Elastic phase transitions in inhomogeneous media

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Abstract
We investigate how the statics and dynamics of elastic phase transitions are influenced by defects which locally increase the transition temperature. The theory is developed both for isolated and for a finite concentration of randomly distributed defects. It is based on a one-dimensional Ginzburg-Landau functional with short-range defects, thereby modelling a grain boundary or a thin intermediate layer in the three-dimensional case. Analogously to distortive transitions, we find the phenomenon of local condensation at the defect when the temperature T reaches the “local transition temperature” T1 c. However, the dynamical features concerning the soft acoustic phonons are significantly different. Instead of localized modes, there is a localized quasi-resonant vibrational part in each of the scattering states. When T approaches T 1 c , the transmission amplitudes of the long-wavelength acoustic phonons vanish and the vibrational part “condenses” at the defect. If a local condensation is forced upon the crystal by application of an external stress, which is removed immediately afterwards, the corresponding deformation decays into a boost of acoustic phonons, the relaxation time diverging in the vicinity of T 1 c . The phonon-phonon response function can be calculated analytically. Averaging over the defect positions we obtain the response and correlation functions for a random multi-defect system to first order in the defect concentration n. These display the development of a pronounced central peak when the temperature approaches Tc (n) = T 1 c + O(n). In contrast to T 1 c , however, at Tc (n) a global phase transition induced by the “softening” defects takes place. Below Tc (n) an inhomogeneous order sets in.