Projection method for 4d non-orthogonal hyperlattices

Abstract

The Wigner-Seitz cell of a lattice in n-dimensional space displays the complete point group of such a lattice. The vertices of the cell when projected onto pseudo space can serve as the outer shape of acceptance domain or motif. This general procedure leads to acceptance domain or motif identical to those discussed in literature for primitive orthogonal hyperlattices. Example of 4d non-orthogonal hyperlattices corresponding to 12-fold symmetry will be considered. It will be shown that the first Wigner-Seitz cell degenerates into more than one shape in 2d pseudo space and can serve as a natural partition of the motif. Following a parallel procedure, the consequence of projection of first 4d Brillouin zone will also be discussed.