Why exactly do radioactive elements decay in half every set amount of time? What is happening on an atomic scale?
In: Physics
The fixed “half life” is a product of the underlying maths. It comes from the fact that the chance of decaying across any particular time period doesn’t depend on how much time has already passed, only on how long the time period is. What has happened in the past doesn’t matter.
To use an analogy, think about tossing a bunch of coins. If you have tossed a fair coin and got 10 tails in a row, what is the probability the next toss will be a tail? The answer is 1/2. Same as if it was the first time you tossed the coin, same as if you had tossed it 1,000 times and got all tails. What came before doesn’t affect the next probability.
Extending that to decay; imagine we have a huge pile of fair coins and we keep tossing them all. After each toss we remove all the ones that gave us “tails” – these are the ones which have “decayed.”
So after the first toss we should get rid of half of them (as about half will be heads, about half will be tails).
After the second toss we expect to get rid of half of the remaining ones (as the fact they all landed heads the first time doesn’t change the chance of getting a head or tail the second time).
And after the third toss, we’ll probably get rid of another half of what’s left (so down to 1/8th of what we started with).
While each individual coin is random, if we have enough of them we can make pretty solid predictions. Each toss we get rid of half, so we have a “half-life” of 1 toss.
If we did this with 6-sided dice instead, and threw out the ones that gave us a six, we’d have different numbers but the same effect; getting rid of 1/6th each time. If we do the numbers we’d expect the number of dice to half every 3.8 rolls (so half after 3.8 rolls, a quarter left after 7.6 rolls and so on) – obviously we couldn’t actually do 3.8 rolls, but after 19 rolls we’d expect to have 1/32nd of our dice left). Our half-life now is 3.8 tosses.
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To get into more physics/maths:
Radioactive decay is generally modelled as a *stochastic* process. It is random (in the sense that it is impossible to tell when any single thing will decay) but it is predictable (in that it follows mathematical models for probabilities quite well).
Let’s pick a particular thing that will decay; a neutron. A neutron, on its own, will eventually decay into a proton and some other stuff.
This will happen *randomly*, but in a way we can model. Given any particular time period (say 1 minute) there will be a certain probability that any one neutron will decay (around 6.6% for a neutron). This is impossible to predict for any one neutron, but if we have lots of them we can start making predictions (there is some maths that tells us this, but let’s not get into that for now).
If we have 1 neutron, over 1 minute it may or may not decay. We don’t know, and we can’t tell.
But if we have 1,000,000 neutrons (still a tiny amount of stuff), in one minute, we would expect 6.6% of them to decay – so around 66,000 of them. Now, we probably won’t get exactly that number – we might get 66,500 or 65,500, but we’ll probably get close to 66,000, and in a way that we can model and predict. [Disclaimer; this doesn’t quite work for 1 million neutrons as they will interact with each other; once some have decayed, later decays will cause some of the decayed ones to turn back into neutrons – these numbers only work if all the neutrons are on their own.]
So, if there is a certain probability any one neutron will decay in 1 minute, we should be able to invert that and find the *time* that will give us a 50% probability of decaying. And for a neutron, if we do the numbers, we get a time of about 10.2 minutes. So if you leave a neutron alone for 10.2 minutes there is a 50% chance it will decay.
Now let’s go back to our 1,000,000 neutrons. In 10.2 minutes we expect 50% of our neutrons to decay. So we will have 500,000 decayed and 500,000 left (plus or minus a bit, given the random nature of things, but a relatively small bit). What about in the next 10.2 minutes? Well, now we are starting with 500,000, and but we still expect half of them to decay in a 10.2 minute period. So we’ll lose another 250,000. And for the next 10.2 minutes, now we have 250,000, and in a 10.2 minute period we expect half to decay taking us down to 125,000.
So each 10.2 minutes we lose half of our neutrons, but *only counting the neutrons we started that 10.2 minute period with*. This 10.2 number (like the 6.6% number) doesn’t depend on how many neutrons we have. However many we have, in any 10.2 minute period we expect half of them to decay.
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