Superspace groups and Landau theory. A physical approach to superspace symmetry in incommensurate structures

A new formulation of superspace symmetry in incommensurate structures which stresses its physical basis and avoids the "supercrystal" picture is presented. Superspace groups are introduced through the invariance properties of the Landau free-energy expansion for these types of materials, without making any reference to the characteristics of their diffraction patterns. In this way, the concept of superspace symmetry is directly related with the invariance of the material free energy for an arbitrary shift of a certain number of phases of the soft-mode coordinates. The dimension of the so-called "internal space" is then given by the number of independent phasons in the structure. Under this approach, the theoretical background that permits one to assign a superspace group to the incommensurate phase resulting from the onset of an order parameter has been developed. The equations obtained simplify previous methods and can be interpreted as a simple generalization of the equations used to determine the space group of a distorted phase in a commensurate case. As an example of their efficiency, the superspace symmetry corresponding to the incommensurate phase exhibited by ${\mathrm{Ba}}_2$ ${\mathrm{NaNb}}_5$ ${\mathrm{O}}_{15}$ is investigated. The formulation is then extended to include the analysis of superspace symmetry for secondary modes. It is shown by means of Landau theory that in this respect a certain von Neumann principle is satisfied for superspace symmetry, which proves the coherency of associating a superspace group to an incommensurate distorted phase from the knowledge of the corresponding order parameter, irrespective of the secondary modes also contributing to the total distortion. Finally, the symmetry of macroscopic properties in incommensurate structures is discussed under this point of view and its definite relationship with the corresponding superspace group is established.