1 Introduction

The half-doped manganites R0.5Sr0.5MnO3 (R = Pr, Nd, La) are specially interesting due to their colossal magnetoresistance (CMR) together with their particular phase diagram with respect to other doping concentrations. From room temperature (where they are both in a paramagnetic insulator state) downward, a ferromagnetic metallic phase (FM) is developed at about 250 K, while at lower temperature, they both become antiferromagnetic insulators (AFM) [1, 2]. If CMR were simply described by the double-exchange mechanism (DE), it would be expected that the system in the pure ferromagnetic metallic state had no orbital ordering and showed an isotropic behavior. But the study of the spin dynamics of these materials by neutron scattering measurements has shown the existence of the static dx2-y2-type orbital ordering in both the paramagnetic and ferromagnetic states [3], implying that there is an anisotropic behavior which has been confirmed in the spin wave dispersion in the FM state. Besides, the antiferromagnetic ordering, it is the CE-type AFM spin structure accompanied with charge ordering for Nd0.5Sr0.5MnO3 [4, 5].

The phase diagram is even a little more complex for the case of Nd0.5Sr0.5MnO3 as it has been demonstrated that there is a coexistence of an A-type antiferromagnetic phase with the CE-type below TN using different techniques [1, 4, 6, 7] and that there is also a coexistence of AFM phase in the FM region (between TC and TN) [6, 8, 9]. Concerning this last proposed phase coexistence, Kawano-Furukawa et al. [3] found A-type AFM spin wave excitations in the FM state, which they attributed to a canted AFM ordering instead of a phase coexistence.

These new properties have undergone a new start to the perovskite materials, especially the magnetic oxides (AMnO3 or the AFeO3). The magnetic properties are important factors for better understanding of the oxide behavior. The study of perovskite NdMnO3 has been motivated by reporting that Nd0.6Sr0.4MnO3 has the largest Colossal magnetoresistance (CMR) effect among the manganites [10, 11].

In this work, we are interested in the theoretical work on the magnetization as a function of temperature with different magnetic fields. The simulation provides magnetocaloric properties such as change of magnetic entropy, relative cooling power and change of thermal capacity.

2 Experimental

The Nd0.6Sr0.4MnO3 compound was prepared using precursors of Sr2O3 (4 N purity), Nd2O3 (4 N) and MnO2 by solid-state reaction [12]. The raw powders were preheated at T = 200 °C during 2 h before weighing and mixing. These powders were then mixed in a required atomic ratio with an agate mortar and pressed, to reduce the grain size of the order of nonometric and give a homogeneous compound. The pellets were heated at 1200 K for 6 h, this choice of heat treatment temperature, to remove all the impurities. The sample structure was characterized by X-ray diffraction with Cu Ka radiation (λ = 1.5406 Å) by step scanning (0.02°) in the range 20° ≤ θ ≤ 100°.

Magnetic measurements were performed in a BS2 magnetometer developed in Louis Neel Laboratory of Grenoble. The magnetization curves were obtained under different applied magnetic fields with a temperature ranging from 4 to 350 K.

3 Results and Discussion

3.1 X-Ray Analysis

The phase identification and structural analysis were carried out by X-ray diffraction (XRD) technique with Cu K radiation at room temperature. The data were analyzed by the Rietveld method using Fullprof program [13].

Figure 1 shows the refinement of XRD patterns for Nd0.6Sr0.4MnO3 sample, and this refinement has revealed the coexistence of two structures attributed to the Pnma orthorhombic and the R-3c rhombohedral space group. However, we note the secondary phase attributed to the unreacted Mn3O4. Related refined cell parameters, unit cell volume, selected interatomic distances and angles are given in Table 1.

Fig. 1
figure 1

X-ray diffraction pattern and Rietveld refinement result for Nd0.6Sr0.4MnO3 at room temperature. The points are the observed profile, the solid line and the calculated one. Positions for the Bragg reflection are marked by vertical bras. The line curve at the bottom of the diagram gives the difference between observed and calculated profiles. The inset is an expanded view of the crystal structure (Color figure online)

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Table 1 Results of refinement and the grain size of the compound Nd0.6Sr0.4MnO3
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The refined structural parameters including the lattice parameters, positional parameters, unit cell volume, bond lengths and bond angles are summarized for the sample in Table 1. The crystal structure for Nd0.6Sr0.4MnO3 sample based on the refined atomic positions was represented graphically as shown in Fig. 1. From this figure, it can be seen that the sample consists of MnO6 octahedral residing in the lattice formed by Sm and Sr. From Table 1, it can be seen that the orthorhombic phase has a ratio c/a < √2 and there are one long Mn–O bond and two short ones in MnO6 octahedral, which reveal the presence of the Jahn–Teller distortion [14,15,16].

3.2 Scanning Electron Microscope

The morphology and particle size of the Nd0.6Sr0.4MnO3 are observed with SEM, as shown in Fig. 2. The spherical particles can be clearly distinguished, and all the observed particles connect with each other. In addition, we can clearly observe that the grain size is calculated by ImageJ software as listed in Table 1. Moreover, we can also calculate the average grain size (D) from the XRD peaks using the Scherrer formula:

$$\mathop D\nolimits_{\text{SC}} = \frac{0.9\lambda }{\beta \cos \theta }$$

where β is the breadth of the observed diffraction line at its half intensity maximum and λ is the X-ray wavelength used. The significance of the broadening of the peaks evidences grain refinement along with the large strain associated with the powder. The value of (DS) is presented in Table 1. Thus, it is clear that the values calculated from XRD data are significantly lower than those shown by the SEM micrograph. This difference is due to the fact that each particle observed by SEM consists of several crystallites domains; this can be explained by the internal stress or defects (vacancies, dislocations) in the particle [17].

Fig. 2
figure 2

SEM image and the particle size distribution image at Nd0.6Sr0.4MnO3 (Color figure online)

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3.3 Magnetic Characterization

The temperature (T) dependence of magnetization (M) reveals that Nd0.6Sr0.4MnO3 oxide shows a magnetic transition from a ferromagnetic (FM) state to a paramagnetic (PM) transition at Curie temperature (TC = 245 K). The evolution of dM/dT versus temperature (T) is reported in inset of Fig. 3. Analysis gives evidence of the presence of two peaks. The first one, observed at a temperature around 46 K, is ascribed to the existence of the minor ferrimagnetic secondary phase Mn3O4 [18, 19]. The second main peak is attributed to the ferromagnetic–paramagnetic transition at the Curie temperature (TC).

Fig. 3
figure 3

Plots of the FC magnetization as a function of temperature measured under an applied magnetic field of 0.05 T. Inset shows the variation in the dM/dT as a function of temperature (Color figure online)

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Figure 4 shows the temperature dependence of zero-field-cooled (ZFC) and field-cooled (FC) magnetization, taken at 0.05 T of Nd0.6Sr0.4MnO3 in the temperature range of 0–300 K. A cusp at 130 K can be clearly seen in the ZFC branch for this compound, which is generally related to a spin-glass or a cluster-glass state [20,21,22,23]. This suggests that Sr substitution for Nd dilutes the double-exchange (DE) mechanism and shows a typical spin-glass and insulating behavior. Many authors studied Ln1−xAxMnO3, Sr-doped manganites with 1 − x and found an antiferromagnetic interaction between Mn ions [24,25,26].

Fig. 4
figure 4

Temperature dependence of magnetization in ZFC and FC mode in a applied magnetic field of 0.05 T for Nd0.6Sr0.4MnO3 sample (Color figure online)

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It is well known that in the paramagnetic region, the relation between χ and temperature T should follow the Curie–Weiss law. This is given by the following relation:

$$\chi = \frac{C}{{T - \theta_{\text{WC}} }}$$

where θP is the Weiss temperature and C is the Curie constant defined as:

$$C = \frac{{N_{\text{A}} \, \mu_{\text{B}}^{2} }}{{3 \, K_{\text{B}} }}\mu_{\text{eff}}^{2}$$

where NA = 6.023 × 1023 mol−1 is the number of Avogadro; \(\mu_{\text{B}}\) = 9.274 × 10−24 (Am2) is the Bohr magneton; and kB = 1.38016 × 10−23 J K−1 is the Boltzmann constant

From fitting the linear paramagnetic region, Curie–Weiss C and θP parameters were obtained (Table 3). The positive θ P value indicates the presence of a ferromagnetic interaction between the spins. From the determined C parameter, we have deduced the \(\mathop \mu \nolimits_{\text{eff}}^{ \exp }\) values. As summing orbital momentum to be quenched in Nd3+, Sr2+, Mn3+ and Mn4+ the theoretical paramagnetic effective moment can be written as:

$$\mu_{\text{eff}}^{\text{the}} = g\sqrt {J(J + 1)}$$

where \(g = 1 + \frac{J(J + 1) + S(S + 1) - L(L + 1)}{2(J(J + 1)}\) is the Landé factor; \(J = \left| {S + L} \right|\), total moment; \(L = \sum {m_{\text{l}} }\), orbital moment; and \(S = \sum {m_{\text{s}} }\), spin moment.

The values of total moment, orbital moment, spin moment and effective moment of species present in this compound are listed in Table 2.

Table 2 Values of total moment, orbital moment, spin moment and effective moment
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The theoretical paramagnetic effective moment can be written as

$$\mathop \mu \nolimits_{\text{eff}}^{{{\text{cal}}^{2} }} = 0.6\mathop \mu \nolimits_{\text{eff}}^{2} \left( {{\text{Mn}}^{3 + } } \right) + 0.4\mathop \mu \nolimits_{\text{eff}}^{2} \left( {{\text{Mn}}^{4 + } } \right) + 0.6\mathop \mu \nolimits_{\text{eff}}^{2} \left( {{\text{Nd}}^{3 + } } \right)$$

Table 3 summarizes the temperature dependence of TC, θP, \(\mathop \mu \nolimits_{\text{eff}}^{\text{the}}\) and \(\mathop \mu \nolimits_{\text{eff}}^{\exp }\) for an applied magnetic field of 0.05 T. Thus, we notice an agreement between the theoretical value of the magnetic moment and the value expected for an isolated paramagnetic system. This result shows that for a field of 0.05 T, all spins are aligned. The variation in the inverse of susceptibility (1/χ) as a function of temperature is shown in Fig. 5. It is clear from this figure that χ−1 (T) does not follow the usual Curie–Weiss (CW) law above TC. So, this phase which is above TC is not a pure paramagnetic (PM) region. Here, it is shown that this inhomogeneous phase above TC has characteristics similar to Griffiths phase (GP), which is depicted as downturn on the temperature dependence of magnetization below a certain temperature denoted as the Griffiths temperature, TG.

Table 3 Transition temperature (TC, TG, θP, \(\mathop T\nolimits_{C}^{\text{Rand}}\)), the effective moment experimental and theoretical and the exponent (λ) for Nd0.6Sr0.4MnO3 compound
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Fig. 5
figure 5

Variation in the inverse susceptibility obtained from magnetization measurements in the field of 0.05 T (Color figure online)

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Below the temperature where (χ−1) deviates from Curie–Weiss behavior, FM clusters emerge in the PM matrix, as is described in a Griffiths phase system [27]. Theoretical modeling of the Griffiths phase for the manganites is present in reports [28, 29]. The TG for the studied sample is presented in Table 3. The Griffiths phase is termed by Bray [30] as the phase in the temperature range TC < T < TG.

The Griffiths singularity is usually characterized by the (χ−1) exponent (λ) (or by M−1 exponent, which is very close to χ−1 one) and obtained from the following relation [31]

$$\mathop \chi \nolimits^{ - 1} \infty \mathop M\nolimits^{ - 1} \infty \mathop {\left( {T - \mathop T\nolimits_{C}^{\text{R}} } \right)}\nolimits^{1 - \lambda }$$
(1)

where 0 ≤ λ ≤ 1 is the exponent characterizing the strength of “Griffiths phase” and \(\mathop T\nolimits_{C}^{\text{R}}\) is the critical temperature of the random ferromagnet. Here, for determining \(\mathop T\nolimits_{\text{C}}^{\text{R}}\), one uses the method already used and described by Jiang et al. [32]. To calculate (λ), we have fitted Eq. (1). In the pure PM region, λ is expected to be zero.

Thus, in order to verify the existence of Griffiths-like phase in the sample in the form of FM cluster system within a PM matrix, we have presented the Arrott plot in Fig. 5. It is clear that no positive intercepts are found for the Arrott plot at different temperatures for the sample when T > TC, indicating the absence of spontaneous magnetization and the presence of short-ranged finite-sized FM clusters taking place in this temperature region [32]. It is also known that a high magnetic field generally suppresses the GP due to the polarization of spins outside the cluster or in the term that the ferromagnetic signal is masked by increasing the paramagnetic signal as was already proposed by Pramanik et al. [31].

The behavior of the spin-glass type is generally due to the existence of ferromagnetic and antiferromagnetic interactions. It can also be due simply to a random antiferromagnetic distribution. However, at low temperature, such behavior makes the existence of a long-range magnetic order impossible. The competition between ferromagnetic and antiferromagnetic interactions results in a spin-glass behavior. A spin-glass system corresponds to the presence of ferromagnetic nano-domains coupled by complex interactions (ferromagnetic and antiferromagnetic interactions). Generally, the quenching of antiferromagnetic spin fluctuations promotes ferromagnetic correlations among Mn spins.

3.4 Theoretical Aspects

Using the phenomenological model, described in [33], M(T) has been generated according to: \(M = \left( {\frac{{M_{\text{i}} - M_{\text{f}} }}{2}} \right)\tanh \left[ {A\left( {T_{\text{C}} - T} \right)} \right] + BT + C\) (1)where Mi/Mf is an initial/final value of magnetization at ferromagnetic–paramagnetic transition as shown in Fig. 6, B is the magnetization sensitivity \(\frac{{{\text{d}}M}}{{{\text{d}}T}}\) at ferromagnetic state before transition; \(A = \frac{{2\left( {B - S_{\text{C}} } \right)}}{{\left( {M_{\text{i}} - M_{\text{f}} } \right)}}\), SC is the magnetization sensitivity \(\frac{{{\text{d}}M}}{{{\text{d}}T}}\) at Curie temperature TC, and \(C = \left( {\frac{{M_{\text{i}} + M_{\text{f}} }}{2}} \right) + BT_{\text{C}}\).

Fig. 6
figure 6

Variation in the magnetization M versus temperature for Nd0.6Sr0.4MnO3 compound under constant applied field (Color figure online)

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The temperature dependence of ΔS under different applied magnetic fields has been calculated using Eq. 2 and is depicted in Fig. 7, where it shows a ΔS maximum (ΔSmax) at Tc with magnetic field dependence. Around magnetic transition, the magnetization increases rapidly at low fields and shows a tendency to saturate at higher field values, as is typical for FM materials. At Tc, the FM–PM transition increases the magnetic spins disorder that reaches a maximum at Tc resulting in ΔSmax. In order to evaluate the MCE, they can be calculated using the following expression [34]:

Fig. 7
figure 7

Experimental and theoretical magnetic entropy changes for Nd0.6Sr0.4MnO3 sample under applied fields ranging from 1 to 5 T: the solid lines are predicted results, and symbols represent experimental data (Color figure online)

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$$\Delta S_{\text{M}} = \left\{ { - A\left( {\frac{{M_{\text{i}} - M_{\text{f}} }}{2}} \right)\text{sech}^{2} \left[ {A\left( {T_{\text{C}} - T} \right)} \right] + B} \right\}\mu_{0} H_{\hbox{max} }$$
(2)

Using Eq. 2, the maximum magnetic entropy change \(\Delta S_{\hbox{max} }\) (where T = TC) can be determined by the following equation [35]:

$$\Delta S_{\hbox{max} } = \left[ { - A\left( {\frac{{M_{\text{i}} - M_{\text{f}} }}{ 2}} \right) + B} \right]\mu_{ 0} H_{\hbox{max} }$$
(3)

The relative cooling power is the negative of the product of the maximum magnetic entropy change, and the full-width at half-maxima (\(\delta T_{\text{FWHM}}\)) can be carried out using Eq. 5 [34, 35]:

$$\delta T_{\text{FWHM}} = \frac{ 2}{A}{\text{sech}}\left[ {\sqrt {\frac{{ 2A(M_{\text{i}} - M_{\text{f}} )}}{{A(M_{\text{i}} - M_{\text{f}} ) + 2B}}} } \right]$$
(4)

Another very important parameter for magnetic refrigeration is the relative cooling power (RCP) pointing to the transferred heat between the cold and the hot reservoirs in a refrigerator during one ideal thermodynamic cycle [36,37,38], and it is defined as:

$${\text{RCP}} = - \delta T_{\text{FWHM}} \times \Delta S_{\hbox{max} } \left( {T,\mu_{ 0} H_{\hbox{max} } } \right) = \left( {M_{\text{i}} - M_{\text{f}} - 2\frac{B}{A}} \right)\mu_{ 0} H_{\hbox{max} } \times {\text{sech}}\left[ {\sqrt {\frac{{ 2A(M_{\text{i}} - M_{\text{f}} )}}{{A(M_{\text{i}} - M_{\text{f}} ) + 2B}}} } \right]$$
(5)

The change of specific heat associated with a magnetic field variation from zero to \(\mu_{ 0} H\) is given by [39,40,41]:

$$\Delta C_{\text{p}} (T,\mu_{ 0} H) = C_{\text{p}} (T,\mu_{ 0} H) - C_{p} (T, 0) = T\frac{{\partial \left[ {\Delta S_{\text{M}} ({\text{T}}, \, \mu_{ 0} {\text{H}})} \right]}}{\partial T}$$
(6)

Based on this model, \(\Delta C_{\text{p}} (T, \mu_{ 0} H)\) can be defined as:

$$\Delta C_{\text{p}} (T, \mu_{0} H) = - TA^{2} \left( {M_{\text{i}} - M_{\text{f}} } \right) \, \text{sech}^{2} \left[ {A\left( {T_{\text{C}} - T} \right)} \right]\tanh \left[ {A\left( {T_{\text{C}} - T} \right)} \right]\mu_{0} H_{\hbox{max} }$$
(7)

Numerous works pertaining to the field dependence of the magnetic entropy change (\(\Delta S_{\text{M}}\)) of manganites at the ferromagnetic–paramagnetic transition have been conducted. The field dependence of \(\Delta S_{\text{M}}\) can be expressed as follows [42]:

$$\Delta S_{\text{M}} = a\left( {\mu_{0} H} \right)^{n}$$
(8)

where a is a constant and the n exponent depends on the magnetic state of the sample. It can be locally calculated as follows [43, 44]:

$$n = \frac{{{\text{d}}Ln\left| {\Delta S_{\text{M}} } \right|}}{{{\text{d}}Ln\left| {\mu_{0} H} \right|}}$$
(9)

In recent years, Franco and co-workers suggested that the \(\Delta S_{\text{M}} \left( T \right)\) curves modeled with different maximum applied fields should collapse onto a single universal curve in the case of a second-order phase transition [45, 46]. The construction of this phenomenological master curve necessitates the normalizing of all the \(\Delta S_{\text{M}}\) (T, µ0H) curves with their respective peak entropy change \(\Delta S_{\text{M}}^{\hbox{max} }\) and to rescale the temperature axis as.

$$\begin{aligned} & \theta = {{\left( {T_{\text{C}} - T} \right)} \mathord{\left/ {\vphantom {{\left( {T_{\text{C}} - T} \right)} {\left( {T_{r1} - T_{\text{C}} } \right),\quad T \le T_{\text{C}} }}} \right. \kern-0pt} {\left( {T_{r1} - T_{\text{C}} } \right),\quad T \le T_{\text{C}} }} \\ & \theta = {{\left( {T - T_{\text{c}} } \right)} \mathord{\left/ {\vphantom {{\left( {T - T_{\text{c}} } \right)} {\left( {T_{r2}^{{}} - T_{\text{C}} } \right),\quad T > T_{\text{C}} }}} \right. \kern-0pt} {\left( {T_{r2}^{{}} - T_{\text{C}} } \right),\quad T > T_{\text{C}} }} \\ \end{aligned}$$
(10)

where θ is the rescaled temperature; Tr1 and Tr2 are the temperature values of the two reference points that have been selected as those corresponding to \(0.5\Delta S_{\text{M}}^{\hbox{max} }\).

From this phenomenological model, it can be easily assess the values of \(\left| {\Delta S_{\text{M}}^{\hbox{max} } } \right|\),\(\delta T_{\text{FWHM}}\), RCP and \({{\Delta C_{\text{p}}^{\hbox{max} } } \mathord{\left/ {\vphantom {{\Delta C_{\text{p}}^{\hbox{max} } } {\Delta C_{\text{p}}^{\hbox{min} } }}} \right. \kern-0pt} {\Delta C_{\text{p}}^{\hbox{min} } }}\) of this compound under magnetic field variation. Moreover, the order of phase transition in the present system will be clarified by utilizing this model.

3.5 Model Application

In order to confirm the ferromagnetic behavior at low temperature and to apply the phenomenological model, for the understanding of the magnetic properties, we traced the magnetization as a function of the temperature in different magnetic fields between 1 and 5 T for Nd0.6Sr0.4MnO3 in Fig. 8. The symbols represent experimental data, while the solid curves represent data modeled using the model parameters given in Table 4. These parameters were determined from the experimental data. It is obvious that the results of the calculation are in good agreement with the experimental results.

Fig. 8
figure 8

Temperature dependence of magnetization for Nd0.6Sr0.4MnO3 compound in different applied magnetic field shifts. The solid curves are the modeled results, and symbols represent the experimental data (Color figure online)

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Table 4 Model parameters for Nd0.6Sr0.4MnO3 compound in different applied magnetic fields
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Our theoretical calculations of the magnetocaloric effect in Nd0.6Sr0.4MnO3 sample are presented. Numerical calculations were made with parameters as displayed in Table 5. These parameters were determined from experimental data. The isothermal magnetization curves as a function of magnetic field, with μ0H = 0–5 T and thermal interval 5 K, are shown in Fig. 9. While the solid curves represent modeled data using model parameters given in Table 5, the symbols represent the experimental data [47]. It is noteworthy to mention that there is a good agreement between the experimental and the calculated results.

Table 5 Model parameters for Nd0.6Sr0.4MnO3 compound at several magnetic fields
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Fig. 9
figure 9

Temperature dependence of ΔCP under different magnetic field variations for Nd0.6Sr0.4MnO3 sample (Color figure online)

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It can be seen that the results of the calculation are in accordance with the experiment [38]. Furthermore, the magnetic entropy change depends on the applied magnetic field change. The maximum magnetic entropy change exhibits a linear rise with the increase in the field, as shown in Table 5. This indicates a much larger entropy change to be expected at higher magnetic field signifying that the effect of spin–lattice coupling is associated with the changes in the magnetic ordering process in the sample [48].

The specific heat changes ΔCP as a function of temperature for Nd0.6Sr0.4MnO3 sample under different applied magnetic fields derived from \(\left| {\Delta S_{\text{M}} } \right|\) curves by using Eq. 7 are depicted in Fig. 9. This figure shows the variation in ΔCP as a function of temperature. The positive or negative values of ΔCP closely above or below TC may strongly alter the total specific heat.

Using Eq. 8, we obtain the value of n as a function of temperature as exhibited in Fig. 10.

Fig. 10
figure 10

Temperature dependence of the exponent n for Nd0.6Sr0.4MnO3 for different magnetic fields (Color figure online)

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All the n(T) curves follow the universal behavior illustrated by Franco et al. [48]. In this context, the n exponent approaches 1 and 2 far below and above TC, respectively. Furthermore, the n exponent exhibits a moderate decrease with the increase in temperature, with a minimum value in the vicinity of the transition temperature, sharply increasing above TC. This directly reveals the paramagnetic entropy change, which is governed by the Curie–Weiss law. The minimum n value of 0.35 is similar to those obtained for other magnetic materials containing rare-earth metals [49, 50].

The normalized entropy change (\({{\Delta S_{\text{M}} } \mathord{\left/ {\vphantom {{\Delta S_{\text{M}} } {\Delta S_{\text{M}}^{\hbox{max} } }}} \right. \kern-0pt} {\Delta S_{\text{M}}^{\hbox{max} } }}\)M) as a function of the rescaled temperature (θ) for the magnetic ordering transitions of Nd0.6Sr0.4MnO3 compound is shown in Fig. 11. It is evident that all normalized entropy change curves collapse into a single curve, which proves that the paramagnetic–ferromagnetic phase transition observed for our sample is of second order [51].

Fig. 11
figure 11

Normalized entropy change (ΔSM/ΔS maxM ) versus rescaled temperature (θ) for different applied magnetic fields for Nd0.6Sr0.4MnO3 sample (Color figure online)

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4 Conclusion

In this work, we have investigated the structure and magnetic properties of a perovskite Nd0.6Sr0.4MnO3 compound. The sample crystallizes in an orthorhombic structure with the Pnma space group. The magnetic measurement shows a transition ferro–paramagnetic at a temperature TC. The inverse of the susceptibility shows a clear downturn for this sample above TC, which is attributed to the occurrence of the Griffiths phase. The ZFC/FC shows the existence of a cluster-glass state.

Finally with the help of the phenomenological model, a detailed investigation of magnetic and magnetocaloric properties has been conducted. The extracted data affirm that this phenomenological model is useful for the prediction of magnetocaloric properties.

Building on this model, we can calculate the values of the magnetic entropy change, full-width at half-maximum and magnetic specific heat change for the sample from the data of magnetization as a function of temperature under different external magnetic fields.