Broadening and shifting of Bragg reflections of nanoscale-microtwinned LT-Ni₃Sn₂

Abstract

The effect of nanoscale microtwinning of long-range ordered domains in LT-Ni₃Sn₂ on its diffraction behaviour was studied by X-ray powder diffraction and electron microscopy. LT-Ni₃Sn₂ exhibits a Ni₂In/NiAs-type structure with a superstructure breaking the symmetry relative to the hexagonal high-temperature (HT) to the orthorhombic low-temperature (LT) phase, implying three different twin-domain orientations. The microstructure was generated by annealing HT-Ni₃Sn₂ considerably below the order–disorder transition temperature, establishing the LT phase avoiding too much domain coarsening. High-resolution electron microscopy reveals domain sizes of 100–200 Å compatible with the Scherrer broadening of the superstructure reflections recorded by X-ray diffraction. Whereas the orthorhombic symmetry of the LT phase leads in powder-diffraction patterns from coarse-domain size material to splitting of the fundamental reflections, this splitting does not occur for the LT-Ni₃Sn₂ with nanoscale domains. Instead, a (pseudo)hexagonal indexing is possible giving hexagonal lattice parameters, which are, however, incompatible with the positions of the superstructure reflections. This can be attributed to interference between X-rays scattered by the differently oriented, truly orthorhombic domains leading to merging of the fundamental reflections. These show pronounced anisotropic microstrain-like broadening, where the integral breadths on the reciprocal d-spacing scale of a series of higher order reflection increase in a non-linear fashion with upward curvature with the reciprocal d-spacings of these reflections. Such a type of unusual microstrain broadening appears to be typical for microstructures which are inhomogeneous on the nanoscale, and in which the structural inhomogeneities lead to small phase shifts of the scattered radiation from different locations (e.g. domains).

Keywords:

• X-ray diffraction
• domain structure
• transmission electron microscopy
• intermetallic phases

Acknowledgement

The authors thank Prof. Dr. E.J. Mittemeijer (MPI for Intelligent Systems, formerly Metals Research, Stuttgart, Germany) for his continuous support of and interest in this work. We are grateful to also Dr. M. Knapp (Darmstadt University of Technology, Germany; present address: Karlsruhe Institute for Technology, Germany) and Dr. C. Baehtz (Darmstadt University of Technology, Germany; present address: Helmholtz Zentrum Dresden Rossendorf, Germany) assistance with the synchrotron measurements performed at DESY, Hasylab Beamline B2, as well as to Prof. D. Rafaja (TU Freiberg, Germany) for discussion. Electron microscopy was done at the electron microscopy centre of the ETH Zurich (EMEZ).

Notes

1. A series of higher order reflections is characterized by a constant direction of the diffraction vector in the crystal frame of reference, which can be given by a unit vector x^hkl = d₀^hkl(ha^∗ + kb^∗ + lc^∗), where a^∗, b^∗ and c^∗ are the basis vectors of the reciprocal lattice. For many purposes, it is convenient to express x^hkl by its components x₁, x₂ and x₃ (leaving out the upper index hkl for brevity) with respect to a Cartesian coordinate system spanned by the vectors e₁, e₂ and e₃. This Cartesian coordinate system is defined in some specific way with respect to the crystallographic coordinate system spanned by a, b c. If corresponds to the first-order reflection along such a direction x^hkl, holds for a reflection of the (integer) order l, where l > 1 indicates a higher order reflection. This proportionality implies that a variation of with will imply a variation of β_ℓ with ℓⁿ, which can be applied to Eq. (1).

2. , where is an occasionally used measure for the extent of microstrain [3].

3. The term microtwinning is used here to indicate that a single grain of the high-symmetry phase is transformed into many domains of the low-temperature phase. The term “micro” does not give a measure for the domain size.

4. Note that the Eqs. (2)–(3) have been formulated with respect to lattice parameters a_orth = ½a_LT, b_orth = b_LT and c_orth = c_LT, which had been used for an easier comparison of the LT phase with an incommensurate LT′ phase occurring at lower Ni contents.

5. The refinements are in general done by minimizing ranging over the data points i and with and being the observed and calculated (from the model) intensities. Thereby, unit weighting implies w_i = 1 instead of the value based on the counting statistics.

6. The line widths of to the HT-Ni₃Sn₂ powder batch HT1023 were slightly larger than those of LaB₆ SRM660a standard. That additional broadening was too small to be reasonably analysed and it was assumed that this broadening constitutes an intrinsic contribution by the alloy (e.g. due to composition variations), which is also present for the LT-Ni_1.50Sn in the powder batches LT573 and LT583. For the case of the sealed-tube measurements (see section 3.2.2), the profiles due to LaB₆ SRM660a standard were adopted as instrumental profile because at that time the powder batch HT1023 was already used up for other experiments. This leaves a somewhat larger physical line broadening for all reflections. The effect of the use of the data from LaB₆ instead of the data from batch HT1023 as instrumental standard is somewhat significant for the relatively narrow fundamental reflections but negligible for the broad superstructure reflections. Therefore, the line broadening data for the fundamental reflections from the sealed-tube powder diffraction data are not explicitly analysed in this work.

7. Due to time restrictions during the synchrotron-radiation measurements, the 00l_HT reflections and other reflections with large l (showing little line broadening) were not measured in detail.

8. The employed model to describe the anisotropy of domain-size broadening derives from a model for the anisotropy of microstrain broadening applicable to incommensurate structures. That model is applicable to describe anisotropy of domain-size broadening if only first-order satellites are treated. In the present commensurately modulated structure, all superstructure reflections showing significant intensity in the X-ray diffraction data can be regarded as first-order satellites (described, however, in terms of conventional hkl triplets), i.e. they are analogs of first-order satellite reflections of the incommensurate LT′ and LT′′ phases occurring for lower and higher Ni contents in the Ni-Sn system.

9. An incomplete degree of long-range order in large LT phase domains may also reduce the values of orth based on “true” lattice parameters. This occurs if large-domain size LT-Ni₃Sn₂ is equilibrated at 773 K, i.e. just below the order–disorder transition (unpublished). Furthermore, incompatibility stresses between the domains due to the hexagonal–orthorhombic transition may lead to elastic strains, which in turn lead to average lattice parameters, which lead to smaller values of orth. Therefore, it is not claimed that e.g. the “true” (within a domain) value of orth amounts to 0.009–0.010 like in coarse-grained LT-Ni₃Sn₂ equilibrated at 473 K or 673 K [13].

10. Thereby, it is not implied that the symmetry of the LT phase is lower than orthorhombic. The twinning is a phenomenon of microstructure, which constitutes some extrinsic feature in the atomic structure.

11. Note that the configuration in Figure 7(a) resulting from twinning by a mirror operation is different from such a configuration resulting by leaving out one of the three orientation variants in Figure 7(b).