Study of α-Particle Scattering By Means of
The Rutherford Experiment
G.S. Sagoo
With Laboratory Partner: Nilesh Agnihotri
University College London
15th December 2009
Abstract: This experiment is a modernised version of Rutherford’s classical scattering
experiment which investigates the angular dependence on α-scattering through a gold
foil. The purpose of this investigation was to investigate the angular dependence of αparticles scattering from a thin gold foil verifying the Rutherford Scattering Law where
n=4 [1]. This value was outside our experimental uncertainty of n = 4.58 ± 0.03
suggesting that the methodology used was precise but not accurate and that there were
systematic errors not taken into consideration.
1
Introduction
The primary objective for this experiment is to investigate the angular dependence of αparticles scattered from a thin gold foil in order to confirm the Rutherford Scattering Law
[1](1).
The scattering of α-particles is due to a Coulomb force experienced between the nucleus
of gold atoms and the incident α-particles. Since they are both positively charged, upon
approaching each other a repulsion force is exerted and the α-particle is scattered in a
dI
per atom of an incident beam of intensity,
random direction. In this case the fraction,
I
I, scattered at an incidence angle, φ, into the detector normal to area, dS, at distance, R,
from the scattering source is given by:
d2
dI
⎛ ϕ ⎞ dS
= N t t 0 cos ec 4 ⎜ ⎟ 2
(1)
I
16
⎝2⎠R
Where:
Nt = number of atoms per unit volume in the scattering foil (m-3)
t = thickness of foil (m)
do = collision diameter (m)
By keepind a constant incident energy and scattering geometry, the counting rate of flux
incident at angle φ, one would expect:
⎛ϕ ⎞
Rate ∝ cos ec n ⎜ ⎟
(2)
⎝2⎠
By equating equations (1) and (2) it is expected that n=4 for this experiment.
Method
Figure 1 shows a schematic diagram of the apparatus used in the setup. By using a
vacuum chamber and blocking out the light with a thick opaque lid, the α-particles could
propagate with no collisions with air molecules and the detector would not count photons.
The apparatus first had to be calibrated for the optimum discriminator bias [2]. Although
in this experiment the value on the dial was arbitrary, rotating the dial would change the
discriminator bias from 0.00 to 4.00V. The purpose of the discriminator bias was to
calibrate the Signal Processing Unit (SPU) so that any electrical noise is normalized.
Usually this occurs at a range of voltages and thus a discriminator bias within this range
was used.
2
Figure 1: Diagram Showing Schematic of Apparatus
Upon finding the optimum discriminator bias, the source was rotated +90°so that it was
screened from the detector. The lid was placed on the equipment and a measurement of
background counts was taken for 1000 seconds. A repeat of this reading was made at 90°. These readings were taken to correct for any background counts being detected.
Once the apparatus had been calibrated the source was returned back to 0°.
The detector was left at 0° and a time interval set so that the number of counts were larger
than 100. The reasoning behind the 100 counts is that assuming the decay process fits a
Poisson Distribution, the random error on the counts, C, is
ΔC =
1
C
(3)
To maintain an acceptable error of ~10%, while maintaining precision 100 counts was
decided. It was also decided to limit the maximum time interval per degree of rotation to
1000s due to time constraints during lab sessions. Upon recording the number of counts
for the set interval, the source was rotated and the process repeated. Data was collected
for incidence angles, φ, from -17° to +18° shown in Figure 3.
Results and Analysis
Figure 2 shows how count rate changes with discriminator bias. In the range of 0.13-0.3
of the discriminator bias, a well was formed. From theory this should not happen,
however since the discriminator value on the dial was arbitrary it could have been that the
dial had not been calibrated. From the region of 0.66-3 there was a plateau suggesting in
that range most electrical noise was eliminated. By taking the median value of the range,
a discriminator bias value of 2.03 was selected.
3
Over a period of 1000 seconds the background count was found to be 0.006Bq in the
positive angular direction and 0.01Bq in the negative angular direction. By taking the
average of the two values our overall background count contribution was found to be
0.008Bq.
Since the data collected for the radiation counts represented a Poisson distribution, this
meant that the error on the counts themselves had to be propagated for the error on the
count rate, which was calculated by:
2
2
⎛ ΔC ⎞ ⎛ Δt ⎞
ΔR = ⎜
⎟ +⎜ ⎟ ×R
⎝ C ⎠ ⎝ t ⎠
Where R is count rate, C is counts and t is time and Δx is the associated error with x.
(4)
However since Δt << t, it meant that the associated total error on time was effectively
zero. By substituting equation X, the error on the count rate was infact:
C
(5)
×R
C
Figure 3 shows a graph of how count rate per second depends on incidence angle.
Theoretically this graph should be symmetrical about the y-axis, meaning that the zero
error, φc, needed to be determined. The zero error represents the true axis position of the
α-particle beam. At small angles, due to the fact that at these small angles, the α-particles
could have simply passed through the gold foil rather than being scattered and due to this
it is expected that the power law will not follow the Rutherford Equation. By drawing a
line of best fit along the pseudo linear region on Figure 3 and extrapolating the data, the
zero error was found to be 0.75°±0.25°. Unlike small angles, at large angles the count
rate reduced dramatically and due to time constraints the extreme incidence angles were
limited to -17° to +18°.
ΔR =
Upon calculating the zero angle the collected data needed to be compared with the
Rutherford Scattering Law. To do this the collected incidence angles needed to be
corrected for the zero error. Figure 4 shows how the count rate depends on the corrected
angle. This gradient should represent the power in equation 4, however because there are
large systematic errors at smaller angles, the graph began to plateau at
⎛
⎛ ϕ − ϕc ⎞⎞
ln⎜⎜ cos ec⎜
⎟ ⎟⎟ > 3.5 due to the reasons explained above., data points were ignored
⎝ 2 ⎠⎠
⎝
in this plateau region and the line of best fit was taken of points before the plateau.
Since there were errors associated with both the incidence angle and the zero error, they
needed to be taken into consideration when taken the gradient of the line in Figure 4. By
propagating these errors the overall error was found to be:
⎛
⎛ ϕ − ϕc ⎞ ⎞ 1 ⎛ ϕ − ϕc ⎞ 2
2
Δ ln⎜⎜ cos ec⎜
⎟ ⎟⎟ = cot⎜
⎟ ϕ + ϕc
⎝ 2 ⎠⎠ 2 ⎝ 2 ⎠
⎝
(6)
4
Using the MATLAB function ‘llsfitcol’ which deduces the least square plot and the
associated error, the power in the Rutherford Scattering Law was found to be:
n = 4.58 ± 0.03
Conclusion
⎛ϕ ⎞
Equation (2) states that R ∝ cos ec n ⎜ ⎟ ; n was found to be 4.58 ± 0.03 . The accepted
⎝2⎠
value of 4 [1] is not even close to the extremes of our experimental uncertainty
suggesting that our experiment had been influenced by systematical errors and that the
methodology needs to be improved.
Since the collimating slit is of a finite size, large systematical are introduced as the beam
of α-particles could pass straight through the gold foil rather than being scattered. By
making reducing the size of the collimating slit so that only a single beam can pass
through it will significantly reduce this error as the majority of the beam will be scattered
by the gold foil.
Even though 10% accuracy was ensured of the random decay of the α-particles, by taking
more counts, the experiment will be more accurate. Due to time constraints this was not
done in the current methodology
Although the value of n = 4.58 ± 0.03 was precise it was not accurate and that some
errors were not taken into consideration. By making the suggested improvement the
current value of n = 4.58 ± 0.03 will tend towards the accepted value of n=4 [1].
References
1. The Scattering of α and β Particles by Matter and the Structure of the Atom, E.
Rutherford , Philosophical Mag, Volume 6, 1909
2. Experiment R2 Lab Script, Dept Physics and Astronomy, UCL, course PHAS2440, (2003).
5
Appendix
A graph showing how the Log of the Count rate per 10s changes with the discriminator bias
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Log Count rate per 10s
5
4
3
2
1
0
0
1
2
3
4
5
6
Discriminator Bias
Figure 2: Graph Showing How Count Rate Changes with Discriminator Bias
180.00
160.00
140.00
Count Rate (s-1)
120.00
100.00
80.00
60.00
40.00
20.00
0.00
-20
-15
-10
-5
0
5
10
15
20
Incidence Angle(°)
Figure 3: Graph Showing How Count Rate Changes With Incidence Angle
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6.00
5.00
4.00
ln(count rate)
3.00
2.00
1.00
0.00
1.50
2.50
3.50
4.50
5.50
6.50
7.50
-1.00
-2.00
-3.00
ln(cosec(angle-correction/2))
Figure 4: Graph Showing How Count Rate Changes With The Corrected Angle
7