Symmetry of Magnetically Ordered Three Dimensional Octagonal Quasicrystals

The theory of magnetic symmetry in quasicrystals, described in a companion paper [Acta Crystallographica AXX (2003) xxx-xxx], is used to enumerate all 3-dimensional octagonal spin point groups and spin space-group types, and calculate the resulting selection rules for neutron diffraction experiments. 1. Introduction We enumerate here all three-dimensional octagonal spin groups and calculate their selection rules for neutron scattering, based on the theory developed in a companion paper (Lifshitz & Even-Dar Mandel, 2003) where we have provided the details for the extension to quasicrystals (Lifshitz, 1998) of Litvin and Ope-chowski's theory of spin groups (Litvin, 1973; Litvin & Ope-chowski, 1974; Litvin, 1977). We assume the reader is familiar with the cmpanion paper, where we have also given experimental as well as theoretical motivation for carrying out a systematic enumeration of spin groups for quasicrystals. This is the first complete and rigorous enumeration of spin groups and calculation of selection rules for any 3-dimensional quasicrystal. Other than the pedagogical example of 2-dimensional octagonal spin groups given in the companion paper, past calculations are scarce. Decagonal spin point groups and spin space-group types in two dimensions have been listed by Lifshitz (1995) without providing much detail regarding the enumeration process. All possible lattice spin groups Γ e for icosahedral quasi-crystals have been tabulated by Lifshitz (1998) along with the selection rules that they impose, but a complete enumeration of icosahedral spin groups was not given. We intend to continue the systematic enumeration of spin groups in future publications by treating all the other common quasiperiodic crystal systems (pentagonal, decagonal, dodecagonal, and icosahedral), though we shall not provide complete details of the calculations as we do here. Although the enumeration in three dimensions is more elaborate , it proceeds along the same lines as the two-dimensional example given in the companion paper (Lifshitz & Even-Dar Man-del, 2003). Familiarity with the calculation of ordinary (non-magnetic) octagonal space groups (Rabson et al., 1991) may also assist the reader in following the calculations performed here, although knowledge of that calculation is not assumed. We begin in Sec. 2. with a description of the two rank-5 octagonal Bravais classes, a reminder of the octagonal point groups in three dimensions (summarized in Table 1), and a summary of the effect of the different point group operations on the generating vectors of the two lattice types (Table 2). In Sec. 3. we enumerate the octagonal spin point groups by …