Original paper
Elastic anomalies in minerals due to structural phase transitions
Carpenter, Michael A.; Salje, Ekhard K.H.
European Journal of Mineralogy Volume 10 Number 4 (1998), p. 693 - 812
335 references
published: Jul 10, 1998
manuscript accepted: Feb 7, 1998
manuscript received: May 13, 1997
Abstract
Abstract Landau theory provides a formal basis for predicting the variations of elastic constants associated with phase transitions in minerals. These elastic constants can show substantial anomalies as a transition point is approached from both the high-symmetry side and the low-symmetry side. In the limiting case of proper ferroelastic behaviour, individual elastic constants, or some symmetryadapted combination of them, can become very small if not actually go to zero. When the driving order parameter for the transition is a spontaneous strain, the total excess energy for the transition is purely elastic and is given by: which has the same form as a Landau expansion. In this case, the second-order elastic constant Cik softens as a linear function of temperature with a slope in the low-symmetry phase that depends on the thermodynamic character of the transition. If the driving order parameter, g, is some structural feature other than strain, the excess energy is given by: In this case, the effect of coupling, described by the term in λemQn, is to cause a great diversity of elastic variations depending on the values of m and n (typically 1, 2 or 3), the thermodynamic character of the transition and the magnitudes of any non-symmetry-breaking strains. The elastic constants are obtained by taking the appropriate second derivatives of G with respect to strain in a manner that includes the structural relaxation associated with Q. The symmetry properties of second-order elastic constant matrices can be related to the symmetry rules for individual phase transitions in order to predict elastic stability limits, and to derive the correct form of Landau expansion for any symmetry change. Selected examples of "ideal" behaviour for different types of driving order parameter, coupling behaviour and thermodynamic character have been set out in full in this review. Anomalies in the elastic properties on a macroscopic scale can also be understood in terms of the properties of acoustic phonons. These microscopic processes must be considered if elastic anomalies due to dynamical effects are to be accounted for correctly. Such additional anomalies are characterised by softening of the form ACik = Aik|T-Tc|K as the transition is approached from the high-symmetry side. The A coefficient is a property of the material, and K depends on how the branches of the critical acoustic mode soften in three dimensions. Adopting this approach allows the quantitative description of elastic variations in "real" systems. Albite provides a likely example of proper ferroelasticity in minerals, and values for the required coefficients, extracted from experimental data, yield a complete picture of the expected elastic properties. The ß ⇌ a transition in quartz provides an example of co-elastic behaviour. Data for TeO2, BiVO4 and KMnF3 (a perovskite) have been reviewed to illustrate the full range of elastic anomalies that should be expected at structural phase transitions in natural minerals.
Keywords
phase transitions • elastic constants • ferroelasticity • Landau theory • albite • quartz • perovskite