Range of Alpha Particles in Air
-William Walshe
Abstract:
In this experiment, the range of 4.88MeV alpha particles in air was investigated. Their
range was found to be 34±1mm, which agrees with the accepted value of 34mm. This
value was obtained first by assuming the detector and source were point-like, and then
developing this model to account for their respective dimensions.
Using the measured value and the Bragg-Kleeman rule, an estimate of 52.8μm for the
range of alpha particles in water was then obtained.
Introduction:
In this experiment, we attempted to find the distance from a 209Po source at which the
intensity of alpha radiation dropped to zero, i.e., the range of a 4.88 MeV α-particle in
air.
Alpha emission occurs due to Coulomb repulsion between particles in a nucleus. The
Coulomb force increases with Z2, whereas nuclear binding energy increases
approximately with A, so alpha emission becomes important in very heavy nuclei. The 2
proton – 2 neutron structure is favoured for its high stability and binding energy.
Classically, an α-particle does not have the energy to escape from a nucleus, but the
phenomenon of quantum tunnelling allows their emission.
Once emitted, alpha particles lose energy via ionising collisions with the electrons of the
atoms constituting the medium they are travelling through. Since the α-particle mass is
much greater than that of an electron, the particles are not scattered by any significant
angle during collisions in a very low-density medium like air. They continue in a straight
line until they lose all their energy through these ionising interactions, eventually slowing
down and forming neutral helium atoms.
The apparatus we used was made up of two parts: a circular 209Po source measuring
5.3mm in diameter, and a detector with the same aperture radius directly underneath
consisting of a ZnS scintillator above a photomultiplier.
When the scintillation powder is stuck by an energetic particle, it becomes excited to a
higher energy level. It then de-excites, emitting a photon which is detected by the
photomultiplier (the construction of the scintillator is such that is transparent to its own
radiation). The pulses reaching the photomultiplier pass via an amplifier and single
channel analyser to a pulse counter, allowing us to measure the number of alpha particles
reaching the detector in a given amount of time.
Due to the fact that both the source and detector are finite in extent, the count rate is not
simply a linear function of distance. Intuitively, the alpha particles are presented with a
larger target when they are close to the source, whereas the detector will “appear” smaller
at larger distances and consequently it will be less likely that an alpha will hit it. This idea
is encapsulated in the solid angle considerations in the analysis.
Procedure & Results:
The apparatus was set up as described in lab handout. The source was placed at its closest
displacement from the detector and the number of counts measured in five minutes was
recorded. The source was then moved further away in 2mm steps and the counts
measured in 5 mins recorded at each point. The following results were obtained:
d (mm)
±0.5
3
6
8
11
13
16
18
21
23
26
28
31
33
36
38
41
43
46
Counts in 300s
927±30
678±26
405±20
330±18
240±15
180±13
135±12
117±11
90±9
62±8
30±5
11±3
6±2
0
0
0
0
0
Count Rate
(counts/minute)
185±6
136±5
81±4
66±4
48±3
36±3
27±2
23±2
18±2
12±2
6±1
2±.3
1±.3
0
0
0
0
0
Analysis:
(1)
Assuming a point source and detector, we can generate the following graph:
Count Rate V Distance
250
Count rate (counts/min)
200
150
100
50
0
0
5
10
15
20
25
30
35
40
45
50
-50
Distance (mm)
This allows us to estimate the range of alpha particles as approximately 33± 1mm.
However, our source-detector model is invalid, both the source and the detector are finite
in extent, so the count rate is not simply a function of distance. We correct for this in part
in the next section.
(2) Examining Count Rate per Solid angle (ie assuming detector is finite)
We now model the alpha source as a point at the centre of a sphere of radius R, and the
detector as a spherical cap of height h and base radius 5.3mm. The source-detector
distance we have been measuring corresponds to (R-h) in the above diagram.
To correct our readings for the detectors finite extent, we want to calculate the count rate
per solid angle for every distance d. We know a solid angle of 1str is subtended by a
spherical cap of area one square unit, so a spherical cap of area A in a sphere of radius R
subtends an angle of
A
Ω= 2 [where Acap  2Rh ]
R
which allows us to calculate the following:
d (mm)
± .05
3
6
8
11
13
16
18
21
23
26
28
31
33
36
38
41
43
46
Surface Area
of Cap (mm^2)
118 ±20
101 ±8
96 ±6
93 ±4
92 ±4
91 ±3
90 ±3
90 ±2
89 ±2
89 ±2
89 ±2
89 ±1
89 ±1
89 ±1
89 ±1
89 ±1
89 ±1
89 ±1
Ω (str)
13
2.8
1.5
0.77
0.54
0.35
0.28
0.203
0.169
0.132
0.114
0.092
0.082
0.068
0.061
0.053
0.048
0.0418
±4
±0.5
±0.2
±0.07
±0.04
±0.02
±0.02
±0.009
±0.007
±0.005
±0.005
±0.003
±0.002
±0.002
±0.002
±0.001
±0.001
±0.0009
C/Solid Angle
(counts/min)
14 ±5
49 ±10
54 ±10
86 ±13
89 ±12
103 ±14
96 ±14
113 ±15
107 ±16
91 ±19
53 ±11
22 ±4
12 ±4
0
0
0
0
0
Count Rate per Solid angle V Distance
140
Count rate per solid angle
(counts/min)
120
100
80
60
40
20
0
-20
0
10
20
30
40
50
Distance (mm)
This gives us a value of 35±1mm for the range of alpha particles in air, which agrees with
the accepted value of 34mm. For the purposes of fitting a line to the data points however,
the count rate values of zero do nothing to help. Re-plotting without these values gives a
better result:
From this graph we get the more accurate value of 34±1mm.
However, our model of the source and detector is still not correct. It is also necessary to
make a minor correction for the fact that the source too is finite in extent.
(3) Examining Count Rate per Actual Solid Angle (i.e. assuming finite detector & source)
Seeing as the geometric situation we are dealing with here is one where 2 finite disks are
facing each other, neither the point source nor spherical cap assumptions of the previous
section are completely accurate (becoming increasingly less valid as separation
decreases). Disk 1 has a radius S, disk 2 a radius R and they are separated by a distance d.
Calculating the actual solid angle subtended between two such disks is an arduous task
necessitating the use of the integral:
S
(R / d , S / d ) 
  ( d , R , x ) xdx
0
S
 xdx
0
where x is the radius to a particular point on the disks surface.
Thankfully, there is a simplification when R=S and from the table in the appendix of the
lab handout, we can plot the following graph
Solid Angle V R/d
4
3.5
Solid Angle (str)
3
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
-0.5
R/d
which (in theory) allows us to read off values for the ratio R/d and find the corresponding
solid angles. Since the correction is only really significant for small displacements, this
section of the analysis does little other than blur the results by introducing proportionally
astronomical human error in all other values of d. With the table provided, it is simply not
practical to visually distinguish between an R/d value .140 and .148 (for example). For
this reason the (horrible) CR/Solid Angle graph produced by this correction is not
reproduced here.
Further Considerations:
(1)
The range of an alpha particle in water can be estimated using the Bragg-Kleeman rule,
which states
where R = range of alpha particle in a particular medium
ρ = density of medium
A= mass number of medium.
We have to make several assumptions to apply this rule: We will take ρair = 1.299 kg/m3
and ρwater = 1000 kg/m3. Since air is 71% nitrogen, we will use 7 for Aair. Water is
generally assigned an “effective mass number” of 10 (8+1+1) in this type of calculation.
This gives us an estimated range of ~52.8μm for an alpha particle in water.
(2)
Alpha particles and other heavy charged particles interact strongly with matter, so it is
valid to apply the notion of range to them. Electrons, however, interact very weakly with
matter and are much more likely to pass through it unaffected. So instead of a cut off
point, their intensity falls off very gradually, making the notion of range difficult to apply
to them.
Analysis of Uncertainty:
Uncertainty in counts was obtained from the standard uncertainty rule for a random
emission system (i.e. C  C ).
Since the two plates we were measuring distance between in this experiment were
nowhere near perfectly parallel, it was attempted to take measurements with them at the
same orientation and from the same point on their circumference each time. This,
however, was a fundamentally inexact process, so it was decided to include an
uncertainty of .5mm in these distance measurements, despite the callipers used being in
themselves accurate to .02mm. Improving this part of the apparatus would improve our
results.
Our data points fall far too steeply from the last non-zero count rate to zero. While our
measurement of alpha particle range agrees with the accepted value, there is an
upsettingly large gap in our graph at the critical point were the intensity falls to zero.
Were we to repeat this experiment, we would definitely make many times more readings
in the 30-40mm range.
Further to the previous point: 5 minutes was not an ideal timeframe to measure counts in
the lower intensity ranges. At these low levels, just because there were no counts
recorded in 5 minutes does not necessarily mean there would be none in six. If time
constraints were not a factor, it would be preferable to leave the detector running for
around a half hour in this range.
The table provided for obtaining actual solid angles between the source and detector
proved insufficient to apply to this experiment with any kind of accuracy. If the time
were available, this experiment could have been improved by writing some mathematica
to obtain solid angles from a given R/d value.
Conclusions:
We found the range of alpha particles in air to be 34±1mm, which agrees with the
accepted value of 34mm. Using the Bragg-Kleeman rule, a value of 52.8μm was
estimated for their range in water.