The stone that pulls ash

Around 315 BC Theophrastus described a stone he called lyngourion that attracted straws and bits of wood; he thought it formed from the urine of a wild animal, which Pliny later identified as the lynx. Many historians think the stone was tourmaline (Lang 2005). The property was rediscovered in Europe when tourmaline from Ceylon reached the Netherlands at the start of the eighteenth century: Dutch gem cutters noticed that a stone placed in a fire first attracted ash from the coals and then repelled it, as Johann Georg Schmidt reported in 1707. Franz Aepinus studied the effect experimentally in the eighteenth century, and in 1824 David Brewster gave it the name it still has, pyroelectricity (Lang 2005).

Why tourmaline and not quartz

A crystal is pyroelectric if it has a spontaneous polarisation, a built-in electric dipole moment per unit volume, which changes with temperature. That requires a polar crystal class, one with a single direction that is not equivalent to its reverse. Ten of the 32 classes are polar. Tourmaline belongs to the trigonal class $3m$, which is polar along the crystal’s $c$ axis; quartz, class 32, is not polar, so it is piezoelectric but not pyroelectric (Chapter 1).

In a crystal at constant temperature, the polarisation’s bound charges at the two ends are soon covered by free charges from the air and from surface conduction, and nothing can be detected. When the temperature changes by $\Delta T$, the polarisation changes by

$$\Delta P = p\,\Delta T,$$

where $p$ is the pyroelectric coefficient, and for a while the surfaces carry an uncompensated charge density $\sigma = p\,\Delta T$.

# A 1 cm × 1 cm × 1 cm tourmaline crystal, c axis vertical, warmed by ΔT.
eps0 = 8.854e-12
p = 4.0e-6           # C m⁻² K⁻¹, magnitude for iron-poor tourmaline
eps_r = 7.0          # relative permittivity along c (order of magnitude)
A, L = 1e-4, 0.01    # face area (m²) and length along c (m)
e = 1.602e-19

for dT in (1, 2, 5):
    sigma = p * dT                       # surface charge density, C/m²
    E_air = sigma / eps0                 # field just outside an isolated face
    V = sigma * L / (eps0 * eps_r)       # open-circuit voltage across the crystal
    Q = sigma * A
    print(f"ΔT = {dT} K: σ = {sigma * 1e6:4.0f} µC/m², field at face ≈ {E_air / 1e6:4.2f} kV/mm, "
          f"V ≈ {V / 1e3:4.1f} kV, charge {Q * 1e9:.1f} nC = {Q / e:.1e} e")
print("dielectric breakdown of dry air ≈ 3 kV/mm")
# ΔT = 1 K: σ =    4 µC/m², field at face ≈ 0.45 kV/mm, V ≈  0.6 kV, charge 0.4 nC = 2.5e+09 e
# ΔT = 2 K: σ =    8 µC/m², field at face ≈ 0.90 kV/mm, V ≈  1.3 kV, charge 0.8 nC = 5.0e+09 e
# ΔT = 5 K: σ =   20 µC/m², field at face ≈ 2.26 kV/mm, V ≈  3.2 kV, charge 2.0 nC = 1.2e+10 e
# dielectric breakdown of dry air ≈ 3 kV/mm

These are real and impressive numbers. A warming of a few degrees produces a field at the crystal’s ends approaching the breakdown strength of air, which is why a warmed tourmaline attracts dust and ash, and why its ends can give small sparks when heated strongly. In a vacuum, where no air ions can neutralise the charge, a heated pyroelectric crystal of lithium tantalate has produced fields strong enough to accelerate ions to nuclear fusion in a laboratory demonstration (Naranjo et al. 2005).

Charge is not a supply of ions

Tourmaline is widely sold as a source of “negative ions”, in jewellery, textiles, hair dryers and mattress pads. Two facts limit what it can do. First, charge is produced only while the temperature is changing; at steady temperature, for example against skin at a stable 33 °C, the production stops. Second, the charge is small, and in ordinary air it is mostly neutralised by attracting ions that already exist rather than by creating new ones.

# How much air ionisation could that charge account for?
e = 1.602e-19
Q = 4.0e-6 * 5 * 1e-4            # C, the 5 K case from the previous program
ions_once = Q / e
room = 30.0                      # m³, a small bedroom
natural_rate = 10 * 1e6          # ion pairs per m³ per second (≈ 10 per cm³ per s indoors)
print(f"at most {ions_once:.1e} ions, once, for a 5 K change")
print(f"natural ionisation of the room makes {natural_rate * room:.0e} pairs per second")
print(f"→ equivalent to {ions_once / (natural_rate * room):.0f} s of background ionisation,")
print("  and nothing at all while the temperature stays constant")

# Iron lowers the coefficient (Hawkins et al. 1995, 296 K, FeO in wt%).
for name, feo in (("iron-free (pink elbaite)", 0.0), ("intermediate", 7.0),
                  ("iron-rich", 14.6), ("schorl end-member", 20.46)):
    print(f"{name:25s} FeO {feo:5.2f} %  p ≈ {3.949 - 0.094 * feo:.2f} µC m⁻² K⁻¹")
# at most 1.2e+10 ions, once, for a 5 K change
# natural ionisation of the room makes 3e+08 pairs per second
# → equivalent to 42 s of background ionisation,
#   and nothing at all while the temperature stays constant
# iron-free (pink elbaite)  FeO  0.00 %  p ≈ 3.95 µC m⁻² K⁻¹
# intermediate              FeO  7.00 %  p ≈ 3.29 µC m⁻² K⁻¹
# iron-rich                 FeO 14.60 %  p ≈ 2.58 µC m⁻² K⁻¹
# schorl end-member         FeO 20.46 %  p ≈ 2.03 µC m⁻² K⁻¹

Even if every elementary charge produced by a 5 K warming became a free ion, which it does not, the total would equal less than a minute of the room’s natural ionisation, once. Iron-rich black schorl, the cheapest and most commonly used tourmaline in consumer products, has the lowest coefficient of all.