Application of the Landau Theory to Elastic Phase Transitions

Abstract

At an elastic phase transition, a crystal structure change takes place by a homogeneous lattice deformation, the associated soft mode being a zone centre acoustic phonon. As such a transition is approached, the mode softening is manifested as a reduction in the second order elastic constant which acts as the quadratic invariant and so the higher order terms become increasingly important: a pure elastic transition will be dominated by vibrational anharmonicity of the soft acoustic mode. For a first order, elastic transition, a quantitative understanding of this anharmonic behaviour must rest upon measurements of the third order elastic constants which constitute the cubic invariant in the order parameter (which is a strain tensor component). To obtain the forms of these physically significant invariants in the Landau free energy expansion up to third order in the strain eigenvectors, the elastic free energy is transformed from finite strain space to the irreducible strain tensor space keeping the free energy invariant. In-Tl alloys undergo a f.c.c.-f.c.t. elastic phase transition whose order has been difficult to establish experimentally, although the Landau theory suggests that the transition should be first order because there is a cubic invariant of the order parameter. These alloys provide a suitable vehicle for a quantitative asessment of the applicability of the Landau theory to a first order elastic phase transition in which the atomic displacements are small. The transition is accompanied by softening of the N[110], polarisation U[1-10] zone centre acoustic phonons: the second order invariant ½(C11 - C12) goes to zero at Tc. The third order invariants obtained from third order elastic constant measurements made on In-Tl alloys are discussed. The hydrostatic pressure derivative ∂(½(C11 - C12))/∂P is negative for those f.c.c. alloys which undergo the transition. As suggested by the Landau theory, the third order invariant (1/8)(C111 - 3C112 + 2C123) of the order parameter is small, in agreement with the near second order character of the transition: there is only a slight discontinuity in the strain at Tc. The first order character of the transition is evidenced further by a marked hysteresis in ultrasonic wave velocities as crystals are taken through the transition by pressure or temperature cycles.