Piezoelectric coefficients of α -quartz are derived from symmetry arguments based on Neumann’s
principle using three different methods: Fumi, Landau–Lifshitz and Royer–Dieulesaint. While the Fumi
method is tedious and the Landau–Lifshitz method requires additional physical principles to evaluate
the piezoelectric coefficients, the Royer–Dieulesaint method is the most elegant and most efficient
of the three techniques.