Incorrect units when calculating cosmic ray muon flux

[Sciencemaster]

TL;DR

I recently used a single 3D scintillator to collect cosmic ray muons. I'd like to find the flux, but because my detector is 3D, I end up with something of units different from flux.

I recently performed an experiment that involved using a cylindrical scintillator to detect cosmic ray muons by observing the amount of particles that decayed within 20 microseconds over a long period of time. I'd like to use this to find the flux of muons at my scintillator so that I can compare the results to other experiments done at different altitudes. However, because my experimental setup uses a 3D detector, I get something in terms of $\frac{Hz}{cm^3}$, whereas every other experiment seems to get the expected units of $\frac{Hz}{cm^2}$. Is there some operation I can perform on my result to find flux in terms of $\frac{Hz}{cm^2}$ despite having used a single 3D detector? Could I even find an accurate flux with this data, as I'm only considering particles that actually decayed within the scintillator?

 

[Vanadium 50]

I think you need to think about what you are trying to do. If all cosmic rays were directly down, if your units were Hz/c,³ doubling the detector thickness would double the cosmic ray rate. Does that make sense to you?

 

[mfb]

What fraction of the muons passing the detector do you catch? How does that fraction depend on the angle and thickness? Ultimately every detector is three-dimensional, treating it as having a very small height is just an approximation.

 

[Sciencemaster]

I think you need to think about what you are trying to do. If all cosmic rays were directly down, if your units were Hz/c,³ doubling the detector thickness would double the cosmic ray rate. Does that make sense to you?

That does make sense. However, I don't think that all cosmic rays are straight down. I'm pretty sure that the incident intensity is a function of $\cos^2{(\theta)}$ where $\theta$ is the incident angle of particles. Given this, I am inclined to think that there could be more to do than multiplying by the thickness of the detector.

 

[Sciencemaster]

What fraction of the muons passing the detector do you catch? How does that fraction depend on the angle and thickness? Ultimately every detector is three-dimensional, treating it as having a very small height is just an approximation.

I'm not sure exactly the fraction of total muons that went through the detector were actually detected. I know how many *decayed* within the detector, but I'm not quite sure how to tell how many muons left the detector without decaying, as they were indistinguishable from other background radiation which is filtered out. As for the angle and thickness, as Vanadium50 said, if the particles all moved directly downward, the cosmic ray rate would be proportional to the detector thickness. However, I'm pretty sure the incident cosmic ray rate is proportional to the cosine squared of the incident angle.

 

[mfb]

The probability of a decay in flight is tiny, so you are studying muons that were stopped in the detector. Assuming the flux is the same throughout your detector, how is the detection rate related to the volume?

 

[Sciencemaster]

The probability of a decay in flight is tiny, so you are studying muons that were stopped in the detector. Assuming the flux is the same throughout your detector, how is the detection rate related to the volume?

I would think that detection rate is proportional to volume. If the flux is homogeneous within, then doubling the volume would mean that there's twice as many particle detections.

 

[mfb]

I agree.

Keep in mind that muon flux experiments typically measure muons that just pass through the detector, not muons that decay in the detector. The latter is a small fraction of the former, and to make things worse that fraction also depends on the muon energy.

 

[Sciencemaster]

I agree.

Keep in mind that muon flux experiments typically measure muons that just pass through the detector, not muons that decay in the detector. The latter is a small fraction of the former, and to make things worse that fraction also depends on the muon energy.

That's fair. Is there a way I can use the information I have to figure out the actual flux of muons, despite these difficulties?
Alternatively, perhaps I should look for another, similar experiment. I would imagine that the amount of muon decays per unit volume is related to the muon flux at that position.

 

[mfb]

If you want the total flux, look for muons that cross the detector. Use a second detector to only count simultaneous hits, that gets rid of most background.

Captures depend too much on the energy.

 

[Sciencemaster]

Unfortunately, I only had access to a single scintillator for this experiment, and as such I had to use the decay time as a part of background removal. The experiment only output decay times and counts, so as far as I'm aware, I don't have the energy either.

If I can't directly find total muon flux with the data I have, perhaps I can just compare this experiment to another that also measures muon decays in a detector. I would imagine the amount of decays detected would be proportional to the flux of muons at a given position.

 

[Vanadium 50]

Unfortunately, I only had access to a single scintillator for this experiment

Then you are likely looking only at noise.

 

[Sciencemaster]

Then you are likely looking only at noise.

Well, the idea is that by discriminating out any signals that aren't within 40 #\mu s# of each other, the apparatus pretty much only detects muon decays, with only some background that can be filtered out. Assuming that what the device detected were muon decays, I'd imagine it's mostly from cosmic rays.

 

[Vanadium 50]

Nature doesn't care what you think you are looking at. From your description, you are looking at noise. PMT noise is measured in kilohertz. Stopping cosmic rates are measured in millihertz.

 

[Sciencemaster]

Nature doesn't care what you think you are looking at. From your description, you are looking at noise. PMT noise is measured in kilohertz. Stopping cosmic rates are measured in millihertz.

I don't think it's just noise. By calculating the mean lifetime from the average time between pulses, I got something close to 2.2 #\mu s#. That seems unlikely if what I'm looking at is just noise.

 

[Sciencemaster]

Maybe the detections were just noise, maybe they weren't. Under the assumption that what was detected is comsic ray muons, is there a way to use my data to compare muon flux between my experiment and another done at a different position.

 

[Vanadium 50]

A few points to consider:

• "I got the answer in the back of the book" is a heckuva way to do science. Well, "science" anyway.
• If you have ~450 kHz of Poisson noise you will see pulses on average 2.2 us apart.
• 2.2 us is the wrong answer! The correct number is closer to 2.0 for reasons not discussed on this thread. If you are getting 2.2, you are doing something wrong.
• Oh. and one more: Calling yourself a "Science Master" doesn't make you one.

 

[Sciencemaster]

Alright, thank you for your time and help! I really appreciate it!