Nuclear Fusion: Mass into energy?

Updated: Apr 28

I'm a big fan of Dan Carlin and his Hardcore History, where he pulls apart a topic in history in all of its gory details. I am toying with the idea of doing something similar, a "Hard Physics", where I explore ideas in more depth than they are covered in general discussions.

For the first topic, I was thinking of "The Physics of Nuclear Weapons". Growing up in the UK in the 1980s, I was well aware of the "Four Minute Warning" and that the nearby Llandarcy Oil Refinery was a potential target. In 1984, I watched Threads on TV (and the much more family friendly American The Day After). Still somewhat haunted by the cover of the Radio Times.

But how do nuclear weapons work? In my physics undergraduate, I studied nuclear physics, understanding that the nuclear strong force is much stronger than the electromagnetic force, so nuclear reactions are much more energetic than chemical reactions, so nuclear explosions are more energetic than chemical explosions. There is something about binding energy per nucleon and mass deficits, and that's why it works.

Like a lot of things, you're happy with an explanation until you really have to think about it. And once you do, you realise the picture is not as simple as you thought it was.

I'll deal with nuclear fission, where big atomic nuclei are split into smaller nuclei at a later date. Here I'm going to talk about nuclear fusion, where lighter elements are bound together into heavier elements. This is the source of energy in the Sun, and is the process that the first elements were formed in the very early universe. It is also the source of energy in hydrogen bombs, the most powerful of nuclear weapons.

This is the mushroom cloud from Tsar Bomba. Tested in 1961, this explosion was equivalent to more than fifty million tons to TNT, but in theory the yield could be as large as you want. Thankfully, it was realised such immense weapons, whilst frightening, are of little use strategically, and the focus has been on smaller weapons (not that this is a good thing either).

So, we know the energy comes from nuclear fusion. To achieve the conditions for this, a fission bomb is used to compress and heat the fusion fuel, a mix of lighter elements. Details of the explosion sequence can be found over at wikipedia.

But what is happening at nuclear level? To understand this, we need to think about the forces involved. The atomic nuclei are made of neutral neutrons and positively charged protons, so they repel one another. The strong nuclear force, however, is very short range (the reason for that is itself interesting, but we can get to that at another time), and nuclei have to get within about 10^-15 m for the strong force to bind them together. The high temperatures and densities are give nuclei enough speed to overcome their electromagnetic repulsion and get them close enough to fuse via the strong force. So, we have a picture something like this; here, two isotopes of hydrogen, deuterium and tritium, are fusion to produce helium and a neutron.

So, the story goes that if you add together the mass of the deuterium and tritium nucleus before the fusion, and subtract off the mass of the helium nucleus and the neutron after the fusion, some mass has gone missing. Through Einstein's famous E=mc^2, that missing mass has been turned into energy. This energy is the kinetic energies of the helium and the neutron which fly out and hit surrounding material. This is nicely described over at the excellent hyperphysics.

At this point, most people are happy that they understand what is going on, and move onto the next physics topic. But here's a question - just which mass has been converted into energy? It can't be missing from the neutron as that's just a neutron and they all have the same mass. So, maybe its from the particles in the helium nucleus? But does it come from one of the protons and neutrons inside the helium, or are they all a bit lighter by a bit?

What if I told you that the protons and neutrons inside the helium had the same mass as individual protons and neutrons everywhere. So, the question is - what is the source of the missing mass, and just where did the energy come from?

Let's consider a different kind of fusion - two deuterium nuclei.

This is not the only route for two deuterium nuclei to fuse. Half the time the fusion products are a tritium nucleus and a proton. But let's think about the physics here.

At low temperatures, the two deuterium nuclei zip towards each other, but the electromagnetic repulsion stops them in their tracks before reversing their path. As the temperature is raised they get closer and closer, till eventually they are close enough for strong force to grab them. This yanks them together, so if they were at rest just as they touched, they (almost) instantly have a lot of kinetic energy and are whizzing about each other. The "potential energy" of the strong force has been converted into kinetic energy.

So, the two fused deuterium nuclei have energy up as a helium nucleus, but 4He, with two protons and two neutrons. This helium has a lot of energy and so is in an excited state, and, like an electron in an excited state, will shed the excess energy to end up in its ground state. The 4He can do this via the strong force, ejecting either one of the protons or neutrons with the result of tritium of 3He in its ground state. It can also do it by emitting a photon in the same way as an atom. Because the strong force timescales are much shorter than the electromagnetic (as the force is stronger), it's normally the ejection of the proton or neutron that is seen, but the photon emission does occur now and again (image from understandingscience).

So, the course of events was that in the act of fusion, strong force potential energy got converted into kinetic energy, with an excited state of 4He, which shed this energy through emission of a proton, neutron or photon.

Notice that nowhere does mass get "converted" to energy!

Now, you might be scratching your head over this. The mass deficit is definitely there, so where did the mass go? The issue is that the relationship between mass and energy is more implicit than you might imagine.

Einstein considered this in his second relativity paper, also published in 1905.

So, the mass of an object, in terms of its inertia (which is related to how hard you have to push something to get it to accelerate) depends not only on the mass, but also the energy content. So, a hot object contains more energy than a cold object, and so you need more of a push to accelerate a hot object than a cold object.

This is well known in nuclear and particle physics, where excited states are more massive than the ground states. There is a nice table over at wikipedia that shows various particle resonances, showing that an excited version of a proton or neutron can be considerably more massive than the ground state, due to the extra energy content.

The same is true for for individual atoms, in that an excited atom is more massive than an atom in its ground state because it has more energy. But for atoms, the energy differences are in the scale of eV, about a millionth of what is happening at the nuclear scale, so we can generally ignore them.

Great, so things can be more massive if we add energy, but our mystery remains as we are looking to lose mass in fusion. What we have failed to do is to account for all of the energy involved, namely the binding energy due to the strong force, and this energy is negative, and so represents a negative mass. This is why the final mass is lower.

I know some people don't like this, but it might be easier to think in atomic terms. When a proton captures an electron to become a hydrogen atom, energy has to be given up if the proton is to hold onto the electron. The energy is radiated away as photons and the electron is bound to the proton. If you want to get the electron out, you have to put energy in to overcome the binding. It is the same in nuclear fusion, except the energy scales are much larger.

I also know that people get grumpy about all of this negative energy stuff, but really its all relative and it depends where you draw you baseline. We typically take this to be zero for well separated particles, but we would take it from the most bound state possible then all energy levels would be positive. The results would all be the same as it's always the change of energy that matters.

Alright, this feels like enough scribbling for a Sunday. Before I finish, let's ask a deeper question - we take the zero level for the strong force binding as well separated particles, as that what we see in the universe. Most matter is in the form of protons yet to be fused into heavier elements. This is great for the universe (and us) as the natural fusion reactors in stars put out plenty of energy, allowing there to be life on this planet.

But why weren't all these protons fused together in the nuclear fires at the start of the universe, so all particles would be in the ultimate ground state? This would be catastrophic for life in the universe as there would be no stars in the universe. This is a bigger question which we'll just have to leave for another time, but to get a flavour, have a read of a recent paper I wrote.

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