Spin arrangements of the first family line groups

Magnetic properties of materials are, partly or in some cases completely, determined by the ground state spin arrangements. Their symmetry is, in some cases based on the magnetic groups, but this approach is incapable to include many classes of helimagnets, and a generalization of spin space groups was introduced. On the other hand, geometrical symmetries of regular quasi one-dimensional systems, including various types of nanotubes, nanowires, or polymers, are gathered into line groups. Concerning magnetic order, it turns out that the atoms carrying spins in helimagnets usually single out quasi one-dimensional substructure. Therefore spin line groups are necessary tool in description and classification, as well as deep understanding of properties of wide class of helimagnets. Here we derive spin groups associated to the first family line groups, and compare spin arrangements compatible with these and the corresponding magnetic groups. Presented theory is illustrated by several examples of quasi one-dimensional helimagnets: cuprates, hexaferrites, etc.