This may be a somewhat philosophical question, but here goes.

Wikipedia claims that several nuclides (e.g. hydrogen-5) have half-lives shorter than $10^{-22}$ seconds. This is on the same order of magnitude as the amount of time that it takes light to cross a nuclear diameter - i.e., the shortest possible time that (it seems to me that) it even makes any sense to talk about the logical possibility of a bound state.

Given this, is it really meaningful to call something like hydrogen-5 an "atomic nucleus", as opposed to just "one proton and four neutrons that happen to momentarily pass very close by each other, interact via the strong interaction, and have the proton and two of the neutrons fuse into a tritium (hydrogen-3) nucleus while the other two neutrons go on their merry way"?

Clearly once we get to nuclides with half-lives in the seconds and minutes, it becomes useful to think of them as metastable bound states (and indeed, for the purpose of applications like atomic or molecular physics or chemistry, to abstract them out to new "primitive particles"). But is there any kind of sharp (or even semi-sharp) notion of a "nucleus" for the incredibly unstable nucleides - e.g. any specific phenomenological signatures that differs from just colliding a bunch of protons and neutrons together? Or does a concept like "a hydrogen-5 nucleus" really just summarize "the complicated scattering process that ensues when we collide together one proton and four neutrons"?

  • 1
    $\begingroup$ From physics.stackexchange.com/a/468493/123208 "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". $\endgroup$
    – PM 2Ring
    Dec 25, 2023 at 19:38
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    $\begingroup$ IUPAC defines an element to exist if its lifetime is longer than $10^{−14}$ second, which is the time it takes for the atom to form an electron cloud. $\endgroup$
    – Farcher
    Dec 25, 2023 at 21:47
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    $\begingroup$ @Farcher True, but that's their convention for defining when a new (transuranic) element has been created, not a nuclide, which is what my question was about. The Wiki article lists 37 nuclides with half-lives shorter than $10^{-14}$ seconds. $\endgroup$
    – tparker
    Dec 25, 2023 at 23:12

2 Answers 2


The answer to your question is yes, in most cases. The unstable bound states that you refer to are known as resonances and are represented by non-normalizable (even in the dirac delta sense) solutions to the Schrodinger equation. In scattering theory these solutions show up as poles of the cross section in the lower-complex energy plane. Due to the analyticity of the cross section, these poles lead to peaks on the real energy line which we can detect in experiments.

In fact, this is precisely how new particles are found by experimentists.

Tong has a very nice chapter about this on his free online notes. Check page 221 for an example of unstable nucleii showing up as poles in the scattering cross section.

  • $\begingroup$ By "peak", do you just mean any local maximum in the total scattering cross-section as a function of center-of-mass energy? And the more unstable the bound state, the broader the peak (and therefore the less "peak-y")? $\endgroup$
    – tparker
    Dec 25, 2023 at 21:14
  • $\begingroup$ Does the theory of QCD predict that some combinations in inbound particles will product a strictly monotonic cross-section function? Or is there always some peak, that gets gets unmeasurably tiny for large numbers of particles? $\endgroup$
    – tparker
    Dec 25, 2023 at 21:16
  • $\begingroup$ And is a pole in the lower complex plane the only way to get a local maximum on the real energy line? $\endgroup$
    – tparker
    Dec 25, 2023 at 21:40
  • $\begingroup$ By analyticity and causality (using the Kramers-Kronig relations) the poles in the lower complex plane completely determine the scattering amplitude (up to a phase) for a fixed number of particles. There are also poles on the real line corresponding to thresholds. $\endgroup$
    – tobi_s
    Dec 26, 2023 at 7:57
  • $\begingroup$ States in 5He, 5Li and 8Be (including ground states) are also populated via beta-delayed particle decay chains from 9Li, 9C, and 8Li/8B. (Thank goodness for long-lived 8Be!) $\endgroup$
    – uhoh
    Dec 26, 2023 at 20:54

Any nucleus consisting of parts that have less energy than the sum is unstable.

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    $\begingroup$ The OP is asking for the lower bound on what counts as an "unstable nuclide", not the upper bound. Note that they are contrasting hydrogen-5 with hydrogen-3, not hydrogen-1 or -2. $\endgroup$
    – No Name
    Dec 26, 2023 at 7:44

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