The polycrystalline RGa ( , Gd, Tb, Dy, Ho, Er, Tm) compounds were prepared by arc melting R and Ga in argon atmosphere. And the purity of the starting elements is more than 99.9%. In order to make the ingots homogenous, they were turned over and melted several times. Then the samples were wrapped in molybdenum foils, sealed in a high-vacuum quartz tube and annealed at 800 C for 7 days. After annealing they were quenched into liquid Nitrogen. To confirm purity and crystal structure of the samples, we carried out powder x-ray diffraction (XRD) experiments with Cu radiation. The XRD patterns of RGa ( , Tb, Dy, Er, (Gd, Er)) compounds are shown in Fig. 1.^[32,34] And the inset of Fig. 1 is the rietveld refined powder XRD patterns of TbGa compound. Results show that all of the peaks can correspond to the Bragg positions of the orthorhombic CrB-type structure (space group ), which indicates that the prepared compounds are of single phase. The crystal structure of RGa compounds has been described before,^[20,22] but we can also describe it from another view. The detailed crystal structure of PrGa compound is shown in Fig. 2. All of the atoms distribute in a form of layers perpendicular to a-axis, and Pr and Ga atoms are connected in a form of hexagon rings within each layer. Other RGa compound has the same crystal structure.

Fig. 1.

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Fig. 1. (color online) The x-ray powder diffraction patterns of binary RGa ( , Tb, Dy, Er) compounds and pseudo-binary Gd x Er Ga ( ) compounds. The inset is the observed (cross symbols) and Rietveld fitted (lines) powder diffraction patterns of TbGa compound. The calculated positions of the Bragg peaks are shown as vertical bars just below the plots of the observed and calculated intensities. The differences between the observed and calculated intensities are shown at the bottom of the plot.[32]

Fig. 2.

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	Fig. 2. (color online) The Crystal and magnetic structure of PrGa compound. Distribution of magnetic moments in each unit cell and the distribution of atoms connected in form of the hexagon ring in yz crystal plane.[33]

High resolution neutron powder diffraction (NDP) experiment was then employed to study the magnetic structure of PrGa compound. The NPD experiments were performed at 295 K and 4.5 K, respectively. Fitting and calculations were carried out by Rietveld refinement method afterwards. The observed and the calculated neutron powder diffraction patterns at 295 K and 4.5 K are shown in Figs. 3(a) and 3(b). NPD results at 295 K show that PrGa compound has a pure phase and it crystalizes in a CrB-type orthorhombic structure (space group #63 cmcm). Both the nuclear and the magnetic structure models are involved in the calculation at 4.5 K. The magnetic moments are FM-ordered along crystallographic c axis and the ordered magnetic moment is determined to be 2.6 at 4.5 K.^[33] Figure 4 shows the isothermal magnetization curve at 11K and the inset of Fig. 4 shows the temperature dependence of up to 300 K, where χ is the magnetic susceptibility. The linear relationship of and temperature in the high temperature range indicates that the Curie–Weiss law is appropriate here. According to the Curie–Weiss law and the definition of χ, is calculated to be 3.6 from the slope of the linear fitting curve.^[33] The MH curve at 5 K (not shown in Fig. 1) has also been measured and it is almost coincided with the MH curve at 11 K in high field range. For the isothermal magnetization curve at 5 K and 11 K, the magnetization increases with field increasing and reaches its saturation value rapidly, indicating that it is reasonable to calculate saturated magnetization from the MH curve at 5 K or 11 K. ʼs at these two temperature are calculated to be the same value of 2.0 , which is the average value per Pr atom.^[33] The value of is similar to the ordered magnetic moment obtained from NPD experiments at 4.5 K but considerably smaller than that of .

Fig. 3.

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Fig. 3. (color online) The neutron powder diffraction pattern of PrGa at 295 K (a) and 4.5 K (b), respectively. The observed data and the calculated patterns are plotted in cross and solid line, respectively. The solid line at the bottom is the difference between the observed and calculated data. The Bragg positions for the nuclear and magnetic structure are marked as short vertical lines. Only nuclear model is considered in the refinement at 295 K and both nuclear model and magnetic model are considered in the refinement at 4.5 K.[33]

Fig. 4.

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	Fig. 4. (color online) Isothermal magnetization curve of PrGa compound up to 5 T at 11 K for PrGa compound and the inset is the temperature dependence of the magnetic susceptibility inverse.[33]

In order to study the magnetic ground state of PrGa compound at low temperature, we investigated the degree of magnetic order, which is known as magnetic entropy ( ). The total entropy can be calculated from the integration of the heat capacity data:

(1)

(2)

(3)

(4)

(5)

Fig. 5.

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	Fig. 5. (color online) The entropy contributed from magnetic moments. The temperature dependence of the magnetic entropy of PrGa compound.[33]

Considering that is much smaller than and at 10 K is considerably larger than zero, we can assume that the magnetic moments point to different directions but with a fixed deviation angle from c axis and the value of 2.5 is the projection of the magnetic moments on c axis.^[33] If we draw all of the magnetic moments at the same atom site, we can see that they are distributed randomly within a conical surface. The sketch of the magnetic structure of PrGa is shown in Fig. 6. The real magnetic moment is and every magnetic moment can be resolved into two components. M _z and M _xy are the components along the c axis and within the ab crystal plane, respectively.^[33] Furthermore, M _z is long-range FM-ordered but M _xy is distributed randomly without any long-range order. , M _z , and M _xy have the following relation:

(6)

Fig. 6.

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	Fig. 6. (color online) The cone-surface distribution of the actual magnetic moments in PrGa compound and the vector decomposition of the magnetic moments.[33]

2.2. The magnetic properties and magnetic transitions of Ga compounds

Spin reorientation has been observed in some of the RGa compounds, such as PrGa, SmGa, HoGa, and ErGa according to the results of Mössbauer spectrum experiments.^[20] Here we studied the magnetic properties of RGa compounds in detail including PrGa, GdGa, TbGa, DyGa, HoGa, ErGa, and TmGa compounds. First of all, we analyze the magnetic properties and magnetic transitions of heavy-rare-earth RGa compounds. Figure 7 is the thermal magnetization curves of GdGa, TbGa, DyGa, HoGa, and ErGa compounds under a field of 0.01 T. And the inset of Fig. 7 shows the zero-field-cooled (ZFC) thermal magnetization and field-cooled (FC) thermal magnetization curves of TmGa compound under a field of 0.04 T. All of the M–T curves show hight-variability twice as temperature increases. Besides of the drastic change around Curie temperature ( , another obvious change of magnetization is observed in low temperature ranges, which is described as SR transition and the transition temperature is marked as .^{[19,32,34,38]} The transition temperatures for SR are determined to be 66 K, 31 K, 25 K, 20 K, and 15 K for GdGa, TbGa, DyGa, HoGa, and ErGa compounds, respectively. The Curie temperatures are determined to be 181.9 K, 154 K, 113 K, 69 K, and 30 K for GdGa, TbGa, DyGa, HoGa, and ErGa compounds, respectively. It is clearly that decreases steadily as the atomic number of R atoms increases, but the alteration of is not as obvious as R atom changes. The value of spin quantum number (S) decreases with the atomic number of R atoms increases, and is positively correlated with S. Figure 8(a) shows the isothermal magnetization curves at different temperatures for ErGa compound. All of the M–H curves at the temperature range above show typical FM characteristic. That is, magnetization increases as magnetic field goes up and reaches its saturation value quickly. Therefore, the SR magnetic transition is also ferromagnetic (FM) to FM transition. TmGa compound exhibits two successive magnetic transitions: ferromagnetic (FM) to antiferromagnetic (AFM) transition at K and AFM to paramagnetic (PM) transition at K.^[39] The ZFC and FC curves are well overlapped above , indicating that thermal hysteresis is very small as observed in other MCE materials with a second-order magnetic transition.^[16] However, there is an obvious bifurcation at temperatures below and it is likely due to a domain wall pinning effect as described in literature.^[40] All of the transition temperatures are listed in Table 1.

Fig. 7.

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	Fig. 7. (color online) The temperature dependences of magnetization curves of RGa ( , Tb, Dy, Ho, Er) compounds at a field of 0.01 T. The inset is the zero-field-cooling and field-cooling curves under a field of 0.01 T for TmGa compound.

Fig. 8.

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	Fig. 8. (color online) (a) The isothermal magnetization curves up to 5 T at a temperature range from 3 K to 50 K for HoGa compound. (b) The isothermal magnetization curves up to 5 T obtained with field increasing and decreasing at a temperature range from 13 K to 33 K.

Table 1.

Table 1.

Table 1. The transition temperatures and magnetocaloric parameters of RGa ( , Gd, Tb, Dy, Ho, Er, Tm) compounds and Gd x Er Ga ( ) compounds.[34] . Materials /K /K MCE_0–2 T MCE_0–5 T Ref. /(J/kg K) /K RC/(J/kg) /(J/kg K) /K RC/(J/kg) PrGa 27 37 7.0 10.6 59.3 10.7 18.1 158.7 [41] GdGa 66 181.9 2.7 99.8 182.6 5.3 159.4 616.1 [34] TbGa 31 154 3.7 135.9 181.8 8.3 155.5 620.6 [32] DyGa 25 113 2.3 101.2 92.8 5.8 112.7 381.9 [32] HoGa 20 69 8.8 16.7 113.2 17.1 83 455 [38] ErGa 15 30 10.9 22.6 166 21.3 30.9 494 [19] TmGa 12 15 20.6 9.9 149 34.3 14.2 359.7 [39] Gd Er Ga 20.6 49 4.8 48.6 184.7 11.4 56.2 535.2 [34] Gd Er Ga 21.4 76.4 4.9 74.4 244.8 10.9 80.9 659.4 [34] Gd Er Ga 22.9 98.7 3.2 95.9 222.7 7.3 108.3 685.3 [34] Gd Er Ga 21 115.8 2.6 94.1 170.4 5.7 129.3 629.9 [34] Gd Er Ga 26.7 146.8 3.2 153.6 250.4 6.5 162.8 707.2 [34] Gd Er Ga 28.8 167.7 2.7 168.3 254 5.6 185.5 715.3 [34]

Table 1. The transition temperatures and magnetocaloric parameters of RGa ( , Gd, Tb, Dy, Ho, Er, Tm) compounds and Gd x Er Ga ( ) compounds.[34] .

Figure 8(b) shows that the M–H curves measured in field increasing and decreasing mode almost coincide, which means good magnetic reversibility. The ZFC and FC of ErGa compound also show no thermal hysteresis around . According to the Banerjee criterion,^[42] a magnetic transition is expected to be of the first order when the slope of M ² versus plot is negative, whereas it will be of the second order when the slope is positive. The relationship between and M ² has been calculated for ErGa compound and the arrott plot is not shown here. The positive slopes of plot around confirm that the FM–PM transition is of second order,^[42] which accords well with the case where both thermal and magnetic hysteresis are absent. Further results show that all of the RGa ( , Gd, Tb, Dy, Ho, Er, Tm) compounds are of second order transition materials.

The thermal magnetization curves of Gd _x Er Ga ( ) compounds are measured at a field of 0.1 T and the temperature dependence of the magnetization is shown in Fig. 9. Sharp decrease is observed on each M–T curve around , and another anomaly is observed in low temperature range for all the samples. In the compounds with a low content of Gd, the magnetic properties are similar to ErGa compound. And when the content of Gd is large, such as Gd Er Ga and Gd Er Ga, the magnetic properties show similar feature to GdGa compound.^[34] According to previous analysis, the transition in low temperature range is SR transition for most Gd _x Er Ga ( ) compounds and we marked that transition temperature as . For Gd Er Ga and Gd Er Ga compounds, the transition in low temperature range is an AFM to FM transition, which will be discussed later. And we mark that transition temperature as in this case. In fact, the SR transition of ErGa and GdGa has been studied carefully by Mössbauer spectrum.^[20] The magnetic moment points to different directions at different temperatures and it results from the competition between exchange interaction and crystal field interaction. The magnetic moments of GdGa show a direction with or a direction with below , but they point to an identify direction with above .^[20] In other words, the moments rotate towards the b axis in the ab plane with temperature increasing. The direction of the moments for ErGa below is complex. But it is proved that the moments also rotate toward b axis with temperature increasing.

Fig. 9.

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	Fig. 9. (color online) The temperature dependences of magnetization curves under a field of 0.1 T for Gd x Er Ga ( ) compounds.

Isothermal magnetization curves up to 5 T at 5 K for Gd _x Er Ga ( ) compounds are shown in Fig. 10. The magnetization goes up and comes to its saturation value quickly with magnetic field increasing for Gd _x Er Ga ( , 0.1, 0.2, 0.6, 0.8, and 1.0) compounds, and it is the typical characteristic of the FM ground state. For Gd Er Ga and Gd Er Ga compounds, the magnetic exchange interaction is not large enough to counterbalance the crystal field interaction below , and Er and Gd magnetic moment each points to the favorite directions on their own.^[34] That is to say, the ground state is AFM below for Gd Er Ga and Gd Er Ga compounds. As a result, the magnetization at 5 K increases linearly with increasing magnetic field in low field ranges (see Fig. 10). Besides, it is found that the magnetization exhibits a sharp increase when the applied field exceeds a certain value, indicating that the field-induced metamagnetic transition from AFM to FM state occurs. The critical field required for metamagnetic transition for Gd Er Ga and Gd Er Ga, which is determined from the maximum of , is found to be 0.98 T and 1.03 T, respectively.^[34] The transition temperatures of Gd _x Er Ga ( ) compounds, which are determined from derivative of M–T curves, are all listed in Table 1. We can see that increases from 30 K to 181.9 K as the content of Gd increases from 0 to 1. The reason is that the average value of spin quantum number (S) increases with the content of Gd, and increases with S increasing. The and are listed in the same column, and results show that the change of or is not as large as . The lowest and highest values of are 15 K and 66 K respectively for the Gd _x Er Ga compounds.

Fig. 10.

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	Fig. 10. (color online) The isothermal magnetization curves up to 5 T at 5 K for Gd x Er Ga ( ) compounds.[34]

The SR transition due to the competition between the magnetic exchange interaction and the crystal field interaction has been observed in NdGa, SmGa, HoGa, ErGa, and GdGa compounds by Mössbauer spectrum experiments.^[20] With decreasing temperature, the magnetic moments rotate gradually from the bc plane toward the crystallographic a axis. For HoGa compound, the azimuth angle ϕ does not change with temperature and the rotation of the magnetic moments occurs in the ac plane when only the angle θ is changed. For GdGa compound, in the temperature range 5 K To 100 K, the ferromagnetic structure of the GdGa compound is noncollinear. At 5 K the magnetic moments of the Gd ions point in two distinct directions with respect to the crystallographic a axis ( and ).^[20] High-resolution neutron powder diffraction was also employed to study the SR transition of HoGa compound.^[29] In the temperature range between and , the magnetic structure comprises ferromagnetic order of the Ho sublattice along the c axis. Upon cooling below , the Ho magnetic moments cant away from the c axis towards the ab plane. At 3 K, the Ho moment is 8.8(2) and the Ho magnetic moments point in the direction and with respect to the crystallographic c axis. The observation of an ab plane component at around 50 from the a axis is in contrast with the suggested magnetic structure reported on the basis of a ¹¹⁹Sn Mössbauer spectroscopy study of an Sn-doped HoGa sample.^[20,29] The SR transition GdGa compound is also studied by ¹⁵⁵Gd Mössbauer spectroscopy and neutron powder diffraction.^[30] Between and , the magnetic structure is characterized by ferromagnetic order of the Gd moments along the b axis. On cooling below , the Gd 4c magnetic moments split into two groups (2:2). At 3.6 K, the Gd moment is 6.7(4) , and the Gd magnetic moments are in the bc plane, canted by 84(3) and 46(4) with respect to the crystallographic b axis. This splitting into two magnetically inequivalent sites is confirmed by 5 K ¹⁵⁵Gd Mössbauer results.^[30]

The thermal magnetization curves at different magnetic fields and isothermal magnetization curves are measured for PrGa compound, which are shown in Fig. 11. From M–T curves at 0.05 T, we can see that the magnetization curve experiences two drops at 28 K and 38 K respectively. The sharp changes are corresponding to two magnetic transition temperatures marked as and . The transition at lower temperature is similar to the FM to AFM transition and it seems to be supported by isothermal magnetization results shown in Fig. 11(b). In the temperature range between and , the magnetization curve shows a linear increase and then a sudden jump with magnetic field increasing, which appears to indicate the ground state is AFM and there is a metamagnetic transition from AFM to FM phase.^[41,43] As the applied field is added, moves to higher temperature zone. It indicates that the temperature region of metamagnetic transition gets larger when the applied field increases. To make it clear, partially enlarged view is shown in the inset of Fig. 11(a). When the applied field is high enough, metamagnetic transition occurs in the whole temperature region between and , thus the two drops of magnetization almost become one. The M–T curves of PrGa are very similar to those of LaFe Al .^[44] The critical field of metamagnetic transition determined by the maximum value of is also calculated. The result indicates that the critical field increases with temperature. In other words, the temperature zone where metamagnetic transition happens extends with applied field increasing. It is consistent with the conclusion obtained from M–T curve.

Fig. 11.

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	Fig. 11. (color online) (a) The thermal magnetization curves of PrGa compound under the field of 0.05 T, 0.1 T, 0.2 T, 0.4 T, 0.6 T, 0.8 T, and 1 T, respectively. (b) The isothermal magnetization curves up to 5 T at different temperatures in the range from 11 K to 75 K for PrGa compound.[41]

The temperature dependence of heat capacity at different magnetic field for YGa, PrGa, ErGa, and TmGa compounds are measured and the C _H –T curves are shown in Fig. 12. The YGa compound is non-magnetic and has the same crystal structure with other RGa compounds. The C _H –T curve of the YGa compound obeys the typical heat capacity change rule as temperature increases and there is no peak observed on the curve (see Fig. 12(a)). Unlike the YGa compound, peaks are observed on the C _H –T curves of magnetic PrGa, ErGa, and TmGa compounds. The heat capacity results show similar change rules with M–T curves. Two peaks are clearly shown on the C _H –T curve when the applied field is zero for the PrGa compound (Fig. 12(b)). And the two peaks are corresponding to and respectively. The peak at lower temperature moves toward higher zone with applied field increasing. When the applied field exceeds a certain value, the two peaks turn into a large one, such as those in the case of 1, 2, and 5 T.^[41] The C _H –T curves of the ErGa compound for 0 T, 1 T, 2 T, and 5 T are shown in Fig. 12(c). Two sharp peaks are observed at 15 K and 30 K under zero-field, corresponding to the SR and FM to PM transitions, respectively. With the application of magnetic field, the intensity of the peak at is reduced greatly, and the peak shifts toward higher temperatures with field increasing. On the other hand, the peak at spreads to a broad anomaly with a much lower magnitude when a field larger than 1 T is applied, and it almost disappears under a field of 5 T. It may also be noted from Fig. 12(c) that the values of heat capacity in applied fields above are much larger than those at zero-field in a wide temperature range. The abnormal heat capacity of ErGa may be related to the existence of short-range FM correlations in the PM state .^[19] Figure 12(d) displays the heat capacity curves for TmGa under the fields of 0, 1, 2, and 5 T, respectively. It clearly shows two successive magnetic transitions at K and at K, which is fully consistent with the magnetic measurements.^[39] With the increase of magnetic field, the two peaks are not only turn a broader and lower, but also shifts toward higher temperature, which is the typical characteristic of FM material.^[2] It is known that the heat capacity peak is caused by the absorption of heat which is utilized in randomization of magnetic moments around transition temperature.^[39] With the application of field, the randomization of moments would spread out over a wide temperature region, and the maximum peak moves towards higher temperature.^[5] In fact, the performance of C _H –T curve is the inevitable result of thermal magnetization data. Now we discuss the form of the magnetic contribution to heat capacity approximately. The magnetic component of internal energy can be expressed as , where M is the magnetic moment of the compound and is the total magnetic field in the compound. According to the theory of mean field, the total magnetic field can be expressed as , where is the external field, α is the mean field constant, and represents the internal field.^[31,45] The magnetic contribution to heat capacity can be obtained by , thus we can see that . According to this formula, a sudden drop of magnetic moment on the M–T curve can definitely cause a sharp peak on the C _H –T curve. Every magnetic transition is accompanied by a sudden drop of magnetic moment in magnetic RGa compound. As a result, obvious peaks are observed on C _H –T curves near transition temperatures.

Fig. 12.

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Fig. 12. (color online) (a) The temperature dependence of heat capacity for YGa compound at zero-field. (b) The temperature dependence of heat capacity for PrGa compound at a field of 0 T, 0.5 T, 1 T, 2 T, and 5 T, respectively. (c) The temperature dependence of heat capacity for ErGa compound at a field of 0 T, 1 T, 2 T, and 5 T, respectively. (d) The temperature dependence of heat capacity for TmGa compound at a field of 0 T, 1 T, 2 T, and 5 T, respectively.[19,33,39]

According to the magnetic and heat capacity measurements, ErGa compound undergoes an SR transition and an FM to PM transition as temperature increases.^[19] And the SR transition has been confirmed by ¹¹⁹Sn Mössbauer spectroscopy study.^[20] Two magnetic transitions are also observed according to magnetic and heat capacity measurements for PrGa compound.^[41] However, the no SR transition is observed according to ¹¹⁹Sn Mössbauer spectroscopy experiment.^[20] So it is still not clear that what kind of magnetic transition it is at and whether the ground state at the temperature range between and is AFM. High resolution neutron powder diffraction (NPD) is a good way to study the magnetic properties. The magnetic structure of PrGa comound at 4.5 K has been determined in the above section. Then the change of magnetic structure as temperature increasing and magnetic transition of PrGa compound can also be studied by NPD method. All the magnetic structures at different temperatures were solved out. The detailed results including crystal lattice parameters, ordered magnetic moments and errors of refinement are listed in Table 2. The temperature dependence of the ordered magnetic moment and the sketch of the magnetic structures are shown in Fig. 13. As temperature increases, the magnetic moment shows a sudden decrease around K, whereas the direction of the ordered magnetic moment remains the same. That is to say, the only change of magnetic structure at is the variation of angle of cone. At K the ordered magnetic moment decreases to zero, which indicates an FM to PM transition. Therefore we can conclude that PrGa compound undergoes an FM to FM magnetic transition at and an FM to PM transition at as temperature increases. According to the results of Mössbauer spectroscopy experiments, a magnetic spin reorientation transition from bc plane toward the a axis with decreasing temperature was observed for NdGa, SmGa, HoGa, and ErGa compounds.^[18] But for PrGa compound, there is no such reorientation of the ordered magnetic moments around , though the crystal structure of PrGa compound is almost the same as other RGa compounds. Furthermore, lattice parameters a, b, and c change as temperature increases, but only c shows an obvious response to both the FM to FM transition and the FM to PM transition, because the change of magnetic structure at transition temperatures occurs mainly along the c axis.

Fig. 13.

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	Fig. 13. (color online) Temperature dependence of the ordered magnetic moment obtained from the powder neutron diffraction experiment. The ordered magnetic moments are along the crystallographic z axis. M z and are represented in red and blue arrows, respectively.[33]

Table 2.

Table 2.

Table 2. Refined structural parameters of PrGa compound. Space group , Atomic positions: Pr (0, y, 0.25); Ga (0, y, 0.25). a, b, c, and V are lattice constant and volume of unit cell. M, M x , M y , and M z are the total ordered magnetic moment and the individual component of moment along a, b, and c axes, respectively. ϕ and θ are the angle between the ordered magnetic moment and x, z axes, respectively. [34] . Atoms Parameters 4.5 K 15 K 20 K 25 K 28 K 30 K 33 K 36 K 40 K 295 K a/Å 4.4545(3) 4.4545(3) 4.4545(3) 4.4538(3) 4.4533(3) 4.4522(3) 4.4515(4) 4.4506(3) 4.4504(4) 4.4494(4) b/Å 11.2533(9) 11.2530(9) 11.2527(9) 11.2547(9) 11.259(1) 11.261(1) 11.264(1) 11.265(1) 11.267(1) 11.312(1) c/Å 4.1878(3) 4.1880(3) 4.1877(3) 4.1875(3) 4.1865(3) 4.1860(3) 4.1861(3) 4.1862(3) 4.1857(3) 4.1953(4) V/Å3 209.93(4) 209.93(4) 209.91(4) 209.90(4) 209.91(4) 209.87(4) 209.90(5) 209.88(4) 209.89(5) 211.15(5) Pr y 0.1416(2) 0.1419(2) 0.1418(2) 0.1421(2) 0.1420(2) 0.1425(3) 0.1424(3) 0.1426(3) 0.1429(3) 0.1441(3) 0 0 0 0 0 0 0 0 0 0 2.56(4) 2.43(4) 2.22(4) 1.90(4) 1.05(6) 0.6(1) 0.6(1) 0.65(9) 0 0 /( ) 0 0 0 0 0 0 0 0 0 0 Ga y 0.4301(2) 0.4301(2) 0.4305(2) 0.4302(2) 0.4303(2) 0.4304(2) 0.4306(2) 0.4304(2) 0.4306(2) 0.4304(2) 6.01 6.07 6.12 5.85 7.39 7.29 7.29 7.08 χ 2 1.363 1.274 1.300 1.281 1.292 1.276 1.313 1.224 1.246 1.458

Table 2. Refined structural parameters of PrGa compound. Space group , Atomic positions: Pr (0, y, 0.25); Ga (0, y, 0.25). a, b, c, and V are lattice constant and volume of unit cell. M, M x , M y , and M z are the total ordered magnetic moment and the individual component of moment along a, b, and c axes, respectively. ϕ and θ are the angle between the ordered magnetic moment and x, z axes, respectively. [34] .

2.3. The magnetocaloric effects of Ga compounds

The magnetic entropy change of RGa ( , Tb, Dy, Ho, Er, Tm) compounds was calculated from magnetization isotherms by using the Maxwell relation . The curves calculated from isothermal magnetizations for a field change of 0 T–2 T and 0 T–5 T are presented in Figs. 14(a) and 14(b), respectively. The insets of Fig. 14 are the enlarged view of the curves. Two peaks are found to be centered at and for the RGa ( , Tb, Dy, Ho, Er) compounds. The two peaks are corresponding to the SR transition and PM to FM transition. The peaks around for TbGa, DyGa, HoGa, and ErGa compounds can be detected clearly in magnetic measurements. Although there are also two magnetic transitions as temperature increases for TmGa compound, only one peak can be observed on the curve. The maximal values of with a field change of 0 T–5 T for RGa ( , Tb, Dy, Ho, Er, Tm) compounds are calculated to be 5.3 J/kg K, 8.3 J/kg K, 5.8 J/kg K, 17.1 J/kg K, 21.3 J/kg K, and 34.3 J/kg K, respectively. The refrigerant temperature width is calculated to be 159.4 K, 155.5 K, 112.7 K, 83 K, 30.9 K, and 14.2 K, respectively. The RC is 616.1 J/kg, 620.6 J/kg, 381.9 J/kg, 455J/kg, 494 J/kg, and 359.7 J/kg, respectively. All of the MCE parameters with a field change of 0 T–2 T and 0 T–5 T are listed in Table 1. It can be seen that for some compounds the two peaks are far from each other, and for some compounds the two peaks are close to each other. The two magnetic transitions are so near that only one peak can be seen for TmGa compound. Also it is found that the peak value around is comparable to that around for ErGa compound. Figure 15 shows the curves of all the Gd _x Er Ga ( ) compounds for a field change of 0 T–2 T and 0 T–5 T, respectively. Each curve has two peaks, though the peak near is not obvious enough for GdGa. The location of the two peaks is corresponding to or and , respectively. It indicates that both SR transition and PM to FM transition contribute to MCE. As the content of Gd increases, the position of the peak near or changes not much, but the peak near moves to higher temperature obviously. As the content of Gd increases, the value of near or decreases from 16.5 J/kg K to 3.0 J/kg K and the value of near decreases from 21.3 J/kg K to 5.3 J/kg K. Oesterreicher and Parker^[47] have proposed the relationship between near and other magnetic parameters as follows: . Where R is universal gas constant, is Lande factor, is Bohr magneton, J is the total angular quantum number, H is external magnetic field, is Boltzmann constant, and is Curie temperature. For Gd _x Er Ga ( ) compounds, the value of J decreases and the value of increases with the content of Gd increasing, so the near shows downtrend. For Gd Er Ga and Gd Er Ga, the value of is positive below (inset of Fig. 15), and that is because the magnetic ground state in low temperature zone is AFM. And positive is always observed in that case, which is due to the disordered magnetic sub-lattice antiparallel to the applied magnetic field.^[48,49]

Fig. 14.

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Fig. 14. (color online) (a) The temperature dependence of magnetic entropy with a field change of 0 T–2 T for RGa ( , Tb, Dy, Ho, Er, Tm) compounds respectively. The inset is the enlarged view of the curves in low temperature range. (b) The temperature dependence of magnetic entropy with a field change of 0 T–5 T for RGa ( , Tb, Dy, Ho, Er, Tm) compounds respectively. The inset is the enlarged view of the curves in low temperature range.[46]

Fig. 15.

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Fig. 15. (color online) (a) The temperature dependence of magnetic entropy with a field change of 0 T–2 T for Gd x Er Ga ( ) compounds respectively. The inset is the whole curves for Gd Er Ga and Gd Er Ga compounds. (b) The temperature dependence of magnetic entropy with a field change of 0 T–5 T for Gd x Er Ga ( ) compounds respectively. The inset is the whole curves for Gd Er Ga and Gd Er Ga compounds.[34]

The value of can be calculated from magnetization data according to the Maxwell relation and from heat capacity data. The –T curves for PrGa, ErGa, and TmGa compounds obtained from the two methods are shown in Fig. 16(a), Fig. 16(b), and Fig. 16(c), respectively. Before we give the details of calculating the of PrGa compound from heat capacity data, we need to review the standard methods to extract from C _H data. Take the for example, we need three steps to achieve the result:^[48]

Fig. 16.

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Fig. 16. (color online) (a) Magnetic entropy change curves obtained from heat capacity measurement (solid lines) and magnetic property measurement (empty symbols). (b) Temperature dependences of magnetic entropy change calculated from magnetizations and heat capacity data for field changes of 0 T–2 T and 0 T–5 T, respectively. (c) Temperature dependences of magnetic entropy change in TmGa for different magnetic field changes.[19,39,41]

The first step is to obtain the total entropy at zero-field

(7)

(8)

(9)

However, heat capacity needs be measured at extremely low temperature, so now the question is whether we may use the above standard method. Considering the fact that the starting temperature of heat capacity measurement for PrGa compound is 10 K and the magnetic structure is complex in low temperature range, the standard method to calculate from C _H data will not hold true in this case anymore. In fact, the problem can be solved as long as we make some modifications to this method. The key point of modification is to replace the reference temperature of 0 K with 72.5 K.^[41] In this case, the value of is not zero, but we can acquire its value from the calculations of M–H data. The reason why we choose 72.5 K as the reference temperature is that the magnetic ground state and magnetic transition are clearly known here far from . And we can see that the curves obtained from the two methods are consistent well with each other above 25 K. Large deviation below 25 K may result from the complex magnetic structure and magnetic transition in PrGa compound.^[41] The temperature dependences of for ErGa and TmGa compounds calculated from magnetizations and heat capacity data for field changes of 0–2 and 0 T–5 T are shown in Fig. 16(b) and Fig. 16(c), respectively. The results obtained by using the two techniques are in good agreement with each other.

As another important parameter to evaluate the MCE of magnetocaloric materials, the adiabatic temperature change was calculated from the C _P versus T curves by using the method suggested by Pecharsky and Gschneidner,^[50] and it is summarized below. Firstly, is obtained according to formula (7) and S(72.5 K) is set to be zero. Secondly, is obtained from and , which is calculated from M–H data. Thirdly, we can obtain the curves of and , which are the inverse functions of and respectively. Finally, we make a difference according to the formula – and then the –T curves are obtained.

The curves of PrGa compound for a field change of 0 T–0.7 T, 0 T–2 T, and 0 T–5 T are shown in Fig. 17(a) and the inset is the enlarge view of the curves. As two most important parameters to characterize MCE, and curves show similar change rules. When the magnetic field change is small such as 0.7 T, two peaks are observed on the curves near and respectively. That is to say, both the FM to FM transition and the PM to FM transition contribute to MCE. We can also see that the peak near is much higher than that near . That is because the FM to FM transition is of first order and accompanied by volume transition, which always plays an important role in enhancing MCE.^[51,52] With the field change increasing, the peak in lower temperature zone shifts towards higher temperature region. That is because the temperature zone where the FM to FM transition occurs extends. When the field change is high enough such as 5 T, the two peaks are merged together and a wide platform is observed on the curve from 29.5 K to 37.5 K. This kind of material with nearly constant and in a wide temperature range has many practical applications, because people can use as few materials as possible in the limited space of magnetic refrigerator to realize refrigeration. The magnetic property and MCE of PrGa are very similar to those of LaFe Al .^[44] Some other RGa compounds have the same crystal structures as PrGa compounds, but no platforms are found on their –T curves.^[19,29,32] The main difference is that the transition temperature does not shift toward higher zone obviously with applied field increasing for other RGa compounds. Though the maximum value of and for PrGa is not large enough (( is 10.7 J/kg K and ( is 2.9 K for a field change of 5 T), the feature of nearly constant and is valuable for researches compared with other RGa compounds.^[41] Anyway, the future work is still necessary to improve the maximum value of and keep the platform on the curves at the same time. Positive value and negative value are observed around 35.5 K for the field changes of 0.7 T, and similar results have been reported more than once.^[48,49] The maximum values of for TmGa compound are 3.2, 5, and 9.1 K for the field change of 1, 2, and 5 T, respectively (Fig. 17(b)). The MCE of TmGa, especially under the low magnetic field change, is comparable with or even larger than those of other magnetocaloric materials around the liquid hydrogen temperature. Therefore, TmGa compound appears to be a very attractive candidate material for use in a magnetic refrigerator working in low temperature.

Fig. 17.

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	Fig. 17. (color online) (a) Temperature dependence of adiabatic temperature change Δ for PrGa compound under a field change of 0 T–0.7 T, 0 T–2 T and 0-5 T, respectively. The inset is the enlarged view of the curve for 0 T–0.7 T. (b) Temperature dependence of adiabatic temperature change for TmGa compound under a field change of 0 T–1 T, 0 T–2 T and 0 T–5 T, respectively.[39,41]

To evaluate the applicability of RGa compounds as magnetic refrigerant materials in a reasonable way, we have estimated the RC by using the approach suggested by Gschneidner et al.^[53] The RC is defined as , where T ₁ and T ₂ are the temperatures corresponding to both sides of the half maximum value of peak, respectively. Calculations show that the maximal value of RC is 158.7, 616.1, 620.6, 381.9, 455, 494, 359.7 J/kg with a magnetic field change of 0 T–5 T for RGa ( , Gd, Tb, Dy, Ho, Er, Tm) compounds, respectively. Especially, the value of RC for TbGa is found to be 900 J/kg with K (temperature of the cold end) and K (temperature of the hot end) for a field change of 0 T–7 T.^[32] Although the TbGa compound has a moderate , its RC value is much larger than those of other magnetic refrigerant materials in a wide temperature range. The large RC of TbGa originates from the combined contribution from SR and FM–PM transitions, which enlarges the temperature span of the MCE. It is noted that the value of for RGa increases monotonically with the decrease of as the R atomic number increases. The results indicate that the working temperature and temperature span of the MCE could be tuned by the substitution between different R elements. All the magnetic parameters and MCE results of Gd _x Er Ga ( ) compounds are also listed in Table 1. We can see that though the maximum value of decreases with the content of Gd increasing, the value of RC is greatly improved compared with ErGa. Also we find that the greatly extends as the content of Gd increases. Since both transitions (PM to FM transition and SR transition) contribute to MCE, there is a positive correlation between – and T ₂–T ₁. According to the discussions on transition temperatures, increases much faster than with Gd-content increasing; therefore, the increasing of Gd-content enhances . And the extending of plays an important role in improving the value of RC according to the calculation method. The exact value of RC fluctuates with the content of Gd increasing, but Gd Er Ga and Gd Er Ga compounds show better performances than others in this series by considering both and RC. Another important result is that table-like curves are observed for certain compounds, such as those in the case of , 0.2, 0.3, and 0.4. This kind of curves has been observed several times and such a feature of curves is very useful for practical applications of refrigeration.^[54,55]

As we mentioned before, the isothermal magnetization curves showed a typical AFM-like characteristics in the temperature range between and . However, neutron diffraction results show that PrGa compound is an FM-ordered state in this temperature zone. We take the representative isothermal magnetization curve at 32 K to reinterpret the magnetizing process in PrGa compound. The isothermal magnetization curve at 32 K is quoted and the magnetic structure sketches in different field zones have been plotted in Fig. 18.

Fig. 18.

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	Fig. 18. (color online) Isothermal magnetization curve of PrGa compound at 32 K and the corresponding magnetic structure in every magnetizing stage. M z and are symbolized in red and blue arrows, respectively. Domain walls under zero-field are represented by the black lines in the inset.[33]

The magnetizing process can be divided into two stages by the critical field ( , which is determined by the maximum of .^[33] For PrGa compound, is the critical field where the metamagnetic transition begins and is calculated to be 0.6 T at 32 K. In the range of , there are many magnetic domains with different FM-ordered directions, so the macroscopic magnetization is zero at the beginning. The magnetization increases gradually with increasing magnetic field. In this stage, magnetizing process is mainly achieved by means of domain-wall displacement. Inside of every magnetic domain, the projection of every magnetic moment on z axis does not really change. Since the ordered magnetic moment is rather small at 32 K (see Fig. 13), the magnetization changes slowly in this stage. In the range of , the domain-wall displacement reaches its limits. As magnetic field continues to increase, the magnetizing process is achieved by means of magnetic moment rotation. That is, the direction of every magnetic moment tends to align with z axis gradually. The most remarkable difference between the above two stages is that the magnetizing process is irrelevant to the degree of magnetic order in the first stage, but it is closely related in the second stage.

The curves calculated from the Maxwell relation of PrGa compound with a field change of 0 T–0.3 T and 0 T–5 T are shown Fig. 19, respectively. Figure 19 also shows that only for a large field change can we observe the plateau. When or , the value of is relatively small no matter the field change is large or small. The temperature range between and is the most important zone for the formation of the plateau. Again we take 32 K data to interpret the cause of the plateau.^[33] When the field change is 0 T–0.3 T, the magnetizing process is in the first stage. As we have mentioned before, magnetizing process is achieved by domain-wall displacement, and hardly changes in this stage. The value of is almost zero and no plateau is observed for a field change of 0 T–0.3 T. When the magnetic field exceeds , the magnetizing process steps into the second stage and all the magnetic moments begin to rotate toward z axis. changes significantly and thus a large value is observed for a field change of 0 T–5 T. It can be seen that the formation of a plateau is closely related to the magnetic structure, magnetic transition, and magnetizing process in PrGa compound.

Fig. 19.

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	Fig. 19. (color online) The temperature dependence of magnetic entropy change curves for PrGa compound under a field change of 0 T–0.3 T and 0 T–5 T, respectively.[33]

The magnetic field dependence of MR at various temperatures has been measured and calculated (shown in Fig. 20). It is observed that the MR drops significantly with the increase of field when the magnetic field smaller than T around , then retains a large value with the further increase of field. It is worth noting that the MR value is nearly as large as 30% at 28 K when a relatively small field (1 T) is applied, which is beneficial for the potential technology applications of the present sample. For the PrGa compound, the values of MR are found to be % and % at 28 K and 30 K in a magnetic field of 5 T, respectively. The maximal MR in PrGa is comparable to or much larger than those of some best known R-based intermetallic compounds, such as SmMn₂Ge₂ (8%),^[56] Ce(FeRu)₂ (20%),^[57] Gd₂In (29%),^[58] Gd₅(Si₂Ge (26%),^[59] and Gd₅(Si Ge (20%).^[60] Theoretical study based on the independent scattering mechanism shows that in metallic systems, the difference in resistivity owing to the magnetic structure modification. The electrical resistivity in magnetic materials includes residual resistivity (ρ , resistivity due to electron–electron scattering (ρ , resistivity due to electron–phonon scattering (ρ and resistivity due to the electron–magnon interactions (ρ .^[59] Among them, magnetic field mainly affects the ρ term. ρ increases with increasing and reaches to its maximum value when the temperature exceeds .^[61] Since the FM-ordered magnetic component is along z axis, spin polarization of electrons exists. The component in the xy crystal plane is in the disordered state, therefore the electron motion within the xy plane is almost forbidden. The only motion path of electrons is thus along z axis, and the obstacle of the movement mainly comes from the disordered M _xy . Therefore, we assume that the resistance caused by the magnetic moments is proportional to the projected area of the circular cone in the xy crystal plane. The above assumption can be expressed in the following formula:^[33]

(10)

Fig. 20.

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	Fig. 20. (color online) (a) The field dependence of magnetoresistance up to 5 T for PrGa compound at the temperatures from 10 K to 28 K. (b) The field dependence of magnetoresistance up to 5 T for PrGa compound at the temperatures from 30 K to 50 K.[33]

Since MR mainly comes from magnetic contribution in PrGa compound, at the fixed temperature of T, the change of electrical resistivity with different magnetic fields can be written as follows:

(11)

(12)

(13)

(14)

(15)

Then it is the key question to determine and . is the FM-ordered component of magnetic moments at zero magnetic field and it cannot be detected from the macroscopic magnetic measurement because of the existence of magnetic domains. While can be determined accurately by NPD results at zero field. The deviation of obtained from bulk compound and powder sample has been discussed. In order to study MR in bulk PrGa compound, we revised the obtained from NPD experiment by taking M _z (0, 5 K) as a reference. The expression of is very complex, because it is not only temperature-dependent but also field-dependent. The whole temperature range is divided into three parts by K and T= 38 K:^[33]

Case 1   K

The magnetizing process is generally realized by domain-wall displacement and magnetic moment rotation sequentially as magnetic field increases. T, is a macroscopic physical quantity, and if T, is no more than , we can conclude that the magnetizing process is only in the first stage and M _z does not really change even though the magnetic field is as high as 5 T. There is almost no difference between and T, in this temperature range. So, we have the following expression:

(16)

Cases 2  17 K K

In this temperature range, T, is considerably larger than , which indicates that the magnetizing process experiences both stages as magnetic field increases from 0 T to 5 T. Also it should be noted that is obtained in this temperature range. As we discussed on the magnetizing process at 32 K, is the field separatrix of the two magnetizing stages. And can be considered as the measure of , when the magnetic field exceeds . So can be expressed in the following piecewise function:

(17)

Case 3   K

In this temperature range, PrGa compound is in the PM state and every magnetic moment distributes randomly at zero field. When the magnetic field is applied, the macroscopic magnetization is the ordered component of the magnetic moment along z axis. So, we have the following expression:

(18)

Putting Eqs. (16)–(18) into Eq. (15), we can obtain . In order to simplify the expression of MR, we make the following variable transformation:

(19)

(20)

(21)

Case 1)   K

(22)

Case 2)  17 K K

(23)

Case 3)   K

(24)

On one hand, can be calculated according to Eq. (22)–(24). On the other hand, the experimental results of can be calculated from and . The curves for X dependence of MR at K, 20 K, 26 K, 27 K, 32 K, and 38 K were plotted in Fig. 21. The dotted lines are the calculated results and the scattered symbols are the experimental results. The experimental results are in good accordance with the calculated results in all three temperature zones. This indicates that the assumption about expressed in Eq. (10) and the calculation of MR are reasonable. And also we can see that MR is indeed closely related magnetic structure, magnetic transition and magnetizing process in PrGa compound.

Fig. 21.

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	Fig. 21. (color online) Relationships between MR and X at K, 20 K, 26 K, 27 K, 32 K, and 38 K, respectively. . The scattered symbols are the experimental data. The dotted lines are the calculated results. The quadratic dependence of was plotted in solid lines.[33]

Many works on MR in granular system have been reported, and it indicates that MR and X have the quadratic relationship in granular materials.^[63–65] So it is necessary to discuss whether the quadratic relationship is still appropriate in bulk PrGa compound. From Eqs. (22)–(24) and Fig. 21, it is clear that the MR for K does not obey the quadratic relationship and the result for K matches the quadratic relationship strictly. The relationship of MR and X has more complex form in the temperature range between 17 K and 38 K according to Eq. (23). The MR results at K, 26 K, 27 K, and 32 K shown in Fig. 21(b) and Fig. 21(c) are in this temperature range. To make it easy to compare, the function of is also plotted in Fig. 21(b) and Fig. 21(c) in solid line. It shows that the experimental result for K can be considered to obey the quadratic relationship approximately. However, the experimental results for K, 26 K, and 27 K show considerable deviations from the relation. In order to find out why the approximation is effective in Fig. 21(c) and why the approximation is ineffective in Fig. 21(b), we need to review the derivation formula of MR in the temperature range between 17 K and 38 K.^[33] According to the second part of Eq. (23), if can be approximated as zero, MR can be expressed as . It shows that M ₀(20 K), M ₀(26 K), and M ₀(27 K) are 1.7 , 1.3 , and 1.0 , respectively, while M ₀(32 K) is only 0.5 which is relatively smaller than each of the above three. As an approximation, we treat M ₀(32 K) as zero, and then obtain the quadratic relationship in the second part of Eq. (23). Further, if the at 32 K is neglected, MR(32 K) can be expressed as approximately. However, at 20 K, 26 K, and 27 K is relatively large and it cannot be approximated as zero. Therefore, MR curves deviate noticeably from the quadratic relationship shown in Fig. 21(b).

3.1. The crystal structure and basic magnetic properties of Ni ( , Dy, Ho, Er) compounds and Ho Er Ni compounds

Binary RNi ( , Dy, Ho, Er) compounds and pseudo-binary Ho _x Er Ni ( ) compounds were fabricated and powder x-ray diffraction was carried out to confirm the phase purity. The XRD patterns of Ho _x Er Ni ( ) compounds are shown in Fig. 22. The patterns of GdNi and DyNi are not shown literature.^[77,78] All of the XRD patterns have been fitted and refined. All of the RNi compounds is single phase without any impurity phases. The vertical bars on the bottom of Fig. 22 are Bragg positions. It can be seen that almost all of the peaks can be corresponding to the Bragg positions. Actually, results of refinement show that GdNi compound crystallize in the orthorhombic CrB-type structure (space group , #63), which is the same with RGa compounds. Both Gd and Ni atoms are in the same crystallographic sites with the C _2v point symmetry.^[22,23] While other RNi compounds crystallize in the orthorhombic FeB type structure (space group Pnma, #62).^[69,77,78] The unit cell of FeB-type crystal structure is shown in Fig. 23. Both R and Ni atoms occupy the site of the unit cell. The structure is made up of trigonal prisms with R at the corners and Ni at the center.^[77] The lattice parameters of GdNi are determined to be Å, Å, Å. The lattice parameters of DyNi, HoNi, ErNi are Å, 7.018±0.001 Å, 6.992±0.001 Å, Å, 4.137±0.001 Å, 4.115±0.001 Å, Å, 5.435±0.001 Å, 5.418±0.001 Å, respectively.^[77,78] As the R atom number increases, the radius of R atom decreases. And as a result, the crystal parameters also show a decreasing tendency.

Fig. 22.

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	Fig. 22. (color online) The x-ray powder diffraction pattern of pseudo-binary Ho x Er Ni ( ) compounds. The positions of the Bragg peaks are shown as vertical bars below the observed intensities.[76]

Fig. 23.

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	Fig. 23. (color online) The crystal structure and unit cell of RNi compounds.

The thermal magnetization curves of RNi ( , Dy, Ho, Er) compounds were measured and the field-cooling curves of GdNi, HoNi, DyNi, and ErNi are shown in Fig. 24(a), Fig. 24(b), Fig. 24(c), and Fig. 24(d), respectively.^[77,78] As the temperature increases, the magnetization of GdNi compound shows a sudden decrease at K, which is determined by the maximum of . In the low temperature range, the magnetization of GdNi compound almost keeps a stable value. The above result is in accordance with previous work.^[66] From the M–T curve, it can be concluded that GdNi is a simple FM material, and it undergoes an FM to PM transition as temperature increases around . HoNi compound has been known as a multi-phase-transition material, and the magnetic transitions have been observed by magnetic measurements and neutron diffraction technique.^[66,67] From Fig. 24(b), it can be seen that as temperature increases the magnetization undergoes drastic change twice at 13.5 K and 35.5 K, respectively. The two temperatures are corresponding to the SR transition and FM to PM transition respectively. The magnetic properties of DyNi and ErNi compounds have been studied before, and drastic change of the magnetization due to the spin flopping was observed for both ErNi and DyNi compounds.^[72] The magnetic structure of DyNi compound is non-collinear, and the magnetization of DyNi compound shows several increase due to spin flopping with external field increase.^[77] DyNi and ErNi compounds have been treated as simple FM materials in previous work, and only one FM to PM transition has been discussed for DyNi and ErNi compounds.^[78] However, from Fig. 24(c) and Fig. 24(d), it is clearly shown that there are two drastic change of magnetization below as temperature increases for DyNi and ErNi compounds, which indicates that SR transition may exist both for DyNi and ErNi compounds. The value of for DyNi and ErNi is determined to be 20 K and 6.5 K, respectively. The for DyNi and ErNi is determined to be 59 K and 11 K, respectively.^[77] All of the transition temperatures of RNi compounds ( , Dy, Ho, Er) are shown in Table 3. As the atom number increases, Curie temperature shows a decrease tendency. That is because the spin angular momentum quantum number (S) increases with atom number for heavy rare earth elements. And S is positively correlated with Curie temperatures of R-based compounds.

Fig. 24.

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	Fig. 24. (a) The temperature dependence of magnetization curve for GdNi compound at a field of 0.02 T. (b) The temperature dependence of magnetization curve for HoNi compound at a field of 0.01 T. (c) The temperature dependence of magnetization curve for DyNi compound at a field of 0.05 T. (d) The temperature dependence of magnetization curve for ErNi compound at a field of 0.01 T.[77,78]

Table 3.

Table 3.

Table 3. The transition temperatures and magnetocaloric parameters of RNi ( , Dy, Ho, Er) compounds and Ho x Er Ni ( ) compounds. . Materials /K /K MCE_0–2 T MCE_0–5 T Ref. /(J/kg K) /K RC/(J/kg) /(J/kg K) /K RC/(J/kg) GdNi – 69 8.7 19.9 133.3 16.8 31.7 396.8 [78] DyNi 20 59 9.7 20.6 150.5 18 31 429.8 [77] HoNi 13.5 35.5 7.8 33.4 198.7 16.6 41.6 556.7 [76] Ho Er Ni 11 32 7.5 32.7 179.8 16.1 40.1 487.3 [76] Ho Er Ni 9 27 6.9 27.8 145.1 16.1 34.2 431.1 [76] Ho Er Ni 4 20 10.4 16.3 148.8 20.7 26.3 418 [76] Ho Er Ni 5 16.5 12.2 13.2 119.7 25.3 21.3 407.7 [76] Ho Er Ni 6 13.5 15.2 9.3 128.4 28.3 18.2 392.3 [76] Ho Er Ni 9 11 18 7 104.2 34 14.2 366.6 [76] ErNi 7 11 14.2 10.7 122.4 29.6 15.6 350 [76]

Table 3. The transition temperatures and magnetocaloric parameters of RNi ( , Dy, Ho, Er) compounds and Ho x Er Ni ( ) compounds. .

The isothermal magnetization curves of HoNi compound were measured up to 5 T and the curves at 2 K, 12 K, 18 K, 24 K, 30 K, 36 K, 42 K, 48 K, and 55 K are shown in Fig. 25, respectively. For the M–H curve obtained below , the magnetization increases as magnetic field increases, and reaches its saturation values quickly. That is the typical feature of FM materials. The two magnetic transitions HoNi compound undergoes as temperature increases have been confirmed by magnetic measurement and neutron diffraction technique. Then it can be concluded that the transition around is FM to FM transition, because the M–H curves obtained below and above are both FM type. The M–H curve at 36 K shows a nearly linear change feature, which is the characteristic of PM state. So it can be concluded that there is little short range FM order above for HoNi compound.

Fig. 25.

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	Fig. 25. (color online) The field dependence of magnetization curve for HoNi compound in the temperature range from 2 K to 55 K.

Arrott plot was obtained from the isothermal magnetization curves for HoNi compound and it is shown in Fig. 26. According to the Banerjee criterion a magnetic transition is expected to be of the first-order when the slope of Arrott plot is negative, whereas it will be of the second-order when the slope is positive.^[42,79] From the Arrott plots in the temperature range from 2 K to 45 K, it can be seen that none of them show a negative slope. The positive slope of Arrott around and indicates that the SR transition and FM to PM transition are both of second order. Generally to say, the magnetocaloric effect materials of second order are more valuable for actually use, because there are little magnetic hysteresis and heat hysteresis in the process of magnetic transition.^[19]

Fig. 26.

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	Fig. 26. (color online) The Arrott plots of HoNi compound in the temperature range from 2K to 45 K.

Figure 27 shows the ZFC and FC curves of HoNi compound at a field of 0.02 T. It is clearly that HoNi compound show good thermal reversibility both around and . The results of ZFC and FC are in accordance with the results of Arrott plots. In addition, the ZFC and FC curves also clearly show that HoNi compound undergoes two magnetic transitions as temperature increases at K and K, respectively. The magnetic transition of HoNi will be discussed in detail in the following section.

Fig. 27.

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	Fig. 27. (color online) The temperature dependence of zero-field-cooling magnetization and field-cooling magnetization curves for HoNi compound under a field of 0.02 T.

In order to study the influence of Ho atom on the magnetic property of ErNi compound, Ho _x Er Ni ( ) compounds were fabricated and the thermal magnetization curves at a field of 0.01 T were measured.^[76] The M–T curves of Ho _x Er Ni ( ) compounds are shown in Fig. 28. The inset of Fig. 28 is the enlarge view of M–T curve of ErNi compound. The magnetization undergoes a sharp decrease around for all of the Ho _x Er Ni ( ) compounds. And it is clear that there is also another significant decrease in low temperature range for Ho _x Er Ni ( ) compounds, which indicates that these compounds undergo another magnetic transition in low temperature zone. Considering of the SR transition of HoNi, the above transitions are probably SR transition as well. Spin slopping was observed with field increasing for ErNi compound, but ErNi compound has been treated as a simple FM material in previous researches.^[72,78] But from the enlarge view of M–T curve of ErNi compound, it can be seen that there is also another decrease of magnetization around K, although the decrease is not so obvious. Actually, the decrease has been observed in work given by Kumar et al.^[78] So it can be concluded that ErNi compound also under goes an SR transition and an FM to PM transition as temperature increases, which is the same case as HoNi compound. After the detailed study of the ZFC and FC, it found that Ho Er Ni compound also undergoes two magnetic transitions. But for Ho Er Ni compound, no drastic change of magnetization can be observed obviously except for the one at . AC susceptibility measurements show that Ho Er Ni compound also undergoes an SR transition around 9 K.^[76] All of the magnetic transition temperatures have been determined by the position of the peak on the curves. The transition temperatures are shown in Table 3. It can be seen that the Curie temperature increases from 11 K to 35.5 K as the content of Ho atoms increases, showing a steadily increasing tendency. But the SR transition temperature only changes between 4 K and 13.5 K, which is relatively stable compared with . That is because the Curie temperature is positively correlated with S, and the average S of Ho _x Er Ni compound increases as the content of Ho increases. But for , it is affected by crystal field and crystal structure. Considering that the Ho _x Er Ni compounds crystallize in the identify crystal structure, thus changes not so much for different content of Ho atoms. It can also be seen that 10% of Ho-substitution for Er atoms obviously shortens the temperature interval between SR transition and FM to PM transition. The isothermal magnetization curves up to 5 T at 2 K for Ho _x Er Ni ( ) compounds were also measured and have been shown in Fig. 29. All of the M–H curves show that the magnetization increases as magnetic field adding up and arrives at its stable value quickly, which is the typical feature of FM ground state. It indicates that the ground state of Ho _x Er Ni ( ) compounds in low temperature range is FM and it can be speculated that the SR transitions are FM to FM transition for Ho _x Er Ni ( ) compounds. The saturation magnetization for HoNi compound shows the largest value, because the total angular quantum number of HoNi compound is the largest. The saturation magnetizations for other Ho _x Er Ni compounds show a confused order in value, which indicates the magnetizing process is affected by other factors, such as crystal field.

Fig. 28.

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	Fig. 28. (color online) The temperature dependence of magnetization curves for Ho x Er Ni ( ) compounds at a field of 0.01 T. The inset is the enlarged view of M–T curve for ErNi compound.[76]

Fig. 29.

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	Fig. 29. (color online) The field dependence of magnetization curves up to 5 T at 2 K for Ho x Er Ni ( ) compounds, respectively.

The magnetic transition of HoNi compound has been studied according to many kinds of experiments such as magnetic measurements and neutron diffraction technique.^{[66,67,78,80–82]} It indicates that HoNi compound undergoes an SR transition and an FM to PM transition as temperature increases. The neutron diffraction studies show that the magnetic moments in HoNi have ferromagnetic components along the a axis and antiferromagnetic components along the c axis in the temperature range between and .^[67,78] Below , a change in symmetry has been observed due to the development of the ferromagnetic component along the b axis which grows with decreasing temperature.^[78] From the behavior of the M–Tcurve of HoNi, the presence of a canted structure below can be inferred.^[78] But the magnetic and magnetic transition of HoNi compound still needs to be studied further. In the following section the magnetic structure and magnetic transition of HoNi compound will be studied systemically by employing powder neutron diffraction experiments.^[76]

Firstly, the magnetic structure of HoNi compound at low temperature needs to be determined. And before determining the magnetic structure, the crystal structure was analyzed and determined. The powder neutron diffraction pattern of HoNi compound at 295K is shown in Fig. 30(a). The fitted lines, differences and Bragg positions are also shown there. Since the Curie temperature is around 35 K, HoNi compound is in PM state at 295 K. Thus only nuclear model is considered in the process of refining and fitting. All of the peaks can be indexed to the FeB-type orthorhombic crystal structure (space group pnam, #62). The crystal structure has been shown in Fig. 23 and the related parameters are as follows: Å, Å, Å, , , . The powder neutron diffraction pattern at 5.3 K is shown in Fig. 30(b). The fitted lines, differences and Bragg positions are also shown in the figure. At 5.3 K, HoNi compound is in FM-ordered magnetic state. Thus nuclear and magnetic structures are both considered. The Bragg positions for nuclear structure and magnetic structure are marked in Fig. 30(b), respectively. The related parameters are as follows: Å, Å, Å, , , . Results indicate that the only Ho atoms are magnetic ordered, and the Ni atoms contribute nothing to the magnetic moment. According to the refined magnetic structure at 5.3 K, Ho atoms are FM-ordered along a axis and b axis, and they are AFM-ordered along c axis. The component of ordered magnetic moment for x, y, z is determined to be 7.66(7) , 4.4(1) , and 3.80(5) , respectively. And the total ordered magnetic moment is calculated to be 9.63(8) , which is close to the effective moment of Ho ion. It indicates that almost no magnetic component is quenched for Ho ion in HoNi compound. The magnetic structure of HoNi compound is presented in Fig. 31, where it is viewed from b axis. Since it is viewed from b axis, the y component of ordered magnetic moment cannot be plotted.

Fig. 30.

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Fig. 30. (color online) The neutron powder diffraction pattern of HoNi at 295 K (a) and 5.3 K (b), respectively. The observed data and the calculated patterns are plotted in cross and solid line, respectively. The solid line at the bottom is the difference between the observed and calculated data. The Bragg positions for the nuclear and magnetic structure are marked as short vertical lines. Only nuclear model is considered in the refinement at 295 K and both nuclear model and magnetic model are considered in the refinement at 5.3 K.[76]

Fig. 31.

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	Fig. 31. (color online) The magnetic structure of HoNi compound viewed from the crystal y direction. The black arrows are corresponding to the ordered magnetic moments of Ho atoms.

From Fig. 31 it can be seen that the Ho atoms are connected in the form of six-member ring and the Ni atoms are in the center of the six-member ring. Half of the Ho atoms point to one direction with a positive x component and a positive z component, and the other half of Ho atoms point to another direction with a positive x component and negative z component. The x component is the same for all the Ho atoms, which indicates that Ho atoms are FM ordered in HoNi compound. The absolute value of z component of all the Ho atoms is also uniform, which means that AFM order of Ho atoms exists in HoNi compound. Just as what has been mentioned before, there is also an FM order of Ho atoms along b axis in HoNi compound, which cannot be presented in Fig. 31.

Secondly, the neutron powder diffraction patterns at different temperatures were also obtained and the magnetic structures at different temperatures were determined according to the refinement and fitting. The specific parameters of crystal structure, ordered magnetic moments, and error information are all listed in Table 4.

Table 4.

Table 4.

Table 4. (a) Refined structural parameters of HoNi compound. Space group , Atomic positions: Ho (x, 0.25, z); Ni (x, 0.25, z). a, b, c, and V are lattice constant and volume of unit cell. M, M x , M y , and M z are the total ordered magnetic moment and the individual component of moment along a, b, and c axes, respectively. ϕ and θ are the angle between the ordered magnetic moment and x, z axes, respectively. . Atoms Parameters 5.3 K 8 K 10 K 12 K 13 K 15 K 20 K 25 K a/Å 6.9765(1) 6.9763(2) 6.9765(2) 6.9768(2) 6.9765(2) 6.9769(2) 6.9764(1) 6.9766(2) b/Å 4.15480(3) 4.15477(4) 4.15481(3) 4.15486(4) 4.15482(4) 4.15490(4) 4.15478(3) 4.15482(4) c/Å 5.4004(1) 5.4008(1) 5.4007(1) 5.4010(2) 5.4010(2) 5.4015(2) 5.4016(1) 5.4019(2) V/Å3 156.534(5) 156.542(6) 156.545(5) 156.562(6) 156.556(6) 156.582(6) 156.566(5) 156.582(6) Ho x 0.1820(3) 0.1816(3) 0.1817(3) 0.1813(4) 0.1818(3) 0.1812(3) 0.1814(3) 0.1808(4) z 0.1313(6) 0.1327(7) 0.1327(6) 0.1332(7) 0.1318(7) 0.1307(8) 0.1332(6) 0.1336(7) 7.66(7) 7.63(8) 7.73(6) 7.58(7) 7.57(8) 7.42(7) 6.98(5) 5.93(7) 4.4(1) 3.9(1) 2.7(1) 1.5(3) 1.9(2) 1.3(3) 0 0 –3.80(5) –3.74(6) –3.69(5) –3.70(6) –3.73(6) –3.65(6) –3.39(4) –3.06(6) 9.63(8) 9.34(9) 8.99(7) 8.57(9) 8.65(9) 8.38(8) 7.76(6) 6.67(7) 113.2(2) 113.6(3) 114.2(3) 115.6(3) 115.6(3) 115.8(3) 115.9(3) 117.3(5) 30.1(6) 27.0(8) 19.6(9) 11(2) 14(2) 10(2) 0 0 Ni x 0.0358(3) 0.0354(3) 0.0355(3) 0.0357(3) 0.0364(3) 0.0857(3) 0.0362(2) 0.0362(3) z 0.6282(5) 0.6268(5) 0.6261(5) 0.6257(5) 0.6271(6) 0.6287(6) 0.6269(4) 0.6261(5) /% 0.0673 0.0816 0.0705 0.0854 0.0838 0.0793 0.0609 0.0796 /% 0.0803 0.0982 0.0837 0.1016 0.0996 0.0957 0.0736 0.0967 χ 2 2.795 1.346 1.984 1.396 1.364 1.335 2.271 1.262

Table 4. (a) Refined structural parameters of HoNi compound. Space group , Atomic positions: Ho (x, 0.25, z); Ni (x, 0.25, z). a, b, c, and V are lattice constant and volume of unit cell. M, M x , M y , and M z are the total ordered magnetic moment and the individual component of moment along a, b, and c axes, respectively. ϕ and θ are the angle between the ordered magnetic moment and x, z axes, respectively. .

Table 4.

Table 4.

Table 4. (b) Refined structural parameters of HoNi compound. Space group Pnma, Atomic positions: Ho (x, 0.25, z); Ni (x, 0.25, z). a, b, c, and V are lattice constant and volume of unit cell. M, Mx, My, and Mz are the total ordered magnetic moment and the individual component of moment along a, b, and c axes, respectively. ϕ and θ are the angle between the ordered magnetic moment and x, z axes, respectively. . Atoms Parameters 30 K 32 K 34 K 36 K 38 K 40 K 42 K 45 K 295 K a/Å 6.9769(2) 6.9774(2) 6.9760(2) 6.9758(2) 6.9767(2) 6.9768(2) 6.9779(2) 6.9784(2) 7.0162(2) b/Å 4.15490(4) 4.15500(4) 4.1559(1) 4.1557(1) 4.1555(1) 4.1552(1) 4.1546(1) 4.1536(1) 4.1405(1) c/Å 5.4022(2) 5.4011(2) 5.3999(2) 5.4003(2) 5.4003(2) 5.4008(2) 5.4011(2) 5.4019(2) 5.4359(1) V/Å3 156.601(6) 156.584(6) 156.552(6) 156.553(6) 156.565(6) 156.570(6) 156.580(6) 156.578(6) 157.916(5) Ho x 0.1804(3) 0.1797(3) 0.1802(3) 0.1797(3) 0.1797(3) 0.1797(3) 0.1798(3) 0.1803(3) 0.1800(2) z 0.1315(8) 0.1316(7) 0.1316(7) 0.1332(7) 0.1340(7) 0.1318(7) 0.1329(7) 0.1334(6) 0.1313(5) 4.27(6) 3.08(7) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 –2.00(7) –1.33(9) 0 0 0 0 0 0 0 4.71(7) 3.36(7) 0 0 0 0 0 0 0 115.1(8) 113(1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Ni x 0.0367(3) 0.0360(3) 0.0367(3) 0.0365(3) 0.0364(3) 0.0364(3) 0.0364(3) 0.0363(3) 0.0372(2) z 0.6277(6) 0.6285(6) 0.6291(5) 0.6286(5) 0.6276(5) 0.6282(5) 0.6271(5) 0.6275(5) 0.6285(4) /% 0.0667 0.0671 0.0642 0.0677 0.0639 0.0632 0.0609 0.0660 0.0384 /% 0.0816 0.0811 0.0797 0.0832 0.0787 0.0783 0.0755 0.0804 0.0475 χ 2 1.087 0.9613 0.9177 0.8818 0.9514 0.8915 0.8803 0.9041 0.9758

Table 4. (b) Refined structural parameters of HoNi compound. Space group Pnma, Atomic positions: Ho (x, 0.25, z); Ni (x, 0.25, z). a, b, c, and V are lattice constant and volume of unit cell. M, Mx, My, and Mz are the total ordered magnetic moment and the individual component of moment along a, b, and c axes, respectively. ϕ and θ are the angle between the ordered magnetic moment and x, z axes, respectively. .

Results show that the x component of ordered magnetic moment exists in the temperature range from 5.3 K to 32 K, and it changes from 7.66(7) to 3.08(7) . The z component shows the similar case to the x component and it changes from 3.80(5) to 1.33(9) as temperature increases from 5.3 K to 32 K. So the temperature around 32 K is corresponding to magnetic order to disorder temperature point, which is known as Curie temperature. Also it is found that the y component of ordered magnetic moment changes from 4.4(1) to 1.3(3) as temperature increases and it disappears from K. It indicates that the there is a magnetic transition and the transition temperature is around 15 K. As a matter of fact, the magnetic transition of HoNi compound in low temperature range has been observed according to magnetic measurements.^[66,78]

Figure 32(a) shows the thermal magnetization curve of HoNi compound at a field of 0.01 T. As temperature increases, the magnetization shows drastic decrease at and , respectively. The value of and is determined to be 13.5 K and 35.5 K, respectively, according to the maximum of (see the inset of Fig. 32(a)). The results of ordered magnetic moments and the components along different axis according to NPD fitting are also shown in Fig. 32(b). It is clear that the FM x component and AFM z component both undergo an order to disorder transition around 32 K, and the FM y component decreases as temperature increases and falls down to zero at 20 K. As the total ordered magnetic moment, M shows sudden decrease both at and .

Fig. 32.

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Fig. 32. (color online) (a) The field dependence of magnetization curve for HoNi compound at a field of 0.01 T. The inset is the plot of first order derivation of M–T curve. (b) The x component, y component, z component and total value of ordered magnetic moment in HoNi compound. (c) The temperature dependence of the angle between x axis and the projection of ordered magnetic moment in xy plane.[76]

The angle between the x axis and the xy component of the ordered magnetic moment was also calculated for every temperature and it was marked as ψ. The temperature dependence of the ψ curve is shown in Fig. 32(c). As temperature increases, ψ decreases from 30 at 5.3 K to 10 at 15 K, and it falls down to zero at 20 K, which means that the y component disappears. From the change tendency of the three components of ordered magnetic moment and the value of ψ as temperature increases, it can be seen that the HoNi compound undergoes a spin reorientation process at except for the ferromagnetic to paramagnetic transition at as temperature increases. Besides, an obvious change of crystal lattice parameters was also observed according to the neutron powder diffraction experiments. The temperature dependences of parameters a, b, c, and V are shown in Fig. 33. Parameters a, b, c, and V all show a change corresponding to the spin reorientation transition around 15 K, which means that the disappearing of the y component affects the crystal structure from three directions. As temperature increases, all of the above parameters show sharp variation at the temperature , which indicates that the ferromagnetic to paramagnetic transition affects the crystal structure more than the spin reorientation transition does. Although the spin reorientation transition has been observed and studied before, the present work shows a more complete sketch of that magnetic transition for the HoNi compound.

Fig. 33.

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	Fig. 33. (color online) (a) The temperature dependence of lattice constant a. (b) The temperature dependence of lattice constant b. (c) The temperature dependence of lattice constant c. (d) The temperature dependence of the volume of unit cell.

3.3. The magnetocaloric effect of Ni compounds

The of RNi ( , Dy, Ho, and Er) compounds was calculated based on the isothermal magnetization data (not shown here) by using the Maxwell relation . The curves as a function of temperature for RNi ( , Dy, Ho, Er) compounds with a field change of 0 T–2 T and 0 T–5 T are shown in Fig. 34(a) and Fig. 34(b), respectively. The data for GdNi and DyNi compounds are cited from literatures.^[77,78] and the data for HoNi and ErNi compounds are newly obtained in this work. The MCE curve obtained for GdNi shows a steady increase in the value with temperature and attains maximum value around its .^[78] The maximum value of , , and RC for a field change of 0 T–5 T for DyNi compound are estimated to be 18 J/kg K, 31 K, and 429.8 J/kg, respectively. The large MCE values in DyNi can be attributed to its metamagnetic nature.^[77] It is possible that the gradual flopping of moments towards the direction of the applied field is responsible for the reduction in magnetic entropy, which results in MCE.^[77] Spin-flop process has indeed been found to be responsible for the anomalous magnetoresistance observed along the c axis in single crystals of DyNi compound.^[77,83] The curve for ErNi shows a sharp increase in the value near its exhibiting the character of a typical ferromagnet whereas there is another peak around 15 K in the case of HoNi. The difference in the MCE behavior of ErNi and HoNi implies that though these compounds are non-collinear, they have different arrangements of moments in the magnetic sub-lattice.^[78] The peak in low temperature range in HoNi compound is corresponding to the SR transition. It also can be seen that the maximum value at is comparable to that at , which indicates that there is a very large change of magnetic entropy in the process of spin reorientation for HoNi compound. The two peaks on the curve for HoNi compound are corresponding to the two transition temperatures. As is discussed based on the thermal magnetization curves, there are also two magnetic transitions for DyNi and ErNi compounds. But no obvious magnetic entropy change is observed on the curve around for DyNi and ErNi compounds, which indicates that the SR transition is very weak for the DyNi and ErNi compounds. All of the MCE parameters are listed in Table 3. It is also clear that the maximum for ErNi compound shows the largest value among the RNi compounds.

Fig. 34.

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	Fig. 34. (color online) (a) The temperature dependence of magnetic entropy change curves under a field change of 0 T–2 T for RNi ( , Dy, Ho, Er) compounds, respectively. (b) The temperature dependence of magnetic entropy change curves under a field change of 0 T–5 T for RNi ( , Dy, Ho, Er) compounds, respectively.

The isothermal magnetization curves at different temperatures were measured and the was calculated for all of the Ho _x Er Ni ( ) compounds by using Maxwell relation .^[84] The temperature dependence of is shown in Fig. 35. It is found that the ( of Ho Er Ni is the largest among Ho _x Er Ni ( ) compounds. The value of ( for ErNi and Ho Er Ni compound is 29.6 J/kg K and 34 J/kg K, respectively, i.e., 10% of Ho-substitution brings as high as % ((34–29.6)/29.6 = 0.149) enhancement on ( . The refrigerant capacity (RC) of Ho _x Er Ni ( ) compounds were also calculated by using the approach suggested by Gschneidner et al.^[53] The RC is defined as , where T ₁ and T ₂ are the temperatures corresponding to both sides of the half-maximum value of ( , respectively. The refrigerant temperature width ( is defined as the temperature span of the full width at half maximum value of ( . Then can be obtained by T ₂–T ₁.

Fig. 35.

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	Fig. 35. (color online) (a) The temperature dependence of magnetic entropy change curves under a field change of 0 T–2 T for Ho x Er Ni ( ) compounds, respectively. (b) The temperature dependence of magnetic entropy change curves under a field change of 0 T–5 T for Ho x Er Ni ( ) compounds, respectively.

All the MCE parameters including ( , , and RC for Ho _x Er Ni ( ) compounds are listed in Table 3. As the content of Ho increases, the ( first increases and then decreases. Among all the Ho _x Er Ni ( ) compounds, Ho Er Ni compound shows the largest ( . Ho Er Ni compound has proved to be a second-order-transition material according to Arrott plots, but the ( (34 J/kg K for 0 T–5 T) is comparable to the representative low temperature first-order-transition MCE material ErCo₂ compound (36.8 J/kg K for 0 T–5 T).^[85] The changing trend of is opposite to ( . Among all the Ho _x Er Ni ( ) compounds, Ho Er Ni compound shows the smallest . As for RC, it increases monotonously from 350 J/kg to 556.7 J/kg as the content of Ho changes from 0 to 1. For Ho–Er–Ni compounds, Ho Er Ni exerts ideal ( but with ordinary and RC. But it is valuable to discuss the MCE enhancement (peak value) with 10% of Ho-substitution compared with ErNi.

In fact, the MCE enhancement in Ho–Er–Ni compounds is closely related to the temperature interval between SR transition and FM to PM transition. Magnetic materials are in the non-steady magnetic state around transition temperatures. When a field is applied around transition temperature, obvious MCE can be observed, because the magnetic entropy is easy to change in this state. As a result, more than one peaks can be observed in multi-phase-transition MCE materials, such as ErGa.^[19] and Ho₂In.^[86] compounds. According to thermal magnetization and ac susceptibility measurements, all of the Ho _x Er Ni ( ) compounds undergo two magnetic transitions as temperature increases: SR transition and FM to PM transition. More specially, the temperature interval between the two transitions is only 2 K for Ho Er Ni, which is the smallest value among the Ho _x Er Ni ( ) compounds. When the temperature interval between the magnetic transitions is large, the corresponding two peaks are separate, such as HoNi, Ho Er Ni and Ho Er Ni compounds (see Fig. 35). But for other samples, the two peaks are hard to distinguish, because they overlap and merge together, which is similar to Ho₁₂Co₇ compound with large field change.^[18] In this case, can also be used to evaluate the level of overlap. It is conspicuous that the is the smallest for Ho Er Ni compound, indicating the two peaks are almost entire-overlapping. Therefore, Ho Er Ni compound shows the largest (

4.1. The magnetic transition of ₁₂Co₇ compounds

The temperature ( dependence of magnetization ( was measured in both zero field-cooled (ZFC) and field-cooled (FC) processes in order to determine the thermal hysteresis and the magnetic transition temperature.

Figure 37 shows the thermomagnetic M–T curves of R ₁₂Co₇ ( , Dy, Ho, Er) compounds measured under an external magnetic field of 0.1 T. The inset of the Fig. 37 is the ZFC and FC curves of Tb₁₂Co₇ compound under a field of 0.01 T. Apparently, the ZFC and FC magnetization curves are almost same for Gd₁₂Co₇ compound. The large drop of M–T curves corresponds to ferromagnetic to paramagnetic phase transition. The of the compound is defined as the temperature where reached the peak point. In this work, of Gd₁₂Co₇ is about 163 K, the compound shows soft-ferromagnetic behavior below 163 K.^[90] The ZFC and FC magnetization curves of Tb₁₂Co₇ compound are separated from each other below 100 K and there is a cusp in the ZFC curve, which is due to the temperature-dependent coercivity.^[89] The compound shows ferromagnetic behavior. The Curie temperature of Tb₁₂Co₇ is about 100 K, as obtained from the minimum of the first derivative of the FC M–T curve in a magnetic field of 0.01 T.^[89] One can find that Dy₁₂Co₇ experiences an FM–PM transition around the Curie temperature K, defined as the minimum value of curve.^[91] It can also be seen from Fig. 37 that the ZFC and FC curves are completely reversible in the vicinity of as shown usually in magnetic materials with a second-order magnetic transition. Meanwhile, a significant thermal irreversibility between the ZFC and FC branches is clearly observed below under 0.1 T. Like many rare earth-metal compounds,^[94] Dy₁₂Co₇ may have strong anisotropy due to its large Dy content. It has been reported that in materials with high anisotropy energy and low exchange energy, the domain wall width could be comparable to that of lattice spacing, thus leading to a large pinning effect.^[94] Considering the magnetic anisotropy and low for Dy₁₂Co₇, the thermomagnetic irreversibility is likely attributed to the narrow domain wall pinning effect.^[91] When the sample is cooled under a zero field, it freezes in an irregular domain structure where the direction of the spontaneous magnetization is only determined locally by the magnetic anisotropy.^[91] When a small field is switched on and the sample is warmed up, the walls begin to move and in every domain the direction of magnetization turns into the energetically most favorable direction which now also depends on the direction of the applied field.^[95] So we can see that the ZFC magnetization is low below and reaches a maximum at low temperature. Whereas in FC mode, the magnetic field during the cooling prevents the pinning effect, and therefore the magnetization at low temperature is higher than that in ZFC mode and gradually reaches saturation.^[91] According to the M versus T characteristics, it is suggested that the Ho₁₂Co₇ compound undergoes three magnetic transitions in sequence with decreasing temperature. The transition at the highest temperature corresponds to a change from paramagnetic state to ferromagnetic state at K. The rest two phase transitions are ferromagnetic to antiferromagnetic transition at K and antiferromagnetic to antiferromagnetic transition at K, which will be confirmed by the isothermal magnetization curves.^[18] The transition temperatures for Er₁₂Co₇ compound is determined to be 6.5 K and 13.5 K, respectively. All of the transition temperatures of R ₁₂Co₇ compounds are listed in Table 5.

Fig. 37.

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	Fig. 37. (color online) The temperature dependence of zero-field-cooling and field-cooling magnetization curves at a field of 0.1T for R 12Co7 ( , Dy, Ho, Er) compounds. The inset is the temperature dependence of zero-field-cooling and field-cooling magnetization curves at a field of 0.01 T for Tb12Co7 compounds.[18,89–92]

Table 5.

Table 5.

Table 5. The transition temperatures and magnetocaloric parameters of R 12Co7 ( , Tb, Dy, Ho) and Ho x Er Co7 ( , 4, 6, 8, 10, 12) and Gd Tb x Co7 ( , 4, 8) compounds. . Materials /K /K MCE_0–2 T MCE_0–5 T Ref. /(J/kg K) /K RC/(J/kg) /(J/kg K) /K RC/(J/kg) Gd12Co7 123 163/160.8 4.6 37.4 123.3 8.8 71.7 462.8 [90,92] Gd8Tb4Co7 – 140.8 – – – 8.2 51.8 316 [92] Gd4Tb8Co7 – 118.9 – – – 7.1 38.5 202.2 [92] Tb12Co7 – 100 3.1 8.7 20.5 – – – [89] Dy12Co7 – 64 5 21.7 81.5 10 42.1 303.7 [91] Ho12Co7 9/17 30 9.2 29.4 206.2 19.2 37.4 555 [18] Ho10Er2Co7 9/19 26.3 6.2 20.9 94.8 16.1 32.5 396 [93] Ho8Er4Co7 11 22 7.8 22.8 137.8 17.0 31.8 422 [93] Ho6Er6Co7 12 18 9.7 19.7 150.5 18.6 28.5 411 [93] Ho4Er8Co7 8.6 15 9.6 16.5 122.6 18.0 24.7 342 [93] Er12Co7 6.5 13.5 10.2 14.5 103.6 18.3 22.8 317 [93] Ho6Er6Co7 42% Ho12Co7 58% – – – – – 16.7 38.0 553 [93]

Table 5. The transition temperatures and magnetocaloric parameters of R 12Co7 ( , Tb, Dy, Ho) and Ho x Er Co7 ( , 4, 6, 8, 10, 12) and Gd Tb x Co7 ( , 4, 8) compounds. .

The M–T curves of Ho₁₂Co₇ compound under different external magnetic field are shown in Fig. 38. One can find that T ₁ increases monotonically with magnetic field increasing from 0.01 T to 0.3 T. However, almost no change is observed for T ₂ and with magnetic field change. It indicates that a field-induced metamagnetic transition from AFM to FM state occurs below T ₂, and T ₁ is pushed towards higher temperature with the applied magnetic field increasing. Similar phenomenon has been observed in PrGa compound.^[43] Figure 39(a) shows the isothermal magnetization curves of Ho₁₂Co₇ in a temperature range of 3 K–50 K under the magnetic fields up to 7 T. The isothermal magnetization curves in some selected temperature ranges at low magnetic fields are shown in Figs. 39(b), 39(c), and 39(d), respectively. One can find that the isothermal magnetization curves between K and K, show FM characters (Fig. 39(d)). The magnetization remains a linear dependences of the magnetic field at low field range below = 17 K (Figs. 39(b) and 39(c)), indicating the existence of AFM ground state.^[18] Now we can confirm the transition around T ₂ is corresponding to FM to AFM transition, and the one around T ₁ is corresponding to AFM to AFM transition. The magnetization curves deviate from the linear relationship, when the applied field exceeds a certain value, showing a field-induced metamagnetic transition from AFM to FM state. The critical field determined from the maximum of for Ho₁₂Co₇ is found to be 0.12 T at 6 K and 0.28 T at 16 K.^[18] The result indicates that the Ho₁₂Co₇ compound is a weak antiferromagnetic material below T ₂, and a small magnetic field can destroy the AFM structure. One can also find that the isothermal magnetization curves obtained well above show strong curvatures at low fields from Fig. 39(a). Similar results have been observed in some other intermetallic compounds.^[19,96,97] It may result from the existence of short-range ferromagnetic correlations in the PM state.

Fig. 38.

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	Fig. 38. (color online) The temperature dependence of magnetization curves for Ho12Co7 compound at a field of 0.01 T, 0.05 T, 0.1 T, 0.2 T, and 0.3 T, respectively.[18]

Fig. 39.

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Fig. 39. (color online) (a) The field dependence of magnetization curves up to 7 T for Ho12Co7 compound at the temperatures from 3 K to 50 K. (b) The isothermal magnetization curves in the low field range at 6 K, 9 K, and 11 K. (c) The isothermal magnetization curves in the low field range at 13 K, 16 K, and 19 K. (d) The isothermal magnetization curves in the low field range at 21 K, 24 K, and 27 K.[18]

The S-shaped Arrott plots (not shown here) below T ₂ confirm the existence of the field-induced first-order AFM–FM transition. The positive slope of the Arrott plot above indicates a characteristic of a second-order PM–FM transition. The temperature dependences of the heat capacity in different magnetic fields are shown in Fig. 40. The peaks around 30 K, which are obvious for T and 0.5 T, are corresponding to . It is found that there are obvious differences of heat capacity below T ₂ for different magnetic field. The reason why the applied magnetic field affects heat capacity so greatly is that a magnetic transition from the weak AFM ground state to the FM state occurs. There is no obvious peak related to the AFM to AFM transition temperature for zero-field heat capacity curve, but an obvious peak around 13 K appears on the heat capacity curve for an applied field of 0.5 T.^[18] As is illustrated in the M–T curves, the peak is corresponding to AFM to AFM transition and it has been pushed by the applied field from T ₁ ( K) to higher temperature ( K). In the temperature range between T ₂ and , the differences of heat capacity for different magnetic fields are not as clear as those in lower temperature range. That is because Ho₁₂Co₇ compound is FM state in this range, and the influence of applied field on heat capacity is small. The heat capacity data demonstrates that the statement of magnetic transition on the basis of M–T and M–H curves is reasonable.

Fig. 40.

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	Fig. 40. (color online) (a) The temperature dependence of heat capacity curves for Ho12Co7 compound at a field of 0 T and 0.5 T, respectively. (b) The temperature dependence of heat capacity curves for Ho12Co7 compound at a field of 2 T and 5 T, respectively.[18]

Tb₁₂Co₇ compound has been treated as a simple ferromagnetic material and it shows typical soft ferromagnetic features below =100 K. But from the isothermal magnetization (M–H) curves up to 5 T at 5 K and 80 K, which is shown in Fig. 41, it seems that Tb₁₂Co₇ compound shows weak AFM ground state at 5 K, which is very different from the FM type curve obtained at 80 K. Considering that Ho₁₂Co₇ compound with the same structure with Tb₁₂Co₇ compound undergoes complex magnetic transition as temperature increases, it is possible that Tb₁₂Co₇ compound undergoes another magnetic transition in low temperature range besides the FM to PM transition at . The reason why the low-temperature transition was not observed according to M–T curves may be the AFM ground state in low temperature range is too weak. The temperature (T) dependences of magnetization (M) of Gd Tb _x Co₇ ( , 4, and 8) alloys under a 0.05-T applied field in zero-field cooling (ZFC) and field cooling (FC) modes are shown in Fig. 42. The Curie temperature , which is defined as the temperature at the maximum of versus T plot, for Gd₁₂Co₇, Gd₈Tb₄Co₇, and Gd₄Tb₈Co₇ are 160.8, 140.8, and 118.9 K, respectively. All alloys undergo an FM to PM order transition. The magnetization changes rapidly with the change of temperature in the vicinity of , indicating that a large entropy change may be observed near Curie temperature. Furthermore, a spin reorientation behavior with =123 K is observed for the (Gd Tb Co₇ alloy with , which should result from the competition between magnetic exchange and the crystalline electric field.^[98] It should be noted that no SR transition is observed on the M–T curve of Gd₁₂Co₇ compound according to Chen et al.^[90] The FC and ZFC data branch out from each other in the vicinity of Tc, indicating thermomagnetic irreversibility for this alloy.

Fig. 41.

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	Fig. 41. (color online) The field dependences of magnetization curve up to 5 T for Tb12Co7 compound at 5 K and 80 K, respectively.[89]

Fig. 42.

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	Fig. 42. (color online) Temperature dependencies of magnetization curves for Gd Tb x Co7 ( , 4, and 8) compounds under a field of 0.05 T.[92]

The zero-field-cooled (ZFC) and field-cooled (FC) magnetization curves for Ho _x Er Co₇ ( , 4, 6, 8, 10, 12) compounds were measured^[93] and the ZFC curves are shown in Fig. 43. The field dependence of magnetization (M–H) curves were also measured in detail for Ho _x Er Co₇ ( , 4, 6, 8, 10, 12) compounds. For the Ho _x Er Co₇ ( , 4, 6, 8) compounds, two drastic changes are observed at different temperature range on the M–T curves, which indicates that there are two magnetic transitions. M–H measurements show that the magnetic ground state of the three temperature zones divided by the two transition temperatures is antiferromagnetic (AFM), ferromagnetic (FM), and paramagnetic (PM), respectively. That is, Ho _x Er Co₇ ( , 4, 6, 8) compounds undergo an AFM to FM transition and an FM to PM transition in sequence as temperature increases. The transition temperature ( of AFM to FM transition is 6.5 K, 8.6 K, 12 K, and 11 K and the Curie temperature ( is 13.5 K, 15 K, 18 K, and 22 K for Ho _x Er Co₇ ( , 4, 6, 8) compounds, respectively. All of the transition temperatures of Ho _x Er Co₇ ( , 4, 6, 8, 10, 12) compounds are listed in Table 5. From the results of transition temperature, it is found that as the content of Ho atoms increases, the number of transitions grows from two to three and the magnetic property becomes more and more complex. The transition at T ₁, whether the AFM to FM transition for Ho _x Er Co₇ ( , 4, 6, 8) compounds or the AFM to AFM transition for Ho _x Er Co₇ ( , 12) compounds, means the competition of two ordered magnetic states. From Table 5, the value of T ₁ fluctuates not more than 5.5 K as the content of Ho atoms changes. That is because the competition is affected by crystal structure and crystal field, and the crystal structure of Ho _x Er Co₇ ( , 4, 6, 8, 10, 12) compounds is consistent with one another. The results are similar to the changing characteristics of spin reorientation transition temperature in Gd _x Er Ga compounds.^[34] However, the Curie temperature of Ho _x Er Co₇ ( , 4, 6, 8, 10, 12) compounds shows a steadily increasing trend as the content of Ho atoms increases. That is because the Curie temperature and spin angular number (S) are positively correlated. Since the S of Ho is larger than Er, more Ho atoms means larger average S and thus the Curie temperature increases step by step as the content of Ho goes up.

Fig. 43.

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	Fig. 43. (color online) The temperature dependence of magnetization curves at a field of 0.01 T for Ho x Er Co7 ( , 4, 6, 8, 10, 12) compounds, respectively.[93]

The magnetic entropy change of R ₁₂Co₇ ( , Tb, Dy, Ho, Er) compounds can be calculated from isothermal magnetization data by using Maxwell relation .

Figure 44 displays the values of for R ₁₂Co₇ ( , Tb, Dy, Ho, Er) compounds as a function of temperature for a field change of 0 T–2 T. The maximum value of for a field change of 0 T–2 T is calculated to be 4.6 J/kg K, 3.1 J/kg K, 5J/kg K, 9.2 J/kg K, and 10.2 J/kg K for R ₁₂Co₇ ( , Tb, Dy, Ho, Er) compounds, respectively. The maximum value of nearly shows an increase tendency as the R atom number increases. The change tendency of is not as obvious as maximum of . All of the MCE parameters including MCE for 0 T–5 T are all listed in Table 5. For Dy₁₂Co₇ compound, the distinct “ ”-type of –T curves as well as the immovable position of the peak of with the increase of field change further confirms the SOPT (second-order phase transition) nature of Dy₁₂Co₇. The maximum values of reach 5 J/kg K and 10.0 J/kg K for the field changes of 0 T–2 T and 0 T–5 T, respectively. These values are comparable with or larger than those of some magnetocaloric materials with a SOPT in a similar temperature range under the same field change. For example, for the field change of 0 T–5 T, the maximum values of TbCoAl ( K) compound and Er(Co Si ( K) compound are 10.5 J/kg K and 8 J/kg K, respectively.^[99,100] In addition, refrigerant capacity (RC), another important parameter to quantify the heat transferred between the hot and cold sinks in an ideal refrigeration cycle, is estimated by using the approach , where T ₁ and T ₂ are the temperatures corresponding to the both sides of half maximum value of peak.^[53] Thus, the RC values of R ₁₂Co₇ compounds are obtained for the field changes of 0 T–2 T and 0 T–5 T, respectively.

Fig. 44.

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	Fig. 44. (color online) The temperature dependence of magnetic entropy change curves under a field change of 0 T–2 T for R 12Co7 ( , Tb, Dy, Ho, Er) compounds, respectively.

The of Ho₁₂Co₇ was calculated from isothermal magnetization data by using the Maxwell relation and from heat capacity data through the expression as well.^[50] Figure 45(a) shows the as a function of temperature for different magnetic field changes. Figure 45(b) shows the comparison of curves obtained from the two different ways. The curves obtained using magnetization data are not in good agreement with the corresponding curves obtained using heat capacity. Considering that the actual combined relative error in the calculated from magnetization data can reach % and that from heat capacity data can be %,^[3,50] the deviations between the value of calculated from the two methods are in the error range with the maximal error being 17% and 11%, for the field change of 0 T–2 T and 0 T–5 T, respectively. Although there are differences between the curves obtained from the two methods, the shape of the curves matches well with each other. It is clear that there are two peaks on the –T curves, which indicates that both AFM–FM and PM–FM transitions contribute to MCE. Figure 45(c) shows that when the field change is small, for example 0 T–0.2 T, the value of is positive in certain temperature range, and as the field change increasing, the value of changes into negative. That results from the occurrence of metamagnetic transition, because the positive or negative value of is related to AFM or FM state. Besides, with the field change increasing, the first peak corresponding to AFM–FM transition moves towards higher temperatures, while the second peak related to PM–FM transition almost centers at K, which is in accordance with the results of M–T and C _p –T measurements. When the field change is high enough, for example 0 T–2 T and 0 T–5 T, the first peak is so close to the second one that they almost change into a larger peak. On the basis of magnetization measurement, the maximal value of for Ho₁₂Co₇ is found to be 19.2 J/kg K around for a field change 0 T–5 T. Compared with the refrigerant materials in a similar temperature range, the value of Ho₁₂Co₇ compound is smaller than that of ErGa (21.3 J/kg K at 30 K),^[19] DyCuAl (20.4 J/kg K at 28 K),^[16] and ErCo₂ (33 J/kg K at 36 K)^[101] compounds, but it is larger than that of DyNiAl (19 J/kg K at 32 K),^[102] TbCoC ₂ (15.3 J/kg K at 30 K),^[103] and GdNi₅ (11.5 J/kg K at 32 K).^[104]

Fig. 45.

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Fig. 45. (color online) (a) The temperature dependence of magnetic entropy change curves for Ho12Co7 compound under a field change of 0 T–0.5 T, 0 T–1 T, 0 T–2 T, 0 T–3 T, 0 T–4 T, 0 T–5 T, 0 T–6 T, and 0 T–7 T, respectively. (b) Temperature dependences of magnetic entropy change calculated from magnetization and heat capacity data for field changes of 0–0.5 T, 0 T–2 T, and 0 T–5 T, respectively. (c) Temperature dependences of magnetic entropy change calculated from magnetization only for field changes of 0–0.2 T, 0–0.3 T, 0–0.4 T, and 0–0.5 T, respectively.[18]

One can also see from Fig. 45(b) that both the peak value and peak width of –T curves depend on the magnetic field change. When the field change is 0 T–0.5 T, the maximum value of the two peaks are small and they are far apart from each other. However, when the applied field increases to 2 T and 5 T, the two peaks become one large peak. That is to say, both of the maximal value and the width of the peak have an obvious increase. The peak shape is very helpful to improvement of the value of RC in Ho₁₂Co₇ compound. The RC value of Ho₁₂Co₇ is estimated to be 206.2 J/kg with K (temperature of the cold reservoir) and K (temperature of the hot reservoir) for a field changing from 0 T to 2 T. And the RC value is estimated to be 554.9 J/kg with K and K for a field changing from 0 T to 5 T. As a result, Ho₁₂Co₇ compound has an outstanding refrigerant capacity among the magnetocaloric materials in a similar temperature range, such as ErGa ( J/kg with K and K),^[19] DyCuAl ( J/kg with K and K),^[16] ErCo₂ ( J/kg with K and K),^[101] DyNiAl ( J/kg with K and K),^[102] TbCoC ₂ (RC= 354 J/kg with K and K),^[103] and GdNi₅ ( J/kg with K and K),^[104] although ErGa, DyCuAl, and ErCo₂ exhibit larger values. Here, the RC values of some compounds are estimated from the temperature dependence of in the literatures, respectively. The large value of and RC suggests that Ho₁₂Co₇ can be an appropriate candidate for magnetic refrigerant in low temperature ranges.

Figure 46 shows the temperature dependencies of under the applied magnetic field change from 0 T to 5 T. And the inset of Fig. 46 shows the curves of Tb₁₂Co₇ compound under the field change of 0 T–1 T and 0 T–2 T, respectively. The peak values of are found to be 8.8, 8.2, and 7.1 J/kg K for the alloys with , 4, and 8, respectively.^[92] The RC values are calculated to be 478.4 J/kg, 327.2 J/kg, and 160.2J/kg for Gd Tb _x Co₇ alloys with , 4, and 8, respectively.^[92] It should be noted that no obvious peak is found around the of Gd₁₂Co₇ compound. There are two possible reasons for the absence of the peak in low temperature range: firstly, the magnetic measurement was not carried out as low as possible; secondly, the SR transition is too weak.

Fig. 46.

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	Fig. 46. (color online) The temperature dependence of magnetic entropy change curves under a field change of 0 T–5 T for Gd12Co7, Gd8Tb4Co7, and Gd4Tb8Co7 compounds, respectively. The inset is the temperature dependence of magnetic entropy change curves for Tb12Co7 compound under a field change of 0 T–1 T and 0 T–2 T, respectively.[92]

The (T) curves of Ho _x Er Co₇ ( , 4, 6, 8, 10, 12) compounds for a field change of 0 T–2 T and 0 T–5 T are shown in Fig. 47, respectively.^[93] The maximum value of for each Ho _x Er Co₇ ( , 4, 6, 8, 10, 12) compounds is listed in Table 5. Two peaks are observed obviously on the curve of Ho₁₂Co₇ compound. Actually, this kind of multi-peaks curves has been observed in many magnetocaloric materials with several transitions, such as ErGa, HoNi, TbGa, DyGa, etc.^[19,32,78] For other Ho _x Er Co₇ compounds, only one peak around can be observed and in low temperature zone there is a small convex instead of a peak. For Er₁₂Co₇ compound the small convex can even hardly be observed.

Fig. 47.

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	Fig. 47. (color online) (a) The temperature dependence of magnetic entropy change curves under a field change of 0 T–2 T for Ho x Er Co7 ( , 4, 6, 8, 10, 12) compounds, respectively. (b) The temperature dependence of magnetic entropy change curves under a field change of 0 T–5 T for Ho x Er Co7 ( , 4, 6, 8, 10, 12) compounds, respectively.[93]

Another important parameter to evaluate the magnetocaloric materials is refrigerant capacity (RC) and it has been also calculated. All of the parameters, , RC, and , are listed in Table 5. As the content of Ho atoms goes up, the of Ho _x Er Co₇ ( , 4, 6, 8, 10, 12) compounds increases steadily. The change characteristics of are closely related to the transition temperatures. According to the analysis of transition temperatures, T ₁ changes slightly and increases rapidly as the content of Ho atoms goes up. Therefore, shows an increase with the content of Ho atoms. The RC shows a similar changing trend to , but Ho₁₀Er₂Co₇ compound is an exception. Though the maximum of is not so large, the and RC show good performance for Ho _x Er Co₇ ( , 4, 6, 8, 10, 12) compounds.

For the actual application of magnetocaloric materials in refrigerators, the materials with table-like curves have more important value. That is because a much wider temperature range of cooling can be realized by this kind of magnetocaloric materials. The magnetocaloric materials with a plateau on curve are so useful that people have made efforts to design them by mixing different kinks of materials, such as R ₃Ni₂ and RFeSi.^[105,106] Considering that each Ho _x Er Co₇ ( , 4, 6, 8, 10, 12) compound shows advantages on MCE at different temperature zones, the Ho _x Er Co₇ compounds are mixed at different weight proportions with each two as one group. It is found that when Ho₁₂Co₇ and Ho₆Er₆Co₇ compounds are combined at the weight ratio of 58:42, the mixed magnetocaloric material shows the best performance (Fig. 48). The maximum of is 16.7 J/kg K, and the and RC are as high as 38 K and 553 J/kg, respectively. The RC of newly obtained material is comparable to Ho₁₂Co₇ compound and the is even better. The most important is that there is a plateau on the curve, which means that the mixed compounds of Ho₁₂Co₇ and Ho₆Er₆Co₇ with a weight ratio of 58:42 has a larger using value. The excellent MCE performance indicates its potential use for cooling in the temperature range around 20 K–30 K.

Fig. 48.

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	Fig. 48. (color online) The temperature dependence of magnetic entropy change curves under a field change of 0 T–5 T for the Ho12Co7 compound, Ho6Er6Co7 compound and the mixture of the two compounds with different ratio.[93]

5.1. The magnetic properties of ₃Co compounds and Gd _x Co compounds

Polycrystalline R ₃Co ( , Tb, Dy, Ho, Er, Tm) and Er Gd _x Co ( ) compounds were prepared by arc melting in a high-purity argon atmosphere. The purities of starting materials were 99.9% for R and Co. The sample was turned over and remelted several times to ensure its homogeneity. The ingot obtained by arc melting was wrapped by molybdenum foil, sealed in a quartz tube of high vacuum, annealed at 873 K for 4–8 days and then quenched to room temperature.^[49,119] X-ray diffraction (XRD) measurements on powder samples were performed by using CuKα radiation to identify the crystal structure. Magnetizations were measured as functions of temperature and magnetic field by using a superconducting quantum interference device (SQUID) magnetometer and vibrate sample magnetometer (VSM). Both the ac susceptibility and the specific heat were measured by using a physical property measurement system (PPMS) from Quantum Design. The powder XRD patterns of R ₃Co ( , Tb, Dy, Ho, Er, Tm) and Er Gd _x Co ( ) compounds obtained at room temperature were refined by the Rietveld method. The refinement shows that the prepared sample is of single phase, crystallizing in the orthorhombic Fe₃C-type crystal structure (space group Pnma #62). The crystal structure of R ₃Co compounds is shown in Fig. 49. R atoms occupy two nonequivalent positions [ ( ) and ( )] and Co atoms are located at the position within the trigonal prisms formed by Tm atoms. Results of all the structural parameters are determined by the Rietveld technique using GSAS program.^[123] For Er Gd _x Co ( ) compounds, with increasing Gd concentration, the diffraction peaks move to the left and the unit cell volume increases linearly. It is attributed to the contraction law of rare-earth intermetallic compound (the bigger atomic radius of the Gd atom than Er).^[122]

Fig. 49.

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	Fig. 49. (color online) The crystal structure of R 3Co compounds.

The temperature dependences of zero-field-cooling (ZFC) and field-cooling (FC) magnetization ( for Tb₃Co, Ho₃Co, Er₃Co, and Tm₃Co compounds were measured and they are shown in Fig. 50(b), Fig. 50(d), Fig. 50(e), and Fig. 50(f), respectively. The applied field is different and the field is 0.005 T, 0.01 T, 0.05 T, and 0.01 T for Tb₃Co, Ho₃Co, Er₃Co, and Tm₃Co compounds, respectively. The ZFC curves of Gd₃Co and Dy₃Co compounds are shown in Fig. 50(a) and Fig. 50(c), respectively and the applied field is 0.01 T. Although the crystal structure of R ₃Co ( , Tb, Dy, Ho, Er, Tm) compounds is the same, the magnetic properties of them differs from each other a lot. Results show that Gd₃Co is a simple AFM material and it undergoes an AFM to PM transition at K as temperature increases.^[122] According to neutron diffraction investigation by Baranov et al.,^[118] Tb₃Co undergoes two successive transitions at temperatures =72 K (FM to AFM transition) and =82 K (AFM to PM transition), corresponding to the maximum slopes in the M–T curve (Fig. 50(b)). The neutron diffraction investigation suggested that the incommensurate FM state, at for a first-order magnetic phase transition, develops to a modulated AFM state that exists below .^[118] On the other hand, reversible behavior above 50 K in ZFC and FC curves at a magnetic field of 1 T (not shown here) indicates that Tb₃Co experiences a second-order FM to PM phase transition at K.^[119] Arrott plots also confirm its second-order nature. It can be deduced that a small magnetic field, less than 1 T, can induce a metamagnetic transition from AFM to FM state.^[119] It is found that the Ho₃Co compound undergoes two successive magnetic phase transitions at low temperatures (see Fig. 50(d)), corresponding to the two maximal values of M–T curve. As revealed by the neutron diffraction studies,^[125] the Ho₃Co compound exhibits a PM to AFM transition at K, followed by a spin–reorientation transition at K. The anomaly at about 30 K in the M–T curve may be attributed to the existence of a minor amount of impurity phase, because no anomaly of magnetization at the same temperature was observed based on the results reported by Baranov et al.^[126] and Nagai et al. ^[110]

Fig. 50.

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Fig. 50. (color online) (a) The temperature dependence of magnetization curve for Gd3Co compound under the field of 0.01 T. (b) The ZFC and FC curves for Tb3Co compound under the field of 0.005 T. (c) The temperature dependence of magnetization curve for Dy3Co compound under the field of 0.01 T. (d) The ZFC and FC curves for Ho3Co compound under the field of 0.01 T. (e) The ZFC and FC curves for Er3Co compound under the field of 0.05 T. (f) The temperature dependence of magnetization curve for Tm3Co compound under the field of 0.01 T.[49,119–123]

It can be seen from Fig. 50(e) that Er₃Co undergoes a magnetic transition from FM to PM states, and its Curie temperature is determined to be 13 K, which is in good agreement with the value reported by Saito et al.^[127] In addition, it is also observed that there is no hysteresis in ZFC and FC curves above . Below , the difference between the ZFC and FC curves may be related to the domain-wall pinning effect.^[121] It can be seen that there is almost no thermal hysteresis between the ZFC and FC curves for Tm₃Co compound when the temperature increases above 6.5 K from Fig. 50(f). The small bifurcation may be related to the domain-wall pinning effect below 6.5 K. The versus T curve reveals that the Tm₃Co undergoes an AFM to FM transition at K and an FM to PM transition at = 6.5 K, which is corresponding to the minimum value of .^[123] The temperature dependence of specific heat ( ) in a zero-field for Dy₃Co was measured and is shown in Fig. 51. The magnetic property and magnetic transition of Dy₃Co compound will be discussed based on Fig. 50(c) and Fig. 51. The results show that Dy₃Co has two successive magnetic transitions as reported by Baranov et al. previously.^[112] The anomaly at lower temperature may be associated with the phase transition from one AFM to another AFM state.^[112] The transition temperature ( K), determined by the position of the low-temperature peak on the specific heat curve, is fully consistent with the value derived from the –T curve.^[49] The transition at higher temperature corresponds to a change from AFM to PM state as temperature increases. The Neel temperature is determined to be 44 K, which is in good agreement with the previous results.^[112,113] The temperature dependences of the magnetization at different magnetic fields have also been measured. It is observed that the magnetic fields can drive magnetic transition temperatures and to lower temperature and a field-induced metamagnetic transition from AFM to FM state is present below .^[49] When the applied magnetic field is higher than about 1.2 T, the magnetization as a function of temperature for Dy₃Co exhibits step-like behavior above , which corresponds to the FM to PM transition.^[49]

Fig. 51.

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	Fig. 51. (color online) The temperature dependence of heat capacity at zero-field for Dy3Co compound.[49]

The temperature (T) dependences of magnetization (M) data measured in an applied field of 0.01 T under zero-field-cooling (ZFC) and field-cooling (FC) conditions for Er Gd _x Co ( , 0.5, 1, 1.5, 2, 2.5, 3) compounds are shown in Fig. 52. The compound Er₃Co shows an FM-like transition at 13 K due to the thermomagnetic irreversibility between ZFC and FC curves.^[122] It is obvious that the compound with shows two successive magnetic transitions at K which shows the transition from PM to AFM, and K which coincides to the transition from AFM to FM-like order.^[122] However, two transition behaviors seem to disappear instead of a magnetic plateau emerges below for the compounds with . Only a sharp AFM transition is observed and linearly shifts to higher temperature with x.^[122] No thermal hysteresis is observed between ZFC and FC curves around for the compounds with , which is a characteristic of the second-order magnetic transition.^[122]

Fig. 52.

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	Fig. 52. (color online) The temperature dependences of magnetization curves at a field of 0.01 T for Er Gd x Co ( , 0.5, 1, 1.5, 2, 2.5, 3) compounds, respectively.[122]

All of the magnetic transition temperatures of R ₃Co ( , Tb, Dy, Ho, Er, Tm) and Er Gd _x Co ( , 0.5, 1, 1.5, 2, 2.5, 3) compounds have been listed in Table 6. From the table it can be seen that the magnetic property and magnetic transition for R ₃Co compound are different from each other. It undergoes two magnetic transitions as temperature increases for Tb₃Co, Dy₃Co, Ho₃Co, Tm₃Co, and Er Gd Co compounds. It undergoes only one magnetic transition as temperature increases for Gd₃Co, Er₃Co, Er₂Gd₁Co, Er Gd Co, Er₁Gd₂Co, and Er Gd Co compounds. The Curie temperature or Neel temperature decreases as the rare earth atom number increases for R ₃Co ( , Tb, Dy, Ho, Er, Tm) compounds. That is because the spin angular moment quantum number (S) decreases as atom number increases for Lanthanum series and S is positively correlated with order-disorder transition temperature. With the same reason, it is found that the order–disorder transition temperature increases as the content of Gd atoms in Er Gd _x Co ( , 0.5, 1, 1.5, 2, 2.5, 3) compounds.

Table 6.

Table 6.

Table 6. The transition temperatures and magnetocaloric parameters of R 3Co ( , Tb, Dy, Ho, Er, Tm) and Er Gd x Co ( ) compounds. . Materials /K /K MCE_0–2 T MCE_0–5 T Ref. /(J/kg K) /K RC/(J/kg) /(J/kg K) /K RC/(J/kg) Gd3Co – 129 – – – 11.1 60 505.4 [122] Tb3Co 72 82 8.5 26.2 177.1 18 39.3 532 [119] Dy3Co 29 44 5 33.3 111 13.9 45 498 [49] Ho3Co 8 21 6.5 29.3 151.6 14.5 38.5 440 [120] Er3Co – 13/13 12.5/– 16.5/– 163/– 24.5/24.6 24.8/25.1 476/480 [121] Tm3Co 4.5 6.5 11.6 8.1 72 19.9 15.3 230.9 [123] Er Gd Co 13 28 – – – 13.1 44.6 472 [122] Er2Gd1Co – 50 – – – 11.1 68.7 619.5 [122] Er Gd Co – 70.2 – – – 9.8 83.2 624.5 [122] Er1Gd2Co – 90 – – – 10.3 77.4 591.4 [122] Er Gd Co – 110.7 – – – 11 63.2 518.3 [122]

Table 6. The transition temperatures and magnetocaloric parameters of R 3Co ( , Tb, Dy, Ho, Er, Tm) and Er Gd x Co ( ) compounds. .

5.2. The magnetocaloric effects of ₃Co compounds and Gd _x Co compounds

The temperature dependences of for R ₃Co ( , Tb, Dy, Ho, Er, Tm) compounds calculated by using Maxwell relation formula under a magnetic field change of 0 T–2 T and 0 T–5 T are shown in Fig. 53(a) and Fig. 53(b), respectively.

Fig. 53.

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	Fig. 53. (color online) (a) The temperature dependences of magnetic entropy change under a field change of 0 T–2 T for R 3Co ( , Dy, Ho, Er, Tm) compounds, respectively. (b) The temperature dependences of magnetic entropy change under a field change of 0 T–5 T for R 3Co ( , Tb, Dy, Ho, Er, Tm) compounds, respectively.[46]

For Tb₃Co compound, the maximum of is , , , and J/kg K, respectively, for the magnetic field changes of 2, 3, 5, and 7 T.^[119] The large , J/kg K achieved for a low magnetic field change of 2 T, is beneficial to application.^[119] The peaks of at 84 K are just corresponding to the FM to PM phase transition. It is clearly observed that the –T curves of Dy₃Co exhibit two maximum values at 29 K and 44 K, respectively, which corresponds to the two transitions at and . The maximum value of for Dy₃Co is found to be J/kg K around for a magnetic-field change from 0 T to 5 T.^[49] It is apparent that the large in Dy₃Co is associated with the field-induced metamagnetic transition below . The refrigerant capacity (RC) of Dy₃Co compound has been also calculated by numerically integrating the area under the –T curve, with the temperatures at half maximum of the peak used as the integration limits.^[53] The RC value thus obtained for Dy₃Co is 498 J/kg with K (temperature of the cold reservoir) and K (temperature of the hot reservoir) for a field changing from 0 T to 5 T.^[49] The large RC originates from the combined contribution of two successive magnetic transitions, which enlarge the temperature span of large MCE. The full width at half maximum of the peak under a magnetic field of 5 T attains K for Dy₃Co due to the superposition of the AFM-to-AFM and AFM-to-PM phase transitions.^[49]

The values of for Ho₃Co are positive at temperatures below 7 K and at lower magnetic field, but they change to a negative value with the increase of applied field due to the field-induced magnetic transition from AFM to FM state.^[120] The positive (negative MCE) is often observed in magnetocaloric materials with the first order magnetic transitions,^[9] which results from the mixed exchange interaction, and the applied magnetic field leads to a further spin-disordered state near the transition temperature, thereby increasing the entropy.^[10] A small positive value of also indicates a weak AFM coupling in Ho₃Co at low temperatures. It can be seen from Fig. 53 that the peak of the –T curve of Ho₃Co shows a broad distribution due to its successive magnetic transitions. The maximal values of for Ho₃Co are found to be J/kg K and J/kg K for field changes of 0 T–2 T and 0 T–5 T, respectively.^[120] The large results from the high saturation magnetization and is caused by the field-induced AFM-to-FM metamagnetic transition.

For Er₃Co compound, both the peak and the width of depend on the applied magnetic field, and increase obviously with increasing field. The temperature dependences of exhibit a -shape profile as shown usually in magnetic materials with a second-order magnetic transition. A giant , which results from a second-order FM-to-PM transition, is observed in Er₃Co around . The maximal value of for Er₃Co is found to increase monotonously with applied magnetic field and reaches a value of 24.5 J/kg K at 16 K for a field change of 0 T–5 T.^[121] It is interesting to note that a large reversible for Er₃Co is as high as 12.5 J/kg K for a field change of 0 T–2 T, which is beneficial to the practical applications because a magnetic field of 2 T can be supplied by a permanent magnet.^[121] Calculations show that the maximal value of RC for Er₃Co is 476 J/kg for a magnetic field change of 0 T–5 T. In particular, a high value of RC is also obtained to be 163 J/kg for a relatively low field change of 0 T–2 T.^[121] This is attributed to the appreciably large values of obtained at for Er₃Co.

The maximum value of for Tm₃Co compound is found to increase monotonically with the increase of applied magnetic field and reaches a value of 19.9 J/kg K for a magnetic field change from 0 T to 5 T. In particular, the maximum value of is found to be 11.6 J/kg K under a low magnetic change of 2 T, which can be realized by permanent magnet and be advantageous for applications.^[123] The large of Tm₃Co compound origins from the field-induced metamagnetic transition from AFM to FM. Moreover, the does not die out even at temperatures well above , possibly due to the presence of spin fluctuations.^[124] All of the MCE parameters for R ₃Co ( , Tb, Dy, Ho, Er, Tm) compounds are listed in Table 6.