Virtual segmentation of a small contact HPGe detector: inference of hit positions of single-site events via pulse shape analysis

Abstract

Exploring hit positions of recorded events can help to understand and suppress backgrounds in rare event searches. We propose a pulse shape analysis method to discriminate single-site events (SSEs) in the inner and outer layer of a small contact P-type germanium detector (HPGe). SSEs in the inner and outer layer have different pulse shape features, of which the rise time of the $(T_{Q})$ and current pulse $(T_{I})$ are selected for discrimination. A 500 Bq Thorium-228 (Th-228) source is used to determine the boundaries between the two layers. The double escape peak events from 2614.5 keV $\gamma$-ray are selected as typical SSEs, their numbers in the two layers are used to calculate the volumes and shapes of those layers. Considering the statistical and systematic uncertainties, the inner layer volume is evaluated to be 47.2% ± 0.26%(stat.) ± 0.18%(sys.) ± 0.22%(sys.) of the total sensitive volume. Selecting the inner layer as the analysis volume can reduce the external background in the signal region of Ge-76 neutrinoless double beta (0$\nu \beta \beta )$ decay. We use the Th-228 data to validate the inner layer model and evaluate the background suppression power in the 0$\nu \beta \beta$ signal region $(Q_{\beta \beta }=2039$ keV). The virtual segmentation further reduces the background from the external Th-228 source by about 10%. The virtual segmentation could be used to efficiently suppress surface background like electrons from Ar-42 decay in 0$\nu \beta \beta$ experiments using germanium detectors immersed in liquid argon.

1 Introduction

Small contact high purity germanium (HPGe) detectors are widely used in searching for rare events from physics beyond Standard Model, for instance the neutrinoless double beta $(0\nu \beta \beta )$ decay [1,2,3]. Those searches need an extremely low background level in the signal region to achieve sufficient sensitivity. The discrimination of background and signal via pulse shape analysis is a powerful background suppression technology and is widely used in HPGe based experiments [4,5,6,7].

The energy depositions from $0\nu \beta \beta$ decay events are typically within about a millimeter and are regarded as single-site events (SSEs). On the other hand, backgrounds can be single-site or multi-site events (MSEs), depending on their origination. Small contact HPGe detectors, such as point contact Ge (PCGe) and broad energy Ge (BEGe), have been demonstrated to have SSE and MSE discrimination capability utilizing pulse shape analysis [3, 5,6,7]. After the SSE/MSE discrimination, signals are still mixed with SSE-like backgrounds, such as single Compton scattering of incoming $\gamma$ or direct energy depositions from beta decay electrons penetrating the surface layer of the detector. Signals are expected to have a uniform distribution in the detector, while the backgrounds tend to be close to the detector surface. Therefore, inference of the SSE position can help to understand and suppress the SSE-like backgrounds.

Previous studies [8,9,10] have demonstrated that the charge collection time in a small contact HPGe detector depends on the energy deposition position. Past work [8, 9] has shown that the rise time of the event pulse can be used to estimate the distance of energy deposition from the contact in a PCGe detector.

In this paper, we propose a pulse shape analysis method to discriminate SSEs in the inner and outer layer of a small contact P-type HPGe detector. SSEs are discriminated with time features of the charge and current pulse, the start and end times of which are optimized to have a better discrimination power and a larger inner layer volume. As the search for Ge-76 $0\nu \beta \beta$ decay focuses on controlling background in the signal region $(Q_{\beta \beta }=2039$ keV), the shape and volume of the inner layer are modeled, determined, and validated in a series of Th-228 irradiation experiments. We also discuss the background suppression potential of this method towards possible application in future $0\nu \beta \beta$ experiments, for instance, the LEGEND [11] and CDEX-300 experiments [3].

This paper is organized as follows: the HPGe detector and the experimental setup are introduced in Sect. 2; the pulse shape discrimination methods for SSE/MSE and inner/outer layer events are reported in Sect. 3; the modeling of the inner layer is in Sect. 4 and followed by the determination of the model parameters in Sect. 5; the statistical and systematic uncertainties of the model are assessed in Sect. 6; the background suppression potential is discussed in Sect. 7; a summary is given in Sect. 8.

2 Experimental setup

The detector used in this work is a small contact p-type HPGe detector produced by ORTEC. The detector crystal has a height of 42.6 mm and a diameter of 80.0 mm, and the thin p+ contact is about 3.1 mm in diameter and is implemented in a 1 mm deep hole on the bottom surface of the crystal. The n+ surface of the detector crystal, formed by the lithium diffusion, contains an inactive layer and reduces the sensitive mass of the detector. The thickness of the inactive layer is evaluated to be $(0.87\pm 0.067)$ mm in our previous work [12]. Subtracting the inactive layer, the total sensitive mass of the detector is $(1052.3\pm 0.4)$ g.

Fig. 1

Schematic diagram of the DAQ system

Fig. 2

Experimental setup at CJPL

Fig. 3

a An example of shaping amplifier pulse, the blue region indicates the integral of the pulse after subtracting the baseline, and it is used as the energy estimator; b an example of smoothed preamplifier pulse and the extracted current pulse. Pulse time parameters $T_Q,$ $T_I,$ and parameter “A” in the A/E discriminator are also illustrated. The current pulse is rescaled for demonstration

As shown in Fig. 1, the data acquisition (DAQ) system is based on commercial NIM/VME modules and crates. The detector is operated under 4500 V bias voltage provided by a high voltage module. The output signal from the p+ contact is fed into an resistance-capacitance (RC) preamplifier. The RC-preamplifier provides two identical output signals. One is loaded into a shaping amplifier with a gain factor of 10 and shaping time of 6 $\upmu$s. The output of the shaping amplifier and the other output of the RC-preamplifier are fed into a 14-bit 100 MHz flash analog-to-digital convertor (FADC) for digitalization. The digitalized waveforms are recorded by the DAQ software on a PC platform.

A detector scanning device is built in China Jinping Underground Laboratory (CJPL) [13]. As shown in Fig. 2, the detector and the liquid nitrogen (LN) Dewar are installed with the scanning device. A Th-228 source with an activity of 500 Bq is mounted on the source holder with a step motor controlling the source position.

3 Pulse processing and event discrimination

3.1 Digital pulse processing

Typical pulses from the shaping amplifier and preamplifier are illustrated in Fig. 3. After subtracting the baseline, the integration of the shaping amplifier pulse is used to estimate the event energy (as shown in Fig. 3a). Energy calibration is performed by the measured Th-228 spectrum with characteristic $\gamma$-ray peaks from decays of radionuclides in the Th-228 decay chain.

The pulses from the preamplifier are used to estimate the time features of the event (as shown in Fig. 3b). The charge drift time $(T_Q)$ is defined as the time between the moments when charge pulse reaches 0.2% and 10% of its maximum amplitude. The current pulse is extracted from the charge pulse by a moving average differential filter with a 50 ns filter window, and the current rise time $(T_I)$ is the time between the moments when the current pulse reaches 0.2% and 20% of its maximum amplitude. The start and end points in $T_Q$ and $T_I$ definitions are optimized to achieve a better discrimination between SSEs in the inner and outer detector layers.

3.2 Single and multi-site event discrimination

The single/multi-site event discriminator (A/E) is defined as ratio of the maximum amplitude of the current pulse (A) and the reconstructed energy (E). It has been discussed in various literature [5, 7, 14, 15] that SSE tends to have higher A/E value than MSE in a small contact HPGe detector. Therefore, we apply a cut on A/E to select the SSEs. The acceptance region of the A/E cut is determined by the double escape peak (DEP) events from a measured Th-228 spectrum. The DEP events are mixed with some MSEs from multi-Compton scattering of high energy $\gamma$-rays. The A/E distribution of DEP events is the combination of a low A/E tail from MSEs and a high A/E peak from SSEs. Therefore, we use a Gaussian function to fit the A/E distribution of DEP events to determine the mean $(\mu _{SSE})$ and standard deviation $(\sigma _{SSE})$ of A/E parameter for SSEs. As shown in Fig. 4, the cut threshold is set to $\mu _{SSE}-5\sigma _{SSE},$ leading to about 80% survival fraction of DEP events above this cut and 9% survival fraction of single escape peak events (typical MSEs). The bottom panel of Fig. 5 demonstrates the distribution of A/E versus energy in Th-228 source data, the SSEs form a nearly horizontal line in the A/E-E map, and the same A/E cut is applied to all events in Fig. 5.

Fig. 4

A/E distributions of DEP and SEP events in Th-228 calibration data. The dashed line is the A/E cut threshold $(\mu _{SSE}-5\sigma _{SSE})$

Fig. 5

Typical Th-228 spectra before and after the A/E cut. The characteristic peaks from decay daughters of Th-228 (Tl-208, Bi-212) and other radionuclides (K-40, and Bi-212) are labeled in the spectra. The double-escape peak (DEP) of Tl-208 2614.5 keV $\gamma$-ray is marked in red. Bottom panel is the distribution of A/E versus energy, the single-site event acceptance region is marked in red

Fig. 6

Discrimination of linear and nonlinear events. Data in the figure are from DEP events $(1592.5\pm 5$ keV, after A/E cut) in a Th-228 calibration experiment (source placed at the center of detector top surface). a Distribution of $T_Q$ and $T_I.$ The blue dashed line is the fitted linear function of $T_Q$ and $T_I.$ Red dashed line is the cut limit for inner layer events; b histogram of event linearity index L,  and the Gaussian fit of linear (blue line) and nonlinear (red line) events; c $T_Q$ Histogram for nonlinear events selected by L cut in b. The black dashed lines in b and c are the cut limit for inner layer events

Upper panel of Fig. 5 shows the Th-228 spectra before and after the A/E cut. Main characteristic peaks from the Th-228 source and radionuclides in the surrounding materials are labeled in the spectra. The full-width-at-half-maximum (FWHM) of the double escape peak (1592.5 keV) before (after) the A/E cut is $2.19\pm 0.05$ keV $(2.18\pm 0.03$ keV). The FWHM of the 2614.5 keV peak before (after) the A/E cut is $2.51\pm 0.01$ keV $(2.46\pm 0.02$ keV). A slight improvement in the energy resolution is observed after the A/E cut, as the MSEs have more charge drift paths than SSEs, where they suffer more charge trapping leading to a worse energy resolution.

3.3 Linear and nonlinear event discrimination

Figure 6 demonstrates the $T_Q$ vs $T_I$ distribution of the SSE samples from the DEP events of Th-228 $\gamma$ lines. The $T_Q$ vs $T_I$ distribution of the SSE sample exhibits two types of events: events gathered in a rodlike region in Fig. 6a are referred to as linear events, and other events gathered in a cluster are referred to as nonlinear events. The physical origin of the two types of events will be explored in Sect. 4.1 As shown in Fig. 6, the charge drift time $(T_Q)$ and a linearity index (L) are used to discriminate the linear and nonlinear events. The linearity index is defined as:

$$\begin{aligned} L=T_I-\left( k\times T_Q+b\right) , \end{aligned}$$

(1)

where fit parameters k and b are calculated via fitting $T_Q$ and $T_I$ of typical linear events with the function $(T_I=k\times T_Q+b).$ First, initial values of fit parameters $(k_0$ and $b_0)$ are calculated by fitting events with $T_Q$ and $T_I$ below 500 ns. Then events with linearity $L=T_I-(k_0\times T_Q+b_0)$ in $[-50, 50]$ ns are fitted to give the final value of k and b. As shown in Fig. 6b, the distribution of linearity index L is fitted with two Gaussian functions corresponding to linear and nonlinear events, respectively. The cut limit is set to $(\mu _{L,linear}-3\sigma _{L,linear}),$ where $\mu _{L,linear}$ and $\sigma _{L,linear}$ are the mean and standard deviation of L distribution for linear events. The distribution of $T_Q$ for nonlinear events selected by linearity index L is fitted with a Gaussian function, and the cut limit is set to $(\mu _{T,nonlinear}-3\sigma _{T,nonlinear}),$ where $\mu _{T,nonlinear}$ and $\sigma _{T,nonlinear}$ are the mean and standard deviation of $T_Q$ distribution for nonlinear events as shown in Fig. 6c. The red dashed line in Fig. 6a shows the discrimination limit set by the linearity index L and the charge drift time $T_Q.$

Fig. 7

Pulse shape simulation for SSEs in different positions of the detector. a Charge drift time $(T_Q)$ for SSE as a function of the interaction position; b current rise time $(T_I)$ for SSEs as a function of the interaction position; c distribution of $T_Q$ and $T_I$ for pulses in a and b, those events are gathered in two clusters with a linear and nonlinear relationship between $T_Q$ and $T_I.$ Red crosses mark the positions of four selected SSEs

4 Detector segmentation model

4.1 Demonstration of spatial distribution of linear and nonlinear events via pulse shape simulation

We perform a pulse shape simulation (PSS) for the HPGe detector to investigate the spatial distribution of the linear and nonlinear events. The electric field and weight potential field in the detector are calculated using the $mjd\_fieldgen$ package [16, 17], assuming a linear impurity profile in the Z-direction with an impurity density of $3.7\times 10^9~\text {cm}^{3}$ and $8.0\times 10^9~\text {cm}^{3}$ at the top and bottom surface of the crystal. SSEs with 1 MeV energy deposition are generated at different positions in the crystal. The corresponding charge pulses are calculated via the SAGE-PSS package [18] and electronic noise extracted from the data is added to the simulated pulses.

Figure 7 demonstrates the $T_Q$ and $T_I$ as a function of the interaction position. As shown in Fig. 7a and b, SSEs close to the p+ contact have shorter $T_Q$ and $T_I.$ With the distance to contact increasing, the $T_Q$ and $T_I$ of induced pulses increase simultaneously, for instance, the SSE-3 and SSE-4. These events are typical linear events in Fig. 7c. However, when SSEs near the top and side surfaces of the detector, their $T_Q$ and $T_I$ are not sensitive to their positions. Those SSEs, such as SSE-1 and SSE-2 are typical nonlinear events. It can be explained by the Shockley–Ramo theory [19]: when SSEs deposit energy near the outer surface of the detector, the induced charge and current pulses will not exceed the 0.2% of their maximum amplitude as charge carriers drift in the weak electric and weight potential field area near the surface, the $T_Q$ and $T_I$ of those SSEs are not sensitive to the energy deposition position. Therefore, the start points in $T_Q$ and $T_I$ corresponds to the shape and volume of the inner layer. Setting smaller start points could enlarge the inner layer but be limited by the electronic noise. The noise level (defined as the standard deviation of the baseline) in our experiment is about 1 keV. As our analysis focuses on DEP (1592 keV) and $0\nu \beta \beta$ (2039 keV) events, we set the 0.2% start points to keep the signal-to-noise ratio $>3$ for events with energy over 1500 keV.

4.2 Parameterized segmentation model

According to the pulse shape simulation, the linearity between $T_{Q}$ and $T_{I}$ of the SSE can be used to infer its hit position. We segment the detector into two layers referring to the positions of linear and nonlinear SSEs. The boundary between the two layers is related to the inhomogeneity of the electric field within the detector, which affects the drift trajectory of charge carriers and thereby the pulse shape of the output signal. And due to the lack of precise knowledge of the impurity profile within the Ge crystal, we cannot rely on the PSS to calculate the shape of the two layers but take it as a reference. Therefore, we take an empirical approach to build a segmentation model with 14 parameters to described the boundary.

As shown in Fig. 8, the boundary of the inner layer is the linear connection of 8 spatial points. It is worth noting that the number of spatial points in the model is arbitrary, and it will be demonstrated later that the 8 points model is sufficient for this study. Table 1 lists the bound for each model parameter. In our model, the top of the inner layer can be either on the detector surface or be surrounded by the outer layer. Therefore, the first spatial point $(r_{1},z_{1})$ could be on the top surface or the central axis. To determine the value of each model parameter, we design and conduct a Th-228 scanning experiment.

Fig. 8

Parameterized segmentation model of the detector, where H and R are the height and radius of the crystal. The top spatial point $(r_1, z_1)$ could be on the top surface $(z_1=H)$ or on the central axis $(r_1=0)$ of the crystal. The green shadow region is the inner layer in the segmentation model, and the gray shadow is the inactive layer in the n+ surface

Table 1 Bounds for segmentation model parameters, R and H are the radius and height of the Ge crystal

Fig. 9

Schematic of Th-228 source positions in calibration experiments. The red points indicate the position of the Th-228 source. The red, blue, and green dashed boxes mark the selected measurements for sub-datasets in the uncertainty assessment. The Th-228 source is mounted on a source holder. The carbon fiber vacuum cryostat and the copper crystal holder are also shown

5 Optimization of segmentation model parameters

5.1 Th-228 source scanning experiment

A Th-228 source with about 500 Bq activity is used to perform a scan of the detector top and side surfaces at 19 different positions as shown in Fig. 9. Events in the DEP energy region $(1592.5\pm 5$ keV) are selected as SSE candidates. After removing MSEs by the A/E cut, the linear events in the remaining SSEs are selected using the method in Sect. 3.3. A background measurement is conducted to evaluate SSEs not from the calibration source. The ratio of linear events from the Th-228 source $(R_{L,DEP})$ is then calculated by:

$$\begin{aligned} R_{L,DEP}=\frac{N_{L,S}-N_{L,B}\cdot t_S/t_B}{N_{T,S}-N_{T,B}\cdot t_S/t_B}, \end{aligned}$$

(2)

where $N_{T,S}$ and $N_{T,B}$ are total numbers of selected single-site DEP events in Th-228 and background measurements, respectively. $N_{L,S}$ and $N_{L,B}$ are numbers of selected linear events. The live time of background measurement $(t_B)$ is 42.6 h, and the live times of source measurements $(t_S)$ are at least 7.5 h. The uncertainty of $R_{L,DEP}$ is calculated by propagating the Poisson uncertainties of event counts in Th-228 and background measurement through Eq. (2). Figure 10 shows the linear event ratio of SSEs in the DEP region as a function of Th-228 source positions. The $R_{L,DEP}$ decreased from 33.3 to 24.0% as the source moved from the top center to the edge of the detector. About 2.9% changes in $R_{L,DEP}$ is observed when moving the source along the detector side surface.

Fig. 10

Ratio of the linear event ($(R_{L,DEP})$) in selected DEP events as a function of Th-228 source positions. Error bars indicate the 1$\sigma$ uncertainty. The red squares are the $R_{L,DEP}$ calculated by Eq. (3) using the optimized model parameters in Sect. 5.3

5.2 Spatial distribution of DEP events

Fig. 11

$\delta _D$ histogram for simulated DEP events with the Th-228 source is placed at the center of the top detector surface

Fig. 12

Spatial distribution of simulated SSEs in DEP region. a Th-228 source in the center of the top surface; b Th-228 source on the side of the detector. The labels of the color bar represent the distribution density (arbitrary unit). The Th-228 source holder and scanning device are not shown for visualization

As the linear events are located in the inner layer of the segmentation model, the linear event ratio $R_{L,DEP}$ can be modeled by:

$$\begin{aligned} R_{L,DEP}= & {} \iint _{}^{}M(r,z\mid \theta )F_{DEP}(r,z) \cdot {\textrm{d}}{r} {\textrm{d}}{z}, \end{aligned}$$

(3)

$$\begin{aligned} M(r,z\mid \theta )= & {} \left\{ \begin{matrix} 1 &{}(r,z)\in \text { inner layer}\\ 0 &{} (r,z)\in \text {outer layer} \end{matrix}\right. , \end{aligned}$$

(4)

where $M(r,z\!\mid \!\theta )$ is the select function for the inner event using the segmentation model, $\theta$ represents the model parameters in Table 1, $F_{DEP}(r,z)$ is the spatial distribution of SSEs in the DEP region. The energy deposition of $\gamma$ emitted by the Th-228 source is simulated by Geant4 [20]. We build a detailed geometry model including the germanium detector, the scanning device, and the source holder in the simulation. The energy depositions occurring in the inactive layer of the detector are not recorded in the simulation. The single-site events are selected by the $\delta _D$ parameter. $\delta _D$ is the average distance between the energy deposition points to the charge center of the event:

$$\begin{aligned} \delta _D= & {} \frac{1}{n}\sum _{i=0}^{n}\sqrt{(x_i-{\hat{x}})^2+(y_i-{\hat{y}})^2+(z_i-{\hat{z}})^2}, \end{aligned}$$

(5)

$$\begin{aligned} {\hat{x}}= & {} \sum _{i=0}^{n}x_i\frac{E_i}{E_{tot}}, {\hat{y}}=\sum _{i=0}^{n}y_i\frac{E_i}{E_{tot}}, {\hat{z}}=\sum _{i=0}^{n}z_i\frac{E_i}{E_{tot}}, \end{aligned}$$

(6)

where n is the number of steps in one event, $(x_i,y_i,z_i)$ and $E_i$ are the hit position and energy deposition of the i-th step. $({\hat{x}},{\hat{y}},{\hat{z}})$ and $E_{tot}$ are the charge center and total energy deposition of the event. Events with $\delta _D<\delta _{D,SSE}$ are selected as SSEs, where $\delta _{D,SSE}$ is determined by matching the survival fraction of DEP events in simulation with that of the A/E cut in the experiment. Figure 11 demonstrates a typical $\delta _D$ distribution of simulated DEP events when the Th-228 source is at the top center of the detector. The charge center of the selected SSE is then used to obtain the spatial distribution $F_{DEP}(r,z).$ Figure 12 shows the simulated $F_{DEP}(r,z)$ for the Th-228 source at two different positions.

5.3 Optimization of model parameters

As shown in Fig. 12, the position of the Th-228 source affects the spatial distribution of DEP events and therefore leads to different observed linear event ratios in Fig. 10. Thus, we use a minimum-$\chi ^2$ method to calculate the model parameters $(\theta ),$ in which $\chi ^2$ is defined as:

$$\begin{aligned} \chi ^2=\sum _{k=1}^{19} \frac{\left( R_{k,exp}-\iint _{}^{}M(r,z\mid \theta ) F_{DEP}(r,z){\textrm{d}} r {\textrm{d}} z \right) ^2}{\sigma _k^2}, \end{aligned}$$

(7)

where $R_{k,exp}$ is the measured linear event ratio for Th-228 source at position k $(k=1,2,\ldots 19),$ $\sigma _k$ is the corresponding uncertainty of $R_{k,exp}.$ $F_{DEP,k}(r,z)$ is the simulated spatial distribution of single-site DEP events for the Th-228 source at position k. The minimization of $\chi ^2$ is implemented by the genetic algorithm using a python-based calculation package Geatpy [21]. Figure 13 shows the optimized results. The volume of the inner layer is 47.2% of the total sensitive volume of the detector. The linear event ratios calculated by Eq. 3 using the optimized model parameters are shown as the red squares in Fig. 10. The fit result agrees well with the measurements, the p-value of the $\chi ^2$ fit is 0.701.

6 Uncertainty assessment and model validation

Uncertainties of the shape and volume of the inner layer in the optimized model mainly consist of four parts:

Fig. 13

Optimized result of the segmentation model, the volume of the inner layer is 47.2% of the total sensitive volume

• (1) Statistical uncertainty of the linear event ratio $(R_{L,DEP})$ propagated by the $\chi ^2$-method is evaluated using a toy Monte Carlo method. 3000 Monte Carlo datasets are generated assuming a Gaussian distribution for the $R_{L,DEP}$ with the mean and standard deviation equal to the measured value and uncertainty, respectively. Model parameters are recalculated for each dataset following the same analysis in Sect. 5.3. The distribution of inner layer shapes and volumes for the 3000 samples are illustrated in Fig. 14. The distribution of inner layer volume is fitted with a Gaussian function, and the standard deviation, ± 0.26%, is adopted as the statistical uncertainty.

• (2) Systematic uncertainty from the inactive layer thickness: the inactive layer thickness is measured to be $0.87\pm 0.067$ mm [12], which corresponds to a total sensitive volume of $198.0\pm 0.76~\text {cm}^3$ and contributes a $(\pm$ 0.18%) uncertainty to the inner layer volume.

• (3) Systematic uncertainty due to the choice of dataset: we divide the measured data in Fig. 10 into three sub-datasets. Sub-dataset I and II each consists of ten measured data (marked by red dashed boxes for sub-dataset I, and blue dashed boxes for sub-dataset II in Fig. 9). Sub-dataset III consists of six measured data (green dashed boxes in Fig. 9). The fitting of model parameters are performed in each sub-dataset, and the largest difference in inner layer volume between all sub-datasets and the full dataset (Fig. 16a) is ± 0.22% as a systematic uncertainty.

• (4) Systematic uncertainty due to the construction of the segmentation model: we reconstruct the segmentation model using 6 spatial points (10 free parameters) and 10 spatial points (18 free parameters) and calculate the model parameters using the full dataset. Figure 15b shows the optimized results for the reconstructed models. The overall shape and volume of the inner layer are similar in the three models, and the largest difference in inner layer volume is 0.02%, which is about 10 times smaller than the other three uncertainties and thereby negligible. This indicates the 8-point segmentation model is sufficient in this study.

Including the statistical and systematic uncertainties discussed above, the volume of the inner layer is given as 47.2% ± 0.26%(stat.) ± 0.18%(sys.) ± 0.22%(sys.).

Fig. 14

a Inner layer shapes of the 3000 Monte Carlo datasets. The green, yellow, and blue shadow bands are corresponding to 68%, 95%, and 99.7% quantiles, respectively. The gray shadow is the inactive layer on the n+ surface. b Distribution of inner layer volumes. The red line is the fit of inner layer volumes using a Gaussian function, $\mu$ and $\sigma$ are the mean and standard deviation, respectively

Fig. 15

a Optimized results using different datasets, full dataset (black line) consists of all measured data, sub-dataset I, II, III are selected from the full dataset. b Optimized results for three different models, the chi-square $(\chi ^2)$ and p-value are given to demonstrate the fit goodness of each model. The gray shadow regions in both figures are the inactive layer on the detector n+ surface

Fig. 16

Comparison of simulation and experiment for Th-228 source placed on the side of the detector. a The linear event ratio as a function of energy, The uncertainty band for simulation (the green shadow) consists of uncertainty from the inner layer shape (68% quantile region in Fig. 14a) and statistical uncertainty in simulation. The normalized residuals are shown in the bottom figure, b measured and simulated spectra in 1400–2100 keV region

The measured Th-228 spectra are compared to the simulated spectra to validate the segmentation model. The energy depositions of the $\gamma$-rays emitted from the Th-228 source are simulated via Geant4 and convoluted with the energy resolution of the detector. The SSEs are selected using the $\delta _D$ parameter defined in Sect. 5.2, and the inner layer events are selected by their charge center positions defined by Eq. (6). The measured background spectrum is scaled by the live time of measurement and added to the simulated spectra.

Figure 16 compares the spectra and ratio of inner layer SSE events between simulation and experimental results for one of Th-228 source measurements. The gray band in Fig. 16a is the statistic uncertainty of experiment data, the green band is the combination of the statistic and systematic uncertainties in the simulation. In this case, the systematic uncertainty is taken as the discrepancy between linear event ratios corresponding to the innermost and outmost shape of the 68% quantile of the inner layer (the green region in Fig. 14a). Figure 16b is the comparison of measured and simulated spectra, it demonstrates that the $\delta _D$ cut in the simulation is a good approximation for the A/E cut, and the spectra of inner layer events also show a good agreement between the simulation and measurement in the 1400–2100 keV energy region. For < 1400 keV energy region, our method is limited by the electronic noise as it affects the determination of $T_Q$ and $T_I.$

7 Background suppression performance of virtual segmentation

In the search for Ge-76 $0\nu \beta \beta$ decay using HPGe detectors, backgrounds, mostly $\gamma$-rays and electrons from outside the detector, have to penetrate the outer layer of the detector to deposit their energy in the inner layer. Thus, the outer layer in the virtual segmentation could act as a shielding for the inner layer, and a lower background level of the inner layer may improve the detection sensitivity.

We use the Th-228 scanning data to evaluate the background suppression power of the virtual segmentation. The count rates in spectra are normalized to unit sensitive mass to include the mass loss due to the analysis volume selection. The masses of the detector are 1.052 kg and 0.496 kg for the total sensitive volume and the inner layer, respectively. Figure 17 demonstrates spectra before and after A/E cut and inner layer event selection when the Th-228 source is placed on the side of the detector. First the whole detector is selected as the analysis volume and the A/E cut is applied to removes multi-site events (gray and blue regions in Fig. 17). Then the inner layer of the virtual segmentations is selected as the analysis volume, a further reduction on the event rate is shown in Fig. 17 (red region). It is expected that the SSEs mostly come from the single Compton scattering of high energy $\gamma$-rays emitted from the source and are clustered near the surface of the detector. Thereby the inner layer has a lower background level in the detector.

Fig. 17

Measured spectra for the Th-228 source on the side surface of the detector. cpkkd represents counts per kg per keV per day, $Q_{\beta \beta }$ is the energy of Ge-76 $0\nu \beta \beta$ signal

Fig. 18

Event rate in $0\nu \beta \beta$ signal region (1900–2100 keV) as a function of Th-228 source position. The left and right figures show the event rate for the Th-228 source placed on the top and side surface of the detector, respectively

Figure 18 shows the event rate in the $0\nu \beta \beta$ signal region (1900–2100 keV) as a function of the Th-228 source positions. The highest background suppression power is achieved when the Th-228 source is at the side of the detector. In this case, the A/E cut reduces the event rate by 62%, and the virtual segmentation yields a further reduction of 12% on the basis of the A/E cut. We also perform our method on the background data, the A/E cut and inner layer selection reduce the background in the $0\nu \beta \beta$ signal region from 83.0 cpkkd to 29.6 cpkkd and 22.0 cpkkd, respectively. The virtual segmentation reduces the background by 9.2% in the background measurement, the reduction is within those measured in the Th-228 scanning (5–12%), indicating that the Th-228 source is a good proxy for background from high energy $\gamma$-rays from primordial radioisotopes.

In future $0\nu \beta \beta$ experiments using small contact HPGe detectors, this method might be used to further suppress background in the signal region. Especially for experiments using a liquid argon (LAr) veto system where the HPGe detector is directly immersed in LAr, such as GERDA [1], the LEGEND [11], and CDEX-300$\nu$ experiments [3]. The background from K-42 (daughter of cosmogenic Ar-42 in LAr, with a Q-value of 3525 keV) beta-decay is mainly located in the surface of the detector, therefore might be suppressed if the inner layer is selected as the analysis volume. It should be noted that the balance between a lower background and the loss in detector sensitive mass should be considered in the searching for the $0\nu \beta \beta$ signal. Also, a joint analysis of the inner and outer layer data may help to discriminate the surface background while not losing the exposure in the outer layer, and may consequently improves the sensitivity.

Furthermore, the discrepancy between the inner and outer layer SSE spectrum could be used to infer the location of the background source. A more precise background model could be built by fitting the spectra of events in the inner and the outer layer simultaneously.

8 Summary

In this study, we develop a virtual segmentation model for a small contact HPGe detector and demonstrate its background suppression capability in the Ge-76 $0\nu \beta \beta$ signal region. The HPGe detector is virtually segmented into two layers, and a selection algorithm based on charge pulse drift time $(T_{Q})$ and current rise time $(T_{I})$ is established to identify the position of the single-site event. The shape and volume of the inner layer in the segmentation model are determined using the DEP events in a series of Th-228 source calibration experiments. The volume of the inner layer is evaluated to be 47.2% ± 0.26%(stat.) ± 0.18%(sys.) ± 0.22%(sys.) of the total sensitive volume of the detector.

The background suppression power of the virtual segmentation in Ge-76 $0\nu \beta \beta$ signal region is evaluated by the Th-228 scanning data. Choosing the inner layer as the analysis volume, a further 12% reduction of background is achieved when the Th-228 source is on the side of the detector. Other backgrounds in the $0\nu \beta \beta$ signal region, especially those clustered on the surface of the detector, such as Ar-42 in future $0\nu \beta \beta$ experiments, could also be reduced by the virtual segmentation.

The principle of the virtual segmentation can be extended to other small contact HPGe detectors, for instance, point-contact Ge (PCGe) and broad energy Ge (BEGe) detectors.