The Muggles call Einstein the Father of the Atom Bomb. We, their intellectual superiors, know better. When someone (probably a reporter from E!) first confronted Einstein with the prospect that E = mc^2 seemed to make possible the release of prodigious amounts of energy and maybe not in a good way, Albert famously retorted: Daran habe ich gar nicht gedacht! If I recall my high school German, that's Hochdeutsche for GTFO. On another occasion, Albert opined that splitting the atom was akin to shooting at birds in the dark in a place where there aren't very many birds. Basically: it's not possible and if it is I never thought of that also.
Einstein was in good company. Ernest Rutherford --- the guy credited with first splitting the atom -- went to his grave thinking it was impossible. (Clarification: Rutherford believed the large-scale release of nuclear energy was impossible. Doing so in the laboratory was fair dinkum.) Enrico Fermi -- the guy who built the world's first nuclear reactor -- told the US military an atomic bomb would be as big as the sun. It would take a World War and the threat of a Nazi bomb to whip the physics establishment into a frenzy sufficiently Promethean to steal the nuclear fire. This gave us Hiroshima and Nagasaki and the Cold War and what all else. The crisis de jour is a nuclear ambitious Iran. One might point out Iran was kept well in check until the Bush administration decided to whack Saddam Hussein for reasons that were never explained beyond "shut your cakehole." But I digress.
Our topic today is Einstein's famous equation describing mass-energy conversion, E = mc^2. Here, E is energy, m is mass, and c is the speed of light. The speed of light is a very large number, and c^2 is larger still. The quantity of energy produced, prodigious by any measure, steals the spotlight. I dare say it wraps the physics in such enigmatic air the mind reels. From whence cometh this great and terrible power?
Mass-energy conversion. What does that even mean? "Mass" is simple enough. That's just "stuff." "Conversion" is also not hard to parse: after conversion the stuff is gone. But what is "energy" in this context? It seems to me that gets glossed over in the textbooks. When a narrator solemnly narrates this destruction was wrought by the energy in a few pounds of uranium over stock footage of razed Hiroshima, what exactly did uranium convert to call down such havoc? It seems supernatural. As I hope to convince you, it is not. There is no mystery here; so much ado about nothing. Lipstick on a pig, my Southern cousins say, and let's hope that remains metaphorical.
If you are laboring under the impression that "energy" is some etherial demon lurking in matter, ready to spring forth under the proper nuclear circumstances, I'm afraid LabKitty is about to snuff some whimsy from your worldview. The quantum mechanoids may cluck their tongues and go on about liquid drop models and even/odd nucleons and the curve of binding energy, but at its heart the conversion of mass to energy is entirely pedestrian. Potato physics. If you could watch it happen you would shrug. Dur-hey. What did you expect?
We have a nucleus. The nucleus contains a bunch of protons. The protons are all positively charged so they are mutually repulsive. They must be held together by something. That something is called the strong nuclear force. If you break a nucleus in half, the strong force still holds the protons in each half together, but it no longer holds one half to the other. The two halves are mutually repulsive. Ergo, the two halves fly apart. There are now things flying apart where there wasn't previously. Flying apart things have energy (kinetic energy, to be precise). That's it. That's the E in E = mc^2.*
* well, almost. See below.
At first glance, this is no more mysterious than compressing a spring and tying it down. Cut the string and the spring snaps back into shape. If there had been something resting on the spring when it snapped -- a ping pong ball for example -- then the ping pong ball would be launched skyward. There is now kinetic energy whereas before there was none.
The added mystery is that a compressed spring and a released spring weigh the same. The before nucleus and the after nucleus do not. There is a mass deficit. The sum of the mass of the two fission fragments (as the halves are called) is less than the mass of the nucleus we started with. Multiply this deficit by the square of the speed of light, and it is approximately equal to the kinetic energy you create.
I can't help but wonder if physicists had been observing atoms splitting long before Einstein came along, would E = mc^2 have been a big deal? The equation fits the experiment, but it is not necessary for the experiment. One day, a grad student thinks to weigh the fragments and discovers the mass deficit fits E = mc^2. That's nice, the physicists say. Go fetch me a sammich.
There is, however, a little more to the story.
A Little More to the Story
I have told you truth but not the whole truth.
When you split a nucleus, other things come out in addition to the fission fragments. First and foremost is one or more neutrons. Fission is usually achieved by hitting a nucleus with a neutron. If you hope to sustain a chain reaction, be it for energy production or for something more nefarious (yeah, looking at you Mr. Rouhani), you need to produce at least one new neutron to replace the one you used to split the nucleus in the first place (for technical reasons, you actually need, on average, more than one. But that is a story for another time). The neutron that comes out has kinetic energy and so it also represents a little bit of the energy in E = mc^2.
Additionally, radiation comes out. In general we say a photon comes out and it's the wavelength of the photon that determines what we ultimately call it. In order of increasing wavelength: Gamma radiation, X-rays, UV, visible light, infrared. The shorter the wavelength of the photon the higher the energy and the badder the news for anyone that gets in its way. The photon also represents a little bit of the energy in E = mc^2.
Finally, because of the astronomical number of nuclei participating in any real world fission event, you are statistically certain to bump into a handful of rogue nuclei who go off script and do weird things instead of fissioning. An important weird thing is neutron capture, in which a nucleus swallows a neutron but does not fission. Instead, the neutron churns in its belly for awhile like bad Taco Bell and eventually the nucleus either barfs out mutated helium in an event called alpha decay or a proton in the nucleus (usually) turns into a neutron and an electron gets ejected in an event called beta decay. Alpha and beta decay writ large become alpha and beta radiation. These are not as dangerous as gamma radiation, but can still create problems. Notably, plutonium undergoes prodigious spontaneous alpha decay, which is one reason you shouldn't eat it, in case you were unclear on that. In any event, these events also represent a little bit of the energy in E = mc^2.
To demonstrate I am not simply making this up, here's a table I found in Arya's Elementary Modern Physics:
Which assigns the fission fragments 162/195 = 83% of the total energy, or 162/174 = 93% of the immediate energy, as the energy of the decay products is released over a time scale of seconds to years.
Hence the take home: Fission energy is distributed among a number of products, but most of the energy is the kinetic energy of the fission fragments. The E in E = mc^2.
The Rest of the Story
We have converted mass into kinetic energy of fission fragments, but you curious little Georges probably want to know how we get from there to city-sized smoldering vacant lots. I would be a cruel Juliet to leave thoust so unfulfilled, and I'd bet a beer Shakespeare's first draft had Romeo suggesting something a bit more carnal than the exchange of true love's vows. Here's the rest of the story.
For fission to be useful, you need a lot of fissions. This is the idea behind a chain reaction. You fission a nucleus and arrange for the neutrons that come out to fission other nuclei. They tell two friends, and they tell two friends, and so on. Along the way, the E from each fission adds and eventually good (or bad) things happen.
Footnote: It turns out not all neutrons are equally effective in convincing a nucleus to fission. Very fast and (for weird quantum mechanical reasons) very slow are effective; middle speeds not so much. Clearly, the neutron a fission liberates must be the kind that can fission other nuclei otherwise a chain reaction isn't possible. Unfortunately (or fortunately, I guess) when most elements fission they spit out neutron(s) that are the wrong speed. There's tricks for slowing neutrons, which works for a reactor, but there's no easy way to make them go faster, which you need for a bomb. The way you get around this problem is to use one of the half dozen or so elements that will fission with its own neutrons. The two most common are plutonium (Pu239) and the bomby kind of uranium (U235), which is why you hear about these anytime nuclear proliferation makes the news.
Something like 80 generations of neutrons are generated in the detonation of a nuclear bomb. So the total number of fissions is something like 1 + 2 + 4 + ... + 2^79 + 2^80. That's a lot of fissions. Oddly enough, the chain reaction doesn't stop because all of the fuel gets used up (the Hiroshima and Nagasaki bombs fissioned less than 10% of their uranium and plutonium, respectively); it stops because fission generates heat. Things expand when they get hot, and expansion increases the distance between nuclei. Eventually, the distance increases to the point where a neutron flying off in some random direction will likely escape from the surface of the bomb without ever encountering a nucleus. Once there are more escaping neutrons than fissioning neutrons, the chain reaction halts.
What's hard to get your head around (well, it is for me) is that 80 generations of fission happen in about a microsecond. A millionth of a second. A microsecond is to a second like a second is to a million seconds, and a million seconds is eleven and a half days. A microsecond is a very brief time indeed.
The other mind-boggling twist is the temperature increases by some 5 million degrees. The hot core of the bomb heats the surrounding air which then expands as a shock wave (the same process by which lightning generates thunder). Here, then, are the primary destructive effects of a nuclear weapon: heat and blast. However, these are preceded by additional, invisible destructive effects: radiation and neutrons. Whereas the shock wave propagates at the speed of sound, radiation propagates at the speed of light and neutrons at about 10% of the speed of light. Radiation is lethal in sufficient quantities. Neutrons are lethal in sufficient quantities. Neutrons can also transmute elements, producing radioactive debris which alpha and beta decays over the coming days, weeks, and years.
In a nuclear reactor, the chain reaction is kept on a short leash by inserting control rods into the fuel that absorb just the right number of neutrons to keep the chain reaction humming along at a steady pace. The heat generated is prodigious but not catastrophic, and it's used to make steam which turns a turbine and generates electricity. The electricity goes into the power grid and comes out of the wall socket in your garage and recharges your Tesla, which you then drive to the nuclear energy protest. But I kid the hippies.
Footnote: An interesting/creepy detail is the control rods in a nuclear reactor aren't there to control neutrons produced by fission -- which are permanently suppressed below the level necessary to sustain a chain reaction -- but rather a handful of surplus neutrons produced by the radioactive decay of the fuel (more properly called delayed neutrons). This is because neutron levels resulting from fission can fluctuate over microseconds, and there's no mechanical system on Earth that can respond fast enough to control such a beast.
Epilogue
Whether you're using it to generate electricity or foreign policy, fission is nothing more than an exercise in energy conversion. It's not magic; it's physics. Superb physics, Robert Oppenheimer once called the atomic bomb, when he wasn't calling it the destroyer of worlds. It's not even particularly complicated physics. In every country that has tested an atomic bomb, the design worked the first time. And there's hundreds of nuclear power plants across the globe that operate every day without incident. The whole business is an imposing feat of engineering, but one with an outstanding safety record as long as you don't build your reactor in a tsunami zone (Fukusihma) or double purpose it to make plutonium (Chernobyl).
And at the heart of it all is the kinetic energy of the fission fragments.
The E in E = mc^2.
Einstein was in good company. Ernest Rutherford --- the guy credited with first splitting the atom -- went to his grave thinking it was impossible. (Clarification: Rutherford believed the large-scale release of nuclear energy was impossible. Doing so in the laboratory was fair dinkum.) Enrico Fermi -- the guy who built the world's first nuclear reactor -- told the US military an atomic bomb would be as big as the sun. It would take a World War and the threat of a Nazi bomb to whip the physics establishment into a frenzy sufficiently Promethean to steal the nuclear fire. This gave us Hiroshima and Nagasaki and the Cold War and what all else. The crisis de jour is a nuclear ambitious Iran. One might point out Iran was kept well in check until the Bush administration decided to whack Saddam Hussein for reasons that were never explained beyond "shut your cakehole." But I digress.
Our topic today is Einstein's famous equation describing mass-energy conversion, E = mc^2. Here, E is energy, m is mass, and c is the speed of light. The speed of light is a very large number, and c^2 is larger still. The quantity of energy produced, prodigious by any measure, steals the spotlight. I dare say it wraps the physics in such enigmatic air the mind reels. From whence cometh this great and terrible power?
Mass-energy conversion. What does that even mean? "Mass" is simple enough. That's just "stuff." "Conversion" is also not hard to parse: after conversion the stuff is gone. But what is "energy" in this context? It seems to me that gets glossed over in the textbooks. When a narrator solemnly narrates this destruction was wrought by the energy in a few pounds of uranium over stock footage of razed Hiroshima, what exactly did uranium convert to call down such havoc? It seems supernatural. As I hope to convince you, it is not. There is no mystery here; so much ado about nothing. Lipstick on a pig, my Southern cousins say, and let's hope that remains metaphorical.
If you are laboring under the impression that "energy" is some etherial demon lurking in matter, ready to spring forth under the proper nuclear circumstances, I'm afraid LabKitty is about to snuff some whimsy from your worldview. The quantum mechanoids may cluck their tongues and go on about liquid drop models and even/odd nucleons and the curve of binding energy, but at its heart the conversion of mass to energy is entirely pedestrian. Potato physics. If you could watch it happen you would shrug. Dur-hey. What did you expect?
We have a nucleus. The nucleus contains a bunch of protons. The protons are all positively charged so they are mutually repulsive. They must be held together by something. That something is called the strong nuclear force. If you break a nucleus in half, the strong force still holds the protons in each half together, but it no longer holds one half to the other. The two halves are mutually repulsive. Ergo, the two halves fly apart. There are now things flying apart where there wasn't previously. Flying apart things have energy (kinetic energy, to be precise). That's it. That's the E in E = mc^2.*
* well, almost. See below.
At first glance, this is no more mysterious than compressing a spring and tying it down. Cut the string and the spring snaps back into shape. If there had been something resting on the spring when it snapped -- a ping pong ball for example -- then the ping pong ball would be launched skyward. There is now kinetic energy whereas before there was none.
The added mystery is that a compressed spring and a released spring weigh the same. The before nucleus and the after nucleus do not. There is a mass deficit. The sum of the mass of the two fission fragments (as the halves are called) is less than the mass of the nucleus we started with. Multiply this deficit by the square of the speed of light, and it is approximately equal to the kinetic energy you create.
I can't help but wonder if physicists had been observing atoms splitting long before Einstein came along, would E = mc^2 have been a big deal? The equation fits the experiment, but it is not necessary for the experiment. One day, a grad student thinks to weigh the fragments and discovers the mass deficit fits E = mc^2. That's nice, the physicists say. Go fetch me a sammich.
There is, however, a little more to the story.
A Little More to the Story
I have told you truth but not the whole truth.
When you split a nucleus, other things come out in addition to the fission fragments. First and foremost is one or more neutrons. Fission is usually achieved by hitting a nucleus with a neutron. If you hope to sustain a chain reaction, be it for energy production or for something more nefarious (yeah, looking at you Mr. Rouhani), you need to produce at least one new neutron to replace the one you used to split the nucleus in the first place (for technical reasons, you actually need, on average, more than one. But that is a story for another time). The neutron that comes out has kinetic energy and so it also represents a little bit of the energy in E = mc^2.
Additionally, radiation comes out. In general we say a photon comes out and it's the wavelength of the photon that determines what we ultimately call it. In order of increasing wavelength: Gamma radiation, X-rays, UV, visible light, infrared. The shorter the wavelength of the photon the higher the energy and the badder the news for anyone that gets in its way. The photon also represents a little bit of the energy in E = mc^2.
Finally, because of the astronomical number of nuclei participating in any real world fission event, you are statistically certain to bump into a handful of rogue nuclei who go off script and do weird things instead of fissioning. An important weird thing is neutron capture, in which a nucleus swallows a neutron but does not fission. Instead, the neutron churns in its belly for awhile like bad Taco Bell and eventually the nucleus either barfs out mutated helium in an event called alpha decay or a proton in the nucleus (usually) turns into a neutron and an electron gets ejected in an event called beta decay. Alpha and beta decay writ large become alpha and beta radiation. These are not as dangerous as gamma radiation, but can still create problems. Notably, plutonium undergoes prodigious spontaneous alpha decay, which is one reason you shouldn't eat it, in case you were unclear on that. In any event, these events also represent a little bit of the energy in E = mc^2.
To demonstrate I am not simply making this up, here's a table I found in Arya's Elementary Modern Physics:
Table 14.1 -- Energy Distribution [in MeV] after Fission of U235
Fission fragments: 162
Fission neutrons: 5
Gamma rays: 7
Decay products: 21
----
Total: 195
Fission fragments: 162
Fission neutrons: 5
Gamma rays: 7
Decay products: 21
----
Total: 195
Which assigns the fission fragments 162/195 = 83% of the total energy, or 162/174 = 93% of the immediate energy, as the energy of the decay products is released over a time scale of seconds to years.
Hence the take home: Fission energy is distributed among a number of products, but most of the energy is the kinetic energy of the fission fragments. The E in E = mc^2.
The Rest of the Story
We have converted mass into kinetic energy of fission fragments, but you curious little Georges probably want to know how we get from there to city-sized smoldering vacant lots. I would be a cruel Juliet to leave thoust so unfulfilled, and I'd bet a beer Shakespeare's first draft had Romeo suggesting something a bit more carnal than the exchange of true love's vows. Here's the rest of the story.
For fission to be useful, you need a lot of fissions. This is the idea behind a chain reaction. You fission a nucleus and arrange for the neutrons that come out to fission other nuclei. They tell two friends, and they tell two friends, and so on. Along the way, the E from each fission adds and eventually good (or bad) things happen.
Footnote: It turns out not all neutrons are equally effective in convincing a nucleus to fission. Very fast and (for weird quantum mechanical reasons) very slow are effective; middle speeds not so much. Clearly, the neutron a fission liberates must be the kind that can fission other nuclei otherwise a chain reaction isn't possible. Unfortunately (or fortunately, I guess) when most elements fission they spit out neutron(s) that are the wrong speed. There's tricks for slowing neutrons, which works for a reactor, but there's no easy way to make them go faster, which you need for a bomb. The way you get around this problem is to use one of the half dozen or so elements that will fission with its own neutrons. The two most common are plutonium (Pu239) and the bomby kind of uranium (U235), which is why you hear about these anytime nuclear proliferation makes the news.
Something like 80 generations of neutrons are generated in the detonation of a nuclear bomb. So the total number of fissions is something like 1 + 2 + 4 + ... + 2^79 + 2^80. That's a lot of fissions. Oddly enough, the chain reaction doesn't stop because all of the fuel gets used up (the Hiroshima and Nagasaki bombs fissioned less than 10% of their uranium and plutonium, respectively); it stops because fission generates heat. Things expand when they get hot, and expansion increases the distance between nuclei. Eventually, the distance increases to the point where a neutron flying off in some random direction will likely escape from the surface of the bomb without ever encountering a nucleus. Once there are more escaping neutrons than fissioning neutrons, the chain reaction halts.
What's hard to get your head around (well, it is for me) is that 80 generations of fission happen in about a microsecond. A millionth of a second. A microsecond is to a second like a second is to a million seconds, and a million seconds is eleven and a half days. A microsecond is a very brief time indeed.
The other mind-boggling twist is the temperature increases by some 5 million degrees. The hot core of the bomb heats the surrounding air which then expands as a shock wave (the same process by which lightning generates thunder). Here, then, are the primary destructive effects of a nuclear weapon: heat and blast. However, these are preceded by additional, invisible destructive effects: radiation and neutrons. Whereas the shock wave propagates at the speed of sound, radiation propagates at the speed of light and neutrons at about 10% of the speed of light. Radiation is lethal in sufficient quantities. Neutrons are lethal in sufficient quantities. Neutrons can also transmute elements, producing radioactive debris which alpha and beta decays over the coming days, weeks, and years.
In a nuclear reactor, the chain reaction is kept on a short leash by inserting control rods into the fuel that absorb just the right number of neutrons to keep the chain reaction humming along at a steady pace. The heat generated is prodigious but not catastrophic, and it's used to make steam which turns a turbine and generates electricity. The electricity goes into the power grid and comes out of the wall socket in your garage and recharges your Tesla, which you then drive to the nuclear energy protest. But I kid the hippies.
Footnote: An interesting/creepy detail is the control rods in a nuclear reactor aren't there to control neutrons produced by fission -- which are permanently suppressed below the level necessary to sustain a chain reaction -- but rather a handful of surplus neutrons produced by the radioactive decay of the fuel (more properly called delayed neutrons). This is because neutron levels resulting from fission can fluctuate over microseconds, and there's no mechanical system on Earth that can respond fast enough to control such a beast.
Epilogue
Whether you're using it to generate electricity or foreign policy, fission is nothing more than an exercise in energy conversion. It's not magic; it's physics. Superb physics, Robert Oppenheimer once called the atomic bomb, when he wasn't calling it the destroyer of worlds. It's not even particularly complicated physics. In every country that has tested an atomic bomb, the design worked the first time. And there's hundreds of nuclear power plants across the globe that operate every day without incident. The whole business is an imposing feat of engineering, but one with an outstanding safety record as long as you don't build your reactor in a tsunami zone (Fukusihma) or double purpose it to make plutonium (Chernobyl).
And at the heart of it all is the kinetic energy of the fission fragments.
The E in E = mc^2.
Disclaimer: I suppose I should have prefaced all this with a disclaimer that I'm not a nuclear wonk; I just find this part of physics to be interesting. Indeed, if we'd like to get our jaded youth interested in science and mathematics, teaching them about how physics can be used to blow the bejezus out of stuff might be a good place to start.
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