the mass-distribution of the metal electrodes (or in the case of non-planar resonators, the quartz crystal thickness itself). Energy trapping turns the crystal turns into an acoustic lens and the wave is focused to the center of the crystal. Energy trapping is necessary in order to be able to mount the crystal at the edge without excessive damping. Energy trapping slightly distorts the otherwise planar wave fronts. The deviation from the plane thickness-shear mode entails flexural contribution to the displacement pattern. Flexural waves emit compressional waves into the adjacent medium, which is a problem when operating the crystal in a liquid environment. Overtones Planar resonators can be operated at a number of overtones, typically indexed by the number of nodal planes parallel to the crystal surfaces. Only odd harmonics can be excited electrically because only these induce charges of opposite sign at the two crystal surfaces. Overtones are to be distinguished from anharmonic side bands (spurious modes), which have nodal planes perpendicular to the plane of the resonator. The best agreement between theory and experiment is reached with planar, optically polished crystals for overtone orders between n = 5 and n = 13. On low harmonics, energy trapping is insufficient, while on high harmonics, anharmonic side bands interfere with the main resonance. Amplitude of motion The amplitude of lateral displacement rarely exceeds a nanometer. More specifically one has with u0 the amplitude of lateral displacement, n the overtone order, d the piezoelectric strain coefficient, Q the quality factor, and Uel the amplitude of electrical driving. The piezoelectric strain coefficient is given as d = 3.1·10‑12 m/V for AT-cut quartz crystals. Due to the small amplitude, stress and strain usually are proportional to each other. The QCM operates in the range of linear acoustics. Effects of temperature and stress The resonance frequency of acoustic resonators depends on temperature, pressure, and bending stress. Temperature-frequency coupling is minimized by employing special crystal cuts. A widely used temperature-compensated cut of quartz is the AT-cut. Careful control of temperature and stress is essential in the operation of the QCM. AT-cut crystals are singularly rotated Y-axis cuts in which the top and bottom half of the crystal move in opposite directions (thickness shear vibration) during oscillation. The AT-cut crystal is easily manufactured. However, it has limitations at high and low temperature, as it is easily disrupted by internal stresses caused by temperature gradients in these temperature