Signal and noise simulation of CUORE bolometric detectors
aa r X i v : . [ phy s i c s . i n s - d e t ] J u l Preprint typeset in JINST style - HYPER VERSION
Signal and noise simulation of CUORE bolometricdetectors
M. Carrettoni a and M. Vignati b ∗ a Dipartimento di Fisica, Università di Milano-Bicocca, Milano I-20126, Italy b Dipartimento di Fisica, Sapienza Università di Roma and Sezione INFN di Roma, RomaI-00185, ItalyE-mail: [email protected] A BSTRACT : Bolometric detectors are used in particle physics experiments to search for rare pro-cesses, such as neutrinoless double beta decay and dark matter interactions. By operating at cryo-genic temperatures, they are able to detect particle energies from a few keV up to several MeV, mea-suring the temperature rise produced by the energy released. This work focusses on the bolometersof the CUORE experiment, which are made of TeO crystals. The response of these detectorsis nonlinear with energy and changes with the operating temperature. The noise depends on theworking conditions and significantly affects the energy resolution and the detection performancesat low energies. We present a software tool to simulate signal and noise of CUORE-like bolome-ters, including effects generated by operating temperature drifts, nonlinearities and pileups. Thesimulations agree well with data.K EYWORDS : Bolometer, Detector modeling and simulations, Neutrinoless Double Beta Decay,Dark Matter Interactions. ∗ Corresponding author. ontents
1. Introduction 12. Description of the detector 23. Signal and noise features 44. Signal model 6
5. Estimation of the signal model 106. Noise generation 127. Simulation and validation with data 138. Applications 16
1. Introduction
Bolometers are detectors in which the energy from particle interactions is converted to heat andmeasured via their rise in temperature. They provide excellent energy resolution, though theirresponse is slow compared to electronic or photonic detectors. These features make them a suit-able choice for experiments searching for rare processes, such as neutrinoless double beta decay(0 n DBD) and dark matter (DM) interactions.The CUORE experiment will search for 0 n DBD of
Te [1, 2] using an array of 988 TeO bolometers of 750 g each. It may also be sensitive to DM interactions [3]. Operated at a temperatureof about 10 mK, these detectors exhibit an energy resolution of a few keV over an energy rangeextending from a few keV up to several MeV. In this range the response function is found to benonlinear [4]. The conversion from signal amplitude to energy is complicated and the shape ofthe signal depends on the energy itself. Moreover, the amplitude of the signal depends on thetemperature of the detector, which is very difficult to keep stable with current cryostats within thefew ppm level, the level that would not perturb the energy resolution. The noise of the detectoris dominated by thermal fluctuations induced by vibrations, and significantly affects the energyresolution at low energies [5].In this paper we present a method to simulate signal and noise of CUORE-like bolometers. Thesimulation should be able to reproduce all the features of the data and can be used, for example,to estimate detection efficiencies and to test analysis algorithms. The shape of the signal and thenoise are estimated from the data. The nonlinearities of the signal are reproduced using a model ofthe thermal sensor of the bolometer [4]. – 1 – . Description of the detector A CUORE bolometer is composed of two main parts, a TeO crystal and a neutron transmutationdoped Germanium (NTD-Ge) thermistor [6, 7]. The crystal is cube-shaped (5x5x5 cm ) and heldby Teflon supports in copper frames. The frames are coupled to the mixing chamber of a dilu-tion refrigerator, which keeps the system at a temperature of ∼
10 mK. The thermistor is glued tothe crystal and acts as thermometer (Fig. 1). A Joule heater is also glued to most crystals. It isused to inject controlled amounts of energy into the crystal, to emulate signals produced by parti-cles [8, 9]. When energy is released in the crystal, the crystal temperature increases and changesthe thermistor’s resistance according to the relationship [10]: R ( T ) = R exp ( T / T ) g (2.1)where R and T are parameters that depend on the dimensions and on the material of the thermistor.For CUORE bolometers values are about 1 . W and 3 . g can be considered constant and equal to 1 / Heat bath ~ 10 mK(copper)Thermal coupling(PTFE)Thermistor(NTD-Ge)
Absorber Crystal(TeO ) Figure 1.
Sketch of a CUORE-like bolometer (left) and a photograph of a bolometer (right). The TeO crystal is held by Teflon supports, the thermistor is glued to the crystal and its wires are attached to thecopper frame. The supports and the thermistor wires thermally couple the crystal to the copper frame, whichact as heat bath. To read out the signal, the thermistor is biased in differential configuration with a bipolar volt-age generator ± V bias connected to a pair of load resistors, R L ’s, finally connected to the thermistor’sterminals. The resistance of the thermistor varies in time with the temperature, R ( t ) , and the voltageacross it, V R ( t ) , is the bolometer signal. The value of R L ’s is chosen to be much higher than R ( t ) so that V R ( t ) is proportional to R ( t ) . Since from Eq. (2.1) positive temperature variations inducenegative resistance variations, the polarity of V bias is chosen to be negative in order to obtain pos-itive signals. The connecting wires add in parallel to the thermistor a parasitic capacitance c p . Aschematic of the biasing circuit is shown in Fig. 2. In the figure, the series of the two load resistorsis represented as a unique resistor R L . The signal V R ( t ) is amplified, filtered with a 6-pole activeBessel filter, and then digitized with an 18-bit analog-to-digital converter (ADC). To fit the signalin the range of the ADC, which is [-10.5,10.5] V, a programmable offset voltage V h is added to V R ( t ) . The front-end electronics, which provide the bias voltage, the load resistors, the amplifierand the offset voltage, are placed outside of the cryostat, at ambient temperature [13].At 10 mK the value of R ( T ) is of order 100 M W , R L is chosen as 54 G W (27 +
27 G W ) and V bias as ∼ c p value depends on the length of the wires that carry the signal out of the– 2 – L R ( t ) V bias c p V R ( t ) Figure 2.
Biasing circuit of the thermistor. A voltage generator V bias biases the thermistor resistance R ( t ) in series with a load resistance R L . The bolometer signal is the voltage V R ( t ) across R ( t ) . The wires used toextract V R ( t ) from the cryostat have a non-negligible capacitance c p . cryostat, typically it is of order 400 pF. The amplifier gain, the Bessel filter frequency bandwidth,the duration of the acquisition window and the sampling frequency are set typically at 5000 V / V,12 Hz, 5.008 s and 125 Hz, respectively.The data analyzed in this paper come from test bolometers operated by the CUORE collabora-tion at the Gran Sasso underground laboratory (LNGS) in Italy [14]. The bolometers were exposedto a
Th calibration source which, together with an a line generated by Po contamination inthe crystal, allows the analysis of an energy range up to 5407 keV.To simplify the description of this work we will focus our analysis on a single bolometer. Thesignal rate on that bolometer was 133 mHz, that has to be combined to the rate of heater pulses thatwere fired at an energy of 1885 keV every 300 seconds (3.3 mHz). The energy spectrum acquiredin about 3 days is shown in Fig. 3.
Energy (keV) c oun t s / . ( k e V ) Figure 3.
Energy spectrum. All lines are generated by the
Th calibration source except for the line at5407 keV, arising from
Po contamination in the TeO crystal. Heater pulses were fired at an energy of1885 keV. – 3 – . Signal and noise features Examples of signals generated by a 2615 keV g -ray and by the heater, as acquired by the ADC, areshown in Fig. 4. The baseline voltage of the pulses is related to the thermistor temperature in staticconditions, and the amplitude is related to the energy released. The shape of heater pulses is foundto be slightly different from that of particle pulses: the rise time of the particle (heater) pulse in thefigure, computed as the time difference between the 10% and the 90% of the leading edge, is 55(54) ms while the decay time, computed as the difference between the 90% and 30% of the trailingedge, is 220 (255) ms. Time (s) A m p lit ud e ( m V ) Time (s) A m p lit ud e ( m V ) Figure 4.
Pulse shapes of a 2615 keV g -ray (left) and heater (right). The baseline is related to the temperatureof the thermistor before the particle interaction or the heater shot. The amplitude carries information on theamount of energy released. As already observed in Ref. [4] several nonlinearities are present:1. The rise and the decay times of a pulse depend on the energy (Fig. 5).2. The amplitude of the pulse depends on the base temperature, which varies during the dataacquisition (Fig. 6).
Amplitude (mV) R i s e ti m e ( m s ) heater Amplitude (mV) D eca y ti m e ( m s ) heater Figure 5.
Pulse shape parameters versus energy. The correlation with energy is negative for the rise time(left) and positive for the decay time (right). Heater pulses are marked in red. – 4 – aseline (mV) A m p lit ud e ( m V ) Figure 6.
Amplitude of heater pulses versus baseline. A change in the bolometer temperature also changesits response, degrading the energy resolution.
Energy [keV] E [ k e V ] D -10-50510152025 Figure 7.
Residuals obtained using a linear calibration function for the well-identified peaks in the
Thsource spectrum. Considering that the energy resolution is ∼ D E betweenthe estimated energies and the true peak energies is not compatible with zero. The error bars refer to theuncertainty on the estimated peak position that depends on the FWHM and on the number of events N in thepeak as FWHM / ( . √ N ) .
3. The amplitude dependence on energy is not linear. The deviation from linearity of the datais estimated by comparing the result of a linear calibration function,Energy = constant · Amplitude , (3.1)to the true energy of the source peaks. The residuals evaluated on the peaks generated bythe Th source (see Fig. 3) are shown in Fig. 7. The 5407 keV line is not considered herebecause a particles have a quenching factor different from g and b particles [5].The noise of the bolometer in the signal frequency region (0-10 Hz) is a result of vibrationsinside the cryostat, and depends on the mechanical setup of the experimental apparatus. The noisepower spectrum was estimated as: N ( w k ) = < | n ( w k ) | > (3.2)– 5 –here n ( w k ) is the k-th component of the discrete Fourier transform (DFT) of an acquired wave-form not containing signals, and <> denotes the average over a large number of waveforms. Atypical noise waveform and the estimated power spectrum are shown in Fig. 8. The peaks in thepower spectrum are due to the crystal friction against its frame, and the residual common modecontribution from the vibration of the connecting wires (readout by the differential preamplifier).Which of the two sources is dominant may depend on the set-up. The white noise contribution athigh frequency is due to the ADC digitization.In the next sections we will describe the procedure we developed to simulate the signal shape,the nonlinearities and the noise. Time (s) A m p lit ud e ( m V ) Frequency [Hz] / H z ] P o w e r [ m V -4 -3 -2 -1 Figure 8.
Noise of a bolometer at the output of the acquisition chain. A sample waveform (left) and thepower spectrum estimated from a large number of waveforms (right).
4. Signal model
To simulate the data we developed a model of the bolometer signal that reproduces the pulse shapeand the nonlinearities. Starting from the energy release in the crystal, the model is divided intothree main stages: the thermal response of the bolometer, the response of the thermistor and of itsbiasing circuit, and the simulation of the electronics. All parameters are detector variables, exceptfor the thermal model parameters, which we determine from fits to the data.
A bolometer is a thermal system composed of a crystal, crystal supports, thermistor, and their cou-pling elements [15] (Fig. 9). The crystal capacitance C c is connected to the thermistor throughthe glue spots with conductance K g and to the supports through a contact conductance K cs . Thethermistor can be represented as a two-stage system composed of a lattice and an electron gas, eachwith its capacitance and conductance, and inter-connected by a conductance K ep . The lattice ca-pacitance, not shown in the figure, is negligible and it discharges through the gold wires connectedto the main heat bath K Au . The electron gas capacitance and conductance are labeled as C e and K e ,respectively. The left side of the circuit shows the crystal supports with their capacitance C s andtheir conductance to the main bath K s . The main bath acts as the reference ground.We are interested in the expression of the temperature variation of the thermistor electron gas(node 4 in Fig. 9) as a function of the time D T ( t ) after that an amount of energy E is released in– 6 – s K cs K g K ep C e K s C c K Au K e crystal support crystal thermistor Figure 9.
Thermal circuit of a CUORE-like bolometer. Each of the elements is identified in the text. the crystal. The energy release is effectively instantaneous, because the phonons produced by aparticle thermalize in a time much shorter than the rise time of acquired pulses [16]. Under theseconditions the analytical expression of D T ( t ) is found to be: D T ( t ) = A T (cid:16) − e − t t r + a e − t t d + ( − a ) e − t t d (cid:17) , (4.1)where all the parameters are complicated functions of the elements of the thermal circuit. Theformula indicates that the thermistor temperature increases with one rise time constant, t r , anddecreases with two decay constants, t d and t d . The parameter a weighs the two exponentialdecays and satisfies the condition 0 ≤ a ≤
1. The amplitude of the thermal pulse, A T , is also afunction of the thermal elements and is directly proportional to E .In principle the parameters in Eq. 4.1 could be known if one were able to measure the un-derlying thermal elements. Previous investigators have measured the thermal parameters [15], andobtained heat capacitances and heat conductances of order 10 − a J / K and 10 − b W / K with a in therange 9 to 10 and b in the range 9 to 11. Such measurements are of little utility for us for severalreasons. First, the capacitances and conductances depend on the temperature and should be mea-sured in the bolometer working temperatures, that depend on the setup. Second, several thermalelements vary from bolometer to bolometer. For example, the contact heat conductance betweenthe crystal and its supports ( K cs ) changes with the detector configuration, and the heat conduc-tance of the glue ( K g ) varies because of the weak reproducibility of the glue deposition. Third, weshould know the parameters with a precision of the order of the present energy resolution (0 . K e and C e , and therefore does not affect the configuration of thethermal circuit and the form of Eq. 4.1. – 7 – .2 Thermistor and biasing circuit models When the thermistor temperature varies, its resistance varies according to Eq. 2.1: D R ( D T ) = R exp (cid:18) T T B + D T (cid:19) g − R B (4.2)where T B is the initial temperature of the bolometer and R B = R ( T B ) . Because we do not know theparameters R , T and T B with the desired accuracy, we adopt an approximation valid for the small D T that will occur [4]: D R ( D T ) ≃ R B (cid:2) exp ( − h D T / T B ) − (cid:3) , (4.3)where h = (cid:12)(cid:12)(cid:12)(cid:12) d log Rd log T (cid:12)(cid:12)(cid:12)(cid:12) = g log R ( T ) R . (4.4) h is the sensitivity of the thermistor, which has value of order 10 but is not known with precision.The advantage of Eq. 4.3 is that the parameter R B can be measured with precision and that the un-known h / T B is just a scale factor applicable to the temperature variation. We obtain the expressionfor the resistance variation after an energy release by substituting Eq. 4.1 into Eq. 4.3 : D R ( t ) = R B n exp h − A (cid:16) − e − t t r + a e − t t d + ( − a ) e − t t d (cid:17)i − o (4.5)where A = h A T / T B . This expression absorbs the unknown parameter h / T B with the unknownthermal amplitude A T . The thermistor model therefore does not change the number of unknownparameters and adds the measurable parameter R B .The relationship between the voltage across the thermistor V R ( t ) and its resistance R ( t ) can beobtained from the differential equation describing the thermistor’s biasing circuit (see Fig. 2): (cid:20) R L + R ( t ) R ( t ) (cid:21) V R ( t ) − V bias + R L c p dV R ( t ) dt = . (4.6)The model we are building is based on variations of the resistance from the measured value of R B ,which in turn generate voltage variations from the corresponding voltage V BR : V BR = V bias R B R B + R L . (4.7)By splitting R ( t ) and V R ( t ) into time-independent and time-dependent contributions, R ( t ) = R B + D R ( t ) V R ( t ) = V BR + D V R ( t ) , (4.8)we obtain the differential equation relating resistance and voltage variations: " R L + R B + D R ( t ) R B + D R ( t ) V bias R B R B + R L + D V R ( t ) − V bias + R L c p d D V R ( t ) dt = . (4.9)Given the form of D R ( t ) in Eq. 4.5, D V R ( t ) cannot be obtained in an closed form. In our bolometermodel we solve Eq. 4.9 numerically using the Runge-Kutta method [17].– 8 –he thermistor and the biasing circuit are the only sources of nonlinearities of our model, andshould be able to describe the nonlinearities observed in the data. If we assume that the thermalcircuit responds linearly, the amplitude A in Eq. 4.5 is directly proportional to the energy E releasedin the bolometer, A = c · E . (4.10)The corresponding resistance variation, however, is not proportional to A , because the exponentialdependency in Eq. 4.5 does not transform linearly the shape and the amplitude of the pulse. More-over the voltage variation is not strictly proportional to the resistance variation. This model descrip-tion should be sufficient to generate the shape dependence on energy and the nonlinear calibrationfunction (Figs. 5 and 7). The amplitude dependence on the baseline in Fig. 6 can be generated fromEq. 4.5 by varying the baseline voltage of the pulse and hence the thermistor resistance R B . The front-end electronics amplifies the bolometer signal D V R ( t ) by G , a parameter that is measuredwith a precision better than 0 . . D V G ( t ) = D V R ( t ) · G , (4.11)is fed into a six-pole Bessel filter, whose transfer function is B ( s ) = s + s + s + s + s + s + . (4.12)In the above equation s is the normalized Laplace variable which can be expressed in terms of thefrequency w as: s = w . f b , (4.13)where f b is the filter cutoff (12 Hz in our case).The signal is filtered by multiplying its DFT, D V G ( w ) , by B ( w ) , removing the “DFT wraparoundproblem” with the method described in Ref. [18]. The output of the filter is then obtained as: D V ( t ) = F − [ D V G ( w ) · B ( w )] , (4.14)where F − denotes the inverse DFT.In summary , the model of the signal, from the energy release in the crystal E to the signalacquired by the ADC D V ( t ) , is obtained using the thermal model in Eq. 4.1, the thermistor modelin Eq. 4.5, the voltage across the thermistor from Eq. 4.9, the amplifier and Bessel filter effects inEqns. 4.11 and 4.14: D V ( t ) = E −−−−→ Eq : 4 . D T ( t ) −−−−→ Eq . . D R ( t ) −−−−−−→ Eq . . D V R ( t ) · Eq . . G ⊗ Eq . . B ( t ) . (4.15)– 9 –he baseline voltage of the signal, V B , is the sum of two components, the thermistor voltage V BR in Eq. 4.7 scaled by the electronics gain G , and the offset voltage V h added by the electronicsitself: V B = V BR · G + V h = V bias G R B R B + R L + V h . (4.16)To reproduce a real waveform, V ( t ) , we add the baseline voltage to the pulse in Eq. 4.15 and wealso account for the onset time t of the pulse, which starts about 1 s after the beginning of thewaveform: V ( t ) = V B + Q ( t − t ) D V ( t − t ) , (4.17)where Q ( t ) is the Heaviside step function. Since the thermistor temperature is not stable, theresistance R B and the baseline voltage V B are not fixed parameters of the model. V B is measured ondata by averaging the first 0.8 s of the waveform. The corresponding value of R B is then computedfrom Eq. 4.16 and used in the signal model. In Tab. 1 we list the parameters of the model andindicate whether they are measured, or determined from fits to signal waveforms. Table 1.
Parameters of the signal model in Eqns. 4.15, 4.16 and 4.17.
Parameter Name Equation Estimation t r Thermal rise time 4.1 Fit a Weight of the two thermal decay constants 4.1 Fit t d Fast thermal decay constant 4.1 Fit t d Slow thermal decay constant 4.1 Fit c Energy to thermal amplitude conversion 4.10 Fit R B Thermistor resistance at the pulse baseline 4.5 Measured V bias Bias voltage 4.9 Measured R L Load resistor 4.9 Measured c p Parasitic capacitance 4.9 Measured G Electronics gain 4.11 Measured f b Bessel filter cutoff frequency 4.14 Measured V h Electronics offset voltage 4.16 Measured V B Baseline voltage 4.17 Measured t Onset time of the pulse 4.17 Fit
5. Estimation of the signal model
We intend that the model we developed accounts for all the nonlinearities of the signal. The un-known thermal parameters in Tab. 1 are expected to be independent of the energy, except for theamplitude A , which should be proportional to the energy (Eq. 4.10). The energy independenceallows us to determine unmeasured parameters from fits to pulses at a single energy, and apply theresulting model over the entire range of energies. We performed fits on particle pulses occurring inan energy window of 30 keV around the 2615 keV g peak. We performed a separate set of fits onheater pulses, because their shape differs from the shape of particle pulses.– 10 – ime (s)0 1 2 3 4 5 A m p lit ud e ( m V ) F it - D a t a ( m V ) -4-2024 Time (s)1 1.02 1.04 1.06 1.08 1.1 1.12 A m p lit ud e ( m V ) A m p lit ud e ( m V ) F it - D a t a ( m V ) -3-2-1012 Time (s)1.1 1.12 1.14 1.16 1.18 1.2 F it - D a t a ( m V ) -3-2-1012345 Figure 10.
Fit of a 2615 keV pulse. Data (black solid lines) with superimposed fit function (Eq. 4.17) (bluedashed lines) are shown in the left column, fit residuals are shown in the right column. The two bottom rowsdisplay a zoom of the rise and of the maximum of the pulse, respectively. The fit region ends when the pulsefalls to within 3 standard deviations of the baseline level.
Table 2.
Average parameters fitted on particle (2615 keV g ) and heater pulses and the c / ndf of the fits. Parameter Particle Heater t r ( ms ) . ± .
05 18 . ± . a . ± .
002 0 . ± . t d ( ms ) . ± . . ± . t d ( ms ) ±
20 970 ± c ( / MeV ) . ± . . ± . t ( s ) . ± . . ± . c / ndf 2 . ± . . ± . g -ray pulse, Tab. 2 reports the parameters averagedover 25 fits of particle and heater pulses. The average c is about twice the number of degreesof freedom, i.e. about twice the value one expects for a perfect model. Shortcomings of the modelare also apparent in the fit residuals on the right of the figure, where mismatches are evident inthe leading edge and in the vicinity of the maximum of the pulse. The voltage amplitude of the fit– 11 –unction is slightly biased, and found to be, on average, higher than the amplitude of the pulse by0 . ± .
02 % for particle pulses and 0 . ± .
02 % for particle heater pulses.The fit error can be ascribed to an incompleteness of the thermal model, that would probablybenefit from the inclusion of second order effects like the nonlinearities of the thermal elementsand the electrothermal feedback. For our purposes, however, the model performs well, reproducingthe signal at the per mil level.
6. Noise generation
A complete and predictive model for the noise of CUORE bolometers is missing. The main sourcesof fluctuations that spoil the energy resolution have different origins and, to estimate the overallcontribution, each of them should be propagated with the transfer function of each step of theacquisition chain. Examples of these sources are the Johnson noise of the load resistors, vibrationsof the experimental apparatus that dissipate energy in the bolometer and instabilities of the cryostattemperature. All these effects contribute to the power spectrum shown in Fig. 8.Although the noise from the load resistors and amplifiers is predictable, a model for the noisefrom vibrations is not in hand. Moreover, it varies from bolometer to bolometer, because of theweak reproducibility of the assembly. On this account we adopted a statistical model for the noise.We simply require that, on average, the simulated time series behave like the experimental one,namely that the average power spectrum of the simulated baselines is as close as possible to data.We followed an approach [19] based on an application of the Carson’s theorem [20] to adiscrete time series. A random waveform n ( t i ) can be represented as a superposition of independentpulses of fixed shape g ( t i ) and amplitude A , distributed in time according to a Poisson process ofrate l : n ( t i ) = A (cid:229) l g ( t i − t l ) , (6.1)where the differences between consecutive t l ’s follow an exponential distribution with mean 1 / l .The theorem states that N ( w k ) and G ( w k ) , the average power spectra of n ( t i ) and g ( t i ) , respectively(see Eq. 3.2), satisfy the relationship: N ( w k ) = l TA G ( w k ) , (6.2)where T is the length of the time series (5.008 s in our case). To utilize the theorem one must finda shape g ( t i ) for which the power spectrum is proportional to the average power spectrum of thenoise time series to be simulated. A and l are adjustable parameters with product fixed by Eq. 6.2.These parameters set the aspect of the noise in the time domain, i.e. the same power spectrum canbe produced with a small rate of pulses with large amplitude and vice versa. Once g ( t i ) and, say, l are determined the method is fully specified and n ( t i ) is obtained from Eq. 6.1.We built g ( t i ) as the inverse DFT of g ( w k ) = p N ( w k ) e q k , (6.3)where q k is a phase randomly sampled within [ , p ] . The reality of g ( t i ) is guaranteed by imposingthe constraint g ( w k ) = g ∗ ( − w k ) . l was chosen equal to the Nyquist frequency (62.5 Hz), i.e. themaximum rate producing distinguishable pulses.– 12 –he left panel of Fig. 11 shows a simulated and a measured noise waveform, and their appeare-ance is similar. The right panel of the figure shows the agreement between simulated and measuredpower spectra. Time (s) A m p lit ud e ( m V ) Frequency [Hz] / H z ] P o w e r [ m V -4 -3 -2 -1 Figure 11.
Comparison of noise simulation (solid red line) with data (dashed black line / dots). Samplewaveforms (left) and the power spectrum estimated from a large number of waveforms (right).
7. Simulation and validation with data
We built a simulation engine that is able to generate particle, heater, and pure noise waveforms. Ituses as input the parameters of the signal model in Tab. 1 and the measured noise power spectrum.To correct the small error of the model in reproducing the pulse shape (see Sec. 5), the engine scalesthe simulated signals by the estimated bias, so as to match the amplitude of measured signals. Thesimulated noise is summed to the signal assuming that it is purely additive. The additivity issupported by the fact that the absolute energy resolution for heater pulses is independent of theenergy release, and by the residuals in Fig. 10 that do not seem correlated with the time evolutionof the pulse.The simulation can be highly customized. One can choose the energy distribution of the eventsto be generated, the baseline distribution, and the distribution of the time interval between events.The features of the signal described in Sec. 3 should be automatically reproduced by the model,and signals close in time are summed to reproduce pileups.To validate the simulation engine, we show the results of a simulation configured to reproducethe data shown in Sec. 3. Signals were generated sampling their energy from the spectrum inFig. 3, the baseline was generated in the range of Fig. 6, the time delay between particle pulseswas generated following an exponential distribution with mean 1 / (
133 mHz ) , and heater pulseswere generated every 300 s. Particle and heater pulses were generated using the fitted parametersin Tab. 2, and the noise was generated from the power spectrum in Fig. 8.The comparison of the simulation with the data shows good agreement. The shape of thepulses with noise added reproduces well the acquired waveforms (Fig. 12). The distributions of therise and decay times shown in Fig. 13 also confirm the agreement. We attribute a small mismatchin the rise time of heater pulses to imperfections of the model (see Sec. 5). The correlation betweenpulse amplitude and baseline is very well reproduced (Fig. 14). The dependence of amplitudeon energy follows the data well at high energies (Fig. 15). The mismatch of order 1% at zero– 13 –nergy is another manifestation of the imperfections of the model. In most applications this erroris ignorable, nonetheless we incorporated an option to generate waveforms by sampling from theamplitude spectrum instead of the energy spectrum, in which case the mismatch is suppressed byfiat. Time (s) A m p lit ud e ( m V ) Time (s) A m p lit ud e ( m V ) Figure 12.
Comparison of the simulation (solid red line) of a 2615 keV g pulse with data (black dots). Fullwaveform (left) and expansion of the leading edge of the signal (right). Amplitude (mV) R i s e ti m e ( m s ) heater Amplitude (mV) D eca y ti m e ( m s ) heater Figure 13.
Comparison of the rise time (left) and decay time (right). The black, gray, red, and blue dotsrepresent measured particle pulses, simulated particle pulses, measured heater pulses, and simulated heaterpulses respectively. – 14 – aseline (mV) A m p lit ud e ( m V ) Figure 14.
Comparison of the amplitude of heater pulses versus baseline from data (solid black) and simu-lation (open gray).
Energy [keV] A m p lit ud e d i ff e r e n ce [ % ] -0.8-0.6-0.4-0.200.2 Energy [keV] E [ k e V ] D -10-50510152025 Figure 15.
Difference between the amplitudes of simulated and real pulses as a function of the energy (left),and deviation from linearity of the calibration function (right) for data (black circles) and simulation (redtriangles). – 15 – . Applications
The simulation can be used to test and tune analysis algorithms, comparing the results with the socalled “Monte Carlo truth”, i.e. the generated values of baseline, amplitude, energy and the timeof each signal. In same cases, in fact, the signal identification is not straightforward and the energycould be wrongly estimated.For example the data analysis can be complicated by pileups, that alter the baseline and theshape of the signals. A simulated sequence of close signals is shown in Fig. 16, where it can beseen how a signal can be modified by other signals, and how its identification is complicated. Inthis case the simulation can be used to improve the analysis algorithms and to estimate the error onthe results.A potential application is the estimation of detection efficiencies at low energy. Recently it hasbeen demonstrated [3] that CUORE may have an energy threshold of only a few keV, thus beingsensitive to Dark Matter interactions and rare nuclear decays. Figure 17 shows simulations of verysmall pulses. In this regime noise can both mask signal and mimic signal. Simulations producedby our engine may be used to test the immunity of analysis algorithms to both types of error. Whena heater is available it is used to produce controlled pulses and estimate the detection efficiencies.The simulation can correct for the difference in pulse shape between particle and heater generatedenergy deposit. With no heater the simulation could serve on its own to estimate the detectionefficiencies.
Acknowledgments
We thank the members of the CUORE collaboration, in particular G. Pessina for fruitfuldiscussions on the signal model and on the electronics setup, and O. Cremonesi for suggestions onthe noise generation. We thank F. Bellini, P. Decowski, F. Ferroni, F. Orio, C. Rosenfeld andC. Tomei for comments on the manuscript.
Time (s) A m p lit ud e ( m V ) Figure 16.
Simulated pileups. The pulse at ∼ – 16 – ime (s) A m p lit ud e ( m V ) Time (s) A m p lit ud e ( m V ) Figure 17.
12 keV (left) and 6 keV (right) simulated pulses.
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