Quartz Crystal Microbalance - Quantification of Dissipative Processes

Quantification of Dissipative Processes

For advanced QCMs, both the resonance frequency, f_r, and the bandwidth, w, are available for analysis. The latter quantifies processes which withdraw energy from the oscillation. These may include damping by the holder and ohmic losses inside the electrode or the crystal. In the literature some parameters other than w itself are used to quantify bandwidth. The Q-factor (quality factor) is given by Q = f_r/w. The “dissipation”, D, is the inverse of the Q-factor: D = Q−1 = w/f_r. The half-band-half-width, Γ, is Γ = w/2. The use of Γ is motivated by a complex formulation of the equations governing the motion of the crystal. A complex resonance frequency is defined as f_r* = f_r + iΓ, where the imaginary part, Γ, is half the bandwidth at half maximum. Using a complex notation, one can treat shifts of frequency, Δf, and bandwidth, ΔΓ, within the same set of (complex) equations.

The motional resistance of the resonator, R₁, is also used as a measure of dissipation. R₁ is an output parameter of some instruments based on advanced oscillator circuits. R₁ usually is not strictly proportional to the bandwidth (although it should be according to the BvD circuit; see below). Also, in absolute terms, R₁ – being an electrical quantity and not a frequency – is more severely affected by calibration problems than the bandwidth.