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THE GEIGER-MULLER TUBE AND THE STATISTICS OF RADIOACTIVITY
This experiment examines the Geiger-Muller counter, a device commonly used for detecting and
counting ionizing radiation. Various properties of the counter are measured, and the analysis of
counting experiments in general is studied.
Theory:
The Geiger-Muller Counter
The counter consists of a cylindrical chamber (tube) with a wire stretched along its longitudinal
axis and insulated from its walls. The chamber walls act as the cathode, and a positive voltage is
applied to the wire (usually tungsten), making it the anode. The cylinder is filled with a low
pressure gas mixture of argon and ethyl alcohol.
When an ionizing particle passes through the gas in the counter it liberates electrons by collision,
and these are attracted toward the centre wire (anode). As the electrons approach the high electric
field near the centre wire (within a few diameters) they begin to pick up energy between
collisions with gas atoms. Provided the applied voltage between cathode and anode is above the
threshold value, the electrons pick up enough energy to ionize the atoms. The electrons produced
in these ionizing collisions can also participate in the ionization process, causing an 'avalanche' of
ion formation to occur. Atoms are excited by collisions with high-speed electrons at the avalanche
site. The atoms generate photons as they decay to a lower energy state. These photons can ionize
other alcohol molecules, and the resulting photoelectrons can cause further avalanches. Thus the
discharge spreads along the tube, resulting in a positive ion sheath around the wire anode.
Eventually the ion sheath reduces the electric field at the wire to such an extent that no further
electron 'multiplication' can occur. When this happens, production of new avalanches ceases. The
electrons striking the anode and the change in tube voltage due to the ion sheath produce a pulse
at the anode which is amplified and counted.
The ion sheath moves toward the cathode. During this time, the counter is inoperative because of
the reduced tube voltage due to the ion sheath. The time after one discharge has occurred until
another can occur is called the dead time of the counter and is typically of the order of hundreds
of microseconds.
As the ion sheath moves toward the cylinder, collisions between argon ions and alcohol
molecules result in the formation of neutral argon atoms and alcohol ions. The reverse process is
energetically impossible. Thus when the sheath reaches the cylinder wall it consists only of
alcohol ions. On striking the cylinder, the alcohol ion neutralizes and dissociates into two
uncharged atomic groups. If, on the other hand, argon ions had been able to reach the cylinder,
secondary electrons would have been produced. These electrons could then trigger another
discharge, and the counter would recycle indefinitely. Counters containing polyatomic gases such
as alcohol are called 'self-quenching', since the action of the gas prevents recycling, or continuous
discharge, both by interaction with the argon ions and by absorption of photoelectrons. If,
however, the operating voltage of the counter is set too high, the counter will go into continuous
discharge because of the greater number of ions formed, not all of which can be neutralized by
the alcohol before they reach the cathode.
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Once the ion sheath has dissipated to an extent that the counter voltage rises above the threshold
value, passage of another radioactive particle through the tube will cause another discharge. If the
ion sheath from a previous discharge has not completely dissipated when another discharge
forms, the resulting voltage pulse will be smaller than if the tube had been in a neutral state. The
time during which output pulses are smaller than normal is called the recovery time of the
counter. Figure 1 shows a schematic diagram of the Geiger counter; and the output of the
counter, over a period of time, as displayed on an oscilloscope.
Figure 1
The operating characteristics of a G-M tube are commonly displayed by plotting counts per unit
time (count rate) vs. anode voltage for a given source. The voltage at which the tube begins to
count is called the starting voltage. The flat portion of the curve is called the plateau and
corresponds to the Geiger region, the range of voltages over which "avalanching" occurs without
continuous discharge (recycling) of the counter. A good counter has a plateau at least 100 V long
with a small slope, thus fluctuations in the supply voltage will not affect the count rate. As the
tube ages, the plateau shortens and its slope increases due to the increased probability of recycling
as the alcohol gas is used up.
A number of factors must be taken into account when analyzing counting experiments. Three of
these factors are background, counter losses, and statistics.
Background: the term applied to counts that are registered even when no radiation source is
present. This is due to contamination in the lab from other experiments, building materials, the
soil, and cosmic rays. The background counting rate must be subtracted from the total rate to
obtain that due to the radiation source alone.
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Counter losses: As already mentioned, during the development of a discharge the voltage at the
anode is lowered to such an extent that a second particle passing through at that time will not be
counted. This dead time is denoted 𝜏𝐷 . If 𝑚′ is the observed counting rate, the actual rate is given
approximately by
m=
m′
1 − m′τ D
(1)
Since 𝜏𝐷 is small, of the order of 100 microseconds, this correction usually need only be applied
for counting rates higher than 100/sec (i.e. when 𝑚′𝜏𝐷 is significant compared to 1, say
𝑚′𝜏𝐷 > 0.01). It is possible to determine the dead time of a counter by means of an operational
technique. Suppose two sources S1 and S2, when placed one at a time a certain distance from the
counter, give true counting rates m1 and m2. The counter, however, registers 𝑚1′ < 𝑚1 and
𝑚2′ < 𝑚2 due to dead time losses. When both sources are present the counter registers
𝑚12 ′ < 𝑚1 ′ + 𝑚2 ′ because of the higher counting rate and hence higher count loss to dead time.
However, the actual rates should obey 𝑚12 = 𝑚1 + 𝑚2 . Using equation (1)
𝑚12 = 𝑚1 + 𝑚2
′
m12
m1′
m2′
=
+
′ τ D 1 − m1′τ D 1 − m2′τ D
1 − m12
which can be solved for 𝜏𝐷 :
τD =
′ (m1′ + m2′ − m12
′ ) / m1′m2′
1 − 1 − m12
′
m12
(2)
(3)
(4)
Since 𝑚1 ′ + 𝑚2 ′ – 𝑚12 ′ is small compared to 𝑚1 ′ , 𝑚2 ′ , and 𝑚12 ′, this may be expanded to
yield
τD =
′
m1′ + m2′ − m12
2m1′m2′
(5)
Statistics of Radioactivity
The count rate obtained in any time interval will fluctuate from the average counting rate over a
long period of time according to the laws of probability. Counting experiments involving
radioactive decay, or gamma radiation absorption obey a Poisson statistical distribution. For a
single measurement of n counts the standard deviation is equal to n , and the result of a
measurement is quoted as n ± 2 n since a 95% confidence corresponds to 2 σ . To obtain a
relative uncertainty of 1%, for example, one must have n /n = 0.01, or n = 10,000 counts. One
must always count long enough to obtain sufficient counts in order to have the desired accuracy.
If N amount of runs are made under the same conditions, the results are given by n ± n / N
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where n is the mean. That is, the standard deviation of the mean of N runs, n , is reduced by a
factor of N . The probable error for a single measurement is 0.67
. This means that if a
series of measurements was made under the same conditions and the deviation from the mean
(𝑛 − 𝑛�) was calculated, half of the deviations would be greater than 0.67
and half would be
less.
The Gaussian approximation to the Poisson distribution states that when counting radioactive
decays (and other similar random events), the results of a series of count measurements repeated
under identical conditions should be distributed according to the function:
f (n) = Ae
where
− (n − n ) 2 / 2n
 (n − n ) 2 
= A exp −

2n 

n =
mean (average) number of counts measured
A=
height of the distribution (Ideally, this should correspond to the
'number of occasions' that the average number of counts was
observed, i.e. the largest # of occasions should occur for a count of n .)
f(n) = predicted 'number of occasions' for a count of n to be observed
Let
then
where
N=
total number of count measurements taken (i.e. number of
count intervals)
pi =
number of occasions that a count ni was obtained
n=
1
N
∑ p n of
i
i
i
i denotes count measurements with non-zero 'number of occasions'.
Applications:
The scientific applications of the GM counter are limited since it does not have any energy
resolution. However, in situations where the radioactive species are already known, the GM
counter can be used to monitor ambient radiation levels for safety purposes.
A solid understanding of radiation statistics is important for anyone doing research in nuclear or
particle physics. A reported value is only as good as its uncertainty, and the statistics of
radioactive processes are an integral component of error calculations.
Apparatus:
The apparatus consists of the Geiger-Muller counter tube mounted above adjustable-height
shelves on which the source may be placed, a SPECTECH ST360 counter/power
supply/interface, and a personal computer (PC).
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The ST360 supplies the high voltage for the G-M tube and counts the pulses from the anode. The
high voltage is adjustable from within the ST360 software.
In addition to simply being counted, output pulses from the Geiger-Muller tube are stored in the
PC for further analysis. In this experiment, the PC will be used to record the activity (count rate)
of a radioactive source during a series of measurements made under identical conditions in order
to investigate the statistical distribution of the count rates.
A 36Cl radiation source will be used in this experiment. It will be used to determine the starting
and operating voltages and plateau curve for the G-M tube and to measure the dead time of the GM tube. The PC will be used to study the statistical distribution of count rates from the 36Cl
source. The 36Cl disk source should be handled with the tweezers provided.
Procedure and Experiment:
As mentioned in the theory, G-M counter tubes have finite lifetimes due to consumption of the
alcohol gas, and excessive voltage will damage the counter.
DO NOT EXCEED 880 VOLTS.
DO NOT TOUCH THE COUNTER TUBE, IT IS FRAGILE
AND MAY COLLAPSE IF PINCHED.
Determination of Starting and Operating Voltages
1. Turn on the PC and monitor and login (there is no password).
2. Turn on the ST360.
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3. Open the ST360 software program by double-clicking on the desktop icon.
4. Place the 36Cl source in the second shelf below the counter tube.
5. The starting and operating voltages will be determined by doing a series of 1 minute counts at
increasing voltages. Go to the Setup menu and set the HV Setting (high voltage) to 780 V and
enable a Step Voltage of 5 V. Go to the Preset menu and set the Time to 60 s and the Number
of Runs to 21. Click on the start button (the green diamond) to acquire data. Data acquisition
will stop automatically after the 21st measurement.
6. Save the data as a tab-separated values file by clicking on the Save button. Give the data file a
descriptive name.
7. Open the data file in Excel and plot count rate versus tube voltage.
8. Choose the operating voltage of the G-M counter slightly lower (say 10 V) than that
corresponding to the middle of the plateau region. If you are not sure where to set the voltage,
consult your instructor.
9. Set the operating voltage by going to the HV Setting item in the Setup menu. Also be sure to
set the Step Voltage to 0 and the Step Voltage Enable to OFF.
10. Remove the source and determine the background radiation by making a 5 minute count
measurement. (Go to the Preset menu and set the Time to 300 s and the Number of Runs to 1.)
Measurement of Dead Time
11. Use the following two source method (see equation (2) in Theory) to measure the dead time.
Note that your 36Cl source consists of two pieces. Measure m1′ (the counts in 60 s for one
half), then measure m12′
 (the counts in 60 s for both halves), then measure
m2′ (the counts
in 60 s for the other half). The two halves must not be moved relative to the counter tube
(other than to place them in the holder or remove them) in order to prevent errors due to
altering the source geometry with respect to the counter tube. Taking n as the absolute error
in a count n, determine the error in the dead time.
Study of Statistics of Radioactivity
12. Place the 36Cl source in the fifth shelf below the G-M tube.
13. Data will now be acquired for 3600 trials of 1 s count measurements. Go to the Preset menu
and set the Time to 1 s and the Number of Runs to 3600.
14. Data acquisition will stop automatically following completion of the 3600th trial. Note that
due to lag time between the ST360 and the PC it takes about 90 minutes to acquire the 3600
trials.
15. Save the data as a tab-separated values file by clicking on the Save button. Give the data file a
descriptive name.
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16. Close the ST360 software program and turn off the ST360.
17. Open the data file in Excel.
18. Sort the data from lowest to highest according to the Counts column. In a blank column of the
spreadsheet, enter the numbers starting with the lowest number of counts and ending with the
highest number of counts, in increments of 1.
19. From the Tools menu, select Data Analysis…, Histogram. For the input range select your
Counts data, for the bin range select your lowest to highest number of counts, and click OK.
20. The new worksheet that is created contains the number of occurrences (Frequency) of each 1 s
count (Bin) that you measured.
21. This histogram data is now analysed using the radstats2007.xls spreadsheet.
Analysis:
As mentioned in the theory, it is expected that the distribution of a series of radioactivity
measurements made under identical conditions will follow a Poisson distribution. This will be
tested by fitting a Poisson curve to the data, and by calculating various parameters from the data
and seeing how well they compare with the corresponding Poisson parameters.
An Excel spreadsheet that performs a complete analysis of the data (i.e. data consisting of a
series of radioactivity count measurements made under identical conditions) is available for your
use. In order to do the analysis, the spreadsheet requires that you input the lowest count with nonzero occasions, the highest count with non-zero occasions, and the number of occasions for each
count from lowest to highest.
The spreadsheet is obtained from the lab manual web page:
http://physics.usask.ca/~bzulkosk/modphyslab/phys381.htm :
1. Load the spreadsheet radstats2007.xls.
Once the file has loaded and if you are using a university computer, save it to your home
directory (likely on the h: drive) by selecting Save As... from the File menu.
The columns of the spreadsheet contain the following values:
column A:a ‘counting’ column (numbers increasing in sequence from 1)
column B:counts obtained in the selected time interval (1 sec in this case) from lowest counts
with non-zero occasions to highest counts with non-zero occasions.
column C:number of occasions that the count listed in column B occurred. (Obtained from
your data histogram and entered manually or by cut and paste.)
column D:theoretical number of occasions predicted by the Gaussian-approximated Poisson
distribution, normalized to fit the total number of occasions and the average
counts obtained experimentally.
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Columns F through J contain the results of ‘intermediate’ calculations and need not be
printed.
column F:product of counts and number of occasions (column B entry times column C entry).
column G:difference of actual counts and average counts
column H:negative of number of occasions, if column G entry is greater than 0.67 times
theoretical standard deviation.
column I:number of occasions, if column G entry is less than 0.67 times theoretical standard
deviation.
column J:product of column C entry and square of column G entry (used to calculate the
experimental standard deviation).
2. The data required by the spreadsheet is entered as follows:
cell C4:lowest counts with non-zero occasions (as soon as the value is entered in cell C4,
column B is updated to the proper range of count values).
cell C7:highest counts with non-zero occasions
column C:number of occasions for all counts values between these two values, inclusive.
Once the data has been entered, the theoretical distribution is normalized to the
experimental data to obtain the best fit. This normalization is done as follows:
Using only integer values for ‘# of occasions for peak of theoretical fit’ (cell D117), which
corresponds to A, the height of the distribution, in the equation
f (n) = Ae − ( n − n )
2
/ 2n
 (n − n ) 2 
= A exp −
,
2n 

the value for cell D117 is chosen by trial and error such that the ‘Total # of Occasions
for Theoretical fit’ (cell D115) matches the ‘Total # of Expt. Occasions (# of 1 s trials)’
(cell D114) as closely as possible.
The spreadsheet automatically generates a graph of the experimental and theoretical (fitted,
Gaussian-approximated, Poisson) count distributions. (Click on the Chart1 tab to view the
graph.)
The numerical results of the spreadsheet analysis are presented in rows 114 to 122.
Remember to save your file.
The standard deviation, , of a Poisson distribution is  = n . Compare this theoretical value
(cell D122) with the experimental standard deviation, exp, (cell D121) as calculated by the
spreadsheet using
1
σ exp =
∑ p ( n − ni ) 2
N −1 i i i
For a Poisson distribution, the most probable error is 0.67 n (= 0.67). If the data obeys a
Poisson distribution, half of the deviations from the mean should be greater than 0.67 n and
half should be less.
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Compare cell D119, the total number of occasions for which |n – n | < 0.67 n , with cell D118,
the total number of occasions for which |n – n | > 0.67 n . Are these numbers approximately the
same? Discuss the result in terms of the previous paragraph.
View the graph of the experimental and theoretical (fitted, Gaussian-approximated, Poisson)
count distributions as produced by the spreadsheet. Qualitatively, how well does the theoretical
curve fit your experimental data? Do you feel justified in assuming that radioactive decay count
measurements obey a Poisson distribution?
NOTE:No error calculations are required for this part of the experiment (the statistics of counting
experiments). The purpose is to determine whether or not the theoretical statistical
uncertainty in a Poisson distribution ( n ) can be applied to experimental results
obtained in counting experiments.
If it is decided that counting experiments do obey Poisson statistics, then if a counting
measurement is made once, and the value n is obtained, this value n is the best estimate for the
expected mean count (if the experiment were repeated) and the best estimate for the standard
deviation is n . That is, if one measurement is made of the number of events in some time
interval, and the value obtained is n, then the value for the expected mean count for that time
interval is n ± n .
References:
Bleuler & Goldsmith, Experimental Nucleonics, QC 784
Fretter, Introduction to Experimental Physics, QC 41
Halliday, Introductory Nuclear Physics, QC 173
Melissinos, Experiments in Modern Physics, QC 33
Taylor, An Introduction to Error Analysis, QA 275