Matt Strassler [May 8, 2012]
One fact that mystifies many people when they first learn about the nature of ordinary material is that the nucleus of any atom heavier than hydrogen contains both protons and neutrons, but neutrons decay (i.e., disintegrate into other particles) in about 15 minutes, on average! How can carbon and oxygen and nitrogen and silicon nuclei be so stable if the neutrons out of which they are constructed can’t survive on their own?
The answer to this question is in the end very simple, once you understand how energy works; it is straightforward bookkeeping. But understanding energy is not entirely simple. You should first read my article about interaction energy and binding energy (which are related, but are not the same thing). Before you do that you may need to read the first parts of my article about energy, momentum and mass. You’ll need to be reasonably clear on these concepts before you’ll understand the answer to this question.
Ok. If you’ve read the above-mentioned article about interaction energy, you know that a hydrogen atom consists of a proton and an electron that because of a negative binding energy cannot escape from one another — they are trapped together inside the atom. The negative binding energy is from a negative interaction energy, partially balanced by positive motion energy of the electron (and a bit from the proton) as it moves within the atom. The interaction energy comes from the effect of the electron on the electric field near the proton (and vice versa). [If what I just said didn’t make sense to you, you should definitely read or review that article.]
What I’m going to do in this article is explain to you why a neutron is stable in the next-simplest of the atomic nuclei: the deuteron, the nucleus of “heavy hydrogen”, also known as “deuterium”. The deuteron consists of just one neutron and one proton — very simple, in principle, not so different from a hydrogen atom with one electron and one proton. Once you understand why a neutron is stable inside a deuteron, you will understand the basic principle by which neutrons can be stable inside of all stable atomic nuclei. The punchline is this: the interaction energy among the protons and neutrons is negative, and of sufficient magnitude that in some nuclei, a neutron decaying would cause the energy of the system (the leftovers from the nucleus after the neutrons’ decay, and any other particles emitted in the decay) necessarily to increase, thus violating the principle of the conservation of energy. Since energy is conserved, that makes the decay impossible.
I’m not going to describe the forces between a neutron and a proton, which are due to the strong nuclear force, and are much more complicated than the electric (and a bit of magnetic) forces between the proton and electron that make up the hydrogen atom. [This is partly because the force is not an elementary force, but a rather complicated composite force — it is a bit similar to the way the electromagnetic force can bind two hydrogen atoms into a hydrogen molecule, even though both atoms are electrically neutral. But there are some more important details that aren’t captured by that analogy. Nuclear physics is a long story of its own!]
Fortunately we won’t need to know about these complications here. All we need to know is that these forces do create a negative interaction energy for the system of the proton, neutron, and the various complicated fields that allow them to interact with each other. And that results in a stable deuteron. Just as a hydrogen atom cannot spontaneously fall apart into an electron and a proton, a deuteron cannot spontaneously fall apart into a neutron and a proton.
[Note this is not to say that a deuteron or a hydrogen atom is unbreakable. You can “ionize” a hydrogen atom (i.e., kick the electron far away from the proton) if you add some energy from outside, say, in the form of a sufficiently energetic photon. A similar method could be used to break up deuterium, and kick the neutron away from the proton. But energy must be obtained from outside the system; neither a hydrogen atom nor a deuteron will just break apart on their own.]
The Neutron Can Decay
To set the stage, let’s recall a necessary (though not sufficient) condition for an object to decay (i.e. disintegrate) is that the original object must have a mass which exceeds the sum of the masses of the objects to which it decays. Where does this condition come from? It comes from energy conservation. We’ll see how and why in just a moment, or you can look at the reasoning given in the above-linked article. [As is true for all modern particle physicists, by the word “mass” I always mean “rest mass”; all electrons have the same mass, 0.000511 GeV/c2, no matter what they are doing or how fast they’re moving.]
Let’s check that this condition is true for a neutron, which can decay to a proton, an electron and an electron-type anti-neutrino. The decay is shown in Figure 2; the neutron spontaneously converts into these three particles. [I’ve drawn the neutron and proton as larger not because they are heavier but because they are actually larger in size than the electron and anti-neutrino — though the drawing is still not to scale, in any sense. A neutron or proton has a measured diameter of about a billionth of a trillionth of a meter (100,000 times smaller than an atom), while the diameter of an electron or neutrino is known to be at least 1000 times smaller than that.]
In Figure 3, we see the energy bookkeeping (see Figure 1). Before the neutron decays, the energy of the whole system is just the mass-energy (E= m c2 energy) of the neutron. The mass of the neutron is measured to be 0.939565… GeV/c2
The dots remind us that this number isn’t exact, but we don’t need to know it any better than this for today. And the mass-energy of the neutron is therefore
- 0.939565… GeV
After the neutron decays, what’s the energy of the system now? Well, energy is conserved, and since no energy came in from outside, the energy of the system must still be the same as before : 0.939565 GeV!
But how is this energy distributed?!
First, there is no interaction energy. This isn’t instantly obvious, but it’s very important. That’s because all of the forces between protons, electrons and anti-neutrinos become very small as these particles move far apart.
Second, there is mass-energy for each of the three particles. How much mass-energy is there?
- the mass-energy of a proton is 0.938272… GeV
- the mass-energy of an electron is 0.000511… GeV
- the mass-energy of an anti-neutrino is so minuscule that it makes no difference what it is. That’s a good thing, because we don’t yet know the masses of neutrinos — but we do know their mass-energies are much, much smaller than 0.000001 GeV.
So the total mass-energy is
- (0.938272… + 0.000511… + 0.000000…) GeV = 0.938783… GeV
which is less than the neutron’s mass-energy that we started with, by 0.000782… GeV. So we haven’t seen yet how energy is conserved; the mass-energy of the neutron is not entirely converted into mass-energy of the proton, electron and neutrino. The excess energy is indicated in yellow in Figure 3.
We can make up the difference, however, with motion-energy. Motion-energy is always positive. As long as we distribute the excess 0.000782 GeV among the motions of the three particles in a way that preserves momentum conservation (which we can, trust me) then we’re done: energy is conserved, because the mass-energy of the neutron was converted to mass-energy and motion-energy of the proton, electron and neutrino.
I haven’t specified the exact amounts of motion energy that go to the proton, the electron and the neutrino separately, because in fact the energy will be distributed differently for each neutron decay, just by random chance (thate’s quantum mechanics for you.) Only the total motion-energy is always the same: 0.000782 GeV.
The Deuteron is Stable
Ok, now let’s turn to the deuteron. The deuteron, like a hydrogen atom, has total energy made from the positive mass-energy for its two constituents (i.e. the proton and the neutron), positive motion-energy for its two constituents, etc., and negative interaction-energy that more than cancels off the motion-energy. Moreover, as for any particle or system, the mass of the deuteron is just its total energy (to be precise — the total energy you measure it to have when it is not moving relative to you) divided by c2, the speed of light squared. Consequently for a deuteron that isn’t moving relative to you, based on the mass that it is measured to have (1.875612… GeV/c2) you’d say its energy is
Deuteron mass-energy = 1.875612… GeV =
- proton mass-energy + neutron mass-energy
- + proton motion-energy + neutron motion-energy
- + interaction energy (negative and bigger than the motion-energy)
< proton mass-energy + neutron mass-energy
= 0.938272… GeV+ 0.939565… GeV = 1.877837… GeV
Thus the binding energy of the deuteron is
1.875612… GeV – 1.877837… GeV = -0.002225… GeV.
The fact that there is negative binding energy implies (just as for a hydrogen atom) that the deuteron cannot just fall apart into a neutron and a proton, as shown in Figure 4; this would violate the conservation of energy, which implies that a decaying particle must be more massive than the particles to which it decays. As shown in Figure 5, there’s no way to make energy be conserved; the neutron and proton together have more mass-energy than the deuteron does, and there’s no source of negative energy to cancel off the deficit of energy, since there’s no interaction-energy between a well-separated proton and neutron, and no way for motion-energy to be negative. This means that the process shown in Figure 4 simply cannot occur.
The Neutron Inside a Deuteron Cannot Decay
We have one last step to go, and given the steps already taken, it’s an easy one. The question is: why can’t the neutron inside the deuteron decay?
Suppose it did: what would be left over? Well, there would now be two protons, an electron and an anti-neutrino; see Figure 6. Two protons repel each other (they both have positive electric charges, so the electric force between them pushes them apart; and the strong nuclear force between them, which tries to pull them together, is not as strong as for a neutron and a proton, the end result being that the sum of the two types of forces is repulsive.) The effect of this repulsive force is to push the two protons far apart. Meanwhile the electron and anti-neutrino would also depart the scene.
With all four particles far from one another (as indicated roughly in Figure 6, but imagine them much further apart than shown), there would be no substantial interaction energy among the particles; the energy of the system would consist only of the sum of the particles’ mass-energies and any motion-energy they might have. Since the motion-energy is always positive, the minimum energy the four particles could have is the sum of their mass-energies. But this energy is larger than the deuteron mass-energy (Figure 7)! Already the mass-energy of two protons, 1.876544… GeV, is larger than the mass-energy of the deuteron; the additional 0.000511 GeV from the electron’s mass-energy just adds unnecessary insult to injury.
And therefore the neutron inside the deuteron cannot decay; the interaction energy that helps hold the deuteron together pulls the mass of the deuteron down — down far enough that for the neutron inside a deuteron to decay would violate the conservation of energy!
Other Atomic Nuclei
So it is with all the stable nuclei of nature. But you shouldn’t get the idea that it is always true that when you combine neutrons and protons, the result is a stable nucleus! Stable nuclei are actually very rare.
If you take Z protons and N neutrons, and try to make a nucleus out of them, for most values of Z and N you will fail. Most such nuclei will instantly fall apart; they don’t even form at all. Roughly speaking, the attractive force among the Z protons and N neutrons is strongest when Z and N are about the same size. On the other hand, the protons all repel each other because of the electromagnetic force. That force grows large when Z is large. The competition between these two effects implies that nuclei are most likely to be stable when Z is somewhat less than N; and the larger are Z and N, the bigger the difference between Z and N has to be. You can see this in Figure 8. Only the nuclei shown in black are stable; they lie in what is poetically called “the valley of stability.”
What are the nuclei shown in color? It turns out that there are quite a few nuclei that, like a neutron, do fall apart eventually but can survive for quite a while. [We often call these objects “unstable”, though things that are rather long-lived are often called“metastable”; the word-usage is often context-dependent, so you have to pay attention.] The neutron lasts 15 minutes; but there are nuclei that survive for milliseconds, days, decades, millenia, even billions of years. These nuclei are what we call radioactive; they are the dangerous aftermath of radiation accidents or weapons, and the tools used in smoke detectors and in fighting cancer, among other applications.
There are a number of different ways that these nuclei can fall apart, but some of them decay by having a neutron convert to a proton inside a nucleus. (We know this because the charge of the nucleus increases, and an electron comes flying out, along with an anti-neutrino.) Others even decay by having a proton convert into a neutron! (We know this because the charge of the nucleus decreases, and a positron [an anti-electron] comes flying out). Figuring out how long a given type of nucleus will survive, and how it will decay, is very complicated nuclear physics — I won’t give you a course in that here (indeed I’m no expert).
Suffice it to say that negative interaction energy among particles, combined with the conservation of energy, can change the game in a big way, making impossible certain processes that are normally possible — and vice versa.