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If we start with a particular unstable isotope and this begins a decay chain, and if at some steps along the way the parent nuclide can decay by either beta or alpha decay, why is it that we end up with the same isotope at the end in all cases?

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  • $\begingroup$ Consider the alternative - multiple possible end isotopes. But, that would require that they all are equally stable, or else decay could continue. $\endgroup$
    – Jon Custer
    Commented Sep 18, 2017 at 13:27
  • $\begingroup$ I doubt that the assertion is true. Do you have evidence for this? I would certainly be willing to believe that in most cases, all the chains terminate in the same isotope, and that in nearly all cases, there is a high probability of terminating at the same isotope. $\endgroup$
    – user4552
    Commented Sep 18, 2017 at 15:53
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    $\begingroup$ In particular any decay chain that involves a spontaneous fission almost certainly has multiple end states. $\endgroup$ Commented Sep 18, 2017 at 16:54

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The assertion that radioactive decay chains terminate at the same isotope regardless of different decay paths is not correct. Potassium-40, for example, can decay to calcium-40 via $\beta^-$ decay, or to argon-40 either via electron capture or via $\beta^+$ decay. Both calcium-40 and argon-40 are stable nuclides.

In addition to the above example, decay chains that involve fission inevitably end up with a number of stable end states.

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Some decay chains "terminate" in the same isotope, but this is part coincidence and part a testament to the stability of that particular isotope. The decay chain of silicon-22 (simplified map below), for example, has several branches which terminate with neon-21.s-22 decay chain, simplified

The neon-21 is generated through the decay of two different elements (from beta+ decay both times) with extremely different half-lives. But the chains "stop" here because neon-21 is stable, and not for any other reason. The fact that two different chains start at different isotopes but terminate at the same one is not because of some property of these chains, it is because of a property of the final isotope (its stability). And it's important to note that our definition of "stable" usually means that we have not yet learned an isotope's half-life. All isotopes will eventually decay into what is basically just particle soup. All chains will end up the same at the end of the universe!

Matter is constantly trying to reach the lowest-possible-energy configuration (most stable isotope, here). Some ways of getting there are better than others, and some isotopes are more stable than others. These are the ways that are preferred probabilistically. As a result, many chains - but not all - "pass through" or even terminate at the same isotope. A counterexample for the idea that all chains pass through the same isotopes would be the comparison of something like gold-196 with silicon-22; the gold stops at a stable mercury and platinum isotopes, and never makes it down to isotopes with atomic numbers as low as silicon. @David Hammen's answer gives a similar counterexample with isotopes much closer together in mass. And as @Patrick Weith touched on in their answer to your question above, isotopes of similar mass or atomic number are generally more likely to have similar decay chains, just because there aren't that many options for how they can decay.

In conclusion, it's sort of just coincidence that certain chains end together, and also kind of fate.

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If you look at a table of nuclides (http://www.creation-science-prophecy.com/u238series.gif) and the possible paths, you'll realize that the decay processes are all diagonal steps, either to top-left or down-left. In the case of alpha decay it is a double step to down-left. So there are 2 types of processes, one type reduces protons and neutrons (alpha and ß+) and another type that turns a neutron into a proton (ß-). For each atom the stable isotope lies somewhere in the middle and if you go to high neutron numbers, the processes that turns neutrons into protons have a high rate (-> low half-life), where on the other side, low neutron numbers, the process that reduces neutrons and protons at the same time dominates regarding rates (-> has low half-life). Now take into account that the curve of stable isotopes is a bit curved, so this means, it is possible with these processes to return to a stable isotope and the direction towards stable isotopes is favored.

This answer is for over 99% of the mass of an isotope. There are always other possible decay processes with really low rates (high half-life), that also happen but are not important for over 99% of involved mass. Always consider half-life ranges from microseconds to thousands of years. Whenever there is a process happening on the seconds scale, we don't care about the process happening in years, but in fact it is there and offers for a different path along the nuclides. So if you take into account all small fractions, then the statement that all isotopes end at the same isotope is not true. But for the vast majority of atoms this is true, since moving away from the stable isotopes means the processes pushing back to the stable isotopes becomes faster than the process pushing away (left side: alpha/ß+, right side: ß-)

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