Abstract
A mathematical description of crystal structure is proposed consisting of two parts: the underlying translational periodicity and the distinct atomic positions up to the symmetry operations in the unit cell, consistent with the International Tables for Crystallography. By the Cauchy-Born hypothesis, such a description can be integrated with the theory of continuum mechanics to calculate a derived crystal structure produced by solid-solid phase transformation. In addition, the expressions for the orientation relationship between the parent lattice and the derived lattice are generalized. The derived structure rationalizes the lattice parameters and the general equivalent atomic positions that assist the indexing process of X-ray diffraction analysis for low-symmetry martensitic materials undergoing phase transformation. The analysis is demonstrated in a CuAlMn shape memory alloy. From its austenite phase (L2(1) face-centered cubic structure), it is identified that the derived martensitic structure has orthorhombic symmetry Pmmn with the derived lattice parameters a(d) = 4.36491, b(d) = 5.40865 and c(d) = 4.2402 Å, by which the complicated X-ray Laue diffraction pattern can be well indexed, and the orientation relationship can be verified.