Radioactivity Simulation
Names:_______________________________________________________________________
Last week, we used a Geiger counter to examine half-lives (time needed for 50% of a radioactive
substance to decay) and penetrating powers of various radioactive sources. This week, we’re going to
do something very similar through computer simulation. The simulations have some advantages over
real life: they require no cleanup and are much quicker (in our case) than the actual experiment. Of
course, we need to make sure they reflect real life, or we’re just wasting our time.
What may have been the strangest result from the physical experiment last week was the fact that
the chance for an atom to decay is independent of its age. This is definitely not like what you are used
to with biological organisms. After maturity, the chance that an organism will die in a particular year
generally increases with age, sometimes rapidly. Today’s exercises are meant to demonstrate that
this unexpected feature of radioactive decay is just a consequence of the randomness of it. We will
use a computer’s ability to “flip a coin” to build up this exponential pattern of decay.
These simulations are written using Excel, and are based on a few very simple ideas. First, we assume
that the particles decay (or are absorbed) in a completely random way; the particles are all equally
likely to decay in a particular time interval or to be absorbed in a particular thickness of material.
Notice that we are definitely not saying that the chance of decay is independent of the time interval.
If we watch an atom for 10 seconds, it is 10x as likely to decay as if we only watch it for one second.
What we are saying is that the chance of it decaying in the first 10 seconds after its creation is the
same as the chance it will decay in the 10 seconds immediately after its one billionth birthday.
Excel generates what are known as pseudorandom numbers. Actual radioactive decay is truly
random; there is absolutely no way to predict when a particular nucleus will decay, or which one in a
batch of them will decay next. Pseudorandom numbers are generated by a computer program
through a complicated series of mathematical operations. They’re called pseudorandom because
someone with knowledge of the internal workings of the program (Excel) could easily predict the
“random” numbers it will give you, but if you don’t have that information, the numbers will seem to
be completely random. Similarly, if you were to write down the last digit of everyone’s phone
number from the phone book, the list of numbers would be pseudorandom. You wouldn’t see a
pattern, but if someone else had a copy of the phonebook, they would know what numbers come
next. Excel’s (pseudo) random number function is called RAND() and gives a number from 0-1.
The half-life spreadsheet (located at http://www.chemistry.armstrong.edu/baird/halflife.sim.xls) has
only two user-adjustable cells. They are highlighted in blue, and are the half-life of the material in
seconds and the time step (also in seconds). Start this by hitting the Reset button, which will clear any
data in the sheet. Note that when you click on Reset, all of the cells in column A are filled with “1”s
and turn green. This indicates atoms that have not yet decayed (known as parents). Once the atom
represented by an individual cell decays, the 1 will be replaced with a 0 and the cell color will change
to pink. This indicates a daughter atom.
1
You could set both the time step and the half-life to the same value. In that case, the computer would
essentially be flipping a coin for every atom and calling the heads decayed and tails undecayed. The
program would handle this by drawing a random number and if it is greater than 0.5, call that heads.
If not, it’s tails. We could do this to keep things simple, but with a population of only a thousand
atoms, we would get down to a small number of them in just a few steps and our curve fit will not be
great. If we instead make the time step smaller than the half-life, all we have to do is compare our
1 (𝑇𝑖𝑚𝑒 𝑠𝑡𝑒𝑝⁄ℎ𝑎𝑙𝑓−𝑙𝑖𝑓𝑒 )
random number to ( )
2
with the same condition (greater = heads, less = tails).
To make sure we get enough points, we will, set the half-life to be 15 seconds and the time step to be
3 seconds. Start clicking the “Time Forward” button, and time will move forward 3 seconds per click.
Your total elapsed time is in cell G2 and columns B & C will keep a record of the number of
radioactive atoms remaining from the original 1000 atoms. Keep moving time forward until you’re
down to about 10 atoms or so. If you let time go on until you have 0 undecayed atoms left, Excel
(appropriately) will not be able to find an exponential curve that includes zero. If you run out of
parent atoms, the trendline will disappear.
The spreadsheet should now contain a graph of counts detected vs. time, as well as the trendline.
Record your R2 value and trendline equation, and calculate your predicted half-life (remember from
last week that t1/2 = ln(2)/ where  is the coefficient of x in your exponential trendline). How does it
compare to your known value of 15 seconds? Remember that our description of radioactive decay is
as a statistical process, which means it relies on large numbers of atoms. We can’t predict the decay
of a single atom at all, and if you’re down to just a handful, things are almost as bad.
R2______________________ Trendline_____________________ T1/2______________________
Now, move time forward again. Make a note of the time when the last atom decays (first occurrence
of zero in the column C). Hit the Reset button and move time forward again until you’re out of atoms,
recording the time of the last decay. Do this for a total of 5 times. Are all your numbers the same?
What you should see above is the evidence of random error. While we are here using Excel’s
pseudorandomness, the same thing would happen in a physical experiment and the outcome would
be truly random. Click on reset and move time forward one step (3 seconds). Make a note of the
number of atoms which have decayed (found using 1000 – number remaining), and repeat the
process (reset, then one time step forward). Do this 10 times and record the results below.
Number of decays during first 3 seconds
2
(Would the number of decays after one time step be any different, except for random error, if we
used t1/2 = 5 seconds and 1 second time steps instead of using t 1/2 = 15 seconds and 3 second time
steps?) Y
N (Circle one)
When we did the lab with actual sources and a Geiger counter, we found that the standard deviation
of many trials was equal to the square root of the average number of decays. In a new spreadsheet,
record the 10 values from the table above and have Excel give you the average and the standard
deviation ( “=AVERAGE(A1:A10)” and “=STDEV(A1:A10)”. Record both of those below, and compare
the standard deviation to the square root of the average (add it below, too)
Mean_______________ Mean ________________ Std. Dev. ___________________
Now, download http://www.chemistry.armstrong.edu/baird/penetration.depth.sim.xls .
This spreadsheet illustrates the fact that, when using relatively low-energy  rays, we can think of any
absorbing material as being made of a large number of very thin sheets of material, each of which has
the same probability of letting the radioactive particle through (known as the transmission
probability). Different materials have different transmission probabilities, but we’ll notice as we add
layers of material that this gives us an exponential curve for the probability of penetration as
thickness increases.
Looking at the new spreadsheet, column A holds numbers from 1 to 100 representing layers of
material. Column B is where Excel’s random number function comes in – a zero in this column means
the particle has made it through that particular layer and goes on to the next one, while a one means
the particle was absorbed. You’ll notice that there may be more than one “1” in column B, which
makes no sense. If the particle is absorbed in layer 5, for example, it can’t be absorbed again in layer
9. This is a side effect of the way the spreadsheet is written. The program that runs when you click
the “Generate Decay(s)” button takes care of this by only counting the first occurrence of a “1”. From
there, it just keeps a running total of how many particles have been absorbed by each layer. We can
then plot these by layer number (representing thickness) and we’ll find the same exponential curve
we had last week. Of course, we have to pay attention to Excel’s problems with recalculating trend
lines, and we also may have to adjust the size of our data set as we did before.
To begin, set your transmission probability to 90% and pick 500 decays per click. Click on the
“Generate Decay(s)” button and watch the curve appear. Write down the R 2 value and the
coefficient of the exponential in your trendline equation (your data set should be adjusted so that
your graph contains about 30 points).
R2______________________ Coefficient of exponential in trendline________________________
Now, keep clicking “Generate Decay” until your total (cell I2) is 5000. Again, write down the R2 value
and the coefficient of the exponential in your trendline equation.
R2______________________ Coefficient of exponential in trendline________________________
3
Predict what the coefficient will be if we change the transmission probability to 94%. Hit reset,
change the probability (cell H2), and generate 5000 more decays. Were you right? Put your prediction
and the actual answer below.
R2______________________ Coefficient of exponential in trendline________________________
Write down the values in cells F2 through F5 below. How do they compare to 0.94 x for various values
of x?
0.94x
Spreadsheet value
0.941=
0.942=
0.943=
0.944=
F2=
F3=
F4=
F5=
The half-value layer (HVL) is the amount of material required to block 50% of the incoming
radioactive particles. How can you find the half-value layer from your trendline equation? What is it
in this case? Compare this value to the one found by looking at your data in column F.
HVL = _______________________ Found by ____________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
Your transmission probability should still be set to 94%. What should it be if you want your half-value
layer to be twice as large? Verify your guess & put the equation and HVL below.
Prob._________ Equation___________________________________ HVL_____________________
Finally, reset the sheet and set your transmission probability to 99.95%. Generate 1000 decays and
write down your trendline equation and your R2 coefficient. Now add another trendline – this time,
just a regular linear fit. Write the equation for this line down, as well as the R 2 value. Notice any
similarities?
R2______________________ Coefficient of exponential in trendline________________________
R2______________________ Linear trendline equation__________________________________
4