Specific heat and susceptibility of PrNi2

Solid State Communications, Vol. 91, No. 3, pp. 259-263, 1994 Elsevier Science Ltd Printed in Great Britain 0038-1098/94 $7.00 + .00

Pergamon

0038-1098(93)E0271-C

SPECIFIC HEAT AND SUSCEPTIBILITY OF PrNi2 P. Javorsk~ Charles University, Department of Metal Physics, Ke Karlovu 5, 121 16 Prague 2, Czech Republic and G. Schaudy, T. Holubar and G. Hilscher Institut for Experimentalphysik, TU Wien, A-1040 Wien, Austria

(Received 27 October 1993; in revisedform 25 February 1994 by P. Burlet) The specific heat of the cubic PrNi2, YNi2 and LuNi 2 intermetallic compounds has been measured in magnetic fields up to 11 T in the temperature range 1.6-80 K and in zero field up to room temperature. The comparison of the measured susceptibility data with those of the static crystal field calculation yields an overall agreement but allows no decision with respect to a singlet or doublet ground state. Our specific heat data show that the singlet rl is the ground state in PrNi2. The excess specific heat observed below 20K can be attributed to a broadening of the first excited level. PrNi2 BELONGS to the group of RNi2 (R = rare paramagnetic isostructural compounds YNi2 and earth or Y) compounds, which crystallize in the cubic LuNi2. The specific heat measurements were done on ,-~ 1 g MgCu2-type structure (space group 07) [1]. The magnetic properties of PrNi2 have been studied by bulk polycrystalline samples in an adiabatic caloriinelastic neutron scattering (INS) [2, 3], specific heat meter employing the step heating technique in the [4, 5], volume magnetostriction [6] and susceptibility temperature range 1.6-300 K. PrNi2 was measured in [7] measurements. T h e nickel atoms in RNi2 magnetic fields of 0, 3, 6 and 11 T at temperatures up compounds are essentially nonmagnetic, which is to 80 K. The accuracy of Cp data in the temperature consistent with results of band structure calculations region 80-120 K was insufficient in our experimental [8]. Magnetic susceptibility and magnetization setup. The measured Cp(T) dependencies are shown measurements indicated a mixed electronic-nuclear in Figs. 1, 2 and 3. Susceptibility measurement has magnetic phase transition at 0.27K [9]. The cubic been performed with SQUID-magnetometer in the crystal field (CF) removes the degeneracy of the temperature region 4.2-300 K. The following expression was used to analyze the multiplet ground state of the Pr 3+ ions. However, the specific heat data: nature of the ground state is still a controversial matter. The F3 doublet ground state has been Cp = C N --}-Cel "l- Cph "Jr"CSch. (1) concluded from the results of INS measurements [2, 3], but the Fl singlet ground state was not ruled out Here, Cph and Cet represent the phonon and completely. On the other hand, the Fi singlet ground electronic specific heat, Csch is the Schottky specific state was proposed in [4, 5], as inferred from specific heat. The nuclear specific heat CN is important only heat data. The aim of the present work is to study the at very low temperatures (below 1 K) and can be energy spectrum of Pr 3+ ions in PrNi2 by means of neglected above 1.5 K. The influence of the magnetic field was taken into specific heat and susceptibility measurements in external magnetic fields. In order to have the account using the formula [10]: . possibility to subtract the phonon and electronic Cp=(1/1089)(440C[1oo] + 352C[t 10] + 297C[111]), (2) contributions from the specific heat of PrNi2, we have measured also the specific heat of the Pauli where Cil00 ], CIll0] and Ciltl ] is the specific heat 259

260

SPECIFIC HEAT AND SUSCEPTIBILITY OF PrNi2 .

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Vol. 91, No. 3 ,

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tic ph . . . . S(6tW 148K),4:)0°

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r..3=' 2C 0 (3 50

100

150

200

250

300

0

T [K]

0

10

20

30

50

40

60

T [K]

Fig. 1. Specific heat of PrNi 2 and LuNi2; inset: C . / T vs T 2 plot; A, V experimental points, calculated data Cp = 'TT + ~T 3.

Fig. 2. Comparison of measured specific heat and calculated nonmagnetic contributions•

calculated for B in [100], [110] and [111] direction, respectively. To separate the magnetic contribution of the specific heat data one has to subtract the electronic and lattice contributions. These can be obtained by analyzing the specific heat of the nonmagnetic compounds YNi 2 and LuNi 2. We assume that the observed specific heat of YNi2 and LuNi2 consists of the lattice and electronic contributions only. The electronic part due to conduction electrons can be expressed as

where R is the gas constant, 0D and 0e; are the characteristic Debye and Einstein temperatures, respectively• The analysis of the specific heat data of YNi2 and LuNi2 shows that only one characteristic Einstein temperature 0e can be used, with sufficient accuracy, for all six optic branches• From a C p / T ( T 2) plot (inset in Fig. l) we have obtained the values of '7 = 6.0mJmol -l K -2 and 7 = 6 . 4 m J m o l - l K -2 for YNi 2 and LuNi2, respectively, which are in agreement with the values of'7 = 6.0 mJ mol -l K -2 [5] and '7 = 5.2 mJ mo1-1 K -2 [11] determined for YNi2 from the specific heat measurements• The values of 0D and 0e were then determined by the simplex least square method comparing equation (4) with the experimental data up to 60 K. The results are given in Table 1. The specific heat calculated using these values is shown in Figs. 1 and 2. To obtain the parameters '7, 0D and 0e for PrNi2 (or other RNi2 compounds), we propose the following interpolation scheme:

Cet = "yT.

(3)

In a crystalline solid the phonon spectrum consists of 3 acoustic and 3 ( p - l) optic branches (p is the number of atoms per primitive cell; in RNi2 compounds p -- 3). In order to obtain a satisfactory expression for Cph, we applied the Debye model to 3 acoustic branches, and the Einstein model to 6 optic branches of the phonon spectrum. The nonmagnetic specific heat is then expressed by c = Cet + Cph

Oo/T I x4eX = 'TT + 9R(T/0D) 3

(~ex _ 1) dx

(4)

0 6 ~Oe~/T + R Z(Oei/T)2 ET/ i=1 (e0 T 1)2'

(a) The "7 value is taken to be 6.0mJmol -l K -2. (b) The characteristic Debye temperature 0D is calculated from that of LuNi2 using equation [12]:

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0o(RN12)- 0D(R Nl2)

Table 1. Proposed values of'7, 0o, and Oe R

7 [mJmol-I K-2]

0D [K]

0e [K]

Y Pr Lu

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182 160 148

265 247 235

(M~)

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~

3/2 1/3 )

°~5 )

The value of 00 for YNiz can be obtained also directly by fitting the experimental data of YNi2 to equation (4). The difference between this value and the value of 0D obtained for YNi2 from the equation (5) is less than 1%. (c) A linear dependence between the value of 0e and the molar mass is assumed•

Vol. 91, No. 3 20

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SPECIFIC HEAT AND SUSCEPTIBILITY OF PrNi2 ,-

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T [K] Fig. 3. Low temperature specific heat of PrNi2; calculated curves: - x = 1.0, w = -3.2 K (singlet ground state), - - - x = 0.64, w = -1.973 K (doublet ground state). Analogous approximations have been successfully used for RNiAI compounds [13]. The parameters 3', OD and 0e derived from our model are given in Table 1. Our value of 0z~ for YNi2 cannot be compared with the value (OD = 264 K) given in [5] because the latter has been obtained from the C / T vs T 2 plot in the temperature range 1.5K up to 10K which implies that the Debye model has been applied to all nine phonon branches. This is consistent with our data yielding a good description of Cp(T) up to 12K. However, a single Debye function (with, e.g. OD = 300K) cannot account for the data up to room temperature yielding substantial deviations between 20 and 250K. This is also the case for LuNi 2. Thus, we use the combination of the Debye and Einstein functions applied to the three acoustic and six optic branches, respectively. The cubic CF splits the 3H4 ground state multiplet of the Pr 3+ ions into four levels (two triplets, one doublet, one singlet) [14]. This splitting can be described in the L L W formalism [14] by the Hamiltonian:

at 72 K (F1 singlet) and 97 K (I"5 triplet). On the other hand, the F~ singlet is claimed to be the ground state [4, 5], inferred from specific heat. The F4 triplet at 45 K and F3 doublet at 77 K (which corresponds to the CF parameters x to be close to unity and w m 3.2 K) are proposed in [4]. In the former ease, the F3 doublet ground state should be split in the magnetic field, and a Schottky anomaly around T = 2 K should be observed in the specific heat (inset in Fig. 3). Our measurements show no such anomaly (Fig. 3). The Cp(T) dependence in zero field is almost linear below 30 K. The decrease of the low temperature specific heat (below 6 K) in the external magnetic fields indicate that the excited levels are shifted to higher energies. The calculation with the Fl singlet ground state (x = 1.0, w = -3.2 K) is also shown in Fig. 3. The calculated specific heat in zero field exceeds the measured data above 10K. Better agreement with the experiment can be obtained by taking into account the broadening of the first excited level (the F4 triplet), which shifts a part of the magnetic entropy to the lower temperatures. Such a broadening can result from the exchange interaction between Pr 3+ ions and is also discussed in [3, 5]. In ease of the F3 doublet ground state, the calculated zero field Cp(T) is in rather good agreement with the experiment but only in the limited regime up to 10 K and deviates substantially above. A satisfactory agreement at high temperatures cannot be achieved by taking the broadening of the levels into account as will be shown below. With regard to the incompatibility of the high field C1, data with the doublet ground state the agreement at zero field and low temperatures is thus presumed to be accidental.

15 E

t d°s ~

1 -Ixl

o

(6)

where O~' are the Stevens equivalent operators, w determines the energy sealing of the CF splitting, x weights the potential terms of the fourth and sixth order, F(4) and F(6) are the numerical factors. The INS results were interpreted in terms of the F3 doublet ground state [2, 3]. The CF parameters x = 0.64 and w = -1.973 K proposed in [2] lead to the first excited state I"4 triplet at 30 K, and the next levels

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o

(0° + 504) + F(6------~

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261

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Fig. 4. Experimental and calculated specific heat of PrNi2 in zero magnetic field; corresponding energy level schemes are schematically shown in the inset (calc A: singlet ground state, calc B: doublet ground state).

262

Vol. 91, No. 3

SPECIFIC HEAT A N D SUSCEPTIBILITY OF PrNi2

In order to account for dynamic and/or exchange effects additionally to the static CF splitting we incorporate the broadening of the first excited level [see Fig. 4(a)] similar to that proposed in [5]. The lower (upper) band has the width 61 (62) and contains 15 equally-spaced levels. The density of states is linearly decreasing in the lowest (highest) third of this band. The calculations with bands containing more than 15 levels gave the same results. The degeneracy of the 1"4 triplet, of course, has been considered in the calculation. The best agreement with the experimental data in zero magnetic field was obtained for 81 = 35 K. The influence of the width 62 on the specific heat below 20K is negligible. The calculated specific heat for 61 = d~2= 35 K is shown in Fig. 4 as calc A. The possibility of the splitting of the 1-'3 doublet ground state in zero field due to the Jahn-Teller effect was also taken into account. However, no value A of such a splitting (and broadening of the excited levels) can give better agreement with experimental data than for the case of the Pl singlet ground state. For illustration, the Cp (T) dependence calculated using the energy level scheme given in [2] with a splitted doublet ground state (A = 20K) and broadened excited levels ( 6 1 = 6 2 = 2 0 K , 6~ = 6 ~ = 2 0 K ) is shown in Fig. 4 as calc B. The magnetic contribution to the heat capacity after subtracting the phonon (On = 160 K, 0e = 247K) and electronic (7 = 6 .0mJmol-I K-2) contributions is shown in Fig. 5. The corresponding magnetic entropy S is shown in the inset. The overall slope of S(T) is hardly changed by external field up to 11 T and only a small part of the entropy is shifted to higher temperatures. This is in agreement with the F1 singlet ground state while for the doublet ground state a significant part of the entropy should appear 12

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T [~] Fig. 6. Susceptibility of PrNi2. at low temperatures due to the Zeeman splitting (see Fig. 3). At 80 K the magnetic entropy exceeds the theoretical high temperature value for both the doublet (Rln4.5) and singlet (Rln9) ground state and increases further up to 300 K (see Fig. 1). This discrepancy at high temperatures might arise from the subtraction of an improper phonon contribution but cannot be sufficiently removed by using either Cp(T) of LuNi2 or YNi 2. The strong coupling of the crystalline field excitation to a low energy phonon proposed in [15] may be another source of the additional heat capacity at high temperatures. The temperature dependence of the susceptibility between 15 K and 300 K follows approximately the Curie law for the free Pr 3+ ion and is compared with the calculation for the Fl singlet (x = 1.0, w = --3.2 K) and F3 doublet (x = 0.64, w -- - 1.973 K) ground state in Fig. 6. The calculation demonstrates that only at low temperatures up to 15K a discernible difference between susceptibility data with the Fl or the F3 as the ground state can be resolved while for higher temperatures both curves merge one another. The overall enhancement of the experimental susceptibility with respect to the calculation has been observed also in [7] and was discussed in terms of an exchange coupling using a temperature dependent molecular field constant A (X -l = Xc~ - A). The phenomenological correction yields at T = 4.8 K A values of 45 and 57 kOe/#s for the singlet and doublet ground state, respectively, which is comparable with A = 68 kOe/# B at 3.8 K for the doublet ground state [7]. This analysis indicates that the static CF splitting provides a reasonable overall description of the experimental data but allows no decision of whether the ground state is the F1 singlet or the F3 doublet. By analogy with the specific heat analysis we presume that the consideration of the level broadening will improve the agreement between the susceptibility calculation

Vol. 91, No. 3

SPECIFIC HEAT AND SUSCEPTIBILITY OF PrNi 2

and the experiment. This, however, needs a more sophisticated calculation which is in progress. In conclusion, specific heat measurements in external fields up to 11 T provide significant evidence for a CF splitting in PrNi2 with the F1 singlet as the ground state since in case of the F3 doublet ground state the expected Schottky anomaly due to the Zeeman splitting of the doublet is not observed experimentally. Furthermore, the anomalous low temperature behaviour of the heat capacity can be successfully explained by incorporating the broadening of the first excited levels to the static CF splitting.

Acknowledgements - - This work was supported by the Austrian Science Foundation Funds under project No. 7620 and Charles University Grant Agency (Grant No. 288 and 312). The authors thank M. Divig (Charles University, Prague) for fruitful discussion and R. Resel (TU Wien) for the sample preparation and X-ray diffraction.

3. 4.

5. 6. 7. 8. 9. 10. 11. 12.

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