# How a quartz crystal microbalance works — the Crystal Lab

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This page is "The Crystal Lab": six interactive experiments that explain how a quartz crystal microbalance (QCM) weighs a molecule. Below is the same content as text, with the equations and the constants used by the simulations.

Constants for AT-cut quartz: density ρq = 2648 kg/m³, shear modulus μq = 2.947·10¹⁰ Pa, acoustic impedance Zq = √(ρq·μq) = 8.83·10⁶ kg/(m²·s), shear-wave velocity vq = √(μq/ρq) = 3336 m/s.

## 01 · The plate and its overtones

A QCM sensor is a thin AT-cut quartz disc with gold electrodes on both faces. Quartz is piezoelectric: an alternating voltage makes it shear. The plate resonates when its thickness tq holds half a wavelength of the shear wave: f_n ≈ n · vq / (2·tq) (real overtones sit slightly off n·f1). A 5 MHz crystal is about 334 µm thick, a 10 MHz crystal about 167 µm. Only odd overtones (n = 1, 3, 5, 7, 9) can be excited electrically: for even n the two faces move in the same direction, no net charge reaches the electrodes and there is no resonance.

## 02 · Resonance: many ways to listen

The resonance is a peak with frequency f_r and half-bandwidth Γ. Dissipation D = 2Γ/f_r = 1/(π·f_r·τ), where τ is the decay time after the drive is switched off (ring-down). openQCM Q-1 and NEXT find the peak by sweeping frequencies around it, like a scalar network analyser: a peak-detection algorithm gives f_r, and the width at a calibrated level below the peak gives D. A 5 MHz crystal in air: Γ ≈ 25 Hz, D ≈ 1·10⁻⁵, Q ≈ 100 000, τ = 1/(2πΓ) ≈ 6.4 ms. In water at 25 °C: Γ ≈ 700 Hz, D ≈ 2.8·10⁻⁴, τ ≈ 0.23 ms.

Ways to read a crystal: frequency sweep (scalar or vector network analysis; f and D on any overtone; openQCM Q-1 and NEXT); oscillator (Pierce, Colpitts, Miller or bridge: the crystal sits in the feedback loop of an amplifier and the circuit oscillates where the loop phase is zero, slightly above the series resonance because of the load capacitance; a counter measures f; D only with automatic gain control; openQCM Wi2, SpaceBug, TWIN and Holder); ring-down (drive, switch off, fit the decay: f and D from τ; several commercial QCM-D); phase-locked loop (tracks the zero-phase point: f and resistance continuously); impedance analyser with an equivalent-circuit (Butterworth-Van Dyke) fit (R, L, C; the reference lab method). In a liquid the motional resistance rises roughly thirtyfold, and an oscillator needs enough loop gain to keep running.

## 03 · Mass: the Sauerbrey equation

For a thin, rigid film the frequency shift is proportional to the areal mass: Δf_n / n = −C_f · Δm, with C_f = 2·f0² / √(ρq·μq). C_f = 56.6 Hz·cm²/µg at 5 MHz and 226 Hz·cm²/µg at 10 MHz (0.0566 Hz per ng/cm² at 5 MHz). Examples at 5 MHz: a protein layer (≈ 250 ng/cm²) gives ≈ −14 Hz; a supported lipid bilayer (≈ 450 ng/cm², rigid: ΔD < 0.5·10⁻⁶) ≈ −25 Hz; 1 nm of gold (1930 ng/cm²) ≈ −109 Hz. Sauerbrey (1959) holds while the film is rigid and thin (shift well below 2% of f0).

## 04 · Liquids: Kanazawa–Gordon

A liquid in contact with the crystal lowers the frequency without any adsorbed mass (Kanazawa and Gordon, 1985): Δf = −√n · f0^(3/2) · √(ρL·ηL / (π·ρq·μq)), and the half-bandwidth rises by the same amount. The shear wave in the liquid decays within the penetration depth δ = √(ηL / (π·f·ρL)). At 5 MHz and 25 °C: water (ρ 997 kg/m³, η 0.89 mPa·s) Δf ≈ −670 Hz, δ ≈ 240 nm; 80% glycerol (ρ 1206 kg/m³, η 45.9 mPa·s) Δf ≈ −5.3 kHz, δ ≈ 1.6 µm. Liquid measurements are therefore taken relative to a stable baseline in the same liquid.

## 05 · Soft films and dissipation (QCM-D)

Soft films (polymers, hydrogels, many biolayers) are not rigid: the top lags the bottom, energy is stored and lost, and the overtones disagree. The signatures are a spread of Δf/n between overtones and an increase in dissipation ΔD. The direction of the spread depends on the medium: in air a soft film looks heavier at higher overtones; under water (the usual QCM-D case) Δf/n becomes less negative at higher overtones and the Sauerbrey mass underestimates the film ("missing mass"). The figure uses the exact small-load result for a Voigt film (Johannsmann et al., Sensors 2021): Δf̃_n = (i·f0/(π·Zq))·(Z_load − Z_L), Z_load = Z_f·(Z_L + i·Z_f·tan(k_f·h_f))/(Z_f + i·Z_L·tan(k_f·h_f)), Z_f = √(ρ_f·Ĝ), k_f = ω_n·√(ρ_f/Ĝ), Ĝ = G′ + iω_n·η_f, Z_L = √(iω_n·ρ_L·η_L) under water and 0 in air, Δf̃ = Δf + iΔΓ. Multi-overtone QCM-D data are analysed with models such as Voigt to recover thickness, shear modulus and viscosity. openQCM NEXT reads overtones n = 1, 3, 5, 7, 9 in every cycle, one after another, and shows them together.

## 06 · Temperature

The AT cut has a nearly flat, cubic frequency–temperature curve near room temperature, but a few tenths of a ppm per °C is still a hertz or two per °C at 5 MHz. In liquids the effect is larger: water viscosity changes by about 2% per °C, so the Kanazawa–Gordon shift drifts by about 7–8 Hz per °C at 5 MHz. This is why openQCM NEXT controls the measurement chamber temperature with a Peltier stage (±0.01 °C class stability).

## Further

- Calculator: Sauerbrey, Kanazawa–Gordon, Voigt forward and multi-overtone inverse fit, in the browser: https://openqcm.com/qcm-calculator
- QCM-D fundamentals guide (PDF, free with email): https://openqcm.com/technology#guide

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