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**Extra info for Worked Examples in the Geometry of Crystals**

**Example text**

1 (k; X∗ ) = (h; X∗ )(X P X) (12) Equation 12 describes the way in which plane normals are affected by the deformation (X P X). From equation 11e, it can be shown that −1 (X P X) = I − mq[X; d](p; X∗ ) (13) where 1/q = det(X P X) = 1 + m(p; X )[d; X]. The inverse of (X P X) is thus another invariant–plane strain in the opposite direction. ∗ Example 11: Tensile tests on single–crystals A thin cylindrical single-crystal specimen of α iron is tensile tested at −140◦ C, the tensile axis being along the [4 4 1] direction (the cylinder axis).

This larger shear is of course inconsistent with the hypothesis that the favoured twinning mode involves the smallest shear, and indeed, this mode of twinning is not observed. To obtain the smallest shear, the magnitude of the vector v must also be minimised; in the example under consideration, the correct choice of v will connect a lattice site of plane +1 with the projection of its nearest neighbour lattice site on plane −1. The twinning direction is therefore expected to be along [1 1 2]. It follows that the operative twin mode for the FCC 1 lattice should involve a shear of magnitude s = 2− 2 on {1 1 1} < 1 1 2 >.

By measuring such tilts it is possible to deduce s, which has been experimentally confirmed to equal half the twinning shear. In FCC crystals, the close-packed planes have a stacking sequence . . ABCABCABC . ; the passage of a single Shockley partial causes the sequence to change to . . ABA . . creating a three layer thick region of HCP phase since the stacking sequence of close–packed planes in the HCP lattice has a periodicity of 2. This then is the physical manner in which the transformation occurs, the martensite having a {1 1 1}γ habit plane – if the parent product interface deviates slightly from {1 1 1}γ , then it will consist of stepped sections of close–packed plane, the steps representing the Shockley partial transformation dislocations.