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Thermal phonon influence on the magnetism of the Invar alloys Fe65Ni35 and Fe72Pt28
PhysicsLettersA North-Holland
184 (1994) 315-317
PHYSICS
LETTERS
A
Thermal phonon influence on the magnetism of the Invar alloys Fee5NiJ5 and Fe72Pt28 V.P. Silin and V.M. Zverev P.N. Lebedev Physical Institute, Leninsky Prospekt 53, Moscow 117333, Russian Federation
Received 10 October 1993; accepted for publication 2 1 October 1993 Communicated by V.M. Agranovich
It is shown that the influence of thermal phonons on the Curie constant, on the temperature dependence of the magnetization, on the specific heat and on the bark derivative of the Curie temperature is anomalously strong in the Invar alloys Fer2Pt2s and FesSNi3r due to the magnetoelasticity.
The magnetization dependence of the phonon spectrum due to magnetoelastic phenomena may essentially change the magnetic properties of magnets. This hypothesis was made by Kim [ 11. In a previous article it was shown [2] how to construct the necessary consistent theory. The quantitative influence of the magnetoelasticity on some properties of real ferromagnets was investigated in ref. [ 31. In this communication we describe the quantitative influence of thermal phonons on the Curie constant, the temperature dependence of the magnetization, the jump of the specific heat at the phase transition and the baric derivative of the Curie temperature in the Invar alloys Fe72Pt28 and Fes5NiS,. We use the following expression for the magnetic part of the free energy that was obtained in ref. [ 2 1,
+ 8’M2ql( T/8) .
(1)
Here M is the magnetization, V is the volume, T is the temperature, a, ( V, T) and a3( V) are determined by the electrons and 19is the Debye temperature. The function q(x) is connected with the usual Grtineisen function f(x) [ 41 by the relation [ 21 V(X) =f(x) -x?-‘(x) -f(O)
*
Curie temperature. Expression ( 1) takes into account the dependence of the real Debye temperature on the magnetization due to the magnetoelasticity in accordance with the law e, = e+ 8242 .
The magnetization dependences of the bulk modulus K,+pK+K’M2 and of the shear modulus GM= G+G’M’ give rise to the following relation for the
coefficient 8’ in the model of an isotropic medium
[51, (4) If the temperature T is higher than the Debye temperature it is possible to use the Debye approximation for the function pp(Tf e), ~(T/a=cPh(T/~--~)
0375-9601/94/$
3
(5)
where Cph= ~KN, is the high-temperature limit of the phonon specific heat, K is the Boltzmann constant and NA is the number of crystal atoms. In accordance with relations ( 1)- (5 ) we have the following equation for the magnetic induction of a ferromagnet,
(2)
Equation ( 1) is good enough in some applications to weak ferromagnets or near the phase transition 07.00 0 1994 Elsevier Science B.V. All rights reserved.
SSDI 0375-9601(93)E0915-W
(3)
2c,hB’ (T-@))M+a,M’, ve
(6)
315
Volume 184, number 3
where B is the magnetic induction, Near the Curie temperature T, it is possible to use the following expansion, a,(v, T)=al(T,)+a;(T,)(T-T,)
3
(7)
where a’, ( T,)= (da,/aT)v.The Curie temperature is determined in accordance with the equation ul(Tc)+~(T&')=O.
(8)
Then the equation of the magnetic state (6) obtains the form B=;(T-Tc)M+uaM3,
(9)
which is usual in the Landau theory of phase transitions. Here C is the Curie constant and & ~a=u;(T,)+-
2&e’ ve
.
Here the second summand is connected with the influence of phonons through the magnetoelastic effects. It is easy to see this influence in the temperature derivative of the magnetization in the ferromagnetic state (B= 0), ($$),=-;(a',(Tc)+F),
(11)
and in the jump of the specific heat at the constant volume 6CV at the Curie point, 6C,
-=T,
10 January 1994
PHYSICS LETTERS A
1 2Q
(12)
.
It is evident that the phonon contribution to relations ( 11 ), ( 12) is connected with the phonon influence on the Curie constant (see eq. ( 9 ) ) . To define the relative contribution of the thermal phonons to ( lo)-( 12) it is sufficient to determine the magnitude of the parameter
x= 2 cc,, et/
ve .
(13)
Experimental data permit us to determine this parameter for such popular Invar alloys as disordered Fe72Pt28 and FessN135. The magnetoelasticity of these alloys was investigated in refs. [ 6,7]. We also use the results of ref. [ 8 ] where the analysis of the experimental data [ 91 for the disordered alloy Fe,2Pt2s was performed. In table 1 we give the numerical data of refs. [ 6-91 on the elasticity and magnetoelasticity of two Invar alloys. The Curie temperatures of the disordered alloy Fe72Pt2s, T,=367 K [ 91, and of the alloy FesSNi3S, T,= 500 K [ 7 1, are higher than the Debye temperatures of these alloys, respectively 8= 320 K [ 61 and & 350 K [ lo]. Therefore we can use approximation ( 5 ) . The necessary values of the Debye specific heat Cph= ~KZV~,the experimental values of the Curie constant [ 111 and the molar volume Vmol[121 are given in table 2. We see in table 2 that the obtained values of the parameter X are not small compared with unity. Therefore the influence of the magnetoelasticity is very important for the Curie constant as well as for the specific heat jump ( 12) and for the temperature derivative of the magnetization. Moreover it is necessary to stress that the calculated values of the parameter X are negative. This means that the electron contribution a; (T,)must be large enough to compensate the destructive influence of the thermal phonons. It is easy to see that Cu‘,(T,)=2.9 in the case Fe72Pt28 and Cu; (T,)= 1.7 in the case of Fes5Ni,S. Without the knowledge of the temperature dependence al (T) it is difftcult to discuss the quantitative influence of the thermal phonons on the Curie temperature, it is only possible to calculate the value of a, ( T,).On the other hand eq. (7) permits one easily
Table 1 K
K’
et/e
( 10’ dyn/cm* G2)
G (10” dyn/cm’)
G’
(10” dyn/cm’)
( 10’ dyn/cm’ Gz)
( 10-8G-2)
15.9 [6] 14.1 [7]
1.0 [6], 1.3 [8] 0.73 [7]
7.1 [6] 6.1 [7]
-1.1 [6] -0.74 [7]
-5.4 -4.2
Volume 184, number 3
10 January I994
PHYSICS LETTERS A
Table 2 Alloy
Fed’tzs Pes5Nt,,
8.0 [ 121 6.9 [12]
to define the temperature,
baric
c Ph ( 10’ erg/m01 K)
c (emu K)
X
dlnf?/dIn
24.9 24.9
0.575 [ll] 0.232 [ll]
-1.9 -0.7
-2.1 [13] -2.3 [13]
derivative
of the
Curie
K dlnT, Bdp=-
-j(l-$(1-$$+~],
(14)
where & is the bulk modulus at constant magnetic induction. The first summand on the right side of eq. ( 14) contains the phonon influence through the Curie constant (see eq. ( 10) ). The second summand of ( 14) is proportional to the parameter X which describes the phonon influence. Till now we have no experimental information about the value of d In V/d In V. We know the value of d In B/d In V that is obtained on the base of the experimental data of ref. [ 131. Table 2 gives a value of Xd In 8/d In V of about (0.5-0.35) xd In TJd In V, so we may consider that the phonon influence is important for the real determination of the baric derivative of the Curie temperature of the Fe72Pt28 Invar alloy. Thus we have shown that thermal phonons play a very important role in the competition with the electron interaction which determines the magnetic properties of such Invar alloys as Fer2Ptz8 and Fee5NiJ5. This conclusion is not connected with any
V
d In T,/d In V 8-11 [11,13] 7-9 [11,13]
detailed electron model of ferromagnets.
This work is partly supported by the Russian Foundation of Fundamental Researches, the German Science Foundation (DFG) and by the Committee for Promotion of the Japan-FSU Collaboration.
References [l] D.Y. Kim, Phys. Rep. 171 (1988) 129. [2] V.M. Zverev and V.P. Silin, Sov. Phys. JETP 66 (1987) 401; Physica B 159 (1989) 43. [ 31 V.P. Silin, V.M. Zverev, M. Thon and D. Wagner, Phys. Lett. A154(1991)45. [4] L.D. Landau and E.M. Lifshitz, Statistical physics (Pergamon, Oxford, 1980). [ 51 V.M. Zverev and V.P. Silin, Sov. Phys. Solid State 30 (1988) 1148. [ 61 G. Hausch, J. Phys. Sot. Japan 3 7 ( 1974) 8 19. [7] G. Hausch, Phys. StatusSolidi (a) 15 (1973) 501. [ 81 D. Wagner, V.M. Zverev, V.P. Silin and M. Thon, JETP Lett. 56 (1992) 595. [ 9 ] L.I. Manosa, G.A. Saunders, H. Rahdi, V. Kawald, J. Pelzl and H. Bach, Phys. Rev. B 45 ( 1992) 2224. [lo] G. Hausch and H. Warlimont, Z. Metallkd. 63 (1972) 547. [ 111 J. Inoue and M. Shimizu, Phys. Lett. A 90 ( 1982) 85. [ 12 ] M. Shimizu, Rep. Prog. Phys. 44 (1981) 329. [ 131 G. Oomi and N. Mori, J. Phys. Sot. Japan 50 ( 1981) 2924.
317
184 (1994) 315-317
PHYSICS
LETTERS
A
Thermal phonon influence on the magnetism of the Invar alloys Fee5NiJ5 and Fe72Pt28 V.P. Silin and V.M. Zverev P.N. Lebedev Physical Institute, Leninsky Prospekt 53, Moscow 117333, Russian Federation
Received 10 October 1993; accepted for publication 2 1 October 1993 Communicated by V.M. Agranovich
It is shown that the influence of thermal phonons on the Curie constant, on the temperature dependence of the magnetization, on the specific heat and on the bark derivative of the Curie temperature is anomalously strong in the Invar alloys Fer2Pt2s and FesSNi3r due to the magnetoelasticity.
The magnetization dependence of the phonon spectrum due to magnetoelastic phenomena may essentially change the magnetic properties of magnets. This hypothesis was made by Kim [ 11. In a previous article it was shown [2] how to construct the necessary consistent theory. The quantitative influence of the magnetoelasticity on some properties of real ferromagnets was investigated in ref. [ 31. In this communication we describe the quantitative influence of thermal phonons on the Curie constant, the temperature dependence of the magnetization, the jump of the specific heat at the phase transition and the baric derivative of the Curie temperature in the Invar alloys Fe72Pt28 and Fes5NiS,. We use the following expression for the magnetic part of the free energy that was obtained in ref. [ 2 1,
+ 8’M2ql( T/8) .
(1)
Here M is the magnetization, V is the volume, T is the temperature, a, ( V, T) and a3( V) are determined by the electrons and 19is the Debye temperature. The function q(x) is connected with the usual Grtineisen function f(x) [ 41 by the relation [ 21 V(X) =f(x) -x?-‘(x) -f(O)
*
Curie temperature. Expression ( 1) takes into account the dependence of the real Debye temperature on the magnetization due to the magnetoelasticity in accordance with the law e, = e+ 8242 .
The magnetization dependences of the bulk modulus K,+pK+K’M2 and of the shear modulus GM= G+G’M’ give rise to the following relation for the
coefficient 8’ in the model of an isotropic medium
[51, (4) If the temperature T is higher than the Debye temperature it is possible to use the Debye approximation for the function pp(Tf e), ~(T/a=cPh(T/~--~)
0375-9601/94/$
3
(5)
where Cph= ~KN, is the high-temperature limit of the phonon specific heat, K is the Boltzmann constant and NA is the number of crystal atoms. In accordance with relations ( 1)- (5 ) we have the following equation for the magnetic induction of a ferromagnet,
(2)
Equation ( 1) is good enough in some applications to weak ferromagnets or near the phase transition 07.00 0 1994 Elsevier Science B.V. All rights reserved.
SSDI 0375-9601(93)E0915-W
(3)
2c,hB’ (T-@))M+a,M’, ve
(6)
315
Volume 184, number 3
where B is the magnetic induction, Near the Curie temperature T, it is possible to use the following expansion, a,(v, T)=al(T,)+a;(T,)(T-T,)
3
(7)
where a’, ( T,)= (da,/aT)v.The Curie temperature is determined in accordance with the equation ul(Tc)+~(T&')=O.
(8)
Then the equation of the magnetic state (6) obtains the form B=;(T-Tc)M+uaM3,
(9)
which is usual in the Landau theory of phase transitions. Here C is the Curie constant and & ~a=u;(T,)+-
2&e’ ve
.
Here the second summand is connected with the influence of phonons through the magnetoelastic effects. It is easy to see this influence in the temperature derivative of the magnetization in the ferromagnetic state (B= 0), ($$),=-;(a',(Tc)+F),
(11)
and in the jump of the specific heat at the constant volume 6CV at the Curie point, 6C,
-=T,
10 January 1994
PHYSICS LETTERS A
1 2Q
(12)
.
It is evident that the phonon contribution to relations ( 11 ), ( 12) is connected with the phonon influence on the Curie constant (see eq. ( 9 ) ) . To define the relative contribution of the thermal phonons to ( lo)-( 12) it is sufficient to determine the magnitude of the parameter
x= 2 cc,, et/
ve .
(13)
Experimental data permit us to determine this parameter for such popular Invar alloys as disordered Fe72Pt28 and FessN135. The magnetoelasticity of these alloys was investigated in refs. [ 6,7]. We also use the results of ref. [ 8 ] where the analysis of the experimental data [ 91 for the disordered alloy Fe,2Pt2s was performed. In table 1 we give the numerical data of refs. [ 6-91 on the elasticity and magnetoelasticity of two Invar alloys. The Curie temperatures of the disordered alloy Fe72Pt2s, T,=367 K [ 91, and of the alloy FesSNi3S, T,= 500 K [ 7 1, are higher than the Debye temperatures of these alloys, respectively 8= 320 K [ 61 and & 350 K [ lo]. Therefore we can use approximation ( 5 ) . The necessary values of the Debye specific heat Cph= ~KZV~,the experimental values of the Curie constant [ 111 and the molar volume Vmol[121 are given in table 2. We see in table 2 that the obtained values of the parameter X are not small compared with unity. Therefore the influence of the magnetoelasticity is very important for the Curie constant as well as for the specific heat jump ( 12) and for the temperature derivative of the magnetization. Moreover it is necessary to stress that the calculated values of the parameter X are negative. This means that the electron contribution a; (T,)must be large enough to compensate the destructive influence of the thermal phonons. It is easy to see that Cu‘,(T,)=2.9 in the case Fe72Pt28 and Cu; (T,)= 1.7 in the case of Fes5Ni,S. Without the knowledge of the temperature dependence al (T) it is difftcult to discuss the quantitative influence of the thermal phonons on the Curie temperature, it is only possible to calculate the value of a, ( T,).On the other hand eq. (7) permits one easily
Table 1 K
K’
et/e
( 10’ dyn/cm* G2)
G (10” dyn/cm’)
G’
(10” dyn/cm’)
( 10’ dyn/cm’ Gz)
( 10-8G-2)
15.9 [6] 14.1 [7]
1.0 [6], 1.3 [8] 0.73 [7]
7.1 [6] 6.1 [7]
-1.1 [6] -0.74 [7]
-5.4 -4.2
Volume 184, number 3
10 January I994
PHYSICS LETTERS A
Table 2 Alloy
Fed’tzs Pes5Nt,,
8.0 [ 121 6.9 [12]
to define the temperature,
baric
c Ph ( 10’ erg/m01 K)
c (emu K)
X
dlnf?/dIn
24.9 24.9
0.575 [ll] 0.232 [ll]
-1.9 -0.7
-2.1 [13] -2.3 [13]
derivative
of the
Curie
K dlnT, Bdp=-
-j(l-$(1-$$+~],
(14)
where & is the bulk modulus at constant magnetic induction. The first summand on the right side of eq. ( 14) contains the phonon influence through the Curie constant (see eq. ( 10) ). The second summand of ( 14) is proportional to the parameter X which describes the phonon influence. Till now we have no experimental information about the value of d In V/d In V. We know the value of d In B/d In V that is obtained on the base of the experimental data of ref. [ 131. Table 2 gives a value of Xd In 8/d In V of about (0.5-0.35) xd In TJd In V, so we may consider that the phonon influence is important for the real determination of the baric derivative of the Curie temperature of the Fe72Pt28 Invar alloy. Thus we have shown that thermal phonons play a very important role in the competition with the electron interaction which determines the magnetic properties of such Invar alloys as Fer2Ptz8 and Fee5NiJ5. This conclusion is not connected with any
V
d In T,/d In V 8-11 [11,13] 7-9 [11,13]
detailed electron model of ferromagnets.
This work is partly supported by the Russian Foundation of Fundamental Researches, the German Science Foundation (DFG) and by the Committee for Promotion of the Japan-FSU Collaboration.
References [l] D.Y. Kim, Phys. Rep. 171 (1988) 129. [2] V.M. Zverev and V.P. Silin, Sov. Phys. JETP 66 (1987) 401; Physica B 159 (1989) 43. [ 31 V.P. Silin, V.M. Zverev, M. Thon and D. Wagner, Phys. Lett. A154(1991)45. [4] L.D. Landau and E.M. Lifshitz, Statistical physics (Pergamon, Oxford, 1980). [ 51 V.M. Zverev and V.P. Silin, Sov. Phys. Solid State 30 (1988) 1148. [ 61 G. Hausch, J. Phys. Sot. Japan 3 7 ( 1974) 8 19. [7] G. Hausch, Phys. StatusSolidi (a) 15 (1973) 501. [ 81 D. Wagner, V.M. Zverev, V.P. Silin and M. Thon, JETP Lett. 56 (1992) 595. [ 9 ] L.I. Manosa, G.A. Saunders, H. Rahdi, V. Kawald, J. Pelzl and H. Bach, Phys. Rev. B 45 ( 1992) 2224. [lo] G. Hausch and H. Warlimont, Z. Metallkd. 63 (1972) 547. [ 111 J. Inoue and M. Shimizu, Phys. Lett. A 90 ( 1982) 85. [ 12 ] M. Shimizu, Rep. Prog. Phys. 44 (1981) 329. [ 131 G. Oomi and N. Mori, J. Phys. Sot. Japan 50 ( 1981) 2924.
317