Tried looking it up and it got complicated real quick.
In: Physics
One nice model to think about radioactive decay is to imaging having a massive bucket of dice. You roll the whole bucket and then remove any dice that landed on a “1,” then collect all the dice back in the bucket and repeat.
If you were rolling a bunch of standard 6-sided dice then you’d expect 1/6 of them to roll a “1” on any given roll. After one roll you have 5/6 of the original number of dice (83%), after two you have 25/36 (69%), three rolls reduces this to 57%, and a fourth roll takes things to 48%. You could keep doing this over and over again and the number of remaining dice would get lower and lower. Eventually each die will roll a “1,” but there’s no guarantee of how long it’ll take and the more dice you start with the more likely it is that one of the dice has a super long lucky streak.
However, regardless of the number of dice you start out with it’ll always be about four rolls to halve the number of dice remaining, at least on average (statistical abnormalities are more common with smaller numbers of dice, but roll a bucket of a billion dice and the results are likely to be pretty close to average). We could say that these six-sided dice have a half life of about four rolls and we could even start describing things in terms of fractional rolls if we’re not too afraid of logarithms (the actual half life of D6 dice is about 3.80 rolls).
By comparison, we might repeat the experiment with 20-sided dice. As before, only a 1 results in the die being removed. Here we see that after one roll we still have 95% of the dice remaining. 10 rolls in and we still have about 60% sticking around, and somewhere around 13-14 rolls we finally hit the halfway point. Thus we see that a D20 has a much longer half life than a D6.
In both of these setups we arrived at a half life by looking at how many dice we returned to the bucket and kept going until the bucket was half as full as it started. However, suppose you start with a bucket with a million dice inside and you don’t know how many sides they have. You roll them and find that 10 dice turned up with a 1. You don’t have to put all the dice back into the bucket to have a good idea about how many sides these dice must have–it’s probably about 100,000, to give ten rolls of 1 out of a million attempts. As soon as you know how many sides the dice have it’s just a matter of some arithmetic to find out how many rolls it would take to come down to half the original population–about 69,000 in this case. Here it’s a lot easier to count the 10 dice we removed than to count the 999,990 that we put back in the bucket.
Turning back to radioactive half lives, much of the same logic still applies. Of course, radioactive decay doesn’t happen in discrete steps like the dice throwing game, but it does follow similar probabilistic patterns. For some fast-decaying isotopes it’s sufficient to just start with a sample and wait until some of it has decayed. You could wait for half of it to be gone, or with a more precise scale you could get away with letting just 1% decay, or less.
However, for a slowly-decaying isotope like U-238 waiting for even 1% of a sample to decay is wildly impractical, so instead of “tracking the dice in the bucket” we “count the ones that were removed.” This could take the form of putting a known quantity of the isotope into a position where we can count the decays. Even with a 4.5 billion year half life there will still be some decays–a billion atoms is nothing, and a billion billion is just starting to get into units that make sense at a macroscopic scale (that would be about half a milligram of U-238). In fact, this decay counting approach is only attractive for long lived isotopes since the decays quickly get too numerous to count for something that decays faster.
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