Muon lifetime determination

Question

My colleagues and I performed several experiments to determine the lifetime of the muon (from secondary cosmic rays) using scintillator detectors coupled to multi-channel analysers. The results invariably showed a muon lifetime lower than the standard 2.2 microseconds. Apart from poor statistics,and assuming no faults in the equipment used, what other factors could be responsible for the discrepancy?

Answer 1

As you suggested in your comment, the $\mu^-$ and $\mu^+$ that stop in matter do not have the same lifetimes. The $\mu^+$ come to rest between the atoms of your stopper (eg: scintillator?) and decay into $\nu_{\mu}e^+\nu_e$ with the standard 2.2 usec lifetime. However, the $\mu^-$ get captured into Bohr orbits about the stopper nuclei. The $\mu^-$ then transistions down to n=1 L=0 orbit by emitting Auger electrons and x-rays. In this closest orbit there is an overlap between the $\mu^-$ wave function and the nucleus, and therefore some rate to interact with the protons and neutrons. So, the seen decay rate of the $\mu^-$ is a sum of the nuclear interaction rate and the natural decay rate of the muon. $$ \frac{1}{\tau_{Seen}}=\frac{1}{\tau_{Nuclear}}+\frac{1}{\tau_{Natural}} $$ The nuclear interaction rate increases with the Z of the nucleus because the orbital radii are smaller and there are more nucleons as Z increases. The lifetimes $\tau_{Nuclear}\approx\tau_{Natural}$ for $Z\approx 10$. There is an extensive review of all this in Physics Reports 354 (2001) 243-409. Table 4.2 shows some $\tau_{Seen}$ for $\mu^-$ stopping in different elements.

In summary, your number of decays versus time is the sum of two exponentials. One for $\mu^+$ with a 2.2 usec lifetime, and one for $\mu^-$ with a lesser lifetime that depends on the elements in your stopper. The ratio of the number of positive muons to negative muons at sea level in cosmic rays is about 1.2 . It is reasonable that you are measuring <2.2 usec for the overall lifetime, but for quantitative sense you will have to fit two exponentials and use the $\tau_{Nuclear}$ for your stopper elements.

Answer 2

I know that you are explicitly asked about not equipment related answers. But when I learned something from experimental physics then that you should always consider equipment flaws.

I could imagine a scenario where the events on which you trigger to start/stop the clock have different rise times depending on where they take place in the scintillator, creating an observational error.

One other thing you could check is that your measured lifetimes are normally distributed. If they are not, you can think of things like applying a power transformation to your data.