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Magnetic properties of the induced moment system TmNi2
Journal
116
MAGNETIC A.F. DEUTZ,
PROPERTIES
OF THE INDUCED
MOMENT
H.B. BROM, C.D. WENTWORTH,
Kamerlingh Onnes Laboratorium
der Rijksuniversiteit
of Magnetism
and Magnetic
SYSTEM
W.J. HUISKAMP,
Leiden, Postbw
Materials 78 (1989) 176-182 North-Holland, Amsterdam
TmNi,
L.J. de JONGH
9506, 2300 RA Leiden, The Netherlands
and K.H.J. BUSCHOW Philips Research Laboratories, Received
22 September
5600 JA Eindhoven,
The NetherIan&
1988
We present measurements of the ac susceptibility, magnetization, resistivity and specific heat of the intermetallic Laves phase compound TmNi,. The specific heat of LuNi, is measured separately from 1.7 to 100 K, enabling an analysis of the magnetic specific heat of TmNi, over the temperature range 45 mK - 100 K. This material shows a ferromagnetic transition temperature of 1.1 K and a crystal field singlet ground state. Several sets of Hamiltonian parameters are discussed, taking into account the recently reported distortion from cubic symmetry in TmNi,.
1. Introduction The intermetallic compound TmNi, shows interesting magnetic and structural properties. The crystal field ground state is a singlet and in such a situation there is a competition between the crystal field splitting and the exchange interaction, which tend respectively to prohibit and promote the magnetic ordering [l-3]. The magnetic moment is an induced moment. Previous specific heat experiments showed no anomaly at 1.1 K, the temperature of the magnetic transition [4]. Reported susceptibility experiments above 1.5 K showed no magnetic ordering and were explained in terms of Van Vleck paramagnetism at the lowest temperature [5-81. Mijssbauer experiments by Gubbens et al. [9] revealed a quadrupole splitting at temperatures of 40 K and below. The results of a structural analysis of TmNi, at temperatures of 4.2 and 300 K have confirmed the presence of a distortion from the cubic Laves phase structure [lo]. This paper includes previously published specific heat and susceptibility data [4]. The
specific heat measurements of TmNi, have been extended from 30 to about 100 K and its analysis in terms of a crystal field Hamiltonian is reconsidered. Two possible sets of crystal field parameters are proposed. Finally, resistivity experiments on TmNi, are presented.
2. Experimental The polycrystalline TmNi, sample was prepared by arc-melting in an atmosphere of purified argon, using starting materials of at least 99.9% purity. The sample was wrapped in Ta foil and sealed in an evacuated quartz tube. Vacuum annealing was performed for 1 week at about llOO” C. After annealing the sample was rapidly cooled by quenching in water. The ac susceptibility above 1.2 K was measured via the mutual inductance technique [ll] at a frequency of the oscillating magnetic field of 82.9 Hz. The amplitude of this field was kept below 0.2 mT. The ac susceptibility in the temperature region below 1.2 K was determined in another apparatus, which also utilized the mutual inductance
0304-8853/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
177
A . F Deutz et al. / Induced moment system TmNi 2
technique. For cooling below 1.2 K we used a magnetic cooling salt. For the specific heat the heat pulse technique was applied. Above 1.2 K the temperatures were measured with a calibrated germanium thermometer, while a mechanical heat switch provided the thermal contact. In the temperature region from 45 m K to 1.5 K CMN- and germanium-thermometry were used. A tin heat-switch connected the sample thermally to the magnetic cooling salt. The low-field magnetization (up to 5 T) was measured in a vibrating sample magnetometer. The high field magnetization data were obtained in the pulsed field magnet of the Kamerlingh Onnes Laboratory. In these experiments the sample was ground to a very fine powder (100 /~m) and we did not observe any significant dependence of the signal on all/at, although the duration of the pulse is approximately 20 ms only. The dc resistivity was measured via a standard four-lead technique. The average voltage which was obtained after commuting the leads has been used as a measure for the resistance.
3. Results
The ac susceptibility of TmNi2 is shown in fig. 1. A maximum is visible at a temperature of i
I
'
I
'
+++~+ o x xxxxx~
*
I
l
I
16 #
+ + +
+ ÷
~" E
12
%
+
J
+.
rbJ UGI
# 4
++ am xx f
O0
J
a D xxx
I
2
,
~ I
+ J
4 6 TEMPERATURE (K)
a I 8
i 10
Fig. 1. AC susceptibility versus temperature for T m N i 2. The plusses correspond to the data in zero magnetic field, while the squares and crosses represent the data in a magnetic field of 0.23 and 0.58 T respectively.
1.0 0.5
c~ 0.2
%
o°
u
0.1
o
0.05
OO2
0.05 O.T
Q2
0.5
1.0
2.0
TEMPERATURE
5.0 10.0 20.0
50.0100.0
(K)
Fig. 2. Specific heat, c, versus temperature for T m N i 2. The specific heat of LuNi 2 (fig. 4) has already been subtracted. The drawn line corresponds to the calculated specific heat using the parameters of set 1 in table 2 and eq. (12). The gas constant is denoted by R.
(1.1 + 0 . 1 ) K. The height of this maximum is almost equal to the estimated value of the reciprocal demagnetizing factor. In low applied fields (0.23 and 0.58 T), the maximum broadens considerably, while its height is greatly reduced and the temperature at which it occurs is shifted to a higher value. In the temperature region from about 4 to 250 K the susceptibility shows a Curie-Weiss behavior. A least-squares fit of a straight line to the curve of the inverse susceptibility in that temperature range versus the temperature leads to the values of 19= 1.05 K and C = 9.0 × 10 -5 K m a / m o l (corresponding to the effective moment of a free T m 3÷ ion) for the Curie-Weiss temperature and Curie constant respectively. In figs. 2 and 3 we present the specific heat of TmNi 2. At the lowest temperatures, the specific heat increases with decreasing temperature. Apart from this upturn, no anomaly is present in the whole temperature range from 45 m K to 94 K of our specific heat data. From the specific heat data in various applied fields (fig. 3) it appears that the magnetic field has a large influence on the shape of the curve around 1.1 K, the temperature of the maximum in the susceptibility. The specific heat of the isomorphous compound LuNi 2 (fig. 4) has already been subtracted from the experimental data of TmNi 2 in fig. 2. We expect the specific heat of LuNi 2 to be a good approximation to the contributions from the lattice phonons and the
A.F. Deutz et a L / Induced moment system TmNi 2
178 I
I
I
0.5
I
x xxxxx
Table 1 P a r a m e t e r s a i o b t a i n e d f r o m a least-squares fit of expression (3) t o t h e e x p e r i m e n t a l specific heat c / R of L u N i 2 in t h e t e m p e r a t u r e region 1 . 7 4 - 1 0 3 K (fig. 4)
xXXX~xXXX×x a =o ~a
o°o =ol~ . ° °.o*,~'°.~.+÷~.~+÷+ .~°= a:
..~+°°° + no + °
"o 0.1
xx
**+.o=
0.2
°
x
xx
xx xl x
~ xx xx xx~ xx
0.O5 I
I
Q5
1.0
I
2.0 TEMPERATURE (K)
I
Parameter
Value
a0 aI a2 a3 a4 a5 a6 a7
-0.1651711 + 0.2895353 - 0.9124620 +0.1739811 -0.5168171 + 0.7004915 -0.4610181 +0.1185855
X 10 . 2 x 10- 2 x 10 - 3 x 10 . 3 x 10 . 5 X 10-6 X 10 - 9 x 10-H
10.0
5.0
Fig. 3. Specific heat, c, versus t e m p e r a t u r e f o r T m N i 2 in a p p l i e d fields of 0.23 T (squares) a n d 1.0 T (crosses) respectively. The d a t a in zero-field (plusses; see fig. 2) are i n c l u d e d f o r c o m p a r i s o n . The gas c o n s t a n t is d e n o t e d b y R .
and
conduction electrons in TmNi 2. In the literature the specific heat of LaNi 2 has been reported only for temperatures below 7 K [12,13]. The available values of the specific heat for YNi 2 [14,15] (also included in fig. 4) are somewhat lower than those for LuNi 2- For LuNi 2 we have obtained the values of 7 = ( 5 - 4 + 1 . 8 ) m J / m o l K 2 and @ D = ( 2 4 1 _ 14) K from a least-squares fit of the experimental data in the temperature range 1.74-103 K, using the expressions
for the conduction electron contribution and lattice contribution respectively. The number of atoms per formula unit is denoted by n (n = 3) and D ( O D / T ) is the (normalized) Debye function. The gas constant is represented by R. The result of this fit is also included in fig. 4 and it appears that some deviations between the calculated and experimental specific heat of LuNi 2 occur. Therefore we have made an additional fit to the same experimental data by a polynomial in the temperature T of the form
CE TT
c L = 3nRD(OD/T
(1)
=
7
(2)
)
T
i
f(T)~ i~=oai(--ii-~) .
(3)
I0
1.0
6 nu
O.I
O.OI
~3 ~2
O.OOI ' ~ ....
,~
2'0 ' TEMPERATURE (K)
~o
'
i8o
Fig. 4. Specific heat, c, versus t e m p e r a t u r e f o r L u N i 2 a n d Y N i 2. The plusses represent the L u N i 2 data, w h i l e t h e Y N i 2 d a t a (circles) are t a k e n from [14]. T h e d r a w n line c o r r e s p o n d s to the values of 7 = 5 . 4 mJ/molK 2 and Oo=241 K as o b t a i n e d from a fit o f t h e e x p e r i m e n t a l d a t a to eqs. (1) a n d (2). R is the gas c o n s t a n t .
z
0
5
10
15 FIELD (T)
20
2'.5
30
Fig. 5. M a g n e t i z a t i o n in / ~ a / T m - a t o m versus f i e l d f o r T m N i 2 at 2 K. The d r a w n line is c a l c u l a t e d b y u s i n g the H a m i l t o n i a n p a r a m e t e r s o f s e t I in table 2.
A.F. Deutz et al. / Induced moment system TmNi e 20
1.5
/ '~0 v
%
l.O
,
o.5 /
/
7°°I °°°I /
t ooolj T (K) 50
I OO 150 TEMPERATURE (K)
200
250
Fig. 6. Temperature derivative do/dT of the resistivity# of TmNi2 versus temperature. The measured resistivity, from which the derivative is obtained, versus the temperature T is represented in the inset.
This expression describes the experimental data within the error (estimated to be 2% for all data points). In table 1 we have listed the fitted parameters a i from eq. (3). The magnetization of TmNi 2 is presented in fig. 5. The high-field data were measured in pulsed magnetic fields up to 30 T, the low-field data (up to 5 T) were measured in a vibrating sample magnetometer. In the lowest fields, a rapid increase up to about 4 # B / T m 3+ is observed. In higher fields a gradual increase remains, which at 30 T is already close to the maximum value of 7/~B/Tm 3+. The resistivity data for TmNi 2 are shown in fig. 6. The derivative of the resistivity with respect to the temperature has been calculated from the measured data (in the inset of fig. 6). The most important feature in the resistivity curve is the change in slope at a temperature of about 60 K. Above this temperature the resistivity increases almost linearly with the temperature.
4. Discussion The behavior of the ac susceptibility of TmNi 2 (fig. t) can be explained by the occurrence of ferromagnetic ordering at T~ = (1.1 + 0.1) K [41. The height of the observed maximum in the susceptibility decreases rapidly from the value of the reciprocal demagnetizing factor in zero-field to
179
much lower values in low applied fields. The decrease in the zero-field susceptibility in the ordered region may be ascribed to the combined effect of the anisotropy, taking the polycrystalline composition of our sample into account, and the measuring frequency. This has not been investigated any further. The sensitivity of the specific heat to low applied fields in the temperature region around and below 1.1 K (fig. 3) also supports the explanation in terms of a ferromagnetic ordering at 1.1 K. Next we shall discuss the absence of an anomaly in the specific heat of TmNi 2 around Tc. This behavior of the specific heat can be explained in terms of an induced moment ordering in a singlet ground state system [1-3]. When the crystal field ground state is non-degenerate, a magnetic moment only arises in this state from the admixture of higher lying states due to the interionic exchange interaction. An important parameter in this competition between the exchange and the crystal field is the ratio of the exchange interaction and the crystal field splitting. If this ratio does not exceed a certain threshold, no magnetic ordering will occur. For values of the ratio just above the threshold, the ordering temperature is rather low when compared to the energy splitting of the singlet ground state and the next higher lying state. In such a situation the entropy left at T~ is quite small, leading to the absence of a noticeable anomaly in the specific heat at that temperature. However, in the zero-field ac susceptibility the ordering is clearly visible. It may thus be concluded that TmNi 2 is an example of an ordered singlet ground state system with a transition temperature which is well below the energy splitting of the lowest crystal levels. The upturn in the specific heat at the lowest temperatures (fig. 2) is now explained as the high temperature tail of the Schottky anomaly due to hyperfine interaction between the electroniCand nuclear moments. In the ordered phase, this hyperfine interaction can be represented by ~nv = -AI~(Jz),
(4)
where (J~) is the thermal average of the moment operator (assuming ordering along the z-axis) and A / k = 18.7 m K [16] represents the hyperfine con-
A.F. Deutz et al. /lnduced moment system TmNi 2
180
stant. The nuclear spin I for T m equals 1 / 2 and from the experimental specific heat an energy splitting of about 70 m K between the nuclear substates has been deduced. This corresponds to
+ ~o l( -O Ixl
_ 21064)] ,
(5)
where we used the Stevens operator equivalents O~ according to the definitions of Hutchings [18]. The crystal field parameters are denoted by x and W, while F 4 and F6 are numerical factors. An orthorhombic distortion can be represented by: M'D, o = B°O ° + B202,
(6)
where B ° and B E are related to the size of the distortion. In the case of a trigonal distortion, another set of second order operators may be used
~ D . t = BxyPxy + ByzPyz + BzxP~x,
(7)
where exy=½(JxJy..l_JyJx)=
i02
2
eyz=½(JYJz'~-JzJY)= 2 ' ezy = ½(J~ Jx + JxJ~) = - -'~-- •
'
(8)
(9) (10)
In expression (7) the parameters Bij (i, j = x, y, z) represent the size of the distortion. When an attempt is being made to find a set of Hamiltonian parameters for TmNi2, the conclusions of the structural analysis should be taken into account. This then leads to a total Hamilto-
nian composed of five parts, each corresponding to a different T m site (two cubic, an orthorhombic and two trigonal sites). The total Hamiltonian consists of a linear combination of the eqs. (5), (6) and (7), with the appropriate weight factors. It is obvious that one single specific heat curve (fig. 2) does not contain enough experimental information in order to allow a fitting with 18 parameters. A fit for the case of only 3 sites (neglecting the differences between sites of the same symmetry; total 11 parameters) already resulted in a very close agreement between the experimental and calculated specific heat in the temperature region 0.2-50 K. On the basis of the above discussion we have decided to approximate the T m site symmetry in TmNi 2 for all sites by the same orthorhombic symmetry. The Hamiltonian used in the scanning and fitting of Hamiltonian parameters is •~'~1 =~CEF + J~'D.o-
(11)
Initially the 4-dimensional (x, W, B °, B22) parameter space was scanned by varying the parameters independently in small steps. The scans were restricted to the regions 0.3 < x < 0.8, 0 < W / k < 3.0 K, I B°/kl < 3:0 K and 0 < B 2 / k < 3.0 K. This choice of values is based on a comparison with e.g. TmA12 [19] and isomorphous Pr Laves phase compounds. In each step, the calculated specific heat was compared to the experimental specific heat in the temperature range 0.2-100 K. If a reasonable agreement was reached, then the corresponding parameters were used as start values in a least-squares fitting procedure. In this way five possible parameter sets have been found. In a next step, the molecular terms have to be added to the Hamiltonian:
~ 2 = ~ 1 - X(gJl~n)2(~Jz)Jz - ½(jz)2),
(12)
where ~ 1 is defined in eq. (11). The molecular field constant is denoted by ~, and gj and #s represent the g-factor and Bohr magneton respectively. For each of the five sets found from the scans, the molecular field constant ?~was adjusted for a transition temperature of 1.1 K. From the previous five sets only two sets remained, the other sets resulted in a ground state magnetic
A.F Deutz et aL / Induced moment system TmNi 2
Table 2 Proposed sets of Hamiltonian parameters for TmNi2. The sets have been obtained from fitting the calculated specific heat to the experimental specific heat as discussed in the text, restricting the calculation to only one Tm site with orthorhombic symmetry. The presented sets of parameters lead to a better agreement between the calculated and experimental data, especially at the lowest temperatures, when compared with the earlier reported parameters [4] Set 1 2
x
0.38 0.71
W/k
B~/k
B~/k
X
(K)
(K)
(K)
(1021T/Am2)
0.~ 0.65
-1.2 -1.2
1.0 2.1
9.1±0.2 9.3±0.3
moment which was much too low. The parameters for these sets are listed in table 2. The calculated specific heat is very similar for the two sets (fig. 2). The ground state magnetic moment ( J z ) which is calculated from the parameters in table 2 amounts to 2.4. The calculated expectation value of the second order 020 operator for a temperature of 4.2 K ( ( O ° ) -- 28 for the sets 1 and 2) agrees within 20% with the value which was obtained from the quadrupole splitting in the M~Sssbauer data of Gubbens et al. [9]. A further selection between these sets from a comparison between the calculated and experimental values for the magnetization or susceptibility is not feasible. The calculated curves do not show any significant difference. The calculated magnetization for set 1 is included in fig. 5 and agrees with the experimental data. In view of the afore mentioned limitation to only a single T m site of orthorhombic symmetry, we have to consider the parameter sets in table 2 as a first approximation. The correspondence of the values for W, B2° and X for the two sets should be considered as coincidental. The energy levels schemes for the sets are also very similar. We ascribe this to the fact that we have used experimental data over a very large temperature region. This has probably led to only one possible arrangement o f crystal field levels. For each set the energy splitting between the ground state and the first excited level is 3.0 K, the next higher level having an energy of 14 and 15 K for set 1 and 2, respectively. The overall crystal field splitting amounts to 208 K for set 1 and 253 K for set 2.
181
The similarity of the level schemes excludes selection of a set on the basis of the resistivity data. Despite the approximation, the sets show a reasonable agreement between calculated and experimental specific heat (fig. 2). The deviations at the highest temperatures (above 50 K) can be ascribed partly to the subtraction of the L u N i 2 specific heat from the measured specific heat. The contribution from LuNi 2 amounts to about 90% of the total specific heat at around 100 K, leading to an uncertainty of 20-30% in the data of fig. 2 (assuming an error of 2% in the measured data for T m N i 2 and LuNi2). Another cause of the deviation can be that at these temperatures the specific heat of LuNi 2 can no longer be considered to account for the non-magnetic contributions to the specific heat of T m N i 2. It should be noted that even for the fit with 11 parameters (mentioned above), a similar deviation is present at the highest temperatures. In the temperature region around 1 K the calculated specific heat is too high. This is ascribed to the limitation to a single site. In the case of a calculation which includes more sites, the weight factor reduces the height of the m a x i m u m around 1 K.
5. Conclusion F r o m the experimental data T m N i 2 can be classified as a singlet ground state system which shows an induced m o m e n t ferromagnetic ordering at 1.1 K. An analysis of the present specific heat data for T m N i 2 has provided two possible sets of crystal field parameters (table 2). These parameters have been obtained assuming the presence of only one T m site with orthorhombic symmetry. It was recently shown that more sites are present [10], but the approach presented appears to lead to a reasonable agreement between calcuYation and experiment.
Acknowledgements We like to acknowledge L. Koene, G.M.J. van Soest and A.P. Verloop for their assistance during the experiments.
182
A.F. Deutz et al. / Induced m o m e n t system T m N i 2
This work is part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (Foundation for Fundamental Research on Matter) and was made possible by financial support from the Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek (Netherlands Organization for the Advancement of Pure Research).
References [1] B. Bleaney, Proc. Roy. Soc. A276 (1963) 19. [2] B.R. Cooper, in: Magnetic Properties of Rare Earth Metals, ed. R.J. Elliott (Plenum Press, London, 1972) p. 17. [3] R.J. Birgeneau, in: AlP Conf. Proc. 10, part 2: Magnetism, and magnetic materials, eds. C.D. Graham and J.J Rhyne (American Institute of Physics, New York, 1973) p. 1664. [4] A.F. Deutz, H.B. Brom, H. Deelen, L.J. de Jongh, W.J. Huiskamp and K.H.J. Buschow, Solid State Commun. 60 (1986) 917. [5] J. Farrel and W.E. Wallace, Inorg. Chem. 5 (1966) 105. [6] R.K. Day and J.B. Dunlop, J. Magn. Magn. Mat. 15-18 (1980) 651.
[7] M.R. Ibarra, J.I. Arnaudas, P.A. Algarabel and A. del Moral, J. Magn. Magn. Mat. 46 (1984) 167. [8] P.C.M. Gubbens, A.M. van der Kraan and K.H.J. Buschow, J. Less-Common Metals 111 (1985) 301. [9] P.C.M. Gubbens, A.M. van der Kraan and K.H.J. Buschow, J. Phys. F14 (1984) 2195. [10] A.F. Deutz, R.B. Helmholdt, A.C. Moleman, D.B. de Mooij and K.H.J. Buschow, to be published. [11] A.F. Deutz, R. Hulstman and F.J. Kranenburg, Rev. Sci. Instr., in print. [12] H.H. Neumann, S. Nasu, R.S. Craig, N. Marzouk and W.E. Wallace, J. Phys. Chem. Solids 32 (1971) 2788. [13] A. Sahling, P. Frach and E. Hegenbarth, Phys. Stat. Sol. (b) 112 (1982) 243. [14] D. Bloch, D.L. Camphausen, J. Voiron, J.B. Ayasse, A. Berton and J. Chaussy, C.R. Acad. Sc. Paris 275 (1972) 601. [15] H. Mori, T. Satoh, H. Suzuki and T. Ohtsuka, J. Phys. Soc. Japan 51 (1982) 1785. [16] B. Bleaney, in: Magnetic Properties of Rare Earth Metals, ed. R.J. Elliott (Plenum Press, London, 1972) p. 383. [17] K.R. Lea, M.J.M. Leaks and W.P. Wolf, J. Phys. Chem. Solids 23 (1962) 1381. [18] M.T. Hutchings, in: Solid State Physics, eds. F. Seitz and D. Turnbull (Academic Press, New York, 1964) p. 227. [19] A.F. Deutz, H.B. Brom, W.J. Huiskamp, L.J. de Jongh and K.H.J. Buschow, Solid State Commun. 68 (1988) 803.
116
MAGNETIC A.F. DEUTZ,
PROPERTIES
OF THE INDUCED
MOMENT
H.B. BROM, C.D. WENTWORTH,
Kamerlingh Onnes Laboratorium
der Rijksuniversiteit
of Magnetism
and Magnetic
SYSTEM
W.J. HUISKAMP,
Leiden, Postbw
Materials 78 (1989) 176-182 North-Holland, Amsterdam
TmNi,
L.J. de JONGH
9506, 2300 RA Leiden, The Netherlands
and K.H.J. BUSCHOW Philips Research Laboratories, Received
22 September
5600 JA Eindhoven,
The NetherIan&
1988
We present measurements of the ac susceptibility, magnetization, resistivity and specific heat of the intermetallic Laves phase compound TmNi,. The specific heat of LuNi, is measured separately from 1.7 to 100 K, enabling an analysis of the magnetic specific heat of TmNi, over the temperature range 45 mK - 100 K. This material shows a ferromagnetic transition temperature of 1.1 K and a crystal field singlet ground state. Several sets of Hamiltonian parameters are discussed, taking into account the recently reported distortion from cubic symmetry in TmNi,.
1. Introduction The intermetallic compound TmNi, shows interesting magnetic and structural properties. The crystal field ground state is a singlet and in such a situation there is a competition between the crystal field splitting and the exchange interaction, which tend respectively to prohibit and promote the magnetic ordering [l-3]. The magnetic moment is an induced moment. Previous specific heat experiments showed no anomaly at 1.1 K, the temperature of the magnetic transition [4]. Reported susceptibility experiments above 1.5 K showed no magnetic ordering and were explained in terms of Van Vleck paramagnetism at the lowest temperature [5-81. Mijssbauer experiments by Gubbens et al. [9] revealed a quadrupole splitting at temperatures of 40 K and below. The results of a structural analysis of TmNi, at temperatures of 4.2 and 300 K have confirmed the presence of a distortion from the cubic Laves phase structure [lo]. This paper includes previously published specific heat and susceptibility data [4]. The
specific heat measurements of TmNi, have been extended from 30 to about 100 K and its analysis in terms of a crystal field Hamiltonian is reconsidered. Two possible sets of crystal field parameters are proposed. Finally, resistivity experiments on TmNi, are presented.
2. Experimental The polycrystalline TmNi, sample was prepared by arc-melting in an atmosphere of purified argon, using starting materials of at least 99.9% purity. The sample was wrapped in Ta foil and sealed in an evacuated quartz tube. Vacuum annealing was performed for 1 week at about llOO” C. After annealing the sample was rapidly cooled by quenching in water. The ac susceptibility above 1.2 K was measured via the mutual inductance technique [ll] at a frequency of the oscillating magnetic field of 82.9 Hz. The amplitude of this field was kept below 0.2 mT. The ac susceptibility in the temperature region below 1.2 K was determined in another apparatus, which also utilized the mutual inductance
0304-8853/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
177
A . F Deutz et al. / Induced moment system TmNi 2
technique. For cooling below 1.2 K we used a magnetic cooling salt. For the specific heat the heat pulse technique was applied. Above 1.2 K the temperatures were measured with a calibrated germanium thermometer, while a mechanical heat switch provided the thermal contact. In the temperature region from 45 m K to 1.5 K CMN- and germanium-thermometry were used. A tin heat-switch connected the sample thermally to the magnetic cooling salt. The low-field magnetization (up to 5 T) was measured in a vibrating sample magnetometer. The high field magnetization data were obtained in the pulsed field magnet of the Kamerlingh Onnes Laboratory. In these experiments the sample was ground to a very fine powder (100 /~m) and we did not observe any significant dependence of the signal on all/at, although the duration of the pulse is approximately 20 ms only. The dc resistivity was measured via a standard four-lead technique. The average voltage which was obtained after commuting the leads has been used as a measure for the resistance.
3. Results
The ac susceptibility of TmNi2 is shown in fig. 1. A maximum is visible at a temperature of i
I
'
I
'
+++~+ o x xxxxx~
*
I
l
I
16 #
+ + +
+ ÷
~" E
12
%
+
J
+.
rbJ UGI
# 4
++ am xx f
O0
J
a D xxx
I
2
,
~ I
+ J
4 6 TEMPERATURE (K)
a I 8
i 10
Fig. 1. AC susceptibility versus temperature for T m N i 2. The plusses correspond to the data in zero magnetic field, while the squares and crosses represent the data in a magnetic field of 0.23 and 0.58 T respectively.
1.0 0.5
c~ 0.2
%
o°
u
0.1
o
0.05
OO2
0.05 O.T
Q2
0.5
1.0
2.0
TEMPERATURE
5.0 10.0 20.0
50.0100.0
(K)
Fig. 2. Specific heat, c, versus temperature for T m N i 2. The specific heat of LuNi 2 (fig. 4) has already been subtracted. The drawn line corresponds to the calculated specific heat using the parameters of set 1 in table 2 and eq. (12). The gas constant is denoted by R.
(1.1 + 0 . 1 ) K. The height of this maximum is almost equal to the estimated value of the reciprocal demagnetizing factor. In low applied fields (0.23 and 0.58 T), the maximum broadens considerably, while its height is greatly reduced and the temperature at which it occurs is shifted to a higher value. In the temperature region from about 4 to 250 K the susceptibility shows a Curie-Weiss behavior. A least-squares fit of a straight line to the curve of the inverse susceptibility in that temperature range versus the temperature leads to the values of 19= 1.05 K and C = 9.0 × 10 -5 K m a / m o l (corresponding to the effective moment of a free T m 3÷ ion) for the Curie-Weiss temperature and Curie constant respectively. In figs. 2 and 3 we present the specific heat of TmNi 2. At the lowest temperatures, the specific heat increases with decreasing temperature. Apart from this upturn, no anomaly is present in the whole temperature range from 45 m K to 94 K of our specific heat data. From the specific heat data in various applied fields (fig. 3) it appears that the magnetic field has a large influence on the shape of the curve around 1.1 K, the temperature of the maximum in the susceptibility. The specific heat of the isomorphous compound LuNi 2 (fig. 4) has already been subtracted from the experimental data of TmNi 2 in fig. 2. We expect the specific heat of LuNi 2 to be a good approximation to the contributions from the lattice phonons and the
A.F. Deutz et a L / Induced moment system TmNi 2
178 I
I
I
0.5
I
x xxxxx
Table 1 P a r a m e t e r s a i o b t a i n e d f r o m a least-squares fit of expression (3) t o t h e e x p e r i m e n t a l specific heat c / R of L u N i 2 in t h e t e m p e r a t u r e region 1 . 7 4 - 1 0 3 K (fig. 4)
xXXX~xXXX×x a =o ~a
o°o =ol~ . ° °.o*,~'°.~.+÷~.~+÷+ .~°= a:
..~+°°° + no + °
"o 0.1
xx
**+.o=
0.2
°
x
xx
xx xl x
~ xx xx xx~ xx
0.O5 I
I
Q5
1.0
I
2.0 TEMPERATURE (K)
I
Parameter
Value
a0 aI a2 a3 a4 a5 a6 a7
-0.1651711 + 0.2895353 - 0.9124620 +0.1739811 -0.5168171 + 0.7004915 -0.4610181 +0.1185855
X 10 . 2 x 10- 2 x 10 - 3 x 10 . 3 x 10 . 5 X 10-6 X 10 - 9 x 10-H
10.0
5.0
Fig. 3. Specific heat, c, versus t e m p e r a t u r e f o r T m N i 2 in a p p l i e d fields of 0.23 T (squares) a n d 1.0 T (crosses) respectively. The d a t a in zero-field (plusses; see fig. 2) are i n c l u d e d f o r c o m p a r i s o n . The gas c o n s t a n t is d e n o t e d b y R .
and
conduction electrons in TmNi 2. In the literature the specific heat of LaNi 2 has been reported only for temperatures below 7 K [12,13]. The available values of the specific heat for YNi 2 [14,15] (also included in fig. 4) are somewhat lower than those for LuNi 2- For LuNi 2 we have obtained the values of 7 = ( 5 - 4 + 1 . 8 ) m J / m o l K 2 and @ D = ( 2 4 1 _ 14) K from a least-squares fit of the experimental data in the temperature range 1.74-103 K, using the expressions
for the conduction electron contribution and lattice contribution respectively. The number of atoms per formula unit is denoted by n (n = 3) and D ( O D / T ) is the (normalized) Debye function. The gas constant is represented by R. The result of this fit is also included in fig. 4 and it appears that some deviations between the calculated and experimental specific heat of LuNi 2 occur. Therefore we have made an additional fit to the same experimental data by a polynomial in the temperature T of the form
CE TT
c L = 3nRD(OD/T
(1)
=
7
(2)
)
T
i
f(T)~ i~=oai(--ii-~) .
(3)
I0
1.0
6 nu
O.I
O.OI
~3 ~2
O.OOI ' ~ ....
,~
2'0 ' TEMPERATURE (K)
~o
'
i8o
Fig. 4. Specific heat, c, versus t e m p e r a t u r e f o r L u N i 2 a n d Y N i 2. The plusses represent the L u N i 2 data, w h i l e t h e Y N i 2 d a t a (circles) are t a k e n from [14]. T h e d r a w n line c o r r e s p o n d s to the values of 7 = 5 . 4 mJ/molK 2 and Oo=241 K as o b t a i n e d from a fit o f t h e e x p e r i m e n t a l d a t a to eqs. (1) a n d (2). R is the gas c o n s t a n t .
z
0
5
10
15 FIELD (T)
20
2'.5
30
Fig. 5. M a g n e t i z a t i o n in / ~ a / T m - a t o m versus f i e l d f o r T m N i 2 at 2 K. The d r a w n line is c a l c u l a t e d b y u s i n g the H a m i l t o n i a n p a r a m e t e r s o f s e t I in table 2.
A.F. Deutz et al. / Induced moment system TmNi e 20
1.5
/ '~0 v
%
l.O
,
o.5 /
/
7°°I °°°I /
t ooolj T (K) 50
I OO 150 TEMPERATURE (K)
200
250
Fig. 6. Temperature derivative do/dT of the resistivity# of TmNi2 versus temperature. The measured resistivity, from which the derivative is obtained, versus the temperature T is represented in the inset.
This expression describes the experimental data within the error (estimated to be 2% for all data points). In table 1 we have listed the fitted parameters a i from eq. (3). The magnetization of TmNi 2 is presented in fig. 5. The high-field data were measured in pulsed magnetic fields up to 30 T, the low-field data (up to 5 T) were measured in a vibrating sample magnetometer. In the lowest fields, a rapid increase up to about 4 # B / T m 3+ is observed. In higher fields a gradual increase remains, which at 30 T is already close to the maximum value of 7/~B/Tm 3+. The resistivity data for TmNi 2 are shown in fig. 6. The derivative of the resistivity with respect to the temperature has been calculated from the measured data (in the inset of fig. 6). The most important feature in the resistivity curve is the change in slope at a temperature of about 60 K. Above this temperature the resistivity increases almost linearly with the temperature.
4. Discussion The behavior of the ac susceptibility of TmNi 2 (fig. t) can be explained by the occurrence of ferromagnetic ordering at T~ = (1.1 + 0.1) K [41. The height of the observed maximum in the susceptibility decreases rapidly from the value of the reciprocal demagnetizing factor in zero-field to
179
much lower values in low applied fields. The decrease in the zero-field susceptibility in the ordered region may be ascribed to the combined effect of the anisotropy, taking the polycrystalline composition of our sample into account, and the measuring frequency. This has not been investigated any further. The sensitivity of the specific heat to low applied fields in the temperature region around and below 1.1 K (fig. 3) also supports the explanation in terms of a ferromagnetic ordering at 1.1 K. Next we shall discuss the absence of an anomaly in the specific heat of TmNi 2 around Tc. This behavior of the specific heat can be explained in terms of an induced moment ordering in a singlet ground state system [1-3]. When the crystal field ground state is non-degenerate, a magnetic moment only arises in this state from the admixture of higher lying states due to the interionic exchange interaction. An important parameter in this competition between the exchange and the crystal field is the ratio of the exchange interaction and the crystal field splitting. If this ratio does not exceed a certain threshold, no magnetic ordering will occur. For values of the ratio just above the threshold, the ordering temperature is rather low when compared to the energy splitting of the singlet ground state and the next higher lying state. In such a situation the entropy left at T~ is quite small, leading to the absence of a noticeable anomaly in the specific heat at that temperature. However, in the zero-field ac susceptibility the ordering is clearly visible. It may thus be concluded that TmNi 2 is an example of an ordered singlet ground state system with a transition temperature which is well below the energy splitting of the lowest crystal levels. The upturn in the specific heat at the lowest temperatures (fig. 2) is now explained as the high temperature tail of the Schottky anomaly due to hyperfine interaction between the electroniCand nuclear moments. In the ordered phase, this hyperfine interaction can be represented by ~nv = -AI~(Jz),
(4)
where (J~) is the thermal average of the moment operator (assuming ordering along the z-axis) and A / k = 18.7 m K [16] represents the hyperfine con-
A.F. Deutz et al. /lnduced moment system TmNi 2
180
stant. The nuclear spin I for T m equals 1 / 2 and from the experimental specific heat an energy splitting of about 70 m K between the nuclear substates has been deduced. This corresponds to
+ ~o l( -O Ixl
_ 21064)] ,
(5)
where we used the Stevens operator equivalents O~ according to the definitions of Hutchings [18]. The crystal field parameters are denoted by x and W, while F 4 and F6 are numerical factors. An orthorhombic distortion can be represented by: M'D, o = B°O ° + B202,
(6)
where B ° and B E are related to the size of the distortion. In the case of a trigonal distortion, another set of second order operators may be used
~ D . t = BxyPxy + ByzPyz + BzxP~x,
(7)
where exy=½(JxJy..l_JyJx)=
i02
2
eyz=½(JYJz'~-JzJY)= 2 ' ezy = ½(J~ Jx + JxJ~) = - -'~-- •
'
(8)
(9) (10)
In expression (7) the parameters Bij (i, j = x, y, z) represent the size of the distortion. When an attempt is being made to find a set of Hamiltonian parameters for TmNi2, the conclusions of the structural analysis should be taken into account. This then leads to a total Hamilto-
nian composed of five parts, each corresponding to a different T m site (two cubic, an orthorhombic and two trigonal sites). The total Hamiltonian consists of a linear combination of the eqs. (5), (6) and (7), with the appropriate weight factors. It is obvious that one single specific heat curve (fig. 2) does not contain enough experimental information in order to allow a fitting with 18 parameters. A fit for the case of only 3 sites (neglecting the differences between sites of the same symmetry; total 11 parameters) already resulted in a very close agreement between the experimental and calculated specific heat in the temperature region 0.2-50 K. On the basis of the above discussion we have decided to approximate the T m site symmetry in TmNi 2 for all sites by the same orthorhombic symmetry. The Hamiltonian used in the scanning and fitting of Hamiltonian parameters is •~'~1 =~CEF + J~'D.o-
(11)
Initially the 4-dimensional (x, W, B °, B22) parameter space was scanned by varying the parameters independently in small steps. The scans were restricted to the regions 0.3 < x < 0.8, 0 < W / k < 3.0 K, I B°/kl < 3:0 K and 0 < B 2 / k < 3.0 K. This choice of values is based on a comparison with e.g. TmA12 [19] and isomorphous Pr Laves phase compounds. In each step, the calculated specific heat was compared to the experimental specific heat in the temperature range 0.2-100 K. If a reasonable agreement was reached, then the corresponding parameters were used as start values in a least-squares fitting procedure. In this way five possible parameter sets have been found. In a next step, the molecular terms have to be added to the Hamiltonian:
~ 2 = ~ 1 - X(gJl~n)2(~Jz)Jz - ½(jz)2),
(12)
where ~ 1 is defined in eq. (11). The molecular field constant is denoted by ~, and gj and #s represent the g-factor and Bohr magneton respectively. For each of the five sets found from the scans, the molecular field constant ?~was adjusted for a transition temperature of 1.1 K. From the previous five sets only two sets remained, the other sets resulted in a ground state magnetic
A.F Deutz et aL / Induced moment system TmNi 2
Table 2 Proposed sets of Hamiltonian parameters for TmNi2. The sets have been obtained from fitting the calculated specific heat to the experimental specific heat as discussed in the text, restricting the calculation to only one Tm site with orthorhombic symmetry. The presented sets of parameters lead to a better agreement between the calculated and experimental data, especially at the lowest temperatures, when compared with the earlier reported parameters [4] Set 1 2
x
0.38 0.71
W/k
B~/k
B~/k
X
(K)
(K)
(K)
(1021T/Am2)
0.~ 0.65
-1.2 -1.2
1.0 2.1
9.1±0.2 9.3±0.3
moment which was much too low. The parameters for these sets are listed in table 2. The calculated specific heat is very similar for the two sets (fig. 2). The ground state magnetic moment ( J z ) which is calculated from the parameters in table 2 amounts to 2.4. The calculated expectation value of the second order 020 operator for a temperature of 4.2 K ( ( O ° ) -- 28 for the sets 1 and 2) agrees within 20% with the value which was obtained from the quadrupole splitting in the M~Sssbauer data of Gubbens et al. [9]. A further selection between these sets from a comparison between the calculated and experimental values for the magnetization or susceptibility is not feasible. The calculated curves do not show any significant difference. The calculated magnetization for set 1 is included in fig. 5 and agrees with the experimental data. In view of the afore mentioned limitation to only a single T m site of orthorhombic symmetry, we have to consider the parameter sets in table 2 as a first approximation. The correspondence of the values for W, B2° and X for the two sets should be considered as coincidental. The energy levels schemes for the sets are also very similar. We ascribe this to the fact that we have used experimental data over a very large temperature region. This has probably led to only one possible arrangement o f crystal field levels. For each set the energy splitting between the ground state and the first excited level is 3.0 K, the next higher level having an energy of 14 and 15 K for set 1 and 2, respectively. The overall crystal field splitting amounts to 208 K for set 1 and 253 K for set 2.
181
The similarity of the level schemes excludes selection of a set on the basis of the resistivity data. Despite the approximation, the sets show a reasonable agreement between calculated and experimental specific heat (fig. 2). The deviations at the highest temperatures (above 50 K) can be ascribed partly to the subtraction of the L u N i 2 specific heat from the measured specific heat. The contribution from LuNi 2 amounts to about 90% of the total specific heat at around 100 K, leading to an uncertainty of 20-30% in the data of fig. 2 (assuming an error of 2% in the measured data for T m N i 2 and LuNi2). Another cause of the deviation can be that at these temperatures the specific heat of LuNi 2 can no longer be considered to account for the non-magnetic contributions to the specific heat of T m N i 2. It should be noted that even for the fit with 11 parameters (mentioned above), a similar deviation is present at the highest temperatures. In the temperature region around 1 K the calculated specific heat is too high. This is ascribed to the limitation to a single site. In the case of a calculation which includes more sites, the weight factor reduces the height of the m a x i m u m around 1 K.
5. Conclusion F r o m the experimental data T m N i 2 can be classified as a singlet ground state system which shows an induced m o m e n t ferromagnetic ordering at 1.1 K. An analysis of the present specific heat data for T m N i 2 has provided two possible sets of crystal field parameters (table 2). These parameters have been obtained assuming the presence of only one T m site with orthorhombic symmetry. It was recently shown that more sites are present [10], but the approach presented appears to lead to a reasonable agreement between calcuYation and experiment.
Acknowledgements We like to acknowledge L. Koene, G.M.J. van Soest and A.P. Verloop for their assistance during the experiments.
182
A.F. Deutz et al. / Induced m o m e n t system T m N i 2
This work is part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (Foundation for Fundamental Research on Matter) and was made possible by financial support from the Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek (Netherlands Organization for the Advancement of Pure Research).
References [1] B. Bleaney, Proc. Roy. Soc. A276 (1963) 19. [2] B.R. Cooper, in: Magnetic Properties of Rare Earth Metals, ed. R.J. Elliott (Plenum Press, London, 1972) p. 17. [3] R.J. Birgeneau, in: AlP Conf. Proc. 10, part 2: Magnetism, and magnetic materials, eds. C.D. Graham and J.J Rhyne (American Institute of Physics, New York, 1973) p. 1664. [4] A.F. Deutz, H.B. Brom, H. Deelen, L.J. de Jongh, W.J. Huiskamp and K.H.J. Buschow, Solid State Commun. 60 (1986) 917. [5] J. Farrel and W.E. Wallace, Inorg. Chem. 5 (1966) 105. [6] R.K. Day and J.B. Dunlop, J. Magn. Magn. Mat. 15-18 (1980) 651.
[7] M.R. Ibarra, J.I. Arnaudas, P.A. Algarabel and A. del Moral, J. Magn. Magn. Mat. 46 (1984) 167. [8] P.C.M. Gubbens, A.M. van der Kraan and K.H.J. Buschow, J. Less-Common Metals 111 (1985) 301. [9] P.C.M. Gubbens, A.M. van der Kraan and K.H.J. Buschow, J. Phys. F14 (1984) 2195. [10] A.F. Deutz, R.B. Helmholdt, A.C. Moleman, D.B. de Mooij and K.H.J. Buschow, to be published. [11] A.F. Deutz, R. Hulstman and F.J. Kranenburg, Rev. Sci. Instr., in print. [12] H.H. Neumann, S. Nasu, R.S. Craig, N. Marzouk and W.E. Wallace, J. Phys. Chem. Solids 32 (1971) 2788. [13] A. Sahling, P. Frach and E. Hegenbarth, Phys. Stat. Sol. (b) 112 (1982) 243. [14] D. Bloch, D.L. Camphausen, J. Voiron, J.B. Ayasse, A. Berton and J. Chaussy, C.R. Acad. Sc. Paris 275 (1972) 601. [15] H. Mori, T. Satoh, H. Suzuki and T. Ohtsuka, J. Phys. Soc. Japan 51 (1982) 1785. [16] B. Bleaney, in: Magnetic Properties of Rare Earth Metals, ed. R.J. Elliott (Plenum Press, London, 1972) p. 383. [17] K.R. Lea, M.J.M. Leaks and W.P. Wolf, J. Phys. Chem. Solids 23 (1962) 1381. [18] M.T. Hutchings, in: Solid State Physics, eds. F. Seitz and D. Turnbull (Academic Press, New York, 1964) p. 227. [19] A.F. Deutz, H.B. Brom, W.J. Huiskamp, L.J. de Jongh and K.H.J. Buschow, Solid State Commun. 68 (1988) 803.