Thermomagnetic Properties and Magnetocaloric Effect of R 2 Fe 17 C (R=Dy, Nd, Tb, Gd, Pr, Ho, Er) Compounds

: We present a mean-field analysis, using the two-sublattice model, for the thermomagnetic and magnetocaloric properties of the R 2 Fe 17 C compounds, where R=Dy, Nd, Tb, Gd, Pr, Ho, Er and C is carbon. The dependence of magnetization, magnetic heat capacity, magnetic entropy and isothermal entropy change ∆S m , are calculated for magnetic fields up to 5T and for temperatures up to 700 K . Direct magnetocaloric effect is present for all compounds with maximum ∆S m between 6.13-10.95 J/K. mole for an applied field change of 5T. It is found that Pr 2 Fe 17 C compound has the highest ∆S m of 10.95 J/K. mole at ∆H=5T and T c =375 K. The inverse MCE is found in ferrimagnetic compounds, e.g. Gd 2 Fe 17 C, with ∆S m = 6.13 J/K mol at critical temperature T c =623K and ∆S m = 0.12 J/K mol at Neel temperature T N =136 K. The calculated Arrott plots confirmed that the magnetic phase transitions in these compounds are of second order. The mean-field model proves its suitability for calculating the properties of the compounds under study


Introduction
In the course of search for new permanent magnetic materials, several studies have been reported on using the two-sublattice molecular field model for calculating the magnetic properties of the rare-earth Fe-rich system R2Fe17C[1].
. The R2Fe17C compounds crystallize in the rhombohedral Th2Zn17-Type crystal structure of the space group P63/mmc [2]. No studies, up to our knowledge, have been done, experimentally or theoretically, on the magnetocaloric properties of these alloys. . In the present work, we focus our efforts on calculating magnetic properties of R2Fe17C, where R stands for Nd, Pr, Gd, Er, Dy, Tb and Ho, such as: the temperature-dependence of magnetization, magnetic specific heat, magnetic entropy and magnetocaloric effect, ∆ using the two-sublattice molecular field theory MFT [3,4].
. The magnetic specific heat can be calculated from the differentiation of the magnetic energy with respect to temperature. Also, the magnetic entropy could, then, be calculated by numerically processing the magnetic specific heat data in a certain range of temperatures and magnetic fields. Using the Landau and Lifshitz theory for phase transitions [5,6], we can calculate the Arrott plots [7] which are used in determining the nature of the ferromagnetic phase transition and in estimating the critical temperature. The M 2 versus H/M and the M 2 versus ∆ [8,9] plots are calculated at several temperatures in the vicinity of the critical temperature, for different magnetic fields, where H and M are the applied field and magnetization respectively. Belove [10] showed that the positive slope of the M 2 versus H M curves is an indicative of a second order phase transition, whereas the negative slope indicates the presence of a first order transition [10,11] .

Model and analyses
Using the two-sublattice molecular field theory MFT, we will calculate, first, the temperaturedependence of magnetization M(T). The total effective field of R and Fe sublattices in 2 17 compounds can be expressed, respectively, as follows : where H is the external field in A/m in the SI system of units. The temperature dependence of each sublattice moment is determined by the Brillouin functions [ 12,13]: (3) ( ) and (0) ( ( ) and (0)) are the magnetic moments, in Bohr magneton , of R ion (Fe ion) at temperatures T and 0 K, respectively. The MFT coefficients are nFF , nRR and nRF. In our calculation, these MFT coefficients are from the work of Wu et al [1]. The factor = N A μ B ρ/A converts the moments per unit volume from μ B to A/m, where ρ is the density of R 2 Fe 17 C in kg/m 3 , the Avogadro ' s number and A is the formula weight of R 2 Fe 17 C in kg. JR and JFe are the individual angular moments of R and Fe , respectively. The resultant magnetization is given by In order to calculate the magnetic specific heat, we start off with the magnetic energy of a binary magnetic compound The magnetic specific heat is determined by : and the magnetic entropy is calculated from : Both of the specific heat and entropy are in units of J/K. mole The isothermal magnetic entropy change can be calculated in two ways: i) Direct subtraction of the entropies at two different fields but at the same temperature i.e.: Arrott plots are calculated in accordance with the equation of state [l4] where 1 and 1 are set of adjustable parameters Its remarked that ∆ is proportional to M 2 from the mean field model [8,9], so where is the magnetization per unit mass.

Magnetization
The total magnetization, as function of temperature, and in zero field, for all the compounds under study is shown in Fig.1.The total magnetization, at T=0 K, is the largest for Pr2Fe17C and Nd2Fe17C namely: 32.27 and 28.64 .
( 1)respectively. Figs 2 and 3 show the temperature dependence of magnetization of the rare-earth and Fe sublattices as well as the total magnetization for the Nd2Fe17C, Pr2Fe17C and Gd2Fe17C. It is evident that the intra-sublattice interaction of the iron sublattice is the largest one. This is due to that the number of iron atoms involved in this compounds is greater than that of rare earth atoms.

Magnetic heat capacity and entropy
The different sublattice contributions to the magnetic specific heat of Pr2Fe17C , at zero field, are shown in Fig(5). The largest contribution is that of the Fe-sublattice. Figure (6) displays the fielddependence of the magnetic specific heat of the same compound as a function of temperature. This behavior is typical for ferromagnetic compounds[1].  Figures (7, 8) illustrate the total entropy and its sublattice contributions, in zero field, for Er2Fe17C and Gd2Fe17C respectively. The total entropy saturates, at T=Tc to about 200 and 190 J/K.mole for these two compounds respectively. The largest contribution to magnetic entropy is that of the Fe-Fe interaction, whereas the other three interactions contribute only about 25% to the total magnetic entropy. The calculated total magnetic entropy at 0 and 10 T for Pr2Fe17C ( Gd2Fe17C) compound is shown in figs. 9(10). Table 1, displays the calculated total magnetization, magnetic entropy and magnetic specific heat data at zero temperature and magnetic field for all compounds. The shown field-dependence of magnetic entropy, at a given temperature, reflects the role of magnetic field in aligning the moments and hence reducing the magnetic entropy [14].

Magnetocaloric effect
We have calculated the magnetocaloric effect ∆ ( , ∆ ) by using two different methods, as mentioned before, e.g. using either Eq.9or 10. The temperature dependence of the magnetic entropy change, using Eq.9 , for R 2 Fe 17 C where R= Pr, Nd, Tb, Dy, Ho, Gd and Er in magnetic field changes of 1, 2, 3 and 5T, are shown in Figs. (11-17). .

Figure 11
. Isothermal entropy change vs T in ferromagnetic Pr2Fe17 C compound for applied magnetic fields from 1 to 5 T .        For the sake of comparison between the results of the two methods used for calculating |∆ ( , ∆ )|, we display them in fig.18 for R=Dy. It is clear that the results of the two methods are rather similar, however the maximum value of ∆ ( , ∆ ) is relatively higher for the direct subtraction method.     (Fig.19) for an applied fields from 1 to 5T. Fig.20 displays the magnetization first-derivative with respect to temperature. The maximum values of this derivative is the integrand in eq.10. Figure 19. the maximum Isothermal entropy change vs T in R2Fe17C compounds for applied magnetic fields from 1 to 5 T.

Conclusions
Thermomagnetic properties have been calculated for R2Fe17C ; R= Nd, Pr, Gd, Er, Dy, Tb and Ho using the two sublattice molecular field theory. Magnetization calculation have shown that the compounds Pr2Fe17C and Nd2Fe17C are ferromagnetic while the other compounds are ferrimagnetic. The magnetic specific heat and the magnetic entropy at different magnetic fields between 0 and 5 T showed that the phase transition involved in these compounds is of the second type. Furthermore, the contribution of the Fe-Fe exchange interaction to the magnetic entropy and heat capacity is the highest compared to other sublattice interactions.
. Arrott plots and the ∆ . M 2 relations have supported our conclusion concerning the type of the phase transition in these compounds i.e. second order phase transition. The Tc values calculated from Arrott plots are fairly close to those calculated from the temperature dependence of magnetization. Both ordinary and inverse magnetocaloric effect are present for all the compounds except those with R=Nd and Pr, where only direct ∆ is present. The highest ordinary ∆ is calculated for R=Pr around 11 J/mol.K at Tc = 375 K and the highest inverse ∆ of 3.45 J/mol.K is calculated for R= Er, for a 5T magnetic field change. The magnetocaloric effect ∆ has been determined using two methods i.e. the direct subtraction and using Maxwell relation. The two methods nearly produce the same temperature dependence except that the direct-subtraction method yields a relatively higher ∆ at Tc. . We may conclude that the magnetocaloric effect in these materials is relatively high in particular for R=Pr, which has a Curie temperature not very far from room temperature. The mean-field analysis proved to be suitable for calculating the magnetothermal and magnetocaloric properties of the compounds studied.