Large-mass bolometers are calorimeters working at low temperatures widely used in particle physics experiments to search for rare events. These devices are able to detect particles energies from few keV up to several MeV, but their response function is not linear and depends on the operating temperature. This study was performed with the bolometers of the CUORE experiment. In a previous work the non-linearities were attributed to the thermistor and to its biasing circuit and an algorithm was derived to obtain a linear response. In this work an improvement of the algorithm is proposed and the possibility of developing a more complete model of the response function, which accounts for the electrothermal feedback induced by the thermistor, is evaluated

Di Biagio, Valeria

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DOI 10.1393/ncc/i2011-10994-6Communications: SIF Congress 2010IL NUOVO CIMENTO Vol. 34 C, N. 5 Settembre-Ottobre 2011Study of the bolometers thermal response for the searchof rare eventsV. Di Biagio(∗)Dipartimento di Fisica, Sapienza Universita` di Roma - Rome, Italy(ricevuto il 30 Dicembre 2010; approvato il 14 Luglio 2011; pubblicato online il 5 Ottobre 2011)Summary. — Large-mass bolometers are calorimeters working at low temperatureswidely used in particle physics experiments to search for rare events. These devicesare able to detect particles energies from few keV up to several MeV, but their re-sponse function is not linear and depends on the operating temperature. This studywas performed with the bolometers of the CUORE experiment. In a previous workthe non-linearities were attributed to the thermistor and to its biasing circuit and analgorithm was derived to obtain a linear response. In this work an improvement ofthe algorithm is proposed and the possibility of developing a more complete modelof the response function, which accounts for the electrothermal feedback induced bythe thermistor, is evaluated.PACS 29.40.Vj – Calorimeters.1. – IntroductionBolometers are detectors in which the energy of the particle interactions is convertedinto phonons and measured via the resulting temperature variation. Owing to theirexcellent energy resolution, bolometers are used in particle physics experiments searchingfor rare events, like Neutrinoless Double-Beta Decay and Dark-Matter interactions.The CUORE experiment (Cryogenic Underground Observatory for Rare Events) willsearch for Neutrinoless Double-Beta Decay in 130Te [1], using crystals of TeO2 as bothemitters and absorbers and NTD-Ge semiconductor thermistors [2, 3] as temperaturesensors. The experiment is in progress at Laboratori Nazionali del Gran Sasso (LNGSof the INFN), in Italy, and tests are in the meanwhile conducted. Bolometers are hostedin a cryostat operating at about 10mK and have an energy response extending from fewkeV up to several MeV. Over this range the energy resolution is kept almost constant atfew keV, but the response function is non-linear. The conversion from signal amplitude toenergy is not direct and requires the use of a calibration function. The signal amplitude(∗) E-mail: valeria.dibiagio@roma1.infn.itc© Societa` Italiana di Fisica 129130 V. DI BIAGIOFig. 1. – General scheme of a bolometer (left) and a CUORE bolometer (right).depends also on the working temperature, which cannot be mantained strictly constant,so that the detector resolution can be worsened. Moreover, the shape of the signalsdepends on their energy, making hard the pulse shape discrimination. Understandingthe sources of non-linearities is therefore necessary to improve the detector performance.Electrothermal models of bolometers have been already proposed [4-6]. However, theyrely on parameters difficult to be measured in working conditions, so they provide onlyqualitative descriptions of the detector features, like the signal shape. On the contrary,this article proposes a model based on few measurable parameters and usable to linearizethe bolometers response function.2. – Description of the detectorA CUORE bolometer is made up of two main parts, a TeO2 crystal and a NTD-Gethermistor. The cube-shaped crystal is held by teflon supports on copper frames, whichare connected to the mixing chamber of a diluition refrigerator. The thermistor, that actsas a thermometer, is glued to the crystal and its biasing wires are glued to the copperframes (see fig. 1). The crystal weighs 750 g and its heat capacitance C at the workingtemperature of 10mK is of order 2 nJ/K [7]. When an amount of energy E is releasedon the inside, the crystal temperature increases by E/C and then returns back to theequilibrium through the thermal conductance of the supports and of the thermistor. Thislatter converts the temperature, T , into resistance, R, according to the relationship(1) R = R0 exp[T0T]γ,where R0 and T0 are parameters that depend on the dimensions and on the materialof the thermistor [8]. Their value is determined experimentally and is about 1.1Ω and3.4K, respectively. At 10mK, the parameter γ can be considered constant and equalto 12 [9, 10], and the value of R(T ) is of order 100MΩ. The bolometer temperaturevaries with time, hence the resistance of the thermistor is a function of the time via thetemperature.To read out the signal, the thermistor R(t) is inserted in a biasing circuit, whichincludes in series a constant voltage generator VB and a load resistance RL. The valueof RL is much higher than R(t), so that the voltage signal V (t) at the two ends of thethermistor is almost proportional to R(t). The biasing wires of the thermistor have anon-negligible parasitic capacitance cp, between them and with respect to the ground,that relies on the material and on the lenght of the wires (see fig. 2(a)). Typical values ofSTUDY OF THE BOLOMETERS THERMAL RESPONSE FOR THE SEARCH OF RARE EVENTS 131Fig. 2. – Biasing circuit of the thermistor (a) and a typical signal read through it (b).the circuit components are: VB ≈ 5V, RL ≈ 50GΩ, cp ≈ 400 pF. The signal V (t) is thenamplified (gain = 5000V/V), filtered by a six-poles anti-aliasing Bessel filter (frequencybandwith νB = 12Hz) and sampled by an analog-to-digital converter (sampling frequencyνADC = 125Hz, duration of the acquisition window Δt = 5.008 s).An example of an acquired signal is shown in fig. 2(b). Its amplitude gives informationabout the quantity of energy released in the bolometer, whereas the baseline depends onthe temperature of the detector before the interaction.3. – Non-linearities of the response functionThe main features of the recorded signal are the amplitude and the shape. The shapeparameters used in this study are the rise time and the decay time, defined as the timedifference between the 10% and the 90% of the pulse height and between the 90% and30%, respectively. The analysed data come from calibration measurements realized facinga 232Th external source to a set of 26 crystals. Some crystals were equipped with onethermistor and others with two, so that the total number of channels to be analysed is36. Hereafter channels are labeled as in the electronics setup, that ranges from 1 to 72.The resulting energy spectrum has a range from few keV up to about 3MeV. In thisconfiguration the amplitude of the pulses is about 0.1mV/MeV× gain, the rise time andthe decay time are of order 80ms and 500ms, respectively. The conversion from signalamplitude to energy is complicated, but feasible through the calibration procedure.Nevertheless the signal exhibits other non-linearities:1. Rise time and decay time depend on the energy of the pulse and are anticorrelatedand correlated with the energy, respectively (see fig. 3(a) and fig. 3(b)). This effectmakes the pulse shape discrimination in the data analysis difficult.2. The amplitude of a pulse with fixed energy relies on the baseline (see fig. 4) andthen on the working bolometer temperature. This leads to a worsening of theenergy resolution.Previous studies [11] have traced back these non-linearities predominantly to the thermis-tor behaviour and to the biasing circuit one. Consequently, we can neglect the furthereffects of the entire thermal circuit constituted by the bolometer, the thermistor andthe supports, whose capacitances and conductivities are potentially dependent on thetemperature and in general hard to measure with precision and independently one fromthe other.132 V. DI BIAGIOFig. 3. – Pulse shape parameters vs. energy in channel 1: rise time (a) and decay time (b) areanticorrelated and correlated with energy, respectively.Fig. 4. – Slope amplitude vs. baseline for 2615 keV γ’s of 208Tl (from the 232Th decay chain),channel 35.In particular, the thermistor behaviour explains the trend of the decay time and thedependence of the amplitude of the pulse with the baseline, whereas the rise time trendis ascribed to the biasing circuit.4. – Thermal response analysis: a first algorithmIn order to linearize the data, an algorithm has been already developed [11]. To derivethe algorithm, first of all we have to separate two contribution in R(t) and V (t): theformer, due to the baseline temperature T , and the latter, due to the signal(2) R(t) = R(T ) + ΔR(t), V (t) = V (T ) + ΔV (t).Solving the biasing circuit in fig. 2(a), we obtain the differential equation relating resis-tance and voltage variations:(3)[RL + R(T ) + ΔR(t)R(T ) + ΔR(t)] [VBR(T )R(T ) + RL+ΔV (t)]− VB + RLcpΔV ′(t) = 0.STUDY OF THE BOLOMETERS THERMAL RESPONSE FOR THE SEARCH OF RARE EVENTS 133Fig. 5. – Pulse shape parameters vs. energy in channel 1: rise time (a) and decay time (b), inSTD (dark gray) and TR (light gray) analysis.From the previous equation we can extract ΔR(t) from ΔV (t) and other measurablequantities:(4) ΔR(t) = − ΔV (t)[R(T ) + RL] + RLR(T )cpΔV′(t)ΔV (t) + RLcpΔV ′(t) + VBRL/[R(T ) + RL].Considering that the resistance variation ΔR(t) in terms of the temperature variationΔT (t) can be approximated as(5) ΔR(t)  R(T )[exp[−ηΔT (t)T]− 1],where η is the thermistor sensitivity, defined as η = |d logR|/|d log T | ≈ 10, we can obtainthe thermal response of the detector, that is a quantity proportional to the temperaturevariation:(6) ΔS =ηTΔT (t) = − log[1 +ΔR(t)R(T )].This algorithm converts the acquired samples ΔV (t) in signals proportional to the tem-perature variation ΔT (t) on the thermistor. It has also the advantage to be reliant onlyon parameters easily measurable, while the unknown quantities are hidden in the scalefactor η/T , which is ininfluent in the perspective of removing the non-linearities.The application of the thermal response algorithm on the data gives the results repre-sented in figs. 5, 6 and 7. The thermal response (TR) data, labeled by light gray points,are compared with the data already shown in fig. 3 and fig. 4, labeled by dark-graypoints, that can be defined as the standard (STD) data. The shape of the TR pulsesis less dependent on the energy, especially in the case of the decay time, and also thedependence of the amplitude from the baseline has been reduced. We also notice thatthe rise time of the TR data is lower than the STD data, because the low pass filter ofthe biasing circuit has been deconvoluted (fig. 5(a)).Nevertheless, there are some residual non-linearities in the response function. Almostall the analysed channels have an amplitude slope of the 2615 keV γ’s of 208Tl which is134 V. DI BIAGIOFig. 6. – Slope amplitude vs. baseline for 2615 keV γ’s of 208Tl in STD (dark gray) and TR(light gray) analysis, channel 35.Fig. 7. – Residual non-linearities: amplitude slope for all channels according to TR analysis (a)and the pulse shape of channel 47 for STD and TR analysis (b).contained in a ±0.01 keV/keV range, but there are some outliers, like the channels 47, 57,63 and 64 (fig. 7(a)). The same channels have also a particular shape of the TR signal.After reaching its maximum, the TR pulse goes down and then up again (see fig. 7(b)).This behaviour is ascribed to the electrothermal feedback phenomenon. According tothis, the biasing current produces a power dissipation which warms up the thermistorand reduces its resistance; when the temperature increases, the variation of the powerdissipated on the thermistor is negative and decreases the signal. In the STD pulse thiseffect is masked by the low-pass filter originated from the the thermistor resistance andthe parasitic capacitances.5. – Insertion of a RC-filter with constant cutoffA first improvement of the algorithm proposed in the previous section can be achievedapplying a low-pass filter with constant cutoff to the TR data. In this way we try toget a shape of the transformed pulse more similar to the original one, mantaining at thesame time the non-linearities removal.STUDY OF THE BOLOMETERS THERMAL RESPONSE FOR THE SEARCH OF RARE EVENTS 135If we solve the biasing circuit approximately, using RL  R and RL  ΔR(t), weobtain the equation(7) ΔV (t) + [R(T ) + ΔR(t)]cpΔV ′(t) = −VBRLΔR(t).This expression has the form of a low-pass filter, with a cutoff that depends on the inputsignal ΔR(t): the higher is its absolute value, the higher is the cutoff, making the pulsefaster. Since R(T )cp ∼ 30ms, this effect is prevailing on the rise time, while on thedecay time there is a possible competition with the opposite effect determined by thethermistor, as mentioned in sect. 3. Considering eq. (7), if we delete the dependenceof the cutoff on ΔR, which is the non-linearity source, we obtain the expression of aRC-filter with constant cutoff(8) ΔV (t) + R(T )cpΔV ′(t) = − VBRLΔR(t),whose more general form is(9) Vout(t) + RcpdVout(t)dt= Vin(t).This equation can be easily discretized. Assuming that the samples of the input andoutput are taken at evenly spaced points in time separated by Δt, let the samples of Vinbe represented by the sequence (x1, x2, . . . , xn) and let the samples of Vout be representedby the sequence (y1, y2, . . . , yn), which corresponds to the same points in time. After thesesubstitutions and the rearrangement of the terms, we can obtain the recurrence relation(10) yi = xi(ΔtRC +Δt)+ yi−1(RCRC +Δt),where the former addend represents the input contribution, while the latter constitutesthe inertia from the previous output: as the time constant RC increases, the output sam-ples (y1, y2, . . . , yn) respond more slowly to a change in the input samples (x1, x2, . . . , xn).The results from the application of the new algorithm (10) implemented as a filteron TR data are represented in fig. 8. The pulse resulting from this analysis, called“TR+RC”, does not exhibit the TR trend and it reproduces the original pulse shape(fig. 8(b)). In addition, the residual slope has been reduced (fig. 8(a)). Notice that it isthe amplitude which was modified, while the baseline is unchanged.6. – The electrothermal feedbackIn order to develop a more complete model of the bolometer response function, weneed to make explicit the contribution due to the electrothermal feedback to which weattribute the shape of the TR signal. The feedback phenomenon is a potential source ofnon-linearities, so it has to be investigated and possibly removed.In the electrothermal models cited in sect. 1, a temperature variation ΔT into abolometer, that has a heat capacitance C and is connected with a thermal bath through136 V. DI BIAGIOFig. 8. – Analysis with RC inserted: amplitude slope for all channels according to TR andTR+RC analysis (a) and the pulse shape of channel 47 for STD, TR and TR+RC analysis (b).a conductance K, is traced back to an external incident power (the signal) W and to thepower dissipated into the detector by the thermistor ΔP , according to the equation(11) CΔT ′(t) + KΔT (t) = W (t) + ΔP (t).In our analysis we have that ΔT = (T/η)ΔS, therefore the power attributed to the signalcan be expressed as(12) W (t) =CTηΔS′(t) +KTηΔS(t)−ΔP (t).Rescaling this equation by the factor KT/η, we obtain the further formula for the thermalresponse W ∗(13) W ∗(t) = ΔS(t) +CK(ΔS′(t)− ηTCΔP (t)),where the former term is the thermal response derived in sect. 4 and the latter representsthe additional share due to the feedback.The ΔS′ quantity can be immediately calculated from ΔS, while ΔP can be calculatedas follows. Defining ΔP (t) = P (t) − P (T ), we have that the power dissipated in staticconditions is(14) P (T ) =(VBRL + R(T ))2R(T ),while in dynamic conditions the power can be calculated from the biasing circuit expres-sion (3) and it assumes the form(15) P (t) =V (t)2R(t)= P (T )⎧⎪⎨⎪⎩[1 + ΔV (t)VB(1 + RLR(T ))]21 + ΔR(t)R(T )⎫⎪⎬⎪⎭ .STUDY OF THE BOLOMETERS THERMAL RESPONSE FOR THE SEARCH OF RARE EVENTS 137Therefore in eq. (13) also ΔP (t) is a computable quantity. Nevertheless, in the sameequation the two parameters C/K and C/K ∗ η/(TC) = η/(TK) include the quantitiesC, K, T that are not known with precision. In eq. (11) C and K are referred generically tothe bolometer, whereas we have the entire thermal circuit composed by the bolometer, thethermistor and the supports, whose capacitances and conductances cannot be measuredindependently.We are working to develop procedures to estimated the two parameters C/K andη/(TK) a posteriori from the data, in such a way that the detector response is opti-mized, that is, the non-linearities are reduced. From CUORICINO measurements [5],preliminary reckonings are C/K ∼ 0.1 s, η/(TK) ∼ 1013 W−1. Since the order of magni-tude of the C/K value is comparable to the rise time one, we can expect a correction tothe trend of this shape parameter.Finally, the effects of the thermal response algorithms on the calibration functionwill be studied. The calibration function is usually parametrized with a third-orderpolynomial function in the standard analysis of TeO2 detectors. To parametrize thefunction, the detector is exposed to the calibration source so that the entire range ofenergies is populated. In CUORE the calibration procedure could take long time (up toone week per month), reducing the live time of the experiment. If the calibration functionwere reduced to a line, the calibration procedures of CUORE could be substantiallyshortened, thus increasing the sensitivity of the experiment to 0νDBD.∗ ∗ ∗I am very grateful to the members of the CUORE-Roma Collaboration. I wish tothank in particular F. Ferroni andM. Vignati for their support and for the supervisionof this work.REFERENCES[1] Ardito R. et al., arXiv:hep-ex/0501010, unpublished.[2] Wang N., Wellstood F. C., Sadoulet B., Haller E. E. and Beeman J., Phys. Rev.B, 41 (1990) 3761.[3] Itoh K. M., Haller E. E., Hansen W. L., Beeman J. W., Farmer J. W., RudnevA., Tikhomirov A. and Ozhogin V. I., Appl. Phys. Lett., 64 (1994) 2121.[4] Jones R. C., J. Opt. Soc. 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Study of the bolometers thermal response for the search of rare events

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Study of the bolometers thermal response for the search of rare events

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