An analytical treatment of diffraction in quasiperiodic superlattices

The author studies the diffraction properties of a class of quasiperiodic superlattices described by the substitution rules A to A p B, B to A where p is a positive integer. These can be obtained by a projection method with a characteristic irrational sigma , e.g. for the Fibonacci lattice (p=1) A to AB, B to A, sigma =(1+ square root 5)/2. It is shown that the diffraction peak positions K k.r can be labeled by two integers k, r and are given by the expression K k.r =2 pi Lambda -1 r sigma ' k where sigma ' are the so called precious means. It is shown that the Fibonacci lattice has the unique property that sigma = sigma '.