Calibration of a Germanium Detector

Bachelor thesis by Laura van der Schaafsupervised by Prof. Laura Baudis and Manuel Walter

August 23, 2012

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Acknowledgements

I would like to thank Prof. L. Baudis for the opportunity to work in her laboratory,give special thanks to Manuel Walter for all the hours he assisted me,

and thank the whole group for the support and the tips I received.

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Contents

1 Introduction 4

2 Germanium Semiconductor Detectors 42.1 Working Principle of a Pure Semiconductor . . . . . . . . . . . . . . . . . 42.2 Detector Temperature and Leakage Current . . . . . . . . . . . . . . . . . 42.3 Gamma Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Doped Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 Germanium in Comparison to Silicon . . . . . . . . . . . . . . . . . . . . . 8

3 The GeMini Detector Design at UZH 93.1 The Germanium Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 The Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Waveform Analysis 114.1 Description of the Waveform . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 Analysis Overwiew . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.3 Moving Average Window . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.4 Influence of the Voltage Supply . . . . . . . . . . . . . . . . . . . . . . . . 144.5 Cuts for Pileup Rejection . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.6 Influence of the Moving Average Window on the Cut Parameters . . . . . 18

5 Calibration 195.1 Energy Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2 Bug . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.3 The Resolution Dependence on the Bias Voltage . . . . . . . . . . . . . . 225.4 Determination of the Resolution of the Spectral Lines . . . . . . . . . . . 225.5 Detector Resolution and a Comparison of the Data taken with MCA and

ADC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.6 Long Term Stability of the Detector Response . . . . . . . . . . . . . . . . 26

6 Slow Control System 286.1 Liquid Nitrogen Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.2 Detector Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

7 Conclusion and Outlook 30

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1 Introduction

The properties of neutrinos as well as the nature of the dark matter are intensely inves-tigated in astroparticle physics [1][2]. Many of these experiments are using high puritygermanium detectors [3][4][5], which are known for their excellent energy resolution[6].

To interpret the data it is essential to know as much as possible about the behaviour ofgermanium detectors in the respective environments. Some information can be obtainedby small scale detectors imitating the respective experiments. These small scale detectorsprovide an environment to study current or future detectors and shieldings.

The goal of this Bachelor project is the calibration of a germanium detector at UZH. Itwill be used as a test detector for the GERDA experiment. The main focus is on theenergy calibration/resolution and the long term stability.

2 Germanium Semiconductor Detectors

This chapter provides an overview on the working principle of semiconductor detec-tors and the interactions which produce the signals. It concentrates on properties ofgermanium (Ge). Apart from section 2.3 which follows [9] this chapter follows [6].

2.1 Working Principle of a Pure Semiconductor

Semiconductors consist of a periodic atom lattice, with the electrons in the crystalforming energy bands. The outermost band in which the electrons are in a boundstate is called valence band, the band lying above the valence band is called conductionband. Electrons in the conduction band are free charge carriers. For semiconductors theband gap between the valence and the conduction band is approximately 1 eV.

By applying an electric field, the electrons in the conduction band drift to the cathodeand the resulting current can be measured. The current depends on the number ofelectrons in the conduction band. Electrons can jump from the valence band to theconduction band by thermal excitation, the current due to this process is called leakagecurrent. Electrons can also be excited by particles interacting with the semiconductor.Electrons that jump from the valence to the conduction band leave behind a hole. Dueto the electric field the hole will travel in the opposite direction to the electron.

2.2 Detector Temperature and Leakage Current

The leakage current can imitate small signals due to its statistical nature – it is thus ofinterest to keep it as small as possible. The temperature of the crystal has a mayor influ-

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ence on the leakage current and thus the energy resolution. The probability for electronsof the crystal to be found in a certain state is given by the Fermi-Dirac statistics

ni =gi

e(Ei−µ)/kB ·T + 1, (1)

where ni is the mean number of electrons in state i with energy Ei, µ is the chemicalpotential, kB the Boltzmann factor, gi the degeneracy (number of states with energy Ei)and T the temperature of the crystal. The probability for an electron to be in a state Eiabove the chemical potential µ (=in the conduction band) grows with the temperatureT thus the leakage current can be suppressed by cooling the crystal.

Band gaps on the surface of the crystal can differ from the ones found in the bulk andcontaminations on the surface can create leakage paths which are the main contributorsto the leakage current [6].

2.3 Gamma Interactions

For the energy calibration presented in section 5.1 gamma sources were used. There arethree main processes involved in the interaction of gamma rays with matter. They areshortly introduced in this section. In Fig.1a the probability for the different interactionsdepending on the incoming γ energy is given.

Photoelectric Effect The photoelectric effect describes the absorption of a γ ray andsubsequent emission of an orbital electron. The cross-section grows with Z5 and dropswith Eγ , where Z designates the atomic number.

Compton Scattering In a Compton scatter process an incident γ ray scatters onan electron. The cross-section per electron for photon scattering on free electrons isgiven by the Klein-Nishina-Formula. The energy of the outgoing electron depends on thescattering angle, the maximum energy is given for a scattering angle θ = π the minimumenergy for θ = 0. The resulting Compton spectrum has a characteristic Compton edgefor the highest possible kinetic energy of the scattered electron (Fig.1b). The totalcross-section for Compton scattering is proportional to Z.

Pair Production In the process of pair production an incident gamma ray photon isconverted into an electron-positron pair. This process can only take place in the vicinityof the Coulomb field of a nucleus. The minimum photon energy for this process to takeplace is two times the rest energy of the electron (positron), 2 ·me = 1.022 MeV.

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(a) (b)

Figure 1: (a) For the full-energy peak the whole incident γ energy is absorbed. Theplot shows which interactions lead to the absorption in dependence on the incoming γenergy. Fig. from [6]. (b) Energy distribution of the outgoing electron for a Comptonscatter, Fig. from [9].

2.4 Doped Semiconductors

Real materials are never pure, there are always some impurities left after their pro-duction. Intentional impurification of a material is called doping. For semiconductordetectors doping is essential. The dopants used for the detector studied in this projectare boron (B) and diffused lithium (Li) [7].

2.4.1 n-type and p-type doping

Diffused Li doping is an example of n-type doping. The Li atom will take an interstitialsite in the Ge lattice. The energy of Li electron lies in the forbidden gap just below theconduction band, for room temperature almost all the Li is ionised. The added Li is aso-called donor impurity.

The added B is an example of a p-type doping. The dopant has one electron less inthe valence band than the surrounding Ge. Therefore one covalent bond is missing nearthe dopant. This vacancy can be filled by an electron (from the conduction band orby thermal excitation from a neighbouring atom) to form a covalent bond. This bond,however, is weaker than between two atoms with four valence electrons - it takes lessenergy for the electron to jump into the conduction band. The energy of these acceptorimpurities lies just above the valence band energy.

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(a) (b)

Figure 2: (a) Li as example of a n-type dopant and B as an example of a p-type dopant.(b) charged pn-junction, the electron from the Li drifted to the B to form a covalentbond.

2.4.2 The pn-junction

Figure 3: Radial crystal doping (figure adapted from [6])

Junctions between n- and p-type semiconductor materials are common in semiconduc-tor detectors. In case of the GeMini detector at the University of Zurich, the detectorstudied within this project, the crystal is n-type doped with a p-type layer on the out-side. An intrinsic semiconductor is not doped. In Fig.3 ni and pi represent the intrinsicconcentration of electrons and holes, respectively. As the holes are produced by thermalexcitation the two concentrations have to be equal. ND represents the donor concen-tration throughout the crystal. NA stands for the added p-type layer on the outside ofthe crystal. n (p) gives the resulting electron (hole) concentration. This constellationis not in equilibrium, the free electrons from the n-type region will travel across thejunction to the p-type region (Fig.2b). This diffusion process causes an electric field due

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to the different net charges which will grow with the number of travelled electrons/holes.Finally the field will be strong enough to prevent further diffusion. The charged regionis called depletion region.

This region makes the detection of small charge concentrations much easier. Let usassume that a gamma ray excites some electrons in the junction. Due to the electricfield these electrons will drift to the n-type region and the simultaneously created holesto the p-type region producing an electrical signal.

In order to suppress noise and incomplete charge collection, an additional external volt-age is applied with the negative pole at the p-type region and the positive pole at then-type region. The reverse bias voltage enhances the depletion region.

2.5 Germanium in Comparison to Silicon

The two most commonly used elements for semiconductor detectors are germanium andsilicon [6]. Here the two elements are compared to each other.

The band gap has an influence on the energy needed to produce an electron-holepair. A small band gap allows a better energy resolution as the electrons need lessenergy to be excited. On the other hand, the leakage current grows with a decreasingband gap (compare equation (1)). Ge has an indirect band gap1 of 0.7 eV, Si of 1.1eV. The energy resolution of Ge is thus better than for Si if the detector is cooled down(commonly cooled down to 77 K, with liquid nitrogen as coolant). However, Ge detectorshave to be operated at low temperatures otherwise signals would be predominated bythe leakage current. Si with a band gap 1.6 times larger can be operated at roomtemperature.

The cross-sections for gamma interactions depends on the atomic number (seesubsection 2.3). Ge (Z = 32) has a larger atomic number than Si (Z = 14) and is thusbetter suited for gamma detection.

Neutrinoless double beta decay (0ν2β) experiments need a detector with goodenergy resolution and a source that decays via the double beta decay. For Ge an isotopeexists (76Ge), which decays via the double beta decay. Experiments looking for theneutrinoless double beta decay can therefore use Ge as source and detector.

1An electron jumping from the conduction band to the valence band by an indirect band gap producesor annihilates a phonon, in case of a direct band gap no phonon is produced or annihilated.

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(a) (b)

Figure 4: (a) Sketch of the Ge crystal at UZH. (b) Scheme of the setup.

Costs must be kept as low as possible in detector design. Si is much cheaper than Ge,therefore it is better suited for large particle tracking detectors, where the energy of theparticle is not of interest.

3 The GeMini Detector Design at UZH

The GeMini detector is a germanium detector located in Zurich, it was calibrated duringthis project. In this section the setup as well as the electronics used for data acquisitionwill be explained.

3.1 The Germanium Crystal

The n-type Ge crystal has a closed-ended coaxial configuration with a length of 40 mmand an outer diameter of 40 mm. The inner hole’s diameter is 9 mm with a depth of35 mm. The p-type doping (dopant is boron [7]) is added to the outer surface of thecrystal, the n-type doping (dopant is diffused lithium [7]) to the inner contact. The twodoped surfaces are reverse biased. The recommended operating high-voltage has a valueof −3000V [10]. The surface connecting the outer and the inner cylinder is insulated(grey area in Fig.4a).

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3.2 The Setup

The GeMini detector allows for data taking in two different configurations. In the firstone, referred to as vacuum mode, the crystal is contained in vacuum. In the second one,called immersion mode, the crystal is operated directly in liquid nitrogen or argon. Inthis project the data for the calibration has been taken in vacuum mode only.

In this mode the crystal as well as the FET (field effect transistor) are contained inan aluminum container (Fig.4b). The container is isolated and evacuated (in our case∼ 10−5 mbar). The vacuum is necessary to protect the crystal surface from contamina-tion (examples are condensing water or ice crystals). The FET would break if operatedat high temperatures [11], typically the crystal can be operated up to 130 K withoutbeeing damaged [6]. Low temperatures (∼ 100K in case of vacuum mode) are obtainedby sticking a cold finger into liquid nitrogen, the dewar containing the fluid has a vol-ume of 30 L which corresponds to refilling approximately every 2 weeks. The thermalcoupling between the cold finger and the crystal is provided by copper braids.

In immersion mode the vacuum container is dismounted and the crystal is immerseddirectly into liquid nitrogen. The dewar used in this case has a volume of 5 L. Thewhole configuration is turned upside down with the cold finger sticking into the air. Thethermal coupling between the crystal and the cold finger is counterproductive in thisconfiguration and thus removed. A heater is situated next to the Ge crystal. It is usedto warm the detector after usage to prevent surface contaminations, from for exaple icecrystals.

3.3 Data Acquisition

The signal from the pre-amplifier is stored for further analysis with an ADC (analogueto digital converter, Fig.5a). Below a certain threshold events are not interesting asthey are dominated by noise. Furthermore, it would take a lot of memory to save them.Therefore a trigger signal has to tell the ADC when to store the data. This trigger signalis obtained by connecting the second output signal which is exactly equivalent to thefirst one to a discriminator. If the signal coming from the pre-amplifier reaches a certainheight, the discriminator sends a logical square signal to the ADC. This signal triggersthe ADC.

The ADC writes the signal to a circular buffer. If no trigger signal comes the data isoverwritten without being saved. The ADC has eight 12 bit channels and a samplingrate of 250 MHz [12]. It has been adjusted such that a +2 V input signal will reachchannel 4098, to exploit the full range of the ADC.

If the data is taken with the MCA (multi channel analyser) the waveforms are notstored. In this case the signal coming from the pre-amplifier is processed by a shapingamplifier. The shaping amplifier integrates the signal over an adjustable time. Theheight of the integrated signal corresponds to the sum of the electrons excited by an

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(a)

(b)

Figure 5: (a) DAQ with ADC. (b) DAQ with MCA.

incident particle in the crystal and thus to the energy of the particle. This signal issplit into two equivalent signals, one is connected immediately to the MCA, the otheris connected to a discriminator to provide the trigger signal. The MCA histograms theincoming signal depending on its height (scheme in Fig.5b).

4 Waveform Analysis

The data presented in this section was acquired using the ADC readout (Fig.5a). Startingpoint is thus a set of waveforms. From this set those events which can be assignedto single γ events should be selected. The energies have to be reconstructed from thewaveform. The obtained energy spectrum will be treated in the next section. This sectionconcentrates on the determination of the energies of the events and the properties of thewaveforms. The development of this analysis was done in the course of this bachelorproject.

4.1 Description of the Waveform

A typical gamma interaction signal in a Ge crystal (as shown in Fig.6a) can be un-derstood quite easily: The pre-amplifier contains a capacitor, which is charged by theincoming signal and discharged over a resistor. If the capacitor would be fully chargedbefore starting to discharge, the peak maximum (Fig.6a) would characterize the energyof the incoming particle. The tail (Fig.6a) has the typical form given by capacitor andresistor, it can be described by an exponential with a characteristic time constant. Theform of the signal can be described by the convolution of a Gaussian with an exponen-tial, which is plotted in Fig.6a in red. As the deviation from the measured values ismaximal for the peak value, the mathematical description is not used in the following.Weather the maximum of the fitted function would provide a better value for the energycalibration was not tested.

Fig.6b shows a pileup event - two γ interactions with a time difference such that their

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(a) (b)

Figure 6: Waveforms taken with the GeMini detector at UZH. Data acquired with theADC readout (Fig.5a). (a) shows a typical waveform, (b) a pileup event. The baselinewas not stable and had to be subtracted to obtain the energy of the event.

signals overlap. The number of pileups depends on the rate of gammas. The used sourceshad a fixed strength and the rate of γ rays could be influenced by the distance of thesource to the detector. Which sources were used and where they were placed will beexplained in section 5.

4.2 Analysis Overwiew

Goal of the analysis is to obtain an energy spectrum without pileup events. The energy ofpileup events cannot be reconstructed with the used analysis method. In this subsectionthe procedure to obtain an energy spectrum is explained. The impact of the differentused parameters will be treated in the following subsections. The code described herewas written in matlab.

The script is divided into two parts. The first part is responsible for the characteristics ofthe baseline. For every waveform the slope of the baseline, the mean baseline position andthe standard deviation of the baseline are written into a histogram. These histogramsare used to choose cuts (range of a parameter, as for example the baseline position, whichis requested for a waveform to participate in the further analysis). Cuts are introducedto discriminate events with improper waveforms (pileup events, events with too muchnoise; see section 4.5). The second part determines which waveforms pass the cuts andcreates the energy spectrum. The energy of the proper events is determined by thesubtraction of the mean baseline from the maximum value of the waveform (compareFig.6a). As the waveform shows some noise, the data was first smoothed out with amoving average window (see section 4.3). The maximum value slightly decreases as it isnot given by random fluctuations anymore (Fig.7a).

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(a)(b)

Figure 7: (a) Noise before and after smoothing, see section 4.3. (b) FWHM of threecalibration peaks depending on moving window width. In order to visualize the datapoints better, different offsets were added to the gamma energies. The FWHM getsbetter with window width until ∼300 ns, afterwards it gets worse again.

4.3 Moving Average Window

Goal of the moving average window is to get rid of electronic noise. The moving averagewindow integrates for every data point of the waveform over a time window around thedata point and divides afterwards by the width of the window. The window is chosento be symmetric around the data point and of constant width for all data points of thewaveform. If the baseline noise behaves periodically it can be smoothed out by a timewindow width equaling the noise time period.

The width of the moving average window was chosen by comparing the FWHM (fullwidth at half maximum) of different calibration peaks and different window widths (seeFig.7b). The figure shows that the FWHM decreases with the chosen window widths.If a window width of 356 ns is used the FWHM for the peak at ∼570 keV decreases by0.52 keV, which corresponds to a reduction of 19 percent. The FWHM for the peak at∼1060 keV is reduced by 0.4 keV (14 percent) and the peak at ∼1460 keV by 0.6 keV(19 percent). Aparently the moving average window has an influence on the energydependence of the resolution. This effect has not been further investigated. Altough theFWHM for the two lower peaks is better in case of a window width of 356 ns, the resultsfor 356 ns and 236 ns are comparable (0.1 keV difference). For the peak at ∼1460 keVthe FWHM is better than in case of 356 ns integration time. For the further analysis awindow width of 236 ns is used. If the window width is increased to 716 ns the FWHMand its error gets even worse than for the raw data.

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4.4 Influence of the Voltage Supply

To study the influence of the high voltage supply data has been taken with two differentHV supplies. In Fig.8a and 8b the baseline positions for two measurements with differentsupplies are shown. For the data taken with the CAEN voltage supply the baseline wasinstable. The data taken with the Canberra supply shows one peak for the baselinepositions (Fig.8a). It is therefore advised to use the Canberra voltage supply. The datashown in Fig.8b is analysed in more detail in the next section.

(a) (b)

Figure 8: Histograms of the baseline positions measured with two different voltage sup-plies. The baseline position is given in keV, the calibration is taken from section 5 withan offset such that the main peak is around zero. (a) Histogram of the baseline positionstaken with the voltage supply from Canberra, model 3125. The baseline is stable.(b)Histogram of the baseline positions for the data taken with the voltage supply fromCAEN, model N1470. The histogram shows two peaks next to the main peak.

4.5 Cuts for Pileup Rejection

Generally cuts are applied to distinguish between ”proper” and ”improper” events. Theyhave to make sure that only those events are used in the analysis, which are understoodand useful. Information on the kind of interacting particle, place of interaction (e.g.electron cloud/nucleus, centre of detector/surface, multiple scatter) and wether it is asingle or pileup event can be seen from the waveform [13]. Chances for interactionsby other particles than gamma rays are slim and trying to eliminate those would gobeyond this project. The cuts used in this analysis are chosen to distinguish betweenpileup and single events as the energy of pileup events cannot be reconstructed by thechosen analysis. As the effects are very small the data taken with the CAEN model

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(a)(b)

Figure 9: (a) Histogram of the slopes fitted into the baselines. Most of the slopes areslightly neagtive, this can be explained by precedent events. The influence of a precedentevent on the energy determination of a current event is neglectable for a sufficiently largeslope. (b) Histogram of the standard deviation of the baselines, overflow peak shown.The standard deviation shows that the baseline is not noise dominated.

N1470 power supply is used. With this voltage supply structures in the histogram ofthe baseline positions were observed (section 4.4) and the effect of cuts on the baselineposition is easily observable (shown in this section). The data used in this section wassmoothed out with a moving average window as described in section 4.3. This processhas no infuence on the results of this chapter (see section 4.6) and makes the waveformplots used in this section more clearly.

For the identification of pileup events the following three possible cut parameters wereintroduced. The corresponding histograms are shown in Fig.8b or Fig.8a, 9a and 9b.The baseline was chosen from 0.8 to 7.2 µs.

• For different waveforms a linear function is fitted to the baseline, the slope shouldyield information on pileup events. A negative slope implies an interaction beforethe beginning of the waveform, a positive slope a pileup event with a peak containedin the baseline.

• The mean value of the baseline is used to specify the baseline position. Changesin baseline position are indicators for changes in leakage current or electronicsbehaviour.

• The standard deviation of the baseline gives information on the noise level.

To get a feeling which waveforms contribute to which parts of the histograms some plotswith waveforms for different cuts were made.

In Fig.10a the 1282 events with their baseline position below -4 keV or above 6 keV areshown (compare Fig.8b). Events with very high energies end up almost for the entire

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(a) (b)

Figure 10: Effect of the cut on the baseline position. The baseline is given in keV. Themain peak is chosen to be at 0 keV. (a) Baseline position above 6 keV or beneath -4 keV,the peak at very low energy is a remnant from smoothing. Plotted are all 1282 events.(b) Baseline position between -4 and 6 keV. Plotted are 2000 events. With cuts on thebaseline position the effects from the voltage supply shown in (a) can be rejected.

(a) (b)

Figure 11: (a) σ between 0 keV and 1.4 keV, 2000 events plotted. This cut has rejectedmost of the small events with slowly rising signals. These signals origin from slow chargecollections, the energy is therefore determined too low. (b) Slope between -0.8 eV/nsand 0.8 eV/ns, 2000 events plotted. For this cut the baseline is very proper, howeverthe slowly rising events are not rejected.

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Figure 12: Slope between -0.8 eV/ns and 0.8 eV/ns, σ between 0 keV and 1.4 keV andbaseline position between -4 and 6 keV, 2000 events plotted. The baseline is very clean.In the tail one pileup event can be seen. Pileup events in the tail are estimated to berare (less than 0.1 per cent of the events recorded). The events saturating the ADCcome from interactions depositing high energies, as for example muons.

acquisition time in the highest ADC channel. They saturate the ADC (for examplemuons). The events in Fig.10b are part of the ones which can be found in Fig.8bbetween -4 keV and 6 keV. In comparison to Fig.10a there are less pileup events in thebaseline. In Fig.11a, 2000 events with a standard deviation between 0 keV and 1.4 keVare plotted (of Fig.9b). These cuts rejected 808 events from 54’000. In Fig.11b, 2000events with a slope between −8 · 10−4 keV/ns and 8 · 10−4 keV/ns are shown. Thesecuts reject only 62 events from 54’000. Fig.12 shows 2000 events with the cuts on theslope and sigma as introduced above and a cut on the baseline position from -4 keV to6 keV. The baseline seems to be very clean now.

In Fig.12 a pileup event can be seen in the tail. An exponential fit into the tail wasintroduced to find such events. This fit however has not worked, it returned always thestart value for the fit. The χ2/ndf for the fit is shown in Fig.13, it shows that the fitdidn’t work, too. The number of pileup events with the pileup in the tail is estimated tobe smaller than 0.1 percent by plotting the waveforms, and thus no time has been spenton repairing the fit.

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Figure 13: Histogram of χ2/ndf of the exponential fits. The fit did not work, this isvisible form the rather large χ2/ndf.

(a)(b)

Figure 14: (a) shows the histograms for the standard deviation for different windowwidths. Smoothing the data results in less noise and as expected a smaller sigma. (b)shows the effect of smoothing on the slope. The plot shows that for the chosen windowwidths the fitted slope is not affected.

4.6 Influence of the Moving Average Window on the Cut Parame-ters

In section 4.5 the data was smoothed out with a moving average window before it wasanalysed. This section shows, that the process has no influence on the detection efficiencyof pileup events.

The baseline position is determined by taking the mean value over the baseline. The

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(a) (b)

Figure 15: (a) Example of a 228Th spectrum used for the determination of the detectorresolution (section 5.4) and the long time stability of the setup (section 5.6). The ”bugpeak” is treated in section 5.2. The data was taken for 20 min, the source located ∼45cm from the Ge crystal. (b) For the calibration a combined spectrum with 22Na, 137Csand 207Bi was taken, the sources where measured at the same time to account for drifts.

moving average window takes for every point the mean value over a fixed window width.Wether the baseline position is determined with the raw or the smoothed out data hasthus no influence.

The moving average window was introduced to suppress electronic noise (section 4.3). Itis therefore expected, that the standard deviation decreases with moving window width(unless it gets wide enough for the signal to have an influence). In Fig.14a the standarddeviation for raw and smoothed data is shown. It behaves as expected. The structureof the histograms stays approximately the same, therefore the process has no influenceon the detection efficiency of pileup events.

The slope of the smoothed out data should not be influenced as long as the movingaverage window is small enough. Fig.14b shows that the moving average window usedin section 4.5 has no influence on the structure or the amount of the slope.

5 Calibration

This section covers the main goal of the project. By using known radioactive gammasources an energy calibration of the detector can be performed (section 5.1), the resolu-tion of the detector can be tested (section 5.4), and the long time stability of the setupcan be checked (section 5.6). The effect of the bias voltage is shown in section 5.3.

A typical spectrum is shown in Fig.15a. For low energies an exponentially decreasing

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background is observed. The sudden drop at low energies is due to the trigger threshold.The 228Th spectra were used for the analysis of the detector response in time stability,whereas the 22Na, 137Cs and 207Bi spectra were used for the energy calibration.

5.1 Energy Calibration

For the energy calibration, a combined spectrum with 207Bi, 137Cs and 22Na was taken for20 minutes (Fig.15b). Spectral lines from 40K and 208Tl are dominant in the backgroundand were used for the calibration too. The distance between the sources and the detectorwas ∼ 45 cm. The calibration data was taken about 24 h after the high voltage andpre-amplifier were turned on to account for possible initial drifts. The respective energiesfor the calibration sources are taken from [14] and shown in table 1.

source peak energy [keV] Ig [%]207Bi 569.702(2) 97.74(3)

1063.662(4) 74.5(2)137Cs 661.657(3) 85.1(2)22Na 511 99.944(14)

1274.53(2) 99.944(14)40K 511 11(1)

1460.83(1) 11(1)208Tl 2614.533(13) 99(1)

Table 1: Sources used for the calibration, the 40K and 208Tl lines come from the labbackground. Ig gives the probability, that the indicated gamma will be emitted in adecay of the isotope. Data from [14].

The positions of the peaks in the measured spectra are obtained from a Gaussian fit,with linear or constant offset (Fig.16) based on a minimization of χ2. The peak positionsin channels versus the peak positions in keV were plotted. To obtain the energy calibra-tion a linear and a quadratic function where fitted. The resulting calibration functionsare:

energy[keV] = −4.7(3) + channel · 0.6881(1) (2)

energy[keV] = −1.7(6) + channel · 0.6846(6) + channel2 · (8(1)) · 10−7 (3)

As can be seen in Fig.17a the residuals for a linear calibration are not compatible withzero, yielding a χ2/ndf of 7.1. If a quadratic term is added the result looks much better(χ2/ndf = 0.5). For the calibration of the GERDA detectors one needs a small quadraticterm, too.

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Figure 16: 40K line at 1461 keV, fit function: f(x) = p0 + p1 · exp(−(x−p2)2

2·p23).

(a)(b)

Figure 17: (a) Residuals for the linear calibration, χ2/ndf = 7.1. (b) Residuals forthe quadratic calibration, χ2/ndf = 0.5. The residuals of the linear calibration are notcompatible with zero, a quadratic term has to be added for the energy calibration.

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5.2 Bug

The bug peak appears as soon as the moving average window is introduced (compareFig.18c and 18d). If the window width is increased the peak shifts to higher values. Truepeaks always shift to the left, as the peak maximum is decreased by averaging. The plotof the 160 waveforms contained in the bug peak is done for a window width of 240 ns,the waveforms look the way they should (Fig.18a). The obtained spectra look apartfrom the extra peak the way they should, the bug peak was present in all spectra afterintroducing the moving average window. It is assumed that the peak has no influenceon the obtained results of this project.

5.3 The Resolution Dependence on the Bias Voltage

The normally used −3 kV bias voltage ensures a fully depleted crystal. The depletionvoltage is estimated to be 2.3 kV [11]. Reducing the voltage below the depletion voltageresults in a worse resolution. To check weather the crystal is depleted, measurementswith a 207Bi source for different bias voltages were taken (each 20 min). The results areshown in Fig.19. As expected no effects are observable. If not indicated otherwise themeasurements presented here were taken with a bias voltage of −3 kV.

5.4 Determination of the Resolution of the Spectral Lines

The FWHM of a Gaussian peak is proportional to the square root of the varianceσ2:

FWHM = 2.35482 · σ. (4)

σ is given by the fit (parameter p3 in Fig.16). In case of a linear calibration curve thetransformation would be easy, σ[keV] = σ[channels] · s where s denotes the slope of thecalibration curve. There is however a quadratic term. This is taken into account by alinearisation of the corresponding peak position (the quadratic term is small):

PeakPos[keV] = a0 + a1 · PeakPos[channels] + a2 · PeakPos2[channels] (5)

slope(PeakPos) = a1 + 2 · a2 · PeakPos[channels] (6)

σ[keV] = slope(PeakPos) · σ[channels]. (7)

The error ∆FWHM on the FWHM is proportional to the error on σ[keV ] with the sameproportionality constant as for the conversion of the FWHM. The error on σ[keV ] isgiven by Gaussian error propagation:

∆σ[keV] =√

(∆slope(PeakPos) · σ[channels])2 + (slope(PeakPos) ·∆σ[channels])2

(8)

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(a)

(b)

(c) (d)

Figure 18: (a) The 160 events in the bug peak, from 1361 keV to 1396 keV for a windowwidth of 240 ns (compare Fig.15a). They show no noticeable features. (b) No smoothing,the spectra looks exactly the way it is expected to look. (c) Window width 20 ns, bugpeak appears. (d) Window width 240 ns, bug peak at higher energies.

∆σ[channels] is given by the fit (error on p3 in Fig.16). ∆slope(PeakPos) can be calcu-lated with the covariance matrix Vij :

∆slope(PeakPos) =

√∑i,j=1,2

Vij ·∂2slope(PeakPos; a1, a2)

∂ai∂aj|PeakPos (9)

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Figure 19: FWHM dependence on bias voltage. As expected the crystal is for the chosenvoltage range depleted. In order to visualize the data points better, different offsets wereadded to the gamma energies.

The covariance matrix is given by the calibration fit routine.

5.5 Detector Resolution and a Comparison of the Data taken withMCA and ADC

Traditionally, spectra are taken with an MCA (multi channel analyser, circuit introducedin section 3.3, Fig.5b). However, the data presented in this project is taken with an ADC(analog to digital converter, Fig.5a in section 3.3). Taking data with the ADC ratherthan the MCA allows to introduce cuts to reject pileup events (section 4.5). Here theresolution obtained by the analysis introduced in this project so far is compared tothe one obtained by an MCA. Further it is analyzed which parameters influence theresolution.

For the data taken with the MCA a linear calibration is used, as the measurements wereonly performed with a 207Bi source and only three peaks are available (one additionalfrom 40K). The error is therefore given by equation (8), where ∆slope(PeakPos) =∆slope is taken from the fit.

In Fig.20a the results are plotted. For an integration time of 1.5 µs the resolution is 20percent better than for an integration time of 1.0 µs, data taken with the MCA. Theenergy resolution for the data taken with the ADC is somewhere in between. Comparedto typical resolutions (Fig.20b) the GeMini detector has a very good resolution.

Now we want to analyze which components of the detector influence the resolution. Thisis done by fitting a model function for the energy resolution into the ADC data. Theenergy resolution is determined by three factors: efficiency of charge collection, statistical

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(a)

(b)

Figure 20: (a) Comparison of the FWHM taken with the ADC and the MCA. Thedata taken with the ADC is comparable to the one taken with the MCA, consideringthat the analysis used is rather rough and the FWHM for the data taken with theADC could be improved. A possible improvement of the resolution for the ADC couldprobably be obtained by integrating, which improves the resolution in case of the MCA.(b) Typical resolutions for Ge detectors (the GeMini is a closed-ended coaxial detector),from http://www.canberra.com/products/465.asp, (2012). The resolution of the GeMinidetector at UZH is very good.

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(a) (b)

Figure 21: Data from two measurements, 207Bi and 228Th, taken immediately after oneanother. (a) Fit of the model function describing only the statistical spread. Fit function:f(E) = p0√

E, χ2/ndf = 27 and p0 = 8.4(4) · 10−3. (b) Fit of a model function describing

the statistical spread and noise of the electronics. Fit function: f(E) =√p0+p1·EE ,

χ2/ndf = 2.4 and the resulting parameters are p0 = 9.9(9) and p1 = 0.1(7) · 10−3.

spread, and electronic noise. For small detectors the efficiency of charge collection ishigher.

If incomplete charge collection is neglected the relative resolution is expected to followthe curve:

FWHM

Eγ=

(p0 + p1 · Eγ)1/2

Eγ(from [6]). (10)

The constant term p0 is added for the effects of the electronics. For the fit in Fig.21a itis only accounted for the statistical spread, p0 is thus set zero. In this case the χ2/ndf is27 and the curve is obviously a bad model for the energy dependence of the resolution.In Fig.21b p0 is left as a free fit parameter. The fit represents the data much better,with a resulting χ2/ndf of 2.4. The discrepancy to the data can be explained by theneglected charge collecting term and the influence of the moving average window on theenergy dependence of the resolution (section 4.3, Fig.7b). Weather the data taken withthe MCA would behave as expected is not investigated due to few data points.

5.6 Long Term Stability of the Detector Response

If data is taken over a long time the stability of the detector is important. For GeMinithe stability of the uncalibrated peak positions was monitored. The 228Th source wasmeasured for 1 h on several days (compare Fig.15a). The high-voltage was switchedon on the first day and not switched off between measurements. As mentioned before

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Figure 22: Drift in time of the measured spectral lines. The 238 keV peak driftedbeneath the trigger threshold on the 10th day.

the performance of Ge detectors depends on the operation temperature of crystal andelectronics, therefore the temperature of the detector and the lab were recorded, too.Furthermore, the slow control (described in section 6) provided continuous measurementsof leakage current and trigger threshold. The obtained data was calibrated with theenergy calibration from the second day. In Fig.22, the calibrated peak positions relativeto the peak position on the second day are plotted. A drift in time is observed. Peakswith higher energies tend to drift less. The effect is of the order of 1 percent for the firstday and decreases afterwards to the order of 1 permil. On the 10th day the data pointfor the 238 keV peak is missing, as it drifted below the trigger threshold.

The detector and room temperature were constant during the time the data was taken.A downward jump in the leakage current could be observed on the third day, from 175 pA

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to 23 pA, afterwards it stayed constant.

After turning on the high voltage a drift is observed on the first day, afterwards there is aperiod where the detector is stable. The observed drift is not induced by the temperatureof the electronics or the crystal. It is recommended not to take data on the first day andto calibrate the detector between measurements (once a day is sufficient).

6 Slow Control System

The Slow Control System monitors temperature and leakage current of the crystal, thetrigger threshold, and the liquid nitrogen level in the cooling dewar. It has been set upby M.Tarka and additional information can be found in [11].

In the course of my bachelor project I checked the temperature measurement of the detec-tor and the measurement of the liquid nitrogen level in the dewar. The measurementsconcerning the leakage current and the trigger threshold were performed by M.Tarka[11].

A Pt100 temperature sensor is used to determine the liquid nitrogen level as well asthe detector’s temperature. The resistance of the Pt100 changes with the temperature,a table with the calibration of resistance to temperature is provided in [15]. The datapoints were plotted with matlab and the resulting linear calibration curve is:

T = 2.5036 ·R+ 23.8761 (11)

A linear fit was chosen because only the temperature for the cooled crystal is of inter-est, if the temperature rises above 120 K the detector should not be operated and theprecise temperature is thus not of interest (typically up to 130 K no lasting damage isexpected [6]). Additionally, the measurement of the detector’s temperature is not veryaccurate, the noise is around 2 K, which makes the small discrepancies of a linear fitneglectable.

6.1 Liquid Nitrogen Level

For the liquid nitrogen level measurement three Pt100 where connected in series (schemein Fig.23a). Depending on how many resistors are inserted in the liquid the resistancechanges. With the resistance of the resistor at the bottom of the dewar RB, the onein the middle RM , and the one at the top RT the measured resistance Rtot can becalculated:

Rtot = RT +RM +RB. (12)

The two limiting cases are that all resistors are inserted into the nitrogen or at roomtemperature. With equation (11) the respective resistances are: Rnitrogen ≈ 22 Ω

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(a) (b)

Figure 23: (a) Liquid nitrogen level measurement circuit. (b) Detector temperaturemeasurement circuit.

Figure 24: Resistance of the three Pt100 temperature sensors while filling the dewar.

and Rroom ≈ 110 Ω. With equation (12) the corresponding measured resistances are:Rtot,nitrogen ≈ 66 Ω and Rtot,room ≈ 330 Ω.

In Fig.24 the resistance curve for the cooling procedure is shown. The warming upprocess takes ∼2 weeks, no calibration curve has been taken so far.

6.2 Detector Temperature

The temperature of the detector is determined with a Pt100 in the vicinity of the crystal.A current source provides a constant current of 0.68 mA, the current flows through thePt100 across which the voltage is measured and then amplified with a gain factor of 10(Fig. 23b). The amplified voltage is digitalized with an ADC (1024 channels / 5 Volt).The expected Pt100 resistance to measured voltage dependence is thus:

R[Ω] =Umeas[channels]

10 · I· 5V

1024[channels](13)

≈ 0.72 · Umeas[channels]. (14)

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Figure 25: Measured resistance dependence on channels with a linear fit.

This relationship was tested by connecting different resistors instead of the Pt100 andmeasuring the obtained voltage. The result is plotted in Fig.25. The fit gives thefollowing result:

R[Ω] = 0.7202 · Umeas[channels] + 0.9321. (15)

The measured slope in equation (15) is in very good accordance with the expectation.The offset comes probably from the wires. The fitted curve (equation (15) combinedwith equation (11)) gives the temperature in Kelvin. This has been implemented intothe Slow Control System.

7 Conclusion and Outlook

The energy calibration of the detector is linear with a very small quadratic term. Intable 2 the expected and measured FWHM is listed [10].

peak energy specification sheet measured vacuum mode

122 keV 0.95 keV -1173 keV - 1.26(5) keV1332 keV 1.40keV 1.14(4) keV

Table 2: FWHM given in the specification sheet for vacuum mode and measured invacuum mode. The 122 keV peak was below the trigger threshold.

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The FWHM for vacuum mode is in agreement with the one obtained in the project con-sidering that the FWHM distribution did not follow the expected curve, this is probablydue to the neglected incomplete charge collection factor. As expected, the bias voltagehas no effects on the resolution for the chosen voltage range (2.5kV to 3.0kV) as thecrystal is depleted. The MCA data seems to result in similar FWHM as the one takenwith the ADC.

Next steps could be to look for effects if the bias voltage is lowered even more, to changethe method of the moving average window (example: instead of averaging everywherewith the same window width, the window width could be made position dependent) orto use the maximum of a fit function instead of the smoothed data.

References

[1] W. Rau, arXiv:1103.5267v1 [astro-ph.CO] (2011)

[2] A.S. Barabash, arXiv:1107.5663v1 [nucl-ex] (2011)

[3] The MAJORANA Collaboration, arXiv:1109.4790v1 [nucl-ex] (2011)

[4] Anatoly A. Smolnikov, arXiv:0812.4194v1 [nucl-ex] (2008)

[5] The EDELWEISS Collaboration, arXiv:1207.1815v1 [astro-ph.CO] (2012)

[6] Glenn F.Knoll,Radiation Detection and Measurement, John Wiley & Sons, 4th Edi-tion, 2010

[7] Standard Electrode Coaxial Ge Detectors (SEGe), Canberra Industries, 2009

[8] J. Eberth, J. Simpson,http://dx.doi.org/10.1016/j.ppnp.2007.09.001 (2007)

[9] C. Amsler,Kern- und Teilchenphysik, vdf Hochschulverlag AG an der ETH Zurich,2007

[10] Detector Specification Sheet, Canberra Industries, 2008

[11] M.Tarka, Development of a low neutron emission calibration source for the GERDAexperiment, 2012, PhD Thesis UZH Zurich

[12] Technical Information Manual MOD. V1720, CAEN Nuclear Physics, 2008

[13] Bakalyarov A.M., Balysh A.Ya., Belyaev S.T., Lebedev V.I., Zhukov S.V.,arXiv:hep-ex/0203017v1 (2002)

[14] S.Y.F. Chu, L.P. Ekstrom and R.B. Firestone, The Lund/LBNL Nuclear DataSearch, February,1999, http://nucleardata.nuclear.lu.se/toi/

[15] Cooled Lithium-Drifted Silicon and Germanium Detectors User’s Manual, 2006,Canberra Industries

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