By C. Lu, A.W. Czanderna
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Extra resources for Applications of Piezoelectric Quartz Crystal Microbalances
However, the understand ing of the mass induced resonant frequency shift was limited to a qualitative nature and this phenomenon had not been carefully investigated until the late 1950s. 28 C. LU y QUARTZ Mn FILM M f (c) mmdmL QUARTZ M VJ Fig. 7. A simplified model of a quartz crystal microbalance. (a) At resonance, the wavelength is equal to half of the quartz plate thickness, (b) An increase in the q u a r t z plate thickness results in a decrease in the resonant frequency (an increase in the wavelength), (c) The mass of a deposited film is t r e a t e d as an equivalent amount of the quartz mass.
15). In theory, the FILM 3 ^ % FILM 2 >% % FILM 1 QUARTZ CRYSTAL *f> p Mq Pq fi l^\ M M t Fig. 15. A quartz crystal resonator with sequential accumulation of different m a t e r i a l s . THEORY AND PRACTICE 51 one-dimensional composite resonator model can be extended to multiple layers of different physical properties. Unfortunately, the complexity of mathematical analysis increases rapidly with respect to the number of layers involved. Additionally, in order to deter mine the areal density of a subsequent layer, the thickness, density, and acoustic impedance of each layer already deposited on the quartz crystal must be known.
Thus, for all practical purposes, Eq. (44) can be considered as a general mass-frequency equation for the QCM. B. Computation and Estimation of Acoustic Impedance When Eq. (44) is employed for the areal density (or film thickness) determination with a QCM, it is necessary to know the shear-mode acoustic impedance of the deposited material as well as that of the quartz crystal. The acoustic impedance of the quartz crystal can be calculated from the relation Z = p q v q . Since the shear wave velocity in a single crystal depends on the direction of wave propagation, the acoustic impedance of a quartz crystal plate depends on its crystallographic orientation.