Compton Scattering of γ-rays
G.S. Sagoo
With Laboratory Partner: F.Wahhab
Department of Physics and Astronomy
University College London
14th November 2010
Abstract: By scattering γ-rays from Caesium-137 source, the rest mass of electron was
found to be (8.8  0.2)  10 31 kg using the Compton Kinematic Scattering Equation, fitted
by χ2, with angular range varied from -120° to +120° and a Duralumin scattering rod.
The accepted value for the rest mass of an electron, 9.1  10 31 kg [1], is outside
experimental uncertainty suggesting that the methodology was not accurate. A number of
possibilities could have led to the loss in accuracy, some being The secondary objective
was to confirm the angular dependence of the Klein-Nishina formula; however this was
unsuccessful.
Intrroduction
n
The primary obbjective for tthis experim
ment was to verify the reest mass of an electron using
the C
Compton K
Kinematic Sccattering Eqquation by sccattering γ-rrays from Caesium-137
C
7
(Cs--137) sourcee through a Duralumin rod. The seecondary obbjective wass to verify thhe rest
mass of an electtron by anguular variatioon of the diffferential crooss section.
Com
mpton scatteering occurs when high energy phootons are inccident upon free electroons. In
this experimentt the high ennergy photonns emitted bby the Cs-1337 are inciddent upon thee free
electrons of the Duraluminn, resulting in a change of momentuum; the scatttered photoon.
Figuure 1 shows this principple.
By eequating thee energy andd momenta oof the scatteered photon before and after collidding
withh the free eleectron it cann be shown that:
1 1
1  cos  

  
E '  E m0 c2  m0 c 2
Wheere E’ = Scaattered photton energy
E = Maxximum photton energy
 = Scatttering angle
(1)
m0 = Resst mass of ellectron
c = Speeed of light
Figure 1: Diagram
D
showin
ng the principlle of Compton
n scattering [2]
The Klein-Nishhina formulaa relates the differentiall cross sectioon with scatttering angle and
is giiven [3] by:
2
2
2
2
 

1

1  coss   
d 1  e 2  
2

 
 1  cos   
 
(2)
d 2  m0 c 2   1   1  cos    
1   1  coos   
E
and e = ccharge of electron.
Wheere  
m0 c 2
Method
Figuure 2 shows a schematicc diagram of
o the apparaatus used in the setup. S
Since the
scinntillator outpputs digital ssignal assocciated to eacch of its channnels, beforre any readinngs
coulld take placee the apparaatus needed to be calibrrated so thatt channel nuumber from the
multti-channel-aanalyser (MCA) to enerrgy [2].
Figu
ure 2: Schemattic diagram of apparatus setu
up [2]
t CASSYL
Lab softwaree package, w
with the pow
wer supply sset to
The MCA was connected to
V and 1024 channels acctivated. Foor 15 minutees the backgground counnt was takenn and a
610V
histoogram of nuumber of couunts againstt channel nuumber was rrecorded. A calibration
sourrce consistinng of strontiium-90 (Sr-90) and Cs--137 was theen placed innto the sourcce
holdder, set to 0°° scattering angle, and tthe number of counts reecorded per channel forr 15
minuutes. The baackground counts
c
were then removved from thee calibrationn count by
subttraction thuss presentingg a two photto peak γ-rayy spectrum. The lower energy photto
peakk representeed the 59.5kkeV energy oof Sr-90 whhile the highher peak reprresented thee
662kkeV energy from CS-1337. Using C
CASSYLab’s inbuilt Gaaussian fit fuunction, thee
corrresponding cchannel num
mber to photto peak wass determinedd thus calibrrating the
appaaratus.
Caesium-137 w
was placed innto the holdder and a durralumin scattering rod placed
p
into
posiition. Varyinng the anglee from -120°° to +120°, the number of counts oover time weere
recoorded with aand without the scatterinng rod. The reason for tthis is at larrge angles thhe
backkground raddiation domiinates the tootal counts. A
At the end oof each sesssion the
equiipment was recalibratedd to check fo
for channel ddrift. This occurs randoomly due to how
the pphotomultipplier works [2] and leadds to a shift in channel aat corresponnding energiies.
By ttaking the difference off counts withh and withoout the scatteering rod, thhe true scatttering
γ-rayy spectrum is given. In addition foor larger anggles the counnt rate decreeases as a
funcction of  , hhence longerr time was nneeded for laarger angless. Upon takiing the diffeerence
a Gaaussian fit oof the photo peak was taaken to deduuce scatterinng energy.
Ressults and A
Analysis
Wheen fitting thee Gaussian using CASS
SYLab on thhe angle datta, the equattion of the fit
f was
2
provvided. Usingg this equatiion, for eachh angle the χ value for the Gaussiaan fit on thee raw
dataa around the photo peakk was reduceed while chaanging the m
mean energyy using a
Microsoft Exceel add on callled “SOLV
VER”. Doingg this meantt that a betteer fit was
obtaained and ultimately a m
more accuraate E’.
1
against coss  one cann determine the mass off electron inn two differeent
E'
wayys. Initially ffor positive and negativve angles, thheir associatted lines of bbest fit begaan to
diveerge. It was suspected thhat 0° was nnot truly 0. T
Thus all anggular data neeeded to be offset
by -1°, in effectt rotating thee system.
By pplotting
mparing figuure 3 with eqquation 3 it can be seenn equation 3 is in the foorm:
Com
y  mx  c
(3)
1
1
1
1
With y  ; m 
; x  coss  ; and c  
2
E'
E m0 c 2
m0 c
Therrefore by taaking the graadient and inntercept resppectively m0 can be dettermined. Fiigure
1
4 shhows how
changes with
w cos  . Since scatteering energiies were recorded for booth
E'
posiitive and neggative anglees, taking thhe gradient and
a interceppt yielded 4 separate vallues
for tthe mass of an electron.
By ttaking the w
weighted meean of the 4 values, the electron maass was founnd to be:
m0  8.8  0.2   10 31 kg
Fiigure 3: Graph
h Showing How
w 1/E of Scatterred Gamma Raays Changes w
with Scattering Angle Offset b
by -1°
In adddition to thhe kinematicc scattering equation ann attempt to find the eleectron mass using
the K
Klein-Nishiina formula was made, however thiis attempt was
w unsucceessful. Know
wing
the ttotal counts, the numbeer of photonns N’ scattereed through aangle ϕ is:
N '
C E 
 E  E P E 
Wheere C(E) is tthe rate of ccounts (s-1) aand  E ,  E , PE  arre correctionn factors [3]].
In adddition to eqquation 4:
d
N '
d
By eequating (4)), (5) and (6):
C E 
1  e2
 k 
 E  E PE 
2  m0 c 2



2


1


 1   1  cos  
2
(4)
(5)

 2 1  cos 2 
1  coos 2   

  (6)
1   1  cos 

Wheere k = consstant of propportionality..
with m0 inittially set as
Havving measureed the count rate, an arbbitrary k waas assigned w
d
8.8xx10-31 kg. Haaving plottedd N’ vs ϕ toogether withh
vs ϕ, χ2 was calcuulated. An
d
attem
mpt was theen made to uuse “SOLVE
ER”, minim
mising χ2 whhile changinng the mass of the
electron used too calculate α
α. This was uunsuccessfuul as “SOLV
VER” repeattedly set m0 to
0kg. Due to tim
me constraintts further prrogression w
was halted annd a graph of
o N’ vs coss(ϕ)
d
togeether with
vs cos(ϕ
ϕ) and was pplotted for both
b
positivee and negatiive angles ass
d
show
wn in figuree 4.
Figure 4:: Graph Showiing How Differrential Cross Seection Varies With
W Cos(φ) Ussing the Klein--Nishina Formu
ula and The
Number of Photons Scattered
S
N’ was veryy close to thhe
Looking at figuure 4 it is cleear to see that for negattive angles, N
maintaining the same treend and lyinng on the booundary of tthe
diffeerential crosss section, m
error bars. However for positive angles, the respective differential cross section values
are far from it. At 41° there was a sharp dip in N’ suggesting anomalous result. This could
be due to a neighbouring experiment in which they too were carrying out the same
procedure. However the dip suggests a loss in counts which could be due to scintillator
drift [2].
Conclusion
In conclusion the mass of an electron was found to be 8.8  0.2   10 31 kg using the
Compton kinematic scattering equation. The accepted value of 9.1  10 31 kg [1] lies
outside the experimental uncertainty suggesting that the methodology of the experiment
was correct and carried out precisely but not accurately. There are many possible sources
of error that could have lead to this inaccuracy, one being that a similar experiment was
being carried out nearby. The neighbouring experiment could have introduced noise to
our data as they recorded scattering angles facing our apparatus. Although 2 inch thick
lead blocks were used in shielding each experiment, some radiation could have leaked
through to our photo multiplier. In future the experiment should be carried out in an
isolated, well shielded room.
In addition problems with software lead to time wasting when χ2 the Klein-Nishina
formula with N’. In future m0 should be deduced by iterating over set values. The
programme would iterate over set values, with the values becoming more precise as χ2
decreases, much like the Bisection method when root finding.
Implementing the aforementioned would yield better results and make the experiment
more accurate and precise.
References
1. Kaye and Laby Online
http://www.kayelaby.npl.co.uk/units_and_fundamental_constants/1_2/1_2_3.html
14th November 2010
2. Experiment R3 Lab Script, Dept Physics and Astronomy, UCL, course 2B40,
(2007)
3. Verification of Compton collision and Klein Nishina formulas – an
undergraduate laboratory experiment, R.P Singhal and A.J. Burns, Am J. Phys.
46(6), June 1978.
Append
dix A
Example histogram from CAS
SSYLab showing background count, Srr-90 amd Cs-137peaks once background
d has been removed from calibration so
ource
counts