A general method to determine twinning elements

Abstract

The fundamental theory of crystal twinning 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 twinning elements. This paper proposes a direct and simple method -valid for any crystal structure and based on the minimum shear criterion -to calculate various twinning elements from the experimentally determined twinning plane for Type I twins or the twinning direction for Type II twins. Without additional efforts, it is generally applicable to identify and predict possible twinning modes occurring in a variety of crystalline solids. Therefore, the present method is a promising tool to characterize twinning elements, especially for those materials with complex crystal structure.

Twin relationships of 5M modulated martensite in Ni–Mn–Ga alloy

“…Based upon the minimum shear criterion, the complete twinning elements (K 1 -the twinning plane; g 1 -the twinning direction; K 2 -the reciprocal or conjugate twinning plane; g 2 -the reciprocal or conjugate twinning direction; P -the plane of shear; s -the amount of shear) of the above three types of twins were unambiguously determined using a general method recently developed by the present authors [30], as displayed in Table 2. It is seen from Table 2 that type I (rational K 1 and g 2 ) and type II twins (rational K 2 and g 1 ) have the same shear s, but the K 1 and K 2 , and g 1 and g 2 are interchanged.…”

Multiple twinning in pure hexagonal close-packed titanium

“…Nespolo & Ferraris, 2000, 2003Nespolo, 2004). Several authors of this paper have also applied mathematical methods imported from number theory, notably Bezout's theorem (Zhang et al, 2010), to develop a general method to determine twinning elements. The purpose of the present paper is to study mechanical twinning in titanium, which belongs to the space group P6 3 /mmc, with cell parameters a = b = 2.9508 Å , c = 4.6855 Å , c/a = 1.58787, = = 90 and = 120 .…”

Crystallographic orientation of modulated martensite in epitaxially grown Ni–Mn–Ga thin film

“…It is clearly seen that in the low contrast zone, the Type-I twin interfaces are dominant, whereas in the high contrast zone, the Type-II twin interfaces are dominant. The complete twinning elements -K 1 , K 2 , η 1 , η 2 , P and s -of the above three types of twins were derived using a general method [26,27], as shown in Table 1. The twinning modes are exactly the same as those in bulk materials [16,27].…”