The martensitic transformation in TiNi-based shape memory alloys
has been investigated using various theoretical methods. The direct
method for ab initio lattice dynamics is used for calculating phonon
dispersion relations in austenite (B2) and martensites (B19, B19′).
Normal mode symmetries in a B2 crystal are determined. Vibrational
modes in the B2 crystal are labelled according to symmetry
properties of their polarisation vector. Normal modes with imaginary
frequencies are observed at the locations of experimentally observed
soft modes. While labelling normal modes, the published mode symmetry
label of a soft transverse acoustic (TA2) mode with a [Mathematical formula appears here. To view, please open pdf attachment]
polarisation at [Mathematical formula appears here. To view, please open pdf attachment] is corrected to Σ2. Irreducible representations
of the soft modes are used for determining the order parameters
for various martensitic transformation paths experimentally observed
using group theoretical methods. Accuracy of the interpolated frequencies
in the phonon dispersion relations is assessed by using three
different supercell sizes and by calculating the elastic constants from
the slopes of the acoustic branches at the Brillouin zone.
Born et al. derived a relation between the force constants and
elastic constants for a general crystal. Further simplification of this
relation is possible for a B2 crystal because of its symmetry. This
simplification is derived by applying elementary group theory. Elastic
constants calculated with the simplified relation are in agreement
with the elastic constants calculated from the slopes of the dispersion
relations.
The self-consistent ab initio lattice dynamics (SCAILD) formulation
is derived for an ordered compound. Precautions that need to be
taken while practically implementing this formulation are discussed.
The effect of Hf or Zr addition to a binary Ti-Ni system is studied
by considering one configuration at each of five compositions from 0
to 25 at.% Hf or Zr. Elastic constants are calculated following a
homogeneous deformation method. Selected normal mode frequencies
are calculated using the direct method.