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This paper presents the theoretical derivation for a method of accurately determining the complex coefficients of single crystal piezoelectric materials. The vibration equations are analytically solved for a piezoelectric plate and a piezoelectric thin rod in an arbitrary crystallographic orientation. The solutions provide the distributions of the stresses and the vibration velocities inside the sample and the electrical impedance between the electrodes. Based on the analysis of the solutions, the complex dielectric, elastic, and piezoelectric coefficients are determined from the impedance or admittance measurements near the resonance frequencies. The measurement for LiNbO/sub 3/ single crystals is used as an example to demonstrate the experiment and calculation procedures and to indicate the comparisons with previous methods.