the resonance frequency and its changes by using the crystal as the frequency-determining component of an oscillator circuit. His method continues to be used as the primary tool in quartz crystal microbalance experiments for conversion of frequency to mass. Because the film is treated as an extension of thickness, Sauerbrey’s equation only applies to systems in which (a) the deposited mass has the same acoustic properties as the crystal and (b) the frequency change is small (Δf / f < 0.05). If the change in frequency is greater than 5%, that is, Δf / f > 0.05, the Z-match method must be used to determine the change in mass. The formula for the Z-match method is: kF is the wave vector inside the film and dF its thickness. Inserting kF = 2·π·f /cF = 2·π·f·ρF / ZF as well as dF = mF / ρF yields Viscoelastic film For a viscoelastic film, the frequency shift is Here ZF is the acoustic impedance of the film (ZF = ρFcF = (ρFGf)1/2)= (ρF/Jf)1/2), kF is the wave vector and dF is the film thickness. Jf is the film's viscoelastic compliance, ρF is the density. The poles of the tangent (kF dF = π/2) define the film resonances. At the film resonance, one has dF = λ/4. The agreement between experiment and theory is often poor close to the film resonance. Typically, the QCM only works well for film thicknesses much less than a quarter of the wavelength of sound (corresponding to a few micrometres, depending on the softness of the film and the overtone order). Note that the properties of a film as determined with the QCM are fully specified by two parameters, which are its acoustic impedance, ZF = ρFcF and its mass per unit area, mF = dF/ρF. The wave number kF = ω/cF is not algebraically independent from ZF and mF. Unless the density of the film is known independently, the QCM can only measure mass per unit area, never the geometric thickness itself. Viscoelastic film in liquid For a film immersed in a liquid environment, the frequency shift is The indices F and Liq denote the film and the liquid. Here, the reference state is the crystal immersed in liquid (but not covered with a film). For thin films, one can Taylor-expand the above equation to first order in dF, yielding Apart from the term in brackets, this