Quartz belongs to point group 32 — a 3-fold axis combined with three perpendicular 2-fold axes. That trigonal symmetry makes the in-plane piezoelectric coefficient repeat every 120° and flip sign every 60°, so cutting the wafer at angle θ from the crystallographic X-axis scales the usable coefficient as:
d_eff(θ) = d11 · cos(3θ) (d11 ≈ 2.3 pC/N for quartz)
Q_static = d_eff · F (direct piezoelectric effect)
The wafer also behaves as a mechanical resonator in thickness-shear mode. Its resonant frequency is set purely by thickness, using the AT-cut frequency constant N ≈ 1.66 MHz·mm:
f_s = N / t
Electrically, a quartz resonator is modelled with the standard Butterworth–Van Dyke (BVD) equivalent circuit: a motional branch (R1, L1, C1) in parallel with the static shunt capacitance C0. The coupling coefficient k² set by the cut angle fixes the motional-to-shunt capacitance ratio (k² ≈ (π²/8)(C1/C0)), which in turn fixes L1 and R1 for a chosen quality factor:
C1 = C0·(8/π²)·k² L1 = 1/(ωs²C1) R1 = √(L1/C1)/Q
Y(f) = 1/(R1 + jωL1 + 1/(jωC1)) + jωC0
Sweeping the drive frequency across |Y(f)| reproduces the crystal's real behaviour: a sharp admittance peak at the series resonance f_s, and a dip at the slightly higher parallel (antiresonance) frequency f_p = f_s·√(1+C1/C0). The quality factor here is set to an illustrative Q≈400 so the peak is visible on screen — real AT-cut quartz reaches Q≈10⁴–10⁶, which would render as an invisibly thin spike at this scale.
- Lattice pane — a 2D cross-section of the SiO₂ tetrahedral network; the Si/O sublattices shift apart along the polar axis under compression, visualizing the charge separation behind Q_static (direction flips when d_eff changes sign).
- Resonance pane — |Y(f)| around f_s and f_p, with the current drive point marked.
- Scope pane — a stylized (slowed-down) waveform whose amplitude tracks the real |Y| at the current drive frequency; the actual device oscillates in the MHz range, far too fast to animate on screen.