Sauerbrey equation

From Wikipedia, the free encyclopedia

The Sauerbrey equation is used in quartz crystal microbalance measurements. It gives the change $Δ f$ in the oscillation frequency of a piezoelectric quartz crystal as a function of the mass $Δ m$ added to the crystal:

$\Delta f = { -2\Delta m f_0^2 \over A \sqrt{ \rho_q \mu_q } } = -\frac{2f_0^2}{A \rho_q v_q}\Delta m$ .

Here, $f 0$ is the resonant frequency of the crystal, $A$ is the active area of the crystal (between electrodes), $ρ q$ is the density of quartz, $μ q$ is the shear modulus of quartz, and $v q$ is the shear wave velocity in quartz.

[edit] Limitations

The Sauerbrey equation was developed for oscillation in air and only applies to rigid masses attached to the crystal. It has been shown that quartz crystal microbalance measurements can be performed in liquid, in which case a viscosity related decrease in the resonant frequency will be observed:

$\Delta f = { -\ f_0^{3/2} ( \eta_l \rho_l / \pi \rho_q \mu_q )^{1/2} }$