Advanced lab course for bachelor students in physics

Transcription

1 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

2 Table of Contents 1 Gaseous Ionisation Detectors Principle of the Gaseous Ionisation Detector Operating Regions of a Gaseous Ionisation Detector The Ionisation Chamber (Example of a Simple Gaseous Ionisation Detector) The Proportional Counter The Geiger-Müller Counter Probability Distributions and Statistics Binomial, Poisson and Gaussian Distribution Spread and Measurement Uncertainty The χ 2 Fit The χ 2 Test Procedure Measurements with the Geiger-Müller Counter Measurements on Statistics Measurement of the Characteristic Curve of a Proportional Counter In Case This Part of the Experiment is Done Before the Geiger-Müller Counter Analysis Measurements with the Geiger-Müller Counter Measurements on Statistics Measurements with the Proportional Counter A Properties of the Radioactive Samples Used 17 B Bethe-Bloch Formula 17 Bibliography [1] Gerd Otter, Raimund Honecker: Atome Moleküle Kerne, Band I: Atomphysik, Band II: Molekül- und Kernphysik, Bo 184 [2] William R. Leo: Techniques for Nuclear and Particle Physics Experiments, Dr 155 [3] Konrad Kleinknecht: Detektoren für 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 Übungsbuch, Cu 202 [8] Strahlenschutzverordnung, [9] The Particle Detector Brief Book, [10] Detecting Particles: How to see without seeing, lecture6.ppt [11] Properties of argon-ethane/methane mixtures for use in proportional counters, Nuclear Instruments and Methods, vol. 188, issue 3, pp , (10/1981) 2

3 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 the ionisation 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-Müller 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 the cathode. Is the electric field near the anode very strong (proportional and Geiger-Müller counter), then the electrons are accelerated in the immediate vicinity of the anode. They themselves then produce more electrons and ions by impact ionisation. The motion of all charges causes a current pulse in the external resistor and therefore causes a negative voltage pulse. This pulse is forwarded to the readout electronics via 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: β particle). 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 that the created electrons and ions recombine decreases. Region II: Above a voltage U 1, recombination no longer occurs. The measured charge stays constant for a certain voltage range and is equal to the sum of the primary ionisation charges. The ionisation chamber operates in this region. Region III: Starting from a voltage U 2, the electrons are accelerated so strongly near the anode, that they can ionise other gas atoms in collisions. Charge avalanches are generated. The total charge still remains proportional to the number of primary ionisation charges. A counter operating in this region is therefore called a proportional counter. The multiplication of the primary ionisation charge caused by impact ionisation is called gas amplification. 3

4 Region IV: Above the voltage U 3, the proportionality of the total charge to the number of primary ionisation charges is limited. Region V: The Geiger region starts at U 4. A property of the Geiger region is that all ionising particles cause the same voltage pulse, independent of their type and energy. A counter operating in this region is called a Geiger-Müller counter. The gas amplification ( 10 9 ) is substantially larger for a Geiger-Müller counter than for a proportional counter, meaning even without additional electronic amplification, pulses of several volts can be generated. Region VI: The pulses grow larger for increasing voltages, until ultimately a continuous discharge occurs, which damages the counter. 1.3 The Ionisation Chamber (Example of a Simple Gaseous Ionisation Detector) The ionisation chamber is one of the oldest detection devices for radiation and is still often used today because 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, independent of the type and energy of the ionising particle. If, for example, the incident radiation deposits 1 MeV of energy in the chamber, then roughly electron-ion pairs are produced, which corresponds to a charge of roughly C for each sign. Depending on the construction, the charges created by the ionising particle are integrated to a constant current (current mode) or generate a voltage pulse for every individual particle (pulse mode). For a typical value for the capacitance of C = F between both electrodes of the ionisation chamber, a chamber operated in current mode with a load resistor of e.g. R = Ω has a time constant of 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 longer times. Such chambers are used e.g. for dose or dose rate measurements for radiation protection and for monitoring and controlling the intensity at particle accelerators. The currents or charges to be measured are exceptionally small, so a good electrical insulation is required for such a chamber. Another problem are recombination processes of the gas, which can interfere with the linear relation between absorbed energy and measured current. For a chamber operated in pulse mode, the resistance is on the order of R = 10 6 Ω (to be confirmed), so the pulse length for the same capacitance is on the order of 1 µs. 4

5 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 100 µm in diameter) across a resistor (R 1 MΩ). Because of its cylindrical geometry, the proportional counter possesses an electric field, which increases strongly near the anode (E 1/r), with field strengths of 10 4 to 10 5 V/cm, such that gas amplification occurs near the anode wire. The height of the measured pulse is proportional to the energy of the particle, so the proportional counter can be used for energy measurements. The gas amplification factor (10 4 to 10 6 ) increases exponentially with the applied voltage. In addition to impact ionisation, UV photons are emitted by excited atoms. These photons free photoelectrons from the gas or the 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 to the large amount of photoelectrons. The proportional region is therefore very small and poorly suited for practical applications. If polyatomic gases, e.g. methane, are used, then the proportional region becomes large. The UV photons are absorbed by the molecules, i.e. the molecules dissociate or vibrations are excited. The quanta emitted by the vibrating states are generally less energetic and can no longer free photoelectrons. Mixtures of noble gases and polyatomic molecules are often used. Using a proportional counter, 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 are tightly sealed with gas-tight walls, which short-ranged radiation cannot penetrate. A methane-gas-flow proportional counter is shown schematically in figure 4. At atmospheric pressure, an argon-methane mixture (10 % methane) flows through the counter volume, in which a thin wire loop serves as the anode and 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 of time) on the voltage U applied. For a proportional counter, this is the curve shown in figure 5 if both α and β particles are simultaneously present. 5

6 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 U T ), the minimum pulse height, which depends on the input sensitivity of the subsequent amplifier, is reached. Because of the larger number of produced electron-ion pairs, 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 choosing the voltage correspondingly, the proportional counter can therefore distinguish primary ionisations of different sizes, i.e. α and β radiation. 1.5 The Geiger-Müller Counter The Counter Discharge in the Geiger Region When the field strength in a proportional counter is increased, more and more UV photons are produced in the avalanche, increasing the probability that photons generate electrons through the photoelectric effect at other, more distant locations in the counter. These electrons induce new avalanches, causing the discharge to propagate along the counter wire. At the end of the avalanche formation, the counter wire is 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 reduces the field strength and therefore, no new avalanches can be triggered. The positive ions wander to the cathode, where they are neutralised at the counter tube. This could free secondary electrons from the 6

7 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 counter circuit. When a particles passes through, the voltage drop across the resistor is so large, that the voltage across the counter falls below the operating voltage. For this to work, the time constant of the counter circuit must be so big, that the voltage drop persists for long enough, for all of the ions to arrive at the counter tube. In self-quenching counters, a so-called quench gas (e.g. methane or carbon dioxide) is mixed in with the fill gas. These gases have wide absorption bands for the absorption of UV photons, which result from the gas amplification process. Additionally, there will be impacts between the positive fill gas ions with the quench gas molecules, where the charge of the fill gas ions is transferred to the quench gas due to the different electron affinities. Only quench gas ions arrive at the counter tube. Because of the lower ionisation potential of the quench gas ions, they are unable to free secondary electrons at the tube, and the discharge process stops. Characteristic Curve of the Geiger-Müller Counter If one exposes a Geiger-Müller counter to a constant amount of radiation and records the pulse rate as a function of the voltage U applied to the counter using a sufficiently sensitive pulse counter, then the resulting curve would look like the one shown in figure 6. This is the characteristic curve of the counter. At the threshold voltage U T, the counter is not yet in the Geiger region. The pulse height depends on the energy of the particles and only the largest pulses are counted. As the voltage is increased, the pulses grow larger and an ever larger portion of the pulses is registered. When the Geiger threshold U G is reached, all pulses are equally large. From this voltage on, the pulse rate should be practically constant (plateau), in reality however, it increases with increasing voltages due to increasingly frequent multiple discharges. The slope of the plateau of good counters does not exceed 1 % pulse rate change per 100 V voltage change. If the voltage is increased further, a continuous discharge occurs, which damages the counter. To on the one hand be sure one is not operating on the rising portion of the curve and on the other hand protect the counter, one chooses the operating voltage U = U G V for this experiment. The choice of operating voltage generally depends strongly on the type of counter and the gas mixture. Depending on the gas mixture, it is possible that the plateau is very short or practically non-existent. Figure 6: Characteristic curve of a Geiger-Müller counter. Dead Time and Recovery Time of the Counter After a particle has caused a discharge in a Geiger-Müller counter, it takes a certain amount of time for the positive ions to wander to the cathode due to their relatively low mobility. During this time, the space charge of the positive ions weakens the field near the counter wire. A particle which passes through the counter 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 T d and the recovery time T r. The dead time is the time required after a discharge for the counter to be able to detect the next particle. After this time, 7

8 the ion tube has moved far away enough from the counter wire for gas amplification to be able to occur again. The recovery time has passed, when the ion cloud has reached the cathode. After this, the pulses reach their original height again. The dead time and recovery time of a counter can be visualised and measured with an oscilloscope using a method specified by H. Guyford Stever: the output of the counter is handed over to an oscilloscope and the trigger threshold U t is set, such that the fully formed pulses are shown at the start of the time axis. 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 superimposing multiple images (afterglow of the oscilloscope), one gets a series of pulses, with an envelope from which the 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 which is large enough to be registered by the pulse counter. This so-called resolution time τ depends on the amplification of the pulse counter and on the chosen threshold of the amplifier and lies between T d and T r. Dead Time Correction If a counter with resolution time τ measured a pulse rate of n particles per second, then the counter was insensitive for a fraction n τ of the time. If a total of N ionising particles per second passed through the counter, then N n τ of those particles were not detected. The measured pulse rate n is the difference between 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 τ i after an initial counter pulse passes through. An artificial dead time of known or measurable length τ i is generated. If one measures the counter rate n 1 for a stage dead time τ 1, which is considerably larger than the resolution time τ of the counter, then the true pulse rate N is: N = n 1 1 n 1 τ 1 (τ 1 τ) If one repeats this measurement with another dead time τ 2, which is significantly smaller than the resolution time τ of the counter, then the measured counter rate n 2 is determined by the resolution time of the counter: N = n 2 (τ 2 τ) 1 n 2 τ Exercise: Derive a formula for determining the resolution time of a counter by using a dead time stage. 8

9 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 sample constitute a very large statistical sample. The observed decay rate of the sample is set by the decay probability of the single atomic nuclei, which is a constant of nature and is virtually uninfluenced by external conditions. The decays of different nuclei are independent and as such lead to the observed probability 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 n identical 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): ( ) n P (k) = p k (1 p) n k (k = 0, 1, 2,..., n) (1) k The expectation value of this probability distribution is calculated to be k = n k P (k) = np, k=0 for the standard deviation one gets σ = k 2 k 2 = n k 2 P (k) Poisson Distribution k=0 n (k P (k)) 2 = np(1 p). If n is very large, such that k is small compared to n, then the limit n, p 0 with np = const = µ yields the Poisson distribution (see figure 9) as an approximation to the binomial distribution (1): k=0 P (k) = µk e µ k! (k = 0, 1, 2,...) (2) The expectation value of the Poisson distribution is the standard deviation is given by k = k P (k) = np = µ, k=0 σ = np = µ. 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, as an approximation, to the Gaussian or normal distribution (see figure 10): P (k) = 1 ) (k µ)2 exp ( 2π σ 2σ 2 9 (3)

10 The Gaussian distribution is, in contrast to the binomial and Poisson distributions, a continuous distribution, i.e. it is defined for all real numbers k. It is symmetrical around the mean µ with a width σ. In the special case, that the Gaussian distribution results from approximating the Poisson distribution, µ and 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 rate A = 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 the amount of time that has passed since the previous decay is irrelevant. Therefore, the probability that a nucleus 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 case an exponential distribution: ρ( t) = A exp ( A t) The expectation value of the exponential distribution is A 1, the standard deviation is also A Spread and Measurement Uncertainty The fact that, for the Poisson and special Gaussian distributions, the relation (4), i.e. that the standard deviation is set by just the mean, is extremely important. In practice, this means that the spread of a measured number of pulses N (and every measured quantity which is known to result from a Poisson process) is known from just a single measurement, because, as long as N is not too small, N N and σ 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 of all data points, which lie within a given interval [N a, N + a], is given by the integral of the probability distribution (3) over the interval, that is N+a N a P (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 χ 2 Fit Given is a sequence x i of n normally distributed data points with errors σ i and a sequence of values X i = X i ( a), which are expected based on a model and depend on the parameters a = (a 1, a 2,...). The parameters are to be estimated in such a way, that the model matches the data points as accurately as possible. The function χ 2 ( a) is a measure of the deviation of the measured values from the expected values, where each deviation is weighted using the error on the data point: χ 2 ( a) = n (x i X i ( a)) 2 i=1 The summation is over all measurements i. The best match between the data points and the model corresponds to the minimum of χ 2 ( a). Thus, the χ 2 fit consists of varying the parameters a of the model, until the minimum of χ 2 has been reached. The associated values of the parameters are the optimal estimates. σ 2 i 10

11 Figure 8: Binomial distribution with n = 10, p = 0,3. The probability that an unstable nucleus emits a γ quantum when it decays, is 30 %. How many of ten observed decays show a γ emission? Figure 9: Poisson distribution with µ = 1,2. A radioactive sample contains unstable nuclei, each of which decays with a probability of within the next second. How many decays are observed per second? Figure 10: Special Gaussian distribution with µ = 4320, σ = The measurement from figure 9 is performed for an hour. How many decays are observed within one hour? 11

12 Figure 11: Measurement of ten pulse rates N i Figure 12: Minimum of the function χ 2 Aside from the optimal estimates of the parameters, the uncertainty on these estimates is of interest for the analysis of experiments as well. The statistical error on the estimated parameter values is obtained from the χ 2 fit by increasing the value of χ 2 ( a) from χ 2 min to χ2 min + 1 (see figure 12). The simplest example is the measurement of 10 pulse rates x i = N i shown in figure 11. As one expects the same value µ of the pulse rate for every one of the ten measurements, the theoretical expectation is given by X i (µ) = µ. In this case, it depends on just a single parameter µ. If the function χ 2 (µ) is plotted against µ, the result is a parabolic curve like in figure 12 with a minimum at µ = N. The mean N is the optimal estimate of the parameter. This example only serves to demonstrate the χ 2 fit. A useful application is of course found in cases of complicated theoretical dependencies, e.g. the Gaussian distribution with the parameters µ and σ. 2.4 The χ 2 Test Also of importance is the probability, that a repetition of the experiment yields a larger value for χ 2, despite identical experimental conditions. In statistics, it can be shown that, for large statistical samples, the probability density associated with χ 2 (the so-called χ 2 distribution) is of the following form: ( P n (χ 2 1 χ ) = 2 Γ ( 2 ) n 2 2 ) n 2 2 exp ) ( χ2 2 The number of degrees of freedom n is equal to the number of independent equations involving the parameter vector a = (a 1, a 2,...) and the observation vector x = (x 1, x 2,...). One can determine the number of degrees of freedom by subtracting the number of variable parameters from the number of data points, where in many cases one additional degree of freedom is lost due to the normalisation constraint on the theoretical distribution. Shown in figure 13 is the probability distribution P (χ 2 ) for n = 1, 2, 3, 5, 10 and 20. The curves P n (χ 2 ) take on their maximum slightly before χ 2 = n. The integral F n (χ 2 ) = χ 2 P n ( χ 2 ) d χ 2 is the probability, that a larger value of χ 2 is found when the experiment is repeated. Therefore, F n (χ 2 ) is a measure of the goodness of the hypothesis, that the experimental data are correctly described by the existing theory. Very small probabilities (F < 0,05) indicate a poor agreement between experiment and theory. The function F n (χ 2 ) has been plotted for several values of n in figure

13 Figure 13: χ 2 distribution P n (χ 2 ) for different values of n. Figure 14: The function F n (χ 2 ) for several values of n. 13

14 3 Procedure 3.1 Measurements with the Geiger-Müller Counter Put the Geiger-Müller counter into operation and observe the pulses using the oscilloscope. The voltage applied to the Geiger-Müller counter should be at most 700 V! Using the NIM modules, construct a setup which can be used to measure count rates. 1. Measurement of the counter s characteristic curve: insert the 90 38Sr sample into the lead chamber. Measure the pulse rate of the Geiger-Müller counter as a function of the high voltage applied. Determine the threshold voltage U T and the Geiger threshold U G. 2. Choose an operating voltage of U = U G V. From here on, this setting should not be changed any 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- Müller 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-Müller counter should now pass through the dead time stage before reaching the counter. Measure the count rates for a small and a large dead time of the dead time stage and determine from this the dead time of the counter. Make sure to measure a sufficiently large 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 to produce a Gaussian distribution later on. 2. Remove the sample from the lead chamber and set the measurement time, such that the average event count per measurement is at most 2. Again take enough measurements to be able to produce a Poisson distribution later on. (Can be combined with the background measurement of the Geiger- Müller counter.) 3. (recommended for bachelor students) Measure the times between two consecutive events. Use for this the 90 38Sr sample and the oscilloscope again. Choose a suitable time scale, such that sufficiently many events (but less than 80) are visible in a single image, and save it as a CSV file. Repeat this step until a satisfactory amount of time distances is available. Create a histogram of the measured times and discuss the result. (Hint: this part of the experiment can be done both using the Geiger- Müller counter as well as the proportional counter, in which case a different source should of course be used.) 4. Alternatively, the time distances can also be recorded by hand using the counter. Afterwards, create a 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-methane mixture). Set the gas flow adequately: roughly 4 divisions on the flow meter, bubbles should rise in the viewing glass. The maximal voltage for this counter is 2 kv. Please do also pay attention to the switch next to the voltage control, which additionally multiplies the set voltage with a certain factor. 14

15 1. Insert the Am 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. Choose suitable voltage steps. Why are the pulses not all equally large, even though the particles are monoenergetic? 2. Repeat the same measurement with the 14 6C sample. 3. Check the intensity of the background: measure the pulse rate without a source in the counter. It makes sense to use the same voltage steps used before In Case This Part of the Experiment is Done Before the Geiger-Müller Counter If you have some time left, you can already do point 2 from the measurements on statistics using the proportional counter. For this purpose, pick a voltage on the α plateau (cf. measurement with the Am sample). 4 Analysis 4.1 Measurements with the Geiger-Müller Counter 1. Calculate the resolution time using the dead time stage and compare this with the results obtained using 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 mean and the standard deviation. From each of these, calculate the expected distribution and plot it together with the measured distribution. (Hint: in case the data for the measurement of the time distances were recorded with the oscilloscope, use the data points from the file mydata???ms ascii.txt for this analysis (see below).) 3. Test the goodness of your prediction of the measured distribution by doing a χ 2 test for each distribution. To this end, calculate the χ 2 between the expected and measured distributions and quantify the probability of the hypothesis using figure 14. Only consider intervals containing more than 4 entries. 4. Using a program (e.g. Origin, Maple, Mathematica, Root,...), perform χ 2 fits to the measurements. For each distribution, compare the mean, standard deviation and χ 2 with point 2 and (Hint for the analysis of the time differences) Analyse the saved oscilloscope data (.CSV) using the BASH script findpeaks.sh and the program findpeaks.exe, which can be downloaded from the web 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: 15

16 ./findpeaks.exe myfilelist.txt -0.1 The (negative) threshold of -0.1 is needed to require a minimum size of the negative counter pulses (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 new files in your folder: First, there are images (mygraph osc??.png) of every saved oscilloscope display (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. In case you prefer doing the χ 2 fits with a program other than ROOT, use one of the ASCII files containing 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 your choice. Discuss the curves. 16

17 Appendix A Properties of the Radioactive Samples Used Am: 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: 3 µm of Gold (ρ = 19,3 g/cm 3 ). Decay chain: Am α (5,65 MeV) T 1/2 : 433 y Np α (4,96 MeV) T 1/2 : 2, y Pa β (0,57 MeV) T 1/2 : 27,0 d U α (4,91 MeV) T 1/2 : 1, y Th Th α (5,17 MeV) T 1/2 : 7932 y Ra β (0,36 MeV) T 1/2 : 14,9 d Ac α (5,94 MeV) T 1/2 : 10,0 d Fr α (6,46 MeV) T 1/2 : 286 s At At α (7,20 MeV) T 1/2 : 32,3 ms Bi β (1,42 MeV) T 1/2 : 45,6 m Po α (8,54 MeV) T 1/2 : 3,72 µs Pb β (0,64 MeV) T 1/2 : 3,25 h Bi 90 38Sr: Enclosed sample; perspex disk mounted in sheet steel. Decay chain: 90 38Sr β (0,55 MeV) T 1/2 : 28,9 y 90 39Y β (2,28 MeV) T 1/2 : 64,1 h 90 40Zr 14 6C: Enclosed sample, decay: 14 6 C β (0,16 MeV) T 1/2 : 5700 y 14 7 N B Bethe-Bloch Formula To calculate the energy loss of α particles in air, the following form of the Bethe-Bloch formula can be used: de dx = κ ρ Z z 2 [ ( 2me c 2 γ 2 β 2 )] A β 2 ln (5) I Where κ = 2π N 0 r e 2 m e c 2 = 0,154 MeV cm 2 /g, ρ = 1, g/cm 3 is the density of air on the surface, Z is the nuclear charge number of air (mostly nitrogen), m e c 2 = 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 = E α / 1 2 m αc 2 is the velocity of the traversing particle in units of the speed of light (E α is the energy and m α the mass of the α particles) and γ = 1/ 1 β 2 is the Lorentz factor. 17