Constant plane shift model: Structure analysis of martensitic phases in Ni50Mn27Ga22Fe1 beyond non-modulated building blocks

Acta Materialia 255 (2023) 119042 Available online 30 May 2023 1359-6454/© 2023 The Authors. Published by Elsevier Ltd on behalf of Acta Materialia Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Constant plane shift model: Structure analysis of martensitic phases in Ni 50 Mn 27 Ga 22 Fe 1 beyond non-modulated building blocks M. Vinogradova a , * , A. Sozinov a , L. Straka b , P. Veˇ rt´ at b , O. Heczko b , e , M. Zelený c , R. Chulist d , E. L¨ ahderanta a , K. Ullakko a a Department of Physics, LUT University, Lappeenranta, Finland b FZU - Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic c Institute of Materials Science and Engineering, Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic d Institute of Metallurgy and Materials Science, Polish Academy of Sciences, Krakow, Poland e Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic ABSTRACT Martensites of Ni-Mn-Ga-based alloys consist of hierarchical twinning domains spanning from microto nanoscale. This affects the diffraction pattern and thus can decrease the accuracy of the determination of the crystal structure. We propose a method to obtain different martensitic phases in Ni-Mn-Ga-Fe alloy with simplified variant microstructures and domain sizes of more than 2 micrometers. The use of simplified variant microstructures allows the influence of nanometer-scale domains on diffraction line position to be circumvented and enabls the comparison of the lattice parameters of non-modulated (NM), five-layered modulated (10M), and seven-layered (14M) phases in the same temperature range due to the large hysteresis of the intermartensitic transformations. It is found that the short crystallographic axes in NM, 14M, and 10M martensites at the same temperature have different lengths. As a result, equilibrium NM structure building blocks cannot be used to build the crystal structures of 14M and 10M martensites. Instead, we introduce a constant plane shift model with identical shift values of the nearest planes (110) along [110] or [110] as a replacement for the tetragonal building blocks model. The work demonstrates that plane shift values differ dramatically between martensites, which agrees with ab initio calculations. The application of the constant plane shift and hard sphere models in modulated lattices for atomic-level twinning considerations is discussed. 1. Introduction Many fascinating features have been reported in Ni-Mn-Ga alloys, such as a large magnetocaloric effect, giant magnetic field-induced strain of 6–12%, and extremely low twinning stress of 0.02–1 MPa [1–8]. Twinning stress has been proven to be temperature sensitive and dependent on the crystal structure and type of twin boundaries [9–13]. The crystal structures observed in Ni-Mn-Ga are the typical Heusler cubic L2 1 structure and low-symmetry five-layered modulated (10M), seven-layered modulated (14M), and non-modulated tetragonal (NM) martensites [14]. In the context of martensites, it is worth distinguishing martensitic transformation as transformation between cubic austenite and low symmetry martensite phases and intermartensitic transformation between various martensites. Numerous studies have been focused on martensitic transformations [15–18], however, fewer research has been devoted to intermartensitic transformations [19–23], and their nature thus remains less clear. The significance of the precise determination of crystal structure of Ni-Mn-Ga-based magnetic shape memory alloys becomes evident considering the uniquely high twin boundary mobility and functionality of the material in magnetic fields. The crystal structure of modulated 10M and 14M martensitic phases in Ni-Mn-Ga alloys remains a subject of research and discussion, and the influence of lattice modulation, which can be commensurate or incommensurate [24–29], on alloy properties is not yet fully clear. Khachaturyan et al. [30,31] were the first to suggest that modulated martensitic phases can occur in alloys with very low twin boundary energy by nanotwinning from a more stable tetragonal phase. This concept has been applied to a structure proposal for Ni-Mn-Ga martensites, in which the modulated structures 10M and 14M are described by an alternating sequence of nanotwins comprising of NM tetragonal building blocks (TBBs) twinned on (101) lattice planes. We refer to such description of structure as the TBB model. The 14M structure, denoted as (52) 2 , comprises five lattice planes in one orientation and two planes in the other orientation. This is repeated twice to fulfill the atomic ordering. Similarly, the 10M structure can be considered as alternating * Corresponding author. E-mail address: [email protected] (M. Vinogradova). Contents lists available at ScienceDirect Acta Materialia journal homepage: www.elsevier.com/locate/actamat https://doi.org/10.1016/j.actamat.2023.119042 Received 10 February 2023; Received in revised form 4 May 2023; Accepted 29 May 2023 Acta Materialia 255 (2023) 119042 2 nanotwins of width three and two lattice planes, denoted as (32) 2 . Therefore, each martensitic structure can be built from basic perfectly tetragonal units or their mirroring parts forming nanotwin boundaries [32–35]. Tetragonal parameters of these blocks for all martensitic structures are equal to a/2 and c/2 lattice parameters of NM structure. The simplicity of TBB model enabled its widespread application as an initial approximation in ab initio calculations before structural optimization [36–39]. The TBB model has also been discussed in experimental crystal structure investigations, but its application to the incommensurate modulated structures is challenging due to limited number of degrees of freedom in basic building units [26,29,40–43]. Additionally, the traditional structure determination methods have been challenged by the influence of hierarchical twin microstructures ranging from microto nanoscale [44–47] in the Ni-Mn-Ga system, which originates from the twin boundaries with high-mobility and low-energy [48]. To overcome the shortcomings of the TBB model, we analyze the average lattice of the martensitic phases and the application of a nanotwinnig model to Ni-Mn-Ga-based alloys in this study. The basic or average lattice description is widely used in crystallography of modulated lattices. The determination of the average lattice involves the exclusion of modulation satellites from the diffraction pattern, and the consideration of solely the main reflections [49–53]. The average lattice concept is widely accepted for Ni-Mn-Ga-based alloys because it easily explains the magnitude of deformation during stress-induced or magnetic field-induced crystal reorientation from lattice parameters [3–5, 8–10,23,33]. The average lattice has been used in a comprehensive analysis of twinning in modulated Ni-Mn-Ga martensites [11], and the conclusions of analysis has been empirically confirmed for 10M [11,54] and 14M [55] martensites. Appendix A explains the details of different phases in Ni-Mn-Ga-based systems and their lattice parameters in different coordinate systems commonly used in the literature. To alleviate the influence of hierarchical twinning, we examined martensites in a Ni-Mn-Ga-Fe single crystal with simplified variant microstructures at various temperatures ranging from 328 K to 98 K. Large thermal hysteresis of intermartensitic transformations allowed comparison of the lattice parameters of the non-modulated and modulated 10M and 14M phases in the same temperature range. Based on experimental results, we propose a constant plane shift (CPS) model as a significant improvement of the TBB concept. We also discuss how the hard sphere (HS) model can be used to describe the lattice properties more acurately. 2. Method The Ni 50 Mn 27 Ga 22 Fe 1 alloy was chosen for the investigation due to its large thermal hysteresis in the intermartensitic transformation [56]. Thus, the lattice parameters of all three martensitic structures could be measured at the same temperature. The single crystal ingots were grown by directional solidification using [001] oriented seed crystal in AdaptaMat Ltd. (Helsinki, Finland). A heat treatment at 1300 K for four days was performed to make the ingots chemically homogenous. Slow cooling to ambient temperature at the rate of 50 K/h was applied to guarantee complete atomic ordering of the Heusler-type structure and to avoid the appearance of any cracks. Cuboid samples with a size of 1.0 × 3.0 ×20 mm and {100} faces of the parent cubic phase were cut at 353 K from the ingots by wire electrical discharge machining. Samples were ground and any residual surface deformation was removed by electropolishing. The transition temperatures were determined by a low-field ( μ 0 H ≈10 mT) AC magnetic susceptibility measurement. Curie temperature was T C =380 K. The specimens exhibited sharp martensitic transformation at T M =(M s +M f ) / 2 =330 K and reverse transformation at T A =(A s +A f ) / 2 =335 K, where M s ≈M f , A s ≈A f are martensite and austenite start and finish temperatures, respectively. The transition temperatures are comparable to those reported in [56]. Intermartensitic transformations 10M-14M-NM were identified during cooling and heating of the investigated alloy using lattice parameter measurement and reciprocal space mapping (RSM). A PANalytical Empyrean X-ray diffractometer equipped with Cu tube, λ =1.5406 Å (K α 1 ), polycapillary X-ray lens, Anton Paar domed cooling stage DCS 500, Eulerian cradle, and a Pixel3D detector was used for the lattice parameters measurement and RSM. A hybrid monochromator was used for some measurements. For NM martensite, the shortest axis was determined as a =b and c >a =b. For the 10M and 14M martensites c is the shortest axis, c <b < a. To simplify variant microstructures formed after intermartensitic transformations, we constrained the short c-axis (cubic coordinate system) of the initially 10M single-variant state as shown in Fig. 1. The constraint allowed the sample to contract its short-axis upon temperature decrease but prohibited new variants with long crystallographic axes along the constrained direction. The sample was placed in a 0.05 mm thick copper foil that was folded over at both ends of the sample. In this configuration, the short crystallographic axis was parallel to the longest side of the sample and perpendicular to the diffraction plane. The short crystallographic axis was fixed and could not reorient during intermartensitic transformations, which means that new variants with a long axis along the longest side of the sample could not appear. Thus, we investigated long axes in all martensites using the same diffraction plane without any need for sample tilting or rotation. Optical observations of twin-variant microstructures were taken with the constrained sample installed in a thermal chamber with a transparent window (Linkam DSC 600). The sample was cooled from 413 K to 120 K at a rate close to 4 K/min. The differential interference (Nomarski) contrast in a Zeiss AxioImager.Z1 microscope showed different surface tilts of individual twin domains as regions with different colors. 3. Results The presumption of simplified variant microstructure in the constrained state was first validated by optical observations. The micrographs of the sample surface at selected temperatures are shown in Fig. 2. Initially, the 10M sample at room temperature consisted of a nearly single variant, which contained one narrow residual (modulation) twin, Fig. 2a. This 10M nearly single variant with respect to the caxis (modulation and a/b twins still existed) transformed into a twovariant 14M martensite. In micrographs, the variants differ by their colors. All twin domain traces were parallel to the constrained short axis with the domain size ranging from 2 to 100 μ m. Such variant microstructure simplified considerably the interpretation of diffraction results. Under further cooling, the two-variant 14M martensite started to transform into a two-variant NM structure, Fig. 2e. Due to the temperature limit, we were not able to complete the transformation of the NM phase under optical observation. However, in the partial 14M-NM transformation, we observed similar twin domain scale and the same trace orientation along the constrained axis. This gives reasonable Fig. 1. Schematic illustration of 10M sample with constrained c-axis, c <a, b. Axes a and b are not distinguished due to a/b twinning. M. Vinogradova et al. Acta Materialia 255 (2023) 119042 3 confidence that the final NM structure would consist of only two variants. In order to provide more reliable evidence, the RSM was used to verify the crystal structure and orientation of the sample (Fig. 2b, d, f). Two-variant microstructures were confirmed for 14M and NM martensites. A nearly single modulation direction was found in the initial Fig. 2. Optical images of the sample surface (100) showing simplified martensite microstructures. The orientation of the short axis in the image plane is indicated by the arrows. (a) 10M (303 K), (c) 14M (223 K), (e) a mixture of 14M and NM martensites (120 K). Reciprocal space maps of a plane perpendicular to the short axis for (b) 10M (303 K), (d) 14M (223 K), and (f) NM (98 K) martensites after intermartensitic transformations during cooling. Fig. 3. (a) Temperature dependence of lattice parameters in cubic coordinates measured at the constrained condition for 10M, 14M, and NM martensites with the short axis perpendicular to the diffraction plane. (b) Temperature dependence of the short crystallographic axis length for 10M, 14M, and NM martensites. M. Vinogradova et al. Acta Materialia 255 (2023) 119042 4 10M martensite structure and this modulation direction was inherited by the 14M phase. For 10M martensite (Fig. 2b), (400) peak (and (040) peak due to a/b twins) is visible in the center of the marked area (red rectangle), which corresponds to the sample orientation with the a,b-axis vertical in the diffraction plane. Four satellites (S 1 400 , S 2 400 , S -2 620 , S -1 620 ) are clearly visible between the (400) and (620) peaks, confirming the 10M crystal structure. Lower-intensity satellites are visible in a perpendicular direction, indicating the presence of modulation twins in a small amount. The transformation from 10M to 14M martensite (Fig. 2d) results in the disappearance of the 10M (400) peak and the appearance of two new peaks. These two peaks correspond to two 14M variants v1 and v2, identified as twin-related peak pairs (400) v1 and (040) v2 , and (620) v1 and (260) v2 , based on 2θ angles of both peaks and the angle between the Fig. 4. Temperature dependence of (a) 2θ angles for (444) and ( 444) peaks, and (b) d-spacings for individual martensitic phases. Fig. 5. Schematic representation of the CPS model for a martensite with modulated lattice. (a) Comparison of CPS and TBB models; color distinguishes type of atoms in three component alloy. (b) View of the two nearest (110) planes with a plane normal out of paper; color distinguishes the arrangement of atoms in planes not the type of atom. The shift between the nearest (110) planes is dx. 3D view of the nearest (110) planes: (c) in a cubic lattice, (d) after the tetragonal distortion along the caxis, and (e) after the tetragonal distortion along c-axis of 10M or 14M martensite lattice (c <a), and plane shift. M. Vinogradova et al. Acta Materialia 255 (2023) 119042 5 planes. The direction of each plane normal is in the diffraction plane and is determined by the value of ω (angle between the incident beam and horizontal in the diffraction plane) and 2θ angles. The rest of the peaks are satellites of both variants. The distance between the main peaks of variant v1 and variant v2 is almost equal to the distance between the nearest satellites. Two-variant 14M martensite transforms to twovariant NM martensite (Fig. 2f). These two variants were identified as peak’s pairs (004) v1 and (400) v2 , and (206) v1 and (602) v2 , based on 2θ angles of both peaks and the angle between the planes. Satellites connected to the lattice modulation in 10M and 14M martensites disappear in the NM phase. Figure 3a shows the temperature dependence of the lattice parameters in cubic coordinates for all martensitic phases, measured under constrained c-axis condition. The presented data is a combination of cooling and heating measurements. The temperature range of the instrument limits the data for NM martensite at low temperatures, while NM-14M intermartensitic transformation limits data at high temperatures. The data for 14M martensite is limited by 14M-NM at low temperatures and 14M-10M intermartensitic transformations at high temperatures. The data for 10M is limited by the 10M-14M intermartensitic transition at low temperatures and the 10M-cubic martensitic transformation at high temperatures. The thermal hysteresis of intermartensitic transformations is considerably higher in comparison to martensitic hysteresis, which is about 5 K between austenite and martensite. For 10M-14M transformation, the hysteresis is about 60 K, and for 14M-NM transformation 90 K. Direct determination of the aand b-lattice parameters from (400) and (040) in 10M martensite is problematic due to their small difference. To overcome this, we measured 2θ for (602), (062) and (004) peaks to calculate the aand b-axis lattice parameters. The 2θ angle positions of (602) and (062) peaks are high and do not depend on lattice monoclinic distortion, and the calculation is thus sufficiently precise and straightforward. The difference between aand bparameters increased with decreasing temperature, in correspondence with previous reports (e.g., [17]), but the peaks merged at temperatures below 263 K. Incommensurate modulation [56] or a/b twinning [44], are considered as possible mechanisms for such a diffraction effect. The 2θ angle of peaks (004) for modulated martensites and (400) for NM were measured to determine the short axis lattice parameter. For the NM martensite short-axis measurements were done in variant v2 of the two-variant state during heating from 98 K. For the short-axis measurements in 10M martensite, the sample was reoriented, thus the short crystallographic c-axis was in the diffraction plane. Measurements were done during cooling from 328 K. After 10M-14M intermartensitic transformation at 193 K one variant of the 14M two-variant state retained direction of the c-axis in the diffraction plane. This way, the short-axis of the 14M martensite was measured in the temperature range of the 14M phase stability. Figure 3b shows that the short axis changes abruptly during the intermartensitic transformations. For 10M and 14M the c-axis is the shortest, for NM the shortest are two axes a =b. The length of the short axis is different for all three phases at the same temperature and increases with temperature rise. The (444) and (444) peak positions determine d-spacings d 111 and d111 with high accuracy due to the high 2θ angles. Figure 4a shows the temperature dependence of the 2θ angles for (444) and (444) peaks. The 2θ difference of the (444) and (444) peaks positions is highly sensitive to lattice monoclinic distortion. The difference and corresponding monoclinic distortion are highest in the 14M phase and decline with increasing temperature for both 14M and 10M phases. There is no monoclinic distortion in the tetragonal NM phase, thus the (444) and (444) peaks overlap. The experimental data for the c-axis and d-spacings d 111 and d111 were used to determine d110 and d110, based on geometrical considerations of 10M and 14M crystal lattices: d110 =c  (c d111)2 −1 √(1) d110 =c  (c d111)2 √−1 (2) The corresponding temperature dependences of these d-spacings are given in Fig. 4b. In 10M and 14M modulated martensites, d 110 /2 corresponds to the distance between the closest planes. The (110) planes are shifted relative to each other, causing lattice modulation. Using the cubic coordinate system (Appendix, Table A1) the c-axis is the shortest in 10M and 14M, but the longest in NM. This provides confusion when comparing the structures. Therefore, to avoid it, the (101) spacing in NM, which is directly related to the concept of constant plane shift, was calculated by replacing d 110 with d 101 in Eq. (1). Such an operation allows us to compare the d-spacings of the shifting planes for all martensite structures. In NM and 14M martensite, the temperature dependency of d 110 (14M) and d 101 (NM) is negligible, whereas it is weak for 10M. Similar negligible dependence of d 110 on the structure type was observed in TEM study [57]. 3.1. Constant plane shift (CPS) model It is apparent from our measurements that the application of a TBB concept to 14M and 10M martensites does not work with respect to the lattice parameters. The main argument is that the short crystallographic axis is different for each martensite at the same temperature (Fig. 3b). This means that the equilibrium tetragonal c/a ratio for NM martensite cannot be used to build modulated phases by simple geometrical Fig. 6. Temperature dependence of the plane shift dx for individual martensitic phases. Table 1 Space diagonals SD, SD short along [111] and SD long along [111] in different phases of Ni-Mn-Ga alloy. They are equal in cubic and NM tetragonal lattice, and slightly different in monoclinic 10M and 14M lattices. T =273 K T =193 K SD, Å SD, Å Cubic 10.099* 10.088* NM 10.161* 10.157 SD long , Å SD short , Å SD long , Å SD short , Å 10M 10.131 10.076 10.133* 10.061* 14M 10.137 10.065 10.137 10.050 *Extrapolated values. M. Vinogradova et al. Acta Materialia 255 (2023) 119042 6 operations as suggested in [32–35]. It is possible to consider various kinds of a lattice relaxation different in each martensitic structure [58], but then the simplicity of the TBB concept would be lost. Thus, there is still missing a unified model, which is able to describe various modulations of martensitic structures by a single parameter. Therefore, we introduce a constant plane shift (CPS) model described in Fig. 5, as an extension of the TBB concept. The model proposes a constant shift dx of the plane (110) along [110] or [110] directions in accordance with the well-known (32) 2 sequence for 10M martensite and (52) 2 sequence for 14M martensite, as validated by transmission electron microscope (TEM) structural studies [57,59]. The CPS model, unlike the TBB concept, does not assume nor require the same short axis length in the NM, 14M, and 10M martensites. Figure 5a shows the difference between the TBB and CPS models. In the TBB model, the modulated lattice of 10M or 14M martensites is made from the building block of NM lattice, which is shown in Fig. 5a as a rectangle with sizes a NM /2, c NM /2. Mirroring building block at the nanotwin boundaries creates the modulated lattice. The CPS model takes one feature from TBB model: the constant absolute value of the plane (110) shift, |dx|, further referred to as constant dx shift or just dx. Other TBB model properties, like b NM =c 10M and dx NM =dx 10M, do not match up with lattice parameter measurements, as shown in Fig. 3b and additionally discussed below. Furthermore, our calculations demonstrate that the angle of building blocks β’ differs from the value of 90◦by about 0.15◦to 0.6◦. This variation depends on the temperature and the type of modulated lattice. The advantage of the CPS model is that the dx shift can be found experimentally, using the measured lattice parameters a, b, and c, and d-spacing d 110 and d110. In current investigations, we calculated the d-spacings from measured 2θ angles for (444), (444), and (004) using Eqs. (1) and (2). The method for determining the dx shift for all martensites is shown below. The spacings d 110 and d110 are calculated from the measured caxis and d 111 and d111 based on Eqs. (1) and (2). The (52) 2 sequence for 14M martensite and (32) 2 sequence for 10M martensite determine the angle between diagonals in the parallelogram build using the lattice parameters a and b (or a and c for NM) (see Appendix A, Fig. A2) as: γ10M diag = π 2−tan−1(2dx 5d110)(3) γ14M diag = π 2−tan−1(6dx 7d110)(4) NM martensite also can be described using CPS as: γNM diag = π 2−tan−1(2dx d101)(5) The lengths of the half diagonals (D1, D2) in the parallelogram build using the lattice parameters a and b are determined by the equations: D1=d110 sin(γdiag)(6) D2=d110 sin(γdiag),(7) where γ diag is substituted by γ10M diag ,γ14M diag or γNM diag and d 110 is substituted for d 101 for NM. Equations (3)-(7) determine the values of the half diagonals D1 and D2, as well as the angle γ diag between them. They depend on the shift dx and the respective sequence, (32) 2 for 10M and (52) 2 for 14M martensite. The known D1, D2, and γ diag allow us to calculate the lattice parameters a and b (a and c for NM) using known trigonometric relations. By a reverse procedure, i.e., by minimizing the difference between the calculated and measured values of lattice parameters, we found dx for each temperature and type of martensite. Figure 6 shows the temperature dependence of the plane shift dx. It can be seen that dx increases with decreasing temperature for all martensites. The plane shift is significantly different in different martensites. It is lowest in 10M martensite, increases threefold upon 10M-14M transformation, and finally increases by further 30% during 14M-NM transformation. We compared the experimentally determined dx with the corresponding shifts calculated from lattice constants obtained by ab initio calculations with the electron delocalization correction [60]. The TBB model was used to create initial structures, which were further fully optimized to obtain the lowest energy. However, the details about character of modulation of fully optimized structures were not discussed in Ref. [60]. The ab initio values of dx calculated with help of Eqs. (3)–(5) are 0.042 for 10M, 0.282 for 14M and 0.346 for NM structure. The agreement is very good for all three martensitic structures although the calculations were performed for stoichiometric Ni 2 MnGa composition and equilibrium lattice parameters corresponded to conditions at 0 K. Using the ab initio lattice constants calculated without delocalization correction, the values of dx are nearly the same, in range 0.486–0.52, for all three structures [60]. This comparison supports the necessity of the electron delocalization correction for ab initio calculations of Ni-Mn-Ga alloys. Moreover, it also suggests that reported differences in dx are basic properties of martensitic lattices and are independent on temperature or alloy composition. Corrected ab initio calculations also provide nearly constant values of interlayer distances d 110 (10M, 14M) and d 101 (NM) calculated with help of Eq. (7). They change for all three martensitic structures only from 4.22 Å to 4.23 Å, in a good agreement with similar nearly constant dependency in experiment, Fig. 4b. We found that the CPS model and our observations can be linked directly with well-known hard sphere (HS) model for BCC-type lattice. The HS model was previously used, e.g., for description of twinning in magnesium [61]. In HS model, the highest Young’s modulus is in the directions of space diagonals marked as red lines in Fig. 5c. The nearest atoms touch one another along cube diagonals [111], and unit cell length a and atomic radius R are related through 4R/ 3 √in cubic lattice. Our calculations, based on explicit expressions presented in Ref. [62] Eqs. (4) and ((5) in the reference), and measured elastic coefficients [63–67], confirm the highest Young’s modulus E 111 in cubic as well as in martensitic phases of the Ni-Mn-Ga alloys. Additional link between CPS and HS models comes from our lattice parameters measurements. We calculated lattice space diagonal SD along the [111] direction for cubic and NM phases, as well as for the monoclinic lattices 10M and 14M with two different diagonals, short SD short along the [111] and more long SD long along the [111]. Data for Table 1 were determined at two temperatures, 273 K and 193 K. For the 14M martensite, the lattice parameters were measured directly at both temperatures. For the 10M martensite, parameters at 273 K were measured and parameters at 193 K were calculated using linear extrapolation. For the NM martensite, parameters at 193 K were measured and parameters at 273 K were calculated using linear extrapolation. The cubic phase exists above 333 K; thus, to compare this phase with others, lattice parameters were measured within the range 333 – 723 K, which was followed by linear extrapolation to 193 K. It is seen from the Table 1 that the difference in SD values between phases is very small, less than 0.5%, which indicates that the short diagonal is difficult to compress further, and strongly supports the validity of the HS model. For the HS model, a small tetragonal distortion of the cubic lattice with the constant shortest distance between atoms results in equal expansion of the lattice in two perpendicular directions. This is shown in Fig. 5c and d. Shifting of the (110) planes relative to each other results in shorter distance between the planes, d 110 /2 <D1/2, because planes with constant shortest interatomic distances are inclined from 90◦, Fig. 5e. Thus, the HS model considerations give a plausible physical explanation why the experimental results show that d 110 <d110 for lattices with modulations (Fig. 4b). Even though the (110) plane is shifted, the D1 M. Vinogradova et al. Acta Materialia 255 (2023) 119042 7 (see Fig. 5e) does not change since the hard directions in the plane keep the plane shape. Furthermore, the coupling of the CPS and HS models allows for the prediction of Poisson coefficients. The stress-induced deformation of a modulated lattice along the c-axis reflects as approximately the same deformation perpendicular to c-axis, indicated as D1 in Fig. 5e. Thus, for this pair of directions, the Poisson ratio will be close to one. In contrast, there will be a minimum deformation along the plane (110) normal, due to (110) plane shifting, and the corresponding Poisson ratio will be close to zero. In addition to this, the periodic shifting of the (110) planes creates a periodic change in position of atoms in the a-c or b-c twinning plane (101) or (011) in the direction of the [110]. The twin planes, either (101) or (011), change from a smooth flat surface to being a sawtooth-shaped modulated surface. This modulation wave should interact with the twin boundary motion by changing locally dx value for the (110) planes. It means that a higher number of atoms is involved in a/c or b/c twin boundary motion in modulated structures in comparison with lattices without modulation. More research on this issue is needed to understand why Ni-Mn-Ga-based systems have such low twinning stress, making them one of the most interesting magnetic shape memory materials. 4. Conclusion •An experimental method is proposed that simplifies variant microstructure after intermartensitic transformations by constraining the shortest crystallographic axis. In Ni-Mn-Ga-Fe alloy, the initial near single-variant 10M structure with constrained c-axis transformed into 14M and then further to NM two-variant microstructures with domain sizes ranging from 2 to 100 μ m. •The simplified variant microstructure enabled confident determination of lattice parameters for 10M, 14M, and NM martensites within the temperature range between 328 K and 98 K. It is shown that all principal axes change discontinuously during the intermartensitic transformations. •We introduced a constant plane shift (CPS) model to reflect precisely the continuous and discontinuous changes in the lattice parameters with temperature. The model proposes equal shift value of the nearest planes (110) along [110] or [110] directions. •The plane shift is significantly different in different martensites. It is lowest in 10M martensite, increases about three times upon 10M14M transformation, and finally increases by a further 30% during 14M-NM transformation. •We propose that the combination of CPS and hard sphere model can be useful for atomic-level considerations of twinning and understanding of elastic properties in modulated lattices. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was financially supported by the Academy of Finland (grant number 325910). The authors from the Czech Republic acknowledge the funding from the Czech Science Foundation grant no. 21-06613S and from the ESIF and MEYS projects MatFun – CZ.02.1.01/ 0.0/0.0/15 003/0000487 and SOLID21 – CZ.02.1.01/0.0/0.0/16_019/ 0000760. The international mobility of Alexei Sozinov was supported by ESIF and MEYS project MOBILITY FZU 2 – CZ.02.2.69/0.0/0.0/18_053/ 0016627. Robert Chulist acknowledges the projects 2021/42/E/ST5/ 00367 of the National Science centre of Poland. We thank Silvia Sedl´ akov´ a for the kind assistance with the operation of Linkam DSC600 thermal chamber. Appendix A. Average lattices in Ni–Mn–Ga-based systems This Appendix explains the different phases in Ni-Mn-Ga-based systems and their lattice parameters in different coordinate systems commonly used in the literature. The cubic phase observed in Ni 50 Mn 27 Ga 22 Fe 1 is the typical Heusler L2 1 structure. The cubic phase is described in Fig. A1. The lattice parameters of the low-symmetry fivelayered modulated (10M), seven-layered modulated (14M), and nonmodulated tetragonal (NM) martensites [14] at 273 K are described in detail in Table A1. For the 10M and 14M phases, the lattice parameters were measured directly. The cubic phase and NM phase parameters were calculated using linear extrapolation. After sample cooling from the cubic phase, the modulation in 10M was commensurate, but it became incommensurate below 263 K and remained in this state after heating from low temperatures to temperatures close to the reverse martensite transformation (see details in [56]). We used the average lattice (or identically average structure) of the martensitic phases to analyze the application of a nanotwinnig model to Fig. A1. Cubic Heusler structure Ni 2 MnGa. (a) – Schematic representation of the L2 1 unit cell. (b) – The relation between the cubic and diagonal coordinates; (c) – The projection of atomic structure to the plane. M. Vinogradova et al. Acta Materialia 255 (2023) 119042 8 Ni-Mn-Ga-based alloys and to develop the CPS model in this study. The average lattice (average structure) is widely used in crystallography for the description of both commensurate and incommensurate (C and IC) modulated structures. The average lattice is typically determined from the diffraction pattern considering only the main reflections and ignoring the satellites related to modulation. Introducing the superspace [49,50] restores the translational symmetry in the crystals lacking the 3D periodicity (IC structure). The same approach can be used for the C structure. The average lattice approach is extremely beneficial for the case of the studied material, in which the modulation changes with temperature between C and IC. It significantly simplifies the considerations of most physical mechanisms related to the lattice geometry, such as lattice reorientation or twinning. We note that in the case of commensurate structure, crystal lattice can alternatively be described by a new larger unit cell, i.e., the crystal consists of identical sections of modulated original cells. These unit cells are commonly referred to as long periodical (LP) cells or supercells. In this case, all the reflections of the diffraction pattern are indexed using three indices, whereas more indices are required for IC structures [49–52]. The average lattice for modulated martensites 10M and 14M in cubic coordinates, coinciding with [100], [010], and [001] of L2 1 Heusler superstructure, is monoclinic and widely used for investigations of NiMn-Ga-based martensites. The largest stressor magnetic field-induced strain corresponds to the difference in lattice parameters of the average lattices [3–5,8–10,23,33]. After phase transformation, the orientation of the average lattice axes differs only by a few degrees from the cubic phase directions [100], [010], and [001] in twin variants for 10M and 14M modulated martensites. As a result, the maximum stress during crystal reorientation is recorded along cubic coordinates. After a detailed investigation of twinning in modulated Ni-Mn-Ga [11], employing average lattice in cubic coordinates and empirical proof for 10M [11,54] and 14M [55] martensites, the average lattice approach turns even more useful. Average lattice in cubic coordinates can be Table A1 Lattice parameters of martensitic phases of Ni 50 Mn 27 Ga 22 Fe 1 alloy at 273 K in different coordinate systems. 10M 14M NM Average lattice in cubic (black) and diagonal coordinates (red). Cubic: a ≅b >c a =5.989 Å b =5.942 Å c =5.560 Å (1 - c/a) =0.072 γ cubic =90.45 deg Diagonal: D1 =4.235 Å D2 =4.202 Å γ diag =89.55 deg Point group: 2/m Cubic: a >b >c a =6.146 Å b =5.803 Å c =5.529 Å (1 - c/a) =0.10 γ cubic =90.58 deg Diagonal: D1 =4.248 Å D2 =4.205 Å γ diag =86.71 deg Point group: 2/m Cubic: a =b <c a =b =5.503 Å c =6.529 Å (c/a −1) =0.19 γ cubic =90 deg Space group: I4/mmm (No.139) Diagonal (2M [35,69]): D1 =D2 =4.269 Å γ diag =80.26 deg Commensurate structures in diagonal coordinates. Atoms in 2D structures are projection of two nearest planes. Dashed lines indicate nanotwin boundaries in TBB model. Direction indices show orientation of cubic coordinate system. Long-periodical (3 2) 2 |dx| =0.0828 Å a LP =D1 =4.235 Å b LP =5*D2 =21.009 Å c =5.560 Å γ diag =89.55 deg Long-periodical (5 2) 2 |dx| =0.2829 Å a LP =D1 =4.248 Å b LP =7*D2 =29.436 Å c =5.529 Å γ diag =86.71 deg 2M monoclinic[35,69] |dx| =0.3609 Å D1 =4.269 Å D2=4.269 Å b =5.503 Å γ diag =80.26 deg Commensurate structure in conventional crystallographic coordinates (b-axis perpendicular to both aand c-axis). Long-periodical a =4.235 Å b =5.560 Å c =21.009 Å β =89.55 deg Space group: C2/m (No.12) Long-periodical a =4.248 Å b =5.529 Å c =29.436 Å β =86.71 deg Space group: C2/m (No.12) 2M monoclinic a =c =4.269 Å b =5.503 Å β =80.26 deg M. Vinogradova et al. Acta Materialia 255 (2023) 119042 9 transformed to a diagonal coordinate system, with the lattice axes coinciding with diagonals in a parallelogram built on cubic aand b-axes for modulated lattices (see Fig. A2). The diagonal coordinate system is beneficial for examining long periodic lattices with commensurate modulations that are five and seven times longer in the direction of modulation for 10M and 14M, respectively. NM martensite in cubic coordinates is simply tetragonally distorted L2 1 along [001] direction. We made a specific decision on the diagonal coordinates for the NM martensite. We consider diagonals D1 and D2 in the plane including a-axis and c-axis for seamless comparison with layered structures. Such presentation of NM lattice is well known in the literature as 2M "monoclinic’’ [35,68,69]. This method enabled us to provide plane shift dx for the NM lattice in a similar way as for layered structures - by minimizing the difference between the calculated and measured values of lattice parameters a and b (a =b for NM martensite in cubic coordinates). The relation of average lattice axes with conventional crystallographic coordinates system, where the b-axis is perpendicular to the aand c-axis and monoclinic angle is β, presented in Tables A1 and A2. The tables also provide details regarding the space groups proposed in various studies, including for 10M and 14M monoclinic martensite lattices. Both structures may be commensurate or incommensurate [24, 25]. For commensurate structures atomic coordinates are the same for all temperatures, whereas for incommensurate structures modulation vector is temperature dependent [28,56], thus the atomic positions are also affected by temperature. Table A2 shows the average incommensurate lattice parameters of 10M martensite in different coordinates. At low temperatures the difference between a and b axes disappears (see Figs. 3a and 6). When a =b in cubic coordinates, D1 is perpendicular to D2 in diagonal coordinates, but not equal. Simultaneously, the structure becomes incommensurate, as evidenced by the satellite positions detected by Veˇ rt´ at et al. [56] and verified by our present research (data not presented in the text). Current investigation suggests a simple algorithm to calculate atomic coordinates in modulated commensurate structures 10M and 14M, however, the case of incommensurate structures is out of the scope of this work. References [1] P.J. Webster, K.R.A. Ziebeck, S.L. Town, M.S. Peak, Magnetic order and phase transformation in Ni 2 MnGa, Philos. Mag. B. 49 (1984) 295–310, https://doi.org/ 10.1080/13642817408246515. [2] K. Ullakko, Magnetically controlled shape memory alloys: a new class of actuator materials, J. Mater. Eng. Perform. 5 (1996) 405–409, https://doi.org/10.1007/ BF02649344. [3] O. Heczko, N. Scheerbaum, O. Gutfleisch, in: J.P. Liu, E. Fullerton, O. Gutfleisch, D. J. Sellmyer (Eds.), Nanoscale Magnetic Materials and Applications, Springer, Boston, 2009, pp. 399–439. https://doi.org/10.1007/978-0-387-85600-1. [4] M. Acet, Ll. Ma˜ nosa, A. Planes, Magnetic-field-induced effects in martensitic Heusler-based magnetic shape memory alloys, in: K.H.J. Buschow (Ed.), Handb. Magn. Mater., Elsevier, 2011, pp. 231–289, https://doi.org/10.1016/B978-0-44453780-5.00004-1. [5] M. Chmielus, X.X. Zhang, C. Witherspoon, D.C. Dunand, P. Müllner, Giant magnetic-field-induced strains in polycrystalline Ni–Mn–Ga foams, Nat. Mater. 8 (2009) 863–866, https://doi.org/10.1038/nmat2527. [6] H. Hosoda, S. Takeuchi, T. Inamura, K. Wakashima, Material design and shape memory properties of smart composites composed of polymer and ferromagnetic shape memory alloy particles, Sci. Technol. Adv. Mater. 5 (2004) 503–509, https:// doi.org/10.1016/j.stam.2004.02.009. [7] I. Karaman, B. Basaran, H.E. Karaca, A.I. Karsilayan, Y.I. Chumlyakov, Energy harvesting using martensite variant reorientation mechanism in a NiMnGa magnetic shape memory alloy, Appl. Phys. Lett. 90 (2007), 172505, https://doi. org/10.1063/1.2721143. [8] I. Aaltio, A. Sozinov, Y. Ge, K. Ullakko, V.K. Lindroos, S.P. Hannula, Giant magnetostrictive materials, in: S. Hashmi (Ed.), Ref. Module Mater. Sci. Mater. Eng., Elsevier, 2016, B9780128035818019000, https://doi.org/10.1016/B978-012-803581-8.01830-0. [9] A. Sozinov, A.A. Likhachev, N. Lanska, K. Ullakko, Giant magnetic-field-induced strain in NiMnGa seven-layered martensitic phase, Appl. Phys. Lett. 80 (2002) 1746–1748, https://doi.org/10.1063/1.1458075. Table A2 Average lattice for 10M incommensurate structure. Coordinate system Cubic Diagonal Conventional crystallographic Lattice parameters a =b =5.967 Å c =5.560 Å γ cubic =90.43 deg D1 =4.235 Å D2 =4.203 Å c =5.560 Å γ diag =90 deg a =4.235 Å b =5.560 Å c =4.203 Å β =90 deg Superspace group Immm(00γ)s00 [24] Fig. A2. Relations between lattice parameters in cubic and diagonal coordinates for 10M, 14M and NM martensites. D1 and D2 are half diagonals in the parallelogram build using the lattice parameters. (a) 10M and 14M modulated martensites; (b) NM martensite. The indices of d-spacings are in cubic coordinates. M. Vinogradova et al.