RADIOACTIVE DECAY AND COUNTING STATISTICS
I. Introduction:
In this experiment, you will study the radioactive decay law by analyzing data for the purpose of
measuring the disintegration rate of a short-lived radioisotope as a function of time. You will also
investigate the statistical nature of radioactive decay.
The nature of radioactive decay is determined by the fundamental fact that the probability per unit time
that a radioactive nucleus will undergo decay is equal to some positive constant, , called the decay
constant. The value of this constant depends on the type of decay and on certain properties of the
nucleus undergoing decay (the parent nucleus) as well as the nucleus which remains after the decay has
taken place (the daughter nucleus). From this fundamental relation it follows that of a sample containing
N radioactive nuclei at time t, the number decaying per unit time is Nand the number dN which decay
in time dt is Ndt. Since the nuclei which decay in time dt represent a decrease in the number of parent
nuclei present in the sample, one writes the change in N as dN = – Ndt. Integrating this expression
from t = 0 to some later time t yields the radioactive decay law
N = Noe– t
(1)
where No is the number of parent nuclei present at t = 0. The activity, or number of disintegrations per
second, is given by
R = – dN/dt = Noe– t = Roe– t
(2)
where Ro is the initial activity. The unit of activity is the Becquerel, defined as 1 disintegration/sec.
Also used is the Curie (Ci), where 1 Ci = 3.7 x 1010 Becquerels. When the natural logarithm of activity
is plotted versus time, a straight line results.
The decay rate of a radioactive isotope is normally characterized by its half-life, t1/2, which is defined as
the time required for one-half of a given number of nuclei to decay. Equivalently, t1/2 is the time
interval during which the activity of a radioactive sample decreases by a factor of two.
N/No = 1/2 = e- t1/2 or
(3)
t1/2 = ln 2/ = 0.693/
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II. Required Equipment:
PASCO Beta-Gamma Nuclear Sensor, Science Workshop Interface, Macintosh, radioactive sources,
small steel tray, corkboard to support steel tray, post and clamps.
III. Procedure:
A. Radioactive Decay
The half-life of metastable 137Ba (Barium) nuclei
will be measured. These nuclei are products of the
radioactive decay of 137Cs (Cesium). The decay
scheme is represented in the energy-level diagram
shown.
The half-life of 137Cs is 30 years. When 137Cs decays, a neutron in this nucleus changes to a proton with
the emission of a negative  -ray. Most of the time, the daughter nucleus 137Ba is created in the
metastable 137mBa state. The half-life of this state is 2.55 minutes, and it decays to the ground (lowest
energy) state of 137Ba by emitting a gamma ray of energy 662 keV. Only 6% of the time does the 137Cs
decay directly to the stable ground state of 137Ba.
An isotope generator containing 137Cs is used to produce the radioactive nuclei. When one drop of dilute
HCl is passed through the generator, some Ba atoms go into solution and are removed from the
generator. The instructor uses a circle of green felt moistened with this solution to measure the half-life
of the 137mBa nucleus.
1) The instructor places the felt moistened by the solution under the Beta/Gamma Nuclear Sensor. The
instructor will turn on a file in Data Studio. He will use the file to keep track of the counts accumulated
during each 30-second time period. He will do this counting for at least 19 thirty-second time periods.
2) When the instructor has finished the measurement of count rate versus time, you will have 15 free
minutes during which you may begin Part B of this experiment. After the 15 minutes have elapsed, the
instructor will measure the background radiation. By then, the 137mBa activity will have decayed to a
negligible level and will not significantly affect the measurement. The instructor will enter the mean
background count into the Text Window of the Graphical Analysis file. The instructor will copy the
Graphical Analysis file to your hard disk.
3) Open the Graphical Analysis 3 file. Double click in the top half (the "label" rectangle) of the
heading of the first column. Change the “Name” and “Short Name” to "Time" and the units to
“seconds”. Similarly, change the name heading on the second column to “Counts” (no units). The plot
must place the points at the middle of each time interval. To accomplish this, select New Calculated
Column from the Data Menu. Name this new column Time-15 (units = seconds), select the variable
Time, and type –15 into the equation box. Then, click on the X-axis of the plot and select “Time-15”.
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4) Let's say your mean background count was 25.5 (see above). From the Data Menu, select New
Calculated Column and name the new column Counts-Bkgd (no units). Select the variable Counts, and
type –25.5 into the equation box. Then, click on the Y-axis of the plot and select “Counts-Bkgd”.
5) From the Data Menu, select New Calculated Column and name the new column Ln(Counts-Bkgd)
(no units). Select the function ln() (the natural logarithm of the value) and choose the variable CountsBkgd. Click on the Y-axis of the graph and select “Ln(Counts-Bkgd)”. The resulting plot should be a
straight line, since the radioactive decay process involves an exponential function of the time. Click on
the graph and then on the “R=” (linear fit) button to get the slope and intercept of the straight line. The
slope will be the negative of the decay constant in seconds. Double-click on the box and select
“Standard Deviation of the Slope” to display the uncertainty in  coming from the fitting procedure.
Move the box to a convenient place on the plot. Select Print Graph from the FILE menu. Include a copy
of this graph in every student’s lab report. Compute t1/2 and its error from  and its uncertainty. The
fractional error in t1/2 is equal to the fractional error in  (see page ME-6 [Sec. 4] in the lab. manual).
Compute the count rate at t=0, which is exp(intercept) divided by 30 (Why?).
6) The least-squares fitting method in GA3 assumes that the uncertainties for all of the points on the
graph are equal, so it will not quite give the best straight-line fit if you include all of your points in the
calculation. In particular, the points measured near the end when the source has become weak may have
relatively large errors and possibly should not be included in your least-squares calculation. You can
edit them out if you wish. Compare the value of (and error in) t1/2 obtained from your leastsquares fit with the accepted value of 2.55 minutes, and discuss possible sources of experimental
error. How many atoms of 137mBa were on the felt pad at time t=0?
7) A word about statistical errors. The net number of counts, n, accumulated during a given interval is
equal to the difference between the measured counts with the source in place and the background counts
accumulated during an equal interval of time.
n = c - b,
where c = measured counts and b = background counts.
To obtain the uncertainty in n in terms of the uncertainties in c and b, one uses the Gaussian errorpropagation relations which are given on page ME-6, Sec. 4. It is a result of statistical theory that the
uncertainty or probable error, N, in a number of counts, N, which are randomly occurring over a given
time interval, can be taken to be equal to the standard deviation, , which is equal to √N for a normal or
Gaussian distribution. Thus the uncertainties in the measured counts, c, and the background counts, b,
should be √c and √b, respectively. The fitting program ignores this and uses a constant uncertainty
instead…which is why the lowest count-rate bins will have larger deviations from the straight-line fit.
B. Counting Statistics
In this part of the experiment you will use the relatively long-lived radioisotope 90Sr (Strontium) (t1/2 =
28 years) to study the statistical nature of the radioactive decay process. You will measure the decay rate
a large number of times and observe the statistical fluctuations in the value of this decay rate. Since the
half-life is so much longer than the time it will take you to do all the measurements, there will be no
perceptible decrease in the counting rate during the duration of the experiment. (The number of nuclei
which decay during this time will be completely negligible compared to the number of nuclei present in
the sample).
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90Sr
decays into 90Y (Yttrium) by beta-ray emission. The maximum electron energy from this decay is
546 keV. The daughter nucleus 90Y in turn decays into stable 90Zr (Zirconium) by  -emission with a
half-life of 64 hours. The maximum electron energy for this decay is 2.27 MeV.
Open Data Studio. On the left side of the Experiment Setup window, in the Sensors column, scroll
down and select Geiger Counter; its icon is yellow. Double click the Geiger Counter. The
Experiment Setup window now shows a copy of the yellow icon connected to Digital Channel 1.
Double click either yellow icon again. The Sensor Properties window appears. Click on the Fast
button. Increase the Sample Rate to 10Hz by clicking on the upper, small triangle until the number 10 is
displayed. Click on OK.
Next, from the Experiment menu at the top of the screen, select Set Sampling Options. At the top of
the window that appears, click on the tab named Automatic Stop. Select Time. Change the display to
60 seconds. Click on OK.
Below the Sensors column on the left-hand side of your screen is the Displays column. In the Displays
column, double click Table.
Turn the switch on your Beta Gamma Nuclear Sensor to AUDIO. Use the light yellow lines to center the
sensor’s active area on the center of the green Strontium-90 source. It is also helpful to turn on the audio
while moving the sensor to the position with a large count rate. (Turn the audio to the ON position
again as soon as the sensor’s active area is centered on the source, since the audio is annoying in the
extreme!). Be careful NOT to move the sensor during the course of a run, as this would introduce nonstatistical fluctuations in the count rate.
Locate the Start button near the top of your screen. Click on the Start button. The Beta-Gamma Nuclear
Sensor will now count for 60 seconds. The number of counts in a given time bin should be greater
than 25. If not, try repositioning the sensor and making a new run.
Each student in the group should repeat this procedure, moving the sensor a bit in between runs so that
the average count rate per time bin is different for each run.
Since Data Studio took periodic samples at the rate of "10 Hz", your data are in "units" of counts per
tenth of a second. After the data have been taken, each student should COPY the data from his or her
best run and PASTE it into Graphical Analysis 3 (see below). (Hint: To select all the data from a given
run, move the cursor in the heading box until a down arrow appears, then click).
Next, open Graphical Analysis 3. Click on the box under X and paste in the data. Double click on the
X heading and change the “Name” and “Short Name” to TIME and the units to “0.1 seconds”. Change
the Y headings to COUNTS (no units). You should now have a graph of COUNTS vs. TIME. (If not,
generate it by clicking on the Y-axis and selecting COUNTS, and then on the X-axis selecting TIME).
Click on the Analyze and then on Statistics, which will give you the mean, standard deviation, and
number of points in the data set (among other information). Drag the statistics box to a convenient place
on the graph. Next, click on INSERT and select ADDITIONAL GRAPHS and then HISTOGRAM. A
histogram plot should pop up, giving the number of times a certain value occurred vs. the value. Double
click on the histogram, make sure that the BIN options are a bin size of 1 starting at 0, and choose
COUNTS under “Column”. Note that two “computed” columns giving the bin number and bin count
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will also appear in the data set. Click on the name of the histogram and move it to a convenient place in
the window (you may need to resize the histogram plot). Click on the histogram and select PRINT
GRAPH under FILE to print out a copy. Repeat for the Counts vs. Time graph. Then plot the Histogran
Bin vs Histogram Count graph. The Graph can be made from the calculated columns created by the
Histogram. The graph should look approximately Gaussian. Then click the Curve Fit button on the
toolbar (or go to ANALYZE >> Curve Fit..) and fit this graph with a Gaussian. You should get the
RMSE (Root Mean Square Error) from the graph itself. The plots for a given student’s run should be
included in that student’s lab report.
The observation of the statistical variations in the numbers of counts is the whole point of this
experiment. Determine the number of data points in your histogram that lie more than one and also the
number that lie more than two standard deviations away from the mean value (the sum of the counts
both above and below the mean). You can either look at the HISTOGRAM computed columns referred
to above or use “Examine” on the ANALYZE menu to display the counts in a given bin (selected using
the mouse) on the histogram plot itself. (NOTE: Since the bin numbers are integers and the mean and
standard deviation will likely not be integers, you will have to approximate to the nearest integer values).
Compare the fractions of your measurements lying more than one and more than two standard deviations
away from the mean value with the fractions expected from a normal (Gaussian) distribution. These are:
31.7% and 4.6% of the total number of points, respectively. Also, compare the value of the standard
deviation of the distribution with the value given by pure counting statistics, i.e., √N, where N is the
mean number of counts per measurement. What does this tell you concerning the random nature of
radioactive decay?
NOTE: Before you leave the lab, all measurements must be complete and each lab partner must have a
complete data sheet, initialed by your TA.
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