An important value in radioactive decay is the decay constant (lambda).

I'm interested to know how you lot would define this quantity... I will weigh in once I've seen a few replies

Well N(t)=No*Exp(-λ*t) and dN(t)/dt=-λ*No*Exp(-λ*t) (No the number of initial radioactive particles at t=0)

So (dN[t)/dt)/N=-λ

Naturally you can see it as the ratio of activity (activity how many decays per unit of time, the rate of decays) divided by the number of radioactive particles that can decay that you have available.

So the fractional decays per unit of time. Like what % of your material you lose to decays every second etc.

For example for Uranium-238 with 4.468 billion years half life with 4.468 billion years

λ=log2/T_1/2

or λ=4.915*10^-18 so you could say you lose 5 billionths of a billionth per second to decays in your original material of pure U238.

Compare with say Technetium-99m used in diagnostic systems in medicine https://en.wikipedia.org/wiki/Technetium-99m (most commonly used medical radioisotope) with half life of 6h for its gamma emission or internal conversion from its excited state (produced from Mo-99) to its ground state Technetium-99 (that has 211000 years half life in comparison).

There λ=1/31200 so you lose 1 in 31200 of your remaining Technetium-99m per second.

I'm not really sure what the question is, tbh.

lambda is a constant that's related to the half life of the radioactive material (the time it takes half of the stuff to decay). It's also the inverse of the average time it takes any particle to decay. The relationship is defined as

Trolly you're right of course, masque has made an error I think...

In what? You asked for a natural definition of λ. The half life is not a natural definition because it is log2/T_1/2.

The constant itself is not the inverse of the time it takes to have decay 50% of your radioactive material (just 1/e). So this is not a natural definition of it unless you say its the inverse of the time it takes to have 1/e of your original sample decay. But the activity is naturally introducing it as the fractional loss of radioactive material per unit of time (ie the rate of fractional loss).

Experimentally how on earth are you going to easily know when the sample has lost 50% of its initial population???

But you can always have a rate of decay it seems ie the activity. All you need for it is to count how many decays you measure per second and some knowledge of your initial sample size (or nuclei % in it) .

You do not need to wait 4.5 bil years for Uranium or even a few years to find the new number of U238 nuclei left to decay (ie to get to the half life connection i mean). All you need is to count how many decays you have in a given sample per minute say and you obtain λ. Use a Geiger counter etc.

λ=(dN/dt)/N is very simple to observe if you are not losing detections or you can estimate how much you are losing vs what you record. Then on course you can find the half life too by the above equation.


Even if you do not have the original size, ie N that you started with, you can still measure activity at different times and see how it changes and model it from there if it is not a very slow decay thing. In that case however that it is very slow you can estimate its mass and number of nuclei in other ways

I doubt there's anything incorrect in it. It's just neither one us really knows what your asking for.

"The constant itself is not the inverse of the time it takes to have decay 50% of your radioactive material (just 1/e). So this is not a natural definition of it. But the activity is naturally introducing it as the fractional loss of radioactive material per unit of time (ie the rate of fractional loss).

Experimentally how on earth are you going to easily know when the sample has lost 50% of its initial population???

But you can always have a rate of decay it seems ie the activity. All you need for it is to count how many decays you measure per second and some knowledge of your initial sample size (or nuclei % in it) . "

This is the problem. It implies that the decay constant is equal to a probability of a given nucleus decaying per unit time (the second you use for your unit of time, which makes sense if we measure activity in Bq). This is often given as a definition in textbooks which is why I brought this up. What happens if you have a decay constant of microseconds?

Sorry half life of microseconds

Imagine that you have an unknown number of dice, but you know that all the dice are 6-sided fair dice. All the dice are rolled at once and you find out that there are 10,000 sixes that were rolled. Can you estimate the total number of dice?

The problem is conceptually not much more difficult than this. The math is a little bit more complicated because you're talking about a continuous decay model, but there's no reason to think that transitioning from a discrete problem to a continuous problem is going to a challenge that's impossible to overcome.

My guess is that you would need more sensitive equipment to measure it accurately.

What do you think happens?

Measuring radioactive decay is super easy, just stand real close and count the xrays.

- Marie Curie

It's not clear if you're asking about the definition of what a half life is or if you're asking how half-life is measured.

Microsecond measurements are no problem at all, incidentally. Your computer processor is doing operations a thousand times faster than that.

If you have a microseconds half life then the fractional loss will have meaning in the proper time frame it has. Basically you do not need to talk in decays per second you can talk in decays per microsecond or something.

All derivatives lose their meaning if you take the per unit of time literally as lasting 1 unit of time because you are no longer at the local point the derivative is evaluated.

If you have something that decays so fast then you need different equipment to measure it than something that has response time much larger than the half time to begin with.

You never physically have a large scale sample then ie in grams type size because you can almost never prepare it that way.


I tried to connect it with what you usually have in experiments available to you ie the activity.

If you want to go purely physically and try to see what it means for the individual radioactive system (ie nuclei) then you can

see that 1/λ=the avg time <t> it takes for a radioactive system to decay.

https://en.wikipedia.org/wiki/Exponential_decay (half life would be the time it takes for the probability of decay to be 1/2)

https://en.wikipedia.org/wiki/Exponential_distribution

Probably should be in its own thread but I will ask it here.

I understand that we don't know "why" something decides to decay when it does. But do we know why it has the half life that it does? Or at least why its half life is a certain percentage of something else's half life?

I guess i don't really understand the question but i'll take a shot.

A half-life is a period of time in which each individual nucleus has a 50% chance of decaying.

You like numbers so let's look at it this way: How many two-dice rolls does it take for someone to have a 50% chance of rolling a 12? If we start with 10 million people, after that number of rolls we'd expect that we'd have about 5 million left. If we have them continue rolling the dice, we'd expect to have about 2.5 million left after that number of rolls again, etc.


FYI in the pharmacy world we work with T90s more than T50s. The T90 (the time when 90% of the drug molecules have not broken down) is usually your drug's expiration date.

ETA: to the first question, the half life is just a descriptor of how stable the nucleus is. Less stable nuclei are more likely to decay and thus have shorter half-lives.

Yes. Some arrangements of protons and neutrons are inherently unstable. You can plug a certain arrangement into a model and guess that some arrangements will decay very quickly and others not so much.

Generically yes, Fermi's Golden Rule can be used to calculate average (say) muon lifetimes and the results are as close as any theory to what is observed. Of course if you can't use perturbation theory or don't really understand the Feynman diagrams for the decay you won't be able to calculate it, like with proton decay.

My original questions was to see if people would come up with a common misconception (which you will find in textbooks) on this forum, and address it.

The error I referred to is in this statement "There λ=1/31200 so you lose 1 in 31200 of your remaining Technetium-99m per second."

It's not not easy to see in this example but imagine a decay constant of 1/5. Would you lose 1 in 5 of your nuclei per second? That would be 20%, when in fact you would lose only just over 18%.

The misconception I'm referring to was probably started by Rutherford and is the definition of the decay constant as being the "probability that a nucleus will decay per unit time" which I have seen in many a textbook and even when I complained about it to an exam board they still said they were happy with that definition. So when I asked the question I was seeing if anybody would quote that.

Masque seemed to use it in that statement at the end, unless he meant to say you would have "roughly" 1/31200 left after 1 second....

It feels like you're trying to be overly clever to make an irrelevant point.

If you think it's irrelevant, ok, but I don't.

I was trying to highlight a commonly given definition in many textbooks which is (I think) incorrect and see what people think.

Widely perpetuated misconceptions in Physics are certainly not irrelevant to me. And I'm not trying to be clever.

I mean, the definition is not "incorrect"....as in anomalous. You can use it to do what it was designed to do. If you think there is a more natural one, write it down.

It's hard for me to believe you when you create a thread with very little guidance and then proceed to claim that there's some widely perpetuated misconception in physics that you're here to correct. Right now, it feels more like you're playing a game. But I'll give you a shot.

1) Precisely define what you think the radioactive decay constant is and how you think it is obtained.
2) Precisely describe the "widely perpetuated misconception" that you think exists.
3) Precisely define what you would propose as the alternative that avoids this error.

My suspicion at this time is that you are the one who is in error. I believe it's that you are mistaking a statement of units as a statement of an average.

1) It is the inverse of the decay time or a constant equal to ln(2) / half life. It can be measured experimentally by determining the half life.
2) The definition I have seen is that the decay constant is "the probability that a given nucleus will decay per unit time".
3) That the the decay constant not be referred to as a probability. This is the error I'm getting at as I don't think it is a probability at all. It approximates a probability if the decay constant is very small compared to the unit of time but that doesn't make it a probability.

Has nobody else here come across this definition for the decay constant? And are you happy with it?

Congrats on figuring out that people were saying for over a century that nuclei could have over 100% probability of decaying if you waited long enough. Nobel incoming for sure.

erm... how can you have a probability over 100%?