# I Xenon 124 -- How did they calculate the half life? (trillions of years)

#### idea2000

Summary
How to calculate half life
How did they calculate the half life of xenon 124 to be longer than the age of the universe if they only observed one decay??? Is there some way to estimate half life???

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#### Orodruin

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You measure it. Half-life is a statistical quantity and tells you the time it would take for half the particles to decay. If you collect enough particles you can measure very long half-lives. If you observe N particles during a time t, then you would expect to see $N(1-1/2^{t/T})$ decays if the half-life is T. If t is much smaller than T, this is approximately given by $Nt\ln(2)/T$. Thus, even if the observation time t is much less than the half-life, you will see decays if you collect enough particles, ie, you take a large enough N.

#### idea2000

Yes, however, the half life of xenon 124 is 1 trillion times the current age of our universe and was just reported to be observed yesterday for the very first time, ever. How did they calculate the half life based on one observation???

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If you have a box of 100 atoms and 50 of them decay in a year, what is the half-life?
If you have a box of 100 atoms and 5 of them decay in a year, what is the half-life?
If you have a box of 1000000 atoms and 5 of them decay in a year, what is the half-life?
If you have a box of 1023 atoms and 5 of them decay in a year, what is the half-life?

#### fresh_42

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I understand this principle. What makes me wonder is, that they get enough isotopes, especially if they are created synthetically.

#### DrClaude

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I understand this principle. What makes me wonder is, that they get enough isotopes, especially if they are created synthetically.
They use natural xenon, 2 tonnes of it, with an abundance of 1 kg 124Xe per tonne.

#### Dr.AbeNikIanEdL

if they only observed one decay
In addition to what was already said, where does this come from? According to the article (https://www.nature.com/articles/s41586-019-1124-4), this is not just based on one event.

that they get enough isotopes, especially if they are created synthetically.
If I understand correctly, it is just the natural abundance in the detector. You only get ~ 0.1% Xenon 124, but then you take a ton of Xenon.

#### Orodruin

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get enough isotopes
There is only one isotope of relevance, 124-Xe. The issue is getting enough nuclei of that isotope.

Yes, however, the half life of xenon 124 is 1 trillion times the current age of our universe and was just reported to be observed yesterday for the very first time, ever. How did they calculate the half life based on one observation???
For the future, you should remember to include a reference to the original paper when you have a particular issue that you want to discuss. In this case, the appropriate reference would have been the Nature paper.

The paper quotes a measured number of events of $N_0 = 126\pm 29$ over 177.7 days of data taking. They have an isotopic abundance of about $10^{-3}$ and the efficiency is almost one. This means that they have
$$N = 10^{-3} \frac{M N_A}{m}$$
124-Xe atoms, where $M$ is the total xenon mass, $N_A$ the Avogadro number, and $m$ the molar mass of xenon (ca 131 g/mol), leading to $N \simeq 5\cdot 10^{24}$. The estimate of the half-life is therefore (solving from the expression in #2)
$$T \simeq \frac{N t \ln 2}{N_0} \simeq 10^{12} T_{\rm univ},$$
where $T_{\rm univ}$ is the age of the Universe.

#### fresh_42

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There is only one isotope of relevance, 124-Xe. The issue is getting enough nuclei of that isotope.
I had encountered another element recently with an equally absurd half time and it wasn't natural, or extremely rare. Unfortunately I have forgotten which one. I only remember that I asked myself the same question, and a high number of isotopes didn't seem to provide a solution. The only other possibility was an extremely long observation time, but it made me wonder, whether there are other methods, maybe theoretical calculations.

#### mfb

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There are just a few isotopes with measured extremely long lifetimes. 130Te with 8*1020 years and 128Te with 2*1024 years are notable examples, both make up ~1/3 of natural tellurium each, however (list of isotopes). The first lifetime is measurable in the lab, the second one is measured based on decay ratios of the two isotopes in very old rocks.

Every isotope with a measured lifetime larger than 1018 years has a natural abundance of at least 0.1%. List by half life

There are theoretical calculations for a lot of unmeasured lifetimes, of course, and theoretical calculations are not limited to naturally occurring isotopes. Long-living isotopes typically exist in nature, however. They might live too long for us to detect the decay but we won't suddenly find an isotope that lives so long but doesn't occur in nature.

"Xenon 124 -- How did they calculate the half life? (trillions of years)"

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