Experimental
setup:
This experiment
must be performed with a three or four paddle configuration, as follows:
Note that the
top paddle must be in channel one, the middle paddle must be in channel two,
and the bottom paddle must be in channel three.
The paddles are placed one atop another, with or without some greater
degree of separation.
Lead may be
placed between or under the paddles to give the muons a place to stop and
decay. Suggested amount is 3 to 4 cm of
lead.
NOTES
ON LEAD:
To date, there is insufficient data to accurately determine
precisely how the placement of lead affects the data.
Suggested arrangements are to have several centimeters of
lead below the third paddle (allowing only for upwarddecaying muons), or
between paddles two and three (allowing for upward or downwarddecaying
muons).
This should provide the muons a place to stop and
decay. In this experiment, lead was
placed between paddles two and three.
A fourpaddle experiment with lead between two and three has
not been tried, but would be particularly interesting for measuring branching
ratios for upwarddecaying and downwarddecaying muons.
Program Configuration:
The following configurations must be performed to measure
muon lifetime.
I.
In the main menu, type 1. Control Register.
Enable at least three paddles (1,2, and 3 were used with this data).
Any
three or fourpaddle configuration is acceptable, but
the top paddle must be in the lowestnumber channel, the
middle paddle must be in the nextlowest channel, and so
forth.
Set Coincidence level to 2.
Return to the main menu.
II.
In the main menu, type 8. Configure This Program.
Select option (c) Select Data to
Report and type ‘n’ (for no) for
each question. It
will now read (FALSE) for option c.
Select option (d) Configure for
Muon Lifetime Run (TRUE)
Exit to the main menu.
Part II:
Running the Experiment
Before running your experiment,
type 3 to do a Single Counter Rate test to ensure that all paddles are
functioning properly.
In the main
menu, type 5. Begin Custom Run (use control register)
Select the
time you wish to run.
Potential muon lifetime events are
few and far between, so longer runs (at least 12 hours) are desired. Separating the paddles drastically decreases the
muon rate, so a run over several days or weeks would be beneficial.
Part III:
Data Analysis
1.
Copy the data into Excel, obtaining five columns,
like this:
Run Started at Wed Jul 23 



Hit number 
Time of Event 
Coincidence 
DoubleHit 
Time Difference 
1 
47814.9543 
(,2,3,) 
(,,3,) 
1.5625E07 
2 
53082.5047 
(,2,3,) 
(,,3,) 
3.2446E06 
2. Scan the data to make sure that there are
no obvious bugs in data collection (i.e. zillions of hits in one second).
3. Use Data à Sort à First Column: Coincidence
Second Column: DoubleHit
Third Column: Time Difference
3. Analyze the data to determine the frequencies of hit types. Discard any hit times with insufficient population. (This number will vary depending upon the length of your data run. In a 13hour run, with 1500 total events, we discarded any hit type with less than 50 events.)
It is also useful to determine the relative frequency of the hit types you do decide to keep. This information may be useful for determining the effects of lead placement on the final results. For example:
Step 3: Data > Sort > Coincidences then
Double Hits then Time 


Hit Type 




(,2,3,) (,,3,) 
112 Hits 



(,2,3,) (,2,,) 
131 Hits 



(,2,3,) (,2,3,) 
98 Hits 



(1,2,,) (,,3,) 
110 Hits 



(1,2,,) (,2,,) 
268 Hits 



(1,2,,) (1,2,,) 
131 Hits 



(1,2,3,) (,,3,) 
114 Hits 



(1,2,3,) (,2,,) 
93 Hits 



(1,2,3,) (1,,,) 
102 Hits 



(1,2,3,) (1,2,3,) 
85 Hits 

4. Create a histogram for each hit type.
First, create a column to use as bin sizes. We used bin sizes of 1.00E7 seconds, but you may use whatever you deem is appropriate. The program reports times in microseconds (us), so you should create your bin sizes in units of microseconds. Example (starting with 1.00E07 seconds, or 0.1 microseconds):
Bin Sizes for Histogram (us) 
0.10 
0.20 
0.30 
0.40 
0.50 
0.60 
0.70 
0.80 
A good range of data is from 3.00E07 to 1.00E05 seconds. Sometimes when graphing it is better to shorten this range due to decreased detector efficiency at these extremes of time differences.
Second, Use Tools à Data Analysis à Histogram to create histograms for each hit type. Use the “Bin Sizes for Histogram” that you just created, and select the appropriate data for each hit type. (This will create many histograms – one for each hit type. You have to manually create each one.).
It is generally a very good idea to create either an XY scatter or a bar graph from this data, to be used in step five, below. Do not include the “more” cell from the histogram.
5. Determine usefulness of each hit type.
Some hit types, such as a coincidence in (1,2,3,) followed by a (,,3,), are likely candidates for muon decay. (In this case, a coincidence in three paddles, followed by a secondary hit (or “Double Hit”) in paddle three. This will be abbreviated in the future as 1233.) Other hit types, such as a 121 or 1232, are more likely to be noise.
Common sense can be used to discard certain hit types, such as 132, as probably not muon decays. Others, however, are more difficult to determine (such as 123123).
To determine if these event types are likely muon decays, examine the graph for each type. If the histogram graph appears to be an exponential decay, then that event type is likely a good candidate for ‘good’ data.
An example of good data is our 233 run.
This data is reasonably exponential, and 233 physically fits what we might expect from a muon decaying in the table and the electron coming back up through paddle three, so we call it ‘good’.
An example of bad data is our 232 run.
It is neither a physically probable situation, nor is it a good
exponential graph. It looks more like a
‘spike’ at ~1 microsecond. (The
characterization and cause of this spike is not yet well understood – it shows
up on several of our ‘rejected graphs).
Running with the new board may eliminate this problem.
Finally, the third type of graph you will get is simply noise, such as our 2323 graph.
2323 is neither a likely occurrence (with the lead between paddles two and three), and the graph looks like random noise. Thus, it is discarded.
2. Create Composite Histogram
Using only the ‘good’ data, create a histogram and graph as before.
3. Fitting the Composite Histogram
Excel’s exponential best fit line is inadequate to the task of fitting this data well. The first problem is that any bins with zero hits in this histogram will make Excel completely unable to make an exponential fit. Furthermore, even if you do not have any zerovalue bins Excel’s fit is still pretty bad. Thus, a (somewhat) more sophisticated fit is needed.
Next to your histogram data, create a column for a Prediction Function.
Next to this, create three more cells. These cells will be manual parameters used to create a best fit line by eyeballing it.
The first cell, Noise Constant, is the number of muons you have at the end of the exponential curve. In our example this appears to be approximately 3.
The second cell,
The third constant, Muon Lifetime, is the slope of the inverse graph, and is also the muon lifetime we are attempting to determine!
In the top cell of the ‘Prediction
Function’ column, create the equation C + Ae^{x/t},
where C = Noise Constant, A =
Bin 
Frequency 
Prediction Function 

Noise Constant 
Muon peak 
Muon Lifetime 




3 
70 
1.55 
0.30 
55 
6.07E+01 




0.50 
35 
53.6994264 




0.70 
33 
47.5620384 




0.90 
43 
42.1676081 




1.10 
54 
37.4261973 




1.30 
63 
33.258755 




1.50 
40 
29.5957999 




1.70 
33 
26.3762617 




1.90 
15 
23.5464626 




2.10 
21 
21.0592232 




2.30 
18 
18.8730749 




2.50 
12 
16.9515696 




2.70 
11 
15.2626709 




2.90 
15 
13.7782207 




For example, the equation in cell C3 is
= $E$2 + $F$2*EXP(A3/$G$2)
that is
=
Noise Constant +
Next, click on your histogram graph, then go to Chart à Add Data and select the entire data range in your ‘Prediction Function’ column. This should create a pink best fit line in your graph. (The fit will probably not be very good at first).
Manually alter the three fit
parameters (Noise Constant,
This graph is the results of the example data used to create this
procedure. The Muon Lifetime was determined
to be 1.55E06 seconds.
Further Study for Advanced
Students:
If desired, a Chi^{2} analysis can be performed on the prediction function. Then
Tools à Solver can be used on the three fit parameters to minimize Chi^{2} and maximize the Chi^{2} probability. This will create the ‘best’ best fit more precisely than eyeballing the curve. This will not work in Excel if you have a zerovalue frequency in any of your bins.
Another statistical tool that seemed to work well (regardless of zerovalue bins) is a Poisson distribution. Obtain the probability for each data point using “=POISSON(Frequency, Prediction, FALSE)”, multiply the probabilities obtained in this manner using “=PRODUCT(probabilities column)”, and then take the log of this sum using “=LOG(product cell)”. Finally, use Tools à Solver to maximize the number given. This should give the ‘best’ best fit curve, and seems to work better than a Chi^{2} analysis.
Example:
As an example for the poisson probability, assume the frequency column starts at cell B3, the prediction column starts at cell C3, and the probability column starts at D3, and that all columns end at row 100.
1. In cell D3 you would type “=POISSON(B3,C3,FALSE)”. Then drag it down the entire column.
2. In cell E3 you would type “=PRODUCT(B3:B100)”.
3. In cell F3 you would type “=LOG(E3)”.
4. Then you would use Tools à Solver, select to maximize cell F3 by changing the cells with your three fit parameters (noise constant, muon peak, and lifetime).