Representation analysis of the magnetic structures

 

Jacques Schweizer

CEA-Grenoble, DSM/DRFMC/SPSMS/MDN, 38054 Grenoble Cedex 9, France

 

 

 

            In a crystalline material, the magnetic moments are submitted to exchange interactions, with an energy U0 which, expanded at the order two, can be written as:

 

                                                                                         (1)

 

where the   represent the components of the magnetic moments , l and l' labelling the crystal cells, j and j' the magnetic atoms in the cell, and a and b the axes x, y or z, and where the   are the exchange interactions between the components of the magnetic atoms. At higher temperatures, in the paramagnetic state, there is no long range order of the magnetic moments, but only magnetic fluctuations which represent a certain tendency to short range order. When cooling down the material, the range of one of these fluctuations increases and, below a caracteristic temperature, one of them transforms into a long range order : a magnetic structure has been established. Such a structure can be described in terms of propagation vectors k and Fourier components  (one Fourier vector  per atom j in the unit cell).

 

                                                                                                                (2)

 

where k is the propagation vector k (mono-k structure), or one of the propagation vectors (multi-k structure).

 

            With the expression of  given by (2), the magnetic energy becomes

 

                                                                       (3)

 

where  is the Fourier transform of  and can be defined as:

 

                                                                                              (4)

 

            In practice, the propagation vector k is determined from neutron diffraction by indexing the magnetic diagramme. Then, the magnetic structure can be determined by comparing the intensities of the magnetic reflections to those which are expected from the possible arrangements of the magnetic moments in the unit cell. The number of these possible arrangements is considerably reduced when restricting to those which are compatible with the symmetry of the crystal. Particularly, when the magnetic order establishes from the paramagnetic state through a second order phase transition, it is very fruitful to apply the Landau theory for phase transitions of this type. This approach classifies the magnetic fluctuations according to the symmetry of the different irreducible representations of the little group Gk (the group of vector k). Actually, it states that, in order to keep the magnetic energy (3) invariant under all the symmetry operations of Gk, the magnetic structures must be built from basis vectors belonging to only one of the irreducible representations of Gk.

 

            In this lecture we shall review :

 

            -what is a group

            -what is the representation of a group

            -what is the magnetic representation

            -what is an irreducible representation

            -how to decompose the magnetic representation into its irreducible representations

            -how to find the basis vectors of the irreducible representations

 

            Then, we shall discuss the action of the time reversal operator  on the magnetic energy. This operator, reversing the velocities, and then the directions of the electric currents, reverses the direction of the magnetic moments () which are axial vectors. But, being an antilinear operator, it also conjugates the Fourier components which are complex vectors . Therefore, instead of working with representations and irreducible representations, as for linear operators, when introducing the time reversal operator, we are obliged to use the theory of corepresentations and irreducible corepresentations introduced by Wigner.