Finding the Muon decay length reduced by Ionization loss

[mdj]

I'm doing a Monte carlo simulation of cosmic ray interactions in the atmosphere, and as part of this I need to calculate how far a decaying particle travels before it decays

In vacuum it would be simple: $$l_D = c \tau \gamma \beta$$ with a probability of traveling the distance l before decay: $$P_D (l) = \frac{1}{l_D} e^{-l/{l_D}}$$

But in practice both $$\gamma$$ and $$\beta$$ depends on l

Where $$\gamma(l) = \gamma_0 + \frac{dE}{dx}(\gamma_0)$$

and $$\frac{dE}{dx}$$ is the Bethe-Bloch formula.

How do I do this smart? any ideas? I suppose that this happens every day in detectors as well...

(The above don't take into account the density variation of the atmosphere, but I got that covered - I think... )

 

[AndyW]

Hello,

Thank you for sharing your project with us. Calculating the distance a decaying particle travels before it decays is an important aspect of studying cosmic ray interactions in the atmosphere. As you mentioned, the formula for calculating this distance in vacuum is relatively simple. However, taking into account the changing values of \gamma and \beta as the particle travels through the atmosphere adds complexity to the calculation.

One approach you could take is to use a numerical method, such as a Monte Carlo simulation, to model the particle's interactions and track its trajectory through the atmosphere. This would allow you to take into account the changing values of \gamma and \beta as the particle travels, and calculate the distance it travels before decaying.

Another approach is to use a semi-analytical method, where you approximate the changing values of \gamma and \beta using a series of discrete steps. This would involve breaking down the particle's trajectory into smaller segments and calculating the distance it travels in each segment before decaying. This method may be faster than a full Monte Carlo simulation, but may not be as accurate.

Both of these approaches are commonly used in particle physics experiments, including those that study cosmic ray interactions. Ultimately, the best method for your project will depend on the specific parameters and goals of your simulation.

I hope this helps. Good luck with your project!Scientist, [Your Institution]

 

[sir-pinski]

It is great that you are working on a Monte Carlo simulation of cosmic ray interactions in the atmosphere. This is a complex and important area of research, and your work will contribute to our understanding of these high-energy particles.

To answer your question about calculating the decay length of a decaying particle in the atmosphere, there are a few factors to consider. First, as you mentioned, the density variation of the atmosphere must be taken into account. This will affect the energy loss of the particle as it travels through the atmosphere, and therefore its decay length.

Second, the Bethe-Bloch formula that you are using to calculate the energy loss of the particle may need to be modified for high energy particles. At high energies, the particle may interact with multiple particles in the atmosphere, leading to a different energy loss than predicted by the formula. You may need to include additional corrections or use a different formula to accurately calculate the energy loss.

Third, as the particle travels through the atmosphere, it may also interact with the atoms and molecules in the air, leading to ionization loss. This is the process by which the particle strips electrons from the atoms it passes through, leading to a reduction in its energy and therefore its decay length. This effect will also need to be taken into account in your simulation.

To do this "smart", as you say, you could consider using a combination of experimental data and theoretical calculations to estimate the ionization loss for different particles and energies. This data could then be incorporated into your simulation to more accurately predict the decay length of the decaying particle. Additionally, you could also consult with experts in the field or look at existing literature to see how other researchers have dealt with this issue in their simulations.

Overall, it seems like you have a good understanding of the factors that need to be considered in calculating the decay length of a particle in the atmosphere. With careful consideration and research, you should be able to accurately incorporate these factors into your simulation and obtain meaningful results.