Incommensurate magnetic structure

Incommensurate magnetic structure

Structure magnétique incommensurable

Definition

An  incommensurate magnetic structure is a structure in which the magnetic moments are
ordered, but without  periodicity that is commensurate with that of the nuclear structure
of the crystal. In particular, the magnetic moments have a spin density with wave vectors
that have at least one irrational component with respect to the reciprocal lattice
of the atoms. Or, in the case of localized moments, the spin function  S( n+rj)
(where the $j$th atom has position {\bf r}$_j$ in the unit cell) has Fourier components
with irrational indices with respect to the reciprocal lattice of the crystal.

 Details

When the atoms of the basic structure are at positions n+rj, where  rj
is the position of the jth atom in the unit cell, then the spin function for
an incommensurate magnetic structure is

[math]{\bf S}({\bf n}+{\bf r}_j)~=~\sum_{\bf k} \hat{\bf S}({\bf k})_j \exp \left(2\pi i {\bf k}.({\bf n}+{\bf r}_j)\right),~~{\bf k}=\sum_{i=1}^n h_i {\bf a}_i^*~~(h_i ~~{\rm integer}).[/math]

This spin structure is incommensurate if one component of the basis vectors  ai*
is irrational. Incommensurate magnetic structures may be linear, but occur as quite
complicated, like fan structures etc. as well. Especially, in rare-earth compounds very
complicated magnetic phase diagrams have been found.