Furthermore, the interplanar spacing of the twinning plane K 1 can be easily calculated by the scalar product of OA and m: where m denotes the unit vector in the direction normal to the twinning 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 twinning elements (η2, 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 twinning direction η1 and g M a vector in the conjugate twinning direction η2, with reference to the parent lattice basis. Applying the twinning operation by a shear γ along η1, g M is transformed into g M′, as shown schematically in Fig. 3 ▶. Since η2 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 twin lattice 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 η1. 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 In crystallography, a lattice plane P with a given Bravais lattice is usually described by the Miller indices (hkl), i.e. a set of three integers with the greatest common divisor . Assume , then . If an arbitrary lattice vector u with the Miller indices [uvw] lies in the plane P, it has As , one can find two integers and that satisfy the following relation according to Bézout’s theorem: 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 and are the Bézout coefficients of equation (7). With the Euclidean algorithm, and can be easily calculated. According to the fundamental law of the reciprocal lattice (Authier, 2001 ▶), 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: Let OA be the lattice vector with the Miller indices [−2u −2v −2w] and K 1 the invariant plane with the Miller indices (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. 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. Let e 1 be the shorter vector between the two base vectors, i.e. . Then, the new base vectors are derived from If , and deliver the reduced basis vectors. Otherwise, is rounded into the nearest integer and the above procedure is repeated until . We are experimenting with display styles that make it easier to read articles in PMC. Our first effort uses eBook readers, which have several ‘ease of reading’ features already built in. These PMC articles are best viewed in the iBooks reader. You may notice problems with the display of certain parts of an article in other eReaders. Source.