# On the critical energy required for homogeneous nucleation in bubble chambers employed in dark matter searches

## Abstract

Two equations for the calculation of the critical energy required for homogeneous nucleation in a superheated liquid, and the related critical radius of the nucleated vapour bubble, are obtained, the former by the direct application of the first law of thermodynamics, the latter by considering that the bubble formation implies the overcoming of a barrier of the free enthalpy potential. Compared with the currently used relationships, the present equations, still allowing for reversible processes only, lead to thermodynamic energy thresholds of the bubble chambers employed in dark matter searches that are closer to the experimental values.

## 1 Introduction

Bubble chambers using superheated liquids have been widely employed in high-energy physics for several decades after the invention of Glaser dated back to 1952 [1]. Recently, variants of such detectors are exploited in the search for dark matter in the form of weakly interacting massive particles (WIMPs), the main difference from the standard bubble chambers being the fact that the target liquid is continuously maintained in the metastable superheated state, instead of for just a few milliseconds [2, 3, 4, 5, 6, 7].

In both applications, bubble nucleation is the result of a highly localized deposition of at least the minimum amount of energy required for the formation of a bubble of critical size, as postulated by Seitz in his “thermal spike” theory [8], which is the model currently accepted as the best explanation available for radiation-induced nucleation in superheated liquids. The minimum amount of energy to be released as a thermal spike to produce a bubble nucleation, typically called critical energy, is generally expressed as the sum of a number of terms, this number varying with the assumptions made by each investigator. Moreover, also the value of the critical bubble radius, which enters directly into the calculation of the critical energy, depends on the assumptions made for its evaluation. Indeed, very often the theoretical values of the critical energy, i.e., the thermodynamic energy thresholds, are lower, sometimes drastically, than the corresponding experimental values. On the other hand, the relatively low threshold needed for WIMP-recoil detection asks to be the most accurate as possible in the prediction of the critical energy required for bubble nucleation, which also helps to provide a correct explanation for why the calibration results give higher thresholds than thermodynamic calculations.

In this general framework, a reasoned review of the critical energy equations readily available in the literature, and the related expressions of the critical bubble radius, is carried out. A pair of relationships for the determination of the critical energy and bubble radius are then proposed and discussed.

## 2 Critical energy for bubble nucleation

Notice that, strictly speaking, the metastable liquid state of coordinates \((\mathrm {T_{L}},\mathrm {p_{L}})\), which apparently falls in the vapour region, could not be displayed in the \(\mathrm {p}\mathrm {T}\) phase diagram, wherein only stable equilibrium states can be represented. Of course, the degree of metastability of the superheated liquid can be expressed either in terms of superheat, \(\,\mathrm {\Delta }\mathrm {T}=\mathrm {T_{L}}-\mathrm {T_{V}}\), or in terms of underpressure, \(\,\mathrm {\Delta }\mathrm {p}=\mathrm {p_{V}}-\mathrm {p_{L}}\).

Terms in the \(\mathrm {E_{c}}\) equation proposed by different authors

Author(s) | Year | Vaporization | Surface formation | Expansion | Kinetic energy | See also Refs. |
---|---|---|---|---|---|---|

Pless and Plano [9] | 1956 | \(\frac{4}{3}\pi \mathrm {R_{c}}^{3}\mathrm {\rho _{V}}\mathrm {\lambda }\) | \(4\pi \mathrm {R_{c}}^{2}\mathrm {\sigma }\) | \(\frac{4}{3}\pi \mathrm {R_{c}}^{3}\mathrm {p_{L}}\) | – | |

Seitz [8] | 1958 | \(\frac{4}{3}\pi \mathrm {R_{c}}^{3}\mathrm {\rho _{V}}\mathrm {\lambda }\) | \(4\pi \mathrm {R_{c}}^{2}\mathrm {\sigma }\) | – | – | |

Bugg [10] | 1959 | \(\frac{4}{3}\pi \mathrm {R_{c}}^{3}\mathrm {\rho _{V}}\mathrm {\lambda }\) | \(4\pi \mathrm {R_{c}}^{2}(\mathrm {\sigma }-\frac{\mathrm {d}\mathrm {\sigma }}{\mathrm {d}\mathrm {T}}\mathrm {T_{L}})\) | \(-\frac{4}{3}\pi \mathrm {R_{c}}^{3}(\mathrm {p_{V}}-\mathrm {p_{L}})\) | – | |

Norman and Spiegler [11] | 1963 | \(\frac{4}{3}\pi \mathrm {R_{c}}^{3}\mathrm {\rho _{V}}\mathrm {\lambda }\) | \(4\pi \mathrm {R_{c}}^{2}(\mathrm {\sigma }-\frac{\mathrm {d}\mathrm {\sigma }}{\mathrm {d}\mathrm {T}}\mathrm {T_{L}})\) | – | \(2\pi \mathrm {\rho _{L}}\mathrm {R_{c}}^{3}\mathrm {v_{r}}^{2}\) | |

Tenner [12] | 1963 | \(\frac{4}{3}\pi \mathrm {R_{c}}^{3}\mathrm {\rho _{V}}\mathrm {\lambda }\) | \(4\pi \mathrm {R_{c}}^{2}(\mathrm {\sigma }-\frac{\mathrm {d}\mathrm {\sigma }}{\mathrm {d}\mathrm {T}}\mathrm {T_{L}})\) | \(\frac{4}{3}\pi \mathrm {R_{c}}^{3}(1-\frac{\mathrm {\rho _{V}}}{\mathrm {\rho _{L}}})\mathrm {p_{L}}\) | – | |

Peyrou [13] | 1967 | \(\frac{4}{3}\pi \mathrm {R_{c}}^{3}\mathrm {\rho _{V}}\mathrm {\lambda }\) | | \(\frac{4}{3}\pi \mathrm {R_{c}}^{3}\mathrm {p_{L}}\) | – | |

Bell et al. [14] | 1974 | \(\frac{4}{3}\pi \mathrm {R_{c}}^{3}\mathrm {\rho _{V}}\mathrm {\lambda }\) | \(4\pi \mathrm {R_{c}}^{2}\mathrm {\sigma }\) | \(-\frac{4}{3}\pi \mathrm {R_{c}}^{3}(\mathrm {p_{V}}-\mathrm {p_{L}})\) | \(2\pi \mathrm {\rho _{L}}\mathrm {R_{c}}^{3}\mathrm {v_{r}}^{2}\) |

All in all, the equation proposed by Bugg [10] (third line of Table 1), which includes the subtractive expansion term, disregarding at the same time the kinetic energy term, seems to be essentially equivalent to the proposed equation (16), at least as long as the values of the specific internal energy variation and the mass density of the saturated vapour calculated at temperature \(\mathrm {T_{L}}\) are not too different from those calculated at temperature \(\mathrm {T_{V}}\), i.e., the superheat degree is not too high.

## 3 Radius of the critically-sized nucleated vapour bubble

Thus, a more realistic approach is required which should be able to reflect that the critical size represents a condition of absolute instability for the vapour bubble. In fact, should the critically-sized vapour bubble lose just a tiny amount of matter, say one molecule, which gets back to be part of the surrounding liquid, then the bubble will literally implode, vanishing, due to the loss of the mechanical equilibrium. Conversely, should the critically-sized vapour bubble gain just a tiny amount of matter, taken away from the surrounding liquid, then the bubble will spontaneously grow, becoming detectable.

## 4 Discussion

First of all, it is worth observing that the procedure followed to obtain (29) by determining \(\,\mathrm {\Delta }\mathrm {H}(\mathrm {r})\) and \(\,\mathrm {\Delta }\mathrm {S}(\mathrm {r})\), and then substituting their expressions in (19), intrinsically demonstrates the validity of (16). In fact, should the heat injection required to nucleate a vapour bubble have been derived from a relationship different from (16), then a relationship different from (29) would have been achieved for the critical radius \(\mathrm {R_{c}}\), and neither (18) nor (17) could have been obtained for low degrees of metastability.

Furthermore, it must be pointed out that the calculation of the critical radius \(\mathrm {R_{c}}\) by the way of (17) or (18) leads to values lower than that expressed by (29), which is a direct consequence of the fact that, since the vapour pressure curve is concave upwards, the temperature derivative of the saturation pressure at temperature \(\mathrm {T_{V}}\) is lower than the corresponding increment ratio \((\mathrm {p_{V}}-\mathrm {p_{L}})/(\mathrm {T_{L}}-\mathrm {T_{V}})\). Of course, the discrepancy increases as the degree of metastability is increased, as shown in Fig. 6, in which a number of distributions of the relative difference \(\mathrm {\delta _{R}}=(\mathrm {R_{c}}-\mathrm {R_{c}^{*}})/\mathrm {R_{c}}\) between the results obtained applying (18) instead of (29) are plotted against the superheat degree \(\,\mathrm {\Delta }\mathrm {T}\) for \(\hbox {C}_{{3}}\hbox {F}_{{8}}\) using the liquid temperature \(\mathrm {T_{L}}\) as a parameter, where \(\mathrm {R_{c}}\) and \(\mathrm {R_{c}^{*}}\) are the values of the critical radius given by (29) and (18), respectively. Even higher discrepancies are obtained if (17) is applied rather than (18).

Accordingly, the critical energy obtained through (16) in which \(\mathrm {R_{c}}\) is calculated by (29) is higher than the critical energy derived applying, for example, the equation proposed by Bugg [10] using (18) to calculate \(\mathrm {R_{c}}\).

A set of distributions of the relative difference \(\mathrm {\delta _{E}}= (\mathrm {E_{c}}- \mathrm {E_{c}^{*}})/\mathrm {E_{c}}\) between the results obtained applying the Bugg’s equation in combination with (18), instead of (16) in combination with (29), are plotted in Fig. 7 against the superheat degree \(\,\mathrm {\Delta }\mathrm {T}\) for \(\hbox {C}_{{3}}\hbox {F}_{{8}}\) using the liquid temperature \(\mathrm {T_{L}}\) as a parameter, where \(\mathrm {E_{c}}\) and \(\mathrm {E_{c}^{*}}\) are the values of the critical energy given by (16) and by the Bugg’s equation, respectively. It is apparent that when the degree of metastability of the superheated liquid is high enough, the relative difference between the two values becomes significantly high, almost regardless of the liquid temperature \(\mathrm {T_{L}}\).

Finally, it seems interesting to examine the theoretical prediction of the combination of (16) and (29) using experimental data available in the literature. To this end, we chose an experimental result recently obtained for liquid Xenon, which seems to be more indicated than the other usual target liquids to test a novel critical energy theoretical equation. In fact, at any recoil energy, an ion of Xenon travelling in pure liquid Xenon has a stopping force quite higher than, for example, that of \(^{12}\)C or \(^{19}\)F in liquid \(\hbox {C}_{{3}}\hbox {F}_{{8}}\), which means that the additional condition for bubble nucleation requiring that the critical energy must be released inside a volume of characteristic dimension \(\mathrm {R_{c}}\) is more accurately satisfied. On the other hand, the use of a single-atom target eliminates the uncertainties on how to account for the relative contribution of the ions of a multi-atom molecule in determining the threshold.

The cited threshold measurement was performed by Baxter et al. [38] using a 30-g Xenon bubble chamber operated at 30 psia and − 60 \(^{\circ }\)C, whose corresponding critical energy calculated by the Bugg’s equation in combination with (18) would be 8.3 keV. Indeed, the observed single and multiple bubble rates consequent to a 3.1 h exposure to a \(^{252}\)Cf neutron source were consistent with the absolute rates predicted by a Monte Carlo simulation of the equipment executed using the MCNPX-POLIMI package assuming that the minimum nuclear recoil energy required to nucleate a vapour bubble was \(19 \pm 6\) keV where, according to the authors, the range was dominated by the 30% uncertainty in their source strength. Conversely, the application of the relationships proposed for the calculation of \(\mathrm {E_{c}}\) and \(\mathrm {R_{c}}\), i.e., (16) and (29), results in a critical energy equal to 20.2 keV, with an uncertainty that could tentatively be assumed to be of the same order cited earlier.

Although we are aware that only reversible processes are considered in the calculation of the thermodynamic energy threshold, the difference between the measured value of \({19\pm 6}\) keV and the predicted value of 8.3 keV seems too high to be interpreted as the energy which goes into irreversible processes, even including the energy losses to scintillation. In fact, according to the measurements performed on liquid Xenon by different research teams – see for example Akerib et al. [39] – the scintillation yield for nuclear recoils between 1 and 100 keV is widely lower than 15%, which lies well inside the indicated 30% uncertainty range. Of course, the comparison with the Xenon result is not an evidence for our calculation, yet it represents a concordance with the earlier statement that our approach may be regarded as an upper theoretical limit of the thermodynamic energy threshold, encouraging enough to lead us to schedule future investigations on this same topic.

## 5 Conclusions

The relationships currently available for the calculation of the critical energy required for homogeneous nucleation in a superheated liquid, \(\mathrm {E_{c}}\), and the corresponding critical radius of the nucleated vapour bubble, \(\mathrm {R_{c}}\), show a number of inconsistencies, which has motivated the present study. Based on the procedure followed to obtain them, the pair of equations proposed here for the calculation of \(\mathrm {E_{c}}\) and \(\mathrm {R_{c}}\) turn out to be more consistent with the physical facts, the first being based on the application of the first law of thermodynamics, the second being derived under the assumption that the extreme instability condition represented by the critically-sized vapour bubble must correspond to a maximum of the difference between the free enthalpies of the metastable liquid and the stable vapour phases. An encouraging good agreement has been found between our theoretical prediction and an experimental result recently reported for Xenon at 30 psia and − 60 \(^{\circ }\)C. Further investigations on this topic are scheduled to be conducted in the next future.

## Notes

### Acknowledgements

The authors are grateful to Donald Cundy for the valuable discussions and suggestions and for his help in reviewing the manuscript.

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