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Sublattice magnetization and magnetization process in TbCu2
Journal of Magnetism and Magnetic Materials 70 (1987) 279-281 North-Holland, Amsterdam
279
SUBLATTICE MAGNETIZATION AND MAGNETIZATION PROCESS IN TbCuz N. IWATA, T. K I M U R A , T. S H I G E O K A Faculty of Science, Yamaguchi UnmersiO', Yamaguchi 753, Japan
and Y. H A S H I M O T O Department of Pt~vsics, Faculty of Education, Fukuoka University of Education, Munakata Fukuoka 811-41, Japan The TbCu 2 compound exhibits antiferromagnetism and has two nonequivalent Tb sites below 54 K ( - T y). The magnetization for the a-axis increases discontinuously at critical field and saturates at high fields. The magnetization process and magnetic moments have been analyzed in terms of the molecular field theory including the crystal-field interaction.
The compound TbCu 2 crystallizes in the orthorhombic CeCu 2 type structure. The atoms Tb and Cu occupy 4e and 8h sites, respectively. Magnetic and neutron diffraction measurements have been made on single crystal [1] and powdered samples [2]. The compound is antiferromagnetic below 54 K and the magnetic unit cell is three times as large as the chemical unit cell as shown in fig. 1. The magnetic moments lie along the a-axis, and there are two magnetic sites A and B (Tb A and TbB) in the antiferromagnetic region. The magnetic moments for Tb A and Tb B atoms have been determined by Sima et al. [2], and fig. 2 shows the data by broken curves. The magnetic moment of the Tb A atom decreases faster than that of TbB atom with increasing temperature for 0.3TN < T < TN. The magnetic moment at 4.2 K has been estimated to be 8.SktB for both atoms, which is close to the theoretical value of 9/~ B for free the Tb 3+ ion. The magnetization process has been measured by one of the present authors [1], and the results at 4.2 K are shown in fig. 3. The magnetization along the a-axis increases very slowly at small fields, and increases discontinuously at 19 kOe and reaches the saturation value. The magnetic properties of the compound may be well described by a model in which the crystalfield interaction and the long range indirect exchange interaction via conduction electron are essential. In the molecular field approximation, the exchange fields acting on Tb A and Tb B atoms are given for the antiferromagnetic structure as follows:
H ( A ) - ( g - 1)2 {(JBB - - J A B ) ( J ( A ) ) g/~B +JAB(J(B)) ) and H(B) - ( g - 1)2 { 2 J A B ( J ( A ) ) + ~ B B ( J ( B ) ) } g/~ B
Where g is the g-factor, and J(A) and J ( B ) ' i r e the total angular momenta of the respective atoms. The exchange constants JAB and o~BB can be given as and
J A . = J 3 - J4 - Js + J6 J..
= J, - J . -
•
? .
" --~?-,-; .
.
"/ , , ,,
~ :," i-~
',i ' ,., . . . .
.
.
.361 ,.!/ ' , ,,
,
,
*
.~
:/
.~,w
.
.
.
.
.
.
.
.
;i .
10
!
";.
b
7 I 1 _ ~ , --,....... ._l!j
-%~__,___~
,,,
(2)
J. + J,.
/
! ,-"
/
;,
/4
,
3Q
Fig. 1. Crystal structure and magnetic structure of TbCu 2. The Tb moments are indicated by arrows. The dark spheres represent Cu atoms.
0304-8853/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
280
N. lwata et al. / Sublattice ma~;netization in T b C u ,
I0
I
I
I
i
I
06 -,,,,,,
:3_ 4
---
exp.
--
cal.
~,~,
\
co
2 0
J
~/
i 20
0
I
J 40
' ~| 60
T(K) Fig. 2. The b r o k e n curves represent magnetic moments for T b A and T b ~ f r o m the w o r k of Sima et al. [2]. The solid curves represent calculated magnetic moments.
with
J, = Y'~ Y'~'J(m, j; n, i), m
(3)
/
where J ( m , j; n, i) is the exchange constant between the j t h atom in the ruth magnetic unit cell and the ith atom in the nth magnetic cell. The summation m includes all magnetic unit cells and £'~ is a sum over 12 atoms except for j = i when m = n, and J9 = "]5, J l o = ']6, J l l = J3 and J12 = J 4 ' The effective Hamiltonians of the respective atoms are given #of)o +B2022, Jt°(i) = - g b t B H ( i ) J ( i +-2"-'2 (i = A, B).
The thermal averages ( J ( i ) ) are obtained from the stable equilibrium condition that makes the free energy minimum. The best fit to the temperature dependence of the sublattice moments resulted from the values for JA~ and JBB which are given in table 1. The values of the crystal-field parameters B~~ and B~ have already been obtained by one of the present authors [1] as given in table 1. The magnetic moments at 4.2 K are calculated to be 8.960/x1~ for Tb A and 8.975/~ B for Tb~. These values are slightly larger than the experimental value of 8.8/za but close to 9/~ B for free the Tb 3+ ion. Magnetization process has been studied. In the presence of a magnetic field H, four Tb sites (A, A', B and B'-site) are to be considered for Hjla, while for Hllb- or c-axis there are two Tb sites. The values of the calculated magnetization and critical field at low temperatures are not sensitive to the choice of the exchange parameters provided that JAR and JBB have the values given in table 1. Solid curves in fig. 3 resulted from Jl + J~ = 3.2 K, J 2 + J 7 = - 1 . 7 K, J 3 + J 6 = 2 . 7 K and J4 4-,Is = - 1 . 0 K. For ferromagnetic arrangement there is only one magnetic site, the exchange field H(f) is given as follows: H(f)-
1 2 (g-)-o,~0(J(f)), gta B
(5)
where (4)
o¢'~ =J1 + J 2 + 2J3 + 2 J 4 + 2Js + 2,16 +-17 + J s -
(6) 10
I
8
i oOOO
o
I
o
o
o
o
The value of o¢"o was determined from the fact that the free energies for the ferromagnetic- and the antiferromagnetic-arrangement cross at 19 kOe. The value of o¢~0 is given in table 1. Kimura [3] has studied the stability of the antiferromagnetic structure and determined a possible range of the exchange constants which stabilizes the antiferromagnetic structure. He has
ao
l
5 s •,-
j
::3,,
io
i 2
0
4.2K
0
10
I
20
c
30
40
50
H (kOe) Fig. 3. Magnetization process measured at 4.2 K; (©)a-. (zx)band (E3)c-axis after ref. [1[. The solid curves represent calculated magnetizations.
Table 1 Exchange constants and crystal-field parameters (K)
JA.
.-eBB
%
B~'
B~
4.85
3,65
4.76
1.25
1.25
N. l w a t a et al. / Sublattice magnetization in T b C u 2
pointed out that J(m, 2; m, 1) < 0 (antiferromagnetic) and J(m, 8; m, 1) > 0. He has also shown that J(m, 6; rn, 1) plays an important role in the magnetization process. Some of the present exchange parameters seem to follow his prediction.
281
References [1] Y. Hashimoto, H. Fujii, H. Fujiwara and T. Okamoto, J. Phys. Soc. Japan 47 (1979) 67. Y. Hashirnoto, J. Sci. Hiroshima Univ. Ser. A43 (1979) 157. [2] V. Sima, Z. Smetana, B. Lebech and E. Gratz, J. Magn. Magn. Mat. 54-57 (1986) 1357. [3] 1. Kimura, J. Magn. Magn. Mat. 52 (1985) 199.
279
SUBLATTICE MAGNETIZATION AND MAGNETIZATION PROCESS IN TbCuz N. IWATA, T. K I M U R A , T. S H I G E O K A Faculty of Science, Yamaguchi UnmersiO', Yamaguchi 753, Japan
and Y. H A S H I M O T O Department of Pt~vsics, Faculty of Education, Fukuoka University of Education, Munakata Fukuoka 811-41, Japan The TbCu 2 compound exhibits antiferromagnetism and has two nonequivalent Tb sites below 54 K ( - T y). The magnetization for the a-axis increases discontinuously at critical field and saturates at high fields. The magnetization process and magnetic moments have been analyzed in terms of the molecular field theory including the crystal-field interaction.
The compound TbCu 2 crystallizes in the orthorhombic CeCu 2 type structure. The atoms Tb and Cu occupy 4e and 8h sites, respectively. Magnetic and neutron diffraction measurements have been made on single crystal [1] and powdered samples [2]. The compound is antiferromagnetic below 54 K and the magnetic unit cell is three times as large as the chemical unit cell as shown in fig. 1. The magnetic moments lie along the a-axis, and there are two magnetic sites A and B (Tb A and TbB) in the antiferromagnetic region. The magnetic moments for Tb A and Tb B atoms have been determined by Sima et al. [2], and fig. 2 shows the data by broken curves. The magnetic moment of the Tb A atom decreases faster than that of TbB atom with increasing temperature for 0.3TN < T < TN. The magnetic moment at 4.2 K has been estimated to be 8.SktB for both atoms, which is close to the theoretical value of 9/~ B for free the Tb 3+ ion. The magnetization process has been measured by one of the present authors [1], and the results at 4.2 K are shown in fig. 3. The magnetization along the a-axis increases very slowly at small fields, and increases discontinuously at 19 kOe and reaches the saturation value. The magnetic properties of the compound may be well described by a model in which the crystalfield interaction and the long range indirect exchange interaction via conduction electron are essential. In the molecular field approximation, the exchange fields acting on Tb A and Tb B atoms are given for the antiferromagnetic structure as follows:
H ( A ) - ( g - 1)2 {(JBB - - J A B ) ( J ( A ) ) g/~B +JAB(J(B)) ) and H(B) - ( g - 1)2 { 2 J A B ( J ( A ) ) + ~ B B ( J ( B ) ) } g/~ B
Where g is the g-factor, and J(A) and J ( B ) ' i r e the total angular momenta of the respective atoms. The exchange constants JAB and o~BB can be given as and
J A . = J 3 - J4 - Js + J6 J..
= J, - J . -
•
? .
" --~?-,-; .
.
"/ , , ,,
~ :," i-~
',i ' ,., . . . .
.
.
.361 ,.!/ ' , ,,
,
,
*
.~
:/
.~,w
.
.
.
.
.
.
.
.
;i .
10
!
";.
b
7 I 1 _ ~ , --,....... ._l!j
-%~__,___~
,,,
(2)
J. + J,.
/
! ,-"
/
;,
/4
,
3Q
Fig. 1. Crystal structure and magnetic structure of TbCu 2. The Tb moments are indicated by arrows. The dark spheres represent Cu atoms.
0304-8853/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
280
N. lwata et al. / Sublattice ma~;netization in T b C u ,
I0
I
I
I
i
I
06 -,,,,,,
:3_ 4
---
exp.
--
cal.
~,~,
\
co
2 0
J
~/
i 20
0
I
J 40
' ~| 60
T(K) Fig. 2. The b r o k e n curves represent magnetic moments for T b A and T b ~ f r o m the w o r k of Sima et al. [2]. The solid curves represent calculated magnetic moments.
with
J, = Y'~ Y'~'J(m, j; n, i), m
(3)
/
where J ( m , j; n, i) is the exchange constant between the j t h atom in the ruth magnetic unit cell and the ith atom in the nth magnetic cell. The summation m includes all magnetic unit cells and £'~ is a sum over 12 atoms except for j = i when m = n, and J9 = "]5, J l o = ']6, J l l = J3 and J12 = J 4 ' The effective Hamiltonians of the respective atoms are given #of)o +B2022, Jt°(i) = - g b t B H ( i ) J ( i +-2"-'2 (i = A, B).
The thermal averages ( J ( i ) ) are obtained from the stable equilibrium condition that makes the free energy minimum. The best fit to the temperature dependence of the sublattice moments resulted from the values for JA~ and JBB which are given in table 1. The values of the crystal-field parameters B~~ and B~ have already been obtained by one of the present authors [1] as given in table 1. The magnetic moments at 4.2 K are calculated to be 8.960/x1~ for Tb A and 8.975/~ B for Tb~. These values are slightly larger than the experimental value of 8.8/za but close to 9/~ B for free the Tb 3+ ion. Magnetization process has been studied. In the presence of a magnetic field H, four Tb sites (A, A', B and B'-site) are to be considered for Hjla, while for Hllb- or c-axis there are two Tb sites. The values of the calculated magnetization and critical field at low temperatures are not sensitive to the choice of the exchange parameters provided that JAR and JBB have the values given in table 1. Solid curves in fig. 3 resulted from Jl + J~ = 3.2 K, J 2 + J 7 = - 1 . 7 K, J 3 + J 6 = 2 . 7 K and J4 4-,Is = - 1 . 0 K. For ferromagnetic arrangement there is only one magnetic site, the exchange field H(f) is given as follows: H(f)-
1 2 (g-)-o,~0(J(f)), gta B
(5)
where (4)
o¢'~ =J1 + J 2 + 2J3 + 2 J 4 + 2Js + 2,16 +-17 + J s -
(6) 10
I
8
i oOOO
o
I
o
o
o
o
The value of o¢"o was determined from the fact that the free energies for the ferromagnetic- and the antiferromagnetic-arrangement cross at 19 kOe. The value of o¢~0 is given in table 1. Kimura [3] has studied the stability of the antiferromagnetic structure and determined a possible range of the exchange constants which stabilizes the antiferromagnetic structure. He has
ao
l
5 s •,-
j
::3,,
io
i 2
0
4.2K
0
10
I
20
c
30
40
50
H (kOe) Fig. 3. Magnetization process measured at 4.2 K; (©)a-. (zx)band (E3)c-axis after ref. [1[. The solid curves represent calculated magnetizations.
Table 1 Exchange constants and crystal-field parameters (K)
JA.
.-eBB
%
B~'
B~
4.85
3,65
4.76
1.25
1.25
N. l w a t a et al. / Sublattice magnetization in T b C u 2
pointed out that J(m, 2; m, 1) < 0 (antiferromagnetic) and J(m, 8; m, 1) > 0. He has also shown that J(m, 6; rn, 1) plays an important role in the magnetization process. Some of the present exchange parameters seem to follow his prediction.
281
References [1] Y. Hashimoto, H. Fujii, H. Fujiwara and T. Okamoto, J. Phys. Soc. Japan 47 (1979) 67. Y. Hashirnoto, J. Sci. Hiroshima Univ. Ser. A43 (1979) 157. [2] V. Sima, Z. Smetana, B. Lebech and E. Gratz, J. Magn. Magn. Mat. 54-57 (1986) 1357. [3] 1. Kimura, J. Magn. Magn. Mat. 52 (1985) 199.