Half Life of Cesium-137
The occurrence of radioactive decay follows
a normal gaussian distribution. That means the mean and standard deviations
of the decay progress can be calculated in a straight forward manner using
standard techniques. Because the decay is normal (random) we can express
the probability that a given number of atoms ( dN ) disintegrating
in a time interval (dt) as:
|
dN / dt
= -  N |
(1) |
where N is the number of atoms present
in the radioactive sample,
is the decay constant and dN / dt is the rate of decay or the number
of disintegrations in a time interval. This quantity is also called the
Activity
of the sample. Radioactive decay is sort of like popcorn popping. You don't
know which kernel will pop, but the rate (once the system starts) is fairly
simple to describe. First a few pop, then more, and avalanche of pops and
then tapering off. From our perspective, radioactive decay rate is uniform
throughout the decay and the amount disintegrating depends on the total
amount in the sample. (So there is a difference.)
Integrating (1) gives us the standard exponential
formula for radioactive decay:
|
N(t) = No
e- t |
(2) |
where N(t) is the number of radioactive
atoms left at time t in the sample and No is the number
present at time t = 0. Since the activity is proportional to the number
of radioactive atoms present, we can write:
|
A(t) = Ao
e- t |
(3) |
We define half-life, t1/2
, as the time required for the number of radioactive atoms in a sample
to decay by 1/2. In terms of activity this is
|
A(t1/2) = Ao
/ 2 = Ao e- t |
(4) |
We can solve for t1/2 by taking
the natural logarithm of both sides of the equation (you fill in details)
to show that:
|
t1/2
= ln(2) / = .693 / |
(5) |
Cesium-137 has a half life of 30 years
on the shelf. It is constantly decaying so that every 30 years half of
the amount present at the beginning of the 30 year period is gone. Cs decays
to Barium-137. Internally, a neutron in the nucleus changes into a proton.
In conserving charge, an electron is emitted (beta particle) A high energy
gamma ray carries off some energy in the process. Now, 30 years in a long
time over which to make these measurements. We can, for practical purposes,
chemically elute some of the material that has decayed and concentrate
it.In the process Cs-137 decays to a metastable Ba-137 which to become
stable, emits a gamma ray. It is this gamma ray our counters will measure.
The process looks like:
| 55Cs |
-------> |
-1
+ 56Ba (meta) |
-------> |
56Ba (stable)
+  (662 KeV) |
|
(t1/2=30yr) |
|
(t1/2=2.6 min) |
|
Objective:
To determine the half-life of this process.
Apparatus:
-
Cesium Mini-generator
-
Detector, Computer sensor
Procedure:
-
Turn on the instrument and allow it to warm
up for five minutes. It will already be assembled and connected to your
computer.
-
Record background activity. The detector picks
up (not necessarily radioactive sources) high energy photons that come
from a variety of sources. Some are even from outer space, called cosmic
rays. The background radiation is unavoidable. Do this for say, three or
four time periods. You'll want to get the number of counts per minute.
Later, when you are sampling the radioactive source, you'll have to subtract
background values. You'll note that even the background is hardly constant,
varying as you'd expect a random source to do. You'll need to calculate
an average background. Remember, each measured value has some uncertainty
in it.
-
Your instructor will prepare a radioactive
source for you. Since the half-life is 2.6 minutes, you'll need to be ready
for it. Don't dawdle!
-
The computer does a nice job of nearly automatically
collecting the data for you. Even so, your job will be to interpret the
results and what they mean.
-
Measure the activity in cpm (counts per minute)
for every minute for about 15 minutes. Since the half-life is 2.6 minutes,
in 15 minutes your sample will have gone through about 6 half lives; i.e.,
its activity will have decreased by a factor of 26 or 64 times! You hardly
be able to see any difference from background values. In fact, this level
is so little that federal rules treat this material as no longer radioactive.
-
Use Excel to plot the Activity (y-axis)
as a function of time (x-axis). remember that you must subtract the background
from the counts to get the activity.
-
From the plot in 6 graphically determine the
half-life. Basically, all you have to do is look at the graph and read
the time it takes for the activity to decrease by a factor of two (i.e.
to one half the starting value.)
-
Compare this value with 2.6 minutes given.
Find % error.
-
From equation (5) you can determine the decay
constant for this process.
-
Discuss Limitations, Analyze Data, Discuss
Results, Summarize Experiment, Conclusions, Recommendations, etc.
