- Email: [email protected]
Magnetostriction of the SmCo5 hexagonal intermetallic
Journal of Magnetism and Magnetic Materials 157/158 (1996) 521-522
C•
journal of magnetism and magnetic materials
ELSEVIER
Magnetostriction
of the SmCo 5 hexagonal
intermetallic
P.A. Algarabel a, *, A. del Moral a, R. Krewenka a,b, J.A. Solera a, R. GrSssinger b Dpto. de Magnetismo, Dpto. de Fisica de la Materia Condensada-ICMA, Universidadde Zaragoza-CSIC, Zaragoza 5009, Spain b Institut3~rExperimentalphysilc, TU-Vienna,A-1040 Vienna, Austria Abstract Magnetostriction measurements of powder-sintered samples using pulsed magnetic fields up to 15 T and between 25 and 300 K have been performed. The strains were measured parallel, perpendicular and at 45 ° from the c-axis, with the field parallel and perpendicular to these directions. To explain the thermal dependence of the irreducible strains, the standard magnetostriction model has been extended to take into account the excited spin-orbit coupling multiplets of the Sm 3+ ion.
Keywords: Magnetostriction; Permanent magnets
We have studied the magnetostriction of the SmCo 5 alloy in a powder sintered sample, aligned along the easy c-axis. The measurements performed allows a separation of the equilibrium irreducible strains (e.i.s.) of hexagonal symmetry: volume distortion (•~1), c / a ratio variation (•~2), basal plane distortion ( • ~ ) and shear distortion of the c-axis (•~). Sm intermetallics offer a new situation where to study magnetostriction in the presence of excited • 6 6 IJM) multlplets, as HT/2 and H9/2 states are only up to 6 • • 3200 K above the ground state Hs/2. In such a SltUatmn one has to resort to use Racah operators, Uff, in order to express the crystal electric field (CEF), HcEF, and magnetoelastic (MEL), HMEL, Hamiltonians. The unperturbed Hamiltonian is for the Sm 3+ site (D6h) [1], Hsm=
E
BkU° + 21~BHex" S + ~L" S
k= 2,4,6 + /zB(L + S ) - n .
(1)
In the exchange Hamiltonian we have only considered the Co-Sin interaction, ~ ( = 590 cm - I ) is the spin orbit coupling parameter and B~ are the CEF parameters. The Sm 3+ sublattice free energy (Fsm) is calculated by the diagonalisation of the (LSJMIHsmlLSJ'M') matrix, bec o m i n g f s m : - k B T In Zsm, with Zs~ the partition function. The Co sublattice free energy is Fco = K 1 s i n e 0 ~co "H, with K 1 the anisotropy constant and 0 the angle between the cobalt ( ~ c o ) moment and the c-axis [2]. Minimization of Fs~ + Fco gives the equilibrium 0 values, allowing us to determine the eigenvectors and eigenvalues of the Sm 3# ion Hamiltonian for the full magneti-
* Corresponding author. Email: [email protected]; fax: + 34-76-553773•
zation equilibrium position, M = n s m "q- n c o . For the Sm 3+ ion, the CEF single-ion and the exchange two-ion MEL Hamiltonians, can be written, in terms of Racah operators Uq [3], as HIEL = _ ( m ~ 2 • or1 q_ mff2, =2)uO - M ' 2 ( • ~ V 2 z + e;/~2z) - iMC2(ezCVl - e(O2~) --(Mr4•
~1
q-Mr4•a2)U:-M'4(•~V:
--Me4'(•;U:
q-•;f.~:)
-- i M f 4 ( • 2 ¢ f 2
-1- i ~ @ ) -- • ( 0 2 ) ,
HIEL = -- (O;0" + D;0• 2)(SS" Sg) -- ( D r 2 ,
+ D;2•
:)(3S
sSzg
--
where U~'~= ( 1 / 2 ) ( U m + U~-m) and 0 ~ = ( 1 / 2 i ) ( U ~ UZ~); Mirj and D~ j are the single ion and exchange MEL coupling parameters, respectively. The expression of the elastic energy, F~I, in terms of the hexagonal symmetry elastic constants, CijF , is given in Ref. [3], Minimization of I L) + (HMz II L) + F~I, gives the e.i.s. thermal averages (H~m The expressions for the e.i.s., in terms of the MEL coupling parameters, elastic constants and the ( U q ) averages, are so obtained in a straightforward way [4]. For example, we present the expressions for the • " and • C strains,
~ = ( M ' 2 / C ' ) ( U ~ ) + ( M ' 4 / C ' ) ( U 4 z) +( M ' 4 ' / C ' ) ( U 2 ) , ~:(= ( M~2/CC)(U21) + ( M¢4/C¢)(U1). We remark that • ~ strains bear on CEF and exchange contributions, while the • ¢ and • ~ strains only depend on the CEF contribution.
0304-8853/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. SSDI 0 3 0 4 - 8 8 5 3 ( 9 5 ) 0 1 0 0 7 - 6
P.A. Algarabel et al. / Journal of Magnetism and MagneticMaterials 157/158 (1996) 521-522
522
By Ref. [3] the magnetostriction can be written as
100
1
1
S0 =
6o
:~
20
~
-
~
~
+ A[a,b] + A[a,c]},
0
I
(b)
•~
60
~
~ -
•
•
~
.
~,~(~)
, "
8 c~'2 (~
•
0
~
1
I
(c)
,O ~
t
-~ T e;,(d)
25
¢aO
.
"
app
20 ~
~ ( a ) = --~{A[a,a] - A[a,b},
where ¢e and ~ are the direction cosines of the magnetic field and the strain measurement directions respectively. ~ 1 ( c ) and ~ 2 ( c ) have the same expressions as ~ " a ( a ) and ~ 2 ( a ) , substituting a by c. The a, b, c, d and e directions correspond to the [001], [010], [001], [1,0,1] and [~,0,1] crystallographic directions. W e have obtained the Sm 3+ sublattice magnetostriction subtracting, from each A[ ¢e,~ ] mode, the corresponding for the YCo 5 compound [5]. In Fig. 1 we present the experimental results obtained for the thermal dependence of the e.i.s, together with the obtained theoretical fits. The used CEF parameters were B e = - 2 0 0 K, B 4 = - 5 0 K, B 6 = 0 and / z B H ~ = 200 K. These values are close to the ones of Ref. [6], obtained from magnetization measurements. From the theoretical fits it was possible to obtain the ratios between the magnetoelastic coupling parameters and the elastic constants. As an example we give below the values obtained from the fits of the e [ and e ( modes, which clearly show the importance of the fourth order contributions to the magnetostriction; these values are (M(2/C £) = - - 2 8 0 × 10 -6, (M~4/C ~) = --1900 X 10 -6, and (Me2/C e) = 100 X 10 -6, (Me4/C ") = - 3 0 0 0 X 10 -6. For all the modes the agreement with experiment is reasonable considering the error propagation in the combinations made to obtain the irreducible strains. Summarizing, the agreement attained with experiment signals the relevance of the excited multiplets, in order to explain the maguetostriction in SmCo 5, and this work
~,~(~)
H =15 T 40
•~
....
I
• • " 8o
~:°~2(a)= V ~ { A [ a , c ] - l ( A [ a , a ] + A[a,b])},
- A[d,e]},
I
_
<]
i • •
100
=
1 ~((d) = ~-{A[d,d]
~
~4 0
The dependence of the magnetostriction with the magnetization direction ( a ) is introduced via the mean values of the Racah operators appearing in the mentioned e.i.s. expressions. Combining the A[ a ,/3 ] modes we separate the e.i.s., i.e., = ~-{A[a,a]
,
SmCo5 Happ =15 T
..,.
+ vff ~x~y~; + v~ t3~t3z~( + vff t 3 y ~ .
~l(a)
i
(a)
1 1 A[ if,,/3 ] = __~.if3 + _ ~ _ {2132 __ ffi2 __ ]~2}~c~2
+
xiO6
1 •
•
"
I
I SmCo5 Ha p p =15 T
•
•
•
0
-25
-50
~
0
~
50
I
I
100 150 200 Temperature (K)
I
250
300
Fig. 1. Thermal variation for the Sm sublattice of: (a) the volume irreducible strain e~'l; (b) the c/a distortion e~2; (c) the basal plane strain, e~, and c-axis shear, e(. The lines are theoretical fits. supposes a step forward beyond the standard magnetostriction Callen & Callen framework [4]. Acknowledgement: The Spanish authors acknowledge the financial support of the CICYT under grant PB92-0095. References
[1] S.G. Sankar, V.S.U. Rao, E. Segal, W.E. Wallace, W.G.D. Frederick and HJ. Garret, Phys. Rev. B 11 (1975) 435. [2] Q. Lu, Ph.D. Thesis, University of Grenoble (1981), unpublished. [3] E. du Tremolet de Lacheisserie, Ann. Physique 5 (1970) 267. [4] E.R. Callen and H.B. Callen, Phys. Rev. 129 (1963) 578; 139 (1965) A455. [5] A. del Moral, P.A. Algarabel and M.R. Ibarra, J. Magn. Magn. Mater. 69 (1987) 285. [6] Zhao Tie-song, Jin Han-min, Guo Guang-hua, Han Xiu-feng and Chen Hong, Phys. Rev. B 43 (1991) 8593.
C•
journal of magnetism and magnetic materials
ELSEVIER
Magnetostriction
of the SmCo 5 hexagonal
intermetallic
P.A. Algarabel a, *, A. del Moral a, R. Krewenka a,b, J.A. Solera a, R. GrSssinger b Dpto. de Magnetismo, Dpto. de Fisica de la Materia Condensada-ICMA, Universidadde Zaragoza-CSIC, Zaragoza 5009, Spain b Institut3~rExperimentalphysilc, TU-Vienna,A-1040 Vienna, Austria Abstract Magnetostriction measurements of powder-sintered samples using pulsed magnetic fields up to 15 T and between 25 and 300 K have been performed. The strains were measured parallel, perpendicular and at 45 ° from the c-axis, with the field parallel and perpendicular to these directions. To explain the thermal dependence of the irreducible strains, the standard magnetostriction model has been extended to take into account the excited spin-orbit coupling multiplets of the Sm 3+ ion.
Keywords: Magnetostriction; Permanent magnets
We have studied the magnetostriction of the SmCo 5 alloy in a powder sintered sample, aligned along the easy c-axis. The measurements performed allows a separation of the equilibrium irreducible strains (e.i.s.) of hexagonal symmetry: volume distortion (•~1), c / a ratio variation (•~2), basal plane distortion ( • ~ ) and shear distortion of the c-axis (•~). Sm intermetallics offer a new situation where to study magnetostriction in the presence of excited • 6 6 IJM) multlplets, as HT/2 and H9/2 states are only up to 6 • • 3200 K above the ground state Hs/2. In such a SltUatmn one has to resort to use Racah operators, Uff, in order to express the crystal electric field (CEF), HcEF, and magnetoelastic (MEL), HMEL, Hamiltonians. The unperturbed Hamiltonian is for the Sm 3+ site (D6h) [1], Hsm=
E
BkU° + 21~BHex" S + ~L" S
k= 2,4,6 + /zB(L + S ) - n .
(1)
In the exchange Hamiltonian we have only considered the Co-Sin interaction, ~ ( = 590 cm - I ) is the spin orbit coupling parameter and B~ are the CEF parameters. The Sm 3+ sublattice free energy (Fsm) is calculated by the diagonalisation of the (LSJMIHsmlLSJ'M') matrix, bec o m i n g f s m : - k B T In Zsm, with Zs~ the partition function. The Co sublattice free energy is Fco = K 1 s i n e 0 ~co "H, with K 1 the anisotropy constant and 0 the angle between the cobalt ( ~ c o ) moment and the c-axis [2]. Minimization of Fs~ + Fco gives the equilibrium 0 values, allowing us to determine the eigenvectors and eigenvalues of the Sm 3# ion Hamiltonian for the full magneti-
* Corresponding author. Email: [email protected]; fax: + 34-76-553773•
zation equilibrium position, M = n s m "q- n c o . For the Sm 3+ ion, the CEF single-ion and the exchange two-ion MEL Hamiltonians, can be written, in terms of Racah operators Uq [3], as HIEL = _ ( m ~ 2 • or1 q_ mff2, =2)uO - M ' 2 ( • ~ V 2 z + e;/~2z) - iMC2(ezCVl - e(O2~) --(Mr4•
~1
q-Mr4•a2)U:-M'4(•~V:
--Me4'(•;U:
q-•;f.~:)
-- i M f 4 ( • 2 ¢ f 2
-1- i ~ @ ) -- • ( 0 2 ) ,
HIEL = -- (O;0" + D;0• 2)(SS" Sg) -- ( D r 2 ,
+ D;2•
:)(3S
sSzg
--
where U~'~= ( 1 / 2 ) ( U m + U~-m) and 0 ~ = ( 1 / 2 i ) ( U ~ UZ~); Mirj and D~ j are the single ion and exchange MEL coupling parameters, respectively. The expression of the elastic energy, F~I, in terms of the hexagonal symmetry elastic constants, CijF , is given in Ref. [3], Minimization of I L) + (HMz II L) + F~I, gives the e.i.s. thermal averages (H~m The expressions for the e.i.s., in terms of the MEL coupling parameters, elastic constants and the ( U q ) averages, are so obtained in a straightforward way [4]. For example, we present the expressions for the • " and • C strains,
~ = ( M ' 2 / C ' ) ( U ~ ) + ( M ' 4 / C ' ) ( U 4 z) +( M ' 4 ' / C ' ) ( U 2 ) , ~:(= ( M~2/CC)(U21) + ( M¢4/C¢)(U1). We remark that • ~ strains bear on CEF and exchange contributions, while the • ¢ and • ~ strains only depend on the CEF contribution.
0304-8853/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. SSDI 0 3 0 4 - 8 8 5 3 ( 9 5 ) 0 1 0 0 7 - 6
P.A. Algarabel et al. / Journal of Magnetism and MagneticMaterials 157/158 (1996) 521-522
522
By Ref. [3] the magnetostriction can be written as
100
1
1
S0 =
6o
:~
20
~
-
~
~
+ A[a,b] + A[a,c]},
0
I
(b)
•~
60
~
~ -
•
•
~
.
~,~(~)
, "
8 c~'2 (~
•
0
~
1
I
(c)
,O ~
t
-~ T e;,(d)
25
¢aO
.
"
app
20 ~
~ ( a ) = --~{A[a,a] - A[a,b},
where ¢e and ~ are the direction cosines of the magnetic field and the strain measurement directions respectively. ~ 1 ( c ) and ~ 2 ( c ) have the same expressions as ~ " a ( a ) and ~ 2 ( a ) , substituting a by c. The a, b, c, d and e directions correspond to the [001], [010], [001], [1,0,1] and [~,0,1] crystallographic directions. W e have obtained the Sm 3+ sublattice magnetostriction subtracting, from each A[ ¢e,~ ] mode, the corresponding for the YCo 5 compound [5]. In Fig. 1 we present the experimental results obtained for the thermal dependence of the e.i.s, together with the obtained theoretical fits. The used CEF parameters were B e = - 2 0 0 K, B 4 = - 5 0 K, B 6 = 0 and / z B H ~ = 200 K. These values are close to the ones of Ref. [6], obtained from magnetization measurements. From the theoretical fits it was possible to obtain the ratios between the magnetoelastic coupling parameters and the elastic constants. As an example we give below the values obtained from the fits of the e [ and e ( modes, which clearly show the importance of the fourth order contributions to the magnetostriction; these values are (M(2/C £) = - - 2 8 0 × 10 -6, (M~4/C ~) = --1900 X 10 -6, and (Me2/C e) = 100 X 10 -6, (Me4/C ") = - 3 0 0 0 X 10 -6. For all the modes the agreement with experiment is reasonable considering the error propagation in the combinations made to obtain the irreducible strains. Summarizing, the agreement attained with experiment signals the relevance of the excited multiplets, in order to explain the maguetostriction in SmCo 5, and this work
~,~(~)
H =15 T 40
•~
....
I
• • " 8o
~:°~2(a)= V ~ { A [ a , c ] - l ( A [ a , a ] + A[a,b])},
- A[d,e]},
I
_
<]
i • •
100
=
1 ~((d) = ~-{A[d,d]
~
~4 0
The dependence of the magnetostriction with the magnetization direction ( a ) is introduced via the mean values of the Racah operators appearing in the mentioned e.i.s. expressions. Combining the A[ a ,/3 ] modes we separate the e.i.s., i.e., = ~-{A[a,a]
,
SmCo5 Happ =15 T
..,.
+ vff ~x~y~; + v~ t3~t3z~( + vff t 3 y ~ .
~l(a)
i
(a)
1 1 A[ if,,/3 ] = __~.if3 + _ ~ _ {2132 __ ffi2 __ ]~2}~c~2
+
xiO6
1 •
•
"
I
I SmCo5 Ha p p =15 T
•
•
•
0
-25
-50
~
0
~
50
I
I
100 150 200 Temperature (K)
I
250
300
Fig. 1. Thermal variation for the Sm sublattice of: (a) the volume irreducible strain e~'l; (b) the c/a distortion e~2; (c) the basal plane strain, e~, and c-axis shear, e(. The lines are theoretical fits. supposes a step forward beyond the standard magnetostriction Callen & Callen framework [4]. Acknowledgement: The Spanish authors acknowledge the financial support of the CICYT under grant PB92-0095. References
[1] S.G. Sankar, V.S.U. Rao, E. Segal, W.E. Wallace, W.G.D. Frederick and HJ. Garret, Phys. Rev. B 11 (1975) 435. [2] Q. Lu, Ph.D. Thesis, University of Grenoble (1981), unpublished. [3] E. du Tremolet de Lacheisserie, Ann. Physique 5 (1970) 267. [4] E.R. Callen and H.B. Callen, Phys. Rev. 129 (1963) 578; 139 (1965) A455. [5] A. del Moral, P.A. Algarabel and M.R. Ibarra, J. Magn. Magn. Mater. 69 (1987) 285. [6] Zhao Tie-song, Jin Han-min, Guo Guang-hua, Han Xiu-feng and Chen Hong, Phys. Rev. B 43 (1991) 8593.