Incommensurate structure with no average lattice : an example of a one-dimensional quasicrystal

We study the ground state of a simple one-dimensional model describing an incommensurate modulation of the vacancy density of a periodic lattice. We show that this structure, though its Fourier spectrum is always discrete, cannot be interpreted as an average crystal with a superimposed periodic displacive distortion except for a discrete sequence of particular values of the vacancy density. The absence of any average periodic lattice permits to consider those structures as genuine one-dimensional quasi-crystals different from the standard one-dimensional incommensurate structures.