Nearly free-electron band structures in a logarithmically periodic solid

A logarithmically periodic solid is an ideal solid, based on a supercluster model for quasicrystals. In this model, icosahedral subclusters are joined by edge–vertex–edge sharing with neighbouring subclusters, so as to construct–with unique orientation–greater icosahedral structures. The model is infinitely extensible. For the ideal structure, an electronic band structure can be derived, based on quasi-Bloch functions. The ideal structure corresponds to real images obtained by optimum-defocus high-resolution electron microscopy, and the structure is consistent with fortuitous convergent-beam electron diffraction. The model has been successfully used to simulate, correctly, experimental diffracted beam intensities. Like the model, and like the ideal band structures, the diffraction pattern is generally logarithmic. In this it differs from Bragg diffraction, as it does also in other ways.