Determination of Beta Decay Ground State Feeding of Nuclei of Importance for Reactor Applications
V. Guadilla, J. L. Tain, A. Algora, J. Agramunt, D. Jordan, M. Monserrate, A. Montaner-Pizá, S. E. A. Orrigo, B. Rubio, E. Valencia, J. A. Briz, A. Cucoanes, M. Estienne, M. Fallot, L. Le Meur, A. Porta, T. Shiba, A. -A. Zakari-Issoufou, J. Äystö, T. Eronen, D. Gorelov, J. Hakala, A. Jokinen, A. Kankainen, V.S. Kolhinen, J. Koponen, I.D. Moore, H. Penttilä, I. Pohjalainen, J. Reinikainen, M. Reponen, S. Rinta-Antila, K. Rytkönen, V. Sonnenschein, A. Voss, L. M. Fraile, V. Vedia, E. Ganioğlu, W. Gelletly, M. Lebois, J. N. Wilson, T. Martinez, E. Nácher, A. A. Sonzogni
aa r X i v : . [ nu c l - e x ] M a y Determination of Beta Decay Ground State Feeding of Nuclei of Importance forReactor Applications
V. Guadilla, ∗ J. L. Tain, A. Algora, † J. Agramunt, D. Jordan, M.Monserrate, A. Montaner-Piz´a, S. E. A. Orrigo, B. Rubio, and E. Valencia
Instituto de F´ısica Corpuscular, CSIC-Universidad de Valencia, E-46071, Valencia, Spain
J. A. Briz, A. Cucoanes, M. Estienne, M. Fallot, L. Le Meur, A. Porta, T. Shiba, and A. -A. Zakari-Issoufou
Subatech, IMT-Atlantique, Universit´e de Nantes, CNRS-IN2P3, F-44307, Nantes, France
J. ¨Ayst¨o, T. Eronen, D. Gorelov, J. Hakala, A. Jokinen, A. Kankainen, V.S. Kolhinen, ‡ J. Koponen, I.D. Moore, H.Penttil¨a, I. Pohjalainen, J. Reinikainen, M. Reponen, S. Rinta-Antila, K. Rytk¨onen, V. Sonnenschein, § and A. Voss University of Jyv¨askyl¨a, FIN-40014, Jyv¨askyl¨a, Finland
L. M. Fraile and V. Vedia
Universidad Complutense, Grupo de F´ısica Nuclear, CEI Moncloa, E-28040, Madrid, Spain
E. Ganio˘glu
Department of Physics, Istanbul University, 34134, Istanbul, Turkey
W. Gelletly
Department of Physics, University of Surrey, GU2 7XH, Guildford, UK
M. Lebois and J. N. Wilson
Institut de Physique Nucl`eaire d’Orsay, 91406, Orsay, France
T. Martinez
Centro de Investigaciones Energ´eticas Medioambientales y Tecnol´ogicas, E-28040, Madrid, Spain
E. N´acher
Instituto de Estructura de la Materia, CSIC, E-28006, Madrid, Spain
A. A. Sonzogni
NNDC, Brookhaven National Laboratory, Upton, NY 11973-5000, USA (Dated: May 19, 2020)In β -decay studies the determination of the decay probability to the ground state of the daughternucleus often suffers from large systematic errors. The difficulty of the measurement is related to theabsence of associated delayed γ -ray emission. In this work we revisit the 4 πγ − β method proposedby Greenwood and collaborators in the 1990s, which has the potential to overcome some of theexperimental difficulties. Our interest is driven by the need to determine accurately the β -intensitydistributions of fission products that contribute significantly to the reactor decay heat and to theantineutrinos emitted by reactors. A number of such decays have large ground state branches. Themethod is relevant for nuclear structure studies as well. Pertinent formulae are revised and extendedto the special case of β -delayed neutron emitters, and the robustness of the method is demonstratedwith synthetic data. We apply it to a number of measured decays that serve as test cases anddiscuss the features of the method. Finally, we obtain ground state feeding intensities with reduceduncertainty for four relevant decays that will allow future improvements in antineutrino spectrumand decay heat calculations using the summation method. ∗ Present address: Faculty of Physics, University of Warsaw, 02-093 Warsaw, Poland † Institute of Nuclear Research of the Hungarian Academy of Sci-ences, Debrecen H-4026, Hungary ‡ Present address: Cyclotron Institute, Texas A&M University,College Station, Texas 77843, USA § Present address: Faculty of Engineering, Nagoya University,Nagoya 464-8603, Japan
I. INTRODUCTION
The measurement of ground state (g.s.) β -decay feed-ing probabilities is hampered by the absence of associated γ radiation. In β − decays the energy released is sharedbetween the electron and the antineutrino leading to con-tinuous energy distributions, extending from zero to themaximum decay energy Q β . This makes it difficult todetermine precisely the number of β particles. The prob-lem arises because of the difficulty of disentangling thefeatureless continuum associated with all of the decays toexcited states from that to the g.s. through β -spectrumdeconvolution. In addition, electrons are easily absorbedor scattered by any material surrounding the β detector,and this effect must be properly taken into account inthe response function of the β detector. This explainswhy β -decay probabilities to the g.s. are often obtainedindirectly.The most common approach is to determine both thetotal number of decays and the number of decays pro-ceeding to excited states, since the difference is due todecays to the g.s.. Usually the total number of decaysis measured using a β detector and the decays to ex-cited states are obtained from high-resolution (HR) γ -ray spectroscopy and conversion electron spectroscopy tobuild the decay level scheme. Assigning the correct inten-sity for the decay to the g.s. is equivalent to determiningabsolute γ intensities. The limited efficiency of HPGedetectors results in many weak transitions from levelsat high-excitation energy remaining undetected, the so-called Pandemonium effect [1]. This shifts the appar-ent β intensity, obtained from the intensity balance ateach level, to levels at low-excitation energies. This initself is not the real problem for g.s. feeding determina-tion but the fact that part of the missed transitions canfeed the g.s. directly, thus introducing a systematic er-ror in the determination of absolute γ intensities. In astrict sense the g.s. feeding probabilities obtained by thismethod should be considered as upper limits.Greenwood and collaborators [2] proposed the useof the Pandemonium -free total absorption γ -ray spec-troscopy (TAGS) technique [3] in combination with a β detector to determine accurately the β intensity to theg.s., in a way that will be explained later. The method,termed the 4 πγ − β method, was applied subsequentlyto determine the g.s. feeding probabilities for 34 fissionproducts (FP) [4, 5].The TAGS technique aimed initially at the determi-nation of (relative) β intensities to excited states. It re-lies on the use of large close-to-4 π γ calorimeters madewith scintillation material to detect the full de-excitation γ cascade rather than the individual transitions. Anideal total absorption spectrometer would have 100% γ -cascade detection efficiency and should be insensitive to β particles. It turns out that the TAGS technique can beused to extract the g.s. β intensity directly as is explainedbelow.The first spectrometer designs emphasized the condi- tion of insensitivity to β particles, either by placing a β detector outside the spectrometer [6] or placing a low-Zabsorber material behind the β detector [7–9] to minimizethe penetration of the electrons or their bremsstrahlungradiation (in short β penetration) in the scintillation vol-ume. This had the undesirable effect of reducing the γ -peak detection efficiency. However, the total detec-tion efficiency for γ cascades of multiplicity 2 or higherremains close to 100% if the solid angle coverage is rea-sonably close to 4 π . The rationale behind these initialdesigns is that β penetration distorts the spectrometer γ response by introducing a high energy tail. The spec-trometer response to decays is needed in the TAGS anal-ysis of real spectrometers to deconvolute the measuredspectrum [10]. The response must be obtained by MonteCarlo (MC) simulations [11] and the consensus at thattime was that an accurate simulation of β -particle in-teractions is more difficult than the simulation of γ in-teractions. Modern MC simulation codes like Geant4[12], have greatly improved the description of low-energyelectron interactions and provide a variety of trackingparameters to optimize the simulation and improve theaccuracy (see for example [13]). Thus the newest spec-trometer designs [14–16] do not make any special effortto minimize β penetration and have a sizable responseto g.s. decays. In this way the deconvolution of TAGSspectra provides, in a natural manner, the intensity of de-cays to the g.s. The first example of this was the decayof Tc for which the TAGS analysis [17, 18] confirmedthe value of 92 . β penetration in the spectrometer (seediscussion in Ref. [21]) 2) the indeterminacy that canarise for particular decay intensity distributions, as willbe shown later for the case of Tc, 3) the loss of sen-sitivity with decreasing g.s. β intensity, 4) the difficultyof separating transitions to states at very low excitationenergy, due to the limited energy resolution, and 5) theproper quantification of the systematic uncertainties.In the present work we revisit the 4 πγ − β method,which, as will be seen, is essentially free from problems1) and 2) listed before and has different systematic un-certainties from the TAGS analysis. These differencesmainly arise from the integral character of the 4 πγ − β method (that uses the total number of counts in the spec-tra) that contrasts with the TAGS deconvolution, sensi-tive to the features of the shape of the experimental spec-trum. As will be shown later, this minimizes the effect ofthe lack of knowledge on the precise de-excitation paths(a relevant source of systematic uncertainty in the TAGSanalyses) in the results of the 4 πγ − β method.Our interest in this topic arises from the importanceof g.s. feeding probabilities for nuclear structure stud-ies and reactor applications. In particular, it was re-cently renewed by the need to obtain accurate antineu-trino energy spectra emitted by fission products (FP)using the β -intensity distributions I β ( E x ) to weight theindividual ¯ ν e spectra for each β end-point Q β − E x .This is the basis of the summation method to obtainthe spectrum of antineutrinos emitted by a nuclear reac-tor [28], which is calculated by weighting the spectrumfor each FP by the cumulative (or evolved individual) fis-sion yield and the contribution of each fissile isotope. Asit happens a number of fission products of significance informing the reactor antineutrino spectrum have a strongor very strong g.s. decay branch [29]. Some of themare Rb (95.2(7)%),
Y (95.5(5)%),
Cs (56(5)%),
Nb (50(7)%),
Cs (35.9(17)%) or Rb (35(3)%).Here the quoted g.s. feeding probability in brackets iscoming from the Evaluated Nuclear Structure Data File(ENSDF) [30]. The importance of direct measurementsof the g.s. feeding and the impact on antineutrino spec-trum calculations can be illustrated with the example of Rb, the top contributor above E ¯ ν e = 5 MeV [20, 29],with Q β = 8 . . γ -rayintensities [34]. From the deconvolution of the measuredTAGS spectrum we obtained a value of 87 . Pandemonium -free value obtained with the TAGS tech-nique was reported [20, 22].An accurate knowledge of the antineutrino spectrum iskey to the analysis of reactor antineutrino oscillation ex-periments. The standard method to obtain this spectrumis to apply a complex conversion procedure to integral β spectra for each of the main fissile isotopes in a reactor[35]. A re-evaluation of the conversion procedure [36, 37]led to the discovery of a deficit of about 6% between ob-served and estimated antineutrino fluxes [38]. This wastermed the reactor antineutrino anomaly. Whether itindicates the existence of sterile neutrinos is a topic ofvery active investigation [39]. The summation methodallows an exploration of the origin of the anomaly froma different perspective, and recently it was shown thatthe consistent inclusion of our newest TAGS decay datareduces the discrepancy with the measured flux to thelevel of the estimated uncertainties [40]. On the otherhand the high statistics spectrum of detected antineutri-nos obtained by the Daya Bay collaboration [41] showswithout doubt shape deviations with respect to the con-verted spectra. Such shape deviations were also seen bythe Double Chooz [42] and Reno [43] collaborations. Theorigin of this shape distortion is unclear but the use ofsummation calculations allows one to explore a numberof possibilities [44]. Moreover, fine structure has been ob-served in the Daya Bay spectra that has been ascribed to a few nuclear species with large g.s feeding on the basis ofsummation calculations [45]. This opens the unlooked-for possibility of doing reactor ¯ ν e spectroscopy.There is a related application in which the role of g.s.feeding values is also of great relevance: the evaluation ofthe energy released in nuclear reactors by the radioactivedecay of the FP, known as decay heat. The decay heatrepresents the dominant source of energy when a reac-tor is powered off and its proper determination is essen-tial for safety reasons. It is usually evaluated by meansof summation calculations that use the same ingredientsmentioned above: fission yields, β -intensity distributionsand β spectra as a function of end-point energies to com-pute the evolution of the reactor decay heat with time.Some important decays for the determination of the re-actor decay heat exhibit relevant g.s. β branches (manyof them are common cases with the reactor antineutrinospectrum explained before). The accurate determinationof the decay heat is thus constrained by the availabilityof reliable g.s. feeding probability values.This paper is organized as follows. The 4 πγ − β methodis presented in Section II, including a correction of theoriginal formulae in Ref. [2]. In addition a modificationof the formulae for the case of β -delayed neutron emittersis introduced. In the Appendix we provide a demonstra-tion of the method using synthetic data generated byrealistic MC simulations. The 4 πγ − β method is appliedto TAGS data taken at the Accelerator Laboratory ofthe University of Jyv¨askyl¨a (JYFL-ACCLAB) in SectionIII. It gives a summary of relevant experimental detailsand presents the results obtained, first for a number ofrelevant test cases and then for a number of isotopes con-tributing significantly to reactor antineutrino spectra anddecay heat. The g.s. intensities obtained are comparedwith the TAGS deconvolution results and the literature.The last Section summarizes the conclusions. II. THE πγ − β METHOD
The method is based on a comparison of the numberof counts detected in the β detector N β and the numberof counts registered in coincidence in both the β detectorand the total absorption spectrometer N βγ . These canbe written in terms of the number of decays f i feedinglevel i , with i = 0 representing the ground state, and theyare related to the β intensity I iβ and the total number ofdecays N d : N β = ε β f + X i> ε iβ f i = ε β I β N d + X i> ε iβ I iβ N d N βγ = ε βγ f + X i> ε iβγ f i = ε βγ I β N d + X i> ε iβγ I iβ N d (1) ε iβ is the probability of detecting a signal in the β de-tector for decays to level i and ε iβγ the probability ofregistering simultaneously signals in the β detector andthe total absorption γ -ray spectrometer. As seen in Eq. 1we have separated explicitly the g.s. contribution. Let usdefine average β efficiencies for decays to excited statesonly, both in singles ¯ ε ∗ β and in coincidence with the totalabsorption γ -ray spectrometer ¯ ε ∗ βγ ¯ ε ∗ β = P i> ε iβ f i P i> f i = P i> ε iβ I iβ − I β ¯ ε ∗ βγ = P i> ε iβγ f i P i> f i = P i> ε iβγ I iβ − I β (2)The reason for this somewhat artificial definition isthat they are well determined from a TAGS analysis evenin the specific cases when the spectrometer is insensitiveto g.s. β penetration as will be shown later. Using theseaverage efficiencies Eq. 1 can be rewritten as: N β N d = ε β I β + ¯ ε ∗ β (1 − I β ) N βγ N d = ε βγ I β + ¯ ε ∗ βγ (1 − I β ) (3)from which I β can be determined: I β = 1 − N βγ N β ¯ ε ∗ β ¯ ε ∗ βγ N βγ N β ε β − ¯ ε ∗ β ¯ ε ∗ βγ − ε βγ ¯ ε ∗ βγ (4)This Eq. 4 can be compared with the equivalent one(Eq. 13) in the original publication of Greenwood et al.[2] that can be rewritten using our nomenclature as: I β = 1 − N ∗ βγ N β ε ∗ γ N ∗ βγ N β ε β − ¯ ε ∗ β ¯ ε ∗ β ¯ ε ∗ γ (5)In the conversion we have used the following equiva-lence to the notation of Greenwood et al.: ¯ ε ∗ γ = 1 − L , ε β = f gs ω β , and ¯ ε ∗ β = f ex ω β . Even assuming that thefactorization ¯ ε ∗ βγ = ¯ ε ∗ β ¯ ε ∗ γ is valid, there are differencesbetween this expression and Eq. 4. A correction term ismissing in the denominator and N ∗ βγ represents N βγ cor-rected by the β penetration for g.s. decays (with proba-bility ε βγ ) and for decays to excited states where the γ cascade is not detected in the spectrometer (with proba-bility ˜ ε iβγ ) N ∗ βγ = N βγ − ε βγ f − P i> ˜ ε iβγ f i . We show inthe Appendix, using synthetic data, that Eq. 5 producesinconsistent results.The application of the 4 πγ − β method requires thedetermination of the experimental ratio R = N βγ /N β and the estimation of three correction factors, a , b and c ,that are ratios of β efficiencies I β = 1 − aR bR − c (6)From its expression (compare Eq. 6 with Eq. 4) one cansee that correction factor a is close to (but larger than)one, correction factor b is a small number and correctionfactor c is a relative measure of β penetration for decaysto the g.s. To estimate accurately the correction fac-tors we need to know the dependency of β efficiency withendpoint energy and the β -intensity distribution with ex-citation energy (see Eq. 2). Notice that only the relative β intensity to excited states is required. We also needto know the β -penetration probability in the total ab-sorption γ -ray spectrometer. Since γ rays interact alsoin the β detector we must take this effect into account,which implies that we must have a knowledge of decay γ cascades. These are also needed to obtain the β − γ detection efficiency. Conversion electrons are readily de-tected in the β detector affecting both the β counts andthe decay detection efficiency, and this is another effectthat must be considered. As a matter of fact all of thisinformation is required for the analysis of TAGS data oris the result of such analysis (see Section III). The ac-curacy of the 4 πγ − β method depends on the accuracyof the ratio of counts R and on the accuracy with whichwe can determine the correction factors. The integratedcounts N β and N βγ can be obtained from the measured β and β − γ spectra but corrections for contaminantsshould be applied. The identification and quantificationof contaminants is an important ingredient of the TAGSanalysis, therefore providing the necessary informationfor the evaluation of this correction. In summary the4 πγ − β method relies on the deconvolution of TAGSdata and becomes a natural extension of it.The decay of β -delayed neutron emitters requires spe-cial consideration. In this case the β -intensity distribu-tion is the sum of two contributions I β ( E x ) = I βγ ( E x ) + I βn ( E x ). The first one I βγ ( E x ) refers to decays that pop-ulate levels in the daughter nucleus that then de-exciteby emission of γ rays. This is the one determined by theTAGS analysis. The second one I βn ( E x ) refers to decaysthat populate levels above the neutron separation energy S n which is then followed by the emission of one or moreneutrons and eventually γ rays in the de-excitation of thefinal nucleus. This component can be obtained from themeasured β -delayed neutron spectrum and a knowledgeof the branching probability to the different levels in thefinal nucleus (see [21] for further details). By separatingthe two components in the second row of Eq. 1 we obtain: N βγ = ε βγ I β N d + X i> ε iβγ I iβγ N d + X i> ε iβnγ I iβn N d (7)The last term represents the counts coming from theinteraction of the β -delayed neutrons with the total ab-sorption γ -ray spectrometer which is another source ofcontamination in the TAGS analysis that must be cor-rected for as explained later. After eliminating thiscontribution the second row in Eq. 1 becomes N βγ = ε βγ I β N d + P i> ε iβγ I iβγ N d . With a re-definition of thecoincidence detection efficiency averaged over excited lev-els (second row of Eq. 2):¯ ε ∗ βγ = P i> ε iβγ I iβγ − I β (8)we arrive formally to the same formula Eq. 4 to calculate I β . Notice however that now ¯ ε ∗ β is calculated with thetotal β -intensity distribution I iβ while ¯ ε ∗ βγ is calculatedwith the partial intensity distribution I iβγ . The latteris normalized to 1 − P n instead of 1. The extension ofthe formulae for β -delayed neutron emitters was not dis-cussed in the work of Greenwood et al. .The validation of the method with synthetic data isleft for the Appendix. In the next section we apply themethod to experimental data for a number of selected iso-topes that either show some particularities in the use ofthe method or are important in determining the reactorantineutrino spectrum and/or the reactor decay heat. III. EXPERIMENTAL RESULTS
A campaign of TAGS measurements was carried outin 2014 at the upgraded Ion Guide Isotope SeparatorOn-Line IGISOL IV facility [46] at the University ofJyv¨askyl¨a. One of the motivations for these measure-ments was to improve both reactor decay heat and an-tineutrino spectrum summation calculations, by provid-ing data free from the
Pandemonium effect for some nu-clei having significant g.s. feeding values. In the experi-ment we employed the 18-fold segmented NaI(Tl) DecayTotal Absorption γ -ray Spectrometer (DTAS) [15] in co-incidence with a 3 mm-thick plastic scintillation detector.This β detector was located at the centre of DTAS and infront of a movable tape for the implantation of the nucleiof interest and the removal of the daughter activity (see[47] for more details about the experiment). The meanefficiency of the β detector is around 30% for endpointenergies above 2 MeV (see Fig. 6 in the Appendix fora similar detector), while the efficiency of DTAS for β -particles ranges from 8% at 3 MeV β end-point energyto 44% at 8 MeV [48]. We provide in the following a brief description ofTAGS data analysis for the reader’s better understand-ing. The β -gated total energy deposited in DTAS wasreconstructed off-line from the signals of the individualdetector modules as described in [48], with threshold val-ues of 90 keV for DTAS modules and 70 keV for the β detector. The coincidence between DTAS and the β de-tector allowed us to get rid of the environmental back-ground. Other sources of contamination need to be ac-counted for. These include in general the activity of thedescendants. For each descendant that contributes signif-icantly we measure the shape of its energy spectra or, inthe case of well known decays, we obtain it through MCsimulations using the available decay data. If possiblethe normalization of these spectra is obtained by adjust-ment to salient features on the measured parent decayspectra that can be identified as due to the descendantactivity, otherwise from the relation of parent-descendanthalf-lives. In the case of β -delayed neutron emitters, aswas mentioned above, the β -delayed neutron branch in-troduces an additional contamination that includes theinteraction of neutrons with DTAS. The shape of the con-taminant spectrum is obtained by MC simulation follow-ing a special procedure detailed in [21] and the normaliza-tion is obtained by adjustment to the measured spectra,if possible, otherwise it is given by the P n value. We alsotake into account the electronic pulse summing-pileup ef-fect that contributes to the distortion of the spectra. Theneed to consider two components (summing and pileup)is particular to multi-detector systems. The pileup orig-inates in the superposition of different event signals inthe same detector module within the analog-to-digitalconverter (ADC) time gate. The summing is due to thesum of signals corresponding to different events that aredetected in different modules within the same ADC gate.The summing-pileup contribution is calculated by meansof a MC sampling method [48] specifically developed forsegmented spectrometers. The calculated spectrum isnormalized from the detection rate and the length of theADC gate [48].In order to determine the β intensities from TAGS ex-perimental spectra, we followed the method developedby the Valencia group [10, 11, 49]. For this, one has tosolve the inverse problem d i = P j R ij ( B ) f j + C i , where d i represents the number of counts in channel i of thespectrum, f j is the number of events that feed level j inthe daughter nucleus, R ij is the response function of thespectrometer, which depends on the branching ratios ( B )for the different de-excitation paths of the states popu-lated in the decay, and C i is the sum of all contaminantsat channel i .To build the spectrometer response to decays we needthe response to individual γ rays and β particles [11].These are obtained from MC simulations using a verydetailed description of the measurement setup (includ-ing electronic thresholds), carefully benchmarked withlaboratory sources. We also need the branching ratiomatrix describing the de-excitation pattern as a func-tion of level excitation energy, including the conversionelectron process. This is obtained from the HR spec-troscopy level scheme at low excitation energies supple-mented with the predictions of the Hauser-Feshbach nu-clear statistical model above a given excitation energywhere the levels are treated as a binned continuum [49].The statistical model provides a realistic description ofthe electromagnetic cascade energy and multiplicity dis-tribution, that in modern segmented spectrometers canbe tested as well [26, 27, 50] and eventually modified.Finally the TAGS spectrum deconvolution is carriedout by applying a suitable algorithm, which in the presentcase is the expectation maximization (EM) algorithm, toextract the β -feeding distribution [10].In order to apply the 4 πγ − β method to obtain I β we must determine the experimental number of counts N βγ and N β . N βγ is obtained from the number of countsin the β -gated DTAS spectrum after correction for thecounts due to the contaminants. As we mentioned in Sec-tion II this correction follows closely the one applied toTAGS spectra for deconvolution since the contaminationcounts are determined by integration of the correspond-ing TAGS contamination spectra that we just described.In this line, the counts due to the activity of the de-scendants and, if needed, the β -delayed neutron branchcontribution are subtracted. In the case of the summing-pileup contribution the counts are added since each countin the summing-pileup spectra represents the loss of twoevents. The uncertainty of N βγ is estimated by consid-ering the uncertainties in the normalization factors ofthe different components, taken from the TAGS analysis.Note that in all cases we integrate the full β -gated DTASspectra, since experimental thresholds are already takeninto account when requiring the β -gating condition, asmentioned above. The experimental thresholds are alsotaken into account in the MC efficiencies employed forthe determination of coefficients a , b and c of Eq. 6. N β is calculated as the number of counts in the spec-trum of the β plastic detector above the threshold with-out any coincidence condition. In addition to the countsdue to descendants, which are estimated from the TAGSanalysis, in this case we also need to subtract envi-ronmental background counts, although this is a smallamount. They are obtained from measurements withoutbeam and normalized by the relative measurement times.The counts lost by electronic pulse pileup are added tothe result. In this case, since we are dealing with a singledetector, the calculation and normalization of the pileupcontribution is performed as in Ref. [51]. The uncertaintyon N β comes from the uncertainties in the normalizationfactors of the contaminants. For the environmental back-ground we take a 20% uncertainty, which is the maximumdeviation observed in tests with laboratory sources, whilefor the rest of the contaminants we take the same uncer-tainties used for the TAGS analysis.Finally we should mention that in general the ra-tio N βγ /N β needs to be corrected for differences in thedata acquisition dead-times. In the present case this is Isotope I β [%]ENSDF TAGS 4 πγ − β Rb ≤ . +0 . − . -0.2(42) Nb 50(7) 46 +16 − Nb - 42 . +9 . − . Tc 93.3(1) 93.9(5) 92.8(5)
Tc 34(8) - 45 . +1 . − . I 45.2(5) 50 . +2 . − . Cs 35.9(17) 39.0 +2 . − . πγ − β method arecompared with results from the TAGS analysis and from eval-uations in the ENSDF database [30]. not necessary because of the way our acquisition systemworks: every acquisition channel is gated with a com-mon gate signal which is an OR of all individual detectortriggers.The results of the application of the 4 πγ − β method topart of the data obtained in the 2014 measurement cam-paign are presented below. Firstly cases of particularinterest showing how the method works (sections III Aand III B) and secondly (section III C) cases that areof relevance for reactor antineutrino summation calcu-lations. They are summarized in Table I. A. The case of Tc One of the cases studied is the decay of the 5 / + g.s.of Tc into
Ru with a Q β value of 2663(10) keV [31].This TAGS measurement was assigned first priority forthe prediction of the reactor decay heat with U/Pu fueland second priority for Th/U fuel by the IAEA [52]. Inaddition to the 3 / + g.s. of the daughter nucleus, thedecay also populates a 5 / + state at only 2.81(5) keVexcitation energy, as observed in a previous HR spec-troscopy experiment [53]. Since we are not able to sepa-rate the two close-lying states we refer to their summeddecay intensity as the g.s. intensity I β . The DTAS and β -detector spectra for this measurement are shown inFig. 1 and Fig. 2 respectively, together with the spec-tra of the contaminants. Since Ru is very long lived( T / = 39 . Tc is a special case. In the TAGSanalysis presented in [54], we found that the reproductionof the measured DTAS spectrum is almost insensitive tothe value of the g.s. feeding intensity, which is an unusualsituation. To illustrate this in Fig. 3 we show the relative β intensities to excited states of Ru when the g.s. β intensity is fixed in the TAGS analysis to values between0% and 90% in steps of 10%. As can be observed they Energy (keV) 0 1000 2000 3000 C oun t s β Q Tc -gated β DTAS Summing-pileup g.s → MC response g.s.
FIG. 1. (Color online) Experimental β -gated spectrum (solidblack) compared with the MC response of DTAS for the tran-sition to the g.s. of Ru (dotted blue). The normalized sum-ming pileup contribution is also shown (grey).
Energy (keV) 0 1000 2000 3000 C oun t s − β Q Tc Beta detectorPileupBackground
FIG. 2. Spectrum in singles of the plastic β detector for thedecay of Tc (solid black line). The contaminants are shownwith the appropriate normalization. are almost unchanged in the range 0% to 80%. For avalue of 90% there is a sizable effect on the β intensity tostates below 300 keV. The different I β values introduce achange of around 15% in the χ of the TAGS analysis (seeFig. 4), with a minimum at 70 − πγ − β method. We obtain a Energy (keV) 0 500 1000 1500 2000 2500 ( % ) β R e l a t i v e I − − − (%) β I 0102030405060708090
FIG. 3. (Color online) Relative β intensities populating theexcited states of Ru in the decay of
Tc. The β intensitiesobtained from the TAGS analysis for different fixed values of I β are normalized to 1- I β . ratio of counts R for the decay of Tc of 0.495(5). Weuse Eq. 4 to obtain I β using correction factors calculatedwith the β -intensity distributions shown in Fig. 3. Asin the case of the synthetic data (see Appendix), we ob-tain very stable results with the 4 πγ − β method. InFig. 4 the I β values obtained with this method for eachfixed I β in the TAGS analysis are presented, showing verysmall variations when the values vary between 0% and90%. The uncertainties for each I β determined with the4 πγ − β method are obtained from the uncertainty in theratio of the number of counts R , which combines statis-tical uncertainties and uncertainties in the correction forcontaminations, and the uncertainties in the correctionfactors. We use a set of β -intensity distributions, a totalof 17 solutions of the TAGS analysis compatible with theexperimental data, to obtain different correction factorsand evaluate the dispersion of I β values determined withthe 4 πγ − β method. It was found to be very small, whichis related to the fact that the uncertainties in the averageefficiencies defined in Eq. 2 are small, as discussed in theAppendix. As explained in detail in [21] each of these β -intensity distributions obtained in the TAGS analysistakes into account the effect of several systematic un-certainties related to the branching ratio matrix used tobuild the spectrometer response, to the accuracy of MCsimulations, to the normalization of contaminants andeven to the deconvolution method. We would like to notethat errors quoted in Table I for the 4 πγ − β values arethe quadratic sum of the uncertainty in the calculation ofthe ratio of counts R (the dominant one) and the smallcontribution coming from the application of the methodwith all solutions used to estimate the error budget ofthe TAGS analyses.The mean of all results in Fig. 4 gives a value of45 . +1 . − . %. The uncertainty is a conservative estimatebased on the lowest and highest uncertainty deviations.This result is compatible with the value 41(10)% ob-tained in the HR spectroscopy work of Niizeki et al. [53].However, it is larger than the 34(8)% value reportedin the ENSDF evaluation [55] based also on HR spec-troscopy studies. The difference between both is relatedto the adopted intensity of the 346.4 keV γ ray used fornormalization: Niizeki et al. uses an intensity of 16% forthis γ ray, whereas the ENSDF evaluation uses a value of18.4% obtained in a fission yield measurement [56]. Thisexample shows one of the difficulties faced when assigningg.s. feeding probabilities in HR spectroscopy. fixed (%) β I0 10 20 30 40 50 60 70 80 90 ( % ) β - γ π f r o m β I χ FIG. 4. (Color online) I β values obtained with the 4 πγ − β method for different TAGS analyses performed with I β fixedto values between 0 and 90 % (solid black). The χ of eachTAGS analysis is shown in dotted red. The horizontal dashedblue line shows the mean I β value. The upper and lower limitsconsidered for the uncertainty are represented by horizontaldotted gray lines. We also use the decay of
Tc to perform an illustra-tive exercise showing the dependence of the uncertaintyof I β in the 4 πγ − β method with I β for a fixed value of theuncertainty in R . The β -intensity distributions obtainedin the TAGS analysis with fixed I β values between 10%and 90% (shown in Fig. 3) have been used as input toMC simulations of DTAS and plastic scintillation spec-tra (see also the Appendix). The 4 πγ − β method wasthen applied to these spectra. The uncertainty in theratio of counts R was fixed to 1%, the value of the cur-rent measurement. As shown in Fig. 5, the relative errorin the determination of the g.s. feeding probability withthe 4 πγ − β method varies between 10% at I β = 10%and 1% at I β = 90%. Thus the precision of the methodis severely limited by statistics at low values of the g.s.feeding intensity. fixed (%) β I10 20 30 40 50 60 70 80 9000.020.040.060.080.1 β I ) β (I σ FIG. 5. Relative uncertainty of the I β values obtained withthe 4 πγ − β method applied to MC simulations. The TAGSresults obtained with I β fixed to different values from 10 to90 % in the analysis of the decay of Tc have been usedas input for the event generator employed in the simulations.The uncertainty in the ratio of counts R was kept fixed at theexperimental 1% value. B. Cases with extreme values of the g.s. feedingintensity
The 4 πγ − β method has been applied to other testcases measured in the same DTAS experimental cam-paign. In those cases the TAGS analyses did show asensitivity to the g.s. to g.s. transition, thus allowing usto compare the I β determined from the deconvolutionwith the value obtained by means of the 4 πγ − β count-ing method presented here. In particular two cases areincluded here due to the extreme character of their g.s.feeding probability and importance: the decay of Rb,a β -delayed neutron emitter where the first forbidden5 / − → / + g.s. to g.s. transition is hindered, and thedecay of Tc, dominated by the large Gamow-Teller1 + → + g.s. to g.s. branch.In the first case, the decay of Rb, we obtain an almostzero g.s. to g.s. feeding from TAGS analysis, in agreementwith previous HR spectroscopy measurements [27] (seeTable I). Nevertheless, our TAGS analysis shows that theHR data is affected by a strong
Pandemonium effect [27].The 4 πγ − β method determines also a I β value that isalmost zero, though the relative uncertainty is large, ascan be expected from the discussion in the previous Sec-tion. In fact (see Table I) in this case the uncertaintyis much larger than that determined by TAGS spectrumdeconvolution. Due to the relatively small fission yieldof Rb, the impact of our TAGS results in reactor an-tineutrino spectrum calculation is less than 1% between7 and 9 MeV. For the same reason, in spite of being as-signed first priority for the U/Pu fuel decay heat [52], theimpact of these TAGS results is also subpercent on theelectromagnetic component of the reactor decay heat of
Isotope 3-4 MeV [%] 4-5 MeV [%] 5-6 MeV [%] U Pu U Pu U Pu Nb 3.5 4.5 5.3 7.6 5.8 9.0
Nb 0.7 1.5 0.7 1.7 0.4 1.0
I 1.7 1.6 2.2 2.3 2.0 2.3
Cs 2.8 2.9 3.3 3.7 2.5 3.0TABLE II. Contribution in % of the selected cases to thereactor antineutrino spectra of
U and
Pu at differentenergy ranges (based on the Nantes summation method [20]).
U and
Pu for times shorter than 1 s [27].In the second case, the decay of
Tc, of interest fornuclear structure, a large I β value of 93.9(5)% is deter-mined from the TAGS spectrum deconvolution. As de-scribed in [57] a different β detector was employed in thismeasurement, which consists of a vase-shaped thin plas-tic scintillator with close-to-4 π solid angle coverage. Thevalue of I β obtained with the TAGS technique is com-patible with the previous value from HR measurements(see Ref. [24] for a detailed discussion). The 4 πγ − β method gives a value of 92.8(5)% in agreement with thevalue from the TAGS analysis, thus confirming this im-portant result. In this case the value quoted by ENSDFis in agreement within the uncertainties with both re-sults. The β -intensity distribution of Tc decay servesas a benchmark for theoretical estimates of the nuclearmatrix elements (NME) in the A=100 system that en-ter into the calculation of the double β -decay process in Mo. NME represent the largest uncertainty in thehalf-life estimate of the neutrino-less branch, thus lim-iting our ability to extract information on this processbeyond the Standard Model.
C. Reactor antineutrino spectrum cases
The remaining cases presented here are decays of fis-sion fragments contributing significantly to the reactorantineutrino spectrum:
Nb,
Nb,
I and
Cs.Table II provides the percent contribution of the four iso-topes to the total antineutrino spectrum for both
Uand
Pu fission in three E ¯ ν e energy intervals coveringthe range from 3 to 6 MeV. These percentages were cal-culated using the Nantes summation method [20]. Allcases listed in Table II have been assigned a first priorityfor TAGS measurement in the IAEA report [52], while I and
Nb are also considered high-priority casesfor the reactor decay heat by the IAEA [52].As can be observed in Table I the relative uncertaintyin I β obtained by TAGS spectrum deconvolution is ratherlarge in these four cases. In particular in the case of Nb, estimated to be one of the largest contributorsin the region of the spectral distortion around 5 MeV,reaches 35%. In the case of
Nb it is 24%. In both cases the TAGS analysis is strongly affected by the un-certainty in the contamination of the parent activity (seeRef. [26] for more details). The characteristic of the4 πγ − β method of being almost insensitive to the actual β -intensity distribution obtained in the TAGS analysiscan be of advantage here.As can be seen in Table I an overall good agreement isfound between the g.s. feeding probabilities obtained withthe 4 πγ − β method and those determined in the TAGSanalyses. The 4 πγ − β method, however, produces re-sults with much smaller relative uncertainties comparedto the TAGS analysis for the two Nb cases: 15% and6% respectively. Smaller uncertainties are also obtainedfor I and
Cs. The central values are in agreementwithin uncertainties for both methods. However, observ-ing all the values in the Table I one could also claim thatthe 4 πγ − β method tends to produce results systemat-ically smaller than TAGS spectrum deconvolution, withthe exception of Nb. Whether this is true and couldbe related to some systematic error in one of the twomethods should be studied further.Compared to the values in the ENSDF database [30]we observe (see Table I) that the 4 πγ − β method is inclose agreement for I and
Cs, and 20% smaller for
Nb although in agreement within uncertainties. Novalue is available for
Nb in the ENSDF database.
IV. SUMMARY AND CONCLUSIONS
In this work we have addressed the determination ofthe β -decay intensity to the g.s. of the daughter nucleus I β by means of a β - γ counting method. This approach,initially proposed by Greenwood et al. [2], relies on theuse of a high-efficiency γ calorimeter in coincidence witha β detector. The original 4 πγ − β method has been re-vised and some inconsistencies in the formulae were foundand corrected. Furthermore we extended the formulae tothe particular case of β -delayed neutron emitters, to takeinto account the fraction of decays proceeding by neutronemission. We have shown that the method becomes anextension of and relies on the total absorption γ -ray spec-troscopy technique. The analysis performed using thistechnique provides the information needed to calculatethe quantities required by the 4 πγ − β method as wellas the accurate determination of contaminant contribu-tions, which results in an improved overall accuracy. Therobustness of the method is demonstrated using syntheticdecay data obtained from MC simulations. It was shownthat statistics becomes a limiting factor for determiningwith precision the decay probability to the g.s. as thisprobability becomes smaller.We have applied the 4 πγ − β method to a number ofcases measured in our last experimental campaign withthe DTAS spectrometer at the IGISOL IV facility. Themain goal of the campaign was to measure accurately the β -intensity distribution in the decay of FP of importancein determining the antineutrino spectrum and the decay0heat from reactors, several of which have a large decayto the g.s. The TAGS analysis of one case, Tc, turnedout to be insensitive to the value of the g.s. feeding proba-bility, and the 4 πγ − β counting method was the only wayto determine its rather large value of about 45%. Eventhough Tc is a special case, this shows one of the po-tential issues when determining the g.s. β -decay intensityfrom the deconvolution method. For the remaining cases,with I β values ranging from 0 to more than 90%, goodagreement between the g.s. feeding probabilities deter-mined in the TAGS analysis and those obtained with the4 πγ − β method was observed. This provides a confirma-tion of previous g.s. feeding probabilities obtained by theTAGS deconvolution method and in particular confirmsthe accuracy of the simulation of the shape of the β pen-etration, to which the 4 πγ − β method is not sensitive.For the cases studied we found that, with the exceptionof the negligible intensity of Rb, the uncertainties inthe 4 πγ − β method are smaller. Besides case-specificreasons this is related to the small effect of our lack ofknowledge of level de-excitations in the daughter nucleuson this method, whilst it represents a significant fractionof the error budget in the TAGS analysis. In particularthe uncertainties for the important contributors to thereactor antineutrino spectrum Nb and
Nb arereduced by factors 2.5 and 4, respectively, resulting inmore precise antineutrino spectra for these nuclei with acorresponding improvement in future summation calcu-lations.In conclusion, the 4 πγ − β method represents an alter-native, generally superior, approach to the TAGS spec-trum deconvolution to determine g.s. feeding probabili-ties. The potential of this tool to provide accurate andprecise I β values, which is hampered by the lack of as-sociated γ -ray emission, was demonstrated in this work.Ground state feeding probabilities are needed to deter-mine the absolute value of the decay intensity to excitedstates and carry important information on the nuclearstructure. In addition, due to the significant influence ofthe β -decay branches to the g.s. on the reactor antineu-trino spectrum and decay heat, our capacity to betterdetermine such transitions will help us understand thechallenging puzzle of reactor antineutrinos, while improv-ing decay heat predictions. ACKNOWLEDGMENTS
This work has been supported by the Spanish Min-isterio de Econom´ıa y Competitividad under GrantsNo. FPA2011-24553, No. AIC-A-2011-0696, No.FPA2014-52823-C2-1-P, No. FPA2015-65035-P, No.FPI/BES-2014-068222, No. FPA2017-83946-C2-1-P, No.RTI2018-098868-B-I00 and the program Severo Ochoa(SEV-2014-0398), by the Spanish Ministerio de Ed-ucaci´on under the FPU12/01527 Grant, by the Eu-ropean Commission under the FP7/EURATOM con-tract 605203, the FP7/ENSAR contract 262010, the SANDA project funded under H2020-EURATOM-1.1contract 847552, the Horizon 2020 research and inno-vation programme under grant agreement No. 771036(ERC CoG MAIDEN), by the Generalitat Valencianaregional funds PROMETEO/2019/007/, and by the
Junta para la Ampliaci ´ on de Estudios Programme(CSIC JAE-Doc contract) co-financed by ESF. We ac-knowledge the support of the UK Science and TechnologyFacilities Council (STFC) Grant No. ST/P005314/1 andof the Polish National Agency for Academic Exchange(NAWA) under Grant No. PPN/ULM/2019/1/00220.This work was also supported by the Academy of Fin-land under the Finnish Centre of Excellence Programme(Project No. 213503, Nuclear and Accelerator-BasedPhysics Research at JYFL). A.K. and T.E. acknowledgesupport from the Academy of Finland under ProjectsNo. 275389 and No. 295207, respectively. This work hasalso been supported by the CNRS challenge NEEDS andthe associated NACRE project, the CNRS/in2p3 PICSTAGS between Subatech and IFIC, and the CNRS/in2p3Master projects Jyvaskyla and OPALE.
APPENDIXApplication to synthetic data
The synthetic data emulate the decay of Br, Br,and Rb, that we have previously investigated with theTAGS technique [21]. From the deconvolution of TAGSspectra we obtained ground state feeding intensities of10 . +1 . − . % and 4 . +1 . − . % for Br and Br respec-tively. By comparison, ENSDF assigns a decay intensityto the g.s. of 12 . . s . → g . s . de-cay in Rb corresponds to a third forbidden transitionwith negligible intensity. These three nuclei are β -delayedneutron emitters with neutron emission probabilities of2 . . . β -intensity distribution (normalized to 1) is used to cal-culate the correction factors in Eq. 4 (Section II) and weneed to scale by 1 − P n the I β obtained in order to com-pare with the true value, defined as the input value forthe simulation.The measurement was performed using a compact 12-fold segmented BaF total absorption spectrometer withcylindrical shape and a thin Si detector placed close tothe source position subtending a solid angle fraction of ≈
30% as described in [21]. Spectra were simulated withthe Geant4 Simulation Toolkit [12] implementing the de-tailed description of the setup and using decay cascadesproduced by the DECAYGEN event generator [49]. Theinputs to the event generator are the branching ratiomatrix used in the TAGS analysis and the resulting β -intensity distribution that was adopted as the final solu-1tion in [21]. This provides a very realistic simulation ofthe decay and its detection. In the simulation, as well asin the analysis, we assume that all β -energy distributionshave an allowed shape. The reconstruction of the energydeposited in the event mimics that of the experiment.The experimental low energy threshold of 65 keV, is ap-plied to each spectrometer segment before summing toobtain the total energy deposited. Similarly a threshold(in both MC and experiment) of 105 keV is applied tothe Si detector before gating on the spectrometer signals. energy (MeV) c oun t s : TAGS: TAGS Si-gated: Si detector Br (MeV) x E e ff i c i en cy ( % ) : Si detector: TAGS Si-gated FIG. 6. (Color online) Top panel: Simulated Si detector spec-trum (red dashed line) and TAGS spectra with (gray line) andwithout (black line) gating on Si signals for the decay of Br.The instrumental resolution is not included in these spectra.Bottom panel: Simulated β efficiency without (black sym-bols) and with (gray symbols) a gating condition on the γ spectrometer. See text for further details. In the top panel of Fig 6 the spectrum of energy de-posited in the spectrometer, with and without the β -gating condition, and the spectrum of energy depositedin the Si detector are shown for the case of the decayof Br. The instrumental resolution is switched-off inthese spectra to show more clearly their features. Onemillion decay events were simulated. The total numberof counts in the ungated TAGS spectrum is 0 . × , Isotope R a b c I β [%] Br 0.8846 1.064 4 . × − Br 0.9444 1.038 − . × − Rb 0.9913 1.008 − . × − R , correction factors a , b and c and calculated β intensity to the g.s for synthetic data. Seetext for further details. and 0 . × in the gated spectrum. The counts in theSi detector spectrum are 0 . × . The lower panelof Fig. 6 shows the simulated detection efficiency of theSi detector as a function of excitation energy togetherwith the simulated probability of registering a signal si-multaneously in the Si detector and the γ spectrometer.The pronounced efficiency drop above E x = 5 MeV isdue to the low energy threshold in the Si detector andthe continuum nature of β spectra. The decrease of thespectrometer-gated Si detector efficiency below 5 MeV incomparison with the ungated efficiency is due to the im-portance in this decay of de-excitations proceeding by asingle γ transition to the g.s., which have a greater prob-ability of escaping detection in the spectrometer. Usingthese efficiency distributions and Eq. 2 we can calculatethe correction factors for each decay and apply Eq. 6 toobtain I β .The results of the calculation for the three isotopesare presented in Table III. As can be observed the valueobtained for the g.s. feeding probability (10.12%, 4.79%,0.11% for Br, Br, Rb respectively) is very close tothe true value (i.e. the input values for the simulation)in all cases (10.10%, 4.72%, 0%). If the original for-mula from Greenwood et al. (Eq. 5) is used, with N βγ corrected for the β penetration, we obtain 8.08%, 3.69%and 0.28%, respectively, which deviate clearly from thetrue values for the bromine isotopes. From the values ofthe correction factor c we can see that in this setup the γ spectrometer is rather sensitive to β penetration fordecays to the g.s., between 48% and 66% of the averageprobability of detecting a decay to an excited state. Theincrease in c with isotope follows the increase in Q β .Another important check that can be performed withthe synthetic data is to test the stability of the resultagainst uncertainties in the deconvolution procedure. Asexplained in [21] the β -intensity distribution obtained inthe TAGS analysis is affected by several systematic un-certainties related to the branching ratio matrix used tobuild the spectrometer response, the accuracy of the MCsimulations, the normalization of contaminants and eventhe deconvolution method. This results in a spread of I β extracted from the deconvolution method which variedbetween 9.16% and 11.29% (14 intensity distributions)for Br, and between 2.60% and 5.82% (13 intensity dis-tributions) for Br. Actually this spread determines thesize of the systematic uncertainty of the g.s. feed obtainedfrom the deconvolution method quoted above. In com-2parison the statistical uncertainty from deconvolution isnegligible (below 0.05%). However if we use the differ-ent β -intensity distributions to calculate the correctionfactors a , b and c and apply the 4 πγ − β method to thesynthetic spectra simulated with the adopted I β distribu-tion (shown in the top panel of Fig. 6 for the Br case)the resulting I β vary very little. For example in the de-convolution of Br data we tested the effect of fixing theg.s. feeding probability to the ENSDF value 12%. Thisresulted in a still acceptable fit to the data, just outsidethe 5% maximum increase in χ that was used to selectthe set of acceptable solutions in the original publication[21]. When using the resulting β intensity to calculatethe correction factors in Eq. 6 we obtain I β = 10 . 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