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There are already some questions about long half life times for radioactive elements, explaining how to calculate the half life time.

Now I am wondering: When you have some radioactive material and you observe the decay, how can you be sure that all individual decays are actually registered by the detectors? Does the background radiation influence the measurement, and how large are the effects? How are experiments for measuring very long half life times (larger than 1 billion years) usually setup?

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  • $\begingroup$ Have you investigated the idea of Activity? $\endgroup$ – Declan Apr 22 '16 at 19:16
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    $\begingroup$ This report on Uranium Half-Lives: A Critical Review, done at Brookhaven (1981) will provide some basic answers. The fundamental approach begins with activity, as @Declan suggested. $\endgroup$ – Peter Diehr Apr 22 '16 at 19:28
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    $\begingroup$ A billion years is still quite fast as you then only need a small amount of the substance to get a large number of decays per second. It becomes a lot more difficult for half lives longer than $10^{18}$ years, see e.g. here. $\endgroup$ – Count Iblis Apr 22 '16 at 19:33
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    $\begingroup$ You are basically asking how one designs and calibrates non-trivial nuclear experiments that have to deal with low count rates among non-trivial background. The best answer to these kinds of questions is to grab the detailed descriptions of these experiments in published articles and, even better, in the design reports of their collaborations. Instrument design is "learning by doing", you have to look at what others have done and then you can refine it. $\endgroup$ – CuriousOne Apr 22 '16 at 19:34
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    $\begingroup$ Re, "How can you be sure that all individual decays are actually registered by the detectors?" You can't be sure of that because they won't all be registered. Some of the particles emitted in the interior of the sample will never escape: They'll be absorbed in the sample material itself. Some of the particles will follow trajectories that do not pass through the active volume of the detector. Some of the particles will pass through the active volume, but not interact with it. Meanwhile, some high energy particles from the outside the experiment will be counted. $\endgroup$ – Solomon Slow Apr 22 '16 at 19:56
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Disclaimer: I've never done the particular class of measurements you ask about, but I have done other low raw-rate, precision measurements (neutrino mixing and weak form-factors).

The focus of experimental work for low count rates is multi-pronged:

  • Maximize the quantity of data. For counting experiments the raw fractional statistical uncertainty goes by $1/\sqrt{N}$ where $N$ is the number of events observed. If your count rates are measured in events-per-day, it can take a long time to get even a third digit of significance.

  • Know your acceptance and efficiency to high precision. You won't get all the events and that isn't a problem as long as you know what fraction you get to a level competitive with either your statistical uncertainty or the desired precision of the measurement. Generally you can measure at least some of this using the same apparatus you're using to collect the data. And that's not actually as circular as it sounds, but it takes care to get right.

  • Toward those ends you simplify the geometry whenever possible. (This is basically something that has to be designed in, BTW, so you worry about it from the very conception of the project.)

  • Finally, the thing you will almost certainly spend an inordinate-seeming amount of time on is characterizing and minimizing the backgrounds. Again, you must beat these down until they no longer dominate the error budget. Again, many of these will be measured as side channels in the experiment. Backgrounds that can't be measured in situ like that are described as "irreducible" and have to be tackled with some combination of prior-planning, secondary measurement using additional instruments or lab work, or in the worst cases rebuilding or re-processing part of your detector.

Of course, each time you really succeed in one of those bullet points the target precision gets higher and the effort shifts to the new worst line in the error budget and around-and-around you go.

None of this is special to half-lives: it's just the stuff you have to do to make precision measurements in nuclear or particle physics.

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