Crystal structure determination of incommensurate modulated martensite in Ni–Mn–In Heusler alloys Haile Yan, a,b Yudong Zhang, b,c Nan Xu, a Anatoliy Senyshyn, d Heinz-Gu ¨ nter Brokmeier, e,f Claude Esling, b,c,⇑ Xiang Zhao a and Liang Zuo a,⇑ a Key Laboratory for Anisotropy and Texture of Materials (Ministry of Education), Northeastern University, Shenyang 110819, China b Laboratoire d’E ´ tude des Microstructures et de Me ´canique des Mate ´ riaux (LEM3), CNRS UMR 7239, Universite ´ de Lorraine, 57045 Metz, France c Laboratory of Excellence on Design of Alloy Metals for low-mAss Structures (DAMAS), Universite ´ de Lorraine, 57045 Metz, France d Forschungsneutronenquelle Heinz Maier-Leibnitz (FRM II), Technische Universita ¨t Mu ¨ nchen, Lichtenbergstrasse 1, D-85747 Garching, Germany e Institut fu ¨ r Werkstoffkunde und Werkstofftechnik, TU Clausthal, 38678 Clausthal-Zellerfeld, Germany f Helmholtz-Zentrum Geesthacht, Max-Planck-Straße 1, 21502 Geesthacht, Germany Received 10 December 2014; revised 14 January 2015; accepted 14 January 2015 Abstract—The crystal structure of modulated martensite in Mn-rich off-stoichiometric Ni 2 Mn 1.44 In 0.56 alloy was determined with high-resolution powder neutron diffraction and synchrotron X-ray diffraction in the frame of (3 + 1)-dimensional superspace theory. The average crystal structure and the modulation wave vector were firstly derived by analyzing the reflection separations induced by the martensitic transformation on the basis of the transformation orientation inheritance. This treatment could be applied to predetermine the modulated structures of materials with displacive structural transformation. The crystal structure of modulated martensite was finally refined by the Rietveld method. Results show that the martensite possesses an incommensurate 6M modulated structure of superspace group I2/m(a0c)00, with lattice parameters a = 4.3919(4) A ˚ , b = 5.6202(1) A ˚ , c = 4.3315(7) A ˚ , and b = 93.044(1)°, and the modulation wave vector q = 0.343(7) c.* The detailed site occupations for extra-Mn atoms with respect to the stoichiometric case were investigated by ab initio calculations. The extra-Mn atoms have a preference to be uniformly dispersed. A threefold layered superstructure in the 3-dimensional space was proposed to approximately describe the incommensurate modulated structure. This 6M super- structure model is considered to be representative for off-stoichiometric Ni–(Co)–Mn–In modulated martensite with martensitic transformation around room temperature. The present study is expected to offer an important basis for reliable crystallographic and microstructural characteriza- tions on Ni–Mn–In alloys, so as to understand the underlying mechanisms of their multifunctional magneto-responsive properties. Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Ni–Mn–In alloy; Modulated martensite; Crystal structure; Rietveld method; Superspace theory 1. Introduction In the past decade, the Ni–Mn–In Heusler-type alloys have attracted increasing attention owing to their rich physical properties, such as magnetic-field-induced shape memory effect [1,2], inverse magnetocaloric effect [3], exchange bias effect [4], Hall effect [5], and magnetoresis- tance effect [6]. The broad spectrum of magneto-responsive behaviors of these materials mainly comes from the cou- pling interactions between crystal structure and magnetic structure during structural transformation under magnetic field (known as magnetic-field-induced inverse martensitic transformation from weak-magnetic martensite to ferro- magnetic austenite). As the multi-functionalities of Ni– Mn–In alloys are intimately related to the crystal structures of constituent phases, the crystal structure determination represents a prerequisite for further exploration of the underlying mechanisms toward property optimization. To date, the crystal structures of martensite phases in Ni–Mn–In alloys have largely been investigated by means of X-ray diffraction (XRD), selected area electron diffrac- tion (SAED) and neutron diffraction (ND) [7–19]. For easy consultation, some results reported in the literature are summarized in Table 1. It can be seen that the crystal struc- tures of martensite in Ni–Mn–In based alloys are sensitive to chemical composition. There exist, in general, two kinds of martensite, namely non-modulated martensite and modulated martensite. For the latter, the modulation type http://dx.doi.org/10.1016/j.actamat.2015.01.025 1359-6462/Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. ⇑ Corresponding authors at: LEM3, CNRS UMR 7239, Universite ´ de Lorraine, 57045 Metz, France (C. Esling), Key Laboratory for Anisotropy and Texture of Materials (MOE), Northeastern Univer- sity, Shenyang 110819, China (L. Zuo); e-mail addresses: claude. esling@univ-lorraine.fr; lzuo@mail.neu.edu.cn Available online at www.sciencedirect.com ScienceDirect Acta Materialia 88 (2015) 375–388 www.elsevier.com/locate/actamat