Spontaneous strain as a determinant of thermodynamic properties for phase transitions in minerals
Carpenter, Michael A.; Salje, Ekhard K.H.; Graeme-Barber, Ann
European Journal of Mineralogy Volume 10 Number 4 (1998), p. 621 - 691
published: Jul 10, 1998
manuscript accepted: Feb 27, 1998
manuscript received: May 13, 1997
ArtNo. ESP147051004002, Price: 29.00 €
Abstract Lattice parameters are geometrical properties of a crystal, but their variations at phase transitions can be formalised for thermodynamic analysis using the concept of spontaneous strain. As with many other physical 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, principal 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 crystal system are also listed. The relationship between any spontaneous strain, e, and the driving order parameter, ß, for a phase transition is explored by first considering a general free-energy expansion of the form: L(Q) is a standard Landau expansion, the coefficient λ describes coupling between Q and e, and the last term describes elastic energies. The exponents m and n depend on the symmetry properties of both Q and e, but, in general, only one coupling term is needed to account for each strain component. Characteristic relationships are then e ∞ Q for symmetry-breaking strains when e and Q have the same symmetry, and e ∞ Q2 for all other strains. The volume strain, Vs, is generally expected to vary linearly with Q2. 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, Pb3(PO4)2, cristobalite, NaMgF3 perovskite, K2Cd2(SO4)3 langbeinite, BiVO4 and BaCeO3 perovskite. The same overall approach applies 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 temperatures (or pressures). When atomic ordering does not lead to a symmetry change (non-convergent ordering), the spontaneous strains are expected to vary as e ∞ Vsoc Q . Landau theory provides a convenient theoretical framework for the quantitative thermodynamic analysis of all these materials.