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1 - Beta decay in a few words
a) Discovery and the Fermi theory
The first type of radioactivity to have been observed as early as 1900, beta radioactivity was the most difficult to explain amoung the three alpha, beta and gamma radioactivities. For nearly 30 years, it was an enigma for the theories of the time, which regularly came up against new experimental observations that failed to describe it. Unlike the radioactivities α and γ, the beta particles produced were not monoenergetic but were distributed along an energy spectrum that challenged the law of energy conservation among others.
This issue had been solved at the beginning of the thirties with the discovery of the neutron and of the neutrino. In the case of beta minus decay, for example, a neutron in a mother nucleus could change into a proton in the daughter nucleus produced and this was accompanied by the emission of two particles: the beta particle (the electron) and an antineutrino. Fermi could now build his theory and calculate the shape of the energy spectrum of the electrons (or antineutrinos produced). With a few approximations, this could be written as follows:
N(p) α p2(Qβ-Te)2F(Z',p)|Mfi|2C
the product of a phase space term (involving the momentum p and the kinetic energy Te of the electron and the Qβ of the decay), with the Fermi function, F, taking into account the effect of the coulomb field of the daughter nucleus (atomic number Z') on the particles produced, and with an interaction term between the initial and final states, involving the nuclear matrix elements Mfi. C is an additional corrective term which has been refined many times since the original Fermi theory. It belongs to the physics topics studied by the SEN group (see e-shape, summation, structure).
Integrating N(p) over the whole possible momenta taken by the emitted electrons gives access to the transition rate, λ, the radioactive decay constant representing the probability of decay, per unit of time, of a radioactive nucleus. The half life (t1/2) of a nucleus being related to lambda by the equation t1/2=ln2/λ, on can then calculate its half life.
b) Why is the study of beta decay interesting?
The process of beta decay is involved in a wide range of fields including physics, nuclear data, chemistry and medicine. The SEN group, through the measurement of certain properties of beta decay and in particular those of fission products, has built up a coherent research activity enabling it to build bridges between several disciplines of fundamental and applied physics. The following decay scheme illustrates the beta minus decay of a mother nucleus AZX towards the AZ+1Y nucleus AZ+1Y, AZX -> AZ+1Y + e- + anti-ν :
|Figure 1 : Beta minus decay scheme of a mother nucleus into a daughter nucleus|
The experimental determination of the beta feeding probabilities, Iβ, of each beta branch and of the energies of the different levels of excitation of the daughter nuclei (the endpoints), allows the group to make the link between its different research fields of study :
- For nuclear structure and nuclear astrophysics, our anchor point lies in the determination of the beta force defined as follows :
- For fundamental and applied neutrino physics, we develop summation calculations using Iβ and endpoints. The same is true for decay heat calculations for nuclear reactors.
- For nuclear data, our measurements allow us to correct or complete the decay data already listed. Furthermore, summation calculations are also of interest to give constraints to fission yield databases and fission models. These calculations are also of interest to constrain the available experimental data on the properties of delayed neutrons.
a) Pandemonium effect and bias in nuclear decay databases
As explained in the section Reactor Neutrino Applications, Residual Power and Summation Calculations, the β- decay properties of fission products are important for simulating the spectrum of antineutrinos emitted by reactors and for calculating the decay heat of shutdown reactors, but the study of these quantities is subject to the accuracy of existing databases which are affected by the so-called "Pandemonium" effect . This effect is due to the fact that the most widely used technique for estimating beta decay properties uses Germanium (Ge) detectors which have an efficiency that falls exponentially with gamma energy, which can lead to an underestimation of the beta decay branches towards the higher energy excited levels in the daughter nucleus. The problem is shown in figure 2: if the Ge detector does not see the high energy gamma called γ1 the feeding that was supposed to go to the higher energy level will be attributed to a lower energy level.
|Figure 2 (from A. Algora) : (left) Illustration of the Pandemonium effect on the energy levels fed into the daughter nuclei. (right) Difference in measurement of a de-excitation cascade between a germanium detector and an TAS.|
By way of illustration, figure 3 shows an energy spectrum of the antineutrinos emitted during the beta decay of 105Mo (calculations performed using the summation method developed by the group).
|Figure3 : Energy spectrum of the antineutrinos emitted during the beta decay of 105Mo. Comparison between the JEFF3.1 database and the data from the TAS experiment.|
The dashed curve shows the spectrum calculated from the decay data present in the JEFF3.1 nuclear database from measurements made with a high-resolution Ge detector. The solid line curve shows the same calculation based on a calorimetric measurement described below performed by the group that allows correction for the Pandemonium effect. At the scale of a given isotope and therefore of a fission product, it can be seen that the mean value of the distribution is almost halved as a result of the correction and the shape of the distributions is also greatly modified. This has a direct impact on nuclear databases, summation calculations for neutrinos and decay heat and beta forces for fundamental nuclear physics.
b) Total Absorption Spectroscopy (TAS) and collaboration
The solution to this problem is the Total Absorption Spectroscopy (TAS) method, which is based on the detection of de-excitation gammas through the use of a high efficiency detector covering a solid angle of almost 4π all around the selected isotope. This method allows direct access to the energy of the excited levels populated by the beta decay in the daughter nucleus. This technique is complementary to the use of Germanium (Ge) detectors for nucleus spectroscopy since it provides high detection efficiency but with lower resolution.
An experimental project started in January 2009, with the participation of members of the SEN group and a team from Valencia (J.-L. Tain and A. Algora) and Surrey (William Gelletly) in a meeting organised by the IAEA in Vienna on the possibility of improving the calculation of the decay heat by re-measuring with the TAS (Total Absorption Spectroscopy) method some selected nuclei. This meeting was the starting point of the collaboration between the two groups to identify and measure fission products of interest for decay heat, antineutrinos and, in most cases, both.
Within the framework of this project, in November 2009, part of the group participated in beta decay measurements of interesting nuclei selected by the Valencia research team and the Subatech team. These measurements were carried out at the accelerator in Jyväskylä (Finland) using the TAS method with a segmented detector composed of 12 BaF2 crystals called Rocinantes (figure 4 - left).
|Figure 4 : The Rocinantes detector (left). The DTAS detector (right).|
A second experiment was carried out in February 2014 again at the Jyväskylä accelerator. During this experiment, 23 nuclei of interest for the calculation of the antineutrino spectrum and the decay heat of reactors were measured . The experiment was performed with a new segmented detector, the DTAS, composed of 18 NaI crystals (Figure 4 - right) .
Before 2009, our Valencian colleagues had developed a TAS detector consisting of a single large NaI crystal (Lucrezia) installed at the ISOLDE accelerator at CERN. The advantage of having then designed segmented detectors opened the possibility of studying the multiplicity of detected gammas of interest in particular for the nuclear structure. The SEN group is thus interested in studying the presence of possible pygmy resonances, the signature of a neutron skin.
The analysis of the TAS data, gives us direct access to the beta feeding to each level of the daughter nucleus by solving the inverse problem d=R*f, where d represents the data, R is the response matrix of the detector and f is the beta feeding that we are trying to measure. To do this we need clean data without contamination and background noise; an accurate knowledge of our measurement system obtained through simulation and minimal knowledge of the properties of the nucleus to calculate the matrix R. For this reason, the TAS method is complementary to the already existing measurements obtained with Ge detectors, since it builds on and complements the knowledge already gained from these measurements.
Through the feeding, the TAS technique gives access to the beta intensity and, therefore, to the beta force which is a microscopic observable that theoretical models can calculate, thus opening a window on the structure of the nucleus. TAS measurements will provide new constraints useful for improving the predictability of models, which are indispensable in other fields of physics such as nuclear astrophysics in nucleosynthesis calculations.
The TAS collaboration has published many results from its two campaigns of experiments in 2009 and 2014, as an illustration, see the articles [4,6]. And the SEN team is a leader in the study of the impact of these new measurements on the calculation of the antineutrino spectrum [7,8].
Recently the SEN team has applied for a grant from the Pays de la Loire Region and made a request to ANR to improve the existing DTAS with one or more crowns of LaBr3, this is the (NA)2STARS project. These new detectors, which combine high efficiency and good energy resolution, will make it possible to measure more exotic and poorly known nuclei, inaccessible to current TAS detectors.
Half of the elements with mass greater than 70 is created by the astrophysical r process, which proceeds via unstable very neutron-rich nuclei in stellar explosions or other violent astrophysical events such as the coalescence of neutron stars. The identification of the location of the r-process remains one of the major challenges of nuclear astrophysics. Recent advances in the description of the interaction of neutrinos with matter and its implementation in the modelling of supernovae explosions tend to show that supernovae explosions only contribute to the production of Z elements below 50. Compact star coalescences are currently considered to be the best candidates for determining the site of the main r-process. Observational confirmation of the occurrence of the r-process in such an astrophysical event was given at the end of 2017 with the multi-messenger detection of the merging of two neutron stars and the measurement of the emitted electromagnetic radiation. Beta decay of nuclei is among the important processes that influence r-element abundances along with neutron capture reactions, photo-dissociation, temperature and density. In particular, measurements of radioactive periods of the progenitors of stable nuclei help to determine their abundance and are thus important ingredients in nucleosynthesis calculations that attempt to reproduce them. With the exception of a few key nuclei that are relatively close to the valley of stability, half-lives must be calculated using theoretical models. In this type of calculation, the strength distribution associated with beta decay as a function of energy must be calculated for all possible end states. This involves determining what proportion of the strength resides in the energy window opened by the beta decay.
The TAGS technique is the preferred technique to perform these measurements, and can also be used to study the presence of low-energy collective modes that influence the paths of the nucleosynthesis process r. The TAGS measurements proposed by the Subatech team, in international collaboration with the IFIC in Valencia, Spain, CIEMAT in Madrid, the University of Surrey in England (and other international laboratories that are coming to participate in the experiments), at state-of-the-art facilities for the production of exotic nuclei such as JYFL in Finland, or ISOLDE at CERN (among others) will provide the necessary tests for the theoretical models.
In a PWR-type reactor, the total number of fissions that occur as a function of time comes essentially from the fission of the 4 isotopes 235U, 238U initially present in the fuel and 239Pu and 241Pu produced by neutron capture and successive beta decay of 238U. Figure 5, which shows this evolution as a function of time (the reactor burn-up), shows that 235U is the main contributor to this fission rate. The increase in the production of 239Pu over time also strongly increases the associated fission rate. The resulting fission products, which are neutron-rich nuclei, are then de-excited by beta or beta-n decay (with the emission of a beta-delayed neutron) and are therefore the source of the considerable flux of antineutrinos that comes out of nuclear reactors.
|Figure 5 : Evolution of the fission rate per second in a PWR-type reactor.|
The beta decay properties of neutron-rich nuclei, in particular of fission products play a major role in the estimation of observables important for nuclear reactor safety as well as neutrino physics, whether fundamental or applied. Recently, the reactor anomaly - a significant discrepancy between predictions made from recent estimates of antineutrino energy spectra and fluxes measured by neutrino experiments within 100m of a reactor core - has attracted the attention of the particle physics and nuclear physics communities . This issue has led to many new neutrino physics experiments in the vicinity of research reactors around the world including the Solid experiment in which the group is involved.
A better knowledge of the spectrum of reactor antineutrinos is required, and for this, a mastery of the underlying nuclear physics is essential. Nuclear physics plays a major role in the two spectrum calculation methods currently under study and the Subatech team is an international leader in the prediction of antineutrino spectra based on the summation method from nuclear data [7,8] in a highly competitive context. For a given isotope, the summation method makes it possible to calculate the expected total energy spectrum of antineutrinos produced in a nuclear reactor. Based on the knowledge of the decay data of individual nuclei, one can first calculate the expected individual antineutrino spectra for the approximately 800 fission products of a given isotope using Fermi theory. The summation method then consists of summing all these individual energy spectra weighted by their fission yield from the corresponding nuclear databases.
Our most recent predictions, of unequalled quality thanks to Total Absorption Gamma-ray Spectroscopy (TAGS) measurements carried out over the last decade, have been published in a high-impact journal , a viewpoint on a PRL of the Daya Bay collaboration on the topic  and in an article in the CNRS/in2p3 newsletter ("Deficit of antineutrinos produced in reactors: nuclear physics provides elements of an answer", with Muriel Fallot's team). These predictions have recently been taken up again in a new publication  to study their global impact on neutrino physics.
The aim of our TAGS experiments and instrumental projects (e-Shape, (NA)2STARS) is to provide a definitive answer to the questions associated with the emission of antineutrinos from reactors. Finally, a common issue with the study of antineutrinos is the study of fission products at the origin of the decay heat of the reactors after their shutdown. This decay heat represents about 7% of the nominal power of a reactor and must be evacuated or else the fuel cladding will melt (cf. the Fukushima accident in 2011). To date, significant differences remain between integral measurements made in experiments within reactors and calculations using nuclear data, even for the decay heat associated with the most common fissile isotopes such as 235U and 239Pu. Our TAGS experiments are aimed to improve significantly this situation in order to reduce the actual uncertainties and obtain reliable decay heat predictions for innovative reactors.
 J. C. Hardy, L. C. Carraz, B. Jonson, and P. G. Hansen, Phyics Letter B 71 (1977) 307.
 V. Guadilla et al., First experiment with the NUSTAR/FAIR Decay Total Absorption γγ-Ray Spectrometer (DTAS) at the IGISOL IV facility, Nucl. Inst. and Meth. B 376 (2016) 334.
 V. Guadilla et al., Characterization and performance of the DTAS detector, Nucl. Instrum. Meth. A 910 (2018) 79-89.
 A.-A. Zakari-Issofou et al., Total Absorption Spectroscopy Study of 92Rb Decay: A Major Contributor to Reactor Antineutrino Spectrum Shape, Phys. Rev. Lett. 115, 102503 (2015);
 V. Guadilla et al., Total absorption γ-ray spectroscopy of the β-delayed neutron emitters 137I and 95Rb, Phys. Rev. C 100 (4) (2019) 044305.
 V. Guadilla et al., Large Impact of the Decay of Niobium Isomers on the Reactor ¯νe Summation Calculations, Phys. Rev. Lett. 122 (2019) 042502.
 M. Fallot et al., New Antineutrino Energy Spectra Predictions from the Summation of Beta Decay Branches of the Fission Products, Phys. Rev. Lett. 109, 202504 (2012).
 M. Estienne et al., Updated Summation Model: An Improved Agreement with the Daya Bay Antineutrino Fluxes, Phys. Rev. Lett. 123, (2019) 022502.
 G. Mention et al., The Reactor Antineutrino Anomaly, Phys. Rev. D 83 (2011) 073006.
 M. Fallot, Viewpoint: Getting to the Bottom of an Antineutrino Anomaly, Physics 10, 66 (2017).
 J. Berryman and P. Huber, Reevaluating Reactor Antineutrino Anomalies with Updated Flux Predictions, Phys. Rev. D 101 (2020) 015008, arXiv:1909.09267.