PART II LABORATORY
13 RUTHERFORD SCATTERING OF ALPHA PARTICLES
13.1 INTRODUCTION
The foundation of modern ideas about atomic structure was laid by Sir Ernest Rutherford in 1911
with his postulates concerning the scattering of alpha particles by atoms of heavy elements. Two
of his co-workers, Geiger and Marsden, had observed large-angle scattering of alpha particles
which could not be explained by existing theories such as the Plum Pudding model1. This data
led Rutherford to speculate on the structure of the atom, and to produce a nuclear model.
According to Rutherford’s model, alpha particles passing through a gold foil are scattered by
electrostatic interactions between the alpha particles and the (massive) gold nuclei. The
probability of alpha particles of a given energy being scattered through a given angle (i.e. the
differential scattering cross section) can be calculated on the basis of the model and Coulomb’s
law, and tested experimentally. The model was thus supported by the experiments of Geiger and
Marsden.
Our experiment, in principle, repeats that test. However because the thickness of the gold foil we
will use is subject to considerable variation, we will use a comparison of measured and theoretical
differential scattering cross section at a number of angles both to test Rutherford’s formula and
estimate the thickness of the gold foil.
Sir Ernest Rutherford was born into a farming community in 1871, in the
picturesque countryside just outside Nelson, New Zealand. Rather than attending
one of the major European or North American universities, he completed a BA
and an MA in mathematics and physics at Canterbury College, Christchurch
(most unusual for a prospective career scientist at that time). Following this he
moved to Cambridge University’s Cavendish Laboratory. It was there he named
both  and  particles.
After returning to Christchurch to marry his university sweetheart, Mary Newton,
they moved Montreal. Rutherford undertook research at McGill University on the
transmutation of elements by radioactive decay that would eventually lead to a
Nobel Prize.
This was not his only honour – in 1914 he was knighted and in 1931 was
awarded the title of Baron, and took the name Lord Rutherford of Nelson in
recognition of his birthplace. Although the house in which he grew up has long
since disappeared, an extensive monument lies on the site.
Ernest Rutherford died without descendants in 1937 at the age of 66, having lost
his only child (a much-adored daughter) at an early age. His ashes are interred in
Westminster Abbey. Rutherford is remembered not only as a brilliant scientist,
but also as “a big, blustering noisy man – kind and much loved”.
13.2 BACKGROUND THEORY
13.2.1 SCATTERING DIMENSIONS
In this experiment we will be using measurements of the scattering angle at different alpha particle
energies to verify Rutherford’s model of atomic structure. A schematic diagram of the scattering
dimensions is given below.
1
The Plum Pudding model of the atom, postulated by J.J. Thomson (discoverer of the electron). In this model, an atom
consists of a number of electrons embedded in a (heavy) cloud of positive charge. The electrons stay in equilibrium
positions unless disturbed (for example by a collision), at which time they vibrate about equilibrium and emit light. This
model does yield frequencies around optical wavelengths, but does not explain spectral lines .
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RUTHERFORD SCATTERING OF ALPHA PARTICLES
Figure 13.1 Schematic of the experiment, showing the scattering dimensions you will
measure.
13.2.2 SOLID ANGLE
In our experiment, we will be measuring the number of alpha particles scattered into our detector.
Solid angle is a way of considering the effective area of the detector, taking into account the
orientation of the detector to the foil and its distance from the foil.
Consider the following system. A
point, P1, is located a distance r away
from a surface of area dS2. Now draw
a series of lines from P1 to the edges
of the surface. You will see that these
lines form the boundary to a volume,
which we can describe by the solid
angle d.
The solid angle can be calculated as:

dS  rˆ
d 
r2
Eq. 13.1
Figure 13.2 Solid Angle.
This useful quantity allows us to
account for the area of the detector.
Pre-lab Question (a) Write down the expression for , the solid angle subtended by the
detector at the foil, in terms of the following quantities:
d = the distance from the centre of the annulus to the detector
S = the area of the detector
R = (R1 + R2)/2 and is the mean radius of the foil annulus
2
Note that the surface area dS is infinitesimal, and therefore can be considered to be flat and so the vector dS fully
describes the orientation of this surface.
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PART II LABORATORY
HINT: Remember that  is defined as a dot product of two vectors. Draw a diagram
showing the directions of these vectors.
13.2.3 THEORETICAL CROSS SECTION
We want to determine the probability that alpha particles with a given energy will be scattered
through a given angle. This probability can be expressed in terms of the number of alpha particles
per second (ds) scattered into a solid angle (d) by a foil of thickness t and containing n nuclei in
the target volume.
d s   i n
d
 d
d
Eq. 13.2
where i is the flux incident on the foil per unit area and d/d() is the differential scattering cross
section (in area units) as a function of the scattering angle, . Nuclear cross sections are usually
quoted in barn (10-28m2).
In terms of quantities that can be measured in this experiment, we can calculate the differential
scattering cross section from the above equation to be
d
    s
d
n i 
Eq. 13.3
where s indicates the (finite) number of alpha particles scattered through  per second and 
is the (finite) solid angle subtended by the detector.
Pre-lab Question (b) The number of atoms, n, in the foil must be known. This can be found by
calculation of the weight of gold in the scattering annulus and knowledge of Avogadro’s
number. As the thickness is unknown, we must settle for
n  c.t atoms
Eq. 13.4
where c is a constant and t is the foil thickness in microns. Calculate the constant and
check the value with your demonstrator.
The theoretical expression for the cross section is
2
2


d
    Zze  14 
d
16 0 E K  sin  2 
Eq. 13.5
where Z is the atomic number of the scattering nuclei; z is the atomic number of the scattered
particles; EK is the initial kinetic energy of the scattered particles; e is the electronic charge and 0
is the permittivity of free space.
Pre-lab Question (c)
For your fixed experimental setup
d
   k4 
d
sin  2 
Eq. 13.6
This constant k must be calculated for your experimental set up to give the final value in
m2. Check this value with your demonstrator.
Pre-lab Question (d) Write down the expression for the scattering angle, , in terms of the foil
to detector distance, d.
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RUTHERFORD SCATTERING OF ALPHA PARTICLES
13.3 APPARATUS
13.3.1 ELECTRONICS
Figure 13.3 Block diagram of electronics
a)
Detector and Pre-amplifier
Solid state silicon detectors are essentially a type of miniature ionisation chamber, the
active volume of which is formed by the depletion region of a large area p-n junction. The
thickness of such a region is controlled by both the resistivity of the original n-type material,
on which a p-type barrier if formed, and the reverse bias applied. A detector with bias
arrangements is shown diagrammatically in Figure 13.4. In the diagram the capacitance C
is the sum of the junction, cable and stray capacitance. The contacts onto the detector
surfaces are usually evaporated films of gold or aluminium.
(a) Detector electronics
(b) Detector voltage pulse
Figure 13.4 Solid State Silicon Detectors
A charged particle, on losing energy in the depletion region, creates electron-hole pairs,
(essentially free charge) which can be collected at the electrodes by the field produced by
the reverse bias voltage, V. The mean energy loss of a charged particle required to produce
one electron-hole pair in silicon is 3.5 eV, hence an average of 2.87 x 105 such pairs will be
produced for every MeV of energy lost.
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PART II LABORATORY
The effect of the collection of the electron-hole pairs is to discharge the capacitor C by an
amount V where V=Q/C, Q being the total charge collected. The capacitor then recharges
through R with a time constant RC. As the energy of the particle absorbed and the charge,
Q, are linearly related, a charge sensitive pre-amplifier is used to produce a signal which is
easily analysed. These pre-amplifiers have an input time-constant much greater that the
charge collection time of the detectors (typically 10-4 sec as against collection times of 10-7
sec) and the first amplification stage has a very high open loop gain (2000 to 10 000) with a
capacitive feed-back ratio very close to unity. The charge sensitivity of the pre-amplifiers
allows changes in the size of the voltage pulse to occur due to change in bias conditions
without changing the final pulse produced by the pre-amplifier, providing the total charge
collected remains constant.
b)
Amplifier
This simply makes the pulse produced by the pre-amplifier larger in the voltage scale, so
the pulse can be detected by the single channel analyser.
c)
Single Channel Analyser (SCA)
The SCA acts as a filter, only passing on pulses within an accepted range to the counter. It
has upper and lower levels as well as “window” or “normal” setting. The lower level always
reflects the lowest value of the range. If “normal” is selected, the upper level represents the
highest value of the range – “lower level 2.0, upper level = 9.0” means pulses greater than
2.0V and less than 9.0V will be passed to the counter. If “window” is selected, the upper
level is the next decimal place to the lower level – “lower level 2.0, upper level 9.0” means
pulses greater than 2.0V but less than 2.90V will be passed to the counter.
d)
Counter
The counter increments by 1 each time an appropriate pulse is sent to it by the SCA. It is
linked to a timer which can be switched on or off, to provide a constant counting time.
Counts follow a Poisson distribution, and the uncertainty in N counts is
 N.
13.3.2 SCATTERING FOIL AND SOURCE
The 241 Am source of 5.48MeV alpha particles is secured in position in the end of the scattering
chamber by a plate which exposes a limited area of the source. To protect users from
contamination, the active material is enclosed between a backing layer of silver and a 3mm thick
gold window layer. This window layer reduces the energy of the emitted particles to 4.66MeV with
a full width at half maximum of 0.46MeV.
The gold foil that you will use was evaporated onto a mylar backing. For scattering purposes, we
can ignore the mylar due to its much lower molecular weight and number than gold.
The gold foil is extremely fragile and must on no account be touched. Great care must be
exercised when removing the insert from the chamber.
There are two inserts that fit between the source and the detector. One has the actual foil
attached, with an annulus cut into an Al backing plate. The second insert is simply an open
annulus which acts as a spacer, to ensure that the source is placed at the same relative distances
as when the foil is in place.
13.3.3 VACUUM SYSTEM
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RUTHERFORD SCATTERING OF ALPHA PARTICLES
Figure 13.5 The Vacuum System and Scattering Chamber
The pressure in the scattering chamber must be reduced since the range of alpha particles in air
at normal pressures is about 3cm. The pump provided will reduce the pressure in the chamber to
about 10-2 torr in 5 to 10 minutes and hence greatly increases their range.
As shown in Figure 13.5, O-ring seals are used at various locations. The shaft slides through Orings, so care should be taken when moving the shaft under vacuum to avoid distortion of the Orings which could cause leaks. At all times turn the detector bias DOWN before moving the
shaft.
13.4 EXPERIMENT: RELATIONSHIP BETWEEN COUNTS AND SOURCE TO
DETECTOR DISTANCE
In this first part of the experiment, you will examine the relationship between the distance (d+l in
Figure 13.1 above) from the source to a detector and N, the number of counts detected.
Pre-lab Question (e)
What relationship do you expect this to follow? Why?
13.4.1 PROCEDURE
Read the safety guidelines in the appendix at the end of these notes.
Place the spacer plate in the chamber and evacuate the apparatus.
Question (a)
Explain why we do this experiment under vacuum. What might happen to
our results, if the experiment were done at atmospheric pressure?
Slowly raise the bias on the detector to 30 volts and view the raw detector output on the CRO with
the detector at a distance of 1cm. Set the upper and lower levels on the SCA and check that the
electronics are working.
Question (b)
How did you choose the upper and lower SCA levels?
Measure the counts reaching the detector over a wide range of detector to “foil” distances, from 1
to 20cm (using 10 second counting times). Using a 5 minute counting time, record the number of
particles reaching the detector at a distance of 1cm.
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PART II LABORATORY
13.4.2 ANALYSIS
Plot the inverse root of the count versus distance. Include your reasoning of the uncertainties in
your data, and the resulting error bars on your graph.
Question (c)
Why do we plot this, rather than distance versus counts directly?
Using your relationship between distance and number of counts, determine the number of counts
you would have got had the detector been sitting at the plane of the foil for the long (5 minute)
measurement.
Pre-lab Question (f)
The incident flux on the foil must be known. This can be calculated after
the number of alpha particles passing to the detector have been measured. The flux incident
on the foil is
2
 i  C0  DS 0  cos r 
Eq. 13.7
ADET  DS  F 
where C0 is the number of counts per second at 0 distance; ADET is the area of the
detector in units of m 2; DS-0 is the distance from the source to zero distance; DS-F is the
distance from the source to the foil and r is shown on the diagram of scattering
dimensions.
With the aid of diagrams, explain why we include the DS-F/DS-0 and cos(r) terms in this
expression.
From your 5 minute measurement, calculate the average incident alpha flux on the foil, as
described in Pre-lab Question (f).
13.5 EXPERIMENT: RUTHERFORD SCATTERING
In this section you will conduct an investigation similar to Rutherford’s famous experiment, from
which one can infer something about atomic structure. By assuming the nuclear model to be
correct, we can also use such scattering measurements to determine the thickness of a scattering
foil.
13.5.1 PROCEDURE
After lowering the bias on the detector and letting the chamber up to air, insert the plate that holds
the gold foil. Due to the thin foil and low source strength, long count times will be required. At least
4 data points must be taken, thus it is recommended that one point be taken in class on the first
day, one point early the following morning, one around lunch time and two during the second day’s
class.
After discussion with your demonstrator, select a range of distances for your measurement. Use
the figure below to help decide how long to count at each distance.
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RUTHERFORD SCATTERING OF ALPHA PARTICLES
Figure 13.6 Use this diagram to decide on counting times for each distance
Set up the required distance and leave the counter on “no automatic stop”. The counter can be
stopped manually when sufficient counts have been detected.
Question (d)
How many counts do you think are sufficient? What effect would a low
count measurement have on your results?
Start taking measurements. In the meantime, consider how you will do the analysis and prepare
the transforms in SigmaPlot.
13.5.2 ANALYSIS
You are now in a position to calculate the foil thickness by equating the theoretical and
experimental cross sections given in the background theory. If the theoretical and experimental
cross sections are in agreement
d
d

d THEORY d EXPERIMENT
k
1  s
 4   
sin  2  t c i 
Eq. 13.8
Therefore, if you plot theoretical cross section against experimental cross-section (as a function of
thickness), you can determine t from the slope of your plot.
Be sure to include a worked solution or sample calculation showing how you converted from your
raw measured data to the theoretical and experimental cross sections.
Question (e)
What thickness do you calculate the foil to be? What is the uncertainty in
this value?
Question (f)
Use your plot to verify Rutherford’s theory of nuclear structure.
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PART II LABORATORY
13.5.3 FURTHER CONSIDERATIONS
Discuss, qualitatively, the effects of the following on your results:
Question (g)
The finite size of the scattering foil, and the corresponding spread in
scattering angle
Question (h)
The effect of alpha particles striking the walls of the chamber.
Question (i)
The change in alpha particle energy as it passes through the foil
Question (j)
The variation in mean path length of the particles through the foil at
different scattering angles
13.6 APPENDIX
13.6.1 SAFETY PRECAUTIONS
a)
Gold foil
As mentioned above, the gold foil is extremely fragile and must on no account be touched.
Be sure to handle the foil frame only by the edges, and be very careful not to scratch or tear
it.
b)
Vacuum system
Working with vacuum systems is a highly specialised art. There is even a Journal of
Vacuum Technology! The vacuum system you will use in this experiment is very simple, but
some general rules apply.
The inside of the evacuated detector chamber remains uncontaminated by keeping it
sealed whenever possible and not touching anything inside it.
Ensure that the exhaust from the pump is adequately ventilated (the exhaust hose should
go outside).
Finally, make sure the whole vacuum chamber is sealed when the pump is running. It is
inefficient and stresses the pump if a valve is left open or there is an air leak in the system.
c)
Detector
The surface of the detector is extremely sensitive and must never be touched. Damage to
the surface would render the device useless, and its replacement cost is several thousand
dollars.
The bias should be applied and taken off in slow, even steps to prevent dielectric
breakdown. Take at least 30 seconds to raise or lower the bias.
There should be no bias on the detector during large changes in pressure. Make sure the
bias is off whenever the chamber is pumped down or let up to air. Also turn the bias off
when moving the detector arm in case of leaks around the sliding O-ring during movement.
Always check the detector arm is clamped in place BEFORE the chamber is pumped down.
d)
Radiation
There is no significant risk of radiation exposure in this experiment (or any others in the Part
2 laboratory). The decay length of alpha particles in air is around 3cm (less than the
distance from the source to the open face of its holder) and the source itself is almost
always sealed inside the vacuum system, out of which virtually no alpha particles can leak.
Common sense in handling the source (especially during the spacer plate to foil
changeover) should still prevail.
Further information about radiation safety can be found on the noticeboards outside the
laboratory, or by asking your demonstrator.
128
RUTHERFORD SCATTERING OF ALPHA PARTICLES
13.6.2 USEFUL DATA
Permittivity of free space 0
1 micron
Energy of  particles emitted by 241Am
Mean energy of 5.48MeV  particles after passing through 3 m
window of Au
Atomic weight of Au
Density of Au
Atomic number of Au
Atomic number of  particle
Avogadro’s Number NA
Approximate thickness of scattering foil
1 electron volt
Electron charge
Derivative of tan-1
13.7 REFERENCES
Physics, Ohanian, W.W. Norton & Co
129
8.8542 x 10-12 f.m-1
10-6 m
5.48MeV
4.66MeV (FWHM =0.46MeV)
196.97g.mole-1
19.3g.cm-3
79
2
6.022x1023 mol-1
1 micron
1.60210x10-19 Joules
1.60210x10-19 C

a
x
arctan    2
2
x
a a x