IUScholarWorks at Indiana University South Bend New Insights into Particle Detection with Superheated Liquids

We report new results obtained in calibrations of superheated liquid droplet detectors used in dark matter searches with different radiation sources (n,α,γ). In particular, detectors were spiked with alpha-emitters located inside and outside the droplets. It is shown that the responses are different, depending on whether alpha particles or recoil nuclei create the signals. The energy thresholds for α-emitters are compared with test beam measurements using mono-energetic neutrons, as well as with theoretical predictions. Finally a model is presented which describes how the observed intensities of particle induced acoustic signals can be related to the dynamics of bubble growth in superheated liquids. An improved understanding of the bubble dynamics is an important first step in obtaining better discrimination between particle types interacting in detectors of this kind. † present address: Department of Physics, McGill University, Montréal, H3A 2T8, Canada ‡ present address: Laboratorium für Hochernergiephysik, Universität Bern, CH-3012 Bern, Switzerland § present address: Department of Physics, University of Oxford, OX1 3PU, Oxford, UK ‖ present address: AECL Chalk River Laboratories, Chalk River ON, K0J 1J0, Canada ¶ present address: Fakultät für Physik, Ludwig-Maximilians-Universität, D-85748 Garching, Germany + present address: Physik Department, Universität Münster, D-48149, Münster, Germany ∗ present address: Department of Physics, Queens University, Kingston, K7L 3NG, Canada ♯ Corresponding author: E-mail address: zacekv@lps.umontreal (V. Zacek) New Insights into Particle Detection with Superheated Liquids 2 PACS numbers: 29.40.-n, 95.35.+d, 34.50.Bw

Abstract. We report new results obtained in calibrations of superheated liquid droplet detectors used in dark matter searches with different radiation sources (n,α,γ). In particular, detectors were spiked with alpha-emitters located inside and outside the droplets. It is shown that the responses are different, depending on whether alpha particles or recoil nuclei create the signals. The energy thresholds for α-emitters are compared with test beam measurements using mono-energetic neutrons, as well as with theoretical predictions. Finally a model is presented which describes how the observed intensities of particle induced acoustic signals can be related to the dynamics of bubble growth in superheated liquids. An improved understanding of the bubble dynamics is an important first step in obtaining better discrimination between particle types interacting in detectors of this kind.

Introduction
The PICASSO dark matter experiment uses the superheated droplet technique, which is based on the operation principle of the classic bubble chamber [1,2,3,4,5]. Detectors of this kind are threshold devices, where the operating parameters (pressure and temperature) determine the energy threshold. Since each temperature, at a given pressure, corresponds to a defined recoil energy threshold, the spectrum of the particle induced energy depositions can be reconstructed in superheated liquids by measuring the rate as a function of temperature.
In the case of PICASSO the active detector liquid is dispersed as droplets of a metastable superheated perfluorobutane, C 4 F 10 , and the detectors are operated in a temperature range such that nuclear recoils in the keV range induced by interactions with Weakly Interacting Particles (WIMPS) could trigger bubble formation. These explosive evaporations are accompanied by acoustic signals, which are recorded by piezoelectric transducers. In previous studies the PICASSO collaboration showed that the acoustic signals contain information about the nature of the primary event [6,7,8]: it was observed that the acoustic signals produced by alpha emitters are more intense than the signals of neutron or WIMP induced recoil events. Recently, this effect was confirmed by the COUPP and SIMPLE collaborations, which used it for a substantial background reduction in their dark matter searches [9,10].
The underlying physics process can be explained by the hypothesis that in the case of alpha emitters the recoiling nucleus and the extended alpha track contribute with at least two vaporization centres to the total signal, whereas the signals of the much more localized nuclear recoils carry the imprint of one single nucleation only. In the following we describe recent efforts to consolidate this hypothesis.
For this purpose PICASSO detectors were spiked with alpha emitters of known types and energies ( 241 Am and 226 Ra) and the emitters were deliberately located outside and inside the droplets. The results, together with data obtained from existing detectors containing relatively large contaminations with alpha emitters, support the proposed model and show that mono-energetic nuclear recoils following alpha decays are detected at lower temperatures than alpha particles. The energy depositions of nuclear recoils are larger than the energy deposition by the Bragg peak of the alpha particles themselves. This latter information can then be used to infer more precise estimates for the parameters of the model proposed by Seitz [11], which still serves as the reference theory to describe the radiation sensitivity of superheated liquids. The results from measurements using alpha emitters are then compared with results from test beam measurements using mono-energetic neutrons with energies between 4.8 keV and 4 MeV and are found to be in good agreement. Data for gamma ray induced nucleations also fit the described model well. Finally a plausible argument is presented to explain the observed alpha-recoil discrimination in terms of the dynamics of bubble growth in superheated liquids. This is largely uncharted terrain, nevertheless some conclusions can be drawn which shed light on the early phase of particle induced bubble formation.

Detection Principle and Theoretical Model
For a phase transition to occur in a superheated liquid the prevailing theoretical model proposed by Seitz [11] predicts that a critical minimum amount of energy E c has to be supplied within a local thermal spike and if the resulting proto-bubble reaches a volume of critical radius R c , it becomes thermodynamically unstable and grows rapidly. Thermodynamics predicts that the growth of the bubble passes through several stages of acceleration and deceleration, which also gives rise to a detectable pressure wave (sect. 10).
Both, R c as well as E c decrease exponentially with temperature and are given by where σ is the surface tension at the liquid-vapor interface, ∆p = p v − p e is the degree of superheat, which is the difference between the vapour pressure p v and the external pressure p e , ρ v is the density of the gas phase, and h lv is the latent heat of evaporation. All of these quantities depend on the temperature of operation T. The first term in expression (2) is the reversible work done during expansion to a bubble of size R c against the pressure of the liquid. The second term represents the energy needed to evaporate the liquid and is necessary to be provided if the subcritical bubble grows faster than energy can be supplied from its liquid vicinity. The third term describes the work needed to create the liquid vapour interface of the proto-bubble. W irr is the work which goes into irreversible processes, like acoustic wave emission; this contribution is small compared to the others. Radiation induced phase transitions imply that the locally deposited kinetic energy of a traversing particle exceeds the critical energy, i.e. E dep ≥ E c (T) and that the stopping power of the particle is large enough to supply this energy as heat over a distance L c such that it is effective in reaching the critical energy within a protobubble of critical size R c [12]: Experimentally relatively good agreement is reported between E dep and E c for several halocarbons and for energies around E c = 17 keV and larger [13]. The situation is different for L c , where a large range of values is quoted. Since R c is the natural length scale of the process, L c is usually given in terms of L c = bR c , where b varies from author to author: L c = 2 R c appears to be intuitively justified and is supported by data for low energy thresholds below 20 keV; L c = 2πR c = 6.28R c is proposed in )  10  1000  280  2540  3600  20  111.1  111  590  1090  30  27.8  60  99  210  40  7.45  35.3  20  40   Table 1. Predictions by the Seitz model for E c (T) and R c (T) in C 4 F 10 using relations (1) and (2) and values for ρ v , h fg , σ and p v compiled by NIST [20]. Also quoted are the ranges R F,C for fluorine and carbon ions with kinetic energies corresponding to the given values of E c .
[14] following arguments of stability of vapour jets in liquids, whereas some authors propose L c = (ρ v /ρ l ) 1/3 R c ≈ 6.6R c , but also values up to 18 have been reported [12,13,14,15,16,17,18,19]. As described in section 8, our recent results from calibration measurements with mono-energetic neutrons and alpha emitters in PICASSO shed additional light on this controversial issue. The active detector material in PICASSO, C 4 F 10 , has a boiling temperature of T b = −1.7 • C at a pressure of 1.013 bar and a critical temperature of T c = 113.3 • C; therefore at ambient temperature and pressure this liquid is in a metastable, superheated state. Table 1 summarizes the predictions of the Seitz model for E c (T ) and R c (T ) using relations (1) and (2) and values for ρ v , h lv , σ and p v compiled by NIST [20]. The contribution of W irr in relation (2) is small and has been neglected. The ranges of the recoiling 12 C and 19 F atoms, R C and R F , are also included in Table 1 for comparison at the quoted values for the threshold energies E c (T ).
It is interesting to note that at e.g. 20 • C most of the work required to create a critical bubble is spent in evaporating the liquid, W ev = 80 keV, the work required to create the liquid vapour interface amounts to W lv = 36 keV and the mechanical work done during bubble expansion amounts to W m = − 4.6 keV. These contributions are affected by errors at the level of 15 -20% due to uncertainties in the thermodynamic parameters (and these uncertainties increase with temperature).
The specific energy losses in liquid C 4 F 10 of the particles used in the calibration measurements discussed below, i.e alpha particles and neutron induced fluorine and carbon recoil nuclei were calculated with SRIM [21]; the results are shown in Figure 1. For the energy range considered here, with E F,C,α < 800 keV, fluorine always has the higher stopping power, followed by carbon and alpha particles. The stopping power at the Bragg peak of alpha particles almost equals the stopping power of fluorine below 100 keV.

Detectors and Read Out
The current PICASSO detector generation consists of cylindrical modules of 14 cm diameter and 40 cm of height [5]. They are fabricated from acrylic and are closed on top by stainless steel lids sealed with polyurethane O-rings. Each detector is filled with 4.5 litres of polymerized emulsion loaded with droplets of C 4 F 10 ; the droplet volume distribution peaks at diameters of around 200µm. The active mass of each detector is typically around 85 g. The active part of each detector is topped by mineral oil, which is connected to a hydraulic manifold. After a measuring cycle the detectors are compressed at a pressure of 6 bar in order to reduce bubbles to droplets and to prevent bubble growth which could damage the polymer. The operating temperature of the modules is controlled with a precision of ±0.1 • C. Each detector is read out by nine piezo-electric transducers. Three transducers are distributed around the detector at each of three different heights. They are flush mounted on a flat spot milled into the acrylic. The transducers are ceramic disks (Pz27 Ferroperm) with a diameter of 16 mm and 8.7 mm thickness and a sensitivity of 27 µV/µbar. The piezoelectric sensors are read out by custom made low-noise preamplifiers that serve a double purpose: providing impedance conversion and strong amplification (gain x 3000 between 0.5 kHz and 130 kHz). The bandwidth of the amplified signal is limited to the range of 1 to 80 kHz using a series of customizable RC filters. The amplified output signal is digitized using a 12-bit analog to digital converter (ADC) with serial output. The maximum amplitude of the digital signal is 2 V. The trigger threshold is individually set for each channel, normally at ±300 mV. The trigger condition requires at least one signal from a detector module crossing the threshold, and in this case will initialize the readout of all channels from this module. The system stores 8192 samples at a sampling frequency of 400 kHz, with 1024 samples before the trigger time. The total recorded signal is 20.48 ms long.

Energy Calibration with the Alpha Emitters 241 Am and 226 Ra
Two detectors were prepared especially to study the response of superheated liquids to alpha decays of known origin. In one detector, the polymer in which the droplets are suspended was first spiked with an aqueous solution of 241 AmCl with an activity of 25.56 Bq and after completion of the measurements this same detector was spiked with 10 Bq of 226 Ra. The other detector was exclusively spiked with 10 Bq of 226 Ra. Both detectors were shortened versions, 1/3 in height of normal PICASSO modules. They were read out in one horizontal transducer plane, i.e. by three piezoelectric sensors arranged under 120 • with respect to each other and mounted on the outside of the detector wall. The observed count rates as a function of temperature are shown in Figure 2.
241 Am decays follow the reaction 241 Am → 237 Np + α + 5.64 MeV. The count rate of the 241 Am-spiked detector exhibits a threshold at 22 • C, reaches a flat plateau at 26 • C and traverses 50% of the plateau rate at 24.5 • C. This temperature corresponds to an energy threshold of E th (24.5 • ) = 71 keV according to neutron calibrations (sect. 5). Since the alpha emitters are predominantly located outside the droplets (sect. 9.2), the only alpha particles which can trigger a phase transition at threshold are those with the highest specific energy deposition, which corresponds to the Bragg peak. At higher temperatures the liquid becomes sensitive to smaller dE/dx on the tracks, but since the detector is already fully sensitive to alpha particles immediately above threshold, the temperature response levels off to a plateau.
After completion of data taking with the 241 Am spike, the same detector was loaded with 226 Ra, this time by injecting locally with a syringe an aqueous solution of RaCl (10 Bq). 226 Ra (T 1/2 = 1602 y) decays into 222 Rn (T 1/2 = 3.8d), which then decays following the sequence of transitions 222 Rn → 218 Po → 214 Pb → 214 Bi → 214 Po → 210 Pb via three α-and two β-decays; the rest of the chain is too long-lived to be relevant here. The energies of the emitted α-particles are 5.49 MeV, 6.0 MeV and 7.69 MeV, respectively, and the half lives are short with respect to the 226 Ra half life (3.8 d, 3 min, 27 min, 19 min, 0.2 ms). By rate measurements and visual inspection of the bubbles formed, the radon induced α-activity could be observed to diffuse slowly within a couple of days over the entire detector volume, including the C 4 F 10 droplets themselves. When equilibrium for this portion of the chain was reached, data were taken as a function of temperature and the results are shown in Figure 2.
The observed threshold is now shifted by about 4 • C towards lower temperatures, i.e Response of detectors spiked with the alpha-emitters 241 Am and 226 Ra. The curve with the higher threshold temperature was obtained after spiking the detector matrix with 241 Am (circles) and only α-particles entering the droplets can induce nucleation. At threshold α-particles trigger nucleation with energy depositions at the Bragg peak. The lower threshold was obtained with two 226 Ra spiked detectors (squares and triangles). Here the recoiling 210 Pb nucleus with the highest recoil energy in the decay chain (E rec = 146 keV) defines the thereshold. The respective count rates of the different measurements are normalized at the plateau. larger energy depositions: it starts at 19 • C and reaches its plateau at 22 • C. However, the observed shift cannot be attributed to the energy deposition of the α-particles emitted by the Ra-chain, since they have the same maximum stopping power at the Bragg peak than those emitted in 241 Am-decay; rather the reduced energy threshold is now caused by 210 Pb nuclei recoiling inside the droplets with energy of 146 keV, since this is the nucleus with the highest recoil energy in the chain.
By further raising the temperature, the detector becomes subsequently sensitive to the lower energetic 214 Pb (112 keV) and 218 Po (101 keV) recoil nuclei. First 214 Pb recoils add to the observed count rate, but once the energy threshold is low enough to allow 218 Po recoils to trigger, the 214 Pb recoils, which follow 218 Po decays (T 1/2 ≈ 3 min.), are no more able to contribute several minutes after run start. This is due to the fact that once a phase transition has occurred in a droplet it is no longer sensitive to subsequent energy depositions. The same situation arises for 214 Po decays which are gradually rendered undetectable following 222 Rn/ 218 Po decays in the same droplet for measuring times exceeding the half lives of the two beta decays after run start (T 1/2 = 27 min and 19 min). This expected asymptotic decrease in count rate due to the depletion of 214 Po decays has been observed for temperatures T > 20 • C and for measuring times lasting up to two hours after run start. In order to eliminate this time dependence, the count rates shown in Figure 2 were always calculated for the same time interval after run start.
The observed threshold curve is characterized by a steep slope, but the data are not precise enough to reveal a step-like substructure which should be caused by the 34 keV energy difference between the 214 Pb and 210 Pb recoils. However, the 146 keV recoil energy of the 210 Pb nuclei at the observed threshold temperature is consistent with the threshold obtained for neutron induced 19 F recoils discussed in sect. 5. By increasing the temperature from 22 • C to 25 • C the 226 Ra spiked detector becomes also sensitive to α-particles, but since the detector is already fully sensitive, the count rate remains unaffected. However, above 25 • C, α-particles contribute to the amplitude of the acoustic signal. This effect will be described in sect. 9.
Although the 241 Am activity is still present after the 226 Ra spike, it does not show up as a step at 22 • C in Figure 2. This is due to the fact that after the Ra-spike the recorded count rate increased by a factor of 110 †. This strong increase in count rate is not yet completely understood. It could possibly be attributed to the combination of two effects: 1) the larger geometric detection efficiency for alpha decays within the droplets and 2) an additional concentration of alpha emitters in the droplets caused by an increased solubility of 222 Rn in fluorocarbons like C 4 F 10 compared to the 222 Rn solubility in the water based polymer.
The second detector used in this study, similar in size and composition, but spiked exclusively with 10 Bq 226 Ra, reproduced the above described threshold results ( Figure  2). Also its count rate, normalized to the active mass of C 4 F 10 was found comparable to that of the Am loaded detector after the Ra spike [22]. Another, similar study of alpha emitters was described by Hahn [23] employing 238 U and 232 Th spikes in CCl 2 F-CClF 2 operated under negative pressure. In those measurements, recoils from 210 Po α-decays could be clearly separated from the 67 keV smaller recoil energies produced in 212 Po decays.

Energy Calibration with Mono-Energetic Neutrons
The dependence of the threshold energy E th on temperature and pressure was studied with neutron induced nuclear recoils. For this purpose extensive calibrations were performed at the Montreal Tandem van de Graaf facility with mono-energetic neutrons ranging from E n = 4.8 keV up to 4 MeV. In the case of nuclear recoils induced by neutrons of low energy the interaction is mainly through elastic scattering on fluorine and carbon nuclei. Inelastic collisions only occur if the centre-of-mass kinetic energies of the neutrons are higher than the first excitation level of the nuclei (1. 5  for 19 F and 12 C, respectively). Absorption of neutrons by the 19 F nucleus followed by alpha particle emission requires a neutron energy of 2.05 MeV.
Assuming neutron elastic scattering on nuclei, the recoil energy, E i R , of the nucleus i is given by where E n and θ are the incident neutron energy and the neutron scattering angle in the centre-of-mass system, m n and m N i are the masses of the neutron and the nucleus N i , respectively. At a given neutron energy E n the recoiling nuclei are emitted with an angular distribution, every angle being associated to a specific recoil energy ranging from 0 keV at θ = 0 up to the maximum energy E i R,max for θ =180 • . At the small energies considered here, the angular distribution is isotropic in the centre-of mass system and the recoil energy spectrum dR i /dE i R has a rectangular, boxlike shape up to E i R,max : The factor f i gives the maximum fraction of the energy of the incident neutron transmitted to the nucleus i, where f F = 0.19 and 0.28 for 19 F and 12 C, respectively. The mono-energetic neutrons used for calibration were produced via nuclear reactions with mono-energetic protons via the 7 Li(p,n) 7 Be and 51 V(p,n) 51 Cr reactions. The measurements with the Li target (10 µg/cm 2 ) cover a range of neutron energies from 100 keV up to 4 MeV and the results obtained are discussed in detail in [18]. With improved proton beam stability these calibrations were recently extended in PICASSO with a 51 V target (9 µg/cm 2 ) down to 4.8 keV. In order to acquire sufficient statistics close to threshold, the proton beam energies were tuned to individual resonances in the 51 V(p,n) 51 Cr reaction cross section [24]. In particular the five resonances quoted in Table 2 have been selected, each of which has an intrinsic width below one keV.
The detectors used for these measurements are smaller in size (63 mL) with an active mass of 1g of C 4 F 10 , but fabricated in the same way and with similar droplet dimensions as the standard 4.5L detectors. For each of the selected neutron energies, data were taken at θ ≈ 0 • with respect to the beam, while ramping the temperature up and down for a given pressure. Since close to threshold the cosmic ray induced n-background can amount to 50% of the total count rate, each neutron run at a fixed temperature was followed by a background run at the same temperature (after 8h of recompression).
For a fixed neutron energy the data have been normalized by the integrated proton current and the count rate of a 3 He counter mounted behind the target was used to compensate for short off-resonance beam energy fluctuations [25]. The measurements at the lowest neutron energy (4.8 keV) were particularly challenging since at threshold and above, the detectors had to be operated between 48 and 60 • C where C 4 F 10 becomes sensitive to the 320 keV gamma rays (T 1/2 = 28 d) following de-excitation of 51 Cr. Therefore this background had to be measured independently during a beam off period after each neutron run and subtracted from the data.
The recorded count rates for the Li and V targets at different neutron energies at ambient pressure and as a function of temperature are compiled in Figure 3. From these measurements, the threshold temperature, T th , can be extracted for a given neutron energy by fitting the data to a theoretical response function including energy losses in the target and the detector matrix and by keeping the intrinsic energy resolution as a free parameter (sect. 7). From this the neutron threshold energy as a function of temperature can be inferred as is shown in Figure 4. For the energies considered here, E n th (T) follows an exponential dependence on temperature. A similar exponential trend was observed by other authors for a series of halocarbons such as C 4 F 8 , CCl 2 F 2 , C 2 H 3 ClF 2 and C 2 Cl 2 F 4 [13].
In practice it is more interesting to know the threshold of the minimum nuclear recoil energy at a given temperature. Due to the composition of the target in use, C 4 F 10 , there are two possibilities in converting neutron energy into recoil energy: 1) assuming that the response depends on the energy which is deposited on the entire recoil track, then the threshold should be attributed to the more energetic carbon nucleus; 2) if the recoil nucleus with the greater dE/dx triggers, then it is fluorine that defines the threshold. In both cases the energy thresholds (at 1 bar) are obtained from the fit to the neutron threshold data and in the case of 19 F follow the relation: The kinematic factor 0.19 relates the measured neutron threshold energy to the maximum respective nuclear recoil energy; if 12 C would trigger the kinematic factor would be 0.28. The error of 3% is largely due to the systematic errors of ± 0.2 • C in the temperature measurement during the test beam runs. Given the temperature range of operation in PICASSO, this translates in the case of 19 F recoils into a range of sensitivity from E F > 2.0 keV at 45 • C to E F > 200 keV at 18.5 • C, respectively. Several observations support the conclusion that indeed fluorine atoms with their higher dE/dx are responsible for defining the threshold: 1) the neutron scattering cross section on 19 F is four to six times larger than the cross section on 12 C in the considered energy range with substantial enhancement in its resonances. If carbon would trigger first, a second threshold should be observed in the threshold curves of Figure 3 at higher temperatures when scattering on 19 F sets in; 2) the detection threshold for the 146 keV recoil 210 Pb nuclei following alpha decays of 214 Po matches the one for 19 F at the same energy; 3) for recoil energies between 10 keV and 1 MeV, the critical energy E c (T) for bubble formation predicted by the Seitz model follows closely the 19 F threshold; 4) measurements with C 4 F 8 and C 2 Cl 2 F 4 show that the recoil energy of the species with the higher dE/dx matches well the predicted E c (T) [13,16]. For threshold energies smaller than 10 keV, data and theory deviate from each other. The cause of this discrepancy is not yet understood, especially since it was observed for halocarbons that E c approaches zero if the temperature attains 90% of the critical energy T c , which would correspond to 74 • C in the case of C 4 F 10 [18]; this temperature is also very close to the limit of stability observed during measurements of the γ-sensitivity discussed in sect. 6. The fact that E c (T) curves up at high temperatures might be due to a still incomplete description of the underlying processes once the critical radius R c approaches the nanometer scale. Thermodynamics also requires that the threshold curve bends up towards infinity at the boiling temperature T b = -1.7 • C.  (2); it is the work required to create a bubble with critical radius R c (T).

Energy Response to Gamma Rays
In contrast to the energy depositions of recoiling nuclei and alpha particles the main interaction process of γ-rays with the detector material occurs via Compton scattering. Because of their very small stopping power, recoiling Compton electrons cannot trigger a phase transition directly in the normal temperature range of operation. Rather the observed sensitivity to gamma rays is attributed to δ-rays or Auger electrons scattered randomly along the tracks of the Compton electrons. These low energy electrons curl up at the end of their trajectory into highly localised clusters of ionisation or hot spots which rarely lead to energy depositions at the keV level. In particular it was found in simulations that the δ -ray energy spectra on tracks of electrons from 5 keV up to 500 keV, and on the tracks of 1 GeV muons are very similar in shape and 50% of the emitted δ rays were found to deposit energies smaller than 0.05 keV [18,26]. Calibrations with γ-rays can therefore give only information about the probability distribution of clusters of energy on the tracks of Compton electrons. Such a study was performed with a 22 Na source (0.7 µCi), which yields 1.275 MeV γ-rays, as well as the two 511 keV photons from e + e − annihilation. Compton scattering of the 1.275 MeV γ-rays produces recoiling electrons in the detector medium with an average energy of 500 keV and with a range of 1.1 mm, whereas the annihilation photons create electrons of 170 keV. Two different detectors were used: a standard 4.5L detector with an active mass of 78.9 ± 8 g was used to explore the low temperature response from 40 to 50 • C and a 10 mL detector with an active mass of 30 mg was used to measure temperatures from 48 up to 72 • C [26,27]. The count rates as a function of temperature are shown in Figure 5 and it was found that the measured sensitivity can be fit over more than 10 orders of magnitude with a sigmoid function with T 0 = 63.6 • C and τ = 1.78 • C. At the plateau, the measured efficiency (∼5%) corresponds roughly to the geometric efficiency that a Compton electron hits a droplet (∼1%). Measurements with other sources ( 57 Co, 60 Co, 137 Cs) showed that the response curves are identical and that the plateau efficiencies are comparable for γ-energies from 127 keV to 1.3 MeV. This is expected, since for an average Z of the detector material of Z ≈ 5.5, the Compton scattering cross section dominates in the energy range from 400 keV to 5 MeV and the mass attenuation coefficient does not vary much. In addition, the stopping power of the scattered electrons does not vary much in this region and since the δ-ray production probability is proportional to the dE/dx on the particle track, the γ-detection efficiency reflects the constant production efficiency of δ-electrons. This scenario was confirmed by simulations, which in addition show that the sigmoid shape of the observed response curve can be reproduced well [18,26].
Above 72 • C the detector becomes sensitive to spontaneous nucleation. An increase in temperature by 1 • leads to an increase in count rate by approximately three orders of magnitude.

Energy Resolution
By inspection of the alpha data in Figure 2 it is apparent, that the detection threshold is a well defined, but not a sharp step function; the count rate increases steeply, but gradually from threshold to full efficiency. The probability P (E dep , E th ), that an energy  Figure 5. Gamma-ray sensitivity as a function of temperature ( • C). The data at the four lower temperature points were recorded with a standard 4.5L detector (triangles); the higher temperature data were taken with a small 10 mL detector. At the plateau the gamma detection efficiency is determined by the geometry and the probability that a Compton electron hits a droplet. Over 10 orders of magnitude in count rate the data are well reproduced by the sigmoid function described by relation (6). After 72 • the detector becomes sensitive to spontaneous nucleation.
deposition E dep larger than the energy threshold E th will generate a nucleation can be approximated by where the parameter a describes the observed steepness of the threshold: the larger a is, the sharper the threshold is defined. This parameter is related to the intrinsic energy resolution and reflects the statistical nature of the energy deposition and its conversion into heat [28]. It has to be determined experimentally for each superheated liquid and for different particle species, respectively. Our measurements with alpha emitters with well defined, mono-energetic recoil nuclei ( 210 Pb) indicate a threshold which can be described best with a = 10 ± 1 at 146 keV; alpha particles depositing their energy at the Bragg peak seem to produce a somewhat less steep threshold described by a = 5.8 ± 0.7 at 65 keV ( Figure 2). This parameter is more difficult to extract from calibrations with mono-energetic and poly-energetic neutrons due to the continuous spectral distributions of the recoiling nuclei. Our data with mono energetic neutrons above 400 keV are compatible with a = 10 ± 5, at lower energies smaller values appear favoured with a = 2.5 ± 0.5 [29]. A more precise study of a suspected temperature dependence of the resolution parameter is the subject of ongoing measurements.

Determination of the Critical Length L c
The relatively good agreement between deposited energy at threshold E dep and the critical energy E c required for nucleation below 40 • C observed in neutron calibrations allows an estimate of the effective ion track length L c = b R c over which the energy deposition is distributed. We follow here the model proposed by d'Errico, which assumes that the vapour cavity formed initially may extend along the charged particle track, before ending up in at least one structure of spherical shape of size R c [2].

L c from alpha emitters
In the case of alpha particles, the threshold energy E th (T) for particles entering from outside the droplets is related to the deposited energy by relation (2) where (dE/dx) Bragg = 210 keV/µm is the maximum specific energy loss at the end of the track of a 5.64 MeV alpha particle emitted in 241 Am decays. From this relation it follows that L c = 0.31 µm, for the critical length L c along which the particle deposits its energy in order to be able to create a critical proto-bubble. Using in addition the prediction by the Seitz model given in (1) for the critical radius R c (T), one obtains L c (24 • ) = 3.6 × R c (24 • ), which yields an estimate of the model parameter b α (24 • ) = 3.6 for this temperature. This value is compared in Figure 6 with those obtained from neutron calibrations.
On the other hand one can use the fact that alpha particles do not trigger phase transitions at 21 • and E dep =146 keV in order to obtain an upper limit on the critical length of L c (21 • C) ≤ 0.45µm which implies b α (21 • C) ≤ 5, as indicated in Figure 6.
Incidentally, these values of L c are close to the range of 19 F at this threshold energy, but definitely smaller than the range of 12 C with R C = 1.16 µm. Since from Figure 1 the stopping power of carbon is always smaller than that of fluorine, its energy deposition would be around 80 keV and therefore smaller than the 146 keV required, which supports the assumption made in sect. 5 that fluorine triggers at threshold.

L c from neutron induced recoils
Applying relation (2) on the data taken during the neutron calibrations reported in sect. 5, two sets of parameters b F (T) and b C (T) result, depending on the assumption whether either 19 F recoils trigger at threshold (lower curve in Figure 6) or 12 C recoils (upper curve in Figure 6). Both curves represent lower bounds on L c . In particular, in the case of b F the area between the two curves is allowed, since above that region carbon would trigger at threshold [28]. The two estimates for L c discussed in the alpha scenario add two independent constraints which favour the lower set of b F values. In  Figure 6. Nucleation parameter b as a function of temperature. This parameter uses L c = bR c to relate the spatial extension of the energy deposition with the critical radius defined in relation (1). If carbon would trigger at threshold, then only nucleation parameters above the upper line would be allowed. If fluorine recoils determine the threshold, then the shaded region between the two curves defines the allowed parameter space. The threshold data of the alpha spiked detectors yield an additional upper limit at 21 • C and an additional prediction at 25 • C (triangles); the alpha data favour the lower part of the shaded parameter space.
summary, the range of the preferred values, together with the observed trend that L c increases with temperature agree well with measurements on several other halocarbons discussed in [13].

Acoustic Signals from Particle Induced Events in Superheated Liquids
It is known that energetic charged particles traversing non-stressed liquids or solids produce acoustic waves during their passage. This so-called thermo-acoustic effect was predicted and described by Askarian et al. [30] and is exploited in high energy neutrino detection in the PeV range by the ANTARES and ICECUBE experiments [31,32]. However, applied to the processes considered here, with energy depositions in the keV range, the emitted sound intensities predicted by the thermo-acoustic effect are undetectable. On the other hand, particle interactions in stressed or superheated liquids, produce detectable acoustic signals which are related to the nature or the extension of the primary event; this suggests that the phase transition process in superheated liquids provides an intrinsic acoustic amplification mechanism with a gain of at least 10 5 [33].

Neutron induced recoils
Calibration data with fast neutrons of AcBe, AmBe and Cf sources showed that the associated waveforms have a short rise time, reaching the maximum amplitude after 20-40µs, with slower oscillations following for several milliseconds. In order to characterize signals of different origins in the detector, a Bessel band pass filter is applied to cut off frequencies below 18 kHz and then the waveform of each transducer is squared and integrated over the signal duration, starting from a fixed pre-trigger time. The resulting values are then averaged over all active transducers for each event in order to mitigate solid angle effects. The logarithm of this averaged acoustic energy is used to define the acoustic energy parameter, so called as it is a measure of the average energy contained in the transducer signals. It shows a well defined distribution, with a centroid which increases smoothly with temperature ‡ (Figure 7). This observation suggests that the fast component of the signal does not depend on the droplet size, but retains information of the very first stage of bubble formation [8]. A plausible explanation of this effect will be given in sect. 10. This property can be used to discriminate particle induced recoil events from non-particle related signals [5,34]. Since WIMP induced recoils are similar to neutron induced recoils, this feature is of prime importance for dark matter searches with superheated liquids.

Alpha decays
Alpha decays in the PICASSO detectors also produce signals with well defined energy, and if fully contained in a droplet, with larger acoustic energy than observed in neutron induced events [6]. This can be explained by the fact that the ranges of neutron induced recoils of keV energies are of sub-µm extension and therefore comparable in size to the critical length L c . Therefore these events are able to trigger only one primary nucleation. However, alpha emitters located within the superheated liquid can trigger at least two vaporizations: one from the recoiling nucleus and the second one or more on the alpha particle track. In both cases the energy released during vaporisation increases with temperature, but stays well defined for a given temperature. Data taken with the alpha spiked detectors described in sect. 4 have been used to investigate this nucleation hypothesis further. In order to detect possible deviations from single bubble nucleations, each alpha measurement at a given temperature was followed by an exposure to an AcBe neutron source. It was found that for the Am spike, where only alpha particles originating from outside the droplets are able to trigger a phase transition, the signal energy distribution coincided with the distribution recorded during the neutron sessions. intrinsic alpha background rates, a different pattern arises when the distribution of the acoustic energy parameter is recorded as a function of temperature ( Figure 8). Between threshold at 21 • C and below 25 • C, only recoil nuclei create a peak which coincides with the location of the neutron induced recoils during exposure with an AcBe source. With further increase in temperature above 25 • , when the detector becomes sensitive to alpha particles a second peak arises at higher acoustic energy and the first peak gradually diminishes. This second peak corresponds to nucleations caused by recoil nuclei plus nucleations caused by the Bragg peak on the alpha track. During this redistribution between the peaks, the sum of the count rates remains constant; the relative contribution to the total count rates are shown in Figure 9. The apparent shift of +2 • C between the threshold data shown in Figure 2 and the data shown in Figures 8 and 9 is due to an equivalent 0.2 bar difference in operating pressure: the spiked detectors ( Figure 2) were operated at 1 bar surface ambient pressure, whereas the data shown in Figures 8 and 9 were taken at the SNOLAB underground site at 1.2 bar ambient pressure. It was also noticed that the degree of separation between recoils and alpha particles depends on the temperature and the frequency content of the signals: at temperatures around 25 • C high-pass filters which accept frequencies above 10 kHz give the best result and the resolution tends to decrease with increasing cut-off frequency. The opposite happens at high temperatures -above 40 • C -where the best results are obtained with cut-offs above 100 kHz and discrimination worsens for lower cut-offs.

Delta-electrons from γ-ray induced events
If detectors are operated at temperatures far below the plateau-γ senistivity (i.e 65 • C) then the clustered energy depositions from Auger-or δ-electrons on the tracks of Compton scattered electrons create events with small multiplicity within a droplet (sec. 6). Therefore the acoustic signals are expected to reproduce those produced by single nucleations on the short tracks of nuclear recoils. In order to verify this hypothesis, data were taken with two different detectors in the presence of a 22 Na and a 137 Cs source, respectively, and compared with the signals induced by fast neutrons from a γ-shielded  Figure 9. The two relative contributions to the acoustic signal energy for alpha contaminations within the droplets: nuclear recoil induced events corresponding to the first peak in Figure 8 (triangles); joint contribution of nuclear recoils and alpha particle induced events corresponding to the second peak in Figure 8 (dots). The first peak coincides with the acoustic energy parameter of neutron induced recoils and dominates between 21 and 25 • C; above that temperature the detector becomes sensitive also to alpha particles which adds to the strength of the signal, but does not change the count rate, since the detector is already fully sensitive. Curves are drawn to guide the eye.
AcBe source. Measurements were performed at 45 and 46 and 50 • C, respectively, and the acoustic energies of the γ-induced signals coincided with those of the neutron induced recoils ( Figure 10). It would be interesting in future to extend these measurements to higher temperatures, where also multiple nucleations might become observable on the Compton electron tracks traversing the droplets (the high temperature data in sec. 6 were taken without neutron reference measurements).

Dynamics of Bubble Growth and Acoustic Signal Formation
The dynamics of bubble growth and the associated sound emission in superheated liquids is a complex phenomenon involving non-linear thermodynamic processes which are still the subject of ongoing research [35]. Particle induced sound generation in superheated liquids was first discussed in [33], but up to now only an approximative and qualitative description can be given of the processes leading to the recorded acoustic signals and the observed alpha recoil discrimination. The early theoretical works of Rayleigh [36], Plesset and Zwick [37] were based on an approximate solution of the Navier-Stokes equation and postulated that the growth of a vapour bubble in a superheated liquid is controlled by three stages: 1) a surface tension controlled stage, 2) followed by a stage where the growth is limited by the inertia of the liquid and where the bubble expands with a constant velocity and 3) an asymptotic stage which is dominated by heat transfer and where the bubble growth is decelerating.
As soon as the bubble radius reaches R > R c , the expansion is driven by the energy stored in the bubble itself and its vicinity and this is described by the Rayleigh-Plesset equation [36]. Its solution implies that the radius increases linearly with time and that the speed of this inertial growth is proportional to the square root of the superheat ∆p defined in sect. 2: Since the superheat increases and the liquid density decreases with temperature, the speed of bubble growth also increases with temperature. In particular we find for C 4 F 10 a prediction of the growth velocity of A(30 • C) = 11.6 µm/µs and A(46 • C) = 13.6 µm/µs.
Due to the expansion of the bubble volume the vapour within the bubble and also the liquid in the vicinity of the bubble walls cools down until it reaches the boiling temperature and after a certain characteristic time τ , further growth is only possible if energy is supplied by heat transfer from more and more distant layers of the liquid. From there on, the growth rate becomes limited by thermal diffusion and it decreases continuously. This is described by the Plesset-Zwick equation [37], which in this regime predicts a much slower increase of the bubble radius proportional to the square root of time: Here κ is the thermal conductivity of the liquid and c pl is its specific heat; the other quantities are as defined in sect. 2. The growth parameter B(T ) can also be expressed as B(T ) = (3κ/2π)J a (T ), where J a is the Jakob number, a dimensionless quantity which characterises the speed of the bubble growth. In particular it was found that the t 1/2law is only valid for 2 < J a < 100 [35,38]. For C 4 F 10 and within the temperature range considered here, J a follows a distribution with a broad peak around 35 • C with J a = 18.5 and which decreases slowly and asymmetrically to J a = 16.5 at 20 • C and J a = 18 at 50 • C, respectively. The growth rate 1 µs after nucleation is predicted by (10) to be ≈ 2.5 µm/µs, which is already smaller than the speed of inertial growth. The time τ , which is the time at which the transition between the two asymptotic solutions occurs, is strongly model dependent and a scope of investigations [35].
The predictions of growth rates by the classical model are, however, idealisations and measurements in superheated liquids showed linear growth rates which were substantially slower than the predicted inertial growth, but still larger than thermal growth up to 100 µs after nucleation [39].
After complete phase transition of a droplet, a freely oscillating vapour bubble is formed. The resulting bubble is a harmonic oscillator, oscillating around its equilibrium radius R b and the ambient equilibrium pressure P 0 with a resonance frequency calculated by Minnaert [40] as: where κ is now the polytropic coefficient of the gas and ρ l the density of the surrounding liquid. For C 4 F 10 at 30 • C the resonance frequency and the bubble radius are related by the simple relation ν R (kHz) = 2.4/R (mm). Typical droplets in PICASSO of 100 µm radius will eventually form bubbles of R b ≈ 0.35 mm radius and are expected to oscillate with a fundamental frequency of ν R ≈ 6 kHz. This frequency is below the 18 kHz cut-off used in the analysis of signals discussed in sect. 9. The pressure of the emitted sound, which is produced in the liquid by an expanding or oscillating spherical bubble of radius R(t), is related to the acceleration of its volume V (t) :  Figure 11. Amplitudes of the measured acoustic signals as a function of temperature compared to theoretical predictions. In the case where bubble growth is driven by thermal diffusion only, the observed signal amplitudes decrease slowly with temperature (dotted); inertial growth predicted by the Rayleigh-Plesset solution of the Navier-Stokes equation predicts a steady rise of the pressure signal with temperature and correspondingly also of the amplitudes (broken). The two theoretical predictions were set to a common value at 20 • C.
where ∆P (t, r) is the pressure change produced in the liquid at a distance r from the source, c is the velocity of sound and ρ l is the density of the liquid [41]. Inserting the solutions for inertial growth R in (t) and for thermal diffusion limited growth R th (t) into (10), one finds the radiated pressure signals for the two modes of asymptotic bubble growth: Piezo-electric transducers are sensitive to the instantaneous pressure ∆P , with sensitivities quoted in terms of µV/µbar (sect. 3). Therefore an analysis of the waveform of the transducer signal allows one to obtain information about the emission process. However, given our present experimental conditions our timing information is severely limited by the sampling frequency (2.5µs/sample) and is distorted especially for times larger than 80µs by reflections and container effects, like sound propagation in the acrylic.
Nevertheless some conclusions about the bubble growth and acoustic signal production can be inferred from the observed amplitudes and their dependence on temperature. Figure 11 compares the measured amplitudes as a function of temperature with those predicted by the two growth models. Apparently only the inertial growth scenario in which the amplitudes increase with temperature shows a trend similar to the data. Since the observed alpha recoil discrimination implies a spatial resolution of two nucleation centres separated by about the length of an alpha track, i.e. L α ∼ 40 µm, it can be concluded that the inertial phase cannot last much longer than about 2 L α / A(T ) : if inertial growth would continue beyond that time, the expanding bubble volumes would have merged completely and all information about multiple nucleation sites would have been washed out. Therefore after that time, i.e. t ≥ 10 µs, and according to the above estimates for A(T ), the decelerating thermal growth phase must have become the dominant effect in order to preserve the information about the spatial extension of the original nucleation volume.
At the moment our alpha -recoil discrimination data indicate the presence of two to three nucleation centres. However, within the scenario discussed above one would expect that with a better timing resolution of the acoustic readout system, more nucleation centres or an extended nucleation region could be resolved along the alpha track at an earlier stage of formation, which would result in a further improvement of the alpha recoil discrimination.

Conclusions
The full sensitivity of superheated liquids to nuclear recoils in the absence of a significant sensitivity to gamma rays or minimum ionising particles has stimulated the interest in this technique for dosimetry, neutron detection in fusion research and recently large scale applications in dark matter searches. However some grey areas exist in the detailed understanding of the underlying radiation detection processes: how precisely does radiation induce phase transitions at the nanometer scale, what are the precise dynamics of bubble growth and what time scales are involved, how are the observed acoustic signals produced, and how much information about the nature of the primary event do they contain?
Our recent studies were able to consolidate some known features and to shed new light on some of the open questions : 1) the energy thresholds predicted by the classic nucleation theory in C 4 F 10 are in good agreement with neutron and alpha calibration data; only at the lowest neutron energy at 4.8 keV does a small discrepancy exist which needs to be clarified; 2) threshold measurements with detectors spiked with alpha emitters allow one to differentiate between energy depositions by the recoiling nuclei and those caused by the Bragg peak of alpha particles; 3) recoil nuclei following alpha decay have a higher energy threshold than alpha particles; 4) signals produced simultaneously by recoil nuclei and alpha particles have more acoustic energy than signals produced by one or the other separately; 5) neutron and alpha data deliver a consistent picture of how the critical interaction length L c evolves in terms of the critical radius R c and with temperature; 6) signal amplitudes increase with temperature which implies that inertial growth contributes to acoustic signal formation; 7) the observed alpha -recoil discrimination requires a fine tuning in the interplay between the inertial bubble growth mode and the asymptotic thermal growth. While our understanding of this interplay is rudimentary at this stage, it is expected that much better discrimination between particles can be achieved by adapting the speed of the acoustic read-out chain to the timescale which carries most of the information about the primary nucleation process.