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Magnetostriction in ferromagnetic shape memory alloys
Journal of Magnetism and Magnetic Materials 234 (2001) 25–30
Magnetostriction in ferromagnetic shape memory alloys V.V. Kokorina, M. Wuttigb,* b
a Institute of Magnetism, Vernadsky 36, 03142, Kiev, Ukraine Department of Material Science and Engineering, University of Maryland, College Park, MD 20742, USA
Received 9 January 2001; received in revised form 22 March 2001
Abstract There is intense interest in the development of new magnetostrictive materials having large strains. Recent experiments [1,2] showed that the ferromagnetic shape memory alloys (SMA) have some perspectives to generate significant magnetic-field-induced deformations. Taking into account the high mobility of the martensite crystal lattice variants, the corresponding magnetic measurements were performed by using the SMA Ni–Mn–Ga and Fe–Pd. It is concluded that the condensation of the soft acoustic phonon mode TA2 q ¼ 13 h1 1 0i takes place at T ¼ Ti > Ms in Ni–Mn–Ga alloy (Ms }martensite start temperature). Magnetic property anomalies accompany this phase transition which can be interpreted as a transition to an intermediate phase. The characteristic temperatures Ms and Ti are close in some alloys. The simultaneous action of the lattice instabilities connected with the formation of martensite and intermediate phases produces the essential decreasing of the elastic constants and as a result the strong growth of the high temperature phase magnetostriction nearly at T ¼ Ms . The behavior of the martensitic phases in magnetic field is also discussed. # 2001 Published by Elsevier Science B.V. Keywords: Martensite; Twins; Lattice instability; Magnetic field induced strain
1. Introduction The ferromagnetic shape memory alloys (FSMA) exhibit a number of interesting mechanical and magnetic properties. The most interesting examples of FSMA are Fe–Ni–Co–Ti, Fe–Pd and Ni–Mn–Ga alloys. The Fe–Pd alloy has SMA properties for a composition near Pd (30% at). This alloy exhibits premartensitic phenomena in X-ray and neutron scattering experiments. In this case a transformation FCC ! FCT (occurs face centered cubic transforms to face centered tetra*Corresponding author. Tel.: +301-405-5212; fax: +301314-9467. E-mail address: [email protected] (M. Wuttig).
gonal crystal lattice). The first measurements showed that the alloys Ni–Mn–Ga and Fe–Pd are perspective to reach the large strains in the moderate magnetic fields [1–3]. A variety of physical property anomalies having martensitic and magnetic nature takes place in the high and low temperature phases of Ni–Mn–Ga alloys [4]. Taking into account the high mobility of martensite transformation twins, the corresponding measurements were performed to determine their ability to induce mechanical strain by the magnetic field in Ni–Mn–Ga alloys [1,2]. The large magnetostriction was detected in the martensitic phase at T5Ms . It was supposed that the twin variants were aligned by a magnetic field giving the macroscopic strain (e 0:2%). Meanwhile a
0304-8853/01/$ - see front matter # 2001 Published by Elsevier Science B.V. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 2 4 4 - X
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measurement at fixed temperature at T > Ms in an austenite showed the magnetic field induced strain nearly ten times less than in the martensite [2]. On the other hand, the softening of elastic moduli observed in austenite prior to the martensitic transformation [5] can give rise to an increase of magnetostriction according to the traditional point of view. This moment will be considered in the paper. The results of measurements of the magnetostriction in an austenite phase of Ni–Mn– Ga alloy will be described in Section 2. Section 3 describes a new concept concerning a selection of the martensitic variants by the magnetic field in FSMA.
2. Premartensitic states and magnetostriction in FSMA The Ni–Mn–Ga alloys were used to study the influence of the premartensitic states on the magnetic field induced strains. These alloys were prepared by induction melting under argon atmosphere and casting into copper mold. The alloy compositions were equal: alloy 1 (at%): Ni–26.6 Mn–24.2 Ga and alloy 2: Ni–23.3 Mn–24 Ga. Single crystals were grown from ingots by the Bridgman method. These crystals were oriented by Laue photographs and cut by a diamond saw. The flat surface coincided with the (1 0 0) plane. The thermal diffuse X-ray intensity was determined using the diffractometer with the two-circle goniometer equipped by a low-temperature closed cycle refrigerator. Monochromatized Cu Ka radiation was used. The martensitic start temperatures (Ms ) were determined to be 165 and 271 K for alloys 1 and 2, respectively. Magnetic field induced strains were measured, using a strain gauge, with fields up to 10 kOe. It has recently been clarified [6] that the softening of the TA2 phonon mode in the alloy with Ms ¼ 165 K was completed by the phase transition (parent transforms to the intermediate phase). This phase transition was considered as a condensation of the TA2 phonon [6] at T ¼ Ti . The atomic vibrations refering to this mode become the static displacements at T ¼ Ti . Temperature dependence of the diffuse spot intensity (I(T)) corresponding the thermal X-ray
scattering by the phonons with wave vector q=1/3 q ¼ 13 h1 1 0i and transversal polarization e || h1 1 0i shows the characteristic anomalies for these phonons. The increase in the diffuse spot intensity during cooling is an evidence of the decreasing corresponding phonon frequency [6]. The dependence I(T ) shows that there is the temperature of the transition (Ti ) from the regime of diffuse scattering by dynamical atom displacements to the regime of temperature dependence of the Bragg reflection intensity. It means that the static wave has appeared with the same wave vector as the mentioned TA2 phonon mode. This process can be considered as an ordering of the displacements, the noncorrelated dynamical atom displacements at T > Ti becoming the correlated ones in the whole crystal at T4Ti . The atoms freeze in new positions corresponding to the crystal lattice of the intermediate phase. The decreasing of the elastic constants was observed in the vicinity of the ordering temperature Ti [6,7]. Meanwhile, the characteristic temperatures Ms and Ti are close in the alloys with Ms 5260 K, these alloys undergo direct transformation (parent phase transforms to martensite). The alloy (1) has two minima of elastic modulus at T ¼ Ti and T ¼ Ms [6], but alloy (2) has one pronounced minimum at T ¼ Ms [8]. It is possible to suppose that simultaneous action of the lattice instabilities, connected with the formation of martensite and intermediate phases at T ¼ Ti Ms in the alloy (2), produces more essential decreasing of elastic constants in comparison with the alloy (1) which has Ti 230 K and Ms 165 K. So, the magnetostriction measurements were performed using the alloy (2) samples as more perspective. Fig. 1 shows the longitudinal magnetostriction induced by magnetic field up to 8 kOe, the direction of field was close to [1 0 0]. The strong growth of the austenite striction can be observed to be nearly at T ¼ Ms (Fig. 1). The meaning e 103 was detected at T Ms . It is worth noting that the magnetic field induced strain (contraction) has the rather high value in relatively weak field H=600 Oe (Fig. 2). The temperature dependence of Fig. 1 resembles the function eðTÞ measured using pulsed magnetic fields [1]. But the largest strain in an austenite [1] is
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the decreasing of shear modulus (c0 ) is not so large to produce this behavior of magnetostriction. Further measurements are required to find an additional reason for determining the dependence eðTÞ (Fig. 1).
3. Selection of the martensite crystal orientation variants by magnetic field
Fig. 1. Magnetic field induced strain (e) as a function of temperature (Ms ¼ 271 K).
Fig. 2. Field dependence of the magnetostriction of a high temperature phase at T Ms .
more than ten times less in comparison with the result of measurement in the stationary magnetic field (Fig. 1).This difference can reflect some time dependence of the crystal reaction.The austenite magnetostriction, when a technical saturation is achieved, can be expressed [9] as e1 0 0 ¼ 2=3A1 =ðc11 c12 Þ;
ð1Þ
where e1 0 0 is the magnetic-field-induced strain along the direction [1 0 0], Hk[1 0 0]; A1 the magnetoelastic constant, c11 , c12 are the elastic moduli. In order to explain the significant growth of the magnetostriction (Fig. 1) at T5Ms we should suppose that the strong anomaly of the shear constant c0 ¼ ðc11 2c12 Þ=2 takes place in the vicinity of Ms . According to the sound velocity measurements [8] such anomaly exists near Ms , but
The nucleation of the martensite crystals is discussed in many papers; for example, one can see the detailed consideration in the review [10]. Magnetic field influence will be essential for the martensite crystal nucleation if the magnetocrystalline anisotropy of a martensite is large enough. In particular, the martensite crystal lattice induced by cooling or mechanical loading in the ferromagnetic alloy Ni–Mn–Ga has the tetragonal symmetry [11] with considerable uniaxial magnetocrystalline anisotropy (K 106 erg/cm3) [12] and the axes of the tetragonal distortion and easy magnetization are directed along the crystallographic axis [1 0 0]. The austenite and martensite phases are both ferromagnetic and the difference of the saturation magnetization at T ¼ Ms is not so large [13]. Small magnetic field influence on the martensitic start temperature for Ni–Mn–Ga alloys [14] proves that the difference between the magnetizations indeed is negligible. The saturation fields for both phases are different, high temperature phase was saturated at field Hs 1 kOe but martensite has Hs 8 kOe [13]. Thus, magnetocrystalline anisotropy will create the energy difference between the martensite inclusions with different crystal lattice orientations at H 8 kOe. We can then write the energy of small single magnetic domain martensite inclusion with uniaxial anisotropy: E ¼ VKef sin2 c V MHcosðj cÞ;
ð2Þ
where V is the volume of a martensite nucleus, Kef the anisotropy constant, M the specific magnetic moment of a martensite, c the angle between an easy axis (e.a.) and magnetic moment vector, j the angle between magnetic field and e.a. The nucleus has a size which is small enough to prevent the magnetic polydomain state. The first
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term (2) is the magnetic anisotropy energy and the second is Zeeman energy which depends on the mutual orientation of the vectors M and H. All tetragonality axis orientations of a martensite have the same probability for H ¼ 0 after cooling below Ms . But, the nucleus energy will be dependent on a vector H orientation in the case of large field. The energy can be written as: E ¼ VðKef MHÞ
ð3Þ
for j ¼ c ¼ p=2 in large enough field, when a magnetic moment rotation process was finished for an inclusion. The energy difference, for the differently oriented nuclei when the direction [1 0 0] of an austenite crystal coincides with H and the martensite crystal lattice tetragonality axes directed also along h1 0 0i, for the saturation field, will be DE ¼ VKef: Martensitic transformation can start when a condition : Df > fel þ f mag
ð4Þ
will be fulfilled, Df ¼ fa fb , fa and fb are the specific free energies of an austenite and martensite respectively; fel the specific elastic energy, fmag the magnetic anisotropy energy. The value Df is increasing during overcooling from the phase equilibrium temperature (T0 ). It is evident that the appearance of an anisotropy energy will require an additional overcooling, the condition (4) will be satisfied at less overcooling for the nuclei with parallel orientation of the tetragonality axes and H. If magnetic field will be oriented along h1 1 1i direction then the angles j will be the same for all martensite nuclei. It means that magnetic field selection of the variants depends on H orientation in an austenite single crystal and will be absent for H kh1 1 1i and e.a. k h1 0 0i. The sample size dependence on j can be used for experimental verification of this concept, taking into account that the preferred martensite crystal orientation gives rise to the shape change of a sample. The energy (En ) of the martensite nucleus which is capable of the spontaneous growth will be less at H k e.a. in the case of fixed overcooling. The nucleation velocity In (see for example [10])
depends on the temperature as: In expðEn =kT Þ;
ð5Þ
where k is the Boltzmann constant. The energy En can be written in case of unfavorable orientation (j ¼ c ¼ p=2) as En ¼ E0 þ Kef Vc ;
ð6Þ
where E0 is the critical nucleus energy at parallel orientation of H and e.a. Vc the volume of the critical nucleus, which can be taken Vc ¼ 1018 cm3 . Anisotropy energy Kef Vc ¼ 1012 ergs at Kef ¼ 106 erg=cm3 . This simple evaluation demonstrates that the nucleation of a nucleus with j ¼ c ¼ p=2 will be blocked by the multiplier expðKef Vc =kTÞ because Kef Vc 4kT. Therefore, the crystals with close to parallel orientations of an easy axis and magnetic field will mainly nucleate and grow during cooling. The formation of every martensite crystal produces the deformation of the sample shape. Mutual compensation of the shear deformations takes place at H ¼ 0 because all orientations are possible. But the variants with less anisotropy energy will dominate at saturation field, giving rise to the shear strain of a sample. The change of the sample length was observed [2] during cooling of the Ni–Mn–Ga alloy in magnetic field 10 kOe (Fig. 3). The strain at T ¼ Ms was 1.5 103, the length change took place in narrow temperature interval DT ¼ 2 K. The elongation was measured by means of a strain gauge, magnetic field was applied along the [1 0 0] direction of the high temperature phase crystal lattice. This result can be explained by using the concept of the selection of the martensite crystal lattice orientations during a nucleation in magnetic field. It is known that the martensitic phases, which take place during cooling of an austenite of Ni– Mn–Ga alloy with the same composition as was used for magnetic field cooling [2], have long period crystal lattice structures [11]. In this case, it is possibly to consider a martensite crystal as a sandwich with a multistage twinning, the primary twins are the very thin (one atomic layer) lamellae of two variants of the tetragonal phase. Such situation was named as an adaptive martensite
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Fig. 3. Strain vs. temperature in the magnetic fields H ¼ 0 and H ¼ 10 kOe.
[15]. According to the recent experiments [16], a single martensite twin variant with favorably oriented easy axis can be realized as a result of magnetic field effect (strain e 102 was fixed in this case). The multistage untwinning should be observed during magnetic field increasing as a consequence.
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striction was measured at T ¼ 213 K (alloy 2). Fig. 4 shows the field dependence of the strain of the martensite. The magnetostriction has a meaning e 2 103 at H ¼ 8 kOe. This value is consistent with e 1:9 103 observed at T ¼ 265 K for a martensite of similar alloy [2]. Meanwhile, the spontaneous transformation between different martensitic phases were found in this alloy [17]. The long period crystal lattice structure with 5 layer modulation transforms to the low temperature martensite phase with tetragonal crystal lattice without any modulation at Tm 110 K [17]. This type of martensitic transitions can also give rise to large magnetic field induced strains. Nucleation of martensite with the tetragonal crystal lattice inside an initial martensite of cooling in magnetic field will be accompanied by the selection of the martensite variants as it was considered in Section 3. The low temperature martensite has more magnetization [13] in comparison with the martensite which was formed in the vicinity of Ms temperature, the difference is about 20%. It is also important that the transformation heat for the intermartensitic transition is considerably less (ten times) than the heat of the parent phase to martensite transition [18]. In this case it is possibly to expect that the intermartensitic transition will be induced by a moderate magnetic field. If we use the Clapeyrone–Clausius thermodynamic equation (see for example [14]) for the estimation of the transition temperature shift DT in magnetic field, we will get DT 5210 K for magnetization M ¼ 80 emu=g [12], H ¼ 104 Oe,
4. Martensite phases of FSMA in magnetic field The martensitic phases of Ni–Mn–Ga alloys are sensitive as to different external influences (mechanical stresses, magnetic field, deviations of a composition). It was supposed [2,3] that the main reason explaining the high magnetic field induced strain in martensite of FSMA for the isothermal condition is the growth of the favorably oriented transformation twins. The martensite magneto-
Fig. 4. Strain vs. applied field in the martensite phase at T ¼ 213 K.
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Ms ¼ 3 102 K, transformation heat L¼ 120:5 J=g [19]. The temperature shift DT can be more than 5–10 K taking into account an inaccuracy of transformation heat measurement [19] for relatively small values of L in case of intermartensitic transformations.
5. Conclusions Ferromagnetic shape memory alloys have a mobile internal martensitic structure which gives rise to the high sensitivity of a sample size as a function of magnetic field. The premartensitic temperature range can be characterized by the increasing the austenite magnetostriction in the vicinity of Ms . The growth of the parent phase magnetostriction is explained by the elastic moduli anomaly preceding the martensitic transformation. Magnetic field nucleation of the martensitic crystals with high magnetocrystalline anisotropy was discussed as a possible reason for the very large magnetic field induced strain. Selection of the martensite crystal lattice variants can take place both due to cooling through the temperature interval of martensite transition in magnetic field and at a permanent temperature on account of field induced intermartensitic transitions during magnetic field increase.
References [1] A.N. Vasiliev, S.A. Klestov, R.Z. Levitin, V.V. Snegirev, V.V. Kokorin, V.A. Chernenko, JETP 82 (3) (1996) 524.
[2] K. Ullakko, J.K. Huang, C. Kantner, D.C. O’Handley, V.V. Kokorin, Appl. Phys. Lett. 69 (1996) 1966. [3] R.D. James, M. Wuttig, Phil. Mag. A 77 (5) (1998) 1273. [4] P.J. Webster, K.R.A. Ziebeck, S.L. Town, M.S. Peak, Phil. Mag. B 49 (3) (1984) 295. [5] A.N. Vasiliev, V.V. Kokorin, Yu.I. Savchenko, V.A. Chernenko, JETF 98 (4) (1990) 1437. [6] V.V. Kokorin, V.A. Chernenko, J. Pons, C. Segui, E. Cesari, Solid State Commun. 10 (1) (1997) 7. [7] L. Manoza, A. Gonzales-Comas, E. Obrado, A. Planes, V.A. Chernenko, V.V. Kokorin, E. Cesari, Phys. Rev. B 55 (17) (1997) 11068. [8] V.A. Chernenko, V.V. Kokorin, A.N. Vasiliev, Yu.I. Savchenko, Phase Transitions 43 (1993) 187. [9] S.V. Vonsovsky, (Ed.) Nauka, Moscow, 1971, p. 1032. [10] L. Kaufman, M. Cohen, Thermodynamics and kinetics of martensitic transformations, in: Progress in Metal Physics, Pergamon Press Oxford, Vol. 7, 1958. [11] V.V. Kokorin, V.V. Martynov, V.A. Chernenko, Scripta Met. 26 (2) (1992) 175. [12] R. Tickle, R.D. James, J. Magn. Magn. Mater. 195 (1999) 627. [13] V.V. Kokorin, V.T. Valkov, S.M. Konopliuk, V.A. Chernenko, E.A. Hapaliuk, Fiz.tverdogo tela (Solid State Physics) 37 (12) (1995) 3718. [14] I. Dikstein, V. Koledov, V. Shavrov, A. Tulaikova, A. Cherechukin, V. Buchelnikov, V. Khovailo, M. Matsumoto, T. Takagi, J. Tani, IEEE Trans. Magn. 35 (5) (1999) 3811. [15] A.G. Khachaturyan, S.M. Shapiro, S. Semenovskaya, Phys. Rev. B 43 (1991) 10832. [16] R. Tickle, R.D. James, T. Shield, M. Wuttig, V.V. Kokorin, IEEE Trans. Magn. 35 (5) (1999) 3811. [17] V.V. Kokorin, Proceedings of the International Conference on Solid–Solid Phase Transformations 99, The Japan Institute of Metals, 1999, p. 1100. [18] A.N. Vasilev, A. Kaiper, V.V. Kokorin, V.A. Chernenko, T. Takagi, J. Tani, JETF pisma (JETP Lett.) 58 (4) (1993) 297. [19] V.A. Chernenko, C. Segui, E. Cesari, J. Pons, V.V. Kokorin, Phys. Rev. B 57 (5) (1998) 2659.
Magnetostriction in ferromagnetic shape memory alloys V.V. Kokorina, M. Wuttigb,* b
a Institute of Magnetism, Vernadsky 36, 03142, Kiev, Ukraine Department of Material Science and Engineering, University of Maryland, College Park, MD 20742, USA
Received 9 January 2001; received in revised form 22 March 2001
Abstract There is intense interest in the development of new magnetostrictive materials having large strains. Recent experiments [1,2] showed that the ferromagnetic shape memory alloys (SMA) have some perspectives to generate significant magnetic-field-induced deformations. Taking into account the high mobility of the martensite crystal lattice variants, the corresponding magnetic measurements were performed by using the SMA Ni–Mn–Ga and Fe–Pd. It is concluded that the condensation of the soft acoustic phonon mode TA2 q ¼ 13 h1 1 0i takes place at T ¼ Ti > Ms in Ni–Mn–Ga alloy (Ms }martensite start temperature). Magnetic property anomalies accompany this phase transition which can be interpreted as a transition to an intermediate phase. The characteristic temperatures Ms and Ti are close in some alloys. The simultaneous action of the lattice instabilities connected with the formation of martensite and intermediate phases produces the essential decreasing of the elastic constants and as a result the strong growth of the high temperature phase magnetostriction nearly at T ¼ Ms . The behavior of the martensitic phases in magnetic field is also discussed. # 2001 Published by Elsevier Science B.V. Keywords: Martensite; Twins; Lattice instability; Magnetic field induced strain
1. Introduction The ferromagnetic shape memory alloys (FSMA) exhibit a number of interesting mechanical and magnetic properties. The most interesting examples of FSMA are Fe–Ni–Co–Ti, Fe–Pd and Ni–Mn–Ga alloys. The Fe–Pd alloy has SMA properties for a composition near Pd (30% at). This alloy exhibits premartensitic phenomena in X-ray and neutron scattering experiments. In this case a transformation FCC ! FCT (occurs face centered cubic transforms to face centered tetra*Corresponding author. Tel.: +301-405-5212; fax: +301314-9467. E-mail address: [email protected] (M. Wuttig).
gonal crystal lattice). The first measurements showed that the alloys Ni–Mn–Ga and Fe–Pd are perspective to reach the large strains in the moderate magnetic fields [1–3]. A variety of physical property anomalies having martensitic and magnetic nature takes place in the high and low temperature phases of Ni–Mn–Ga alloys [4]. Taking into account the high mobility of martensite transformation twins, the corresponding measurements were performed to determine their ability to induce mechanical strain by the magnetic field in Ni–Mn–Ga alloys [1,2]. The large magnetostriction was detected in the martensitic phase at T5Ms . It was supposed that the twin variants were aligned by a magnetic field giving the macroscopic strain (e 0:2%). Meanwhile a
0304-8853/01/$ - see front matter # 2001 Published by Elsevier Science B.V. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 2 4 4 - X
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measurement at fixed temperature at T > Ms in an austenite showed the magnetic field induced strain nearly ten times less than in the martensite [2]. On the other hand, the softening of elastic moduli observed in austenite prior to the martensitic transformation [5] can give rise to an increase of magnetostriction according to the traditional point of view. This moment will be considered in the paper. The results of measurements of the magnetostriction in an austenite phase of Ni–Mn– Ga alloy will be described in Section 2. Section 3 describes a new concept concerning a selection of the martensitic variants by the magnetic field in FSMA.
2. Premartensitic states and magnetostriction in FSMA The Ni–Mn–Ga alloys were used to study the influence of the premartensitic states on the magnetic field induced strains. These alloys were prepared by induction melting under argon atmosphere and casting into copper mold. The alloy compositions were equal: alloy 1 (at%): Ni–26.6 Mn–24.2 Ga and alloy 2: Ni–23.3 Mn–24 Ga. Single crystals were grown from ingots by the Bridgman method. These crystals were oriented by Laue photographs and cut by a diamond saw. The flat surface coincided with the (1 0 0) plane. The thermal diffuse X-ray intensity was determined using the diffractometer with the two-circle goniometer equipped by a low-temperature closed cycle refrigerator. Monochromatized Cu Ka radiation was used. The martensitic start temperatures (Ms ) were determined to be 165 and 271 K for alloys 1 and 2, respectively. Magnetic field induced strains were measured, using a strain gauge, with fields up to 10 kOe. It has recently been clarified [6] that the softening of the TA2 phonon mode in the alloy with Ms ¼ 165 K was completed by the phase transition (parent transforms to the intermediate phase). This phase transition was considered as a condensation of the TA2 phonon [6] at T ¼ Ti . The atomic vibrations refering to this mode become the static displacements at T ¼ Ti . Temperature dependence of the diffuse spot intensity (I(T)) corresponding the thermal X-ray
scattering by the phonons with wave vector q=1/3 q ¼ 13 h1 1 0i and transversal polarization e || h1 1 0i shows the characteristic anomalies for these phonons. The increase in the diffuse spot intensity during cooling is an evidence of the decreasing corresponding phonon frequency [6]. The dependence I(T ) shows that there is the temperature of the transition (Ti ) from the regime of diffuse scattering by dynamical atom displacements to the regime of temperature dependence of the Bragg reflection intensity. It means that the static wave has appeared with the same wave vector as the mentioned TA2 phonon mode. This process can be considered as an ordering of the displacements, the noncorrelated dynamical atom displacements at T > Ti becoming the correlated ones in the whole crystal at T4Ti . The atoms freeze in new positions corresponding to the crystal lattice of the intermediate phase. The decreasing of the elastic constants was observed in the vicinity of the ordering temperature Ti [6,7]. Meanwhile, the characteristic temperatures Ms and Ti are close in the alloys with Ms 5260 K, these alloys undergo direct transformation (parent phase transforms to martensite). The alloy (1) has two minima of elastic modulus at T ¼ Ti and T ¼ Ms [6], but alloy (2) has one pronounced minimum at T ¼ Ms [8]. It is possible to suppose that simultaneous action of the lattice instabilities, connected with the formation of martensite and intermediate phases at T ¼ Ti Ms in the alloy (2), produces more essential decreasing of elastic constants in comparison with the alloy (1) which has Ti 230 K and Ms 165 K. So, the magnetostriction measurements were performed using the alloy (2) samples as more perspective. Fig. 1 shows the longitudinal magnetostriction induced by magnetic field up to 8 kOe, the direction of field was close to [1 0 0]. The strong growth of the austenite striction can be observed to be nearly at T ¼ Ms (Fig. 1). The meaning e 103 was detected at T Ms . It is worth noting that the magnetic field induced strain (contraction) has the rather high value in relatively weak field H=600 Oe (Fig. 2). The temperature dependence of Fig. 1 resembles the function eðTÞ measured using pulsed magnetic fields [1]. But the largest strain in an austenite [1] is
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the decreasing of shear modulus (c0 ) is not so large to produce this behavior of magnetostriction. Further measurements are required to find an additional reason for determining the dependence eðTÞ (Fig. 1).
3. Selection of the martensite crystal orientation variants by magnetic field
Fig. 1. Magnetic field induced strain (e) as a function of temperature (Ms ¼ 271 K).
Fig. 2. Field dependence of the magnetostriction of a high temperature phase at T Ms .
more than ten times less in comparison with the result of measurement in the stationary magnetic field (Fig. 1).This difference can reflect some time dependence of the crystal reaction.The austenite magnetostriction, when a technical saturation is achieved, can be expressed [9] as e1 0 0 ¼ 2=3A1 =ðc11 c12 Þ;
ð1Þ
where e1 0 0 is the magnetic-field-induced strain along the direction [1 0 0], Hk[1 0 0]; A1 the magnetoelastic constant, c11 , c12 are the elastic moduli. In order to explain the significant growth of the magnetostriction (Fig. 1) at T5Ms we should suppose that the strong anomaly of the shear constant c0 ¼ ðc11 2c12 Þ=2 takes place in the vicinity of Ms . According to the sound velocity measurements [8] such anomaly exists near Ms , but
The nucleation of the martensite crystals is discussed in many papers; for example, one can see the detailed consideration in the review [10]. Magnetic field influence will be essential for the martensite crystal nucleation if the magnetocrystalline anisotropy of a martensite is large enough. In particular, the martensite crystal lattice induced by cooling or mechanical loading in the ferromagnetic alloy Ni–Mn–Ga has the tetragonal symmetry [11] with considerable uniaxial magnetocrystalline anisotropy (K 106 erg/cm3) [12] and the axes of the tetragonal distortion and easy magnetization are directed along the crystallographic axis [1 0 0]. The austenite and martensite phases are both ferromagnetic and the difference of the saturation magnetization at T ¼ Ms is not so large [13]. Small magnetic field influence on the martensitic start temperature for Ni–Mn–Ga alloys [14] proves that the difference between the magnetizations indeed is negligible. The saturation fields for both phases are different, high temperature phase was saturated at field Hs 1 kOe but martensite has Hs 8 kOe [13]. Thus, magnetocrystalline anisotropy will create the energy difference between the martensite inclusions with different crystal lattice orientations at H 8 kOe. We can then write the energy of small single magnetic domain martensite inclusion with uniaxial anisotropy: E ¼ VKef sin2 c V MHcosðj cÞ;
ð2Þ
where V is the volume of a martensite nucleus, Kef the anisotropy constant, M the specific magnetic moment of a martensite, c the angle between an easy axis (e.a.) and magnetic moment vector, j the angle between magnetic field and e.a. The nucleus has a size which is small enough to prevent the magnetic polydomain state. The first
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term (2) is the magnetic anisotropy energy and the second is Zeeman energy which depends on the mutual orientation of the vectors M and H. All tetragonality axis orientations of a martensite have the same probability for H ¼ 0 after cooling below Ms . But, the nucleus energy will be dependent on a vector H orientation in the case of large field. The energy can be written as: E ¼ VðKef MHÞ
ð3Þ
for j ¼ c ¼ p=2 in large enough field, when a magnetic moment rotation process was finished for an inclusion. The energy difference, for the differently oriented nuclei when the direction [1 0 0] of an austenite crystal coincides with H and the martensite crystal lattice tetragonality axes directed also along h1 0 0i, for the saturation field, will be DE ¼ VKef: Martensitic transformation can start when a condition : Df > fel þ f mag
ð4Þ
will be fulfilled, Df ¼ fa fb , fa and fb are the specific free energies of an austenite and martensite respectively; fel the specific elastic energy, fmag the magnetic anisotropy energy. The value Df is increasing during overcooling from the phase equilibrium temperature (T0 ). It is evident that the appearance of an anisotropy energy will require an additional overcooling, the condition (4) will be satisfied at less overcooling for the nuclei with parallel orientation of the tetragonality axes and H. If magnetic field will be oriented along h1 1 1i direction then the angles j will be the same for all martensite nuclei. It means that magnetic field selection of the variants depends on H orientation in an austenite single crystal and will be absent for H kh1 1 1i and e.a. k h1 0 0i. The sample size dependence on j can be used for experimental verification of this concept, taking into account that the preferred martensite crystal orientation gives rise to the shape change of a sample. The energy (En ) of the martensite nucleus which is capable of the spontaneous growth will be less at H k e.a. in the case of fixed overcooling. The nucleation velocity In (see for example [10])
depends on the temperature as: In expðEn =kT Þ;
ð5Þ
where k is the Boltzmann constant. The energy En can be written in case of unfavorable orientation (j ¼ c ¼ p=2) as En ¼ E0 þ Kef Vc ;
ð6Þ
where E0 is the critical nucleus energy at parallel orientation of H and e.a. Vc the volume of the critical nucleus, which can be taken Vc ¼ 1018 cm3 . Anisotropy energy Kef Vc ¼ 1012 ergs at Kef ¼ 106 erg=cm3 . This simple evaluation demonstrates that the nucleation of a nucleus with j ¼ c ¼ p=2 will be blocked by the multiplier expðKef Vc =kTÞ because Kef Vc 4kT. Therefore, the crystals with close to parallel orientations of an easy axis and magnetic field will mainly nucleate and grow during cooling. The formation of every martensite crystal produces the deformation of the sample shape. Mutual compensation of the shear deformations takes place at H ¼ 0 because all orientations are possible. But the variants with less anisotropy energy will dominate at saturation field, giving rise to the shear strain of a sample. The change of the sample length was observed [2] during cooling of the Ni–Mn–Ga alloy in magnetic field 10 kOe (Fig. 3). The strain at T ¼ Ms was 1.5 103, the length change took place in narrow temperature interval DT ¼ 2 K. The elongation was measured by means of a strain gauge, magnetic field was applied along the [1 0 0] direction of the high temperature phase crystal lattice. This result can be explained by using the concept of the selection of the martensite crystal lattice orientations during a nucleation in magnetic field. It is known that the martensitic phases, which take place during cooling of an austenite of Ni– Mn–Ga alloy with the same composition as was used for magnetic field cooling [2], have long period crystal lattice structures [11]. In this case, it is possibly to consider a martensite crystal as a sandwich with a multistage twinning, the primary twins are the very thin (one atomic layer) lamellae of two variants of the tetragonal phase. Such situation was named as an adaptive martensite
V.V. Kokorin, M. Wuttig / Journal of Magnetism and Magnetic Materials 234 (2001) 25–30
Fig. 3. Strain vs. temperature in the magnetic fields H ¼ 0 and H ¼ 10 kOe.
[15]. According to the recent experiments [16], a single martensite twin variant with favorably oriented easy axis can be realized as a result of magnetic field effect (strain e 102 was fixed in this case). The multistage untwinning should be observed during magnetic field increasing as a consequence.
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striction was measured at T ¼ 213 K (alloy 2). Fig. 4 shows the field dependence of the strain of the martensite. The magnetostriction has a meaning e 2 103 at H ¼ 8 kOe. This value is consistent with e 1:9 103 observed at T ¼ 265 K for a martensite of similar alloy [2]. Meanwhile, the spontaneous transformation between different martensitic phases were found in this alloy [17]. The long period crystal lattice structure with 5 layer modulation transforms to the low temperature martensite phase with tetragonal crystal lattice without any modulation at Tm 110 K [17]. This type of martensitic transitions can also give rise to large magnetic field induced strains. Nucleation of martensite with the tetragonal crystal lattice inside an initial martensite of cooling in magnetic field will be accompanied by the selection of the martensite variants as it was considered in Section 3. The low temperature martensite has more magnetization [13] in comparison with the martensite which was formed in the vicinity of Ms temperature, the difference is about 20%. It is also important that the transformation heat for the intermartensitic transition is considerably less (ten times) than the heat of the parent phase to martensite transition [18]. In this case it is possibly to expect that the intermartensitic transition will be induced by a moderate magnetic field. If we use the Clapeyrone–Clausius thermodynamic equation (see for example [14]) for the estimation of the transition temperature shift DT in magnetic field, we will get DT 5210 K for magnetization M ¼ 80 emu=g [12], H ¼ 104 Oe,
4. Martensite phases of FSMA in magnetic field The martensitic phases of Ni–Mn–Ga alloys are sensitive as to different external influences (mechanical stresses, magnetic field, deviations of a composition). It was supposed [2,3] that the main reason explaining the high magnetic field induced strain in martensite of FSMA for the isothermal condition is the growth of the favorably oriented transformation twins. The martensite magneto-
Fig. 4. Strain vs. applied field in the martensite phase at T ¼ 213 K.
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Ms ¼ 3 102 K, transformation heat L¼ 120:5 J=g [19]. The temperature shift DT can be more than 5–10 K taking into account an inaccuracy of transformation heat measurement [19] for relatively small values of L in case of intermartensitic transformations.
5. Conclusions Ferromagnetic shape memory alloys have a mobile internal martensitic structure which gives rise to the high sensitivity of a sample size as a function of magnetic field. The premartensitic temperature range can be characterized by the increasing the austenite magnetostriction in the vicinity of Ms . The growth of the parent phase magnetostriction is explained by the elastic moduli anomaly preceding the martensitic transformation. Magnetic field nucleation of the martensitic crystals with high magnetocrystalline anisotropy was discussed as a possible reason for the very large magnetic field induced strain. Selection of the martensite crystal lattice variants can take place both due to cooling through the temperature interval of martensite transition in magnetic field and at a permanent temperature on account of field induced intermartensitic transitions during magnetic field increase.
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