# Physical limit for the shortest possible half-life?

This question comes from observation that there are no known half-lives in range;
$$1\times 10^{-10}$$ seconds to $$1\times 10^{-21}$$ seconds.
(Except Beryllium-8, which has a half-life of o $$7\times 10^{-18}$$ seconds.)

As the isotopes are mostly produced with a neutron Flux, which practically means, that neutrons are colliding with a certain velocity to the target particle, I became an idea what this actually means.

A neutron is very similar to a proton, and Proton diameter is said to be $$0.84\times 10^{-15}$$ meters. If I calculate with typical Neutron speed which is present in Nuclear-fission; $$1960000$$ m/s, it would take at least $$4\times 10^{-22}$$ second, for a Neutron to travel away from its position which is farther away than it's own size.

But as radioactive decay is happening all the time, more presenting speed would be that of Thermal Neutrons $$2200$$ m/s. This means that Neutron needs $$4\times 10^{-19}$$ second to change is position more than its size is.

Calculation with the speed of ultra-cold Neutrons; speed $$<200$$ m/s, gives for a time $$4\times 10^{-18}$$ seconds. This simple rule would mean that only Beryllium-8 would have long-enough half-life to be an independent nucleus, compared to pure neutron collision. But looking this isotope, shows, that it decays with $$\alpha$$-decay. Which in this case means that it would split in two equal Helium-4 nucleus.

QUESTION;
Have such a theoretical limit for an independent isotope established? ..And if yes, how is it explained that some isotopes with just $$23\times 10^{-24}$$ like Hydrogen-7 are considered to be something else than just colliding neutrons?

• – Lewis Miller Oct 17 '16 at 1:28
• @LewisMiller Thanks. This gives something real. Ie. if the "electronmass-energy; 0.5 MeV is calculated with this, the associate lifetime would be 1.3x 10^-21 sec and this shows that high energy can have shorter lifetime. and ie. Neutron Energy 938 MeV gives associate lifetime of 0.7 x10^-24 seconds. – Jokela Oct 17 '16 at 6:25
• Are neuton, neutrons, and neurons, the same thing? – lcv Mar 25 at 9:24
• @lcv Thanks for your comment. I do have problems with writing due to reasons stated in my profile. It's certainly easier to type clean, if you only need to use one language. Then you can even use a program which corrects everything for you, and you can feel particularly smart. But If you write > 4 languages in daily basis. You simply cant use them any more. But then, understanding is way above words&typos. I actually feel sorry for you, when you needed to ask this. – Jokela Mar 25 at 10:00
• I edited the question for you @PM2Ring – Rick Mar 25 at 13:05

First of all, the Wikipedia article you linked to is not complete and doesn't claim to be. (As of the time of writing, the header says "This list is incomplete; you can help by expanding it.") For example, magnesium-19 has a half-life of about $$4 \times 10^{-12}$$ s as measured by Mukha et al.

The main reason we can't observe isotopes with femtosecond half-lives is experimental. Very short-lived isotopes are normally detected by smashing nuclei into pieces using high-energy collisions, then tracking the products using magnetic separation, using an instrument like the BigRIPS separator. However, the speed of light becomes a limiting factor: in $$10^{-15}$$ s, a nucleus cannot travel more than 300 nm. Accurately tracking nuclei over such short distances is very difficult (even the CMS tracker at LHC is only accurate to around 10 $$\mu$$m). In theory, relativistic time dilation might help a little, but on the other hand, isotope separation becomes harder for very fast-moving, short-lived nuclei because magnetic fields barely affect their motion.

But to answer your question, the theoretical limit on very short half-lives is the fact that information cannot travel faster than the speed of light. Obviously, a nucleus must be bound before we can discuss its half-life. But it makes no sense to say that a nucleus is bound if it decays before all of its constituent protons and neutrons "know" that it has formed. Therefore, since nuclei must be larger than the radius of a proton, a hard lower limit on nuclear half-lives is $$r_\text{proton} / c \approx 2.9 \times 10^{-24}$$ s. (Not coincidentally, the strong nuclear interaction has a characteristic time scale on the same order of magnitude.) Indeed, the half-lives of hydrogen-6 and hydrogen-7 are within an order of magnitude or two of this limit.

• Thanks for the answer, I agree with this lower hard limit. Do you have any idea / explanation why there are no isotopes existing with half-lifes in the wide range 2*10^-10...1*10^-21 ? I agreee that the list is not complete, but I actually went through the all the isotopes from their independent pages, like; en.wikipedia.org/wiki/Isotopes_of_neodymium#List_of_isotopes ...and didn't find any. – Jokela Mar 25 at 7:24
• @JokelaTurbine This was discussed in my answer. :) Lots of isotopes exist with half-lives in this range. For example, magnesium-19 has a half-life of $4\times 10^{-12}$ s. – Thorondor Mar 25 at 7:56
• Thanks, Yes. Sorry, my eyes didn't catch the small difference. So now the cap is from 4x10^-12 to 8x10^-18 which still means there is a half-life free time range of 1000 000 magnitude. And if this Beryllium-8 is like 2x helium, it could be considered as single anomaly with observational difficulties, which means the range is 4x10^-12 to 1.3x10^-21, so the observed magnitude gap spreads to 1 000 000 000. – Jokela Mar 25 at 8:13
• @Jokela again, the reason is just limitations of available detection methods. Particle tracking works down to about $10^{-12}$ s, while for extremely short half-lives below about $10^{-20}$ s, the decay energy is so high that it is possible to determine the half-life by directly measuring the missing mass-energy in the decay products. (The decay energy is inversely proportional to the level width.) The diagram on page 4 of arxiv.org/pdf/1111.0482.pdf might help clarify this for you. – Thorondor Mar 25 at 10:21
• Thanks again. So this might also be sort of an Observational bias. I mean before asking I actually really went through all pages in wekipedia and checked the values. It's notable that most of the isotopes have some known value. So if this gap is covered by some new values, these must come from new observations, like it's in the case of Magnesium-19. – Jokela Mar 25 at 13:30

A decay time of ${10^{-24}}$ s is characteristic for a process mediated by the strong nuclear interaction.

Also an intuitive picture like colliding nucleons is not a viable approach here. In particular you should not think of the neutron as a sphere with a certain diameter.

• Thanks for the answer. Even if I use the "Strong interaction range, the dimensions remains very same. en.wikipedia.org/wiki/Strong_interaction I also understand that high speeds produces relativistic effects. -But this is not about high speeds- – Jokela Oct 16 '16 at 16:09
• No this is not about speeds. Neutrons do not physically move and collide inside the nucleus. – polwel Oct 16 '16 at 16:11
• en.wikipedia.org/wiki/Neutron_emission ..in which a neutron is simply ejected from the nucleus, and en.wikipedia.org/wiki/Neutron_capture ....is a nuclear reaction in which an atomic nucleus and one or more neutrons collide and merge to form a heavier nucleus. – Jokela Oct 16 '16 at 16:16
• Uhm, why the downvote? You asked if there was an explanation for why nuclei could decay this quickly. The answer is strong interaction. And no, you cannot explain it via colliding neutrons. Neutrons leaving the nucleus with some kinetic energy does not mean that the reaction is triggered by kinetic collisions in the nucleus. – polwel Oct 16 '16 at 16:21
• Sorry, I can vote you up somewhere else. Thanks for answer, but I asked what is the physical limit for shortest possible half-life. -A theory- which ie. explains that the shortest possible half-life is 10^-30 seconds -because of some reason. Or can it be ie. 10^-90 seconds? – Jokela Oct 16 '16 at 16:30