Why exactly do radioactive elements decay in half every set amount of time? What is happening on an atomic scale?

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Why exactly do radioactive elements decay in half every set amount of time? What is happening on an atomic scale?

In: Physics

It’s not exactly a set process.

Think of it like this; you have 100 apples, and every minute you have a 50/50 chance of selling one apple.

The half life is how long *on average* it takes to sell half of your apples.

In a similar fashion, radioactive materials have a chance to lose electrons and break down into other materials. This loss of electrons is what we call radiation.

At an atomic scale, the decay is random. Each atom has a chance to decay in any given period of time that stays constant.

But because there are *so many* atoms, the behavior at observable scales doesn’t seem random. If you flip one coin, it can come up either way. But if you flip a trillion coins, you will get very, *very* close to 500 billion heads (even though each individual coin can still land either way). The number of atoms in a macroscopic sample of radioactive material is more than a trillion trillion, so large that the randomness is more-or-less invisible.

To the best of my knowledge we don’t know the full answer to that question. What we do know is this:

* Whether or not an atom of a radioactive element decays at any given moment is essentially random.
* Different elements are more or less susceptible to decay (i.e., some elements have a higher probability of decaying at any given moment than other elements have).

Imagine you have 100 atoms of a radioactive element. Every second you roll one 6-sided die for each atom, and on a roll of 1 that atoms decays. When you pick up your big handful of dice after the first second and roll them you can reasonably expect that roughly 1/6th of all rolls will be 1s, leaving you with around 83 undecayed atoms. For the next second, you pick up your handful of dice and roll them and can reasonable expect that 1/6th of all rolls will be 1, leaving you with around 69 undecayed atoms. Repeat again and you get 58, and then 48, and then 40, and so on. The half-life would be somewhere between the third and fourth seconds, so this element would have a half-life of about 3.5ish seconds.

Notice that the number of atoms that decay in each iteration gets smaller each time (17, 14, 11, 10, 8, …) because you’re losing a consistent *percentage* of atoms, even though every atom has a consistent probability of decaying during any given second: each second it has a 1-in-6 chance of decaying. It’s kind of a weird property of probability that at large enough scales random events become very predictable, and half-life is an example of that.

The thing we *don’t* know, as far as I’m aware (and someone please correct me if I’m wrong here), is what governs whether or not an atom decays at any given moment. It’s probably not *literally random*, it’s probably caused by something in the atom’s structure or the fabric of spacetime or… who knows what. But there’s a lot about how the universe works at that scale that we’re still trying to puzzle out.

So radioactive atoms are unstable. This instability makes them decay. Some are more unstable than others so they decay faster or slower. Imagine for a moment that each second a dice is rolled for each atom. If the dice is on a right number it will decay and if not it won’t. The more unstable the more correct numbers there are. With large numbers of atoms you will expect a given percent to decay on each dice roll based on how unstable the atom is. The half life describes the expected amount of time for half a sample to decay. Understand that as more decay you are rolling fewer dice but the percent of dice that result in a decay will always be the same for a given element. This also means that for a single atom there is no way to know when it will decay just like there is no way to know the outcome of a single dice.

Hope this helps and feel free to ask for clarification.

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.

It is the mathematical result if each atom has some probability of decaying each unit of time.

If each atom has a 1% each of decaying each second that result in a calculable half-life. The atoms have no memory so each second they exit the probability is the same

1% chance of decay means that 99% survived

Start with 10 000 atoms and after 1 s you 10 000*0.99 = 9900 left. After 2 seconds you have 9900*0.99= 9801 atoms left.

But 9900*0.99 = 10 000*0.99*0.99 = 10 000* 0.99^2

if you expand it you get to after n seconds you have 10 000* 0.99^n atoms left.

If you solve the equation 10 000* 0.99^n= 5000 and if you put it into for example wolfram alpha you get n=68.97 less call it 69 seconds.

The result is if 1% of all atoms decay each second the half-life is 69 seconds.

Half- life is a simple thing to use in the calculation you can for example direct know that after two half-life you have 1/4 of the original amount so 2500 after 138 s.

You can even calculate the amount that you have after 20 s The formula is:

initial amount*(1/2)^( time/ half-life)

So you get 10000*(1/2)^(20/69) = 8179 atoms

Probabilities like this work if you have a lot of atoms and there is thousand of billion of a billion atom if a gram of matter.
The do not work well if you have very few. You cant tell the time it takes for a single atom to decay but you can say that if you have 1 billion you only have half a billion after a half-life.

Atoms decay because of the quantum effect in the interaction of the week and strong force in the nucleus of an atom. The result of it that each atom has a percentage chance of decaying in a unit of time.

You can do the exact same thing for an increase. If you have 100 in a bank account and 6% interest each year you can calculate the [Doubling_time](https://en.wikipedia.org/wiki/Doubling_time) that will be 11.9 years
You can calculate the amount you have after any amount if time with the formula:

initial amount*(2)^( time/ doubling time)

So if the doubling time is 11.9 year s and you start with 100 you have after 1 year 100*2^(1/11.9)=105.9977 = 106 So a 6% annual interest. IT is not exact because 11.9 year is founded to 2 decimals places it is more exactly 11.8956610459…..

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So you get a half-life if you lose a percentage after a unit of time and a doubling time if it increases by a percentage after a unit of time.

The effect of band account interest and radioactive decay are mathematically almost identical

Imagine you start with a million coins. You flip all of them, and only keep the ones that came up ‘heads’. Then you flip again, and so on. After one flip, you’ll have about half of your original coins remaining. After another, you halve the number of coins again, etc. until there are zero coins left.

Now let’s repeat the experiment but let’s use loaded coins instead of regular ones. These coins come up heads an average of 99 every 100 times. Starting with a million of these coins, it will take a lot longer to get to 0.5 million: about 69 flips. But guess what: it will take another 69 flips to halve that number again and get to 0.25 million. And another 69 to get to 0.125 million, and so on. So even though the probabilities are no longer 50:50, there is still a set (average) time that it takes for the number of remaining coins to be reduced by half. This “coin half-life” is a convenient way to describe the speed of the coin discarding process. However, note that we could equally use the probability of heads or tails (0.99 or 0.01) to convey the same information. One determines the other: if you know the probabilities, you can work out the half-life, and vice versa.

It is worth emphasizing that these are all averages. It will not take exactly 69 flips to have 0.5 million coins left, every time you run this experiment. It is, after all, random. But if you repeat the experiment many times, the average number of flips required will be 69.

Radioactive decay is just like a biased coin toss. In any given period of time (say, a nanosecond), each nucleus of your radioactive material (say, Uranium-235) has some fixed probability of decaying. It is useful to know the speed of decay of different elements, and you could use probabilities for this. However, note that unlike the coin tosses in my example, the universe doesn’t run in discrete time steps*. So if you want to quantify the probability that an atom will decay, you need to specify the time window that you’re talking about (e.g. “each second, a rubsebium-331 atom has a 0.001 probability of decaying”). So now you have to use two numbers to describe the speed of decay: a time window and a probability. Whereas, using the half-life, you can convey the same information using just a single number: the time it takes (on average) for 50% of atoms in a given sample of your material to decay.** It’s also arguably a more easily interpretable format, if you want to get an idea of how quickly something decays. If you tell me the half life of an element, it’s reasonably easy to get a feel for how long it will take before most of a sample of this stuff will be decayed away. Whereas, if you told me the “nanoseconddecayprobability” of that same element, I’d be hard-pressed to convert that to a similar intuition.

(*Actually, there are some theories that speculate time might be discrete, but this isn’t certain, and in any case we don’t treat or experience time that way.)

(**To be fair, this is sort of cheating because there is actually another number involved, namely 50%. It’s just that this second number has been agreed upon beforehand, so you don’t need to say it every time. But we could equally have decided on a standard time window to report decay-probabilities instead, e.g. “the nanoseconddecayprobability of rubsebium-331 is 0.001”.)