Things to D o and Results to be Reported:
1. Set up and establish an operating point for the GM detector.
(a) Obtain and set up the Geiger-Müller counter apparatus. Mount the GM tube such that
it can be positioned approximately 2 cm above the table. Prior to connecting the HighVoltage cable ensure that the Power Supply voltage is set to V=0 Volts. With the
power supply turned off, connect the red SHV cable from the High Voltage Power
Supply to the SHV connector on the resistor network box. Connect the cable from the
GM tube to the connector marked GM on the resistor network box. Connect the
Counting Electronics Board to the appropriate connector of the resistor network box.
Connect a USB cable between the electronics board and the computer. Obtain a sealed
radioactive source.
NEVER SET VOLTAGE ABOVE 600V. DOING SO MAY DESTROY
THE TUBE!!!!
(b) Figure 2 shows the Geiger-Müller counter output count rate as a function of
applied potential Vs. The "plateau" region lies in the range V1  VS  V2 where the
output count rate remains approximately constant independent o f V S . The
operating point OP should be in the plateau region, but far removed from the "knee" of
the curve and from the region of spontaneous ionization.
Figure 2: Geiger-Müller counter "plateau" region.
Measure the counting rate r as a function of the power supply voltage V S , for
VS  600V < 600 Volts. Perform a linear fit of the data in the plateau region, and
make a residual graph of the data in the plateau region as a test of linearity.
(Discuss this with your instructor.) Determine the slope and the operating point
OP.
2. Gaussian Distribution
From a or  -radiation source, measure the number of counts v in a time T such that
the mean number of counts   50 . Perform N = 100 trials, then
(a) Compute the mean value, , the sample standard deviation,   and the standard
deviation of the mean,   of your measurements.
1
(b) Group the data into equal bins of size  and make a histogram. Label  and
  on this histogram.
(c) Make a second graph showing the probability d ensity in each of your equal bins.
Fit your graph of probability density versus v to a Gaussian function of the form
y( x ) =
From the fit parameters, m 1 and m 2, determine  and   . Compare to the values
you calculated above.
(d) Perform a x2-test of the Gaussian assumption by dividing the data into four bins:
i.
     

ii.

iii.
        
iv.

      
    
using the values for  and   that you calculated in part (a). By comparing your
result with the expectation for a Gaussian, compute  2 and the chance probability
or p-value of your measurements under the Gaussian hypothesis.
1
As a guide to choosing histogram bin size, Reference [2] suggests the prescription
1
W
3.49
N 1/ 3
where W is the bin width,  the standard deviation of the distribution and N the total number of histogram
entries. As a practical matter, it may be preferable to give an integer width to your bins, close to the W
recommended by the equation.
3. Poisson Distribution
In the absence of a source, your Geiger-Müller counter will detect background radiation
at a rate of a count every few seconds. The low rate is suitable for counting statistics
studies in the Poisson regime.
Determine a time interval T such that the mean number of background counts   5
Perform N = 100 trials, then
(a) Count the number of trials N0 in which you measure 0 counts, N1 in which you
measure 1 count, et cetera. Plot the results in a histogram.
(b) Compute the mean value, , the sample standard deviation,   and the standard
deviation of the mean,   of your measurements. Compare the computed value
of   with the Poisson expectation, given  .
(c) Compute the experimental probabilities Pi = Ni / N, and plot these on a graph.
Include uncertainties. On the same graph, plot the Poisson probability
distribution with mean µ = (using  as computed in part (b) above).
(d) Comment on agreement between observed and expected distributions.
4. Binomial Distribution
Choose a time interval T such that the mean number of observed background counts
µ is about 0.2. According to the Poisson probability distribution, you should observe
zero counts about 82% of the time under such circumstances.
Since the probability p of observing zero is not small, the statistics governing the
number of observed zeroes will be binomial, not Poissonian as one might expect.
To test this, perform the following procedure under the conditions listed above:
(a) Take 10 measurements of the rate. Record the observed number of zeroes.
(b) Repeat the above 10 times.
(c) Compute the standard deviation of the number of zeroes.
(d) Compare your observed standard deviation with the predictions of Binomial
statistics and Poisson statistics.
References
[1] J. R. Taylor, An Introduction to Error Analysis, 2nd Ed., University Science Books,
Sausalito CA (1997).
[2] D. Scott, "On optimal and data-based histograms", Biometrika 66 605 (1979).