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- Add neutrino mass parameter to SpectrumConfig and CLI interface - Update PhaseSpace component to handle neutrino mass effects properly - Update CSV headers to show basic endpoint and all transition types - Add --csv-verbosity option to CLI with 'full' and 'concise' options - Fix comparison plots to properly show linear scale variations - Add comprehensive tests for neutrino mass functionality
562 lines
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562 lines
43 KiB
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This is a self-archived version of an original article. This version
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may differ from the original in pagination and typographic details.
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Kostensalo, Joel; Suhonen, Jouni
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Author(s):
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gA-driven shapes of electron spectra of forbidden β decays in the nuclear shell model
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Title:
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Year: 2017
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Version:
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Published version
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Copyright: © 2017 American Physical Society. Published in this repository with the kind permission of the publisher.
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Rights:
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Rights url:
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Please cite the original version:
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Kostensalo, J., & Suhonen, J. (2017). gA-driven shapes of electron spectra of forbidden β decays
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in the nuclear shell model. Physical Review C, 96(2), Article 024317.
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https://doi.org/10.1103/PhysRevC.96.024317
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PHYSICALREVIEWC96,024317(2017)
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g -drivenshapesofelectronspectraofforbiddenβ decaysinthenuclearshellmodel
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A
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JoelKostensalo* andJouniSuhonen†
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DepartmentofPhysics,UniversityofJyvaskyla,P.O.Box35,FI-40014,Finland
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(Received10May2017;published21August2017)
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The evolution of the shape of the electron spectra of 16 forbidden β− decays as a function of g was
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A
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studiedusingthenuclearshellmodelinappropriatesingle-particlemodelspaceswithestablished,well-tested
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nuclearHamiltonians.Theβspectraof94Nb(6+)→ 94Mo(4+)and98Tc(6+)→ 98Ru(4+)werefoundtodepend
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stronglyong ,whichmakesthemexcellentcandidatesforthedeterminationoftheeffectivevalueofg withthe
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A A
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spectrum-shapemethod(SSM).Astrongg dependenceisalsoseeninthespectrumof96Zr(0+)→ 96Nb(6+).
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A
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Thisdecaycouldbeusedfordeterminingthequenchingofg insixth-forbiddendecaysinthefuture,whenthe
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A
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measurementofthespectrumbecomesexperimentallyfeasible.Thecalculatedshell-modelelectronspectraofthe
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ground-state-to-ground-statedecaysof87Rb,99Tc,and137Csandthedecayof137Cstotheisomeric11/2−state
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in137Bawerefoundtobeinexcellentagreementwiththespectrapreviouslycalculatedusingthemicroscopic
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quasiparticle-phononmodel.ThisisfurtherevidenceoftherobustnatureoftheSSMobservedintheprevious
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studies.
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DOI:10.1103/PhysRevC.96.024317
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I. INTRODUCTION valueoftheweakcouplingconstants.IntheSSMtheshapes
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of computed and experimental (normalized) electron spectra
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At the nuclear level β decay can be considered as a
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are compared in order to find the ratio g /g for nonunique
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mutual interaction of the hadronic and leptonic currents forbiddenbetaβ−decays,forwhichthes A hap V efactorsdepend
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mediated by a massive vector boson W± [1]. The leptonic
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ong /g inaverynontrivialway.InRef.[21]theSSMwas
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and hadronic currents can be expressed as a mixture of both A V
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applied to the fourth-forbidden ground-state-to-ground-state
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vector and axial-vector components [2–4]. The weak vector
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transitionof113Cdusingthemicroscopicquasiparticle-phonon
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andaxial-vectorcouplingconstantsg andg enterthetheory
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V A model (MQPM) [22,23], the nuclear shell model (NSM)
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when the hadronic current is renormalized at the nucleon
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[24–26], the microscopic interacting boson-fermion model
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level[5].Theconservedvector-currenthypothesis(CVC)and
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(IBFM-2) [27,28], and experimental spectrum of Ref. [29].
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partially conserved axial-vector-current hypothesis (PCAC)
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The closest match between the theoretical and experimental
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yield the free-nucleon values g =1.00 and g =1.27 [6]
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V A spectra was found when g /g ≈0.92 for all three nuclear
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but inside nuclear matter the value of g is affected by A V
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A models.Thehalf-lifecomparisonmethod,ontheotherhand,
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many-nucleon correlations and a quenched value might be
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gave very different results when different nuclear models
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neededtoreproduceexperimentaldata[1].Preciseinformation
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were used. In Ref. [30] it was noticed that the shape of the
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ontheeffectivevalueofg iscrucialwhenpredictinghalf-lives
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A spectrum was largely unaffected by modest changes in the
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of neutrinoless double beta decays since the half-lives are
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MQPM wave function, even though the predicted half-life
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proportionaltothefourthpowerofg [7,8].
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A wasaffectedsignificantly.Theobservationsoftheseprevious
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The effective value of g has earlier been probed using a
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A studiessuggestthattheSSMisveryrobust,quiteinsensitiveto
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half-lifecomparisonmethod,inwhichthepredictedandexper-
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thedetailsoftheadoptednuclearmeanfieldandHamiltonian.
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imentalvaluesarecomparedfordifferentvaluesofg .Thishas
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A Itisthusareliabletoolforprobingtheratiog /g .
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beendoneforGamow-Tellerandfirst-forbiddendecaysusing A V
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The g -driven evolution of the shapes of several exper-
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nuclear matrix elements calculated with the proton-neutron A
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imentally measurable forbidden nonunique decay spectra in
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quasiparticlerandom-phaseapproximation(pnQRPA)[9–13].
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medium-heavyodd-AnucleiwasstudiedusingtheMQPMin
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Thehalf-lifecomparisonmethodpredictsthatthevalueofg
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A Ref.[30].Onlyfourgoodcandidatesfortheapplicationofthe
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is quenched significantly in these transitions. Also in the old
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spectrum-shape method have been found thus far. These are
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studies[14–17]offirst-forbiddenβdecays,strongquenchings
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theground-state-to-ground-statedecaysof87Rb,99Tc,113Cd,
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werepredicted.Inarecentshell-modelstudy[18]ofr-process
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and 115In [20,30]. In the present paper we set out to find
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waiting-pointnuclei,astrongquenchingofg wasconfirmed.
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A potential new candidates for the application of the SSM in
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Thehalf-lifecomparisonmethodcouldbeusedtofindoutif
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light to medium-heavy nuclei using the nuclear shell model.
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similar quenching of g is seen in highly forbidden unique
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A We also compare the NSM and MQPM β spectra of 87Rb,
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beta decays by using the half-lives calculated in Ref. [19],
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99Tc,and137Cstoseeifthemodelindependenceobservedin
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onceexperimentaldatabecomeavailable.
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Ref.[21]appliesforthesedecaysaswell.Thedecaysof137Cs
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In Ref. [20] the spectrum-shape method (SSM) was tothe3/2+groundstateandthe11/2−isomericstateof137Ba
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introduced as a complementary way to study the effective
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are particularly interesting for comparing the differences in
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the MQPM and NSM spectra, since the MQPM calculations
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predictag -independentspectrumshapeforthesedecays.
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A
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*joel.j.kostensalo@student.jyu.fi This article is organized as follows. In Sec. II we give
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†jouni.suhonen@phys.jyu.fi the theoretical background behind the β spectrum shape. In
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2469-9985/2017/96(2)/024317(7) 024317-1 ©2017AmericanPhysicalSociety
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JOELKOSTENSALOANDJOUNISUHONEN PHYSICALREVIEWC96,024317(2017)
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Sec.IIIwedescribetheapplicationofthenuclearshellmodel θ beingtheCabibboangle.Inordertosimplifytheformalism
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C
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tothespectrumshapeofforbiddenβ− decays.InSec.IVwe it is usual to introduce unitless kinematic quantit(cid:2)ies w
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e
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=
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presentourresultsandinSec.Vwedrawtheconclusions. W /m c2,w =W /m c2,andp =p c/(m c2)= w2−1.
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e e 0 0 e e e e
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Withtheunitlessquantitiestheintegratedshapefactorcanbe
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II. THEORETICALFORMALISM
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expressedas
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We begin the description of the β− decay by making the (cid:3)
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so-calledimpulseapproximation,inwhichattheexactmoment C˜ =
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w0
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C(w )pw (w −w )2F (Z,w )dw . (6)
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of decay the decaying nucleon only interacts via the weak e e 0 e 0 e e
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1
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interaction and the strong interaction with the other A−1
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nucleons can be ignored. Since the vector boson W− has a The shape factor C(w e ) of Eq. (6) contains complicated
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largemassandthuspropagatesonlyashortdistance,theflow combinationsofboth(universal)kinematicfactorsandNMEs.
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lines of the nucleons, i.e., the hadronic current, and the flow In this paper we use the nuclear shell model to calculate the
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lines of the emitted leptons, i.e., the leptonic current, can be one-bodytransitiondensitiesrelatedtotheNMEsoftheshape
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considered to interact at a pointlike weak-interaction vertex factor. The details of the shape factor and the constitution
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with an effective coupling constant G , the Fermi constant. of its NMEs can be found from Refs. [32,33]. As in the
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F
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The parity nonconserving nature of the weak interaction is previous SSM studies [20,21,30] we go beyond the earlier
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reflected in the fact that the hadronic current can be written studies[32,33]andtakeintoaccountthenext-to-leading-order
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at the quark level (up u and down d quarks) as a mixture of
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vectorandaxial-vectorcomponentsas[2–4]
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Jμ =u¯(x)γμ(1−γ )d(x), (1)
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H 5
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whereγμ aretheusualDiracmatricesandγ =iγ0γ1γ2γ3.
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5
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Renormalization effects of strong interactions must be taken
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intoaccountwhenmovingfromthequarkleveltothehadron
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level.Thehadroniccurrentatthenucleonlevel(neutronnand
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protonp)canthenbewrittenas[5,6]
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Jμ =p¯(x)γμ(g −g γ )n(x), (2)
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H V A 5
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where g and g are the weak vector and axial-vector
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V A
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couplingconstants,respectively.Theconservedvector-current
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hypothesisofthestandardmodel(CVC)givesthefree-nucleon
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valueg =1.0fortheweakvectorcouplingconstantandthe
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V
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partially conserved axial-vector-current hypothesis (PCAC)
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gives the free-nucleon value g =1.27 for the weak axial-
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A
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vector coupling constant. Inside the nuclear matter the value
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of g is affected by many-nucleon correlations and so the
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A
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free nucleon value might not be the one to use in practical
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calculations [6]. The present paper is a step towards solving
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theproblemofwhatvaluetouseforg insidefinitenuclei.
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A
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Intheimpulseapproximationtheprobabilityoftheelectron
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beingemittedwithkineticenergybetweenW andW +dW
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e e e
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is
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G 1
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P(W )dW = F C(W )
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e e (h¯c)62π3h¯ e
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×p cW (W −W )2F (Z,W )dW , (3)
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e e 0 e 0 e e
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where p is the momentum of the electron, Z is the proton
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e
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number, F (Z,W ) is the Fermi function, and W is the
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0 e 0
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end-point energy of the β spectrum. The nuclear structure
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informationisburiedintheshapefactorC(w ).
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e
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Thehalf-lifeofaβ decaycanbewrittenas
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κ
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t = , (4)
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1/2 C˜
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whereC˜ istheintegratedshapefactorandtheconstantκ has FIG.1. Normalized electron spectra for the second-forbidden
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thevalue[31] nonunique β− decays of 36Cl [panel (a)] and 60Fe [panel (c)], and
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thefourth-forbiddennonuniquedecayof50V[panel(b)].Thevalue
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κ = 2π3h¯7ln2 =6147s, (5) g V =1.00 was adopted and the color coding represents the value
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m5c4(G cosθ )2 ofg .
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e F C A
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024317-2
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g -DRIVENSHAPESOFELECTRONSPECTRAOF ... PHYSICALREVIEWC96,024317(2017)
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A
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termsoftheshapefunction.Thedetailsofthenext-to-leading tocalculatetheelectronspectraofthesedecays.TheGXPF1A
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ordershapefactorsarediscussedin[21]. interactionisverygoodatpredictingthepropertiesofpf-shell
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The shape factor C(w ) can be decomposed into vector, nuclei. However, since the computational burden increases
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e
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axial-vector, and mixed vector-axial-vector parts. In this dramatically when the number of particles approaches half
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decompositiontheshapefactoris of the maximum in the valence space, this interaction could
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| | | | | | | | | | 60Fe(0+)→ | | 60Co(2+) | | | |
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| --- | --- | ------ | --- | --- | --- | --- | ------ | -------- | --------- | --- | -------- | --- | ----- | ----------- |
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| | | | | | | | not be | used for | the | | | | decay | transition. |
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| | )=g | 2C )+g | 2C | )+g | | | | | | | | | | |
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C(w e V (w e A (w e V g A C VA (w e ). (7) Instead, a model space spanning the π0f , ν1p , ν0f ,
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| | | V | A | | | | | | | | | 7/2 | 3/2 | 5/2 |
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| --- | --- | --- | --- | --- | --- | --- | ------ | ------------------------------------------ | --- | --- | --- | --- | --- | --- |
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| | | | | | | | andν1p | orbitalswasadoptedwiththeHorie-Ogawainter- | | | | | | |
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1/2
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IntegratingEq.(7)overtheelectronkineticenergy,wegetan
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| | | | | | | | action [40,41]. | | pfg | shell: | the jj44b | interaction | | developed |
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| ---------------------------------------------- | --- | -------- | ----- | ---- | ---- | --- | --------------- | ------- | ------------ | --------- | --------- | ----------- | -------- | --------- |
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| analogousexpressionfortheintegratedshapefactor | | | | | | | | | 9 / 2 | | | −1p−0g | | |
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| | | | | | | | by Brown | and | Lise t s kiy | for | the 0f | | | model |
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| | | | | | | | | | | | | 5/2 | | 9/2 |
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| | | | | | | | space (see | endnote | on | Ref. [28] | of | [42]) | was used | for the |
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| | | C˜ =g2C˜ | +g2C˜ | +g g | C˜ , | (8) | | | | | | | | |
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V V A A V A VA calculation of the 87Rb(3/2−)→ 87Sr(9/2+) transition. For
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A=94–99
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wherethefactorsC˜ inEq.(8)arejustconstants,independent the decays of the nuclei, the effective interac-
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i
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| | | | | | | | tion of | Gloeckner | [43] | in the | model | space | spanning | the |
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| --- | --- | --- | --- | --- | --- | --- | ------- | --------- | ---- | ------ | ----- | ----- | -------- | --- |
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oftheelectronenergy.
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| | | | | | | | π1p , | π0g | , ν2s | , and 2d | was | used. | sdgh | shell: |
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| --- | --- | --- | --- | --- | --- | --- | -------- | ------- | ----------------- | -------- | ---------- | ----- | --------- | ------ |
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| | | | | | | | 1/2 | 9/2 | 1/2 | | 5/2 | | | 11/2 |
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| | | | | | | | | | | | 126Sn(0+)→ | | 126Sb(2+) | |
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| | | | | | | | finally, | for the | decay transitions | | | | | and |
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III. ADOPTEDMODELSPACESANDNUCLEAR 137Cs(7/2+)→ 137Ba(11/2−,3/2+) the Sn100pn interaction
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HAMILTONIANS
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| | | | | | | | [44]inthe0g | 7/2 | 2s1d0h | 11/2 modelspacewasadopted. | | | | |
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| --- | -------- | ------- | ----- | --------- | -------------- | ---- | ----------- | --- | ------ | -------------------------- | --- | --- | --- | --- |
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| The | electron | spectra | of 16 | forbidden | β− transitions | were | | | | | | | | |
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g
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| calculated | for | different | values | of the coupling | constant | A | | | | | | | | |
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| ---------- | --- | --------- | ------ | --------------- | -------- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
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IV. RESULTSANDDISCUSSION
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| using | the nuclear | matrix | elements | produced | by the | nuclear | | | | | | | | |
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| ----- | ----------- | ------ | -------- | -------- | ------ | ------- | --- | --- | --- | --- | --- | --- | --- | --- |
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shellmodel.Thecalculationsweredoneusingtheshellmodel Below we present our results: the electron spectra of
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code NuShellX@MSU [34], with appropriate model spaces 16 forbidden β− decays (Figs. 1–5) and their integrated
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and well established Hamiltonians chosen for each studied shape factors (Table I). The electron spectra are discussed
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decay individually. No additional truncations, beyond those inSec.IVAandtheintegratedshapefactorsinSec.IVB.
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usedintheoriginalworks,wereintroduced.Thecalculations
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wererunonadesktopcomputerwitha3.3GHzprocessor,so
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| | | | | | | | | A. Electronspectraandtheeffectivevalueofg | | | | | | |
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| ---------------------- | --- | ------ | --- | ----------- | ------------ | ---- | ------------------------------------- | ----------------------------------------- | --- | --- | --- | --- | --- | --- |
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| the computational | | burden | had | to be taken | into account | when | | | | | | | | A |
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| choosingthemodelspace. | | | | | | | Thestudiedspectrafallintothreegroups. | | | | | | | |
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TheadoptedmodelspacesandHamiltoniansareasfollows. GROUP 1. The first group, for which the spectra can be
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sdshell:forthecalculationofthewavefunctionsandone-body found in Figs. 1 and 2, consists of six transitions for which
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| transitiondensitiesrelatedtothedecay36Cl(2+)→ | | | | | 36Ar(0+) | | | | | | | | | |
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| --------------------------------------------- | --- | --- | --- | --- | -------- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
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theSSManalysishasnotbeendonepriortothisstudy.These
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weadoptedthe0d–1smodelspaceandthephenomenological decaysarepotentiallyinterestingforthepracticalapplication
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USDBinteraction[35].pfshell:asintheearliershell-model ofthespectrum-shapemethod,sincemostofthemaresecond-
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studiesregardingthehalf-livesofthetransitions48Ca(0+)→
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| | | | | | | | forbidden | nonunique | transitions | | with | branching | ratios | above |
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| --- | --- | --- | --- | --- | --- | --- | --------- | --------- | ----------- | --- | ---- | --------- | ------ | ----- |
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48Sc(4+,5+,6+)[36]and50V(6+)→ 50Cr(2+)[37],the0f– 90%. The only exception is the undetected fourth-forbidden
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1pmodelspacewiththeinteractionGXPF1A[38,39]wasused β− decay branch of 50V, for which the branching ratio has
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TABLEI. UnitlessintegratedshapefunctionsC˜ ofthestudiedtransitionsandtheirvectorC˜ ,axial-vectorC˜ ,andmixedcomponents
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| | | | | | | | | | V | | | A | | |
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| -------------------------------------- | --- | --- | --- | ----------------------------------------- | --- | --- | --- | --- | --- | ----- | --- | --- | --- | --- |
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| C˜ .ForthetotalintegratedshapefactorC˜ | | | | | | | | | =g | =1.0. | | | | |
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| VA | | | | thevaluesofthecouplingconstantsweresettog | | | | | V A | | | | | |
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| | | | | | | C˜ | | C˜ | | | C˜ | | | C˜ |
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| Transition | | | | Type | | | | | | | | | | |
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| | | | | | | V | | A | | | VA | | | |
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137Cs(7/2+)→137Ba(11/2−) 1stuniq. 9.681×10−11 8.322×10−6 3.761×10−8 8.359×10−6
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36Cl(2+)→36Ar(0+) 6.081×10−9 3.152×10−10 −2.746×10−9 3.650×10−9
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2ndnonuniq.
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60Fe(0+)→60Co(2+) 2.347×10−13 5.232×10−11 −7.000×10−12 4.556×10−11
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2ndnonuniq.
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94Nb(6+)→94Mo(4+) 2ndnonuniq. 1.598×10−8 1.469×10−8 −3.058×10−8 1.029×10−10
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98Tc(6+)→98Ru(4+) 2ndnonuniq. 2.723×10−8 2.544×10−8 −5.254×10−8 1.207×10−10
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99Tc(9/2+)→99Ru(5/2+) 2ndnonuniq. 2.240×10−9 2.130×10−9 −4.361×10−9 8.777×10−12
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126Sn(0+)→126Sb(2+) 1.422×10−8 7.125×10−9 2.011×10−8 4.145×10−8
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2ndnonuniq.
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137Cs(7/2+)→137Ba(3/2+) 4.211×10−6 2.836×10−6 6.879×10−6 1.392×10−5
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2ndnonuniq.
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87Rb(3/2−)→87Sr(9/2+) 3rdnonuniq. 1.185×10−13 2.082×10−14 −9.734×10−14 4.202×10−14
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48Ca(0+)→48Sc(4+) 4thnonuniq. 8.946×10−28 5.934×10−29 −4.606×10−28 4.934×10−28
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50V(6+)→50Cr(2+) 4thnonuniq. 1.024×10−23 9.137×10−25 2.131×10−24 1.329×10−23
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96Zr(0+)→96Nb(4+) 3.176×10−27 3.170×10−28 −2.006×10−27 1.486×10−27
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4thnonuniq.
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48Ca(0+)→48Sc(5+) 2.577×10−30 3.931×10−25 1.334×10−27 3.945×10−25
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4thuniq.
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96Zr(0+)→96Nb(5+) 4thuniq. 1.765×10−29 1.551×10−24 6.844×10−27 1.558×10−24
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48Ca(0+)→48Sc(6+) 6thnonuniq. 1.124×10−32 3.498×10−34 3.947×10−33 1.554×10−32
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96Zr(0+)→96Nb(6+) 5.811×10−33 7.494×10−33 −1.318×10−32 1.234×10−34
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6thnonuniq.
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024317-3
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| JOELKOSTENSALOANDJOUNISUHONEN | | | | PHYSICALREVIEWC96,024317(2017) | | |
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| ----------------------------- | --- | --- | --- | ------------------------------ | --- | --- |
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FIG.2. SameasFig.1butforthesecond-forbiddennonunique FIG.3. Same as Fig. 1 but for the decays 48Ca(0+)→
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decaysof94Nb[panel(a)],98Tc[panel(b)],and126Sn[panel(c)]. 48Sc(4+,5+,6+)[panels(a),(b),and(c)].
|
||
beenpredictedtobe≈2%byarecentshell-modelcalculation
|
||
| | | | beindependentofthevalueofg | A ,whilethedependencewas | | |
|
||
| --- | --- | --- | -------------------------- | ------------------------ | --- | --- |
|
||
[36]. An earlier study [30] of the g -driven evolution of the significant for 87Rb and 99Tc. The shell-model and MQPM
|
||
A
|
||
electronspectraofodd-Anucleidemonstratedthatthesecond spectra of these decays are compared in order to see if the
|
||
andespeciallythefourth-forbiddennonuniqueβ-decayspectra 113Cd 115In
|
||
| | | | previously observed | model independence | of and | |
|
||
| --- | --- | --- | ------------------- | ------------------ | ------ | --- |
|
||
dependstronglyontheeffectivevalueoftheweakaxial-vector spectra[21]holdsalsoforthesedecays.
|
||
couplingconstant. The spectra presented in Figs. 1 and 2 (GROUP 1) fall
|
||
GROUP 2. The second group consists of the fourth- into two subgroups. The spectra of 36Cl, 50V, and 60Fe
|
||
48Ca(0+)→ 48Sc(4+,5+,6+) [Fig.1,panels(a),(b),and(c)],and126Sn[Fig.2,panel(c)]
|
||
| to-sixth-forbidden | decays | | and | | | |
|
||
| ------------------------- | ------ | --- | --- | --- | --- | --- |
|
||
| 96Zr(0+)→ 96Nb(4+,5+,6+), | | | | | | |
|
||
for which the beta spectra are do not exhibit noticeable dependence on the value of the
|
||
presented in Figs. 3 and 4. The decays of 48Ca and 96Zr axial-vector coupling constant. This is very surprising since
|
||
thedecayof50Visfourthforbidden,andallthesevenstudied
|
||
appeartobedominatedbythetwo-neutrinodoublebetadecay
|
||
mode [36,45,46], so the study of these decays is somewhat fourth-forbidden decays in Ref. [30] depend heavily on g .
|
||
A
|
||
challenging experimentally. However, they can shed light on The spectra of the second subgroup, 94Nb and 98Tc [Fig. 2,
|
||
whether the spectra of highly forbidden beta decays are g panels (a) and (b)], are significantly g dependent. The g
|
||
| | | | A | | A | A |
|
||
| --- | --- | --- | --- | --- | --- | --- |
|
||
dependentornot. driven evolution of the spectra is not only very similar for
|
||
GROUP 3. The third group consists of four de- these two decays, but also nearly identical to the one of
|
||
cays: 137Cs(7/2+)→ 137Ba(11/2−,3/2+), 99Tc(9/2+)→ 99Tc, shown in Fig. 5, panel (c), even though the Q value
|
||
| 99Ru(5/2+), | 87Rb(3/2−)→ | 87Sr(9/2+). | | | | |
|
||
| ----------- | ----------- | ----------- | --- | --- | --- | --- |
|
||
and The electron of these decays varies between 300 keV and 450 keV. A
|
||
spectra of these decays was studied previously in [30] using similargroupingphenomenonwasseenforseveraldecaysin
|
||
the MQPM. The spectra of the 137Cs decays were found to [30]:forexampletheshapesofthespectraoffourth-forbidden
|
||
024317-4
|
||
|
||
g -DRIVENSHAPESOFELECTRONSPECTRAOF ... PHYSICALREVIEWC96,024317(2017)
|
||
A
|
||
| | | | in | the present | paper | and | the | fourth-forbidden | | ones | studied |
|
||
| --- | --- | --- | ----------- | ------------ | ------------- | ------------- | -------- | ---------------- | --------- | -------- | -------- |
|
||
| | | | in | [30]. | The branching | | ratio | is, however, | | so small | that the |
|
||
| | | | measurement | | of | this spectrum | | is not | currently | within | reach |
|
||
| | | | of | experiments. | | It is | somewhat | striking | that, | from | the 38 |
|
||
differentforbiddendecaysstudiedinthepresentpaperandin
|
||
| | | | Ref.[30],strongg | | | dependenceisseenonlyineven-forbidden | | | | | |
|
||
| --- | --- | --- | ---------------- | --- | --- | ------------------------------------ | --- | --- | --- | --- | --- |
|
||
A
|
||
decays.
|
||
| | | | | In Fig. | 5 the | shell-model | | and | MQPM | spectra | of the |
|
||
| --- | --- | --- | --- | ------- | ----- | ----------- | --- | --- | ---- | ------- | ------ |
|
||
experimentallyinterestingdecaysfoundinRef.[30]areshown
|
||
| | | | for | comparison. | | For 87Rb | [panels | (a) | and (b)], | which | is the |
|
||
| --- | --- | --- | ------------------------------------------------- | ----------- | --- | -------- | ------- | --- | --------- | ----- | ------ |
|
||
| | | | bestcandidateforthedeterminationofthequenchingofg | | | | | | | | in |
|
||
A
|
||
| | | | third-forbidden | | | decays, | the MQPM | | and shell-model | | spectra |
|
||
| --- | --- | --- | --------------- | --- | --- | ------- | -------- | --- | --------------- | --- | ------- |
|
||
99Tc
|
||
| | | | are | practically | | identical. | For | | [panels | (c) and | (d)] the |
|
||
| --- | --- | --- | --------- | ----------- | --------- | ---------- | ------- | ------ | ------- | ------------- | -------- |
|
||
| | | | evolution | | predicted | by | the two | models | is | also similar, | but |
|
||
| | | | in | the range | g | =0.9–1.1 | there | is | some | difference | in the |
|
||
A
|
||
shape,leadingtoapotentialsmalluncertaintyinthevalueof
|
||
| | | | g | when | comparing | to | the experimental | | spectrum, | | hopefully |
|
||
| --- | --- | --- | --- | ---- | --------- | --- | ---------------- | --- | --------- | --- | --------- |
|
||
A
|
||
availableinthe(near)future.Thetwostudieddecaysof137Cs
|
||
| | | | [panels | (e) | and | (f)] were | found | to be | g A independent | | when |
|
||
| --- | --- | --- | ------- | --- | --- | --------- | ----- | ----- | --------------- | --- | ---- |
|
||
usingtheMQPMmodelinRef.[30].Theshell-modelspectra
|
||
| | | | calculatedforthesedecayswerenotonlyalsog | | | | | | | independent, | |
|
||
| --- | --- | --- | ---------------------------------------- | --- | --- | --- | --- | --- | --- | ------------ | --- |
|
||
A
|
||
butoverlapwiththeMQPMspectraperfectly.Measurements
|
||
| | | | of | these | spectra | would | by very | desirable, | | since they | would |
|
||
| --- | --- | --- | --- | ----- | ------- | ----- | ------- | ---------- | --- | ---------- | ----- |
|
||
shedlightonthequalityofthecalculatedspectra,independent
|
||
ofthechosennuclearmodel.
|
||
| | | | | | | B. IntegratedshapefactorC˜ | | | | | |
|
||
| --- | --- | --- | --- | --------------------------- | --- | -------------------------- | --- | ----------------------- | --- | --- | --- |
|
||
| | | | | TheintegratedshapefactorsC˜ | | | | andtheirdecompositionto | | | |
|
||
thecomponentsofEq.(8)arepresentedinTableI.Forallthe
|
||
studieddecaysthesignofthevectorandaxial-vectorcompo-
|
||
nentsispositiveandthesignofthemixedvector-axial-vector
|
||
componentvaries.Forthesecond-forbiddennonuniquedecays
|
||
| | | | of | 94Nb and | 98,99Tc | which | have | very | similar | spectrum-shape | |
|
||
| --- | --- | --- | --- | -------- | ------- | ----- | ---- | ---- | ------- | -------------- | --- |
|
||
evolution,thevector-andaxial-vectorcomponentsareroughly
|
||
equalandthevector-axial-vectorcomponentisroughlytwice
|
||
96Zr(0+)→
|
||
| FIG.4. | Same as Fig. 1 | but for the decays | | | | | | | | | |
|
||
| ------ | -------------- | ------------------ | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|
||
96Nb(4+,5+,6+)[panels(a),(b),and(c)]. aslarge,butwithanegativesign.Theresultingtotalintegrated
|
||
| | | | shape | factor | is two | magnitudes | | smaller | than | its components. | |
|
||
| --- | --- | --- | ----------------------------------- | ------ | ------ | ---------- | --- | ------- | ----------------- | --------------- | --- |
|
||
| | | | Forthesixth-forbiddendecay96Zr(0+)→ | | | | | | 96Nb(6+)theaxial- | | |
|
||
ground-state-to-ground-stateβ−decaysof97Zr,101Mo,117Cd,
|
||
vectorcomponentis30%largerthanthevectorcomponentand
|
||
and119Inhaveanalmostidenticalg evolution. thevector-axial-vectorcomponentisagainroughlythesumof
|
||
A
|
||
Decays of GROUP 2, 48Ca(0+)→ 48Sc(4+,5+,6+) and thetwoothercomponentsandnegativeinsign.
|
||
| 96Zr(0+)→ | 96Nb(4+,5+,6+),werestudiedinordertoseeif | | | 87Rb | | | | | | | |
|
||
| --------- | ---------------------------------------- | --- | --- | ---- | --- | ------------- | --- | ---------- | --- | -------- | ------ |
|
||
| | | | | For | the | decomposition | | is similar | to | the case | of the |
|
||
C˜
|
||
thereappearssomeinterestingsystematicbehavior.Sincethe MQPM: the vector component is the largest, the axial-
|
||
V
|
||
shape of the spectra of fourth-forbidden nonunique decays vectorcomponentC˜ isabout20%ofC˜ ,andthevector-axial-
|
||
| | | | | | | A | | | V | | |
|
||
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|
||
of odd-A nuclei depends on g in a significant way, it vectorcomponentC˜ isthesmallestandhasanegativesign.
|
||
| | | A | | | | VA | | | | | |
|
||
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|
||
is interesting to see if this holds for the even-A nuclei. However,thecomponentsandthetotalintegratedshapefactor
|
||
Since the spectrum on 50V is largely independent of the are approximately 30% larger for the MQPM. In the case of
|
||
valuesoftheweakcouplingconstants,thedependenceisnot 99TcthedifferenceismuchlargerandtheMQPMresultsare
|
||
strong for at least some fourth-forbidden decays. The shape roughlytwicetheshellmodelones.Forthetwostudieddecays
|
||
48Ca(0+)→ 48Sc(4+,5+,6+)
|
||
of the electron spectra of and of137Csthereisnog dependence.Thedecaytotheisomeric
|
||
A
|
||
96Zr(0+)→ 96Nb(4+,5+)transitionsisindependentofg ,as 11/2−stateisuniqueandthus(practically)g independent.On
|
||
| | | A | | | | | | | A | | |
|
||
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|
||
showninFig.3,panels(a),(b),and(c)andinFig.4,panels(a) theotherhand,thedecaytothe3/2+groundstateisnonunique
|
||
and(b),revealingasecondexception,48Ca(0+)→ 48Sc(4+), independentsincethethreecomponentsofC˜
|
||
| | | | butg | A | | | | | | | havethe |
|
||
| --- | --- | --- | ---- | --- | --- | --- | --- | --- | --- | --- | ------- |
|
||
totheg -dependentfourth-forbiddennonuniquedecays.The samesignandnointerferencebetweenthecomponentsoccurs.
|
||
A
|
||
sixth-forbidden decay 96Zr(0+)→ 96Nb(6+) [Fig. 4, panel ForthesetwodecaysthenormalizedshellmodelandMQPM
|
||
C˜
|
||
(c)],ontheotherhand,exhibitsastrongg A dependence.The electron spectra overlap perfectly but the difference in is
|
||
evolutionisverydifferentfromtheoneofthe94Nband98Tc considerable. Again, the relative fractional decomposition is
|
||
[Fig.2,panels(a)and(b)],and99Tc[Fig.5,panel(c)]studied almost identical for the two nuclear models, but the absolute
|
||
024317-5
|
||
|
||
| JOELKOSTENSALOANDJOUNISUHONEN | | | | | | PHYSICALREVIEWC96,024317(2017) | | | |
|
||
| ----------------------------- | --- | --- | --- | --- | --- | ------------------------------ | --- | --- | --- |
|
||
FIG.5. Presentlycalculatedshell-modelspectraandtheMQPMspectra,firstpublishedinRef.[30],fortheground-state-to-ground-state
|
||
decaysof87Rb[panels(a)and(b)],99Tc[panels(c)and(d)],and137Cs[panel(f)],andthedecayof137Cstotheisomeric11/2−statein137Ba
|
||
[panel(e)].Thecolorcodinginthespectraof87Rband99Tcreferstothevalueofg .Theshapeoftheelectronspectraofthetwo137Csdecays
|
||
A
|
||
isg independent,andthecolorcodinginthesedecaysreferstotheadoptednuclearmodel.Thefree-nucleonvalueg =1.00wasusedinall
|
||
| A | | | | | | | V | | |
|
||
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|
||
calculations.
|
||
valueoftheMQPMC˜ is40-foldtheshellmodeloneforthe objective of the study was to find experimentally detectable
|
||
decaytothe11/2−state,and8-foldforthedecaytothe3/2+ transitionsforwhichtheshapeoftheelectronspectradepends
|
||
state.DuetotheC˜−1dependenceofthehalf-life,thehalf-lives sensitively on the value of the weak coupling constants.
|
||
predicted by the two models do not agree with each other at Comparingthecalculatedandmeasuredspectratheeffective
|
||
all.Thisisastrongindicationthatthespectrum-shapemethod valueofg canbeextracted.
|
||
A
|
||
| | | | | | The electron | spectra | of 94Nb(6+)→ | 94Mo(4+) | and |
|
||
| --------------------------------------------------- | --- | --- | --- | --- | ------------ | -------- | ------------ | -------- | --- |
|
||
| isamorerobusttoolfordeterminingtheeffectivevalueofg | | | | A | | | | | |
|
||
| | | | | | 98Tc(6+)→ | 98Ru(4+) | | | |
|
||
thanjustasimplehalf-lifecomparison. depend significantly on the effective
|
||
| | | | | | value of | g and these transitions | are | thus excellent | new |
|
||
| --- | --- | --- | --- | --- | -------- | ----------------------- | --- | -------------- | --- |
|
||
A
|
||
| | | | | | candidates | for the spectrum-shape | method | (SSM). | Both |
|
||
| --- | --- | --- | --- | --- | ---------- | ---------------------- | ------ | ------ | ---- |
|
||
V. CONCLUSIONS
|
||
| | | | | | of these | decays have almost | a 100% | branching | ratio, |
|
||
| --- | --- | --- | --- | --- | -------- | ------------------ | ------ | --------- | ------ |
|
||
The evolution of the electron-spectrum shapes of 16 and have been experimentally observed. The shell-model
|
||
| | β− | | | | spectra of | the previously | found candidates, | the | ground- |
|
||
| --------- | -------------- | ------ | ------------------------- | --- | ---------- | -------------- | ----------------- | ----- | ------- |
|
||
| forbidden | decays, driven | by the | value of the axial-vector | | | | | | |
|
||
| | | | | | | | 87Rb | 99Tc, | |
|
||
coupling constant g , were studied using nuclear matrix state-to-ground-state decays of and agree well
|
||
A
|
||
elements derived from the nuclear shell model. Established with the microscopic quasiparticle-phonon model predicted
|
||
| | | | | | spectra of | a previous study, | which makes | the experimen- | |
|
||
| --------------- | ------- | ------------ | --------- | -------- | ---------- | ----------------- | ----------- | -------------- | --- |
|
||
| and well-tested | nuclear | Hamiltonians | were used | in these | | | | | |
|
||
investigations.Intheβ-decayshapefactorstheusuallyomitted tal measurement of these spectra very desirable. The g A -
|
||
next-to-leading-ordertermsweretakenintoaccount.Themain independent shell model and MQPM spectrum shapes of the
|
||
024317-6
|
||
|
||
g -DRIVENSHAPESOFELECTRONSPECTRAOF ... PHYSICALREVIEWC96,024317(2017)
|
||
A
|
||
137Cs(7/2+)→ 137Ba(11/2−,3/2+) A nuclei with A(cid:2)98 have now been studied. The potential
|
||
| | | | transitions overlap | per- | | | | |
|
||
| --- | --- | --- | ------------------- | ---- | --- | --- | --- | --- |
|
||
fectly.Themeasurementofthesespectrawouldshedlighton new candidates for the spectrum-shape method are heavier,
|
||
thequalityofthecalculatedspectra. and the study of these decays requires the adoption of some
|
||
othernuclearmodelduetothelargecomputationalburdenand
|
||
| Thetransitionsforwhichaheavyg | | | A dependencehasbeen | | | | | |
|
||
| ----------------------------- | ------------ | ------------------- | ------------------- | ----------- | ------------------ | ---------------- | ------------ | --------- |
|
||
| | | | | | the other problems | with shell-model | calculations | for heavy |
|
||
| recorded | thus far are | all even forbidden. | Unlike | in the case | | | | |
|
||
ofthefourth-forbiddentransitionsinodd-Anucleistudiedin deformednuclei.
|
||
| Ref. [30], | which all | showed a strong | g dependence, | such a | | | | |
|
||
| ---------- | --------- | --------------- | ------------- | ------ | --- | --- | --- | --- |
|
||
A
|
||
dependenceisnotseeninthecorrespondingtransitionsinthe
|
||
ACKNOWLEDGMENTS
|
||
studiedeven-Anuclei.Thusfarallthefoundtransitionswith
|
||
averystronglyg -dependentspectrumshapeareinthemass ThisworkhasbeenpartiallysupportedbytheAcademyof
|
||
A
|
||
A=94–119.
|
||
region Most of the experimentally accessible Finland under the Finnish Centre of Excellence Programme
|
||
forbiddennonuniquetransitionsinodd-AnucleiwithA(cid:2)169
|
||
| | | | | | 2012-2017 (Nuclear | and Accelerator | Based | Programme at |
|
||
| ------------------------------------------------------ | --- | --- | --- | --- | ------------------ | --------------- | ----- | ------------ |
|
||
| andsecondorhigher-forbiddennonuniquetransitionsineven- | | | | | JYFL). | | | |
|
||
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| --------------------------------- | --- | --- | --- | --- | ------------- | --- | --- | --- |
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||
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| ------------- | --- | --- | --- | --- | ------------- | --- | --- | --- |
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| ------- | --- | --- | --- | --- | --------------------------------------------------- | --- | --- | --- |
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| -------------- | --- | --- | --- | --- | ---------------------------------------- | --- | --- | --- |
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