Which threshold value provides the most accurate results for muon decay time?

[tryingtolearn1]

Homework Statement

Determining if my results are accurate

Relevant Equations

$N(t) = N_0 \exp(- \frac{t}{\tau}) + \text{bg},$ where $bg$ is the background counts, $t$ is time interval of the muons in the detector, $\tau$ is the decay time that needs to be determined.

I am trying to understand my results for my muon experiment. I conducted the experiment using a plastic scintillator photomultiplier detector. I have four different data sets, with different discriminator thresholds: 148 mV, 190 mV, 260 mV and 550 mV. I made a histogram of the counts of all four datasets and I fitted a best fit line in order to determine the decay time using $N(t) = N_0 \exp(- \frac{t}{\tau}) + \text{bg}$ as my model function. For the $148 mV$ I got the decay time value of $1.97\mu s$, for $190mV$ I got $2.13\mu s$, for $260mV$ I got $2.05\mu s$ and for $500mV$ I got $2.06\mu s$.

Therefore based on my results it seems like the $500mV$ threshold gave the most accurate results since the mean decay time for muon is around $2.2\mu s$ but how do I confirm that the $500mV$ threshold is actually the best threshold value? I also, took the logarithm of the counts and plotted it and it provided a linear relationship therefore I used a slope equation for the best fit line and it gave me the following results $3.45\mu s$ for $148mV$, $2.56\mu s$ for $190mV$, $2.36\mu s$ for $260mV$ and for $500mV$ I got $2.15\mu s$ so again $500mV$ provided the most accurate results.

Should I be confident that $500mV$ is indeed the best threshold value since my data suggests that? Or is there some there missing information that I am not aware of that I can use to determine which threshold value will provide the most accurate results?

 

[Vanadium 50]

With no description of what your setup is, it's impossible to tell. That said, if someone "picked the best setting" by comparing with the known value of what they were trying to measure, I would give them a very low grade.

 

[tryingtolearn1]

With no description of what your setup is, it's impossible to tell. That said, if someone "picked the best setting" by comparing with the known value of what they were trying to measure, I would give them a very low grade.

For the plastic scintillator detector the PMT High Voltage (HV) was set between 1100 and 1200 Volts. The HV remained in that constant range however the discriminator mV was changed each time which I gave the values above.

But hmm why would you give me a low grade for lol? I plotted all the histogram dataset corresponding to each discriminator setting mV, I found the best fit line to determine the decay constant, I even took the log of counts to make sure that my decay constant that was fitted from the histogram corresponded to the slope of the log of counts, I then even took the mean of the measurements which provided an estimate to the decay time constant and it also gave that the 500mV discriminator voltage as the most accurate result.

Having done all that I just wanted to confirm what the general outcome when you have a constant PMT HV but with different discriminator settings what the outcome will be? I know with a lower discriminator setting you get more noise since the muons are more energetic therefore the higher the discriminator setting the less noise but if you have it too high then there won't be any muon detection.

 

[Vanadium 50]

If you refuse to descriube your setup, I can't help you. Nobody can.

 

[berkeman]

If you refuse to descriube your setup, I can't help you. Nobody can.

I think it is from his other thread on this. @tryingtolearn1 is that correct?

Summary:: Plastic scintillator

When detecting a muon, we can use a plastic scintillator which consists of the components in the diagram attached. I am trying to understand what exactly the "two-stage amplifier" does which is located in the top left corner in the image attached? I am unable to find any information on it online in regards to the particle detector. I understand what the scintillator and the photomultiplier does, but not sure what the two-stage amplifier does.

 

[Vanadium 50]

Thanks @berkeman . One shouldn't have to fish around PF for a clear description of the question.

If you have one PMT, I am surprised you are not dominated by noise. I am also surprised that you can see the muon lifetime by looking at a sample where most muons go straight through.

 

[tryingtolearn1]

Thanks @berkeman . One shouldn't have to fish around PF for a clear description of the question.

If you have one PMT, I am surprised you are not dominated by noise. I am also surprised that you can see the muon lifetime by looking at a sample where most muons go straight through.

Ops, sorry, I thought I attached that image in the original post. But not sure what you mean by one PMT? The set up in that diagram is a very popular TeachSpin instrument that is used in many universities: https://www.teachspin.com/muon-physics.

The test was run for over a day. The data 148mV definitely has a lot of noise but the other mV settings does not have as much noise.

 

[Vanadium 50]

Normally this is done with multiple counters. For example, suppose you have three - A (above the main counter), B (the main counter) and C (below). If I have simultaneous hits in A and C, I know a muon went through B. I can adjust the threshold until it's 90% efficient, or 95%, or whatever I want.

To do the measurement, I start the clock when I have simultaneous signals in A and B, but not C, and I stop it whenever I have a signal in B and not A or C. If you don't use a coincidence technique, I don't see why you aren't dominated by noise - hundreds of Hz - and through-going muons (1 Hz).

If you have 100 Hz of noise, and look for 20 microseconds (what their plot shows), there will be 12x as many accidental double-hits as real signal. If it's 200 Hz, that number jumps to 48x. To get this to work, the tube has to be surprisingly quiet. (And what they show is crazy quiet - to reproduce the figure, the non-stopping signal rate needs to be below 3 Hz. And one of those hertz is a through-going muon)

Further, the plot they show has 22000 muons in it, and they say they get one stopped muon per minute. Did they run this for two weeks?

I don't get how this works as well as it does.

 

[mfb]

Cross check: The bins are 1 us wide, they have ~50 events in each bin, or ~1000 background events in the plot.
22700 minutes are about two weeks.

To appear in the plot we need two signals within 20 us, at a random background rate of x we expect to get these events with a rate of x² * 20 us. Setting that equal to 1000/(2 weeks) I get x=sqrt(1000/(2 weeks*20us)) = 6 Hz.

If the scintillator produces enough light then we can at least discriminate between particles and noise from the PMT, but we still get muons that are not stopped and radioactive decays. It's surprising to see a background level that low.

 

[Vanadium 50]

I'm wondering if the threshold might be very, very high - so high as to be nearly blind to mips. Even at the very lowest sittings. Nearly stopping muons are ionizing a lot, and Michel electrons are, well, electrons. Might let you get away with a cheaper tube as well.

 

[mfb]

Keep both high energy muons and beta decays under the threshold? That's an interesting idea.

Having done all that I just wanted to confirm what the general outcome when you have a constant PMT HV but with different discriminator settings what the outcome will be? I know with a lower discriminator setting you get more noise since the muons are more energetic therefore the higher the discriminator setting the less noise but if you have it too high then there won't be any muon detection.

As you said, both thresholds that are too high and too low will make the detection difficult, and will increase the uncertainty. There should be a threshold where the resulting uncertainty is minimal. Ideally you determine this with a dataset independent of your actual result, to avoid biasing yourself. And certainly you shouldn't use the comparison to the literature lifetime to make that choice.

 

[tryingtolearn1]

Normally this is done with multiple counters. For example, suppose you have three - A (above the main counter), B (the main counter) and C (below). If I have simultaneous hits in A and C, I know a muon went through B. I can adjust the threshold until it's 90% efficient, or 95%, or whatever I want...

Oh I wasn't aware of this. But yes, when running the instrument, it did take days to accumulate accurate results. Here is a better manual that discusses the instrument more in depth:

http://www.phys.utk.edu/labs/Muon Physics/Coan & Ye Muon Physics Manual.pdf

@mfb Can you please elaborate on this, " Ideally you determine this with a dataset independent of your actual result, to avoid biasing yourself. And certainly you shouldn't use the comparison to the literature lifetime to make that choice." Are you saying even though the 500mV threshold provided the best comparison to the lifetime, doesn't mean that it is the most efficient settings compared to the other settings?

 

[Vanadium 50]

You should read what the manual says about setting the discriminator. It's on a page number that is a power of 2.

 

[mfb]

@mfb Can you please elaborate on this, " Ideally you determine this with a dataset independent of your actual result, to avoid biasing yourself. And certainly you shouldn't use the comparison to the literature lifetime to make that choice." Are you saying even though the 500mV threshold provided the best comparison to the lifetime, doesn't mean that it is the most efficient settings compared to the other settings?

Imagine you repeat the same measurements on four days. On Monday you measure 3.45 us, on Tuesday you measure 2.56 us, on Wednesday you measure 2.36, on Thursday you measure 2.15 us. Does that mean Thursday is the best day to measure? Does that mean you can measure very accurately if you only take data on Thursday? Of course not. You picked the "best" day based on your prior expectation. What if your prior expectation was wrong? Even if it was right - you didn't perform an independent measurement, if you will always pick the value that's the closest to previous experiments then your result is basically useless.

What you would do for a "real" experiment: Set different thresholds, measure the approximate background and muon rate each time. Feed that into a simulation that randomly produces events based on a few different values for the muon lifetime and the measured muon and background rates. Fit each curve, see if the extracted values match the inputs (especially for the lifetime, obviously). Fix problems if these don't match. Then pick a threshold that leads to a good agreement between input and output lifetime in the simulation while minimizing the uncertainty. Pick that threshold for data-taking. Maybe take shorter runs at other thresholds to check that your procedure works.
As additional advantage of this approach: You can determine in advance how long you need to measure to reach a given uncertainty.

 

[Vanadium 50]

Only measure muons in months with R's in them. And always wait an hour before swimming. Or something like that.

@mfb's procedure will get you the optimal discrimination threshold. But you don't need to be exactly optimal to do the experiment. You can get close by using your head - if the threshold is too high, I see no signal. If the threshold is too low, I see a lot of noise. What does that tell me about finding a working point?

 

[Vanadium 50]

For the $148 mV$ I got the decay time value of $1.97\mu s$, for $190mV$ I got $2.13\mu s$, for $260mV$ I got $2.05\mu s$ and for $500mV$ I got $2.06\mu s$.

Therefore based on my results it seems like the $500mV$ threshold gave the most accurate results since the mean decay time for muon is around $2.2\mu s$

Apart from the issues @mfb and I brought up, why do you think 2.06 is closer to 2.2 than 2.13 is?

 

[tryingtolearn1]

Apart from the issues @mfb and I brought up, why do you think 2.06 is closer to 2.2 than 2.13 is?

I actually figured out the issue with my dataset. The issue was that i forgot to subtract off the background noise which skewed my data a lot and gave me the weird relationship where the the mV values decay times where inconsistent with the mV values. But once I removed the background noise, I noticed a relationship where the increasing mV provided much better results and this is due to the detector not filtering enough noise due to the low mV values. The 550mV was the highest threshold used but it wasn't too high where no muons entered.

Thanks for all the help!

Also, the reason why I thought 2.06 is closer to 2.2 than 2.13 was because of the chi-squared value, p-value and mean value that was computed. The 2.06 gave me a very high uncertainty whereas the 2.13 gave me a good fit. So I knew that the 2.06 value was not reliable.