Gamma Spectroscopy. Calligaris Luca Massironi Andrea Presotto Luca. Academic Year 2006/2007

Transcription

1 Gamma Spectroscopy Calligaris Luca Massironi Andrea Presotto Luca Academic Year 2006/2007 Abstract Here we propose the results of a number of experiments with gamma rays. In the first part we concentrated on the behavior of the detectors while in the second part we analyzed some physical phenomena like absorption of photons in matter and gamma ray angular correlation in nuclear decays. 1 Introduction We want to study gamma rays emission using HPGe and NaI detectors. The goal of the first part of this article is to calibrate the detectors, analyze their pulse shape, study the dependence of detectors efficiency versus energy of gamma rays, finding their best working condition (shaping time) and determining NaI and HPGe Fano Factor. Then we performed an experiment about gamma ray absorption in different materials and three coincidence measurements: it was verified the back to back propagation of the two photons coming from electron-positron pair annihilation using a Na β + source; it was determined the decay scheme of 60 Ni (coming from a 60 Co) studying the angular correlation between the two deexcitation photons and it was analysed the decay of 208 T l. 2 Description of the equipment We used three detectors: Teledyne Isotope S 44 I/2 NaI scintillator NaI scintillator Ortec HPGe solid state detector Ortec 276 Photomultiplier base with preamp We had a crate containing the following modules: Ortec 570 Amplifier-Shaper Two H.V. power supplies for the two detectors (an Ortec 659 and a Silena mod. 7720) When we need to make the coincidence measurement we first added a 7616 Silena AMP-TISCA and then LeCroy 688AL Level Adapter Lecroy 4608 Octal Threshold Discriminator Caen N145 Quad Scaler Cern TU 277 Timer We acquired data via an PCI ADC/MCA card installed in a pc with an acquisition software, Maestro. The acquisition of the signal followed this path: both the photomultiplier of the NaI and the HPGe had a preamplified output, this output was connected to the Amp-Shaper were the signal was given a gaussian shape and then it was fed into the input of the ADC/MCA card in the computer. 3 NaI Detector As mentioned before we had two NaI detectors, but since we soon discovered that Teledyne Isotopes S 44 I/2 NaI didn t had enough resolution in the energy range of interest we have always used NaI scintillator. 3.1 Calibration In order to calibrate the NaI detector we analyzed the spectrum from 60 Co plus 57 Co plus 22 Na. In 1

2 the experiments we mainly used two different operating voltages: in the initial runs we were working with 660V power supply, but after a quick study of the resolution vs HV we decided that the best working conditions were at 600V. Calibration is done with a source of 60 Co plus 22 Na using the following fitting function (see Fig.1): E = A + B Ch (1) where E is the Energy, A and B are the fit parameters and Ch is the number of channels recorded from the ADC. Figure 2: Linearity Study of NaI A = 11.1 ± 0.9 kev B = ± kev/channel Figure 3: Linearity Study of NaI Figure 1: Calibration of NaI with 22 Na, 57 Co and 60 Co Then we look at the difference between the values we get from this function and the theoretical ones. In the Fig.2 we see the absolute difference in kev and in Fig.3 the relative difference. All the values are within a 3 kev range and 2 % of relative displacement. We can conclude that the detector behaves linearly in the energy range of interest. However, as shown soon in section Temperature Analysis, the gain of the detector as a whole is connected to the temperature of the environment, hence when we need the coefficients A and B we usually recalibrate the apparatus each time. 3.2 Pulse Shape The NaI scintillator has two outputs. A preamplified one and a direct signal from the anode of the photomultiplier. The anodic signal should have a pulse shape developed from the circuit 4. In a scintillator electrons become excited instantaneously and then they fall on their ground state with a characteristic time τ r. The current from the anode of the photomultiplier will be proportional to the number of deexcitating electrons and we ll have I = I 0 e t τr and the voltage will than be V = V 0 ( e t τr e t τ f ) Measuring the rise time of the pulse will then give a measure of the scintillation time of this scintillator. τ f will otherwise give a measurement of the RC. We fit the data with this function: ( ) V = a + b e t t 0 τ f e t t 0 τr (2) e In Fig.5 we see the direct anodic signal: τ anode r τ anode f = (250.1 ± 0.1)ns = (320.6 ± 0.1)ns 2

3 Figure 4: Circuit for Anodic Signal From [1] we expect a scintillation time of about 230 ns. In Fig.6 we can see superimposed on the same graphic the anodic signal and the preamplified one. Even for the preamplified signal the circuit is similiar to Fig.4 thus from image 6 we get a good fit of the preamp fall time: Figure 6: NaI Anodic Pulse and Pre τ pre fall = (58.49 ± 0.02)µs Fig.7 focuses on the rise of the preamp signal to be compared with the anodic pulse length: τ pre rise = (315 ± 8)ns Figure 7: NaI Anodic Pulse and Pre Figure 5: NaI Anodic Pulse 3.3 Shaping Time The amplifier is used to amplify the signal and to shape it with a gaussian with adjustable shaping time. We made an analysis for NaI detector with a source of 57 Co and 22 Na fitting the Resolution versus Shaping time. The result is in Fig.8 where the black points stand for 122 kev photons and red points for 511 kev photons. We have chosen low energy photons since at low energy the resolution should we characterized by electronic noise that we can reduce with a choice of the best shaping time. For both series we can see that the resolution is quite linear and so the Shaping time doesn t influence the resolution. The shaping time we have chosen is 0.5 µs. 3.4 Resolution vs Energy We have analized some spectra with NaI to measure the dependence of resolution versus Energy. We have chosen a 238 U source in order to have a good number of peaks to fit. So we have obtained a graph of Resolution, defined as: R = F W HM E E The overall energy resolution achieved in a NaI detector is determined by a combination of at least two factors: the inherent statistical spread in the 3

4 Resolution Resolution vs Energy χ 2 / ndf / 4 b ± c 6.716e-007 ± a ± Figure 8: Shaping Time analysis 57 Co number of charge carriers R I and contribution of electronic noise R E. There should be also a contribution due to the photomultiplier and we schematize it as a constant factor. R I = F W HM N N So we have R P M constant = 2.35 N N = 2.35 W = 2.35 N E R E 1 E R = R P M R I R E Thus we have fit this graph with this function: ( R = a 2 + b2 c ) 2 E + E From the fit (see Fig.9) we obtain the following results: a = ± b = 1.6 ± 0.4 kev c = ± 30 kev We have found that Resolution contribution due to electronic noise is negligible as we have previously noticed in the shaping time analysis. 3.5 Fano Factor For a detector we have a correction of Resolution called Fano factor since the charge formation isn t a purely poissonian process. The Real Intrinsic Resolution can be written in function of Fano factor F as Energy (kev) Figure 9: Graph of Resolution vs Energy of 238 U detected with NaI R I = F F W HMN N = 2.35 F F W E = 2.35 F N N = N = 2.35 For a scintillator Fano factor should be about 1, so we can calculate how much is W from the result of the fitting of Resolution vs Energy: F W 2.35 E = b E W = b 2 F (2.35) 2 We obtain W = 0.46 ± 0.11 kev. We expected W 100 ev and we found a result about 5 times bigger, so we suppose that this difference is caused by worse quantic efficiency and light collector efficiency. 3.6 Temperature Analysis We have performed with NaI detector 72 runs with a 238 U source which lasted about 1 hour each in order to see if there is any dependence of the gain of the apparatus from the temperature. We have used also a temperature detector which however hasn t got enough sensitivity to detect small changes in temperature (less than 0.5 C). Data in Fig.11 have been shifted in order compare different peaks. As pictures 10 and 11 show there is a temperature dependence of the peak position and thus of the gain. Peaks position seem to have a period of 24 hours and they seem to have a decreasing trend: from the temperature detector we see that the temperature was increasing. From Fig.11 we can see that the relative variation of peak position is 2% whereas we measured a change in 4

5 Figure 10: Temperature analysis with 238 U Figure 12: Number of Dinodes However we have done an oversimplification, because as suggested in [1] δ = α V β (5) Figure 11: Peaks variation and Temperature temperature of about 4.5 C. We should consider in our successive measurements that this is an instrinsic limit in our precision about peaks position and resolution and that if we make long runs we should take into account this displacement in the peaks. 3.7 Number of Dinodes We have changed the voltage of the photomultiplier (PM) and we have calculated the gain for each voltage. Since the gain of the PM varies as G = δ n (3) where δ is the number of photoelectrons produced at each dinode and n the number of dinodes in the PM, if we suppose that δ varies linearly with the voltage δ = α V we have G = (α δ) n (4) If we plot G vs V we can find the number of dinodes n. Number of Dinodes = ± Thus G V β n (6) Thus the number we have found is β n. The number of pins we connect to the PM is 14 so we can suppose that there are actually 12 dinodes (one pin is the anode and another is for the photocatode). With this assumption we obtain β = 0.67 that is similar to the value suggested by [1] 4 HPGe Detector 4.1 Calibration The HPGe detector always worked with a 4 kv bias voltage. We never changed the gain and the shaping time of the Shaper however we noticed some slight variation of the conversion factors from one run to another. So we decided to calibrate with two different radiation spectra. The first one was made with a 238 U source and the second one with a 232 T h source. This detector was very powerful: with the uranium source we have been able to recognize 26 peaks, while with the thorium we were able to fit 45 peaks. We fit data with this function: E = A + B Ch (7) 5

6 where E is the Energy, A and B are the fit parameters and Ch is the number of channels recorded from the ADC. A = 3.7 ± 0.1 kev B = ± kev/channel Figure 14: Linearity Study of HPGe Figure 13: Calibration of HPGe: 232 T h and 238 U Then we look at the difference between the values we get from this function and the theoretical ones. In the Fig.14 we see the absolute difference in kev and in Fig.15 the relative difference. All the values are within a 1 kev range and they are within the error bars. Fig.14 shows that data from two different sources stay sistematically over or under the mean value, hence we see that there is instability in the apparatus even if it is always within the experimental bar errors. In the relative difference graph it appears that the lowest energy peaks stay somehow systematically below the average, but this is still within the error bars and the effect is smaller than 1%. We can conclude that the detector behaves linearly in the energy range of interest. 4.2 Pulse Shape The only output is the preamplified signal. Since the current generation is a bit more complex than in a NaI detector and should be calculated through Ramo theorem, we fit only the the fall of the signal, which should go like V = V 0 e t t 0 τ f From this fit Fig.16 we get τ fall = (69 ± 1)µs Figure 15: Linearity Study of HPGe 4.3 Shaping Time We have performed the same experiment of NaI in order to find the best shaping time for HPGe detector. From Fig.17 we see that for Ge detector the value of resolution changes varying the shaping time. We have fitted the points with function A Resolution = τ + Bτ Studying the graphic we can see a minimum of the function at shaping time 1 µs. The minimum represents the best resolution for the detector and so for all measurements with HPGe we have chosen this shaping time. 4.4 Resolution vs Energy We have analized some spectra with HPGe detector to measure the dependence of resolution versus 6

7 So we have R = R C R I R E Thus we have fit this graph with this function: ( R = a 2 + b2 c ) 2 E + E From the fit (see Fig.18) we obtain the following results: a = ± Figure 16: HPGe Fall Time b = ± kev c = 1.23 ± 0.01 kev Resolution vs Energy Resolution) Figure 17: Shaping Time analysis with a source of 57 Co recorded with HPGe Energy. We have chosen a 232 T h source in order to have a good number of peaks to fit. So we have obtained a graph of Resolution, defined as: R = F W HM E E The overall energy resolution achieved in a germanium detector is determined by a combination of three factors: the inherent statistical spread in the number of charge carriers R I, variation in the charge collection efficiency R C and contribution of electronic noise R E. R I = F W HM N N = 2.35 N N = 2.35 W = 2.35 N E R C E E = costant R E 1 E Energy (kev) Figure 18: Graph of Resolution vs Energy of 232 T h detected with HPGe 4.5 Fano Factor Also for a HPGe detector there is a correction to resolution due to charge formation and its statistic: Fano factor. F W R I = 2.35 E So we can calculate the Fano factor for Ge detector from the result of the fitting of Resolution vs Energy: F W 2.35 E = b E F = b 2 W (2.35) 2 The calculated value of Fano factor for HPGe is F F ano = ± using W = 2.9 ev. 7

8 4.6 Efficiency versus Energy The measure of relative efficiency in relation of energy was made with a 232 T h source. We have a source of 232 T h with all nuclei from decay in equilibrium, so we can consider that gamma sources have the same activity. We have considered a relative efficiency, defined as ε rel = N 2σ I % where N 2σ is number of counts for each peak in 2 sigma of gaussian distribution and I % is the Intensity of emission in percent from the table of gamma decays [3] (see Fig.19). 600keV the NaI detector, the green line, has a weak efficiency at full energy peak, while the HPGe still recognizes very well many different peaks. Then we can zoom around the 300keV Fig.21: the peaks are greatly sharper for HPGe. We can also see a huge number of low intensity peaks when analyzing HPGe spectra while in NaI spectrum there is only a smooth background. Log Efficiency Efficiency Vs. Energy χ 2 / ndf / 12 p ± p ± Log Energy Figure 20: Energy Spectrum of 238 U with HPGe and NaI Figure 19: Graph of Relative Efficiency vs Energy with 232 T h source recorded with HPGe detector We have obtained the Relative efficiency dividing the efficiency of each peak by efficiency of the highest energy one. In the graph we can see the linear correlation between log of Efficiency vs log of Energy. So we can see that we have the worst efficiency corresponding with highest energy. We have fitted the points with the function: log ε = A + B log E E 0 (8) as suggested in [1]. Putting E 0 = 1 kev we obtain from the graph A = 5.8 ± 0.3 B = ± Superimposed spectra It is interesting to see superimposed spectra we see when measuring the same source with two different detectors. It can be seen in Fig.20 that over Figure 21: Energy Spectrum of 238 U HPGe and NaI Zoom Then we compared two spectra detected with the HPGe Fig.22. One was taken with the detector and the source in free air, while the second was taken with everything inside a Pb shield. There are mainly three differences that can be easily seen: the big peak around 1460 kev that comes from the decay of 40 K, the 2614 kev of the 208 T l and a huge amount of background mainly in the low energy range. 8

9 Fig.23 with the 1460 kev line of 40 K. Figure 22: Spectra of 238 U in a shielded and non shielded enviroment 6 Cross Section We want to study the absorption of gamma rays in several materials. We put varying quantities of different substances between the source and the detector and we measure how the rate of photons decreases with the increasing of the thickness of the material. We have measured the number of counts we have around each peak subtracting the background and then we divided all by the live time of the apparatus. Data has been fitted by an exponential function: Rate = Rate(0) e µx (9) Where x stands for the thickness of material and µ is the mass absorption coefficient. We have performed the experiment with a source of 238 U and 60 Co. Since when we measure the counts of 238 U there is also a natural signal, we have fitted those data with an exponential function plus a background. Figure 23: Attenuation in Water 238 U and 60 Co Then we obtained the gamma cross section for a molecule of water with equation 10 σ = µa N a d (10) Where A is the atomic mass unit of H 2 O, d is density of water and N a is the Avogadro number. We have put on the same graph experimental cross section at various energies (red points) and data from [4] and we have obtained a good agreement (see Fig.24) 6.1 Water We have put a hollow cylinder over the top of the HPGe detector and then we filled it with water in 100 ml step. The source ( 238 U and 60 Co) was put over the top of this cylinder and this kept the source at constant distance. Image 23 shows how the rate decreases when we increase the length of water that a gamma ray may go across. We have also noticed that ambiental peaks remain constant while varying the thickness of various material as shown in Figure 24: Cross Section Water 9

10 6.2 Lead and Copper When measuring the cross section in Pb and in Cu we first needed to put the source further from the detector to avoid an excessive count rate. So we used a piece of wood that kept the source at a distance about 18 cm. Then we added an increasing number of disks of the material we were analyzing. This modified very slightly the distance of the source from the detector by a factor that has been used in the analysis. Since the intensity goes like I 1 r we have that 2 dn N = 1 dr 2 r where N that stands for the number of gamma rays per unit of time, r is about 18 cm and dr is less than a cm (it depends on how many disks have been used). Figure 26: Attenuation in Copper 238 U Figure 27: Cross Section Lead 7 Coincidences and Angular Correlation Figure 25: Attenuation in Lead 238 U The analysis of the data is the same as for water, thus we obtain the mass absorption coefficient µ and then the cross section for atom σ and we compare the experimental results with data from [4] (see Figs. 25, 26, 27, 28). The purpose of this experiment is to use the Coincidence mode to perform a study about the decay of several radioactive sources and the direction of the emitted gamma rays. The experimental setup is described in Fig.29: the source was positioned between NaI detector and HPGe detector, then NaI was free to move around the source maintaining the distance from that. To achieve this aim we have blocked the scintillator on a rod which was able to move around a pivot on which we placed the source. The rod had also a goniometer attached on in or- 10

11 Figure 28: Cross Section Copper der to read the angle. The distance of the HPGe detector was fixed with a blocked rod. Figure 31: Schematic Setup for Window Energy Calibration for 22 Na Figure 29: Setup for Coincidences Experiments The gate has been calibrated using the coincidence technique between NaI and itself (Fig.31): we have shaped the signal with 0.5 µs and we have delayed with an internal delayer of TI-SCA and we have gated with the logic pulse produced from the same original signal. In such a way we had been able to select the useful energy window around 511 kev Na We have set up the following experiment in order to verify the back to back production of two gamma rays in β + decay of 22 Na. Figure 30: Energy Levels 22 Na 22 Na decays β + on 22 Ne which decays emitting a 1274 kev (see Fig.30). The electronic setup is shown in Fig.32: the signal of the HPGe amplified and shaped is accepted by the software only if the NaI has detected a gamma ray of 511 kev and has produced a signal which has been converted into a logic pulse. The logic pulse is long about 1 µs and the signal from HPGe is accepted only if the logic pulse last at least 0.5 µs after the peak of the signal. In all the experiments we have checked the timing of the signals with an Oscilloscope. Figure 32: Schematic Setup for 22 Na The HPGe detector was placed at a fixed distance of 8.5 cm from the source and was shaped at 1 µs, while the NaI detector was at a distance of 9 cm and shaped at 0.5 µs. We have performed several runs at different angles. We have measured the number of counts around 511 kev and around 1274 kev. The former are the sum 11

12 of gamma rays emitted together with the gammas which have opened the gate and possible accidental coincidences, the latter are a probe to test that the apparatus hasn t changed geometry. The measurement of coincidence at an angle of 90 degrees between the two detector was used as a monitor of accidental coincidence: thus we have subtracted this value to all the rates measured at all the angles. We have tested that the 1274 kev rate remains about constant, so we can neglect small displacements in the apparatus. considered the possibility of a Compton scattering in the detector with the leak of the recoiled gamma ray. From the geometry of the detector this physical event is localized near the edge of the detector so we can say that gamma ray which undergoes a Compton scattering in about 8 mm from the edge of the detector has a big probability of escaping from the detector without leaving any trace. In order to verify this hypothesis we have put in the same graph the simulation and the sum of all the counts of the detector due to 511keV photon (see Appendix A): thus we have both the full energy peak and the Compton spectrum. Figure 33: 22 Na Angular Acceptance of NaI through The graph 33 shows that we have about 20 of angular acceptance. We have also done a Montecarlo simulation of the apparatus as shown in Fig.33. The probability of interaction in the detector has been taken proportional to 1 e µx where x is the penetration thickness and µ the Ge absorption coefficient. In the simulation we have used geometrical parameters such as distances and dimensions of the detectors, but in order to fit the experimental data to the simulated data we had to reduce the radius of the detectors of about 0.8 cm. This imperfection is due to the low level physical approximation of the simulation, because we have set that the probability of interaction of a gamma ray with the detector is 1 e µx but we haven t Figure 34: Simulation and Compton 22 Na As Fig.34 shows, data fit well with the simulation. In all this calculations we have neglected the background in the energy spectrum due to natural radioactivity and its accidental coincidences Co Using a source of 60 Co which decays β in 60 28Ni we have analyzed the angular correlation between the two photons emitted from 60 28Ni disexcitation (Fig.35). The experimental setup is similar to the one used for 22 Na. Since we have a source which is not perfectly a point and has a layer of material which can 12

13 tered around 1332 kev. ST HPGe 1 µs ST NaI 0.5 µs Figure 35: Energy Levels of 60 Co absorb a part of the radiation we have to rotate not only the NaI detector but also the source in order to maintain the symmetry between the two detectors at each angle (Fig.36) Logic pulse 1.5 µs Distance HPGe 11.0 cm Distance NaI 13.0 cm Using this apparatus we can count the events around 1173 kev and 1332 kev and then normalize all with the number of times that the gate has been opened (Scaler) R = n Scaler The typical spectrum we see is the one shown in Fig.38 and Fig.39. Figure 36: Schematic Setup for 60 Co We have added a Scaler in the circuit in order to measure the rate of the events that opened the gate. As shown in Fig.36 the signal from NaI had to be doubled: one had to open the gate and the other had to go to a Scaler. Due to different standards usage of the instruments the electronic setup is a bit more complex as shown in Fig.37. Figure 38: Energy Spectrum 60 Co Figure 37: Electric Setup for 60 Co The Energy window for the gate has been cen- Figure 39: Energy Spectrum 60 Co Log Scale 13

14 The peak at 1332 kev is a measurement of accidental coincidences and is actually much smaller than 1173 kev, below the 10% as total number of events. As for 22 Na we can see at different angles the rate at 1332 kev is constant. The rate at 1173 kev has been measured with equation 11 thus we subtract the accidental coincidences. The number must be corrected by the efficiency ε of the detector and the intrinsic intensities (branching ratios BR) of the gammas through a comparison of the rate between the two peaks without the coincidence apparatus. R nogate 1173 BR(1173) ε(1173) = R nogate 1332 BR(1332) ε(1332) N acc 1173 = N 1332 BR(1173) ε(1173) BR(1332) ε(1332) = N 1332 R1332 nogate R1173 nogate Thus the number of accidental coincidences at R 1173 kev is equal to N nogate so that R1173 nogate R 1173 = N 1173 N 1332 Scaler R nogate 1332 R nogate 1173 (11) From [2] we have that the angular correlation function between the direction of the two photons of 60 28Ni is W (ϑ) = 1 + l a n cos 2n ϑ n=1 Where 2l is the order of the lowest multipole in the cascade. Thus if gamma-rays are quadrupoles W (ϑ) = 1 + a 1 cosϑ + a 2 cos 2 ϑ Table 1: Coefficients for the angular correlation with the spin of the ground state J 1 = 0 D = Dipole and Q = Quadrupole J 2 J 3 Multipoles a 1 a D-D D-D -1/ D-D -1/ Q-D -1/ Q-D 3/ Q-D -3/ D-Q -3/ D-Q 3/ D-Q -1/ Q-Q Q-Q 5-16/3 2 2 Q-Q -15/13 16/3 3 2 Q-Q 0-1/3 4 2 Q-Q 1/8 1/24 If we fit experimental data with different coefficients a i we can find which is the J of the two excited states of 60 28Ni (see Fig.41). From Fig.41 we see that the excited states of 60 28Ni have J 2 = 4 and J 1 = 2 as we can check on [4] T h We have tried to analyze the coincidences of 208 T l, a daughter of 232 T h. Using an apparatus similar to 22 Na we have studied the correlation between different gamma rays. We have gated on a 2647 kev and we expect to find several peaks in our spectrum (see Fig.42). We have used HPGe detector for the gate since NaI detector can t resolve 2647 kev peak very well. ST HPGe 1 µs ST NaI 0.5 µs Figure 40: 60 Co decay From the same article [2] we obtain the coefficients a i for different transitions. Logic pulse 1.2 µs Distance HPGe 7.8 cm Distance NaI 8.4 cm 14

15 Figure 43: Energy Spectrum 232 T h Figure 41: Angular Correlation 60 Co Figure 42: Energy Levels of 232 T h We found three peaks as shown in Fig.43 that correspond to the three transitions in Fig.42. We have also changed the angle between the two detectors and we have seen that the rate of the three peaks changes, but the angular correlation analysis this time is too complicated and we limit ourselves to a qualitative result (see Fig.44). Figure 44: Angular Correlation of 232 T h 8 Conclusions We have learned how to use a gamma ray detector: we have characterized two different detectors and we have used them in some nuclear physics experiments. The experiments worked out well. We found the angular correlation of the 69 Co and of the positronium annihilation from the 22 Na decay to be compatible with the theoretical one. 15

16 References [1] Knoll Radiation Detection and Measurements [2] F.L. Brady and M. Deutsch Angular correlation of Succesive Gamma-Rays Physical Review,voI. 78, June 1950 [3] List of Gamma Rays [4] NIST National Institute of Standards and Technology Appendix A Image 45 shows the spectrum we analyzed a bit simplified and the two contributes to the total spectrum due to two different gamma rays. C = Compton F = Full Energy peak We can suppose that F 1 F 2 = C1 C2. Then F 2 = γ F 1 = β δ α + β + δ = C1 + C2 + F 1 C2 = C1 F 1 F 2 If we suppose that β region is the superimposition of the 1 full energy peak and a part of the Compton spectrum of peak 2 and if we imagine that over the region δ and β the Compton of the highest peak is constant we can say that F 1 β δ with δ = δ x x Simple calculation leads to C1 = α + δ + δ β δ + γ (β δ ) Thus the sum of Compton and Full Energy peak is F 1 + C1 = α + β + δ + γ β δ + γ (β δ ) 16

17 Figure 45: Compton and Full Energy Peak 17