The Crookes Radiometer: The Spinning Vanes That Fooled Physicists

The apparatus and the paradox

A Crookes radiometer is deceptively simple. Inside a sealed glass bulb, four thin mica or aluminium vanes are mounted like the sails of a windmill on a light horizontal cross that pivots on a fine needle — a near-frictionless spindle. One face of each vane is blackened (soot or lampblack), the other left reflective. The bulb is evacuated to a partial vacuum, not a hard one: roughly 1 pascal, about a hundred-thousandth of atmospheric pressure.

Shine light on it and the cross spins, the black faces always moving away from the source, at anything from a few revolutions per minute in dim room light to several hundred rpm in bright sunlight. Sir William Crookes stumbled on the effect around 1873 while weighing samples in an evacuated balance and noticing that warm objects seemed to be pushed; he built and described the light-mill to the Royal Society in 1874–1875. The device was an instant curiosity — and a scientific scandal, because nobody could agree on why it moved.

The wrong answer: it is not the pressure of light

The seductive explanation is radiation pressure. Light carries momentum: an electromagnetic wave delivers a pressure equal to its intensity divided by the speed of light, P = I/c (the time-averaged Poynting flux S divided by c). Maxwell had predicted exactly this, and it sounds like a perfect fit for a spinning light-mill.

It fails on two counts. First, the magnitude is absurdly small. In full sunlight the intensity is about 1360 W·m⁻², so P = I/c ≈ 4.5 µPa — on a 1 cm² vane that is a force near 5×10⁻¹⁰ N, roughly a hundred thousand times weaker than the radiometric force actually observed at the working pressure. Second, the direction is backwards. A reflecting surface returns a photon's momentum and so feels twice the push of an absorbing surface (2I/c versus I/c). Radiation pressure would therefore shove the shiny faces hardest and spin the mill with the silver sides retreating — the opposite of what happens. As Arthur Schuster demonstrated in 1876 by hanging the whole bulb on a torsion fibre and watching the glass case recoil opposite to the vanes, the driving force is internal to the bulb (gas pushing on vane, vane pushing on gas) — not an external light pressure. Real radiation pressure is real, but you must go to a hard vacuum to see it: Ernest Nichols & Gordon Hull and, independently, Pyotr Lebedev measured P = I/c in 1901 with torsion balances pumped hard enough to kill the radiometric effect that had been masking it.

Why 'hotter molecules hit harder' is not enough

The next guess is thermal, and it is nearly right. The black face absorbs light and warms up; the reflective face stays cooler, so the black face may run several kelvin — in strong illumination tens of kelvin — hotter than the shiny one. Gas molecules that strike the hot face rebound faster (their re-emission samples a higher-temperature Maxwell–Boltzmann distribution), and faster rebounds mean more momentum delivered, hence more pressure on the black side. Push the black side, black retreats. Neat — but wrong as stated.

The flaw, spotted by Maxwell and Reynolds, is that in the steady state the gas next to a large flat plate is not accelerating, so the excess push on the hot face is balanced by the gas and produces no net force over the flat interior of the vane. If the whole face contributed, the force would scale with the vane area and grow with gas density right up to atmospheric pressure — yet radiometers refuse to spin near 1 atm. Something localizes the force, and averages the face contribution away. That something lives at the edges.

The real mechanism: thermal creep and the edge force

The resolution came in 1879 from Osborne Reynolds and James Clerk Maxwell — the latter in On stresses in rarefied gases arising from inequalities of temperature, one of the last papers he wrote before his death that year. Reynolds had discovered thermal transpiration: when a hot region and a cold region of gas are connected through fine pores or narrow channels, gas creeps from cold to hot until, in the free-molecular limit, a pressure difference builds up with p_hot/p_cold = √(T_hot/T_cold) — the temperature-gradient cousin of Graham's effusion law.

Maxwell derived the underlying thermal creep: along any solid wall carrying a tangential temperature gradient, the gas acquires a slip velocity directed toward the hotter end,

u_creep = (3/4)·(μ / ρT)·(dT/ds),

where μ is the gas viscosity, ρ its density, T the temperature, and s the coordinate along the surface. Molecules arriving from the hot direction carry more tangential momentum than those from the cold direction, so a thin layer of gas near the wall drifts hotward — and by Newton's third law the wall is pushed coldward.

Now apply this at a radiometer vane. Around its thin edge there is a steep temperature gradient tangent to the surface, running from the cool silvered face to the warm black face. Gas creeps around the rim from cold to hot (toward the black side); the reaction on the vane points from the hot black face toward the cool silver face. Summed around the perimeter of all four vanes, this edge force is the torque that spins the mill with the black faces retreating. Albert Einstein revisited the problem in 1924, estimating that the force is concentrated within about one mean free path of each edge and scales with the perimeter length and the tangential temperature difference — explaining why thin, small vanes with a large edge-to-area ratio are efficient light-mills. Modern direct-simulation Monte Carlo (DSMC) studies and micro-scale torsion experiments (Selden, Ketsdever, Gimelshein and coworkers, c. 2009) resolve both the dominant edge term and a smaller area contribution, confirming the Reynolds–Maxwell picture in quantitative detail.

The Goldilocks pressure: it lives in the transition regime

The single most telling fact about a radiometer is that it works only in a narrow window of pressure. The controlling number is the Knudsen number, Kn = λ/L, the ratio of the gas mean free path λ to the vane size L. The mean free path scales inversely with pressure, λ = k_BT / (√2·π d² p): for air it is about 68 nm at 1 atm, so at 1 Pa (≈10⁻⁵ atm) it stretches to roughly 7 mm — comparable to a centimetre-scale vane. That puts the device squarely in the transition/slip regime, Kn ≈ 0.1–1, which is exactly where thermal creep is strong and there is still appreciable gas to push.

• Too much gas (near 1 atm, Kn ≪ 1): the creep layer is a negligible sliver of the edge, viscous drag on the moving vanes is enormous, and rapid heat conduction through the dense gas erases the temperature difference before it can drive a flow. The mill sits still.
• Too little gas (hard vacuum, Kn ≫ 1): there are simply too few molecules to carry momentum, and the radiometric force vanishes as the density does. The mill stalls — which is precisely why radiation-pressure experiments must be run in high vacuum, to strip away the radiometric masking force.
• The sweet spot (≈ 1 Pa, Kn ≈ 0.1–1): mean free path ≈ vane size, and the net force peaks. A radiometer that develops a slow leak and rises toward atmospheric pressure gradually slows and stops — a diagnostic every collector eventually notices.

Measurements, look-alikes, and where the physics is used

The observable numbers are modest but consistent: a black–silver temperature difference of a few kelvin (tens of kelvin under a bright beam), an effective radiometric pressure many orders of magnitude above the ~5 µPa of sunlight, and rotation from a few rpm up to several hundred rpm as illumination brightens. Crucially, the direction — black retreating — is the fingerprint that rules out radiation pressure and confirms a thermal-creep origin.

The same physics turns up under other names. Thermophoresis is thermal creep acting on a suspended particle rather than a plate: in a gas with a temperature gradient, dust drifts from hot toward cold, which is why a dust-free 'dark space' forms around a hot body (noticed by Tyndall) and why thermal precipitators and semiconductor cleanrooms exploit the effect to keep particles off hot wafers. Reynolds's thermal transpiration is the basis of the Knudsen pump (Knudsen compressor) — a gas pump with no moving parts that drives flow through micro-channels using only a temperature difference — and of proposed radiometric micro-actuators and micro-thrusters for MEMS and spacecraft, an area revived by DSMC modelling in the 2000s. It even matters for satellites and dust grains skimming the rarefied upper atmosphere, where radiometric and thermophoretic forces compete with genuine solar radiation pressure.

That last contrast is the moral of the story. Radiation pressure is real and is now routinely harnessed — in optical tweezers, laser cooling, and solar sails — but it is a feeble, hard-vacuum phenomenon. The lively little light-mill on the windowsill, by contrast, runs on the collective statistical mechanics of a nearly-empty bulb: a reminder that in the rarefied-gas regime, edges, temperature gradients, and the finite mean free path can matter far more than the momentum of light.

What actually turns the vanes: radiation pressure versus the radiometric (thermal-creep) force, and why the naive pictures fail

Effect / regime	Physical origin	Direction of push	When it dominates
Radiation pressure (Poynting)	EM field momentum S/c; a reflecting face returns 2× the momentum	Pushes the shiny face → black would advance (WRONG way)	~4.5 µPa in full sun; only in a hard vacuum
Radiometric / thermal-creep force	Tangential T gradient at the vane edges makes gas creep cold→hot; reaction on the vane	Pushes the warm black face back → black retreats (observed)	~1 Pa, Kn ≈ 0.1–1 (transition flow)
Naive 'hot molecules hit harder' (face)	Faster rebound off the warm black face	Would push black back	Cancels in the steady-state gas interior — not the true net force
Continuum limit (~1 atm, Kn ≪ 1)	Conduction shorts out ΔT; huge viscous drag	Negligible	Never — a radiometer will not spin at atmospheric pressure
Free-molecular limit (hard vacuum, Kn ≫ 1)	Too few molecules to transfer momentum	Negligible	Never — the mill stalls in high vacuum