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!