In this work, we determine the intrinsic mechanical energy dissipation limit for single-crystal resonators due to anharmonic phonon-phonon scattering in the Akhiezer (Ωτ << 1) regime. The energy loss is derived using perturbation theory and the linearized Boltzmann transport equation for phonons, and includes the direction and polarization dependent mode-Grüneisen parameters in order to capture the strain-induced anharmonicity among phonon branches. This expression is the first to reveal the fundamental differences among the internal friction limits for different types of bulk-mode elastic waves. For cubic crystals, 2D-extensional modes have increased dissipation compared to width-extensional modes because the biaxial deformation opposes the natural poisson contraction of the solid. Additionally, we show that shear-mode vibrations, which preserve volume, have significantly reduced energy loss because dissipative phonon-phonon scattering is restricted to pure-shear phonon branches, indicating that Lamé or wine-glass mode resonators will have the highest upper limit on mechanical efficiency. Finally, we employ key simplifications to evaluate the quality factor limits for common mode shapes in single-crystal silicon devices, explicitly including the correct effective elastic storage moduli for different vibration modes and crystal orientations. Our expression satisfies the pressing need for a reliable analytical model that can predict the phonon-phonon dissipation limits for modern resonant microelectromechanical systems (MEMS), where precise manufacturing techniques and accurate finite element methods can be used to select particular vibrational mode shapes and crystal orientations.

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