Quasisymmetry (P-symmetry) in crystals

Abstract

The concepts of semi-direct product, quasi semi-direct product and the method of constructing quasisymmetry (P-symmetry) groups [Krishnamurty, Prasad & Rama Mohana Rao (1978). J. Phys. A, 11, 805-811; (1980). J. Phys. A, 13, 1947-1956] have been explored and a general method of constructing quasisymmetry (P-symmetry) groups with the crystallographic space groups as generators is suggested. The study is restricted to only the cubic system for the chosen boundary condition Tx2 = Ty2 = Tz2 = E. The minor quasisymmetry cubic space groups so obtained are associated with the one-dimensional complex and two-dimensional real irreducible representations of the generator groups using the ideas of little groups and their one-dimensional allowable irreducible representations. The symmorphic cubic space groups F23, F432 and the non-symmorphic cubic space groups Fd3, Fd3m are exemplified. For the rest of the cubic space groups the results obtained are tabulated. Some suggestions have been made as to the possible studies in which the groups obtained here can be applied.