Coherent pair production by energetic gamma rays incident on quasicrystals

We develop a Born-approximation theory of coherent pair production (CPP) of electrons by energetic gamma rays incident on an icosahedral quasicrystal, described by a schematic model (K model) that includes phonon and phason disorder. Our main result is a formula for the cross-section dσ_cpp/dε₊ for CPP, differential with respect to the positron energy ε₊ and of order α² in the fine-structure constant α ≈ 1/137, but which is otherwise exact. We discuss results of numerical calculations of dσ_cpp/dε₊ versus y = ε₊/k for gamma rays of energies k = 20 MeV, 200 MeV, and 3 GeV, incident on icosahedral Al-Mn-Si, described as a special case of the K model (vertex model). This consists in placing an Mn atom at each vertex of the relevant Ammann tiles. Our calculations include CPP of types A and B. Both types exhibit vertical intensity drops at irregularly distributed y-values, many of these drops being so large that they should be observable experimentally. They are analogous to the large intensity drops exhibited by coherent bremsstrahlung in quasicrystals. We predict that CPP drops also occur for realistic models of i-Al-Mn-Si at the same y-values as for the vertex model, but whose magnitudes may differ from those predicted by this model.