# Incommensurate magnetic structure

### From Online Dictionary of Crystallography

Structure magnétique incommensurable (*Fr*). Inkommensurable Magnetstruktur (*Ge*). Struttura magnetica incommensurabile (*It*). 非整合磁気構造 (*Ja*). Estructura magnética inconmensurable structure (*Sp*).

## 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**+**r**_{j})
(where the *j*th atom has position **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**+**r**_{j}, where **r**_{j}
is the position of the *j*th atom in the unit cell, then the spin function for
an incommensurate magnetic structure is

[math]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\, \mathrm{integer})[/math].

This spin structure is incommensurate if one component of the basis vectors **a**_{i}^{*}
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.