Radioactive Decay
Ryan Berg and Charlie Watson
University of the Fraser Valley, Abbotsford, BC V2S 7M8, Canada
(Dated: May 6, 2010)
Abstract
Radioactive decay is an ever-present process, responsible for an endless number of products
and processes, from Carbon-14 dating to Helium production. As such a prevalent and important
process, it has been, and continues to be, thoroughly investigated. A selection of experiments were
performed and analyzed, through which an understanding of radioactive decay and its measurement
was gained. First, the dead time of a Geiger-Müller tube was found to be τ = 21.1 µs ± 14.3 µs.
With this, a technique for correcting erroneous measurements, we examined the effects of the
backscattering of beta radiation, as well as the energy of beta particles emitted from a Strontium90 source.
I.
between the electrodes, but no current can
INTRODUCTION
flow, as the gas is nonconductive. When raRadioactive decay occurs in three diation enters the tube through the transflavours, each with its own type of name and parent window in one end, some gas atoms
particle. Alpha particles are Helium-4 nuclei are ionized by the radiation. The released
ejected from a nucleus. Beta particles are electrons, and the remaining ions, accelerhigh-energy electrons (or positrons) emitted ate towards the anode and cathode, respecwhen a neutron is converted into a proton (or tively. While travelling, the electrons collide
a proton into a neutron). Gamma rays are with other atoms, causing further ionization.
high-energy photons, emitted by a variety of The result is an avalanche, or cascade, of
nuclear processes, and are produced and par- charged particles. When this cascade reaches
ticipate in a variety of particle interactions. the electrode, it registers as a current, which
is counted.
One device commonly used to measure radioactive decay is the Geiger-Müller (GM)
Through use of a GM tube, different ef-
tube. A GM tube is an airtight cylinder, fects can be explored. One such effect is scatcontaining an easily ionized gas and two elec- tering. As the effects involve photons, electrodes. A potential difference is established trons and nuclei, one must consider the struc1
ture of matter on an atomic (or subatomic) is left in a state of ionization. Until the tube
scale. Consider a travelling electron (beta is returned to a state of neutrality, no more
particle) incident upon a substance (a piece decays can be detected. The process of reof metal, say). The surface is not continuous, turning to neutrality is called quenching, and
and is in fact a lattice of atoms. As the decay can be accomplished in several ways. Whatparticle is at most of a comparable size to the ever the method, it is not instantaneous, and
atoms, it may pass through the lattice with no counts can be deteced until the tube is
little or no disturbance. If the particle does quenched. This resolving time, or dead time,
collide with an atom, it may be deflected, a is then quite important to take into account
process called scattering. The frequency and for higher count rates.
distribution of scattering strongly depends on
The calculation of the dead time is ac-
the atomic structure of the surface, such as tually quite simple, and the equation inthe size and spacing of the atoms.
volved can be derived both formally, from the
In this lab, we began by measuring the re- Poisson distribution,1 or qualitatively. We
solving time of the GM tube. The results of present a qualitative argument. Consider
this investigation make up Procedure II. Af- an experiment with a GM tube with a (unter finding the resolving time, we used this known) dead time of τ . If r is the count rate
information in the later procedures. Next, in for a time t, then in a time t, the detector will
Procedure III, we investigated the effects of be “dead” for a time rtτ , and the useful meascattering of beta particles. Lastly, in Proce- surement time will be only t−rtτ = t(1−rτ ),
dure IV, we investigated the energy of emit- where rτ is dimensionless. To get the true
ted beta particles from a Strontium-90 beta count, then we must then scale the measured
source. Lastly, in Section V, we draw con- count, so
clusions from the set of experiments we perR=
formed.
II.
r
.
1 − rτ
(1)
Now, we can find τ . Consider a source
PROCEDURE: DEAD TIME
that can be split into two pieces. Then, in
A.
addition to the total count rate, the rate of
Theory
each piece could be measured. We call the
A GM tube, as described above, is use- total rate rT , and the piece rates r1 and r2 .
ful for counting a single decay, but the tube If we assume that the background radiation
2
rate, b, is the same in each case, we have
(r1 + b) + (r2 + b) = rT + b,
We then need only measure the total and
(2)
piece count rates to find the dead time of the
GM tube.
or
r1 + r2 = rT − b.
(3)
It’s worth noting that at very small count
rates, the dead time does not effect the over-
These are observed rates, not true rates,
however, and the relationship is not correct.
We can convert these to true rates with our
equation for correcting counts (1) to get the
true relationship,
all counts whatsoever. For example, since the
dead time is expected to be in the microsecond range, count rates that make the product
rτ much smaller than 1 will not cause a noticeable correction to the true count rate.
r1
r2
rT
b
+
=
+
. (4)
1 − r1 τ 1 − r2 τ
1 − rT τ 1 − bτ
Solving this for τ , we have a cubic equation, due to the b term. If the background
rate is small compared the total rate, then
B.
Apparatus
rT − b ≈ rT , and the b term vanishes. We are
left with a quadratic equation,
r1 r2 rT τ 2 − 2r1 r2 τ + r1 + r2 − rt = 0.
We used “The Nucleus” Model 500 Nu(5)
clear Scaler, with an attached GM tube.
We used two Sr-90 samples as beta-decay
This can be further simplified by noting sources.
that τ is on the order of microseconds, and
so τ 2 is negligibly small. This only holds for
small count rates; as our count rate increases,
τ is only negligibly small when rτ is much less
than 1, so that our correction factor of
1
1−rτ
C.
Data Collection
doesn’t get unreasonably large. With this
in mind, we must not register a very large
amount of counts, else we need to use the
quadratic formula, and the following equation does not hold. We then have a linear
data, we needed to verify the validity our
theoretical assumption, b rT . That is,
we needed to make measure the counts when
there was no source actually present. We did
equation, and
r1 + r2 − rT
τ=
.
2r1 r2
Before we could collect the necessary
this a number of times, with the results tab(6)
3
ulated in Table I.
TABLE I: Background Count
TABLE II: Count for Both Sources
Trial Number Count in 30s
Trial Number Count in 30s
1
3
1
1208
2
4
2
1273
3
6
3
1237
4
8
4
1183
5
6
5
1183
TABLE III: Count for Left Source
Trial Number Count in 30s
From these data, we calculated the av-
1
747
2
772
3
807
In the next step, we deviated from the sug-
4
754
gested apparatus.2 It was suggested that we
5
782
erage count rate for background radiation,
b = 11 ± 3 counts/min.
use two semidisc-shaped sources. Then half
could be removed and replaced with dummy
material. This would preserve the geometry
of the source, as well as the shielding from
TABLE IV: Count for Right Source
Trial Number Count in 30s
unwanted sources. We did not have such a
source available, and so we used two disc-
1
503
shaped sources, with the point of contact di-
2
468
rectly beneath the detector.
This allowed
3
474
the preservation of the geometry, but not the
4
459
shielding.
5
473
The procedure was largely identical, with
an individual source being removed and re-
From the data in Tables II-IV, we obtained
placed as needed. The results are collected average count rates for each of the individual
in Tables II-IV.
cases.
4
Originally, we thought that the geome-
TABLE V: Configuration Averages
Configuration
Average Counts
Right
951 ± 30 counts/min
Left
1545 ± 43 counts/min
Both
2434 ± 69 counts/min
try might be an issue. The two lab manuals that we checked with similar experiments
used various semi-discs to maintain the geometry of a disc under the GM tube.23 In
our case, using two discs while maintaining
the geometry of the situation gave us a dead
From these average counts we used Equation (6) to determine τ to be
τ = 21 µs ± 14 µs.
time in the range we were expecting, albeit
with a large error. This suggests that the er-
(7)
ror is unlikely to stem from the lack of shielding.
D.
Discussion
III.
The dead time of the GM tube was
PROCEDURE:
BACKSCATTER-
ING
found to lie in the 1µs to 100µs range, as
was expected.23 However, the error in this
A.
Theory
value was quite large. This may have resulted
from the large standard deviations in the de-
Radiation from a typical decay source
tection count measurements. Another factor is emitted isotropically, but the window of a
that, upon further investigation, was a big GM tube subtends only a small solid angle.
contributor, was what is used in Equation 6 Thus, most radiation is not emitted towards
to determine the dead time. The numerator the GM tube, and one would not expect to
is quite close to zero; the sum r1 + r2 is com- detect it. However, when emitted radiation
paratively quite close to the vale for rT , and collides with matter, it can be deflected, or
thus is very sensitive to small differences in scattered. When passing through any sigthese values. This effect propagated through nificant amount of matter, a decay particle
the error, and was likely the greatest source is scattered many times, and follows a semiof error. The counts were the only variables, random walk.
and the individual errors were quite large and
Thus, a particle’s path through matter
Equation 6 was very sensitive to the count need not be even remotely linear. Indeed, it
rates, producing a great deal of uncertainty can reverse its direction, and leave the sub(δτ was 68% of τ ).
stance through the same surface it entered, in
5
a form of reflection. This is called backscat- thickness is not easily predicted,45 and we
tering, and is shown in FIG. 1. In this case, shall explore it empirically.
the re-emitted particle may enter the GM
tube, increasing the count.
B.
Apparatus
For this procedure, we used the same “The
Nucleus” Model 500 Nuclear Scaler, with the
attached GM tube, as well as the same Sr-90
beta emitter. We used various thicknesses of
absorbers, made of aluminium, lead, copper,
and polyethylene to investigate the effect of
thickness and atomic number.
FIG. 1: Backscattering of a beta particle.
C.
Data Collection
The amount of backscattering depends
greatly on the atomic structure of the solid.
To begin, we measured the backscatter-
Several factors contribute, such as atomic ing effects of different substances. To do this,
number, density and thickness. The atomic we took several sheets of different absorbers;
number will affect the spatial distribution of those used were lead, copper, aluminium, and
charge within the atom, and so affect how the polyethylene. We placed these absorbers diparticle (in our case, a beta particle) inter- rectly beneath the beta source, being careful
acts with the atom. The density will affect to keep the source always a constant distance
the number of collisions, as will the linear away from the detector, in order to minimize
thickness. Generally, these two parameters count rate fluctuation from 1/r2 falloff. Usare combined into the so-called “areal” thick- ing a linear thickness of approximately 0.16
ness. Areal thickness is the product of the cm for each of the absorbers, we measured
density and linear thickness, and measures the count rate. The average count rate for
the quantity of matter the particle must pass each substance is recorded in Table I.
through. We will refer to the areal thickness
simply as the thickness.
We then measured the effect of varying thickness by using polyethylene sheets
The relationship between the amount of of varying sizes as absorbers.
We placed
backscattering and the atomic number and the source directly under the GM tube, two
6
TABLE VI: Counts for Constant Linear Thick-
glean any significant insight into the effect of
backscattering. The error in the count rate
ness
Element
Counts Recorded per Minute
No absorber
2346 ± 43
Aluminium
2357 ± 46
Lead
2421 ± 66
Copper
2381 ± 53
Polyethylene
2325 ± 44
was simply too high to be able to yield any
strong results. However, we invested a lot of
time into what sort of effects may have contributed to our lack of conclusive data.
First, the inverse square effect with respect to distance from the GM tube was
a large obstacle to overcome.
shelves down from the top. The position of
the source was marked, so as to ensure the
distance between the source and the detector did not change.
The count varies by
an inverse-square law, and so at this distance scale (1-3 cm), small changes in distance cause a significant change in count. We
Very small
changes in the distance from the detector had
drastic effects on the detection rate. Moving
the source from the second shelf to the third,
a distance of about one half centimetre, cut
the counts in half. Given this, even a change
of one millimetre had a significant effect.
Next, we considered whether or not the
radiation was preferentially emitted from the
summarize the data in Table VII.
top or bottom of the disc in which it is stored.
TABLE VII: Counts for Polyethylene Absorber
By flipping the source, we found that the
Thickness Counts/min Corrected
count dropped by several hundred (by as
(mm)
counts
much as 50%) . This could be caused by
1
2341 ± 43
2463
an uneven coating of plastic on the bottom,
2
2312 ± 42
2431
which provided preferential shielding, or by
3
2304 ± 44
2422
noncentral positioning of the source in the
4
2420 ± 46
2550
disc. As mentioned above, any change in the
5
2302 ± 40
2420
distance had a significant effect. This would
not account for a lack of backscatter, however, as it would not decrease the number of
D.
emissions through the bottom.
Discussion
To see if it mattered which side of the
This experiment was plagued with puck was placed upwards in the backscattercomplications.
From our data, we cannot ing effect, we did a few trial runs with the
7
source upside down with a few of our types ment is certainly the length of our measureof absorbers. However, we saw the same non- ments. Increasing the length would decrease
variation from absorber to absorber. Upon the relative error due to the standard devitesting a few extreme cases, we decided that ation, with no increase in complexity of the
although the radiation detected is in fact dif- experiment. Overall, the uncertainty in the
ferent depending on which side of the puck count is a fairly pronounced effect, and within
is facing upwards, this is not a backscatter- this uncertaintly, we couldn’t generate any
ing effect. If it were, the different absorbers conclusive data about the effect of backscatwould have effected the detection of radia- tered radiation.
tion, but within our error, we saw no noticeable effect.
IV.
PROCEDURE: BETA PARTICLE
As another method to attempt to pick up
ENERGY
a backscattering effect, we cut small slabs of
lead of roughly the same area of the source
A.
Theory
disc, and placed enough of them on top of
the source to completely block any emissions.
When beta particles are emitted from
This ensured that any non-scattered emis- a radioactive source, they can have an entire
sions did not enter the GM tube, and we only spectrum of energy. This is because the beta
counted the backscattered emissions. How- particle itself is not the only product in the
ever, the recessed position of the detector, decay that produces it; a neutron decays into
and the width of the lead slabs made it un- a proton, as well as a neutrino and a beta
likely (geometrically speaking) to expect any particle. The distribution of energy can be
counts from scattering. This was reflected in over all of the decay products, and thus the
our results, which showed nothing above the energy of the electron (beta particle) can take
expected background count.
on a large range of values.
In the end, if the backscattered radiation
As the spread of energy in this decay pro-
accounts for much of the overall detected cess is randomly distributed, the beta particount, then either our geometry did not en- cles emitted from a source will all have differable detection, or our measurement intervals ent energies. However, there is an upper limit
were too short. There are many configura- on the amount of energy that emitted beta
tions of source and absorber that we did not particles can have, and this simply occurs
investigate, but the most significant improve- when the beta particle ends up with most
8
of the energy in the decay process. Thus, ness required to drop the GM tube’s counts
for beta particles emitted from a radioactive to zero will enable us to find an upper limit
source, there is a maximum amount of energy on the energy of beta particles emitted by a
that the particles can actually carry.
Strontium-90 source.
When beta particles are allowed to pass
through an absorber, they lose some of their
B.
Apparatus
energy. If the absorber is thick enough, the
particles will lose so much energy that they
As with the previous procedures, we used
won’t be able to pass all of the way through. ”The Nucleus” Model 500 Nuclear Scaler,
For greater thicknesses, fewer particles will with the attached GM tube, as well as the
have the energy to pass all of the way through same Sr-90 beta emitter. We used various
the absorber. Thus, since beta particles emit- thicknesses made of polyethylene to deterted from a radioactive source have an up- mine the thickness required to entirely shield
per limit on the energy they can have, there the beta source.
will be a thickness absorber that will be sufficient to stop essentially all beta particles
C.
Data Collection
from passing through.
In this section of the lab, we investigated
To measure the upper limit on the
the upper limit on the energy of beta particles energy emitted from the radiation source
emitted from a Strontium-90 beta source. To (in our case, Strontium-90), we passed
do this, we needed to determine the thick- beta particles through various thicknesses
ness of absorber was required in order to of polyethylene, in order to determine the
halt essentially all beta particles from pass- ’range’ of the beta particles emitted. This
ing through. In the lab manual we followed,2 is related to the energy of the emitted beta
the equation for the maximum energy of an particles. Particles with higher energy make
emitted beta particle E (measured in MeV) it through more layers of shielding, so by dewas given by
termining the number of particles that are
counted through different layers of shielding,
E = 1.84 T + 0.212,
(8) we are able to obtain the upper limit of the
where T is the maximum thickness (in g/cm2 ) energy of the beta particles emitted from our
that the emitted beta particles can travel source.
through.
So, finding the maximum thick9
To this end, we used shielding with vary-
ing ’thickness’, as described in Section IIIA.
Because of this, the material used as shield-
TABLE VIII: Count Variation as a function of
Thickness For Beta Particle Absorption
ing isn’t particularly relevant; a slab of lead
Thickness Counts Recorded Corrected
that is thin can be as comparatively ’thick’
(mg / cm2 )
per Minute
Counts
0
3957
4315
10
3670
3976
20
3542
3826
sheets of varying thickness, and the thickest
43
2961
3157
polyethylene sheet was sufficient to reduce
73
2826
3004
the detection count to nearly zero.
151
2149
2250
We measured the detected counts for var-
224
1559
1612
ious thickness of polyethylene sheets. The
305
1086
1111
thickness of the individual sheets is listed in-
378
756
768
side the container where they were stored.
456
479
484
The results are collected in Table VIII. We
529
271
272
needed, then, to correct for the dead time of
610
124
124
as a slab of polyethylene that is compensatingly wider. However, our lead shielding was
actually too strong to get any counts from,
and we only used a collection of polyethylene
the Geiger counter in the number of counts.
To do this, we can use the equation
C = 4175.82 e−0.0045 T ,
(10)
r
,
(9) where C is the number of counts as deter1 − rτ
where R is the true count of the Geiger mined by the GM tube, and T is the thickR=
counter, r is the detected count as measured ness of the polyethylene sheets.
by the Geiger counter, and τ is the dead time
We then solved Equation 9 for the thick-
(as found in Procedure I) of the counter. The ness T for when the number of counts was
corrected count is then tabulated in Table less than one. This yielded a thickness of
T = 1853 mg/cm2 , which is what was re-
VIII as well.
From the data in Table VIII, we con- quired to use Equation 8 to determine the
structed a plot of the Counts per minute ver- maximum energy of the emitted beta partisus thickness of the shielding. This plot is cle.
shown in FIG. 2; the best fit line for the data
we used was determined to be
However, this is where we ran into a problem. Using the value for the cutoff thickness
10
tions it necessarily holds. Also, after some
investigation, we were unable to find a different equation to use that would relate cutoff
thickness to maximum beta particle energy,
and thus have been stopped in our tracks.
Given more time, or a more thourough search
for a correct equation relating thickness and
energy, the previous data may be used to find
the maximum energy of an emitted beta particle from an Sr-90 source.
FIG. 2: Counts as a function of Thickness.
As another source of potential error is
found previously, Equation 8 tells us that what was used as the cutoff thickness, at
the maximum energy of a beta particle emit- which no beta particles made it through. The
ted from a Sr-90 source is E = 3.62 MeV; model that was generated was used to prethis is wildly off from the accepted value for dict when only on average one particle would
the energy, given as E = 0.546 MeV.2 This be detected; this is at a thickness far above
raises some serious questions about what that which we actually used. This then may
actually be somewhat unrealistic to expect,
we’ve done here.
and may provide some extra source of error
D.
in what we find to our energy.
Discussion
The value we obtained for the maximum V.
energy of beta particles emitted from Sr-90
was E = 3.62 MeV, whereas the accepted
CONCLUSION
We found the dead time of our GM
value is E = 0.546 MeV.2 What went wrong tube to be τ = 2.1 × 10−5 s ± 1.4 × 10−5 s.
is something to definately investigate. To be- This does fall within the expected range for
gin, the equation that we used was simply a GM tube, but with a very large error. We
stated as given in the lab manual’s procedure did not discover any backscattering effect, dethat we followed,2 and there was no explaina- spite several different styles of approach.
tion as to where it came from. This poses
The primary lesson of these experiments
a problem; from Equation 8, we don’t know has been the power of the error in the counts.
how this was derived, or under what condi- In any future attempts, we would need to find
11
a way to reduce the size of the error in the fixed value over a long enough period. As
counts. One way to accomplish this would the half-life of Sr-90 is over 28 years, the debe to increase the measurement interval. We cay rate wouldn’t change appreciably over a
would expect the count rate to approach a matter of minutes or hours.
1
2
R. Bridges, Physics Education 25 (1), 60 1990.
Introduction to Modern Physics, 6th Ed.
doi: 10.1088/0031-9120/25/1/010.
(McGraw-Hill, New York, NY, 1969), pp. 651-
USD - Spectrum Techniques, Student Lab
653.
5
Manual, July 2002.
Melissinos, Experiments in Modern Physics,
3
Ortec - AN34 Experiment 2 - Geiger Counting.
(Academic Press, New York, NY, 1966), pp.
4
Richtmeyer, Kennard and Cooper, Interna-
152-165.
tional Series in Pure and Applied Physics:
12