We consider ferroelastic first-order phase transitions with NOP
order-parameter strains entering Landau free energies as invariant polynomials,
that have NV structural-variant Landau minima. The total free energy
includes (seemingly innocuous) harmonic terms, in the n=6−NOP {\it
non}-order-parameter strains. Four 3D transitions are considered,
tetragonal/orthorhombic, cubic/tetragonal, cubic/trigonal and
cubic/orthorhombic unit-cell distortions, with respectively, NOP=1,2,3 and 2; and NV=2,3,4 and 6. Five 2D transitions are also considered, as
simpler examples. Following Barsch and Krumhansl, we scale the free energy to
absorb most material-dependent elastic coefficients into an overall prefactor,
by scaling in an overall elastic energy density; a dimensionless temperature
variable; and the spontaneous-strain magnitude at transition λ<<1.
To leading order in λ the scaled Landau minima become
material-independent, in a kind of 'quasi-universality'. The scaled minima in
NOP-dimensional order-parameter space, fall at the centre and at the NV
corners, of a transition-specific polyhedron inscribed in a sphere, whose
radius is unity at transition. The `polyhedra' for the four 3D transitions are
respectively, a line, a triangle, a tetrahedron, and a hexagon. We minimize the
n terms harmonic in the non-order-parameter strains, by substituting
solutions of the 'no dislocation' St Venant compatibility constraints, and
explicitly obtain powerlaw anisotropic, order-parameter interactions, for all
transitions. In a reduced discrete-variable description, the competing minima
of the Landau free energies induce unit-magnitude pseudospin vectors, with NV+1 values, pointing to the polyhedra corners and the (zero-value) center.Comment: submitted to PR