Is half-life a statistical average of variable decay times?

1,690

Solution 1

Half life is, by definition, the amount of time until half of an infinitely large sample would decay. That's precisely equivalent (according to the frequentist interpretation of probability, if that matters to you) to the time until an individual particle's probability of decay reaches one half. The half life is a theoretical quantity that doesn't depend on the actual number of particles you're dealing with.

If you actually put 8 particles in a box and watch how long it takes for half of them to decay, you could consider that a measurement of the half life of the particles. As with any measurement, the value you measure will not, in general, be the same as the true (theoretical) value. So yes, there will be fluctuations, and once the number of particles remaining drops to two or one or zero, those fluctuations will be very very large. But what is fluctuating is your measurement of the half life, not the true theoretical half life itself.

Solution 2

Yes, it is a statistical average in the sense that the measured half life will approach a single value of a true half life if you do lots of measurements.

In other words, if you did the experiment many, many times you would find that on average you had 4 particles left after a half-life had passed.

For any individual experiment, the results would vary.

Each atom has a probability of surviving intact after a time $t$ according to $$p = \exp(-\lambda t)$$ where $\lambda$ is the decay constant and the half life $t_{1/2} = \ln 2/\lambda$.

If you wait 4 half lives then $t = 4\ln 2/\lambda$ and the probability of an individual particle surviving is $\exp(-4\ln 2) = 0.0625$.

In practice, you have to have an integer number of particles, so the most likely outcomes are either 1 or zero intact atoms remain.

If you have 8 atoms and the probability that any of them will have decayed is $p=0.0625$, then one can use the binomial probability distributionto work out the probability that any number $n$ will survive from a population of $N$ is $$ P(n) = \frac{N!}{n! (N-n)!} p^{n}(1-p)^{N-n}$$

So $P(0)= 0.597$, $P(1) = 0.318$, $P(2)= 0.037$ and so on.

Now, if your aim is to estimate the half life based on a single experiment with these 8 atoms, then I see (at least) two possibilities.

(i) If you measure the time it takes for the 4th decay to occur, then you can calculate $P(4)$ as above, but calculate it for a range of possible values of $\lambda$. This will give you a probability distribution for $\lambda$ from which you can find the maximum likelihood value or a confidence interval.

(ii) If you have the individual decay times of each decay, then for each atom you can calculate a probability that it would have decayed in less than its observed decay time, given an assumed $\lambda$, which is $P_i(\lambda) = (1- \exp[-\lambda t_i])$. You can also include any atoms that haven't decayed, $P_i(\lambda) = \exp[-\lambda t_i]$. You then form the product of these probabilities $P(\lambda)= \prod P_i(\lambda)$ to give you an overall likelihood distribution for $\lambda$, from which you can estimated a maximum likelihood value for $\lambda$ and a confidence interval.

Share:
1,690

Related videos on Youtube

Skyler
Author by

Skyler

Updated on June 10, 2020

Comments

  • Skyler
    Skyler over 3 years

    Is the half life of a material only accurate as long as you are still in a macroscopic regime? If I had 8 particles in a box would I observe a fluctuation in half lives, and what would occur within the 4th half life?

    • MSalters
      MSalters over 8 years
      It helps to realize that a single particle cannot half-decay. At t=0 you observe the original state and at some t>T you observe the decayed state. This time T is just the time it took the particle to decay from the moment you started observing it, and it's not the half life. This shows that you cannot extrapolate the half life concept to a single observation.
    • Josef says Reinstate Monica
      Josef says Reinstate Monica over 8 years
      But what if the particles where created just in the moment I start to observe them @MSalters ? (e.g. by decay of another particle)
    • MSalters
      MSalters over 8 years
      @Josef: Particles have no memory. "Just created" doesn't give them a longer life expectancy.
    • Skyler
      Skyler over 8 years
      @MSalters at least in their rest frame
    • MSalters
      MSalters over 8 years
      @Skyler: The "no memory" property does apply in all rest frames.
    • Skyler
      Skyler over 8 years
      @MSalters, but if they are moving then in a frame outside their rest frame the decay rate would be attenuated by relativistic effects. The example that comes to mind is very high energy muon decay.
  • Skyler
    Skyler over 8 years
    Thanks. If you have anymore input on the decay time of small sets of particles I would be happy if you could elaborate on that. I guess I should reorient my question to focus more on what I'm interested in, the decay dynamics of small systems.
  • David Z
    David Z over 8 years
    @Skyler what exactly would you want to know? If you're curious about the distribution of measurements of the half life, I could edit that in, but if it goes beyond that, it might be the case that you should be asking a separate question.
  • Taemyr
    Taemyr over 8 years
    Who defines half life in this matter. What does it mean that half of an infinitely large sample has decayed?
  • JiK
    JiK over 8 years
    It looks like the answer is confusing likelihoods and posterior distributions in (ii). $P_i(\lambda)$ is the probability that $i$th data point is what was observed assuming that the decay constant is $\lambda$. Thus $P(\lambda)$ is the likelihood of $\lambda$, i.e., probability of data assuming that the decay constant is $\lambda$. It is however true that with (improper) uniform prior for $\lambda$, the posterior pdf for $\lambda$ would be $\mathrm{const.}\times P(\lambda)$, but either state it explicitly or don't call it "probability distribution for $\lambda$".
  • David Hammen
    David Hammen over 8 years
    @Taemyr - It's one of many consequences of the law of large numbers (google that phrase). Suppose you put two particles in a box and measure the time until one has decayed. Repeat that experiment and you'll get a very different answer. You'll get a much smaller variance on repeated experiments if you start with eight particles in the box and measure the time until four have decayed. You'll get an even smaller variance with one hundred particles, and even smaller variance with a million particles. The variance becomes vanishingly small in the limit of an infinitely large sample.
  • ProfRob
    ProfRob over 8 years
    @Jik Yes, I'm sure you are right, the distribution would have to be normalised.
  • JTP - Apologise to Monica
    JTP - Apologise to Monica over 8 years
    Given that a cubic cm of stuff is close to 6.02*10^23 isn't that enough to consider a fair sample? By infinite, do you just mean that single digits are too small to consider, and we need lots of zeros to have an observable qty?
  • David Z
    David Z over 8 years
    @JoeTaxpayer I mean that the more particles you include in the experiment, the greater the probability that the observed measurement of the half life is close to the true half life. That's basically what David Hammen's comment said. Whether single digits matter or not depends on your definition of "close" (except that you won't get a proper measurement of the half life from just one particle).
  • Skyler
    Skyler over 8 years
    @DavidZ, so would it be accurate to say that half-life is undefined for one particle since it does not have a half-way point. The elaboration I was asking about is what the microscopic decay property is that when you reach the statistical regime averages out to half-life, but I can see that being a whole different question in it's self.
  • David Z
    David Z over 8 years
    @Skyler no, it would not be accurate. As I said, half life is a theoretical property that has nothing to do with how many particles you actually have. The fact that a single particle cannot half-decay only means that you can't measure the half life with just one particle.
  • ProfRob
    ProfRob over 8 years
    At the risk of confusing everyone, even an observation of one particle gives you a probability distribution for the half life!
  • Taemyr
    Taemyr over 8 years
    @DavidHammen I am aware of the law of large numbers. It never talks about infinite samples. The variance vanishes as the sample size goes to infinitiy, but going to infinity and being infinte is two very different things.
  • David Hammen
    David Hammen over 8 years
    @Taemyr - Re but going to infinity and being infinite are two very different things. Not to a physicist. Physicists aren't quite as rigorous in their mathematical statements as are mathematicians.