Why Isotopes?

In the process of writing my lengthy unfinished article about how chemistry works, a side question came up for me: Why do the chemical elements have isotopes? Or more broadly, why do atomic nuclei always contain both protons and neutrons? All combinations consisting of anything but a single proton by itself (which acts as a nucleus when it forms a hydrogen atom) contain a number of neutrons approximately equal to the number of protons — often a slightly larger number, especially in the heavier elements. Why do you never see two protons stuck together with no neutrons, or vice versa?

It turns out that the answer is beta decay. This process is based on the nuclear “weak force”, which is shared between the two main classes of massy fundamental particles, the quarks and the leptons. (The “strong force” applies to quarks only. It’s what holds nuclei together.)

Protons and neutrons, and most of the exotic heavy particles created in accelerators, are not fundamental, but are composed of quarks. But electrons are leptons, and are fundamental. There are, in the currently accepted standard model, six quarks and six leptons, of which only the lightest varieties are common and stable — the heavy ones decay in very brief spans of time. For instance, the average lifespan of a c or b quark is about a picosecond, and the t quark’s duration is so sort we have no measurement of it. The tau lepton also decays in a fraction of a picosecond.

The heavier an exotic particle is, the faster it decays. This is because its mass represents excess energy, and unless there is a conservation law which prevents it, that energy wants to be released. When it gets out, it creates lighter particles in place of the heavier one, with any excess usually going into kinetic energy, which ends up as heat. The lighter particles that result have to fit the same conservation rules as the heavier one, such as having the same net electric charge. More available energy makes it more eager to decay, so lighter particles don’t decay as quickly. For example, the middleweight s quark has an average lifespan of around twelve nanoseconds, which is thousands of times longer than the duration of the much heavier c. And the muon, the lepton that falls between the electron and the tau in mass, lives for about two microseconds.

The lightest quark is the u, and the second lightest is the d. (If you’re wondering, the six letters stand for up, down, strange, charm, bottom, and top. These names are completely arbitrary and at first were just placeholders, but stuck when nothing better came along. Officially, the particle names are just lowercase letters. The Greek letters stuck onto some other particles are equally arbitrary.) An odd fact about quarks is that although most of them have huge differences in mass, the difference in mass between the u and the d is small — maybe half a percent of the mass of a proton. This means that for a d to decay into a u is relatively rare, and the average lifespan of the d is about fifteen minutes — an eternity in the world of fundamental particles.

Not that their mass as isolated particles means much, because quarks cannot exist in isolation. In fact, their isolated mass cannot be measured, only estimated from theory, because they can exist only when bound into groups. The simplest groups are pairings, but because these must consist of one quark and one antiquark, there is nothing to stop the whole thing from decaying. So groupings of three are much more abundant. (Clumps of five, or other exotic numbers, are theoretically possible and recently have been observed.) Because of “color charge” conservation (a topic we will not digress into here) and other quantum rules, these triple combinations can be stable. The two lightest triple quark combinations are the proton and the neutron. Heavier triplets will generally leave one of these behind after decaying.

Since the proton is the lightest triple-quark particle, it cannot decay into anything else. The neutron, however, can decay into a proton. This happens when a d quark inside it decays into a u. An unbound neutron, outside of a nucleus, will do so after an average time of fifteen minutes. But to do so, it has to balance the electric charge, since the proton has a positive charge while the neutron has none. To give the proton a positive charge, the decay has to give something else a negative charge that compensates for it. The lowest energy way to do this is to create an electron, which is very lightweight compared to quark-based particles.

Because the quarks in a proton carry a net positive charge, there is some mutual electrostatic repulsion between them. This adds further energy to the proton, besides the masses of the quarks and the strong-force binding energy that they are stuck together with (which far outweighs the quarks themselves). The electrostatic field between the quarks is compressed like a spring, storing energy. Because of this, and because of mass-energy equivalence, the proton is slightly heavier than the combination of quarks might suggest, and the difference in mass between a neutron and a proton is only about half as much as the estimated difference between a d and a u quark. The difference is two and a half times the mass of an electron, or just a quarter percent of a free proton’s mass. So during decay, though the excess is tiny as a portion of the total mass involved, it is more than enough to create an electron with.

A side note — you might think that if a proton is the lightest triple quark, it would consist of three u quarks, but that’s not the case. The proton consists of two u and one d, while the neutron is one u and two d. The reason a particle of three u quarks is heavier than a proton is because it has a double electric charge, adding way too much electrostatic repulsion, and therefore excess energy. (The Pauli exclusion principle is another factor, but that’s a topic for another day.) This particle, which is known as delta++, decays rapidly. This issue of extra repulsion will be relevant to our question about isotopes.

So a neutron can divide into a proton and an electron. But there’s a problem with creating a brand new electron: electrons are stable, and have a quantum property which is conserved. It doesn’t have any measurable effect, like electric charge or quantum spin do, but it does mean that the net count of leptons minus anti-leptons in the universe cannot change. Electrons cannot be created unless this elusive property is balanced. And this is possible thanks to neutrinos.

I mentioned that there were six leptons, but I only named three: electron, muon, and tau. The other three are neutrinos: they are known respectively as the electron neutrino, the muon neutrino, and the tau neutrino. They have no electric charge and their mass is so small that we can’t give a precise number for it, but they do have a full share of the mysterious lepton-ness quality which prevents electrons from decaying. When a neutron decays into a proton, it has to produce both an electron to balance the electrical charge, and an electron anti-neutrino to balance the lepton-ness of the new electron. The remaining excess energy kicks the new electron away at high speed, creating what is traditionally called a beta ray — one of several forms of dangerous radiation which can be produced by radioactive materials. These “beta rays” are nothing but ordinary electrons with an excess of speed.

So bare neutrons undergo beta decay to turn into a proton, an electron, and an anti-neutrino. Now why doesn’t this happen inside a nucleus? In there, neutrons can last forever.

Well, in some nuclei they don’t. Certain radioactive isotopes are known for undergoing beta decay, spitting out an electron and turning a neutron into a proton. For instance, the isotope carbon-14 undergoes gradual beta decay to turn into nitrogen-14, which has seven protons and seven neutrons, whereas the carbon had six protons and eight neutrons. This happens because the nitrogen nucleus is slightly lighter in weight, so a bit of energy is released, in just the same way as with a bare neutron, though more slowly. But in carbon-12, with six neutrons, beta decay never happens, nor does it happen to nitrogen-14. What prevents it?

Excess electric charge pressure is what prevents it. If a carbon-12 were to beta-decay into a nitrogen-12, the increase in the positive electric charge of the nucleus would actually make it heavier, despite it being made of particles that would be lighter in isolation. Nitrogen-14 has less electrostatic repulsion energy because the additional neutrons help space the protons apart, making the nucleus larger. When two of the neutrons are absent, the pressure between the protons increases just enough that there is now an energy gain from replacing one of the protons with a neutron. In fact, when nitrogen-12 exists, it decays back into carbon-12 by (drum roll) reverse beta decay. This is a process in which, rather than a neutron emitting an electron, a proton absorbs an electron which is nearby. Or alternatively, if sufficient excess energy is available, it can emit an anti-electron, a.k.a. a positron. In either case it also emits an electron neutrino, the opposite of the antineutrino produced by forward beta decay.

The fact that beta decay can happen in both directions is what makes some isotopes stable. Isotopes with too many neutrons allow them to decay into protons, because that replaces a heavier particle with a slightly lighter one. Isotopes with too few neutrons allow some of their protons to be promoted into neutrons, to relieve the pressure of too much electric charge in too tight a space. Isotopes are stable when neither move will reduce the total mass or energy bound into the nucleus. Stability occurs when the extra mass of a neutron and the extra charge energy of a proton are in close balance. And if it weren’t for a series of apparent coincidences, in which the masses of the various particles happen to be separated by a narrow gap that’s just enough to do these decay reactions with, isotopes wouldn’t even be a thing.

You may wonder if this explanation is oversimplified. It absolutely is. I could go into a ton more detail, and still be oversimplifying. Only working through the actual math gives the complete picture. But I think I’ve been accurate enough so that the explanation above, though well short of the whole truth, is not false.

Switching between protons and neutrons is not the only means of nuclear decay. If an isotope’s neutron excess is especially pronounced, it may simply eject one of them, or in extreme cases, two or more at a time. If there’s enough energy to gain, this is simpler and faster than juggling neutrinos via the “weak force”.

Heavy isotopes from the bottom parts of the periodic table (and light elements if they’re whacked hard enough) can also decay by ejecting a cluster of two protons and two neutrons as a unit. This is called alpha decay. An “alpha ray” is such a clump — a helium-4 nucleus — moving at high speed. They come out in groups like this because this 2+2 arrangement is especially stable and robust. (For reasons I won’t get into here, nuclei tend to find it energetically favorable to have an even number of particles, and better yet is to have an even number of each type, and alpha decay preserves that condition for both pieces.) In the very heaviest elements, this decay is so rapid that such nuclei are not found in nature, and if made artificially, have a very short lifespan. Once you get to a triple digit number of protons, it’s pretty much impossible for a nucleus to hold together at all.

And finally, a select few heavy elements can decay by simply splitting in half. Usually this does not occur spontaneously, but only if an extra neutron happens to fall into it. This is called nuclear fission, and is what nuclear power plants get their energy from. The balance of how many protons and neutrons end up in each half is random, and there are usually a few neutrons left over. These excess neutrons can fall into another nucleus of the same kind, making it much more likely that it will also undergo fission, thereby creating a chain reaction. Without the reabsorption of neutrons, fission would be a super-slow process, hardly occurring at all. For instance, the average lifespan of a nucleus of uranium-235 is 700 million years, and when it does decay it’s usually by alpha emission. This is why this isotope can still be found in nature instead of having to be made artificially.

Whereas fission involves some of the heaviest elements, the opposite process, fusion, generally involves the lightest. This process can’t really be called decay. It consists of sticking together small nuclei to make bigger ones — a process that requires extreme heat to push them together hard enough. This is the process that powers the sun, and might someday run a new kind of nuclear reactor which is much safer and cleaner than the fission plants we build now. In the sun, the process is slow because the gas from which the sun is made, being mostly hydrogen, has tons of protons but very few neutrons. It takes many steps of adding protons to get a stable heavier nucleus, and about half of those steps have to wait for reverse beta decay before they can proceed further, in order to create the neutrons that are needed for the built-up nuclei to be stable. For this reason, the sun produces a huge number of neutrinos. They flood the whole solar system, and are passing straight through you right now, having no effect. The most difficult step is probably to get the first two protons to stick together… it might not even work unless the reverse beta reaction is practically simultaneous. The protons have to hit each other dead-on with enough energy to not only overcome their electrostatic repulsion, but to also generate a positron and neutrino. (The positron is then annihilated when it encounters an electron, yielding two gamma ray photons. Gamma rays are simply a very high energy form of light, and when generated deep in the sun, they are soon reabsorbed and just end up creating heat.)

In a hydrogen bomb, on the other hand, they select isotopes which are already rich in neutrons, so no reverse beta decay needs to occur. This makes the reaction extremely rapid, hence the explosive power. We will probably have to use those same isotopes in fusion power plants, so they can consume their fuel as quickly as it’s fed in, instead of taking billions of years as the sun does. (We can thank the slowness of stellar fusion for the fact that life had time to evolve.)

One way that a sun can speed up the fusion process is to undergo a huge explosion such as a supernova, or a massive collapse such as the formation of a neutron star, which from the outside can also appear explosive. This works best in big stars. It is thought that these processes are responsible for creating much of the universe’s stock of heavy elements from the middle parts of the periodic table. Small stars like our sun don’t produce such elements, or do so only in their final stages… and even when they do make some, they mostly hold onto them instead of spraying them outward. All the heavier metals and minerals that we have around us, making up most of the Earth, had to be inherited from other solar systems of the past that had bigger suns, and violent enough ends to scatter the resulting materials.

And scientists are now moving toward a theory which says that the truly heavy elements, like lead or platinum or uranium, are not created by fusion.  It had always been a puzzle how stellar fusion could create elements heavier than iron… and a new consensus is starting to agree that stellar production does indeed top out with elements not much past iron in the periodic table, such as copper and zinc. The rest were most likely produced when middleweight elements (the most abundant being iron) absorbed extra neutrons and then underwent beta decay. The same stellar explosions that fused a lot of these middle elements also produced huge floods of neutrons, which could then push those elements further down the periodic table. The biggest neutron floods occur not in supernovas, but in neutron star events, the most extreme being when one neutron star falls into another. So without beta decay we probably wouldn’t be able to mine substances like gold or neodymium.

Speaking of stellar explosions and collapses, there is another circumstance in which neutrons are stable, and that’s when they’re under extreme gravitational pressure. Every star eventually runs out of fuel and goes cold. Without the great heat produced by fusion, the gas pressure which gives the star a large volume diminishes, and it shrinks to a much smaller size. For a small star like our sun, the result is a white dwarf, which slowly cools to become a red dwarf. The universe is full of red dwarfs, and they are so faint that only the nearest ones can be seen. The shrinkage intensifies the already huge gravitational force, and crushes the matter inside the dwarf to extraordinary density.

For a star somewhat larger than our sun, this crushing reaches a point where the inside of the star becomes nearly as dense as an atomic nucleus. Under these conditions, the electrostatic repulsion of protons becomes unfavorable, and the whole bulk of the star undergoes reverse beta decay. The flood of neutrinos arising from this lasts only about a minute and carries away a gigantic amount of energy, though normally neutrinos are so insubstantial that the energy they carry is insignificant. The result of this massive decay is what we call a neutron star, because except for a thin surface layer, the whole thing is made of practically nothing but neutrons. Such stars are so dense that they fit the mass of a sun into the size of a minor asteroid — the diameter of the star shrinks from like two million kilometers to twenty kilometers. They’re as dense as matter can be before collapsing into a black hole. If you consider it to be a nucleus, that is sure one heavy isotope.

The universe is certainly trending toward turning more and more of its protons and electrons into neutrons and neutrinos, but this doesn’t continue without limit: once most matter is in the form of non-hydrogen elements, the process of neutron formation is going to slow way down. For instance, in red giant stars, which form when a sun’s original stock of hydrogen runs out, helium is fused into carbon and oxygen without requiring any new neutrons. The final state of the universe will probably still contain lots of the common materials we’re familiar with today, such as carbon and iron, and even water. They’ll just be colder.