Tensor parameters of ferroic phase transitions I. Theory and tables

The main purpose of this work is to describe the changes of material proper ties, expressed by tensors of up to the fourth rank, induced by ferroic pha se 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 a cross 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 grou ps and of their irreducible representations (ireps). The usual Schonflies a nd Hermann-Mauguin symbols are embellished by subscripts which define the o rientation of groups. Matrix form of irreducible representations is associa ted with so-called typical variables which transform in a well defined way. The transformation properties of material property tensors up to fourth ra nk 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 pa rameters which describe the tensor allowed by the parent symmetry G. Other combinations, so-called covariant tensor components, are the potential tens or parameters of ferroic transitions G double down arrow F-i, where F-i is the set of conjugate low symmetry groups. The main tables contain vital inf ormation about the change of tensors up to fourth rank for ail ferroic phas e transitions. To each symmetry descent between crystallographic point grou ps 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 d omain state of the ferroic symmetry. These parameters are distinguished by ireps of the parent group which refer to their transformation properties un der the action of the parent group. This is an important point because para meters belonging to different ireps change in a different manner from one d omain state to the other. To each such set of parameters there is given inf ormation about its symmetry in the first domain state, about the total numb er of domain states and about the numbers of ferroelectric and of ferroelas tic domains. Information about interactions is given in the form of the ext ended integrity bases, faint interactions, electric and elastic switching i nteractions. The tables facilitate writing of the invariant potentials, cla ssification of domain states and investigation of the relationship between domain states.