The Muon Lifetime Analysis Lab
Example analysis at: http://www.pas.rochester.edu/~ksmcf/muonAnalysis.xls
Often when taking experimental data one will record events which you do not wish to include in your final analysis. In most experiments in particle physics data samples are contaminated with such events, which are often referred to as background noise. For example, in an experiment looking production of muons in colliding particle beams, cosmic ray muons are a source of background noise!
The usual technique for correcting for background noise is to measure an estimate of this noise and then subtract it before data analysis. The first thing to do is to decide what data sample we want to keep for the analysis, in this case, the downward muon count. Since muons originate from the atmosphere are absorbed in the earth, downward going events are most likely real stopping cosmic ray muons, whereas upward going events are most likely to be background. We want to remove all the upward counts, and also use these as an estimate of the background noise in the downward sample (assuming, of course, that the background is equal in the two cases). It is easy to separate upward from downward as the DAQ program provides the necessary information to determine the order of the hits. We then divide the muon hits into two groups, up and down, and subtract downward going events from upward going in each time bin.
The board works in such a way that it can identify upward and downward muons and it can also identify a muon-like hit caused by noise. The muon lifetime will work with 2, 3, or 4 paddles. It assumes the layouts shown below where the diagrams represent the physical configuration of the paddles. (E.g., in the 2 Paddle Layout, paddle 1 is assumed to be above paddle 2.) The code will not work with a different paddle arrangement!
2 Paddle Layout
---------- paddle 1
---------- paddle 2
3 Paddle Layout
---------- paddle 1
---------- paddle 2
---------- paddle 3
4 Paddle Layout
---------- paddle 1
---------- paddle 2
---------- paddle 3
---------- paddle 4 (code assumes 4th paddle is a veto)
The board can be placed in muon lifetime mode using option 8 (“Configure this program”) on the main menu. Then select option f (“Enable Muon lifetime code”). Now select option 4 (“Begin Coincidence Run”), from the main menu. The rate for muon decays is quite low, so you will need to run for a significant amount of time. A suggested data run for a good data sample is at least 12 hours, but this is an absolute minimum.
The DAQ records information from the board in a number of ways. First, it receives the raw data line which is not displayed. The on screen display gives an interpretation of that data line which looks something like this:
6 1609.000 (1,2,-,-) 2 169.000002 .000002012
This is interpreted as the following:
6 = event number, in this case the 6th event recorded
1609.00 = time in seconds the event occurred form the first event recorded
“(1,2,-,-)”= hit description. This tells you which paddles fired, 1&2 in this case
2 = the paddle number (1, 2,3, or 4) where the decay electron was seen.
169.000002 = the time in seconds since the first event that the decay was seen
.000002012 = the time in seconds from the coincidence hit until the electron from the muon decay was seen.
The DAQ records this data into 4 files, depending on characteristics of the data line. The program creates the following files:
The .filename extension is the date stamp of the run unless you specify a unique file name for the run. The muon file records all information in the run. The downward file records all downward (from the atmosphere towards earth) muons. The upward (from earth towards the atmosphere) ones are largely noise. The noise file contains events that cannot be interpreted as a muon stopping, such as ones where the 1st and 3rd paddles are hit, but the 2nd is not, or ones where the 1st, 2nd and 3rd paddles are all hit, but the decay electron is observed in the 2nd paddle.
In addition to recording the raw data line the program also produces histograms of the data in two ways. These histograms are located in the same file, but after the raw data information. The first histograms the data is a “fine” histogram. Here the data is divided in bins which are one “clock tick” wide, the smallest division of the data possible. The second histogram is a “coarse” histogram, in which the bin width is ten clock ticks wide. Each time bin is then filled with the number of muon decay events recorded to have that life time. You can use either the fine or coarse histogram information to do the lifetime, although the coarse histogram is easier to use with lower statistics. It is this data that you import into Excel to do the lifetime analysis. You need both the upward and downward files for a good analysis.
1) Background subtraction in Excel
Form two adjacent columns of downward and upward going counts, and simply subtract the downward from upward columns to get the “background subtracted” result (Column D in the example spreadsheet). This may give you negative numbers. Don’t worry! This is expected and will be handled in error calculations.
2) Calculate s (uncertainty)
We now calculate the uncertainty in the background subtracted data. This is done using the following equation:
Here alpha is the number of upward counts and beta is the number of downward counts (Column E in the example spreadsheet). This was then plotted as the half-length of the symmetric error bar when displaying the data. It is important to remember that uncertainty means the measured number could fluctuate either positive or negative (which is why the error bar covers both sides of the measurement).
3) Fitting the curve
We want to model the observed time distribution with an exponential function representing the decay probability of the muon versus time. We divide this analysis into a number of different steps below: identifying where to fit, the functional form of the fit and normalizing your fitting function.
a) Where to fit?
The board response to the second hit indicating muon decay is unfortunately not perfectly uniform in time, particularly in the first bins. You can see this is so because the “background noise” samples, such as the upward muons, don’t populate the first bins of the analysis. Correcting for this isn’t easy because we don’t have a model for the source, so the safest thing to do is to remove data with small times.
We chose where to cut based on where the upward going muon data starts to look flat, which in the data in the sample spreadsheet is bin 4 (coarse binning) or clock tick 40. In a normal distribution we want the sum of all probabilities to be 1. Wherever you chose this cut-off to be, the analysis will still work, unless of course you include suspect data or you remove the entire exponential curve!
Due to the finite lifetime of the muon, we expect the rate to fall exponentially with time. We will fit to a functional form
The point of normalization is that we want the total predicted number of events to be the same as the number achieved in the data. This is done by adjusting the parameter, N0. As a result of following this procedure, the fit function will only depend on a single free parameter, the muon lifetime, which greatly simplifies the procedure of determining the lifetime from the data.
Calculating the normalization is essentially determining the area under the curve. Since these are discrete measurements we will NOT use calculus and perform an integral. Instead, we perform a discrete sum over the bins included in our calculation:
bi = initial bin # (integer value)
bf = final bin # (integer value)
t = muon lifetime in units of bin widths
We will now manipulate this statement into a form we can use. A finite summation can be re written as a difference of two infinite summations:
Remove the bi and bf +1 terms and factor the common summation:
Finally, it is known from the study of infinite series that:
This can be easily derived but is omitted here. Appling this to the last summation we obtain:
Now multiply (both top and bottom) thought by:
And you get:
This provides the area under the curve for the corrected data, now we simply divide by the sum of the difference between downward going and upward going events in each bin (Column D of the sample spreadsheet). Thus our normalization constant is given by:
where Du-d = The difference between up an down for a bin
Recall that bi and bf are integer bin numbers, not “times”. This allows us to make a helpful substitution. The integer 1 in the above equation can be written as the difference of to adjacent bins. We will use the substitution of 1= bf – (bf -1), indicating the last and next to last bin. Placing this in the equation we get:
c) The Fit Function
Now that the hard part is done we can actually do a fit. The complete fit function is:
where N(t) is the predicted value in the bin number b, t(b) is the time of the bin and t is the muon lifetime.
You may have noticed that we never specified what the muon lifetime is numerically. That is because this is what we are trying to find! As a first method, take a guess and try to make your graph match as closely as possible to the data. You may also have noticed there are still negative data points after the subtraction. That is okay, as long as the negative values are consistent with 0. .
4) Making Sure you have the Best Fit
A more sophisticated analysis can be done using a chi-squared test for the best fit. The chi (pronounce to rhyme with why) squared fit (c2) is a standard test for fitting and the idea is to minimize its value. The equation:
O(t) = Down-up counts
N(t) = predicted counts from previous calculation
sO = calculated error for the bin
The sum is taken over all bins used in the analysis.
After this is calculated adjust your muon lifetime to minimize c2. (There is a special tool in Excel called “Goal Seek” which will allow you to do this automatically. Just tell it that you want to set a bin in which the chi-squared is calculated to zero by varying the cell with the muon lifetime value.) This will give you your best curve.
 This is the solution you get if you assume that the decay rate for an individual muon is constant in time. Then for a population of muons, N(t), the decay rate dN/dt is proportional to N(t). This is a simple first-order differential equation,
The solution to this equation (plug it in and show that it works!) is
where t is the lifetime of the muon.