Investigating the magnetic and magnetocaloric behaviors of LiSm(PO₃)₄

Abstract

We report a detailed study on the magnetic behaviors and magnetocaloric (MC) effect of a single crystal of lithium samarium tetraphosphate, LiSm(PO₃)₄. The analyses of temperature-dependent magnetization data have revealed magnetic ordering established with decreasing temperature below T_p, where T_p is the minimum of a dM/dT vs. T curve and varies as a linear function of the applied field H. The Curie temperature has been extrapolated from T_p(H) data, as H → 0, to be about 0.51 K. The establishment of magnetic-ordering causes a substantial change in the heat capacity C_p. Above T_p, the crystal exhibits paramagnetic behavior. Using the Curie–Weiss (CW) law and Arrott plots, we have found the crystal to have a CW temperature θ_CW ≈ −36 K, and short-range magnetic order associated with a coexistence of antiferromagnetic and ferromagnetic interactions ascribed to the couplings of magnetic dipoles and octupoles at the Γ₇ and Γ₈ states. An assessment of the MC effect has shown increases in value of the absolute magnetic-entropy change (|ΔS_m|) and adiabatic-temperature change (ΔT_ad) when lowering the temperature to 2 K, and increasing the magnetic-field H magnitude. Around 2 K, the maximum value of |ΔS_m| is about 3.6 J kg⁻¹ K⁻¹ for the field H = 50 kOe, and ΔT_ad is about 5.8 K for H = 20 kOe, with the relative cooling power (RCP) of ∼82.5 J kg⁻¹. In spite of a low MC effect in comparison to Li(Gd,Tb,Ho)(PO₃)₄, the absence of magnetic hysteresis reflects that LiSm(PO₃)₄ is also a candidate for low-temperature MC applications below 25 K.

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1. Introduction

The magnetocaloric (MC) effect is an intrinsic property of any magnetic material. It is related to a temperature change of a magnetic material under an adiabatic process when an external magnetic field is applied. This phenomenon is generated from a decreased magnetic entropy that is compensated by the lattice-entropy change because the total entropy of the system under an isentropic process is unchanged. Together with the isentropic process, an adiabatic process also takes place irreversibly, after in the entropy–temperature (S–T) diagram.¹ In other words, the isentropic change of the material leads to an adiabatic temperature change (ΔT_ad), and when the magnetization is isothermally carried out, a magnetic-entropy change (ΔS_m) causes an isothermal total entropy change.^1–3 Using the adiabatic demagnetization and paramagnetic (PM) salts, typically Gd₂(SO₄)₃·8H₂O, (Dy,Ce)(C₂O₅SO₄)₃·9H₂O and (NH₄)₂SO₄MnSO₄·6H₂O, one has been given the birth of cryogenic and sub-Kelvin cooling.^1,4,5 The successful fabrication of adiabatic demagnetization refrigerators can somewhat solve the shortage and high price of ³He helium isotope.⁶

Comparing with the vapor-compression refrigeration cycle, the MC-based refrigeration has been proven to gain a higher efficiency,⁷ and does not cause environmental challenges because of using recyclable solid refrigerants composed of non-toxic elements.^1,8 Apart from using PM salts, it has been seeking for H₂O-free MC materials applicable for sub-Kelvin/cryogenic cooling devices^1,4,5 and space technology,⁹ particularly for the liquefaction of hydrogen and helium isotopes at below 20 and 4.2 K, respectively. These materials need to ensure some application requirements, such as low magnetic-ordering temperature, large |ΔS_m| and ΔT_ad values, no thermal hysteresis phenomenon, and no degrading at high temperatures and ultra-high vacuum. Until now, it has been introduced many materials with magnetic-ordering temperatures below 20 K satisfying with the above requirements, such as Re(V, P)O₄ zircon-type oxides (Re is a rare-earth element),^9–12 ReMO₃ perovskite-type oxides (M is a transition metal),^13–16 EuRe₂O₄ spinel-type oxides,^17,18 Re₃CrGa₄O₁₂ garnets,^19,20 KBaYb(BO₃)₂,²¹ and many alloys (including ErMn₂Si₂,²² Er₂Ni_1.5Ga_2.5,²³ Er₃Ni₆Al₂,²⁴ Ho₂Ni_0.95Si_2.95,²⁵ and YbPt₂Sn²⁶), in which YbPt₂Sn²⁶ could work from 2 K down to 0.2 K. Recently, Petrov et al.²⁷ have found a large MC effect in LiRe(PO₃)₄ single crystals (Re = Gd, Tb, and Dy), a material system has been widely used for optoelectronic applications.^28–30 Intriguing optical and magnetic properties of these LiRe(PO₃)₄ materials are related to Re³⁺ ions confined in the ReO₈ dodecahedral crystal field. In crystallography, along the b axis, edge-shared ReO₈ dodecahedra and LiO₄ tetrahedra in turn arrange, while PO₄ tetrahedra form helical chains. Additionally, the ReO₈ and LiO₄ chains share their corners with the helical PO₄ chains that constitute a 3D framework.^31,32 Together with the 3D framework, a large nearest Re–Re distance (∼5.6 Å^31,33) is thought to obstruct exchange interactions between Re ions, leading to a large MC effect of LiRe(PO₃)₄ at low temperatures. Investigating LiRe(PO₃)₄ materials with Re = Gd, Tb, Dy and Pr, Petrov et al.^27,32 found their |ΔS_m| and ΔT_ad values at 2 K are about 9.8–27.6 J kg⁻¹ K⁻¹ for the field H = 50 kOe and 3–13 K for H = 20 kOe, respectively.

To learn more about this LiRe(PO₃)₄ material system, we have prepared a single crystal of lithium samarium tetraphosphate, LiSm(PO₃)₄, and investigated in detail its structural characterization, magnetic behaviors and MC effect. To the best of our knowledge, no previous work on the magnetocaloric behaviors of LiSm(PO₃)₄, though its electronic and optical properties have been widely studied.^34–36 Herein, the magnetic behaviors have studied through the magnetization (M) and heat-capacity (C_p) measurements versus the temperature and magnetic field. Based on these M and C_p data, and thermodynamic relations, the characteristic parameters of the MC effect (|ΔS_m| and ΔT_ad) have been evaluated in comparison to the previous works on low-temperature MC materials.

2. Experimental details

A single crystal (SC) of LiSm(PO₃)₄ was grown by using the flux method.³⁷ After fabrication, its crystal structure was analyzed by using an X-ray diffractometer in Bragg–Brentano geometry (Malvern Panalytical) equipped with an X-ray wavelength of 1.5406 Å. To record the magnetization and heat-capacity data dependent on the temperature and magnetic field, M(T, H) and C_p(T, H), respectively, we used a Quantum Design Physical Properties Measurement System (PPMS-9). Specifically, M(T, H) data were recorded at temperatures T = 2–300 K and magnetic fields H = 0–50 kOe, while C_p(T, H) data were recorded at T = 2–300 K and H = 0–20 kOe. After collecting M(T, H) and C_p(T, H) data, |ΔS_m(T, H)| and ΔT_ad were calculated by using Maxwell's relations and thermodynamic equations.

3. Results and discussion

The LiSm(PO₃)₄ SC after fabricated has pale yellow (see the inset of Fig. 1), which was roughly ground to examine the structural properties by using the X-ray diffraction (XRD) method. The X-ray data analysis based on the profile matching mode (Le Bail fit) belonging to the FullProf structural refinement software³⁸ has revealed that the diffraction peaks can be fitted successfully only with a monoclinic phase with the C2/c symmetry, indicating a single-phase nature of our sample. The refined lattice parameters, a = 16.3784(7) Å, b = 7.0373(6) Å, c = 9.6895(5) Å, and β = 126.071(3)°, are well consistent with those obtained from previous works on LiSm(PO₃)₄.^39,40

Fig. 1 Room-temperature XRD pattern of LiSm(PO₃)₄. Experimental data (red symbols) and calculated profile (blue lines) are shown. Vertical blue ticks denote the nuclear reflections of the monoclinic C2/c phase. The inset shows an image of LiSm(PO₃)₄ single crystals studied in this work.

Following the structural characterization, we have studied the magnetic behaviors of LiSm(PO₃)₄ upon analyzing M(T, H) and C_p(T, H) data. In Fig. 2, it graphs M(T) data measured for the sample in powder according to the increasing and decreasing directions of temperature for an applied field H = 1 kOe. Similar to Li[Gd,Tb,Dy](PO₃)₄ (ref. 27) and (Er,Yb)VO₄ (ref. 9) materials, we have also observed a gradual decrease in M when T increases from 2 to ∼300 K. Though T reached 2 K, no trace of magnetic-ordering transition temperatures (Néel/Curie temperatures, T_N/T_C) was observed. The performance of the magnetic susceptibility (χ⁻¹ = H/M) from these M(T) data, and the extrapolation of the high-temperature Curie–Weiss (CW) law would obtain the CW temperature (θ_CW) of about −36 K, as shown in Fig. 2. A negative θ_CW value reflects that antiferromagnetic (anti-FM) super-exchange interactions take place between Sm³⁺ ions in SmO₈ chains of monoclinic LiSm(PO₃)₄ mediated by O and P ions. In other words, LiSm(PO₃)₄ is anti-FM with the Néel temperature (T_N) lower than 2 K. Measuring a full M(H) loop at 2 K, we have found no magnetic hysteresis. M tends to saturate as H > 10 kOe, with a saturation value (M_s) of ∼22.4 emu g⁻¹ (or ∼1.97 μ_B/f.u.), see the inset of Fig. 2. This effective moment value is fairly greater than the theoretical value of g_J[J(J + 1)]^1/2 = 0.85μ_B for a free Sm³⁺ ion in the ground state (4f⁵, ⁶H_5/2, with S = 5/2, L = 5 and J = L − S), and experimental values obtained for Sm³⁺ in van-Vleck-paramagnetic compounds of Sm₃Sb₃Zn₂O₁₄ (ref. 41) and SmZnAl₁₁O₁₉.⁴² Such differences could be due to the fact that Sm³⁺ is a Kramers ion and occupies in various crystal fields. In the SmO₈ dodecahedral field, there could be a degenerate ground state with several types of degrees of freedom. After the ground term ⁶H_5/2, the next higher order term, J = L − S + 1 (⁶H_7/2), has a moment value of 3.32μ_B. Energy levels of these terms seem close to each other, and can mix as a function of T and H. Additionally, due to the crystalline-electric-field effect, a J = 5/2 ground state can split into a Γ₇ doublet and a Γ₈ quartet. The Γ₇ state has a magnetic dipole while the Γ₈ state has a magnetic dipole, electric quadrupoles and magnetic octupoles.⁴³ The interactions of magnetic dipoles and octupoles could result in a large value of magnetic moment of LiSm(PO₃)₄. As studying the materials having strongly-correlated electron states via hybridization with electrons, it has also been found large magnetic moments of 1.2–1.7μ_B.⁴³ Measuring a series of M(T) curves at higher fields H = 2–50 kOe and T = 2–50 K, as shown in Fig. 3(a), we have found a gradual increase of paramagnetic background signals, and M becomes saturated at M_s ∼ 22.4 emu g⁻¹ as H ≥ 10 kOe. There is a temperature range (named a saturation region) from 2 K that M is less changed and approximately equal to M_s. It broadens towards high temperatures when H increases above 30 kOe, see a dashed rectangle plotted in Fig. 3(a). After the saturation region, M would gradually decrease with increasing T. Interestingly, having performed dM(T)/dT data, we have observed the minimum that starts appearing as H ≈ 5 kOe at the so-called T_p temperature. T_p tends to shift towards higher temperatures as increasing H, see Fig. 3(b) and its inset. This proves an H-dependent ferromagnetic–paramagnetic (FM–PM) separation at T_p. These T_p(H) data can be described a linear function of T_p = 0.51 + 0.48 × H, as shown in Fig. 4. The zero-field extrapolation, H → 0, would obtain the Curie temperature (T_C) of LiSm(PO₃)₄ at 0.51 K. A negative θ_CW value together with these results suggest a coexistence of FM and anti-FM interactions induced by Sm³⁺ ions (probably associated with interactions between magnetic dipoles and octupoles of the Γ₇ and Γ₈ states) in LiSm(PO₃)₄ at low temperatures and fields. The high-field application promotes FM-ordering establishment, leading to the FM–PM transition at T_p. It is necessary to measure M(T, H) data at lower temperatures (<2 K) to identify exactly LiSm(PO₃)₄ being a ferromagnet and/or antiferromagnet.

Fig. 2 M(T) and χ⁻¹(T) curves of LiSm(PO₃)₄ in the field H = 1 kOe, and the straight line is plotted according to the CW law. The inset is an M(H) loop recorded at T = 2 K.

Fig. 3 (a) M(T) and (b) dM(T)/dT data of LiSm(PO₃)₄ in different applied fields of H = 2–50 kOe. The inset plots an enlarged view of the dM(T)/dT data for H = 6–30 kOe showing the minima at T_p.

Fig. 4 H-Dependent T_p data of LiSm(PO₃)₄ showing a FM–PM phase separation.

The FM-ordering establishment in the LiSm(PO₃)₄ sample can be further confirmed upon C_p(T) measurement at different applied fields for a bulk sample. The C_p(T, H) data for H = 0–20 kOe graphed in Fig. 5 indicate a gradual decrease of C_p as decreasing T from 300 K to T_m, which is defined as the point that a decrease in temperature below it will enhance C_p. T_m is about 3.7 K for H = 0 and increases to ∼6.7 K for H = 20 kOe. An enhancement of C_p as lowering T below T_m is assigned to the magnetic contribution to heat capacity,⁴⁴ demonstrating the establishment of magnetic ordering in this material. Due to the limit of low-temperature measurement, we could not obtain the onset of magnetic ordering (i.e., T_N or T_C) corresponding to a (Schottky) peak in the C_p(T, H) curves such as phenomena observed in materials GdVO₄,⁹, (Tb,Dy,Ho)₃CrGa₄O₁₂,¹⁹ Li(Gd,Tb,Dy)(PO₃)₄,²⁷ and ErRuSi.⁴⁴ It should be noticed that the magnetic contribution to C_p is mainly at temperatures below T_m. Above T_m, the phonon contribution (C_ph) to C_p plays a dominant role, consequently C_p is less dependent on H. We have found that the C_p(T) data for H = 0 and T = 2–100 K can be described by the combination of one Debye and one Einstein terms for phonons:⁴⁵

see the inset of Fig. 5, where θ_D is the Debye temperature, θ_E is the Einstein temperature, and m and n are the fit parameters. The sum m + n was fixed to the total number of atoms in the formula unit of LiSm(PO₃)₄. Herein, the obtained values of θ_D and θ_E are about 217.6 and 439.4 K, respectively.

Fig. 5 C_p(T) data of LiSm(PO₃)₄ in the log–log scale for H = 0, 10, and 20 kOe. The insets plots C_p(T) data for H = 0 fitted to a combination of Debye–Einstein functions, eqn (1).

Magnetic behaviors of LiSm(PO₃)₄ can be further understood as considering M(H) isotherms recorded at T = 2–50 K. Typical M(H) curves shown in Fig. 6(a) indicate the change in their curvature when T increases from 2 to 40 K, particularly at fields H = 0–20 kOe. In other words, low-temperature nonlinear M(H) curves become linear as increasing T above 30 K. This indicates the collapse of magnetic ordering, and the material becomes paramagnetic at high temperatures. Similar to M(T) data shown in Fig. 3(a), below 8 K, M goes to saturate as raising H above 30 kOe. As performing M²(H/M) plots, see Fig. 6(b), it comes to our attention that these curves have positive slopes, and are not parallel straight lines. These features reflect that LiSm(PO₃)₄ undergoes a second-order nature in the investigating temperature range (according to Banerjee's criteria⁴⁶) and exhibiting short-range magnetic order (according to Arrott and Noakes^47,48). Short-range magnetic order in LiSm(PO₃)₄ is ascribed to a coexistence of FM and anti-FM interactions between Sm³⁺ ions via O and P atoms. Additionally, under the impacts of the crystal and magnetic fields, Kramers doublet on Sm³⁺ ions could have more (2J + 1)-fold degeneracy, leading to multiplets.³² The jj-coupling and/or spin population on these multiplets could also cause short-range magnetic order in LiSm(PO₃)₄.

Fig. 6 (a) Representative M(H) and (b) M²(H/M) data of LiSm(PO₃)₄ at temperatures T = 2–40 K, where temperature increments are maintained at 2 K.

Together with the magnetic behaviors, we have also studied the MC effect through the parameters |ΔS_m| and ΔT_ad calculated by the following expressions:¹

Fig. 7(a) shows |ΔS_m| data of LiSm(PO₃)₄ at temperatures T = 2–50 kOe in magnetic fields H = 5–50 kOe, with magnetic-field increments (ΔH) fixed at 5 kOe. One can see that |ΔS_m(T)| increases with increasing H, particularly at T < 25 K. Due to magnetic saturation, |ΔS_m(T)| less changes at high fields (>30 kOe). At T = 2 K, |ΔS_m| is largest (|ΔS_max|) of ∼3.6 J kg⁻¹ K⁻¹ for H = 50 kOe. This value is fairly smaller than the |ΔS_m| values (9.8–27.6 J kg⁻¹ K⁻¹, H = 50 kOe) obtained for Li(Gd,Tb,Dy,Pr)(PO₃)₄ single crystals.^27,32 It is reasonable because magnetic moment of Sm³⁺ is lower than that of Gd³⁺, Tb³⁺, Dy³⁺, and Pr³⁺.⁴⁹ Considering |ΔS_m(H)| data at T = 2 K, we have found that these data could be described by a power law of y = a × Hⁿ, with a = 1.05 and n = 0.31. This n value falls in the range of Li(Gd,Tb,Dy)(PO₃)₄ materials with n = 0.12–0.64,²⁷ but smaller than the value of mean-field theory (MFT) n = 0.67.⁸ According to Franco et al.,⁸ n has a relationship with another parameter N(T, H) characteristic for magnetic ordering, which is calculated as follows:

Fig. 7 (a) |ΔS_m(T)| data of LiSm(PO₃)₄, and (b) |ΔS_m(H)| data at T = 2 K fitted to a function y = a × Hⁿ.

For ferromagnets obeying the MFT (long-range ferromagnets), N goes to 1 and 2 as T ≪ T_C and T ≫ T_C, respectively. It achieves the minimum at T = T_C, and n = N(T_C) = 0.67, which is independent of H.⁸ In our work, though N tends to 2 at high temperatures (corresponding to the paramagnetic region), the minima of N(T, H) are about 0.1–0.2 and strongly dependent on H, which are much lower than the MFT value, as shown in Fig. 8.

Fig. 8 N(T, H) data of LiSm(PO₃)₄ for H = 5–50 kOe calculated by using eqn (4).

This deviation is ascribed to (i) |ΔS_m(H)| and N(T, H) values calculated at temperatures higher than T_N/T_C, and (ii) the absence of long-range magnetic order in LiSm(PO₃)₄. If combining |ΔS_m(T)| and C_p(T) data for H = 0–20 kOe, we would evaluate ΔT_ad(T) values upon eqn (3). As seen in Fig. 9, ΔT_ad increases gradually when T decreases below 12 K. The maximum ΔT_ad for LiSm(PO₃)₄ are about 4 and 5.8 K at 2 K, for applied fields of 10 and 20 kOe, respectively, which is higher than the maximum ΔT_ad (∼2.8 K) of LiPr(PO₃)₄ (ref. 32) at the same conditions. For Li(Gd,Tb,Dy)(PO₃)₄, they have fairly higher maximum ΔT_ad values of 11.9–12.9 K at 2 K and H = 20 kOe.²⁷ These differences are mainly related to the ground-state degeneracy of RE ions due to spin–orbit coupling and impacts of crystal and magnetic fields, depending on each rare-earth ion type.

Fig. 9 ΔT_ad(T) calculated from C_p(T, H) data of LiSm(PO₃)₄ for H = 10 and 20 kOe, using eqn (3).

Together with |ΔS_m| and ΔT_ad, it is also additionally assessed the relative cooling power (RCP) and/or the refrigeration capacity (RC) by using the following expressions:^1,8

RCP = |ΔS_max| × δT, (5)

where δT is the full-width-at-half maximum of a |ΔS_m(T)| curve. Meanwhile, T₁ and T₂ are defined as the cold and hot ends, respectively, of an ideal thermodynamic cycle that are usually selected at the middle points of a |ΔS_m(T)| curve. With |ΔS_m(T)| curves shown in Fig. 7(a), one can see their linewidth (δT) increases with increasing H, and δT ≈ 23 K for H = 50 kOe, see Fig. 10(a). In the investigating H range, δT(H) dependence is almost linear (described by a function y ≈ 0.5 × H^0.98). An enhanced δT is expected to increase RCP and RC with respect to H. As shown in Fig. 10(b), both RCP and RC increases according to power functions of H (y = b × H^m, with m = 1.31 and 1.24 for RC and RCP, respectively), in which RCP is about 1.2–1.4 time higher than that of RC. For H = 50 kOe, RCP (RC) is about 82.4 (57.4) J kg⁻¹.

Fig. 10 (a) δT(H) and (b) RCP(H) and RC(H) data of LiSm(PO₃)₄ for H = 0–50 kOe fitted to a power law of y = b × H^m, with the values of b and m labeled in the figures.

For comparison, we have summarized the MC behaviors (|ΔS_max| and RCP/RC values) of LiSm(PO₃)₄ compared with those of some typical materials with T_N (T_C) < 10 K, for the field H = 50 kOe. These materials are potential magnetic coolants useable for liquefying helium and hydrogen gases. The data shown in Table 1 reveals that LiSm(PO₃)₄ has MC values of |ΔS_max| = 3.6 J kg⁻¹ K⁻¹ and RCP (RC) = 82.4 (57.4) J kg⁻¹. Belonging to the same family, Li(Gd,Tb,Dy)(PO₃)₄ have larger MC values, with |ΔS_max| = 11.7–27.6 J kg⁻¹ K⁻¹ and RCP (RC) = 174–334 (133–254) J kg⁻¹,²⁷ which are comparable to MC behaviors of DyVO₄ (ΔS_max| = 19.8 J kg⁻¹ K⁻¹ and RCP ≈ 300 J kg⁻¹)⁵¹ and ErRu₂Si₂ (ΔS_max| = 17.6 J kg⁻¹ K⁻¹ and RCP = 278 J kg⁻¹).⁵³ It has been found large MC values as studying NaGdF₄ (|ΔS_max| = 51.2 J kg⁻¹ K⁻¹ and RCP (RC) = 331 (250) J kg⁻¹),⁵⁰ and EuTiO₃ (|ΔS_max| = 40.4–42.2 J kg⁻¹ K⁻¹ and RCP = 300–353 J kg⁻¹).¹⁴ Studying EuSe, it has found its MC values being very large with |ΔS_max| = 37.5 J kg⁻¹ K⁻¹ and RCP = 580 J kg⁻¹.⁵³ Though Sm³⁺-related MC materials have been less studied, it is clear that LiSm(PO₃)₄ has fairly humble MC values. We think that spin–orbit interaction (L–S coupling) and the J = 5/2 ground-state degeneracy into the Γ₇ and Γ₈ states lead to a competition of FM and anti-FM interactions between Sm³⁺ ions, meaning short-range magnetic order, consequently low MC effect in LiSm(PO₃)₄.

Table 1 MC behaviors of LiSm(PO₃)₄ and typical MC materials with T_N (T_C) < 10 K at H = 50 kOe

Compound	TN/TC (K)	|ΔSmax| (J kg^−1 K^−1)	RCP/RC (J kg^−1)	Ref.
LiSm(PO3)4	/0.51	3.6	82.4/57.4	This work
LiGd(PO3)4	—	27.6	174/133	27
LiTb(PO3)4	—	15.9	334/254	27
LiDy(PO3)4	—	11.7	236/193	27
NaGdF4	2.3	51.2	331/250	50
DyVO4	3.5	19.8	∼300	51
EuSe	4.6	37.5	580	52
EuTiO3	4.4–5.6	40.4–42.2	300–353	14
ErRu2Si2	5.5	17.6	278	53

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4. Conclusion

We recorded XRD, dc magnetization M(T, H), and heat capacity C_p(T, H) data to investigate the magnetic and MC behaviors of LiSm(PO₃)₄. Rietveld refinement demonstrated a single phase in the C2/c monoclinic structure of LiSm(PO₃)₄. The analyses of M(T), M(H) and C_p(T, H) data demonstrated the magnetic-ordering establishment at low temperatures, and a coexistence of FM and anti-FM interactions with θ_CW ≈ −36 K and T_C ≈ 0.51 K. The FM phase tends to widen towards high temperatures as increasing H. MC assessments based on |ΔS_m| and ΔT_ad indicated an increase of these parameters as lowering T down to 2 K, and increasing the H magnitude. At 2 K, the maximum values of |ΔS_m|, and ΔT_ad are about 3.6 J kg⁻¹ K⁻¹ for H = 50 kOe, and 5.8 K for H = 20 kOe. Thus, the corresponding values of RCP and RC are about 82.5 and 57.4 J kg⁻¹ for H = 50 kOe, respectively. It should be noticed that the MC effect of LiSm(PO₃)₄ is fairly smaller than that of Li(Gd,Tb,Dy)(PO₃)₄ having the same structure. Strong L–S coupling, the J = 5/2 ground-state degeneracy into the Γ₇ and Γ₈ states, and the competition of FM and anti-FM interactions between Sm³⁺ ions, causing short-range magnetic ordering, are thought to lead to low MC effect of LiSm(PO₃)₄.

Author statement

T. A. Tran, Dimitar N. Petrov, N. T. Dang, and T. L. Phan: conceptualization, methodology and writing; T. P. Hoang, H. C. Tran, B. D. Tu, Yu S. Koshkid'ko, B. Weise, and V. M. Tien: investigation and data analyses; and J. Ćwik, and H. N. Nhat: reviewing and editing.

Conflicts of 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.

Acknowledgements

This work belongs to the project grant no: T2022-17 funded by Ho Chi Minh City University of Technology and Education (Vietnam).

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