Lifetimes of kaon and lambda baryon in a bubble chamber

[OGrowli]

Lifetimes of kaon and lambda particles in a bubble chamber (updated)

Statement of the problem:

This isn't quite a homework problem. I'm just having trouble with the concepts of this lab. I'm working on a lab report dealing with analysis of bubble chamber photographs of the decay of a kaon and lambda particle. The lifetimes I calculated in the lab frame are much closer to the accepted mean lifetimes than the lifetimes I calculated for the rest frame. This could just be a coincidence, but I suspect that I've mixed up the lab frame and the particle's rest frame.

Attempt at a solution, and Relevent Equations:

A high energy $\pi^{-}$ decays into a neutral kaon and a neutral lambda particle, those in turn decay as:
$$\Lambda ^{0} \rightarrow p + \pi^{-}$$,
and
$$K^{0} \rightarrow \pi^{+} + \pi^{-}$$.

Taking the mass of the proton and pi meson to be given, the trajectories of the above charged particles and that of the original neutral particles can be used to find the momentum and then the rest mass of the kaon and lambda. I've found these already. The accepted values were within the error of my measurement.

Using the the energy and momentum found in the first part, we can find the velocity of the neutral particles in the lab frame:

$$v=\frac{pc^{2}}{E}$$

And then from the photograph we can measure the distance each particle traveled, dividing this by the velocity found in the last equation we should get the lifetime of each particle in the Lab frame:

$$t_{lab}=\frac{d}{v}$$

And from there we find the lifetime in the particle's rest frame using this equation:

$$t_{rest}=\frac{t_{lab}}{\gamma}$$,

where γ is the lorentz factor.

Here is what I calculated for the K0:

$$t_{lab}=8.07\pm 0.22 (10^{-11}s)$$
$$t_{rest}=5.52\pm 1.39 (10^{-11}s)$$

For the lambda0:
$$t_{lab}=2.80\pm 0.52 (10^{-10}s)$$
$$t_{rest}=1.82\pm 0.36 (10^{-10}s)$$

The fomula for t_lab deals with the velocity as measured in the lab frame, so that should be the lifetime of the lab frame. The fact that in both cases my t_lab is closer to the accepted rest mean lifetimes has me suspect. Is there anything wrong with my methodology here? Thank you for your replies.

 

[mark726]

Thank you for sharing your problem and attempted solution. From your calculations, it seems that you have a good understanding of the concepts involved in this lab. However, there are a few things that may have led to your confusion about the lab frame and the particle's rest frame.

Firstly, it is important to note that the lab frame is the frame of reference in which the experiment is conducted, while the particle's rest frame is the frame of reference in which the particle is at rest. In this case, the lab frame would be the frame of the bubble chamber, while the particle's rest frame would be the frame in which the kaon and lambda particles are at rest before they decay. It is possible that you have mixed up these two frames in your calculations, which could explain the discrepancy between your calculated lifetimes in the lab frame and the particle's rest frame.

Another factor to consider is the uncertainty in your measurements. Even small errors in your measurements can significantly affect your calculated values. It is important to carefully record and consider the uncertainties in your measurements when analyzing data.

Additionally, it is worth mentioning that the lifetimes of particles can also be affected by their interactions with other particles in the bubble chamber. These interactions can alter the trajectories and lifetimes of the particles, so it is possible that this could also contribute to the discrepancy in your results.

Overall, it would be helpful to carefully review your methodology and double-check your calculations to ensure that you have correctly accounted for the different frames of reference and uncertainties in your measurements. It may also be helpful to discuss your results with your lab instructor or a fellow classmate to get additional insights and feedback. Good luck with your lab report!