Saturday, 21 July 2018

radiation - What does the decay constant mean?


In my curriculum, the decay constant is "the probability of decay per unit time"


To me, this seems non-sensical, as the decay constant can be greater than one, which would imply that a particle has a probability of decaying in a time span that is greater than 1.



Can someone explain this?



Answer



You're missing two things. First, that the decay constant is the probability of decay per unit time. That part is important. The actual decay probability over a short time period is equal to the probability per unit time, multiplied by the time period:


P=λΔt


λ can be as large as you like, but for a small enough interval Δt, you'll still have P<1. So there's no contradiction there.


The other thing you're missing is that λ is only the probability per unit time given that the nucleus has not already decayed. That's also important. You have to start with an undecayed nucleus.


So let's say you have an undecayed nucleus at t=0.


P0(decayed)=0P0(undecayed)=1


After some short time Δt, the probability that it will have decayed is λΔt, as above.


P1(decayed)=λΔtP1(undecayed)=1λΔt



Now consider the next time interval, from t=Δt to t=2Δt. If the nucleus didn't decay in the first time interval, it has a probability λΔt of decaying in this second interval. But if the nucleus did decay in the first time interval, the probability that it will have decayed by the end of the second time interval is 1. So overall, the probability that it has decayed by t=2Δt is


P2(decayed)=P1(undecayed)λΔt+P1(decayed)(1)=(1λΔt)λΔt+λΔt=(2λΔt)λΔtP2(undecayed)=P1(undecayed)(1λΔt)=(1λΔt)2


You can probably see the pattern from here:


P3(decayed)=P2(undecayed)λΔt+P2(decayed)(1)=(1λΔt)2λΔt+(2λΔt)λΔt=(33λΔt+(λΔt)2)λΔtP3(undecayed)=P2(undecayed)(1λΔt)=(1λΔt)3


In particular, at t=nΔt,


Pn(undecayed)=(1λΔt)n


Now, in the limit where Δt is short, and n is large, as it must be if T=nΔt is going to be a normal-scale time interval, you may recognize this as an exponential:


lim


So the equation for exponential decay emerges naturally from the fact that the decay constant is the decay probability per unit time for an undecayed nucleus. (Or of course the same argument applies to any other system that undergoes exponential decay, not just nuclei.)


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