  We propose a mathematical description of crystal structure: underlying
translational periodicity together with the distinct atomic positions up to the
symmetry operations in the unit cell. It is consistent with the international
table of 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, we
generalize the expressions for orientation relationship between the parent
lattice and the derived lattice. 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), we
identify that the derived martensitic structure has the orthorhombic symmetry
Pmmm with derived lattice parameters a_dv = 4.36491 \AA, b_dv = 5.40865 \AA and
c_dv = 4.2402 \AA, by which the complicated X-ray Laue diffraction pattern can
be well indexed, and the orientation relationship can be verified.
