Study of the Magnetocaloric Effect and Critical Behavior in Double Perovskite Manganese Oxides Pr 1.5 A 0.5 Mn 2 O 6 (A=Mg, Ba)

Study of the Magnetocaloric Effect and Critical Behavior in Double Perovskite Manganese Oxides Pr 1.5 A 0.5 Mn 2 O 6 (A=Mg, Ba) | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Study of the Magnetocaloric Effect and Critical Behavior in Double Perovskite Manganese Oxides Pr 1.5 A 0.5 Mn 2 O 6 (A=Mg, Ba) Huiqin Yun, Ze Li, Xiang Jin, Jianjun Zhao, Jingshun Liu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4141743/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 7 You are reading this latest preprint version Abstract In this study, polycrystalline samples of Pr 2 Mn 2 O 6 (parent phase) and Pr 1.5 A 0.5 Mn 2 O 6 (A = Mg, Ba) were prepared using the high-temperature solid-phase reaction method. The effects of Mg and Ba doping on magnetocaloric properties and critical behavior of the parent phase were systematically investigated. Under a magnetic field of 7 T, the relative cooling power (RCP) values for this sample series were approximately 483.46 J·kg − 1 , 428.22 J·kg − 1 , and 479.88 J·kg − 1 , respectively. The critical behavior analysis revealed that the parent phase showed short-range exchange interactions, while Pr 1.5 A 0.5 Mn 2 O 6 (A = Mg, Ba) exhibited long-range exchange interactions. The temperature dependence of the order parameter n was studied under different magnetic fields, confirming the phase transition types and validating the accuracy of the critical exponents obtained. The research findings suggest that both the parent phase and Ba-doped ceramics at the A-site hold promise as magnetic refrigeration materials. Magnetocaloric effects Magnetic phase transition Second-order phase transition Critical behavior Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 1 Introduction In recent years, magnetocaloric refrigeration (MR) based on the magnetocaloric effect (MCE) has emerged as a promising technology to replace traditional gas compression-expansion techniques[ 1 ] [ 2 ]. Researchers are actively seeking more suitable magnetocaloric materials, and a considerable focus has been placed on perovskite manganites of the form Ln 1 − x A x MnO 3 (Ln = La, Pr, Nd, ..., A = Sr, Ca, Ba, ...) due to their extensive studies on electrical and magnetic properties with potential applications[ 3 ]. Phan and Yu have provided a comprehensive overview of the observed magnetocaloric effects in manganese oxides[ 4 ]. The investigation of the LnMnO 3 system is particularly intriguing, characterized by significant MCE values and adjustable phase transition temperatures. These manganese oxides offer convenient synthetic routes, and their Curie temperatures ( T C ) are reasonable under various doping conditions. Consequently, the study of the magnetocaloric effect in manganese oxide composites has become a new developmental trend. Manganese oxides are generally classified into first-order phase transition materials and second-order phase transition materials based on the type of phase transition. It is well-known that first-order phase transition materials often exhibit significant magnetocaloric effect (MCE). However, they are accompanied by challenges such as a narrow cooling temperature range, relatively small refrigeration capacity, and large magnetic hysteresis[ 5 ]. On the other hand, second-order phase transition materials, while having a wider cooling temperature range and smaller magnetic hysteresis, face the drawback of a slightly lower maximum magnetic entropy change, making them less practical for applications[ 5 ]. Law et al. [ 6 ] have pointed out that ideal magnetocaloric materials may lie in a critical state between first and second-order phase transitions. Therefore, the focus of research in the field of magnetocaloric refrigeration should be on the study of MCE and phase transition control to identify suitable magnetocaloric materials. This study focuses on modulating the properties of Pr 2 Mn 2 O 6 through the doping of Mg and Ba atoms. The introduction of Mg and Ba may induce lattice distortions and adjust magnetic interactions, thereby influencing the magnetocaloric effect and phase transitions of Pr 2 Mn 2 O 6 . We anticipate that the results of this study will not only contribute to a deeper understanding of the complex magnetic behavior of Pr 2 Mn 2 O 6 but also provide new insights for the development of functional materials with superior magnetic properties. 2 Experimental methods The polycrystalline samples of perovskite manganese oxides Pr 1.5 A 0.5 Mn 2 O 6 (A = Mg, Ba) were prepared using the traditional solid-state reaction method. High-purity (> 99.99%) Pr 6 O 11 , MgO, BaCO 3 , and MnO 2 dry solid powder raw materials (from Alfa Aesar, America) were accurately weighed according to stoichiometric ratios. The weighed samples were then thoroughly mixed and ground using a ball mill (QM-QX04, full range planetary ball mill, Nanjing Nanda Instruments Co., Ltd., China) for 6 hours. Subsequently, the well-ground samples were first dried and then subjected to decarbonization treatment at 950°C for 12 hours in a box-type sintering furnace (TM KSL-1700X, Hefei Kejing Materials Technology Co., LTD., China). After a second grinding, the samples were calcined again at a high temperature of 1350°C for 24 hours in the box-type sintering furnace, followed by a controlled cooling from 1350°C to 800°C over 5 hours, and then naturally cooled to room temperature. This process yielded well-formed phase samples of Pr 2 Mn 2 O 6 , Pr 1.5 Mg 0.5 Mn 2 O 6 , and Pr 1.5 Ba 0.5 Mn 2 O 6 (hereinafter referred to as Sample-A, Sample-B, and Sample-C, respectively). The samples' crystal structure and phase purity were analyzed using a PANalytical X'Pert Powder X-ray diffractometer. Their magnetization as a function of temperature ( M ( T )) was assessed in a 0.05 T magnetic field, while magnetization versus magnetic field strength ( M ( H )) was evaluated over a 0–7 T range using a Quantum Design PPMS-9 multifunctional physical property measurement system. 3 Results and analysis Figure 1 presents the refined X-ray diffraction spectra of polycrystalline samples. No significant impurity peaks were observed in any of the samples, indicating that they possess good single-phase properties and belong to the orthorhombic crystal system with the space group Pbnm. Table 1 lists the cell parameters and refinement parameters for each sample, obtained after refinement using MDI Jade 6. As can be seen from Table 1 , the lattice parameters a, b, and c, as well as the cell volume V, decrease with Mg doping and increase with Ba doping. This is due to the smaller ionic radius of Mg (0.65 Å) compared to Pr (1.13 Å), and the larger ionic radius of Ba (1.49 Å) compared to Pr. Table 1 Lattice parameters of Sample-A, Sample-B, and Sample-C. samples a (Å) b (Å) c (Å) V (Å 3 ) χ 2 R Bragg Sample-A 5.471 5.590 7.681 234.920 1.5 9.8 Sample-B 5.470 5.470 7.788 233.090 1.2 9.7 Sample-C 5.506 5.531 7.781 236.990 1.6 9.8 The Fig. 2 illustrates the zero-field (ZFC) and field-cooled (FC) magnetization curves for three samples under an applied magnetic field of 0.05 T, with the inset displaying the d M /d T - T curves. It is observable from the graph that all three samples exhibit paramagnetic behavior at relatively higher temperature ranges. As the temperature decreases, the magnetic nature of the samples transitions from paramagnetic (PM) to ferromagnetic (FM) at their respective T C ( T C(Sample−A) ≈ 91 K, T C(Sample−B) ≈ 77 K, and T C(Sample−C) ≈ 152 K). It is evident that doping with Mg and Ba respectively decreases and increases the magnetization strength and T C of the samples. The doping with Mg leads to a reduction in the Mn-O-Mn bond angle, which in turn increases the lattice distortion in the samples, weakening the double-exchange interaction between Mn 3+ and Mn 4+ , resulting in a lower T C and reduced magnetization strength[ 1 ]. Conversely, doping with Ba has the opposite effect[ 8 ]. Additionally, a distinct bifurcation in the magnetization curves of these samples at low temperatures is observed, which may be attributed to the presence of domain-wall pinning effect[ 9 ] or spin reorientation[ 10 ] . We performed measurements to evaluate the effect of magnetic field variability near the T C on the magnetization of the samples. The M – µ 0 H curves were plotted as illustrated in Fig. 3 . The data show that above T C , M increases in a linear fashion with µ 0 H , which indicates the presence of an unsaturated paramagnetic state. Below T C , the plots show a sharp increase in M with µ 0 H , tending toward saturation and exhibiting ferromagnetic behavior. The partial saturation that has been observed at low temperatures can be attributed to the competition between antiferromagnetic and ferromagnetic clusters within the sample[ 11 ]. In all samples, no significant hysteresis was observed in the M – µ 0 H curves below and around T C , suggesting the possible presence of a magnetic field driven second-order phase transition in the system[ 12 ] . To investigate the magnetocaloric effect of this series of samples, we calculated the isothermal magnetic entropy change (|-Δ S M |) of the samples using the Maxwell thermodynamic equations, as shown in the following relationship[ 13 ] : $$\begin{array}{c}\left|-{\Delta }{S}_{\text{M}}\right|={\int }_{0}^{{H}_{\text{m}\text{a}\text{x}}}{\left(\frac{\partial M}{\partial T}\right)}_{\text{H}}dH\#（1）\end{array}$$ Figure 4 shows the temperature dependence of the |-Δ S M | in the sample. The |-Δ S M | increases with the applied magnetic field. This can be explained by the strengthening of magnetic order with the increase in the external magnetic field. As the magnetic field enhances, the magnetization near the T C increases, leading to a higher |-Δ S M | value. Under a 7 T magnetic field, the maximum magnetic entropy changes are approximately |- ∆ S M max | (Sample−A) ≈ 4.40 J·kg − 1 ·K − 1 , |- ∆ S M max | (Sample−B) ≈ 3.66 J·kg − 1 ·K − 1 , and |- ∆ S M max | (Sample−C) ≈ 3.72 J·kg − 1 ·K − 1 . Notably, although the |- ∆ S M max | decreases with Mg and Ba doping, the refrigeration temperature range (FWHM) broadens (FWHM (Sample−A) ≈ 110 K, FWHM (Sample−B) ≈ 117 K, FWHM (Sample−C) ≈ 128 K), suggesting a more continuous phase transition in doped samples. The reduction in |- ∆ S M max | is consistent with the broadening of the FM-PM transition[ 14 ] . From a refrigeration perspective, the Relative Cooling Power (RCP) is a crucial parameter for evaluating the cooling capacity of materials. The equation is as follows[ 15 ] : $$\begin{array}{c}RCP=\left|-\varDelta {S}_{\text{M}}^{\text{m}\text{a}\text{x}}\right|\bullet \left(\varDelta {T}_{\text{F}\text{W}\text{H}\text{M}}\right)\#\left(2\right)\end{array}$$ Herein, ∆ T FWHM denotes the full-width at half-maximum of the |-∆ S M |( T ) profile. It is observed that at a magnetic field strength of 7 T, the RCP values are approximately RCP (Sample−A) ≈ 483.46 J·kg − 1 , RCP (Sample−B) ≈ 428.22 J·kg − 1 , and RCP (Sample−C) ≈ 478.88 J·kg − 1 . This demonstrates that the reduction in |-∆ S M max | due to doping with Mg and Ba surpasses the broadening of the cooling temperature range. Additionally, Table 2 presents the magnetocaloric parameters for PrMnO system refrigeration materials. The findings indicate that Sample-A in this study is more apt as a magnetocaloric refrigerant. Table 2 Comparison of the magnetocaloric parameters of Sample-A, Sample-B, and Sample-C with some other magnetic refrigerants in magnetic fields of 5 T. Samples T C / K |- Δ S M |/ J·kg − 1 ·K − 1 RCP / J·kg − 1 µ 0 Δ H/ T Ref Sample-A 91 4.40 483.46 7 This work 3.35 301.85 5 Sample-B 77 3.66 428.22 7 This work 2.84 269.35 5 Sample-C 152 3.72 479.88 7 This work 3.00 311.53 5 Pr 0.87 Ca 0.13 MnO 3 - 4.608 460.77 7 [ 16 ] Pr 2 CoMnO6 - 1.862 126.62 7 [ 17 ] Pr 0.55 Sr 0.45 MnO 3 291 1.70 143.60 3 [ 18 ] In addition, the Normalized Refrigeration Capacity (NRC) serves as an alternative metric for the selection of magnetic refrigeration materials. The formula for NRC is[ 19 ] : $$\begin{array}{c}NRC=\frac{1}{{\Delta }{\mu }_{0}H}{\int }_{{T}_{C\text{o}\text{l}\text{d}}}^{{T}_{\text{H}\text{o}\text{t}}}\left|\varDelta {S}_{\text{M}}\left(T\right)\right|dT\#\left(3\right)\end{array}$$ where T Hot and T Cold represent the high and low-temperature ends, respectively. As depicted in Fig. 5 (a), the plots of the maximum NRC (NRC max ) for temperature intervals of 4 K under external magnetic fields of 2 T and 5 T are shown. It is evident that NRC max escalates with the increase in Δ T H−C , the temperature differential between the hot and cold ends. Notably, at 2 T, both Sample-A and Sample-C exhibit enhanced magnetic refrigeration efficiency. The Temperature-Averaged Entropy Change (TEC) is another indicator used to quantify the MCE of magnetic refrigeration materials within a specific temperature range. It is calculated using the following formula[ 20 ] : $$\begin{array}{c}TEC\left(\varDelta {T}_{\text{H}-\text{C}}\right)=\frac{1}{\varDelta {T}_{\text{H}-\text{C}}}max\left\{{\int }_{{T}_{\text{m}\text{i}\text{d}}-\frac{\varDelta {T}_{\text{H}-\text{C}}}{2}}^{{T}_{\text{m}\text{i}\text{d}}+\frac{\varDelta {T}_{\text{H}-\text{C}}}{2}}\left|\varDelta {S}_{\text{M}}\right|\text{d}T\right\}\#\left(4\right)\end{array}$$ where T mid represents the average temperature at the midpoint of the entropy change curve, and Δ T H−C corresponds to the temperature difference between the hot and cold ends. A Δ T H−C value of 4 K is used in this context. Figure 5 (b) illustrates the |-∆ S M max |( µ 0 H ) curves and TEC( µ 0 H ) curves for the series of samples at different magnetic fields. The overlapping nature of the curves for the three samples indicates a good alignment. This suggests that the series of samples can effectively serve as magnetic refrigerants within a working range of 4 K, for instance, in applications like Active Magnetic Regeneration (AMR) cycles[ 21 ]. This enables the achievement of performance close to their maximum potential. Similar research results provide favorable references for future exploration of novel magnetic heat materials. The type of phase transition in a material is generally divided into first- and second-order phase transitions. Therefore, we use Arrott's plot to further identify the type of phase transition in the sample. By the criterion of Banerjee [ 22 ], a negative slope or "S" shaped curve in the Arrott plot indicates a first-order phase transition, while a positive slope indicates a second-order phase transition. Figure 6 illustrates that all Arrott curves display positive slopes, indicating that the series of samples experience field-driven second-order phase transitions. Through the analysis of the relationship between magnetic entropy change and magnetic field, the magnetic ordering of the sample is further examined, with the formula given as follows[ 23 ] : $$\begin{array}{c}n=\frac{\text{d}(\text{l}\text{n}\varDelta {S}_{\text{M}})}{\text{d}\left[\text{ln}\left({\mu }_{0}H\right)\right]}\#\left(5\right)\end{array}$$ Figure 7 (a) illustrates the relationship n ( T , µ 0 H ) for this series of samples. It is noteworthy that the spin-ordered states of the sample system increase with the increase in magnetic field, leading to significant variations in the n values under different magnetic fields[ 24 ]. Figure 7 (b) clearly shows whether the sample exhibits portions exceeding the numerical value "2," as indicated by the black region. This finding serves as a criterion for identifying first-order phase transitions[ 25 ]. These results also corroborate the phase transition phenomena observed in the Arrott plot. Additionally, the mean field theory and 3D-Heisenberg theory explain that the minimum value of n at the critical temperature ( T = T C ) is projected to be around 0.67 and 0.637[ 26 ]. From Fig. 7 (a), it can be observed that the minimum values of n for the samples are approximately 0.619, 0.656, and 0.649, respectively. This suggests that Sample-A is close to the 3D-Heisenberg Model, while Sample-B and Sample-C are close to the Mean Field Model. The critical exponents β , γ , and δ characterizing the second-order phase transition of the samples were determined using the Kouvel-Fisher (K-F) method. The equations are expressed as follows[ 27 ] : $$\begin{array}{c}{M}_{\text{S}}\left(T\right){\left[\text{d}{M}_{\text{S}}\left(T\right)/\text{d}T\right]}^{-1}=\left(T-{T}_{\text{C}}\right)/\beta \#\left(6\right)\end{array}$$ $$\begin{array}{c}{\chi }_{0}^{-1}\left(T\right){\left[\text{d}{\chi }_{0}^{-1}\left(T\right)/\text{d}T\right]}^{-1}=\left(T-{T}_{\text{C}}\right)/\gamma \#\left(7\right)\end{array}$$ $$\begin{array}{c}\delta =1+\left(\gamma /\beta \right)\#\left(8\right)\end{array}$$ where M S and χ 0 represent spontaneous magnetization and initial susceptibility, respectively. The critical exponents β and γ for this series of samples were determined through M S (d M S / d T ) −1 - T and χ 0 −1 (d χ 0 −1 / d T ) −1 - T as shown in Fig. 8 . The slopes of the fitted lines correspond to 1/ β and 1/ γ . The results indicate that β (Sample−A) ≈ 0.388 ± 0.005, γ (Sample−A) ≈ 1.056 ± 0.005, β (Sample−B) ≈ 0.460 ± 0.005, and γ (Sample−B) ≈ 1.025 ± 0.005, β (Sample−C) ≈ 0.435 ± 0.005, and γ (Sample−C) ≈ 0.940 ± 0.005. Comparing with the critical exponents of different models in Table 3 , it can be observed that the critical exponents of Sample-A are close to the theoretical values of the 3D-Heisenberg Model, while Sample-B and Sample-C are closer to the Mean Field Model. This suggests that near the T C , Sample-A exhibits short-range exchange interactions in the system, while Sample-B and Sample-C demonstrate long-range exchange interactions[ 28 ] . For the doped samples, the β value has increased significantly from 0.39 to 0.46. This implies that Mg and Ba doping contribute to the formation of long-range ferromagnetic order in sample-A. As can be seen in Table 3 , we can notice a slight discrepancy in the critical exponents of this series of samples from those of the Mean Field Model and the 3D Heisenberg Model. In the Mean Field Model, spin transitions of Mg and Ba ions at T C or in hole-poor regions can lead to changes in the β value[ 29 ]. Doping and then high temperature annealing in manganese oxides significantly affects the host lattice, changing the magnetic order and the strength of magnetic interactions. Therefore, the inherent inhomogeneity and coexistence of multiple phases in these materials make them non-universal. Table 3 The values of critical exponents for the samples and the predicted critical exponents of the various theoretical models. Sample Method β γ δ Ref Mean field Model - 0.5 1 3 [ 5 ] 3D-Heisenberg Model - 0.365 1.386 4.8 [ 5 ] Tricritical Model - 0.25 1 5 [ 5 ] Sample-A K-F 0.388 ± 0.005 1.056 ± 0.005 3.722 ± 0.002 This Work Sample-B K-F 0.460 ± 0.005 1.025 ± 0.005 3.228 ± 0.002 This Work Sample-C K-F 0.435 ± 0.005 0.940 ± 0.005 3.161 ± 0.003 This Work The scaling hypothesis states that the magnetic equation of state can be defined through the correlation between M ( H , ε ), H , and T , expressed as[ 30 ] : $$\begin{array}{c}M\left(H, \epsilon \right)={\epsilon }^{\beta }f\pm \left(H/{\epsilon }^{\beta +\gamma }\right)\#\left(9\right)\end{array}$$ Here, f represents a conventional analytic function, with f + and f − applicable respectively for T > T C and T < T C . ε is the reduced temperature ( ε= ( T − T C )/ T C ). The rescaled graph of M /| ε | β is a function of H /| ε | β + γ , showing two independent branches near T C [ 31 ]. Figure 9 shows the two branches corresponding to T > T C and T < T C . The results provide confirmation that the critical parameters are consistent with the scaling hypothesis. In addition, the separate ln M/|ɛ| β -ln H/|ɛ| β+γ plot in the inset of Fig. 9 further demonstrates the accuracy of T C and the critical exponents. 4 Conclusion This study successfully prepared polycrystalline samples of Pr 2 Mn 2 O 6 and Pr 1.5 A 0.5 Mn 2 O 6 (A = Mg, Ba) through high-temperature solid-phase reaction. The influence of Mg and Ba doping on the magnetocaloric effects and critical behavior of these materials was investigated. The results show that the samples exhibit excellent single-phase characteristics in the orthorhombic crystal system (space group Pbnm). Under a 7 T magnetic field, the maximum magnetic entropy change values for the samples are |- ∆ S M max | (Sample−A) ≈ 4.40 J·kg − 1 ·K − 1 , |- ∆ S M max | ( Sample−B) ≈ 3.66 J·kg − 1 ·K − 1 , and |- ∆ S M max | ( Sample−C) ≈ 3.72 J·kg − 1 ·K − 1 . The relative cooling powers are RCP (Sample−A) ≈ 483.46 J·kg − 1 , RCP (Sample−B) ≈ 428.22 J·kg − 1 , and RCP (Sample−C) ≈ 479.88 J·kg − 1 , indicating significant magnetic refrigeration potential. The series of samples undergo a second-order phase transition with a small magnetic hysteresis, crucial for magnetic refrigeration applications. Using the Kouvel-Fisher method, a detailed analysis of critical behavior was conducted to determine critical exponents ( β , γ , and δ ). The results suggest that Sample-B and Sample-C are ferromagnets with a small amount of weakly itinerant electrons, exhibiting long-range exchange interactions near T C , while Sample-A exhibits short-range exchange interactions near T C . In summary, the Pr 2 Mn 2 O 6 ceramics investigated in this study exhibit significant potential as materials for magnetic refrigeration, offering valuable insights for research in magnetic refrigeration technology. Declarations Acknowledgements This project was supported by the National Natural Science Foundation of China (NSFC) under grant nos.52061035, 51871124, 51561026 and 51401111, Young Leading Talent of “Grassland Talents” Project of Inner Mongolia Autonomous Region, Inner Mongolia Natural Science Cultivating Fund for Distinguished Young Scholars (no. 2020JQ05), Science and Technology Planning Project of Inner Mongolia Autonomous Region (no. 2020GG0267), Program for Innovative Research Team in Universities of Inner Mongolia Autonomous Region (no. NMGIRT2211), Inner Mongolia University of Technology Key Discipline Team Project of Materials Science (no. ZD202012). The authors declare no conflict of interest. References Xu P, Hu L, Zhang Z Q, Wang H F, Li L W. Electronic structure, magnetic properties and magnetocaloric performance in rare earths(RE) based RE 2 BaZnO 5 (RE = Gd, Dy, Ho, and Er) compounds[J]. Acta Materialia, 2022, 236: 118114. Zhang Y K, Li S, Hu L, Wang X H, Li L W, Yan M. 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Critical field analysis and magnetocaloric effect of A-Site double-doped manganese oxide La 0.9 EuSr 0.1 MnO 3 [J]. Journal of Superconductivity and Novel Magnetism, 2021, 34: 2651-2666. Kouvel J S, Fisher M E. Detailed magnetic behavior of nickel near its Curie point[J]. Physical Review, 1964, 136(6A): A1626. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 29 Apr, 2024 Reviews received at journal 09 Apr, 2024 Reviewers agreed at journal 01 Apr, 2024 Reviewers invited by journal 26 Mar, 2024 Submission checks completed at journal 22 Mar, 2024 Editor assigned by journal 22 Mar, 2024 First submitted to journal 21 Mar, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4141743","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":282966240,"identity":"0793af5f-c203-439e-a23f-da3a9ea5579a","order_by":0,"name":"Huiqin Yun","email":"","orcid":"","institution":"Inner Mongolia University of Technology","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Huiqin","middleName":"","lastName":"Yun","suffix":""},{"id":282966242,"identity":"242a1d64-aa8a-45c6-9069-3e5510509f67","order_by":1,"name":"Ze Li","email":"","orcid":"","institution":"Inner Mongolia University of Technology","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Ze","middleName":"","lastName":"Li","suffix":""},{"id":282966244,"identity":"edc935fc-e3ef-4986-8baa-14e057c46b4b","order_by":2,"name":"Xiang Jin","email":"","orcid":"","institution":"Baotou Teachers’ College","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Xiang","middleName":"","lastName":"Jin","suffix":""},{"id":282966247,"identity":"fb4a9717-4952-489e-a9cd-01c48c9b076a","order_by":3,"name":"Jianjun Zhao","email":"","orcid":"","institution":"Baotou Teachers’ College","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Jianjun","middleName":"","lastName":"Zhao","suffix":""},{"id":282966248,"identity":"c6477ddc-57dd-4971-9958-96f5794ba4a1","order_by":4,"name":"Jingshun Liu","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA1ElEQVRIiWNgGAWjYBACxmYQ0QAk2IHkBwMbOxK0MDM2MM4oSEsm0iqwFiDi+XAIzMYLmNt5D7/8ueNwHgMzc5u0jcEBZgb2w0c34HcYX5o175nDxUCHtUnnGNzhY+BJS7uBXwuPmTFj2+HEBoiWZ8wMEjxmBLUY/oRpsTA4zNhAhBbjB7wwLQxEajFj5m1LB2lptuwxSEtmI+QXw/4zxh9/tlknNrC3P7zx44+NHT/74WP4tTQwsEmAGPYHoCJs+JSDgDwwaj4QUjQKRsEoGAUjHAAA0YRE3j3Z7P0AAAAASUVORK5CYII=","orcid":"","institution":"Inner Mongolia University of Technology","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Jingshun","middleName":"","lastName":"Liu","suffix":""}],"badges":[],"createdAt":"2024-03-21 08:03:04","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4141743/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4141743/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":53451078,"identity":"77d01ae2-e923-4a86-870a-e99644beddc5","added_by":"auto","created_at":"2024-03-26 06:40:13","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":36359,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eThe refined X-ray diffraction (XRD) patterns of Sample-A, Sample-B, and Sample-C.\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-4141743/v1/2d122835cb2ee812f73e037e.png"},{"id":53451080,"identity":"e54bfbfb-e231-40d7-84ad-9113a92a374a","added_by":"auto","created_at":"2024-03-26 06:40:13","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":4508883,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eThermomagnetic curves \u003c/strong\u003e\u003cem\u003e\u003cstrong\u003eM\u003c/strong\u003e\u003c/em\u003e\u003cstrong\u003e(\u003c/strong\u003e\u003cem\u003e\u003cstrong\u003eT\u003c/strong\u003e\u003c/em\u003e\u003cstrong\u003e) for Sample-A, Sample-B, and Sample-C in a magnetic field of 0.05 T, the inset shows d\u003c/strong\u003e\u003cem\u003e\u003cstrong\u003eM/\u003c/strong\u003e\u003c/em\u003e\u003cstrong\u003ed\u003c/strong\u003e\u003cem\u003e\u003cstrong\u003eT\u003c/strong\u003e\u003c/em\u003e\u003cstrong\u003e(\u003c/strong\u003e\u003cem\u003e\u003cstrong\u003eT\u003c/strong\u003e\u003c/em\u003e\u003cstrong\u003e) curves.\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-4141743/v1/cc3b0ffa8a7a56101fc5beb2.png"},{"id":53451079,"identity":"c71c71ee-a183-4cd8-8b96-fff8a966b50b","added_by":"auto","created_at":"2024-03-26 06:40:13","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":234866,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003e\u003cstrong\u003eM\u003c/strong\u003e\u003c/em\u003e\u003cstrong\u003e-\u003c/strong\u003e\u003cem\u003e\u003cstrong\u003eH\u003c/strong\u003e\u003c/em\u003e\u003cstrong\u003e curves of Sample-A, Sample-B, and Sample-C at different temperatures.\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-4141743/v1/85284631e35017513c9ae88d.png"},{"id":53451084,"identity":"5ede0436-3355-49a0-a368-c863df4ba301","added_by":"auto","created_at":"2024-03-26 06:40:13","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":284485,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eThe |-Δ\u003c/strong\u003e\u003cem\u003e\u003cstrong\u003eS\u003c/strong\u003e\u003c/em\u003e\u003cstrong\u003eM| of Sample-A, Sample-B, and Sample-C measured under a field change from\u003c/strong\u003e\u003cem\u003e\u003cstrong\u003e \u003c/strong\u003e\u003c/em\u003e\u003cstrong\u003e0 to 7 T.\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-4141743/v1/2075c7e5e707019cd12bfc51.png"},{"id":53451085,"identity":"f09f1c22-68c6-4b20-ad86-3a48a2799f29","added_by":"auto","created_at":"2024-03-26 06:40:13","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":6749053,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003e(a) NRC\u003c/strong\u003e\u003csub\u003e\u003cstrong\u003emax\u003c/strong\u003e\u003c/sub\u003e\u003cstrong\u003e plots and (b) \u003c/strong\u003e\u003cem\u003e\u003cstrong\u003e|-∆S\u003c/strong\u003e\u003c/em\u003e\u003csub\u003e\u003cstrong\u003eM\u003c/strong\u003e\u003c/sub\u003e\u003csup\u003e\u003cstrong\u003emax\u003c/strong\u003e\u003c/sup\u003e\u003cem\u003e\u003cstrong\u003e|\u003c/strong\u003e\u003c/em\u003e\u003cstrong\u003e(\u003c/strong\u003e\u003cem\u003e\u003cstrong\u003eμ\u003c/strong\u003e\u003c/em\u003e\u003csub\u003e\u003cstrong\u003e0\u003c/strong\u003e\u003c/sub\u003e\u003cem\u003e\u003cstrong\u003eH\u003c/strong\u003e\u003c/em\u003e\u003cstrong\u003e) and TEC(\u003c/strong\u003e\u003cem\u003e\u003cstrong\u003eμ\u003c/strong\u003e\u003c/em\u003e\u003csub\u003e\u003cstrong\u003e0\u003c/strong\u003e\u003c/sub\u003e\u003cem\u003e\u003cstrong\u003eH\u003c/strong\u003e\u003c/em\u003e\u003cstrong\u003e) curves for Sample-A, Sample-B, and Sample-C.\u003c/strong\u003e\u0026nbsp;\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-4141743/v1/1a45a0e797a6a3f9e1b5383d.png"},{"id":53451081,"identity":"28aedcb8-03a9-4626-9764-dcc89e4c591e","added_by":"auto","created_at":"2024-03-26 06:40:13","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":6985339,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eArrott curves of Sample-A, Sample-B, and Sample-C.\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-4141743/v1/7dd47677741a435067f7282f.png"},{"id":53451086,"identity":"a6b1d8c4-4834-47e7-8b88-2bdfea8a7ccd","added_by":"auto","created_at":"2024-03-26 06:40:14","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":12659444,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eField and temperature dependence of the exponent \u003c/strong\u003e\u003cem\u003e\u003cstrong\u003en\u003c/strong\u003e\u003c/em\u003e\u003cstrong\u003e for\u003c/strong\u003e \u003cstrong\u003eSample-A, Sample-B, and Sample-C.\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-4141743/v1/7410d0e56e93cdf829f57579.png"},{"id":53451087,"identity":"2d3363a6-3af0-4543-b4e0-8d8440dab341","added_by":"auto","created_at":"2024-03-26 06:40:14","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":5538623,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eKouvel-Fisher plots for the spontaneous magnetization \u003c/strong\u003e\u003cem\u003e\u003cstrong\u003eM\u003c/strong\u003e\u003c/em\u003e\u003csub\u003e\u003cstrong\u003eS\u003c/strong\u003e\u003c/sub\u003e\u003cstrong\u003e(\u003c/strong\u003e\u003cem\u003e\u003cstrong\u003eT\u003c/strong\u003e\u003c/em\u003e\u003cstrong\u003e) and for the inverse initial susceptibility \u003c/strong\u003e\u003cem\u003e\u003cstrong\u003eχ\u003c/strong\u003e\u003c/em\u003e\u003csub\u003e\u003cstrong\u003e0\u003c/strong\u003e\u003c/sub\u003e\u003csup\u003e\u003cstrong\u003e-1\u003c/strong\u003e\u003c/sup\u003e\u003cstrong\u003e(T) of Sample-A, Sample-B, and Sample-C\u003c/strong\u003e.\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-4141743/v1/e1d923390dd383806ceff38f.png"},{"id":53451082,"identity":"5b85ce20-f30b-4294-9ef1-c9281d8725ad","added_by":"auto","created_at":"2024-03-26 06:40:13","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":6421063,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eScaling plots of Sample-A, Sample-B, and Sample-C on two universal curves above and below \u003c/strong\u003e\u003cem\u003e\u003cstrong\u003eT\u003c/strong\u003e\u003c/em\u003e\u003csub\u003e\u003cstrong\u003eC\u003c/strong\u003e\u003c/sub\u003e\u003cstrong\u003e, with insets showing the logarithmic scale plots.\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-4141743/v1/ac16ce81ce111c326d722771.png"},{"id":53451625,"identity":"4fa56f74-a86b-40a3-a23d-5536b2458312","added_by":"auto","created_at":"2024-03-26 06:48:14","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1494613,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4141743/v1/4541ca23-3dbd-4723-b132-728323fa123b.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Study of the Magnetocaloric Effect and Critical Behavior in Double Perovskite Manganese Oxides Pr 1.5 A 0.5 Mn 2 O 6 (A=Mg, Ba)","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eIn recent years, magnetocaloric refrigeration (MR) based on the magnetocaloric effect (MCE) has emerged as a promising technology to replace traditional gas compression-expansion techniques[\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e] [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. Researchers are actively seeking more suitable magnetocaloric materials, and a considerable focus has been placed on perovskite manganites of the form Ln\u003csub\u003e1\u0026thinsp;\u0026minus;\u0026thinsp;\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003eA\u003csub\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003eMnO\u003csub\u003e3\u003c/sub\u003e(Ln\u0026thinsp;=\u0026thinsp;La, Pr, Nd, ..., A\u0026thinsp;=\u0026thinsp;Sr, Ca, Ba, ...) due to their extensive studies on electrical and magnetic properties with potential applications[\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. Phan and Yu have provided a comprehensive overview of the observed magnetocaloric effects in manganese oxides[\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. The investigation of the LnMnO\u003csub\u003e3\u003c/sub\u003e system is particularly intriguing, characterized by significant MCE values and adjustable phase transition temperatures. These manganese oxides offer convenient synthetic routes, and their Curie temperatures (\u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e) are reasonable under various doping conditions. Consequently, the study of the magnetocaloric effect in manganese oxide composites has become a new developmental trend.\u003c/p\u003e \u003cp\u003eManganese oxides are generally classified into first-order phase transition materials and second-order phase transition materials based on the type of phase transition. It is well-known that first-order phase transition materials often exhibit significant magnetocaloric effect (MCE). However, they are accompanied by challenges such as a narrow cooling temperature range, relatively small refrigeration capacity, and large magnetic hysteresis[\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. On the other hand, second-order phase transition materials, while having a wider cooling temperature range and smaller magnetic hysteresis, face the drawback of a slightly lower maximum magnetic entropy change, making them less practical for applications[\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. Law et al. [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e] have pointed out that ideal magnetocaloric materials may lie in a critical state between first and second-order phase transitions. Therefore, the focus of research in the field of magnetocaloric refrigeration should be on the study of MCE and phase transition control to identify suitable magnetocaloric materials.\u003c/p\u003e \u003cp\u003eThis study focuses on modulating the properties of Pr\u003csub\u003e2\u003c/sub\u003eMn\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e6\u003c/sub\u003e through the doping of Mg and Ba atoms. The introduction of Mg and Ba may induce lattice distortions and adjust magnetic interactions, thereby influencing the magnetocaloric effect and phase transitions of Pr\u003csub\u003e2\u003c/sub\u003eMn\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e6\u003c/sub\u003e. We anticipate that the results of this study will not only contribute to a deeper understanding of the complex magnetic behavior of Pr\u003csub\u003e2\u003c/sub\u003eMn\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e6\u003c/sub\u003e but also provide new insights for the development of functional materials with superior magnetic properties.\u003c/p\u003e"},{"header":"2 Experimental methods","content":"\u003cp\u003eThe polycrystalline samples of perovskite manganese oxides Pr\u003csub\u003e1.5\u003c/sub\u003eA\u003csub\u003e0.5\u003c/sub\u003eMn\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e6\u003c/sub\u003e (A\u0026thinsp;=\u0026thinsp;Mg, Ba) were prepared using the traditional solid-state reaction method. High-purity (\u0026gt;\u0026thinsp;99.99%) Pr\u003csub\u003e6\u003c/sub\u003eO\u003csub\u003e11\u003c/sub\u003e, MgO, BaCO\u003csub\u003e3\u003c/sub\u003e, and MnO\u003csub\u003e2\u003c/sub\u003e dry solid powder raw materials (from Alfa Aesar, America) were accurately weighed according to stoichiometric ratios. The weighed samples were then thoroughly mixed and ground using a ball mill (QM-QX04, full range planetary ball mill, Nanjing Nanda Instruments Co., Ltd., China) for 6 hours. Subsequently, the well-ground samples were first dried and then subjected to decarbonization treatment at 950\u0026deg;C for 12 hours in a box-type sintering furnace (TM KSL-1700X, Hefei Kejing Materials Technology Co., LTD., China). After a second grinding, the samples were calcined again at a high temperature of 1350\u0026deg;C for 24 hours in the box-type sintering furnace, followed by a controlled cooling from 1350\u0026deg;C to 800\u0026deg;C over 5 hours, and then naturally cooled to room temperature. This process yielded well-formed phase samples of Pr\u003csub\u003e2\u003c/sub\u003eMn\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e6\u003c/sub\u003e, Pr\u003csub\u003e1.5\u003c/sub\u003eMg\u003csub\u003e0.5\u003c/sub\u003eMn\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e6\u003c/sub\u003e, and Pr\u003csub\u003e1.5\u003c/sub\u003eBa\u003csub\u003e0.5\u003c/sub\u003eMn\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e6\u003c/sub\u003e (hereinafter referred to as Sample-A, Sample-B, and Sample-C, respectively). The samples' crystal structure and phase purity were analyzed using a PANalytical X'Pert Powder X-ray diffractometer. Their magnetization as a function of temperature (\u003cem\u003eM\u003c/em\u003e(\u003cem\u003eT\u003c/em\u003e)) was assessed in a 0.05 T magnetic field, while magnetization versus magnetic field strength (\u003cem\u003eM\u003c/em\u003e(\u003cem\u003eH\u003c/em\u003e)) was evaluated over a 0\u0026ndash;7 T range using a Quantum Design PPMS-9 multifunctional physical property measurement system.\u003c/p\u003e"},{"header":"3 Results and analysis","content":"\u003cp\u003eFigure \u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e presents the refined X-ray diffraction spectra of polycrystalline samples. No significant impurity peaks were observed in any of the samples, indicating that they possess good single-phase properties and belong to the orthorhombic crystal system with the space group Pbnm. Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e lists the cell parameters and refinement parameters for each sample, obtained after refinement using MDI Jade 6. As can be seen from Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, the lattice parameters a, b, and c, as well as the cell volume V, decrease with Mg doping and increase with Ba doping. This is due to the smaller ionic radius of Mg (0.65 \u0026Aring;) compared to Pr (1.13 \u0026Aring;), and the larger ionic radius of Ba (1.49 \u0026Aring;) compared to Pr.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eLattice parameters of Sample-A, Sample-B, and Sample-C.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"No\" id=\"Taba\" border=\"1\"\u003e \u003ccolgroup cols=\"7\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003esamples\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ea\u003c/em\u003e(\u0026Aring;)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cem\u003eb\u003c/em\u003e(\u0026Aring;)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003ec\u003c/em\u003e(\u0026Aring;)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u003cem\u003eV\u003c/em\u003e(\u0026Aring;\u003csup\u003e3\u003c/sup\u003e)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u003cem\u003eχ\u003c/em\u003e\u003csup\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003e\u003cem\u003eR\u003c/em\u003e\u003csub\u003eBragg\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSample-A\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e5.471\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e5.590\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e7.681\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e234.920\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e1.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e9.8\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSample-B\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e5.470\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e5.470\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e7.788\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e233.090\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e1.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e9.7\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSample-C\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e5.506\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e5.531\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e7.781\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e236.990\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e1.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e9.8\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e illustrates the zero-field (ZFC) and field-cooled (FC) magnetization curves for three samples under an applied magnetic field of 0.05 T, with the inset displaying the d\u003cem\u003eM\u003c/em\u003e/d\u003cem\u003eT\u003c/em\u003e-\u003cem\u003eT\u003c/em\u003e curves. It is observable from the graph that all three samples exhibit paramagnetic behavior at relatively higher temperature ranges. As the temperature decreases, the magnetic nature of the samples transitions from paramagnetic (PM) to ferromagnetic (FM) at their respective \u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e (\u003cem\u003eT\u003c/em\u003e\u003csub\u003eC(Sample\u0026minus;A)\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;91 K, \u003cem\u003eT\u003c/em\u003e\u003csub\u003eC(Sample\u0026minus;B)\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;77 K, and \u003cem\u003eT\u003c/em\u003e\u003csub\u003eC(Sample\u0026minus;C)\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;152 K). It is evident that doping with Mg and Ba respectively decreases and increases the magnetization strength and \u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e of the samples. The doping with Mg leads to a reduction in the Mn-O-Mn bond angle, which in turn increases the lattice distortion in the samples, weakening the double-exchange interaction between Mn\u003csup\u003e3+\u003c/sup\u003e and Mn\u003csup\u003e4+\u003c/sup\u003e, resulting in a lower \u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e and reduced magnetization strength[\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]. Conversely, doping with Ba has the opposite effect[\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]. Additionally, a distinct bifurcation in the magnetization curves of these samples at low temperatures is observed, which may be attributed to the presence of domain-wall pinning effect[\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e] or spin reorientation[\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e] .\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eWe performed measurements to evaluate the effect of magnetic field variability near the \u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e on the magnetization of the samples. The \u003cem\u003eM\u003c/em\u003e\u0026ndash;\u003cem\u003e\u0026micro;\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e\u003cem\u003eH\u003c/em\u003e curves were plotted as illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. The data show that above \u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e, \u003cem\u003eM\u003c/em\u003e increases in a linear fashion with \u003cem\u003e\u0026micro;\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e\u003cem\u003eH\u003c/em\u003e, which indicates the presence of an unsaturated paramagnetic state. Below \u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e, the plots show a sharp increase in \u003cem\u003eM\u003c/em\u003e with \u003cem\u003e\u0026micro;\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e\u003cem\u003eH\u003c/em\u003e, tending toward saturation and exhibiting ferromagnetic behavior. The partial saturation that has been observed at low temperatures can be attributed to the competition between antiferromagnetic and ferromagnetic clusters within the sample[\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e]. In all samples, no significant hysteresis was observed in the \u003cem\u003eM\u003c/em\u003e\u0026ndash;\u003cem\u003e\u0026micro;\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e\u003cem\u003eH\u003c/em\u003e curves below and around \u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e, suggesting the possible presence of a magnetic field driven second-order phase transition in the system[\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e] .\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eTo investigate the magnetocaloric effect of this series of samples, we calculated the isothermal magnetic entropy change (|-Δ\u003cem\u003eS\u003c/em\u003e\u003csub\u003eM\u003c/sub\u003e|) of the samples using the Maxwell thermodynamic equations, as shown in the following relationship[\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e] :\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$$\\begin{array}{c}\\left|-{\\Delta }{S}_{\\text{M}}\\right|={\\int }_{0}^{{H}_{\\text{m}\\text{a}\\text{x}}}{\\left(\\frac{\\partial M}{\\partial T}\\right)}_{\\text{H}}dH\\#（1）\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e shows the temperature dependence of the |-Δ\u003cem\u003eS\u003c/em\u003e\u003csub\u003eM\u003c/sub\u003e| in the sample. The |-Δ\u003cem\u003eS\u003c/em\u003e\u003csub\u003eM\u003c/sub\u003e| increases with the applied magnetic field. This can be explained by the strengthening of magnetic order with the increase in the external magnetic field. As the magnetic field enhances, the magnetization near the \u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e increases, leading to a higher |-Δ\u003cem\u003eS\u003c/em\u003e\u003csub\u003eM\u003c/sub\u003e| value. Under a 7 T magnetic field, the maximum magnetic entropy changes are approximately \u003cem\u003e|-\u003c/em\u003e∆\u003cem\u003eS\u003c/em\u003e\u003csub\u003eM\u003c/sub\u003e\u003csup\u003emax\u003c/sup\u003e\u003cem\u003e|\u003c/em\u003e\u003csub\u003e(Sample\u0026minus;A)\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;4.40 J\u0026middot;kg\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e\u0026middot;K\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e, \u003cem\u003e|-\u003c/em\u003e∆\u003cem\u003eS\u003c/em\u003e\u003csub\u003eM\u003c/sub\u003e\u003csup\u003emax\u003c/sup\u003e\u003cem\u003e|\u003c/em\u003e\u003csub\u003e(Sample\u0026minus;B)\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;3.66 J\u0026middot;kg\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e\u0026middot;K\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e, and \u003cem\u003e|-\u003c/em\u003e∆\u003cem\u003eS\u003c/em\u003e\u003csub\u003eM\u003c/sub\u003e\u003csup\u003emax\u003c/sup\u003e\u003cem\u003e|\u003c/em\u003e\u003csub\u003e(Sample\u0026minus;C)\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;3.72 J\u0026middot;kg\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e\u0026middot;K\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e. Notably, although the \u003cem\u003e|-\u003c/em\u003e∆\u003cem\u003eS\u003c/em\u003e\u003csub\u003eM\u003c/sub\u003e\u003csup\u003emax\u003c/sup\u003e\u003cem\u003e|\u003c/em\u003e decreases with Mg and Ba doping, the refrigeration temperature range (FWHM) broadens (FWHM\u003csub\u003e(Sample\u0026minus;A)\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;110 K, FWHM\u003csub\u003e(Sample\u0026minus;B)\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;117 K, FWHM\u003csub\u003e(Sample\u0026minus;C)\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;128 K), suggesting a more continuous phase transition in doped samples. The reduction in \u003cem\u003e|-\u003c/em\u003e∆\u003cem\u003eS\u003c/em\u003e\u003csub\u003eM\u003c/sub\u003e\u003csup\u003emax\u003c/sup\u003e\u003cem\u003e|\u003c/em\u003e is consistent with the broadening of the FM-PM transition[\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e] .\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFrom a refrigeration perspective, the Relative Cooling Power (RCP) is a crucial parameter for evaluating the cooling capacity of materials. The equation is as follows[\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e] :\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\n$$\\begin{array}{c}RCP=\\left|-\\varDelta {S}_{\\text{M}}^{\\text{m}\\text{a}\\text{x}}\\right|\\bullet \\left(\\varDelta {T}_{\\text{F}\\text{W}\\text{H}\\text{M}}\\right)\\#\\left(2\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eHerein, ∆\u003cem\u003eT\u003c/em\u003e\u003csub\u003eFWHM\u003c/sub\u003e denotes the full-width at half-maximum of the |-∆\u003cem\u003eS\u003c/em\u003e\u003csub\u003eM\u003c/sub\u003e|(\u003cem\u003eT\u003c/em\u003e) profile. It is observed that at a magnetic field strength of 7 T, the RCP values are approximately RCP\u003csub\u003e(Sample\u0026minus;A)\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;483.46 J\u0026middot;kg\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e, RCP\u003csub\u003e(Sample\u0026minus;B)\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;428.22 J\u0026middot;kg\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e, and RCP\u003csub\u003e(Sample\u0026minus;C)\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;478.88 J\u0026middot;kg\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e. This demonstrates that the reduction in |-∆\u003cem\u003eS\u003c/em\u003e\u003csub\u003eM\u003c/sub\u003e\u003csup\u003emax\u003c/sup\u003e| due to doping with Mg and Ba surpasses the broadening of the cooling temperature range. Additionally, Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e presents the magnetocaloric parameters for PrMnO system refrigeration materials. The findings indicate that Sample-A in this study is more apt as a magnetocaloric refrigerant.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eComparison of the magnetocaloric parameters of Sample-A, Sample-B, and Sample-C with some other magnetic refrigerants in magnetic fields of 5 T.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSamples\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003eC\u003c/em\u003e\u003c/sub\u003e \u003cem\u003e/\u003c/em\u003eK\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cem\u003e|-\u003c/em\u003eΔ\u003cem\u003eS\u003c/em\u003e\u003csub\u003eM\u003c/sub\u003e \u003cem\u003e|/\u003c/em\u003e J\u0026middot;kg\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e\u0026middot;K\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eRCP\u003cem\u003e/\u003c/em\u003eJ\u0026middot;kg\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u003cem\u003e\u0026micro;\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003eΔ\u003cem\u003eH/\u003c/em\u003eT\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eRef\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eSample-A\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e91\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.40\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e483.46\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eThis work\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3.35\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e301.85\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eSample-B\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e77\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3.66\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e428.22\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eThis work\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2.84\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e269.35\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eSample-C\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e152\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3.72\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e479.88\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eThis work\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e311.53\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePr\u003csub\u003e0.87\u003c/sub\u003eCa\u003csub\u003e0.13\u003c/sub\u003eMnO\u003csub\u003e3\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.608\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e460.77\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e[\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePr\u003csub\u003e2\u003c/sub\u003eCoMnO6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.862\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e126.62\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e[\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePr\u003csub\u003e0.55\u003c/sub\u003eSr\u003csub\u003e0.45\u003c/sub\u003eMnO\u003csub\u003e3\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e291\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.70\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e143.60\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e[\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eIn addition, the Normalized Refrigeration Capacity (NRC) serves as an alternative metric for the selection of magnetic refrigeration materials. The formula for NRC is[\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e] :\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equc\" name=\"EquationSource\"\u003e\n$$\\begin{array}{c}NRC=\\frac{1}{{\\Delta }{\\mu }_{0}H}{\\int }_{{T}_{C\\text{o}\\text{l}\\text{d}}}^{{T}_{\\text{H}\\text{o}\\text{t}}}\\left|\\varDelta {S}_{\\text{M}}\\left(T\\right)\\right|dT\\#\\left(3\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eT\u003c/em\u003e\u003csub\u003eHot\u003c/sub\u003e and \u003cem\u003eT\u003c/em\u003e\u003csub\u003eCold\u003c/sub\u003e represent the high and low-temperature ends, respectively. As depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e(a), the plots of the maximum NRC (NRC\u003csub\u003emax\u003c/sub\u003e) for temperature intervals of 4 K under external magnetic fields of 2 T and 5 T are shown. It is evident that NRC\u003csub\u003emax\u003c/sub\u003e escalates with the increase in Δ\u003cem\u003eT\u003c/em\u003e\u003csub\u003eH\u0026minus;C\u003c/sub\u003e, the temperature differential between the hot and cold ends. Notably, at 2 T, both Sample-A and Sample-C exhibit enhanced magnetic refrigeration efficiency.\u003c/p\u003e \u003cp\u003eThe Temperature-Averaged Entropy Change (TEC) is another indicator used to quantify the MCE of magnetic refrigeration materials within a specific temperature range. It is calculated using the following formula[\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e] :\u003cdiv id=\"Equd\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equd\" name=\"EquationSource\"\u003e\n$$\\begin{array}{c}TEC\\left(\\varDelta {T}_{\\text{H}-\\text{C}}\\right)=\\frac{1}{\\varDelta {T}_{\\text{H}-\\text{C}}}max\\left\\{{\\int }_{{T}_{\\text{m}\\text{i}\\text{d}}-\\frac{\\varDelta {T}_{\\text{H}-\\text{C}}}{2}}^{{T}_{\\text{m}\\text{i}\\text{d}}+\\frac{\\varDelta {T}_{\\text{H}-\\text{C}}}{2}}\\left|\\varDelta {S}_{\\text{M}}\\right|\\text{d}T\\right\\}\\#\\left(4\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eT\u003c/em\u003e\u003csub\u003emid\u003c/sub\u003e represents the average temperature at the midpoint of the entropy change curve, and Δ\u003cem\u003eT\u003c/em\u003e\u003csub\u003eH\u0026minus;C\u003c/sub\u003e corresponds to the temperature difference between the hot and cold ends. A Δ\u003cem\u003eT\u003c/em\u003e\u003csub\u003eH\u0026minus;C\u003c/sub\u003e value of 4 K is used in this context. Figure\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e(b) illustrates the |-∆\u003cem\u003eS\u003c/em\u003e\u003csub\u003eM\u003c/sub\u003e\u003csup\u003emax\u003c/sup\u003e|(\u003cem\u003e\u0026micro;\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e\u003cem\u003eH\u003c/em\u003e) curves and TEC(\u003cem\u003e\u0026micro;\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e\u003cem\u003eH\u003c/em\u003e) curves for the series of samples at different magnetic fields. The overlapping nature of the curves for the three samples indicates a good alignment. This suggests that the series of samples can effectively serve as magnetic refrigerants within a working range of 4 K, for instance, in applications like Active Magnetic Regeneration (AMR) cycles[\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e]. This enables the achievement of performance close to their maximum potential. Similar research results provide favorable references for future exploration of novel magnetic heat materials.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe type of phase transition in a material is generally divided into first- and second-order phase transitions. Therefore, we use Arrott's plot to further identify the type of phase transition in the sample. By the criterion of Banerjee [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e], a negative slope or \"S\" shaped curve in the Arrott plot indicates a first-order phase transition, while a positive slope indicates a second-order phase transition. Figure\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e illustrates that all Arrott curves display positive slopes, indicating that the series of samples experience field-driven second-order phase transitions.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThrough the analysis of the relationship between magnetic entropy change and magnetic field, the magnetic ordering of the sample is further examined, with the formula given as follows[\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e] :\u003cdiv id=\"Eque\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Eque\" name=\"EquationSource\"\u003e\n$$\\begin{array}{c}n=\\frac{\\text{d}(\\text{l}\\text{n}\\varDelta {S}_{\\text{M}})}{\\text{d}\\left[\\text{ln}\\left({\\mu }_{0}H\\right)\\right]}\\#\\left(5\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e(a) illustrates the relationship \u003cem\u003en\u003c/em\u003e (\u003cem\u003eT\u003c/em\u003e, \u003cem\u003e\u0026micro;\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e\u003cem\u003eH\u003c/em\u003e) for this series of samples. It is noteworthy that the spin-ordered states of the sample system increase with the increase in magnetic field, leading to significant variations in the \u003cem\u003en\u003c/em\u003e values under different magnetic fields[\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e]. Figure\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e(b) clearly shows whether the sample exhibits portions exceeding the numerical value \"2,\" as indicated by the black region. This finding serves as a criterion for identifying first-order phase transitions[\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e]. These results also corroborate the phase transition phenomena observed in the Arrott plot. Additionally, the mean field theory and 3D-Heisenberg theory explain that the minimum value of \u003cem\u003en\u003c/em\u003e at the critical temperature (\u003cem\u003eT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e) is projected to be around 0.67 and 0.637[\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e]. From Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e(a), it can be observed that the minimum values of \u003cem\u003en\u003c/em\u003e for the samples are approximately 0.619, 0.656, and 0.649, respectively. This suggests that Sample-A is close to the 3D-Heisenberg Model, while Sample-B and Sample-C are close to the Mean Field Model.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe critical exponents \u003cem\u003eβ\u003c/em\u003e, \u003cem\u003eγ\u003c/em\u003e, and \u003cem\u003eδ\u003c/em\u003e characterizing the second-order phase transition of the samples were determined using the Kouvel-Fisher (K-F) method. The equations are expressed as follows[\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e] :\u003cdiv id=\"Equf\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equf\" name=\"EquationSource\"\u003e\n$$\\begin{array}{c}{M}_{\\text{S}}\\left(T\\right){\\left[\\text{d}{M}_{\\text{S}}\\left(T\\right)/\\text{d}T\\right]}^{-1}=\\left(T-{T}_{\\text{C}}\\right)/\\beta \\#\\left(6\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equg\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equg\" name=\"EquationSource\"\u003e\n$$\\begin{array}{c}{\\chi }_{0}^{-1}\\left(T\\right){\\left[\\text{d}{\\chi }_{0}^{-1}\\left(T\\right)/\\text{d}T\\right]}^{-1}=\\left(T-{T}_{\\text{C}}\\right)/\\gamma \\#\\left(7\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equh\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equh\" name=\"EquationSource\"\u003e\n$$\\begin{array}{c}\\delta =1+\\left(\\gamma /\\beta \\right)\\#\\left(8\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eM\u003c/em\u003e\u003csub\u003eS\u003c/sub\u003e and \u003cem\u003eχ\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e represent spontaneous magnetization and initial susceptibility, respectively. The critical exponents \u003cem\u003eβ\u003c/em\u003e and \u003cem\u003eγ\u003c/em\u003e for this series of samples were determined through \u003cem\u003eM\u003c/em\u003e\u003csub\u003eS\u003c/sub\u003e(d\u003cem\u003eM\u003c/em\u003e\u003csub\u003eS\u003c/sub\u003e\u003cem\u003e/\u003c/em\u003ed\u003cem\u003eT\u003c/em\u003e)\u003csup\u003e\u0026minus;1\u003c/sup\u003e-\u003cem\u003eT\u003c/em\u003e and \u003cem\u003eχ\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e(d\u003cem\u003eχ\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e\u003csup\u003e\u0026minus;1\u003c/sup\u003e\u003cem\u003e/\u003c/em\u003ed\u003cem\u003eT\u003c/em\u003e)\u003csup\u003e\u0026minus;1\u003c/sup\u003e-\u003cem\u003eT\u003c/em\u003e as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e. The slopes of the fitted lines correspond to 1/\u003cem\u003eβ\u003c/em\u003e and 1/\u003cem\u003eγ\u003c/em\u003e. The results indicate that \u003cem\u003eβ\u003c/em\u003e\u003csub\u003e(Sample\u0026minus;A)\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;0.388\u0026thinsp;\u0026plusmn;\u0026thinsp;0.005, \u003cem\u003eγ\u003c/em\u003e\u003csub\u003e(Sample\u0026minus;A)\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;1.056\u0026thinsp;\u0026plusmn;\u0026thinsp;0.005, \u003cem\u003eβ\u003c/em\u003e\u003csub\u003e(Sample\u0026minus;B)\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;0.460\u0026thinsp;\u0026plusmn;\u0026thinsp;0.005, and \u003cem\u003eγ\u003c/em\u003e\u003csub\u003e(Sample\u0026minus;B)\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;1.025\u0026thinsp;\u0026plusmn;\u0026thinsp;0.005, \u003cem\u003eβ\u003c/em\u003e\u003csub\u003e(Sample\u0026minus;C)\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;0.435\u0026thinsp;\u0026plusmn;\u0026thinsp;0.005, and \u003cem\u003eγ\u003c/em\u003e\u003csub\u003e(Sample\u0026minus;C)\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;0.940\u0026thinsp;\u0026plusmn;\u0026thinsp;0.005. Comparing with the critical exponents of different models in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, it can be observed that the critical exponents of Sample-A are close to the theoretical values of the 3D-Heisenberg Model, while Sample-B and Sample-C are closer to the Mean Field Model. This suggests that near the \u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e, Sample-A exhibits short-range exchange interactions in the system, while Sample-B and Sample-C demonstrate long-range exchange interactions[\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e] .\u003c/p\u003e \u003cp\u003eFor the doped samples, the \u003cem\u003eβ\u003c/em\u003e value has increased significantly from 0.39 to 0.46. This implies that Mg and Ba doping contribute to the formation of long-range ferromagnetic order in sample-A. As can be seen in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, we can notice a slight discrepancy in the critical exponents of this series of samples from those of the Mean Field Model and the 3D Heisenberg Model. In the Mean Field Model, spin transitions of Mg and Ba ions at \u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e or in hole-poor regions can lead to changes in the \u003cem\u003eβ\u003c/em\u003e value[\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e]. Doping and then high temperature annealing in manganese oxides significantly affects the host lattice, changing the magnetic order and the strength of magnetic interactions. Therefore, the inherent inhomogeneity and coexistence of multiple phases in these materials make them non-universal.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eThe values of critical exponents for the samples and the predicted critical exponents of the various theoretical models.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSample\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMethod\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cem\u003eβ\u003c/em\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003eγ\u003c/em\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u003cem\u003eδ\u003c/em\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eRef\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMean field Model\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e[\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3D-Heisenberg Model\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.365\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.386\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e4.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e[\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTricritical Model\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e[\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSample-A\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eK-F\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.388\u0026thinsp;\u0026plusmn;\u0026thinsp;0.005\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.056\u0026thinsp;\u0026plusmn;\u0026thinsp;0.005\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e3.722\u0026thinsp;\u0026plusmn;\u0026thinsp;0.002\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eThis Work\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSample-B\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eK-F\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.460\u0026thinsp;\u0026plusmn;\u0026thinsp;0.005\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.025\u0026thinsp;\u0026plusmn;\u0026thinsp;0.005\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e3.228\u0026thinsp;\u0026plusmn;\u0026thinsp;0.002\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eThis Work\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSample-C\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eK-F\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.435\u0026thinsp;\u0026plusmn;\u0026thinsp;0.005\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.940\u0026thinsp;\u0026plusmn;\u0026thinsp;0.005\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e3.161\u0026thinsp;\u0026plusmn;\u0026thinsp;0.003\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eThis Work\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe scaling hypothesis states that the magnetic equation of state can be defined through the correlation between \u003cem\u003eM\u003c/em\u003e(\u003cem\u003eH\u003c/em\u003e,\u003cem\u003eε\u003c/em\u003e), \u003cem\u003eH\u003c/em\u003e, and \u003cem\u003eT\u003c/em\u003e, expressed as[\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e] :\u003cdiv id=\"Equi\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equi\" name=\"EquationSource\"\u003e\n$$\\begin{array}{c}M\\left(H, \\epsilon \\right)={\\epsilon }^{\\beta }f\\pm \\left(H/{\\epsilon }^{\\beta +\\gamma }\\right)\\#\\left(9\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eHere, \u003cem\u003ef\u003c/em\u003e represents a conventional analytic function, with \u003cem\u003ef\u003c/em\u003e\u003csub\u003e+\u003c/sub\u003e and \u003cem\u003ef\u003c/em\u003e\u003csub\u003e\u0026minus;\u003c/sub\u003e applicable respectively for \u003cem\u003eT\u003c/em\u003e\u0026thinsp;\u0026gt;\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e and \u003cem\u003eT\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e. \u003cem\u003eε\u003c/em\u003e is the reduced temperature (\u003cem\u003eε=\u003c/em\u003e(\u003cem\u003eT\u003c/em\u003e\u0026thinsp;\u0026minus;\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e)/\u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e). The rescaled graph of \u003cem\u003eM\u003c/em\u003e/|\u003cem\u003eε\u003c/em\u003e|\u003csup\u003e\u003cem\u003eβ\u003c/em\u003e\u003c/sup\u003e is a function of \u003cem\u003eH\u003c/em\u003e/|\u003cem\u003eε\u003c/em\u003e|\u003csup\u003e\u003cem\u003eβ\u003c/em\u003e+\u003cem\u003eγ\u003c/em\u003e\u003c/sup\u003e, showing two independent branches near \u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e[\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e]. Figure\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e shows the two branches corresponding to \u003cem\u003eT\u003c/em\u003e\u0026thinsp;\u0026gt;\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e and \u003cem\u003eT\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e. The results provide confirmation that the critical parameters are consistent with the scaling hypothesis. In addition, the separate ln\u003cem\u003eM/|ɛ|\u003c/em\u003e\u003csup\u003e\u003cem\u003eβ\u003c/em\u003e\u003c/sup\u003e-ln\u003cem\u003eH/|ɛ|\u003c/em\u003e\u003csup\u003e\u003cem\u003eβ+γ\u003c/em\u003e\u003c/sup\u003e plot in the inset of Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e further demonstrates the accuracy of \u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e and the critical exponents.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"4 Conclusion","content":"\u003cp\u003eThis study successfully prepared polycrystalline samples of Pr\u003csub\u003e2\u003c/sub\u003eMn\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e6\u003c/sub\u003e and Pr\u003csub\u003e1.5\u003c/sub\u003eA\u003csub\u003e0.5\u003c/sub\u003eMn\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e6\u003c/sub\u003e (A\u0026thinsp;=\u0026thinsp;Mg, Ba) through high-temperature solid-phase reaction. The influence of Mg and Ba doping on the magnetocaloric effects and critical behavior of these materials was investigated. The results show that the samples exhibit excellent single-phase characteristics in the orthorhombic crystal system (space group Pbnm). Under a 7 T magnetic field, the maximum magnetic entropy change values for the samples are \u003cem\u003e|-\u003c/em\u003e∆\u003cem\u003eS\u003c/em\u003e\u003csub\u003eM\u003c/sub\u003e\u003csup\u003emax\u003c/sup\u003e\u003cem\u003e|\u003c/em\u003e\u003csub\u003e(Sample\u0026minus;A)\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;4.40 J\u0026middot;kg\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e\u0026middot;K\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e, \u003cem\u003e|-\u003c/em\u003e∆\u003cem\u003eS\u003c/em\u003e\u003csub\u003eM\u003c/sub\u003e\u003csup\u003emax\u003c/sup\u003e\u003cem\u003e|\u003c/em\u003e\u003csub\u003e( Sample\u0026minus;B)\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;3.66 J\u0026middot;kg\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e\u0026middot;K\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e, and \u003cem\u003e|-\u003c/em\u003e∆\u003cem\u003eS\u003c/em\u003e\u003csub\u003eM\u003c/sub\u003e\u003csup\u003emax\u003c/sup\u003e\u003cem\u003e|\u003c/em\u003e\u003csub\u003e( Sample\u0026minus;C)\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;3.72 J\u0026middot;kg\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e\u0026middot;K\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e. The relative cooling powers are RCP\u003csub\u003e(Sample\u0026minus;A)\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;483.46 J\u0026middot;kg\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e, RCP\u003csub\u003e(Sample\u0026minus;B)\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;428.22 J\u0026middot;kg\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e, and RCP\u003csub\u003e(Sample\u0026minus;C)\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;479.88 J\u0026middot;kg\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e, indicating significant magnetic refrigeration potential. The series of samples undergo a second-order phase transition with a small magnetic hysteresis, crucial for magnetic refrigeration applications. Using the Kouvel-Fisher method, a detailed analysis of critical behavior was conducted to determine critical exponents (\u003cem\u003eβ\u003c/em\u003e, \u003cem\u003eγ\u003c/em\u003e, and \u003cem\u003eδ\u003c/em\u003e). The results suggest that Sample-B and Sample-C are ferromagnets with a small amount of weakly itinerant electrons, exhibiting long-range exchange interactions near \u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e, while Sample-A exhibits short-range exchange interactions near \u003cem\u003eT\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e. In summary, the Pr\u003csub\u003e2\u003c/sub\u003eMn\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e6\u003c/sub\u003e ceramics investigated in this study exhibit significant potential as materials for magnetic refrigeration, offering valuable insights for research in magnetic refrigeration technology.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAcknowledgements\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis project was supported by\u0026nbsp;the National Natural Science Foundation of China (NSFC) under grant nos.52061035, 51871124, 51561026 and 51401111, Young Leading Talent of \u0026ldquo;Grassland Talents\u0026rdquo; Project of Inner Mongolia Autonomous Region, Inner Mongolia Natural Science Cultivating Fund for Distinguished Young Scholars (no. 2020JQ05), Science and Technology Planning Project of Inner Mongolia Autonomous Region (no. 2020GG0267), Program for Innovative Research Team in Universities of Inner Mongolia Autonomous Region (no. NMGIRT2211), Inner Mongolia University of Technology Key Discipline Team Project of Materials Science (no. ZD202012).\u003c/p\u003e\n\u003cp\u003eThe authors declare no conflict of interest.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eXu P, Hu L, Zhang Z Q, Wang H F, Li L W. Electronic structure, magnetic properties and magnetocaloric performance in rare earths(RE) based RE\u003csub\u003e2\u003c/sub\u003eBaZnO\u003csub\u003e5\u003c/sub\u003e(RE = Gd, Dy, Ho, and Er) compounds[J]. Acta Materialia, 2022, 236: 118114.\u003c/li\u003e\n\u003cli\u003eZhang Y K, Li S, Hu L, Wang X H, Li L W, Yan M. Excellent magnetocaloric performance in the carbide compounds RE\u003csub\u003e2\u003c/sub\u003eCr\u003csub\u003e2\u003c/sub\u003eC\u003csub\u003e3\u003c/sub\u003e(RE= Er, Ho, and Dy) and their composites[J]. Materials Today Physics, 2022, 27: 100786.\u003c/li\u003e\n\u003cli\u003ePhan M H, Franco V, Bingham N S, Srikanth H, Hur N H, Yu S U. 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Physical Review, 1964, 136(6A): A1626.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"journal-of-low-temperature-physics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jltp","sideBox":"Learn more about [Journal of Low Temperature Physics](http://link.springer.com/journal/10909)","snPcode":"10909","submissionUrl":"https://submission.nature.com/new-submission/10909/3","title":"Journal of Low Temperature Physics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Magnetocaloric effects, Magnetic phase transition, Second-order phase transition, Critical behavior","lastPublishedDoi":"10.21203/rs.3.rs-4141743/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4141743/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn this study, polycrystalline samples of Pr\u003csub\u003e2\u003c/sub\u003eMn\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e6\u003c/sub\u003e (parent phase) and Pr\u003csub\u003e1.5\u003c/sub\u003eA\u003csub\u003e0.5\u003c/sub\u003eMn\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e6\u003c/sub\u003e (A\u0026thinsp;=\u0026thinsp;Mg, Ba) were prepared using the high-temperature solid-phase reaction method. The effects of Mg and Ba doping on magnetocaloric properties and critical behavior of the parent phase were systematically investigated. Under a magnetic field of 7 T, the relative cooling power (RCP) values for this sample series were approximately 483.46 J\u0026middot;kg\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e, 428.22 J\u0026middot;kg\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e, and 479.88 J\u0026middot;kg\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e, respectively. The critical behavior analysis revealed that the parent phase showed short-range exchange interactions, while Pr\u003csub\u003e1.5\u003c/sub\u003eA\u003csub\u003e0.5\u003c/sub\u003eMn\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e6\u003c/sub\u003e (A\u0026thinsp;=\u0026thinsp;Mg, Ba) exhibited long-range exchange interactions. The temperature dependence of the order parameter \u003cem\u003en\u003c/em\u003e was studied under different magnetic fields, confirming the phase transition types and validating the accuracy of the critical exponents obtained. The research findings suggest that both the parent phase and Ba-doped ceramics at the A-site hold promise as magnetic refrigeration materials.\u003c/p\u003e","manuscriptTitle":"Study of the Magnetocaloric Effect and Critical Behavior in Double Perovskite Manganese Oxides Pr 1.5 A 0.5 Mn 2 O 6 (A=Mg, Ba)","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-03-26 06:40:07","doi":"10.21203/rs.3.rs-4141743/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2024-04-29T16:27:24+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-04-09T17:20:55+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"8131ca0e-77af-4557-9556-6f13d2bbfd0f","date":"2024-04-01T14:40:52+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2024-03-26T19:07:25+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2024-03-22T10:25:59+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2024-03-22T10:25:59+00:00","index":"","fulltext":""},{"type":"submitted","content":"Journal of Low Temperature Physics","date":"2024-03-21T08:01:52+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"journal-of-low-temperature-physics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jltp","sideBox":"Learn more about [Journal of Low Temperature Physics](http://link.springer.com/journal/10909)","snPcode":"10909","submissionUrl":"https://submission.nature.com/new-submission/10909/3","title":"Journal of Low Temperature Physics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"57f85544-f255-45ae-a811-a189f938e72c","owner":[],"postedDate":"March 26th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2024-06-27T18:43:00+00:00","versionOfRecord":[],"versionCreatedAt":"2024-03-26 06:40:07","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-4141743","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4141743","identity":"rs-4141743","version":["v1"]},"buildId":"7rjqhiLT3MXkJMwkYKINL","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}