Deformation twinning in Ni2MnGa

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Acta Materialia 60 (2012) 3976–3984 www.elsevier.com/locate/actamat

Deformation twinning in Ni2MnGa R.C. Pond a, B. Muntifering b, P. Mu¨llner b,⇑ a

College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter EX2 7JL, UK b Department of Materials Science and Engineering, Boise State University, Boise, ID 83725, USA Received 18 November 2011; received in revised form 21 March 2012; accepted 24 March 2012 Available online 2 May 2012

Abstract Deformation twinning is investigated in the martensitic phase of a Ni46.75Mn34Ga19.25 (at.%) alloy. X-ray and electron diffraction are used to establish the crystallography of the non-modulated tetragonal martensite, and transmission electron microscopy is employed to deduce the twinning parameters. It is convenient to define the twinning parameters with respect to a “monoclinic” unit cell, designated 2M: then K1, g1, K2, and g2 are (0 0 1), [1 0 0], (1 0 0), and [0 0 1] respectively. The Burgers vector of the active twinning disconnections is close to 1/6[1 0 0] and the disconnections are associated with steps of height d(002). These defects are expected to be highly mobile since their motion does not require atomic shuffling. It is shown that periodic arrangements of two layer twins produce modulated crystal structures, such as 14M. Ó 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Transmission electron microscopy; Heusler phases; Deformation structure; Twinning; Dislocations

1. Introduction Alloys with compositions near Ni2MnGa (hereafter NMG) are ferromagnetic shape memory alloys (FSMAs) suitable for device applications [1–4]. For example, in the martensitic state, strains of up to 10% can be induced reversibly by an external magnetic field [5,6]. To exploit the potential of this material, it is important to elucidate the microstructural processes underpinning its useful properties. NMG transforms from the cubic L21 ordered Heusler phase [7] to one of several known martensite structures. The ground state of the martensitic phase is thought to be tetragonal [8], however, several studies have reported more complex forms wherein tetragonal or orthorhombic regions are modulated by thin lamellae of twinned material arranged more or less periodically [9–11]. The objective of the present work is to characterise these twinned regions using transmission electron microscopy (TEM) and to ⇑ Corresponding author. Tel.: +1 208 426 5136.

E-mail address: [email protected] (P. Mu¨llner).

deduce their mechanism of formation. In Section 2 we review the crystallography of NMG phases, and outline the principles of deformation twinning in Section 3. Our experimental observations are set out in Section 4, and interpreted in Section 5. Section 6 is a discussion of the results, and Section 7 is a summary of our conclusions. 2. Crystal structures A unit cell of cubic NMG is depicted in Fig. 1a. This phase is denoted C with lattice parameter ac and spacegroup Fm3m [7]. At lower temperatures a tetragonal form T arises [9] (Fig. 1b), with lattice parameters aT and cT, and spacegroup Fm4 mm. Both of these unit cells comprise 16 atoms: 4 lattice sites, each decorated by a Ni2MnGa motif in the stoichiometric compound. In the present context it is convenient to use an alternative description of the tetragonal form in terms of a “monoclinic” unit cell exhibiting 8 atoms Fig. 1c [12]. This body-centred cell is designated 2M, and has lattice parameters a2M, b2Mp, c2M, and b2M, where a2M = c2M, b2M = bT, c2M = 1/2 (c2T þ a2T ), and b2M = 2tan–1cT/aT. All three unit cells depicted in Fig. 1

1359-6454/$36.00 Ó 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.actamat.2012.03.045

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Fig. 2. Schematic illustration of twinned unit cells 2M and 2Mt projected along [0 1 0]. The twinning parameters K1, g1, K2, and g2 are (0 0 1), [1 0 0], (1 0 0), and [0 0 1], respectively, and the elementary twinning disconnection has b = 2D[1 0 0] and h = d(002). The diagram is constructed for a nominal value of D of 1/12.

Fig. 1. Crystal structures: (a) cubic, C; (b) tetragonal, T; (c) monoclinic, 2M. Mn, Ga and Ni are represented by green, red and blue symbols, respectively. All structures are viewed along their [0 1 0] directions.

are viewed along their [0 1 0] direction. As shown in Fig. 1c, successive (0 0 2)2M/(2 0 2)T planes are relatively sheared with displacement D in the direction [ 1 0 02M =½1 0 1T [12,13]. We subsequently show that the Burgers vectors of twinning disconnections are equal to 2D[1 0 0]. The coordinate transformations P inter-relating the C, T and 2M crystal structures are set out in Appendix A. 3. Deformation twinning The mechanism of deformation twinning in various crystal structures has been reviewed by Christian and Mahajan [14]. It proceeds by the motion of twinning dislocations, referred to here as disconnections, since these defects exhibit both dislocation and a step character [15]. In simple cases the Burgers vectors b of such disconnections are parallel to g1, and their step heights h, are equal to the interplanar spacing of the K1 planes. Motion of a disconnection across each sequential K1 plane produces

Fig. 3. Schematic view along [0 1 0] of the tip of a four layer deformation twin in a 2M crystal. The dislocation symbols mark the disconnections. A half plane of type ð6 0 1Þ2M ends at each disconnection. The offsets of the 200 matrix (red) planes across the twin (blue) changes from one disconnection to the next by +1/3, 1/3, and 0. The offset of 0 after three disconnections is highlighted with a red dotted line. The diagram is constructed for a nominal value of D of 1/12. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

the twin. The twinned form of 2M crystals, designated 2Mt, is depicted in Fig. 2. We take the twinning operation inter-relating 2M and 2Mt to be a mirror reflection across the K1 plane, as defined in Appendix A. In this case the twin is compound, since K1, g1, K2, and g2 are all rational, as depicted in Fig. 2. For simplicity the nominal value of D is taken to be the rational value 1/12, in this work, unless stated otherwise. The (b, h) parameters for twinning disconnections can be obtained from the theory of interfacial defects [16]: for the elementary disconnection we have b = 1/2½1 1  12Mt  1=2½1 1 12M , which can be expressed as b = 2D[1 0 0], and h = d(002) (indexing relates to 2M unless otherwise stated). Motion of this disconnection is expected to occur at low stresses, since all atoms would be sheared into the correct positions without additional shuffling [14]. Deformation

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4. Experimental observations 4.1. Specimen preparation

Fig. 4. Unit cell of 14M NMG showing the periodic intrusion of two layer twinning in a 2M crystal. Twinning planes are indicated by dashed lines.

twins exhibit pile-up configurations of disconnections towards their tips, as illustrated schematically in Fig. 3. These are (b, h) disconnections: if the lower surface of the twin were stepped rather than flat as shown in Fig. 3, the defects there would be (b, h) disconnections (line direction ½0 1 0). Growth of such twins in originally perfect 2M crystal could produce modulated structures [17]. For example, periodic arrangements of two layer twins produce the idealised 14M and 10M structures [17]: when regarded as crystals with large unit cells, these are truly monoclinic with spacegroup I2/m, with the former illustrated in Fig. 4. Using the Zdanov notation [18] the 14M cell is designated (5; 2Þ2 , where the first symbol refers to a sequence of five (0 0 2)2M planes, the second to two ð0 0 2Þ2Mt planes, and the subscript to the fact that this arrangement occurs twice within a 14M unit cell: the 10M cell is designated ( 3; 2Þ2 .

Polycrystalline NMG of nominal composition Ni46.75Mn34Ga19.25 (at.%) was prepared in a Reitel induction furnace from the constitutive metals, 99.9% pure Ni (Alfa Aesar), 99.9% pure Mn (Alfa Aesar), and 99.999% pure Ga (Sigma Aldrich), cast in a copper mould. During the casting 2.5% of the mass of the ingot was lost. Subsequently the composition was determined using a Hitachi S-3400-II scanning electron microscope equipped with an energy dispersive spectrometer and found to be Ni46.17Mn34.04Ga19.80. Similar compositions have been reported to result in non-modulated martensite [19]. The bulk ingot was sectioned and mechanically polished down to 1 lm grit size for X-ray diffraction (XRD) characterisation using a Bruker D8 Discover diffractometer with a Cu Ka source equipped with a Go¨bel mirror and point detector. Hot stage and room temperature XRD were carried out to determine the lattice parameters of the C and T/2M lattice parameters, as recorded in Table 1. For examination by TEM a section of the polycrystalline sample was mechanically thinned to 80–120 lm, from which 3 mm disks were obtained using a disc punch (model 656, Gatan Inc.). Thin foils were then obtained in a TenuPol 3 (Struers) double jet electro-polisher operated at 243 K and 10 V. The electrolyte used was 700 ml of methanol (Aldrich) and 300 ml of 69.9 vol.% nitric acid (Aldrich). The foils were examined at room temperature in a JEOL 2100 HR electron microscope operated at 200 keV with a LaB6 filament. 4.2. TEM In foil regions remote from inter-variant boundaries relatively large areas of non-modulated 2M and 2Mt crystals were observed, giving the selected area diffraction patterns shown in Fig. 5. The spot spacings and positions in these patterns are consistent with the lattice parameters listed in Table 1. The reflections 2 0 2 and 2 0 2 in Fig. 5a and 0 4 0 in Fig. 5b correspond to the tetragonal reflections 4 0 0T, 0 0 4T, and 0 4 0T, respectively, as may be confirmed using matrix (A6) in Eq. (A2) of Appendix A. Profuse twinning was observed in regions near inter-variant boundaries. A typical view showing twins tapering away from an inter-variant boundary is shown in Fig. 6.

Table 1 Lattice parameters of the cubic and tetragonal phases obtained by XRD. ˚) Phase Lattice parameter (A

Cubic Tetragonal “Monoclinic” 2M

Angle (°)

a

b

c

a

b

c

5.89 ± 0.01 5.50 ± 0.02 4.29 ± 0.02

5.50 ± 0.02 5.50 ± 0.02

6.58 ± 0.04 4.29 ± 0.02

90 90 90

90 90 100.2 ± 0.5

90 90 90

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Fig. 5. Selected area diffraction patterns from non-modulated 2M/2Mt crystals. The incident electron beam directions are: (a) ½0 1 02M=2Mt ; (b) ½1 0 12M ; (c) ½2 1 02M=2Mt .

but for clarity not all reflections are shown. A guide to the relative intensities of the reflections can be obtained using the kinematic theory of electron diffraction, although, of course, electron diffraction is strongly dynamic [20]. We define the “motif factor” as the amplitude scattered by the four atoms in a Ni2MnGa motif: these are chosen to lie in a (0 0 1) plane. The internal coordinates of these atoms in 2M and 2Mt unit cells are taken to be Mn 0, 0, 0 (at the centre of symmetry), Ga 0, ½, 0, Ni ½, 1=4 , 0, and Ni ½,3=4 ,0. The motif factor for a reflection (h, k, l)2M in the 2M stoichiometric structure is given by: F m ðh; k; lÞ2M ¼ Ri fi exp 2piðhxi þ ky i þ lzi Þ Fig. 6. Bright field image with g = 2 0 2 of twinned region near an inter-variant boundary. The box highlights a twin tapering away from the inter-variant boundary. The projection of the line direction n of the disconnections is approximately parallel to the reciprocal space vector g = 2 0 2.

The twins were further investigated by forming two beam images of the boxed region indicated in Fig. 6 using three non-coplanar reciprocal space vectors 2 0 2, 1 2 1, and 0 4 0. The micrographs are shown in Fig. 7.

ð1Þ

where fi is the atomic scattering factor, and the summation is taken over the atoms (i = 1–4) in the motif. Similar expressions can be written for the 2Mt structure using the planes in those crystals corresponding to (h, k, l)2M. For simplicity we do not consider the angular dependences of fi, but simply take them to be equal to the atomic numbers of Ni (28), Mn (25) and Ga (31). The structure factor for 2M is given by: F ðh; k; lÞ2M ¼ F m ðh; k; lÞ2M Rj exp 2piðhxj þ ky j þ lzj Þ

ð2Þ

5. Interpretation of the experimental observations

where the summation is over the two lattice sites in the unit cell. We designate this sum L2M, i.e.

5.1. Electron diffraction patterns

L2M ¼ f1 þ exp piðh þ k þ lÞg

Unit cells of the 2M and 2Mt reciprocal lattices are depicted schematically in Fig. 8. These cells are face-centred,

L2M specifies the well-known conditions for systematically absent reflections in body-centred cells: only reflections with even values of (h + k + l) are permitted, and

ð3Þ

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Fig. 7. Bright field two beam images formed using the reflections: (a) g = 2 0 2; (b) g = 1 2 1; (c) g = 0 4 0. The dashed arrow points to the end of the same crystal dislocation in each image as a marker. The solid arrow indicates the g vectors used to form each image.

Table 2 Normalised kinematic intensities of 2M reflections.

Fig. 8. Schematic illustration of the 2M and 2Mt unit cells in reciprocal space. For clarity not all spots are shown. Filled circles represent strong reflections.

then L2M = 2. (Of course, additional absences may arise for reflections where Fm(h, k, l)2M = 0.) The intensity of the h, k, l reflection from a 2M crystal is Iðh; k; lÞ2M / F ðh; k; lÞ2M F  ðh; k; lÞ2M

ð4Þ

where the asterisk denotes the complex conjugate. In a full treatment this intensity is convoluted with the shape factor, which depends on the number of unit cells in the irradiated volume and the deviation s of the incident beam from the Bragg condition [20]. Moreover, the magnitude of D can be deduced to be close to the nominal value from the coincidence of 6 0  22M and 6 0 02Mt in Fig. 5c. Exact coincidence is subject to experimental uncertainty due to the finite size of the diffraction spots. However, for the present purposes we consider s = 0, and ignore the angular dependencies of fi. Some values of I(h, k, l)2M normalised to I(0, 0, 2)2M are listed in Table 2. These calculated values are broadly consistent with the patterns in Fig. 5. An exception is the reflection g = 1 0 1, which is kinematically forbidden for the stoichiometric compound (Table 2) yet appears weak in Fig. 5a. This discrepancy is presumably due to multiple

g2M = h, k, l

Fm(h, k, l)2M

L2M

I(h, k, l)2M

200 002 020 040 101 202 220 1 2 1 1 2 1 222 6 0 2

112 112 0 112 0 112 0 112 112 0 112

2 2 2 2 2 2 2 2 2 2 2

1 1 0 1 0 1 0 1 1 0 1

scattering of the incident beam and deviation from stoichiometry. Similar considerations apply to reflection g = 0 2 0 in Fig. 5b. The diffraction patterns in Fig. 5 confirm the twinning parameters sketched in Fig. 2. The K1 mirror plane (0 0 1) is evident in Fig. 5a and c. Moreover, the magnitude of D can be deduced from the coincidence of 6 0  22M and 6 0 02Mt in Fig. 5a. Using d ðh k lÞ ¼ 

sin b 2

h a2

þ

2

2

k sin b b2



2lh cos b ac

þ

l2 c2

12

ð5Þ

it can be readily confirmed that these two families of planes have equal spacing in monoclinic crystals if a = c, and cos b = –1/6, hence b = 99.59°. Thus they appear as common planes when 2M and 2Mt unit cells are superimposed, as in Fig. 2. (The kinematically forbidden planes 6 0 12M =6012Mt , which are perpendicular to g1, are also shown.) Hence, the nominal magnitude of 2D, which is equal to the intercept of these planes on a2M=2Mt , is 1/6. Thus for this case the elementary disconnection would have b = 1/6[1 0 0] and h = d(002).

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5.2. Image contrast Two types of contrast features are present in the images shown in Fig. 7, dislocation contrast and a-fringes. When both types are present simultaneously in an image the fringe contrast is generally perturbed, thereby accentuating the dislocation image [21]. Dislocation contrast depends on the product gb, the dislocation line direction n, and s [20]. In the present work strong beam conditions were employed, so s  0. The direction of the disconnection lines can be deduced from Fig. 7a. Since the projection of the disconnection lines is approximately parallel to g = 2 0 2, the line direction is approximately perpendicular to the intersection of the (2 0 2) planes with the (0 0 1) twinning plane. Thus n  [1 0 0], so the defects are close to screw orientation. Hence, a relatively weak contrast is expected using 2 0 2, for which |gb| is 1=3, and a weaker contrast using  12 1; where |gb| is 1=6 (assuming D to have the nominal value). Using 0 4 0, the defects are expected to be invisible because gb = 0 and gb  n = 0. These contrast levels are consistent with Fig. 7a–c, respectively. a-Fringe contrast arises when the matrix crystal is oriented for diffraction but a thin twin within the foil is not so oriented. If the diffracting planes in the matrix material above the twin are offset with respect to those below the resulting phase difference between scattered electrons produces fringe contrast [20]. (a-fringes may also arise if both the matrix and twin are oriented for diffraction by a set of planes common to both, and an offset arises from a rigid body relative shift at the interface [21]. In the current study the planes g2M=2Mt ¼ 0 4 0 are of this type, but no fringe contrast is observed (Fig. 7c). Thus we deduce that these planes are continuous through the twin plane.) Offset planes across a thin deformation twin arise systematically at a pile-up of twinning disconnections. Here a sequence of phase differences a, equal to 2p(gb), 2p(g2b), 2p(g3b), etc., is produced as the twin progressively thickens, leading to a corresponding variation in fringe contrast levels [20] (this sequence is independent of whether the twin has a flat lower surface, as in Fig. 3, or is stepped on both surfaces). Thus modulo 2p, assuming D to have the nominal value, the repeating sequences 2p(1/3, 1/3, 0, . . .) and 2p(1/6, 1/3, 1/2, 1/3, 1/6, 0, . . .) arise for reflections 2 0 2 and 1 2  1, respectively, and no contrast is expected using 0 4 0. The resulting three level and six level patterns of fringe contrast are depicted schematically in Fig. 9. Note that the fringe period is halved when (gb) = p [20]. No attempt has been made to include dislocation contrast in this diagram, although it is present in the images. Dislocation strain fields cause s to vary, which modifies the extinction distance locally and hence introduces curvature to the fringes in the vicinity of defects. The spatial extent of these perturbations is similar to dislocation image widths, which are about one-third of the operating extinction distance in strong beam images [20]. Since the observed disconnection separations near twin tips have about the same magnitude as this, around 30 nm, the fringes there become more or

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less continuously curved, rather than straight as in Fig. 9. Nevertheless, some principal features of Fig. 9 are consistent with the experimental observations. For example, the three contrast levels expected in 2 0 2 should correlate with six levels in 1 2 1. This can be clearly seen by comparing the fringe sequences between the twin tip and the crystal dislocation image used as a marker in Fig. 7a and b. Identical conclusions have recently been reached by Ge et al. [22] and Za´rubova´ et al. [23]. 6. Discussion 6.1. Diffraction data xIn Section 5.1 the magnitude of the elementary twinning disconnection Burgers vector was determined to be the rational fraction |b| = a2M/6. This value was deduced on the basis of coincident 6 0 22M and 6 0 02Mt reflections, which is subject to a small experimental uncertainty because of the finite size of diffraction spots in Fig. 5a. The corresponding value of b2M is equal to cos–1 (1/6) = 99.594°. This value falls midway in the range determined by XRD for temperatures between room temperature b1 = 100.2° (Table 1) and b2 = 98.9° at the austenite start temperature. The corresponding values of D = sin(b  90)/2 for these angles are D1 = 0.0885 and D2 = 0.0774, respectively. Further evidence for the magnitude of D comes from the images in Fig. 7. For example, in Fig. 7a the sequence of fringe contrast levels along the twin includes several regions of nominally zero contrast, as depicted schematically in Fig. 9. For an irrational fractional value of D a finite phase change would accumulate, eventually leading to detectable a-fringe contrast. We define d to be the disparity of D from the nominal value, i.e. D = 1/12 + d. Then, the phase difference produced by each dislocation for the reflection g2M = 2 0 2 is 8p(1/12 + d). Hence, from the lack of distinct a-fringe contrast in the region corresponding to 30 dislocations from the end of the twin we can estimate that modulo 2p, 240p(1/12 ± d) < 2p/10, or |d| < 1/1200 [20,21]. Thus we conclude that the magnitude of D is within 1% of the nominal value. The uncertainty of d found using TEM (±0.0008) lies within the range determined by XRD, d1 = +0.0052 and d2 = 0.0059. It is valuable to discuss how a diffraction pattern such as Fig. 5a, which exhibits the complement to b (between 0 0 2 and 2 0 0) for the 2M structure, becomes modified by periodic twinning to become characteristic of the 14M structure. Consider the reflection 2 0 0 for example, it is evident from Fig. 4 that the two layer 2Mt twin material will not scatter in phase with scattering from the 2M material, thereby reducing the intensity of that reflection. On the other hand, the reflection g14M = 2 0 0, previously absent, will become more intense. Its kinematical intensity can be determined from Iðh; k; lÞ14M ¼ F ðh; k; lÞ14M F  ðh; k; lÞ14M

ð6Þ

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Fig. 9. Schematic illustration of the fringe contrast patterns produced by a sequence of eight disconnections with b = 1/6[1 0 0] and reflections 202 and 2 1 1.

with F ðh; k; lÞ14M ¼ F m N 14M L14M

ð7Þ

where Fm is as in Eq. (1) and L14M is as in Eq. (3). FmN14M specifies the scattering amplitude from the atomic motif of the 14M cell, where N 14M ¼ ðexp 2pig 14M  t1 þ exp 2pig 14M  t2 Þ þ ðexp 2pig14M  m1 þ    þ exp 2pig14M  m5 Þ

ð8Þ

The vectors t1 and t2, and m1 to m5 specify the location of the seven 0 0 1 414M planes in the motif, as detailed in Appendix B. Substituting into Eq. (6), one obtains I(2 0 0)14M = 18.54, which can be compared with the normalised intensity I(2 0 0)2M = 49 from seven unit cells of 2M (Table 2), i.e. a relative intensity ratio I(2 0 0)14M/ I(2 0 0)2M = 0.38 for scattering from the same amount of material. Fig. 10 is a schematic illustration of the location of relevant diffraction spots indexed with respect to the 14M, 2Mt, and 2M structures. This diagram has been stretched vertically for clarity. The 0 0 2 reflection of 14M is kinematically forbidden but would appear as a weak superlattice spot through dynamic scattering. From the geometry of Fig. 4 it follows that (b  90)14M = tan–1 ((3tan(b  90)2M)/7). 6.2. Deformation twinning Deformation twinning is evidently easily nucleated and propagated in NMG. Nucleation has not been observed in the present study, but is thought to arise at the austenite–martensite interface. According to the topological theory of martensitic transformations [24] such interfaces comprise semi-coherent terraces where misfit is accommodated by two (or more) sets of defects. One component of the misfit is accommodated by an array of disconnections, whose motion across the interface also produces the transformation. The second component is accommo-

Fig. 10. Schematic illustration of the location of diffraction spots in [0 1 0] patterns from modulated structures. Reflections are indexed with respect to 14M, 2Mt and 2M structures. Only the boxed indices and filled circles correspond to observed reflections.

dated by either slip or twinning in the martensite phase: we presume that the observed deformation twinning is this second mode. The fraction of material that should undergo twinning can be determined from the topological model, and will be addressed in a later paper. A twinning ratio of 5/2 would be statistically consistent with the formation of 14M material, for example, and could explain the observation of a phase other than the 2M (tetragonal) ground state. The excess energy of the martensite phase compared with the ground state depends not only on the twinning ratio, but also on the twin thickness, since this controls the number of twin interfaces present. Thin twins reduce the extent of the austenite–martensite interfacial strain field, but this would increase the overall energy of the product phase. A process of twin coarsening may proceed in regions remote from the transformation interface [25], and is consistent with the observation of tapering twins in the present work. Calorimetric studies [26,27] and ab

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initio simulations [28] suggest that twin boundary energy is very small in NMG, of the order 0.01 J m–2. Finally, we reiterate that the observed twinning disconnections are likely to be very mobile, thereby assisting twin propagation. Glide along the twin plane is expected to occur under the influence of low stresses because no atomic shuffling is involved and |b| and h are small.

0

0

1 C A

ðA7Þ

0 1

1 2=21 C 0 A

ðA8Þ

0

1=7

0 7

0

Financial support from the Department of Energy, Office of Basic Energy Sciences under Contract DEFG-0207ER46396 is acknowledged. Appendix A The direct space basis and reciprocal space vectors of crystals a and b transform co-variantly [29]. Thus

1 0

B P2M2Mt ¼ P1 1 2M2Mt @ 0 0

P2Mt 14M

Acknowledgement P1 2Mt 14Mt

1 B ¼ @0

1=3

1

0

C A

0

5=3

1 1

1

0

C A

0 0

ðA9Þ

ðA10Þ

0 0 7 0 1 1 0 5=21 B C ¼ @0 1 0 A 0 0 1=7

ðA11Þ

Note that the coordinate frame in the twin is left-handed, since the twinning operation is taken to be mirror reflection across (0 0 1)2M. Appendix B

ðA1Þ Table B1 lists the internal coordinates of the 56 atoms in the 14M unit cell.

and ðA2Þ

The basis vectors in reciprocal space and the indices of direct space vectors transform contra-variantly [29], i.e. 0 1 0 1 a a B C 1 B  C @ b A ¼ Pba @ b A ðA3Þ   c a c b and 0 1 0 1 u u B  C 1 B  C @ v A ¼ Pba @ v A w a w b

2=3

Twinned crystals 2M and 2Mt are interrelated by

We have shown that deformation twinning occurs profusely in non-modulated Ni–Mn–Ga. The twinning parameters K1, g1, K2, and g2 are (0 0 1), [1 0 0], (1 0 0), and [0 0 1], respectively, and the Burgers vector of the elementary twinning disconnection is very close to the rational value 1/6[1 0 0] and its step height is h = d(002).

ðh k lÞa ¼ ðh k lÞb Pba

0

B P2M14M ¼ @ 0 1 0 0 0 1 B 1 P2M14M ¼ 1=2@ 0

7. Summary

ða b cÞa ¼ ða b cÞb Pba

1

3983

ðA4Þ

Useful transformation matrices in the present context are listed below. 0 1 1 0 1 1B C 2 0A ðA5Þ PT2M ¼ @ 0 2 1 0 1 0 1 1 0 1 B C P1 1 0 A ðA6Þ T2M ¼ 1=2@ 0 1 0 1

Table B1 Internal coordinates of the 56 atoms in the 14M unit cell, specified with respect to the Mn atom at the origin in Fig. 4. Motif

x14M

y14M

z14M

x2M=2Mt

y 2M=2Mt

z2M=2Mt

Mn Ga Ni Ni 2Mt t1 t2

0 0 1/2 1/2

0 1/2 1/4 3/4

0 0 0 0

0 26/42 3/42

0 1/2 0

0 1/14 2/14

3/42 22/42 41/42 18/42 37/42

0 1/2 0 1/2 0

2/14 3/14 4/14 5/14 6/14

0 0 1/2 1/2 x2Mt 0 1/2 1/6 x2M 0 1/2 0 1/2 0

0 1/2 1/4 3/4 y 2Mt 0 1/2 0 y2M 0 1/2 0 1/2 0

0 0 0 0 z2Mt 0 –1/2 1 z2M 0 1/2 1 3/2 2

0 1/2

0 1/2

0 1/2

5/6

1/2

7/2

2M m1 m2 m3 m4 m5

d1

Lattice d2

The first two (0 0 1 4)14M layers are designated t1 and t2, and the following five are m1 to m5. The vector d1 locates m1 with respect to the origin: the 2Mt and 2M coordinates are converted to 14M coordinates using Eqs. (A11) and (A8), respectively, in Eq. (A4).

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