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Martensitic transformation in ferromagnets: experiment and theory
~ ELSEVIER
Journal of Magnetism and Magnetic Materials 196-197 (1999) 859-860
Journalof m:lneusm magnetic
~i~ materials
Martensitic transformation in ferromagnets: experiment and theory V.A. Chernenko a'*, V.A. L'vov b, E.
Cesari c
alnstitute of Magnetism, Vernadsky str. 36, Kiev, 252680, Ukraine bTaras Shevchenko University, Glushkov str. 2, Kiev, 252127, Ukraine cUniversitat de les llles Balears, Ctra. de Valldemossa, Km 7.5, E-07071, Palma, Spain
Abstract
A phenomenological model relating the magnetization process in the twinned martensitic state of the Fe- and Ni-based alloys to the strain-induced magnetic anisotropy has been analyzed. The estimated values of the magnetic saturation fields, the magnetization jump at the martensitic transformation temperature and the magnetoelastic deformation of a N i - M n - G a shape memory alloy are in a good agreement with the experimental data. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Shape memory alloy; Magnetization; Magnetostriction; N i - M n - G a
It is known that some ferromagnetic alloy systems like N i - M n - G a , F e - N i - C o - T i , Fe-Pd, and F e - P t exhibit the cubic-tetragonal martensitic transformation (MT) (see, e.g., Refs. [1-5]). The martensite formed is characterized by the spatially inhomogeneous elastically self-accommodated substructure (twins, stacking faults, etc.). Concerning the magnetic state, the lowering of point symmetry group results in a considerable enhancement of magnetocrystalline anisotropy of martensite. Due to the magnetoelastic interaction the elastic energy stored by martensite gives rise to the magnetic anisotropy, which can dominate in a magnetization process [6], It was shown experimentally, that MT in N i - M n - G a shape memory alloys is accompanied by (i) abrupt decrease of magnetization measured at nonsaturation fields [1,2], (ii) nearly 40 times drop of the initial magnetic susceptibility [3], (iii) appearance of a maximum in the temperature dependence of the transversal magnetostric-
*Corresponding author. Universitat de les Illes Balears, Departmento de Fisica, Ctra. de Valldemossa, km 7.5, E-07071, Palma-de-Mallorca, Spain; e-mail: [email protected].
tion [4] and (iv) an order of magnitude increase of magnetostriction [5]. The results of theoretical analysis of the (i)- and (iv)-labeled phenomena in the framework of a phenomenological model of the ferromagnetic martensite [6] are reported. In the low magnetic fields H ~< Hcu, (H~ is the field for disappearance of 180 ° walls) both 180 ° and 90 ° magnetic domain walls exists in the martensite and the magnetization process is caused mainly by the displacement of 180 ° walls. The rotations of magnetic moments of the domains separated by 90 ° walls dominate for H > H~ fields. As it was shown in Ref. [6], the magnetic anisotropy of polytwinned martensite is induced mainly by the average deformation of the polytwins and it may be expected that at H > H~ polytwins represent the magnetic domains separated by 90 ° walls. General expressions for the magnetic anisotropy energy F ~u) and average magnetization I of martensite were obtained in Ref. [6]. For the five-layered periodic structure occuring below TM , the temperature of cubic-tetragonal MT, F tij) may be written as F tu~ -- :~56uo(m 2 - 3m2),
0304-8853/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 85 3(9 8 ) 0 0 9 8 0 - 9
(1)
KA. Chernenko et al. / Journal o/" Magnetism and Magnetic" Materials 196-197 (1999) 859-860
860 10080 E {4 60
~ 4o ~ 20
,
,
0
100
,
,
•
,
200 300 Temperature (K)
.
,
.
400
Fig. 1. Temperature dependencies of the magnetization of Ni-Mn-Ga alloy for H = 12 kOe (upper curve, circles) and H = 0.82 kOe (lower curve, traingle).
where 6 = 2±C' is the magnetoelastic constant, 2± and C' are the transversal magnetostriction and shear modulus of austenite, respectively, m is the unit vector directed along the magnetic moment of the polytwin, Uo = 2 ( a - c)/a, a and c are the lattice parameters of the tetragonal unit cell. Subscripts i,j mark the hard and easy magnetization axis, respectively. As far as the symmetry of polytwins is close to orthorhombic, the hard axis is perpendicular to the easy one. The angle ¢ between m and H has to satisfy the condition of energy minimum, i.e., ~(F " j ) - m H ) / ~ = 0, so cos ¢ = cos ~ = H / H , for H < H~ and cos ¢~ = 1 for H / > H, (~ = 1 corresponds to the polytwins with the hard axis directed along the field, H1 = 245u0/5 and = 2 corresponds to the case when both hard and easy axes are perpendicular to the field, H2 = 3H~/4). For the polytwins with the easy axis directed along the magnetic field ~k=0, 7r if H < H M and ~ = 0 for H>/HMr. The average magnetization I of martensite may be expressed as a sum of magnetic moments of polytwins [6]. Therefore, I(H,T) = ~Io(T)(AM + cos ~k~ + cos ~2)
Estimated Hx value coincides with the experimental field for magnetic saturation reported in Ref. [4]. Fig. 1 shows I(T) curves computed using Eq. (2) for HcM = H A = 1 kOe and two values of the external field. Theoretical I(T) dependencies are close to the experimental values (symbols in Fig. 1) taken from Ref. [2]. The field-induced strains arising in N i - M n - G a martensite at T < TM may be related to the magnetoelastic constant 6 bearing in mind that the five-layered martensitic structure transforms into the homogeneous tetragonal phase under the action of uniaxial mechanical stress a, ~ 2 x 108 erg/cm 3 at T M - - T ~ 10 K [7]. The specimen deformation e for stress values a < a, may be estimated from ~: ~ (a/aO(a c)/a. Since the dimensions of magnetic domains (polytwins) exceed by one order of magnitude the period of the five-layered martensitic structure, the magnetic field induces the quasiuniform stress a ~ 5 which partially transforms the five-layered martensite into the tetragonal one. As a consequence, the deformation at T < TM can be obtained: ~
~0.04--~2.4×10 O"t
(I
(3)
3.
O't
This value is close to the experimental magnitude of fieldinduced strain e ~ 2 x 10 -3 reported in Ref. [5]. By contrast, the magnetoelastic strains of the parent cubic phase are of the order of 2 x 10 -5 (see Ref. [4]). The comparatively large filed-induced strain e ~ 10 4 reported in Ref. [5] for the austenite phase could be attributed to the partial cubic-tetragonal stress-induced transformation observed in Ref. [7] Thus, the application of this phenomenological model of ferromagnetic martensite results in a satisfactory explanation of the unusual magnetic and magnetoelastic properties of N i - M n - G a shape memory alloys. V.A.C. is grateful to SAB95-0640).
DGESeIC
(sabbatical
stay
(2)
References where lo(T) = lo(O)y(T), y(T) = t a n h [ T c y(T)/T], AM = H / H M forH < H M and A = 1 for H ~> H M, Tc is the Curie temperature. The magnetization of austenite may be approximated by the standard formula I ( H , T ) = A A I o ( T ) , where AA = H/H)r for H < Hear and A = 1 otherwise (HA is the critical field for disappearance of the magnetic domain structure of austenite). For the N i - M n - G a alloy studied in Ref [4] TM = 285 K, Tc = 376 K, I(0) = 92 G cm3/g, Uo = 0.12, C'~6xI011 erg/cm 3, 2 ± ~ 2 x 1 0 -5 (see Ref. [6]). These experimental values result in the following estimations: ~ ~ 1.2 × 10 7 erg/cm 3, Ha ~ 10 K O e , / / 2 ~ 7.5 kOe.
l-l] P.J. Webster, K.R.A. Ziebeck, S.L. Town, M.S. Peak, Philos. Mag. B 49 (1984) 295. [2] V.V. Kokorin, V.A. Chernenko, V.I. Val'kov, S.M. Konoplyuk, E.A. Khapaliuk, Solid State Phys. 37 (1995) 3718. [3] A.N. Vasil'ev, V.V. Kokorin, Y.I. Savchenko, V.A. Chernenko, Sov. Phys. JETP 71 (1990) 803. [4] A.N. Vasil'ev, S.A. Klestov, R.Z. Levitin, V.V. Snegirev, V.V. Kokorin, V.A. Chernenko, JETP 82 (1996) 524. [5] K. Ullakko, J.K. Huang, C. Kantner, R.C. O'Handley, V.V Kokorin, Appl. Phys. Lett. 69 (1996) 1966. [6] V.A. L'vov, E.V. Gomonaj, V.A. Chernenko, J. phys.: Condens. Matter 10 (1998) 4587. [7] V.V. Kokorin, V.V. Martynov, Fiz. Met. Metall. 9 (19911106.
Journal of Magnetism and Magnetic Materials 196-197 (1999) 859-860
Journalof m:lneusm magnetic
~i~ materials
Martensitic transformation in ferromagnets: experiment and theory V.A. Chernenko a'*, V.A. L'vov b, E.
Cesari c
alnstitute of Magnetism, Vernadsky str. 36, Kiev, 252680, Ukraine bTaras Shevchenko University, Glushkov str. 2, Kiev, 252127, Ukraine cUniversitat de les llles Balears, Ctra. de Valldemossa, Km 7.5, E-07071, Palma, Spain
Abstract
A phenomenological model relating the magnetization process in the twinned martensitic state of the Fe- and Ni-based alloys to the strain-induced magnetic anisotropy has been analyzed. The estimated values of the magnetic saturation fields, the magnetization jump at the martensitic transformation temperature and the magnetoelastic deformation of a N i - M n - G a shape memory alloy are in a good agreement with the experimental data. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Shape memory alloy; Magnetization; Magnetostriction; N i - M n - G a
It is known that some ferromagnetic alloy systems like N i - M n - G a , F e - N i - C o - T i , Fe-Pd, and F e - P t exhibit the cubic-tetragonal martensitic transformation (MT) (see, e.g., Refs. [1-5]). The martensite formed is characterized by the spatially inhomogeneous elastically self-accommodated substructure (twins, stacking faults, etc.). Concerning the magnetic state, the lowering of point symmetry group results in a considerable enhancement of magnetocrystalline anisotropy of martensite. Due to the magnetoelastic interaction the elastic energy stored by martensite gives rise to the magnetic anisotropy, which can dominate in a magnetization process [6], It was shown experimentally, that MT in N i - M n - G a shape memory alloys is accompanied by (i) abrupt decrease of magnetization measured at nonsaturation fields [1,2], (ii) nearly 40 times drop of the initial magnetic susceptibility [3], (iii) appearance of a maximum in the temperature dependence of the transversal magnetostric-
*Corresponding author. Universitat de les Illes Balears, Departmento de Fisica, Ctra. de Valldemossa, km 7.5, E-07071, Palma-de-Mallorca, Spain; e-mail: [email protected].
tion [4] and (iv) an order of magnitude increase of magnetostriction [5]. The results of theoretical analysis of the (i)- and (iv)-labeled phenomena in the framework of a phenomenological model of the ferromagnetic martensite [6] are reported. In the low magnetic fields H ~< Hcu, (H~ is the field for disappearance of 180 ° walls) both 180 ° and 90 ° magnetic domain walls exists in the martensite and the magnetization process is caused mainly by the displacement of 180 ° walls. The rotations of magnetic moments of the domains separated by 90 ° walls dominate for H > H~ fields. As it was shown in Ref. [6], the magnetic anisotropy of polytwinned martensite is induced mainly by the average deformation of the polytwins and it may be expected that at H > H~ polytwins represent the magnetic domains separated by 90 ° walls. General expressions for the magnetic anisotropy energy F ~u) and average magnetization I of martensite were obtained in Ref. [6]. For the five-layered periodic structure occuring below TM , the temperature of cubic-tetragonal MT, F tij) may be written as F tu~ -- :~56uo(m 2 - 3m2),
0304-8853/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 85 3(9 8 ) 0 0 9 8 0 - 9
(1)
KA. Chernenko et al. / Journal o/" Magnetism and Magnetic" Materials 196-197 (1999) 859-860
860 10080 E {4 60
~ 4o ~ 20
,
,
0
100
,
,
•
,
200 300 Temperature (K)
.
,
.
400
Fig. 1. Temperature dependencies of the magnetization of Ni-Mn-Ga alloy for H = 12 kOe (upper curve, circles) and H = 0.82 kOe (lower curve, traingle).
where 6 = 2±C' is the magnetoelastic constant, 2± and C' are the transversal magnetostriction and shear modulus of austenite, respectively, m is the unit vector directed along the magnetic moment of the polytwin, Uo = 2 ( a - c)/a, a and c are the lattice parameters of the tetragonal unit cell. Subscripts i,j mark the hard and easy magnetization axis, respectively. As far as the symmetry of polytwins is close to orthorhombic, the hard axis is perpendicular to the easy one. The angle ¢ between m and H has to satisfy the condition of energy minimum, i.e., ~(F " j ) - m H ) / ~ = 0, so cos ¢ = cos ~ = H / H , for H < H~ and cos ¢~ = 1 for H / > H, (~ = 1 corresponds to the polytwins with the hard axis directed along the field, H1 = 245u0/5 and = 2 corresponds to the case when both hard and easy axes are perpendicular to the field, H2 = 3H~/4). For the polytwins with the easy axis directed along the magnetic field ~k=0, 7r if H < H M and ~ = 0 for H>/HMr. The average magnetization I of martensite may be expressed as a sum of magnetic moments of polytwins [6]. Therefore, I(H,T) = ~Io(T)(AM + cos ~k~ + cos ~2)
Estimated Hx value coincides with the experimental field for magnetic saturation reported in Ref. [4]. Fig. 1 shows I(T) curves computed using Eq. (2) for HcM = H A = 1 kOe and two values of the external field. Theoretical I(T) dependencies are close to the experimental values (symbols in Fig. 1) taken from Ref. [2]. The field-induced strains arising in N i - M n - G a martensite at T < TM may be related to the magnetoelastic constant 6 bearing in mind that the five-layered martensitic structure transforms into the homogeneous tetragonal phase under the action of uniaxial mechanical stress a, ~ 2 x 108 erg/cm 3 at T M - - T ~ 10 K [7]. The specimen deformation e for stress values a < a, may be estimated from ~: ~ (a/aO(a c)/a. Since the dimensions of magnetic domains (polytwins) exceed by one order of magnitude the period of the five-layered martensitic structure, the magnetic field induces the quasiuniform stress a ~ 5 which partially transforms the five-layered martensite into the tetragonal one. As a consequence, the deformation at T < TM can be obtained: ~
~0.04--~2.4×10 O"t
(I
(3)
3.
O't
This value is close to the experimental magnitude of fieldinduced strain e ~ 2 x 10 -3 reported in Ref. [5]. By contrast, the magnetoelastic strains of the parent cubic phase are of the order of 2 x 10 -5 (see Ref. [4]). The comparatively large filed-induced strain e ~ 10 4 reported in Ref. [5] for the austenite phase could be attributed to the partial cubic-tetragonal stress-induced transformation observed in Ref. [7] Thus, the application of this phenomenological model of ferromagnetic martensite results in a satisfactory explanation of the unusual magnetic and magnetoelastic properties of N i - M n - G a shape memory alloys. V.A.C. is grateful to SAB95-0640).
DGESeIC
(sabbatical
stay
(2)
References where lo(T) = lo(O)y(T), y(T) = t a n h [ T c y(T)/T], AM = H / H M forH < H M and A = 1 for H ~> H M, Tc is the Curie temperature. The magnetization of austenite may be approximated by the standard formula I ( H , T ) = A A I o ( T ) , where AA = H/H)r for H < Hear and A = 1 otherwise (HA is the critical field for disappearance of the magnetic domain structure of austenite). For the N i - M n - G a alloy studied in Ref [4] TM = 285 K, Tc = 376 K, I(0) = 92 G cm3/g, Uo = 0.12, C'~6xI011 erg/cm 3, 2 ± ~ 2 x 1 0 -5 (see Ref. [6]). These experimental values result in the following estimations: ~ ~ 1.2 × 10 7 erg/cm 3, Ha ~ 10 K O e , / / 2 ~ 7.5 kOe.
l-l] P.J. Webster, K.R.A. Ziebeck, S.L. Town, M.S. Peak, Philos. Mag. B 49 (1984) 295. [2] V.V. Kokorin, V.A. Chernenko, V.I. Val'kov, S.M. Konoplyuk, E.A. Khapaliuk, Solid State Phys. 37 (1995) 3718. [3] A.N. Vasil'ev, V.V. Kokorin, Y.I. Savchenko, V.A. Chernenko, Sov. Phys. JETP 71 (1990) 803. [4] A.N. Vasil'ev, S.A. Klestov, R.Z. Levitin, V.V. Snegirev, V.V. Kokorin, V.A. Chernenko, JETP 82 (1996) 524. [5] K. Ullakko, J.K. Huang, C. Kantner, R.C. O'Handley, V.V Kokorin, Appl. Phys. Lett. 69 (1996) 1966. [6] V.A. L'vov, E.V. Gomonaj, V.A. Chernenko, J. phys.: Condens. Matter 10 (1998) 4587. [7] V.V. Kokorin, V.V. Martynov, Fiz. Met. Metall. 9 (19911106.