research papers
A general method to determine
elements^{a}Key Laboratory for Anisotropy and Texture of Materials (Ministry of Education), Northeastern University, Shenyang 110819, People's Republic of China, and ^{b}Laboratoire d'Étude des Textures et Application aux Matériaux (LETAM), CNRS FRE 3143, Université Paul Verlaine – Metz, Metz 57012, France
^{*}Correspondence email: lzuo@mail.neu.edu.cn
The fundamental theory of crystal
has been long established, leading to a significant advance in understanding the nature of this physical phenomenon. However, there remains a substantial gap between the elaborate theory and the practical determination of elements. This paper proposes a direct and simple method – valid for any and based on the minimum shear criterion – to calculate various elements from the experimentally determined plane for Type I twins or the direction for Type II twins. Without additional efforts, it is generally applicable to identify and predict possible modes occurring in a variety of crystalline solids. Therefore, the present method is a promising tool to characterize elements, especially for those materials with complex crystal structure.Keywords: twinning; minimum shear; interface structure; transmission electron microscopy; scanning electron microscopy/electron backscatter diffraction.
1. Introduction
Crystal twins are commonly observed during solidification, deformation, solidstate phase transformation and recrystallization in a variety of crystalline solids with low
energy. Often, these features occur on the nanometre to micrometre scale, and they represent a particularly symmetric kind of grain boundary, giving rise to a much lower level of interfacial energy than general grain boundaries. As an underlying mechanism for microstructural changes, crystal has acquired great importance in fields such as metallography, mineralogy, crystallography and physics.Early efforts to define crystal twins were based on the study of deformation K_{1} – the or that is the invariant (unrotated and undistorted) plane of the simple shear; (2) η_{1} – the direction or the direction of shear lying in K_{1}; (3) K_{2} – the reciprocal or conjugate plane, the second undistorted but rotated plane of the simple shear; (4) η_{2} – the reciprocal or conjugate direction lying in K_{2}; (5) P – the plane of shear that is perpendicular to K_{1} and K_{2} and intersects K_{1} and K_{2} in the directions η_{1} and η_{2}, respectively; (6) γ – the magnitude of shear. Moreover, the orientation relationship between two twinrelated crystals can be specified by simple crystallographic operations: a reflection across K_{1} or a 180° rotation about the direction normal to K_{1}; or a 180° rotation about η_{1} or a reflection across the plane normal to η_{1}. According to the rationality of the of K_{1}, K_{2}, η_{1} and η_{2} with respect to the parent lattice, crystal twins are usually classified into three categories: Type I twin (K_{1} and η_{2} are rational), Type II twin (K_{2} and η_{1} are rational) and compound twin (K_{1}, K_{2}, η_{1} and η_{2} are all rational).
By convention, a deformation twin is a region of a crystal that has undergone a homogeneous shape deformation (simple shear) in such a way that the resulting structure is identical to that of the parent (matrix), but differently oriented. A mode is fully characterized by six elements: (1)The classical definition and description of deformation et al., 2009; Wang et al., 2006; Li et al., 2010). In such a case, the twinned martensitic variants always form regular arrays of alternate lamellae with fixed thickness and the twin boundaries are highly glissile, where the detwinning shear determines the shape memory performance.
have been further extended to describe other processes associated with phase transformation and recrystallization. Notably, the concept of transformation is widely adopted for the elucidation of structural changes during martensitic transformation. Although the formation of twinned martensitic variants is driven by a deformation from the parent phase and may not have any relation to the simple shear deformation defined by the shear, the detwinning process can be well predicted by these elements, especially for the newly developed ferromagnetic shape memory alloys (GaitzschFor many years, constant attempts have been made to determine , 1958) and Jaswon & Dove (1956, 1957, 1960) based on the minimum shear criterion, and later completed by Bilby & Crocker (1965) and Bevis & Crocker (1968, 1969). It provides the general expressions – valid for all crystal structures – to predict the elements for both Type I and Type II twins with a known shear. However, in a practical determination of unknown twins, it is only feasible to resolve the possible plane K_{1} for Type I twins or the direction η_{1} for Type II twins by means of (TEM) or scanning electron microscopy/electron backscatter diffraction (SEM/EBSD). In other words, with the given general expressions, one always suffers from insufficient information to derive the unknown elements, especially the shear. As a common practice, laborious geometrical examination of the lattice correspondence of the stacking planes parallel to the plane has to be conducted. Such a process becomes particularly difficult when the plane and the shear plane are irrational and the is complicated. Hence, there exists a substantial gap between the elaborate theory and the practical determination.
elements of crystalline materials from the knowledge of because of their importance for insight into possible modes and resultant orientation relationships of twinned crystals in the context of microstructural manipulation. A systematic theory was developed by Kiho (1954In this paper, we present a complete method to find all K_{1} (Type I) or η_{1} (Type II). As a general method applicable to any it may facilitate future characterization studies of crystal twinning.
elements for the three classical types of twins, based on the assumption that a simple minimum shear operation transforms the lattice points of a crystal into their counterpart twin positions. The initial inputs are simply the and the experimentally determined2. Methodology
2.1. Determination of mode
For a twinned crystal, the crystallographic orientations of the twin and its parent can be experimentally determined with SEM/EBSD or TEM. In the case of SEM/EBSD examination, the orientation of a crystal with respect to the macroscopic sample coordinate system is usually characterized in terms of three Euler angles. The misorientation between the twin and the parent is then calculated from their Euler angles, and expressed by a set of rotation angles and the corresponding rotation axes (Cong et al., 2006, 2007). According to the definition of twin relationships mentioned above, there exists at least one 180° rotation. If the of the plane normal to the 180° rotation axis are rational, the mode belongs to Type I and the plane is the plane K_{1}. If the of the 180° rotation axis are rational, the mode refers to Type II and the direction of the rotation axis is the direction η_{1}. Since a compound twin has two 180° rotations with rational K_{1}, K_{2}, η_{1} and η_{2}, the plane normal to the 180° rotation axis that offers the minimum shear should be the plane K_{1}.
In contrast to the SEM/EBSD examination, the TEM determination process involves examining the spot diffraction image (Nishida et al., 2008). For Type I and compound twins, the diffraction image – obtained on condition that the incident beam is parallel to the K_{1} plane – consists of two sets of reflections that are in mirror symmetry to each other with respect to the K_{1} reflection. Thus, the K_{1} plane can be identified. For Type II twins, the diffraction image – obtained with the incident beam along the η_{1} direction – contains a single visible pattern, i.e. the reflections from two twinrelated crystals overlap each other. The η_{1} direction could also be determined.
Based on the above experimental identification, the other
elements to define a mode can be further derived with the method outlined below.2.2. Determination of elements
2.2.1. Type I and compound twins
According to the classical definition, a Type I or compound twin is related to its parent by a reflection across the K_{1}, where the K_{1} plane is a rational lattice plane with relatively small With this condition as starting point, the possible direction η_{1} and the magnitude of shear γ can be deduced in conformity with the minimum shear criterion, i.e. the shear that moves all parent lattice points to their correct twin positions appears to be the smallest in magnitude. Hereafter, our calculations are conducted in the direct primitive lattice of the parent crystal. For the coordinate transformations between the primitive lattice basis and the conventional basis, we refer to International Tables for Crystallography (Hahn, 1996).
planeAt first, let us choose two basis vectors u_{1} and u_{2} in the plane K_{1} and transform them into the reduced vectors e_{1} and e_{2}, as shown schematically in Fig. 1. The reduced basis vectors e_{1} and e_{2} must be the two shortest translations and the most orthogonal to each other among all possible basis vectors in the plane K_{1}. Note that such a reduced basis is useful for determining the nearest lattice point(s) to a given point (not necessarily lattice site) in the plane K_{1}. The procedures to find the basis vectors u_{1} and u_{2} and to reduce them to e_{1} and e_{2} are detailed in Appendix A and Appendix C, respectively.
Now, we show how to determine the t by use of the reduced basis e_{1} and e_{2}. Let Plane 0 represent the (invariant) plane K_{1} that separates the (above Plane 0) from that of the parent (below Plane 0), as shown schematically in Fig. 2. Since the nearest neighbor plane (Plane −1) of the parent lattice and its counterpart (Plane 1) for the are parallel and in mirror symmetry with respect to the invariant plane K_{1}, the perpendicular projection of Plane −1 onto Plane 1 allows us to identify the possible shear vector. Here, we select a parent lattice vector OA that ends at the lattice point A on Plane −1, and denote by A′ the endpoint of the projection of vector OA on Plane 1. Obviously, the vector t that joins A′ – a point – to its nearest parent lattice point N on Plane 1 defines the direction η_{1} and ensures the smallest magnitude of shear. The procedures for determining the vectors OA and t are described in Appendix B.
shear vectorFurthermore, the K_{1} can be easily calculated by the scalar product of OA and m:
of the planewhere m denotes the unit vector in the direction normal to the plane K_{1}. Thus, the magnitude of shear is given by
Once the shear vector t and the magnitude of shear γ are determined, the other elements (η_{2}, K_{2} and P) can be readily calculated according to the Bilby–Crocker theory (Bilby & Crocker, 1965).
Let I be the unit vector in the direction η_{1} and g_{M} a vector in the conjugate direction η_{2}, with reference to the parent lattice basis. Applying the by a shear γ along η_{1}, g_{M} is transformed into g_{M}′, as shown schematically in Fig. 3. Since η_{2} is defined by a rotated but undistorted lattice line of the shear, g_{M}′ has the same indices (and hence the same length) as g_{M}, if it is referred to the basis. Moreover, g_{M} and g_{M}′ lying in the shear plane P (perpendicular to K_{1}) are in mirror symmetry with respect to the plane that contains the vector V (= ) and is perpendicular to η_{1}. Thus, the three vectors g_{M}, g_{M}′ and g form an isosceles triangle. As g (= ) in the shear direction is divided into two equal lengths by V, we obtain
Notably, g_{M} is not necessarily a lattice vector, and its components – expressed in terms of the parent lattice basis – can always be transformed into rational indices. Once the lattice vector in the η_{2} direction is determined from g_{M}, the shear plane P and the conjugate plane K_{2} can be easily calculated by the vector cross product (Bilby & Crocker, 1965).
2.2.2. Type II twin
By definition, a Type II twin is related to its parent by a 180° rotation about the η_{1} or a reflection across the plane normal to the direction η_{1}. Let us first recall the fundamental relationships between and Every lattice vector in the corresponds to a set of lattice planes normal to this vector in the and vice versa. Thus, the twin relationship of a Type II twin in the can be equivalently expressed by a reflection with respect to the plane that is normal to η_{1} in the or in other words, a Type II twin in the is visualized as a Type I twin in the As the two spaces are strictly linked to each other, we can see that, when the undergoes the is subject to the same deformation (shear in the same direction and with the same magnitude) and verse visa. In this context, the determination of the elements of Type II twins can follow the same procedure as that of Type I, except that all the calculations should be conducted in the Moreover, the resultant directions (planes) in the correspond to the same indexed planes (directions) in the as summarized in Table 1.
direction

3. Conclusions
As a widely observed and intrinsic process, crystal K_{1} for Type I twins or the direction η_{1} for Type II twins and the as input. As a first step, it determines a reduced basis of the invariant lattice plane that serves as the mirror plane (in the for Type I twins and in the for Type II twins) between the parent and twin lattices. Then, a lattice vector – with its origin at the invariant lattice plane and its end at the nearest neighbor lattice plane of the same set – is selected from the parent lattice and projected onto the counterpart lattice plane of the Among the vectors that join the endpoint of the projected lattice vector to the surrounding parent lattice points forming the reduced basis, the shortest vector defines the direction and the shear. Finally, the other elements can be easily calculated using the vector product operations. The present method, as it stands, is highly significant for facilitating the study of in a variety of crystalline materials.
has a broad impact on the microstructures and properties of crystalline materials. So far, the classical theory of has advanced greatly the study of twining, but it often suffers from insufficient information for practical determination of full elements. To progress beyond this state, a general method is elaborated based on the minimum shear criterion, using the experimentally identified possible planeAPPENDIX A
Determination of base vectors u_{1} and u_{2} on a lattice plane
In crystallography, a lattice plane P with a given is usually described by the (hkl), i.e. a set of three integers with the greatest common divisor gcd (h,k,l) = . Assume ; then . If an arbitrary lattice vector u with the [uvw] lies in the plane P, it has
where u, v and w are integers. Since , the following relation holds:
Let , and ; then , and are also integers. Equation (5) can be written as
As , one can find two integers u_{0} and v_{0} that satisfy the following relation according to Bézout's theorem:
Multiplying both sides of equation (7) by (), we obtain
Let and ; then equation (8) becomes
Subtracting equation (9) from equation (6), we have
Since , there exists an integer such that
Rearranging equation (11), we obtain
By definition, the basis vectors are a set of linearly independent vectors such that each vector in the space is a linear combination of the vectors from the set. Therefore, equation (12) proves that the vector (, , ) constitutes the basis vectors of the plane (hkl). Setting and , and and , respectively, we obtain two basis vectors:
where u_{0} and v_{0} are the Bézout coefficients of equation (7). With the Euclidean algorithm, u_{0} and v_{0} can be easily calculated.
APPENDIX B
Determination of lattice vector OA and shear vector t
B1. Lattice vector OA
According to the fundamental law of the ), for an arbitrary vector OA with its origin O at the zeroth plane of a family of lattice planes (hkl), if it intersects the nth plane at the point with coordinates (x, y, z), the following relation holds:
(Authier, 2001Let OA be the lattice vector with the [−2u −2v −2w] and K_{1} the invariant plane with the (hkl), as shown in Fig. 2. Then, we have
The Bézout coefficients u_{0}, v_{0} and w_{0} of equation (15) can be calculated with the Euclidean algorithm, and hence the lattice vector OA.
B2. Shear vector t
Consider a lattice vector OA with its origin at O on Plane 1 and its end at A on Plane −1, as shown in Fig. 4.
Let A′ be the perpendicular projection of the lattice point A on Plane 1. The shear vector t is defined as the shortest vector among all vectors that connect A′ with the surrounding lattice points on Plane 1. Introducing the reduced basis e_{1} and e_{2}, we can derive from Fig. 4 that
where m is the unit vector of the plane normal. By comparing the lengths of O′A′, qA′, pA′ and rA′, the shortest vector t can be easily found.
APPENDIX C
Transformation of basis vectors u_{1} and u_{2} into the reduced basis e_{1} and e_{2}
To find the closest lattice point to the projection A′ and thus the minimum shear of the it is essential to establish a reduced basis, i.e. the two shortest lattice vectors that are most orthogonal to each other (Zuo et al., 1995). With the basis vectors u_{1} and u_{2} determined according to Appendix A as input, the reduced basis e_{1} and e_{2} can be derived using an iterative procedure, as described below.
Let e_{1} be the shorter vector between the two base vectors, i.e. . Then, the new base vectors are derived from
To render the two vectors orthogonal to each other, this yields
If , and deliver the reduced basis vectors. Otherwise, is rounded into the nearest integer and the above procedure is repeated until .
Acknowledgements
This work was supported by the National Natural Science Foundation of China (grant No. 50820135101), the Ministry of Education of China (grant Nos. 707017, 2007B35, IRT0713 and B07015), the PhD Innovation Program of Northeastern University of China (grant No. N090602002), the CNRS of France (PICS No. 4164) and the ANR (OPTIMAG No. ANR09BLAN0382).
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