Harald Scharf, Tobias Voss

VP2: Muon Lifetime

VP2: Muon Lifetime

Contents

1  Introduction
2  The Theory of Cosmic Rays
    2.1  The Primary Component
    2.2  The Secondary Component
3  The Muon's Decay
    3.1  Free Decay
    3.2  Decay in Matter
4  Experimental Set-Up
    4.1  Detector Subsystem
    4.2  Logic Subsystem
    4.3  Data Read-Out
5  The Measurements

1  Introduction

In 1937, Anderson and Neddermeyer discovered a new elementary particle: the muon. It was found out to be of leptonic character, obeying the weak and electromagnetic force only. Its mass is about 207me, and it has the charge �qe. The muon is not so stable, decaying into an electron or positron and two neutrinos after approximately 2.19 ms.
To measure this muon lifetime with a tricky electronic set-up is the goal of the experiment.

2  The Theory of Cosmic Rays

Most particles hitting the earth are scattered by nitrogen and oxygen atoms in the atmosphere, 20km above sea-level. This generates secondary particles and radiation underlying an angular distribution because of their diffent pathlengths in the atmosphere.

2.1  The Primary Component

Created in solar flares, supernovae, the galactic core, and other sources, the highly relativistic, charged particles called �the primary component of cosmic radiation move on spiral trajectories along the magnetic field lines through space. Because of the statistical distribution of the magnetic fields in our galaxy, the cosmic radiation is (almost) isotropic. Before reaching the atmosphere, its components are about:

85% protons
14% a-particles
1% electrons
< 1% heavy nucleons

Their mean energy is 2 GeV, distributed like

N(E) = const ·E-g
(1)
with g � 1 for energies up to 1011 GeV and g > 1.8 for higher energies.

2.2  The Secondary Component

Created by primary component particles (mostly protons) scattering in the earth's atmosphere, the secondary component particles mainly consist of knocked-out nucleons for lower proton energies and neutral and charged p-mesons (pions) for higher proton energies (bremsstrahlung). The neutral pions decay with t � 6·10-15 s as follows:
p0
�
g+ g
(2)
The charged pions decay with t � 2.5·10-8 s as follows:
p+
�
m++nm
(3)
p-
�
m-+
n
 
m 
(4)

This is the main source of cosmic muons. Many of them reach the planetary surface with an angular distribution due to the thickness of the atmosphere, many others decay before that.

3  The Muon's Decay

Muons are instable particles that decay after about 2.19 ms. One can easily calculate that this time is by far not sufficient for a muon generated in 20 km height to reach the earth, even if it were travelling at the speed of light. The explanation for this phenomenon is called relativistic dilatation; travelling at a speed of about 0.999·c, the muons' life-time in the lab system is about [(2.19ms)/( �{1-([0.999·c/ c])2})] � 50 ms, and many of them reach the earth or pass through it.
The decay follows a natural exponential decay function:
dN
dt
 = - lN(t)
(5)
Which transforms to show the time distribution:
N(t) = N(0)·e-lt
(6)
l is the decay probability and t = 1 / l is the mean life time, which this experiment is all about.

3.1  Free Decay

When no other particles are involved, the muon decays "freely" as follows:
m-
�
e-+
n
 
e 
 +nm
(7)
m+
�
e++ne+
n
 
m 
(8)

3.2  Decay in Matter

When the m+ passes through matter, it just gets repelled by the postitively charged nuclei and its mean life time t+ is given only by the free decay. For the m-, however, there is a small chance of being captured by an atom, knocking an electron out off its orbit. Then it either decays in the shell or (if it lives long enough) is attracted and absorbed by the nucleus as follows:
m-+p
�
n+nm
(9)
To get its mean lifetime t- the decay probabilities have to be added:
1
t-
 = 1
t+
 + 1
tabsorbed
(10)
For a nucleus charge Z > Z0 � 10.2 (below which absorption becomes irrelevant),
labsorbed is about proportional to Z4:
labsorbed = l+  �
�
� 		Zeff
Z0
 �
�
� 		4

 
(11)
with the empirical formula for Zeff:
Zeff = Z   �
�
� 		1+ �
�
� 		Z
37.2
 �
�
� 		1.54

 
 �
�
� 		-[1/ 1.54]

 
(12)
The real formula would involve the sum of the wave-functions of all nucleons and is therefore rather complicated.
For our aluminum absorber (with Z = 13), we calculate Zeff = 11.56. With all these numeric values substituted, we get:
1
t-
 = 1
t+
   �
� 		1+ �
� 		1.1 _
3
 
 �
� 		4
 
 �
�
(13)
which results in
t- = 0.83ms
(14)

4  Experimental Set-Up

The experimental set-up consists of

4.1  Detector Subsystem

A lead shielding between the top and second scintillator served to block unwanted particles. Muons cannot interact strongly with matter and the energy loss by electro-magnetic interactions is too low to stop them. If we get a pulse from the first two PMs (at approximately the same time) but none from the third, we can conclude that a muon has been caught in the aluminum. This generates a start signal. The stop signal is generated if we get a pulse from number 2 or 3 within 12.5 ms from the start signal.

4.2  Logic Subsystem

It serves to distinguish between "bad" events and "good" events (and measure the lifetime of the good ones). The bad events are those that don't have a stop signal in reasonable time (12.5 ms) after their start signal. The good events' lifetime is measured by counting the oscillations of a fed-back AND-gate oscillating with a well-defined frequency.
It was very tricky to make all NIM-signals process correctly by choosing the right cable lengths and signal widths. Due to the rare nature of good events, we also spent much time adjusting the correct discriminator levels and high voltages for the PMs.

Figure
Figure 1: our logic set-up

4.3  Data Read-Out

Whenever a valid stop signal occurred, we read out the life-time from the scaler into the VALET-PLUS (our PILS-program did that, of course!) where the corresponding bin was increased. During that transaction, the generation of a new start signal was inhibited. This happened about once per minute. About once per week, we (personnally) stopped the entire process to download the array from the VALET-PLUS to the Macintosh, where it was stored in a text file. At the same occasion, we measured, how many life-time clicks a timeout (12.5 ms) would have. This varied between 582 and 600 and was used to assign time values to the bins when we merged the individual data-sets.
Here is the source code of our program:

10 CONST branch = 1
20 CONST crate = 1
30 INT32 wert1, wert2, zwert1, zwert2, q
40 INT32 a,i, t(1 TO 600)
50 CHAR filename
60 INT32 scaler1, scaler2, outreg, testreg
70 cdreg(scaler1, branch, crate, 12, 1)
80 cdreg(scaler2, branch, crate, 12, 2)
85 cdreg(outreg,branch,crate,3,0)
90 cccz(scaler1)
100 cccz(scaler2)
105 cccz(outreg)
110 zwert1=0
120 zwert2=0
123 cfsa(16, outreg, 1, q)																			! reset
128 cfsa(16, outreg, 0, q)
129 loop
130   cfsa(0, scaler1, wert1, q)
140   cfsa(0, scaler2, wert2, q)
149  ! if (zwert1=wert1) and (wert1<>0) then
150   if zwert1=wert1 then
155     if wert1<>0 then
160        if wert2=1 then
170           t(wert1)=t(wert1)+1
180        endif
190        zwert1=0
200        cccc(scaler1)
201     endif
204     a = a or 1
205     cfsa(16, outreg, a, q)																! reset
207     a = a and 254
208     cfsa(16, outreg, a, q)
210   else
220     zwert1=wert1
230   endif
240 endloop
500 end

5  The Measurements

We merged the measured data into 125 bins, graphed and fitted the decay function for two decays with the "Root" Software:

Figure
Figure 2: Measured events per time-bin, fitted with Root

The mean life-times, according to our formula, are:

t+ = (1.93 �0.16)ms (theoretical value: 2.19 ms)
t- = (0.77 �0.15)ms (theoretical value: 0.83 ms)

The error was calculated using:

Dt = Dl· �
�
� 		�t
�l
 �
�
� 		= Dl· 1
l2
(15)

References

[1]
Particle Physics Data Booklet
[2]
Morita, Beta Decay and Muon Capture, HQ 115
[3]
Janossy, Cosmic Rays, Ko 28
[4]
Hughes and Wu, Muon Physics I-III, Ke 197
[5]
Griffiths, Einführung in die Elementarteilchenphysik

File translated from TEX by TTH, version 2.25.
On 1 Jul 1999, 18:42.