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School of Physics

1.4 Muon Physics

1. Introduction

The muon is a fundamental particle, discovered in 1937 by Anderson and Neddermeyer1, when they

exposed a cloud chamber to cosmic rays. In many ways muons are similar to ―heavy‖ electrons and

carry an electric charge of +e or -e (i.e., they come in two ―flavours‖, μ+ or μ

-). However, the muon

is not stable and decays into an electron (positron), neutrino, and an antineutrino:

μμ vve e

μμ vve e . (1)

The finite lifetime (inverse of the decay rate) was first demonstrated in 1941 by Rasetti2 and is on

the order of microseconds.

The present experiment is based on Reference [1] and utilises the MPA-1Muon Physics instrument

developed by TeachSpin Inc. It will allow the charge averaged mean lifetime of the muon (at rest in

a plastic scintillator) to be measured.

2. Cosmic Rays as the Source of Muons

The Earth is continuously bombarded by a flux of high energy particles that originate outside the

solar system. These primary ―cosmic rays‖ are composed of 85% protons, 12% helium nuclei, 1%

heavier nuclei and 2% electrons. When these high energy particles reach the Earth’s upper

atmosphere they induce nuclear interactions so that at sea level we observe the final products of the

nuclear cascade.

The interaction of the primary protons with oxygen and nitrogen nuclei of the atmosphere produces

a shower of secondary unstable particles, including pions , K

mesons etc., which in turn decay by

the weak nuclear interaction into lighter particles, including muons , electrons, positrons and

neutrinos. Since the Earth’s atmosphere is equivalent to ten nuclear interaction lengths, all strongly

interacting particles are absorbed before reaching sea level3. What is observed will therefore be a

―hard component‖ consisting of muons and a ―soft component‖ consisting of electrons/positrons

and low energy γ-rays. Figure 1 shows an example of a cosmic ray cascade induced by a proton

striking a nitrogen nucleus. Some of the charged pions produced in the cascade interact via the

strong force with nuclei, whilst others will spontaneously decay (via the weak force) into a muon

plus a neutrino or antineutrino (which are not detected in this experiment):

1

S. H. Neddermeyer and C. D. Anderson, Phys. Rev. 51, 884 (1937).This paper can be downloaded from:

http://prola.aps.org/pdf/PR/v51/i10/p884_1

2 F. Rasetti, Phys. Rev., 59, 706 (1941).This paper can be downloaded from:

http://prola.aps.org/abstract/PR/v59/i9/p706_1

3At sea level the total flux per solid angle per unit area around the vertical is 1.1 10

2 m

-2sr

-1s

-1 where 75% of the flux

comprises muons. The angular distribution varies approximately as cos2θ (with = 0 at the zenith). The total flux of

muons per unit area at sea level is 1.8 102 m

-2s

-1 with a mean kinetic energy of about 4 GeV.

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μv

v . (2)

p

+

n

v+

e+

e-e+

e-

+

-

+

ov

-

Figure 1: Cosmic ray cascade induced by a high energy proton striking a nitrogen

nucleus in the upper atmosphere.

3. Time of Arrival of Cosmic Rays

The arrival of comic rays is a random process. When the expected number of events (i.e., arrival of

a cosmic ray) in a given time interval is small, the observed number should obey a Poisson

distribution. Let λ be the average event rate (i.e., the average number of cosmic rays arriving per

second). Then the probability of observing n events in a time interval t is given by:

,n

ettn,P

λtn

!

(3)

For the case of a single event (n =1) occurring in a differential interval of time dt, you should be

able to show that the probability is given by (dt 0)

dP = P(1,dt) = dt (4)

Similarly the probability of no events (n = 0) occurring in the interval t is given by:

P(0,t) = e-t

. (5)

A time interval t between events is defined by requiring no event for the interval t and a single event

at the time t (in the differential dt).Therefore the distribution of the time interval between the arrival

of adjacent events is given by the product of Eqs. (4) and (5):

.edt

dPtD λt (6)

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It is important to note that the distribution (6) is exponential; namely short time intervals between

events are much more probable than longer intervals4. Equation (6) provides the basis for

determining the muon decay time (see Sec. 3.1).

3.1 The Muon Decay Time Distribution

In this experiment we do not have an ensemble of muons whose number we can measure as they

decay. Instead we detect muon decays from muons that enter the detector at random times (typically

one at a time). In Sec. 3 we noted that the decay time distribution D(t) has a simple exponential

form (6). In other words the time-dependent probability that a muon decays in a time interval

between t and t + dt is given by D(t)dt. If we had started with No muons, then the fraction dN/No

that would on average decay in the time interval between t and t + dt is just given by:

D(t) = e-t

.

(7)

where λ is a constant ―decay rate‖ that characterises how rapidly a muon decays; the muon lifetime

τ is the reciprocal of the decay rate, i.e., τ = 1/λ. This is true regardless of the starting value of No.

That is, the distribution of decay times, for new muons entering the detector, is also exponential

with the very same exponent used to describe the surviving population of muons.

Because the muon decay time is exponentially distributed, it does not matter that the muons whose

decays we detect do not originate in the detector but somewhere above us in the Earth’s

atmosphere. An exponential function always ―looks the same‖ in the sense that whether you

examine it at early times or late times, its e-folding time is the same.

4. Detection of Muons

In this experiment cosmic ray muons are detected by using a plastic scintillator, viewed by a

photomultiplier tube. Subsequent sections will describe the detector and readout electronics in

detail.

4.1 Interaction of Muons with Matter

The muons whose lifetime we measure necessarily interact with matter. Negative muons that stop in

the scintillator can bind to the scintillator’s carbon and hydrogen nuclei in much the same way as

electrons do. Since the muon is not an electron, the Pauli exclusion principle does not prevent it

from occupying an atomic orbital already filled with electrons. Such bound negative muons can

then interact with protons before they spontaneously decay. For example,

- + p n + v. (8)

Since there are now two ways for a negative muon to disappear, the effective lifetime of negative

muons in matter is somewhat less than the lifetime of positively charged muons, which do not have

this second interaction mechanism. The muon lifetime measured in this experiment is an average

over both charge species so the mean lifetime of the detected muons will be somewhat less than the

free space value [2]:

= 2.19703 0.00004 s. (9)

4

This provides some justification for the old adage that one disaster is soon followed by another. For an amusing

discussion of coincidences, see B. Eastman and J. Wyndham, Why do Buses come in Threes? Books, London, 1998

V3.1 16/09/10 4

The probability for nuclear absorption of a stopped negative muon by nuclei in the scintillator is

proportional to Z4, where Z is the atomic number of the nucleus [3]. A stopped muon captured in

an atomic orbital will make transitions down to the K-shell on a time scale short compared to its

time for spontaneous decay. Its Bohr radius is roughly 200 times smaller than that for an electron

due to its much larger mass, increasing its probability of being found in the nucleus. From our

knowledge of hydrogenic wavefunctions, the probability density for the bound muon to be found

inside the nucleus is proportional to Z3. Once inside the nucleus, a muon’s probability for

encountering a proton is proportional to the number of protons there and so scales like Z. The net

effect is for the overall absorption probability to scale like Z4. Again, this effect is relevant only for

negatively charged muons.

4.2 μ+/μ

- Charge Ratio at Sea Level

The measurement of the muon lifetime in the plastic scintillator is an average over both negatively

and positively charged muons. We have already noted that negatively charged muons have a

lifetime somewhat smaller than positively charged muons because of weak interactions between

negative muons and protons in the scintillator nuclei. This interaction probability is proportional to

Z4, so the lifetime of negative muons in the scintillator and carbon should be very nearly equal. This

latter lifetime C is measured to be C = 2.043 0.003 s [4].

It is straightforward to determine the expected average lifetime obs of positive and negative muons

in the plastic scintillator. Let -

be the decay rate for negatively charged muons in the plastic

scintillator, and let + be the corresponding lifetime for positively charged muons. If N

– and N

+

represent the number of negative and positive muons incident on the scintillator per unit time,

respectively, the average observed decay rate is given by:

,

N

N

λλN

N

NN

λNλN

11

(10)

where = N+ / N

-. The average lifetime obs =

-1 which from Eq. (10) is given by:

,obs

1

1

111 (11)

where - = 1/

- is the lifetime of negative muons in the scintillator and

+ = 1/

+ is the

corresponding lifetime for positive muons. Due to the Z4 effect,

- = C for the plastic scintillator,

and we can set + equal to the free space lifetime value , since positive muons are not captured by

the scintillator nuclei. Setting = 1 allows us to estimate the average muon lifetime we expect to

observe in the scintillator. We can measure for the momentum range of muons that stop in the

scintillator by rearranging Eq. (11):

.obs

obs

(12)

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4.3 Background Events

The detector responds to any particle that produces enough scintillation light to trigger its readout

electronics. These particles can be either charged, like electrons or muons, or neutral, like photons,

that produce charged particles when they interact inside the scintillator. The detector has no

knowledge of whether a penetrating particle stops or not inside the scintillator and so has no way of

distinguishing between light produced by muons that stop and decay inside the detector, from light

produced by a pair of muons that pass through the detector one right after the other. This important

source of background events can be dealt with in two ways. First, we can restrict the time interval

during which we look for the two successive flashes of light from the plastic scintillator, which is

characteristic of muon decay events. Secondly, we can estimate the background level by looking at

large times in the decay time histogram where we expect few events from genuine muon decay.

4.4 Fermi Coupling Constant GF

The long lifetime for muon decay indicates that the decay does not occur through the strong

(nuclear) interaction, but rather through the weak interaction responsible for the beta decay of

nuclei. The muon decay process is important because it only involves leptons (no strongly

interacting particles participate) and thus it can be used to calculate the Fermi weak interaction

constant, GF, which is a measure of the strength of the weak force. The mean life of the muon is

given by:

,

cm

c

G22

F

3

5

μ

6 192

11

(13)

where m is the rest mass of the muon [2], and the other symbols have their standard meanings.

Measuring allows a value for GF to be calculated.

5. Muon Physics Instrumentation

5.1 Detector

The active volume of the detector is a plastic scintillator in the shape of a right circular cylinder of

15 cm diameter and 12.5 cm height placed at the bottom of the black anodized aluminium alloy

tube. The plastic scintillator consists of a transparent organic material made by mixing together one

or more fluors with a solid plastic solvent that has an aromatic ring structure. A charged particle

passing through the scintillator will lose some of its kinetic energy by ionisation and atomic

excitation of the solvent molecules. Some of this deposited energy is then transferred to the fluor

molecules whose electrons are promoted to excited states. Upon radiative de-excitation, light in the

blue and near-UV portion of the electromagnetic spectrum is emitted with a typical decay time of a

few nanoseconds. A typical photon yield for a plastic scintillator is one optical photon emitted per

100 eV of deposited energy. The properties of the polyvinyltoluene-based scintillator used in the

muon lifetime instrument are summarized in Table 1.

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Table 1. Properties of the Plastic Scintillator.

Mass density 1.032 g/cm3

Refractive index 1.58

Base material Polyvinyltoluene

Rise time 0.9 ns

Fall time 2.4 ns

Wavelength of Maximum

Emission

423 nm

To measure the muon’s lifetime, we are interested in only those muons that enter, slow, stop and

then decay inside the plastic scintillator. Figure 2 summarizes this process. Such muons have a total

energy of only about 160 MeV as they enter the tube. As a muon slows to a stop, the excited

scintillator emits light that is detected by a photomultiplier tube (PMT), eventually producing a

logic signal that triggers a timing clock (see Sec. 5.2 for a discussion of the electronics). A stopped

muon decays into an electron, a neutrino and an anti-neutrino. As noted in Sec. 4.1 negative muons

that stop in the scintillator can bind to the scintillator’s nuclei. Such bound negative muons can then

interact with protons before they spontaneously decay.

Since the electron mass is so much smaller that the muon mass, m /me ~ 210, the electron tends to

be very energetic and to produce scintillator light essentially all along its path length. The neutrino

and anti-neutrino also share some of the muon’s total energy but they entirely escape detection. This

second burst of scintillator light is also seen by the PMT and used to trigger the timing clock. The

distribution of time intervals between successive clock triggers for a set of muon decays is the

physically interesting quantity used to measure the muon lifetime.

Figure 2: Schematic showing the generation of two light pulses (depicted by short

arrows) used in determining the muon lifetime. One pulse originates

from the slowing muon (dotted line), and the second arises from the

decay of the muon into an electron or positron (wavy line).

PMT

Plastic Scintillator

e

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5.2 Electronics

A block diagram of the readout electronics is shown in figure 3. Scintillation light is detected by a

photomultiplier tube (PMT) whose output signal feeds a two-stage amplifier. The amplifier output

then feeds a voltage comparator (―discriminator‖) with adjustable threshold. This discriminator

produces a TTL output pulse for input signals above threshold and this TTL output pulse triggers

the timing circuit of the FPGA. A second TTL output pulse arriving at the FPGA input within a

fixed time interval will then stop and reset the timing circuit; the reset takes about 1 ms during

which the detector is disabled. The time interval between the start and stop timing pulses is the data

sent to the PC via the communications module that is used to determine the muon lifetime. If a

second TTL pulse does not arrive within the fixed time interval, the timing circuit is reset

automatically for the next measurement.

The front panel of the electronics box is shown in figure 4. The amplifier output is accessible via

the BNC connector labelled Amplifier Output. Similarly, the comparator output is accessible via the

connector labelled Discriminator Output. The voltage level against which the amplifier output is

compared to determine whether the comparator triggers can be adjusted using the Threshold

Control knob. The threshold voltage is monitored by using the red and black connectors that accept

standard multimeter probe leads. The toggle switch controls a beeper that sounds when an amplifier

signal is above the discriminator threshold. The beeper should be turned off when you are taking

muon decay data as it can affect the counting electronics.

Figure 3: Block diagram of the readout electronics. The amplifier and discriminator

outputs are available on the front panel of the electronics box (see Figure 4). The

high voltage (HV) power supply is inside the detector tube.

PMT

Plastic

Scintillator

e

LEDLED

Pulser

HV Power

Supply

Discriminator FPGA Timer

Two Stage

Amplifier

Amplifier

Output

Discriminator

Output

LED

Monitor

Variable Time

Delay

PMT

Input

HV Adjust

Serial

Port

USB

Port

PC

Reference

Monitor

Discriminator

Reference

HV Monitor

V3.1 16/09/10 8

Figure 4: Front panel of the MPA-1 Muon Physics electronics box, showing amplifier and

discriminator output, and the PMT input. The threshold voltage is monitoring

using a digital multimeter, and is adjusted using the control knob.

Figure 5 shows the top of the detector cylinder. DC power to the electronics inside the detector tube

is supplied from the electronics box through the connector DC Power. The high voltage (HV) to the

PMT can be adjusted by turning the potentiometer located at the top of the detector tube. The HV

level can be measured by using the pair of red and black connectors that accept standard multimeter

probes. The HV monitor output is 1/100 times the HV applied to the PMT. A pulser inside the

detector tube can drive a light emitting diode (LED) imbedded in the scintillator. It is turned on by

the toggle switch at the tube top. The pulser produces pulse pairs at a fixed repetition rate of 100 Hz

while the time between the two pulses comprising a pair is adjusted by the knob labelled Time Adj.

The pulser output voltage is accessible at the connector labelled Pulse Output.

Figure 5: Top view of the detector, showing the PMT output, HV adjustment

potentiometer and monitoring ports for the PMT and pulser output.

6. Software and User Interface

Software is used to both control the instrument and to record and process the raw data. There is also

software to simulate muon decay data. All software is contained on the CD that accompanies the

instrument. For convenience the relevant files have been saved to the desktop. Source code for the

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user interface and the data fitting software is written in the Tcl/TK scripting language and is

provided. The muon decay simulation software can be run without any of the detector electronics

present. Table 2 lists the folders and describes the software they contain.

Table 2. Important Folders

Folder Name Folder Description

Muon_data Main executable and data files

muon_simu Simulation program and simulated data

muon_util Utility programs

sample_data Sample data file(s)

Tcl User interface source code

USB Driver USB driver software

The program you will probably use the most is muon, located in the folder muon_data. This is the

principal data acquisition program used to collect real data. This program stores its data in one of

two files: the file named data, or a file named with the date in which the data were collected derived

from the year, month, day and time, in that order, when the date is written in numerical form. For

example, the file 03-07-15:39 corresponds to a data file written 15 July 2003 at 3:39pm. Minutes

and hours are written in 24-hour notation. Real data files are automatically written to the folder

muon_data. Files with the extension ―tcl‖ and ―dll‖ are support programs that you do not need to

modify for normal running. However, the Tcl files are useful to read if you want to see how the user

interface and the curve fitter are written in the Tcl/Tk scripting language.

The program muon is the main data acquisition program. After you launch it, you will first see the

user interface, which allows you to set port settings on your PC, to observe various data rates, to

control how data is displayed on the PC, and to fit the collected data.

There are five sections to the main display panel:

Control;

Muon Decay Time Histogram;

Monitor;

Rate Meter;

Muons through Detector.

6.1 Control

The Configure sub-menu allows you to specify which communications port you will connect to the

electronics box. Please note that Com3 should be selected for data acquisition. If you select the

wrong port, an error message will eventually appear after you try to start the data acquisition (see

below), telling you that the port selected cannot be opened.

The maximum x-axis value for the histogram of the muon decay times and the number of data bins

is also set here. There are also controls for reading back data that have already been collected. The

blue coloured Save/Exit switch is used to finalise all your communication and histogram selections.

The Start button in the user interface initiates a measurement using the settings selected from the

configure menu. After selecting it, you will see the ―Rate Meter‖ and the ―Muons through detector‖

graphs, which show the data as it is being collected.

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The Pause button temporarily suspends data acquisition so that the three graphs stop being updated.

Upon selection, the button changes its name to Resume. Data taking resumes when the button is

selected a second time.

The Fit button when selected will prompt the user for a password. Please note that the password is

currently set as: “muon”. Once the password is entered, the data displayed in the decay time

histogram is fitted and the results are displayed in the upper right hand corner of the graph. Data

continues to be collected and displayed. The fitted curve drawn through the data points disappears

once a new data point is collected, but results of the fit remain.

The View Raw Data button opens a window that allows you to display the timing data for a user

selected number of events, with the most recent events read in first. Here an event is any signal

above the discriminator threshold, so it includes data from muons passing through the detector, as

well as signals from muons that stop and decay inside the detector. Each raw data record contains

two fields of information. The first is a time, indicating the year, month, day, hour, minute and

second (reading left to right) in which the data were recorded. The second field is an integer that

encodes two kinds of information. If the integer is less than 40000, it is the time between two

successive flashes, in units of nanoseconds. If the integer is greater than or equal to 40000, then the

position of the units indicates the number of ―time outs‖ (i.e., instances where a second scintillator

flash did not occur within the preset timing window opened by the first flash). See the data file

format below for more information. Typically, viewing raw data is a diagnostic operation and is not

needed for normal data taking.

The Quit button stops the measurement and asks you whether you want to save the data. Answering

No writes the data to a file that is named after the date and time the measurement was originally

started, i.e., 03-07-13-17-26.data. Answering Yes appends the data to the file muon.data. The file

muon.data is intended as the main data file.

6.2 Data File Format

Timing information about each signal above threshold is written to disk and is contained either in

the file muon.data, or a file named with the date of the measurement session. Which file depends on

how the data is saved at the end of a measurement session. The first field is an encoded positive

integer, which is either the number of nanoseconds between successive signals that triggered the

readout electronics, or the number of ―timeouts‖ in the one-second interval identified by the

corresponding data in the second column. An integer less than 40000 is the time, measured in

nanoseconds, between successive signals and, background aside, identifies a muon decay. Only data

of this type are entered automatically into the decay time histogram.

An integer greater than or equal to 40000 corresponds to the situation where the time between

successive signals exceeded the timing circuit’s maximum number of 40,000 clock cycles. A non-

zero number in the units place indicates the number of times this ―timeout‖ situation occurred in the

particular second identified by the data in the first field. For example, the integer 40005 in the first

field indicates that the readout circuit was triggered 5 times in a particular second, but in each case

the timing circuit reached its maximum number of clock cycles before the next signal arrived.

The second field is the number of seconds, as measured by the PC, from the beginning of 1 January

1970 (i.e., 00:00:00 1970-01-01 UTC), a date conventional in computer programming.

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6.3 Monitor

This panel shows rate-related information for the current measurement. The elapsed time of the

current measurement is shown, along with the accumulated number of times from the start of the

measurement at which the readout electronics was triggered (i.e., the number of muons).The Muon

Rate is the number of times the readout electronics was triggered in the previous second. The

number of pairs of successive signals, where the time interval between successive signals is less

than the maximum number of clock cycles of the timing circuit, is labelled Muon Decays, even

though some of these events may be background events and not real muon decays. Finally, the

number of muon decays per minute is displayed as Decay Rate.

6.4 Rate Meter

This continuously plots the number of signals above discriminator threshold versus time. It is useful

for monitoring the overall trigger rate.

6.5 Muons through Detector

This graph shows the time history of the number of signals above threshold. Its time scale is

automatically adjusted and is intended to show time scales much longer than the rate meter. This

graph is useful for long term monitoring of the trigger rate. Strictly speaking, it includes signals

from not only muons passing through the detector, but also from any source that might produce a

trigger. The horizontal axis is time, indicated down to the second. The scale is sliding so that the far

left-hand side always corresponds to the start of the measurement session. The bin width is

indicated in the upper left-hand portion of the plot.

6.6 Muon Decay Time Histogram

This plot is a histogram of the time difference between successive triggers and is the plot used to

measure the muon lifetime. The horizontal scale is the time difference between successive triggers

in units of microseconds. Its maximum displayed value is set by the Configure menu. All time

differences less than 20 μs are entered into the histogram, but may not actually be displayed due to

the choice of menu. You can also set the number of horizontal bins using the same menu. The

vertical scale is the number of times this time difference occurred and is adjusted automatically as

data are accumulated. You can change the vertical scale from linear-linear to log-linear using

Change y scale Linear/Log. The horizontal error bars for the data points span the width of each

timing bin and the vertical error bars are the square root of the number of entries for each bin.

The upper right hand portion of the plot shows the number of data points in the histogram. Again,

due to menu selections not all points may be displayed. If you have selected the Fit button then

information about the fit to the data is displayed. The muon lifetime is returned, assuming muon

decay times are exponentially distributed, along with the χ2

per degree of freedom, a standard

measure of the quality of the fit (see e.g., Ref. [5]).

To do a screen capture of the display hold the ―CTRL‖ and press ―Print Screen‖ keys on the

keyboard. This will copy the screen into memory. Open the ―Paint‖ utility (in Windows) and

execute the ―Paste‖ command under the ―Edit‖ pull-down menu to insert the screenshot into a blank

image file, which you can then save.

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6.7 Manual Data Analysis

Experience has shown that the fit to the data generated by the muon control panel software can be

erroneous. It appears that sometimes spurious ―muon‖ events are generated, presumably from

electrical equipment, and these need to be rejected to get an accurate lifetime measurement.

Additionally the software does not allow previously saved data sets to be loaded for fitting.

You are advised to determine the lifetime using manual data processing techniques using the

software ORIGIN. Data processing consists of the following steps:

1. Importing that data into a workbook in ORIGIN

2. Determine which events are attributable to muon decays. These are events with a timecode

less than 40000.

3. Check to see if spurious events are present.

4. Create a histogram of decay times.

5. Fit the histogram to Eqn 7, corrected for background counts. The fitting equation is

)exp( bxbcay , (14)

Where a is the background count, b is the reciprocal of the decay time i.e., b/1 , and c is a

normalisation constant, y is the number of counts at time x.

Importing the Data

The data file for you experiment is located in the c:\Software Version 2.0\Windows\muon_data\

directory and will be named something like ―09-07-10-14-54.data‖ The file will have a creation

date that is the time the experiment was started.

Open ORIGIN. You will see the work environment and a small window labelled ―Book1‖. This is

where the imported data will be placed. To import your data file do ―File – Import - Single ASCII‖

and click on your data file. The workbook will now contain your data and will show a small plot of

the data at the top of each column of data.

Find the Muon Decays

To find the events attributable to muon decays you will need to search the first column of data,

which is the time between PMT pulses for those events that occurred within 40000 counts of the

FPGA timer (20 s).

Click on the top of column A to select the event times and do ―Worksheet – Extract

Worksheet Data…‖

Click on ―A‖ in the ―extract‖ columns and click on the => to enter A in the ―Select Column

Variable for IF Test‖ window. Ensure ―A‖ and ―B‖ are ticked in the ―extract‖ column.

In the ―Condition‖ window type ―A<40000‖. Click ―Test —select if true‖. ORIGIN will

now search the workbook and find the required events. This may take some time.

Click on ―OK‖, which will generate a new workbook with the found muon data.

Determine if Spurious Events Are Present

It appears that events not attributable to muon decays occasionally appear in the data record of some

experimental runs. It is not known what causes these events, but it is probably from electrical

equipment in the lab.

V3.1 16/09/10 13

The presence of these events can be found by examining the timestamps associated with each muon

event. Real muon events are detected roughly once every 60 seconds, however it is found that the

spurious events are clustered together in time.

To find out whether spurious events are present, produce a histogram of the timestamps of your

―muon‖ events data. Bins that have values much larger than the average will indicate the presence

of the spurious event

To create the histogram of the timestamps to check for spurious events:

Go to the menu item ―Plot – Statistics – Histogram‖. Tick the box next to the ―B‖ in the

―column‖ cell of the window. Click OK. This will select the column of data that corresponds

to the timestamps of your data. The histogram of the timestamp data will now appear in a

new window.

We need to change the number of histogram bins to increase the time resolution. The

software is limited a maximum of 1000 bins. To set the binning properties, double click on

the histogram and go to the ―data‖ tab. Untick ―Automatic Binning‖ and set ―Bin Size‖ to a

number at least 10 times smaller than the current one. Hit the ―Apply‖ button and the

software will calculate the number of bins. You will need to change the bin size value

iteratively until you approach the maximum of 1000 bins. Unfortunately it is not possible to

set the number of bins by direct entry. If you request more than 1000 bins, the bin size will

be reset. Click ―OK‖ when you are finished and the resampled histogram will appear.

To reset the vertical scale, you will need to double-click on the vertical axis numbers which

will bring up control window. Click on the ―Scale‖ tab, and change to the ―To‖ value to an

appropriate value. Also change the ―Increment‖ parameter to around 10 times less than the

―To‖ value to give you a reasonable number of

If you have noticeable peaks, such as in Fig 5 then you will need to filter out the spurious

events from your data before you do any lifetime fitting. Otherwise proceed to the section

below ―Creation of Event Histogram‖

To filter out the spurious events do the following:

Observe the values of the histogram bins by right-clicking on the histogram and selecting

―Go to Bin Worksheet‖.

Scroll through the rows and look for values that are significantly above the average bin

values, which you observed on the timestamp histogram.

Figure 5: Histogram of

timestamps for muon events.

When it is found that a large

number of events occur within a

small time window these events

should be discarded.

V3.1 16/09/10 14

Note down the values of the timestamps that are at the end of each spurious region. Clicking

on the value of the timestamp (column A) will bring up the non-truncated value.

Go back to the original data sheet (“Book 1….”)

Click on ―B‖ in the ―extract‖ columns and click on the => to enter B in the ―Select Column

Variable for IF Test‖ window. Ensure ―A‖ and ―B‖ are ticked in the ―extract‖ column.

In the ―Condition‖ window type a logical condition to exclude the spurious data, based on

your measurements of their timestamps. Eg, to exclude the spurious points in Fig 5. which

has two spurious regions the command would be:

(B<1280301600 OR B>1280303200) AND (B<1.28046e9 OR B>1.28048e9)

Click ―Test —select if true‖. ORIGIN will now search the workbook and find the required

events. This may take some time.

Click on ―OK‖, which will generate a new workbook with the filtered muon data.

Creation of Event Histogram

Select the window with your data, either the original data import window, or the filtered

data window if you have had to filter out events.

To create the histogram of the muon events do ―Plot – Statistics – Histogram‖. Tick the box

next to the ―A‖ in the ―column‖ cell of the window. Click OK. The histogram of the data

will now appear in a new window.

To set the binning properties, double-click on the histogram and go to the ―data‖ tab. Untick

―Automatic Binning‖ and set ―Bin Size‖ to ―400‖, ―Begin‖ to ―0‖ and ―End‖ to ―20000‖.

This will create 50 bins equally spaced over the 20 microsecond collection window and is

recommended for most data sets. Use of too many bins will mean that insufficient counts are

obtained to estimate the background count level, which is observable in bins that are at long

event times (>5 decay lifetimes). Click ―OK‖ and the resampled histogram will appear.

To see the histogram data, right click on the histogram and do ―Go to Bin Worksheet‖.

Select the data in the first two columns, which are the position of the bins and the number of

counts, and select the all the values starting at the second row, and copy them. The first

row of data should be discarded. You will be asked later why you think this is.

Open a new workbook ―File – New – Workbook‖ and paste the data in.

Plot the data by doing ―Plot – Symbol - Scatter‖. Tick the box to make ―A‖ the ―X‖ column

and the box to make ―B‖ the ―Y‖ column. Click ―OK‖ and a graph of the data will appear.

Calculating the fit to the histogram

The decay time is calculated by fitting the histogram data to Eqn 14.

First ensure that the graph of your histogram data is selected. Then to access the fitting

routines, do ―Analysis – Fitting – Nonlinear Curve Fit – Open Dialog‖ and select in the

―Category‖ window ―Muon_decay‖ located below the ―User Defined‖ category.

Unfortunately the fitting is highly non-linear and some initial starting values need to be

provided to allow the fit to converge. Click on the ―Parameters‖ tab and enter in the

―values‖ column the values ―1‖ for ―a‖ (background), ―5x10-4

‖ for ―b‖ (1/[decay time in

ns]). The initial value for ―c‖ should be approximately the value of the counts in the first bin

(around 800) divided by the value for b (5x10-4

).

Click on the ―Fit‖ button. You should now have a line which is a good fit to the points and a

box on the graph with the fitted values. If the data fit is poor, try changing the initial value

of ―b‖ by an order of magnitude and refitting.

V3.1 16/09/10 15

You can adjust the labels and other features on your graph by double-clicking to edit them. Make

sure your graphs are appropriately labelled!

6.8 Muon Decay Simulation

Simulated muon decay data can be generated using the program muonsimu found in the muon_simu

folder. Its interface and general functionality are very similar to the program muon in the

muon_data folder. The simulation program muonsimu lets you select the decay time of the muon

and the number of decays to simulate. Simulated data are stored in exactly the same format as real

data.

6.9 Utility Software

The folder muon_util contains several useful programs that ease the analysis of decay data. The

executable file sift sifts through a raw decay data file and writes to a file of your choosing only

those records that describe possible muon decays. It ignores records that describe timing data

inconsistent with actual muon decay.

The executable file merge merges two data files of your choosing into a single file of your

choosing. The data records are time ordered according to the date of original recording so that the

older the record the earlier it occurs in the merged file. The executable file ratecalc calculates the

average trigger rate (per second) and the muon decay rate (per minute) from a data file of your

choosing. The errors returned by this program are statistical.

The executable program freewrap is the compiler for any Tcl/Tk code that your write or modify. If

you modify a Tcl/Tk script, you need to compile it before running it. On a Windows machine you

do this by opening a DOS window, and going to the muon_util directory. You then execute the

command: freewrap your_script.tcl, where your_script.tcl is the name of your Tcl/Tk script. Do not

forget the tcl extension.

7. Experimental Procedure

Figure 6 shows a photograph of the experimental set-up. The detector is the black aluminium

cylinder, which has been placed in the wooden pedestal for convenience.

Figure 6: Photograph of the MPA-1 Muon Physics instrument, showing the experimental set-up.

V3.1 16/09/10 16

(i) Connect the power cable and signal cable between the electronics box and the detector (see

figures 4 and 5). Connect the communication cable between the back of the electronics box

and the USB connection on the PC.

(ii) Turn on the power to the electronics box - the switch is at rear of the box. The red LED

power light should now be shining steadily. The green LED may or may not be flashing.

(iii) Set the high voltage (HV) between -1100 volt and -1200 volt using the dial at the top of the

detector tube (see Figure 5). The exact setting is not critical; however, the voltage should be

monitored using a digital multimeter. The probe connectors are at the top of the detector

tube.

(iv) Examine the output of the photomultiplier tube (PMT) using the PMT Output on the detector

tube (see figure 5). A two channel digital oscilloscope is provided for this measurement. Be

certain to terminate the input to the oscilloscope with the BNC 50Ω terminator, or your

signal will be distorted. Record and label the oscilloscope trace.

(v) Connect the BNC cable between PMT Output on the detector and PMT Input on the

electronics box (see figure 4). Using a digital multimeter, adjust the discriminator setting on

the electronics box so that it is in the range 180 mV - 220 mV. The green LED on the box

front panel should now be flashing.

(vi) Examine the amplifier output using a digital oscilloscope - the Amplifier Output is located

on the front panel of the electronics box (see Figure 4). Be certain to terminate the input to

the oscilloscope with the BNC 50Ω terminator. Record and label the oscilloscope trace.

(vii) Similarly, examine the output from the discriminator using a digital oscilloscope; the

Discriminator Output connector is located on the front panel of the electronics box. Again, a

BNC 50Ω terminator must be used. It is useful to simultaneously display the amplifier and

discriminator outputs on the one screen using a two channel digital oscilloscope.

7.1 Instrumentation Exercises

(i) Measure the gain of the 2-stage amplifier (using a sinusoidal input), by applying a 100 kHz

100mV peak-to-peak sine wave to the PMT input of the electronics box (see Figure 4).

Measure the amplifier output and take the ratio Vout /Vin. Due to attenuation resistors inside

the electronics box, inserted between the amplifier output and the front panel connector, you

will need to multiply this ratio by the factor 1050/50 = 21 to determine the real amplifier

gain.

❐ Adjust the frequency and plot the transfer function, i.e., Vout /Vin versus frequency. How

good is the frequency response of the amplifier?

❐ Estimate the maximum decay rate you could observe with the instrument.

(ii) Measure the saturation output voltage of the amplifier, by increasing the magnitude of the

input sine wave and monitoring the amplifier output. Does a saturated amplifier output

V3.1 16/09/10 17

change the timing of the FPGA? What are the implications for the ―size‖ of the light signals

from the scintillator?

(iii) Examine the behaviour of the discriminator by applying a sine wave to the PMT input and

adjusting the discriminator threshold. Monitor the discriminator output and describe its

shape.

(iv) Measure the timing properties of the FPGA by using the pulser on the detector (see Figure

5).

❐ Measure the time between successive rising edges on a digital oscilloscope. You can

compare this number with the number shown on the software display (see Sec. 7.2).

❐ Measure the linearity of the FPGA, by altering the time between rising edges, and

plotting your results against the FPGA results. Use time intervals between 1s and 20

s in steps of 2s.

❐ Determine the timeout interval of the FPGA by gradually increasing the time between

successive rising edges of a double-pulse and determine when the FPGA no longer

records results. What does this imply about the maximum time between signal pulses?

❐ Decrease the time interval between successive pulses and try to determine the FPGA

internal timing bin width. What does this imply about the binning of the data? What

does this imply about the minimum decay time you can observe?

(v) Adjust (or misadjust) the discriminator threshold, by increasing the discriminator output rate

as measured by the digital oscilloscope. Observe the raw muon count rate and the spectrum

of ―decay‖ times. N.B. This exercise needs some patience since the counting rate is slow.

(vi) Investigate the effect of the HV setting, by adjusting the HV and observing the amplifier

output. We know that ―good‖ signals need to be at about 200 mV before being input to the

discriminator, so set the discriminator before hand. With fixed threshold, alter the HV and

monitor the raw muon count rate and decay spectrum.

7.2 Measuring the Muon Lifetime

(i) Open the folder/directory muon_data and launch the program muon.exe. Configure the port

on your PC - make sure you select Com3 (see Sec. 6.1). Choose your histogram options and

click on the Save/Exit button. To start data acquisition click on Start. You should see the

rate meter at the lower left-hand side of your computer screen immediately start to display

the raw trigger rate for events that trigger the readout electronics. The mean rate should be

about 6 Hz.

(ii) Set up the instrument for a muon lifetime measurement. Before you start data acquisition it

is critical to ensure you have the correct voltage settings for the PMT (see Sec. 7 (iii) for the

HV setting), and the discriminator (see Sec. 7. (v) for adjusting the discriminator setting).

Start and observe the decay time spectrum.

Note that in order to acquire a statistically significant number of events (>1500) it will

be necessary to acquire data over a period of two or more days. Please discuss this with

your demonstrator.

V3.1 16/09/10 18

Do not connect/disconnect USB items or switch attached equipment (e.g. CRO) on and

off while you are collecting data as this may cause electrical noise which can give rise to

false muon events.

Question. The muons whose decays we observe originate outside the detector and therefore

spend some (unknown) portion of their lifetime outside the detector. So, we never measure

the actual lifetime of any muon. Yet, we claim we are measuring the lifetime of muons.

How can this be?

(iii) Use the lifetime fitter function in the Muon control panel to fit the decay time histogram. A

Screen capture button allows you to produce a plot of the display. Select the button and then

open the Paint utility (in Windows) and execute the Paste command under the Edit pull-

down menu.

(iv) You should also investigate your own fit to the data due to limitations of the Muon software.

Section 6.7 outlines the data fitting process in ORIGIN, which is located on the computer

controlling the muon experiment. It is also possible to analyse the data using other software,

such as Excel or Matlab, using a similar protocol to that outlined in Section 6.7.

❐ Compare your measured value of the muon lifetime with that reported in the literature,

and that reported by the muon lifetime fitter.

Question. Why in the manual data analysis process is it suggested to discard the first bin

in the histogram before fitting?

(v) From your measurement of the muon lifetime calculate the value of the Fermi coupling

constant GF. Compare your value with a literature value.

(vi) Using the approach outlined in Sec. 4.2, measure the charge ratio of positive to negative

muons at sea level. Compare to value in the literature.

Exercise:

Once the muon lifetime is determined, compare the theoretical binomial distribution with an

experimental distribution derived from the random lifetime data of individual muon decays. For

example, let p denote the (success) probability of decay within one lifetime, p = 0.63. The

probability of failure q = 1 - p. Take a data sample of 2000 ―good‖ decay events. For each group of

50 events, count how many have a decay time less than one lifetime. (On average this is 31.5.)

Histogram the number of ―successes.‖ This gives you 40 ―experiments‖. The plot of data from these

40 ―experiments‖ should have a mean of 50×0.63, with a variance 2 = Npq = 500.630.37 = 11.7.

Are the experimental results consistent with theory?

References

[1] T. Coan and J. Ye, Muon Physics MP!-A User’s Manual, TeachSpin Inc. The relevant web

site for this instrument is: http://www.teachspin.com/instruments/muon_physics/index.shtml

[2] A detailed list of the properties of the muon can be found at the Particle Data Group web

site: http://pdg.lbl.gov/2007/listings/s004.pdf

V3.1 16/09/10 19

[3] B. Rossi, High Energy Particles (Prentice Hall, New York, 1952). See also B. Rossi,

Reviews of Modern Physics, 20, 537 (1948). This article can be downloaded from:

http://prola.aps.org/abstract/RMP/v20/i3/p537_1

[4] R. A. Reiter et al. Phys. Rev. Lett. 5, 22 (1960). This article can be downloaded from:

http://prola.aps.org/pdf/PRL/v5/i1/p22_1

[5] P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical

Sciences, 2ed., (McGraw-Hill, New York, 1992).

[6] M. Bahmanabadi, F. Sheidaei, M. Khakian Ghomi, and J. Samimi, The charge ratio of the

atmospheric muons at low energy, Physical Review D 74, 082006 (2006)

[7] J. M. Baxendale, C. J. Hume and M. G. Thompson, Precise measurement of the sea level

muon charge ratio, J Phys G: Nucl Phys, Vol 1, No 7, 1975.

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