PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Condensed Matter, Materials & Soft Matter · Crystal structure, bonding and diffraction
Solid State Physics, Ch Solids without a lattice: glasses and quasicrystals
Problem
A crystallographer, confronted with the sharp electron diffraction pattern of icosahedral Al70Pd21Mn9, whose spot spacings are related by the golden mean φ=(1+5)/2=1.618 and which shows apparent five-fold symmetry, argues as follows: "Any diffraction pattern, however fine its spacings, is only ever measured to finite resolution. So instead of abandoning the lattice, I can always find some ordinary three-dimensional Bravais lattice, with a large enough primitive cell, whose diffraction spots lie close enough to the observed positions to be indistinguishable, at any resolution I can achieve, from a true five-fold pattern."
((a)) State what the crystallographic restriction theorem says about the rotational symmetries a two-dimensional lattice can possess, and confirm that the same restriction to rotations of order n=1,2,3,4,6 holds for a three-dimensional Bravais lattice of any primitive-cell size.
((b)) Using that restriction, say precisely what is wrong with the crystallographer's argument, distinguishing the exact statement it makes about symmetry from the separate question of numerical resolution.