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Space groups
The positions of the objects (atoms or molecules) in the crystal are given by
{$$\mathbf R_{\mathbf n \nu} = n_a\mathbf a + n_b\mathbf b + n_c\mathbf c + \mathbf r_\nu$$}
where {$n_a,n_b,n_c \in {\mathbb Z}$}, the vectors {$\mathbf a,\mathbf b,\mathbf c$} define a primitive or a conventional unit cell of the lattice and {$\nu=1\cdots r$} defines the basis.
For simplicity in the following we think of localized point-like moments, although the symmetry description can be applied to extended objects.
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Shubnikov two-color groups
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Propagation vector (Fullprof method)
In general a magnetic structure can be defined in the appropriate primitive or conventional unit cell of the magnetic lattice by a set of propagation vectors
{$\mathbf k = m_a\mathbf a^* + m_b\mathbf b^* + m_c\mathbf c^*$} by specifying:
- the three dimensional Fourier components of each atomic magnetic moment of the basis, {$S_{\mathbf k \nu, i},\quad i=a,b,c$}, and
- their phases {$\phi_{\nu\mathbf k}$}.
The moments at each position {$\mathbf R_{\mathbf n \nu}$} are given by
{$$\mathbf m_{\mathbf n \nu} = \sum_{\mathbf k} \mathbf S_{\nu \mathbf k}\, e^{i(\mathbf k\cdot \mathbf R_{\mathbf n \nu}+\phi_{\nu\mathbf k})}$$}
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