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Deposition and crystal
Frequency shifts are tiny compared with 5 MHz but measurable to about a hertz, which is nanogram sensitivity.
Film
Model checks
The science
What is modelled
An AT-cut quartz disc vibrates in thickness-shear mode at 5 MHz. A thin, rigid, well-coupled film adds mass and lowers the frequency in proportion to its areal mass, which is how deposition rates are monitored in vacuum coaters.
Equations
- Sauerbrey:
Δf = -2 f0² (Δm/A) / √(ρq μq), sensitivity 56.6 Hz cm²/µg for 5 MHz. - Resonance curve: amplitude
1/√(1 + x²)withx = 2 δ/Γ, bandwidthΓ = f/Q. - Desorption (exact):
h(t) = h0 e-kt + (r/k)(1 - e-kt). - Temperature: an illustrative AT-cut cubic drift term.
Checks run in the page
- 100 nm of gold gives the Sauerbrey shift, and the sensitivity matches the textbook 56.6.
- The full width of the resonance curve at 1/√2 equals f/Q.
- A desorbing film follows exp(-kt) over 100 s.
Honest simplifications
Sauerbrey holds only for thin, rigid, uniform films; soft polymers, liquids and thick films need viscoelastic models (the polymer film here is treated as rigid). The Q-factor losses and the thermal term are empirical, not calibrated to a cut angle. The drawn film thickness is exaggerated.
What to try
- Leave the shutter open with gold and read the frequency drop; close it and the frequency holds.
- Raise the crystal temperature to 70 C and see the thermal part separate from the mass part on the first graph.
- Choose Poor vacuum: the quality factor falls and the resonance curve widens.