‘Their aim is that we accept a capacity of ten thousand separative work units which is equivalent to ten thousand centrifuges of the older type that we already have,’ Ali Khamenei, Iran’s supreme leader, said on 4 June. ‘Our officials say we need 190,000 SWU. Perhaps this is not a need this year or in two years or five years, but this is the country’s absolute need.’

Khamenei’s statement reminds me of some of the answers I used to mark when I was teaching physics, when the student had the right vocabulary but failed to put the words together in a meaningful way. I do not entirely blame Khamenei. The separative work unit – SWU, pronounced ‘swoo’ – is the least intuitive unit I have ever encountered in physics. I doubt most physicists could define it correctly. I will define it, tell you where it came from and then try to adumbrate Khamenei’s statement. It’s important because it may be the key to the outcome of the present nuclear negotiations with Iran.

Let W_{SWU} be the number of SWUs needed to separate a feed with a percentage of uranium-235 *x*_{f} – the rest being uranium-238 – into a product with a U-235 percentage *x*_{p} and a remainder of U-238 containing a small percentage *x*t (a ‘tail’) of U-235. Typically you might begin with a feed of natural uranium, which is more than 99 per cent U-238, and you might be looking for a product in which, say, the U-235 content is 3.5 per cent and the tail is a few tenths of a per cent. Let the value function ‘V(*x*)’ be defined as V(*x*) = (1 − 2*x*)ln[(1 − *x*)/*x*], where ‘ln’ stands for the natural logarithm. Then by definition:

W

_{SWU}= PV(x_{p}) + TV(x_{t}) − FV(x_{f})

Here P, T and F stand for amounts, usually in grams or kilograms. I write this out mainly to show what Khamenei was trying to cope with.

Paul Dirac was one of the greatest physicists of the 20th century. He was among the creators of quantum theory, and when he unified it with the theory of relativity he saw that this led to anti-particles. The first one, discovered in 1932, was the anti-electron: the ‘positron’. Around this time Dirac became interested in the separation of isotopes. He began a collaboration with the Russian physicist Pyotr Kapitza, and together they invented a machine for the purpose. Basically it was a curved tube down which a gas could be injected at high speed. When the gas rounded the curve the centrifugal force on the molecules would in principle produce a separation of isotopes. South Africa used a version of the idea to produce enough U-235 to make half a dozen bombs. But before Dirac and Kapitza got very far Kapitza went to Russia on holiday and was never allowed to return.

The scene shifts to Britain, and two refugee scientists, Rudolf Peierls and Otto Frisch. Niels Bohr had shown in a paper that the fissile isotope of uranium was U-235. Frisch and Peierls investigated how much would be needed to make a bomb. They underestimated the amount by a great deal – the actual amount needed is about 52 kilograms – but the two reports they wrote in 1940 began the quest for the bomb. They realised that the real problem in making a bomb was to produce enough U-235 to constitute a critical mass. They set off to work out how to produce the required amount, Frisch experimentally and Peierls theoretically. Peierls recruited Dirac, a colleague and friend, who had originally studied engineering.

Dirac’s analysis can be applied in the operation of gas centrifuges. In these, a gas is injected into a cylinder that is made to spin at rates faster than the speed of sound. When separating uranium into its isotopes, the gas used is uranium hexafluoride, each molecule of which consists of one uranium atom and six fluorine atoms. It’s a solid at room temperature, which makes it easy to transport; as a gas, it is highly corrosive. Passing the gas through a cascade of centrifuges results in a product that is more and more highly ‘enriched’ with U-235: about 3.5 per cent enrichment is required for use in a nuclear reactor, and more than 90 per cent for a bomb. Dirac invented the separative work unit to quantify both the amount of separation achieved by a given centrifuge and the amount needed to perform some specific task, such as producing a kilogram of 3.5 per cent enriched uranium hexafluoride. He also found there was a theoretical limit to the degree of separation that a given centrifuge could achieve (though it is aspirational since no actual centrifuge can reach it). Dirac wrote a brief note about this; it has never been properly published, but copies exist. Peierls adopted the SWU in his work with Klaus Fuchs, which was published in a classified report. It came to the United States with them and the Russians got hold of it too since Fuchs was a spy. Now someone has tried – with limited success – to explain it to Khamenei.

Tables are now available for calculating the SWU needed to perform particular tasks. I’m using one supplied by the physicist Richard Garwin. Suppose we want to produce a kilogram of uranium hexafluoride enriched to 3.5 per cent. We begin with natural uranium hexafluoride with a U-235 content of 0.7 per cent, and accept ‘tails’ of 0.4 per cent. According to the tables, to perform this task we need about 3.6 SWU per kilogram.

The Iranians haven’t been forthcoming about the SWU capacities of their centrifuges. Khamenei’s remarks are all but incomprehensible. But the International Atomic Energy Agency has learned enough to enable us to put the pieces together. A very good summary of the most recent data can be found in a report issued on 23 May by the Institute for Science and International Security. Since 2007 Iran has produced some 12,000 kilograms of 3.5 per cent enriched uranium hexafluoride. It has also produced about 230 kilograms of 20 per cent enriched uranium hexafluoride. According to my estimates, the total SWU used comes to approximately 45,000. The number of centrifuges used by Iran has increased over the years. At present they have nearly 18,000 IR-1 centrifuges: they seem to be what Khamenei is referring to as the ‘older type’. It’s estimated that these each produce about 1 SWU per year if they operate continuously. The SWU figures given by Khamenei must refer to a single year.

So, is Iran’s present level of SWU production ‘adequate’? That depends on its purpose. A nuclear power reactor of the type Iran has at Bushehr requires about 25,000 kilograms of 3.5 per cent enriched uranium per year to operate. That’s more than twice the total amount currently produced by Iran. In the case of Bushehr that doesn’t matter since the fuel is supplied by the Russians, whose commercial enrichment plants produce millions of SWU per year. To make present and future Iranian nuclear power plants self-sufficient would require a giant expansion of capacity.

But here’s the problem. I’ve said that the critical mass of U-235 required to make a bomb is 52 kilograms. But with good design only about half this amount is actually needed. It takes about 232 SWU per kilogram to enrich natural uranium to 95 per cent: weapons grade. If Iran were to devote all its enrichment capacity to this end, it would in theory be able to produce bombs at the rate of a few per year. That is the spectre that haunts the present negotiations, and it is difficult to see how the matter can be resolved.

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