5. Chain reactions, nuclear reactors, and atomic bombs
Nuclear explosions, cancer, the population bomb, lightning, the spread of viruses (both biological and computer), and
snow and rock avalanches all have something in common: they are based on the principle
of the chain reaction. There are other
According to legend, the game of chess was invented by Grand Vizier Sissa Ben Dahir, who presented it to King Shirham of
"One grain of wheat, representing the first square of a chessboard. Two grains for the second square. Four grains for the next. Then eight, sixteen, thirty two … doubling for each successive square until the 64th and last square is counted."
The king was impressed with the apparent modesty of the request, and he immediately granted it. He took his chess board, removed the pieces, and asked for a bag of wheat to be brought in. But, to his surprise, by the bag was emptied by the 20th square. The king had another bag brought in, but then realized that the entire second bag was needed for just the next square. In fact, in 20 more squares, he would need as many bags as there were grains of wheat in the first bag! And that was only up to square 40. Legend does not record what the king then did to his Grand Vizier.
The number of grains for the last square can be calculated by multiplying 2 times itself 63 times. That doesn't take very long on a pocket calculator. But it is faster if you use the power law key (if you have a scientific calculator), marked yx. Two, multiplied by itself 63 times is 263. So put in 63 as x, and 2 as y. The answer is 9223372036854775808 ≈ 0.922 x1019 ≈ 1019. If you include the grains on the first 63 squares, the sum is about twice as large, 2 x1019 = 264. The calculation can also be done easily on a spreadsheet.
Suggested Figure: illustration of the chessboard (first 12 squares filled) with 1,2,4,8,16,32,64 (=8x8),128 (=8x8x2), 256(=8x8x4),512(=8x8x8),1024 (=8x8x16), 2048 (=8x8x32), 4096 (=8x8x64)
If all the grains for the last square were stacked into a cube, the cube would have about 2 million grains on each edge. If each grain were 1 mm in size, then each edge of the giant cube would be 2,000,000 mm = 2,000 meters ≈ 2 km. The cube would be 2 km long, 2 km wide, and 2 km high. It would take a very large chessboard to hold this cube. If the grains had to fit on a square that was 2 cm x 2 cm, its height would be 20 billion km. At that height, it would take almost a day for light from the top of the pile to reach the bottom. This amount of wheat on the board would exceed the total world production for over a thousand years.
The amazing feature of this problem is that with just 63 steps, each one quite modest (you are only doubling) you get a really huge number. This type of rapid growth is called exponential growth. It is called that because you raise a number (in this case, 2) to a power (63) called the exponent. Exponential growth is the secret behind all the phenomena we discuss in this section.
We had something very similar phenomenon in the chapter on radioactivity, except there it worked backwards. Each half life, the number of remaining atoms was cut in half. That is the opposite of exponential growth, and is called exponential decay.
When a neutron hits a U-235 nucleus, it has a high probability of triggering the breakup of that nucleus into two large pieces, a process called "fission" (named in analogy with the fission of a biological cell). The two large pieces are called "fission fragments" (we discussed these in Chapter 4), and they are arguably the most dangerous kind of radiation that comes from nuclear bombs. In addition to the two large fragments, when U-235 fissions there are and additional two neutrons usually released. These two neutrons make the chain reaction possible. If there are other U-235 nuclei around, then these neutrons might hit them, and cause additional fissions. That doubling process can continue until very large numbers of nuclei are split.
In the first fission (the first "generation"), one atom is split into two. In the second generation, two are split -- then four, eight, etc. In the 64th generation, the number of atoms fissioned will be 1019 -- the same number as we found for the chess problem. The total number of atoms fissioned (including the ones in prior generations) is about twice this: 2 x1019.
How many atoms are there in, say, 10 kg of uranium? (The International Atomic Energy Agency calls that a "significant amount" – an amount that could be used in a nuclear weapon.) The answer is 2.6 x1025, much larger than the number we fissioned in 64 generation. How many generations will it take to reach this number? To find the answer, you could keep on multiplying 2s until you get there. It won't take that long. In an additional 20 generations (84 total), you'll reach 2x1025. In other words, after 84 successive doublings, every atom in 10 kg of uranium will be fissioned. How much energy is released? Each fission of a uranium nucleus releases about 30 million times as much as a molecule of TNT. So the 10 kg will be equivalent to about 30 million x 10 kg of TNT = 300 million kg = 300 kilotons of TNT. That is the fundamental idea behind an atomic bomb. (Many people prefer to use the term "nuclear bomb" today, since it is the nucleus of the atoms that fissions, and it is the nuclear energy that is released.) The first nuclear bomb released the energy of about 20 kilotons tons of TNT, less than what we just calculated, and that shows that not every atom was fissioned before the bomb blew itself apart.
Neutron-induced fission also takes place with plutonium-239, abbreviated Pu-239. But in plutonium, typically 3 neutrons are released per stage. How many states does it take to reach 2x1025? To find the answer, multiply 3 by itself until you get this number. The answer (so you can check yourself) is in the footnote. We'll talk more about nuclear chain reactions at the end of this.
We used a lot of numbers in this section, so let me summarize which are the important ones: 10 kg of U-235 is enough for a nuclear weapon. If the chain reaction proceeded by doubling, it would all be split in only 84 generations. Plutonium takes fewer generations since each fission releases 3 neutrons.
The Fetus: a chain reaction in the womb
You began life as a single cell, the result of the fusion of your father's sperm and your mother's egg. That cell then divided, in a process called fission. Then the two resulting pairs divided again. How many times did the cells have to divide to make up a complete human body? There are about 1011 cells in a human. As you can check, 1011 = 237, so the answer is 37 stages of doubling. Even if each division stage took a day, the whole process would be over in 37 days.
So why does it take 9 months to be born? The answer is that the cells can't keep dividing that rapidly. They have to grow in between divisions, and that takes nutrients. Soon after the process begins, the growth is limited by the body's ability to bring nutrition to the dividing cells. If you did keep doubling in size with each successive generation, then the baby would double its weight in the last day. My wife claims that is exactly what being pregnant felt like.
Cancer: an unwanted chain reaction
When a baby reaches full size, the body "turns off" the chain reaction that is responsible for growth. It appears to have several mechanisms to do this. Many cells maintain the ability to turn the chain reaction back on, if it is needed. For example, if you are wounded or your skin is cut, then the local cells start reproducing again. They can fill the wound with remarkable speed, since they follow the doubling rule of the chain reaction.
The potential danger of unrestricted splitting is so great, however, that our cells have several mechanisms to prevent this from happening when it isn’t needed. If all else fails, the cell can be signaled to kill itself, a process known as apoptosis.
If your cells are unlucky enough to receive several specific mutations, they lose this ability to commit suicide. If that happens, then the cells can grow unregulated, continuing to double and double and double. When that happens, we say that the cells have become a cancer. If the cells stay in one region of the body, they eventually may limited by their ability to get nutrients, and the collection of cells is a benign cancer. However, if the cells are the type that can break off and drift in the blood or other body fluids to a different part of the body, then the cancer is termed malignant. Cancers that can spread in this way reach areas where nutrition is abundant, and they continue to grown in an unlimited chain reaction. Eventually their growth can interfere with vital body functions, and the victim dies.
The reason that cancer can be so devastating, and that people can die from it so quickly, is because it takes advantage of the chain reaction to grow with great rapidity.
The Population Bomb
In 1798, Thomas Rohr Malthus wrote his 'Essay on the Principle of Population'. It is hard to find a more influential essay in the history of public thought. In this (you can read his original essay, if you like) he argued that population growth obeyed the rules of a chain reaction. If each parent has, on the average, 2 children (per parent), then there will be a doubling every generation. If each parent has, on average, 1.4 children, then there will be a doubling every two generations (since 1.4 x 1.4 ≈ 2). The doubling would take longer, but it would still lead to a chain reaction.
Malthus argued that the available food supply will not show a similar exponential growth. It will grow much more slowly, because it is limited by available land, water, and other resources. As a result, the growth of population will always outrun the growth of food. Malthus thought that the only thing that would stop this population bomb was disease and famine. Based on this observation, others argued that famine was not only inevitable, but it served an important purpose. That was a very bleak outlook on life. Malthus' analysis has been so influential that some call economics "The Dismal Science" based on this pessimistic outlook.
Many people today still think this population bomb is the ultimate disaster working on the human population. But there is an alternative to starvation: birth control. It is interesting to note that Malthus thought the bomb was imminent in 1798. We have managed quite a few doublings since then, and we are still well fed. In 1968 Paul Erlich wrote "The Population Bomb" in which he predicted that massive starvation would hit the world in the 1970s. Now he maintains it will happen in the early 2000s.
But there is reason to be more optimistic. According to recent studies, world population is departing from exponential growth. (If you wish, you can read the paper in Nature by Lutz et al., but that isn't required.) The United Nations estimates made in 2003 are that the world population growth will slow, and the total population will not exceed 10 or 12 billion before it starts to decline. The reason for the slowing growth is not understood, but it is possible that humans tend to have fewer children when they become well-fed and secure. If this is true, then the secret to the end of the population bomb would then be a truly delightful one: make everyone in the world wealthy. To avoid polluting the world (when everyone can afford SUVs), it may be necessary to increase energy conservation to match the wealth. This appears to be possible. (If you are interested, see my essay The Conservation Bomb.)
Mass extinction recovery
Sixty five million years ago, the dinosaurs were destroyed, yet the mammals survived. Why? Well, the extinctions were not as simple as many people think. It wasn’t true that all the mammals survived. In fact, it was likely that that 99.99% of all mammals were killed at the same time. For recovery, all it takes is for one or more “breeding pairs” to survive. Imagine, for example, that two rats made it through the bad period. Assume that rats take about a year for rats to grow and to breed. After the extinctions, the number of rats can double every year. After just 56 years (an instant in geologic time) there would be so many rats that they would completely cover the surface of the Earth like a carpet. 
Of course, that kind of massive rat bomb never happened. The growth of rats was limited by the availability of food, by disease, and by competition with other animals. But the example shows that it is difficult to spot great catastrophes in geologic history unless the extinctions are so great that they actually eliminate entire species, i.e. they leave no breeding pairs. The rats recovered; the dinosaurs didn’t. It turns out that 65 million years ago all large animals disappeared. That may be because such animals are rare, and require lots of territory to stay alive. When 99.99% are killed, they are less likely to be able to find mates than are the little guys.
A population explosion sometimes occurs when a new species is
introduced into an environment in which there are no natural predators. Twenty four rabbits were released in
Nobody knows what led to the plague of rats that gave rise to the story of the Pied Piper of Hamelin. If you aren’t familiar with Robert Browning’s poem, I recommend it highly. It is online at http://www.indiana.edu/~librcsd/etext/piper/text.html.
the Polymerase Chain Reaction (PCR)
In every cell of your body, there is a collection of molecules called DNA that contain the information that runs your body. For different people, the molecules are virtually identical. The DNA contains the "genetic code" that tells the cells how to reproduce, how to respire, how to function. But there is a tiny component that is different for different individuals, such as the parts that determine eye color. Half of your DNA came from your father and half from your mother, so your DNA is very similar to theirs, but not identical. (If it were identical, you would be either an "identical twin" or a "clone".)
DNA fingerprinting consists of looking at the parts of the DNA that vary among different people. If you pick enough of these regions, you have a unique identification. Moreover, these regions will consist of parts inherited from your parents, so the fact that you are close biological relatives will be indicated by the presence of some segments that are identical to those of your father. mixed with other segments that are identical to those of your mother.
The difficulty in DNA fingerprinting is that methods to "read" the code, i.e. determine the exact sequence of molecules in the relevant regions of the DNA, do not work with just a few molecules; they requires many billions of copies.
That's where the chain reaction comes
it. DNA is a molecule that is designed
to reproduce; it does that before the cell is split so that replicas can go in
each half. Kary
Mullis realized that if he had even one DNA molecule, he could use the chain reaction to get billions of copies. The procedure involves using chemicals that will trigger duplication of the segment of the DNA molecule that different for different people. The stages in the chain reaction are achieved by temperature cycling of the fluid containing the DNA. When the temperature is cool, the desired part of the DNA makes what is called a "complementary strand" which stays attached to the original DNA. The mixture is then heated (to near boiling) and the two strands separate. When cooled, both the original DNA and the complementary strand duplicate; these are separated by heating, and the cycle is repeated. After 32 cycles (taking less than an hour) there will be 232 = 4.3x109 copies. This gives the scientist enough material that he can determine the exact genetic code of the fragments.
DNA fingerprinting is the method used to identify people
based only a few cells from their body. The method was used to identify the
remains of people at the
Illness and epidemics –
chain reactions of viruses and bacteria
A virus or bacteria duplicating in your body uses the chain reaction to reach enormous numbers. If your body has to devote major parts of its resources to killing the germ, then you feel sick. If it can't succeed in stopping the exponential growth, you die.
The math of a chain reaction also describes the spread of an epidemic. Consider a single person infected by the smallpox virus. That person can spread it to someone else by contact or by saliva droplets spread from the breath. If one person infects two people, and they infect and additional two people, and that pattern continues, then it takes only 33 such stages to infect the entire world (since 233 = 8.6 billion > world population). Worse, suppose the first person infects 10 people, and they each infect 10 others. Then, after only 10 such stages, the number infected will be 10x10x10x10x10x10x10x10x10x10 = 1010, and that is greater than the entire world population. Such spread, in the past, has been limited by the fact that people didn't travel widely, and so an epidemic could be localized. But today, with airplane travel, an infected person could infect thousands.
Note that not all illnesses are chain reactions. When humans are infected with anthrax (as happened during the terrorist attacks in 2001), the disease was not spread from one person to another. Those infected become ill, and some died, but the disease did not spread like a chain reaction.
Computer viruses – an electronic chain reaction.
Computer viruses obey the same laws as other chain reactions. A virus in your computer system can spread to other systems by the copying or sharing of infected programs. Email can spread like a chain reaction if it allows a message to be automatically forwarded, for example, to everyone on your mailing list, or if it contains an attached program that fools you into opening it.
Such computer viruses can spread with remarkable speed, in part, because the multiplication number at each stage can be large. If, for example, an infected computer infects 100 others, then the virus could spread around the work in fewer than 4 stages (since 1004 = 108, and that is greater than the total number of computers in the world), at least in principle. Of course, the infection would not spread to people who were on nobody's email list, or who run anti-virus programs that intercept and “kill” the infection.
a rock or snow chain reaction
A stone falls from a ledge, and knocks out two more stones. Each of them knocks out two, and so on. This is an avalanche. The doubling rule applies.
If each rock knocks out less than one additional rock, then the avalanche will fade away and die. Suppose, for example, that each rock knocks out, on the average, 0.5 additional rocks. Then after 4 stages, the number being knocked out is the number that started times (0.5)4 = 1/16. This typically happens when the avalanche reaches a part of the slope that is no longer steep, so the rocks are more strongly planted in the ground and harder to knock loose.
A snow avalanche can be similar, although the snow doesn't usually exist in discrete objects like rocks. And, like a rock avalanche, it will die when the slope is no longer steep.
a chain reaction of electrons
Compound interest – seen as a chain reaction
Compound interest refers to the fact that you can earn interest on your interest. If you invest money at an annual rate of, say, 5%, then that means that after a year, you have 1.05 times as much as you started with. After two years, the amount is (1.05)x(1.05) = (1.05)2 times greater, and after 14 years, the amount is (1.05)14 ≈ 2 times the original amount. Your money will continue to doubles every additional 14 years. After 28 years it will have grown by a factor of 4 times, and after three doublings (42 years) it will have grown by a factor of 8.
Compound interest is a form of a chain reaction. The doubling creates two amounts each equal to the original, and each of these will continue to double. That's why the math is the same.
Suppose you start with $1000, and would like to become a billionaire. All it takes is a factor of a million ≈ 220. From this math, you can see that a factor of a million takes 20 doublings. At 14 years per doubling period, it would take 280 years, and your billion dollars would be worth a lot only if there was negligible inflation. To really become a self-made billionaire, you have to have a doubling time of no more than one or two years.
The same doubling rule that we see in chain reactions occurs in other phenomena. One of the most famous ones is in computer technology. In 1965, Gordon Moore, one of the founders of the integrated circuit industry, noticed that the number of basic components that could be put on a chip had doubled every year for the previous six years. From what he knew of the technology, he expected the trend to continue, at least until 1975. By that time, instead of 50 components per chip, he predicted there would be 65,000! Below is the plot from his original paper (which was once available in pdf format at http://www.intel.com/research/silicon/moorespaper.pdf).
The Nobel Prize in Physics in 2000 was given to Jack Kilby, who along with Noyce, invented the integrated circuit that allowed this
expansion to take place. But we don't really understand the reason for
Folding paper and
A particularly easy way to study the doubling rule is by folding paper. Suppose you take a sheet and fold it in half. There are then two layers. Fold again for 4 layers, and again for 8 layers. After 7 folds, there will be 27 = 128 layers. An old trick is to challenge someone to fold paper 8 times. It turns out to be essentially impossible to do, because the thickness is so great. Take a newspaper sheet and try it.
An ancient but secret method for increasing the strength of metal used in swords was to fold the metal, hammer it back to the original thickness, and then repeat. This led to great strength, probably because the microscope separation of the metal into thin layers prevented cracks from propagating through the material. The master sword makers would not do this indefinitely, but only about 20 times. What would the thickness of an individual layer be after the metal had been folded and beaten in this way? There would be 220 = 106 layers at that time. If the sword thickness was 1 mm = 0.1 cm, then the thickness of an individual layer would be a million times thinner, about 10-7 cm. That is only slightly larger than the size of an atom! It is possible that the reason that stopped folding and doubling, is that you can't make layers any thinner, so additional folding didn’t create new layers, and therefore didn’t strengthen the sword. Even though the ancients didn't know the size of atoms, they may have inadvertently determined the size in this indirect way, if only they knew how to interpret their limit. (Maybe some of them did. People who work in secret sometimes make discoveries, but if they don’t pass them on publicly, they may get no credit.)
Here is one last example illustrating how the doubling rule can lead, in a small number of steps, to a large number of objects. Suppose a tree has a trunk that divides into three large branches, and each of these divides into three more branches. Suppose the branches continue to divide another 6 times, and then you arrive at three leaves. How many leaves are there on the tree? Do you suppose that Nature uses a trick like this to simplify the code required in the design of trees? Suppose, in addition to the doubling rule, it put in a random process. So, for example, the probability of two branches might be 50%, the probability of branching into three new branches might be 30%, and the probability of dividing into four is 20%. That would make for a more interesting tree. Take a look at actual trees and see what you think.
Next we’ll combine our facts on radioactivity with the math of doubling to discuss the most famous of the chain reactions.
Nuclear Weapons Basics
As soon as it was discovered that a neutron-induced fission
creates more neutrons, it was clear that there was a potential method for
releasing enormous nuclear energy. The
concept of the nuclear chain reaction had actually been patented in
As discussed earlier in this chapter, the chain reaction makes used of the fact that more than one neutron comes out per uranium fission. If those neutrons can be made to hit other uranium nuclei, then soon the doubling rule will result in the fission of nearly every nucleus. It takes only 80 doublings. The key to achieving this is the concept of critical mass.
If the uranium chain reaction is to work, there must be enough material so that the emitted neutron hits another uranium nucleus, instead of escaping in between the nuclei and out of the bomb. If enough uranium surrounds the initial fission that the neutrons will not escape, then we say that we have a "critical mass" of uranium. For many years, the value of this critical mass was highly classified. This was because many people thought it was larger than it turned out to be. The critical mass for a bomb based on uranium fission is different than that for plutonium fission. Part of this is due to the fact that more neutrons are emitted when plutonium fissions.
To make a critical mass, there must be enough material so
that after each fission, more than one of the neutrons
that is emitted must hit another nucleus, to keep the chain reaction
going. A simple calculation indicates
that this requires a sphere of uranium 13.5 cm in radius, weighing 200 kg = 440
was no hope during World War II that so much U-235 could be obtained, and that
may be the reason why the Germans (under the direction of the famous physicist
Werner Heisenberg) abandoned the effort.
The term "critical mass" has worked its way from physics into our everyday language as a metaphor. One or two people, working on a problem, may not be enough. But if you assemble a "critical mass" of people, the progress can be explosive.
The nuclear bomb that destroyed
Plutonium bomb are more difficult (see next section). For that reason, a bomb that uses uranium is the material of choice for a terrorist, since the design is so simple. But such a bomb requires highly enriched U-235, and that is not easy to make. When you dig uranium from the ground, it is 99.3% U-238, and only 0.7% U-235. It is only the rare isotope U-235 that can be used for a bomb. Separating this isotope is extremely difficult to do.
Prior to the
The bomb tested at
The bomb dropped on
The plutonium is often arranged as a hollow shell, with explosives on the outside. The explosives drive the shell into a little blob, and compress it (even though it is solid). The compression pushes the atoms close enough together that neutrons produced in the chain reaction are unlikely to be able to leak in between them. Thus compressed plutonium has a smaller critical mass than uncompressed plutonium.
The explosives often use a special kind of explosive "lens" (a special shape in the explosive that tends to make the explosion converge on a point).
According to the chief nuclear weapons designer of Saddam
Thermonuclear weapon or "Hydrogen Bomb"
In the hydrogen bomb, deuterium and tritium are heated by a plutonium or uranium fission bomb, to the point where they overcome their natural repulsion (the nuclei of both are positively charged) and fuse. This releases energy, and neutrons. The high energy neutrons cause fission in a uranium case (usually just U-238), and that releases even more energy. The biggest hydrogen bomb ever tested (they have never been used in war) released an energy over 50 million tons of TNT. That is million, not thousand!
The "secret" of the hydrogen bomb, kept highly classified until just a few years ago, is that the plutonium bomb emits enough x-rays that they can be used, after bouncing off the uranium cases, to compress and ignite the tritium/deuterium combination. There is a second secret, although this has been public for a longer period. Instead of using tritium, the bomb can contain a stable (not radioactive) isotope of lithium called Li-6. This is a solid, which means that the material is stored at high density. The neutrons from the fission weapon break up the Li-6 to make the tritium. Thus the fuel is created in the same microsecond that the bomb is exploding. The fusion fuel is usually lithium combined with deuterium, called lithium deuteride.
Boosted fission weapon
You can increase the energy of a fission bomb by added a small container with tritium/deuterium gas. In the heat of the explosion, the tritium and deuterium fuse, releasing more energy and more neutrons. The additional neutrons mean that the fission bomb chain reaction becomes more complete, and that increases the yield of the bomb. A boosted fission weapon uses fusion, but it uses the fusion for neutrons to split plutonium, rather than for energy production, so it usually isn’t consider a fusion bomb.
Terminology: Atomic Bombs, Hydrogen Bombs, and all that
A bomb that uses the energy of the nucleus to release energy
can safely be called a “nuclear bomb.”
Some prominent people (starting with President Eisenhower, and lastly
President George W. Bush) pronounced nuclear as “nukular”
– but that is generally considered incorrect.
President Harry Truman referred to the bombs dropped over
The bomb based on fusion of hydrogen is often referred to as a “hydrogen bomb.” A name typically used by scientists is “thermonuclear bomb.” The word thermonuclear refers to the fact that the fusion takes place because of the high temperature (that’s the thermo part).
Much of the danger from nuclear weapons comes from the nuclear fallout. This consists of the fission fragments from the uranium or plutonium in the bomb. Fallout is particularly bad if the bomb is exploded near the ground. (The
A large fraction (over 5%) of the fission fragments are the isotope strontium-90, a highly radioactive material that has a half-life of 29 years, that gets into the food supply. Back in the 1950s, when many people were worried about the long term effects of nuclear testing, the term strontium-90 was well known by the general public.
A nuclear reactor is a device in which a "sustained" chain reaction takes place. It doesn't involve doubling; instead, from each fission, only one of the emitted neutrons (on average) hits another nucleus to cause another fission. It is as if every man and woman had, on the average, two children. Then the population would not grow. The power output from a sustained nuclear reaction doesn’t grow, but is constant.
The power comes out in the form of heat, just as it does when burning coal or gasoline. Frequently the heat is used to boil water into steam. This steam is then used to run a turbine. (A turbine is really just a fan; as the steam expands through it, it makes the fan turn.) Imagine this: a super high-tech nuclear submarine really just uses uranium to boil water!
For their fuel, commercial nuclear reactors use primarily U-235, just as in a nuclear bomb. But the uranium is not enriched to bomb quality. Recall that natural uranium has only 0.7% U-235; the rest is U-238. For use in a bomb, the U-235 has to be enriched to about 80%. But for a nuclear reactor, it only has to be enriched to about 3%. (An exception is the Canadian reactor, called "Candu". We'll discuss this in a moment.)
Why can a reactor use less enriched fuel? There are two reasons: the first is that they don't require that both neutrons hit U-235; only one. So if one of the two neutrons is absorbed, that's ok -- for the reactor, not the bomb. Having a lot of U-238 around isn't so bad.
But there is a more important reason: a nuclear reactor uses a "moderator." A moderator is a chemical mixed in with the fuel that tends to slow down the neutrons without absorbing them. The most popular moderators are ordinary water, H20, heavy water: D2O (the D stands for deuterium, which you recall is a hydrogen atom that has both a neutron and a proton in the nucleus), and graphite (which is nearly pure carbon). The moderators consist of nuclei which are light and don't absorb neutrons. The neutrons hit the moderator, and bounce off, but in the process they lose a little energy. After enough such bounces, the neutrons are no faster than expected from their temperature. They are called "thermal neutrons" to reflect the fact that they have slowed down to thermal velocities.
In the commercial nuclear reactor, the fast neutrons emitted in fission bounce off the moderator and become thermal (slow) neutrons. These neutrons are more readily absorbed on other U-235 nuclei, so the enrichment (concentration) of the U-235 need not be 80%, but only 3%.
Can a reactor turn into an atomic bomb?
In the atomic bomb, they had to use fast neutrons (not moderated) in order to have the entire 80 generations over with before the bomb blew itself apart. After 80 generations, the temperature was many millions of degrees. The only reason is hasn't yet blown apart is that there wasn't enough time! With moderated neutrons, the chain reaction is much slower, since the neutrons are slower.
This is an important fact: commercial nuclear reactors depend on using slow neutrons. The reason this is important, is that if the nuclear reactor begins to “run away”, i.e. if the operator makes a mistake and the chain reaction begins to grow exponentially (doubling), then the slowness of the neutrons limits the size of the explosion. Once the temperature rises to a few thousand degrees K, the atoms are moving faster than the neutrons, and so the neutrons can’t hit them; the chain reaction stops. The energy released will blow up the reactor, but that energy will be about the same that you would get from TNT. It's an explosion, but it is a million times smaller than an nuclear bomb.
A chain reaction that depends on slow neutrons cannot give rise to a nuclear explosion. For that reason, a commercial nuclear reactor cannot blow up like a nuclear bomb. It is important to know this, and to be able to explain the logic to the public, since this fact is not widely known.
There are real dangers from nuclear reactors (see the China Syndrome below). Blowing up like a nuclear bomb is not one of them.
Why are slow neutrons more likely to be absorbed on U-235? The physical reason is simple: a slowly moving neutron feels the nuclear force for a longer time, and is more readily deflected towards the U-235 nucleus.
Of course, the slow neutrons are also more strongly attracted to U-238. But the effect turns out to be much stronger for U-235. So having slow neutrons means you can use 3% enriched U-235 instead of 80%.
In a nuclear reactor, only one of the neutrons from uranium fission is used to produce another fission. The other is absorbed, possibly by control rods (to prevent a growing chain reaction) or possibly by U-238. When U-238 absorbs a neutron, it becomes U-239. This is radioactive, and decays (it emits an electron and neutrino, and has a half-life of about 23 minutes) to an isotope of neptunium: Np-239. This isotope of neptunium is also radioactive. It emits an electron and a neutrino, with a half-life of 2.3 days, to turn into the very famous isotope of plutonium, the one that can be used for a nuclear weapons, Pu-239.
That is how we manufacture plutonium. We make it from U-238 by hitting it with neutrons in a nuclear reactor. The plutonium is a different chemical element from uranium, so when the fuel is removed, the plutonium can be chemically separated. That is not hard to do. The extraction of plutonium is called “uranium reprocessing.” When we give nuclear power plants to underdeveloped countries, we do not allow them to do their own reprocessing, for fear that they would get a supply of plutonium in this way. Of course, we do give them nuclear fuel to run the reactors – but that is a mixture of U-235 and U-238, with too large of a fraction of U-238 for it to work as a bomb.
The Pu-239 is usually not considered nuclear waste, because it can be used itself to run a nuclear reactor. It is nuclear fuel. Moreover, if you put it in a nuclear reactor, you get three neutrons per fission instead of two. In a reactor, operating at constant (not exponentially growing) power, you want only one neutron per fission to produce another fission. What do you do with the extra two neutrons? Answer: put U-238 in the reactor, and make more plutonium.
Thus a reactor can make (out of U-238) more Pu-239 fuel than it consumes! Such a reactor is called a “breeder” reactor. It has the potential of turning all uranium, not just 0.7% of it, into nuclear fuel, and thereby increase the available fission fuel by a factor of 140. The time to double fuel, in a breeder reactor, is about ten years.
There has been public opposition to breeder reactors. The two most common objections are:
1. “The plutonium economy.” Breeder reactors would allow much greater use of nuclear power, but it means that plutonium would be widespread. Besides the fact that plutonium is radioactive, and therefore dangerous, some might be diverted by terrorists to make nuclear bombs. Proponents respond that the dangers of plutonium has been greatly exaggerated, and that terrorists would not be able to make plutonium bombs because it is extremely difficult to get the required implosion to work adequately.
2. Reactor explosion. The most efficient kind of breeder reactor would use fast, not slow, neutrons. This is called a “fast breeder.” But if fast neutrons are used, then the main safely aspect of the ordinary reactor is lost. In a fast breeder the chain reaction could spread uncontrolled, and instead of just a meltdown, the reactor really could explode like an atomic bomb. Proponents respond that they would put in lots of other safely systems that would prevent this from happening.
Dangers of plutonium
Plutonium has been called “the most toxic material known to man.” There is widespread fear of plutonium and of something referred to as a potential “plutonium economy”. Because plutonium is so important in public discussion, it is worthwhile giving some of the physics facts.
The dangers of plutonium are analyzed in detail in a Lawrence Livermore National Laboratory Report that is available on the web at www.llnl.gov/csts/publications/sutcliffe/118825.html.
Here are the key facts: Plutonium is toxic both because of its chemical effects and because of its radioactivity. The chemical toxicity is similar to that of other "heavy metals" and is not the cause for the widespread fear. The dangers are different for ingestion and for inhalation.
Ingestion of plutonium
For acute radiation poisoning, the lethal dose is estimated to be 500 milligrams (mg), i.e. about 1/2 gram. A common poison, cyanide, requires a dose 5 times smaller to cause death: 100 mg. Thus for ingestion, plutonium is very toxic, but five times less toxic than cyanide. There is also a risk of cancer from ingestion, with a lethal doze (1 cancer) for 480 mg.
Inhalation of plutonium dust
For inhalation, the plutonium can cause death within a month (from pulmonary fibrosis or pulmonary edema); that requires 20 mg inhaled. To cause cancer with high probability, the amount that must be inhaled is 0.08 mg = 80 micrograms. The lethal dose for botulism toxin is estimated to be about 0.070 micrograms = 70 nanograms.  Thus botulism toxin is over a thousand times more toxic. The statement that plutonium is the most dangerous material known to man is false. But it is very dangerous, at least in dust form.
How easy is it to breathe in 0.08 mg = 80 micrograms? To get to the critical part of the lungs, the particle must be no larger than about 3 microns. A particle of that size has a mass of about 0.140 micrograms. To get to a dose of 80 micrograms requires 80/0.14 = 560 particles. In contrast, the lethal dose for anthrax is estimated to be 10,000 particles of a similar size. Thus plutonium dust, if spread in the air, is more dangerous than anthrax – although the effects are not as immediate.
When U-235 is enriched, there is some U-238 left over. This is called “depleted uranium.” It is about half as radioactive as ordinary uranium since the U-235 and the radioactive isotope U-234 are gone. The remaining U-238 does decay by emitting an alpha particle, with a half life of 4.5 billon years, roughly the age of the Earth. That’s why there is so much left on the Earth – only half of the original U-238 has decayed.
In contrast, U-235 has a half-life of 0.7 billion years. In the 4.5 billion year age of the Earth, it has gone through 4.5/0.7 = 6.5 half lives. That has reduced its abundance by a factor of 26.5 = 90. That’s why there is so little left.
Depleted uranium is used by the military for certain kinds of weapons, particularly shells that are used to attack tanks and other armored vehicles. Depleted uranium is not used because of its radioactivity, but because of two other features it has: 1. It is very dense. With a density of 19 grams per cubic centimeter, it is almost twice as dense as lead. That is important for penetration. 2. When it hits a metal shield, it tends to form highly concentrated streams, instead of spreading out and splattering. This also helps it to penetrate armor.
People oppose the use of depleted uranium because it leaves radioactive material on the battlefield. Proponents say that the danger of radioactivity is small compared to the damage done by war, and that the alternative (lead) is also highly poisonous.
A nuclear reactor 1.7 billion years ago in
In 1972, the French discovered that the uranium they were
French scientists finally discovered that the U-235 had been destroyed by fission, about 1.7 billion years ago. Back then, the fraction of U-235 had been much larger than it is now. (That’s because it decays faster than U-238.) Instead of 0.7% of natural uranium ( the current value) it would have been over 3%.
3% is large enough to use in a nuclear reactor, provided that
there is water around to serve as a moderator. That is what we now believe
But U-235 was burned, and it dropped below the (then) natural level of 3%. It produced plutonium and fission fragments. Eventually the uranium dropped to a lower level and the reactor turned off. Remarkably, despite abundant groundwater, the plutonium and fission fragments drifted through the rock less than 10 meters over the next 1.7 billion years.
For more information on the
A miner pointing to the uranium ore in
Nuclear reactor Fuel requirements
To get a gigawatt of electric power from a nuclear reactor, for a year, you must consume some uranium. The amount is surprisingly small: about 1 ton of U-235, which (if pure) takes a volume of about a cubic foot. This has to be extracted from ordinary uranium that would fill up a cube 2 meters on a side. If you are interested in how I got this number, you can read the following calculation:
uranium fuel calculation
We want to calculate the amount of U-235 needed to run a 1 gigawatt power plant for a year. We’ll do a simplified calculation that will give us an approximate answer. As we said earlier, each fission of U-235 produces about 200 MeV of energy. Let’s convert that to joules. 1 eV = 1.6 x10-19 joules, so 200 MeV = 200 x 106 x 1.6 x10-19 ≈ 3 x10-11 joules.
How many do we need for a gigawatt-year of energy? A year is 3 x107 seconds. A gigawatt is 109 joules per second. So the number of joules in one year is E = 109 x 3 x107 = 3 x1016 joules.
So the number of fissions needed N is the energy needed divided by the energy per fission: N = (3 x1016 joules)/( 3 x10-11 joules per fission) = 1027 fissions. So we need 1027 atoms of U-235 to produce a gigawatt for a year.
We assumed that all of the energy goes into electric power. But that isn’t true – only about a third does. So we really need 3 x1027 U-235 atoms.
One mole contains 6 x1023 atoms. So we need (3 x1027)/( 6 x1023) = 5000 moles. Each mole weighs 235 grams (since there are 235 protons and neutrons in each atom). So the weight of U-235 that we need is 5000 x 235 ≈ 106 grams = 1 ton of U-235. Uranium has a density of 19 grams per cubic centimeter. So the amount of U-235 needed, 106 grams, is 106/19 ≈ 50,000 cubic centimeters, which is a cube with sides of 37 cm, a little more than a foot. So remember it this way: the amount of U-235 required is about a cubic foot.
This U-235 is found in natural uranium, but it is only 0.7%, i.e. it is 0.007 of the natural uranium. So the amount of natural uranium it takes to run a nuclear reactor for a year is about 1 ton/0.007 = 140 tons = 140 x106 grams. With a density of 19 grams per cubic centimeter, this works out to (140 x106)/(19) = 7.4 x106 cm3, which is a cube with sides of about 2 meters.
The fission fragments from uranium all come from the uranium, so their weight is comparable. Thus, a year of operation of a nuclear power plant will produce about one ton of fission fragments. There may be a comparable amount of plutonium produced, but as discussed above, that is considered valuable fuel, and will be removed by chemical reprocessing. That’s why the plutonium is not usually considered part of the waste. It is also much less radioactive than the fission fragments, since its half-life (24,000 years) is so long.
If they were concentrated, the fission fragments would take up a few cubic feet of volume. But it is expensive to do that, and so they are normally mixed in with larger amounts of unspent fuel, primarily U-238. This fuel with its fission fragments makes up the high level radioactive waste of nuclear energy.
Most of the fission fragments are radioactive. They are the same particles that caused radioactive fallout. Some of them have half-lives of a few seconds. Some have half lives of years. We already discussed Strontium-90, which makes up 5% of the fission fragments, and has a half-life of 28 years.
The following interesting plot shows this decrease. It was made specifically to address the dangers of the lingering radioactivity. So the vertical axis shows the deaths that would occur this radiation were all put inside people. Obviously that can't happen, but the plot is interesting anyway.
Suppose we "turn off" the chain reaction. We can do this by removing the moderator, or by putting in special "control rods" that absorb neutrons. Then the reactor will still produce heat from the radioactive decay of the remaining fission fragments. So the reactor continues to produce power, although the power level continues to decrease.
Look at the dark line that lies on top of all the others. Notice the scale on both axes is not linear; each tick mark increases the amount by a factor of ten. In the units of this plot, right after the material is removed from the reactor there is enough radioactivity to kill between 10 and 100 million people. (Let's take the value to be 50 million.) Check: can you see that on the plot?
One hundred years later, there is still radioactivity. It has dropped to a level of about 1 million. In other words, it has decreased from 50 to 1, i.e. it is down by a factor of 50, to about 2% of its original level.
After 10,000 years, the radioactivity has dropped to about 0.1 million. At this point it is no more radioactive then the original uranium that was taken out of the ground to produce it. So the net effect, if this radioactivity is buried, is that the average radioactivity of the ground has been reduced by the "burning of uranium."
The storage of nuclear waste has become a very contentious
issue. A waste storage location is under
The China Syndrome
The term "China Syndrome" was originally invented by someone with a strange sense of humor to describe the worst possible nuclear reactor accident. (Most people seem to think there is something worse: a reactor becoming a nuclear bomb. But, as described above, that is not possible because the uranium is not sufficiently enriched.)
In the China Syndrome, the water that is usually being boiled by the chain reaction, suddenly leaks away. There is no water to boil. What would happen in this "loss of coolant" accident? Can you guess?
The first thing is surprising to most people: the chain reaction stops. The reason is that the cooling water is also a moderator; it slows neutrons. So when the water is gone, the neutrons are not moderated. That means that most neutrons are absorbed on U-238, which does not give a chain reaction. So the chain reaction stopped.
Interesting flub by Senator: When the Chernobyl Nuclear Reactor underwent a similar accident, the Russians announced that the chain reaction had stopped. The chairman of the Senate Intelligence Committee announced on television that this was a "blatant lie." He was confusing the chain reaction with the decay of the remaining fission fragments. He knew the radioactivity hadn't stopped, but didn't realize that the Soviet's were being completely honest. The fact that the chain reaction had stopped was important; it meant that the level of power being produced had dropped enormously. (Remember this, if you become a Senator!)
The chain reaction stops, but there is still the "waste
heat" from the fission fragments. Without the cooling water, the reactor
gets hotter and hotter. The fuel finally melts. It melts through its containers
and forms a puddle at the bottom of the steel reactor vessel. The fuel puddle
keeps on getting hotter and hotter. The steel reactor vessel melts. The fuel
falls into the ground. It keeps on getting hotter. The soil and rock melts. The
fuel just keeps on going -- all the way "to
No, obviously it won't reach
There is a huge amount of radioactivity in the reactor --
enough to kill 50 million people (if they ate it). Even a small amount leaked
into the atmosphere can do enormous damage. As we stated in Chapter 4, the
number of expected deaths from
24,000 deaths is a pretty frightening number. Is nuclear power worth it? Why not just use something else, such as solar? Well -- people may not want to use solar until it is as cheap as oil. (That will happen sometime in your lifetime -- a Muller prediction.) So in the meantime, let's just use something safe: oil.
Is oil really safe? It pours lots of carbon dioxide into the
atmosphere. The consequences of that are debated, but most people think the
result will be serious global warming. How bad is that? How would you compare
it to 24,000 deaths? Some people might argue that the Afghan war is one of the
consequences of our use of oil. Why would we have bases in
Other things are dangerous too. If you are unfamiliar with
the tragedy of
Nuclear Waste Storage
What should we do with the waste, the fission fragments left
over from burning U-235? Some people say "bury it." Put it back into
the ground. But what if it get into the ground water?
Most people assume that that would be horribly bad. Therefore, they argue, it
must be put in a very stable geologic mine, some place where it will undergo no
disturbance for 10,000 years. Such a location has been prepared in
Other people argue that the danger is greatly exaggerated. If
all the radioactivity were released, in a fire, for
example, then the number of deaths is about 24,000. (It happened --
What do you think? Keep in mind that there is a real public fear of radioactivity that makes it very difficult to make a rational decision. Any governor who accepts radioactive waste storage in his state, is very likely to be challenged by people who feel that any level of radioactivity is too much. As a potential future president, how would you handle this? How can you balance the real risks vs. the perceived risks, and still be reelected?
Some people say, avoid all this complication! Just put the waste into rockets, and send it into the sun! But people who say this are ignoring the possibility of an accident. What is the probability that the rocket will fail, and fall back to earth, releasing all the radioactivity.
Is waste storage a technical problem? Many scientists think it is, and are trying to find a clever technical solution. But it seems that the issue is dominated by the public perception problem. The politician must find a solution that seems safe even to those who are unaware of the natural radioactivity in the environment. It is a very tough political problem.
Coal burning plants bury their waste in the ground. They are not radioactive, but the ashes are very high in carcinogens. What if these get into the ground water? How safe is coal, as an alternative to nuclear?
The big issue for the stockpile now is reduction (through
treaties) and "stockpile stewardship". This refers to the fact that as our weapons get old, some people argue that
they may fail. In olden days, we would assure their functionality through
periodic testing, but we have now entered an era when we have decided no more
testing. (This is largely an attempt to keep other nations from developing nuclear
weapons.) So there is a large program at
A fascinating historical book on this subject is "The Making of the Atomic Bomb" by Richard Rhodes. Another is "The Los Alamos Primer" by Robert Serber, published by the
Most of the material about nuclear weapons was once
classified, but it is not now. The Department of Energy has a site that lists
previously classified material: www.osti.gov/opennet/.
In particular, see the document RDD-7.
See also a paper by Richard Garwin titled "Maintaining
Nuclear Weapons Safe and Reliable under a CTBT", dated
For accurate detailed information, it is hard to beat the web site for the Federation of American Scientists.
The doubling law takes you from small numbers, to extremely high numbers, in a relatively small number of generations (e.g. 64, as in the squares on a chess board). Chain reactions can involve doubling, tripling, or any other factor greater than one (such as 1.4). The classic chain reaction is the one that takes place in an atomic bomb. That is made possible that in a neutron-induced fission, 2 or 3 additional neutrons are released, and these can trigger further fissions. In 64 to 84 generations, an entire mole of material (6E23 atoms) can be split.
Other chain reactions include the
growth of the fetus, the growth of a cancer, and the spread of a virus (both
biological and computer). The population
bomb was thought by Malthus to be similar, but it is
presently slowing down. (Population
explosions do occur after mass extinctions, and when foreign animals are
introduced into a land in which they have no natural enemies, such as rabbits
“Atomic bomb” is the common name for
a nuclear weapons based on the chain reaction of U-235 or Pu-239. U-235 is a rare (0.7%) isotope of uranium
that is difficult to separate. In World
A thermonuclear bomb, also called a “hydrogen bomb”, is a two stage weapon, in which the fission primary ignites a secondary that contains two isotopes of hydrogen (deuterium and tritium). The fission fragments from the primary (and from a uranium shield) are the most dangerous part of the residual radioactivity. If the bomb was exploded at low altitude, then dirt mixed with the fission fragments causes these radioactive pieces to fall out rapidly, and that could cause greater death than the explosion itself. The worst fallout is strontium-90.
At the peak of the “cold war”, the
Nuclear reactors are based on the chain reaction, but they normally work with a neutron multiplication of 1, so the reaction doesn’t grow. Nuclear reactors use moderators to slow the neutrons. This increases the probability that a neutron will be attracted to a nucleus. If the moderator is lost (e.g. the water leaks out) then the chain reaction stops. If the reactor runs away (because of operator error, setting the neutron multiplication number greater than 1), then the reactor will burn or explode, releasing energy roughly equivalent to that of a few pounds of TNT. A nuclear reactor that depends on a moderator cannot explode like an atomic bomb. If the fuel collects at the bottom of the reactor, it will continue to heat from the remaining radioactivity of the fission fragments, and this can lead to the “China Syndrome.”
Nuclear waste consists of the long-lived fission fragments. It takes about 10,000 years for the radioactivity to drop below the level of the original uranium that was removed from the ground. Proponents of nuclear power argue that the nuclear waste is far safer, since it is placed in special locations isolated from ground water, unlike the original uranium.
 In fact, the ancient game of chess was quite different from the modern game. Until about 500 years ago, the Queen could move only one square at a time, and the Bishop could jump over pieces in the manner of a knight.
 but only twice as large. Do you see why? Try it will smaller chessboards, maybe one with only 4 squares. You'll see that you always put one more on the last square than on all the preceding squares added together. If you are mathematically inclined, you might try to prove this as a theorem.
 For example, in Microsoft Excel, put the number 2 in location A1. In location A2, put the equation "=A1*2". Then use the spreading function to extend this to cells down to cell 64. The cell A20 will then contain the value of 220; the cell A64 will contain the value of 264.
There is an even simpler way to do this in Excel. Just put the symbols “=2^20” into any box. The carat ^ means “to the power of” so 2^20 = 220. Excel will then evaluate this value.
 That’s because 2 million cubed = (2 million)3 = (2x106)3 = 8 x1018 ≈ 1019.
 U-235 stands for uranium-235. Uranium is the heaviest atom that is found in relative abundance on the natural Earth. Uranium has atomic number 92, i.e. it has 92 positively charged protons, and 92 negatively charged electrons. U-235 also has 143 neutrons. The atomic "weight" is the sum of 92 protons + 143 neutrons = 235 heavy particles in the nucleus.
 If you have studied chemistry, then you may know how to calculate this number. Since the atomic weight of U-235 is 235, that implies that 235 grams of U-235 will contain one mole. One mole is 6x1023 atoms. (That number is known as "Avogadro’s Number".) Ten kilograms of U-235 will contain (10,000/235) = 42.6 times as many atoms as 235 grams, i.e. it will contain (42.6)x(6x1023) = 2.6 x1025 atoms.
 Plutonium is an artificial element, created in nuclear reactors. It has atomic number 94, i.e. it has 94 protons in the nucleus, and 94 electrons orbiting the nucleus. The Pu-239 nucleus also contains 145 neutrons. The atomic weight 239 is the sum of 94 protons and 145 neutrons.
 The answer is 53 generations. That's even less then the number of squares on a chessboard!
 He said that population would grow in "geometric ratio." By the terminology of the time (still in use in some math classes) that means that it grows by the same factor with each generation. That factor could be 2; then we call it the doubling rule. But any factor greater than 1, e.g. 1.4, will still result in an unlimited chain reaction. With the factor 1.4, the population still doubles; it just takes 2 generations. (That’s since 1.42 ≈ 2.)
 As of 2003, there appears to be enough food to feed everybody alive today. Starvation and hunger in the present world, but not because of lack of food, rather because available food is not reaching them.
 The area of the Earth is 5E18 square centimeters. After 56 generations, the number of rats would be 2^56 = 7E16. That allows a 10cm x 10cm area for each rat, including the oceans.
 See http://rubens.anu.edu.au/student.projects/rabbits/history.html
 DNA stands for deoxyribonucleic acid, if that helps.
 His Nobel Lecture is at http://www.nobel.se/chemistry/laureates/1993/mullis-lecture.html.
In 2003, Governor George Ryan of
 39 = 19,683.
 This calculation is performed in the book “The Los Alamos Primer,” by Robert Serber (ISBN 0-520-07576-5) on page 27.
 The density of plutonium is 20 gm/cm3, so 5 kg = 5000 g would fit in 250 cm3, which is about the volume of a standard cup.
 Hamza eventually defected, and now lives in the
 Unfortunately, for public confidence, the best known commercial nuclear reactor operator is Homer Simpson.
The toxicity of chemicals such as botulism toxin is not well known, since we
don’t do experiments on humans, and many people feel that experiments
on animals is also improper. Some people estimate that the LD50 for
botulism may be as low as 3 ng (rather than 70). An article by John S. Urbanetti
M.D., FRCP, was once online at:
 That’s the number you get if you take 60 seconds per minute, 60 minutes per hour, 24 hours per day, and 365 days per year: 60x60x24x365 = 3.16x107 ≈ 3 x107 seconds.