Modulated Martensite

In the case of 5M and 7M martensites X-ray and electron diffraction experi-ments have shown, that the structure of martensite is actually more complex.

Satellite reflections were observed in diffraction patterns between the main re-flections defined by the non-modulated structure [51, 65]. For the 5M structure four satellites were observed between the main reflections and the 7M marten-sitic structure is characterized by a diffraction pattern with six satellites. These observations were interpreted in terms of a superstructure constituted by five (five-fold modulation, 5M) respectively seven (seven-fold modulation, 7M) unit cells of basic structure. These martensitic superstructures exhibit a periodic shuffling of the(110)atomic planes along the[1¯10]direction (see Fig. 2.2).³

Static Wave Approach The description of the displacive modulation of the atomic layers has been the object of several structural investigations [51, 65, 67–

70]. In the general approach of static displacement a wave-like function is used

3 The coordinate system is referred to crystallographic directions of the austenitic L10unit cell.

Figure 2.2.: Modulated martensitic structure for the 5M phase of Ni2MnGa. The supercell was obtained by first-principles calculations by Zayaket. al. Image taken from [66].

to model the displacement of the atomic layers [64, 68]. In this approach the general position of theith atom in the basic structure is given by:

x_i = ¯x_i+u_i(¯x₄), (2.2) u_i(¯x₄) =

inf

X

n=1

Aⁱ_nsin(2πnx¯₄) +B_nⁱ cos(2πn¯x₄), (2.3) whereu_idefines the modulation function which depends on thex₄superspace coordinate, and indexnindicates the order of the Fourier series. From the ex-perimental side only terms up to third order are considered [51, 68].

The 5M martensitic structure was extensively investigated by elastic neutron scattering and high resolution diffraction experiments performed on powder and single crystal samples [16, 68, 69, 71]. These studies have demonstrated that the five-fold modulation in Ni-Mn-Ga alloys can be commensurate or

2.1. Crystal Structure incommensurate. It was demonstrated that the modulation function for all three elements has the same phase and periodicity. A maximum amplitude of the modulation in the range of 0.28-0.31 Å was found. It is of particular interest that the modulation amplitudes differ for all three elements, being minimal for Mn and maximal for Ni atoms [68].

The five-layered modulated 5M structure was also obtained by first princi-ples calculations performed by Zayaket al.(Fig. 2.2) [66]. A tetragonal crystal structure lattice-distortive strain is stabilized aroundc/a= 0.94with respect to the L21structure when, in addition, modulation shuffles with a period of 5 atomic planes are taken into account. Also the modulation amplitudes of the 3 elements were found to be different and in good agreement with experimental data:0.29Å for Mn and Ga atoms, and0.32Å for Ni atoms.

Stacking Approach Another model than the modulation approach can be used to to explain the diffraction patterns with additional superstructure spots.

It describes the displacement of the atomic planes with a long-order stacking sequence [65, 70, 72]. As in the previous approach, the (110) atomic planes are shifted along the[1¯10]direction, so that the modulation propagates along the[110]direction (Fig. 2.3). The stacking sequences can differ and the most frequently reported sequences are(3¯2)2 for 5M and(5¯2)2 for 7M martensitic phases⁴ [67, 70]. This stacking approach is well known for martensitic mate-rials and is used to describe the modulated structure of Ni-Al and Ni-Mn-Al alloys [65, 73]. It shall be pointed out that the stacking approach is restricted to commensurate structures and uniform long range order stacking periodicity.

A distinction between the two approaches in experimental studies is difficult since most experiments are probing integral properties (e.g. X-ray and electron diffraction). An exact pattern of atomic displacements can not be identified definitely in this way. Whenever an atomic modulation pattern is proposed, it is a fit of the experimental data to some theoretical model. Local experimental

4 More precise names for the 5M and the 7M stacking sequences are 10M or 14M, respectively.

This is due to the fact that the L21chemical Heusler order is identical every 10, respectively, 14 atomic planes. However, we will hang on to the 5M/7M notation in this work.

methods which probe the structure on the atomic scale in real space give a direct evidence of the modulation pattern [67].

Figure 2.3.: The stacking-like 14M structure of Ni2MnGa with a(5¯35¯1)(a)and a(5¯2)2 (b)stacking sequence. The directions are shown according to the con-ventional cubic Heusler structure of Ni2MnGa. Image taken from [74].

Origin of the Modulated Structure

Both presented approaches to describe the modulated structures observed in Ni-Mn-Ga are discussing the martensitic phases as stable thermodynamic phases. The justification for this point of view comes from the observation that the modulated phases often show precursor behavior. For the case of stoichio-metric nearly stoichiostoichio-metric Ni2MnGa a three-layered modulated 3M phase is observed (see Table 2.1) [63]. Premartensitic phenomena were also observed for martensitic transformations in many other (magnetic) shape memory alloys showing modulated martensitic phases such as NiTi or NiAl [75, 76]. The oc-currence of the modulated structures can be related to a specific behavior of an

2.1. Crystal Structure acoustic branch in the phonon dispersion of austenite. This becomes soft with decreasing temperatures when approaching the martensitic transformation.

TA2

Figure 2.4.: (a)The phonon dispersion of stoichiometric Ni2MnGa sample for the high symmetry directions of the L21structure. The symbols indicate the ex-perimental results obtained at room temperature by means of neutron scattering measurements. The lines represent the first-principles phonon dispersion calcu-lations.(b)The TA2[ξξ0]phonon modes of Ni2MnGa for different temperatures.

Image adapted from [77].

In the case of Ni2MnGa it is the transverse acoustic TA2[ξξ0]branch (Fig. 2.4).

The softening behavior and its locations in reciprocal space reveal the marten-sitic transition mechanism and temperature dependence. The austenite L21

structure transforms to the tetragonal modulated martensite structure by shuf-fling the(110)planes in the[1¯10]direction [16, 24, 77, 78]. Figure 2.4(b) clearly shows that the frequency (or energy) lowering of the[ξξ0]TA2phonon branch is restricted to a very narrow range betweenξ = 0.2andξ = 0.5and remains un-changed for higher values ofξ. The frequency softening reaches its minimum for a temperature of260K, before the transformation to the 3M premartensitic state, atξ= 1/3. This value corresponds to the inverse period of the premarten-sitic modulation. First-principles calculations, which refer to the hypothetical 0K properties of the austenitic L21structure, even exhibit imaginary frequen-cies for this phonon branch (solid lines in Fig. 2.4(a)). As a result an instability

of the austenite shows up towards lower temperatures. As a consequence, a phase transition must occur and the resulting low temperature modulated phase can be considered as the condensation of the soft phonon mode [79].

The occurrence of soft phonon mode related modulations are frequently connected to the specific nesting behavior of the Fermi surface and/or the presence of a charge density wave (CDW) in the crystal. The impact of these phenomena, which have their origin in the electronic properties of the material, on the martensitic phase transition will be discussed in section 2.3.