Tag Archives: atom bomb

How not to make an atom bomb

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There are many books on how the atom bomb was made. They are histories of the great men who succeeded at site Y, Los Alamos, usually with a sidelight of the economics and politics in the US at the time. It’s sometimes noted that there was an equally great German group working too, and one in Japan and in Russia, that they didn’t succeed, but it’s rarely discussed what they did wrong. Nor does anyone make clear why so many US scholars were needed. What did all those great US minds to do? The design seems sort-of obvious; it appears in the note Einstein sent to Roosevelt, so what were all these people thinking about all that time, and why did the Germans fail? By way of answer, let me follow the German approach to this problem, an approach that won’t get you anywhere, or anywhere that I’ve seen.

It seems that everyone knew that making a bomb was possible, that it would be fearsomely powerful, and that it would be made using a chain reaction in uranium or plutonium. Everyone seems to have understood that there must be a critical mass: use less and there is no explosion, use more and there is one. The trick was how to bring enough uranium together make the thing go off, and as a beginning to that, there is the concept of “a barn.” A barn is a very small unit of area = 10−24 cm², and a typical atom has a cross-section of a few barns. Despite this, it is generally thought to be very easy to hit an atom at the nucleus, that is, at the right spot, as easy as hitting the board side of a barn (hence the name). The cross section of a uranium atom is 600 barnes at room temperature, or 6×10−22 cm². But each cubic centimeter of uranium holds .5 x 1023 atoms. Based on this, it comes out that a thermal neutron that enters a 1 cm cube of uranium has a virtual certainty of hitting an atom — there are 3 cm² of atoms in a 1 cm² box. You could hardly miss.

Each uranium atom gives off a lot of energy when hit with a neutron, but neutrons are hard to come by, so a practical bomb would have to involve a seed neutron that hits a uranium atom and releases two or more neutrons along with energy. The next neutron has to hit another nucleus, and it has to releases two or more. As it happens uranium atoms, when hit release on average 2.5 neutrons, so building a bomb seems awfully easy.

But things get more difficult as the neutron speeds get greater, and as the atoms of uranium get hotter. The cross-section of the uranium atom goes down as the temperature goes up. What’s more the uranium atoms start to move apart fast. The net result is that the bomb can blow itself apart before most of the uranium atoms are split. At high speed, the cross -section of a uranium atom decreases to about 5 barnes you thus need a fairly large ball of uranium if you expect that each neutron will hit something. So how do you deal with this. For their first bomb, the American scientists made a 5 kg (about) sphere of plutonium, a man-made uranium substitute, and compressed it with explosives. The explosion had to be symmetrical and very fast. Deciding how fast, and if the design would work required a room full of human “computers”. The German scientists, instead made flat plates of uranium and slowed the neutrons down using heavy water. The heavy water slowed the neutrons, and thus, increased the effective size of the uranium atoms. Though this design seems reasonable, I’m happy to say, it can not ever work well; long before the majority of the reaction takes place, the neutrons get hot, and the uranium atoms fly apart, and you get only a small fraction of the promised bang for your bomb.

How fast do you need to go to get things right? Assume you want to fusion 4 kg of uranium, or 1 x 1025 atoms. In that case, hitting atoms has to be repeated some 83 times. In tech terms, that will take 83 shakes (83 shakes of a lamb’s tail, as it were). This requires getting the ball compressed in the time it takes for a high speed neutron to go 83 x 3 cm= 250 cm. That would seem to require 1 x 10-7 seconds, impossibly fast, but it turns out, you can go somewhat slower. How much slower? It depends, and thus the need for the computers. And how much power do you get? Gram for gram, uranium releases about 10 million times more energy than TNT, but costs hardly more. That’s a lot of bang for the buck.

Robert Buxbaum, Mar 29, 2020.

Recycle nuclear waste

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In a world obsessed with stopping global warming by reducing US carbon emissions, you’d think there would be a strong cry for nuclear power, one of the few reliable sources of large-scale power that does not discharge CO2. But nuclear power produces dangerous waste, and I have a suggestion: let’s recycle the waste so it’s less dangerous and so there is less of it. Used nuclear fuel rods, in particular. We burn perhaps 5% of the uranium, and produce a waste that is full of energy. Currently these, semi-used rods are stored in very expensive garbage dumps waiting for us to do something. Let’s recycle.

I’ve called nuclear power the elephant in the room for clean energy. Nuclear fuel produces about 25% of America’s electricity, providing reliable baseline generation along with polluting alternatives: coal and natural gas, and less-reliable renewables like solar and wind. Nuclear power does not emit CO2, and it’s available whether or not the sun shines or the wind blows. Nuclear power uses far less land area than solar or wind too, and it provides critical power for our navy aircraft carriers and submarines. Short of eliminating our navy, we will have to keep using nuclear.

Although there are very little nuclear waste per energy delivered, the waste that there is, is hard to manage. Used nuclear fuel rods in particular. For one thing, the used rods are hot, physically. They give off heat, and need to be cooled. At first they give off so much heat that the rods must be stored under water. But rod-heat decays fractally. After ten years or so, rods can be stored in naturally cooled concrete; it’s still a headache, but a smaller one The other problem with the waste rods is that they contain about 1.2% plutonium, a material that can be used for atomic bombs. A major reason that you can’d just dump the waste into the ocean or into a salt mine is the fear that someone will dig it up and extract the plutonium for an a- bomb. The extraction is easy compared to enriching uranium to bomb-grade, and the bombs work at least as well. Plutonium made this way was used for the bomb that destroyed Nagasaki.

The original plan for US nuclear power had been that we would extract the plutonium, and burn it up by recycling it to the nuclear reactor. We’d planned to burry the rest, as the rest is far less dangerous and far less, long-term radioactive. We actually did some plutonium recycling of this sort but in the 1970s a disgruntled worker named Silkwood stole plutonium and recycling was shut down in the US. After that, political paralysis set in and we’ve come to just let the waste sit in more-or-less guarded locations. There was a thought to burry everything in a guarded location (Yucca Mountain, Nevada) but the locals were opposed. So the waste sits waiting to leak out or be stolen. I’d like to return to recycling, but not necessarily of pure plutonium as we did before Silkwood: there is no guarantee that there won’t be other plutonium thieves.

Instead of removing the plutonium for recycling, I’d like to suggest that we remove about 40% of the uranium in the rod, and all of the “ash”, this is all of the lighter atom elements created from the split uranium atoms. This ash is about 5% of the total. The resultant rods would have about 2% plutonium, 97.5% enriched uranium (about 1% enriched at this stage) plus about 0.5% higher transuranics. This composition would be a far less dangerous than purified plutonium. It would be less hot and it would not be possible to use it directly for atom bombs. It would still be fissionable, though, at the same energy content as fresh rods.

There is an uncommonly large amount of power available in nuclear fuel

Several countries recycle by removing the ash. Because no uranium is removed, the material they get has about half the usable life of a fresh rod. After one recycle, there is not much more they could do. If we remove uranium material is a lot more easily used, and more easily recycled again. If we keep removing ash and uranium, we could get many, many recycles. The result is a lot less uranium mining, and more power per rod, and fewer rods to store under guard.

The plutonium of multiply recycled rods is also less-usable for fission bombs. With each recycle, the rods build up a non-fisionabl isotope of plutonium: Pu 240. This isotope is not readily separated from the fissionable isotope, Pu 239, making multiply used rods relatively useless for fission bombs.

Among the countries that do some nuclear waste recycling are Canada, France, Russia, China, and Germany. Not a bad assortment. I would be happy to see us join them.

Robert Buxbaum September 9, 2019

Disease, atom bombs, and R-naught

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A key indicator of the speed and likelihood of a major disease outbreak is the number of people that each infected person is likely to infect. This infection number is called R-naught, or Ro; it is shown in the table below for several major plague diseases.

R-naught - communicability for several contagious diseases, CDC.

R-naught – infect-ability for several contagious diseases, CDC.

Of the diseases shown, measles is the most communicable, with an Ro of 12 to 18. In an unvaccinated population, one measles-infected person will infect 12- 18 others: his/her whole family and/ or most of his/her friends. After two weeks or so of incubation, each of the newly infected will infect another 12-18. Traveling this way, measles wiped out swaths of the American Indian population in just a few months. It was one of the major plagues that made America white.

While Measles is virtually gone today, Ebola, SARS, HIV, and Leprosy remain. They are far less communicable, and far less deadly, but there is no vaccine. Because they have a low Ro, outbreaks of these diseases move only slowly through a population with outbreaks that can last for years or decades.

To estimate of the total number of people infected, you can use R-naught and the incubation-transmission time as follows:

Ni = Row/wt

where Ni is the total number of people infected at any time after the initial outbreak, w is the number of weeks since the outbreak began, and wt is the average infection to transmission time in weeks.

For measles, wt is approximately 2 weeks. In the days before vaccine, Ro was about 15, as on the table, and

Ni = 15w/2.

In 2 weeks, there will be 15 measles infected people, in 4 weeks there will be 152, or 225, and in 6 generations, or 12 weeks, you’d expect to have 11.39 million. This is a real plague. The spread of measles would slow somewhat after a few weeks, as the infected more and more run into folks who are already infected or already immune. But even when the measles slowed, it still infected quite a lot faster than HIV, Leprosy, or SARS (SARS is a form of Influenza). Leprosy is particularly slow, having a low R-naught, and an infection-transmission time of about 20 years (10 years without symptoms!).

In America, more or less everyone is vaccinated for measles. Measles vaccine works, even if the benefits are oversold, mainly by reducing the effective value of Ro. The measles vaccine is claimed to be 93% effective, suggesting that only 7% of the people that an infected person meets are not immune. If the original value of Ro is 15, as above, the effect of immunization is to reduce the value Ro in the US today to effectively 15 x 0.07 = 1.05. We can still  have measles outbreaks, but only on a small-scale, with slow-moving outbreaks going through pockets of the less-immunized. The average measles-infected person will infect only one other person, if that. The expectation is that an outbreak will be captured by the CDC before it can do much harm.

Short of a vaccine, the best we can do to stop droplet-spread diseases, like SARS, Leprosy, or Ebola is by way of a face mask. Those are worn in Hong Kong and Singapore, but have yet to become acceptable in the USA. It is a low-tech way to reduce Ro to a value below 1.0, — if R-naught is below 1.0, the disease dies out on its own. With HIV, the main way the spread was stopped was by condoms — the same, low tech solution, applied to sexually transmitted disease.

Image from VCE Physics, https://sites.google.com/site/coyleysvcephysics/home/unit-2/optional-studies/26-how-do-fusion-and-fission-compare-as-viable-nuclear-energy-power-sources/fission-and-fusion---lesson-2/chain-reactions-with-dominoes

Progress of an Atom bomb going off. Image from VCE Physics, visit here

As it happens, the explosion of an atom bomb follows the same path as the spread of disease. One neutron appears out of somewhere, and splits a uranium or plutonium atom. Each atom produces two or three more neutrons, so that we might think that R-naught = 2.5, approximately. For a bomb, Ro is found to be a bit lower because we are only interested in fast-released neutrons, and because some neutrons are lost. For a well-designed bomb, it’s OK to say that Ro is about 2.

The progress of a bomb going off will follow the same math as above:

Nn = Rot/nt

where Nn is the total number of neutrons at any time, t is the average number of nanoseconds since the first neutron hit, and nt is the transmission time — the time it takes between when a neuron is given off and absorbed, in nanoseconds.

Assuming an average neutron speed of 13 million m/s, and an average travel distance for neutrons of about 0.1 m, the time between interactions comes out to about 8 billionths of a second — 8 ns. From this, we find the number of neutrons is:

Nn = 2t/8, where t is time measured in nanoseconds (billionths of a second). Since 1 kg of uranium contains about 2 x 1024 atoms, a well-designed A-bomb that contains 1 kg, should take about 83 generations (283 = 1024). If each generation is 8 ns, as above, the explosion should take about 0.664 milliseconds to consume 100% of the fuel. The fission power of each Uranium atom is about 210 MeV, suggesting that this 1 kg bomb could release 16 billion Kcal, or as much explosive energy as 16 kTons of TNT, about the explosive power of the Nagasaki bomb (There are about 38 x10-24 Kcal/eV).

As with disease, this calculation is a bit misleading about the ease of designing a working atomic bomb. Ro starts to get lower after a significant faction of the atoms are split. The atoms begin to move away from each other, and some of the atoms become immune. Once split, the daughter nuclei continue to absorb neutrons without giving off either neutrons or energy. The net result is that an increased fraction of neutrons that are lost to space, and the explosion dies off long before the full power is released.

Computers are very helpful in the analysis of bombs and plagues, as are smart people. The Manhattan project scientists got it right on the first try. They had only rudimentary computers but lots of smart people. Even so, they seem to have gotten an efficiency of about 15%. The North Koreans, with better computers and fewer smart people took 5 tries to reach this level of competence (analyzed here). They are now in the process of developing germ-warfare — directed plagues. As a warning to them, just as it’s very hard to get things right with A-bombs, it’s very hard to get it right with disease; people might start wearing masks, or drinking bottled water, or the CDC could develop a vaccine. The danger, if you get it wrong is the same as with atom bombs: the US will not take this sort of attack lying down.

Robert Buxbaum, January 18, 2019. One of my favorite authors, Issac Asimov, died of AIDS; a slow-moving plague that he contacted from a transfusion. I benefitted vastly from Isaac Asimov’s science and science fiction, but he wrote on virtually every topic. My aim is essays that are sort-of like his, but more mathematical.