Tensor parameters of ferroic phase transitions

Abstract

The main purpose of this work is to describe the changes of material properties, expressed by tensors of up to the fourth rank, induced by ferroic phase transitions. The results also prepare a firm ground for further analysis of the two main problems of “Continuous Bicrystallography”: 1. The tensor distinction of pairs of domain states. 2. The change of tensor properties across a bicrystallographic interface (domain wall or twin boundary). Since it is our aim to present the information in a standardized manner, we begin with the nomenclature of specifically oriented crystallographic point groups and of their irreducible representations (ireps). The usual Schönflies and Hermann-Mauguin symbols are embellished by subscripts which define the orientation of groups. Matrix form of irreducible representations is associated with so-called typical variables which transform in a well defined way. The transformation properties of material property tensors up to fourth rank are presented in tables of tensorial covariants where independent linear combinations of cartesian tensor components transforming like the typical variables are listed. Invariant combinations represent the set of tensor parameters which describe the tensor allowed by the parent symmetry G. Other combinations, so-called covariant tensor components, are the potential tensor parameters of ferroic transitions G ⇓ F_i , where F_i is the set of conjugate low symmetry groups. The main tables contain vital information about the change of tensors up to fourth rank for all ferroic phase transitions. To each symmetry descent between crystallographic point groups there is given a table from which one can read immediately the form of tensors for the parent group and tensor parameters onsetting in the first domain state of the ferroic symmetry. These parameters are distinguished by ireps of the parent group which refer to their transformation properties under the action of the parent group. This is an important point because parameters belonging to different ireps change in a different manner from one domain state to the other. To each such set of parameters there is given information about its symmetry in the first domain state, about the total number of domain states and about the numbers of ferroelectric and of ferroelastic domains. Information about interactions is given in the form of the extended integrity bases, faint interactions, electric and elastic switching interactions. The tables facilitate writing of the invariant potentials, classification of domain states and investigation of the relationship between domain states.

Key Words:

• Ferroic phase transition
• symmetry descent
• parent group
• ferroic subgroup
• tensor parameters
• covariant tensor components
• labelling of covariants
• conversion equations
• fine structure of domain states
• principal and secondary tensor parameters
• primary and secondary (faint) order parameters
• (extended) integrity basis
• faint interactions
• switching interactions.