Advanced lab course for bachelor students in physics

Experiment T7

Gaseous ionisation detectors and statistics

January 2019

Prerequisites

• Passage of charged particles through matter

• Gas-filled particle detectors

• Probability distributions and statistics

Goal of the experiment

• Operation of gaseous ionisation detectors

• Measurement of statistical distributions

Table of Contents

1 Gaseous Ionisation Detectors 31.1 Principle of the Gaseous Ionisation Detector . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Operating Regions of a Gaseous Ionisation Detector . . . . . . . . . . . . . . . . . . . . . 31.3 The Ionisation Chamber (Example of a Simple Gaseous Ionisation Detector) . . . . . . . 41.4 The Proportional Counter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 The Geiger-Muller Counter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Probability Distributions and Statistics 92.1 Binomial, Poisson and Gaussian Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Spread and Measurement Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 The χ2 Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 The χ2 Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Procedure 143.1 Measurements with the Geiger-Muller Counter . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Measurements on Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Measurement of the Characteristic Curve of a Proportional Counter . . . . . . . . . . . . 14

3.3.1 In Case This Part of the Experiment is Done Before the Geiger-Muller Counter . . 15

4 Analysis 154.1 Measurements with the Geiger-Muller Counter . . . . . . . . . . . . . . . . . . . . . . . . 154.2 Measurements on Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3 Measurements with the Proportional Counter . . . . . . . . . . . . . . . . . . . . . . . . . 16

A Properties of the Radioactive Samples Used 17

B Bethe-Bloch Formula 17

Bibliography

[1] Gerd Otter, Raimund Honecker: Atome – Molekule – Kerne,Band I: Atomphysik, Band II: Molekul- und Kernphysik, Bo 184

[2] William R. Leo: Techniques for Nuclear and Particle Physics Experiments, Dr 155

[3] Konrad Kleinknecht: Detektoren fur Teilchenstrahlung, Dr 143

[4] Walter Blum, Luigi Rolandi: Particle Detection with Drift Chambers, Dr 181

[5] Claus Grupen: Teilchendetektoren, Dr 189

[6] Hanno Krieger: Strahlenphysik, Dosimetrie und Strahlenschutz, Du 130

[7] Siegmund Brandt: Datenanalyse, mit Beispiel- und Ubungsbuch, Cu 202

[8] Strahlenschutzverordnung, http://www.bfs.de

[9] The Particle Detector Brief Book, http://physics.web.cern.ch/Physics/ParticleDetector/BriefBook/

[10] Detecting Particles: How to see without seeing,www.physics.ucdavis.edu/Classes/Physics252b/Lectures/252b lecture6.ppt

[11] Properties of argon-ethane/methane mixtures for use in proportional counters, Nuclear Instrumentsand Methods, vol. 188, issue 3, pp. 521-534, (10/1981)

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1 Gaseous Ionisation Detectors

1.1 Principle of the Gaseous Ionisation Detector

For the detection of ionising radiation, gas-filled detectors are often used. The detection is based on theionisation of the gas atoms, which is caused by incident radiation. There are three types of detectors:

1. Ionisation chamber (not part of this experiment)

2. Proportional counter

3. Geiger-Muller counter

An ionising particle produces electrons and positive ions on its way through the gas-filled detector.The electrons move towards the anode very quickly, whereas the ions move relatively slowly towards thecathode. Is the electric field near the anode very strong (proportional and Geiger-Muller counter), thenthe electrons are accelerated in the immediate vicinity of the anode. They themselves then produce moreelectrons and ions by impact ionisation. The motion of all charges causes a current pulse in the externalresistor and therefore causes a negative voltage pulse. This pulse is forwarded to the readout electronicsvia a coupling capacitor.

1.2 Operating Regions of a Gaseous Ionisation Detector

Figure 1 shows the total charge Q arriving at the anode versus the counter voltage U (logarithmic axes).A and B show the curves for two incident particles with different ionising power (e.g. A: α particle, B: βparticle). The pulse amplitude at the output of the counter is proportional to Q.

Figure 1: Total charge versus the counter voltage.

Region I: The pulse amplitude increases with the voltage. As the voltage increases, the probability thatthe created electrons and ions recombine decreases.

Region II: Above a voltage U1, recombination no longer occurs. The measured charge stays constant fora certain voltage range and is equal to the sum of the primary ionisation charges. The ionisationchamber operates in this region.

Region III: Starting from a voltage U2, the electrons are accelerated so strongly near the anode, thatthey can ionise other gas atoms in collisions. Charge avalanches are generated. The total chargestill remains proportional to the number of primary ionisation charges. A counter operating in thisregion is therefore called a proportional counter. The multiplication of the primary ionisationcharge caused by impact ionisation is called gas amplification.

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Region IV: Above the voltage U3, the proportionality of the total charge to the number of primaryionisation charges is limited.

Region V: The Geiger region starts at U4. A property of the Geiger region is that all ionising particlescause the same voltage pulse, independent of their type and energy. A counter operating in thisregion is called a Geiger-Muller counter. The gas amplification (≈ 109) is substantially larger fora Geiger-Muller counter than for a proportional counter, meaning even without additional electronicamplification, pulses of several volts can be generated.

Region VI: The pulses grow larger for increasing voltages, until ultimately a continuous discharge oc-curs, which damages the counter.

1.3 The Ionisation Chamber (Example of a Simple Gaseous Ionisation De-tector)

The ionisation chamber is one of the oldest detection devices for radiation and is still often used todaybecause of its simple setup. It is shown schematically in figure 2.

Figure 2: Schematic depiction of an ionisation chamber.

The energy required for the creation of an electron-ion pair in gases is roughly 30 eV, independentof the type and energy of the ionising particle. If, for example, the incident radiation deposits 1 MeVof energy in the chamber, then roughly 3 · 104 electron-ion pairs are produced, which corresponds to acharge of roughly 5 · 10−15 C for each sign. Depending on the construction, the charges created by theionising particle are integrated to a constant current (current mode) or generate a voltage pulse for everyindividual particle (pulse mode).

For a typical value for the capacitance of C = 10−12 F between both electrodes of the ionisationchamber, a chamber operated in current mode with a load resistor of e.g. R = 1012 Ω has a time constantof 1 s, which is large compared to the time required for the collection of the ions (≈ 1 ms). Therefore,one does not detect the passage of individual particles, but measures the currents integrated over longertimes. Such chambers are used e.g. for dose or dose rate measurements for radiation protection and formonitoring and controlling the intensity at particle accelerators. The currents or charges to be measuredare exceptionally small, so a good electrical insulation is required for such a chamber. Another problemare recombination processes of the gas, which can interfere with the linear relation between absorbedenergy and measured current.

For a chamber operated in pulse mode, the resistance is on the order of R = 106 Ω (to be confirmed),so the pulse length for the same capacitance is on the order of 1µs.

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1.4 The Proportional Counter

Principle and Operation

Figure 3: Schematic representation of a proportional counter.

A proportional counter is shown schematically in figure 3. The cylindrical tube (cathode) is grounded.A positive high voltage (U ≈ 1 kV) is applied to a thin anode wire (20 to 100µm in diameter) across aresistor (R ≈ 1 MΩ). Because of its cylindrical geometry, the proportional counter possesses an electricfield, which increases strongly near the anode (E ∝ 1/r), with field strengths of 104 to 105 V/cm, suchthat gas amplification occurs near the anode wire. The height of the measured pulse is proportional tothe energy of the particle, so the proportional counter can be used for energy measurements. The gasamplification factor (104 to 106) increases exponentially with the applied voltage. In addition to impactionisation, UV photons are emitted by excited atoms. These photons free photoelectrons from the gas orthe chamber walls, which can in turn cause new impact ionisations.

If the detector is filled with pure noble gases, the gain increases very rapidly with the voltage due tothe large amount of photoelectrons. The proportional region is therefore very small and poorly suited forpractical applications. If polyatomic gases, e.g. methane, are used, then the proportional region becomeslarge. The UV photons are absorbed by the molecules, i.e. the molecules dissociate or vibrations areexcited. The quanta emitted by the vibrating states are generally less energetic and can no longer freephotoelectrons. Mixtures of noble gases and polyatomic molecules are often used. Using a proportionalcounter, resolution times of roughly 10 ns can be achieved.

Gas-flow Proportional Counters

In so-called gas-flow proportional counters, the fill gas is continuously exchanged via a regulating valve.These counters are particularly useful for measuring α and low-energy β particles, as well as soft x-rays, because the sample can be put directly inside the counter volume. Counters with a gas-flow aretightly sealed with gas-tight walls, which short-ranged radiation cannot penetrate. A methane-gas-flowproportional counter is shown schematically in figure 4. At atmospheric pressure, an argon-methanemixture (10 % methane) flows through the counter volume, in which a thin wire loop serves as the anodeand the walls as the cathode.

Characteristic Curve of the Proportional Counter

The characteristic curve of a counter is the dependence of the pulse rate (number of pulses per unit oftime) on the voltage U applied. For a proportional counter, this is the curve shown in figure 5 if both αand β particles are simultaneously present.

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Figure 4: Schematic representation of a methane-gas-flow proportional counter.

Figure 5: Characteristic curve of a proportional counter in the presence of both α and β particles.

For a specific voltage (threshold voltage UT ), the minimum pulse height, which depends on the inputsensitivity of the subsequent amplifier, is reached. Because of the larger number of produced electron-ionpairs, only α particles are counted at first and only for higher voltages are the β particles counted as well,so the characteristic curve first reaches the α plateau and then the (α+β) plateau (figure 5). By choosingthe voltage correspondingly, the proportional counter can therefore distinguish primary ionisations ofdifferent sizes, i.e. α and β radiation.

1.5 The Geiger-Muller Counter

The Counter Discharge in the Geiger Region

When the field strength in a proportional counter is increased, more and more UV photons are producedin the avalanche, increasing the probability that photons generate electrons through the photoelectriceffect at other, more distant locations in the counter. These electrons induce new avalanches, causing thedischarge to propagate along the counter wire. At the end of the avalanche formation, the counter wireis surrounded along its entire length by a plasma tube. The counter is operating in the Geiger region.

During the relatively short time (roughly 10 ns) in which the electrons are siphoned off, the ion tube,which moves slowly, shields the electric field of the counter wire. The space charge of the ion tube reducesthe field strength and therefore, no new avalanches can be triggered. The positive ions wander to thecathode, where they are neutralised at the counter tube. This could free secondary electrons from the

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surface, which could restart the discharge process. It should therefore be ensured, that the discharge is

”quenched“.

Non-self-quenching counters induce the quench-process by using very large resistors R in the countercircuit. When a particles passes through, the voltage drop across the resistor is so large, that the voltageacross the counter falls below the operating voltage. For this to work, the time constant of the countercircuit must be so big, that the voltage drop persists for long enough, for all of the ions to arrive at thecounter tube.

In self-quenching counters, a so-called quench gas (e.g. methane or carbon dioxide) is mixed in withthe fill gas. These gases have wide absorption bands for the absorption of UV photons, which result fromthe gas amplification process. Additionally, there will be impacts between the positive fill gas ions withthe quench gas molecules, where the charge of the fill gas ions is transferred to the quench gas due tothe different electron affinities. Only quench gas ions arrive at the counter tube. Because of the lowerionisation potential of the quench gas ions, they are unable to free secondary electrons at the tube, andthe discharge process stops.

Characteristic Curve of the Geiger-Muller Counter

If one exposes a Geiger-Muller counter to a constant amount of radiation and records the pulse rate asa function of the voltage U applied to the counter using a sufficiently sensitive pulse counter, then theresulting curve would look like the one shown in figure 6. This is the characteristic curve of the counter.At the threshold voltage UT , the counter is not yet in the Geiger region. The pulse height depends on theenergy of the particles and only the largest pulses are counted. As the voltage is increased, the pulses growlarger and an ever larger portion of the pulses is registered. When the Geiger threshold UG is reached, allpulses are equally large. From this voltage on, the pulse rate should be practically constant (plateau), inreality however, it increases with increasing voltages due to increasingly frequent multiple discharges. Theslope of the plateau of good counters does not exceed 1 % pulse rate change per 100 V voltage change. Ifthe voltage is increased further, a continuous discharge occurs, which damages the counter. To on the onehand be sure one is not operating on the rising portion of the curve and on the other hand protect thecounter, one chooses the operating voltage U = UG + 100 V for this experiment. The choice of operatingvoltage generally depends strongly on the type of counter and the gas mixture. Depending on the gasmixture, it is possible that the plateau is very short or practically non-existent.

Figure 6: Characteristic curve of a Geiger-Muller counter.

Dead Time and Recovery Time of the Counter

After a particle has caused a discharge in a Geiger-Muller counter, it takes a certain amount of time forthe positive ions to wander to the cathode due to their relatively low mobility. During this time, the spacecharge of the positive ions weakens the field near the counter wire. A particle which passes through thecounter during this time either does not cause a pulse or generates only a small pulse height.

A distinction is made between the actual dead time Td and the recovery time Tr. The dead time isthe time required after a discharge for the counter to be able to detect the next particle. After this time,

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the ion tube has moved far away enough from the counter wire for gas amplification to be able to occuragain. The recovery time has passed, when the ion cloud has reached the cathode. After this, the pulsesreach their original height again.

The dead time and recovery time of a counter can be visualised and measured with an oscilloscopeusing a method specified by H. Guyford Stever: the output of the counter is handed over to an oscilloscopeand the trigger threshold Ut is set, such that the fully formed pulses are shown at the start of the timeaxis. Until the end of the dead time, no more pulses appear on the screen. Within the recovery time,pulses occur whose height increases with time and finally reaches the original value. By superimposingmultiple images (afterglow of the oscilloscope), one gets a series of pulses, with an envelope from whichthe dead time and recovery time can be read (see figure 7).

Figure 7: Stever diagram for determining the dead time and recovery time.

Crucial for the measurement is the time, after which the counter can once again deliver a pulse whichis large enough to be registered by the pulse counter. This so-called resolution time τ depends on theamplification of the pulse counter and on the chosen threshold of the amplifier and lies between Td andTr.

Dead Time Correction

If a counter with resolution time τ measured a pulse rate of n particles per second, then the counter wasinsensitive for a fraction n · τ of the time. If a total of N ionising particles per second passed through thecounter, then N · n · τ of those particles were not detected. The measured pulse rate n is the differencebetween the actual pulse rate N and the number of particles per time N · n · τ which were not detected:

n = N −Nnτ or N =n

1− nτ

In a so-called dead time stage, counter pulses are ignored during an adjustable time interval τi afteran initial counter pulse passes through. An artificial dead time of known or measurable length τi isgenerated. If one measures the counter rate n1 for a stage dead time τ1, which is considerably larger thanthe resolution time τ of the counter, then the true pulse rate N is:

N =n1

1− n1τ1(τ1 τ)

If one repeats this measurement with another dead time τ2, which is significantly smaller than theresolution time τ of the counter, then the measured counter rate n2 is determined by the resolution timeof the counter:

N =n2

1− n2τ(τ2 τ)

Exercise: Derive a formula for determining the resolution time of a counter by using a dead time stage.

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2 Probability Distributions and Statistics

2.1 Binomial, Poisson and Gaussian Distribution

Radioactive decay is a typical example of a stochastic process. The atomic nuclei of a radioactive sampleconstitute a very large statistical sample. The observed decay rate of the sample is set by the decayprobability of the single atomic nuclei, which is a constant of nature and is virtually uninfluenced byexternal conditions. The decays of different nuclei are independent and as such lead to the observedprobability distributions.

Binomial Distribution

If there are n radioactive atoms, each of which decays with a probability p within a fixed time interval,then P (k) is the probability that exactly k atoms decay within this time interval. Because these are nidentical atoms, one can choose (

n

k

)=

n!

k! (n− k)!

different groups of k atoms. Then the probability distribution P (k) sought is the binomial distribution(see figure 8):

P (k) =

(n

k

)pk(1− p)n−k (k = 0, 1, 2, . . . , n) (1)

The expectation value of this probability distribution is calculated to be

k =

n∑k=0

k P (k) = np,

for the standard deviation one gets

σ =

√k2 − k2 =

√√√√ n∑k=0

k2 P (k)−n∑k=0

(k P (k))2

=√np(1− p).

Poisson Distribution

If n is very large, such that k is small compared to n, then the limit n→∞, p→ 0 with np = const = µyields the Poisson distribution (see figure 9) as an approximation to the binomial distribution (1):

P (k) =µke−µ

k!(k = 0, 1, 2, . . .) (2)

The expectation value of the Poisson distribution is

k =

∞∑k=0

k P (k) = np = µ,

the standard deviation is given by

σ =√np =

õ.

The Poisson distribution specifies the probability that k events occur, when the expectation value is µ.

Gaussian Distribution

If µ becomes large (for which µ ≈ 10 already suffices), then the Poisson distribution (2) transitions, asan approximation, to the Gaussian or normal distribution (see figure 10):

P (k) =1√2π σ

exp

(− (k − µ)2

2σ2

)(3)

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The Gaussian distribution is, in contrast to the binomial and Poisson distributions, a continuous distri-bution, i.e. it is defined for all real numbers k. It is symmetrical around the mean µ with a width σ. Inthe special case, that the Gaussian distribution results from approximating the Poisson distribution, µand σ are not independent, but rather the same relation holds as for the Poisson distribution:

σ =√µ. (4)

Exponential Distribution

For a stochastic process, in which on average µ nuclei decay per time interval T , such that the event rateA = µ/T is constant, the time period ∆t between two successive events can sometimes be of interest.For radioactive decay, which is based on a Poisson process, the individual decays are independent, so theamount of time that has passed since the previous decay is irrelevant. Therefore, the probability that anucleus is going to decay within the next time interval T is a constant.

The probability density for the measured time period ∆t between two successive decays is in this casean exponential distribution:

ρ(∆t) = A exp (−A∆t)

The expectation value of the exponential distribution is A−1, the standard deviation is also A−1.

2.2 Spread and Measurement Uncertainty

The fact that, for the Poisson and special Gaussian distributions, the relation (4), i.e. that the standarddeviation is set by just the mean, is extremely important. In practice, this means that the spread of ameasured number of pulses N (and every measured quantity which is known to result from a Poissonprocess) is known from just a single measurement, because, as long as N is not too small, N ≈ N andσ ≈

√N . The laborious determination of σ can be dropped. The larger the number of pulses N is, the

more precise the measurement result is, since for the relative error we get:

∆N

N=σ

µ≈√N

N=

1√N

Assuming that a series of measurements can be described by a Gaussian distribution, the fraction ofall data points, which lie within a given interval [N − a,N + a], is given by the integral of the probabilitydistribution (3) over the interval, that is ∫ N+a

N−aP (N) dN.

The probability that a single measurement falls within the given interval is thus 68.3 % for a = σ, 95.4 %for a = 2σ and 99.7 % for a = 3σ.

2.3 The χ2 Fit

Given is a sequence xi of n normally distributed data points with errors σi and a sequence of valuesXi = Xi(~a), which are expected based on a model and depend on the parameters ~a = (a1, a2, . . .). Theparameters are to be estimated in such a way, that the model matches the data points as accuratelyas possible. The function χ2(~a) is a measure of the deviation of the measured values from the expectedvalues, where each deviation is weighted using the error on the data point:

χ2(~a) =

n∑i=1

(xi −Xi(~a))2

σ2i

The summation is over all measurements i.The best match between the data points and the model corresponds to the minimum of χ2(~a). Thus,

the χ2 fit consists of varying the parameters ~a of the model, until the minimum of χ2 has been reached.The associated values of the parameters are the optimal estimates.

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Figure 8: Binomial distribution with n = 10,p = 0,3. The probability that an unstable nucleusemits a γ quantum when it decays, is 30 %. Howmany of ten observed decays show a γ emission?

Figure 9: Poisson distribution with µ = 1,2. A ra-dioactive sample contains 3 · 1012 unstable nuclei,each of which decays with a probability of 4·10−13

within the next second. How many decays are ob-served per second?

Figure 10: Special Gaussian distribution with µ = 4320, σ =√

4320. The measurement from figure 9 isperformed for an hour. How many decays are observed within one hour?

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Figure 11: Measurement of ten pulse rates Ni Figure 12: Minimum of the function χ2

Aside from the optimal estimates of the parameters, the uncertainty on these estimates is of interestfor the analysis of experiments as well. The statistical error on the estimated parameter values is obtainedfrom the χ2 fit by increasing the value of χ2(~a) from χ2

min to χ2min + 1 (see figure 12).

The simplest example is the measurement of 10 pulse rates xi = Ni shown in figure 11. As one expectsthe same value µ of the pulse rate for every one of the ten measurements, the theoretical expectation isgiven by Xi(µ) = µ. In this case, it depends on just a single parameter µ. If the function χ2(µ) is plottedagainst µ, the result is a parabolic curve like in figure 12 with a minimum at µ = N . The mean N is theoptimal estimate of the parameter.

This example only serves to demonstrate the χ2 fit. A useful application is of course found in casesof complicated theoretical dependencies, e.g. the Gaussian distribution with the parameters µ and σ.

2.4 The χ2 Test

Also of importance is the probability, that a repetition of the experiment yields a larger value for χ2,despite identical experimental conditions. In statistics, it can be shown that, for large statistical samples,the probability density associated with χ2 (the so-called χ2 distribution) is of the following form:

Pn(χ2) =1

2 Γ(n2

) (χ2

2

)n−22

exp

(−χ

2

2

)The number of degrees of freedom n is equal to the number of independent equations involving theparameter vector ~a = (a1, a2, . . .) and the observation vector ~x = (x1, x2, . . .). One can determine thenumber of degrees of freedom by subtracting the number of variable parameters from the number ofdata points, where in many cases one additional degree of freedom is lost due to the normalisationconstraint on the theoretical distribution. Shown in figure 13 is the probability distribution P (χ2) forn = 1, 2, 3, 5, 10 and 20. The curves Pn(χ2) take on their maximum slightly before χ2 = n.

The integral

Fn(χ2) =

∫ ∞χ2

Pn(χ2) dχ2

is the probability, that a larger value of χ2 is found when the experiment is repeated. Therefore, Fn(χ2)is a measure of the goodness of the hypothesis, that the experimental data are correctly described by theexisting theory. Very small probabilities (F < 0,05) indicate a poor agreement between experiment andtheory. The function Fn(χ2) has been plotted for several values of n in figure 14.

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Figure 13: χ2 distribution Pn(χ2) for different values of n.

Figure 14: The function Fn(χ2) for several values of n.

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3 Procedure

3.1 Measurements with the Geiger-Muller Counter

Put the Geiger-Muller counter into operation and observe the pulses using the oscilloscope. The voltageapplied to the Geiger-Muller counter should be at most 700 V! Using the NIM modules, construct asetup which can be used to measure count rates.

1. Measurement of the counter’s characteristic curve: insert the 9038Sr sample into the lead chamber.

Measure the pulse rate of the Geiger-Muller counter as a function of the high voltage applied.Determine the threshold voltage UT and the Geiger threshold UG.

2. Choose an operating voltage of U = UG + 100 V. From here on, this setting should not be changedany more.

3. Check the intensity of the background: measure the pulse rate without a source in the lead chamber.

4. Produce a Stever diagram using the oscilloscope. For this, use the direct output of the Geiger-Muller counter. Estimate the dead time and recovery time of the counter.

5. Use the pulses of the 1-MHz-generator, which are available from the backside of the pulse counter,and verify the electronically generated dead times indicated on the dead time stage. To this end,use the pulse counter and write down the respective pulse rates.

6. The pulses from the Geiger-Muller counter should now pass through the dead time stage beforereaching the counter. Measure the count rates for a small and a large dead time of the dead timestage and determine from this the dead time of the counter. Make sure to measure a sufficientlylarge amount of events per time setting.

3.2 Measurements on Statistics

1. Set the measurement time, such that an average of at least 25 events are recorded per measurement.Measure the number of events per measurement time. Take enough measurements to be able toproduce a Gaussian distribution later on.

2. Remove the sample from the lead chamber and set the measurement time, such that the averageevent count per measurement is at most 2. Again take enough measurements to be able to producea Poisson distribution later on. (Can be combined with the background measurement of the Geiger-Muller counter.)

3. (recommended for bachelor students) Measure the times between two consecutive events. Use forthis the 90

38Sr sample and the oscilloscope again. Choose a suitable time scale, such that sufficientlymany events (but less than 80) are visible in a single image, and save it as a CSV file. Repeat thisstep until a satisfactory amount of time distances is available. Create a histogram of the measuredtimes and discuss the result. (Hint: this part of the experiment can be done both using the Geiger-Muller counter as well as the proportional counter, in which case a different source should of coursebe used.)

4. Alternatively, the time distances can also be recorded by hand using the counter. Afterwards, createa histogram of the measured times and discuss the result.

3.3 Measurement of the Characteristic Curve of a Proportional Counter

The methane-gas-flow proportional counter must be flushed continuously with a fill gas (argon-methanemixture). Set the gas flow adequately: roughly 4 divisions on the flow meter, bubbles should rise in theviewing glass. The maximal voltage for this counter is 2 kV. Please do also pay attention tothe switch next to the voltage control, which additionally multiplies the set voltage with acertain factor.

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1. Insert the 24195Am sample into the sample changer and measure the pulse rate as well as the pulse

height (measure of the gas amplification) of the α particles as a function of the voltage. Choosesuitable voltage steps. Why are the pulses not all equally large, even though the particles aremonoenergetic?

2. Repeat the same measurement with the 146C sample.

3. Check the intensity of the background: measure the pulse rate without a source in the counter. Itmakes sense to use the same voltage steps used before.

3.3.1 In Case This Part of the Experiment is Done Before the Geiger-Muller Counter

• If you have some time left, you can already do point 2 from the measurements on statistics usingthe proportional counter. For this purpose, pick a voltage on the α plateau (cf. measurementwith the 241

95Am sample).

4 Analysis

4.1 Measurements with the Geiger-Muller Counter

1. Calculate the resolution time using the dead time stage and compare this with the results obtainedusing the oscilloscope. Plot the dead time correction.

2. Plot the counter’s characteristic curve including the dead time correction.

4.2 Measurements on Statistics

1. Split the measured values into sensible intervals and present them as histograms.

2. For each of the measurements (Gaussian, Poisson and exponential distribution), calculate the meanand the standard deviation. From each of these, calculate the expected distribution and plot ittogether with the measured distribution. (Hint: in case the data for the measurement of the time di-stances were recorded with the oscilloscope, use the data points from the file mydata ???ms ascii.txtfor this analysis (see below).)

3. Test the goodness of your prediction of the measured distribution by doing a χ2 test for eachdistribution. To this end, calculate the χ2 between the expected and measured distributions andquantify the probability of the hypothesis using figure 14. Only consider intervals containing morethan 4 entries.

4. Using a program (e.g. Origin, Maple, Mathematica, Root, ...), perform χ2 fits to the measurements.For each distribution, compare the mean, standard deviation and χ2 with point 2 and 3.

5. (Hint for the analysis of the time differences) Analyse the saved oscilloscope data (.CSV) using theBASH script ’findpeaks.sh’ and the program ’findpeaks.exe’, which can be downloaded from theweb page of this experiment of the bachelor’s lab course and used under Linux from the CIP-Pool.To do so, follow these steps:

(a) Create a folder and into it, copy your data as well as the files ’findpeaks.sh’ and ’findpeaks.exe’.

(b) Make these files executable:

chmod u+x findpeaks.*

(c) Using ’findpeaks.sh’, create a list of your oscilloscope files:

./findpeaks.sh ./

You should now see a file called ’myfilelist.txt’ in the folder.

(d) Afterwards, you can run the program ’findpeaks.exe’ to start the analysis:

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./findpeaks.exe myfilelist.txt -0.1

The (negative) threshold of -0.1 is needed to require a ’minimum size’ of the negative counterpulses (if necessary, you can always experiment with this...). Based on the generated images(see below), check whether your choice of threshold was suitable.

(e) Should the program have finished without raising an exception, you will find a lot of newfiles in your folder: First, there are images (mygraph osc ??.png) of every saved oscilloscopedisplay (black: original data, red: after noise suppression, blue: reconstructed peak positions)as well as an image of the histogram of the time distances (myhist delta t ????.png). Secondly,a ROOT file (myhists.root) was generated, which contains the abovementioned histogram. Incase you prefer doing the χ2 fits with a program other than ROOT, use one of the ASCII filescontaining the histogram’s data (myhist ???ms ascii.txt) instead.

4.3 Measurements with the Proportional Counter

1. Plot the measured pulse rates of the three samples against the voltage and discuss the differences.

2. Plot the pulse heights of the samples as a function of the voltage in a suitable scale. Justify yourchoice. Discuss the curves.

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Appendix

A Properties of the Radioactive Samples Used

• 24195Am: Open sample, but covered to prevent contact. Substance on the front side, 1 mm in diameter,

the sample is sunk 1,5 mm into the casing; covering foil in front of the radioactive substance: 3µmof Gold (ρ = 19,3 g/cm3). Decay chain:

24195Am

α (5,65MeV)−→T1/2: 433 y

23793Np

α (4,96MeV)−→T1/2: 2,14·106 y

23391Pa

β− (0,57MeV)−→T1/2: 27,0 d

23392U

α (4,91MeV)−→T1/2: 1,59·105 y

22990Th

22990Th

α (5,17MeV)−→T1/2: 7932 y

22588Ra

β− (0,36MeV)−→T1/2: 14,9 d

22589Ac

α (5,94MeV)−→T1/2: 10,0 d

22187Fr

α (6,46MeV)−→T1/2: 286 s

21785At

21785At

α (7,20MeV)−→T1/2: 32,3ms

21383Bi

β− (1,42MeV)−→T1/2: 45,6m

21384Po

α (8,54MeV)−→T1/2: 3,72µs

20982Pb

β− (0,64MeV)−→T1/2: 3,25 h

20983Bi

• 9038Sr: Enclosed sample; perspex disk mounted in sheet steel. Decay chain:

9038Sr

β− (0,55MeV)−→T1/2: 28,9 y

9039Y

β− (2,28MeV)−→T1/2: 64,1 h

9040Zr

• 146C: Enclosed sample, decay:

146C

β− (0,16MeV)−→T1/2: 5700 y

147N

B Bethe-Bloch Formula

To calculate the energy loss of α particles in air, the following form of the Bethe-Bloch formula can beused:

−dEdx

= κ ρZ

A

z2

β2

[ln

(2mec

2γ2β2

I

)](5)

Where

• κ = 2πN0 re2me c

2 = 0,154 MeV cm2/g,

• ρ = 1,225 · 10−3 g/cm3 is the density of air on the surface,

• Z is the nuclear charge number of air (mostly nitrogen),

• mec2 = 0,511 MeV is the mass of the electron,

• I ≈ 15 eV is the mean ionisation potential of air,

• A is the atomic weight of air,

• z is the charge of the traversing particle,

• β =√Eα/

12mαc2 is the velocity of the traversing particle in units of the speed of light (Eα is the

energy and mα the mass of the α particles) and

• γ = 1/√

1− β2 is the Lorentz factor.

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