Determining the Beta Particle Absorption Coefficient of
Aluminum and the Gamma Ray Absorption Coefficient of Lead
H. Potter
(Completed 30 March 2006)
The beta particle absorption coefficient for aluminum was determined to be 2.17
± .05 mm-1. The gamma ray absorption coefficient was determined to be .062 ±
.01 mm-1.
I. Introduction
There are three basic forms of radioactive decay. The first form of radioactive
decay is alpha decay. In alpha decay a helium nucleus is emitted from a radioactive
element’s nucleus. This process changes the identity of the radioactive element because
its atomic number decreases by two. The emitted alpha particle is quite massive, at about
4 amu, and has a positive charge of 2e. Alpha particles, therefore, react very readily after
they are emitted and do not penetrate very far into shielding materials.
The second form of radioactive decay is beta decay. In beta decay a lepton is
emitted, either a positron or an electron, and the radioactive nucleus again changes its
identity because conservation of charge requires its atomic number to either increase by
one (when an electron is emitted) or decrease by one (when a positron is emitted). Beta
particles are less massive than alpha particles by factor of roughly 8000, and only carry a
charge of 1e; consequently, beta particles can appreciably penetrate many potential
shielding materials.
The third form of radioactive decay is gamma decay. In gamma decay
electromagnetic radiation of very high energy is emitted from a radioactive nucleus.
Light waves that are emitted in such a radioactive decay process are deemed gamma rays
to distinguish them from various other sources of electromagnetic radiation. Gamma rays
are simply very energetic photons, and thus they have neither excess charge nor rest
mass; however, due to their high energy they do eventually interact, and some materials
are moderately effective at shielding gamma radiation.
The intensity of all three forms of radiation falls off exponentially as the radiation
penetrate a medium; however, the rate at which this absorption occurs depends upon the
specific form of radiation and upon the medium through which the radiation is traveling.
Accurately determining the rates of this absorption for various materials is important in
determining how thick radioactive shields must be in order to reduce the intensity of the
incoming radiation by a certain specified percentage. Since the intensity of the radiation
is typically measured in terms of counts registered on a radiation detector, the relevant
relationship is
N  N 0 e  d ,
(1)
where N0 is the observed count when shielding is not present, and N is the observed
count given a shielding substance of depth d and absorption coefficient μ. The reason
that μ is called the absorption coefficient is because in a very mathematically exact sense
it gives the ratio of the rate of decrease of radiation intensity as the depth d increases to
the current radiation intensity, as shown in Equation (2).

N
   N 0 e   d  N
d


(2)
II. Experiment
In this experiment the beta particle absorption coefficient of aluminum and the
gamma ray absorption coefficient of lead will be measured experimentally. These
materials and types of radiation were chosen in order to make observations of slowly
varying intensities feasible. The gamma ray absorption coefficient for aluminum, for
example, is far too low to measure easily experimentally, and the beta particle absorption
coefficient of lead is far too high to measure easily experimentally.
In order to appropriately and accurately zero the counts that were to be measured,
a total of 30 minutes was allotted initially to determining an accurate background
radiation count. This count data was then used to obtain an average background count for
a 5 minute interval of observation.
After the background count was determined, an accurate base count was
determined for the beta source by recording the count for a total of 20 minutes with no
shielding material between the beta source and the detector. Several plates of aluminum
were then found and the thickness of each plate was determined using a micrometer.
These plates were then placed between the beta source and the detector and counts were
observed and recorded for intervals of 5 minutes for each different thickness of
aluminum. In order to increase the number of available aluminum thicknesses, the
individual plates were often used in combination with one another.
The same exact process was carried out with the gamma ray source and the iron
shielding material.
III. Results and Data Analysis
The background radiation data, which was used for both the aluminum and the
lead data sets, is given in Table 1. The thicknesses of the various shielding materials
were determined to a precision of 5 micrometers for each individual plate. The precision
of the thickness of combinations of these individual plates is correspondingly less. This
is accounted for in Tables 2 and 3, which display all count data for various thicknesses of
shielding materials for Aluminum and Iron.
Background Radiation: For 5 minutes:
Duration: 30 Minutes
n
Unc.
Count:
748
124.67
4.56
Table 1: Background radiation data.
2
Aluminum:
d (mm)
0.000
0.490
0.610
0.790
1.040
1.100
1.250
1.280
1.400
1.555
1.860
2.005
2.045
2.230
2.460
3.155
Unc.
0.000
0.005
0.005
0.005
0.005
0.007
0.005
0.007
0.007
0.005
0.007
0.005
0.007
0.005
0.005
0.005
n
1728.75
664
582
443
312
305
211
235
218
179
151
147
125
163
148
141
Unc.
20.79
25.77
24.12
21.05
17.66
17.46
14.53
15.33
14.76
13.38
12.29
12.12
11.18
12.77
12.17
11.87
n'
1604.08
539.33
457.33
318.33
187.33
180.33
86.33
110.33
93.33
54.33
26.33
22.33
0.33
38.33
23.33
16.33
Unc.
21.28
25.77
24.12
21.05
17.66
17.46
14.53
15.33
14.76
13.38
12.29
12.12
11.18
12.77
12.17
11.87
ln n'
7.38
6.29
6.13
5.76
5.23
5.19
4.46
4.70
4.54
4.00
3.27
3.11
-1.10
3.65
3.15
2.79
Unc.
0.01
0.05
0.05
0.07
0.09
0.10
0.17
0.14
0.16
0.25
0.47
0.54
33.54
0.33
0.52
0.73
Table 2: Data for aluminum shielding of beta radiation.
Aluminum Beta Absorption
8
y = -1.843x + 6.9646
R2 = 0.5946
6
4
2
0
-2
0.0
0.5
1.0
1.5
2.0
2.5
3.0
d (mm)
3
Lead:
d (mm)
0
0.940
1.690
2.340
2.630
3.280
4.030
4.920
4.970
5.860
6.355
6.610
7.260
7.295
7.550
8.045
8.200
8.695
8.950
8.985
9.635
9.890
10.385
11.275
11.325
12.215
12.965
13.615
13.905
14.555
15.305
16.245
Unc.
0.000
0.005
0.005
0.005
0.007
0.007
0.007
0.005
0.009
0.007
0.005
0.007
0.007
0.007
0.009
0.007
0.009
0.007
0.009
0.009
0.009
0.010
0.009
0.007
0.010
0.009
0.009
0.009
0.010
0.010
0.010
0.011
n
1665
1460
1375
1294
1218
1176
1151
1175
1086
1059
1007
1028
1023
953
942
911
891
905
918
831
872
812
794
826
763
741
752
755
762
662
655
636
Unc.
20.40
38.21
37.08
35.97
34.90
34.29
33.93
34.28
32.95
32.54
31.73
32.06
31.98
30.87
30.69
30.18
29.85
30.08
30.30
28.83
29.53
28.50
28.18
28.74
27.62
27.22
27.42
27.48
27.60
25.73
25.59
25.22
n'
1540.33
1335.33
1250.33
1169.33
1093.33
1051.33
1026.33
1050.33
961.33
934.33
882.33
903.33
898.33
828.33
817.33
786.33
766.33
780.33
793.33
706.33
747.33
687.33
669.33
701.33
638.33
616.33
627.33
630.33
637.33
537.33
530.33
511.33
Unc.
20.91
38.48
37.36
36.26
35.20
34.59
34.23
34.58
33.27
32.86
32.06
32.38
32.31
31.21
31.03
30.53
30.20
30.43
30.64
29.19
29.88
28.86
28.54
29.10
28.00
27.60
27.80
27.85
27.98
26.13
26.00
25.63
ln n'
7.34
7.20
7.13
7.06
7.00
6.96
6.93
6.96
6.87
6.84
6.78
6.81
6.80
6.72
6.71
6.67
6.64
6.66
6.68
6.56
6.62
6.53
6.51
6.55
6.46
6.42
6.44
6.45
6.46
6.29
6.27
6.24
Unc.
0.01
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.05
0.05
0.05
Table 3: Data for lead shielding of gamma radiation.
Lead Gamma Absorption (Thick Set)
Using Equation (1), a linear relationship can be found between the thickness of
7.4 the shield (d) and the natural log of the adjusted count (ln n’). It is
7.2
y = -0.0618x + 7.2063
ln n  ln n0    d .
(3)
7.0
R2 = 0.9644
6.8
6.6 Thus by plotting these quantities and performing a least squares linear regression, the
6.4 coefficient of absorption for both data sets can be found as the negative of the slope of
6.2 the regression line. The lines thus obtained for the two data sets are shown in Figures 1
6.0 and 2.
0
5
10
15
d (mm)
4
Aluminum Beta Absorption
8
y = -1.843x + 6.9646
R2 = 0.5946
ln n'
6
4
2
0
-2
0.0
0.5
1.0
1.5
2.0
2.5
3.0
d (mm)
Figure 1: Aluminum beta absorption linear regression line.
ln n'
Lead Gamma Absorption
7.4
7.2
7.0
6.8
6.6
6.4
6.2
6.0
y = -0.0618x + 7.2063
R2 = 0.9644
0
5
10
15
d (mm)
Figure 2: Lead gamma absorption linear regression line.
Note that the linear regression for lead appears to be very reliable, whereas that
for the aluminum deviates from the initial linear trend as the effective count begins to
approach zero. This can be attributed to the great significance of small random
fluctuations away from the true mean on the value of ln n’ as the count approaches zero.
Since gamma rays are so penetrating, even the thickest lead shield used did not bring the
effective count close to zero; however, the last 4 aluminum data points have an effective
count quite close to zero because beta rays are more easily absorbed. The mathematical
relation ship used the find the uncertainty in ln n’ is given by Equation (4), and as
Equation (5) makes clear, as the effective count goes to zero, the uncertainty in ln n’
becomes infinite.
5
n
 ln n 

n
n 2  nbg 2
n  nbg
lim  ln n  lim
n0
nnb g
 n  
2

n  nbg
n  nbg
n  nbg
nbg

2

n  nbg
n  nbg

(4)
(5)
In order to better determine the beta particle absorption coefficient for aluminum, this
analysis suggests that the last four data points should be omitted from the linear
regression. This should yield a more reliable regression line, and indeed it does, as
shown in Figure 3. Based upon Figures 2 and 3 an estimation of the uncertainty in the
two coefficients of absorption was made. The final results, along with these estimated
uncertainties, are given in Table 4.
Aluminum Beta Absorption (Modified)
8
y = -2.1723x + 7.4268
R2 = 0.9916
ln n'
6
4
2
0
0.0
0.5
1.0
1.5
2.0
d (mm)
Figure 3: Modified aluminum beta absorption linear regression line.
μAl (mm-1)
2.17
μFe (mm-1)
0.062
Unc.
0.05
Unc.
0.01
Table 4: Final results for the beta particle absorption coefficient for aluminum and the
gamma ray absorption coefficient for lead, along with estimated uncertainties.
IV. Conclusion
The beta particle absorption coefficient of aluminum was found to be 2.17 ± .05
mm , and the gamma ray absorption coefficient of lead was found to be .062 ± .01 mm-1.
Thus, making reference to Equation (1), only 2.2mm of aluminum will absorb more than
99% of incoming beta radiation, whereas nearly 75mm of lead would be needed in order
-1
6
to absorb 99% of incoming gamma radiation. Given the fact that lead is a far denser
element than aluminum, and is thus more effective as a radiation shield in general, this
gives a rough indication of how much more penetrating gamma radiation is than beta
radiation. This presents problems for those attempting to shield locations from gamma
radiation, either for sensitive experiments or for the safety of humans, but by measuring
the absorption coefficients of various materials and comparing the results, these concerns
can be addressed quantitatively and the course of action that best combines efficacy and
practicality can be identified and implemented.
7