Magnetization, magnetic anisotropy and magnetocaloric effect of the Tb0.2Gd0.8 single crystal in high magnetic fields up to 14 T in region of a phase transition

Accepted Manuscript Magnetization, magnetic anisotropy and magnetocaloric effect of the Tb0.2Gd0.8 single crystal in high magnetic fields up to 14 T in region of a phase transition S.A. Nikitin, T.I. Ivanova, A.I. Zvonov, Yu.S. Koshkid'ko, J. Ćwik, K. Rogacki PII:

S1359-6454(18)30721-3

DOI:

10.1016/j.actamat.2018.09.017

Reference:

AM 14826

To appear in:

Acta Materialia

Received Date: 2 May 2018 Revised Date:

11 September 2018

Accepted Date: 11 September 2018

Please cite this article as: S.A. Nikitin, T.I. Ivanova, A.I. Zvonov, Y.S. Koshkid'ko, J. Ćwik, K. Rogacki, Magnetization, magnetic anisotropy and magnetocaloric effect of the Tb0.2Gd0.8 single crystal in high magnetic fields up to 14 T in region of a phase transition, Acta Materialia (2018), doi: https:// doi.org/10.1016/j.actamat.2018.09.017. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT Magnetization, magnetic anisotropy and magnetocaloric effect of the

Tb0.2Gd0.8 single crystal in high magnetic fields up to 14 T in region of a phase transition

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Lomonosov Moscow State University, Faculty of Physics, 119991 Moscow, Russia

Institute of Low Temperature and Structure Research, PAS, 50-950, Wroclaw, Poland

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S.A. Nikitin1, T.I. Ivanova1, A.I. Zvonov1, Yu.S. Koshkid'ko2, a), J. Ćwik2, K. Rogacki2

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Abstract

The Tb0.2Gd0.8 single crystal magnetization, magnetic anisotropy and adiabatic temperature change ∆T dependencies on magnetic field and temperature have been studied in high magnetic

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fields (up to 14 T). In this work, we analyze experimental data within the framework of thermodynamic Landau theory for second order magnetic phase transitions to describe the relationship between the magnetization, adiabatic temperature change ∆T and magnetic field in

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the region of a phase transition. Moreover, it is proved that the Landau-Ginzburg equations are applicable in the case of high magnetic fields. Furthermore it is discovered that the adiabatic

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temperature change is proportional to the magnetic field with the relation ∆T ~ (µ0H)2/5 in the region of high magnetic fields.

______________ a)

Electronic mail: [email protected] 1

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1. Introduction

The technology of magnetic cooling has been intensively developed for more than two decades to eliminate the shortcomings of conventional technologies of vapor compression cooling. But this technology is still under investigation. Alternative magnetic cooling devices use

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the method of adiabatic demagnetization, this method is based on the magnetocaloric effect (MCE) phenomenon. Recently, intensive experimental studies of new magnetic materials with large MCE values [1-9] were focused on rare earth metals and their alloys. As a result it was

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found that these materials are one of the most promising refrigerants for magnetic cooling. Moreover, the study of magnetic and magnetocaloric properties of the rare earth alloys in order

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to clarify the magnetization processes in rare-earth magnets is nowadays of a great scientific interest.

Many of these rare earth alloys possess helicoidal magnetic structure and giant values of magnetocrystalline anisotropy. The character of the magnetic phase transitions that appear in these materials significantly depends on the magnitude and orientation of the applied magnetic

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field with respect to the main crystallographic axes. Though there is a large number of scientific publications devoted to the MCE study of rare earth alloys, only some materials based on Gd,

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such as Tb-Gd [10-13] and Gd-Dy [14-16], are of particular interest as potential refrigerants for domestic magnetic cooling devices, including cascade magnetic refrigerators [13,17,18].

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Refrigerants for magnetic cascade refrigerators should have high MCE values and possess TC values only a few dozen degrees lower than that of pure Gd [20]. TbxGd1-x alloys satisfy these requirements, as the increase in the Tb concentration is accompanied by a weak decrease of the Curie temperature, from 293 to 232 K. These alloys form a continuous series of solid solutions with hexagonal closed-packed (HCP) crystal structure. TbxGd1-x alloys with x < 0.94 possess high values of magnetic anisotropy with the hard magnetization axis along the c crystallographic axis. The Tb substitution effect on magnetic and magnetocaloric properties of TbxGd1-x alloys

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has been investigated previously by measuring MCE and magnetization in magnetic fields only up to 6 T [10-12]. In this work we present the results of experimental and theoretical studies of the anisotropy of MCE and magnetization for the Tb0.2Gd0.8.single crystal in high magnetic fields up to 14 T. It

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is important to note that the experimental studies were performed on single crystals with high magnetic anisotropy. Furthermore, the combination of giant magnetic anisotropy and high magnetic fields near magnetic phase transition may cause a deviation from the physical models

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and patterns that previously were used for materials with low magnetic anisotropy. Moreover, the results of magnetic measurements in high magnetic fields for rare-earth elements have not

magnetic phase transitions.

2. Experimental

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been analyzed so far in the framework of the thermodynamic Landau theory for a second order

The Tb-Gd single crystals were grown by the method of pulling crystals from the melt

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(Czochralski method) in crucibles made of tungsten-based alloy [19]. Terbium and gadolinium of high purity (99.99 at.%) were used for the growth of the single crystals. The grown single

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crystal had a form of cylindrical bar of a 6-10 mm diameter and length of 40-60 mm. The samples were cut using a diamond saw blades and spark cutting. The crystal structure was

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controlled by methods of metallography and x-ray analysis (Fig.1). The single crystals were oriented by Laue method. The Tb0.2Gd0.8 single crystal has a HCP crystal structure with P63/mmc space group.

The setup for direct MCE measurements was described in [20]. During the ∆T

measurement, the pre-oriented specimen was rigidly fixed in heat insulating sheath and magnetic field was oriented along the main crystallographic axes b or c. Movement of the sample from/to the region of the maximum magnetic field was enabled by a LinMot® actuator [21]. The magnetization measurements have been implemented on vibration magnetometer [22] in the 3

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temperature range 4.2-300 K. A Bitter-type magnet was used to generate magnetic fields up to14 T.

3. Results and discussion

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3.1. Magnetization The temperature dependencies of magnetization for the Tb0.2Gd0.8 single crystal oriented along the b axis and the c axis in magnetic fields equal to 1 T, 5 T and 14 T are shown in Fig.2. It

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is evident from Fig.2 that the magnetization curves demonstrate a classic ferromagnetic behavior with a sharp decline near the Curie temperature in magnetic field parallel to the b axis (easy

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magnetization axis). In the field parallel to the c axis (hard magnetization axis), the character of magnetization curves is more complex and depends on the value of magnetic field. The maximum on the M(T) curve is observed at T ~ 270 K in field µ0H = 1 T. In µ0H = 5 T the maximum on the M(T) curve is strongly blurred and shifted towards low temperatures ~ 200 K. Finally, this maximum disappears in a strong magnetic field equal to 14 T. And above 100 K, the

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M(T) curves for magnetization along the b and c axes almost completely overlap. Such a complex behavior of magnetization is caused by the appearance of the spin reorientation

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transition in Tb0.2Gd0.8 in a magnetic field H || c [1]. The Tb0.2Gd0.8 single crystal possesses a high magnetic anisotropy and therefore the spontaneous magnetization vector MS rotates to the c

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axis by a magnetic field with big difficulty. As was found in the previous studies [11] a high-symmetry phase (MS parallel to the c axis) appears in the high magnetic fields µ0H > µ0Hcr.

3.2. Magnetic anisotropy The energy of magnet in external magnetic field FA for uniaxial crystal with a hexagonal crystal structure is described by the equation: FA = K 2 sin 2 θ + K 4 sin 4 θ − HM s cosθ ,

(1)

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where θ is angle between the magnetization vector MS and the c axis, H is the magnetic field oriented along the hard c axis, K2 and K4 are the anisotropy constants of second and fourth order which determine the magnetic anisotropy in the bc-plane. For the Tb0.2Gd0.8 single crystal, the relationship between magnetic field H and the magnetocrystalline anisotropy constants K2 and K4

 M2  HM s2 − = 2 K 2 + 4 K 4 1 − 2 , M  Ms 

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was obtained by minimizing the function FA with respect to θ angle: (2)

where M is the magnetization per volume unit in the magnetic field oriented along the c axis, MS

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– the spontaneous magnetization vector. The constants K2 and K4 were determined by the

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Sucksmith and Thompson method [23], using the experimental data obtained for Tb0.2Gd0.8 single crystal. The temperature dependencies of the single-ion magnetic anisotropy constants for Tb0.2Gd0.8 single crystal are shown in Fig.3. According to the Callen's single-ion theory [24], at T < TC the magnetic anisotropy constants are related to the magnetization as follows: l ( l +1) 2

,

(3)

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K l (T ) =m K l (0 )

where m = MS(T)/MS(0) is relative magnetization, l - order of constant. A graphical representation of the Kl(T) dependence:

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K l (T ) =K l (0)ml(l+1 )/ 2 ,

(4)

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for the Tb0.2Gd0.8 single crystal at l = 2 and l = 4 (see Fig.3, solid line) indicates that experimental points are in a good agreement with calculated curve. As it is seen, the constant K2, which determines the anisotropy of the Tb0.2Gd0.8 single crystal in the bc-plane is proportional tо m3. These results confirm that the single-ion theory of magnetic anisotropy is suitable for description of the Tb0.2Gd0.8 single crystal magnetic anisotropy in high magnetic fields up to 14 T in temperature region below TC.

3.3. The magnetic field dependence of magnetization near the Curie temperature 5

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The study of the temperature and field dependencies of the magnetization of the rare-earth ferromagnet in the region of the phase transition in high magnetic fields up to 14 T, was one of the objectives of this work. It should be noted that high magnetic field could fundamentally change the magnetization dependencies on magnetic field near the Curie temperature. This type

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of studies on the Tb0.2Gd0.8 single crystal in magnetic fields up to 14 T has not been carried out yet. The possibility of applying the thermodynamic Landau theory [25] to describe processes of magnetization in ferromagnet near the Curie temperature in high magnetic fields will be

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considered in this chapter. M. Kuzmin [26] proposed the approximate equation of thermodynamic state based on thermodynamic Landau theory of the second-order phase

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transitions. According to Landau theory, the thermodynamic potential of the system near second-order transition can be presented as a function of the order parameter. Near the Curie temperature the thermodynamic potential Φ for an uniaxial ferromagnet in a magnetic field H can be written as [26, 27]:

(5)

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1 1 1 Φ = Φ 0 + α M 2 + β M 4 + γ M 6 − MH , 2 4 6

where a specific magnetization M (order parameter) is determined experimentally; thermodynamic coefficients α, β, γ are dependent on temperature and pressure but independent

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of magnetic field H. The relation for magnetization M near the Curie temperature (forced

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magnetization) is obtained by minimization of the thermodynamic potential Φ with respect to M and can be written in the form [26]:

α + βM 2 + γM 4 = H / M .

(6)

If the expression (6) is limited only to the term with the second power of magnetization, the well known Landau-Ginzburd equation is obtained [27]:

α + βM 2 = H / M,

(7)

where α and β are the coefficients determined from the experimental data. The expression (7) allows us to determine the Curie temperature TC accurately using the Belov-Arrott method [27]. 6

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The α and β coefficients, and TC can be determined by plotting the graph of Eq. (7) at different temperatures. For example, the experimental curves H/M(M2) are presented in Fig.4a for the Tb0.2Gd0.8 single crystal in magnetic field parallel to the b axis and with magnitudes not exceeding 4 T. The similar curves for magnetic fields up to 2 T parallel to the c axis are shown in

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Fig.5a. It should be emphasized that K. Belov et al. performed an extensive research of the magnetization dependence on magnetic field for ferromagnetic and ferrimagnetic materials in the area of the forced magnetization. They have shown that the Eq. (7) is applicable in the region of

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TC for magnetic fields ~2 T [27].

However, for the measurements in high magnetic fields up to 14 T the value of

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magnetization near a phase transition becomes very large, so that γM4 >> βM2. Therefore it is necessary to take into account the term with the fourth power of magnetization instead of term with the second power in the Eq. (6). Hence the relation for magnetization process in a high magnetic field can be written as:

(8).

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α + γM 4 = H / M .

The high magnetic field suppresses the magnetic moments fluctuations in the area of ferromagnetic-paramagnetic phase transition. A graphical representation of Eq. (8) in the form of

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H/M(M4) curves (plotted from experimental data) confirms the applicability of the Eq.(8) for the case of high magnetic fields up to 14 T near a ferromagnetic-paramagnetic phase transition (see

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Fig.4b and Fig.5b).

3.4. Magnetocaloric effect It is well known that MCE phenomenon is the change of a magnetic material temperature

during its adiabatic magnetization or demagnetization. The experimental results of MCE measurements for the Tb0.2Gd0.8 single crystal demonstrate a complex dependence on the value and orientation of magnetic field and also on the temperature (see Fig.6). As it is seen on Fig.6, the maximum of the ∆T(T) curve is observed near the Curie temperature in the both cases of 7

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magnetic field H orientation (parallel to c or b axis). The value of MCE measured in magnetic field applied along the b axis is much larger than that in magnetic field applied along the c axis in both cases: µ0H = 6 T and µ0H = 14 T. However, the negative values of MCE can be observed only on the ∆T(µ0H) curves when magnetic field is applied along the c axis (Fig.7) at low

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temperatures. S. Nikitin et al. declared three main contributions to the magnetocaloric effect of the Tb-Gd single crystal [10,11]. The first contribution is caused by the variation of exchange

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interaction energy (forced magnetization). The second one is caused by a change of magnetic anisotropy energy as a result of rotation of the spontaneous magnetization vector MS from the

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basal plane to the c axis by a magnetic field. This fact explains the appearance of negative MCE at low temperatures as mentioned above. The third contribution in the MCE is explained by the exchange interaction between the magnetic sublattices of Tb and Gd. The contribution due to the

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forced magnetization dominates in the TC region.

3.4.1. Anisotropy of magnetocaloric effect Rotating MCE ∆Trot, which is caused by full rotation of the spontaneous magnetization

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vector of the Tb0.2Gd0.8 single crystal in magnetic field applied along hard magnetization axis

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and exceeding the magnetic anisotropy field, is given by expression [29]:

∆Trot ≈ −

T ∆S rot C p ,H

,

(9)

where ∆Srot is rotating magnetic entropy change that can be written as follows [27]:

∆S rot = −

∂Е a . ∂T

(10)

The energy of magnetocrystalline anisotropy Ea for uniaxial ferromagnet in the absence of an external magnetic field can be written as: Ea = K 2 sin 2 θ + K 4 sin 4 θ .

(11) 8

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Since the sines of the θ angles are close to θ angle value in the high magnetic fields directed along c axis (hard magnetization axis of Tb0.2Gd0.8 single crystal) then the entropy change as a result of the Ms rotation from the basal plane to the c axis looks as follows:

 ∂K ∂K  ∆Srot = −  2 + 4  . ∂T   ∂T

(12)

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Fig.8 depicts the entropy change as a function of temperature ∆Srot(T) which was calculated by the formula (12) using anisotropy constants K2 and K4 (see Fig.3).

As it is seen on the ∆Srot(T) curve, a wide blurred maximum (with 220 K width) near

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150 K is clearly visible. The maximum value of ∆Srot reaches 5.4 J/kg·K, which in turn leads to the giant values of the relative cooling power (RCP) - 1210 J/kg. These high values of the RCP

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are due to the rotation of the spontaneous magnetization vector towards the hard magnetization c axis. It should be noted that obtained RCP value is significantly higher than the RCP value caused by forced magnetization.

The theoretical estimation of the maximum ∆Trot value can be done by using the

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temperature dependence of the heat capacity in the formula (9). The heat capacity as a function of T for the Tb0.2Gd0.8 single crystal shows a peak near TC (see the inset in Fig.8). The numerical calculations show that the ∆Trot maximum value is equal to 3.5 K. This value is in good

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agreement with the results of ∆Trot (3.2 K) obtained by the following formula using ∆Tc and ∆Tb

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direct measurements:

∆Trot = ∆Tc − ∆Tb ,

(13)

were ∆Tc and ∆Tb are the experimental values of adiabatic temperature change measured in magnetic field applied along the c axis and b axes respectively in magnetic fields up to 14 T. It is important to note that up to 210 K the ∆Trot values in magnetic fields 6 T and 14 T coincide and reach a significant value of 3.2 K. This coincidence is due to the fact that the anisotropy field for the Tb0.2Gd0.8 single crystal is lower than the external magnetic field. However, below 210 K the magnetic field of 6 T becomes insufficient to rotate the magnetization vector in the direction of 9

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c axis (see inset in the Fig.6). Therefore, a further decrease of ∆Trot values is observed with a temperature decrease in magnetic field equal to 6 T.

3.4.2. The MCE dependence on magnetic field near the Curie temperature

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Taking into account the thermodynamic feature of MCE, the experimental results obtained will now be treated in the framework of the Landau theory of second-order phase transitions. According to this theory the relationship between the magnetization of a ferromagnet and

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the external magnetic field H near the transition temperature TC for low magnetic fields is described by Eq. (7). The adiabatic temperature change ∆T is given by the following equation: M

 ∂H    dM ,  ∂T  M

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T C M,P 0

∆T = − ∫

(14)

where CM,P – heat capacity at constant magnetic field, M is the specific spontaneous magnetization at a given temperature in magnetic field H. Near the Curie temperature β coefficient is weakly dependent on temperature and the H derivative with respect to the

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temperature from the Eq. (7) equals to:

∂H = α1M . ∂T

(15)

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Substituting (15) in the formula (14) and integrating the obtained expression, we obtain:

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∆T =

α1T 2C M ,P

M 2.

(16)

Thus, the MCE value is proportional to the square of magnetization in a region of forced magnetization:

where k =

α 1T 2C M , p

∆T = kM 2 ,

(17)

.

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The ∆T dependence on magnetic field near the Curie temperature can be obtained by substituting expression (17) in Eq.(7). Therefore, the MCE dependence on magnetic field can be expressed as: ∆T ~ ( µ 0 H ) 2 / 3 .

(18)

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The experimental ∆T vs (µ0H)2/3 curves for Tb0.2Gd0.8 single crystal are presented in Fig.9 in magnetic fields not exceeding 2.4 T and directed along the b axis. As it is seen, a straight line fits the experimental points well. From these results it is concluded that ∆T ~ (µ0H)2/3 relation is

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valid in the case of magnetic fields up to 4 T for Tb0.2Gd0.8 single crystal. Previously it was found [30] that maximum of the ∆T dependence on magnetic field also obeys the ∆T ~ (µ0H)2/3

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relation in the range of low magnetic fields (0.1-2 T).

The experimental studies of the Tb0.2Gd0.8 single crystal have shown that the ∆T magnitude is proportional to the square of magnetization in the range of magnetic fields from 4 T to 14 T, i.e. ∆T = kM2. However, the slope of this curve differs from that of the same curve in the range

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of low magnetic fields (not shown). Thus, the ∆T(H) dependence for high magnetic fields can be obtained by substituting the expression ∆T = kM2 into Eq. (8): ∆T ~ ( µ 0 H ) 2 / 5 .

(19)

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The experimental ∆T vs (µ0H)2/5 dependence for the Tb0.2Gd0.8 single crystal in magnetic fields range 4 - 14 T for 290 K is presented in the inset of Fig.9b. This inset also presents the

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∆Tad values for high purity Gd which are taken from ref. [20]. It is seen from these figures that a linear dependence is obeyed. Thus, for the first time it was discovered that ∆T is proportional to (µ0H)2/5 near the Curie temperature in high magnetic fields up to 14 T. It is necessary to emphasize the practical significance of the results of this research. The Tb0.2Gd0.8 single crystal or Tb0.2Gd0.8-based material with magnetic texture can be used as refrigerant for magnetic cooling devices since its MCE maximum reaches practically the same value of 18 K as for Gd single crystal in 14 T [20]. In addition, by gradually changing the Tb concentration it is possible to vary the Curie temperature value, which is especially important for 11

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materials’ applications in cascade refrigerators. We have obtained the ∆T vs H dependence, which allows calculating the numerical value of ∆T in a wide range of magnetic fields.

Conclusion

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The results of comprehensive measurements of magnetization, magnetic anisotropy and adiabatic temperature change as а function of temperature and magnetic field (up to 14 T) for highly anisotropic Tb0.2Gd0.8 single crystal have been analyzed within the framework of the

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thermodynamic Landau theory. The term with sixth power of order parameter in thermodynamic potential should be taken into account to describe the magnetization processes near TC in high

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magnetic fields. It was shown that the magnetization and adiabatic temperature change dependencies on magnetic field are fairly described by the equations derived from the thermodynamic Landau theory.

It was discovered that the magnetic field dependencies of ∆T in high magnetic fields up to 14 T are determined by the relation ∆T ~ (µ0H)2/5 also according to the thermodynamic Landau

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theory. It was shown that the single-ion theory of magnetic anisotropy is suitable for description of the Tb0.2Gd0.8 single crystal magnetic anisotropy in high magnetic fields up to 14 T in

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temperature region below TC. The giant RCP value in the Tb0.2Gd0.8 single crystal due to rotation of the spontaneous magnetization vector toward the hard magnetization c axis was obtained.

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Summarizing the results of the experimental and theoretical investigations of the adiabatic temperature change ∆T for the Tb0.2Gd0.8 single crystal we can conclude that Landau theory of second-order phase transitions is applicable for ∆T description in high magnetic fields. Previously similar investigations have not been performed for rare earth metals and their alloys.

Acknowledgments

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This research was funded from the budget of the Institute of Low Temperature and Structure Research, and was supported by RFBR Grants №№ 16-02-00472, 16-52-00223, and by the

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National Science Center (Poland) under the program SONATA (2016/21/D/ST3/03435).

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Figure

Fig.1. Back Laue X-ray image of the Tb0.2Gd0.8 single crystal.

TE D

250 200

2

M (Am /kg)

1 T, H II c 1 T, H II b 5 T, H II c 5 T, H II b 14 T, H II c 14 T, H II b

EP

150

Tb0.2Gd0.8

AC C

100 50

0

0

50

100

150

200

250

300

T (K)

Fig. 2. Magnetization of the Tb0.2Gd0.8 single crystal as a function of the temperature in magnetic fields µ0H = 1, 5, and 14 T, oriented perpendicular (closed symbols) or parallel (open symbols) to the c axis.

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4

K4

RI PT

3

Kl (MJ/m )

0 -4

SC

-8

K2 0

Tb0.2Gd0.8 50

100

M AN U

-12

150

200

250

T (K)

Fig. 3. Temperature dependences of the magnetic anisotropy constants K2 and K4.

AC C

EP

TE D

Kl(T) = Kl(0)ml(l+1)/2 - solid line.

2

µ0H/Μ (kgT/Am )

ACCEPTED MANUSCRIPT 0.06 310 K

0.04

290 K 280 K

0.02

270 K

260 K

240 K

250 K

(a) 0

5

2

10

3

15 2

20

4

25

RI PT

0.00

2

SC

M (10 A m /kg )

320 K 310 K 300 K

0.10

b)

290 K

M AN U

280 K

2

µ0H/Μ (kgT/Am )

a)

H II b

300 K

270 K 260 K

0.08 0.06

0.02 0.00

0

TE D

0.04

2

4

4

6

8

250 K

240 K

H II b Tb0.2Gd0.8

8

4

8

10

4

12

14

EP

M (10 A m /kg )

Fig. 4. Belov-Arrot plots for the Tb0.2Gd0.8 single crystal determined from the experimental

AC C

magnetization curves in magnetic field parallel to the b axis: a) µ0H/M(M2) in magnetic fields 0 - 4 T, b) µ0H/M(M4) in magnetic fields 4 - 14 T.

2

µ0H/M (kgT/Am )

ACCEPTED MANUSCRIPT 0.08

320 K 310 K

300 K

0.04

280 K

270 K 260 K 250 K

(a) 0

5

2

10

15

3

2

20

4

2

SC

M (10 A m /kg )

320 K 310 K

(b)

300 K

0.10

290 K 280 K

M AN U

2

µ0H/M (kgT/Am )

25

RI PT

0.00

a)

H II c 290 K

270 K

260 K

0.08 0.06

0.02 0.00

0

TE D

0.04

2

4

4

6

8

250 K

H II c Tb0.2Gd0.8

8

4

8

10

4

12

14

EP

M (10 A m /kg )

Fig. 5. Belov-Arrot plots for the Tb0.2Gd0.8 single crystal determined from the experimental

AC C

magnetization curves in magnetic field parallel to the c axis: a) µ0H/M(M2) in magnetic fields 0 - 2 T, b) µ0H/M(M4) in magnetic fields 2 - 14 T.

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18

14 T 3 2

H II c (14 T) H II b (6T)

1 0

160

200

6 3

280

320

Tb0.2Gd0.8 100

H II c (6T)

150

200

T (K)

250

300

M AN U

0 50

240

T (K)

RI PT

9

6T

SC

12

H II b (14 T)

−∆Trot (K)

∆T (K)

15

Fig. 6. Temperature dependences of ∆T for the Tb0.2Gd0.8 single crystal in the magnetic field of 6 and 14 T, parallel to the c-axis (open symbols) and to the b-axis (closed symbols). Insert: ∆Trot

AC C

EP

TE D

vs. T in the magnetic field of 6 and 14 T.

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3.5

Tb0.2Gd0.8

3.0

µ0H II c 287 K

2.0

RI PT

276 K

1.5

270 K 266 K

1.0 0.5

252 K

0.0

227 K

0.5

1.0

1.5

µ0H (T)

2.0

2.5

M AN U

-0.5 0.0

SC

∆T (K)

2.5

3.0

Fig. 7. Magnetic field dependences of ∆T for the Tb0.2Gd0.8 single crystal measured at different

AC C

EP

TE D

temperatures, for the field oriented parallel to the c-axis.

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Tb0.2Gd0.8

4 400 300

2 1

200 150 100 50 0

0

50

100

150

200

250

T(K)

0

50

100

150

250

M AN U

T (K)

200

300

SC

0

250

RI PT

350

3 Ñ (J/(kgK))

∆Srot (J/(kgK))

5

Fig. 8. Temperature dependence of ∆Srot for the Tb0.2Gd0.8 single crystal. Insert: temperature

AC C

EP

TE D

dependence of the specific heat for the Tb0.2Gd0.8 single crystal.

ACCEPTED MANUSCRIPT 6 20

8

Tb0.2Gd0.8

4

290

T=290 K 2

4

6

8

∆T (K)

H||b

12

0 0

3

18

275

10 12 14

12

Tb0.2Gd0.8

290

10

T=290 K

275

2.2

1

1.2

1.6

2.0

µ0H (T)

3.0

270

(T )

266

227

a)

0

2.8

2/ 5

252

227

0.8

2.6

(µ0H)

266 252

0.4

2.4

2/ 5

2

0.0

H||b

14

2.0

270

µ0H (T)

16

Tb0.2Gd0.8

Gd T=294 K

RI PT

∆T (K)

∆T (K)

5 4

20

Tb0.2Gd0.8

16

2.4 0.0

0.4

0.8

1.2

1.6

b)

2.0

(µ0H) (T ) 2/3

SC

2/3

M AN U

Fig. 9. Magnetic field dependences of ∆T for the Tb0.2Gd0.8 single crystal measured at different temperatures, for the field oriented parallel to the b-axis. a) ∆T vs magnetic field for the Tb0.2Gd0.8 single crystal in low magnetic field for different temperatures. Inset ∆T vs magnetic field up to 14 T at T = 290 K. b) ∆T vs µ 0H2/3 for the Tb0.2Gd0.8 single crystal in low magnetic

TE D

field for different temperatures. Inset ∆T vs µ 0H2/5 in magnetic field from 4 to 14 T at T = 290 K

AC C

EP

for Tb0.2Gd0.8 (open symbols) and for high purity Gd at T = 294 K (filled symbols).

AC C

EP

TE

D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT