27Al and 63Cu NMR studies on intermetallic Kondo compound CeCu3Al2

Physica B 405 (2010) 4691–4695

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27

Al and

63

Cu NMR studies on intermetallic Kondo compound CeCu3Al2

B. Bandyopadhyay , M. Majumder, A. Ghoshray, K. Ghoshray Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700 064, India

a r t i c l e in fo

abstract

Article history: Received 19 July 2010 Accepted 27 August 2010

Detailed 27Al and 63Cu nuclear magnetic resonance (NMR) experiments have been performed in between 4.5 and 295 K on a polycrystalline sample of CeCu3Al2 which has a Kondo temperature  11 K. The NMR spectral features and Knight shift measurements show that in this compound, Cu occupy both c and g lattice sites, while Al occupy only the g sites. From the temperature dependence of Knight shift (K), the hyperfine fields for 63Cu and 27Al at the g sites have been estimated as 2.02(5) and 1:67ð5Þ kOe=mB , respectively, and that for 63Cu at c sites as 3:83ð5Þ kOe=mB . Nuclear spin-lattice relaxation time (T1) measurements indicate that though the relaxation process is governed by fluctuations of magnetic interaction, nuclear quadrupolar interaction also has a significant contribution, especially towards relaxation of 63Cu in both c and g sites. For 27Al, K and T1 have yielded the effective 4f-spin correlation rate 1=t4f . The behavior of its temperature dependence indicates that at low temperatures in CeCu3Al2, the valence state of cerium might not be stable. & 2010 Elsevier B.V. All rights reserved.

Keywords: Intermetallic compound Kondo system Knight shift Spin-lattice relaxation

1. Introduction Rare earth-based intermetallic compounds with localized 4f electron moments are often characterized by a competition between RKKY interaction which facilitates long-range magnetic ordering, and Kondo interaction that screens the localized moments. An easy way of changing or tuning the physical properties of such compounds is the substitution of one or some of the constituent elements. AB5 (A¼rare earths, B¼Co, Ni, and Cu) [1] compounds form hexagonal CaCu5 structure with A in a: 0,0,0; and B in 2c: 7 (1/3,2/3,0), and 3g: 1/2,0,1/2; 0,1/2,1/2; 1/2,1/2,1/2. Many of these compounds show interesting changes in their physical properties when the B element is partially replaced by Al. For example, CeCu5 is a magnetically ordered Kondo compound with Kondo temperature, TK, of  5:5 K and antiferromagnetic transition at  4 K [2]. Al substituted compounds CeCu4Al and CeCu3Al2, both show Kondo behavior with TK 45:5 and  11 K, respectively. However, while in CeCu4Al the possibility at 0.5 K of a magnetic phase transition is not excluded, there is no indication of such a transition in CeCu3Al2 down to 30 mK [2]. SmCu5 is antiferromagnetic below 9 K [3], whereas SmCu3Al2 shows ferromagnetic transition at 12 K [4]. GdCu5 is likely to have a complex magnetic structure [5] and shows antiferromagnetic transition at 12.5 K, but in GdCu3Al2 ferromagnetism is obtained below 20 K [4]. In all these examples, substitution of Cu by Al preserves the hexagonal structure, with Al predominantly in g sites [4].

 Corresponding author. Fax: + 91 33 2337 4637.

E-mail address: [email protected] (B. Bandyopadhyay). 0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.08.065

Though various studies were performed on parent AB5 compounds, there have been very little studies on Al substituted materials, and so far there is no report of microscopic investigations into the effect of Cu/Al substitution. Here we present detailed 27Al and 63Cu nuclear magnetic resonance (NMR) experiments on CeCu3Al2 in between 4.5 and 295 K. This compound shows Curie–Weiss (CW) behavior in magnetic susceptibility at temperatures above 50 K, resistivity maximum at 2.8 K, thermopower maximum below 50 K and Kondo temperature variously estimated as 10–11 K [2,4]. Moreover, an electronic contribution of specific heat (cp/T) showed a broad maximum with the peak value of 490 mJ/mol K2 at 1.9 K, which can be interpreted as the coherence temperature [6] that is characteristic of non-magnetic localized moment systems. Isostructural compound LaCu 3Al2 [4] was used as reference for the analysis of NMR data.

2. Sample preparation and characterization The compounds were prepared from high-purity elements by arc-melting in purified argon atmosphere, and subsequently annealed at 873 K for 7 days. X-ray powder diffraction measurements show the formation of single phase compounds belonging to P 6/mmm space group with a ¼ 0.5260 nm and c ¼0.4177 nm for the Ce compound, and a ¼0.5295 nm and c¼0.4186 nm for the La compound, agreeing well with the published data [4]. Magnetic susceptibility measurements on CeCu3Al2 at temperatures 2r T r 300 K resemble the published results [2,7,4], i.e., CW behavior at temperatures above 50 K with paramagnetic Curie

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1

400

0.95

300

0.9

200

0.85

100

0.8

0 0

50

100

150 T (K)

200

250

27

Cu Ref.

(a)

(b)

0.75 300

Fig. 1. Temperature (T) dependence of inverse of magnetic susceptibility ð1=wÞ (, left y-axis), and wðT þ yp Þ with yp ¼ 41 K (+ , right y-axis) in CeCu3Al2. In 1=w vs T plot, the Curie–Weiss behavior above 50 K is shown by the broken line.

(d) (c)

77.5

temperature ðyp Þ of 41 K and effective moment 2:49mB =Ce ion (mB is the Bohr magneton). Fig. 1 shows the experimental 1=w vs T plot of CeCu3Al2 and the fitting with the CW equation with the above-mentioned parameters. Also shown in this figure is the deviation from CW behavior through the wðT þ yp Þ vs T plot. As temperature is decreased below 50 K, the increase in w first becomes slower than that predicted by CW law, goes through a minimum near 30 K, and then becomes much faster. Such a behavior of magnetic susceptibility suggests the presence of crystal fields, though crystal fields alone cannot explain the result [8]. On the other hand, it appears from the published literature that this kind of susceptibility behavior might have been exhibited earlier in YbAlCu4 [9] and La0.4Yb0.6Cu3Al2 [10], compounds showing signs of Yb valence instability.

3. Nuclear magnetic resonance (NMR) measurements 3.1. NMR spectra and Knight shift 27

63

Al Ref.

χ (T + 41) (mol.K/emu)

1/χ (mol/emu)

4692

Al and 63Cu spectra in CeCu3Al2, narrow at higher temperatures and quite broad at lower temperatures, as shown in Fig. 2, were obtained using appropriate methods [11]. At 295 K, the spectrum of 27Al (spin I¼5/2) in CeCu3Al2 (Fig. 2c) shows a very strong central transition at a resonance frequency higher than that of 27Al reference, and accompanied with comparatively weak but well-separated pairs of satellite transitions on both sides. A theoretical fit of the spectrum involving anisotropic Knight shift and nuclear quadrupolar interaction yields an isotropic shift of 0.118(1)%, quadrupolar coupling constant (QCC) of 0.7 MHz and asymmetry ðZÞ of 0.1. With decrease in temperature, the same spectrum is gradually broadened and its shift increases. Spectral features clearly show that Al in CeCu3Al2 occupy only one type of sites, and it should be the g sites, as indicated by crystallographic studies [4]. LaCu3Al2 yields a similar 27Al resonance line with five transitions and a temperature independent shift of 0.065% in between 80 and 295 K. On the other hand, the spectrum at 295 K of 63Cu (I¼3/2) in CeCu3Al2 consists of two resonance lines, both shifted to high frequency side of the reference position. Unlike 27Al, both the 63Cu

78

78.5 79 Freq (MHz)

79.5

80

Fig. 2. NMR spectra in CeCu3Al2. (a) and (b) are 63Cu spectra in 295 and 160 K, respectively. (c) is 27Al spectrum at 295 K, and (d) is composite 27Al and 63Cu spectrum at 40 K in which the arrow marks the position of overlapping 63Cu(c) line and þ 3=22þ 5=2 line of 27Al. Broken lines joining discreet points in the spectra are guide to the eye. Vertical broken lines mark the positions of 27Al and 63Cu references.

resonance lines are very much broadened due to anisotropic Knight shift and nuclear quadrupolar interaction, and the satellite transitions are not discernible. The shifts are determined at the position of the peak for both the resonance lines. With decrease in temperature, the shift of one of the lines increases, and that of the other decreases, as shown in Fig. 2a and b. Comparing with the behavior of 27Al resonance, we assign the former 63Cu resonance line as 63Cu(g), i.e., arising from Cu in g sites, and the later as 63 Cu(c). At around 40 K, 63Cu(c) resonance line merged with the þ 3=22 þ 5=2 transition line of 27Al (Fig. 2d), and no measurements on 63Cu(c) line were possible below 40 K. LaCu3Al2 yields a single 63Cu resonance line in between 80 and 295 K. The width of this line, when compared with that of 63Cu resonance lines of CeCu3Al2, suggests that it might actually be a superposition of two lines originating from two different Al sites. The difference in their positions is comparable with the width of the lines, so that they appear as a single line with a temperature independent shift of 0.12%. In systems with localized magnetic moments, the total shift (K) is expressed as a sum of a temperature independent part (K0) and a temperature dependent part that is linear with molar magnetic susceptibility ðwÞ due to localized moments, as given by, K ¼ K0 þ ðHhf =N mB Þw:

ð1Þ

In above equation, Hhf is the hyperfine field at the nucleus. Fig. 3 shows the fractional shift of the central transition of 27Al resonance line and the two 63Cu resonance lines as a function of temperature, and also as a function of w with temperature as an implicit parameter. In case of 27Al, K vs w plot is linear in between 35 and 295 K, and Eq. (1) yields K0 ¼0.050(5)% and Hhf ¼ 1:67ð5Þ kOe=mB . A similar plot yield K0 ¼0.075(5)% and Hhf ¼ 2:02ð5Þ kOe=mB for 63Cu(g), for which K vs w seems to be linear almost down to 10 K. In case of 63Cu(c) in between 50 and

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0.006

Al in CeCu3Al2 Cu (g) in CeCu3Al2 Cu (c) in CeCu3Al2 Al in LaCu3Al2 Cu in LaCu3Al2 1/T1 = 0.126T 1/T1 = 0.328T

0.005 0.004

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0.003 0.002 0.001

120 1/T1 (s-1)

Shift (K)

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0.012

0.016

0.02 0

χ (emu/mol)

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100

150

200

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T (K)

0.005

Fig. 4. Spin-lattice relaxation rates (1/T1) for 27Al, 63Cu(g) and 63Cu(c) resonance lines in CeCu3Al2. Also shown are (1/T1) for 27Al and 63Cu in LaCu3Al2 with linear fits to the equations as described in text.

0.004 0.003 Shift (K)

0.002 0.001 0 -0.001 -0.002 -0.003 -0.004 -0.005 0

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150 T (K)

200

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Fig. 3. Shift as a function of temperature (bottom panel) and magnetic susceptibility (top panel) for 27Al ð ,’Þ, 63Cu(g) ðB,~Þ and 63Cu(c) ðn,mÞ resonance lines in CeCu3Al2. The solid and dashed lines are fits to the equations as described in text.

295 K, the linearity of K vs w yields K0 ¼0.205(5)% and Hhf ¼ 3:83ð5Þ kOe=mB . 63 Cu(c) resonance line yields a rather high value of K0 compared to the other two resonance lines originating from g sites; even though it is noted that 63Cu(c) line is very asymmetric, and for this line K measured from the peak position could be much different from the actual isotropic shift. K0 can have contributions from s-contact interaction with conduction electrons and interactions with orbital magnetic moments. Magnetic susceptibility data show that above 50 K, Ce in CeCu3Al2 is trivalent and therefore should be isoelectronic with LaCu3Al2. In that case, the shift due to s-contact interaction and the orbital contributions from Cu-3d and overlapping 5d of La/Ce should be similar in the two compounds. However, more interestingly, Hhf is negative at the c sites. In rare earth intermetallic compounds, Hhf is shown [12] to be positive for ions with J¼L  S and negative for J¼L+ S. While this rule is followed by most f-electron systems, exceptions have been found in quite a few 4f and 5f intermetallic compounds [13–16]. It was suggested by Ohama et al. [17] that in 4f compounds the direct hybridization between 4f and ligand s states is responsible for those exceptions. In CeCu3Al2, the

occurrence of such hybridization is more probable at c site, rather than at g site, because in all AB5 compounds with CaCu5 structure, the distance a–c is always smaller than a–g by about 0.03–0.04 nm [1]. Below 35 K and down to 4.8 K, the K vs w plot of 27Al deviates from linearity and shows a weaker w dependence than that obtained at higher temperatures. The same might have happened in case of 63 Cu(g) below 25 K, but since this linewidth was much larger compared to 27Al linewidth at similar temperatures, the error in shift measurement also became large. And below 12 K, the peak positions could not be determined with any reasonable accuracy. Therefore, even if there was a similar non-linearity in case of 63Cu(g) line, it could not be established. It may be mentioned that at these temperatures, the behavior of w1 vs temperature itself of CeCu3Al2 deviates from Curie–Weiss linearity, while w continues to increase with decreasing temperature. Such a behavior of K vs w has been observed in intermetallic compounds having localized magnetic moments [17,16] and also in impurity Kondo systems [18]. At this point, the reasons for such a behavior in CeCu3Al2 are not clear, though an impurity effect cannot be convincingly denied. The dashed line in the plot (Fig. 3) of K vs w for 27Al in between 4 and 35 K is obtained by a quadratic temperature correction on linear fit as, K ¼ 0:0006þ 0:35w6:7w2 . K obtained from such fitting have been used in the calculations of correlation time mentioned in the following section. 3.2. Nuclear spin-lattice relaxation rate (1/T1) Spin-lattice relaxation time (T1) was measured by selective excitation around a peak position using an appropriate 901 radiofrequency pulse in standard solid-echo pulse sequence 903 t903 -echo where t is the variable delay time during which nuclear magnetization evolves to MðtÞ on its way to thermal equilibrium value M0 following the initial 901 pulse. For the central transition of 27Al spectrum, T1 was estimated by fitting MðtÞ with the equation: 1MðtÞ=M0 ¼ C1 ½0:029expðt=T1 Þ þ 0:178expð6t=T1 Þ þ 0:794expð15t=T1 Þ:

ð2Þ

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Yb undergoes a fluctuation in its valence state. A minimum in temperature dependence of 1=t4f is obtained in intermediate valence compound YbCuAl [22], to which the absolute values of 1=t4f given in energy units as in Fig. 5, may be compared. Such a minimum is obtained also in YbCu5 [23], in which there might be a crossover of 4f electrons state from itinerant fermi liquid state below 4.2 K to the stable Yb3 + state with local moments above 50 K. Cox et al. [24] calculated dynamic susceptibilities and hence magnetic relaxation rates in valence fluctuating Ce compounds. Their calculation showed a minimum in 1=t4f around a characteristic temperature T0 which is the position of the many-body Kondo resonance in the 4f density of states. In the present case, the position of minimum seems to be just below the lower limit of crystal field splitting, which is between 90 and 180 K, estimated from the observation of thermopower maximum [2], and coincides with the position where the magnetic susceptibility deviates from CW behavior. The similarity in the behavior of 1=t4f with that of known valence fluctuating systems and theoretical observation strongly indicates that Ce-4f electron state in CeCu3Al2 might not be stable at low temperatures. Fig. 4 shows that at higher temperatures, 63Cu relaxation rates are considerably higher than 27Al relaxation rates. In fact, the g sites are shared by both Cu and Al, and for a relaxation dominated by magnetic interaction, the ratio, 63(K2T1T)  1/27(K2T1T)  1 should be  1. On the other hand, for a relaxation dominated by nuclear quadrupolar interaction, the ratio of relaxation rates 63 T11 =27 T11 should be close to ð63 eQ =27 eQ Þ2  2:25, where eQ is the nuclear quadrupolar moment. In reference compound LaCu3Al2, localized magnetic moment is absent, and ratio of relaxation rates is  2:6 showing that nuclear quadrupolar relaxation is dominant in this system. In CeCu3Al2 at 295 K, 63(K2T1T)  1/27(K2T1T)  1 is  2 and 63 1 27 1 T1 = T1 is  4, indicating that in this compound Cu nuclei which has a higher nuclear quadrupolar moment compared to Al nuclei, experience a significant contribution in their relaxation from nuclear quadrupolar interaction, over and above that from magnetic interaction. However, Eqs. (2) and (3) that assume dominance of magnetic interaction in the relaxation process, were still used in estimating the T1, as the form of recovery law is rather insensitive to additional relaxation channel, i.e., quadrupolar fluctuations in the presence of predominantly magnetic fluctuations and vice-versa [25]. Due to the above reasons, the absolute values and the behavior of 1=t4f obtained from the NMR data of 63Cu(g) line, shown in Fig. 5, appear to be different from those obtained from 27Al line. However, even if there is no minimum, the fact 1=t4f decreases below about 30 K, as that obtained from 27Al line, may indicate that at low temperatures both the g site nuclei tend to yield similar results. The reasons for the decrease at low temperatures of 1=t4f are not clear at this moment. However, as stated earlier, the electronic specific heat data points to the existence of a coherence temperature at near 2 K, and therefore, the assumption made in derivation of Eq. (3), i.e., no correlation between local moments, may not hold good at low temperatures. A study of NMR in the parent compound CeCu5, a Kondo system that also orders antiferromagnetically, might be useful to gain further insight in these systems.

10

4. Conclusion

For the relaxation of magnetizations corresponding to 63Cu(c) and 63 Cu(g) lines, the following equation was used: 1MðtÞ=M0 ¼ C1 ½0:1expðt=T1 Þ þ0:9expð6t=T1 Þ:

ð3Þ

In above equations, C1 and T1 are the fitting parameters. Fig. 4 shows the temperature dependence of spin-lattice relaxation rates (1/T1) for all the three lines. For 63Cu(c) and 63Cu(g) lines, measurements could be performed only down to 50 and 12 K, respectively, for reasons as stated earlier. With decrease in temperature, (1/T1) decreases slowly until about 50 K, below which (1/T1) decreases rather rapidly. The reference compound LaCu3Al2 does not have localized magnetic moments, and here the relaxation process originates from conduction electrons. The relaxation rate, in this case denoted by 1/T1 K, has a linear relation with temperature. As shown in Fig. 4, 1/T1 K in LaCu3Al2, measured in between 80 K oT o300 K obey Korringa relation: 1/T1 K ¼0.126 T for 27Al, and, 1/T1 K ¼ 0.328 T for 63Cu, and it is assumed that these relations would hold good at lower temperatures too. It might be interesting to compare the 1/T1 behavior of CeCu3Al2 with that of other Kondo systems, such as, CeNiAl4 [19] and YbAgCu4 [20], both having TK  100 K. In both these systems, T1 1 decrease nearly monotonically with decrease in temperature. Relaxation behavior in the two different Al sites in CeNi2Al5 [11], another dense Kondo compound, with TK  4 K and antiferromagnetic transition at 2.6 K, are also different from the present one. On the other hand, the present data have similarities with those of Ce(Cu1  xNix)2 Ge2 [21] systems in which predominance of RKKY interaction and the compensation of localized magnetic moments both are observed at different doping and temperature regimes. The contribution in 1/T1 of local spin fluctuations that is transferred via RKKY interaction from Ce sites to the resonant nuclear sites was estimated as: 1/T1f ¼1/T1  1/T1 K. For the case of uncorrelated spin motion, the effective 4f-spin correlation rate 1=ðt4f Þ can be estimated from 1/T1f as [21,22] ðK 2 T1 TÞ1 w ¼ 2g2N kB t4f :

ð4Þ

Fig. 5 shows the behavior of 1=t4f obtained from shift and relaxation time measurements on 27Al in CeCu3Al2. 1=t4f shows a maximum around 30 K, followed by a broad minimum just above 50 K, and then a sharp increase at higher temperatures. This behavior is very similar to that obtained from Cu nuclear quadrupole resonance (NQR) studies in YbAgCu4 [20], in which

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(h/2π)/τ4f (meV)

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0 0

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Fig. 5. Effective Ce-4f spin correlation rate in energy units h=2ðpÞðt1 4f Þ obtained from 27Al ð Þ and 63Cu(g) ðBÞ resonance lines in CeCu3Al2.

The results of NMR experiments in CeCu3Al2 clearly demonstrate that in this compound, Al occupy only the g sites, and that the hyperfine fields at the g and c sites have opposite signs. The behavior of the temperature dependence of 4f-spin correlation rate indicates the presence of instability in valence state of cerium.

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References [1] J.H. Wernick, S. Geller, Acta Crystallogr. 12 (1959) 662. [2] E. Bauer, D. Gignoux, D. Schmitt, K. Winzer, J. Magn. Magn. Mater. 69 (1987) 158. [3] P. Svoboda, M. Divis, E. Gratz, R. Cerny, L. Dobiasova, Phys. Status Solidi (a) 123 (1991) K149. [4] J.W. Chen, B.K. Wang, J. Alloys Compds. 442 (2007) 146. [5] L.D. Tung, K.H.J. Buschow, J.J.M. Franse, P.E. Brommer, H.G.M. Duijn, E. Bruck, N.P. Thuy, J. Alloys Compds. 269 (1998) 17. [6] C. Lacroix, J. Magn. Magn. Mater. 60 (1986) 145. [7] E. Bauer, G. Amoretti, L.C. Andreani, B. Delley, M. Ellerby, K. McEwen, R. Monnier, E. Pavarini, P. Santini, J. Phys.: Condens. Matter 10 (1998) 4465. [8] B.C. Sales, R. Viswanathan, J. Low Temp. Phys. 23 (1976) 449. [9] D.T. Adroja, S.K. Malik, B.D. Padalia, R. Vijayaraghavan, J. Phys. C Solid State Phys. 20 (1987) L307. [10] D.P. Rojas, L.P. Cardoso, A.A. Coelho, F.G. Gandra, A.N. Medina, Phys. Rev. B 63 (2001) 165114. [11] R. Sarkar, K. Ghoshray, B. Bandyopadhyay, A. Ghoshray, Phys. Rev. B 71 (2005) 104421. [12] V. Jacarino, B.T. Matthias, M. Peter, H. Suhl, J.H. Wernick, Phys. Rev. Lett. 5 (1960) 251.

4695

[13] Y. Kohori, T. Kohara, H. Shibai, Y. Oda, T. Kaneko, Y. Kitaoka, K. Asayama, J. Phys. Soc. Jpn. 56 (1987) 2263. [14] H. Nakamura, Y. Kitaoka, K. Asayama, Y. Onuki, T. Komatsubara, J. Phys. Soc. Jpn. 57 (1988) 2276. [15] K. Kojima, Y. Hukuda, S. Miyata, T. Takabatake, H. Fujii, T. Hihara, J. Magn. Magn. Mater. 104–107 (1987) 49. [16] K. Kojima, A. Harada, T. Takabatake, S. Ogura, K. Hiraoka, Physica B 269 (1999) 249. [17] T. Ohama, H. Yasuoka, D. Mandrus, Z. Fisk, J.L. Smith, J. Phys. Soc. Jpn. 64 (1995) 2628. [18] H. Alloul, Physica 86–88B (1977) 449. [19] K. Ghoshray, B. Bandyopadhyay, A. Ghoshray, Phys. Rev. B 65 (2002) 174412. [20] H. Nakamura, K. Nakajima, Y. Kitaoka, K. Asayama, K. Yoshimura, T. Nitta, Physica B 171 (1990) 238. [21] N. Buttgen, R. Bohmer, A. Krimmel, A. Loidl, Phys. Rev. B 53 (1996) 5557. [22] D.E. MacLaughlin, F.R. de Boer, J. Bijvoet, P.F. de Chatel, W.C.M. Mattens, J. Appl. Phys. 50 (1979) 2094. [23] N. Tsujii, K. Yoshimura, K. Kosuge, Phys. Rev. B 59 (1999) 11813. [24] D.L. Cox, N.E. Bickers, J.W. Wilkins, J. Appl. Phys. 57 (1985) 3166. [25] A. Suter, M. Mali, J. Roos, D. Brinkmann, J. Phys.: Condens. Matter 10 (1998) 5977.