The decagonal plane-wave model for 2D and 3D quasicrystals

We examine the free electron model for some rational approximants of 2D quasicrystals and quasi-lattices that exhibit quasiperiodicity in (x, y) planes and periodicity along the z-direction. Our study is based on a scattering potential with ten symmetrically oriented plane-wave components in the (k x , k y ) plane for the 2D case, and two additional plane-wave components in directions perpendicular to the plane for the 3D case. We calculate the band structure and the density of states for the rational approximants and examine the formation of a gap or a pseudogap in such systems. The relevance of this study to the stability and electrical conduction property of decagonal quasicrystals is discussed.