# Incommensurate magnetic structure

### From Online Dictionary of Crystallography

Structure magnétique incommensurable (Fr.)

## 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 jth atom has position rj 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

$S( n+ r_j)~=~\sum_ {k} \hat{S}( k)_j \exp \left(2\pi i k.( n+ r_j)\right),~~ k=\sum_{i=1}^n h_i a_i^*~~(h_i ~~{\rm integer}).$

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.
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