The article presents the study of the magnetocaloric effect (MCE) for the rhombohedral Laves phase of Pr $ 2$Rh $ 3$Ge, which shows a magnetic order below $ T C=8.5$ K. We have established that the compound exhibits a continuous second-order type of transition which was demonstrated and confirmed by several different techniques, mainly by analyzing universal curves of normalized entropy change as a function of scaled temperature. The observed MCE, in our opinion, is a consequence of an indirect exchange coupling between the magnetic sublattices of the rare earth ions, which, however, does not exclude the potential contribution of sublattice-containing transition metals. In this paper, the procedure to evaluate the MCE from magnetization and specific heat data is described. As a result, important parameters such as the isothermal magnetic entropy change ( $\Delta $*S* $ M$), adiabatic temperature change ( $\Delta $*T* $ a d$), relative cooling power (*RCP*), and the temperature averaged entropy change (*TEC*) were determined. The highest values of $\u2212\Delta S M$, $\Delta T a d$, and *RCP* for a field change ( $\Delta \mu 0H$) of 5 T at $ T C$ are 5.96 J/kg $$K, 3.87 K, and 72.62 J/kg, respectively. These results obtained for Pr $ 2$Rh $ 3$Ge seem to be, however, low compared to the values obtained for the rhombohedral Laves phases, belonging to the group of ternary germanides RE $ 2$Rh $ 3$Ge containing heavy rare earth metals (RE = Gd, Tb, Ho, and Er). Nevertheless, we believe that the results presented in this work extend and complement the current knowledge on the magnetocaloric properties of this family of materials.

## I. INTRODUCTION

The presence of magnetic order in lattice arrangements of robust magnetic moments based on rare earth elements, usually heavy ones, opens the possibility of studying the magnetocaloric properties of such compounds. While the magnetocaloric effect (MCE) has been/is one of the more widely studied and developed branches of condensed matter physics over the last couple of decades, the influence and contribution of strong electronic correlations to the phenomenon of the MCE still remains largely unclear. Enhanced electron correlations in lanthanide-based compounds are generally found in systems exhibiting features of heavy fermion Kondo lattice, i.e., compounds based mainly on Ce and Yb.^{1–3} Frequently, these materials also show long-range magnetic order at low temperatures (usually of an antiferromagnetic nature), which also provides the basis for research and analysis of the MCE, an example is Ce $ 6$Pd $ 12$In $ 5$.^{4}

In the case discussed in this paper, we are dealing with the ternary compound Pr $ 2$Rh $ 3$Ge obtained in the polycrystalline form, which belongs to the rhombohedral Laves phases (space group *R $ 3 \xaf$m*).^{5} The compound is known to be a ferromagnet with magnetic ordering temperature *T* $ C$ = 8.5 K.^{6} Nevertheless, the general picture of the physical properties of the ground state in this instance seems to be quite intriguing, not to say intricate. The reason for this is the observed moderate heavy fermion behavior of itinerant charge carriers at low temperatures ( $\gamma =315$ mJ/Pr mol K $ 2$), with no sign of the spin Kondo effect.^{6} In this situation, the mechanism responsible for mass enhancement leading to the heavy electron ground state is probably related to the dynamic low-lying crystal field excitations. This assumption is based on the theory of excitonic mass enhancement proposed by White and Fulde^{7} to explain the electron mass improvement in Pr ions and subsequently extended to rare earth systems.^{8}

The mentioned ternary rhombohedral Laves phases with the general formula RE $ 2$Rh $ 3$Ge appear to belong to an interesting material category. This initially rather narrow (RE = Y, Ce, Pr, Er^{5,6,9}), but still growing and developing group of germanides, especially in terms of magnetic and magnetocaloric properties (RE = Gd, Tb, Ho, and Er^{10–13}) exhibits quite interesting physical properties of the ground state.

Guided by curiosity and a desire to expand the scope of existing knowledge about the physical properties of Pr $ 2$Rh $ 3$Ge, we present research on the MCE basis of the well-known thermodynamic approaches. The magnetocaloric properties were established mainly about the magnetic entropy change $\Delta $*S* $ M$ and the adiabatic temperature change $\Delta $*T* $ a d$ using the specific heat data and magnetization measurements. We also applied a few methods to verify the order of the magnetic phase transition, to finally confirm its nature.

## II. EXPERIMENTAL ISSUES

The preparation and characterization of the main bulk properties of Pr $ 2$Rh $ 3$Ge were described in the previous work,^{6} so the reader interested in synthesis procedure and physical background of this compound can find all this information in the above article. Briefly, the polycrystalline sample was prepared by argon arc melting and subsequent heat treatment under a high vacuum at 800 $ \xb0$C for a week. The crystal structure was established from room-temperature x-ray diffraction (XRD) data and found to be a rhombohedral Mn $ 2$Ni $ 3$Si-type structure (triple hexagonal cell) with space group *R $ 3 \xaf$m*—see Fig. 1. Refinement of the crystal structure was performed with the program Fullprof^{14} under the WinPlotr shell,^{15} while the crystal structure pictures were generated using the VESTA software.^{16} The obtained refinement parameters are gathered in Table I. In addition, the phase purity and the chemical composition of the sample were probed by using scanning transmission electron microscopy (STEM) equipped with an EDX detector. The stoichiometric ratio of the elemental composition has been established to be close proximity with the nominal one.

Molar mass (g/mol) | 663.1718 | |||

Type of structure | Mn_{2}Ni_{3}Si-type of rhombohedral Laves phase | |||

Space group | R $ 3 \xaf$m, No. 166 | |||

Lattice parameters (Å) | a = b = 5.625(1); c = 11.998(3) | |||

Cell volume (Å^{3}) | 328.764 | |||

Formula units per cell, Z | 3 | |||

Calculated density (g/cm^{3}) | 10.11 | |||

Radiation, λ (Å) | Cu Kα_{1,2}, 1.5405 | |||

Scan range (2θ°) | 10–90 | |||

Scan, step (φ) | 0.017 | |||

Goodness of fit | 1.51 | |||

Atom | Wyckoff | x | y | z |

Pr | 4 c | 0 | 0 | 0.3718 |

Rh | 9 d | 0.5 | 0 | 0.5 |

Ge | 3 a | 0 | 0 | 0 |

Molar mass (g/mol) | 663.1718 | |||

Type of structure | Mn_{2}Ni_{3}Si-type of rhombohedral Laves phase | |||

Space group | R $ 3 \xaf$m, No. 166 | |||

Lattice parameters (Å) | a = b = 5.625(1); c = 11.998(3) | |||

Cell volume (Å^{3}) | 328.764 | |||

Formula units per cell, Z | 3 | |||

Calculated density (g/cm^{3}) | 10.11 | |||

Radiation, λ (Å) | Cu Kα_{1,2}, 1.5405 | |||

Scan range (2θ°) | 10–90 | |||

Scan, step (φ) | 0.017 | |||

Goodness of fit | 1.51 | |||

Atom | Wyckoff | x | y | z |

Pr | 4 c | 0 | 0 | 0.3718 |

Rh | 9 d | 0.5 | 0 | 0.5 |

Ge | 3 a | 0 | 0 | 0 |

Magnetic measurements were recorded using a 7 T SQUID magnetometer (MPMS XL). The magnetization measurements, $M$( $T$) were carried out according to the zero-field-cooling (ZFC) and field-cooling (FC) protocols under $ \mu 0H=0.01$ T and in a temperature range of 1.7–400 K, while isothermal magnetization *M*( $ \mu 0H$) was recorded for different temperatures (below and above *T* $ C$) under a magnetic field up to 7 T. The specific heat measurements, $ C p$( $T, \mu 0H$) were performed on the PPMS-9 T platform from *Quantum Design* employing the relaxation method using the 2- $\tau $ model. The experiment was carried out at a constant pressure in several different applied magnetic fields ranging between 0 and 5 T.

## III. RESULTS AND DISCUSSION

The low-temperature variation of field-cooled (FC) and zero-field-cooled (ZFC) magnetization $M$( $T$) at applied magnetic field of 10 mT for Pr $ 2$Rh $ 3$Ge is shown in Fig. 2. As can be seen both magnetization procedures display fairly large dependence on the magnetic history of the specimen, i.e., FC and ZFC curves diverge, which is primarily due to the appearance of magnetic order implying the hysteretic behavior.^{6} On the other hand, an appreciable thermomagnetic irreversibility visible between ZFC and FC curves in the magnetically ordered state may also indicate a high degree of magnetocrystalline anisotropy. To define the magnetic transition temperature, we plotted in the inset in Fig. 2 the temperature derivative of magnetization. The presence of ferromagnetic-like phase transition is observable at *T* $ C$ = 8.5 K, in the form of a distinct and sharp minimum in d*M*/d*T* vs *T*. The characterization of the paramagnetic state of Pr $ 2$Rh $ 3$Ge from magnetic susceptibility data (not shown here—see Ref. 6) reveals a Curie–Weiss behavior with an effective Pr moment $\mu $ $ e f f=3.74$ $\mu $ $ B$/Pr and a paramagnetic Curie temperature $ \theta p=3$ K. The former is close to the theoretical value for a free trivalent Pr $ 3 +$ (a slightly higher value than the theoretical one of the pure Pr $ 3 +$ ion (3.58 $\mu $ $ B$) may imply that Rh atoms give the additional contribution to the effective magnetic moment), while the positive sign of $ \theta p$ can be understood from the development of ferromagnetic-type correlations.

To better understand magnetic behavior and the nature of magnetic transition, isothermal magnetization curves were measured at various temperatures (above and below *T* $ C$). Figure 3(a) shows the representative *M*( $ \mu 0$*H*) curves at selected temperatures between 2 and 60 K for the investigated compound. *M*( $ \mu 0H$) curves recorded below *T* $ C$ exhibit a conspicuous rise at low fields and tend to saturate at much higher fields, as would generally be expected for ferromagnetic materials. The gradual evolution with increasing temperature of *M*(* $ \mu 0H$*) curves to linear behavior characterizes an increase of the paramagnetic contribution above *T* $ C$. In the previous paper,^{6} we employed the Banerjee criterion^{17} to establish the magnetic phase transition character. According to this criterion, it is assumed that the magnetic transition is of the first-order or second-order depending on the sign of the slope of $ M 2( \mu 0H/M$) dependences. In the case of a first-order transition, the negative contributions from the higher-order terms in the Landau free energy expansion are expected. In other words, in the vicinity of *T* $ C$ the Arrott’s curves should reflect the S-shape. Alternatively, by searching for the presence of a negative (or positive) slope region on the isothermal plots of $ \mu 0H/M$ vs *M* $ 2$ first-order (or second-order) phase transitions can be also recognized.^{18} Accordingly, to carefully verify the nature of the magnetic transition in Pr $ 2$Rh $ 3$Ge, three isotherms $ \mu 0$*H*/*M* vs *M* $ 2$ for *T* $\u2264$ *T* $ C$ were plotted in Fig. 3(b). The applied criterion seems to be clear in this instance. Figure 3(b) shows that the slope of $ \mu 0$*H*/*M* vs *M* $ 2$ curves is positive, contrary to what is expected for a first-order phase transition.

What, however, may draw the reader’s attention is the isotherm at 2 K, where there are some points at low fields which, the way they are depicted, might give the wrong impression. As seen, at low fields the Arrott curve does not show a purely ferromagnetic feature, i.e., where according to the Arrott criterion as well as mean-field theory the plot should be more or less linear.^{19} This situation also applies to the S-like curvature in the *M* $ 2$( $ \mu 0$*H*/*M*) isotherm for $T=2$ K observed earlier,^{6} which could indicate a first-order-like nature or inhomogeneous magnetism. Nevertheless, in our opinion, this is most likely because for $T=2$ K the *M*( $ \mu 0H$) measurement was performed as a full magnetic hysteresis loop (without the previous magnetization history of the sample), and the $ \mu 0$*H*/*M* vs *M* $ 2$ isotherm at 2 K shown in Fig. 3(b) reflects within this context the initial magnetization curve whose shape at low fields somewhat resembles the S-shape.

*S*$ M$ based on

*M*(

*$ \mu 0$H*) and

*$C$*$ p$(

*T, $ \mu 0$H*) measurements. The isothermal magnetic entropy change, which results from the spin ordering under the influence of a magnetic field, can be determined from the isothermal magnetization curves as the field is changed from an initial value $ \mu 0$

*H*$ 1$ to a final value $ \mu 0$

*H*$ 2$ using the integrated Maxwell thermodynamical relation

^{20,21}

*M*(

*$ \mu 0$H*) curves measured at different constant temperatures, the above relation can be approximated to the following expression:

^{22,23}

*S*$ M$,

*M*, $ \mu 0$

*H*, and

*T*are the magnetic entropy, magnetization, applied magnetic field, and temperature of the system, respectively.

*$C$*$ p$(

*T, $ \mu 0$H*) in the proximity of the ordering temperature and a well-defined jump observed in zero magnetic field

*$C$*$ p$(

*T*, 0) at

*T*$ C$, with a peak height $\u223c3.8$ J/Pr mol $$K

^{6}the $\Delta $

*S*$ M$ can be evaluated by a formula

^{22}

*S*$ M$ observed for both experimental methods are found near a paramagnetic-ferromagnetic phase transition. Also in both cases, the maxima of the $\u2212\Delta S M$ curves at higher applied magnetic fields swiftly increase up to

*T*$ C$, and then they show a more monotonous decrease above it. Such behavior seems to be in line with the conventional caloric response of the field-induced ferromagnetic phase. The maximum values of $\u2212\Delta S M max$ around

*T*$ C$ for the external magnetic field change $\Delta \mu 0H=0\u22125$ T from the measurements of

*M*( $ \mu 0$H) and

*$C$*$ p$( $T, \mu 0H$) are in fairly good compliance, but they are lower for the same applied field compared to other ternary rhombohedral Laves phases RE $ 2$Rh $ 3$Ge (RE = Gd, Tb, Ho, or Er

^{11–13}),—this is well visible in Fig. 6(a). The $|\Delta S M max|$ values obtained from magnetic measurements, plotted as a function of $ \mu 0$

*H*up to 7 T are collected in Fig. 5 (see the left-hand axis). Also, to better systematize and enable comparison of individual parameters characterizing the magnetocaloric properties in Pr $ 2$Rh $ 3$Ge, their values are listed in Table II.

Δμ_{0}H
. | $|\Delta S M max | M ( \mu 0 H )$ . | $|\Delta S M max | C p ( T )$ . | ΔT_{ad}
. | RCP
. | TEC(3)
. | TEC(10)
. |
---|---|---|---|---|---|---|

(T) . | (J/kg K) . | (J/kg K) . | (K) . | (J/kg) . | (J/kg K) . | (J/kg K) . |

0−0.1 | … | 0.38 | 0.24 | … | … | … |

0−0.3 | … | 0.69 | 0.48 | … | … | … |

0−0.5 | 0.44 | 0.95 | 0.66 | 3.03 | 0.43 | 0.31 |

0−1 | 1.25 | 1.55 | 1.11 | 9.22 | 1.19 | 0.88 |

0−2 | 2.74 | 2.52 | 1.90 | 23.78 | 2.61 | 2.02 |

0−3 | 3.98 | … | … | 40.04 | 3.82 | 3.06 |

0−4 | 5.04 | … | … | 55.87 | 4.87 | 3.99 |

0−5 | 5.96 | 4.58 | 3.87 | 72.62 | 5.78 | 4.84 |

0−6 | 6.77 | … | … | 92.74 | 6.59 | 5.60 |

0−7 | 7.50 | … | … | 111.47 | 7.37 | 6.35 |

Δμ_{0}H
. | $|\Delta S M max | M ( \mu 0 H )$ . | $|\Delta S M max | C p ( T )$ . | ΔT_{ad}
. | RCP
. | TEC(3)
. | TEC(10)
. |
---|---|---|---|---|---|---|

(T) . | (J/kg K) . | (J/kg K) . | (K) . | (J/kg) . | (J/kg K) . | (J/kg K) . |

0−0.1 | … | 0.38 | 0.24 | … | … | … |

0−0.3 | … | 0.69 | 0.48 | … | … | … |

0−0.5 | 0.44 | 0.95 | 0.66 | 3.03 | 0.43 | 0.31 |

0−1 | 1.25 | 1.55 | 1.11 | 9.22 | 1.19 | 0.88 |

0−2 | 2.74 | 2.52 | 1.90 | 23.78 | 2.61 | 2.02 |

0−3 | 3.98 | … | … | 40.04 | 3.82 | 3.06 |

0−4 | 5.04 | … | … | 55.87 | 4.87 | 3.99 |

0−5 | 5.96 | 4.58 | 3.87 | 72.62 | 5.78 | 4.84 |

0−6 | 6.77 | … | … | 92.74 | 6.59 | 5.60 |

0−7 | 7.50 | … | … | 111.47 | 7.37 | 6.35 |

*T*$ a d$. This can be obtained from the magnetic entropy calculated from the specific heat data. The isoentropic difference between two entropy curves gives the dependence of the adiabatic temperature change,

The adiabatic temperature changes as a result of $ C p(T, \mu 0H)$ measurements are shown in Fig. 4(c). As seen, the increase in the applied magnetic field leads to an increase in $\Delta T a d$ near the phase transition and the maximum value of $\Delta T a d$(*T*) reaches $\u223c3.9$ K for $\Delta \mu 0H=0\u22125$ T.

*S*$ M$ and $\Delta $

*T*$ a d$ values, we can see that for Pr $ 2$Rh $ 3$Ge, these parameters do not give good grounds to classify our compound as a material with promising magnetocaloric properties. The fact is that one of the main criteria that magnetic refrigerators should meet is a large change in magnetic entropy and a large adiabatic temperature change. However, from a practical point of view, an alternatively important quality parameter is also the relative cooling power (

*RCP*), which defines the amount of heat transferred from the cold to the hot end of the refrigeration circuit in a thermodynamic refrigeration cycle.

^{24–26}The

*RCP*is defined as the product of $|\Delta S M max|$ and the full width of half maximum of the peak for the $\Delta S M$(

*T*) curve,

Figure 5 (right-hand axis) shows the variation of *RCP* with an applied magnetic field. As the magnetic field strength increases, the *RCP* reaches rather moderate values, not to say low (111.5 J/kg at $ \mu 0H=7$ T) compared to other ternary germanides RE $ 2$Rh $ 3$Ge based on heavy rare earth metals.^{11–13} For a material exhibiting a second-order transition, as it is foreseen in the framework of the mean-field theory, the field-dependent maximum magnetic entropy change should be $\Delta S M max\u221d \mu 0 H 2 / 3$, while the relative cooling power *RCP* $\u221d$ $ \mu 0 H 4 / 3$.^{27} Hence, in Fig. 5, we present the fittings for the field dependencies of both parameters to test their power law behavior $ \mu 0 H n$. The obtained values of the power-law coefficient *n* are in both cases close to those predicted by the mean-field theory. In addition, in the inset of Fig. 5, we show the plot of $\Delta T a d max$ vs $ \mu 0$*H* by making use of the specific heat data. The observed non-linear increase in $\Delta T a d max$ with the increasing strength of the magnetic field to some extent resembles the behavior of $|\Delta S M max|$ seen in the main panel of Fig. 5.

In Fig. 6, we illustrate in the form of bar charts the position of our Pr-compound against the background of other RE $ 2$Rh $ 3$Ge Laves phases (including one ternary silicide phase crystallizing in the same structure type as germanides, viz Er $ 2$Rh $ 3$Si^{28}) in the context of magnetocaloric parameters such as $|\Delta S M max|$ and *RCP* for $\Delta \mu 0H=0\u22125$ T. As has already been noticed, the magnetocaloric performance of the Pr $ 2$Rh $ 3$Ge and the rest investigated to date the ternary rhombohedral Laves phases based on heavy rare earth metals differ significantly. The reason for this state of affairs can be found, for example, in possible weaker interactions between sublattices of Pr atoms and/or additional contribution of interactions of sublattice containing transition metals. However, it should also be considered that when it comes to Pr $ 2$Rh $ 3$Ge the ground state may differ significantly from the rest of RE $ 2$Rh $ 3$Ge compounds due to the observed enhanced value of the electronic specific heat coefficient, and thus improvement in the thermodynamic effective mass of the fermionic quasiparticles at low temperatures. This is primarily about the likely impact of the crystalline electric field effects, within the so-called dynamic low-lying crystal field fluctuations, which can play a pivotal role in the properties of the ground state of Pr $ 3 +$ ions (*J* = 4),^{7} but inelastic neutron scattering experiments are strongly needed to confirm this point.

*TEC*) proposed by Griffith

*et al.*

^{29}which is a suitable early indicator of the materials utility for magnetocaloric applications

^{30–32}and it can be calculated using the data of isothermal magnetization measurements via the expression

*T*$ l i f t$ represents the selected range of temperatures where a swift response of magnetic field change $\Delta \mu 0$

*H*can be observed,

*T*$ m i d$ is the center of $\Delta $

*T*$ l i f t$ and the above integral is maximized with respect to

*T*$ m i d$. Figure 7(a) shows the results of numerical integration of isothermal $\u2212\Delta $

*S*(

*T*) data (extrapolated to zero kelvin) for a given $\Delta \mu 0$

*H*using Eq. (6). As is evident from Fig. 7(a), a noticeable tendency for

*TEC*to monotonically decrease with increasing temperature span $\Delta $

*T*$ l i f t$ regardless of different values of $\Delta \mu 0$

*H*is observed, which is typical and expected behavior.

^{33,34}As far as we are concerned, we focused mainly on determining the two usually selected

*TEC*values, i.e., at $\Delta T l i f t=3$ and 10 K—see vertical dashed lines in Fig. 7(a) as well as the points shown in Fig. 7(b). Table II also compares both

*TEC*values, among others, with $|\Delta S M max|$ obtained from isothermal magnetization data. While comparing

*TEC*(3) and $|\Delta S M max|$ at various $\Delta \mu 0$

*H*, one can notice a great similarity between these values, and even looking at the individual

*TEC*(10) values, it can be seen that on average they constitute nearly 78 $%$ of $|\Delta S M max|$. This fact shows that despite rather low maximum values of $\Delta $

*S*$ M$, $\Delta $

*T*$ a d$, and

*RCP*our compound can have considerable potential in terms of the magnetic refrigeration effectiveness in a fairly wide temperature range.

^{17}This criterion has been for years the universal and routine method to determine the order of a phase transition using magnetic measurements. Nevertheless, Franco

*et al.*

^{27}have proposed a more accurate method in which a phenomenological universal curve of the field dependence of $\Delta $

*S*$ M$ can correctly distinguish the order of the magnetic phase transition. The universal master curve can be constructed by normalizing all the $\u2212\Delta S M$(

*T*) curves against the respective maximum $\u2212\Delta S M max$, by rescaling the reduced temperature $\theta $ below and above

*T*$ C$, defined as

*T*$ r 1$ and

*T*$ r 2$ are the temperatures of two reference points of each curve which correspond to $\Delta S M( T r 1, T r 2)=\Delta S M max/2$, and assuming that both reference temperatures meet the condition $ T r 1< T C< T r 2$. The normalized entropy change curves as a function of the rescaled temperature, $\Delta S M/\Delta S M max$ vs $\theta $ for Pr $ 2$Rh $ 3$Ge are shown in Fig. 8. As is apparent all normalized curves collapse into a single curve confirming that the paramagnetic-ferromagnetic transition observed in our material is of the second-order. Therefore, the magnetic transition in Pr $ 2$Rh $ 3$Ge should be considered as a continuous second-order phase transition, and not as misinterpreted in the previous work,

^{6}a first-order. The rescaled magnetic entropy change curve can be well-fitted by a Lorentz function,

*a*,

*b*, and

*c*are adjustable parameters. As displayed in Fig. 8, the applied fitting procedure using Eq. (8) provides a pretty good approximation of a master curve, yielding the following values of the parameters: $a=0.91$, $b=0.90$, and $c=\u22120.01$. Inset in Fig. 8 shows the field dependence of the reference temperatures

*T*$ r 1$ and

*T*$ r 2$. It is worth noting that good efficiency of the MCE is observed mainly in the

*T*$ r 1$ region, while around

*T*$ r 2$ it is practically negligible. Hence, it can be assumed that for our compound, as well as presumably for other ferromagnetically ordered ternary RE $ 2$Rh $ 3$Ge rhombohedral Laves phases, the MCE is due to the magnetic entropy variation arising largely from strong interactions of the rare earth sublattice.

*S*$ M$, which can also well reflect the intrinsic relation between the MCE in the studied material and the universality class. It has been found that this dependence can be expressed by a power-law of the external applied field $|\Delta S M|\u221d \mu 0 H n$, where the power-law coefficient

*n*is dependent on magnetic flux density and temperature. To locally estimate the exponent

*n*, we can use the formula

^{35,36}

*n*more directly, i.e., using the linear regression in the following form:

In this situation, the exponent *n* is determined directly from the slope of a straight line (see inset in Fig. 9). In Eq. (10), *C* is a proportionality constant depending on temperature. From Fig. 9, two important observations can be made, namely, (i) that the *n*(*T*) curve exhibits the minimum value of $n=0.66$ in the proximity of $ T C$, which is very close to the mean-field theory value of $n=2/3$ and (ii) in the paramagnetic state $n<2$, another evidence of the transition of the second-order.

## IV. CONCLUSIONS

In summary, our extended analysis regarding the determination of the order of the magnetic phase transition in Pr $ 2$Rh $ 3$Ge clearly points to the second-order type, in contrast to the former report of the first-order type. The magnetocaloric properties were evaluated based on measurements of isothermal magnetization and specific heat. The applied procedure to evaluate the magnitude of the MCE based on the well-known theoretical approaches, including Maxwell’s thermodynamical relations, using experimental *M*(* $ \mu 0$H*) and * $C$* $ p$(*T, $ \mu 0$H*) data has provided rather average estimates for the investigated material. It turned out that in the case of Pr $ 2$Rh $ 3$Ge, the calculated parameters characterizing the MCE at $\Delta \mu 0$*H* = 0 $\u2212$5 T, such as $\u2212\Delta S M(=5.96 J/ kg K$), $\Delta T a d(=3.87 K)$, and $RCP(=72.62 J/ kg)$ are smaller compared to other ternary rhombohedral Laves phases RE $ 2$Rh $ 3$Ge containing heavy rare earth elements, like Gd ( $\u2212\Delta S M=9.4$ J/kg $$K, $RCP=352$ J/kg^{11}), Tb ( $\u2212\Delta S M=9.2$ J/kg $$K, $RCP=320$ J/kg^{12}), Ho ( $\u2212\Delta S M=18.4$ J/kg $$K, $RCP=368$ J/kg^{13}), or Er ( $\u2212\Delta S M=9.2$ J/kg $$K, $RCP=225$ J/kg^{11}). Nevertheless, the optimistic accent seems to be the fact that our compound may have significant potential in the context of magnetic refrigeration effectiveness in a fairly wide temperature range. This may be indicated by the similar values of *TEC*(3) and *TEC*(10) compared to the isothermal $|\Delta S M max|$ values for a broad scope of different $\Delta \mu 0H$. Finally, we also found that the value of the local exponent *n* near *T* $ C$ is close to the theoretical prediction of the mean-field model value of $n=2/3$, which implies that the ferromagnetic-type interactions dominates the critical behavior in Pr $ 2$Rh $ 3$Ge.

## ACKNOWLEDGMENTS

The author wishes to express his gratitude to Professor André M. Strydom for the hospitality at the Highly Correlated Matter Research Group and financial support during the Postdoctoral Fellowship at the Physics Department of the University of Johannesburg between 2012 and 2014.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**M. Falkowski:** Conceptualization (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Visualization (lead); Writing – original draft (lead).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*The Kondo Problem to Heavy Fermions*

*WinPLOTR: A windows tool for powder diffraction patterns analysis*, edited by R. Delhez and E. J. Mittenmeijer [Materials Science Forum, Proceedings of the Seventh European Powder Diffraction Conference (EPDIC 7), 118 (2000)].

*The Magnetocaloric Effect and Its Applications*

*Intermetallic Compounds—Principles and Practica*, edited by J. H. Westbrook and R. L. Fleischer, Vol. 3 (Wiley, New York, 2001).