Nuclear Decay: Half a life is better than none
“Without adventure, civilization is in full decay” – Alfred North Whitehead
In this activity, we’ll observe (from a safe distance and behind
proper shielding) the decay of an unstable (but not necessarily
radioactive) material. An example of this type of nuclear
reaction is the decay of a neutron to a proton plus and
electron and an anti-neutrino: n 0  p  e   e (the last little
critter is needed to conserve momentum and energy, among
other things, but more about that later). The rate of decay of
the neutrons in any sample of material is proportional to the
amount of material present. This type of decay begins at a
high rate – but the rate of decay quickly decreases.
1. Collect data using the program RadDecay.exe
a. Set the initial number of particles N=1000; set Decay Prob/step = 0.05 (that’s a 5% chance
that each particle in the sample will decay in a given step).
b. Press ‘One step’, record the ‘number remaining’ and repeat until the number remaining falls
below 500. You can record the data points (# steps executed, # particles remaining) directly
in GA and graph-as-you-go if you like.
c. Use the ‘10 steps’ button until you are down to just a few particles remaining
d. Graph # remaining vs step number.
e. Find a best fit function of the form y = A e-Cx (you will have to modify GA’s Natural Exponent
function y = A e-Cx + B by deleting the additive constant B: Highlight ‘Natural Exponent’ and
click ‘define function’. Backspace over the constant, click OK, then ‘Try fit’. The constant A
should be very close to the initial N =1000 (the starting sample size is often called N0).
f.
An alternative method to find the decay constant is to plot the #remaining vs. #steps data on
a logarithmic axis: Since we are using y = A e-Cx, it must follow that ln y  ln A  Cx , which
is the equation of a straight line when graphed as ln y vs. x. GA will happily calculate a
column of the natural logs for you.
g. From the value of the decay constant C, determine the half life of these particles (if the
function is N = N0 e-Ct, the half life t1/2 is the time that sets the value of N/N0 = 0.5, ie
t1/2 = – ln(0.5) / C = 0.693 / C), which should match the # of steps it took to reach 500
remaining.
h. Repeat the entire procedure two more times (you will get different results each time, but the
half lives should be similar). Return to your team with 3 values of half life.
2. Teamwork: Your Spokesperson will designate a group of 2 to make the team’s graph.
Obtain the half lives as calculated by each member of your team (18 in all).
3. Construct a histogram of the group half
life results.
A histogram is a graph that shows how
often a measurement occurs, using a
bar graph format. We sometimes call
the histogram’s y axis "number of
counts" or just "frequency".
The raw data must be ‘binned’:
determine an interval size that
adequately represents the range of
data values. Bins must not be too
coarse or too fine, but just right.
Once you decide on a bin size, count the number of observations that fall within each bin.
The bins appear on the x axis.
As an example, given the set of observations: { 5,6,4,5,7,6,4,5,8,6,2}
Using bin size =4: 0<=x<4: 1 4<=x<8: 9 x>=8: 1
this binning is too coarse
Using bin size =2: 0<=x<2: 0 2<=x<4: 3 4<=x<6: 3
6<=x<8:4
x>=8: 1
this is the histogram shown above, which looks ‘just right’.
Using bin size =1, just count the number of times each value in the data set occurs.
More examples of histograms of this data set are shown below.
Making a histogram in GA (can also be done in Excel, but requires some fancy tricks):
a. Click anywhere in your graph. From the menu, select
Insert-> Additional graphs -> Histogram
b. Click inside the histogram window that opens. From the menu, select Options-> Additional
graph options-> Histogram options
c. In the Bin and Counts Options tab,
check the column you want to
histogram and set the bin size. Note
‘bin start’ is a value less than your
smallest data point. This will affect
the appearance of your graph.
d. You can change these values once
the histogram is drawn. Bring this
window back up by right clicking on the histogram.
4. Calculate the mean value from the histogram: This will be a weighted sum (like center of
mass). The weighting is the number of counts; multiply this value by the midpoint of each bin;
sum up these products and divide by the total number of counts. In the bin size 2 example: the
mean is (0*1+3*3+3*5+4*7+1*9) / 11 = 5.55. Here’s a formula for the mean  of N independent
measurements:  
nb
i i
N
, where ni is the frequency for bin center bi. If you have outliers in
your data (perhaps one group calculated something other than half life?), you can easily remove
this by setting a minimum frequency to include in your mean.
5. Calculate the standard deviation from the mean: Like average deviation, but more
appropriate to larger data sets. Subtract each measurement from the mean and square the
difference. Add the squares, divide by the number of measurements and take the square root.
Here’s a formula: s 
Bin size 4 is too coarse
 (x  )
i
N
2
, where  is the mean of N measurements xi.
Bin size 1 is too fine
The two team members who make the histogram must explain the process to the remainder of
the team.