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So, as I understand it, in a substance that is made of radioactive elements, the half-life tells us how long until the half of those atoms decay into their next atom [is there a name for that: the element or isotope that is the result of a prior radioactive decay?]. My question is, is there some sort of pattern to which atoms decay at which time, or is it some miraculous property of quantum mechanics that somehow each atom knows when to decay? or do I just understand radioactive decay incorrectly?

Are the particles entangled or in some otherway attached (aside from their molecular bonds)? If there is no known answers, I would prefer any serious, leading theories.

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  • $\begingroup$ In order for the particles to decay in some other time dependent way, they would have to know when your decay experiment began. What would start that physical clock? A nucleus doesn't know anything about you sitting there with a counter, pencil and a sheet of paper for the diagram. $\endgroup$ – CuriousOne Apr 30 '15 at 5:24
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    $\begingroup$ Essentially the same as Do we really not know why atoms 'decide' to produce a photon?, because emitting a photon is also just a "decay" into a lower energy state. $\endgroup$ – ACuriousMind Apr 30 '15 at 10:01
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    $\begingroup$ Try not to think about atoms as minds, it helps a lot when untangling "mysterious" physics. And don't get stopped on getting an explanation that doesn't actually allow you to anticipate anything different than before - if I tell you that yes, the particles are entangled, what will that allow you to anticipate? You're just getting a convenient label for the same mystery - do not let your confusion disappear just because you added a name for it :)) That said, radioactive decay is so random, we're actually using it to generate truly random numbers - it's completely stateless. $\endgroup$ – Luaan Apr 30 '15 at 11:04
  • $\begingroup$ @Luaan But, if we have a 100 m^3 of a radioactive substance, let it decay, then randomly chop up blocks of it at random time intervals, wouldn't each block still have the same overall decay? At least at macroscopic sizes? Is that just one of those "deal with it" things? $\endgroup$ – Jimmy G. Mar 29 '16 at 12:25
  • $\begingroup$ Yes, each block would still have the same overall decay % per time - but that's really all about how probabilities work in general. The exact same thing would happen with a chopped-up thermodynamics process, or with a chopped-up coin-tossing championship, for example. It's nothing "incomprehensible", it just needs understanding probabilities; the problem is that people suck at that :D If you toss a hypothetical coin, there's 50% chance it will be tails. Another, that's 50% for tails again. It doesn't matter what the previous toss was. Yet, the chance of tossing tails twice in a row is 25%. $\endgroup$ – Luaan Mar 29 '16 at 12:34
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There are no patterns. When a particle decays, the moment when it does so is absolutely random, chosen from the distribution $$ P_{\rm decay}(t\lt T\lt t+dt) = \frac{dt}{t_0}\cdot \exp(-t/t_0) $$ For $t=t_0$, the beginning of time when we knew that the particle still existed, the exponential is equal to one and we see that the "probability of the decay per unit time" is $1/t_0$. As the probability that the particle still exists exponentially decreases, so does the probability that it decays at a later moment.

The randomness of the decay time is just another example of the randomness that quantum mechanics, the basic framework for all the laws of physics since 1925, predicts for every phenomenon in Nature.

In the most widespread description of quantum mechanics, the decaying particle is described by a wave function. And that wave function evolves into a superposition of the undecayed and (various) decayed components, and the probability amplitude (value of the wave function) associated with the undecayed particle decreases as $\exp(-t/2t_0)$. This probability amplitude has to be squared and the result, $\exp(-t/t_0)$, gives us the probability that the particle hasn't decayed yet.

Theories that would try to find some "internal" reason why the particle decayed at the given moment are called "hidden variable theories" and they may be shown incorrect – either incompatible with the experiments about the decay in this case, or with experiments backing the special theory of relativity. So physicists have to embrace the intrinsic randomness of Nature as a fact. The randomness of the decay time is a Nature's perfect random generator, one that can't be fooled or cheated.

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    $\begingroup$ You can't predict what specific particle will decay at a given time, but you can predict the percentage of all the particles that will not have decayed by that time. Is this a correct understanding? Can you determine how stable an element is by its structure, and can you compare stability among elements? $\endgroup$ – Ernie Apr 30 '15 at 5:06
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    $\begingroup$ Yes, the percentage of particles at a given time that will have decayed is computable. But the precise integer may fluctuate. So if you expect $N$ particles to decay in average by a certain moment, what you get in reality will be $N\pm \sqrt{N}$ or so – the error is comparable to the square root of $N$, quite a usual thing in random processes. And yes, also, we have the theories that allow us – at least in principle and often in practice – calculate the half-times for all particles or nuclei etc. It's understood why some isotopes decay more quickly, others less quickly, and others are stable. $\endgroup$ – Luboš Motl Apr 30 '15 at 5:24
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    $\begingroup$ Just to be sure. The percentage of the particles (of the same kind) that have decayed is nothing else than the probability that one particular of them has decayed. $\endgroup$ – Luboš Motl Apr 30 '15 at 5:26
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Is there some sort of pattern to which atoms decay at which time, or is it some miraculous property of quantum mechanics that somehow each atom knows when to decay?

Atoms are dumb. They don't know anything. Radioactive decay is a memoryless process, a process that doesn't depend on history. Consider three atoms of radon 222. One was created a month ago (8 half-lives), another four days ago (~1 half-life), and the third, 12 hours ago (~ 1/8 half-life). Which will be the next to decay? The atoms don't know. Nobody knows; in fact, nobody can know. Each of the three atoms has the exact same small chance of decaying in the next minute.

Radioactive decay is the canonical example of a Poisson process. Knowledge (or state) is not required in a Poisson process. State gets in the way. An ideal Poisson process is stateless and memoryless. While there are lots and lots of examples of processes whose probability distribution is close to Poisson, nothing comes closer to it than does radioactive decay.

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You have just read reasonable answers of knowledgeable people, so now you know that "radioactive decay...- it's completely stateless (@Luaan)", "There are no patterns" (@Luboš Motl), and "Atoms are dumb" (@David Hammen). However, there is a bit more to it. Atoms may be dumb, but they happen to know quantum mechanics much better than we, mere mortals, do. So there may be randomness, but there cannot be perfect randomness. Someone Khalfin showed many years ago that strictly exponential decay is not compatible with quantum theory, there must be some tiny deviations, both at very short and very long times. Please see references to theoretical and experimental work in my answer at Does average lifetime even mean anything?

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