Magnetocaloric Effect in R6Fe23: R = Dy, Ho, Er, and Tm

We present a mean field study on the R6Fe23 system, where R = Dy, Ho, Er, and Tm, to calculate the magnetization, magnetic heat capacity, and the magnetocaloric effect (MCE) (isothermal entropy change (ΔSm) and the adiabatic temperature change (ΔTad)) for different field changes up to 5 T and at temperatures ranging from 0 to 600 K. The maximum ΔSm, using the trapezoidal method, for the R6Fe23 system is in the range 4.9–9.8 J/K mol, and the maximum ΔTad is in the range 9.56–15.17 K for a field change ΔH = 5 T. The largest ΔSm and largest ΔTad are found for Tm6Fe23 to be 9.8 J/K mol and 15.17 K at Curie temperature Tc = 489 K, for ΔH = 5 T. The relative cooling power RCP(S) is in the range 148–560 J/mol for ΔH = 5 T, which is comparable to that of bench-mark materials, e.g., Gd. Also, the RCP based on the adiabatic temperature change, RCP(T) is in the range 449–1092 K2 for ΔH = 5 T, which is comparable also to that of bench-mark materials, e.g., Gd. We investigated the type of phase transition in the light of universal curves, Arrott plots, and the behavior of the magnetic moment, magnetic heat capacity, and MCE (ΔSm, ΔTad), which confirm that the type of phase transition at Tc of this system is second-order phase transition (SOPT). A calculation of some critical exponents adds more evidence that the MFT is fairly suitable to handle the aforementioned properties in the studied systems.


Introduction
The magnetocaloric effect (MCE) has been discussed before for magnetic and rare-earth intermetallic compounds [1][2][3]. There are different functional materials useful for technological applications such as magnetic refrigeration technology, which is hoped to be environmentally safer and a more efficient alternative to traditional refrigeration technology [4][5][6]. Many studies, on MCE, were done by using the mean field theory MFT, e.g., TmFe 2 [7], R 2 Fe 14 B [8], Gd-Co [9], LaMnO [10], and R 3 Co 11 B 4 [11]. Several studies have been carried out on the R 6 Fe 23 compounds, e.g., magnetic properties, magnetostriction, electronic, and transport properties, as well as on crystal structure, lattice vibrations, and x-ray photoemission [12][13][14][15][16][17][18][19]. It is known that R 6 Fe 23 compounds crystallize in Th 6 Mn 23 type structure. Rare earth 4f-transition metals 3d intermetallic compounds show interesting magnetic properties. The two sublattice molecular field theory proved to be fairly suitable for calculating the magnetization of the compounds [12]. There are, however, few studies on the MCE of R 6 Fe 23 ; therefore, our motivation to study in detail the magnetothermal properties and MCE (∆S m , ∆T a d) in the R 6 Fe 23 system is justified. In addition, we calculated the relative cooling powers RCP(S) and RSP(T) as figures of merit. Investigating the order of the magnetic transition, via the temperature and field dependences of the magnetothermal and MCE properties, the universal curve, and the Arrott plots is also done.

Model and Analysis
By using the MFT, the exchange fields of rare earth elements and Fe sublattices can be expressed as follows [12,20]: The symbols in Eqs. (1) and (2) have their usual meaning [12]. The molecular field coefficients n RR , n FeFe , and n RFe are dimensionless. The magnetic moments of rare-earth M R (T) and iron M Fe (T) at temperature T.
The total magnetic moment can be calculated from From the following Maxwell relation, the magnetic entropy change is given by The above integral could be cast into a summation by using the well-known trapezoidal rule [21].
A universal curve [22] is the relation between △S m /△S peak m vs. Ѳ.Where Ѳ is defined from the following: where T r is the reference temperature, it can be chosen such that [22] The total heat capacity C tot is calculated from the sum of the magnetic C m , the electronic C e , and the lattice C l heat capacities [23,24].
First, from the temperature-first derivative of the magnetic energy, we can calculate the magnetic contribution to heat capacity as the following equation: Second, the electronic heat capacity is calculated by [25] (3)  (10) C e (T) = e T = 2 ∕3 N a k 2 N E f T γ e is the electronic heat capacity coefficient, and N (E f ) is the density of states at Fermi energy.
Third, the lattice heat capacity is calculated as follows: where ϴ D is Debye temperature and x = ϴ D / T. The adiabatic temperature change [23] is given by where C tot is the total heat capacity. The Arrott plots (M 2 vs. ∆S m ) and (H/M vs. M 2 ) are used to investigate the order of the phase transition from the sign of the plot's slopes. Namely, positive slopes indicate secondorder phase transition, according to Arrot-Belov-Kouvel (ABK) [26,27].
Ginsburg theory is expressed as follows [28]: From the equilibrium condition at T c , the magnetic equation of state is given by where A(T) and B(T) are Landau's coefficients.
The RCP [29] is considered as figure of merit for the magnetocaloric materials and is defined from magnetic entropy change as: And also from the adiabatic temperature change, Figure 1a-d exhibits the calculated magnetic moment of the two sublattices of rare earth R, Fe, and the total magnetic moment where R = Dy, Ho, Er, and Tm, respectively. Ferrimagnetic coupling is present in these compounds, with compensation points except for Tm 6 Fe 23 . We can show that the magnetic moment for Dy atom at 0 K is M Dy (0) = g Dy J Dy = (4/3) × (15/2) = 10 μ B /atom. The magnetic moments

Magnetization
for R = Ho, Er, and Tm are calculated by Herbst and Croat [12]. As known, the magnetic moment for rare earth R is localized whereas that of the 3d-transition elements Fe in rareearth intermetallic compounds. So, the magnetic moment of Fe sublattice is obtained from the experimental data of the total magnetic moment and the calculated magnetic moments of the rare earth R, for example, in Dy system M Fe (0) = 48.8/23 = 2.12 μ B /atom. Table 1 shows the experimental [12] and theoretical data of both the total magnetic moments and the Curie temperatures; for the R 6 Fe 23 system, also, the percentage difference between the experimentally determined moments and calculated magnetic moment at T = 1 K [12] is shown in Table 1. This difference is only ≤ 2.4%, and the difference in the T c data is ≤ 2.86%. As shown, the mean field theory succeeded in studying the magnetization for R 6 Fe 23 and the rare earth intermetallic compounds such as R 3 Co 11 B 4 [11]. Magnetization calculations showed that, for example, both Dy 6 Fe 23 and Dy 3 Co 11 B 4 are ferrimagnetic compounds with the total magnetic moments 11.6 and 16.6 μ B /f.u, respectively. Figure 2 shows the field dependence of the magnetic heat capacity as function of temperature for Ho 6 Fe 23 at different magnetic fields up to 5 T. The maximum magnetic heat capacity decreases by increasing the applied field around Curie temperature, which is typical for compounds with second-order phase transition, for example, Ho 3 Co 11 B 4 [11] and TmFe 2 [7]. The electronic heat capacity is obtained from Eq. (10), and the coefficient γ e is given from the materials project [30] as shown in Table 3. The lattice heat capacity is calculated from Debye temperature by Eq. (11). The Debye temperatures for most crystals are around room temperature.

The Isothermal Entropy Change (△S m )
△S m has been calculated using the Maxwell relation from Eq. (6) and also using the trapezoidal rule. Figure 3a-d shows (△S m vs. T) at applied fields up to 5 T, for R = Dy, Ho, Er, and Tm, respectively. Both direct and inverse MCE, i.e., two peaks are present: the first peak at T c and the second at a temperature below the compensation temperature. The data of magnetic entropy change △S m using both Maxwell's relation and the trapezoidal rule showed agreement between the two methods, at low field changes, as shown in Table 2. For the sake of comparison with bench-mark materials and other R 6 Fe 23 compounds, we compare our results of ΔS m , which is in the range of 3.9 to 9.8 J/mol K for a field change ∆H = 5 T, with that of Gd metal, i.e., 1.48 J/mol K at ΔH = 5 T as reported by Wang et al. [31], and also with Jemmali et al. [32], where ΔS m of Er 6 Fe 23 is 3.64 J/mol K at ΔH = 1.4 T.

Adiabatic Temperature Change (ΔT ad )
We report in this part ΔT ad . Because of the weak dependence of the total heat capacity on the applied field, around Curie temperature for the R 6 Fe 23 compounds, as shown in Fig. 4 for example Dy 6 Fe 23 system, the term T/C is taken out of the integral in Eq. (12). Figure 5a-c shows the adiabatic temperature change ΔT ad for R = Dy, Ho, and Tm using Eq. (12) for applied fields up to 5 T. The maximum value for ΔT is 15.17 K, as shown in Table 3 for applied field 5 T, and in the case of Tm 6 Fe 23 , the temperature is decreasing by a rate of 3.03 K/T.

Relative Cooling Power (RCP)
RCP is based on the isothermal process RCP(S), which is calculated by Eq. (15), for different field changes. The RCP(S) of Er 6 Fe 23 at a field of 1.4 T is 95.76 J/mol, as reported by Jemmali et al. [32], and also RCP(S) of Gd metal is 150.72 J/mol at field change 5 T as reported by Wang et al. [31]. Also, RCP is based on the adiabatic temperature change RCP(T) by Eq. (16). It has no physical meaning, but it is used for numerical comparison of other MC compounds. A large RCP(T) generally indicates a better magnetocaloric material, as shown in Table 5. The relative cooling power RCP(T) is in the range 449-1092 K 2 for a field change ΔH = 5 T, which is compared with 967 K 2 of Gd at field 6 T, as reported by Gschneidner et al. [3]. The calculations show that the RCP(S) and RCP(T) increase with increasing the applied magnetic field, as shown in Tables 4 and 5. Figure 6 displays the universal curves (ΔS m vs. Ѳ) for Dy 6 Fe 23 compound in applied fields of 1.5, 3, and 5 T. It can be clearly shown in Fig. 6 that the data of different applied fields collapse into a single universal curve, which shows the phase transition in the R 6 Fe 23 system is a second-order phase transition. According to Arrot-Belov-Kouvel (ABK) [26,27], the order of the phase transition involved the second-order SOPT or the first-order FOPT, from the sign of the plots  slopes. Namely, positive slopes indicate SOPT, whereas negative slopes or s-shaped slopes indicate FOPT. Figure 7a  According to the mean field theory, the relation ΔS m vs. (H/ T C ) 2/3 is a criterion for the existence of the SOPT [33,34]. It would be instructive to evaluate some of the critical exponents [35][36][37][38] and compare them with those of the mean-field theory. We have calculated the parameters n, β, δ, and ɤ, where n = 1 + (β − 1) / (β + ɤ) [37]. The parameter δ has been evaluated from the isothermal magnetization curve where M ~ H 1/δ [38]. The mean-field parameters are β = 0.5, ɤ = 1, and δ = 3. Our calculation showed that β and ɤ are, at most, 8% and 16% off the mean-field values, respectively. The exponent δ is at most 30% off the mean-field value. Data Availability Authors agree to make the data available upon request.

Conflict of Interest
The authors declare no competing interests.
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