Crystallographic relationship between the P-type icosahedral quasicrystal and its 1/1 approximant with the so-called T structure based on an atomic density wave model

In the present study, the revised model is based on the assumption that, unlike a previously developed model, both the P-type quasicrystal and its 1/1 approximant are transformed from an assumed normal state with a primitive-cubic lattice in a three-dimensional reciprocal space. The three primitive reciprocal vectors of the assumed normal state, as a framework of the model, can be defined by three basic orthonormal vectors (e₁*, e₂* and e₃*). The order parameters of the transitions are given by six atomic density waves containing a variable t. Note that the values of t = τ and 5/3 correspond to the P-type quasicrystal and its 1/1 approximant, respectively, where τ is the golden ratio. The crystallographic features of the P-type quasicrystal and 1/1 approximant were calculated using this framework and compared with our previously reported experimental data. Good agreement between the calculated and experimental results supported the validity of the assumed normal state hypothesis. This suggests that a quasicrystal and its 1/1 approximant can be identified as a purely incommensurate state with the golden ratio t = τ and a commensurate state with t = 5/3 in three-dimensional real space, respectively.