Group theory for magnetic structure determination: Recent developments and quadrupolar ordering analysis

In this paper, the recent developments in practical applications of symmetry analysis are described. The theoretical basis shortly described in Section 1 has been implemented in several computer applications, one of which is the program 'MODY-win', developed by the authors of the paper. The program calculates the so-called basis vectors of irreducible representations of a given symmetry group, which can be used for calculation of possible ordering modes. Its practical application is demonstrated on some examples, presenting the recent aspects of using the symmetry analysis to description of various types of ordering encountered in solids. The scalar-type ordering (occupation probability) is discussed shortly for occupation of interstitial sites by hydrogen atoms in inter-metallic compounds. The description of vector ordering is demonstrated on the magnetic ordering modes, with special attention focused on the freedom that is left in the structure after imposing all the symmetry constraints. In practice, the final ordering mode usually contains some free parameters that cannot be determined from the symmetry itself. The last application presented in the paper is the description of quadrupolar ordering, recently found in some compounds of 4f (5f) elements. For the latter case, an additional advantage is demonstrated by calculation of possible displacements of neighboring atoms after the establishment of non-zero quadrupolar order parameter on the central atom