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Geiger-Müller Detector and Counting Statistics
*REPORT MUST FOLLOW THE FORMAT DISCUSSED IN LECTURE, NOT THAT OF THIS HANDOUT!*
Note: it is expected that each question below be addressed in your report; e.g. the questions
about how a Geiger counter works should be addressed in your Methods section. Note that
there is not a linear mapping between the topics listed below and where you address them in
your report; it is your responsibility to organize them into a report.
Purpose
To understand how a Geiger-Müller detector works and, by counting radiation events, to test
the model that the Binomial, Poisson and Gaussian probability distributions describe, in
different regions of parameter space, the statistics of counting experiments. We will also use
an investigation of the electronic workings of the detector as a vehicle for learning the
operation of an oscilloscope, an important tool used in many areas of physics.
References. Read before starting the experiment.
Melissinos 3.1, 3.4
Tait pp. 190 – 192; be sure to read the paragraph under “Signal pulse” on pp. 190 – 191.
Introduction
In this experiment we could potentially use α, β, γ, and µ radiation. (The last of these are more
widely known as “muons”.) Since α particles have a very short penetration depth and there is
no easily available source of muons, we will restrict our interest to β and γ. The sources we
have available are 55Fe, 60Co, 137Cs, 133Ba, 14C, 22Na and 204Tl. Research the type of emission,
energy and half life associated with each source. Be sure to cite any references used in
preparing these descriptions. Have any of these sources lost a significant fraction of their
activity since purchase? (Dates of manufacture are on each source.)
In the Introduction of your report, be sure to describe “ionizing radiation” as it applies to this
experiment. In the Theory section, review the properties of the Binomial, Poisson and Gaussian
distributions and their areas of applicability as well as expectations of the circumstances in
which each apply in this experiment. Give the theoretical expression for the mean and standard
deviation for each distribution.
Theory
In addition to analysis of the results, there is a lot of good physics to be seen in the construction
of the Geiger- Müller tube itself. Assume the G-M tube is a cylindrical capacitor whose
dimensions are given in the spec sheet and/or measured by you in the lab. Calculate the
capacitance and compare to the value in the manufacturer’s specification sheet (Fig.1). From
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Gauss’s Law, calculate the electric field inside the G-M tube. According to Tait, what
determines the rise time for the pulse? Calculate the rise time for your tube. (This does not
require physics more advanced than sophomore, but does include a numerical integration.)
Does the rise time depend on the supply voltage or another parameter you control? Compare
to your measurements. Melissinos claims (p. 177) that “…the whole gas becomes ionized, and a
discharge takes place.” Estimate the number of electrons and ions in the tube if this occurs.
(Use the Ideal Gas Law; assume the neon pressure is 75 Torr.) Compare to the number of
electrons you measure in a typical pulse. (How will you estimate the number of electrons in a
typical pulse?) Will this be a topic for discussion in your report? (This task is repeated later in
conjunction with analysis of how a pulse propagates through the circuit. You need do it only
once.)
Available Equipment
Radiation Sources
Geiger-Müller Tube
Electronics board for pulse processing
Junction box: High voltage to G-M tube and pulse output (gray chassis box)
High voltage power supply
Oscilloscope
Computer with automated Counting Software
Digital voltmeter
Experimental Setup
Familiarize yourself with the
setup, starting with a
comparison between the
physical detector (Fig. 1 and
2) and the schematic diagram
(Fig 3.). How does a Geiger-
Müller detector work? What
is its basic construction? How
does it produce a “pulse” to
signal the passage of
radiation? See Appendices 1
and 2 as well as the Wikipedia
article “Geiger-Muller Tube” Figure 1: Geiger experiment general layout
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for details. If you incorporate information from any of these sources in your report you must
cite them! (To paraphrase Barack Obama: “You didn’t think up these words and concepts
yourself.”)
Operation of the oscilloscope is explained in a PowerPoint posted on Canvas. Think of the
oscilloscope as
software: you learn
how it works by
playing with it.
Your report should
contain several
paragraphs
describing the
operation of the
oscilloscope:
enough information
so that the next
time that you have
to use one you can
use your report as a
reference. This
portion (like all others) should also contain sufficient detail to convince the grader that you
understand how an oscilloscope works.
Figure 2: Geiger electronics board
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Figure 3: Geiger counter circuit diagram
What does the initial pulse look like? How do the pulse shape and height depend on power
supply (anode – cathode) voltage? In sophomore physics you learned about the RC time
constant of a resistor-capacitor combination. (Series or parallel? Does it matter?) Exactly how
does the charge collected from the ionized neon atoms in the G-M detector work its way
through the initial parts of the circuit (up to the op amp)? Are the pulse rise time and duration
related to circuit components? How? Do your calculations based on your “circuit analysis”
agree with measurements from your ‘scope trace?
Let us investigate some characteristic times in the pulse generation system. Measure the pulse
duration. (It will be some microseconds.)
Calculate (estimate) the amount of charge in a typical pulse. (The oscilloscope measures the
voltage across what? How do you get a current, and subsequently charge, from this?) It is
claimed that all of the gas in the G-M tube is ionized by the passage of a “particle” of ionizing
radiation. Do your data support or contradict this “model”? At the percent level or within an
order of magnitude?
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Follow the pulse with the oscilloscope as it travels through the circuit. How does the signal
change throughout the circuit? Include images in your report. For a given source-detector
configuration, are all pulses alike? What pulse parameters change with supply voltage?
Document the pulse appearance as it goes through the circuit. Note that the oscilloscope has
[at least] six menu buttons that change the options available on the right of the screen. Items
that appear on the right of the screen are changed by pressing the buttons to their right. An
important option for investigating the long-term behavior of the Geiger counter is found under
the “Display” menu: Set the “Persist” option to “Infinite” with a sweep time of several hundred
µsec and watch the signal develop over several minutes. In this configuration the ‘scope will
give you information about the dead time of the system. In principle you can investigate the
source of the dead time by changing Rs; in practice, since the supply voltage appears across Rs,
this must be done with extreme care or your personal “dead time” will begin very soon and last
very long!) How does the appearance of pulses as seen on the oscilloscope screen depend on
the trigger threshold setting? (Does “threshold voltage” refer to an adjustment on the Geiger
circuit, the oscilloscope or both?) Record for inclusion in your report the pulse shape coming
from (a) the G-M tube, and (b) the counter input. How does the number of counts measured
under fixed conditions of the radiation source and G-M tube juxtaposition, depend on your
setting of the comparator threshold voltage? How should the number of counts depend on this
setting? Find the average of 100 measurements of the count rate when the threshold voltage is
1.0, 5.0 and 10V. Discuss in your report. Your report should include at least a brief description
of what an oscilloscope does and why/how it gives information and why we should trust your
operation of the device. How did you decide on values of important operating parameters for
both the oscilloscope and the Geiger counter circuit? (“That’s the way it was set up when we
got here” is not an acceptable explanation!)
In this experiment we use model 712 G-M tubes from LND Inc. Their specifications are given on
the next page.
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Figure 4: Geiger tube specifications (712)
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Experimental Procedure
For a description of the operation of the Labview data collection program, see Appendix 1.
1. Establish an operating point for the GM detector. Figure 5 shows an artist’s conception of
the Geiger-Müller counter output count rate as a function of applied supply potential, Vs.
The “plateau” region lies in the range , where the output count rate remains
approximately constant independent of . The operating point
OP should be in the plateau region, far removed from the “knee” of the curve and from the
region of spontaneous ionization. How will you determine the plateau region? This is
actually a very important exercise in both particle physics and general experimental design.
Assume that for a given source – Geiger tube configuration you can select a counting time
that gives you about 500 counts. Assume that your plateau region is 150 V wide. (This is a
number you will determine, but let’s start with this guess; it is certainly within a factor of
two.) The manufacturer’s specification on the slope of the plateau is “less than 6% per
100 V, so you would expect to see a maximum increase of 45 counts across the 150 volt
plateau. Obviously if the uncertainty in your measurement of the number of counts at a
given voltage is ± 50 counts, you are going to have a credibility issue trying to claim that you
know the slope is less than 6% per 100 volts. How about ± 5 counts? If your data are
described by a Gaussian distribution, what will be their standard deviation…or is it the
standard deviation of the mean that is relevant? How many times will you have to repeat
the measurement to have 68% confidence that the mean of your measurements lies within
5 counts of the value you would get if you took an infinite number of readings at each
voltage? This is not a trivial concept, but the ability to do the calculation is very important.
Figure 5: Response curve of Geiger-Mueller counter
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Being able to do calculations like this has a huge significance for experimental design. These
considerations are not unique to particle physics: the cutting edge of nearly all experimental
physics is determined by noise in your measuring apparatus; that noise contributes the
same uncertainty as random variations in particle counts. Optical measurements in nano-
physics are often done with single photon counting systems.
First consideration before starting serious data-taking: which source to use and which side
should be facing the detector? We have about a dozen sources available. One day all were
investigated with the same detector at the same supply voltage. The number of counts
measured one day in a certain window varied between 3.8 and 1300. Is the inherent
activity of the source relevant to any consideration in your experiment? If so, you may want
to investigate several options for the source you choose and the side of the source you
place nearest the detector.
DO NOT OPERATE THE DETECTOR OVER 600V, even if you have NOT reached the upward
swing located after V2 on the graph. We have routinely ruined Geiger-Mueller tubes before
the onset of spontaneous ionization. Is there a difference in count rate depending on which
side of the sample faces up or down? To revisit a consideration mentioned above, the
manufacturer specifies a maximum slope for the plateau region. How does this
specification compare to your data? Explain what is meant by “6%/100V”. If you simply
plot your data and fit it, is the slope of your fit equal to this? Explain.
Nest issue. It is generally considered to be a scientific faux pas to publish results which are
characteristic of the measuring apparatus rather than the physical system being studied.
One opportunity for checking such a possibility is to measure the count rate of your source
as a function of the time window for which you measure the number of counts. Obviously
the emission rate of the source could care less for how long you take data. Calculate the
emission rate from a sample in a specific configuration for different measuring times. Do
you always get the same answer? How certain are you that your measurements are
different or the same?
2. Detection of radiation from an active source
For a given radioactive source, measure the count rate numerous times (at least 100) to
create a good distribution of values. Appendix 1 explains how to use the computer to take
data for this part of the experiment. Be sure to record the length of time for which the
detector counts. Adjust the counting time and source-detector distance so that you get
several hundred counts in each measurement. Plot the results in a histogram. Which type
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of statistics do you think best describe this distribution? Try fitting it with the different
statistical distributions. Remember that the statistical distributions describe probability
densities. Refer to Appendix 2 for a suggestion on how to establish bin widths. Was your
prediction correct? What is the reduced of the fit? Do the data support the model or
not? (Obviously if you have one set of data and three different models, not all will be
applicable!) You should be able to determine ̅ ̅ for each of the fits. How do
they compare to the values of ̅ ̅ you obtain by straight calculation? Also
observe ̅ ̅ as a function of the amount of data you take. Do this, e.g., for 2, 5, 10,
20, 50, 100, 200, 500 and 1000 measurements. You are also encouraged to take one data
set overnight or even over a weekend with a very large number of measurements (like
20,000). Discuss how your results vary as functions of how many data points you take.
3. Detection of background radiation
In the absence of a source, your Geiger-Müller counter will detect background radiation at a
rate of a count every few seconds. Measure the count rate numerous times to create a
good distribution of values for the background radiation rate. Plot the results in a
histogram. Which type of statistics do you think can be used to describe this distribution?
Try fitting it with the different possible statistical distributions. Remember that the
statistical distributions describe distributions of probability densities. Was your prediction
correct? What is of the fit? You should be able to determine ̅ ̅ from the fits.
How do they compare to the ̅ ̅ you obtain by straight calculation?
4. [Optional for 10 points extra credit] Binomial Distribution
It is widely known and accepted that the error in a binning experiment is given by the
square root of the number of measurement values in the bin. This follows from the fact
that the standard deviation of a Poisson distribution is given by √ ( ) and if the
probability of the event occurring is small, , so the square root reduces to
√ . But is simply the average value of N, the distribution average, μ. To
investigate a distribution where approximating the error by the square root of the average
is a bad idea because the underlying assumption is incorrect, choose a time interval, , for
measuring the arrival of background radiation such that the mean number of observed
background counts, , is significantly less than one, e.g. 0.2; your time interval will probably
be less than 1 sec. For a given counting configuration, the average number of counts in a
short time will be the same at the average for a long time, which can be determined either
with the manual start/stop and a stopwatch or with the computer. (There is actually a
function in Excel for counting the number of occurrences of some value.) According to the
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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 are claimed to be binomial, not Poissonian as one might expect.
To test this, perform the following procedure under the conditions listed above:
(a) Take 100 measurements of the rate. Record the observed number of zeroes.
(b) Repeat the above step 10 times.
(c) Compute the mean and standard deviation of the number of zeroes in your 10 sets of
measurements.
(d) Compare your observed standard deviation with the predictions of Binomial statistics
and Poisson statistics.
(e) Discuss why, despite the fact that this “experiment” is somewhat contrived, it illustrates
one common circumstances where one can err in accepting the “common knowledge” that
the square root of number of counts in a bin gives a good estimate of its error.
References
[1] J.R. Taylor, An Introduction to Error Analysis, 2nd Ed., University Science Books, Sausalito CA
(1997). Copy in the Undergrad Lab.
*2+ D. Scott, “On optimal and data-based histograms”, Biometrika 66 605 (1979).
[3] C. Melissinos and J. Napolitano, Experiments in Modern Physics, Academic Press, San Diego,
CA (2003). Copy in the Undergrad Lab.
[4] W. H. Tait, Radiation Detection, Butterworths, London (1980). Copy in the Undergrad Lab.
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Appendix 1: Automated data collection from Geiger Detector
One of the primary strengths of computers is to perform a series of actions repeatedly and
accurately. In the context of the Geiger counter experiments, this means automated data
collection of counts.
The computer program is written in Labview and is called MCC4301countNhistogram.vi. This
should be located on the PC in c:/labviewvi. The front panel looks like the image below. The
'total reads' is where to enter the desired number of periods to collect data. The 'period (sec)'
is where to enter the counting duration. The indicator boxes “current count' and 'mean' display
data as the program is run. To run the program, first enter desired values in the total reads and
period boxes then click the arrow button below the “Edit” menu. The program will begin by
asking you for a file name. Start by defining a sub-directory for you and your lab partner,
preferably not on the desktop but rather in the “users” directory. The program will run and
plot a histogram with bin width equal to one during data collection. The counter board will
flash one of the LED's during data transfer to the PC. The button labeled 'stop' stops the
counting and proceeds to the data saving screen. When finished, a pop-up window will ask
where to save data. The resulting text file has 3 columns of data. There are the histogram bin
values, the counts in the bins, and then the raw numbers. These raw numbers can then be used
when plotting a histogram of the data.
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Appendix 2: Selecting a bin width
As a guide to choosing your histogram bin size, Reference [2] suggests the prescription
(4)
Where W is the bin width, the standard deviation of the distribution and N the total number
of histogram entries (measurements). As a practical matter, it may be preferable to give an
integer width to your bins, close to the W recommended by the question. (In counting
experiments this is the only option that makes sense.)
An alternative procedure is provided below. This is clearly more work but should produce a
superior product.
Shimazaki and Shinomoto, Neural Comput 19 1503-1527, 2007
I. Divide the data range into bins of width . Count the number of events that
enter the i'th bin.
II. Calculate the mean and variance of the number of events as
and *
III. Compute a formula*,
IV. Repeat i-iii while changing . Find that minimizes .
*VERY IMPORTANT: Do NOT use a variance that uses N-1 to divide the sum of
squared errors. Use the biased variance in the method.
*To obtain a smooth cost function, it is recommended to use an average of cost
functions calculated from multiple initial partitioning positions.
Finally the procedure probably most commonly used: play with the bin size until your fit is to
your liking. The above option is a mathematical formalism for accomplishing the same task!
Enjoy!