A fiducial-volume cryogenic detector
A detector, sensitive to CNNS, faces two main challenges: an extremely low-energy threshold combined with extraordinarily small background levels. We present a new gram-scale cryogenic detector which combines the possibility of lowest nuclear-recoil thresholds (\({\mathcal {O}}\)(\({\lesssim }10\) eV)) and the advantages of a fiducial-volume device. Those provide active shielding by the outermost regions against external radiation which reduces the background level in the innermost target volume (the fiducial volume). Since an exact spatial position reconstruction of events is difficult to realize in thermal detectors, so far this potential could not be exploited.
Here, a cryogenic detector is presented which realizes a fiducial volume by combining three individual calorimeters: (1) a target crystal (the fiducial volume) with an extremely low threshold of \({\mathcal {O}}\)(\({\lesssim }10\) eV), (2) an inner veto as a 4\(\pi \) veto against surface beta and alpha decays, and (3) a massive outer veto against external gamma/neutron radiation (see Fig. 1). Additionally, the inner veto acts as an instrumented holder for the target crystal allowing to discriminate holder-related events (e.g. from stress relaxations).
Performance model for calorimeters
In order to design the new detector, a simple model was developed to predict the practically achievable performance of calorimeters of different geometry, material and mass [17]. The model is based on experimental results of cryogenic CRESST-type detectors. The main results are derived here, insofar as they drive design-choices for the fiducial-volume cryogenic detector.
The fundamental equation describing a calorimeter is that, for a system in internal thermal equilibrium, the temperature rise
$$\begin{aligned} \varDelta T= \frac{\varDelta E}{C} \end{aligned}$$
(2)
where \({\varDelta E}\) is an energy deposit and C is the heat capacity of the object. Reducing C yields a large increase in temperature and thus a high sensitivity to small energies.
Present cryogenic detectors of \({\sim } 300\) g achieve energy thresholds of \({\sim }300\) eV [18]. In this work we investigate the performance and potential of gram-scale devices.
The fundamental energy resolution \(\sigma _E\) of cryogenic calorimeters is given by irreducible thermal fluctuations between the absorber and the thermal bath [19]:
$$\begin{aligned} \sigma _E^2 \sim k_\mathrm{B}T^2C \end{aligned}$$
(3)
with the absorber’s temperature T, heat capacity C and the Boltzmann constant \(k_\mathrm{B}\). This corresponds to theoretical energy resolutions of \({\mathcal {O}}\)(1 eV) at \({\sim }10\) mK for massive calorimeters with masses of \({\sim }100\) g [20]. Phonon processes in cryogenic calorimeters with thin-film transition-edge-sensors (TESs) as considered in this work are well described by a dedicated thermal model [21].
Equation (2) is valid for a thermometer measuring the temperature of an absorber. In practice, the thermometers of cryogenic detectors can only measure their own temperature. Equation (2) thus changes to
$$\begin{aligned} \varDelta T = E_{\mathrm{abs}}/C_{\mathrm{film}} \end{aligned}$$
(4)
where \(E_{\mathrm{abs}}\) denotes the energy absorbed in the thermometer and \(C_{\mathrm{film}}\) the heat capacity of the thermometer film. In cryogenic calorimeters at very low temperatures (\({\sim } 10\) mK), the energy deposition in the thermometer film happens via the absorption of non-thermal phonons, which propagate ballistically and interact directly with the metallic film electrons. Thus they are not affected by the weak thermal coupling between thermometer phonon and electron systems at such temperatures. To achieve sufficiently low heat capacities, temperatures as low as 10 mK are required for these devices. The strong electron–phonon decoupling in the thermometer film at these temperatures requires a dedicated thermal link to the heat bath. This strongly suppresses the thermal signal, which makes the non-thermal phonon component our dominant information carrier.
The thermometer’s temperature rise can therefore be written as the ratio of the time-constants of the two competing processes that reduce the non-thermal phonon population: (1) the absorption in the thermometer with a time-constant \(\tau _{\mathrm{film}}\), and (2) the thermalization of non-thermal phonons at the crystal surfaces with a time-constant \(\tau _{c}\)
$$\begin{aligned} \varDelta T = \frac{\tau _{c}}{\tau _{\mathrm{film}}}\cdot \frac{\varDelta E}{C_{\mathrm{film}}}. \end{aligned}$$
(5)
It should be noted that this is only valid in the limit \(\tau _{c}\ll \tau _{\mathrm{film}}\), which is equivalent to the statement that collection by the thermometer film does not influence the non-thermal phonon lifetime [21]. All devices considered here operate in this regime. Under these conditions, the temperature signal is not influenced by the presence of the thermometer, and thermometer optimization can be considered separately from a change in absorber parameters.
For the absorber scaling law, we keep only the quantities that depend on absorber properties. The energy threshold of the device is inversely proportional to the temperature rise, so we can write
$$\begin{aligned} E_{\mathrm{th}} \propto \frac{\tau _{\mathrm{film}}}{\tau _{c}}. \end{aligned}$$
(6)
This is the basis for our scaling law which only considers varying absorber material, geometry and mass. Under these changes, \(\tau _{c}\) scales with the average time between surface scatterings of the non-thermal phonons, which can be written
$$\begin{aligned} \tau _{c}\propto \frac{l}{\langle v_g\rangle } \end{aligned}$$
(7)
in terms of the mean phonon free path in the crystal l and the mean phonon group velocity \(\langle v_g\rangle \). For a fixed thermometer surface area, \(\tau _{\mathrm{film}}\) scales with the crystal volume and the mode-averaged absorption rate, like
$$\begin{aligned} \tau _{\mathrm{film}}\propto \frac{V}{\langle v_\perp \alpha \rangle }. \end{aligned}$$
(8)
\(v_\perp \alpha \) is the volume spanned by the phonon modes that cross the thermometer surface per unit time and thermometer area, times the transmission probability into the thermometer. The different dimensionality, (i.e. l vs. V), in the scaling laws, arises from the fact that the crystal surface area scales up with the system dimensions, whereas the thermometer area does not.
In total, the scaling law is
$$\begin{aligned} E_{\mathrm{th}} \propto \frac{V }{l}\cdot \frac{\langle v_g\rangle }{\langle v_\perp \alpha \rangle }. \end{aligned}$$
(9)
The first part is purely geometric, while the second represents material parameters. The threshold of CaWO\(_4\) detectors is expected to be 1.72 higher than Al\(_2\)O\(_3\) of same geometry [17], while Si (1.42) and Ge (1.15) fall between these two. The scaling of two detector geometries as a function of mass are considered here. 1) For cubes of side length d, \(V\propto d^3\) and \(l\propto d\), so that \(E_{\mathrm{th}}\propto d^2\) which yields finally \(E_{\mathrm{th}}\propto m^{2/3}\). 2) For plates of area \(d^2\) and fixed thickness h, \(V\propto d^2\). In the relevant range, \(10\lesssim d/h \lesssim 100\), l(d) rises slowly from \({\sim } 2h\) to \({\sim } 5h\) (from MC simulation). Roughly, we can take \(l\approx \mathrm {const}\), which also gives \(E_{\mathrm{th}}\propto d^2\), but a different mass-scaling \(E_{\mathrm{th}}\propto m\).
With values for l found by Monte Carlo methods for each occurring detector geometry, the model can be used to describe the thresholds of various CRESST-type detectors. Since the model can only predict a scaling under change of absorber properties, the absolute normalization (depending e.g. on the noise level of the setup) has to be taken from the respective experiment. In the following, the noise level of the “benchmark” CRESST setup at LNGS is considered. In Fig. 2, the model predictions for plate and cube detectors are shown as a function of detector mass, fitted to the thresholds achieved in CRESST-II CaWO\(_4\) detectors (green triangles) with a mass of \({\sim }300\) g [18, 22] and a sapphire cube of 262 g used in CRESST-I (blue cross) [23, 24]. The model successfully predicts the energy threshold of CRESST-II light detectors studied in [17] (purple dots), which are sapphire discs with a mass of 2.2 g (diameter 40 mm, thickness 0.45 mm) and also the thresholds of \({\sim }24\) g CRESST-III detectors as expected from a prototype measurement (green error bar) [25]. The capability to extrapolate calorimeter thresholds for different detector geometries and materials over orders of magnitude in mass can be applied to the component design for the fiducial-volume cryogenic detector. Red stars indicate the calculated performance of the calorimeters studied here.
Design of the target calorimeter array
For the research program proposed in this work, the target calorimeter has to fulfill the following requirements:
-
A nuclear-recoil energy threshold \(E_{\mathrm{th}}\) of \({\mathcal {O}}(10\) eV).
-
Rates of \(10^2{-}10^3\)/[kg day] are expected from CNNS at the sites studied here, as will be shown in Sect. 3. Corresponding to this rate, a total target mass of \({\sim }10\) g is needed for the detection of CNNS.
-
Lowest thresholds require a sufficiently low event rate in the calorimeter. To limit the pile-up contribution to a level of \({\mathcal {O}}\)(\(10^{-2}\)), a maximum event rate of \({\mathcal {O}}\)(0.1 Hz) per detector is allowed given the typical (thermal) pulse decay times of \({\sim }100~\)ms [21].
-
The CNNS cross-section is proportional to the target’s neutron number N squared, which highly favors heavy elements. On the contrary, the use of light nuclei facilitates a characterization of neutron backgrounds.
Considering these design features, cubic target crystals with an edge length of 5 mm equipped with a tungsten thin-film TES are ideal. A multi-target approach with a variety of elements is chosen which has great advantages for the separation of signal and background through characteristic interaction strength. Cubes of CaWO\(_4\) (0.76 g), Al\(_2\)O\(_3\) (0.49 g), Ge and Si crystals, which are well-known for their excellent cryogenic detector properties [18, 24], are suitable candidates. The performance model (see Sect. 2.2) predicts energy thresholds of \(E_{\mathrm{th}}\approx 4.0\) eV for Al\(_2\)O\(_3\) and \(E_{\mathrm{th}}\approx 7.0\) eV for CaWO\(_4\) (see red stars in Fig. 2). To obtain the desired total target mass, a \(3\times 3\) detector array is foreseen (see Fig. 3). This corresponds to a total target mass of 6.84 g for the CaWO\(_4\) and 4.41 g for the Al\(_2\)O\(_3\) array, respectively.
For the temperature sensor, a TES is chosen similar to that which is used for the CRESST detectors [25]. The TES consists of a thin W film (thickness 200 nm) with an area of 0.0061 mm\(^2\) and an Al phonon collector with an area of 0.15 mm\(^2\) attached to it (see Fig. 4). The latter increases the collection area for phonons without the penalty of increasing the heat capacity of the sensor [26] yielding an increased pulse height. The TES is weakly coupled to the heat sink via a thin Au stripe (\(0.01\times 7.0~\hbox {mm}^2\), thickness 20 nm) providing a thermal conductance of \({\sim }10\) pW/K (at a temperature of 10 mK). Al and Au wire bonds with a diameter of 25 \(\upmu \)m are used to provide the electrical contacts for the TES (bonded on the phonon collectors) as well as the ohmic heater (separate bond pads), and the thermal link to the heat sink, respectively. Typically, bias currents between 100 nA and 5 \(\upmu \)A are applied to the sensor. The resistance change of the TES can be measured with a SQUID system similar to the one in the CRESST dark matter experiment [27].
Results from a prototype calorimeter
In the framework of this project, a prototype Al\(_2\)O\(_3\) calorimeter of 0.5 g has been produced and equipped with a TES according to the design goals described in the previous section. The detector was installed in a copper holder and mounted in a detector test facility at the Max-Planck-Institut for physics in Munich. It consists of a dilution refrigerator in a surface building without dedicated shielding against ambient radioactivity. Further, no shielding against backgrounds from surfaces in the direct vicinity of the calorimeters is used. A \(^{55}\)Fe X-ray source is placed close to the detector for a calibration of the low-energy region.
In an accompanying paper [9], we present details of a 5.1 h calibration measurement performed with the 0.5 g Al\(_2\)O\(_3\) detector which achieved an energy threshold of \(E_{\mathrm{th}}=(19.7\pm 0.9)\) eV. This is independent of the type of particle interaction since it is a calorimetric device. This is the lowest nuclear-recoil energy threshold reported for massive calorimeters, beyond the fundamental nuclear-recoil reach of ionization-based detectors [28].
The detector operates in the calorimetric mode (see Sect. 2.2), confirmed by the pulse shape. The thermalization times in the crystal and thermometer film are found to be \(\tau _{c}=0.34\) ms and \(\tau _{\mathrm{film}}=2.2\) ms, respectively. This ratio fulfills the condition \(\tau _{c}\ll \tau _{\mathrm{film}}\) but leaves room for improvements (see below).
The measured threshold is higher (by a factor of \(\sim \)5) compared to what is predicted by the performance model for calorimeters (Sect. 2.2). Part of the discrepancy may be explained due to worse noise level in the MPI setup (by factor 1.5–3 [25]) compared to the low-noise benchmark setup used for the calculation of the predictions (Eq. 9). The considered detector, being the first prototype of a gram-scale calorimeter, is expected to improve by further developments and adjustments of the TES sensor. The ratio of \(\tau _{c}/\tau _{\mathrm{film}}\) can be further decreased by reducing the thermometer area and accordingly weakening the thermal link. A corresponding reduction of the Al phonon collectors may improve the transport efficiency of quasi-particles [26]. Furthermore, a moderate reduction of the W-film thickness will reduce the heat capacity of the thermometer without compromising the phonon absorption.
In the calibration measurement a flat background spectrum of \({\sim }1.2\times 10^{5}\) counts/[kg keV day] (7–10 keV) is observed above the calibration peaks [9]. This is expected due to the absence of any shielding and can be considered as an absolute upper limit for the total rate in surface experiments (here it corresponds to \({\sim }0.3\) Hz). It is comparable to typical total account rates observed in \({\mathcal {O}}\)(1 kg) cryogenic detectors operated in underground laboratories [22]. The result clearly demonstrates that gram-scale detectors can be operated in a high-background environment – in particular at surface level – while allowing for low-energy thresholds and stable operating conditions.
The performance of the prototype fulfills the requirements listed in the previous section in terms of energy threshold and operability at surface level. To demonstrate the required background level – for the near future – measurements with further developed CaWO\(_4\) and Al\(_2\)O\(_3\) calorimeters at low-background experimental sites (e.g. a shallow laboratory) are planned. In particular, the target calorimeter(s) will be embedded in the inner and outer cryogenic shieldings which are discussed in detail in the following sections.
Low-threshold inner veto and detector holder
Background from the surfaces of the target crystals and surrounding surfaces is a big challenge for rare-event searches, and can limit the sensitivity at low energies. The inner veto provides an active discrimination against beta and alpha decays occurring on surfaces. Typical Q-values of such decays are between \({\sim }10\) keV and 10 MeV typically shared between 2,3 or more product particles leaving the interaction point in different directions. In a configuration where the target is surrounded by a \(4\pi \) active veto, the total energy of the reaction is detected (apart from the energy transferred to neutrinos in beta decays). In this way, a high fraction of such backgrounds can be rejected by coincident events in the veto. The rejection of surface background is crucial in particular when approaching ultra-low-energy thresholds, as can be seen in experimental data (see e.g. [24]). Figure 5 shows a section view of the inner part of the detector. In the following, the functionality of the relevant detector components is briefly discussed.
-
Target (red): The detector consists of nine target calorimeters (a in Fig. 5) arranged in a \(3\times 3\) detector array. Each crystal is equipped with a TES (see Sect. 2.3).
-
Active components (blue): To realize a 4\(\pi \) veto against surface backgrounds, Si wafers read-out by a TES each are used (b–f). Two of these (b and c) are in contact with the target crystals via pyramidal Si structures on the wafers. The upper one (b) is thin enough (\(200~\upmu \)m) to be flexible – the wafer acts as a spring. Pressed to the target crystals, the thin wafer realizes a spring-loaded holding structure which can compensate for thermal contraction of the various components of the detector. Possible events induced by the detector holder (e.g. by thermal-stress relaxation) can be rejected since they induce also phonon signals in the TESs of b and c.
-
Passive components (green): Two Si slabs (g and h) are used as support structures for the calorimeter array. They are attached to each other by 4 posts (shown in Fig. 3) providing the necessary pressure to hold the target crystals. The lower wafer (h) is equipped with Al (Au) wiring for the electrical (thermal) connection of the target calorimeters and the inner veto devices.
The inset in Fig. 3 shows a prototype Si wafer with a pyramid structure produced at the Halbleiterlabor of the Max-Planck-Society. The structure is defined by photolithography techniques and the pyramid structures are then realized by wet chemical etching.
The rejection power against surface related background was estimated with a dedicated Monte Carlo (MC) study performed with the Geant4 code in version 10.2p1 [29, 30]. We follow the recommendation of the Geant4 Low Energy Electromagnetic Physics Working Group [31] and implement the low-energy behavior of electromagnetic interactions via the Geant4 class G4EmStandardPhysics_option4, a selection of most accurate models. Furthermore, we enabled the atomic de-excitation via emission of fluorescence photons and Auger electrons. With one exception, we applied a production cut of \(250~{\mathrm {eV}}\) throughout our geometry, i.e. for energies above this cut new secondary particles can be created in the simulation, whereas lower energies are directly deposited. The exception are fluorescence photons and Auger electrons which are produced in any case.
Being exemplary for surface contamination, we simulated the \(\beta \)-decay of \(^{210}\hbox {Pb}\) by placing the lead ions at rest on the inner surface of the inner veto, facing one target calorimeter made of \(\hbox {Al}_2\hbox {O}_3\). The source activity is assumed to be \({\mathcal {O}}(1~\hbox {kg}^{-1}~\hbox {keV}^{-1}~\hbox {day}^{-1})\), the maximal external \(\beta \)-activity observed with TUM40, a module with especially low background operated in CRESST-II phase 2 [32]. The black histogram in Fig. 6 shows the background spectrum seen by the target with inactive inner veto, the red histogram shows the spectrum of the remaining background in the case of an active veto with a threshold of \(30~\hbox {eV}\). Clearly a reduction of more than two orders of magnitude is feasible. A more detailed MC study of the complete detector array is under way and intended for future publication.
We note that the step at \({\sim } 100~\hbox {eV}\) (Fig. 6, black histogram) is no artifact of the used production cut. Instead, it is caused by Coster–Kronig transitions as part of the atomic relaxation subsequent to the decay of the \(46.539~\hbox {keV}\)-level of \(^{210}\hbox {Bi}\) to which \(^{210}\hbox {Pb}\) decays in 84% of the cases [33].
Outer-veto detector
Given the smallness of the calorimeter array and the inner veto system, these components can be embedded in a large cryogenic outer veto. We consider cylindrical crystals with a diameter and height of \({\mathcal {O}}\)(10 cm) which are segmented into two (or more) parts with a central cavity to host the inner detector parts (see Fig. 7). Each crystal of the outer veto is instrumented with a TES. It is foreseen to use materials that are known for their excellent phonon properties, such as e.g. Ge and CaWO\(_4\), and that have been demonstrated as cryogenic detectors with masses of \({\mathcal {O}}\)(100 g–1 kg). Thresholds between 300 eV and 1 keV are reached with such devices, in agreement with the prediction of the performance model for calorimeters in Sect. 2.2 (Fig. 2). CaWO\(_4\) is the preferred material: it has the heavy element W which provides a high cross-section for gamma radiation and the relatively light element O for an efficient moderation of neutrons. The simulations below are therefore performed using CaWO\(_4\). However, when scaling up the number of detectors (see Sect. 2.7) larger diameters of CaWO\(_4\) crystals are necessary which currently are not available. In this case, Ge crystals are a promising alternative, since those are readily produced in large diameters (up to 300 mm), with high radiopurity.
It is worth mentioning that the timing information of pulses in the cryogenic detectors is crucial for the efficiency of a coincidence veto. The precision with which the onset of the pulses can be determined defines the dead time in the target calorimeter. We know from neutron scattering experiments that the pulse onset of comparable cryogenic calorimeters can be determined with a uncertainty of \({\pm }5~\upmu \)s [34]. Even an excessive rate of 100 Hz in the veto detector would introduce only a negligible dead time of \(\lesssim \)0.1%.
Also the rejection power of the outer veto was estimated with a MC study. Here a \(\hbox {CaWO}_4\) target was placed inside the nested shields of inner and outer veto. As typical background we investigate gamma rays following the remaining spectrum at the Dortmund Low Background facility [35], a low-background site at the surface which will be discussed in Sect. 3. Figure 8 shows as black histogram the background spectrum observed by an unshielded target, in blue the remaining background in the case of a passive outer veto, and in red the remaining background in the case of an active outer veto with a threshold of \(1~\hbox {keV}\). Even with only a passive veto a background suppression of more than 3 orders of magnitude at low energies is feasible. Activating the outer veto increase the suppression to more than 4 orders of magnitude. Importantly, the expected gamma-induced electron-recoil spectrum remains flat down to energy threshold (see inset of Fig. 8).
For a first estimate of muon-induced neutron backgrounds, a basic MC simulation was performed. Using an active CaWO\(_4\) outer veto, the neutron background is reduced by a factor of \({\sim }10\), independent of the recoil energy (studied in the energy range from 10 eV to 300 keV). By a clever combination of passive shielding elements like borated polyethylene, and active shielding elements like instrumented plastic or liquid scintillators, and LiF crystals, neutron background levels can be further reduced. This concerns shielding systems placed outside the cryogenic setup surrounding the cryostat at all sides. In addition we provide two technologies to further reduce and reject this potentially harmful background: (1) the outer cryogenic veto system described above and (2) the active background discrimination by the multi-target approach. The latter might be a powerful tool to reduce ultimate backgrounds, particularly neutrons. This is described in more detail in Sect. 3. Nevertheless, we conclude that a dedicated MC simulation using measured muon spectra in combination with a calorimeter measurement at the experimental site are necessary. This is beyond the scope of this work and will be subject of a future publication.
Production and scalability
A disadvantage of cryogenic detectors when compared to e.g., scintillation detectors has always been the difficulty to scale up the experiments in size. The new detector concept presented here overcomes most of these problems. In principle, the detector has been designed such that the number of production steps of the individual detector components are independent of the number of target calorimeters involved.
The target calorimeters are produced from wafers with a thickness of 5 mm and variable diameters (CaWO\(_4\) up to 60 mm, Al\(_2\)O\(_3\) up to 200 mm, and Si up to 300 mm). With well-established techniques of the semiconductor industry, as e.g. photolithography, thin-film evaporation, etching or sputtering, the TES sensors are being simultaneously equipped on each target calorimeter, and the wafer is cut only afterwards into the individual \((5\times 5\times 5)\) mm\(^3\) crystals. The same up-scaling is possible for the inner veto (Sect. 2.5) which acts as a detector holder. It is entirely produced by the above-mentioned methods. The cutting of the wafers is done by means of a laser or other automated methods. The cabling for a large amount of TES sensors are implemented by photolithography in combination with sputtering on the inner veto wafers as done for the \(3\times 3\) array. Further, it has been shown (e.g. in [36]) that large amounts of SQUIDs can be realized by SQUID multiplexing.
For the first phase of the experiment, we focus on the production of \(3\times 3\) arrays with moderate requirements of size and channel numbers which is foreseen as sufficient for a discovery of CNNS (see below). In a second step, the technology mentioned above enables experiments up to the kg-scale with energy thresholds of \({\mathcal {O}}\)(10 eV); an exposure allowing precision measurements of the CNNS cross-section and interesting BSM physics. Figure 9 shows a technical drawing of a future calorimeter array of 225 crystals which correspond, using Al\(_2\)O\(_3\), to a total mass of \({\sim }110\) g.