# Why is there no formula to compute the half-life of any given radioisotope?

Calculators like http://www.radprocalculator.com/Decay.aspx rely on large internal tables, which were created by using the results of experiments.

These experiments are all based on simple observations: Count. Wait. Count again. Here’s your half-life.

But why does it have to be that way? We know about the strong and weak nuclear force. We understand quantum mechanics. So why can’t we just take a nucleus with 6 protons and 8 neutrons for example and calculate how these particles interact? And come up with 5730 years without relying on experiments (of course experiments are a great way to confirm our formulas).

Where’s the difficulty?

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## 16 Answers

I don’t think the theories contain enough information about what is actually going on with subatomic particles that leads to those results.

@mattbrowne Are you saying that by molecular weight or number of electron shells or something else, that a half life should be a definitive number of years?

@Tropical_Willie – No, it can’t be just weights, it’s also the ratio of protons and neutrons. The strong nuclear force has to compensate for the electromagnetic force (protons repel each other).

Sounds like a great opportunity for some post graduate work.

Go to it…

Matt, last I checked (which was years ago), we didn’t have any way to get more information about sub-atomic particles than weight, the forces they exert, number, the forces and particles they release in phase changes and reactions, and the larger structures and compounds they form in combination. Our only tools for observing them are other particles and theories that we test about what they do. With what could we measure the properties that are the reasons why they behave the way they do? All our theories are based on observations of behavior, about everything. As soon as a cat levitates and fades out by itself, we’ll need to amend our theories about cats.

So, what happens when a radioisotope decays, is all theory from observation. What else would it be? When scientists sound like they are speaking about absolute knowledge and complete understanding, is when they’ve stopped being scientists and have become dogmatists.

All of that said, I expect that in the last ten years, there have been many cool new observations and theories that I don’t know about, about radioisotopes, so there probably is something someone who knows about that could say to you that would be interesting, but it will always still be a theory based on observation.

And, if what you want to know is half-life, then the math to look up the answer will no doubt be massively simpler (one multiplication) than to model and predict based on some mechanical simulation.

Are you sure that we can’t? I thought that this was one of the triumphs of the standard model – the accuracy with which quantum calculations agree with empirical observation.

In the case of chemistry (i.e., computing electron wave function via Schrodinger eq etc) I believe that helium is much harder to compute than hydrogen, and such calculations for even heavier atoms require various simplifying approximations.

I wonder if the equations for nuclear decay (involving weak interaction) are simply too complex for all but the simplest nuclei? Can physicists calculate the 12-year half-life of tritium (Hydrogen-3)?

There have been a few questions on Fluther about radioactive decay but no clear answers. It is a mysterious process; why should one atom decay in the next minute and an identical one in one million years time? I find that weird.

The Geiger–Nuttall law establishes a relationship between half life and the energy of radioactive emissions but I don’t think anyone has found a way to calculate half life of a radioisotope from first principles as we don’t know what the mechanism of decay is.

I accept that radioactive decay is fundamentally probabilistic, hence not strictly predictable any more than next week’s lottery numbers. The probabilities themselves, however (which determine observed half-life) can be precisely measured. The question is whether these probabilities can also be calculated.

@flutherother ”...we don’t know what the mechanism of decay is.” This is outside my knowledge of physics, but I know that beta decay somehow involves the weak nuclear force & alpha decay involves a combination of strong and electromagnetic forces. I was under the impression that the underlying probabilities were calculable from first principles.

It’s a complicated situation. Electromagnetism and both the strong and the weak nuclear forces become important in the nucleus. As far as I know there are no *ab initio* formulas for important physical and chemical properties of the elements where only electromagnetism really comes into play (ex: melting point). The other two theories are even more difficult to compute consequences with.

@mattbrowne, I’m not an expert, but I have a sneaking suspicion the situation is like the 3-body problem.

To wit: we know Newton’s laws of gravity. We can imagine a system with three known masses (say, the sun, the earth, and the moon). This all seems simple, right? Just 3 objects in space and Newton’s elegant laws. But turns out, it’s hard as *hell* to do the math. Mathematicians only recently approximated it using supercomputers.

An atom seems simple, too—like our hypothetical sun-earth-moon system. But if you think about what an atom is, in terms of quantum mechanics, there’s a lot of moving parts. Each proton and neutron is itself a very complex system of quarks held together with the laws of quantum chromodynamics. Each one of these systems is also constantly interacting with the electrons through electromagnetism. So a carbon atom is actually a very complicated system composed of 12 subsystems all interacting with each other and with the electron cloud that encases it.

So I can understand if we are having trouble deriving the long-term behavior of such a system from first principles.

Thanks, for your answers so far. I really looked hard. All I found was that there are no formulas. The question is why. I wonder how many top-notch scientists are looking for them. Or is it futile?

For the 3-body problem there’s at least an attempt for a mathematical solution as shown in the article. From what I know tiny changes can lead to a totally different outcome. The state of a carbon nucleus is governed by the laws of quantum mechanics. Like for the electrons there’s something like probability clouds telling us where all the protons and neutrons are most likely to be, right?

But there are at least two different sets of laws at work in an atom. There’s chromodynamics (which is *really complicated*) for the strong force; there’s electrodynamics for the way charged particles interact with each other. And then I think there’s also stuff with whatever is going on with the weak force which I’m not even going to pretend to understand.

So it’s not even as simple as the 3-body problem, which is just about gravity.

And protons and neutrons are baryonic matter (systems of particles); they’re not particles in themselves. So I don’t think we can predict the probability of their position/momentum nearly as easily as we can with electrons.

The 3-body problem cannot be precisely solved because it’s a non-linear dynamic system subject to chaos. Those systems have the peculiar property of being totally deterministic yet not calculatable due to “sensitive dependence on initial conditions”, i.e., the butterfly effect.

Radioactive decay, by contrast, is fundamentally probabilistic and non-deterministic. Einstein famously refused to believe that “god plays dice.” So might there be a “hidden layer” lurking beneath our understanding of particles that, once understood (string theories, maybe?) will reveal the probabilistic nature of quantum mechanics to actually be another instance of chaos – deterministic but unpredictable?

Or is this not a valid distinction in the first place?

@Qingu – But we can calculate the probability of the location of a single proton. For example in the interior of stars. The probability at the outer edge is low, but not zero. And this is one important reason why fusion works. Same for alpha particles inside even heavier and older stars. Probabilities, not random chaos, @gasman.

I realize that’s only part of the problem. The weak force is “mysterious” to me as well. Carbon-14 turns into nitrogen-14 in a predictible way. A neutron becomes a proton.

I think I can see the distinction between the unpredictability of chaotic systems and the inherent unpredictability of quantum systems. All the factors are known in chaotic systems, but not in sufficient detail to make a forecast accurate (as in the weather) while quantum theory doesn’t permit certain kinds of knowledge as it says this knowledge doesn’t even exist.

This to me is the mystery of radioactive decay. We can never tell when a particular radioactive atom will decay as it is completely random. It isn’t that we don’t know, it is that this information doesn’t exist. It could be in the next second, or in the next million years. And yet this randomness somehow gives rise to half life which can be measured to an arbitrary degree of precision. How can such order arise out of randomness and how can a number that is impossible to calculate be infinitely precise?

Yes, we can never tell when a particular radioactive atom will decay as it is completely random, but when we observe large amounts of particular radioactive atoms they do decay in very predictable numbers. This is why we can rely on half lives.

In one of the articles I also found this:

“The proton:neutron ratio is not the only factor affecting nuclear stability. Adding neutrons to isotopes can vary their nuclear spins and nuclear shapes, causing differences in neutron capture cross-sections and gamma spectroscopy and nuclear magnetic resonance properties.”

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