Raw Data format and Decorrelation

The signals of the ionization and heat channel of the germanium detectors are continuously recorded during operation as a data stream. The signal of the ionization channel is the four electric potential of the electrodes. The signal of the heat channel is the voltage of the NTD thermal sensor. A data stream consists in time series of these ionization and heat signals. For both the detectors RED80 and REDN1, possessing one NTD and four electrodes, a data stream saves five electric potentials every time step dt. In this work, the sampling frequency is fs = 400 Hz withdt = 2.5 ms. These voltage values are extracted by an analog-to-digital conversion system and thus expressed in Analog-to-Digital Unit (ADU) as introduced in section4.1.8. The data stream is later processed into events with the software NEPAL as described in the section 6.2.3. An event is a time window from a data stream of length Twin.. In this work, this time length is Twin. = 0.5 s. Considering the sampling frequency, an event is a time series of 200 pointstiwith five voltageVCH(ti)withCH ∈ {^heat,A,B,C,D}. As an example, four triggering events are displayed in the figure6.9.

The raw voltage measuredVCH(t)as a function of the timet, is the superposition of a signal functionSCH(t)and a noise functionNCH(t)such that:

∀CH ∈ {^heat,A,B,C,D}^: VCH(t) =SCH(t) +NCH(t) (6.11)

0 500 1000

Vheat/ADU

Exemples of 10.37keV calibration events from the stream tg18l005 in detector RED80:

Bulk event Guard event Guard-Collect

event Frontier event

−60

−40

−20 0

VA/ADU

−60

−40

−20 0

VB/ADU

0 20 40

VC/ADU

0.00 0.25 Time / s 0

20 40 60

VD/ADU

0.00 0.25 Time / s

0.00 0.25 Time / s

0.00 0.25 Time / s

Figure 6.9: Examples of decorrelated triggering events for the detector RED80. An event is a 0.5 s time window of the data stream corresponding to five time series VCH(t) with CH ∈ {^heat,A,B,C,D}. The ionization signals of the bulk, guard and guard-collect events are consis-tent with the expected signatures in RED80 given in equation6.7. The frontier event corresponds to a linear combination of all these signatures.

The noise functionsNCH(t)are characterized by their PSD inADU²/Hz. The modelization of these functions for the heat and ionization channels are discussed in the corresponding chapters 3 and 4. In the case of the ionization channel, the noise function can be separated into two components:

NCH(t) =n^uncorr._CH (t) +n^corr.(t) (6.12) The first component n^uncorr._CH (t) is the uncorrelated noise which is specific to each electrode.

The second component n^corr.(t) refers to the correlated common noise to the four electrodes A,B,C,D. This correlated noise can be subtracted from the measurement by a "decorrelation"

processing step based on combining the ionization signals. The charge conservation in the de-tector is expressed as:

QA+QB+QC+QD =0 ⇔ ^{1 1 1 1}·_Q~ =0 (6.13) with QX the charge induced on the electrode X composing the induced charge vector _{Q. By}~ combining this conservation equation with the expression of the charge vector (eq. 4.55), we obtain the charge conservation expressed with the electric potential of the electrodes:

1 1 1 1

·^Ctot·_V~ =0

⇔ ^∑iC_i1^tot ∑_iC_i2^tot ∑_iC_i3^tot ∑_iC_i4^tot

·_V~ =0

⇔ C^tot,m_AA VA+C^tot,m_BB VB+C^tot,m_CC VC+C_DD^tot,mVD =0

(6.14)

where C^tot and C^tot,m are the Maxwell and mutual capacitance matrices of the considered de-tector with the capacitive cabling according to the equation 4.75. In section 6.2.6, the cabling capacitance affecting all the electrodes is estimated to about 125 pF which is much greater than the self-capacitance termC^m_XXof the detector RED80 and REDN1 evaluated in equations6.3and 6.9. As a result, we can consider:

C^tot,m_AA ≈^Ctot,m_BB ≈^C_CC^tot,m ≈^Ctot,m_DD ^(6.15) The expression of the charge conservation with the electric potential now simplifies to:

VA+VB+VC+VD =0 (6.16)

One should note that this simplification is only valid in this work because of the high cabling capacitanceC^cablingimposed by the current JFET-based electronics. In the future, the use of the HEMT-based electronics aiming at the low cabling capacitance of 5 pF will discard this simplifi-cation. The simplified charge equation6.16indicates that the measurement of one of the poten-tial is redundant as it can be calculated from the others. As illustration, the electric potenpoten-tial of the electrodeAis given by:

VA(t) =−^VB(t)−^VC(t)−^VD(t) (6.17) This redundancy allows a clever change of variables to improve the channel resolution and subtract the correlate noise. We define the decorrelated electric potentialsV_A^decor.(t)as:

(V_A^decor.(t) =α·VA(t) + (1−α)·(−VB(t)−VC(t)−VD(t))

idem for electrodesB,C,D (6.18)

with αa constant. Assuming the same baseline energy resolution σ0 for each of the raw elec-tric potential measurement VX(t), its optimal value is α = ³₄. By replacing the expressions

6.11and 6.12 in the expression of decorrelated measurements 6.18, one can check that corre-lated noise is indeed suppressed. Moreover, the resolution on the decorrecorre-lated measurement is lowerσ₀^decor. = ^√₂³σ0. Although this combination of the electrode signals is advantageous, it requires the charge conservation to be satisfied which is enforced by the "charge conservation cut" discussed in section6.2.8. For the rest of this work, the terms "ionization signal" will refer to these decorrelated signalsV_X^decor.(t). The ionization signals events presented in the figure6.9 are decorrelated.

For each channelCH, the signal functionSCH(t)is modeled as the scaling of a signal template s_CH(t)as:

SCH(t) = ACH·^sCH(t) (6.19)

with the amplitudeACH expressed in ADU. The theorization of the heat and ionization signals are discussed in the chapter3and chapter4respectively. The signal function of the heat channel is model by a linear combination of decaying exponential. The template of the heat signal of unitary amplitude is expressed:

sheat(t) = (1−ǫ−υ)×(exp(−^t/τ1)−^exp(−^t/τth) +ǫ(exp(−^t/τ2)−^exp(−^t/τth))

+υ(exp(−^t/τ3)−^exp(−^t/τth))

(6.20)

with the parameters ǫ,υ,τth,τ1,τ2,τ3 fixed by the thermal properties of the detector. The heat amplitudeA_heatis proportional to the heat energyE_heat. In the "calibration" step of the analysis pipeline, the conversion rate between amplitude and energy is determined experimentally. For the ionization channel, the template signals sX(t) forX ∈ {^A,B,C,D}simply corresponds to the Heaviside function:

∀CH ∈ {^A,B,C,D}^: sCH(t) =^Θ(t) (6.21) Under the assumption of ideal drift of the charge carriers in the germanium crystal, the ioniza-tion amplitudesACHare proportional to the terms of the voltage vector_V~ calculated in equations 6.7and4.47. The corresponding proportionality factor estimates the ionization energy EIon. of the event. The figure6.9 displays four events attributed to different collection volumes of the detector RED80. The signatures of the bulk, guard and guard-collect events are consistent with their modelization given in equation6.7. The frontier event is attributed to a recoil located at the frontier between the collection volumes in the scheme6.2. The charge carriers are separated between the different volumes and induce a signal on all the electrodes.