SPONTANEOUS STRAIN AS A DETERMINANT OF THERMODYNAMIC PROPERTIES FOR PHASE-TRANSITIONS IN MINERALS

Lattice parameters are geometrical properties of a crystal, but their variations at phase transitions can be formalised for thermodynamic an alysis using the concept of spontaneous strain. As with many other phy sical properties, spontaneous strains consist of up to six independent components forming a symmetric second-rank tensor, and are subject to the constraints of symmetry. Technical aspects of reference states, p rincipal strains, scalar strains, volume strains, etc., are summarised in this review, and sets of equations defining the individual strain components in terms of lattice parameters for different changes in cry stal system are also listed. The relationship between any spontaneous strain, e, and the driving order parameter, Q, for a phase transition is explored by first considering a general free-energy expansion of th e form: G(Q,e) = L(Q) + Sigma(i,m,n)lambda(i,m,n)e(i)(in)Q(n) + 1/2 Si gma(i,k)C(ik)(0)e(i)e(k) L(Q) is a standard Landau expansion, the coef ficient lambda describes coupling between e and e, and the last term d escribes elastic energies. The exponents m and n depend on the symmetr y properties of both e and e, but, in general, only one coupling term is needed to account for each strain component. Characteristic relatio nships are then e proportional to Q for symmetry-breaking strains when e and Q have the same symmetry, and e proportional to Q(2) for all ot her strains. The volume strain, V-s, is generally expected to vary lin early with Q(2). The principles of strain analysis are illustrated for phase transitions in a selection of minerals and model systems. These include: As2O5, albite, tridymite, anorthite, leucite, calcite, NaNO3 , quartz, Pb-3(PO4)(2), cristobalite, NaMgF3 perovskite, K2Cd2(SO4)(3) langbeinite, BiVO4 and BaCeO3 perovskite. The same overall approach a pplies whether the transitions occur in response to changing pressure or temperature. It can also be successful when lattice-parameter data for minerals displaying cation order/disorder phenomena are collected at room temperature (and pressure), rather than in situ at high temper atures (or pressures). When atomic ordering does not lead to a symmetr y change (non-convergent ordering), the spontaneous strains are expect ed to vary as e proportional to V-s proportional to Q. Landau theory p rovides a convenient theoretical framework for the quantitative thermo dynamic analysis of all these materials.