Time Dilation and Muon Energy vs. height (Muon Experiment)

[piareround]

Hey guys

The lab manual for the class is found here
http://advancedlab.physics.gatech.edu/labs/Muons/Muons.pdf

One of the things, I was interested in figuring out was what would be the minimum height needed to put a moun detector (currently at ground level) in order to detect a minimum 1% time dilation with at least a muon count of 100-1000 muons. Because I have only one detector and where I live is very level, I would have to present theoretical results to my professor first in my lab report to his help in find a place to get to measure time dilation. Here is the formula, I derived:
t'= \alpha/\rho \ln(\sqrt{{\gamma }^{2} - 1}+\gamma) =\alpha/\rho =\ln(\sqrt{{\rho/\alpha*h }^{2} - 1}+\rho/\alpha*h )
\alpha = m*c/C_{0}


So I tried to use the time dilation formula gave in the book to put the lab frame decay time t' in terms of time and then figure out the a decay time that give me at least something near the moun decay rate of 2.19. However, each time I plug it in I get unusual results for constants like \alpha α .

I have a spanish 1001 final in like 10 minutes, so I was wonder if later I post my questions can someone out there who knows enough about the Teach Spin Muon experiment help me understand what I am doing wrong with special relativity?

 

[GAsahi]

Hey guys

The lab manual for the class is found here
http://advancedlab.physics.gatech.edu/labs/Muons/Muons.pdf

One of the things, I was interested in figuring out was what would be the minimum height needed to put a moun detector (currently at ground level) in order to detect a minimum 1% time dilation with at least a muon count of 100-1000 muons. Because I have only one detector and where I live is very level, I would have to present theoretical results to my professor first in my lab report to his help in find a place to get to measure time dilation. Here is the formula, I derived:
t'= \alpha/\rho \ln(\sqrt{{\gamma }^{2} - 1}+\gamma)

Looking at your .pdf , you definitely have an error in the symbolic result for the integral, the correct formula is:

t'= \alpha/\rho \ln(2 \sqrt{{\gamma }^{2} - 1}+\gamma)

There might be other errors but this is the first one I noticed, so I stopped here.

 

[piareround]

Thanks :)

 

[H_A_Landman]

I've run the Teachspin device for a couple of months at about 1500 meters altitude. Some things to watch out for:

1. You should probably do your own exponential curve fitting, rather than trusting the canned software that comes with it. If you do this, realize that you are not fitting a full (infinite time) exponential, but a truncated exponential over a finite time. This would have been a little messy 10 years ago, but recently the truncated exponential was solved analytically (F.M. Al-Athari, "Estimation of the Mean of Truncated Exponential Distribution", Journal of Mathematics and Statistics 4(4) 284-288 (2008)) so it's not that hard anymore.
   
2. Many people are finding that their lifetime numbers are slightly lower than the "correct" value. There is an attempt to explain this on the Teachspin website, but I can't make any sense of it. In my own data, I saw evidence for a tiny amount of contamination by shorter-lived particles (probably mainly pions and/or Klongs). Although there were only a few of these (about 0.3%), their lifetime was so much shorter (empirically around 28 nS average, versus over 2000 nS) that it could explain about half the discrepancy. NOTE: At sea level you might get different results. Also NOTE: I was only able to curve fit for a sum-of-2-exponentials to separate out these shorter-lived particles because I had a lot of data. If you only get a thousand events, you probably won't be able to do that reliably.
3. The Teachspin is unreliable at recording times below 2 or 3 multiples of the 20 nS FPGA clock. If you look at the number of events in each bin, you can see this. It's probably best to discard the first 3 bins and slide the other bins over by 60 nS. You can do this because the exponential distribution is memoryless. A side-benefit of doing this is that any effect from contamination by pions etc. will be greatly reduced.