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Molecular field theory analysis of (Er, Y)2Fe14B compounds
Journal of Magnetism and Magnetic Materials 184 (1998) 227—230
Molecular field theory analysis of (Er, Y) Fe B compounds 2 14 M. Manivel Raja!, A. Narayanasamy!,*, J. Voiron", R. Krishnan# ! Department of Nuclear Physics, University of Madras, Guindy Campus, Madras 600 025, India " Laboratoire de Magnetisme Louis Neel, C.N.R.S., B.P. 166, 38042-Grenoble Cedex 9, France # Laboratoire de Magnetisme et, d+ Optique Batiment Fermat, 45, Avenue des Etats Unis, 78035 Versailles, France Received 7 November 1997; received in revised form 9 December 1997
Abstract The temperature dependence of the magnetisation of Er Y Fe B compounds was analysed using a simple two 2~x x 14 sublattice molecular field theory. The molecular field coefficients n , n , n were obtained and the theoretically FF RR RF calculated Curie temperatures were found to agree well with the experimental values. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Permanent magnetic materials; Curie temperature; Magnetisation; Molecular field theory
1. Introduction After the discovery of the Nd Fe B intermetal2 14 lic compound a good number of experimental studies using various techniques have been performed in order to elucidate the intrinsic magnetic properties of the R Fe B compounds. Also, a substantial 2 14 progress has been made in the theoretical understanding of the magnetic properties of these compounds. A theoretical model based on the molecular field analysis was developed by Verhoef et al. [1] and its applicability to Er Fe B has been 2 14 tested using pulsed high-field magnetisation data. In the present work, the authors have used a simple
* Corresponding author.
two-sublattice molecular field theory [2—4] to study the exchange interaction between the rare earth and iron sublattices in the Er Y Fe B 2~x x 14 system. The Curie temperature for these compounds has been calculated from the molecular field coefficients, n , n and n , of the rareRR FF RF earth—rare-earth, iron—iron and rare-earth—iron exchange interactions and compared with the experimental values.
2. Experimental details The samples under investigation, Er Y Fe B 2~x x 14 where x"0.0, 0.5, 1.0, 1.5 and 2.0, were prepared by induction melting the constituent elements in vacuum. The details of the sample preparation have been reported in our earlier paper [5]. The phase
0304-8853/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 0 4 - 8 8 5 3 ( 9 7 ) 0 1 1 4 2 - 6
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M. Manivel Raja et al. / Journal of Magnetism and Magnetic Materials 184 (1998) 227—230
purity of the compounds was checked by the X-ray powder diffraction technique using a high resolution Guinier-type X-ray powder diffractometer with Cu K radiation. The Curie temperatures of a1 the samples were determined using a Perkin-Elmer PC Series DSC-7 differential scanning calorimeter under a constant flow of argon gas. The accuracy of the Curie temperature is $0.5 K. For molecular field analysis, the magnetisation of the samples was measured in an applied magnetic field of 5 T in the temperature range 4.2—295 K using the facilities available at Grenoble.
3. Results and discussion The temperature dependence of the magnetisation for the Er Y Fe B compounds is shown 2~x x 14 in Fig. 1. In the figure, the experimental data are represented by symbols and the molecular field theory fits are indicated by the continuous solid lines. According to the molecular field theory, the molecular field at the rare earth sublattice depends on the magnetisation of both the rare earth and iron sublattices. Similarly, the molecular field at the iron sublattice depends on the magnetisation of iron and rare earth sublattices. Now, the molecular fields [3,4,6] at rare earth and iron sublattices in an applied magnetic field H can be ! written as
Fig. 1. Temperature dependence of the magnetisation of the Er Y Fe B compounds. 2~x x 14
where the Brillouin function is given as B "[(2J#1)/2J] coth[(2J#1)x/2J] J !1 coth[(1/2J)x]. 2
(5)
H (¹)"H #d[2n k (¹)#14n k (¹)], (1) R ! RR R RF F H (¹)"H #d[14n k (¹)#2n k (¹)], (2) F ! FF F RF R with d"N k o/A where N is the Avogadro numA B A ber, o is the density and A is the R Fe B formula 2 14 weight. The constant d converts the magnetic moment per formula unit from k into gauss. The B k and k are the R and Fe ionic magnetic moR F ments. The quantities n , n and n are the RR RF FF molecular field coefficients of R—R, R—Fe and Fe—Fe exchange interactions, respectively. The temperature dependence of rare earth and iron moments can be expressed using the Brillouin function as
Here k (0) and k (0) are zero temperature magnetic R F moments and J and J are the total angular moR F menta of the rare earth and iron atoms, respectively. The free ion value (k "gJ) can be used for R k (0), and k (0) is derived from the experimental R F magnetisation data at low temperatures as k (0)"[k $2k ]/14 with the negative sign for F %91 R the light and the positive sign for the heavy rare earth elements. A reasonable value for J ("S ) is F F unity since k (0)+2.0k in most of the rare-earth— F B transition-metal compounds. The molecular field coefficients n , n and RR FF n are obtained by numerically solving Eqs. (3) RF and (4) under the condition that k (¹)" 505 2k (¹)#14k (¹), and making k (¹) compatible R F 505 with the experimental value of k (¹) by minimis%91 ing the percentage deviation:
k (¹)"k (0)B [k k (0)H (¹)/k ¹], R R JR B R R B k (¹)"k (0)B [k k (0)H (¹)/k ¹], F T JF B F F B
R"+ Dk (¹ )!k (¹ )D*100/+ k (¹ ). %91 i 505 i %91 i i i
(3) (4)
(6)
M. Manivel Raja et al. / Journal of Magnetism and Magnetic Materials 184 (1998) 227—230 Table 1 Molecular field coefficients n , n and n and the average FF RR RF disparity R between the calculated and experimental total magnetisations, theoretical ¹ (cal) and experimental ¹ (exp) Curie C C temperatures of Er Y Fe B compounds 2~x x 14 x
n FF
n RR
n RF
R (%) ¹ (cal) (K) ¹ (exp) (K) C C
0.0 0.5 1.0 1.5 2.0
4112 4502 4935 5013 4525
361 419 439 441 —
!570 !669 !704 !716 —
1.3 1.0 0.8 2.2 2.8
542 556 549 552 560
554 557 560 563 568
The molecular field coefficients n , n and n FF RR RF of the Fe—Fe, R—R and R—Fe exchange interactions and the percentage deviation R between the calculated and experimental total magnetisations are given in Table 1. Among the three coefficients, the value of n is found to be larger than the FF values of n and n and also the value of RR RF n increases with the substitution of Y for Er up to FF x"1.5. From Eqs. (3) and (4), a relation between the Curie temperature and molecular field coefficients can be derived in zero applied field: ¹ "1Man #bn C 2 FF RR #[(an !bn )2#4abn ]1@2N FF RR RF
(7)
where a"14k2(0)[k d/k ][(S #1)/3S ], F B B F F
(8)
b"2k2(0)[k d/k ][(J#1)/3J]. R B B
(9)
The Curie temperatures were calculated from the molecular field coefficients using the relation given in Eq. (7) between ¹ and molecular field C coefficients. From Table 1 it is seen that the calculated values of the Curie temperature agree well with the measured values. The good agreement between the calculated and experimental values of Curie temperature shows that the molecular field theory based on the two-sublattice model is quite successful in describing the temperature dependence of the magnetisation of the Er Y Fe B 2~x x 14 compounds.
229
The mean field theory provides us with the relation [2]: 3k ¹ "a #a #[(a !a )2#4a a ]1@2, B C FF RR FF RR RF FR (10) where a "ZJ S (S #1), FF FF F F a "Z J (g !1)(J #1), RR 3 RR J R (11) a a "Z Z J2 (g !1)(J #1). R RF FR 1 2 RF J The quantities a , a and a "a are the exFF RR RF FR change energies and J , J and J are the exFF RR RF change coupling constants for the Fe—Fe, R—R and R—Fe interactions, respectively. The constants Z ("10) and Z ("2.5) are the number of Fe and 1 R-nearest neighbours of Fe atom while Z ("18) 2 and Z ("2.5) are the number of Fe and R nearest3 neighbours of R atom. The exchange interaction energies a , a and FF RR a and the exchange integral coupling constants RF J , J and J of the Fe—Fe, R—R and R—Fe FF RR RF exchange interactions in Er Y Fe B have been 2~x x 14 computed and given in Table 2. From the table, it can be seen that the strength of the Fe—Fe exchange interaction increases with the substitution of Y for Er. The increase in the strength of the Fe—Fe interaction may be responsible for the observed increase in the Curie temperature with Y substitution. It is also seen that the strength of the R—R interaction has the same order of magnitude as that of the R—Fe interaction which is in agreement with the earlier studies on most of the heavy rare-earth compounds [4]. Table 2 Exchange interaction energies a , a and a and coupling FF RR RF constants J , J and J of Fe—Fe, R—R and R—Fe interactions FF RR RF in Er Y Fe B compounds 2~x x 14 x
a FF
a RR
a
RF
(]10~20 J) 0.0 0.5 1.0 1.5 2.0
1.09 1.12 1.11 1.13 1.15
0.14 0.12 0.08 0.04 —
J FF
J RR
J RF
(]10~22 J) !0.37 !0.18 !0.14 !0.11 —
5.44 5.61 5.59 5.65 5.60
3.33 3.76 3.98 3.90 —
1.21 1.36 1.70 1.38 —
230
M. Manivel Raja et al. / Journal of Magnetism and Magnetic Materials 184 (1998) 227—230
4. Conclusions The results of the two-sublattice molecular field theory analysis of the saturation magnetisation of the Er Y Fe B compounds show that the in2~x x 14 crease in the Curie temperature of these compounds with the increase in Y concentration is attributable to the enhancement in the strength of the Fe—Fe exchange interaction. Also, the good agreement between the calculated and experimental values of the Curie temperature shows that the two-sublattice molecular field theory is quite successful in describing the temperature dependence of magnetisation of Er Y Fe B com2~x x 14 pounds.
Acknowledgements This work was supported by the Council of Scientific and Industrial Research (CSIR), Government of India under the research scheme No.
4(114)/91-EMR-II and also by the University Grants Commission (UGC), Government of India under the Special Assistance Programme (SAP). The authors would like to thank Dr. R. Premanand for his assistance in analysing magnetisation data using molecular field theory. One of the authors (M.M.R) would like to thank CSIR for the award of Senior Research Fellowship (SRF) during the course of this work.
References [1] R. Verhoef, R.J. Radwanski, J.J.M. Franse, J. Magn. Magn. Mater. 89 (1990) 176. [2] S. Sinnema, R.J. Radwanski, J.J.M. Franse, D.B. de Mooij, K.H.J Buschow, J. Magn. Magn. Mater. 44 (1984) 333. [3] C.D. Fuerst, J.F. Herbst, E.A. Alson, J. Magn. Magn. Mater. 54—57 (1986) 567. [4] H.S. Li, Z.W. Zhang, M.Z. Dang, J. Magn. Magn. Mater. 71 (1988) 355. [5] M. Manivel Raja, A. Narayanasamy, J. Alloys Compounds 233 (1996) 140. [6] J.F. Herbst, J.J. Croat, J. Appl. Phys. 55 (1984) 15.
Molecular field theory analysis of (Er, Y) Fe B compounds 2 14 M. Manivel Raja!, A. Narayanasamy!,*, J. Voiron", R. Krishnan# ! Department of Nuclear Physics, University of Madras, Guindy Campus, Madras 600 025, India " Laboratoire de Magnetisme Louis Neel, C.N.R.S., B.P. 166, 38042-Grenoble Cedex 9, France # Laboratoire de Magnetisme et, d+ Optique Batiment Fermat, 45, Avenue des Etats Unis, 78035 Versailles, France Received 7 November 1997; received in revised form 9 December 1997
Abstract The temperature dependence of the magnetisation of Er Y Fe B compounds was analysed using a simple two 2~x x 14 sublattice molecular field theory. The molecular field coefficients n , n , n were obtained and the theoretically FF RR RF calculated Curie temperatures were found to agree well with the experimental values. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Permanent magnetic materials; Curie temperature; Magnetisation; Molecular field theory
1. Introduction After the discovery of the Nd Fe B intermetal2 14 lic compound a good number of experimental studies using various techniques have been performed in order to elucidate the intrinsic magnetic properties of the R Fe B compounds. Also, a substantial 2 14 progress has been made in the theoretical understanding of the magnetic properties of these compounds. A theoretical model based on the molecular field analysis was developed by Verhoef et al. [1] and its applicability to Er Fe B has been 2 14 tested using pulsed high-field magnetisation data. In the present work, the authors have used a simple
* Corresponding author.
two-sublattice molecular field theory [2—4] to study the exchange interaction between the rare earth and iron sublattices in the Er Y Fe B 2~x x 14 system. The Curie temperature for these compounds has been calculated from the molecular field coefficients, n , n and n , of the rareRR FF RF earth—rare-earth, iron—iron and rare-earth—iron exchange interactions and compared with the experimental values.
2. Experimental details The samples under investigation, Er Y Fe B 2~x x 14 where x"0.0, 0.5, 1.0, 1.5 and 2.0, were prepared by induction melting the constituent elements in vacuum. The details of the sample preparation have been reported in our earlier paper [5]. The phase
0304-8853/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 0 4 - 8 8 5 3 ( 9 7 ) 0 1 1 4 2 - 6
228
M. Manivel Raja et al. / Journal of Magnetism and Magnetic Materials 184 (1998) 227—230
purity of the compounds was checked by the X-ray powder diffraction technique using a high resolution Guinier-type X-ray powder diffractometer with Cu K radiation. The Curie temperatures of a1 the samples were determined using a Perkin-Elmer PC Series DSC-7 differential scanning calorimeter under a constant flow of argon gas. The accuracy of the Curie temperature is $0.5 K. For molecular field analysis, the magnetisation of the samples was measured in an applied magnetic field of 5 T in the temperature range 4.2—295 K using the facilities available at Grenoble.
3. Results and discussion The temperature dependence of the magnetisation for the Er Y Fe B compounds is shown 2~x x 14 in Fig. 1. In the figure, the experimental data are represented by symbols and the molecular field theory fits are indicated by the continuous solid lines. According to the molecular field theory, the molecular field at the rare earth sublattice depends on the magnetisation of both the rare earth and iron sublattices. Similarly, the molecular field at the iron sublattice depends on the magnetisation of iron and rare earth sublattices. Now, the molecular fields [3,4,6] at rare earth and iron sublattices in an applied magnetic field H can be ! written as
Fig. 1. Temperature dependence of the magnetisation of the Er Y Fe B compounds. 2~x x 14
where the Brillouin function is given as B "[(2J#1)/2J] coth[(2J#1)x/2J] J !1 coth[(1/2J)x]. 2
(5)
H (¹)"H #d[2n k (¹)#14n k (¹)], (1) R ! RR R RF F H (¹)"H #d[14n k (¹)#2n k (¹)], (2) F ! FF F RF R with d"N k o/A where N is the Avogadro numA B A ber, o is the density and A is the R Fe B formula 2 14 weight. The constant d converts the magnetic moment per formula unit from k into gauss. The B k and k are the R and Fe ionic magnetic moR F ments. The quantities n , n and n are the RR RF FF molecular field coefficients of R—R, R—Fe and Fe—Fe exchange interactions, respectively. The temperature dependence of rare earth and iron moments can be expressed using the Brillouin function as
Here k (0) and k (0) are zero temperature magnetic R F moments and J and J are the total angular moR F menta of the rare earth and iron atoms, respectively. The free ion value (k "gJ) can be used for R k (0), and k (0) is derived from the experimental R F magnetisation data at low temperatures as k (0)"[k $2k ]/14 with the negative sign for F %91 R the light and the positive sign for the heavy rare earth elements. A reasonable value for J ("S ) is F F unity since k (0)+2.0k in most of the rare-earth— F B transition-metal compounds. The molecular field coefficients n , n and RR FF n are obtained by numerically solving Eqs. (3) RF and (4) under the condition that k (¹)" 505 2k (¹)#14k (¹), and making k (¹) compatible R F 505 with the experimental value of k (¹) by minimis%91 ing the percentage deviation:
k (¹)"k (0)B [k k (0)H (¹)/k ¹], R R JR B R R B k (¹)"k (0)B [k k (0)H (¹)/k ¹], F T JF B F F B
R"+ Dk (¹ )!k (¹ )D*100/+ k (¹ ). %91 i 505 i %91 i i i
(3) (4)
(6)
M. Manivel Raja et al. / Journal of Magnetism and Magnetic Materials 184 (1998) 227—230 Table 1 Molecular field coefficients n , n and n and the average FF RR RF disparity R between the calculated and experimental total magnetisations, theoretical ¹ (cal) and experimental ¹ (exp) Curie C C temperatures of Er Y Fe B compounds 2~x x 14 x
n FF
n RR
n RF
R (%) ¹ (cal) (K) ¹ (exp) (K) C C
0.0 0.5 1.0 1.5 2.0
4112 4502 4935 5013 4525
361 419 439 441 —
!570 !669 !704 !716 —
1.3 1.0 0.8 2.2 2.8
542 556 549 552 560
554 557 560 563 568
The molecular field coefficients n , n and n FF RR RF of the Fe—Fe, R—R and R—Fe exchange interactions and the percentage deviation R between the calculated and experimental total magnetisations are given in Table 1. Among the three coefficients, the value of n is found to be larger than the FF values of n and n and also the value of RR RF n increases with the substitution of Y for Er up to FF x"1.5. From Eqs. (3) and (4), a relation between the Curie temperature and molecular field coefficients can be derived in zero applied field: ¹ "1Man #bn C 2 FF RR #[(an !bn )2#4abn ]1@2N FF RR RF
(7)
where a"14k2(0)[k d/k ][(S #1)/3S ], F B B F F
(8)
b"2k2(0)[k d/k ][(J#1)/3J]. R B B
(9)
The Curie temperatures were calculated from the molecular field coefficients using the relation given in Eq. (7) between ¹ and molecular field C coefficients. From Table 1 it is seen that the calculated values of the Curie temperature agree well with the measured values. The good agreement between the calculated and experimental values of Curie temperature shows that the molecular field theory based on the two-sublattice model is quite successful in describing the temperature dependence of the magnetisation of the Er Y Fe B 2~x x 14 compounds.
229
The mean field theory provides us with the relation [2]: 3k ¹ "a #a #[(a !a )2#4a a ]1@2, B C FF RR FF RR RF FR (10) where a "ZJ S (S #1), FF FF F F a "Z J (g !1)(J #1), RR 3 RR J R (11) a a "Z Z J2 (g !1)(J #1). R RF FR 1 2 RF J The quantities a , a and a "a are the exFF RR RF FR change energies and J , J and J are the exFF RR RF change coupling constants for the Fe—Fe, R—R and R—Fe interactions, respectively. The constants Z ("10) and Z ("2.5) are the number of Fe and 1 R-nearest neighbours of Fe atom while Z ("18) 2 and Z ("2.5) are the number of Fe and R nearest3 neighbours of R atom. The exchange interaction energies a , a and FF RR a and the exchange integral coupling constants RF J , J and J of the Fe—Fe, R—R and R—Fe FF RR RF exchange interactions in Er Y Fe B have been 2~x x 14 computed and given in Table 2. From the table, it can be seen that the strength of the Fe—Fe exchange interaction increases with the substitution of Y for Er. The increase in the strength of the Fe—Fe interaction may be responsible for the observed increase in the Curie temperature with Y substitution. It is also seen that the strength of the R—R interaction has the same order of magnitude as that of the R—Fe interaction which is in agreement with the earlier studies on most of the heavy rare-earth compounds [4]. Table 2 Exchange interaction energies a , a and a and coupling FF RR RF constants J , J and J of Fe—Fe, R—R and R—Fe interactions FF RR RF in Er Y Fe B compounds 2~x x 14 x
a FF
a RR
a
RF
(]10~20 J) 0.0 0.5 1.0 1.5 2.0
1.09 1.12 1.11 1.13 1.15
0.14 0.12 0.08 0.04 —
J FF
J RR
J RF
(]10~22 J) !0.37 !0.18 !0.14 !0.11 —
5.44 5.61 5.59 5.65 5.60
3.33 3.76 3.98 3.90 —
1.21 1.36 1.70 1.38 —
230
M. Manivel Raja et al. / Journal of Magnetism and Magnetic Materials 184 (1998) 227—230
4. Conclusions The results of the two-sublattice molecular field theory analysis of the saturation magnetisation of the Er Y Fe B compounds show that the in2~x x 14 crease in the Curie temperature of these compounds with the increase in Y concentration is attributable to the enhancement in the strength of the Fe—Fe exchange interaction. Also, the good agreement between the calculated and experimental values of the Curie temperature shows that the two-sublattice molecular field theory is quite successful in describing the temperature dependence of magnetisation of Er Y Fe B com2~x x 14 pounds.
Acknowledgements This work was supported by the Council of Scientific and Industrial Research (CSIR), Government of India under the research scheme No.
4(114)/91-EMR-II and also by the University Grants Commission (UGC), Government of India under the Special Assistance Programme (SAP). The authors would like to thank Dr. R. Premanand for his assistance in analysing magnetisation data using molecular field theory. One of the authors (M.M.R) would like to thank CSIR for the award of Senior Research Fellowship (SRF) during the course of this work.
References [1] R. Verhoef, R.J. Radwanski, J.J.M. Franse, J. Magn. Magn. Mater. 89 (1990) 176. [2] S. Sinnema, R.J. Radwanski, J.J.M. Franse, D.B. de Mooij, K.H.J Buschow, J. Magn. Magn. Mater. 44 (1984) 333. [3] C.D. Fuerst, J.F. Herbst, E.A. Alson, J. Magn. Magn. Mater. 54—57 (1986) 567. [4] H.S. Li, Z.W. Zhang, M.Z. Dang, J. Magn. Magn. Mater. 71 (1988) 355. [5] M. Manivel Raja, A. Narayanasamy, J. Alloys Compounds 233 (1996) 140. [6] J.F. Herbst, J.J. Croat, J. Appl. Phys. 55 (1984) 15.