# What sets the half life of unstable nuclei/nucleons?

A chemist friend asked me what sets the half life for free neutrons, unstable nuclei, etc. He said in chemistry half lives of molecules can often be calculated based on concentrations and molecular speeds, since it is often the bumping of molecules into each other that causes unstable molecules to decay. He asked what the similar first principles-like calculation for nuclear half lives is.

I told him that unstable nuclei and nucleons lie in a local minimum in energy in some sense, and that there is an energy barrier preventing them from decaying to a lower energy, more stable state. However, quantum mechanics and quantum tunneling, so there is a non-zero probability of tunneling through the energy barrier and the half life is related to this tunneling probability.

After saying this, I went online to confirm that I was correct (I felt correct at the time but didn't feel quite correct later.) I found that the half life of a free neutron has been calculated theoretically, and that this can be understood in the context of quantum tunneling and the W- boson's large mass.

However, for unstable nuclei, I started finding that first-principles calculations of nuclear decay rates are either impossible or approximate.

So, I wanted to confirm that the quantum tunneling picture I gave my friend is correct for nuclear decays as well as free neutron decay. Also, any exposition that can be given on how "first principles-y" calculations of nuclear decay rates are.

So, I wanted to confirm that the quantum tunneling picture I gave my friend is correct for nuclear decays as well as free neutron decay.

There are a bunch of different types of nuclear decay, including beta decay, internal conversion, alpha decay, gamma decay, and fission, as well as a variety of less common processes. There is no way that every one of these should be described as a tunneling process, and in particular I don't think it's particularly appropriate to describe neutron decay as a tunneling process. Note that in the linked answer by CR Drost, "tunnel" is in scare quotes.

It's pretty standard to describe alpha decay and fission as tunneling processes, although there are some problems with such a description. In the standard semiclassical undergraduate textbook presentation of alpha decay, we have an alpha particle that has somehow "pre-formed" itself inside the nucleus, and then makes assaults on the barrier with some frequency. This doesn't really work rigorously. For one thing, an alpha particle cluster can't exist within a nucleus without violating the Pauli exclusion principle.

In the case of gamma decay, I simply don't see any reason to try to describe it as a tunneling process. I don't think that's an appropriate description, and I've never seen it described that way. In gamma decay, what sets the half-life is the relevant matrix element (magnetic dipole, electric quadrupole, or whatever) between the initial and final states.

However, for unstable nuclei, I started finding that first-principles calculations of nuclear decay rates are either impossible or approximate.

I haven't heard of any case where a calculation is impossible. There are certainly many cases where the state of the art produces only an order-of-magnitude estimate. Often we can calculate half-lives quite accurately. E.g., if you have a deformed nucleus and you want to calculate the gamma-decay rate of its first excited $2^+$ state (an end-over-end collective rotation), you can typically estimate the deformation pretty accurately from a Strutinsky smearing calculation, and then estimate the half-life quite well from that.

Also, any exposition that can be given on how "first principles-y" calculations of nuclear decay rates are.

The issues involved in calculating nuclear decay rates are not really very different from the issues involved in calculating low-energy nuclear structure. E.g., if you can do a decent description of a particular nucleus in the spherical shell model a la Maria Goeppert-Mayer, then typically you get beta and gamma half-lives for free from the same calculation.

The general difficulty with doing first-principles calculations of nuclear structure is that it's a quantum many-body problem. The classical many-body problem is basically impossible in general, and the quantum version is at least as hard. One can only hope to find approximation schemes that make the problem more tractable. For example, we can approximate groups of three quarks as if they were particles with no internal structure. This helps a lot, but makes the results less first-principles-y. Other common approximations include ignoring relativity, truncating the space of single-particle states to include only a few valence particles, and approximating pairing using the Bogoliubov approximation.

Your chemist friend should review some of the quantum mechanical processes in chemistry. A molecule, or even an atom in a higher energy configuration can decay by luminescence (photon emission) or by ejecting or transferring an electron. One would calculate these in the same way (superficially at least) for nuclear electromagnetic or particle decay and molecular electromagnetic or electron emission or much more commonly, electron transfer.

The analogy to reaction dynamics in solutions made of large quantities of molecules reacting with each other is not really appropriate. In nuclear decay, one generally treats each nucleus as an isolated quantum system. Reaction dynamics in a solution of chemicals is more analogous to a nuclear pile or reactor (or worse), while nuclear decay is more analogous to radiative or electronic transitions of single molecules.

The calculations for these quantum mechanical transitions in molecules are quite involved and approximate in the same way that nuclear transitions are, and in fact many of the same theoretical techniques are used to address both problems.