# Improvement of the energy resolution via an optimized digital signal processing in GERDA Phase I

## Abstract

An optimized digital shaping filter has been developed for the Gerda experiment which searches for neutrinoless double beta decay in \(^{76}\)Ge. The Gerda Phase I energy calibration data have been reprocessed and an average improvement of 0.3 keV in energy resolution (FWHM) corresponding to 10 % at the \(Q\) value for \(0\nu \beta \beta \) decay in \(^{76}\)Ge is obtained. This is possible thanks to the enhanced low-frequency noise rejection of this Zero Area Cusp (ZAC) signal shaping filter.

## 1 Introduction

Gerda (GERmanium Detector Array) [1] searches for neutrinoless double beta decay (\(0\nu \beta \beta \) decay) in \(^{76}\)Ge. The experiment is located at the underground Gran Sasso National Laboratory (LNGS) of INFN, Italy. Crystals made from isotopically modified germanium with a fraction of \(\sim \)86 % of \(^{76}\)Ge for a total mass of \(\sim \)20 kg are operated as source and detector of the process.

Several extensions of the Standard Model of particle physics predict the existence of \(0\nu \beta \beta \) decay, a process which violates lepton number conservation by two units and which is possible if neutrinos have a Majorana mass component. \(0\nu \beta \beta \) decay is therefore of primary interest in the field of neutrino physics. Neglecting the nuclear recoil energy the energy released by a \(0\nu \beta \beta \) event is shared by the two emitted electrons. Both electrons are stopped within \(\sim \)1 mm of germanium and thus all available energy is deposited in a small region inside the detector. Since distortions by bremsstrahlung are expected to be small the \(0\nu \beta \beta \) decay signature is a peak in the energy spectrum at the \(Q\) value of the reaction, \(Q_{\beta \beta }\), amounting to 2039 keV for \(^{76}\)Ge. The most recent result of this process for \(^{76}\)Ge was published by the Gerda collaboration with a 90 % confidence level (CL) limit on the \(0\nu \beta \beta \) half-life of \(T^{0\nu }_{1/2}>2.1\cdot 10^{25}\) year [2].

A new energy reconstruction shaping filter leading to an improved energy resolution has been developed (Sect. 3), that is denoted as Zero Area Cusp (ZAC) filter. The Gerda experiment (Sect. 2), the readout of the data (Sect. 2.1) and the signal processing (Sect. 2.2) are described first. After the optimization of the ZAC filter (Sect. 4) the Phase I data have been reprocessed (Sect. 5).

## 2 The GERDA experiment

The design and the construction of Gerda were tailored to background minimization. The germanium detectors are mounted in low-mass ultra-pure copper holders and are directly inserted in 64 m\(^3\) of liquid argon (LAr) acting as cooling medium and shield against external background radiation. The argon cryostat is complemented by a water tank with 5 m diameter which further shields from neutron and gamma backgrounds. It is instrumented with photomultipliers to veto the cosmic muons by detecting Čerenkov radiation. A further muon veto is provided by plastic scintillators installed on the top of the structure. A detailed description of the experimental setup is provided in Ref. [1].

A first physics data collection, denoted as Phase I, was carried out between November 2011 and June 2013. In Phase I eight p-type semi-coaxial detectors enriched in \(^{76}\)Ge from the Heidelberg–Moscow (HdM) [4] and IGEX [5] experiments and five Broad Energy Germanium (BEGe) detectors were used [6]. Three coaxial detectors with natural isotopic abundance from the Genius Test Facility (GTF) project [7, 8] were also installed. In a second physics run (Phase II) 30 BEGe detectors will be operated in addition to the eight semi-coaxial together with instrumentation to detect the LAr scintillation light to actively suppress background [9, 10, 11].

### 2.1 Signal readout and shaping with germanium detectors

In Gerda Phase I an additional low-frequency disturbance comes from microphonics related to mechanical vibrations of the long wiring (30–60 cm) connecting the detectors to the preamplifiers.

#### 2.1.1 Digital shaping

In Gerda Phase I the signals were digitized with 14 bits precision and 100 MHz sampling frequency [1]. 16384 samples were recorded per pulse (Fig. 3). After a \(\sim \)80 \(\upmu \)s long baseline the charge signal rises up with a \(\sim \)1 \(\upmu \)s rise time followed by a \(\sim \)80 \(\upmu \)s long exponential tail due to the discharge of the feedback capacitor.

#### 2.1.2 Energy resolution

\(\eta \) is the average energy necessary to generate an electron-hole pair (\(\eta =2.96\) eV in Ge) and \(F\) is the Fano factor (\(\sim \)0.1 for Ge [14]). This term contributes with about 1.8 keV at 2039 keV thus imposing a lower limit to the achievable \(\Delta E\);

\(c\) is a parameter related to the quality of the charge collection and integration. An incomplete charge collection can be induced by charge recombination due to a too high impurity concentration or due to a too low bias voltage while a deficient integration of the collected charge can arise if a filter with a too short integration time is employed. The same effect is obtained in all cases resulting in low-energy tails of the spectral peaks. The parameter \(c\) expresses therefore the amplitude of such tails. For the detectors used in Gerda Phase I, the third term of Eq. (4) is usually one order of magnitude lower than the electronic and charge production terms for events with energy up to 3 MeV.

### 2.2 Data collection and processing in Gerda

Definition of data sets. The run ranges and active detectors are listed

Set | Duration | Detector configuration |
---|---|---|

A | 09.11.11–22.05.12 | ANGs \(+\) RGs \(+\) GTFs |

B | 02.06.12–15.06.12 | ANGs \(+\) RGs \(+\) GTF112 |

C | 15.06.12–02.07.12 | ANGs \(+\) RGs \(+\) GTF112 |

D | 08.07.12–21.05.13 | ANGs \(+\) RGs \(+\) GTF112 \(+\) BEGes |

#### 2.2.1 Calibration of the energy spectrum

The calibrations were performed by inserting up to three \(^{228}\)Th sources in proximity of the detectors [15, 16]. The total activity of the sources was about 40 kBq at the beginning of Phase I. The duration of the measurements was between one and two hours. The energy threshold for the calibrations is \(\sim \)400 keV to reduce disk usage. At least ten peaks with energies between 0.5 and 2.6 MeV are visible in the recorded spectra (Fig. 4). While all peaks are exploited for the calibration of the energy scale, only the full energy peaks (FEP) are used in the fit of the FWHM as function of energy. This is necessary because the single escape peak (SEP), the double escape peak (DEP) and the 511.0 keV line are Doppler broadened.

rejection of events which might decrease the precision of the calibration; e.g., coincidences between detectors, wave forms with superimposed events (pile-up events);

search and identification of the peaks;

fit of the peaks and automatic adjustment of the fitting function according to the number of events in the peak and the peak shape;

extraction of the calibration curve;

fit of the FWHM as a function of energy.

#### 2.2.2 Signal processing

- a delayed differentiation of the sampled tracewhere \(x_0[t]\) is the signal height at time \(t\) and \(\delta \) was chosen to be 5 \(\upmu \)s;$$\begin{aligned} x_0[t] \rightarrow x_1[t] = x_0[t]-x_0[t-\delta ] \end{aligned}$$(5)
- the iteration of 25 moving average (MA) operations:$$\begin{aligned} x_i[t] \rightarrow x_{i+1}[t] = \frac{1}{\delta }\sum _{t' = t - \delta }^t x_i[t'] \quad i=1,\ldots ,25 \end{aligned}$$(6)

This pseudo-Gaussian shaping is a high-pass filter followed by \(n\) low-pass filters. The resolution obtained with the pseudo-Gaussian shaping is very close to optimal if the detectors are operated in conditions where the \(1/f\) noise is negligible [12]. This is not the case for Gerda Phase I where the preamplifiers had to be placed at a distance of 30–60 cm from the crystals due to the low background requirements. The diodes and the pre-amplification chain were connected by OFHC copper strips insulated by soft teflon hoses. Hence, a significant low-frequency noise is present for some of the Gerda Phase I detectors.

As described in Sect. 2.1, the ENC depends on the properties of the detector, of the preamplifier and of the connection between them. In Gerda the diodes have different geometries and impurity concentrations resulting in different capacitances \(C_D\) and different \(I_L\). In addition, the non-standard connections between the detectors and the preamplifiers result in different input capacitances (\(C_i\)). It is therefore preferable to adapt the form and the parameters of the shaping filter to each detector separately.

## 3 ZAC: a novel filter for enhanced energy resolution

Several methods have been developed to obtain the optimum digital shaping for a given experimental setup [12, 13, 18, 19]. For series and parallel noise and with infinitely long wave forms it can be proven [18] that the optimum shaping filter for energy estimation of a \(\delta \)-like signal is an infinite cusp with the sides of the form \(\exp {(t/\tau _s)}\) where \(\tau _s\) is the reciprocal of the corner frequency; i.e., the frequency at which the contribution of the series and parallel noise of the referred input become equal. When dealing with wave forms of finite length, a modified cusp is obtained in which the two sides have the form of a *sinh*-curve. If low-frequency noise and disturbances are also present, the energy resolution is optimized using filters with total area equal to zero [20]. In addition, the low-frequency baseline fluctuations (e.g. due to microphonics) are well subtracted by filters with parabolic shape [21]. The best energy resolution for Gerda is achieved if a finite-length cusp-like filter with zero total area is employed. This can be obtained by subtracting two parabolas from the sides of the cusp filter keeping the area under the parabolas equal to that underlying the cusp.

In reality the detector output current is not a pure \(\delta \)-function, but has a width of approximately 1 \(\upmu \)s. If a cusp filter is used, this leads to the effect of a ballistic deficit [22, 23] and consequently to the presence of low-energy tails in the spectral peaks. This can be remedied by inserting a flat-top in the central part of the cusp with a width equal to almost the maximum length of the charge collection in the diode. The resulting filter is a Zero-Area finite-length Cusp filter with central flat-top that will be referred as ZAC from here on.

## 4 Optimization of the ZAC filter on calibration data

the total filter length \(2L+FT\) was varied for only one calibration run between 120 and 163 \(\upmu \)s. As expected [18] the best energy resolution was obtained for the longest possible filter. Given the variability of the trigger time within a 2 \(\upmu \)s range the maximum of the shaped filter can be at one of the extremes of the wave form when the maximum filter length of 163 \(\upmu \)s is used leading to a wrong energy estimation. This effect completely disappears if the filter is shortened by 2 \(\upmu \)s. Hence, the optimization was performed with \(\sim \)161 \(\upmu \)s long filters;

the optimal length of \(FT\) is related to the charge collection time in the detector. For coaxial detectors this is typically between 0.6 and 1 \(\upmu \)s depending on the electric field configuration in the detector and on the location of the energy deposition. For BEGes it is slightly longer due to the slower charge drift. The value of \(FT\) was therefore varied between \(0.5\) and 1.5 \(\upmu \)s in 120 ns steps;

the optimal filter shaping time \(\tau _s\) depends on the electronic noise spectrum as described in Sect. 2.1. Typically, \(\tau _s\) is of order of 10 \(\upmu \)s. The optimization was therefore performed with values of \(\tau _s\) between 3 and 30 \(\upmu \)s in steps of 1 \(\upmu \)s. Since the optimal \(\tau _s\) was not infinite, the noise present in Phase I data had a non negligible parallel component;

the value of \(\tau \) can in principle be calculated knowing the feedback resistance and capacitance. In reality \(\tau \) is modified by the presence of parasitic capacitance in the front-end electronics. Moreover, given the presence of long cables a signal deformation can arise. Therefore, \(\tau \) is normally estimated by fitting the pulse decay tail. This was not possible due to the presence of more than one exponential. Therefore \(\tau \) was varied between 100 and 300 \(\upmu \)s with 5 \(\upmu \)s step size.

Optimized parameters of the ZAC filter for period D. While the filter length \(2L\) is equal for all the detectors \(FT\) varies between 0.6 and 1.2 \(\upmu \)s according to the charge collection properties of each diode

Detector | 2L (\(\upmu \)s) | FT (ns) | \(\tau _s\) (\(\upmu \)s) | \(\tau \) (\(\upmu \)s) |
---|---|---|---|---|

ANG2 | 160 | 600 | 9 | 190 |

ANG3 | 160 | 840 | 16 | 220 |

ANG4 | 160 | 720 | 13 | 250 |

ANG5 | 160 | 960 | 17 | 170 |

RG1 | 160 | 720 | 12 | 210 |

RG2 | 160 | 680 | 8 | 240 |

GD32B | 160 | 1080 | 13 | 220 |

GD32C | 160 | 960 | 16 | 170 |

GD32D | 160 | 840 | 15.5 | 170 |

GD35B | 160 | 1200 | 17 | 135 |

## 5 Results

Average FWHM over the complete Phase I period. The improvement is computed as the difference between the FWHM for the pseudo-Gaussian and that for the ZAC filter. Only the statistical uncertainty due to the peak fit is quoted

Detector | FWHM at 2614.5 keV (keV) | Improvement (keV) | |
---|---|---|---|

Gaussian | ZAC | ||

ANG2 | 4.712 (3) | 4.314 (3) | 0.398 (4) |

ANG3 | 4.658 (3) | 4.390 (3) | 0.268 (4) |

ANG4 | 4.458 (3) | 4.151 (3) | 0.307 (4) |

ANG5 | 4.323 (3) | 4.022 (3) | 0.301 (4) |

RG1 | 4.595 (4) | 4.365 (4) | 0.230 (6) |

RG2 | 5.036 (5) | 4.707 (4) | 0.329 (6) |

GD32B | 2.816 (4) | 2.699 (3) | 0.117 (5) |

GD32C | 2.833 (3) | 2.702 (3) | 0.131 (4) |

GD32D | 2.959 (4) | 2.807 (3) | 0.152 (5) |

GD35B | 3.700 (5) | 2.836 (3) | 0.864 (6) |

While for the coaxial ANG5 a low-energy tail has to be accounted for in the fit (Fig. 7) the amplitude of the tail in the BEGe GD35B is negligible. The tail it therefore automatically removed from the fit (Fig. 8). This is attributed to the smaller dimensions of the BEGe detector and its reduced charge collection inefficiency. In case of ANG5 the tail amplitude \(D\) is strongly reduced when the ZAC shaping is used thanks to the presence of the flat-top that allows for an improved integration of the collected charge.

A deeper understanding of the result is provided by studying the evolution of the FWHM as function of energy which is fitted according to Eq. (4). An example is given in Fig. 9 showing the resolution curve of all calibration runs for ANG5. As expected the major improvement regards the ENC which reduces FWHM\(^2\) at all energies by a constant. For both, the pseudo-Gaussian and the ZAC filter, the charge production term \(w_p^2=2.355^2\eta F\) is compatible with the theoretical value of \(1.64\cdot 10^{-3}\) keV. Finally, the charge collection term \(c^2\) for the ZAC filter is compatible within the uncertainty with the value obtained for the pseudo-Gaussian filter. The large uncertainty of this parameter is due to the lack of peaks above 3 MeV which makes the fit imprecise. This term is the smallest of the three and accounts for maximally 15 % of the width at 2614.5 keV. A consistent behavior is observed for the other detectors as well.

The Phase I average FWHM for the \(^{208}\)Tl line at 2614.5 keV for each detector obtained with the pseudo-Gaussian and the ZAC filter are reported in Table 3. The average improvement was calculated as the difference between the two values. This is about 0.31 keV for the coaxial and 0.13 keV for the BEGe detectors apart from GD35B for which a much larger improvement is obtained as described above.

A more informative estimation of the energy calibration precision is obtained by calculating the uncertainty \(\delta _{E}\) of the calibration curve at a given energy, e.g. at 1524.6 keV. For each calibration run the quantity \(\delta _{E}(E=1524.6\) keV) is calculated by error propagation on the calibration curve parameters. Using Monte Carlo (MC) simulations \(10^5\) events were randomly generated according to a Gaussian distribution with zero mean and \(\delta _{E}(E=1524.6\) keV). The distributions from all Phase I calibration runs are then summed up and the systematic uncertainty of the energy scale at 1524.6 keV is given by the half-width of the 68 % central interval. This results to be between 0.03 and 0.07 keV and is up to \(16\,\%\) smaller for ZAC shaping with respect to the pseudo-Gaussian filter.

## 6 Summary

The presence of low-frequency noise in the signals of Gerda Phase I mostly induced by microphonic disturbance leads to a degraded energy resolution for some of the deployed detectors. Spectroscopic performance close to optimal is obtained by the use of the ZAC shaping filter. This novel Zero Area Cusp filter is obtained by subtracting two parabolas from the sides of the cusp filter keeping the area under the parabolas equal to that underlying the cusp. A selection of calibration runs has been exploited for the optimization of the ZAC filter. All calibration data sets have then been reprocessed using the optimal filter parameters. An average improvement of 0.30 keV in FWHM has been obtained for both coaxial and BEGe detectors. In one case (GD35B) the energy resolution is improved by 0.86 keV with the excellent low-frequency rejection provided by the ZAC filter.

The stability of the filter parameters over time for the same detector configuration in Gerda along with its outstanding low-frequency noise rejection capabilities provides a FWHM improvement of 0.40 (0.30) keV at the \(^{42}\)K line in the Phase I physics data for the coaxial (BEGe) detectors. Any improvement in the energy resolution will increase the sensitivity of the experiment and allow a better understanding of the experimental background.

The Phase I physics data, reprocessed with the ZAC shaping, will be combined with the Phase II data in a future analysis of the \(0\nu \beta \beta \) decay. The optimization of the shaping filter will be performed from the beginning of Phase II following a procedure similar to the one described in the present work.

## Notes

### Acknowledgments

The Gerda experiment is supported financially by the German Federal Ministry for Education and Research (BMBF), the German Research Foundation (DFG) via the Excellence Cluster Universe, the Italian Istituto Nazionale di Fisica Nucleare (INFN), the Max Planck Society (MPG), the Polish National Science Center (NCN), the Foundation for Polish Science (MPD programme), the Russian Foundation for Basic Research (RFBR), and the Swiss National Science Foundation (SNF). The institutions acknowledge also internal financial support. The Gerda collaboration thanks the directors and the staff of the LNGS for their continuous strong support of the Gerda experiment.

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