The spin that sorts by a difference of mass

Turning is an acceleration found that an object going round a circle at constant speed is accelerating towards the centre all the time, and the forces that are not there found that in a frame turning with it the same fact appears as a force pushing outward, the centrifugal force, which can be written as a potential, $-\tfrac12 m\omega^2 r^2$. The surface a spin decides put a liquid in such a frame and found its surface a paraboloid. Later essays followed the Coriolis term into the atmosphere and the oceans, and the ball a turntable will not throw off.

This one puts a gas in the turning frame. A gas in a potential settles into the exponential that thins the air with height, and that essay noted in passing that the same law, with the centrifugal potential in place of mgh, governs a centrifuge. Followed through, the passing remark turns out to contain the reason one of the hardest separations in industry is done by spinning tubes, and a lesson about the difference between a difference and a ratio.

An atmosphere pressed against a wall #

In a frame turning with a rotor at a steady rate ω, a molecule of mass m at distance r from the axis has a centrifugal potential energy of $-\tfrac12 m\omega^2 r^2$. A gas in that frame, at rest relative to the rotor and in thermal equilibrium at temperature T, distributes itself by Boltzmann’s law, the exponential that decides everything:

$$n(r) = n(0)\,\exp\!\left(\frac{m\omega^2 r^2}{2kT}\right).$$

Its density rises from the axis to the wall by the factor $\exp(mv^2/2kT)$, where $v = \omega a$ is the wall’s speed. For uranium hexafluoride, the only compound of uranium that is a gas near room temperature, a molecule weighs 352 mass units, and with the wall moving at 600 metres a second the exponent is 25. The density at the wall is a hundred thousand million times that on the axis. Nearly all the gas is in a thin layer pressed against the wall, and the middle of the rotor is nearly empty. It is the atmosphere of a planet whose gravity is a hundred thousand times the Earth’s, wrapped round the inside of a tube.

A difference in the exponent #

Now let the gas be a mixture of two isotopes. Each obeys the same law with its own mass in the exponent, so the ratio of the two densities changes across the rotor too, and the change depends only on the difference of the exponents:

$$\alpha = \frac{(n_1/n_2)_{\text{axis}}}{(n_1/n_2)_{\text{wall}}} = \exp\!\left(\frac{(m_2 - m_1)\,v^2}{2kT}\right).$$

The gas on the axis is enriched in the lighter isotope by the factor α relative to the gas at the wall. That is the whole of the centrifuge’s principle, and the formula says two things that are not obvious.

The first is what it does not contain. The masses enter only as their difference. Two isotopes of hydrogen, differing by one mass unit on two, and two of uranium hexafluoride, differing by three on 350, separate according to one and three units respectively; the fact that the uranium pair differ by less than one per cent is irrelevant. The second is that the radius has dropped out: only the wall speed matters, so a thin rotor and a fat one spinning to the same rim speed separate equally.

The other way of separating gaseous isotopes, the one used first, is effusion through a porous barrier. The gas that leaves is not the gas inside found that molecules escape through a small hole at a rate proportional to their speed, which goes as one over the square root of their mass, so the gas that comes through is enriched in the lighter molecule by $\sqrt{m_2/m_1}$. That is a ratio. For uranium hexafluoride it is the square root of 352 over 349, 1.0043. A centrifuge with its wall at 600 metres a second gives 1.24 in one pass.

Why the ratio is the problem #

The figure sets the two methods side by side for five pairs of molecules. For hydrogen and its heavy form, a ratio of three to two, effusion is the better method: the square root of a large ratio is a substantial factor, and one mass unit of difference does little in a centrifuge. For every heavier pair the order is reversed, and the reversal grows with the mass. Carbon dioxide made with carbon-13 differs from the ordinary molecule by one part in forty-four, and effusion separates them by about one per cent; argon-36 and -40 are separated by a factor of four times better in a centrifuge than by effusion; for the xenon pair the advantage is fifteenfold, and for uranium hexafluoride nearly forty at this speed.

That is why enriching uranium was so hard in the 1940s and is so much cheaper now. To raise uranium-235 from its natural 0.72 per cent to the three to five per cent a power reactor needs, the diffusion plants built for the Manhattan Project and after it pushed the gas through barriers in thousands of stages in series, each enriching it by a fraction of a per cent, with compressors driving the gas round the cascade. They were among the largest buildings in the world and used electricity on the scale of a city. A cascade of centrifuges achieves the same with a few tens of stages, each stage many machines in parallel, and uses about a fiftieth of the electricity for the same separation. Mixing what is already mixed computed the minimum work any process must spend to unmix isotopes; both methods spend far more than that minimum, and the centrifuge spends far less of the excess.

An idea older than the bomb #

The principle was proposed almost as soon as isotopes were known to exist. Frederick Lindemann and Francis Aston suggested in 1919 that spinning a gas could separate isotopes, and worked out the factor above; the rotors of the day could not spin fast enough to make it useful. Jesse Beams at the University of Virginia built high-speed rotors in the 1930s, spinning them on magnetic and air bearings in vacuum, and in 1936 separated the two isotopes of chlorine by spinning a chlorine compound. When the Manhattan Project needed enriched uranium, centrifuges were tried at industrial scale and abandoned: the rotors of the time shook themselves apart in the long passage through their resonances, and gaseous diffusion, crude as it was, could be built from pipes, pumps and barriers that already existed.

So for twenty years after the war the world’s enriched uranium came from diffusion plants, and the centrifuge’s advantage, clear on paper since 1919, waited for rotors that could reach and hold the speeds the formula asked for. The essay’s figures show why the wait was worthwhile: at the speeds rotors eventually reached, a single machine does in one pass what fifty diffusion stages would.

Multiplying the separation along the rotor #

A single factor of 1.2 between axis and wall would still be hard to use: the gas on the axis is a near-vacuum, and drawing it off would yield little. Working centrifuges multiply the radial separation along the length of the rotor instead. A slow circulation is driven inside the tube — down along the wall and up along the axis, or the reverse — by a temperature difference between the ends or by a scoop that drags on the spinning gas. The gas moving one way near the axis is slightly enriched in the light isotope, the gas moving the other way near the wall slightly depleted, and the radial separation exchanges between them as they pass, just as a distillation column multiplies a liquid’s small preference for the more volatile component by counter-flowing vapour and liquid past each other. The ends of the rotor end up differing by far more than α, and product and tails are drawn off there.

Paul Dirac worked out in the 1940s the best such a counter-current machine can do, the maximum rate at which it can do separative work:

$$\delta U_{\max} = \frac{\pi Z \rho D}{2}\left(\frac{\Delta M\,v^2}{2RT}\right)^2,$$

with Z the rotor’s length, ρD the gas’s density times its self-diffusion coefficient, which for uranium hexafluoride is nearly independent of pressure, and ΔM the molar mass difference. The radius has dropped out again, the length enters linearly, and the wall speed enters as its fourth power. Doubling the speed of a rotor is worth sixteen times as much as doubling its length.

The currency of enrichment #

The quantity Dirac bounded is the one the industry prices: separative work, measured in separative work units. It counts how much unmixing a separation has achieved, weighted so that the work of producing a given product from a given feed is the same however the cascade is arranged, much as entropy counts disorder independently of how it was produced. Producing one kilogram of uranium enriched to four and a half per cent, from natural uranium, leaving tails at a quarter of a per cent, takes about seven units and ten kilograms of natural uranium. A large power reactor’s yearly reload of fuel takes something over a hundred thousand.

The cost of a unit is dominated by energy, and the energy per unit is where the two methods differ most. A diffusion plant spent about two and a half thousand kilowatt-hours per unit, because every stage recompressed all the gas that passed through it to push a little more of it through the next barrier. A centrifuge plant spends about fifty, most of it overcoming friction in the rotors’ bearings and the drag of the gas they spin. The difference is not in the thermodynamic minimum, which is the same for both and is a small fraction of either; it is in how much more than the minimum each must spend, and the centrifuge’s separation factor per stage, raised by the difference in its exponent, is why it spends so much less.

The rotor’s own limit #

Which puts the question of the centrifuge entirely into the rotor. A thin cylindrical shell spinning about its axis must hold its own material in circular motion, and the stress in its wall is $\sigma = \rho v^2$, the material’s density times the square of the rim speed — independent of the radius, for the same reason the gas’s distribution was. The fastest any shell can spin is the speed at which that stress reaches what the material can bear, $v = \sqrt{\sigma/\rho}$. That is exactly the quantity that the mass and where it sits found setting the energy a flywheel can store per kilogram, and for the same reason: both are hoops whose job is to go round as fast as their material allows.

A strong aluminium alloy reaches about 420 metres a second; maraging steel, a very strong low-carbon alloy, about 500; fibre composites, whose strength is high and density low, considerably more in principle. Because separative power goes as the fourth power of the speed, each step in material was a large step in performance. The centrifuges that first made the method practical, developed in the Soviet Union in the 1950s by a group that included the Austrian engineer Gernot Zippe, used short aluminium rotors spinning on a needle bearing at the bottom and held by a magnetic bearing at the top; the machines built since then in Europe and elsewhere moved to longer rotors of stronger materials.

The rotors have other limits than stress. A long thin tube has flexural resonances, frequencies at which it bends like a guitar string, and as it spins up it must pass through them without shaking itself apart, or be kept short enough that its first resonance lies above its working speed. Those are the subcritical and supercritical designs of the industry, and how they are managed is engineering beyond the physics here. The physics sets the ceiling: wall speed, through a material’s strength-to-density ratio, raised to the fourth power.

Every centrifuge is a buoyancy machine #

Most centrifuges are not used at equilibrium at all. A cream separator, a blood centrifuge and a laboratory spinner sorting cells all use the centrifugal field to speed up settling, and what they exploit is not a Boltzmann distribution but a drift. A particle in a liquid feels the centrifugal force on its own mass and a buoyant force from the liquid it displaces, which in the turning frame points inward exactly as buoyancy points up in a gravitational field; the net is the difference, and the particle drifts outward if it is denser than the liquid and inward if it is lighter. That is the rotating-frame version of the weight of the water that is not there, and the same reason a balloon leans the wrong way in a turning car, towards the inside of the bend.

Cream is fat in water, lighter than the milk round it, so in Gustaf de Laval’s separator of 1878 it drifts towards the axis while the skimmed milk crowds outward, and the two are drawn off continuously. Spun at a few thousand times gravity, blood separates in minutes into red cells at the outside, a thin layer of white cells and platelets, and plasma towards the axis. The drift speed is set by a balance between the net force and the drag of the liquid, which for small particles is Stokes’s drag, so the time a separation takes goes as the inverse square of the particle’s size; that is why proteins, far smaller than cells, needed Svedberg’s ultracentrifuge at hundreds of thousands of times gravity.

Weighing a molecule by its spin #

The same exponential, applied to molecules in a liquid, was one of the founding instruments of biochemistry. In the 1920s Theodor Svedberg built an ultracentrifuge, spinning a small cell of solution so fast that dissolved proteins, much heavier than the water round them, settled measurably towards its outer end. Left long enough, the settling is balanced by diffusion and the concentration reaches the Boltzmann profile, with the molecule’s mass reduced by the buoyancy of the liquid it displaces. The logarithm of the concentration is then a straight line against the square of the radius, and its slope gives the molecular mass directly, with no assumption about the molecule’s shape.

At the time it was not settled that a protein had a definite mass at all; the alternative view was that proteins were colloidal aggregates of variable size. Svedberg’s measurements on haemoglobin gave a single, sharp value, about sixty-eight thousand, and so did those on other proteins, which was among the first evidence that proteins are particular molecules with particular structures. The same principle, a Boltzmann distribution in a centrifugal potential, separates uranium for reactors and weighs proteins for biochemists, and in both the quantity read is a mass difference over kT: the protein’s mass minus the liquid it displaces, the isotope’s minus its neighbour’s.

What the equilibrium picture leaves out #

The figures treat the gas as ideal and at equilibrium, rotating rigidly with its rotor at a single temperature. A real rotor’s gas does not quite rotate rigidly, the internal circulation that multiplies the separation also disturbs it, and the temperature varies along the rotor by design. The separation factor drawn is the equilibrium value from axis to wall; what a machine delivers per stage depends on the circulation and is less than Dirac’s bound. The rotor speeds are ideal limits from a material’s strength and density with no margin, and working machines run below them; the stresses quoted for composites in particular depend strongly on how the fibres are wound. And the protein figure ignores the non-ideal interactions between molecules, which in practice are corrected by measuring at several concentrations.

The two scaling claims — that the separation depends on the mass difference and the wall speed alone, and that the separative power grows as the fourth power of the speed — are consequences of the Boltzmann distribution in a rotating frame and survive all of those corrections.

Still open: what limits a rotor in practice #

The ideal $\sqrt{\sigma/\rho}$ for the best carbon fibres is above a kilometre a second, and no production centrifuge is known to approach it. What holds real machines back is a mixture of the composite’s behaviour under sustained stress over years of continuous running, the dynamics of passing through resonances, the bearings, and the cost of a machine expected to run for decades without stopping. How close the industry is to the ceiling is not public in detail, and for good reason: the same machines that enrich uranium for reactors can, in a different arrangement, enrich it for weapons, which is why the technology is closely controlled.

The habit worth carrying away is to ask whether a process acts on a ratio or on a difference. Effusion separates isotopes by $\sqrt{m_2/m_1}$, 1.0043 for uranium hexafluoride; a centrifuge separates them by $\exp(\Delta m\,v^2/2kT)$, 1.24 at a 600 m/s wall, because a Boltzmann exponent subtracts masses where a speed divides them. The difference does not care how heavy the molecules are, and the speed is bought with a material’s strength.