Experiment 21.
Alpha Particles
Updated by MD and IF July 15, 2016
1
Important Notice
Experiment 21 consist of two independent parts: “Alpha Spectroscopy” and “Alpha Scattering”.
The first part “Alpha Scattering” includes a task which needs to be run overnight. The hardware
and software being used for each part of the experiment are independent of each other so you
can decide to do both parts in ‘parallel’, thereby more effectively utilising the time in which
you will be waiting during for data collection to complete. In this case reserve some space in
your logbook for the skipped activity to keep your notes in order.
2
Safety First
During this experiment we are using radioactive sources. All sources are well shielded or are of
low intensity, nevertheless you should avoid any unnecessary exposure by keeping away from
the sources (r −2 dependence) and limit the time of being in close proximity to them.
21–2
S ENIOR P HYSICS L ABORATORY
Alpha Spectroscopy
3
Objectives
The aim of this experiment is to investigate the characteristic spectra of alpha particles sources,
and observe the energy losses when alpha particles pass through a medium.
4
Introduction
Alpha particles consist of two protons and two neutrons bound together by the strong nuclear
forces. It is identical to helium-4 nucleus and can be considered as doubly ionised helium atom.
In our experiment those particles are produced exclusively in nuclear decay reactions and we
will refer to them as alpha particles (shorthand α). After they stop in the matter they capture
two electrons and become helium atoms. An alpha particle is very stable and in our experiment
it can be considered to be a single, doubly charged particle despite its real structure. This also
can be extended to alpha particles inside the parent nucleus. In classical physics, alpha particles
do not have enough energy to escape from the potential well inside the nucleus. The quantum
tunnelling effect however allows alphas to penetrate the potential barrier formed by the strong
nuclear force and the repulsive Coulomb electromagnetic force due to the nucleus’ positive
charge. Alpha decay occurs when the alpha particle penetrates this barrier and is repelled by
the Coulomb force.
5
Alpha Spectroscopy Apparatus
In this experiment we are using typical instrumentation for alpha spectroscopy. It consist of a
vacuum chamber with a semiconductor detector, a pre-amplifier, a multi-channel analyser, and
a computer with data acquisition software.
The vacuum chamber is made of stainless steel and houses an adjustable table to enable an
alpha source to be placed at the required distance from the detector. Vacuum is needed to avoid
energy losses when the alpha particle travels through an air. At atmospheric pressure as little
as 35 mm of air can stop 5 MeV alphas.
Surface barrier detectors are semiconductor devices designed so that a p-n junction is located
close to the detectors’ surface. The junction is reverse-biased so that all electric current carriers
(electrons and holes) are thus removed from the depleted layer and no current flows through
the detector. When an alpha particle hits the detector it loses its energy by ionising atoms in
the depleted layer producing free electrons and holes. At this time a small electric current goes
through the detector, which produces a change of potential across the resistor which supplies
bias voltage to the detector. This small signal is amplified by the pre-amplifier and then fed to
the multi-channel analyser. The detector is Passivated Implanted Planar Silicon (PIPS) model
PD 25-10-500 AM manufactured by CANBERRA. The main parameters of this detector are:
active area 25 mm2 , resolution 10 keV, and depletion depth 500 µm. Maximum bias voltage for
this detector is +100 V, but we are using +80 V which is enough for full depletion.
A LPHA PARTICLES
21–3
The multi-channel analyser (MCA) analyses a stream of voltage pulses and sorts them into
a histogram of the number of events, versus pulse-height. Height of the pulse is related to
energy which is deposited in the detector by charged particle. In case of an alpha particle, most
frequently, total kinetic energy is deposited in the detector and pulse height is proportional
to this energy 1 . In our experiment we are using a MCA of type: UNIVERSAL COMPUTER
SPECTROMETER UCS 30 manufactured by SPECTECH. This MCA has a maximum of 2048
channels (bins of histogram), variable gain, switched polarity of input pulses, and a high voltage
power supply. It also supplies power voltages for the pre-amplifier. UCS 30 is connected to the
computer through a USB port. The application to control settings and receive data from UCS
30 is USX version 1.2 by Spectrum Techniques.
The sources used in the experiment are: AMR33, AMR13, and RA226. AMR33 and AMR13
are especially build that they can be used for calibration of a detector or high resolution spectroscopy. RA226 is sealed by a thin gold layer. The sources are weak and not dangerous when
used carefully, but under any circumstances do not touch their surface to avoid contaminating
yourself by radioactive material.
6
Instrument calibration
As we mentioned in the previous chapter, MCA produces a pulse height histogram of voltage
pulses received from detector. The pulses produced by the detector by the mono-energetic
alphas are not exactly equal and follow an approximately normal distribution. We can assign
the mean of this distribution to the energy of the alpha particles. Process of assigning alpha
particles’ energy to the channel number is called instrument calibration. In our experiment for
calibration we are using AMR13 source which is pure americium-241 isotope.
Procedure
• Place the AMR13 source in the vacuum chamber, check that the air inlet valve is closed,
and evacuate the chamber.
• Switch on the MCA.
• The MCA is controlled and the output displayed using a program called USX. Open
USX. Open the Mode menu and check that PHA Amp In is ticked.
• Go to the Settings menu of USX, open the HighVoltage/Amp/ADC dialogue box, and
check if the settings are those shown on Fig. 21-1.
• After ≈ 15 min of pumping air from the chamber, the pressure should be low enough
to start taking spectrum of the AMR13. Start data acquisition by clicking on the green
circle at the left of the Tool Bar.
• Acquire the spectrum for 20 minutes. Acquisition may be stopped by clicking on the red
circle in the Tool Bar.
• Save the spectrum to the file.
1
Even if all alpha particles, which hit the detector, deposit the same amount of energy, resulting voltage pulses
are not of exactly the same height and usually produce a histogram the shape of which is close to a normal (Gaussian)
distribution. We will assign mean of this distribution (position of the peak) to the alpha particle energy in a process
called ‘instrument calibration’.
21–4
S ENIOR P HYSICS L ABORATORY
• In AMR13 spectrum display find the channel which shows maximum number of counts.
This channel is corresponding to energy of alpha particles equals 5485.56 keV.
• From USX menu select Settings → Energy Calibration → 2 Point Calibrate (or use
“Ctrl” and “2” keys combination). Then follow the calibration procedure using “keV”
as energy units, channel 0 and energy 0 as “Point 1”, and channel of highest number of
counts and energy 5485.56 keV as “Point 2”. The channel number along the x-axis of the
spectrum will be replaced by the energy corresponding to each channel. Your instrument
is now calibrated!
Fig. 21-1 Recommended MCA settings.
C1 ⊲
7
Energy loss in air
An alpha particle moving through a medium interacts with it very strongly because of its electric charge of +2 e and its relatively slow movement even at large energies (6 MeV alphas stop
A LPHA PARTICLES
21–5
in air in less than 50 mm). The mean stopping range of alphas in both air and gold as a function
of energy is given in Table 21-1.
Procedure
•
•
•
•
Stop the vacuum pump and open the air inlet.
Acquire AMR13 spectrum at atmospheric pressure for 20 min.
Save the spectrum to the file.
In AMR13 spectrum find the channel which shows the maximum number of counts.
Write down the energy corresponding to this channel.
• Using data from the Table 21-1, find the mean stopping range corresponding to this energy2 .
• Similarly find the mean stopping range of alphas of energy 5485.56 keV.
• Subtract both ranges. The difference should be equal to the distance between the source
and the detector. Check with a ruler the validity of your result. Under any circumstances
do not touch the surface of the detector which is 1 mm below the detector’s case.
Question 1: Justify the method being used to find the distance between the source and the
detector.
C2 ⊲
8
Spectrum of the RA226 source
8.1
Secular equilibrium of the RA226 source
When the radium source was prepared it was likely to be almost pure 226 Ra. However, since
that time, some of its daughter nuclides reached secular equilibrium. This is because their
half-lives are short compared with the time since source preparation. Secular equilibrium is
defined for a chain of decays as the situation where all the daughter nuclides are produced at
the same rate as they decay. This is possible only if the half-live of the first parent nuclide in
the decay chain is significantly longer than the half-lives of the daughters. If the source is in
secular equilibrium and all the daughters stay together with the parent, activities of isotopes in
the chain should be equal. Decay chain of the 226 Ra is presented in Table 21-2. Note that the
daughter gas radon 222 Rn is kept inside the source by a thin layer of gold covering the source.
Procedure
• Replace the AMR13 source with the RA226.
• Acquire the RA226 spectrum for 30 min.
• Save the spectrum to the file.
2
You can copy this table to the “QtiPlot” application, make a scatter graph of range vs energy, and then use the
interpolation feature of the program to make a more accurate interpolation table and read mean stopping range for
the desired alpha’s energy. This will be especially useful in the next section.
21–6
S ENIOR P HYSICS L ABORATORY
Energy/keV
Range in air/mm
Range in gold/µm
1000
1250
1500
1750
2000
2250
2500
2750
3000
3500
4000
4500
5000
5500
6000
6500
7000
7500
8000
8500
9000
9500
10000
5.29
6.41
7.63
8.96
10.40
11.96
13.62
15.40
17.27
21.35
25.84
30.71
35.97
41.61
47.62
53.99
60.73
67.81
75.24
83.02
91.11
99.58
108.37
1.44
1.76
2.09
2.42
2.77
3.14
3.53
3.94
4.36
5.25
6.21
7.21
8.27
9.39
10.55
11.77
13.03
14.34
15.69
17.08
18.53
20.01
21.54
Table 21-1 Mean stopping range of alpha particle in air (1 bar, 20°C) and solid gold.
Decay process
Half-life
→222 Rn + α(4784.34 keV)
→218 Po + α(5489.48 keV)
218 Po →214 Pb + α(6002.35 keV)
214 Pb →214 Bi + e− + ν̃
214 Bi →214 Po + e− + ν̃
214 Po →210 Pb + α(7686.82 keV)
210 Pb →210 Bi + e− + ν̃
210 Bi →210 Po + e− + ν̃
210 Po →206 Pb + α(5304.33 keV)
1600 year
3.8232 day
3.071 min
26.916 min
19.8 min
162.3 µs
22.23 year
5.012 day
138.3763 day
226 Ra
222 Rn
Table 21-2 Energy of principal α particles emitted by 226 Ra source.
A LPHA PARTICLES
21–7
• Export your RA226 spectrum data to a text file using File → Save... and navigate to your
folder. In the “Save” file selector dialogue box choose “Files of Type: Tab Separated
File” and type file name, for example: “RA226”. You do not need to include any file
extension, as the application will automatically add “.tsv” to the file name.
• Open the “QtiPlot” application and import previously saved data by using File → Import
→ Import ASCII... (or alternatively use “Ctrl” and “K” keys). Ignore first few lines (they
show some extra information about the spectrum which we are not using) until on the
preview you read “Channel[X], Energy[Y], Counts[Y]” as column titles and then press
the “Open” button.
• Click right mouse button on “Energy[Y]” and then “Set As → X”. Select “Energy[X2]”
and “Counts[Y2]” columns and make the scatter plot of the selected data.
• With the plot window selected go to Analysis → Fit Multi-peak → Gaussian... to fit
Gaussian curves to your spectrum.
• Create a new table or open a text editor and then copy in it the fitting results. Print the
spectrum plot and the fitting parameters.
• Analyse your result and comment on secular equilibrium of the RA226 source.
8.2
Thickness of the gold layer covering the RA226 source
As you will surely notice, peaks in the RA226 spectrum which you observe are at lower energies than those expected from Table 21-2. It is the result of the gold film which covers the
source. Using your measurement you can easy calculate the thickness of the gold layer.
Procedure
• Use a similar procedure to that you used to calculate the distance between the source and
the detector in Section 7 using alpha ranges in air, find the thickness of the gold layer.
Use all observed peaks for your calculations.
C3 ⊲
9
High resolution spectroscopy (Optional)
AMR33 is combination of three alpha emitting isotopes: 239 Pu, 241 Am, and 244 Cm. Energy
of the alphas emitted by this source are listed in Table 21-3 and simplified decay schemes are
shown on Fig. 21-2 3 .
The detector which we are using in this experiment is of very good resolution: 9.9 keV at alphas
from 241 Am 4 . To analyse the spectrum of AMR33 more precisely we can export the data to an
external data analysis program and find the position and size of the peaks using an analytical
curve fitting procedure.
3
The data in this table and the decay schemes comes from [2] and is recommended to use for reference. See [2]
for detailed information on radioactive isotopes.
4
This is the detector manufacturer specification. In reality observed resolution depends also on the quality of
the pre-amplifier used, the MCA, and the alpha source.
21–8
S ENIOR P HYSICS L ABORATORY
4
69
02
51.7007 keV
11.87 %
0.
0.
5/2+
24100 a
13.04 keV
17.14 %
0.
3/2+
26 min
704 Ma
10 -
8
500 ps
03
41
191 ps
01
04
7
100 %
1/2+ 0 keV
239
Pu 145
94
1/2+
7/2-
0.0765 keV
70.79 %
0 keV
0.03 %
235
U
92 143
0.
01
0. 81
02
03
100 %
06
0.
31
35
.9
2
5/2-
102.959 keV
13.23 %
59.54092 keV
84.45 %
54 ps
2.144 Ma
12
15
67 ns
7/2-
2.
80 ps
432.6 a
158.497 keV
1.66 %
0. 69
01
95
9/2-
5/20 keV
241
Am146
95
0.
7/2+
5/2+
237
Np144
93
33.19629 keV
0.23 %
0 keV
0.38 %
164 ps
6561 a
0.
2+
02
58
100 %
0+
0 keV
18.11 a
244
Cm
96
148
0+
240
Pu146
94
42.824 keV
23.3 %
0 keV
76.7 %
Fig. 21-2 236 Pu, 241 Am, and 244 Cm decays. Only energy levels with a probability greater than 1% are
presented.
A LPHA PARTICLES
21–9
Parent
Half-life/year
Daughter
Half-life/year
239 Pu
24100
235 U
704000000
241 Am
432.6
237 Np
2144000
244 Cm
18.11
240 Pu
6561
α Energy/keV
5105.81
5143.82
5156.59
5388.25
5442.86
5485.56
5762.65
5804.77
Probability/%
11.87
17.14
70.79
1.66
13.23
84.45
23.30
76.70
Table 21-3 AMR33 source alpha decay data.
Procedure
•
•
•
•
•
•
Change the RA226 source to the AMR33 source.
Evacuate the system and then acquire the spectrum for at least 30 min.
Save the spectrum to the file.
Export your spectrum data to a text file.
Open the “QtiPlot” application and import your data.
From the imported data table delete rows at the beginning and the end which contain a
very small number of counts, not relevant to the three groups of main peaks.
• Add
pa column for the 5error of number of counts, name it as “Error[yEr]” and set its value
to Counts[Y ] + 1 .
• Select “Channel[X]”, “Counts[Y]”, and “Error[yEr]” columns and make a scatter plot of
the selected data 6 .
We are expecting 8 peaks in the data. Probably only 7 of them will be visible as separated
peaks, as 239 Pu peaks of energies 5143.82 keV and 5156.59 keV are placed too close to be
separated visually, especially because of their different intensities. However, we can try our
best to resolve those peaks. By looking carefully at the shape of peaks present in this spectrum,
we can notice that Gaussian shape is not the best choice as the peaks shown are asymmetrical
- lower channels are spread significantly more than those above the mean. This asymmetry
results mainly from the energy lost by alphas when passing through the detector’s electrode or
some of the source’s external atom layers. There are many bell-like asymmetric functions. We
will use one which is a combination of Lorentzian (equation (1a)) and Gaussian (equation (1b))
functions. An argument of this function is x and it can signify channel number (or energy). The
width of the Gaussian function depends on the σ parameter (standard deviation) and the width
of the Lorenzian function depends on the γ parameter (half width). Parameter c is a centre of
the peak, where the shape function reaches its maximum value, a. The width of the peak is
commonly measured by parameter called F W HM (Full Width at Half Maximum), which is
given by a distance between points on the curve at which function reaches half its maximum
5
Usually, for a larger number of counts the error is calculated as the squared root of number of counts (Poisson distribution). We added one to the number of counts to avoid a unphysical situation when Error = 0 (for
Counts = 0).
6
We are not using the data in column “Energy[Y]” to make a plot. However, you can find it useful to identify
approximately the position of the peaks.
21–10
S ENIOR P HYSICS L ABORATORY
value. In case of S(x, {a, c, γ, σ}) function,

aγ 2


γ 2 + (x − c)2
S(x, {a, c, γ, σ}) =

 − (x−c)2
ae 2σ2
x<c
(1a)
x≥c
(1b)
√
and its F W HM = 2 ln 2 σ + γ ≈ 1.17741σ + γ. Lets assume that the peaks in our spectrum
have the same shape, therefore parameters γ and σ are common to all of them. This assumption
is justified, because energies of all peaks are relatively close and the detector’s response to all
alphas should be similar. The whole fitting function is given by equation (2), where index i is
numbering consecutive peaks going from 1 to 8. Function F (x) depends on 18 parameters7 ,
whose values we will find during curve fitting procedure.
F (x) =
i=8
X
i=1
S(x, {ai , ci , γ, σ})
(2)
Procedure
• Start “Fit Wizard” and choose “8AsymmetricSimilarLorentzGauss” user defined function as the analytical model of your data. Go to the fitting section and set starting
values of 16 parameters ai and ci to approximate amplitude and position of observed
peaks (guess those values for the second hidden peak corresponding to the energy of
5143.82 keV). Set also starting values of width of Lorentzian γ (g = 3) and Gaussian
σ (s = 2).
• Start fitting procedure.
• Find two the best defined peaks (with the smallest error of ci , usually the main peak
from 241 Am and the main peak from 244 Cm. Use their positions ci and corresponding
accepted energy to calculate energies of the other peaks.
• Find relative intensity of the peaks for every isotope in the AMR33 source.
• Compare your results with accepted values and comment.
Question 2: Different energies of alphas emitted by the same isotope can be explained by
different energy levels of a daughter nuclide after the decay process. Compare differences
of alphas energies produced by 241 Am with differences of corresponding energy levels of the
daughter 237 Np. Explain the result.
C4 ⊲
7
List of those 18 parameters is omitted on the left side of the equation (2) for clarity.
A LPHA PARTICLES
21–11
Alpha Scattering
10 Objectives
The aim of this experiment is to investigate the angular distribution of alpha particles scattered
on gold foil. Because the count rate for alpha scattering, particularly for large angles, is very
low, it is not feasible to obtain sufficiently accurate data in a single lab session. To collect
larger number of data you will need to share your measurements with other students doing this
experiment. This is achieved by making measurements at one scattering angle, and combining
your results with data acquired by other students at different scattering angles. Collaboration
by sharing data is a feature of modern physics research.
11 Introduction
The experiments involving scattering of alpha particles when passing through thin gold foil was
originally performed by Hans Geiger 8 and Ernest Marsden 9 under direction of Ernest Rutherford 10 . The outcome of the experiments lead Rutherford to a new model of atom, different
than contemporary model by J.J.Thompson 11 . Rutherford’s model of atom stated that almost
all mass and positive charge of atom is located in a very small region of atom, as opposite to
Thompson’s model, where electrons (discovered by J.J.Thompson) were located in a sphere of
positive charge filling the whole volume of an atom.
11.1 Scattering of charged particles in central electrostatic field
From his model of atom, Rutherford derived the formula, which was later experimentally verified for scattering angles up to 150°. For more information on works of the Rutherford-GeigerMarsden team see their original publications, [4] and [5]. A contemporary derivation of the
formula for differential cross section on α scattering is presented in reference [6]. We present
below a short derivation of a simplified formula for counts rate used in our experiment.
According to [6] the differential cross section for α scattering is:
dσ
=
dΩ
Qn Qα
8πǫ0 mv 2
2
csc4
Θ
2
(3)
where:
• σ is the cross section for scattering at angle Θ,
• Ω is the solid angle where alpha particle scatters,
8
Johannes Wilhelm Geiger, German physicist, 30.09.1882 - 24.09.1945.
Sir Ernest Marsden, English-New Zealand physicist, 19.02.1889 - 15.12.1970.
10
Ernest Rutherford, 1st Baron Rutherford of Nelson, New Zealand-born British physicist, 30.08.1871 19.12.1937.
11
Sir Joseph John ”J.J.” Thomson, English physicist, 18.12.1856 - 30.08.1940.
9
21–12
S ENIOR P HYSICS L ABORATORY
Fig. 21-3 Alpha scattering on gold nucleus. The dotted lines in this figure are asymptotes to the hyperbolic trajectory of the alpha particle; they give the directions of the incoming and scattered alpha particle
at a large distance from the nucleus.
• Qn is the charge of the nucleus (79 e for gold),
• Qα is the charge of the alpha particle (2 e),
• ǫ0 = 8.854187817... pF/m is permittivity of free space,
• m the mass of α particle,
• v the velocity of α particle,
• Θ the deflection angle.
Let assume area of detector A is small compared to its squared distance from gold foil r 2 , the
solid angle Ω can be calculated as:
A
(4)
Ω= 2
r
where:
• A is the area of the detector,
• r is the distance between the gold foil and the detector.
For thin gold foil, counting rate is proportional to the rate of all α particles hitting the target,
number of atoms per volume unit, thickness of the foil, and cross section:
R = R0 ntσ
where:
(5)
A LPHA PARTICLES
21–13
• R is the rate of scattered α’s,
• R0 is the rate of all α’s reaching gold foil,
• n is the number of gold atoms per volume unit,
• t is the thickness of the foil.
α particles from radioactive sources are relatively slow and we can use the classical equation
for energy:
mv 2
E=
(6)
2
where:
• E is the kinetic energy of α particle.
Plugging equations: (4), (5), and (6) into (3) gives:
R = AR0 nt
Qn Qα
16rπǫ0 E
2
csc4
Θ
2
(7)
12 Apparatus
In our experiment to investigate α scattering on gold foil we use an apparatus described by P.J.
Ouseph [7]. The figure Fig. 21-4 shows a simplified schematic of Ouseph’s apparatus.
The alpha scattering from a gold foil is observed in an evacuated chamber which, together with
a semiconductor detector mounted within it, can be rotated. An Americium-241 alpha particle
source is located in a lead container mounted inside the rotatable chamber. It is suspended at
the chamber’s axis this way that gravity preserves its vertical position while the chamber is
rotating. A small hole in the top of this container permits a collimated beam of alpha particles
to be directed radially at the centre of the gold foil. Rotating the chamber thus changes the
direction of observation of the scattered alphas. The gold foil is also rotated, but at half the
angle through which the detector is rotated. The thickness of the gold foil t0 = 0.5 µm. The
detector is connected to the pre-amplifier and the amplified signal is fed to the input of the
computer’s sound card, where it can be analysed by means of the “PRA” application [8].
13 Experimental Confirmation of Rutherford’s α Scattering Formula
13.1 Small Deflection Angles
Procedure
• Evacuate the chamber and rotate to 0°.
21–14
S ENIOR P HYSICS L ABORATORY
Fig. 21-4 Simple apparatus to investigate alpha scattering on gold foil.
• Start the “PRA” application.
• From the menu choose View → Audio Input and View → Pulse Height Histogram, then
Action → Start Data Acquisition. Take the data for 60 s (it should stop automatically).
The pulse height histogram should produce a wide peak, which is selected. Place the
cursor inside selection of the peak and read the Gross number of counts.
• Repeat the last measurement for angle ranges -10° to +10° at steps of 1°.
• Fit a Gaussian curve to a plot of your data.12
Question 3: Is the plot of your data similar to the expected R ∼ csc4 (Θ/2) dependence?
Comment on your results.
Question 4: Why is Rutherford’s equation wrong for small angles? Hint: What are the main
differences between Fig. 21-3 and the real situation in our experiment?
C5 ⊲
12
There is not theoretical reason for using Gaussian as fitting function and depending on profile of the beam of
alphas and the shape and the size of the detector our data can be more or less similar to the Gaussian. In any case the
mean of fitted Gaussian will tell us about right position of zero on the angular scale and its width is a good measure
of the beam’s size.
A LPHA PARTICLES
21–15
13.2 Large Deflection Angles
When the gold foil is rotated by Θ/2 angle, its effective thickness, as seen by the α particle, is
increased by a factor of sec(Θ/2). To correct for this effect we put t = t0 / cos(Θ/2) in equation (7). Then by multiplying both sites of the equation by cos(Θ/2) and replacing R cos(Θ/2)
2
Qn Qα
with Rc , where Rc is ‘corrected’ rate and introducing the constant 13 C = AR0 nt0 16rπǫ
0E
our equation (7) will have a simple form (8):
Θ
Rc = C sin−4
(8)
2
To verify this equation at larger angles we have to collect scattering data for a very long time.
We can achieve this by running data acquisition at chosen angle overnight and contributing the
result to the data set which combines results from observations performed by many students
doing this experiment.
Procedure
• Set vacuum chamber to the angle between 10° to 100°.
• In the “PRA” application set ‘Acquisition time’ to 86400 s (to do this choose menu Settings → Data Acquisition and Analysis and then in group ‘Data acquisition limits’ change
time 60 s with 86400 s and press ‘Apply’ button).
• Start data acquisition.
• Next day visit the lab to stop data acquisition and save your results to the file named
“CumulativeData.ods” located on the desktop of the computer.
13.2.1
Data Analysis
By taking the natural logarithm of both sides of equation (8) we get:
Θ
ln Rc = ln C − 4 ln sin
2
(9)
Procedure
• Derive the formula for the error of ln Rc .
• Open a new “QtiPlot” project and copy and paste data from “CumulativeData.ods” to the
table.
• Add 3 new columns and set their values to ln sin(Θ/2) as “X”, ln Rc as “Y”, and error
of ln Rc as “Y error”.
• Make a scatter plot of the calculated data and fit a straight line.
• Comment on the results.
C6 ⊲
13
We assume that energy of scattered α particle does not depend on scattering angle, which in general is not true.
21–16
S ENIOR P HYSICS L ABORATORY
14 Detailed Verification of Rutherford’s Equation (Optional)
In previous chapters we investigated how well the result of our experiment confirms Rutherford’s equation (7). Our investigation was concentrated only on dependence of the counting
rate and csc4 (Θ/2). We replaced the other factors in equation (7) by a constant value. We
checked that for small scattering angles equation (7) is not supported by our measurements.
However, our measurements for small angles are useful. Knowing that at 0°, the counting rate
includes most of α the particles hitting the gold foil 14 , the counting rate at this angle is a good
approximation of R0 . The measured average energy of the α particles is E = 3818 keV 15 , the
distance between gold foil and detector is r = 76 mm, and area of the detector A = 25 mm2 .
The thickness of gold foil is t0 = 0.5 µm.
Procedure
• Calculate expected value of constant ln C 16 .
• Compare this theoretically calculated result to the measured value.
C7 ⊲
References
[1] Glenn F. Knoll. Radiation detection and measurement. John Wiley & Sons, 1989.
[2] Bureau International Des Poids et Mesures, Pavillon de Breteuil, F-92310 Sèvres. Table
of Radionuclides. http://www.nucleide.org/DDEP_WG/DDEPdata.htm
[3] I. Vasilief, QtiPlot, version 0.9.9, 2016.
http://soft.proindependent.com/qtiplot.html
[4] E. Rutherford, F.R.S. The Scattering of α and β Particles by Matter and the Structure of
the Atom. Philosophical Magazine Series 6, vol. 21, May 1911, p. 669-688
[5] H. Geiger, Ph.D., John Harling Fellow, and E. Marsden, Hatfield Scholar, University
of Manchester. On a Diffuse Reflection of the α-Particles. (Communicated by Prof. E.
Rutherford, F.R.S. Received May 19,-Read June 17, 1909.)
[6] http://en.wikipedia.org/wiki/Rutherford_scattering
[7] P.J. Ouseph, Alpha scattering apparatus. Am. J. Phys. 44, 1012-1013, 1976
[8] M. Dolleiser, PRA, version 15, 2016.
http://www.physics.usyd.edu.au/˜marek/pra/
14
From observation of the data points around zero degrees we can make a conclusion that the area of the detector
and the profile of the beam of alphas are well fitted.
15
The energy of α emitted by the 241 Am is in average 5442 keV, but the source used in this experiment is sealed
by a thin metal layer which reduces the energy of emitted α particles.
16
Use gold’s tabulated properties to find n.