beta_shape_pnpi/.llm-wiki/raw/sources/SRC-2026-05-27-003/extracted.md

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Kostensalo, Joel; Suhonen, Jouni Author(s): gA-driven shapes of electron spectra of forbidden β decays in the nuclear shell model Title: Year: 2017 Version: Published version Copyright: © 2017 American Physical Society. Published in this repository with the kind permission of the publisher. Rights: Rights url: Please cite the original version: Kostensalo, J., & Suhonen, J. (2017). gA-driven shapes of electron spectra of forbidden β decays in the nuclear shell model. Physical Review C, 96(2), Article 024317. https://doi.org/10.1103/PhysRevC.96.024317

PHYSICALREVIEWC96,024317(2017) g -drivenshapesofelectronspectraofforbiddenβ decaysinthenuclearshellmodel A JoelKostensalo* andJouniSuhonen† DepartmentofPhysics,UniversityofJyvaskyla,P.O.Box35,FI-40014,Finland (Received10May2017;published21August2017) The evolution of the shape of the electron spectra of 16 forbidden β− decays as a function of g was A studiedusingthenuclearshellmodelinappropriatesingle-particlemodelspaceswithestablished,well-tested nuclearHamiltonians.Theβspectraof94Nb(6+)→ 94Mo(4+)and98Tc(6+)→ 98Ru(4+)werefoundtodepend stronglyong ,whichmakesthemexcellentcandidatesforthedeterminationoftheeffectivevalueofg withthe A A spectrum-shapemethod(SSM).Astrongg dependenceisalsoseeninthespectrumof96Zr(0+)→ 96Nb(6+). A Thisdecaycouldbeusedfordeterminingthequenchingofg insixth-forbiddendecaysinthefuture,whenthe A measurementofthespectrumbecomesexperimentallyfeasible.Thecalculatedshell-modelelectronspectraofthe ground-state-to-ground-statedecaysof87Rb,99Tc,and137Csandthedecayof137Cstotheisomeric11/2−state in137Bawerefoundtobeinexcellentagreementwiththespectrapreviouslycalculatedusingthemicroscopic quasiparticle-phononmodel.ThisisfurtherevidenceoftherobustnatureoftheSSMobservedintheprevious studies. DOI:10.1103/PhysRevC.96.024317 I. INTRODUCTION valueoftheweakcouplingconstants.IntheSSMtheshapes of computed and experimental (normalized) electron spectra At the nuclear level β decay can be considered as a are compared in order to find the ratio g /g for nonunique mutual interaction of the hadronic and leptonic currents forbiddenbetaβ−decays,forwhichthes A hap V efactorsdepend mediated by a massive vector boson W± [1]. The leptonic ong /g inaverynontrivialway.InRef.[21]theSSMwas and hadronic currents can be expressed as a mixture of both A V applied to the fourth-forbidden ground-state-to-ground-state vector and axial-vector components [2–4]. The weak vector transitionof113Cdusingthemicroscopicquasiparticle-phonon andaxial-vectorcouplingconstantsg andg enterthetheory V A model (MQPM) [22,23], the nuclear shell model (NSM) when the hadronic current is renormalized at the nucleon [24–26], the microscopic interacting boson-fermion model level[5].Theconservedvector-currenthypothesis(CVC)and (IBFM-2) [27,28], and experimental spectrum of Ref. [29]. partially conserved axial-vector-current hypothesis (PCAC) The closest match between the theoretical and experimental yield the free-nucleon values g =1.00 and g =1.27 [6] V A spectra was found when g /g ≈0.92 for all three nuclear but inside nuclear matter the value of g is affected by A V A models.Thehalf-lifecomparisonmethod,ontheotherhand, many-nucleon correlations and a quenched value might be gave very different results when different nuclear models neededtoreproduceexperimentaldata[1].Preciseinformation were used. In Ref. [30] it was noticed that the shape of the ontheeffectivevalueofg iscrucialwhenpredictinghalf-lives A spectrum was largely unaffected by modest changes in the of neutrinoless double beta decays since the half-lives are MQPM wave function, even though the predicted half-life proportionaltothefourthpowerofg [7,8]. A wasaffectedsignificantly.Theobservationsoftheseprevious The effective value of g has earlier been probed using a A studiessuggestthattheSSMisveryrobust,quiteinsensitiveto half-lifecomparisonmethod,inwhichthepredictedandexper- thedetailsoftheadoptednuclearmeanfieldandHamiltonian. imentalvaluesarecomparedfordifferentvaluesofg .Thishas A Itisthusareliabletoolforprobingtheratiog /g . beendoneforGamow-Tellerandfirst-forbiddendecaysusing A V The g -driven evolution of the shapes of several exper- nuclear matrix elements calculated with the proton-neutron A imentally measurable forbidden nonunique decay spectra in quasiparticlerandom-phaseapproximation(pnQRPA)[9–13]. medium-heavyodd-AnucleiwasstudiedusingtheMQPMin Thehalf-lifecomparisonmethodpredictsthatthevalueofg A Ref.[30].Onlyfourgoodcandidatesfortheapplicationofthe is quenched significantly in these transitions. Also in the old spectrum-shape method have been found thus far. These are studies[14–17]offirst-forbiddenβdecays,strongquenchings theground-state-to-ground-statedecaysof87Rb,99Tc,113Cd, werepredicted.Inarecentshell-modelstudy[18]ofr-process and 115In [20,30]. In the present paper we set out to find waiting-pointnuclei,astrongquenchingofg wasconfirmed. A potential new candidates for the application of the SSM in Thehalf-lifecomparisonmethodcouldbeusedtofindoutif light to medium-heavy nuclei using the nuclear shell model. similar quenching of g is seen in highly forbidden unique A We also compare the NSM and MQPM β spectra of 87Rb, beta decays by using the half-lives calculated in Ref. [19], 99Tc,and137Cstoseeifthemodelindependenceobservedin onceexperimentaldatabecomeavailable. Ref.[21]appliesforthesedecaysaswell.Thedecaysof137Cs In Ref. [20] the spectrum-shape method (SSM) was tothe3/2+groundstateandthe11/2−isomericstateof137Ba introduced as a complementary way to study the effective are particularly interesting for comparing the differences in the MQPM and NSM spectra, since the MQPM calculations predictag -independentspectrumshapeforthesedecays. A *joel.j.kostensalo@student.jyu.fi This article is organized as follows. In Sec. II we give †jouni.suhonen@phys.jyu.fi the theoretical background behind the β spectrum shape. In 2469-9985/2017/96(2)/024317(7) 024317-1 ©2017AmericanPhysicalSociety

JOELKOSTENSALOANDJOUNISUHONEN PHYSICALREVIEWC96,024317(2017) Sec.IIIwedescribetheapplicationofthenuclearshellmodel θ beingtheCabibboangle.Inordertosimplifytheformalism C tothespectrumshapeofforbiddenβ− decays.InSec.IVwe it is usual to introduce unitless kinematic quantit(cid:2)ies w e

presentourresultsandinSec.Vwedrawtheconclusions. W /m c2,w =W /m c2,andp =p c/(m c2)= w2−1. e e 0 0 e e e e Withtheunitlessquantitiestheintegratedshapefactorcanbe II. THEORETICALFORMALISM expressedas We begin the description of the β− decay by making the (cid:3) so-calledimpulseapproximation,inwhichattheexactmoment C˜ = w0 C(w )pw (w −w )2F (Z,w )dw . (6) of decay the decaying nucleon only interacts via the weak e e 0 e 0 e e 1 interaction and the strong interaction with the other A−1 nucleons can be ignored. Since the vector boson W− has a The shape factor C(w e ) of Eq. (6) contains complicated largemassandthuspropagatesonlyashortdistance,theflow combinationsofboth(universal)kinematicfactorsandNMEs. lines of the nucleons, i.e., the hadronic current, and the flow In this paper we use the nuclear shell model to calculate the lines of the emitted leptons, i.e., the leptonic current, can be one-bodytransitiondensitiesrelatedtotheNMEsoftheshape considered to interact at a pointlike weak-interaction vertex factor. The details of the shape factor and the constitution with an effective coupling constant G , the Fermi constant. of its NMEs can be found from Refs. [32,33]. As in the F The parity nonconserving nature of the weak interaction is previous SSM studies [20,21,30] we go beyond the earlier reflected in the fact that the hadronic current can be written studies[32,33]andtakeintoaccountthenext-to-leading-order at the quark level (up u and down d quarks) as a mixture of vectorandaxial-vectorcomponentsas[2–4] Jμ =u¯(x)γμ(1−γ )d(x), (1) H 5 whereγμ aretheusualDiracmatricesandγ =iγ0γ1γ2γ3. 5 Renormalization effects of strong interactions must be taken intoaccountwhenmovingfromthequarkleveltothehadron level.Thehadroniccurrentatthenucleonlevel(neutronnand protonp)canthenbewrittenas[5,6] Jμ =p¯(x)γμ(g −g γ )n(x), (2) H V A 5 where g and g are the weak vector and axial-vector V A couplingconstants,respectively.Theconservedvector-current hypothesisofthestandardmodel(CVC)givesthefree-nucleon valueg =1.0fortheweakvectorcouplingconstantandthe V partially conserved axial-vector-current hypothesis (PCAC) gives the free-nucleon value g =1.27 for the weak axial- A vector coupling constant. Inside the nuclear matter the value of g is affected by many-nucleon correlations and so the A free nucleon value might not be the one to use in practical calculations [6]. The present paper is a step towards solving theproblemofwhatvaluetouseforg insidefinitenuclei. A Intheimpulseapproximationtheprobabilityoftheelectron beingemittedwithkineticenergybetweenW andW +dW e e e is G 1 P(W )dW = F C(W ) e e (h¯c)62π3h¯ e ×p cW (W −W )2F (Z,W )dW , (3) e e 0 e 0 e e where p is the momentum of the electron, Z is the proton e number, F (Z,W ) is the Fermi function, and W is the 0 e 0 end-point energy of the β spectrum. The nuclear structure informationisburiedintheshapefactorC(w ). e Thehalf-lifeofaβ decaycanbewrittenas κ t = , (4) 1/2 C˜ whereC˜ istheintegratedshapefactorandtheconstantκ has FIG.1. Normalized electron spectra for the second-forbidden thevalue[31] nonunique β− decays of 36Cl [panel (a)] and 60Fe [panel (c)], and thefourth-forbiddennonuniquedecayof50V[panel(b)].Thevalue κ = 2π3h¯7ln2 =6147s, (5) g V =1.00 was adopted and the color coding represents the value m5c4(G cosθ )2 ofg . e F C A 024317-2

g -DRIVENSHAPESOFELECTRONSPECTRAOF ... PHYSICALREVIEWC96,024317(2017) A termsoftheshapefunction.Thedetailsofthenext-to-leading tocalculatetheelectronspectraofthesedecays.TheGXPF1A ordershapefactorsarediscussedin[21]. interactionisverygoodatpredictingthepropertiesofpf-shell The shape factor C(w ) can be decomposed into vector, nuclei. However, since the computational burden increases e axial-vector, and mixed vector-axial-vector parts. In this dramatically when the number of particles approaches half decompositiontheshapefactoris of the maximum in the valence space, this interaction could

									60Fe(0+)→		60Co(2+)
							not be	used for	the				decay	transition.
	)=g	2C )+g	2C	)+g
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 ,
		V	A									7/2	3/2	5/2
---	---	---	---	---	---	---	------	------------------------------------------	---	---	---	---	---	---
							andν1p	orbitalswasadoptedwiththeHorie-Ogawainter-
1/2
IntegratingEq.(7)overtheelectronkineticenergy,wegetan
							action [40,41].		pfg	shell:	the jj44b	interaction		developed
----------------------------------------------	---	--------	-----	----	----	---	---------------	-------	------------	---------	---------	-----------	--------	---------
analogousexpressionfortheintegratedshapefactor									9 / 2			−1p−0g
							by Brown	and	Lise t s kiy	for	the 0f			model
												5/2		9/2
							space (see	endnote	on	Ref. [28]	of	[42])	was used	for the
		C˜ =g2C˜	+g2C˜	+g g	C˜ ,	(8)
V V A A V A VA calculation of the 87Rb(3/2−)→ 87Sr(9/2+) transition. For
A=94–99
wherethefactorsC˜ inEq.(8)arejustconstants,independent the decays of the nuclei, the effective interac-
i
							tion of	Gloeckner	[43]	in the	model	space	spanning	the
---	---	---	---	---	---	---	-------	---------	----	------	-----	-----	--------	---
oftheelectronenergy.
							π1p ,	π0g	, ν2s	, and 2d	was	used.	sdgh	shell:
---	---	---	---	---	---	---	--------	-------	-----------------	--------	----------	-----	---------	------
							1/2	9/2	1/2		5/2			11/2
											126Sn(0+)→		126Sb(2+)
							finally,	for the	decay transitions					and
III. ADOPTEDMODELSPACESANDNUCLEAR 137Cs(7/2+)→ 137Ba(11/2−,3/2+) the Sn100pn interaction
HAMILTONIANS
							[44]inthe0g	7/2	2s1d0h	11/2 modelspacewasadopted.
---	--------	-------	-----	---------	--------------	----	-----------	---	------	--------------------------	---	---	---	---
The	electron	spectra	of 16	forbidden	β− transitions	were
g
calculated	for	different	values	of the coupling	constant	A
----------	---	---------	------	---------------	--------	---	---	---	---	---	---	---	---	---
IV. RESULTSANDDISCUSSION
using	the nuclear	matrix	elements	produced	by the	nuclear
-----	-----------	------	--------	--------	------	-------	---	---	---	---	---	---	---	---
shellmodel.Thecalculationsweredoneusingtheshellmodel Below we present our results: the electron spectra of
code NuShellX@MSU [34], with appropriate model spaces 16 forbidden β− decays (Figs. 1–5) and their integrated
and well established Hamiltonians chosen for each studied shape factors (Table I). The electron spectra are discussed
decay individually. No additional truncations, beyond those inSec.IVAandtheintegratedshapefactorsinSec.IVB.
usedintheoriginalworks,wereintroduced.Thecalculations
wererunonadesktopcomputerwitha3.3GHzprocessor,so
								A. Electronspectraandtheeffectivevalueofg
----------------------	---	------	---	-----------	------------	----	-------------------------------------	-----------------------------------------	---	---	---	---	---	---
the computational		burden	had	to be taken	into account	when								A
choosingthemodelspace.							Thestudiedspectrafallintothreegroups.
TheadoptedmodelspacesandHamiltoniansareasfollows. GROUP 1. The first group, for which the spectra can be
sdshell:forthecalculationofthewavefunctionsandone-body found in Figs. 1 and 2, consists of six transitions for which
transitiondensitiesrelatedtothedecay36Cl(2+)→					36Ar(0+)
---------------------------------------------	---	---	---	---	--------	---	---	---	---	---	---	---	---	---
theSSManalysishasnotbeendonepriortothisstudy.These
weadoptedthe0d–1smodelspaceandthephenomenological decaysarepotentiallyinterestingforthepracticalapplication
USDBinteraction[35].pfshell:asintheearliershell-model ofthespectrum-shapemethod,sincemostofthemaresecond-
studiesregardingthehalf-livesofthetransitions48Ca(0+)→
							forbidden	nonunique	transitions		with	branching	ratios	above
---	---	---	---	---	---	---	---------	---------	-----------	---	----	---------	------	-----
48Sc(4+,5+,6+)[36]and50V(6+)→ 50Cr(2+)[37],the0f– 90%. The only exception is the undetected fourth-forbidden
1pmodelspacewiththeinteractionGXPF1A[38,39]wasused β− decay branch of 50V, for which the branching ratio has
TABLEI. UnitlessintegratedshapefunctionsC˜ ofthestudiedtransitionsandtheirvectorC˜ ,axial-vectorC˜ ,andmixedcomponents
									V			A
--------------------------------------	---	---	---	-----------------------------------------	---	---	---	---	---	-----	---	---	---	---
C˜ .ForthetotalintegratedshapefactorC˜									=g	=1.0.
VA				thevaluesofthecouplingconstantsweresettog					V A
						C˜		C˜			C˜			C˜
Transition				Type
						V		A			VA
137Cs(7/2+)→137Ba(11/2−) 1stuniq. 9.681×10−11 8.322×10−6 3.761×10−8 8.359×10−6
36Cl(2+)→36Ar(0+) 6.081×10−9 3.152×10−10 −2.746×10−9 3.650×10−9
2ndnonuniq.
60Fe(0+)→60Co(2+) 2.347×10−13 5.232×10−11 −7.000×10−12 4.556×10−11
2ndnonuniq.
94Nb(6+)→94Mo(4+) 2ndnonuniq. 1.598×10−8 1.469×10−8 −3.058×10−8 1.029×10−10
98Tc(6+)→98Ru(4+) 2ndnonuniq. 2.723×10−8 2.544×10−8 −5.254×10−8 1.207×10−10
99Tc(9/2+)→99Ru(5/2+) 2ndnonuniq. 2.240×10−9 2.130×10−9 −4.361×10−9 8.777×10−12
126Sn(0+)→126Sb(2+) 1.422×10−8 7.125×10−9 2.011×10−8 4.145×10−8
2ndnonuniq.
137Cs(7/2+)→137Ba(3/2+) 4.211×10−6 2.836×10−6 6.879×10−6 1.392×10−5
2ndnonuniq.
87Rb(3/2−)→87Sr(9/2+) 3rdnonuniq. 1.185×10−13 2.082×10−14 −9.734×10−14 4.202×10−14
48Ca(0+)→48Sc(4+) 4thnonuniq. 8.946×10−28 5.934×10−29 −4.606×10−28 4.934×10−28
50V(6+)→50Cr(2+) 4thnonuniq. 1.024×10−23 9.137×10−25 2.131×10−24 1.329×10−23
96Zr(0+)→96Nb(4+) 3.176×10−27 3.170×10−28 −2.006×10−27 1.486×10−27
4thnonuniq.
48Ca(0+)→48Sc(5+) 2.577×10−30 3.931×10−25 1.334×10−27 3.945×10−25
4thuniq.
96Zr(0+)→96Nb(5+) 4thuniq. 1.765×10−29 1.551×10−24 6.844×10−27 1.558×10−24
48Ca(0+)→48Sc(6+) 6thnonuniq. 1.124×10−32 3.498×10−34 3.947×10−33 1.554×10−32
96Zr(0+)→96Nb(6+) 5.811×10−33 7.494×10−33 −1.318×10−32 1.234×10−34
6thnonuniq.
024317-3

JOELKOSTENSALOANDJOUNISUHONEN				PHYSICALREVIEWC96,024317(2017)
FIG.2. SameasFig.1butforthesecond-forbiddennonunique FIG.3. Same as Fig. 1 but for the decays 48Ca(0+)→
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
------------------------------------------------------	---	---	---	---	------------------	---------------	-----	------------
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