Control values definition

The electrostatic simulation of the detectors allowed for a theoretical estimation of the fiducial volume percentage according to equation 4.48. This equation compares the fiducial volume drawn by the electric field lines to the total crystal volume. The main assumption made for this calculation is the ideal drift stating that electrons and holes propagates in the germanium follow-ing the electric lines. As discussed in section4.1.4, this assumption differs from the reality. The

Figure 6.19: Ionization energiesEAandEC as a function of the heat energyEheatof all the qual-ity events passing the charge conservation cut from the Background data streams of detector RED80. The black curves delimit the 2σmargin centered on 0 keV used to enforce the bulk cut.

main cause is the oblique propagation of the electrons being a consequence of the band structure of the germanium. Another minor cause is the repulsing force between close drifting carriers of the same charge which might also participate in the spread for the trajectories. This paragraph aims at comparing the expected fiducial volume from the RED80 and REDN1 simulations to the experimental results.

The detectors RED80 and REDN1 were operated in the run 57 with multiple different po-larizations listed in the table 6.4. Each polarization has an associated data stream eventually yielding one experimental estimation of the fiducial volume percentage %_{f id}with analysis. This estimation is based on the 10.37 keV electronic recoils of the calibration peak. Indeed, as these events are induced by the neutron activated germanium atoms of the crystal, they have a uni-form distribution in the crystal. As such, the count of these calibration events is proportional to the volume of the Ge crystal. In order to assure a good energy reconstruction, only events pass-ing the quality cuts with charge conservation are considered. The figure6.20displays the total ionization energy E^tot_Ion.versus the heat energyEheat of all these events in the case of the stream tg09l000 for the detector RED80. The 10.37 keV calibration events are tagged according to their heat-ionization signature as explained in section6.2.7.

We define the quantityntotas the total number of such quality 10.37 keV calibration events in a single data stream. We assume that the numberntotis given by a Poisson distribution Pois(λ) of parameterλ = ntot and standard deviationσ(ntot) = √

ntot. This quantity is proportional to the total volume of the Ge crystal.

Figure 6.20: Total ionization energy E^tot_Ion. versus the heat energy Eheat of all the quality cuts passing the charge conservation cut of the stream tg09l000 for the detector RED80. All non-grey events are part of the 10.37 keV calibration event selection countingntotevents. The red events, numberednincomphave an incomplete charge collection with a partial Luke-Neganov boost. The blue events show a complete charge collection, forming the 10.37 keV calibration peak counting ncomp events. Among these blue events, there arenbulkbulk events represented in black.

All thisntot events have varying ionization and heat energies linked to the quality of their charge collection. We define the condition of complete charge collection as:

(Eheat >_{10.37 keVee}−³·^σ(Eheat)(10.37 keV_ee)

E^tot_Ion. >_{10.37 keVee}−³·σ(E^tot_Ion.)(10.37 keVee) ^(6.56) This condition is delimited with the black lines on the figure. Events with incomplete charge collection are colored red while the events with complete charge collection are blue.

The number n_incomplete is the count of quality 10.37 eV calibration events with incomplete collection. As a subset of the quality 10.37 eV calibration events, the numbernincompleteis consid-ered to result from a binomial distribution B(n,p)with the parameters(n,p) = (ntot,^nincomplete_ntot ). The number of quality calibration events with complete collection is simply ncomplete = ntot− nincomplete.

The number of quality 10.37 keV calibration events passing the bulk cut isn_bulk. This bulk cut is described in section6.2.9. By definition, the bulk events have a complete collection. As such these events form a subset of the quality calibration,nbulkbeing associated with a binomial distribution of parameters(n,p) = (ntot,ⁿ_n^bulk

tot ).

By definition, the incomplete, complete and bulk counts are random variables with a bino-mial distribution whose parameters(n,p)are functions of the random variablentot. This induces an overly complex error propagation. For the sake of simplicity, the standard deviationσ associ-ated with the incomplete, complete and bulk events is the same as for the Poisson distribution:

∀^event∈ {incomplete,complete,bulk}^,σ(event) =√ nevent.

The incomplete percentage quantifies the charge collection of the detector and is expressed:

%_incomplete= ^nincomplete

ntot ·^{100 (6.57)}

The fiducial percentage estimates the fiducial volume of the detector and is expressed:

%_{f id} = ^nbulk

ncomplete ·^{100 (6.58)}

In this percentage,nbulkis considered to be proportional to the fiducial volume such thatnbulk = α·^V(^Fiducial). Similarly, it is considered that the count of events with complete collection is pro-portional to the full Ge crystal volume with the same multiplicative factor such thatncomplete = α·^V(Crystal).

We define the efficiency percentage as:

%_{e f f} = ^nbulk

ntot ·^{100 (6.59)}

This percentage quantifies efficiency of the detector in terms of exposure by combining the ef-fects of both the charge collection and the fiducial volume.

By neglecting the covariance, the error propagation yields an overestimated standard devia-tion:

−4 −3 −2 −1 0 1 2 3 4 Vbias / V

10 20 30 40 50 60 70 80

Percentage/%

Fiducial

(RED80 Simulation) Default configuration:

(V_bias= 2 V) Incomplete collection (Experimental) Fiducial (Experimental) Efficiency (Experimental)

Figure 6.21: Experimental percentages of incomplete collection %_incomplete (in blue), of fiducial events %_{f id}(in orange) and of the efficiency %_{e f f} (in green) as functions of the voltage biasV_bias for the detector RED80. The theoretical fiducial volume percentage %_{f id} = 80.75 % estimated with the RED80 simulation is indicated by the horizontal red line.

In the case of the fiducial percentage, we have:

σ(%_{f id}) =100·_n ¹

complete

s nbulk

1+ ^nbulk ncomplete

(6.61)