and 45.5 J/(kg·K) at μ<sub>0</sub>ΔH = 2 T and 2 K, respectively, and can be employed in the same applications. Due to the characteristic PO<sub>4</sub> structural units, phosphate materials exhibit excellent properties for use in low-temperature magnetic cooling systems. However, only a limited number of gadolinium-based phosphates with MCE have been explored, such as GdPO<sub>4</sub> and LiGdP<sub>4</sub>O<sub>12</sub> [15,18–20]. The maximum magnetic entropy

<footnote><sup>a</sup> LPMAT, FSAC, Hassan 2, University of Casablanca, Morocco</footnote>

<footnote><sup>b</sup> IAP-EMINES, Mohammed VI Polytechnic University, Lot 660, Hay Moulay Rachid, 43150 Ben Guerir, Morocco</footnote>

<footnote><sup>c</sup> Physico-chemistry of Materials Team, Abdelmalek Essaadi University, Faculty of Science, Department of Chemistry, 93002 Tetouan, Morocco</footnote>

<footnote><sup>d</sup> Department of Industrial Engineering, University of Salerno, Via Giovanni Paolo II, 132 Fisciano, Italy</footnote>

<footnote><sup>e</sup> Laboratoire de Physique Appliquée, Institut Michel-Eugène Chevreul, F-59000 Lille, France</footnote>

<footnote><sup>f</sup> CNRS, INRAE, Centrale Lille, UMR 8207 – UMET – Unité Matériaux et Transformations, F-59000 Lille, France</footnote>

<footnote>* Corresponding authors.</footnote>

<footnote>E-mail addresses: lhajelhachemi.omari@univh2c.ma (L.E. Hachemi Omari), gviscusi@unisa.it (G. Viscusi).</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 1. (a) Rietveld plot of the GdPO₄ compound. (b) Crystallite size vs peak position.</figcaption></figure>

change values $-\Delta S_M^{\text{max}}$ reach 27.2 and 20.1 J/(kg·K) under a 2 T magnetic field at 2 K, respectively. In this work, we present a detailed investigation of the structural, magnetic properties and reversible magnetocaloric effect of GdPO₄ at very low temperatures. Our experiments reveal a magnetic phase transition near 1.8 K and the GdPO₄ significant value of the magnetic entropy change ($-\Delta S_M$). The maximum values of $-\Delta S_M^{\text{max}}$ are 20.43 and 44.79 J/(kg·K) for a magnetic field variation ($\mu_0\Delta H$) of 2 T and 9 T, respectively.

## 2. Experimental

GdPO₄ in polycrystalline form was prepared using a conventional solid-state reaction route. Stoichiometric amounts of high-purity Gd₂O₃ (99.99 %) and (NH₄)₂HPO₄ (99.9 %) were used as starting materials. After thorough grinding in an agate mortar, the mixture was placed in a platinum crucible. The temperature was gradually increased to 573 K over 5 h to remove volatile components such as H₂O and NH₃. The mixture was then reground, pressed into a pellet, and reheated to 1173 K for calcination over 24 h.

Powder X-ray diffraction (PXRD) was employed to confirm the successful synthesis of GdPO₄. The analysis was conducted on a Bruker D8 diffractometer using Cu-Kα radiation ($\lambda$ = 0.15406 nm) with a scan range of 10° ≤ 2θ ≤ 80°. A step width of 0.05906° and a scan rate of 1 s/step were used.

<table><caption>**Table 1**
Structural refinement details of GdPO₄.</caption><tr><td>GdPO<sub>4</sub>·xH<sub>2</sub>O</td><td>Structural refinement parameters</td></tr><tr><td>Wavelengths (nm)</td><td>0.154056 (K<sub>α1</sub>), 0.154439 (K<sub>α2</sub>)</td></tr><tr><td>2θ range (°)</td><td>10–80</td></tr><tr><td>2θ step (°)</td><td>0.05906</td></tr><tr><td>Crystal system</td><td>Monoclinic</td></tr><tr><td>Space group</td><td>P 2<sub>1</sub>/n (No.14)</td></tr><tr><td>Lattice parameters</td><td>a = 0.66497(5) nm, b = 0.68457(5) nm, c = 0.63327(4) nm,<br/>β = 104.003(3)°</td></tr><tr><td>Cell volume (nm<sup>3</sup>)</td><td>0.27971(3)</td></tr><tr><td>Formula units per unit cell</td><td>Z = 4</td></tr><tr><td>Pseudo-Voigt function: [PV = ηL + (1 - η)G]</td><td>η = 0.74(3)</td></tr><tr><td>Caglioti parameters (u, v, w)</td><td>0.039(10), -0.029(8), 0.012(1)</td></tr><tr><td>No. of reflections</td><td>379/2</td></tr><tr><td>No. of fitted parameters</td><td>40</td></tr><tr><td>d<sub>calc</sub> (g cm<sup>-3</sup>)</td><td>5.989</td></tr><tr><td>Reliability factors</td><td>R<sub>B</sub> = 11.0, R<sub>F</sub> = 13.1, R<sub>p</sub> = 2.41, R<sub>wp</sub> = 3.12, R<sub>exp</sub> = 2.38</td></tr><tr><td>Goodness-of-fit (χ<sup>2</sup>)</td><td>1.71</td></tr><tr><td>Composition</td><td>GdPO<sub>4</sub><br/>xH<sub>2</sub>O</td></tr><tr><td></td><td>82.82 %<br/>17.18 %</td></tr></table>

<table><caption>**Table 2**
Fractional atomic coordinates and isotropic displacement parameters.</caption><thead><tr><th></th><th>x</th><th>y</th><th>z</th><th>B<sub>Iso</sub></th><th>Occ.</th><th>Site</th></tr></thead><tbody><tr><td>Gd<sup>3+</sup></td><td>0.282(7)</td><td>0.154(1)</td><td>0.096(1)</td><td>0.5(1)</td><td>1</td><td>4e</td></tr><tr><td>p<sup>5+</sup></td><td>0.302(3)</td><td>0.157(3)</td><td>0.611(3)</td><td>1.3(4)</td><td>1</td><td>4e</td></tr><tr><td>O1</td><td>0.254(6)</td><td>0.004(7)</td><td>0.422(8)</td><td>2.5(5)</td><td>1</td><td>4e</td></tr><tr><td>O2</td><td>0.377(5)</td><td>0.346(7)</td><td>0.486(6)</td><td>2.5(5)</td><td>1</td><td>4e</td></tr><tr><td>O3</td><td>0.471(5)</td><td>0.114(5)</td><td>0.823(5)</td><td>2.5(5)</td><td>1</td><td>4e</td></tr><tr><td>O4</td><td>0.151(6)</td><td>0.224(5)</td><td>0.727(7)</td><td>2.5(5)</td><td>1</td><td>4e</td></tr></tbody></table>

Magnetic properties were investigated using a PPMS 9 T (Physical Properties Measurement System) equipped with a Vibrating Sample Magnetometer (VSM). Measurements were performed under varying magnetic fields (up to 9 T) across a temperature range of 1.8 K to 70 K.

## 3. Results and discussion

### 3.1. X-ray diffraction (XRD)

X-ray diffraction (XRD) was used to analyze a polycrystalline sample of gadolinium phosphate (GdPO₄) at room temperature. The Rietveld refinement method [21,22] with FullProf software [23] revealed that GdPO₄ crystallizes in a monoclinic structure with space group P2₁/n (No. 14). The unit cell parameters were determined to be a = 0.66497(5) nm, b = 0.68457(5) nm, c = 0.63327(4) nm, and β = 104.003(2)°. The calculated density for GdPO₄ is 5.989 g/cm³. Fig. 1(a) shows the Rietveld refinement plot and the average size of crystallite of GdPO₄ compound. As can be seen, the experimental pattern is represented by red dots, and the calculated pattern is shown by a solid black line, the green vertical marks indicate the Bragg positions of the main phase (top) and the impurity phase (bottom) and the lower blue curve represents the difference diagram. The excellent agreement between the experimental and theoretical patterns indicates a good fit. The crystallographic results obtained from the Rietveld refinement are summarized in Table 1. The quality of the refinement is further confirmed by the low reliability factors obtained: R<sub>B</sub> = 11.0, R<sub>F</sub> = 13.1, R<sub>p</sub> = 2.41, R<sub>wp</sub> = 3.12, R<sub>exp</sub> = 2.38, and goodness-of-fit χ<sup>2</sup> = 1.71 %.

A minor impurity phase corresponding to GdPO₄·3/2H₂O (ICDD# 00-021-0337) was identified using a hexagonal P3₁21 (#152) space group model during the refinement. The estimated content of this
<table><caption>**Table 3**
Selected interatomic distances (Å) and bond valences.</caption><thead><tr><th>Bonds</th><th></th><th>Distances (Å)</th><th>Bond-valences</th></tr></thead><tbody><tr><td rowspan="2">Gd–O1</td><td>(×1)</td><td>2.35(5)</td><td>0.46(7)</td></tr><tr><td>(×1)</td><td>2.41(5)</td><td>0.39(5)</td></tr><tr><td rowspan="2">Gd–O2</td><td>(×1)</td><td>2.73(4)</td><td>0.17(2)</td></tr><tr><td>(×1)</td><td>2.36(5)</td><td>0.45(6)</td></tr><tr><td rowspan="2">Gd–O3</td><td>(×1)</td><td>2.62(3)</td><td>0.23(2)</td></tr><tr><td>(×1)</td><td>2.39(4)</td><td>0.42(4)</td></tr><tr><td rowspan="2">Gd–O4</td><td>(×1)</td><td>2.43(3)</td><td>0.37(3)</td></tr><tr><td>(×1)</td><td>3.21(4)</td><td>0.05(4)</td></tr><tr><td rowspan="2">&lt;Gd–O&gt;<sub>average</sub>, BVS</td><td>(×1)</td><td>2.34(4)</td><td>0.47(5)</td></tr><tr><td>(×1)</td><td>2.53(4)</td><td>0.28(3)</td></tr><tr><td>P–O1</td><td>(×1)</td><td>2.54(1)</td><td>3.29(1)</td></tr><tr><td>P–O2</td><td>(×1)</td><td>1.56(5)</td><td>1.15(2)</td></tr><tr><td>P–O3</td><td>(×1)</td><td>1.66(5)</td><td>0.90(1)</td></tr><tr><td>P–O4</td><td>(×1)</td><td>1.56(3)</td><td>1.18(1)</td></tr><tr><td>&lt;P–O&gt;<sub>average</sub>, BVS</td><td>(×1)</td><td>1.45(5)</td><td>1.55(2)</td></tr><tr><td></td><td></td><td>1.56(2)</td><td>4.79(3)</td></tr></tbody></table>

impurity is around 17.18 wt%, as determined from the XRD pattern. Crystallographic details, including atomic positions and bond parameters, are provided in Tables 1, 2, and 3. The Rietveld analysis revealed that Gd³⁺, P⁵⁺, and O²⁻ ions occupy specific positions within the crystal structure (4e-sites with specific x, y, and z coordinates, listed in Table 2). Selected interatomic distances and corresponding bond valences are presented in Table 3.

Notably, the average <Gd-O> and <P-O> distances obtained (through refinement) are very close to previously reported values for Gd³⁺-O²⁻ (around 2.5105 Å) and P⁵⁺-O²⁻ (around 1.5344 Å) bonds. The Scherrer equation (Eq. (1)) is used to estimate the crystallite size (D) of the material from the full width at half maximum (FWHM noted as β<sub>hkl</sub>) of the of X-ray diffraction peaks broadening [24]:

$$D = \frac{K\lambda}{\beta_{hkl}\cos(\theta)} \quad (1)$$

where D is the crystallite size in nanometers (nm), K is the Scherrer constant, typically around 0.94, λ is the X-ray wavelength (Cu-Kα average = 0.1542475 nm), θ is the Bragg angle of the strongest diffraction peak. In this study, the full width at half maximum (FWHM) values of the most prominent diffraction peaks were analyzed to estimate the average crystallite size using the Scherrer equation (Fig. 1(b)). A linear regression analysis was performed on the calculated crystallite sizes plotted against the peak positions. The results indicate that the average crystallite size is within the range of 57 to 84 nm, confirming the nanostructured nature of the particles.

## 3.2. EDX and SEM microscopy

Fig. 2 presents scanning electron micrographs of the sample at various magnifications, revealing spherical nanoparticles with a tendency to agglomerate. The nanometric size of the particles is evident. Energy-dispersive X-ray spectroscopy (EDX) analysis confirms the expected elemental composition and stoichiometry.

Also, Fig. 2 indicates that the average grain size primarily ranges from 60 to 80 nm. Furthermore, a second critical average grain size of approximately 250 nm is evident, potentially attributed to the length of certain agglomerated grains. These results align well with the properties established previously. Moreover, SEM micrographs also confirm the spherical shape of the grains at a nanometric scale.

## 3.3. Magnetic characteristics

Fig. 3 shows the magnetic inverse of susceptibility (χ⁻¹(T) = (μ₀H/M)) versus the temperature measured under a constant external magnetic field of 0.1 T across a temperature range of 1.8 K to 400 K (inset shows dχ/dT versus T at T = 1.8–400 K).

Interestingly, above 2 K, the magnetic inverse of susceptibility χ⁻¹ of GdPO₄ follows the equation known as the Curie-Weiss law (Eq. (2)):

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** EDX analysis and SEM micrographs of GdPO₄ nano-spherical particles at various magnification scales.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 3. Magnetic inverse susceptibility of GdPO₄ versus temperature at a magnetic field of 0.1 T; in the inset: dχ/dT versus T at T = 1.8–400 K.</figcaption></figure>

$$ \chi^{-1} = (T - \theta_p)/C \quad (2) $$

here, θp is the paramagnetic Curie temperature, and C represents the Curie Weiss constant. A linear fit to high temperatures yields a Curie constant. By fitting the data at high temperatures to a straight line, we obtained a Curie constant (C) of 6.9 emu.K/mol.Oe and a positive paramagnetic Curie-Weiss temperature θp of 2 K, indicating that the dominant magnetic interactions between Gd³⁺ ions in GdPO₄ are ferromagnetic (FM) at low temperatures. This aligns with the expected behavior of Gd³⁺ (4f⁷, J = 7/2), where the half-filled 4f shell minimizes orbital contributions (L = 0), making the system highly isotropic and dominated by spin-spin exchange interactions. The experimentally determined effective magnetic moment (μₐ₈ = 7.43 μₑ) is close to the theoretical free-ion value (μₐ₈ = 7.94 μₑ), confirming that the Gd³⁺ moments remain unquenched by crystal field effects. The slight reduction in (μₐ₈) may arise from weak antiferromagnetic (AFM) super-exchange pathways mediated by phosphate (PO₄³⁻) groups or due to spin-orbit coupling perturbations, as observed in other Gd-based systems [25]. This discrepancy between μₐ₈ and μₐ₈ can also be attributed to the coexistence of antiferromagnetic and ferromagnetic phases at low temperatures, indicating the presence of short-range order interactions.

The data further reveals a ferromagnetic-to-paramagnetic (FM → PM) transition at a Curie temperature (T_c≈1.8 K), identified by the minimum in dχ/dT. This transition is consistent with previous reports on GdPO₄ nanorods by Y. Yu et al. [16], who found a slightly higher T_c≈2.1 K, attributing it to the coexistence of FM and AFM interactions. Such competition between FM and AFM couplings is a hallmark of geometrical frustration, often seen in systems where Gd³⁺ ions occupy multiple crystallographic sites with differing exchange pathways. This scenario is analogous to observations in EuLn₂O₄ polycrystalline materials [26], where competing interactions lead to complex magnetic phase diagrams. The origin of these competing interactions can be traced to several structural and electronic factors:

* Magnetic Dead Layers: Surface effects in nanoscale GdPO₄ (e.g., disordered spins or canted magnetic moments) can introduce weak AFM contributions, as suggested by studies on rare-earth oxides [27];

* Sublattice Complexity: If Gd³⁺ ions reside in inequivalent magnetic sublattices, FM direct exchange within sublattices may compete with AFM super-exchange between them [28];

* Frustration and Short-Range Order: The competition between FM and AFM interactions can lead to spin-glass-like behavior or short-range correlations, as reported in frustrated Gd-based compounds [29].

<figure><img src="image_6.png" /><figcaption>Fig. 4. M vs T at different magnetic field; in the inset: χ⁻¹ versus T at H = 0.1–7 T.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 5. Magnetization plots M vs. μ₀H of GdPO₄ compound.</figcaption></figure>

(M-T) plots in the presence of different applied magnetic fields within the temperature range 1.8 K–70 K is reported in Fig. 4. It is noticed that the nanoscale GdPO₄ has a magnetic transition below 1.8 K. The variations of the magnetization versus temperature and magnetic field have confirmed magnetic ordering at low temperatures. The magnetization measurements on the M(T,H) function have confirmed a coexistence of AFM and FM phases at low temperatures. The analyses have revealed that the sample exhibits short-range magnetic order as a result the competition between AFM and FM interactions which involve Gd³⁺ ions in different sublattices and magnetic layers [16]. The magnetic susceptibility χ=H/M as a function of temperature (T) curves for different applied magnetic fields have been plotted and these plots are shown in the inset of Fig. 4. For the GdPO₄ nanoparticles (in the paramagnetic region), χ vs. T curves are found to fit well with the well-known Curie-Weiss law.
<figure><img src="image_8.png" /><figcaption>Fig. 6. Entropy change (−ΔS<sub>M</sub>) as a function of temperature at various applied fields from 2 to 9 T of GdPO<sub>4</sub> compound.</figcaption></figure>

3.4. Magnetocaloric effect

Magnetic materials can exhibit a temperature change when exposed to a changing magnetic field. The M–T curve provides direct evidence of the magnetic transition, corroborating the Arrott plot analysis and validating the derived magnetocaloric data. This later phenomenon is called the magnetocaloric effect (MCE). To assess this effect in the studied sample (GdPO<sub>4</sub>), we measured its magnetization M(H) at various temperatures across a range of temperatures from 1.8 to 70 K (Fig. 5).

The measurements were performed at regular temperature intervals to obtain a detailed picture of the magnetocaloric response. The MCE represented by the entropy change (−ΔS<sub>M</sub>) of complex materials undergoing 2nd order transitions can be obtained using Maxwell’s relation (Eq. (3) [30]):

$$
\Delta S_M(T, \Delta H)_{\Delta H} = S_M(T, H) - S_M(T, 0) = \int_0^{\mu_0 H_{\max}} \left( \frac{\partial M(H, T)}{\partial T} \right)_H dH \quad (3)
$$

This relation incorporates the maximum applied external magnetic field (μ₀H<sub>max</sub>). For the case where magnetization measurements are performed at specific field and temperature intervals, the MCE can be described in a different form (Eq. (4) [31]):

$$
\Delta S_M(T, H) = \mu_0 \sum_i \frac{M_{i+1}(T_{i+1}, H) - M_i(T_i, H)}{T_{i+1} - T_i} \Delta H_i \quad (4)
$$

here, M<sub>i</sub> and M<sub>i+1</sub> represent magnetization values measured experi-
mentally at neighboring temperatures T<sub>i</sub> & T<sub>i+1</sub>, respectively, under a
constant applied magnetic field μ<sub>0</sub>ΔH<sub>i</sub>. Fig. 6 displays the variation of
(−ΔS<sub>M</sub>) vs. temperature (T) for GdPO<sub>4</sub>.

The peak of the −ΔS<sub>M</sub>(T) curve occurs near the T<sub>C</sub>. Notably, the
maximum peak value of (−ΔS<sub>M</sub><sup>max</sup>) increases from 20.43 to 44.79 J/(kg·K) when the magnetic field is increased from 2 to 9 T, respectively. Similar findings are reported in EuB<sub>4</sub>O<sub>7</sub> [32] and LiErF<sub>4</sub> [33]. This increase in (|ΔS<sub>M</sub><sup>max</sup>|) can be really understood by considering the rise of magnetization. MCE’s magnetic refrigerator application relies heavily on the RCP (Relative-Cooling-Power) as a crucial parameter [34]. To exploit RCP, Eq. (5) was used:

$$
\mathrm{RCP} = - \Delta S_{\mathrm{M}}^{\mathrm{max}}(T, H_{\mathrm{max}}) \times \delta T^{\mathrm{FWHM}} \quad (5)
$$

(−ΔS<sub>M</sub><sup>max</sup>) denotes the maximum of the entropy changes and δT<sup>FWHM</sup>
denotes the peak’s full width at half maximum. It was determined based

<figure><img src="image_9.png" /><figcaption>Fig. 7. The relative cooling power (RCP) of GdPO₄ varies as a function of the magnetic field.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 8. Universal master curves, average of (ΔS<sub>M</sub>/ΔS<sub>M</sub><sup>max</sup>) versus θ for GdPO<sub>4</sub> compound with Lorentz Fit (red line).</figcaption></figure>

on the temperature dependence of the magnetic entropy change ΔS<sub>M</sub>(T), as obtained from the Maxwell relation using isothermal magnetization data. The results presented here suggest that GdPO<sub>4</sub> is a promising candidate for low-temperature magnetic refrigeration. As shown in Fig. 7, the Relative Cooling Power (RCP) of GdPO<sub>4</sub> increases significantly with a stronger magnetic field.

The RCP values range from 112.02 J/kg (for μ₀H = 2 T) to 818.30 J/
kg (for μ₀H = 9T). This improvement in RCP aligns with findings re-
ported for other materials in previous studies [15,20,35]. The observed
increase in RCP can be attributed to two key factors: the rise in (−ΔS<sub>M</sub><sup>max</sup>)
and the broadening of its peak δT<sup>FWHM</sup>. The broadening of the peak
(δT<sup>FWHM</sup>) likely originates from the consideration of the uneven distri-
bution of atoms within the material, which can lead to variations in the
strength of magnetic interactions between Gd<sup>3+</sup> ions [36,37].

3.5. *Phenomenological universal curve*

Franco and Conde [38] proposed a method to analyse the behavior of
(ΔS<sub>M</sub>) in such magnetocaloric materials. This method, based on the
concept of a “universal curve,” assumes a specific scaling behavior near
a 2nd order phase transition [39]. The “universality” refers to the ability
to collapse ΔS<sub>M</sub> data from various magnetic field strengths into a single
<figure><img src="image_11.png" /><figcaption>Fig. 9. Modified Arrott plots (MAP) with 3D Ising model, Mean-field Mode, tri-critical mean-field and 3D-Heisenberg model (β = 0.365, γ = 1.336) model of GdPO₄.</figcaption></figure>

curve. This is achieved by normalizing each $-\Delta S_M$ curve by its peak value ($-\Delta S_M^{max}$) and then scaling both the temperature axis (T) and the $\Delta S_M$ axis (i.e., $\Delta S' = \Delta S_M(T)/(\Delta S_M^{max})$ [40]). The temperature axis is adjusted such that values fall below and above $T_C$. This scaling transformation creates a new variable called $\theta$, as shown in the following Eq. (6) [41]:

$$ \theta = \begin{cases} -(T - T_C)/(T_{r1} - T_C); & T \le T_C \\ (T - T_C)/(T_{r2} - T_C); & T > T_C \end{cases} \qquad (6) $$

with $T_{r1}$ and $T_{r2}$ represent the temperatures corresponding to two points on each curve. In this study, $T_{r1}$ and $T_{r2}$ were chosen because they approximately correspond to each other through the relationship $\Delta S_M(T_{r1,2}) = (1/2)\Delta S_M^{max}$. Fig. 8 display the average normalized change in entropy ($\Delta S'$) plotted against the rescaled temperatures ($\theta$) for GdPO₄ compound.

The data for various magnetic fields (2 T to 9 T) are combined into a single curve. This curve encompasses both the ferromagnetic ($\theta \le 0$) and paramagnetic ($\theta > 0$) regions. Notably, the collapse of the curves across different magnetic field strengths into a single curve strengthens the evidence for a 2nd order FM/PM phase transition in GdPO₄. This finding aligns with the Banerjee criteria, which will be discussed in a future work. As illustrated in Fig. 8, all experimental dots are distributed over a single universal curve. A Lorentz function can also be used to fit this data through the Eq. (7) [42]:

$$ L(x) = y_0 + \frac{2A}{\pi} \times \frac{w}{4(x - x_c)^2 + w^2} \qquad (7) $$

There are four parameters to know about the Lorentzian: A is the area, w is the width, namely “Width of the Lorentzian,” and $x_c$ is the peak’s abscissa. The Lorentzian function used to fit the universal entropy change curve is not arbitrary but physically motivated. Near second-order phase transitions, critical spin fluctuations cause a symmetric,

peak-like behavior in $\Delta S_M(T)$, which can be well-described by a Lorentzian profile. This fitting approach has been widely adopted in magnetocaloric literature [43,44] and serves as an effective empirical model for characterizing the scaling behavior of materials with second-order magnetic transitions.

### 3.6. Critical comportment

Understanding the magnetic phase transition in GdPO₄ provides valuable insights into the material’s magnetic behavior. This analysis can even help identify the model that best describes our system’s properties. As suggested by the scaling hypothesis [45], a 2nd order magnetic transition nearby $T_C$ can be further analyzed. Figs. 9 a-d demonstrate the modified Arrott plots for GdPO₄.

These plots depict isotherms $M^{1/\beta}$ vs. $(\mu_0 H/M)^{1/\gamma}$, where M is magnetization, $\mu_0$ is the permeability of free space, H is the applied magnetic field, and $\beta$ and $\gamma$ are critical exponents. The plots use four different models: (a) 3D-Ising model ($\beta = 0.325$, $\gamma = 1.24$), (b) Mean-field model ($\beta = 0.5$, $\gamma = 1$), (c) tri-critical mean-field model ($\beta = 0.25$, $\gamma = 1$); and (d) 3D-Heisenberg model ($\beta = 0.365$, $\gamma = 1.336$). In all plots, we observe approximately parallel quasi-straight lines in the high-field region, intersecting the origin at the Curie temperature ($T_C$) of 1.8 K in the $M^{1/\beta}$ vs. $(\mu_0 H/M)^{1/\gamma}$ plane. These lines are consistently parallel in the high-field region and also pass through the origin. These positive slopes indicate that GdPO₄ undergoes a 2nd order ferromagnetic to paramagnetic transition, consistent with the Banerjee criterion [46]. The field dependency of the MCE was examined in terms of the local exponent n, which depends on the magnetic state of the sample. The MCE field dependence of materials with a 2nd-order transition, according to both Oesterreicher and Parker [47] can be represented as Eq. (8):

$$ - \Delta S_M \propto \alpha (\mu_0 H)^n \qquad (8) $$
<table><caption>**Table 4**
Critical exponents based on modified Arrott plots using four theoretical models.</caption><thead><tr><th>Composition</th><th>Technique</th><th>T<sub>c</sub><br>(K)</th><th>β</th><th>γ</th><th>δ</th><th>n</th><th>Ref.</th></tr></thead><tbody><tr><td rowspan="2">GdPO<sub>4</sub></td><td>-ΔS<sup>max</sup> α</td><td>1.8</td><td>-</td><td>-</td><td>-</td><td>0.49</td><td rowspan="2">This work</td></tr><tr><td>H<sup>n</sup></td><td>1.8</td><td>-</td><td>-</td><td>2.94</td><td>-</td></tr><tr><td>Mean-field<br>theory</td><td>RCP α<br>H<sup>1+1/δ</sup></td><td>Theory</td><td>-</td><td>0.5</td><td>1</td><td>3</td><td>2/3</td><td>[48]</td></tr><tr><td>3D-Ising<br>model</td><td>Theory</td><td>-</td><td>0.325</td><td>1.24</td><td>4.82</td><td>-</td><td>[49]</td></tr><tr><td>3D-<br>Heisenberg<br>model</td><td>Theory</td><td>-</td><td>0.365</td><td>1.336</td><td>4.8</td><td>-</td><td>[50]</td></tr><tr><td>Tri-critical<br>mean-field</td><td>Theory</td><td>-</td><td>0.25</td><td>1</td><td>5</td><td>-</td><td>[51]</td></tr></tbody></table>

The value of n was calculated and shown in Table 4 and Fig. 10 (a).

In addition, the influence of the magnetic field on the RCP was also investigated. In materials undergoing a 2nd order phase transition, the relationship between RCP and magnetic field strength can often be described by a power law (Eq. (9) [52]):

$$RCP \propto (\mu_0 H)^{1+\frac{1}{\delta}}, \qquad (9)$$

However, this power-law relationship is only valid if the field dependence of the (-ΔS<sub>M</sub>) is also taken into account. Fig. 10(b) depicts the relationship between RCP and the magnetic field. Table 4 illustrates the comparison of critical exponents values of GdPO<sub>4</sub> with the various theoretical models. The δ and n values calculated for our sample are 2.94 and n = 0.49, respectively.

The value of the exponent n in the power law may be related to information about the magnetic order [53]. For a FM with magnetic correlation obeying the mean-field theory (MFT), n(MFT) = 2/3≈0.667. The value n = 0.49 obtained in the present work is not close to the n (MFT) value. These results have confirmed again the existence of short-range magnetic order in GdPO<sub>4</sub> nanospheres particles. The obtained values of n (Fig. 11) have minor deviation from the typical value of n = 2/3 within mean-field model [54]. This deviation can be explained by the presence of local inhomogeneous magnetic state in the aforementioned samples [55].

### 3.7. Spin interaction

Our previous findings confirmed short-range magnetic ordering in GdPO<sub>4</sub>. This research delves deeper to understand the various types of interactions between the magnetic spins (Gd<sup>3+</sup> ions) within the material. These spin interactions can be categorized based entirely on their strength and range. This study explores how spin-spin interaction in GdPO<sub>4</sub> operates. It proposes a model where the interaction strength J(r)

~ 1/r(d+σ). Here, d represents the lattice dimensionality (d=3 in this case), r represents distance and σ signifies the range of possible interactions [41]. Depending on the value of σ, the interaction range can be short or long. For σ = 2, the extended interaction model suggests a faster decay of J(r) proportional to r<sup>-5</sup>. Conversely, for σ = 3/2, the mean-field model applies, where J(r) weakens faster than r<sup>-4.5</sup>. The specific value of σ influences the susceptibility exponent, as shown in the following Eq. (10) [56]:

$$\gamma = 1 + \frac{4}{d} \left( \frac{n+2}{n+8} \right) \Delta\sigma + \frac{8(n+2)(n-4)}{d^2(n+8)^2} \times \left[ 1 + \frac{2G\left(\frac{d}{2}\right)(7n+20)}{(n-4)(n+8)} \right] \Delta\sigma^2 \qquad (10)$$

where n represents the spin dimensionality. We can define a parameter sigma (σ) as the difference between a constant (Δσ) and half the dimensionality (d) of the spins (σ = (Δσ - (d/2))). Additionally, another quantity, G(d/2), is defined by G(d/2) = 3-1/4(d/2)2. The value of (σ) was obtained as equal to 0.79. These results suggest a deviation from the mean-field model (MFM) predictions in our GdPO<sub>4</sub> sample.

### 4. Conclusion

This study investigated the magnetic and magnetocaloric properties of GdPO<sub>4</sub> in detail. Our findings suggest that the sample undergoes a 2nd order magnetism transition from ferromagnetic to paramagnetic constitutions state at a Curie temperature of around 1.8 K. This 2nd order

<figure><img src="image_12.png" /><figcaption>Fig. 11. n(T) curve of GdPO4 compound.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 10. Variation of (ln(-ΔS<sup>max</sup><sub>M</sub>) vs. ln(μ<sub>0</sub> H)) (a) and Variation of (ln(RCP) vs. ln(μ<sub>0</sub> H)) (b) for GdPO<sub>4</sub> compound.</figcaption></figure>
phase transition is confirmed by Banerjee criteria and further supported by the collapse of magnetic entropy change (−Δ*S*<sub>M</sub>) data into a “universal curve”. We calculated the value of (−Δ*S*<sub>M</sub>) using the Maxwell relation method. The magnetocaloric data obtained from the field dependence of (−Δ*S*<sub>M</sub>) was 20.43 J/(kg·K) at 2 T and 44.79 J/(kg·K) at 9 T, demonstrating its high efficiency for cryogenic refrigeration near its ferromagnetic transition temperature. Furthermore, the relative cooling power (RCP) values range from 112.02 J/kg (2 T) to 818.30 J/kg (9 T), highlighting a significant enhancement in cooling capacity under higher magnetic fields. Finally, exchange interaction analysis suggests a short-range magnetic order in GdPO₄ with interaction strength (J(r)) decaying proportionally to r⁻³.⁷⁹. Our analyses also have inferred that the GdPO₄ nanospherical particles exhibit a short-range magnetic order with coexisting ferromagnetic and antiferromagnetic interactions.

### CRediT authorship contribution statement

**Meryam Lassri:** Software, Formal analysis. **Said El Ouahbi:** Methodology, Formal analysis. **Imane Abdelrhafore:** Methodology, Formal analysis, Data curation. **Lhaj El Hachemi Omari:** Writing – review & editing, Writing – original draft, Visualization, Supervision, Resources, Conceptualization. **Abdelhadi El Hachmi:** Investigation, Formal analysis. **Hassan Hamouda:** Validation, Software, Formal analysis. **Benaissa Harradi:** Formal analysis, Data curation. **Yousra Ounza:** Investigation, Formal analysis. **Gianluca Viscusi:** Supervision, Data curation. **Claire Minaud:** Validation, Methodology, Data curation. **Charafeddine Jama:** Investigation, Data curation. **Hassan Lassri:** Investigation, Data curation.

### Declaration of competing interest

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.

### Acknowledgments

The authors gratefully acknowledge the assistance of Ms. Claire Minaud and the CHEVREUL Institute (FR 2638) in the development of this research. The CHEVREUL Institute is supported by the French Ministry of Higher Education, Research and Innovation, the Nord-Pas de Calais Region, and the European Regional Development Fund.

### Data availability

Data will be made available on request.

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