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1A: OPERATING CHARACTERISTICS OF THE GEIGER COUNTER
OBJECTIVE
The objective of this laboratory is to determine the operating voltage for a Geiger tube and to
calculate the effect of the dead time and recovery time of the tube on the counting rate.
INTRODUCTION
Background & Theory
In medical and biological research, radioactive isotopes are utilized in three types of
measurements: studies of the various chemical and physical interactions within a living
organism or with its environment, measurements of the distribution of elements and
compounds in the body and measurements of radioactive isotopes in the living organisms
without taking samples. The techniques used in these measurements depend on the fact that the
radioactive isotopes emit ionizing radiations, which can be detected by their effects on a
photographic emulsion, or by electrical methods.
Gases conduct electricity only when a number of their atoms are ionized, i.e. split up into a
number of free electrons and positive ions. Alpha, beta or gamma radiation emitted by
radioactive materials ionizes atoms with which they collide. Hans Geiger, an associate of
Rutherford used this property to invent a sensitive detector for radiation.
Mica
Window
α, β, or γ
radiation
Outer Cylinder ( - Voltage)
+
Central Wire ( + Voltage)
_
R
Voltage spike
to counter
V
FIGURE 1: SCHEMATIC OF A TYPICAL GEIGER COUNTER Figure 1 above displays a typical setup for a Geiger counter; it usually contains a metal tube
that encloses a thin metal wire along the middle. The space in between is sealed off and filled
with a suitable gas; the wire is set at a very high positive electric potential relative to the tube.
An electron, positive ion, or gamma radiation that penetrates the tube through a mica (an
extremely low density material that minimizes attenuation of radiation as it enters detector)
window, will ionize a number of the atoms in the gas, and because of the high positive voltage
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of the central wire, the electrons will be attracted to it while the positive ions will be attracted
to the wall. The high voltage accelerates the positive and negative charges, and hence they gain
more energy and collide with more atoms to release more electrons and positive ions; the
process escalates into an "avalanche" which produces an easily detectable pulse of current.
With the presence a suitable filling gas, the current quickly drops to zero so that a single
voltage spike occurs across a resistor; an electronic counter then registers this voltage spike. A
typical composition of the gas filling a Geiger counter tube was usually a mixture of argon and
ethanol; more recently, tubes filled with ethyl formate in place of the alcohol are reported to
have a longer life and smaller temperature coefficients than counters filled with ethanol.
A very important property of the Geiger counter is its self-suppressing mechanism. The
counter is triggered by the pulse from the tube and feeds back a square pulse of 300-500 µsec
duration to the central wire. This pulse has an opposite polarity and high enough amplitude to
extinguish the discharge. This allows for the counter to reset as fast as possible in order to
register the next voltage spike induced by the penetrating radiation.
The Geiger detector is usually called a "counter" because every particle passing through it
produces an identical pulse, allowing particles to be counted; however, the detector cannot tell
anything about the type of radiation or its energy/frequency - it can only tell that the radiation
particles have sufficient energy to penetrate the counter.
To improve its sensitivity to alpha and beta particles, the ST150 detector has a very thin mica
window with a superficial density of only 1.5 – 2 mg/cm2. This window is therefore extremely
fragile and if broken cannot be repaired. Never allow any object to touch the window!
The Geiger Tube Voltage Characteristic Curve
The most important information about a particular counter tube is its voltage characteristic
curve. The counting rate due to a constant intensity radioactive source is graphed as a function
of the voltage across the counter; A curve of the form shown in Figure 2 is obtained.
Counting
Rate
Continuous
Discharge
Region
Proportional
Discharge
Region
Start
Voltage
Geiger
Threshold
Voltage
Geiger Plateau
Operating
Voltage
V0
Geiger
Breakdown
Voltage
FIGURE 2: GEIGER TUBE VOLTAGE CHARACTERISTIC CURVE 2
Applied
Voltage
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The counter starts counting at a point corresponding approximately to the Geiger threshold
voltage; from there follows a “plateau" with little change in the counting rate as the voltage
increases. Finally a point is reached where the self-suppressing mechanism no longer works,
and the counting rate rapidly increases until the counter breaks down into a continuous
discharge. In order to ensure stable operation, the counter is operated at a voltage
corresponding approximately to the mid-point of the plateau.
Hence, a flat plateau is regarded as a desirable characteristic in a counter; a long plateau is also
desirable, but is not as important. In practice most counters have a slightly sloping plateau,
partly because of geometrical limitations of the counter design, and partly because of spurious
counts due to an unsatisfactory gas filling or to undesirable properties of the cathode surface.
The correct operating voltage for any particular Geiger-Mueller tube is determined
experimentally using a small radioactive source such as Cs-137 or Co-60. A properly
functioning tube will exhibit a "plateau" effect, where the counting rate remains nearly constant
over a long range of applied voltage; the operating voltage is then calculated roughly as the
voltage value corresponding to the middle of the plateau region.
Dead Time, Recovery Time and Resolving Time
Geiger-Mueller tubes exhibit Dead Time effects due to the recombination time of the internal
gas ions after the occurrence of an ionizing event. The actual dead time depends on several
factors including the active volume and shape of the detector and can range from a few
microseconds for miniature tubes, to over 1000 microseconds for large volume devices.
The counter discharge occurs very close to the wire, and the negative particles, usually
electrons, are collected very rapidly. The positive ions move relatively slowly, so that as the
discharge proceeds a positively charged sheath forms around the wire. This has the effect of
reducing the field around the wire to a value below that corresponding to the threshold voltage,
and the discharge ceases. The positive ion sheath then moves outwards until the critical radius
r is reached, when the field at the wire is restored to the threshold value. This marks the end of
the true "dead time". If another ionizing event triggers the counter at this stage, a pulse smaller
than normal is obtained, as the full voltage across the counter is not operative. However, if the
positive ions reach the cathode before the next particle arrives, the pulse will be of full size.
This effect can be demonstrated with a triggered oscilloscope; as shown in Figure 3, the period
during which only partially developed pulses are formed is termed the Recovery Time. The
effective Resolving Time or insensitive time following a recorded pulse, is determined by both
the dead time and the recovery time, and will depend not only on a number of parameters
associated with the counter dimensions and gas filling, but in principle, also on the operating
voltage of the counter, on the sensitivity of the electronic recording equipment and on the
counting rate. It is necessary to apply appropriate corrections to the observed counting rates to
compensate for this resolving time.
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PULSE AMPLITUDE
Second Geiger
Discharge
Counted
Initial
Geiger
Discharge
Counted
Minimum
Input
Sensitivity
Level of
Counter
Possible
Events Not
Counted
Dead
Time
TIME
Recovery
Time
Resolving Time
FIGURE 3: A PICTORAL REPRESENTATION OF DEAD, RECOVERY & RESOLVING TIME True versus Measured Count Rate
When making absolute measurements it is important to compensate for dead time losses at
higher counting rates. If the resolving time Tr of the detector is known, the true counting rate
Rt may be calculated from the measured counting rate Rm using the following expression:
Rt =
Rm
1 − R m ⋅ Tr
(Eq. 1)
If the detector resolving time is unknown, it may be determined experimentally using two
radioactive sources simultaneously. Maintaining constant counting geometry is important
throughout the experiment; hence a special container carrying both sources would be ideal for
performing the measurement – however, good results may be obtained by careful positioning
the two standard sources side by side. With the operating voltage set for the GM tube, denoting
the measured count rate for the two sources (a+b) side by side as R(a+b), the measured count
rate for source a alone as R(a) and the measured count rate for the source b alone as R(b), the
resolving time is given by:
Tr =
R(a) + R(b) - R(a + b)
2R(a).R(b)
(Eq. 2)
Because of the solid state electronics used in the circuitry of the ST150 Nuclear Lab Station its
own resolving time is very short - one microsecond or less – and so, not significant compared
to that of the GM tube. Therefore, only the resolving time of the GM tube affects the true count
rate.
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EQUIPMENT
•
•
•
The ST150 Nuclear Lab Station (Figure 4 below) provides a self-contained unit that
includes a versatile timer/counter, GM tube and source stand; High voltage is fully
variable from 0 to +800 volts.
Associated software that allows for operation of ST150 and data collection.
Two types of radioactive sources: 137Cs and 60Co.
FIGURE 4: THE ST-­‐150 NUCLEAR LAB STATION PROCEDURE
Part 1: Measuring the Geiger Plateau and Determining Operating Voltage
1. Record the model number of the Geiger counter. Verify that the Geiger counter has
been connected to a power source and to the lab computer.
2. On the desktop, open the Applications folder and open STX; this is the software
program used to operate the ST-150 lab station. Open MATLAB to prepare for data
collection; refer to Video 1A if needed.
3. Switch on the GM Detector. Check if STX is now activated.
4. Get familiar with the software program – Perform a trial experiment that runs:
a) Three trials in a row, automatically
b) Duration of each trial should be 10 seconds
c) Operate at a voltage of 200V.
5. If you are comfortable with using STX, sign out one radioactive source from your TA.
Record the name and activity of the source.
6. Place the source on a source tray and slide it into the closest grating to the window of
the GM detector.
7. Determine the threshold voltage value of the GM detector; starting at around 200V,
increase the high voltage, a minimum of 10V, until radiation events just begin to be
detected.
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8. Start data collection at 10V behind the threshold value, and increase the voltage in
increments of 20V, recording the counting rate (counts/min) at each increment, until
you reach 800V.
9. If the counts increase dramatically before the 800V mark, do not continue to operate
the tube until you consult with your TA - you may irreversibly damage the tube.
Part 2: Acquiring the Resolving Time of the Geiger Tube
1. Set the high voltage to the operating voltage found in part 1 and the time interval to 5
minutes.
2. Record the count rate for the first source (a) you have already used in part 1 as R(a).
3. Sign out a second radioactive source (b) from your TA and record the count rate of the
second source (b) as R(b).
4. Carefully place both sources side by side on the transparent source stand and record the
count rate as R(a+b).
5. Once data collection is complete, set the voltage to 0V and switch off the GM tube.
**When you have completed Lab 1A, return all sources to your TA**
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DATA ANALYSIS (**Refer to Video 1A**)
1. Using the data obtained, make a plot of the counting rate versus the applied voltage;
add the table as well as the labelled graph to your report.
2. Determine the value of your instrument’s operating voltage. The recommended Geiger
operating voltage may be determined as the center of the plateau region. In the
example below, the plateau extends from approximately 350V to 600V. A reasonable
operating voltage in this case would be 500V.
3. Calculate the resolving time of the tube with the values acquired in Part 2.
4. Using the recovery time calculate the true counting rates for the two individual sources.
5. Find the percentage difference between the measured and the true rates for both sources
used – does this difference seem reasonable?
6. Re-plot the graph of counts vs. voltage in Part 1, AND with the adjusted true rates and
determine the value of the normal operating voltage. How does the resolving time
affect the value of this normal operating voltage determined in Part 1?
7. Were there any errors associated with the experiment? Explain.
8. How can you make this experiment more accurate? Explain.
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1B: STATISTICAL NATURE OF RADIATION COUNTING
OBJECTIVE
The objective of this experiment to explore the statistics of random, independent events in
physical measurements and to test the hypothesis that the distribution of counts of random
emission events from a gamma source using a Geiger-Muller detector either fit a Poisson or
Gaussian distribution, depending on the length of time used for those counts and the range of
the mean of those counts.
INTRODUCTION AND THEORY
Radioactive decay and consequently the emission of particles is a randomly occurring process
and therefore any count of the radiation emitted is subject to some degree of statistical
fluctuation. There is a continuous change in the activity of a source from one instant to the next
due to the random nature of radioactive decay and there are also fluctuations in the decay rate
due to the half-life of the radionuclide.
The Poisson distribution is applied, generally speaking, to situations where only a few events
are counted. It is equally valid when large numbers of events are counted, but the Gaussian
distribution can be used then, and it is easier to use.
The Poisson random variable represents the number of “successes” within a specified time
interval or region of space:
𝑃 𝑋 = ! !! ! !
!!
(Eq. 3)
where 𝜇 is the mean number of successes, and x is any non-negative integer. The mean and
variance of the poisson distribution are both equal to 𝜇.
The Gaussian, or Normal random variable is normally distributed between mean 𝜇 and
variance 𝜎 ! , and is described by the equation:
𝑓 𝑋 = !
!
𝑒!
!!
!!! ! /!! !
(Eq. 4)
Please refer to the Appendix for addition information on Statistics Definitions, as well as a
mathematical derivation of the Poisson & Gaussian distributions as outlined in Equations 3 and
4.
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EQUIPMENT
•
The ST150 Nuclear Lab Station provides a self-contained unit that includes a
versatile timer/counter, GM tube and source stand; High voltage is fully variable
from 0 to +800 volts.
FIGURE 5: THE ST-­‐150 NUCLEAR LAB STATION PROCEDURE
Part 1: Background Radiation only, and Short Time Intervals
1. Use of only the background radiation is required
2. Adjust the high voltage to the operating voltage value obtained in the previous
experiment. (Make sure you are using the same instrument!)
3. Adjust the time interval to 10 seconds and obtain 50 separate counts
4. Record the information onto MATLAB, creating a column vector for the run number
and a column vector, background, for the counts observed.
Part 2: 60Co Gamma source and Long Time Intervals
1. Sign out a 60Co source from your TA and place it on the plastic tray in the second
lowest position from the GM mica window.
2. It is assumed that the high voltage has been adjusted to the normal operation value from
part one. Check again that the value of the high voltage is correct.
3. Adjust the recording time interval to 2 minutes and obtain 50 separate counts
4. Record the information onto MATLAB, creating a column vector, gamma, for the
counts observed
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DATA ANALYSIS (**Refer to Video 1B**)
1. Find the mean and standard deviation of both sets of data – investigate what commands
can be used on MATLAB to obtain these parameters for your data.
2. On MATLAB, investigate the command pdf – this command creates a probability
density function of many distributions. Play around with the parameters, and create two
figures that plots a poisson and Gaussian (Normal) distribution – your plots should
resemble Figure 1 in the appendix.
3. On MATLAB, investigate the command hist – this command generates a histogram of
the data, with a parameter to adjust the width of the histogram bars.
4. Using the Data in parts 1 and 2, create a histogram of the counts, and analyse its shape:
5. How well does the shape of your histogram using Part 1 data resemble the
characteristic skewed shape of the Poisson distribution?
6. How well does the shape of your histogram using Part 2 data resemble the expected
shape of the Gaussian distribution?
7. For both data, find out how many points fall within one σ of the mean
8. Give an overall comparison between the two sets of results – why do you think Part 1
and Part 2 differed in the generated distribution profiles?
9. Were there any errors associated with the experiment? Explain.
10. How can you make this experiment more accurate? Explain.
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APPENDIX
LAB #1
STATISTICS DEFINITIONS:
Statistics
Mean
Sample
x
Population
Μ
2
Variance
s
standard deviation
s
The definitions of these symbols are:
Sample Mean:
1
x=
σ2
σ
N
∑x
N
µ = lim x
N →∞
i
i =1
Sample Variance:
s2 =
1 N
∑ x−x
N − 1 i =1
(
2
)
σ 2 = lim s 2
N →∞
(Note the N - 1 in the definition of s2 ; this makes s2 a correct estimator of σ2.)
DERIVATION OF POISSON/GAUSSIAN DISTRIBUTIONS:
Suppose a sample has n radioactive nuclei, with known half-life and probability of counting in
a detector when they decay. What would be the probability of recording k counts during a
given time interval? The answer is provided by the binomial distribution. The Poisson
distribution describes the large-n limit (k large or small), and the Gaussian distribution applies
in the large-n, large-k limit.
Let us denote:
•
•
p - the probability of a single given nucleus of decaying (and so, of being counted)
during a given interval of time. It could be also treated as the “average number of
counts from a single nucleus”.
q - the probability of a nucleus of surviving the radioactive transformation (notdecaying and so of not being counted) during the same set interval of time.
Let us label the n nuclei in the sample “1, 2,…, k, k + 1,…, n”. Then, the probability of the set
of nuclei labeled “1 through k” of decaying and being counted, and of the set of nuclei labeled
“k+1 through n” of surviving and not being counted is:
P(p, k) = p k ⋅ q n −k
(1)
Since decaying takes place randomly, another set of k nuclei could also give the same number
of counts, with the same probability. The total number of “ways” of getting k counts
corresponds to the number of distinct permutations of the labels of the n nuclei. The total
number of such permutations is
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n!
k!(n − k )!
LAB #1
(2)
It follows that the probability of counting k decaying nuclei out of n (given that the probability
of the decaying transformation is p) is:
P(n, p, k ) =
n!
p k q n −k
k!(n − k )!
(3)
This is the general form of the binomial distribution. The binomial distribution function
specifies the number of times (k) that an event occurs in n independent trials where p is the
probability of the event occurring in a single trial. It is an exact probability distribution for any
number of discrete trials. If n is very large, it may be treated as a continuous function. This
yields the Gaussian distribution. If the probability p is so small that the function has significant
value only for very small k, then the function can be approximated by the Poisson distribution.
Therefore, for large number of atoms in the sample and very small probability of decaying (p
→ 0, n >> k), the previous distribution becomes the “Poisson distribution”. Let us denote the
average number of counts λ. Since the probability p had the meaning of the “average number of
counts from a single nucleus”, it follows that the average number of counts from all the nuclei
µ = pn
(4)
According to the definition of the two probabilities p and q,
q=1–p
(5)
And so,
q
n −k
= (1 − p)
n −k
= (1 − p)
1
⋅p ( n − k )
p
(6)
Furthermore, if n is large compared to k (n >> k), n – k ≈ n, and the equation above becomes:
q
n −k
= (1 − p)
n −k
= (1 − p)
1
⋅λ
p
(7)
At the limit when p → 0 (p << 1), this becomes:
lim q n −k = lim (1 − p)
p→0, n >> k
1
⋅p ( n − k )
p
p→0, n >> k
= lim(1 − p)
p →0
Where we have used the definition of the number e:
1
lim(1 + x ) x = e
x →0
12
1
⋅µ
p
1
− ⎞
⎛
⎜
= lim(1 − p) p ⎟
⎜ p→0
⎟
⎝
⎠
−µ
= e −µ
(8)
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RYERSON PHYSICS
LAB #1
n!
⋅ p k ⋅ e −µ
k!(n − k )!
lim P(n, k, p) =
p →0
(9)
We can also, at the limit n >> k, approximate
n!
1 ⋅ 2 ⋅ ...( n − k ) ⋅ (n − k + 1) ⋅ (n − k + 2) ⋅ ... ⋅ n
=
= (n − k + 1) ⋅ (n − k + 2) ⋅ ... ⋅ n ≈ n
⋅ n
⋅ ...
⋅ n = nk

(n − k )!
1 ⋅ 2 ⋅ ... ⋅ (n − k )
k − − times
therefore, we obtain the Poisson distribution:
n!
λk ⋅ e −µ
n >> k , p → 0
k n −k
P(n, k, p) =
p q ⎯⎯⎯⎯→ P(k, µ) =
;
k!(n − k)!
k!
where k = 0, 1, 2, . . .
(10)
For n, k, and n - k all large, and p is no longer approaching zero, we obtain the Gaussian
distribution, given here without derivation:
P( k , λ ) =
1
2πµ
⋅e
−
( k −λ ) 2
2µ
where k = 0, 1, 2, . . .
(11)
Here λ is the most probable number of counts, and k is the actual number counted. This is a
very practical relation, permitting a graphical determination of P in cases where calculating the
terms would take a long time.
If n and therefore k is very large and the argument in equation 11 is no longer discrete but can
be treated as continuous x, equation (11) becomes:
P( x ) =
1
σ 2π
⋅e
1 ⎛ x −µ ⎞
− ⋅⎜
⎟
2 ⎝ σ ⎠
Where, µ and σ are the mean and standard deviation.
13
2
(12)
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Figure 1: Examples of the Poisson and Gaussian distributions.
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