Muon Decay Simulation

About this page

This page provides an explanation of our experiment, background theory, and a brief discussion of our results and conclusions. For examples of code or explanation of the geant4 program, please see one of our other pages.

Background

At any given point in time the earth is being hit by cosmic rays released by the sun. By the time these rays reach the earth most have decayed into muons, and have an energy of about 4 GeV citation needed. The energy distribution of these particles is modeled by formula below:

Where phi is the zenith angle, angle of trajectory measured from the vertical,
and E is a particular energy of a particle.

This distribution produces the following 3D graph:

The zenith angle of the particles follow a cosine squared distribution citation needed. These muons can further decay into positrons or electrons (from muon+, muon- respectively)(need to fix into symbols) and some nutrinos. Any of these particles can be detected in a variety of ways, because they all carry a charge. By moving through certain materials the particles can ionize a path through the material, losing energy as it does so. There are lots of apparatus that can detect this phenomenon or make it visible to the human eye.

Purpose and Expectations

Our simulation was designed to provide some background data to aid in the design of a larger project, namely to build a cloud chamber that will show cosmic muons decaying. To build such a device it is necessary to slow the particles so a larger number of them will decay inside the actual chamber. To achieve this we proposed setting a block of lead above the detector in the hopes this dense material would successfully slow the particles to the point they would decay.

The primary objective of this simulation was to establish a relationship between the thickness of the lead block and the number of particles that decay. Our expectation was that this relationship would be logarithmic in nature, because some arbitrarily thick should stop all particles causing all of them to decay.

Basic Design and Execution

The simulation environment consisted of an envelope filled with air containing two scintillator plates with a lead block sandwiched in between. Ten thousand mu+ were then shot randomly according to the energy and zenith angle distributions found above, assuring that they would pass through the entire block. This did restrict the maximum zenith angle; however, the angles unavailable were outside of one sigma of the mean(maximum angle = 35.6 degrees, mean = 0, sigma = 32.5 degrees). The simulation would then detect if a new positron was created at any point during each shot. The results were recorded in an Ntuple by the simulation, then extracted an compiled into a 2D plot by a ROOT macro.

Results and Brief Discussion

Our basic finding was that muon decay and lead thickness have a positively correlating relationship. Below are two plots illustrating our data and demonstrating that finding. The first was generated as a result of our first simulation where the energy distribution displayed above was used. The second was generated by another simulation where the energy distribution was ignored and all particles were given a single energy(4 GeV). Brief discussions of both results and an explanation of the inspiration for the second run accompany the results.

In this plot the data points are represented by the red points, and the associated error bars are represented by the red bars attached to each points.

Clearly this data demonstrates a complex relationship between muon decay and thickness. It would appear that the data would be best fit by some combination of decaying exponential curves; however no fit was applied here because of its complex nature. The data does clearly demonstrate a positive relationship between muon decay and lead thickness.

As a result of compiling and view this data, we theorized that the complex nature of the relationship was caused at least in part by the fact that we used an energy distribution to choose a particles initial energy. To see if this conclusion was reasonable, we ran a second simulation not using the distribution, but instead setting the starting energy of each particle to four giga-electron-volts(4 GeV). The results of that simulation are demonstrated below.

In this plot the data points are represented by the red points, and the associated error bars are represented by the red bars attached to each points.

This data set shows a positive correlation between muon decay and lead thickness as well, however the relationship is not as complex as the one revealed by the first simulation. It appears that this data set would be easier to fit and would be an excellent continuation of this project: We were not able to fit this data by the time we stopped working on the project. Despite this fact, two significant conclusions can be drawn about the relationship between muon decay and lead thickness: First, the relationship is smooth and positive, and second, there is almost no decay when no lead is present and eventually as lead is added almost all particles will decay. The asymtotic behavior we predicted would seem to be supported, although with no proper fit this is just a general observation and has no evidential support.

Considering our primary goal was to find if there was a relationship between lead thickness and muon decay, the project was a success despite not being able to generate any kind of model about that relationship. There is clearly a lot more research opportunity associated with this project to be worked on. Fitting the data already collected, collecting more and targeted data, and redesigning the simulation to allow for greater variable ranges are just a few ideas. With most of the code already produced a few simple adaptations could add significant depth to this project going forward.