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*(Å3) | *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 Mn3+ and Mn4+, 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*Mmax*|*(Sample−A) ≈ 4.40 J·kg− 1·K− 1, *|-*∆*S*Mmax*|*(Sample−B) ≈ 3.66 J·kg− 1·K− 1, and *|-*∆*S*Mmax*|*(Sample−C) ≈ 3.72 J·kg− 1·K− 1. Notably, although the *|-*∆*S*Mmax*|* 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*Mmax*|* 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*Mmax| 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*0Δ*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 |

Pr0.87Ca0.13MnO3 | - | 4.608 | 460.77 | 7 | [16] |

Pr2CoMnO6 | - | 1.862 | 126.62 | 7 | [17] |

Pr0.55Sr0.45MnO3 | 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 (NRCmax) for temperature intervals of 4 K under external magnetic fields of 2 T and 5 T are shown. It is evident that NRCmax 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*Mmax|(*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.