Icosahedral Incommensurate Crystals

Abstract

One of the basic facts of condensed matter physics is that the symmetry of 3d crystals can be described in terms of space groups composed of Bravais lattices of 3d translational symmetries and point groups of rotation and reflection symmetries. Only 2-fold, 3-fold, 4-fold and 6-fold rotation symmetries are allowed. In a striking experiment on an Mn-Al alloy, however, Shechtman et al.¹ have observed an electron diffraction pattern with a 5-fold symmetry axis, and an overall icosahedral symmetry (Fig. 1). How can this be possible? The diffraction spots are quite sharp, so there is long range positional ordering, but there can not be translational invariance in view of the considerations above. The spectrum has a “scaling” structure: if the pattern is magnified by a factor G (the golden mean) the positions of peaks remain fixed.