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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
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343 lines
20 KiB
Markdown
IL NUOVO CIMENTO Vol. ?, N. ? ?
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Shell-model treatment of the β decay of 99Tc
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M. Ramalho(1) and J. Suhonen(1)(2)
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(1) DepartmentofPhysics,UniversityofJyva¨skyla¨,P.O.Box35,FI-40014,Jyva¨skyl¨a,Finland
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(2) International Centre for Advanced Training and Research in Physics (CIFRA), P.O. Box
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MG12, 077125 Bucharest-Magurele, Romania
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Summary.— Inthepresentpaperwetreatthesecond-forbiddennon-unique(2nd-
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nu)ground-state-to-ground-stateβ−decay99Tc(9/2+)→ 99Ru(5/2+),witha100%
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branchingratio,withintheframeworkofthenuclearshellmodel(NSM).Theenergy
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spectrum of the electrons emitted in this β-decay transition (β-electron spectrum)
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is sensitive to the wave functions of the involved initial (9/2+) and final (5/2+)
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nuclear ground states through the many involved nuclear matrix elements (NME).
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The β-electron spectrum of this transition is potentially indicative of the effective
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value, geff, of the weak axial coupling, g , of crucial importance for extraction of
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A A
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information on beyond-the-standard-model physics from the results of the present
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and future rare-events experiments. We describe the β spectral shape of this de-
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cay by using a state-of-the-art β-decay formalism and compute the many involved
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NME using the well established NSM Hamiltonians jj45pnb and glekpn. We have
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found a strong dependence of the β spectral shape on the value of g making it a
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A
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good candidate for determination of the value of geff through comparison with the
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A
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corresponding experimental β spectral shape. We have also found an interesting
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dependence of the β spectral shape on the value of the so-called small relativistic
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vector NME, sNME, used to match the computed half-life with the measured one.
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1. – Introduction
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Rare-eventsexperimentstypicallylookforbeyond-the-standard-model(BSM)physics
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by,e.g.,measuringrarenuclearβ decays,doubleβ decays,and(anti)neutrinoandWIMP
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(weakly interacting massive particle) scatterings off nuclei in searches for astrophysical
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neutrinos and cold dark matter of the Universe. In recent years, a booming interest in
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these experiments has concentrated on studies of β electrons (electrons emitted in β
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−
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decays) and their energy distributions, the so-called β-electron spectra. Experimental
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and/ortheoreticalinformationonthesespectraiscrucialfor,e.g.,resolvingtheanomalies
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related to the antineutrino flux from nuclear reactors and for resolving the common
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backgroundsintherare-eventsexperimentsthemselves. Also,pinningdowntheeffective
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'Societ`aItalianadiFisica 1
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2 M.RAMALHOand J.SUHONEN
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values of weak couplings is a considerable incentive for those present and future β-decay
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experiments able to tackle the spectral shapes of β electrons, see the review [1].
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Nuclearβdecaysvaryincomplexity: fromallowedtohighlyforbiddenones[2,3]. Like
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in the case of allowed β decays, also in the case of forbidden unique β decays the lepton
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phase space can be separated from the nuclear part, resulting in a universal β spectral
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shape,independentofnuclear-structuredetails[2]. Ofspecialinterestfortherare-events
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experiments are the forbidden non-unique β decays for which the β spectral shapes can
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bestronglynuclear-structuredependentthroughseveralnuclearmatrixelements(NME).
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ThevaluesoftheseNMEarenon-triviallydeterminedbythewavefunctionsoftheinitial
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and final states of a β-decay transition.
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In addition to the many NME, the (partial) half-life of a forbidden non-unique β
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transition depends on the values of the weak vector and axial-vector couplings, g and
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V
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g [3]. In atomic nuclei, the CVC (conserved vector current) hypothesis sets the value
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A
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g =1.0, whereas the PCAC (partially conserved axial-vector current) hypothesis leads
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V
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only to the so-called effective value of g , recently discussed in the reviews [1, 4, 5].
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A
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On the other hand, the so-called bare-nucleon value g = 1.27 stems from data on the
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A
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decay of an isolated neutron. Typically, the effective value of g is quenched relative
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A
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to the bare-nucleon value which can have drastic effects on the sensitivity estimates of
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rare-events experiments trying to detect the neutrinoless double beta decay [6, 7], of
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crucial importance in the search for the BSM physics. Only in rare cases there is an
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enhancement of g present [8].
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A
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As discussed in [9, 10], information on the value of g can be gained by using the
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A
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so-called spectrum-shape method (SSM). Use of SSM requires a β-electron spectrum
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with a notable g dependence in its shape. In this case information on the effective
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A
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value of g can be gained through the comparison of computed template β spectra,
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A
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for different g values, with the measured one in the aim to find a match. Such SSM
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A
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analyses of β-spectral shapes of individual β transitions have been done recently for
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−
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the fourth-forbidden non-unique β decays of 113Cd and 115In in [11, 12, 13].
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Anadditionalingredientinthetheoreticalanalysesofβspectralshapesistheso-called
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small relativistic vector NME, the sNME, used to fix the measured (partial) half-life in
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the so-called enhanced SSM [14, 15]. In spite of its smallness, sNME can influence the
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(partial) half-lives and shapes of β-electron spectra quite strongly, see [12, 16, 17]. The
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sNME gathers its major contributions outside the proton (neutron) valence major shell
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thatcontainstheproton(neutron)Fermisurfacemakingithardtobecalculatedinmany
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nuclear models.
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In the present paper we investigate the g and sNME dependence of the second-
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A
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forbidden non-unique (2nd-nu) β decay of the 9/2+ ground state of 99Tc to the 5/2+
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−
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ground state of 99Ru, with a branching ratio of 100%.
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2. – Theoretical Background
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.
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21. NSM Hamiltonians. – The NSM calculations were performed using the software
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KSHELL [18] with the interactions jj45pnb [19] and glekpn [20] with the set of single-
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particle energies fit for the mass region of A = 94 98. Their model spaces consist of
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−
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the proton 0f (2)-1p -1p -1g orbitals and the neutron 0g -1d -1d -2s -
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5/2 3/2 1/2 9/2 7/2 5/2 3/2 1/2
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0h orbitals for jj45pnb, as well as 0f (1)-0f (2)-1p -1p -1g proton orbitals
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11/2 7/2 5/2 3/2 1/2 9/2
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and 0g (1)-0g -1d -1d -2s neutron orbitals for glekpn. The restrictions in the
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9/2 7/2 5/2 3/2 1/2
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orbitaloccupationsareimposedinordertoreducethecomputationalburden. Webelieve,
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SHELL-MODELTREATMENTOFTHEβ DECAYOF99TC 3
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99Tc 0 0 .7 . 5 5 ( 3 2 5 2 3 2 7 2 − − − )+ ( ( 3 2 9 2 7 2 5 2 ) ) + + + + 1 2 1+ 1 2 5 2 3 2 1 2 3 − − + + 1 2 1+ 3 2 7 2 − + 1 9 2 2 1 + − 9 2−
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0.25
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7 2 + 1 2 5 2 − + 9
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5
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2 +
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+
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1 2−
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7 2−
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2
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5+ 2 0 9 2 + 7 2 + 5 2−
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Exp. jj45pnb glekpn
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)VeM(ygrenEnoitaticxE 99Ru 1 2 3 2+ + 5 2 + 9 2 + 5 3 2+ + 9+ (5 2)+ 9 2 + 2 2 7+ (1 2)+ (5 2 7 2 ) + + 2 1 2 1 2 1+ + 5 2 + (3 2)+ 1 2 +
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7+
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3 2 + 2 7 2 + 3 2 + 3+ 2
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3+
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2 5+ 7+ 5+ 2 3+ 5+ 2 2 2 2
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Exp. jj45pnb glekpn
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Fig.1.: Energylevelschemesfor99Tcand99RuwiththeircorrespondingNSM-computed
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values using the interactions jj45pnb and glekpn. Experimental data are taken from the
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ENDSF [21] evaluation. Levels in blue are the initial and final states relevant for this
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work.
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however,thattheseposedlimitationsdonotaffectmuchthewavefunctionsoftheinitial
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and final ground states, relevant for the present work.
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With the considered truncations we have calculated the excitation spectra for both
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interactions and compare them with the experimental spectrum in Fig 1. It can be
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seen that the jj45pnb Hamiltonian works, overall, much better than the glekpn one, in
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particular for the ground state of 99Tc.
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To further assess the reliability of the Hamiltonians employed, we have analyzed the
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magnetic dipole and electric quadrupole moments of the 99Tc and 99Ru ground states.
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The analysis can be seen in Table I, where once again the jj45pnb Hamiltonian does
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better overall than the glekpn Hamiltonian.
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.
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22. β spectral shapes and half-lives. – The half-life of a β transition can be obtained
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fromt =κ/C˜,whereκisaconstant[14,15]andC˜ istheintegratedshapefunctionC,
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1/2
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(1) Orbital fixed with maximum amount of nucleons, e.g, 8 for 0f .
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7/2
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(2) Orbital with at least 4 particles.
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TableI.: Comparisonofexperimentalandtheoretical(jj45pnbandglekpnHamiltonians)
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nuclear magnetic dipole (in units of nuclear magneton) and electric quadrupole (in units
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of barns) moments for 99Tc and 99Ru, for the experimental ground states and their
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correspondents in theory.
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99Tc(9/2+) 99Ru(5/2+)
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Exp. jj45pnb glekpn Exp. jj45pnb glekpn
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µ (µ ) +5.678(2) +5.662 +5.4160 -0.641(5) -0.2875 +1.0826
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N
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Q (b) -0.129(6) +0.0218 +0.2428 +0.079(4) -0.0018 -0.3222
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| 4 | | | | | | | | M.RAMALHOand | | |
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| --- | --- | --- | --- | --- | --- | --- | --- | ------------ | --- | --- |
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J.SUHONEN
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Table
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II.: Values of the small relativistic NME emerging from the fitting procedure for
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the 99Tc(9/2+) Ru(5/2+) decay. The sNME values (1) and (2) shown reproduce the
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→
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| experimental | | half-life | for | their | given geff. | | | | | |
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| ------------ | --- | --------- | --- | ----- | ----------- | --- | --- | --- | --- | --- |
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A
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| | jj45pnb | | - CVC =−0.1705 | | | | glekpn | - CVC | =−0.0080 | |
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| ---- | ------- | ------- | -------------- | ------- | --- | ---- | ------ | ------- | -------- | ------- |
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| geff | | sNME(1) | | sNME(2) | | geff | | sNME(1) | | sNME(2) |
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| A | | | | | | A | | | | |
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| 0.8 | | 0.0496 | | -0.0856 | | 0.8 | | 0.1042 | | -0.0311 |
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| 0.9 | | 0.0582 | | -0.0772 | | 0.9 | | 0.1091 | | -0.0257 |
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| 1.0 | | 0.0666 | | -0.0687 | | 1.0 | | 0.1140 | | -0.0202 |
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| 1.1 | | 0.0750 | | -0.0601 | | 1.1 | | 0.1189 | | -0.0148 |
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| 1.2 | | 0.0833 | | -0.0515 | | 1.2 | | 0.1237 | | -0.0093 |
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fullofphase-spacefactorsandNMEsinthenext-to-leading-orderexpansion,asdiscussed
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in detail in [9, 10]. Our calculations employ screening, radiative, and atomic exchange
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corrections. The most essential to the current study, due to its small β-decay Q value,
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is the atomic exchange correction which was originally derived for allowed β decays [22];
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the effects of this correction are the most important for electron energies below 50 keV.
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The complexity of the shape function C can, however, be cast in a very simple de-
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| pendence | | on the | weak couplings | | by writing | | | | | |
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| -------- | --- | ------ | -------------- | ----- | ---------- | ---------- | --- | --------- | ---- | --- |
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| | | | | )=g2C | )+g2C | | | | | |
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| (1) | | | C(w e | | V (w e | A (w e )+g | V g | A C VA (w | e ), | |
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| | | | | V | | A | | | | |
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where w is the total (mass plus kinetic) energy of the emitted electron. The variation
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e
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of the shape of the β-electron spectrum with the value of g comes from the subtle
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A
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interference of the combined vector C (w ) and axial C (w ) parts with the mixed
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| | | | | | V | e | | A e | | |
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| ------------ | --- | ---- | ---- | ------ | --- | --- | --- | --- | --- | --- |
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| vector-axial | | part | C (w | ) [9]. | | | | | | |
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| | | | VA e | | | | | | | |
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Forthecurrentlystudied2nd-nudecaytransition99Tc(9/2+) 99Ru(5/2+)wetake
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→
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theCVC-compatiblevalueg =1.0fortheweakvectorcoupling,andexploretheimpact
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V
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ofthequenchedgeff
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| | | | values,rangingfrom0.8 | | | 1.2(seeTableII),ontheβ | | | spectralshape. | |
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| --- | --- | --- | --------------------- | --- | --- | ---------------------- | --- | --- | -------------- | --- |
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| | | | A | | | − | | | | |
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.
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23.SmallrelativisticNuclearElements.–Animportantcontributortotheβ spectral
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shape and the half-life is the small relativistic NME (sNME) [12, 15, 16, 17]. In these
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works, by fitting the value of the sNME, alongside with geff, one is able to reproduce
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A
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both the β spectral shape and the half-life of a β transition. The computation of the
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value of the sNME is a challenge for the ISM. This is why its CVC value (also presented
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in Table II) can be used as a reference [3]. It should be noted, though, that the CVC
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value represents an ”ideal”, pertaining to a perfect many-body theory, which is not the
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case for NSM, mainly due to its restricted proton and neutron valence spaces.
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Following the philosophy adopted in [12, 15, 16, 17], we search for the values of the
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sNME that reproduce the experimental half-life of the 99Tc(9/2+) Ru(5/2+) transi-
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→
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tion. Sincetheβ-decayratehasaquadraticdependencyonthesNME,twosolutionsare
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geff,
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| usually | found | for | each value | of | as presented | in | Table | II. | | |
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| ------- | ----- | --- | ---------- | --- | ------------ | --- | ----- | --- | --- | --- |
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A
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| 3. – | Results | for | the β | spectral | shapes | | | | | |
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| ---- | ------- | --- | ----- | -------- | ------ | --- | --- | --- | --- | --- |
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Theβ-decaytransitionofinterest,99Tc(9/2+) Ru(5/2+),is2nd-nuandissensitive
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→
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to the Q value of the decay. Thus, to ensure reliability of our β spectral-shape calcula-
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SHELL-MODELTREATMENTOFTHEβ DECAYOF99TC 5
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10 3
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−
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×
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8
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6
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4
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2
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0
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0 50 100 150 200 250 300
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Electronenergy(keV)
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).mron(Ed/Nd
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jj45pnb glekpn
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10 3
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−
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×
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12
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8
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4
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0
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0 50 100 150 200 250 300
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Electronenergy(keV)
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Fig. 2.: Computed β spectral shapes for the decay transition 99Tc(9/2+) Ru(5/2+)
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→
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using the Hamiltonians jj45pnb (left panel) and glekpn (right panel). The crossed-blue
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curvescorrespondtothefittedsNMEclosesttotheCVC-valueandthecircled-redtothe
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farthest(seeTableII),withgeff =1.0. Thegray-hatched(gray-crossed)areacorresponds
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A
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to the range geff = 0.8 1.2 with all the fitted sNME values chosen closest to (farthest
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A −
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from)thecorrespondingCVCvalues. Allcurvesarenormalizedtounitareaunderthem.
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tions, we have fixed the energy difference of the initial 9/2+ and final 5/2+ state by the
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experimental Q value. We plot our results for the β spectral shapes of this transition in
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Fig. 2. There the crossed-blue (circled-red) curves correspond to a typical moderately
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quenched value geff =1.0 [4] and the fitted sNME value closer to (farther from) its CVC
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A
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value, see Table II. The gray-hatched (gray-crossed) area, in turn, corresponds to the
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range geff =0.8 1.2 with all the sNME values chosen closest to (farthest from) the cor-
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A −
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respondingCVCvalues,thesNMEvaluesbeingshowninTableII. Interestinglyenough,
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thetwodifferentoptionsforthesNMEvalue(thetwodifferenthatchedregionsinFig.2)
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sometimes produce drastically different shapes, as can be seen for glekpn, a pattern also
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recorded in [17]. As for the interaction jj45pnb, both hatched regions overlap nicely and
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there seems not to be any strong dependency of the β spectral shape on the choice of
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the value of sNME.
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Since the β spectral shapes produced by the two interactions clearly disagree, it is
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hard to judge which one of the shapes is the more realistic one. Based on both the level
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schemesinFig. 1andthemagneticandelectricmomentsofTableIitistemptingtosay
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that the jj45pnb Hamiltonian is better suited for these particular nuclei. Furthermore,
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looking at a recently measured experimental β spectral shape for this transition in [23],
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especially in their Fig. 5, one can see a striking resemblance of it to the spectral shapes
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forbothfittedsNME(1,2) ofthejj45pnb Hamiltonian(seeTableII)andabitlessstriking
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forthesolutionsNME(2) fortheglekpn Hamiltonian. However,onlyapropercomparison
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with data will allow us to draw any definite conclusions concerning the effective value of
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the axial coupling and the proper value of sNME. This we leave as a future exercise.
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| 6 | | | | | M.RAMALHOand | |
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| --- | --- | --- | --- | --- | ------------ | --- |
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J.SUHONEN
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| 4. – Summary | and | conclusions | | | | |
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| ------------ | --- | ----------- | --- | --- | --- | --- |
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We have performed shell-model calculations of the β spectral shape related to the
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second-forbidden non-unique β -decay transition 99Tc(9/2+) 99Ru(5/2+), with a
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−
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→
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100% branching ratio. We use a state-of-the-art β-decay formalism and compute the
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many involved NME using the well established NSM Hamiltonians jj45pnb and glekpn.
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When compared with future available data, the β-electron spectrum of this transition
|
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is potentially indicative of the effective value of the weak axial coupling, g , of crucial
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A
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importance for extraction of information on beyond-the-standard-model physics from
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the results of the present and future rare-events experiments. We also point out the
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important role played by the small relativistic vector matrix element, used to reproduce
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the measured half-life of the transition, in shaping the β-electron spectrum.
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5. – Acknowledgments
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WeacknowledgethesupportbyCSC–ITCenterforScience,Finland,forthegenerous
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| computational | resources, | which | were | essential to | this project. | |
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| ------------- | ---------- | ----- | ---- | ------------ | ------------- | --- |
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REFERENCES
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| ---------- | ------------ | ------- | ------ | --------------- | ------------- | --- |
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[2] Suhonen,J.,FromNucleonstoNucleus: ConceptsofMicroscopicNuclearTheory,Springer,
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| -------------- | ----------- | --- | ------------ | --------- | ----------------- | ------- |
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