What are the longest half-lives we can detect experimentally? What stops us going further? Are we trying to?

Question

Xenon 136, apparently, has a half-life of 2.11×10²¹ years. This strikes me as a humongously long time to run an experiment, clocking in at about 11 orders of magnitude longer than the age of the universe.

This question puts some numbers in and shows that it's really quite reasonable to determine half-lives in the range of 10⁹ years by simply counting the number of decays in a decently short amount of time. However, if you do equivalent numbers for xenon 136 you get numbers which push this sort of thing much harder: 1g of ¹³⁶Xe will produce about 1.5 decays in a year, and this already occupies 160 ml in standard gas phase conditions. To get reasonable statistics, you will need either a lot of isotopically pure xenon or a long time of very efficient, background-free detection, or (probably) both.

So, 10²¹ years is still sort of reasonable, but push in a few more zeros and you start having an untenable experiments in your hands. And, indeed, a quick browse through Wolfram Research's curated IsotopeData suggests that the longest known half-life is that of ¹³⁰Te, at around 5×10²³ years.

I seem to have run out of questions and answered the ones I initially had, so instead I'll push this a bit further: are these walls hard? That is, can we plausibly measure longer half-lives? Are there current experiments trying to do so? Is ¹³⁰Te indeed the longest we know? Given the difficulty in establishing that it decays at all, the different 'observationally stable' isotopes of tellurium seem more observational and less stable than they do at first sight.

Answer 1

are these walls hard? That is, can we plausibly measure longer half-lives?

No, the walls are not hard, and it is certainly possible to experimentally confirm longer half-lives by doing what the question says:

you will need either a lot of isotopically pure xenon or a long time of very efficient, background-free detection, or (probably) both.

That is, get a lot of, say, xenon (on the multi-ton scale, rather than the grams used in the question, in condensed phase) in an underground cavern, away from background radiation, look for signatures of the decay, count the number of decayed atoms, compare against the total number of atoms present, and infer a half-life from it. As of the writing of this answer (May 2019, some four years after the question), the record has indeed been upped from the ${}^{136}\mathrm{Xe}$ half-life by the XENON1T experiment, which used that method to confirm a much longer half-life for an isotope of xenon on the light end of the range $-$ ${}^{124}\rm Xe$, with a measured half-life of $1.8\times 10^{22}\:\mathrm y$. (Also discussed on this site here.)

Presumably there are similar searches going on with isotopes that have slightly longer half-lives, but they'll not be the easiest to find unless one works inside that community.

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That said, these are not the longest half-lives that we can infer from indirect measurements. The tellurium-130 assignment by the Wolfram curated data looks fairly shaky upon closer inspection (and indeed most half-lives in this class probably have a fair degree of scatter and need to be researched carefully before they're taken without a grain of salt), but there's definitely literature describing half-lives on that timescale.

Specifically, the Wikipedia page on the isotopes of tellurium marks tellurium 128 as the longest known half-life at $2.2\times 10^{24}\:\rm y$, though without a clear literature assignment for where that figure comes from and how it was obtained. I'm not an expert in how you sift through this literature, so the best I could find (which is still pretty good) was the paper

Precise determination of relative and absolute ββ-decay rates of ${}^{128}\rm Te$ and ${}^{130}\rm Te$. T. Bernatowicz, et al. Phys. Rev. C 47, 806 (1993)

which pins the ${}^{128}\rm Te$ half-life at $7.7\times 10^{24}\:\rm y$. This is done through geophysical evidence, by doing precise mass spectroscopy to determine the relative abundances of the different isotopes of tellurium and of their gaseous decay products, together with radionuclide dating of the minerals that contain those isotopes (using other elements with better-established, shorter half-lives), which then enables a measurement of their relative half-life of the different isotopes; one then measures the shorter half-lives and infers the longer ones.

Answer 2

In search of proton decays, the limit for the proton half life is now experimentally in the order of $10^{34}$ years. This means that such long half lifes are actually measurable when one want to undertake the effort to do so.