The properties of radioactive decay and the
resulting particles are verified here by observing the
decay of 212Pb to 208Pb and also by using 241Am as an
alpha particle source.
In radioactive decay, an atom goes through a
nuclear reaction which expels an energetic particle from
the atom. The nuclear reaction will occur when the total
energy of the atom can be reduced, for example by
reducing the total mass or changing to a more stable spin
or parity state, but there is a potential barrier to
overcome before the reaction can take place. It is this
potential barrier which determines the reaction rate.
Since the atom can only decay once, the various types of
decay are exclusive to each other and compete if the
nucleus can decay in more than one way. [1]
The three main types of radioactive decay are
alpha beta and gamma decay. Gamma decay is simply
photon emission which happens to occur along with
other nuclear reactions.
Alpha Decay
In alpha decay, a positively charged helium
nucleus is produced.
XN
p  n  e    e (3)
p  e   n   e (4)
Radioactive Decay
A
Z
occurs is the one of these that is energetically favorable.
These reactions can be written as:
n  p  e    e (2)
A 2
Z 2
X
'
N 2
 He 2 (1)
4
2
[2]
In this case the potential barrier is related to the
relative sizes of the coulombic and nuclear forces around
the nucleus, and can be modeled classically with good
results. Most of the energy produced in the decay goes to
the light alpha particle. Quantum mechanics predicts that
the higher the energy released by the decay, the higher
the chance that the system can overcome the potential
barrier to the reaction. Therefore if more than one alpha
decay reaction can take place in a nucleus, a high energy
particle is more likely to be produced than a low energy
one. This means that qualitatively, the branching ratio
between two alpha decays will tend to favor the high
energy alpha particle. In general a factor of 2 in energy
means a factor of 10 24 in half life. [1]
Here there is less relation between the energy
released in the reaction and the final energy of the electron
(the main observable product) since variable amounts of
the energy goes into the neutrino. A rough estimate of
decay rates cannot be obtained classically as can be done
for alpha decay. [1]
Experimental Determination of Decay Rates
While the analytic calculation of the decay rates is beyond
the scope of this report, the decay rates can be measured
experimentally by using the general properties of
radioactive decay.
By the basic assumptions of decay, we have the
exponential law of radioactive decay for the number of
nuclei present at time t. This assumes that the decay is the
only process occurring, and no new nuclei are being
introduced into the system.
N(t) = N 0 e  λt (5)
[1]
where λ is the disintegration decay constant and N 0 is
the initial number of nuclei. The half life is the time for
half of the nuclei to decay, and is found by substituting
N  N 0 2 in equation (6).
t1 2 
ln( 2)

(7)
In an experiment, we would be measuring the
number of decays per unit time at time t rather than the
number of nuclei at time t. We can define the activity A as
this new value. It can be found by differentiating equation
(5) to get
(t )  N 0 e  λt  N (t ) (8)
Beta Decay
In Beta decay, a proton is converted to a neutron
or a neutron to a proton, and a neutrino is ejected along
with either an electron or a positron. The reaction that
If a nucleus decays to another unstable nucleus,
more than one decay process may be going on at the same
time. In this case, the decay rate of the daughter nucleus
depends not only on how fast it is decaying, but also how
fast it is being created by the parent nucleus, and so
equation (8) does not apply. This case can be written as
the differential
dN 2  1 N1dt  2 dt (9)
where N2 is the number of daughter nuclei. The number
of parent nuclei N1 is the initial nucleus in the reaction,
and follows equation (5). We try to solve this with an
equation of the form N 2 (t )  Ae  t  Be  t with the
initial conditions N1(0) = N10 and N2(0) = N20. Solving
for A and B, and multiplying by λ2 to get the activity, we
get a final equation of
1
 2 (t )  N10
2
12  t  t

e  e   2 N 20e  t (10)
2  1
1
2
2
This assumes that the only measured activity comes
from the daughter nucleus.
Decay of 212Pb
5.607 MeV
5.769 MeV
6.05 MeV
6.09 MeV
rather than only alpha decays, since then the branching
ratio might be predicted. 212Bi does have multiple alpha
decays, but they are too close together in energy for
general trends between energy and decay rate to be
apparent.
Apparatus and Calibration
A surface barrier detector is used to detect alpha
particles and their energies. This can be fit inside of a
vacuum chamber, where a sample may also be placed. The
detector signal goes through a pre-amplifier and an
amplifier, and the resulting signal is recorded to a
computer file with a multichannel analyzer unit (MCA).
The detector measures the alpha particle energy
and generates a voltage signal based on it, and the output
from the detector is linear with the alpha particle energy.
When the MCA records the signal from the detector, it
divides the signal into equally sized channels, with
different channels corresponding to different signal
magnitudes. In order to find the correspondence between
channel number and alpha particle energy, the channel
activated by different alpha energies is measured.
A sample of 241Am, which produces alpha particles
of known energy 5.48 MeV, is used. First, the channel
activated by the 241Am is measured. The energy
corresponding to the remaining channels is determined
using fact that the signal is linear with alpha particle
energy. A voltage generator is calibrated so that a
generated signal of magnitude 5.48 in its scale would fall
into the same channel as the output from the detector for
the 241Am source. The voltage generator reading is then the
same as the output of the detector at the corresponding
energy. By setting the voltage generator to different signal
8.78 MeV
8.78 Mev
Figure 1: Decay of
212
Pb [3]
The decay of 212Pb to 208Pb is the first subject of
study in this report. The decay paths are shown in figure
1. 212Pb first decays to 212Bi by beta decay, which then
decays along two separate paths. Along the first path it
decays by beta decay to 212Po which then decays by
alpha decay to 208Pb. Along the second path 212Bi decays
by alpha decay of various energies to 208Tl, which then
emits a photon if it is not at the ground state. The 208Tl
then decays by beta decay to 208Pb.
It is unfortunate that the branching of the
nucleus we use, 212Bi, is between alpha and beta decays
Figure 2 Fit of the calibration data
magnitudes, the channel activated by that can be
measured.
The channel activated at a number of different
signal magnitudes was measured, and then a linear fit
was done to find the exact correspondence. This was
found to be (5.485 ± 0.006) × 10-3 MeV per channel plus
a constant offset of -0.192 ± 0.006 MeV. This error in
energy measurement is negligible compared to the
spread of our measurements.
Preparation of sample and Measurement
A sample of 212Pb is prepared immediately
before measurement since it decays quickly. It is
obtained from the decay of Thorium 232 and its
products. Thorium salts are placed at the bottom of a
glass container. The thorium will eventually decay to
Radon, which is a gas. The radon will diffuse out of the
salts to form a cushion of gas at the bottom of the
container, and some of it will be ionized by the alpha
particles produced in the various radioactive decay
processes going on in the container. The ionized radon
can then be deposited onto a metal collector plate by
applying a 1000 Volt voltage difference across the inside
of the container, where the collector disk is at a negative
voltage. The ionized radon will be attracted to the
collector plate and will then decay to 212Po and then
208
Pb, which stay attached to the plate. The 212Pb was
collected for one hour.
The collector plate with deposited 212Pb is
placed in a vacuum chamber with the particle detector
Figure 3: Decay spectrum for 212Pb showing the beta particle
peak at the beginning, the alpha peaks at 6.05 MeV, then the
last alpha peak at 8.75 MeV
and depressurized to 200 mTorr. The alpha particle counts
over every 5 minute interval are recorded, over a total of
24 hours. This gives 288 alpha energy spectrums. 5 minute
intervals are chosen because this is short enough that the
rate of alpha decay is constant over that time, and long
enough that a statistically relevant amount of counts are
measured.
The total counts from this measurement are shown
in figure 3. The two main peaks from the decay of 212Pb
are seen at 6.05 and 8.78 MeV. The two peaks at 5.59 and
5.75 MeV are too small to be seen. The two main peaks
have a trail of noise of undetermined origin to the left of
them, which appear to form peaks. The large peak near
channel 0 is due to the beta particles. It is expected that
there should be twice as much beta decay as alpha decay
for 212Pb. The peak due to beta decay is indeed bigger than
the sum of the other peaks, but is not twice as big.
Presumably, not all the beta particles were detected, since
some will have energies too low to be detected.
Branching Ratio and Peak Energy
The branching ratio is determined from the relative sizes
of the alpha particle energy peaks produced in the decay of
212
Pb to 208Pb. The size of the peak is measured by
integrating the number of counts in the peak. The location
of the peak is found by fitting to a Gaussian. The results
are shown in tables 1 to 3.
The two smaller peaks at 5.59 and 5.75 MeV were
fit together with a double Gaussian plus a linear term. The
linear term is included because the two small peaks are on
the trailing edge of the much larger peak at 6.05 MeV.
Figure 4: The small energy peaks near 5.607 MeV and 5.769
MeV, fit with a double Gaussian and linear term.
Particle
α
α0 + α40*
α328
α493
Energy (MeV)
8.76 ± 0.05
6.06 ± 0.05
5.77 ± 0.04
5.60 ± 0.04
Accepted Energy (MeV)
8.78
6.06*
5.769
5.607
Table 1: Experimental and accepted values for alpha energy
from decay of 212Pb. The α0 and α40 peaks are combined
since they could not be distinguished. The expected value
in this case is the weighted average of the respective
expected peaks.
Decay path
Bi → 208Tl → 208Pb
Bi → 208Po → 208Pb
212
212
Figure 5: Gaussian fit for the 6.05 and 6.09 MeV energy
peaks. The spread was too large to be able to distinguish the
two.
Branching
ratio (%)
36.8 ± 0.3
63.2 ± 0.3
Accepted
ratio (%)
36%
64%
Table 2: Branching ratio between 212Po and 208Tl pathways of
decay of 212Bi
Particle
α0 + α40*
α328
α493
Branching ratio (%)
97.2 ± 0.2
1.8 ± 0.1
1.0 ± 0.1
Accepted ratio (%)
97.1
1.67
1.08
Table 3: Branching ratios for the various alpha decays from
212
Bi to 208Tl
Rate of Decay
Figure 6: Gaussian fit for the 8.78 MeV energy peak.
The energy peaks at 8.78 MeV and 6.05 MeV were fit to
a simple Gaussian. Some points on the left side of the
peaks were omitted during fitting since the distribution is
slightly skewed due to small amounts of scattering with
the air. Note that there are is in fact also a peak at 6.09
MeV, however, the spread in the experiment was too
great to be able to distinguish it from the 6.05 MeV
peak. The energy of this peak should in fact be measured
to be the weighted average of the two true energy peaks,
and it is.
The rate of decay is determined by measuring the
total alpha particle activity over time of the sample. Since
alpha particles are only produced by the decay of 212Bi and
one of its daughter nuclei, the alpha particle activity
corresponds to the amount of bismuth in the sample.
As discussed in the theory section, equation 10
predicts the activity of the daughter nucleus, which in this
case is 212Bi. The data measuring alpha counts over time
can be fit to this equation to find the parameters λ1, λ2 and
the initial amounts of nuclei. Note that N2(0) would ideally
be 0, but some radioactive decay takes place during the
collection period and also in the short interval before the
measurements begin, so there will be some 212Bi at t=0.
The equation was fit to the counts in the 6.05 and
8.78 MeV peaks in each five minute interval over the total
time elapsed so far, as shown in figure 7. It is interesting
that if we limit the counted region to only the peak near 0
MeV, which represents the beta particle counts, we get a
value closer to the accepted values for λ1 andλ2.
 v2 
2me v 2
dE 4Z i2 e 4 n  

   (11)

ln
dx
me v 2   I 2 (1  v 2 c 2 )  c 2 
where Zi is the material’s atomic number, n is the electrons
per unit volume of material, me is the electron mass and I
is the mean excitation potential of the target.
Experimentally, the energy loss can be measured
as a function of material thickness. If the path length to the
detector is kept fixed, the material thickness can be
changed by changing the density of the material, which is
simple to do for gases. For an ideal gas
MN
Mx
x
P (12)
V
RT
(using the ideal gas law PV  NRT )
t ( x)   air x 
Figure 7: Total activity over time of the 212Pb sample.
Decay constant (s-1)
212
Pb
212
Bi
(1.92 ± 0.06) x 10-5
(1.8 ± 0.2) x 10-4
Calculated
half-life
10.0 ± 0.3
hours
65 ± 7
minutes
Accepted
half-life
10.64
hours
60.60
minutes
where x is the distance into the material, M is the
molecular mass of air, and the other variables are as in the
ideal gas law. By keeping the distance to the detector
fixed, the angular spread of alpha particles hitting the
detector and the average path lengths are kept constant in
different measurements.
Energy Loss due to Scattering
To perform the experiment, an 241Am alpha source
is placed in a vacuum chamber with the detector. For
various pressures, the alpha energy spectrum is recorded.
The residual energy of the alpha particles can then be
measured as a function of target thickness. The result is
shown in figure 8.
As alpha particles travel through material, they
lose energy by ionizing the particles of the material. The
thicker the material, the more energy loss occurs. Above
a certain material thickness, the alpha particles will lose
all of their energy. Additionally, high energy alpha
particles lose less energy than low energy ones when
traveling through material. Intuitively this is because
they spend less time traversing a unit length of distance
in the material, so have less of a chance of interacting
with the material there. Practically, this means that as the
alpha particle slows down, it delivers more and more
energy to the material. This means that the highest
density of energy delivered is right before the alpha
particle comes to a complete stop, when it is traveling
slowly.
The energy loss per unit length is given by the
Blethe-Bloch formula, which shows the dependence on
velocity.
Figure 8: Residual energy of alpha particles after traveling
through a material of some thickness.
Table 4: Accepted and experimental values for the half-lives of
212
Pb and 212Bi.
The accepted value for the half life of 212Bi was
within one standard deviation of our result, but the result
for 212Pb was about two standard deviations away from
accepted. This is expected since it is derived indirectly
from the data, while the data applies directly to the 212Bi
decay rate.
Overall, this experiment was a success. Most of
the values we obtained where close to the accepted values.
The values for which we had the largest error were those
involving small quantities, such as the difference between
the energies of α0 and α40 or the counts in the peaks for α328
and α493 and would all have been improved by doing the
measurement of the lead spectrum more than once.
References
Figure 8: Degree of energy loss per unit length as alpha
particles penetrate the material. The line is only to guide the
eye.
Taking the derivative of this data with respect to
distance, we get the energy loss per unit distance as a
function of depth, shown in figure 9. As the particle
proceeds deeper into the material, it gets slower as it
loses energy to the material. As predicted by equation
(11), the slower particles deliver more energy per unit
length. As can be seen in figure 9, as the particle enters
the material, there is little energy loss, and as it gets
slower it loses more and more energy at a time. The
sudden drop near the end of the plot occurs when the
alpha particle has lost all its energy and come to a
complete stop, and therefore cannot deliver any more
energy to the material.
Conclusion
The accepted values for the alpha particle energies in the
decay of 212Pb were within a standard deviation of the
experimental values found here. There was too much
noise in the data to be able to distinguish the 6.05 and
6.09 MeV alpha particles however. Other than this, the
values found were quite reasonable.
The accepted values for the 212Po and 208Tl
branching ratio were not quite within the statistical error
bounds of our results, but where still close to the actual
values. The accepted values were also specified to fewer
significant digits than our result.
The accepted branch ratio between the different
alpha decays of 212Bi were within the standard deviation
of our results except for α328, where it was 1.3 standard
deviations away.
[1] Krane, Introductory nuclear physics. Wiley, New
York, 1988
[2] Alpha Decay Lab Manual
[3] McGill Physics: Alpha decay Wiki
http://www.ugrad.physics.mcgill.ca/
wiki/index.php/Alpha_Decay