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I'm curious about how to clearly define the sample size when calculating the half-life of particles. My understanding is that the half life is statistical in nature and that for very small samples, the fluctuations in the number of particles that will decay can get quite large.

"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."

But, what defines the sample size? Is the sample just what you have in the jar? If so, why? Why does your sample not include the particles that are in the jar next to the jar you're measuring? Or in the jar in the next room? Or in a jar half way around the world? Where and why does the line get drawn?

Hypothetically, if you were to take your sample into deep space, separating it by a distance (in light seconds or light years or light whatevers) that is greater than the length of the half-life of your sample, so that the decay must be casually disconnected from every other particle of the same type, would the statistical nature of the decay rate still apply? Or would the fluctuations in the number of particles that decay decrease?

For example, let us posit that you have two particles with a half-life of one minute and you've separated them from any other particles of the same time by two light minutes. Traditionally, we would assume that there is a certain probability that one of the particles will decay after one minute has elapsed. There is a lower probability that both or neither of them decayed during that time. But, if ALL of particles within the light-cone (as defined by the half-life) are in your sample, does the statistical nature of the half-life still apply?

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  • $\begingroup$ Remember, it often isn’t hard to get LOTS of atoms. Carbon-14 natural concentration in the atmosphere is about 1 in a trillion. So, 12 grams of carbon (1 mol) has 602000 trillion C-14 atoms in it. $\endgroup$ – Jon Custer Jul 14 at 20:06
  • $\begingroup$ Yeah, this is more of a thought experiment question than anything practical... But you're right, maybe the hard part would be isolating a small enough number of atoms and not the extreme isolation! $\endgroup$ – Thor Jul 14 at 20:20
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Is the sample just what you have in the jar?

It is the amount of atoms in the sample you intend to measure

If so, why?

Because you are planning an experiment and have to define well the quantities entering so that the numbers coming from the experiment have small errors.

Why does your sample not include the particles that are in the jar next to the jar you're measuring? Or in the jar in the next room? Or in a jar half way around the world? Where and why does the line get drawn?

That is called "estimating the background" which you can subtract from the measurements and take care of it in the errors given for the half life.

If your background is large, the error will be large and your measurment will have no meaning. Thus when designing the experiment one should minimize the sources of radiation that are background to the measurement designed.

To start with the experiment,one measures the background, i.e. takes the jar sample far away. That's the easy way, and the background can be subtracted statistically.

In the specific example, one can use lead blocks at the place of the jar to block alpha and gamma and beta radiation, measure the background in situ, and then put in place the jar for measurement.

If the cosmic muons introduce a backround, there are various ways to take care of it, mu metal, underground laboratory. Also muons can be taken care of in analysis if one has the direction of the radiation, by measuring only those particles that go up.

Hypothetically, if you were to take your sample into deep space, separating it by a distance (in light seconds or light years or light whatevers) that is greater than the length of the half-life of your sample, so that the decay must be casually disconnected from every other particle of the same type, would the statistical nature of the decay rate still apply? Or would the fluctuations in the number of particles that decay decrease?

You no longer have a jar in this thought experiment, you have a dispersed sample and cannot measure it.Yes, the half life measured with the jar in the laboratory will apply, because half lives are calculated in the center of mass of the particles , i.e. no motion, which is mainly true to the jar ( thermal motion is very much less than the energy of decays)

For example, let us posit that you have two particles with a half-life of one minute and you've separated them from any other particles of the same time by two light minutes

Each particle in its center of mass will decay with the probability of the lifetime calculated from a large sample ( your jar).

Muons when stopping decay very fast. Cosmic muons live much longer due to the relativistic effect, accoding to their energy. Their lifetime though is calculated in the center of mass of each particle, not in motion.

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