A voltage from pressure

In 1880 the brothers Jacques and Pierre Curie showed that certain crystals, among them quartz, tourmaline and Rochelle salt, develop electric charges of opposite sign on opposite faces when they are compressed (Curie and Curie 1880). The effect was called piezoelectricity, from the Greek for “to press”. A year later the converse effect was predicted from thermodynamics and confirmed: an applied voltage makes the same crystals change shape very slightly.

Whether a crystal can be piezoelectric depends on its symmetry. The polarisation $P$ produced by a stress $\sigma$ is written $P_i = d_{ijk}\,\sigma_{jk}$, where $d$ is a third-rank tensor of piezoelectric coefficients. If the crystal has a centre of symmetry, inverting every coordinate leaves the crystal unchanged but reverses the sign of a third-rank tensor, so $d$ must equal $-d$, which means $d = 0$ (Nye 1985). Of the 32 crystal classes, 21 lack a centre of symmetry, and all but one of those allow piezoelectricity.

# Squeeze a 1 cm quartz cube along its x axis (the d11 direction).
eps0 = 8.854e-12                 # F/m
d11 = 2.31e-12                   # C/N, α-quartz
eps_r = 4.5                      # relative permittivity of quartz along x
side = 0.01                      # m
F = 10.0                         # N, about the weight of 1 kg held in the hand

Q = d11 * F                      # charge on the faces
C = eps0 * eps_r * side**2 / side
V = Q / C
E = Q**2 / (2 * C)
print(f"charge {Q * 1e12:.1f} pC, capacitance {C * 1e12:.2f} pF")
print(f"open-circuit voltage {V:.0f} V, stored energy {E * 1e9:.2f} nJ")

# The charge appears only while the force changes. Squeezing and
# releasing once a second converts at most 2 E per second.
P = 2 * E
print(f"one squeeze per second: {P * 1e9:.1f} nW, "
      f"{P / 100:.0e} of a resting body's ~100 W")

# Calcite (point group -3m) is centrosymmetric: every d_ijk is zero.
print("calcite cube, same force: 0 pC")
# charge 23.1 pC, capacitance 0.40 pF
# open-circuit voltage 58 V, stored energy 0.67 nJ
# one squeeze per second: 1.3 nW, 1e-11 of a resting body's ~100 W
# calcite cube, same force: 0 pC

Fifty-eight volts sounds large, but it is the voltage across a capacitor of less than half a picofarad, holding less than a nanojoule. The charge appears only while the force is changing: a crystal held in a steady grip produces no further charge, and in a hand the charge is quickly neutralised through moist skin, which conducts. Even squeezing and releasing once a second converts about a nanowatt, a hundred-billionth of the power a resting human body uses. Calcite in the same hand produces nothing at all.

The quartz resonator

The converse effect is what makes quartz so useful. An alternating voltage applied to electrodes on a thin plate of quartz makes the plate vibrate, and near a mechanical resonance the plate’s motion feeds back strongly into the electrical circuit. Because quartz loses very little energy per cycle, the resonance is extremely sharp, and an oscillator built around it keeps time to a few seconds a month. The tuning-fork crystal in a quartz watch vibrates 32,768 times a second, which is $2^{15}$, so that fifteen halvings give one tick per second.

Near resonance, a quartz crystal behaves like a simple electrical circuit, the Butterworth–Van Dyke model: a “motional” branch of a resistor $R_1$, an inductor $L_1$ and a capacitor $C_1$ in series, representing the mass, stiffness and losses of the vibrating plate, in parallel with the capacitance $C_0$ of the electrodes (Van Dyke 1928).

import math

# Butterworth–Van Dyke model of a 10 MHz AT-cut quartz resonator:
# a series R1-L1-C1 "motional" arm in parallel with the electrode capacitance C0.
R1, L1, C1, C0 = 10.0, 12.7e-3, 20e-15, 5e-12      # Ω, H, F, F (typical values)

fs = 1 / (2 * math.pi * math.sqrt(L1 * C1))         # series resonance
fp = fs * math.sqrt(1 + C1 / C0)                    # parallel (anti-)resonance
Q = 2 * math.pi * fs * L1 / R1
print(f"series resonance   fs = {fs / 1e6:.4f} MHz")
print(f"parallel resonance fp = {fp / 1e6:.4f} MHz  (fp − fs = {(fp - fs) / 1e3:.1f} kHz)")
print(f"quality factor     Q  = {Q:.1e}")
print(f"bandwidth fs/Q        = {fs / Q:.0f} Hz")

# Ring-down: after the drive stops, the stored energy decays as exp(−2π f t / Q).
tau = Q / (math.pi * fs)                            # amplitude time constant
print(f"amplitude decays to 1/e in {tau * 1e3:.2f} ms ≈ {tau * fs:.0f} cycles")
# series resonance   fs = 9.9863 MHz
# parallel resonance fp = 10.0062 MHz  (fp − fs = 20.0 kHz)
# quality factor     Q  = 8.0e+04
# bandwidth fs/Q        = 125 Hz
# amplitude decays to 1/e in 2.54 ms ≈ 25365 cycles

A quality factor of 80,000 means that the resonance is only about 125 Hz wide at 10 MHz, and that once the drive is removed the vibration rings on for some 25,000 cycles. Quartz resonators reach quality factors of $10^4$ to $10^6$, which is why they are the time base of almost every clock, computer and phone.

What a held crystal does not do

None of this physics supports the idea that holding or wearing quartz affects the body’s electrical activity. A crystal at rest in the hand is not being driven at its resonance, which for a hand-sized stone lies in the tens or hundreds of kilohertz and requires electrodes and an amplifier to sustain. Without a changing force there is no piezoelectric charge, and what charge there is is tiny, momentary and shorted out by the skin. The body’s own electrical signals, such as the roughly −70 mV across every nerve-cell membrane or the millivolt signal of the heart at the skin, are produced by ion pumps and channels driven by metabolism, and nothing in a passive crystal couples to them.