Search for possible structures of cubic approximants of icosahedral quasicrystals

Admissible structures of the approximants are obtained within the model of the dodecahedral local order (DLO) with due regard for the constraints imposed by the space symmetry. It is shown that the number of cubic approximants of icosahedral quasicrystals having a certain order and dodecahedral local order is finite and that the number of positions that can be occupied by the atoms from the unit cells of an approximant of each order is also finite. The corresponding estimates from above are made. In particular, it is shown that there exists only one structure for the 1/−1 approximant. It is also shown that it is possible to determine all the structures for any approximant of any order. The corresponding algorithm of the exhaustive search for these structures is suggested. The implementation of this algorithm provided the determination of all the structures of the 0/1 approximants and also some possible structures of the 0/1 and 1/1 approximants. The tables of possible approximant structures can be useful in the studies of new phases having approximant structures.