Symmetry and Piezoelectricity: Evaluation of $\alpha$-Quartz coefficients

Piezoelectric coefficients of $\alpha$-Quartz are derived from symmetry arguments based on Neumann's Principle with three different methods: Fumi, Landau-Lifshitz and Royer-Dieulesaint. While Fumi method is tedious and Landau-Lifshitz requires additional physical principles to evaluate the piezoelectric coefficients, Royer-Dieulesaint is the most elegant and most efficient of the three techniques.