Muon Lifetime

Laboratory Instructions

Part I: Setup

 

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 upward-decaying muons), or between paddles two and three (allowing for upward or downward-decaying muons). 

 

This should provide the muons a place to stop and decay.  In this experiment, lead was placed between paddles two and three. 

 

A four-paddle experiment with lead between two and three has not been tried, but would be particularly interesting for measuring branching ratios for upward-decaying and downward-decaying 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 four-paddle configuration is acceptable, but

            the top paddle must be in the lowest-number channel, the

middle paddle must be in the next-lowest 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 15:38:01 2003

 

 

Hit number

Time of Event

Coincidence

Double-Hit

Time Difference

1

47814.9543

(-,2,3,-)

(-,-,3,-)

1.5625E-07

2

53082.5047

(-,2,3,-)

(-,-,3,-)

3.2446E-06

 

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: Double-Hit

                                       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 13-hour 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 1.00E-7 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 1.00E-07 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 3.00E-07 to 1.00E-05 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 123-3.)  Other hit types, such as a 12-1 or 123-2, are more likely to be noise. 

Common sense can be used to discard certain hit types, such as 13-2, as probably not muon decays.  Others, however, are more difficult to determine (such as 123-123).

 

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 23-3 run.


This data is reasonably exponential, and 23-3 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 23-2 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 23-23 graph.


 

23-23 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 zero-value 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, Muon Peak, is the apparent peak in this graph.

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 = Muon Peak, t = Muon Lifetime, and x = the bin time.  An example is included on the next page.


 

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 +Muon Peak*e-bin/MuonLifetime

 

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, Muon Peak, and Muon Lifetime) until you arrive at a good fit.  The entry for “Muon Lifetime” should correspond to the decay constant.

 

Example of Final Results

 


This graph is the results of the example data used to create this procedure.  The Muon Lifetime was determined to be 1.55E-06 seconds.

Further Study for Advanced Students:

 

If desired, a Chi2 analysis can be performed on the prediction function.  Then

Tools à Solver can be used on the three fit parameters to minimize Chi2 and maximize the Chi2 probability.  This will create the ‘best’ best fit more precisely than eyeballing the curve.  This will not work in Excel if you have a zero-value frequency in any of your bins.

 

Another statistical tool that seemed to work well (regardless of zero-value 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 Chi2 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).