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The Simple Math of an Iranian Nuclear Bomb

In an excerpt from ‘Nuclear Iran,’ calculating, with scientific precision, just how far Iran has come in its quest for the bomb

Jeremy Bernstein
October 02, 2014
2008 visit by then-Iranian President Mahmoud Ahmadinejad to the Natanz Uranium enrichment facilities.(Photo by the Office of the Presidency of the Islamic Republic of Iran via Getty Images)
2008 visit by then-Iranian President Mahmoud Ahmadinejad to the Natanz Uranium enrichment facilities.(Photo by the Office of the Presidency of the Islamic Republic of Iran via Getty Images)

No single threat to Israeli—and perhaps global—security has generated as much controversy in the last decade as Iran’s nuclear program. Iranian officials insist the program is peaceful. International observers, Israeli intelligence, and U.S. foreign-policy stewards are skeptical. But behind the diplomacy, spycraft, jockeying, and feinting lies the science of nuclear reactors and weapons: the specific engineering hurdles, the physics of fission, the “weaponization” of minerals required to produce as complex a device as a nuclear bomb. In this excerpt from Nuclear Iran, physicist Jeremy Bernstein unpacks what we know about the centrifuges at Natanz to take an informed guess at how likely Iran is to have enough weapons-grade uranium to make a nuclear warhead.


In 1993 the Iranians received 115.5 kilograms of 19.75 percent enriched uranium fuel elements for the Tehran Research Reactor—the TRR from Argentina. We do not know precisely how this reactor has been run. Its maximum power output is 5 megawatts thermal. How many days a year it has been run at this maximum, or how frequently it has been shut down, one does not know. It is also not known how the fuel has been managed. The core seems to contain eighteen fuel elements, each with 1.87 kilograms of enriched uranium, and five control elements, each containing 1.08 kilograms—the core total then being 39.06 kilograms. Thus, the Iranians were supplied enough uranium for more than one core change. Whether they changed the entire core every few years or some of the elements every year, I do not know. Given the type of reactor, it is possible to estimate how many kilograms of 19.75 percent uranium would be required for it to run at full power for a year. The answer is about 7. Therefore, if it ran full time at 5 megawatts thermal, the total fuel the Iranians acquired in 1993 would have run out in about sixteen years, and the last year of its operation would have been 2009. If it has been running with less power or less often, the time would of course be longer. But in any case the end of its present operational cycle was in view. During this time it would have generated a few hundred grams of plutonium. This would be nowhere near enough for a bomb but plenty to study the chemistry of plutonium separation. Seaborg and his people only had milligrams. This means that the TRR fuel must be carefully supervised. In fact, now it is being manufactured in Esfahan from 20 percent enriched uranium produced by the indigenous centrifuges.

In the late summer of 2009 it appeared as if the IAEA might have been able to broker a deal that would be a win-win for all concerned. The idea was to send to Russia most of the 3.5 percent enriched uranium hexafluoride in Natanz. At room temperature, uranium hexafluoride is a powder and is easy to transport. The Russians would regassify it and use their centrifuges to enrich it to 19.75 percent. Then in powder form it would be sent to France. The French use nuclear power to produce most of their electricity, so they have a large nuclear infrastructure. They would manufacture new fuel elements, which they would then send back to Iran to be used in the TRR. As it is, as of the fall of 2013 the Iranians had produced about 240 kilograms of 20 percent enriched uranium, a relatively small fraction of which is being used in the TRR. The accumulation of 20 percent enriched uranium is always a concern because of the relative ease of converting this to weapons-grade.

The Natanz centrifuge facility has been a concern from the beginning. It was built clandestinely in an unlikely location. Natanz is a city of about 40,000 known primarily for its pears and mountain scenery. The facility is a few miles from the town. Nothing might have been known about the centrifuges except for the same dissidents who revealed the Arak reactor. The centrifuges are located in a gigantic facility of some 100,000 square meters built 8 meters underground. It is protected by thick concrete walls and impervious to ordinary bombs. Inside there are two 25,000-square-meter halls, A and B, for the centrifuges. As of August 2013 the Iranians announced that in Hall A there were to be installed 25,000 centrifuges in 144 cascades. One unit was to contain the updated IR-2m centrifuges, of which six cascades had been installed, although apparently none were running. Fifty-four IR-1 cascades were running and producing low enriched uranium. All of these centrifuges were originally of the P1 type with aluminum rotors. They have peripheral speeds of about 350 meters per second. They have separative powers of 2–3 SWU per year, but from their actual performance they seem to be operating below the low end of the range. Up to the time of the 2013 report, a total of 9,704 kilograms of low enriched hex had been produced.

Why is this of concern? Because there is a potential for “breakout,” in which the low enriched uranium could be highly enriched for weapons. Here I want to deal with the question of how much 19.75 percent enriched uranium could be produced if, say, a metric tonne—1,000 kilograms—of 3.5 percent enriched uranium hexafluoride were further enriched. I will make some plausible but speculative assumptions about the centrifuges.

The least speculative assumption is that in the centrifuging process no significant amount of uranium 235 is lost. We begin with a certain amount and we end with the same amount. At first sight one might think that this violates the object of the exercise, which is to enrich uranium. But in the centrifuge we are not creating or destroying uranium 235. We are simply moving it about. If we make this assumption, we can derive a simple equation that relates the amount of the product to the amount of feed. I will first take these to be uranium hexafluoride, and then later I will extract the uranium. Let P be the amount of uranium hexafluoride product in, say, kilograms, and let F be the amount of feed fed into the cascade, also in kilograms. Let NP be the percentage of uranium 235 in the product, let NF be likewise the percentage in the feed, and let NW be the percentage in the waste or “tails.” Then the equation is

P = F((NFNW) / (NPNW)).

For NF we will put in the value .035, and for NP the value .1975. But what about NW? I will take as two sides of the range .0025 and .004. With F = 1,000 kilograms, I find, using the equation, the product to be 167 kilograms in the first case and 160 kilograms in the second. Not much difference.

However, one wants to insert uranium, not uranium hexafluoride, into the fuel elements. We may then ask, What is the percentage of uranium by mass in, say, 19.75 percent enriched uranium hexafluoride? To determine this we must recall that the atomic mass of fluorine is 19 and that there are six fluorine atoms in each uranium hexafluoride molecule. Thus, the ratio we are looking for is given by

((.1975 × 235) + (.825 × 238)) / ((.1975 × 235) + (.825 × 238) + (6 × 19)) ≈ .68.

Therefore, 160 kilograms of 19.75 percent enriched uranium hexafluoride will yield about 109 kilograms of 19.75 percent enriched uranium. Recall that the core of the TRR contains about 32 kilograms of 19.75 percent enriched uranium. If the power is increased from 5 to 7 megawatts thermal, more uranium may be needed in the core, but it is clear that the low enriched uranium hexafluoride at Natanz contains much more than enough uranium to refuel the TRR.

In September 2009 the Iranians confirmed what American intelligence had already learned: they had a second and heretofore undeclared enrichment facility. This one was located near the sacred city of Qom. It is more than likely that the Iranians made this announcement when they did because they knew that the facility had been discovered. It had been under construction since at least 2006 and makes use of a tunnel complex on a military base. The details are somewhat sketchy, but this is what is presently believed. The facility is large enough to accommodate, in round numbers, 3,000 centrifuges. What makes this of special concern is that the Iranians announced their intention to install a new generation of centrifuge here. It will be recalled that the Iranians acquired prototype P1 centrifuges from A. Q. Khan’s network. But they also acquired the plans for its successor—the P2.

The rotor in the P2 was to be made of maraging steel—a special very strong steel made with a minimum of carbon. As originally designed, the rotor was to be in sections joined by “bellows”—flexible steel parts that are very difficult to construct. The Iranians were not able to buy them or construct them, so they went instead to carbon fiber. This had the great advantage that the peripheral speed of such a rotor is so great that it can be built in one section with no bellows and still produce a very high degree of separation. In fact the P2 produces some 5 SWU per year. This is very much less than what the centrifuges produce in the large commercial establishments, but it is still something like a factor of two better than the P1. We can now make an estimate of what 3,000 of these machines can do.

Evidently for a start they can produce 15,000 SWU per year. We may use a SWU calculator to see what in the way of uranium 235 we can produce with this in a year. If we assume that we begin with natural uranium hexafluoride with an enrichment of .7 percent and assume tails of .25 percent, then it requires 232 SWU to produce 1 kilogram of U-235, assuming we have enriched to 95 percent. Thus, in a year about 65 kilograms of uranium 235 could be produced at Qom. How many bombs this represents depends on the skill of the designer, but certainly more than one. That is why the facility at Qom is of concern. Some 20 percent enriched uranium fluoride has been produced there—as of August 2013 about 195 kilograms, all in the first-generation centrifuges.

At Natanz one could envision a different kind of breakout—something that is called “batch recycling.” For purposes of discussion let us assume that the approximately 1.5 tonnes of 3.5 percent enriched uranium hexafluoride have remained in Natanz. We want to exploit them to make highly enriched uranium. The way we can do this with the least restructuring of the centrifuge cascades is to take the product we have and with some rearrangement of the tubes send it back through the cascades. It turns out that the best way to do this is in steps. This is advantageous because SWU are not additive. The number of SWU it takes to enrich from A to C is not equal to the sum it takes to go from A to B and then B to C. In the case at hand, the sum of the parts is less than the whole. It pays to go in steps. The first step in this example is to enrich uranium hexafluoride from .035 to .26. I focus on uranium hexafluoride and not uranium because in this procedure we feed the successive stages with uranium hexafluoride and extract uranium at the end. This stage requires, with the same assumptions as before, 17 SWU per kilogram of produced 26 percent enriched uranium hexafluoride. Let us assume that when this occurs all 8,000 P1 centrifuges are operating and that each puts out 2 SWU per year, so a total of 308 SWU per week. Using our formula above, and assuming that we begin with 1,500 kilograms, we can find out how many kilograms of 26 percent uranium hexafluoride are produced:

P = 1500(.035 − .00225) / (.26 − .0025) ≈ 191 kilograms.

This requires, then, 17 × 191 SWU = 3,247 SWU. This can be produced in about ten and a half weeks.

In the next step we go from 26 percent enrichment to 71 percent enrichment. This takes only 9 SWU per kilogram and produces a product given by

P = 191(.26 − .00225) / (.71 − .00225) kilograms ≈ 70 kilograms.

This requires 17 × 70 SWU = 1,190 SWU, which takes a little less than four weeks to produce. To go from 71 percent to 96 percent enrichment takes 4 SWU and produces a product given by

P = 70(.71 − .00225) / (.96 − .00225) kilograms ≈ 52 kilograms.

This requires 17 × 52 SWU = 884 SWU, which takes a little less than three weeks to produce. Thus, in about 17 weeks of actual running, 52 kilograms of highly enriched uranium hexafluoride has been produced. We must also allow some time—days—for changing the arrangements of the cascade. The amount of uranium 235 that can be extracted is thus multiplied by the percentage of uranium 235, which is

(.95 × 235) / (.95 × 235 + .05 × 238 + 19 × 6) ≈ .64.

Thus, about 33 kilograms of uranium 235 are produced, enough for one bomb. If we had gone directly from 3.5 percent enriched uranium to 95 percent enriched uranium, it would require 81 SWU per kilogram. If we start with 1,500 kilograms, we end up with about 52 kilograms of highly enriched uranium hexafluoride at a cost of 4,121 SWU, which takes about thirteen weeks to produce.

These numbers are meant to be suggestive. They are based on limited information and some guesswork. For example, we do not know if the Iranians have been able to build 3,000 carbon-fiber centrifuges, because all of their centrifuge manufacturing facilities have been off-limits to inspections. But what the numbers do suggest is that if the Iranians ever throw off the international constraints, they could produce in not many months enough fissile material to begin to manufacture nuclear weapons. Any agreement that would enlarge this breakout time would be extremely helpful. In this respect one might note the malware attack that took place first in June and July 2009 and that was repeated the following spring. The infecting agent, identified as “Stuxnet,” is so sophisticated that one assumes that it was the work of governments. It does not take much imagination to point fingers. It seems to have gotten to Natanz via some equipment that came from Siemens in Germany. What the virus did was to cause the control system of the centrifuges to go berserk—alternately speeding them up past their tolerance levels and then slowing them down. The centrifuges self-destructed—in this case, about 1,000 IR-1 centrifuges were destroyed before the Iranians got hold of the situation. This set back their program only for a time, and it has not been repeated since 2010.

In the summer of 2009 there was a very troubled vote for the presidency of Iran. Mahmoud Ahmadinejad was reelected despite accusations that the vote was badly tainted. Thousands of people took to the streets. Many were arrested and tried. There were accusations of torture. Ahmadinejad’s opponent Mir-Hossein Mousavi was effectively silenced. But one thing must be clearly understood: all the political leaders in Iran are in agreement that in some form the nuclear program must go on. When he was prime minister in the 1980s, Mousavi was complicit in Iran’s dealings with A. Q. Khan, as was Akbar Hashemi Rafsanjani, the éminence grise of the Iranian reform movement who was at that time the president of Iran.

Over the years Rafsanjani has said some very disturbing things. For example, he said this at Friday prayers in December 2001: “If a day comes when the world of Islam is duly equipped with the arms Israel has in its possession, the strategy of colonization would face a stalemate because application of an atomic bomb would not leave anything in Israel but the same thing would just produce damages in the Muslim world.” The latter part of this statement persuades me that Rafsanjani has no conception of what a nuclear weapon is. By this time Israel must have several hundred, including quite possibly hydrogen bombs. A retaliatory raid by the Israelis would leave the Iranian cities in rubble. And here is something Rafsanjani said at Friday prayers on July 17, 2009:

Our country should be united against all the dangers that threaten us. They have now upped their ransom demands and are coming forward to take away our achievements in the fields of hi-tech and particularly nuclear technology. Of course, God will not give them the opportunity to do so, but they are greedy. My brothers and sisters, first of all, you all know me, I have never wanted to abuse this platform in favour of a particular faction and my remarks have always concerned issues beyond factionalism. I am talking in the same manner today. I am not interested in any factions. In my view, we should all think and find a way that will unite us to take our country forward and save ourselves from these dangerous and bad effects, and the emerging grudges. We should disappoint our enemies so that they would not covet our country.

This is from a “moderate.”

Excerpted from Nuclear Iran by Jeremy Bernstein. Published by Harvard University Press. Copyright © 2014 by the President and Fellows of Harvard College. Used by permission. All rights reserved.


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Jeremy Bernstein is the author of many books on science for the general reader, most recently A Palette of Particles.

Jeremy Bernstein is the author of many books on science for the general reader, most recently A Palette of Particles.