Did it take a nuclear physics lab to murder Alexander Litvinenko?

Introduction

The news reports on the murder of former KGB agent Alexander Litvinenko with radioactive polonium 210 (Po-210) quote various experts as saying that polonium is an extremely rare and exotic substance. Making the "huge" amount of it needed to murder somebody supposedly would require a nuclear reactor or perhaps a particle accelerator: large, expensive and exotic pieces of equipment far beyond the reach of amateurs. The obvious implication is that the Po-210 was provided by (or stolen from) the Russian nuclear establishment. (The Russians are known to have produced and used Po-210 in various legitimate applications such as heating the Lunokhod lunar rovers during the 2-week lunar night.)

See, for example, the Reuters article Polonium - deadly, hard to make and rare poison:

Radiation and chemistry experts say large-scale equipment, such as a nuclear reactor, would be needed to produce sufficient amounts to cause death. "It is not as simple as the idea that somebody might have broken into a radioactivity cabinet at some local hospital and walked off with some polonium," Dr Andrea Sella, a lecturer in chemistry at University College London, told Reuters. "You can't make this at home. This is in a different league," he added.

But is this true?

Maybe you can't make it at home, but I'm not so sure about the rest. To explain why, I have to start with some physics background.

Polonium 210 is the second to last entry in the decay chain of U-238 (uranium 238), the most abundant (99.275%) isotope of uranium in nature. Unless they have been removed, all of these isotopes, including Po-210, thus appear in small amounts with uranium wherever it is found. Marie Curie discovered and separated polonium from pitchblende (a uranium ore consisting mainly of uranium dioxide, UO2) in 1897 using purely chemical means. This was over a century ago. The first nuclear reactor and even the particle accelerator were still decades away.

Like every other element heavier than iron (the end result of "normal" stellar thermonuclear fusion), the U-238 present today in the earth was created by the explosion of supernovae long before the solar system and earth were formed about 4.5 billion years ago. Because it has a 4.5 billion year half life, about half of the U-238 originally present in the earth at its formation is still here today. In undisturbed rocks and ores, U-238 exists in equilibrium with every isotope in its long decay chain. This chain ends with stable Pb-206 (lead 206), so as time passes lead accumulates and the radioactive elements (including the U-238) slowly disappear.

At equilibrium, each isotope in the decay chain is produced just as fast as it decays. So every radioactive isotope in the decay chain must have exactly the same radioactivity. That is, if a sample of undisturbed, unprocessed uranium ore contains 1,000 Bq (becquerel) of U-238, then it must also contain 1,000 Bq of thorium 234, 1,000 Bq of palladium 234, etc, all the way down to 1,000 Bq of polonium-210. (The becquerel is the SI unit of radioactivity, defined as one disintegration per second.) After another 4.5 billion years, assuming the sample is sealed to prevent the escape of gaseous elements such as radon, it will have only 500 Bq of U-238, 500 Bq of Th-234, etc, and 500 Bq of Po-210. Half of the U-238 present today will become Pb-206.

Although every isotope in the decay chain has the same radioactivity, the number of atoms of each isotope will vary widely because of their vastly different half lives. In fact, the number of atoms of each isotope in the decay chain at equilibrium is directly proportional to its half life. Polonium-210 has a half life of only 138 days, so the number of Po-210 atoms will be only be 138 days / 4.5 billion years or about 10-10 (1 ten-billionth) of the number of U-238 atoms.

Po-210 as a radiological poison

Po-210 decays by 5.407 MeV alpha decay into Pb-206, which is stable.

Alpha particles are highly ionizing. Even a small amount of matter, such as a piece of paper or a few inches of air, will stop alpha particles and absorb their energy. A radioactive substance that emits only alpha particles is therefore easy to shield. But if the unshielded substance is taken into and spread throughout the body, then the alpha particles are absorbed by the adjacent tissues where they deposit their harmful energy. This makes alpha emitters like Po-210 potent radiological poisons, but only when they are injected, ingested or inhaled. Outside the body they cause little or no harm.

Computing a fatal dose

It takes about 5-10 sieverts, equal to 500-1000 rem, of whole-body radiation exposure to cause prompt, severe radiation sickness and probable death within a few days or weeks in previously healthy humans. The sievert is the SI unit of biological radiation dosage; it is related to the gray and rad, measures of absorbed radiation energy in inert materials, by the biological radiation weighting factor, a dimensionless ratio.

One gray is equal to an absorbed energy dosage of 1 joule per kilogram of absorbing material. The biological radiation weighting factor, i.e., the ratio of sieverts to grays, is the product of two values, Q and N. Q depends on the type of radiation (alpha, beta, gamma), and N depends on the particular body tissue. The Q value for alpha particles is 20, reflecting their highly ionizing nature. For human bone marrow, colon, lung and stomach, N=0.12; for several other organs, including the bladder, brain, breast, kidney, liver and small intestine, N=0.05. Taking the larger value of N=0.12 gives a QN product of 2.4. In other words, the deposition of 1 joule of alpha particle energy in 1 kg of human bone marrow (1 gray) would represent a dose of 2.4 sieverts.

To compute the radiation dosage to the body from a given amount of ingested Po-210, we next need to know how much energy is released in its alpha particles. As mentioned above, each Po-210 decay releases an alpha particle with an energy of 5.407 MeV, so we can easily compute the total energy released by the complete decay of a given amount of Po-210 by multiplying the 5.407 MeV alpha energy by the number of polonium atoms. But we can't wait for all of the Po-210 to decay; we'll want to give our victim his fatal dose within, say, one week, and only half of it decays in about 138 days. (It took several weeks for Litvinenko's doctors to figure out what was wrong with him, and by then it was too late.)

Radioactive decay is an exponential process; the fraction of undecayed atoms left after time t is

e-t * ln(2)/h, where h is the half-life.

The fraction of Po-210 remaining after one week is therefore

e-7 * ln(2)/138.376 = 0.9655

This means 1 - 0.9655 = 3.45% of the Po-210 will decay in one week, and we'll need 1/(1-0.9655) = 29 times as much Po-210 to deliver 10 sieverts in 7 days as we'd need to deliver 10 sieverts in infinite time (or 5 sieverts in one half life of 138.376 days).

One gram of Po-210 is 1/210 = 4.762 * 10-3 mole. Multiplying this by Avogadro's number, the number of atoms in a mole, we obtain

4.762 * 10-3 * 6.022 * 1023 = 2.87 * 1021 atoms of Po-210. If we multiply this many atoms by the alpha energy of 5.407 MeV, we obtain a total energy release of

2.87 * 1021 * 5.407 MeV = 1.55 * 1028 eV.

One joule equals 6.24 * 1018 eV, so this is equal to 2.48 * 109 joules, or 2.48 gigajoules.

The average adult human male has a mass of 70 kg. If we assume the Po-210 is uniformly distributed throughout the body, then the whole-body radiation dosage from 1 gram of Po-210 in one week would be

0.0345 * 2.48 gigajoules / 70 kg * 2.4 sievert/gray = 2.9 * 106 sievert. This is a truly gargantuan dose, but 1 gram is a huge amount of Po-210. If we only need 10 sieverts/week to kill our victim, then we only need

10 sievert / (2.9 * 106 sievert/gram) = 3.4 * 10-6 g.

That's 3.4 micrograms.

Uranium ore as a source of Po-210 poison

Let's suppose we wanted to follow in Marie Curie's footsteps and extract our fatal dose of Po-210 from natural uranium ore using purely chemical means. How much ore would we need?

As discussed above, in a sample of natural uranium in equilibrium with its decay chain, the relative abundance of Po-210 to U-238 is the ratio of their half-lives. The half life of Po-210 is 138.376 days; the half life of U-238 is 4.468 * 109 years, or 1.63 * 1012 days (at 365.25 days/year). The Po-210/U-238 half-life (and abundance) ratio is therefore

138.376 / (1.63 * 1012) = 8.48 * 10-11 (or about 1 part in 10 billion, as mentioned earlier).

That's about one atom Po-210 per 11.8 billion atoms of U-238. But we only need 3.4 micrograms of Po-210. That corresponds to

(238/210) * (3.4 * 10-6 g / 8.48 * 10-11 = 4.54 * 104 g = 45.4 kg U-238.

In other words, we only need as much natural uranium ore as contains 45.4 kg of U-238. That's not all that much, is it?

Concluding thoughts

The above calculation leaves out several factors I do not know, such as the speed and yield of the chemical extraction process, the chemical form and biological behavior (uptake, distribution and excretion) of the polonium in Litvinenko's body, and the actual dose that he ingested. Since he was reported to fall ill within hours of his presumed exposure, it is entirely possible that his dose was much greater than the single-digit microgram amounts that I have assumed. This would require a larger and presumably less practical chemical extraction process. On the other hand, one would have difficulty explaining why Litvinenko survived for nearly a month given that it took nearly that long to find out what had poisoned him, and presumably to begin any effective therapies to remove the polonium from his body.

In any event, microgram amounts of Po-210 seem very dangerous indeed, so the utility, at least in theory, of chemical extraction cannot be ruled out.

If the only possible source of the Po-210 that killed Alexander Litvinenko was a large nuclear laboratory, this would point strongly at Russia, either for deliberately murdering him or for failing to secure their inventories of Po-210. But counter to the intuitive claims of the experts in the media, I don't think we can rule out chemical extraction from manageable amounts of uranium ore with a technique that was first developed (and published) over a century ago.

It is also worth pointing out that, unlike a finite stockpile of Po-210 produced in a reactor or particle accelerator, uranium ore is a nearly infinitely renewable source of Po-210. After extraction, a significant amount is replenished in a matter of weeks by further decay of the parent isotopes in the U-238 decay chain. I think this might have some practical implications vis a vis use of pure Po-210 (stolen or allocated) for international political assassinations. Once pure Po-210 is used, it cannot be reused -- but the stock of uranium ore used to breed it could be. Po-210 also decays relatively rapidly -- a consideration if the international smuggling of radioactive isotopes is time consuming (and I have no way of knowing). Uranium ore, on the other hand, would be an assassin's gift that keeps on giving.

Phil Karn, v 1.02, 26 November 2006