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Molecular field theory analysis of R3Co11B4 compounds
Journal of Magnetism and Magnetic Materials 241 (2002) 131–136
Molecular field theory analysis of R3Co11B4 compounds Zhang Xiang-Mua,*, Huang Rui-Wangb, Zhang Zhong-Wuc a
Physics Department, Cang Zhou Normal College, 28 Nan Huan Xi Lu St., Cang Zhou 061001, Hebei, China b School of Physics and Astronomy, The University of Nottingham, Nottingham NG7 2RD, UK c Physics Department, Hebei Normal University, Shijiazhuang 050016, Hebei, China Received 13 March 2001
Abstract The temperature dependence of magnetization of the R3Co11B4 compounds has been analysed using the twosublattice molecular field theory. The molecular field coefficients, nCoCo ; nRCo ; nRR ; have been calculated by a numerical fitting process. The analytic form of the exchange field HR ðTÞ varying with temperature for each of the R3Co11B4 compounds is presented, and some results are discussed. r 2002 Published by Elsevier Science B.V. PACS: 75.30.E; 75.60.E Keywords: Rare-earth intermetallic compounds; Two-sublattice molecular field theory; Exchange interaction; Curie temperature
1. Introduction In the course of search for new permanent magnets, some studies have been performed on the magnetic behaviour of the family of Rn+1Co3n+5B2n(R=Rare-earth) compounds. For example, the magnetic properties of R3Co11B4 alloys (n ¼ 2), which crystallize in a hexagonal-type structure with space group P6/ mmm [1], have also been reported [2–4]. On the basis of the magnetic measurements, the relations between the temperature and the spontaneous magnetization of R3Co11B4 (R=Pr, Nd, Tb, Dy and Ho) compounds were presented by Tetean et al. [4]. For obtaining more information on the magnetic behaviour of R and Co ion and determining the exchange interactions in R3Co11B4 compounds, the two-sublattice molecular field theory (MFT) [5] is employed to describe the temperature dependence of magnetization in this investigation. In addition, for analysing the spin-reorientation phenomenon and the temperature dependence of the magnetocrystalline anisotropy energy in these compounds, the temperature dependence of the exchange field is needed. Up to now, the relation between the temperature and the exchange field could only be obtained by the molecular field theory analysis of the magnetic experimental data.
*Corresponding author. E-mail address: [email protected] (Z. Xiang-Mu). 0304-8853/02/$ - see front matter r 2002 Published by Elsevier Science B.V. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 1 0 5 1 - 4
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132
2. Calculation According to the MFT, the exchange field (i.e. the total effective field) acting on the R sublattice and Co sublattice in R3Co11B4 compounds can be expressed, respectively, as follows: HR ðTÞ ¼ Ha þ d½3nRR MR ðTÞ þ 11nRCo MCo ðTÞ;
ð1Þ
HCo ðTÞ ¼ Ha þ d½3nRCo MR ðTÞ þ 11nCoCo MCo ðTÞ:
ð2Þ
The temperature dependence of each sublattice moment can be described by Brillouin function: MR ð0ÞHR ðTÞ MR ðTÞ ¼ MR ð0ÞBJR ; kT MCo ð0ÞHCo ðTÞ MCo ðTÞ ¼ MCo ð0ÞBJCo ; kT
ð3Þ ð4Þ
where Ha is the applied field, and MR ðTÞ and MR ð0Þ (or MCo ðTÞ and MCo ð0Þ) are the magnetic moments of the R ion (or the Co ion ) at T and 0 K, respectively. d ¼ NA rmB =A converts the moment per R3Co11B4 in mB to Gauss, where r is density of R3Co11B4 in g/cm3, NA is Avogadro’s number and A is the formula weight of R3Co11B4. JR and JCo are the individual angular moments of R and Co. nRR ; nRCo and nCoCo are the molecular field coefficients, which describe the R–R, R–Co and Co–Co magnetic interactions, respectively. In calculations, by taking MR ð0Þ ¼ gR JR ; MCo ð0Þ can be obtained from the experimental data measured at low temperature MCo ð0Þ ¼ ½Mexp ð0Þ83MR ð0Þ=11;
ð5Þ
where ‘’ represents the case that R is a light rare-earth, and ‘+’ the heavy rare-earth. Previously, Pedziwiatr et al. estimated a magnetic moment of 2:4 mB for Pr in PrCo4B [6]. If MPr ð0Þ ¼ 2:4 mB is used in Pr3Co11B4, MCo ð0Þ ¼ 0:4 mB is obtained according to Eq. (5), which is almost equal to the value obtained for Y3Co11B4 [2]. In calculations, MCo ð0Þ ¼ 0:4 mB is used for the light rare-earth compounds and the free R ion moment is for all heavy rare-earth compounds. The coefficients nRR ; nRCo and nCoCo are determined by numerically solving Eqs. (1)–(4), under the condition that the calculated total moments Mtot ðTÞ ¼ j3MR ðTÞ þ 11MCo ðTÞj correspond best with the experimental data. This is done by minimizing the percentage deviation P jMexp ðTi Þ Mtot ðTi Þj P ; R ¼ 100 Mexp ðTi Þ
ð6Þ
ð7Þ
where Mexp ðTi Þ is the experimental total moment at temperature Ti : From Eqs. (1)–(4), the Curie temperature TC in the zero applied field is related to the molecular field coefficients by
TC ¼
ðanCoCo þ bnRR Þ þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðanCoCo þ bnRR Þ2 þ 4abðnCoCo nRR n2RCo Þ ; 2ab
ð8Þ
where a¼
JR k ; ðJR þ 1ÞmB dMR2 ð0Þ
b¼
3JCo k : 2 ð0Þ 11ðJCo þ 1ÞmB dMCo
ð9Þ
Z. Xiang-Mu et al. / Journal of Magnetism and Magnetic Materials 241 (2002) 131–136
133
3. Results and discussion In Figs. 1 and 2, the temperature dependence of the magnetization for R3Co11B4 compounds is plotted. The circles represent the experimental data obtained by Tetean et al. [4], while the MFT results for total moment, R and Co sublattice moment are drawn by solid, dot-dashed and dotted lines, respectively. Obviously, the differences between the curves of MFT results and the experimental data are slight. The experimental values agree quite well with the theoretical values. These prove that the two-sublattice MFT is successful in describing the temperature dependence of magnetization of R3Co11B4 compounds. Table 1 summarizes the MFT coefficients, the values of TC and other related parameters. From Table 1, it is easily seen that the percentage deviation is o7% (Rp6:4%) and the calculated TC value is close to the experimental value for each compound which also showed that MFT results are consistent with the experimental values. Moreover, nCoCo is the largest one in the MFT coefficients (nCoCo > jnRCo j > nRR ; see Table 1) in all instances, implying that the magnetic interactions are dominated by the exchange between 3d electrons principally. For the sake of comparison, Fig. 3 represents the relation between the reduced temperature and magnetization for the R and Co ion in each compound. From Fig. 3, it can be seen that the curves of reduced magnetic moment MR ðTÞ=MR ð0Þ to reduced temperature T=TC are different from one another, which indicated that the magnetic properties of the R ion are different from each other. However, all the MCo ðTÞ=MCo ð0Þ vs. T=TC curves are almost overlapping, which are consistent with the curves obtained from compounds R2Fe14B [7], RFe10V2Nx [8], and R2Co17 [9] previously. This implies that the reduced temperature dependence of the reduced magnetic moments of Co ion is the same as that of Fe ion.
Fig. 1. Moments vs. temperature of Pr3Co11B4 and Nd3Co11B4. Circles represent the experimental data of total moment. MFT results for total moment, R and Co sublattice moment are indicated by solid, dot-dashed and dotted lines, respectively. MFT results correspond to ferromagnetic coupling of the Pr (or the Nd) and Co moments.
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134
Fig. 2. Moments vs. temperature of Tb3Co11B4, Dy3Co11B4 and Ho3Co11B4. Ferrimagnetic coupling was assumed in MFT work. The representative marks are the same as those indicated in the caption of Fig. 1.
Table 1 Factor d; rare-earth gyromagnetic ratio gR and total angular momentum JR used in MFT calculations. nCoCo ; nRCo ; and nRR are the molecular field coefficients R
Pr Nd Tb Dy Ho
gR
4/5 8/11 3/2 4/3 5/4
JR
3 7/2 6 15/2 8
MR ð0Þ (mB)
2.4 2.5 9 10 10
MCo ð0Þ (mB)
0.4 0.4 1.04 1.215 1.475
d (Gs f.u./mB)
41.337 41.515 42.645 42.807 43.010
nRR
1100 2700 445 54 10
nRCo
6900 7720 970 866 790
nCoCo
82400 84480 11900 8006 4800
R(%)
4.8 3.1 1.7 6.4 5.8
TC (K) cal.
exp.
427 432 433 407 381
442 448 431 415 398
Z. Xiang-Mu et al. / Journal of Magnetism and Magnetic Materials 241 (2002) 131–136
Fig. 3. The reduced temperature dependence of the sublattice reduced magnetization of R (or Co) in R3Co11B4.
Fig. 4. Temperature dependence of HR ðTÞ for five R3Co11B4 compounds.
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136
Table 2 Coefficients A, B and C appearing in Eq. (10) R
HR ð0Þ ( 106 Gs )
A
B
C
Pr Nd Tb Dy Ho
1.582 2.250 1.003 0.548 0.456
0.801 0.744 0.495 0.484 0.035
1.898 1.255 0.597 1.706 2.260
5.119 1.405 0.984 2.056 3.056
In the study of the relation between the magnetocrystalline anisotropy energy and the temperature, it is necessary to know the property of exchange field HR ðTÞ: In Fig. 4, the curves of the temperature dependence of exchange field HR for five compounds were plotted. For the sake of application, the numerical results of HR ðTÞ are fitted to an analytic form by the least-squares method HR ðTÞ ¼ HR ð0Þ½1 þ AðT=TC Þ þ BðT=TC Þ2 þ CðT=TC Þ3 :
ð10Þ
The values of A, B, C are listed in Table 2.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
K. Niihara, S. Yajima, Bull. Chem. Soc. Jpn 46 (1973) 770. E. Burzo, V. Pop, C.C. Borodi, R. Ballou, IEEE Trans. Magn. 30 (1994) 628. A. Kowalezyk, V. Ivanov, J. Baszynski, J. Magn. Magn. Mater. 146 (1995) 182. R. Tetean, E. Burzo, J. Magn. Magn. Mater. 157/158 (1996) 633. J.F. Herbst, J.J. Croat, J. Appl. Phys. 53 (1982) 4304; J.F. Herbst, J.J. Croat, J. Appl. Phys. 55 (1984) 3023. A.T. Pedziwiatr, S.Y. Jaing, W.E. Wallace, E. Burzo, J. Magn. Magn. Mater. 66 (1987) 69. Li Huai-Shan, Zhang Zhong-Wu, Dang Mei-Zhen, J. Magn. Magn. Mater. 71 (1988) 355. Ren Shi-Wei, Zhang Zhong-Wu, Liu Ying, J. Magn. Magn. Mater. 139 (1995) 175. Zhang Zhong-Wu, Ren Shi-Wei, Zhang Xiang-Mu, Song Xue-Jun, Han Lai-Ping, Ni Zhen-Cheng, J. Magn. Magn. Mater. 170 (1997) 184.
Molecular field theory analysis of R3Co11B4 compounds Zhang Xiang-Mua,*, Huang Rui-Wangb, Zhang Zhong-Wuc a
Physics Department, Cang Zhou Normal College, 28 Nan Huan Xi Lu St., Cang Zhou 061001, Hebei, China b School of Physics and Astronomy, The University of Nottingham, Nottingham NG7 2RD, UK c Physics Department, Hebei Normal University, Shijiazhuang 050016, Hebei, China Received 13 March 2001
Abstract The temperature dependence of magnetization of the R3Co11B4 compounds has been analysed using the twosublattice molecular field theory. The molecular field coefficients, nCoCo ; nRCo ; nRR ; have been calculated by a numerical fitting process. The analytic form of the exchange field HR ðTÞ varying with temperature for each of the R3Co11B4 compounds is presented, and some results are discussed. r 2002 Published by Elsevier Science B.V. PACS: 75.30.E; 75.60.E Keywords: Rare-earth intermetallic compounds; Two-sublattice molecular field theory; Exchange interaction; Curie temperature
1. Introduction In the course of search for new permanent magnets, some studies have been performed on the magnetic behaviour of the family of Rn+1Co3n+5B2n(R=Rare-earth) compounds. For example, the magnetic properties of R3Co11B4 alloys (n ¼ 2), which crystallize in a hexagonal-type structure with space group P6/ mmm [1], have also been reported [2–4]. On the basis of the magnetic measurements, the relations between the temperature and the spontaneous magnetization of R3Co11B4 (R=Pr, Nd, Tb, Dy and Ho) compounds were presented by Tetean et al. [4]. For obtaining more information on the magnetic behaviour of R and Co ion and determining the exchange interactions in R3Co11B4 compounds, the two-sublattice molecular field theory (MFT) [5] is employed to describe the temperature dependence of magnetization in this investigation. In addition, for analysing the spin-reorientation phenomenon and the temperature dependence of the magnetocrystalline anisotropy energy in these compounds, the temperature dependence of the exchange field is needed. Up to now, the relation between the temperature and the exchange field could only be obtained by the molecular field theory analysis of the magnetic experimental data.
*Corresponding author. E-mail address: [email protected] (Z. Xiang-Mu). 0304-8853/02/$ - see front matter r 2002 Published by Elsevier Science B.V. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 1 0 5 1 - 4
Z. Xiang-Mu et al. / Journal of Magnetism and Magnetic Materials 241 (2002) 131–136
132
2. Calculation According to the MFT, the exchange field (i.e. the total effective field) acting on the R sublattice and Co sublattice in R3Co11B4 compounds can be expressed, respectively, as follows: HR ðTÞ ¼ Ha þ d½3nRR MR ðTÞ þ 11nRCo MCo ðTÞ;
ð1Þ
HCo ðTÞ ¼ Ha þ d½3nRCo MR ðTÞ þ 11nCoCo MCo ðTÞ:
ð2Þ
The temperature dependence of each sublattice moment can be described by Brillouin function: MR ð0ÞHR ðTÞ MR ðTÞ ¼ MR ð0ÞBJR ; kT MCo ð0ÞHCo ðTÞ MCo ðTÞ ¼ MCo ð0ÞBJCo ; kT
ð3Þ ð4Þ
where Ha is the applied field, and MR ðTÞ and MR ð0Þ (or MCo ðTÞ and MCo ð0Þ) are the magnetic moments of the R ion (or the Co ion ) at T and 0 K, respectively. d ¼ NA rmB =A converts the moment per R3Co11B4 in mB to Gauss, where r is density of R3Co11B4 in g/cm3, NA is Avogadro’s number and A is the formula weight of R3Co11B4. JR and JCo are the individual angular moments of R and Co. nRR ; nRCo and nCoCo are the molecular field coefficients, which describe the R–R, R–Co and Co–Co magnetic interactions, respectively. In calculations, by taking MR ð0Þ ¼ gR JR ; MCo ð0Þ can be obtained from the experimental data measured at low temperature MCo ð0Þ ¼ ½Mexp ð0Þ83MR ð0Þ=11;
ð5Þ
where ‘’ represents the case that R is a light rare-earth, and ‘+’ the heavy rare-earth. Previously, Pedziwiatr et al. estimated a magnetic moment of 2:4 mB for Pr in PrCo4B [6]. If MPr ð0Þ ¼ 2:4 mB is used in Pr3Co11B4, MCo ð0Þ ¼ 0:4 mB is obtained according to Eq. (5), which is almost equal to the value obtained for Y3Co11B4 [2]. In calculations, MCo ð0Þ ¼ 0:4 mB is used for the light rare-earth compounds and the free R ion moment is for all heavy rare-earth compounds. The coefficients nRR ; nRCo and nCoCo are determined by numerically solving Eqs. (1)–(4), under the condition that the calculated total moments Mtot ðTÞ ¼ j3MR ðTÞ þ 11MCo ðTÞj correspond best with the experimental data. This is done by minimizing the percentage deviation P jMexp ðTi Þ Mtot ðTi Þj P ; R ¼ 100 Mexp ðTi Þ
ð6Þ
ð7Þ
where Mexp ðTi Þ is the experimental total moment at temperature Ti : From Eqs. (1)–(4), the Curie temperature TC in the zero applied field is related to the molecular field coefficients by
TC ¼
ðanCoCo þ bnRR Þ þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðanCoCo þ bnRR Þ2 þ 4abðnCoCo nRR n2RCo Þ ; 2ab
ð8Þ
where a¼
JR k ; ðJR þ 1ÞmB dMR2 ð0Þ
b¼
3JCo k : 2 ð0Þ 11ðJCo þ 1ÞmB dMCo
ð9Þ
Z. Xiang-Mu et al. / Journal of Magnetism and Magnetic Materials 241 (2002) 131–136
133
3. Results and discussion In Figs. 1 and 2, the temperature dependence of the magnetization for R3Co11B4 compounds is plotted. The circles represent the experimental data obtained by Tetean et al. [4], while the MFT results for total moment, R and Co sublattice moment are drawn by solid, dot-dashed and dotted lines, respectively. Obviously, the differences between the curves of MFT results and the experimental data are slight. The experimental values agree quite well with the theoretical values. These prove that the two-sublattice MFT is successful in describing the temperature dependence of magnetization of R3Co11B4 compounds. Table 1 summarizes the MFT coefficients, the values of TC and other related parameters. From Table 1, it is easily seen that the percentage deviation is o7% (Rp6:4%) and the calculated TC value is close to the experimental value for each compound which also showed that MFT results are consistent with the experimental values. Moreover, nCoCo is the largest one in the MFT coefficients (nCoCo > jnRCo j > nRR ; see Table 1) in all instances, implying that the magnetic interactions are dominated by the exchange between 3d electrons principally. For the sake of comparison, Fig. 3 represents the relation between the reduced temperature and magnetization for the R and Co ion in each compound. From Fig. 3, it can be seen that the curves of reduced magnetic moment MR ðTÞ=MR ð0Þ to reduced temperature T=TC are different from one another, which indicated that the magnetic properties of the R ion are different from each other. However, all the MCo ðTÞ=MCo ð0Þ vs. T=TC curves are almost overlapping, which are consistent with the curves obtained from compounds R2Fe14B [7], RFe10V2Nx [8], and R2Co17 [9] previously. This implies that the reduced temperature dependence of the reduced magnetic moments of Co ion is the same as that of Fe ion.
Fig. 1. Moments vs. temperature of Pr3Co11B4 and Nd3Co11B4. Circles represent the experimental data of total moment. MFT results for total moment, R and Co sublattice moment are indicated by solid, dot-dashed and dotted lines, respectively. MFT results correspond to ferromagnetic coupling of the Pr (or the Nd) and Co moments.
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134
Fig. 2. Moments vs. temperature of Tb3Co11B4, Dy3Co11B4 and Ho3Co11B4. Ferrimagnetic coupling was assumed in MFT work. The representative marks are the same as those indicated in the caption of Fig. 1.
Table 1 Factor d; rare-earth gyromagnetic ratio gR and total angular momentum JR used in MFT calculations. nCoCo ; nRCo ; and nRR are the molecular field coefficients R
Pr Nd Tb Dy Ho
gR
4/5 8/11 3/2 4/3 5/4
JR
3 7/2 6 15/2 8
MR ð0Þ (mB)
2.4 2.5 9 10 10
MCo ð0Þ (mB)
0.4 0.4 1.04 1.215 1.475
d (Gs f.u./mB)
41.337 41.515 42.645 42.807 43.010
nRR
1100 2700 445 54 10
nRCo
6900 7720 970 866 790
nCoCo
82400 84480 11900 8006 4800
R(%)
4.8 3.1 1.7 6.4 5.8
TC (K) cal.
exp.
427 432 433 407 381
442 448 431 415 398
Z. Xiang-Mu et al. / Journal of Magnetism and Magnetic Materials 241 (2002) 131–136
Fig. 3. The reduced temperature dependence of the sublattice reduced magnetization of R (or Co) in R3Co11B4.
Fig. 4. Temperature dependence of HR ðTÞ for five R3Co11B4 compounds.
135
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136
Table 2 Coefficients A, B and C appearing in Eq. (10) R
HR ð0Þ ( 106 Gs )
A
B
C
Pr Nd Tb Dy Ho
1.582 2.250 1.003 0.548 0.456
0.801 0.744 0.495 0.484 0.035
1.898 1.255 0.597 1.706 2.260
5.119 1.405 0.984 2.056 3.056
In the study of the relation between the magnetocrystalline anisotropy energy and the temperature, it is necessary to know the property of exchange field HR ðTÞ: In Fig. 4, the curves of the temperature dependence of exchange field HR for five compounds were plotted. For the sake of application, the numerical results of HR ðTÞ are fitted to an analytic form by the least-squares method HR ðTÞ ¼ HR ð0Þ½1 þ AðT=TC Þ þ BðT=TC Þ2 þ CðT=TC Þ3 :
ð10Þ
The values of A, B, C are listed in Table 2.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
K. Niihara, S. Yajima, Bull. Chem. Soc. Jpn 46 (1973) 770. E. Burzo, V. Pop, C.C. Borodi, R. Ballou, IEEE Trans. Magn. 30 (1994) 628. A. Kowalezyk, V. Ivanov, J. Baszynski, J. Magn. Magn. Mater. 146 (1995) 182. R. Tetean, E. Burzo, J. Magn. Magn. Mater. 157/158 (1996) 633. J.F. Herbst, J.J. Croat, J. Appl. Phys. 53 (1982) 4304; J.F. Herbst, J.J. Croat, J. Appl. Phys. 55 (1984) 3023. A.T. Pedziwiatr, S.Y. Jaing, W.E. Wallace, E. Burzo, J. Magn. Magn. Mater. 66 (1987) 69. Li Huai-Shan, Zhang Zhong-Wu, Dang Mei-Zhen, J. Magn. Magn. Mater. 71 (1988) 355. Ren Shi-Wei, Zhang Zhong-Wu, Liu Ying, J. Magn. Magn. Mater. 139 (1995) 175. Zhang Zhong-Wu, Ren Shi-Wei, Zhang Xiang-Mu, Song Xue-Jun, Han Lai-Ping, Ni Zhen-Cheng, J. Magn. Magn. Mater. 170 (1997) 184.