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Effect of the crystalline electric field on the Néel temperatures of RCu2 compounds
Journal of Magnetism and Magnetic Materials 224 (2001) 30}32
E!ect of the crystalline electric "eld on the NeH el temperatures of RCu compounds Nguyen Hoang Luong *, J.J.M. Franse, Nguyen Hoang Hai
Faculty of Physics, Center for Materials Science, National University of Hanoi, 334 Nguyen Trai, Hanoi, Viet Nam Van der Waals-Zeeman Laboratorium, Universiteit van Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, Netherlands
Abstract Values for the NeH el temperature of the RCu compounds (R"Tb}Tm) have been calculated using a molecular-"eld model including crystalline-electric-"eld e!ect as presented by Noakes and Shenoy. The calculated results show that the unusual behavior, at the NeH el temperatures, of these compounds can be explained on the basis of this model. 2001 Published by Elsevier Science B.V. Keywords: Crystal-"eld e!ect; NeH el temperature; Rare-earth intermetallic compounds
The RCu (R"rare-earth) compounds crystal lize in the orthorhombic CeCu structure. The magnetic properties of these compounds were "rst studied by Sherwood et al. [1] as early as in 1964. Most of the RCu order antiferromagnetically with values for the NeH el temperature, ¹ , below 55 K. , Hashimoto et al. [2] performed magnetization and magnetic susceptibility measurements on singlecrystalline RCu samples. A large magnetic anisot ropy was found to be present. The magnetic properties of the RCu compounds have been extensively studied in recent years (see a review by Luong and Franse [3] and references therein). One of the features of the RCu compounds is that the values of the NeH el temperature for these compounds are not simply proportional to the de Gennes factor (g !1)J(J#1) and reach a maxH * Corresponding author. Fax: #84-4-858-94-96. E-mail address: [email protected] (N.H. Luong).
imum for TbCu . This fact suggests that the Ruder man}Kittel}Kasuya}Yosida (RKKY) interaction alone is not su$cient to fully understand the magnetic interactions in the RCu compounds. In spite of a substantial progress in the study of the magnetic properties of the RCu compounds, the above-mentioned exception to the de Gennes rule remained unexplained. In this paper, we report on our calculations that have been performed in order to explain the anomalous behavior, at the NeH el temperature, of the RCu compounds (R"Tb}Tm). Our calculations are based on the model of Noakes and Shenoy [4]. When considering only the exchange Hamiltonian, the de Gennes rule can be derived in the simple molecular-"eld model: the rare-earth-atom Hamiltonian H "!2C(g !1)J 1J 2 (1) H X X (where the z-axis is de"ned to be the orderedmoment axis and C the exchange parameter) leads to an implicit equation for 1J 2, and the ordering X
0304-8853/01/$ - see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 1 3 9 0 - 1
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N.H. Luong et al. / Journal of Magnetism and Magnetic Materials 224 (2001) 30}32
temperature is obtained from the small 1J 2 limit X as ¹ "2C (g !1)J(J#1)/3. (2) H When the crystalline-electric-"eld (CEF) e!ects are signi"cant, the de Gennes behavior is not to be expected. In this case, CEF Hamiltonian should be added to the H . This leads to the following expression for the ordering temperature: ¹ "2C (g !1)1J(¹ )2 , (3) + H X + !#$ where 1J(¹ )2 is the expectation value of X + !#$ J under the in#uence of CEF Hamiltonian alone X at the temperature ¹ . The exchange parameter, C, + can be evaluated from the ordering temperature of the Gd compound when modeling a series of rareearth compounds, because Gd, an ¸"0 ion, is essentially una!ected by CEF. For calculating values of the NeH el temperatures, ¹ , of the RCu compounds, expression (3) is used , in which ¹ stands for ¹ . For evaluating C in + , these compounds, we take ¹ (GdCu )"¹ " , 41 K [1,5,6]. In the coordinate system of b"z, c"x and a"y, the orthorhombic CEF Hamiltonian of a CeCu -type of structure is given by H "B O #B O #B O #B O !#$ #B O #B O #B O #B O #B O , (4) where BK are the CEF parameters and OK the L L Stevens equivalent operators. As pointed out by Luong and Franse [3], information about the crystalline-electric-"eld interaction in RCu is not complete. Until now, the full set of CEF parameters is available only for NdCu , ErCu and TmCu . Nevertheless, the two lowest-order CEF parameters B and B have been derived for most of the RCu compounds (see Ref. [3] and references therein). In our calculations, we "rst used the two lowest-order terms in the CEF Hamiltonian. Values for B and B were taken for TbCu , DyCu and HoCu from Ref. [2], for ErCu from Ref. [7] and for TmCu from Ref. [8]. In TbCu , DyCu and HoCu the magnetic moments lie along the a-axis, whereas in ErCu and TmCu the magnetic moments are oriented
Table 1 Values for the crystalline-electric-"eld parameters and the NeH el temperatures in the heavy RCu compounds R
B (K)
B (K)
Gd
¹ exp (K) ,
¹ cal (K) ,
40 [5] 41 [1] 42 [6]
Tb
1.23 [2]
1.23 [2]
54 [1] 53.5 [2] 48.5 [5] 48 [14]
46.4
Dy
0.43 [2]
0.72 [2]
24 [1] 31.4 [2] 26.7 [5] 27 [7]
28.7
Ho
0.14 [2]
0.12 [2]
9 [1] 9.8 [2] 9.6 [5] 11 [19]
17.2
Er
!0.28 [7]
!0.22 [7]
11 [1] 13.5 [2] 11.5 [5]
9.1, 11.7
Tm
!0.94 [8]
!1.23 [8]
6.3 [20]
4.3, 6.7
NeH el temperature predicted by the full CEF Hamiltonian with the BK sets from Refs. [7,8] for ErCu and TmCu , respectively. L
along the b-direction [2,9}17]. The ¹ values for , ErCu and TmCu were calculated directly using the CEF Hamiltonian (4) with only the two lowestorder terms. For the RCu compounds with R"Tb, Dy and Ho, we used the CEF Hamiltonian transformed in the new coordinate system of a"z, b"x, c"y as follows [18]: H "()(!B !B )O #()(3B !B )O . (5) !#$ The calculated values of ¹ are compared with , experimental data in Table 1 and also in Fig. 1. As one can see from Table 1 and Fig. 1, addition of CEF interaction enhances ¹ over the de Gennes , values in the RCu compounds. Moreover, calcu lations predict that TbCu has a highest NeH el tem perature, in good agreement with experiments. The calculated values for the NeH el temperatures across the series are in good agreement with experimental ones. Calculations gave the value of ¹ for HoCu ,
N.H. Luong et al. / Journal of Magnetism and Magnetic Materials 224 (2001) 30}32
32
References
Fig. 1. Comparison of experimental and calculated NeH el temperatures for the RCu compounds. The open circles represent experimental data. The solid circles (solid line) represent calculations using a CEF Hamiltonian with two lowest-order terms, and the solid squares * calculations with the full CEF Hamiltonian as discussed in the text. The dashed line represents the de Gennes rule.
to be somewhat higher than the one obtained from experiments. We tried also to derive the values for the NeH el temperature of ErCu and TmCu with the full BK set taken from Refs. [7,8], respectively. The L results of these calculations using the full CEF Hamiltonian (4) are also given in Table 1 and in Fig. 1. As it can be seen, the use of the full CEF Hamiltonian gives better results than the use of the two lowest-order CEF terms only. In conclusion, the magnetic ordering temperatures in the RCu compounds can be explained by a combination of the RKKY interaction and crystalline-electric-"eld e!ects.
[1] R.C. Sherwood, H.J. Williams, J.H. Wernick, J. Appl. Phys. 35 (1964) 1049. [2] Y. Hashimoto, H. Fujii, H. Fujiwara, T. Okamoto, J. Phys. Soc. Japan 47 (1979) 67. [3] N.H. Luong, J.J.M. Franse, in: K.H.J. Buschow (Ed.), Handbook of Magnetic Materials, Vol. 8, North-Holland, Amsterdam, 1995, p. 415. [4] D.R. Noakes, G.K. Shenoy, Phys. Lett. 91A (1982) 35. [5] N.H. Luong, J.J.M. Franse, T.D. Hien, J. Magn. Magn. Mater. 50 (1985) 153. [6] M.K. Borombaev, R.Z. Levitin, A.S. Markosyan, V.A. Reimer, E.V. Sinitsyn, Z. Smetana, Zh. Eksp. Teor. Fiz. 93 (1987) 1517 (Eng. Transl., Sov. Phys. JETP 66 (1987) 866). [7] P.C.M. Gubbens, K.H.J. Buschow, M. Divis, J. Lange, M. Loewenhaupt, J. Magn. Magn. Mater. 98 (1991) 141. [8] P.C.M. Gubbens, K.H.J. Buschow, M. Divis, M. Heidelmann, M. Loewenhaupt, J. Magn. Magn. Mater. 104}107 (1992) 1283. [9] T.O. Brun, G.P. Felcher, J.S. Kouvel, AIP Conf. Proc. 5 (1971) 1376. [10] V. Sima, Z. Smetana, B. Lebech, E. Gratz, J. Magn. Magn. Mater. 54}57 (1986) 1357. [11] B. Lebech, Z. Smetana, V. Sima, J. Magn. Magn. Mater. 70 (1987) 97. [12] Z. Smetana, V. Sima, B. Lebech, J. Magn. Magn. Mater. 59 (1986) 145. [13] Z. Smetana, V. Sima, A.V. Andreev, Phys. Stat. Sol. B 121 (1984) K131. [14] Z. Smetana, V. Sima, Czech. J. Phys. B 35 (1985) 1232. [15] Y. Hashimoto, H. Kawano, H. Yoshizawa, S. Kawano, T. Shigeoka, J. Magn. Magn. Mater. 140}144 (1995) 1131. [16] V. Sima, M. Divis, P. Svoboda, Z. Smetana, Z. Zajac, J. Bischof, J. Phys.: Condens. Matter 1 (1989) 10153. [17] M. Heidelmann, B. Lebech, Z. Smetana, M. Loewenhaupt, J. Phys.: Condens. Matter 4 (1992) 8773. [18] M. Divis, S. Zajac, V. Sima, Z. Smetana, J. Magn. Magn. Mater. 68 (1987) 253. [19] R.R. Birss, R.V. Houldsworth, D.G. Lord, J. Magn. Magn. Mater. 15}18 (1980) 917. [20] Z. Smetana, V. Sima, J. Bischof, P. Svoboda, S. Zajac, L. Havela, A.V. Andreev, J. Phys. F 16 (1986) L201.
E!ect of the crystalline electric "eld on the NeH el temperatures of RCu compounds Nguyen Hoang Luong *, J.J.M. Franse, Nguyen Hoang Hai
Faculty of Physics, Center for Materials Science, National University of Hanoi, 334 Nguyen Trai, Hanoi, Viet Nam Van der Waals-Zeeman Laboratorium, Universiteit van Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, Netherlands
Abstract Values for the NeH el temperature of the RCu compounds (R"Tb}Tm) have been calculated using a molecular-"eld model including crystalline-electric-"eld e!ect as presented by Noakes and Shenoy. The calculated results show that the unusual behavior, at the NeH el temperatures, of these compounds can be explained on the basis of this model. 2001 Published by Elsevier Science B.V. Keywords: Crystal-"eld e!ect; NeH el temperature; Rare-earth intermetallic compounds
The RCu (R"rare-earth) compounds crystal lize in the orthorhombic CeCu structure. The magnetic properties of these compounds were "rst studied by Sherwood et al. [1] as early as in 1964. Most of the RCu order antiferromagnetically with values for the NeH el temperature, ¹ , below 55 K. , Hashimoto et al. [2] performed magnetization and magnetic susceptibility measurements on singlecrystalline RCu samples. A large magnetic anisot ropy was found to be present. The magnetic properties of the RCu compounds have been extensively studied in recent years (see a review by Luong and Franse [3] and references therein). One of the features of the RCu compounds is that the values of the NeH el temperature for these compounds are not simply proportional to the de Gennes factor (g !1)J(J#1) and reach a maxH * Corresponding author. Fax: #84-4-858-94-96. E-mail address: [email protected] (N.H. Luong).
imum for TbCu . This fact suggests that the Ruder man}Kittel}Kasuya}Yosida (RKKY) interaction alone is not su$cient to fully understand the magnetic interactions in the RCu compounds. In spite of a substantial progress in the study of the magnetic properties of the RCu compounds, the above-mentioned exception to the de Gennes rule remained unexplained. In this paper, we report on our calculations that have been performed in order to explain the anomalous behavior, at the NeH el temperature, of the RCu compounds (R"Tb}Tm). Our calculations are based on the model of Noakes and Shenoy [4]. When considering only the exchange Hamiltonian, the de Gennes rule can be derived in the simple molecular-"eld model: the rare-earth-atom Hamiltonian H "!2C(g !1)J 1J 2 (1) H X X (where the z-axis is de"ned to be the orderedmoment axis and C the exchange parameter) leads to an implicit equation for 1J 2, and the ordering X
0304-8853/01/$ - see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 1 3 9 0 - 1
31
N.H. Luong et al. / Journal of Magnetism and Magnetic Materials 224 (2001) 30}32
temperature is obtained from the small 1J 2 limit X as ¹ "2C (g !1)J(J#1)/3. (2) H When the crystalline-electric-"eld (CEF) e!ects are signi"cant, the de Gennes behavior is not to be expected. In this case, CEF Hamiltonian should be added to the H . This leads to the following expression for the ordering temperature: ¹ "2C (g !1)1J(¹ )2 , (3) + H X + !#$ where 1J(¹ )2 is the expectation value of X + !#$ J under the in#uence of CEF Hamiltonian alone X at the temperature ¹ . The exchange parameter, C, + can be evaluated from the ordering temperature of the Gd compound when modeling a series of rareearth compounds, because Gd, an ¸"0 ion, is essentially una!ected by CEF. For calculating values of the NeH el temperatures, ¹ , of the RCu compounds, expression (3) is used , in which ¹ stands for ¹ . For evaluating C in + , these compounds, we take ¹ (GdCu )"¹ " , 41 K [1,5,6]. In the coordinate system of b"z, c"x and a"y, the orthorhombic CEF Hamiltonian of a CeCu -type of structure is given by H "B O #B O #B O #B O !#$ #B O #B O #B O #B O #B O , (4) where BK are the CEF parameters and OK the L L Stevens equivalent operators. As pointed out by Luong and Franse [3], information about the crystalline-electric-"eld interaction in RCu is not complete. Until now, the full set of CEF parameters is available only for NdCu , ErCu and TmCu . Nevertheless, the two lowest-order CEF parameters B and B have been derived for most of the RCu compounds (see Ref. [3] and references therein). In our calculations, we "rst used the two lowest-order terms in the CEF Hamiltonian. Values for B and B were taken for TbCu , DyCu and HoCu from Ref. [2], for ErCu from Ref. [7] and for TmCu from Ref. [8]. In TbCu , DyCu and HoCu the magnetic moments lie along the a-axis, whereas in ErCu and TmCu the magnetic moments are oriented
Table 1 Values for the crystalline-electric-"eld parameters and the NeH el temperatures in the heavy RCu compounds R
B (K)
B (K)
Gd
¹ exp (K) ,
¹ cal (K) ,
40 [5] 41 [1] 42 [6]
Tb
1.23 [2]
1.23 [2]
54 [1] 53.5 [2] 48.5 [5] 48 [14]
46.4
Dy
0.43 [2]
0.72 [2]
24 [1] 31.4 [2] 26.7 [5] 27 [7]
28.7
Ho
0.14 [2]
0.12 [2]
9 [1] 9.8 [2] 9.6 [5] 11 [19]
17.2
Er
!0.28 [7]
!0.22 [7]
11 [1] 13.5 [2] 11.5 [5]
9.1, 11.7
Tm
!0.94 [8]
!1.23 [8]
6.3 [20]
4.3, 6.7
NeH el temperature predicted by the full CEF Hamiltonian with the BK sets from Refs. [7,8] for ErCu and TmCu , respectively. L
along the b-direction [2,9}17]. The ¹ values for , ErCu and TmCu were calculated directly using the CEF Hamiltonian (4) with only the two lowestorder terms. For the RCu compounds with R"Tb, Dy and Ho, we used the CEF Hamiltonian transformed in the new coordinate system of a"z, b"x, c"y as follows [18]: H "()(!B !B )O #()(3B !B )O . (5) !#$ The calculated values of ¹ are compared with , experimental data in Table 1 and also in Fig. 1. As one can see from Table 1 and Fig. 1, addition of CEF interaction enhances ¹ over the de Gennes , values in the RCu compounds. Moreover, calcu lations predict that TbCu has a highest NeH el tem perature, in good agreement with experiments. The calculated values for the NeH el temperatures across the series are in good agreement with experimental ones. Calculations gave the value of ¹ for HoCu ,
N.H. Luong et al. / Journal of Magnetism and Magnetic Materials 224 (2001) 30}32
32
References
Fig. 1. Comparison of experimental and calculated NeH el temperatures for the RCu compounds. The open circles represent experimental data. The solid circles (solid line) represent calculations using a CEF Hamiltonian with two lowest-order terms, and the solid squares * calculations with the full CEF Hamiltonian as discussed in the text. The dashed line represents the de Gennes rule.
to be somewhat higher than the one obtained from experiments. We tried also to derive the values for the NeH el temperature of ErCu and TmCu with the full BK set taken from Refs. [7,8], respectively. The L results of these calculations using the full CEF Hamiltonian (4) are also given in Table 1 and in Fig. 1. As it can be seen, the use of the full CEF Hamiltonian gives better results than the use of the two lowest-order CEF terms only. In conclusion, the magnetic ordering temperatures in the RCu compounds can be explained by a combination of the RKKY interaction and crystalline-electric-"eld e!ects.
[1] R.C. Sherwood, H.J. Williams, J.H. Wernick, J. Appl. Phys. 35 (1964) 1049. [2] Y. Hashimoto, H. Fujii, H. Fujiwara, T. Okamoto, J. Phys. Soc. Japan 47 (1979) 67. [3] N.H. Luong, J.J.M. Franse, in: K.H.J. Buschow (Ed.), Handbook of Magnetic Materials, Vol. 8, North-Holland, Amsterdam, 1995, p. 415. [4] D.R. Noakes, G.K. Shenoy, Phys. Lett. 91A (1982) 35. [5] N.H. Luong, J.J.M. Franse, T.D. Hien, J. Magn. Magn. Mater. 50 (1985) 153. [6] M.K. Borombaev, R.Z. Levitin, A.S. Markosyan, V.A. Reimer, E.V. Sinitsyn, Z. Smetana, Zh. Eksp. Teor. Fiz. 93 (1987) 1517 (Eng. Transl., Sov. Phys. JETP 66 (1987) 866). [7] P.C.M. Gubbens, K.H.J. Buschow, M. Divis, J. Lange, M. Loewenhaupt, J. Magn. Magn. Mater. 98 (1991) 141. [8] P.C.M. Gubbens, K.H.J. Buschow, M. Divis, M. Heidelmann, M. Loewenhaupt, J. Magn. Magn. Mater. 104}107 (1992) 1283. [9] T.O. Brun, G.P. Felcher, J.S. Kouvel, AIP Conf. Proc. 5 (1971) 1376. [10] V. Sima, Z. Smetana, B. Lebech, E. Gratz, J. Magn. Magn. Mater. 54}57 (1986) 1357. [11] B. Lebech, Z. Smetana, V. Sima, J. Magn. Magn. Mater. 70 (1987) 97. [12] Z. Smetana, V. Sima, B. Lebech, J. Magn. Magn. Mater. 59 (1986) 145. [13] Z. Smetana, V. Sima, A.V. Andreev, Phys. Stat. Sol. B 121 (1984) K131. [14] Z. Smetana, V. Sima, Czech. J. Phys. B 35 (1985) 1232. [15] Y. Hashimoto, H. Kawano, H. Yoshizawa, S. Kawano, T. Shigeoka, J. Magn. Magn. Mater. 140}144 (1995) 1131. [16] V. Sima, M. Divis, P. Svoboda, Z. Smetana, Z. Zajac, J. Bischof, J. Phys.: Condens. Matter 1 (1989) 10153. [17] M. Heidelmann, B. Lebech, Z. Smetana, M. Loewenhaupt, J. Phys.: Condens. Matter 4 (1992) 8773. [18] M. Divis, S. Zajac, V. Sima, Z. Smetana, J. Magn. Magn. Mater. 68 (1987) 253. [19] R.R. Birss, R.V. Houldsworth, D.G. Lord, J. Magn. Magn. Mater. 15}18 (1980) 917. [20] Z. Smetana, V. Sima, J. Bischof, P. Svoboda, S. Zajac, L. Havela, A.V. Andreev, J. Phys. F 16 (1986) L201.