beta_shape_pnpi/.llm-wiki/raw/sources/SRC-2026-05-27-004/extracted.md

High precision measurement of the 99Tc 𝛽 spectrum Michael Paulsen, Philipp Ranitzsch, Martin Loidl, Matias Rodrigues, Karsten Kossert, Xavier Mougeot, Abhilasha Singh, Sylvain Leblond, J. Beyer, L. Bockhorn, et al. To cite this version: Michael Paulsen, Philipp Ranitzsch, Martin Loidl, Matias Rodrigues, Karsten Kossert, et al.. High preci- sion measurement of the 99Tc 𝛽 spectrum. Physical Review C, 2024, 110, 055503 (17 p.). ⟨10.1103/Phys- RevC.110.055503⟩. ⟨cea-04796921⟩ HAL Id: cea-04796921 https://cea.hal.science/cea-04796921v1 Submittedon21Nov2024 HAL is a multi-disciplinary open access archive L’archiveouvertepluridisciplinaireHAL,estdes- for the deposit and dissemination of scientific re- tinée au dépôt et à la diffusion de documents scien- searchdocuments,whethertheyarepublishedornot. tifiquesdeniveaurecherche,publiésounon,émanant Thedocumentsmaycomefromteachingandresearch des établissements d’enseignement et de recherche institutionsinFranceorabroad,orfrompublicorpri- français ou étrangers, des laboratoires publics ou vateresearchcenters. privés. HALAuthorization

				PHYSICAL			REVIEW	C 110,055503(2024)
					precisionmeasurementofthe99Tc					β
				High						spectrum
M.Paulsen ,1,2,*P.C.-O.Ranitzsch,3,†M.Loidl ,4M.Rodrigues ,4K.Kossert,3X.Mougeot ,4,‡A.Singh,4,§
S.Leblond ,4J.Beyer,1L.Bockhorn ,3,(cid:2)C.Enss,2M.Wegner ,5,6S.Kempf ,6,5andO.Nähle 3
1Physikalisch-TechnischeBundesanstaltBerlin,Abbestraße2-12,10587Berlin,Germany
2Kirchhoff-InstituteforPhysics,ImNeuenheimerFeld227,69120Heidelberg,Germany
3Physikalisch-TechnischeBundesanstaltBraunschweig,Bundesallee100,38116Braunschweig,Germany
4UniversitéParis-Saclay,CEA,List,LaboratoireNationalHenriBecquerel(LNE-LNHB),F-91120Palaiseau,France
5InstituteforDataProcessingandElectronics(IPE),KarlsruheInstituteofTechnology(KIT),
Hermann-von-Helmholtz-Platz1,76344Eggenstein-Leopoldshafen,Germany
6InstituteofMicro-andNanoelectronicSystems(IMS),KarlsruheInstituteofTechnology(KIT),Hertzstraße16,76187Karlsruhe,Germany
(Received5October2023;revised16August2024;accepted22October2024;published20November2024)
β
Highly precise measurements of the 99Tc spectrum were performed in two laboratories using metallic
magneticcalorimeters.Independentsamplepreparations,evaluationmethods,andanalysesyieldedconsistent
resultsandthespectrumcouldbemeasureddowntolessthan1keV.Consistentβspectrawerealsoobtainedvia
crossevaluationsoftheexperimentaldatasets.Anadditionalindependentmeasurementwithsilicondetectorsin
a4πconfigurationconfirmsthespectrumshapeabove25keV.Detailedtheoreticalcalculationswereperformed
usingthenuclearshellmodelandincludingatomiceffects.Thespectrumshapewasfoundtobesensitivetothe
effective value ofthe axial-vector coupling constant. Combining measurements and predictions, we extracted
		=295.82(16)keV,			ff=0.376(5),		ff=0.574(36),
---	---	---------------	---	---	------------	---	-------------	-----	-----	--------	----------------	---	------	---
	Qβ			ge			and ge	which	seems	to solve	an inconsistency		of the
				V			A
quenchedcouplingconstantsbetweenfirstandhigherforbiddennonuniquetransitions.Furthermore,wederived
themeanenergyoftheβspectrum,Eβ =98.51(23)keV,logf =−0.47660(22),andlogft =12.3478(23).
DOI:10.1103/PhysRevC.110.055503
I. INTRODUCTION magnetic [12,13], scintillation [14], and semiconductor [15]
β β spectrometers [16]. While these setups corresponded to the
spectrometry,		the	precise	shape	of	spectra,	and
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								state of	the art	when they	were	applied	in the	1950–1970s,
their theoretical		description		have received		increased	inter-
								the measurements		suffer	from rather	high	energy	thresholds
est recently	from	different	research		fields,	e.g., radionuclide
								(>50keV)	and	it is expected	that	one	can achieve	signif-
metrology	[1–3],	neutrino	physics		[4–6],	and nuclear	the-
								icantly	higher	energy resolution		with present-day		methods.
ory [7–9].	In	the context	of	radionuclide		metrology,	the
								In addition,	the	β spectrum	shape	of 99Tc	has recently	been
EuropeanmetrologyresearchprojectMetroBeta[10](2016–
predictedin[8]tobeverysensitivetotheeffectivevalueofthe
2019) addressed		the	precise	measurement		and	theoretical
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calculationofseveralβ weakinteractionaxial-vectorcouplingconstantg A ,makingits
spectra.
								high precision		measurement	very	interesting.	The	currently
---------	----------	--------	-----------	--------	-----------	--------	-----------	--------------	-------	------------	--------	------------	----------	-------------
The	ground	state of	99Tc decays		mainly	via pure	β emis-
								accepted	Q-value	297.5(9)keV	[17]	has	a relative	standard
sion [β−,	99.998	55(30)%]		to the	99Ru ground		state [11];
								uncertainty	of	about 0.3%.	The	value is	mainly	influenced
see Fig.	1. The	spectrum	shape	of	this second		forbidden
								by Alburger	et	al. [18]	who used	an iron-free		intermediate-
nonunique	transition	has	been	measured	several	times	using
								image magnetic		spectrometer	to	measure	the decay	energies
of99mTc.ModernmethodssuchasPenningtrapsorcryogenic
microcalorimeters,aspresentedhere,allowformoreaccurate
*Contactauthor:michael.paulsen@ptb.de
measurements.
†Presentaddress:DeutschesZentrumfürLuft-undRaumfahrte.V.
								Theachievedresultfortheaverageβ				energyof99Tchasa
---	---	---	---	---	---	---	---	-------------------------------	---	---	---	----------------	---	---
Hamburg,Hamburg,Germany.
‡Contactauthor:xavier.mougeot@cea.fr direct application for calculating nuclear reactor models and
radioactivewastemanagement.Inarecentstudy,theinfluence
§Present	address:	Physikalisch-Technische			Bundesanstalt		Braun-
--------	--------	-----------------------	---	---	-------------	---	------	----------	----	--------	---------	-----	-------	-----------
								of nuclear	data	on decay	heat from	spent	nuclear	fuel over a
schweig,Bundesallee100,38116Braunschweig,Germany.
								periodof1to105		yearswasassessed[19].Alistofthemost
---	---	---	---	---	---	---	---	--------------	---	-----------------------------------	---	---	---	---
(cid:2)Presentaddress:InstitutfürFestkörperphysik,LeibnizUniversität
significantcontributingradionuclideswasprovided.99Tcwas
Hannover,Appelstraße2,30167Hannover,Germany.
								placed at	the very	top, with	an average	β	energy	that ranges
---	---	---	---	---	---	---	---	---------	--------	---------	--------------------	---	-------	-----------
								from 84	to 95keV	and	a stated uncertainty		of less	than 1%,
PublishedbytheAmericanPhysicalSocietyunderthetermsofthe
Creative Commons Attribution 4.0 International license. Further dependingonthedatalibrary.
β
distribution of this work must maintain attribution to the author(s) In the framework of the MetroBeta project, a spectrum
andthepublishedarticle’stitle,journalcitation,andDOI. of 99Tc was measured with metallic magnetic calorimeters
2469-9985/2024/110(5)/055503(17) 055503-1 PublishedbytheAmericanPhysicalSociety

M.PAULSENetal. PHYSICALREVIEWC110,055503(2024) TABLE I. Setup properties of the MMC experiments at the LNHB and PTB laboratories. The values for the input inductance andheatcapacityarenominalperfabricationandcalculatedvalues, respectively. Laboratory LNHB[10,20,21] PTB[28,29] 3/4Hedilutioninsert 3/4Hedilutionrefrigerator Cryostat in4He(l)bath withtwostagepulsetube forprecooling forprecooling MMCchip MetroBetaV1-M[10] MetroBetaV2-M[29] FIG.1. Decayschemeof99Tc. SQUIDchip SupraconVC1A PTBX1 inputinductance4.5nH inputinductance2nH Au,heatcapacity Au,heatcapacity (MMCs),whichwasfirstpresentedin[20,21].Itfeaturedtwo- Absorber 350pJ/Kat20mK 112pJ/Kat20mK orders-of-magnitude-lower energy thresholds (0.65keV) and matchedtoMMCchip matchedtoMMCchip a greatly improved energy resolution (0.1keV at 383keV) Sample electrodeposited dropdeposited compared to previous measurements. The spectrum was ob- Calibrationsource 133Ba 57Co tained at the Laboratoire National Henri Becquerel (LNHB) Analysiscode Optimalfiltering Optimalfiltering and shows excellent agreement with a corresponding mea- surementatthePhysikalisch-TechnischeBundesanstalt(PTB) inMATLAB inPYTHON using a similar MMC setup, which confirms the spectrum shapeovertheentireenergyrange.Atenergiesabove25keV, the spectrum shapes are further confirmed with a state-of- II. EXPERIMENTALSTUDY the-art passivated implanted planar silicon (PIPS) detector A. MMCmeasurements measurement,alsoperformedatLNHB. MMCs[22–24]arecryogenicmicrocalorimetersthatcon- The remainder of the paper is organized as depicted in sist of a—mostly metallic—particle absorber in strong ther- Fig. 2. In Sec. II, we report on the three independent mea- surements of the 99Tc β spectrum and on the data analyses mal contact with a metallic paramagnet (here, Ag:Er 300ppm ) acting as a temperature sensor. The paramagnet is placed performed to correct for small, but relevant distortions due in a weak magnetic field (≈10mT) and operated at temper- to the detection systems. We next discuss theory in Sec. III atures <100mK. When an energy E is deposited into the thatconnectstheseaccuratemeasurementswithdetailedthe- oretical predictions in order to extract the 99Tc Q-value and absorber,itleadstoatemperatureincrease(cid:4)T.Asthemag- neticsusceptibilityoftheparamagnethasastrongtemperature the effective g coupling constant using the spectrum-shape A method. The average energy of the β spectrum Eβ and the dependence, the temperature increase causes a change in its magnetization, log f value are derived from the resulting theoretical spec- trum. A qualitative comparison of all three measurements ∂M ∂M E with the currently recommended spectrum [15] confirms our (cid:4)M = ∂T ·(cid:4)T = ∂T · C , (1) findings. tot where C denotes the total heat capacity of the absorber tot and the paramagnet. A superconducting coil coupled to the paramagnetpicksupthechangeinmagnetizationasacorre- spondingmagneticfluxchange(cid:4)(cid:6),whichismeasuredwitha superconductingquantuminterferencedevice(SQUID)[25]. Setups using MMCs with the radionuclide source embed- ded in a 4π solid angle absorber geometry have proven to be among the best β spectrometers in terms of energy res- olution and energy threshold, in particular for low-energy β transitions [20,21,26,27]. Both measurements presented here follow that approach, but the technical realization differs in manydetails.ThesearesummarizedinTableIanddescribed inthefollowing.

1. Setupandanalysis(LNHB) ThestartingpointofthedetectorfabricationatLNHBwas thesourcepreparation.Followingaprotocolyieldingmetallic technetium [30], 99Tc was electrodeposited onto a 10-µm- FIG.2. Schematicoverviewofthedata,evaluation,andanalysis thickgoldfoil.Thefoilwasthenrinsedwithwaterinorderto ofthiswork. removesalthavingcrystallizedfromthe99Tcsolutiononthe 055503-2

HIGHPRECISIONMEASUREMENTOFTHE99Tc … PHYSICALREVIEWC110,055503(2024) foil.Somevisiblesaltdepositremainedevenafterrinsing,but inanautoradiographicimageofthesource,severalareaswith- outanysaltdepositbutwithapresenceof99Tcactivitywere found.Theelectrodepositionyieldwaslowandanareaofthe sourcefoillargerthanthetypicalsizeofMMCabsorbershad

to be used	to have	sufficient		activity	in the	MMC	absorber.
Theselectedpieceofsourcefoil(≈0.9×2.5mm)withatrans-
parently	thin metallic		99Tc deposit		was folded	three	times	to
--------	-------------	---	------------	---	----------	-----	-----	---	---	---	---	---	---	---	---
reduceittoasmallenoughsize(≈0.44×0.64×54mm)such
thatitcouldbeenclosedintotheMMCabsorber.Thefolded
foil with	a 99Tc	activity	of ≈5Bq		was sandwiched		between
---------	------	--------	-------	---	--------------	---	-------	---	---	---	---	---	---	---	---
twogoldfoils(0.9×0.9×74mmeach)andthisstackwasdif-
fusionwelded.Sincethesourcecarrierfoilandabsorberfoils
are made	of gold	and	diffusion	welded,	the	source	foil	with
------------------	----------	----------	----------	----------------------	--------------	---------------	-------------	--------	---	---	---	---	---	---	---
the embedded		activity	forms	part of	the	final absorber,		both
geometrically	and	thermally.		This	means	that if	the radiation
deposits	its energy	in	the source	foil	or	the absorber		foil, it
results in	equivalent		signals.	The	final absorber		had	a heat
capacityof≈350pJ/K			at20mK	andwasgluedwithStycast
1266 epoxy	to	one of	the pixels	of	a MetroBeta		V1 M-sized
MMCchip[10].
Duetothecontinuousnatureoftheβ					spectra,withoutany
------------------------------	---	---	---	---	------------------	---	---	---	---	---	---	---	---	---	---
distinct features, energy calibration is essential, in particular MeasuredandcalibratedLNHBMMCspectrumof99Tc
FIG.3.
if the end	point	energy	is to	be determined		from	an experi-
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									with133Ba	externalsourcepeaksofwhichninewereusedforcali-
mentalspectrum.Topreciselydeterminethespectrumshape,
bration:x-raylinesat30.63keV,30.97keV(denotedI,II:directly
checking and correcting for any nonlinearities in energy is from the 133Ba source); 72.80keV, 74.97keV (denoted III, IV: Pb
alsoimportant.Acommonwaytoperformenergycalibration 287.21keV,
									fluorescence	from the	setup material);	escape		lines at
---	---	---	---	---	---	---	---	---	------------	--------	----------------	------	---	--------	---
is to use x-ray and/or gamma ray photons of well-known 289.02keV(denotedV,VI:Auphotonsescapetheabsorber,gener-
energiesfromanexternalradionuclidesourcecollimatedonto
									ated by γ	rays from	the 133Ba	source);	and γ	rays at	302.85keV,
---	---	---	---	---	---	---	---	---	---------	---------	---------	--------	-----	-------	----------
thedetector.Tocoverthefullenergyrangeofthe99Tcβspec-
									356.01keV,	383.85keV	(denoted	VII,	VIII, IX:	directly	from the
---	---	---	---	---	---	---	---	---	----------	---------	--------	----	---------	--------	--------
trum,atLNHBa133Ba sourceofapproximately50kBq was 133Basource).PleaseseeTableIIIforfurtherdetails.Inset:Spectrum
chosen. It emits x-ray and gamma lines between 30.63 and andbackgroundaroundthemaximumβenergy.
383.85keV.Theconversionelectronsthatarealsoemittedare
stoppedina100-µm-thickaluminumfoilplacedbetweenthe
source and the collimator. The 133Ba source was placed at a the systematic error of the experiment is decreased and the
distance of 31mm from the absorber surface. A lead colli- resulting energy resolution is improved. The final spectrum
contains7264451eventsandispresentedinFig.3.
mator was	composed		of a 8-mm-thick		top	part	with a	1mm
---------	--------	---	---------------	---	---	----	------	---	----------	----------	---	--------	----	----------	------
									The energy	resolution	is	constant	over	the entire	energy
boreanda2mmthickbottompartwitha200µmboreplaced
									(cid:4)E	≈
---	---	---	---	---	---	---	---	---	--------	---	---	---	---	---	---
at 2.5mm from the absorber surface. The MMC signal was range, 100eV up to 384keV. The linearity in energy
									was checked	using	the lines	annotated	in Fig.	3 and	listed in
---	---	---	---	---	---	---	---	---	-----------	-----	---------	---------	-------	-----	---------
readoutbyaSupraconVC1ASQUIDlinkedtoaMagnicon
									Table III.	Three line	types were	considered:		(i)	γ lines and
---	---	---	---	---	---	---	---	---	----------	----------	----------	-----------	---	---	-----------
XXF-1electronics.
Thewholesetupwasshieldedagainststraymagneticfields x rays that originate directly from the calibration source, (ii)
by means of a lead cylinder and operated in a liquid he- fluorescent lines due to secondary effects when the calibra-
		3/4He							tionsourceparticlesimpactonthealuminum,gold,andlead
--------------	---	-----	--------	------------	---	-------------	---	---	---------------------------------------------------	------	------------	------	---	------------	-----
lium precooled			dilution	refrigerator		(Cryoconcept)		at
									components	of the	experimental	setup,	and	(iii) escape	lines
T =12mK.Datawereacquiredasacontinuousstreamover
13.7daysat100kS/s;anti-aliasingfilteringwassetto30kHz which emerge when an atom in the gold absorber is hit by a
									photon of	energy E	from the	calibration	source.	This	creates
---	---	---	---	---	---	---	---	---	---------	--------	--------	-----------	-------	----	-------
onaStanfordResearchSystemsSRS560amplifier.
Pulses	were	triggered	in	the data	stream	offline,	using	a
---------	---------	---------	-----------	--------	---------	---------	----------	----	-----------	--------------	---------------	---	------------	---------------	--------
narrow	band-pass	filter.	To minimize		pile-up,	an	extendable
									TABLE	II. Polynomial	calibration		coefficients	for	the LNHB
dead time	was	applied.	Once	the	pulse	positions	were	de-
									(Set 3) and	PTB (Set	2) measurements		using	the calibration	lines
termined,	the pulse	heights		were	estimated	from	raw	data
ofTableIII.
using an	optimal	filter	in a MATLAB		[31]	environment.		Slow
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variations	of the	pulse	heights	for	a given	energy		due to
												LNHB(Set3)			PTB(Set2)
temperaturedriftsofthecryostatwereremovedbyfittingthe
pulse height-vs-time distribution for one densely populated Calibrationline(keV),Elit 356.012900 122.06065
γ
linewithasplinefunctionandapplyingthisfitfunctiontoall Linearcoefficient(keV),k 355.822 121.355
1
									Quadraticcoefficient(keV),k			0.183			0.706
----------------------------------------------------	--------	---	------	----	-------------	---	-----	---	---------------------------	-----	---	-----	---	---	-----
pulse heights.	Spurious		pulses	were	discriminated		based	on			2
thepulseshapesinaχ-square–vs–pulse-heightplot.Bycor-										Eγlit
									Linearityoffset,		−1	0.1%			0.6%
k1
rectingfortemporalandmagneticfluctuationsofthedetector,
055503-3

M.PAULSENetal.					PHYSICALREVIEWC110,055503(2024)
	TABLEIII.	Photonandescapelinesusedtocheckandcorrectforenergynonlinearity.
LNHBmeasurement
(cid:4)E
	Tabulated	Measured	(Set3)					Calibrationline:
----------------	-----------	-----------	------	---	-------------	---	---	----------------
Origin	energy(keV)	energy(keV)	(keV)		Radiationtype			ForSets
133Bacalibration			−0.02
	30.63	30.61			CsXKα2			I:3,4
source	30.97	30.97	0		CsXKα1			II:3,4
	35.05	34.98	−0.07		CsXKβ
	81.00	81.06	0.06		γ
	276.40	276.46	0.06		γ
γ
	302.85	302.88	0.00					VII:3,4
---	------	------	----	---	---	---	---	-------
γ
	356.01	356.01	0					VIII:3,4
-----------------	------	------	-----	--------------	------	-------------------	---	--------
	383.85	383.82	−0.03		γ			IX:3,4
Au/Pbfluorescence	66.99	66.98	−0.01		AuXKα2
from	68.80	68.78	−0.02		AuXKα1
setup/collimator	72.81	72.82	0.02		PbXKα2			III:3,4
	74.97	74.98	0.01		PbXKα1			IV:3,4
	84.94	84.98	0.04		PbXKβ1
				Gammaline(keV)		Escapingphoton(keV)
Escapelines	278.03	278.07	0.04	356.01[133Ba	γ]	77.98[AuKβ1	]
	278.43	278.54	0.11	356.01[133Ba	γ]	77.58[AuKβ3	]
				356.01[133Ba	γ]
	287.21	287.28	0.07			68.80[AuKα1	]	V:3,4
γ]
	289.02	289.08	0.06	356.01[133Ba		66.99[AuKα2	]	VI:3,4
---	------	------	----	------------	---	-----------	---	------
	305.86	305.92	0.06	383.85[133Ba	γ]	77.98[AuKβ1	]
	315.05	315.08	0.03	383.85[133Ba	γ]	68.80[AuKα1	]
	316.85	316.91	0.05	383.85[133Ba	γ]	66.99[AuKα2	]
PTBmeasurement
	Tabulated	Measured	(cid:4)E (Set2)					Calibrationline:
---------------	-----------	-----------	---------------	---	-------------	---	---	----------------
Origin	energy(keV)	energy(keV)	(keV)		Radiationtype			ForSets
57Cocalibration	14.42	14.44	−0.02		γ
γ
source	122.06	122.06	0.00					VI:1,2
------	------	------	----	---	---	---	---	------
γ
	136.47	136.38	0.09					VII:1,2
-----------------	------	------	-----	---	------	---	---	-------
Au/Pbfluorescence	66.99	67.07	−0.08		AuXKα2
from	68.80	68.95	−0.15		AuXKα1
setup/collimator	72.81	72.98	−0.17		PbXKα2			IV:1,2
	74.97	75.14	−0.17		PbXKα1			V:1,2
−0.15
	84.94	85.09			PbXKβ1
-----------	-----	-----	--------------------------	-------------	------	-------------------	---	-------
			Gamma/fluorescentline(keV)			Escapingphoton(keV)
Escapelines	6.47	6.44	0.03	84.45[PbKβ3	]	77.98[AuKβ1	]
	6.96	6.97	−0.01	84.94[PbKβ1	]	77.98[AuKβ1	]
	16.13	16.22	−0.08	84.94[PbKβ1	]	68.80[AuKα1	]	I:1,2
			−0.17	122.06[57Coγ]
	53.26	53.42				68.80[AuKα1	]	II:1,2
			−0.17	122.06[57Coγ]
	55.07	55.24				66.99[AuKα2	]	III:1,2
a hole in the electronic shell of the gold atom that can be energies [32], where, for the escape lines, the x-ray transi-
filled with an electron that transitions from an electron shell tionenergiesfrom[33]wereused.Thedifferencesinenergy
of higher energy. Then, a photon of energy Eph is emitted between the tabulated and measured energies are all smaller
Au
30.97keV
and,ifitescapestheabsorber,theenergyE −Eph isabsorbed or equal to the energy resolution; for the line, the
Au
instead. most intense x-ray line, it is zero. There is no clear trend in
				the difference	between	tabulated and	measured	energies as a
----------------	-----	----------------	---------	--------------	-------	-------------	--------	-------------
First the energy	scale	was set uniquely	using the
356.01keV line, an intense γ line of the 133Ba calibration function of energy, but it could be described with a second-
source lying beyond the endpoint of the 99Tc β spectrum. order polynomial, as provided in Table II. This polynomial
wasthenusedtocorrecttheenergyscalefornonlineareffects.
Then the resulting	energies	of the other	lines from the
------------------	--------	------------	--------------	---	---	---	---	---
Thevalueswereachievedbymakingaquadraticansatzusing
MMC measurement	were compared	with	the recommended
---------------	-------------	----	---------------	---	---	---	---	---
055503-4

HIGHPRECISIONMEASUREMENTOFTHE99Tc … PHYSICALREVIEWC110,055503(2024) thefitfunction, Elit = ffit(a):=k ·a +k ·a2 i i 1 i 2 i fori∈{I,...,N}andN ∈{VII,IX}. (2) Elit denotes the reference literature value of the respective i peak and a := E i exp is the amplitude value of the detector, i Eγlit which corresponds to the pulse energy of the Gaussian fit- ted calibration line. Eexp denotes the measured experimental i energy and Elit is the reference calibration energy used for γ a first energy calibration to set the energy scale, as the de- tector output is in units of voltage. The goal of the fit is to determine the coefficients k and k . Clearly, for a perfectly 1 2 lineardetectork =Elit,k =0andthelinearityoffsetwould 1 γ 2 be0%.Here,thequadraticpartwasrathersmalland,interms of the linearity offset, it was 0.1% and 0.6% in the LNHB and PTB experiment, respectively. The difference is due to the distinct designs, fabrications, and thermalizations of the MMCdetectors. 2. Setupandanalysis(PTB) The measurement setup at PTB is functionally the same FIG.4. Measured and calibrated PTB MMC spectrum of 99Tc as at LNHB, but differs in several details. The difference with57Cocalibrationsourcepeaksofwhichsevenwereusedforcal- with possibly the largest influence on the measured spec- ibration:Escapelinesat16.13keV,53.26keV,55.07keV(denoted trum shape is the method of source preparation. This was I, II,III: Au photons escape the absorber, generated by fluorescent done by microdispensing an aqueous solution of ammonium Pb lines and γ rays from the 57Co source); 72.81keV, 74.97keV pertechnetate (NH 99TcO ) in 0.1molL−1 ammonia (NH ) (denotedIV,V:Pbfluorescencefromthesetupmaterial);andγ rays 4 4 3 withanactivityofA(99Tc)≈5Bqontoasolidgoldabsorber at122.06keV,136.47keV(denotedVI,VIIdirectlyfromthe57Co substrate with a thickness of 90µm. The source is enclosed source).PleaseseeTableIIIforfurtherdetails.Inset:Spectrumand by diffusion welding a second 90µm gold layer to the first backgroundaroundthemaximumβenergy. one,withmoredetailsofthesourcepreparationprocessbeing describedin[34].Theabsorberisattachedtoonepixelofthe datastreamat200kSs−1 toharddiskforthemeasurement’s MMCdetectorwithStycast1266epoxyandthesecondpixel duration of about 20 days. The data processing and analysis isequippedwithasecondabsorberpreparedinthesameway, areperformedwithacustomsoftwarecodewritteninPYTHON but without any radioactive source material. The choice of [35],whichperformsthenextstepsthatcompriseeventtrig- photoncalibrationsourcealsohasanimpactonthespectrum gering, amplitude/energy determination, event classification shape. 57Co with an activity of about 37kBq was used for throughpulse-shapediscrimination,andgaindriftcorrections. the measurement at PTB as its spectrum with several peaks The analysis yields 5 326 682 counts in the spectrum after is easily distinguishable from the β spectrum shape of 99Tc. cuts. The resulting amplitude distribution is calibrated with A 1.5-mm-thick lead collimator with two 250µm apertures several 57Co γ lines, x-ray fluorescence, and x-ray escape blocks the calibration photons outside of the two detector linesusingasecond-orderpolynomial,returningthespectrum pixels.Theaperturesareblockedbetweendetectorandcolli- shown in Fig. 4. The calibration error was estimated with matorwithanapproximately1-mm-thickaluminiumsheetto the uncertainties from the literature values and the statistical reducesecondaryradiationfromtheleadwithoutsignificantly uncertaintiesofthemeasuredpeaksusingorthogonaldistance impactingthehigh-energyphotonfluxfromthe57Cosource. Thedetectorwasoperatedinapulse-tubeprecooled3/4He regression(cid:2)[29,36–38].Thestatisticaluncertaintywasdefined asσ / N ,whereσ denotesthestandarddeviation dilutionrefrigerator(BlueforsLD250)temperaturestabilized Gauss peak Gauss of a Gaussian function that was fitted to the peak and N toT =14.5mK onadetectormoduledescribedin[10].The peak is the number of counts in the peak. The same procedure used MMC (size M) is an update to the one described in was done with the pixel without radioactive source material, the same publication, with some layout changes improving yieldingaspectrumcontainingthe57Cocalibrationspectrum on experimental shortcomings observed with the first design and any additional background. There, a line spectrum with (e.g., the two on-chip heat baths were linked in the new de- 132235countswasextracted. sign to improve thermalization), without changing the core properties of the detector. The setup is completed with a 3. Crossanalysis PTBSQUID(modelC6X114W)forMMCreadout,aMagni- conXXF-1SQUIDelectronics,aStanfordResearchSystems In both institutes, the analog to digital converter (ADC) SR560 low-noise voltage pre-amplifier and band-pass filter, signalsweredirectlystreamedtoharddiskandthedatawere and a 16-Bit waveform digitizer, that was set to save the full saved as binary files containing 16-bit integers. Therefore, 055503-5

M.PAULSENetal. PHYSICALREVIEWC110,055503(2024) TABLE IV. Overview of the cross validation with counts, spectrum, where background has the strongest impact on the FWHM,andenergythreshold(=E )values.Set1onlyconsidered spectrum shape. With this background model, the spectrum TH asubsetoftherecordeddata. shapecannotbeevaluatedatthepositionofγ lines,whichis notabigdrawbackbecauseofthenarrowlinewidths. Set Meas. Analysis Counts FWHM E TH ForthemeasurementatLNHB,thisapproachneedstobe 1 PTB LNHB 3.67×106 63eVat136.47keV 345eV extended because several calibration lines are close to and 2 PTB PTB 5.33×106 72eVat136.47keV 750eV even above the β endpoint region. While the background 3 LNHB LNHB 7.26×106 150eVat302.8keV 1120eV between lines can reasonably be assumed to be constant, the 4 LNHB PTB 5.66×106 108eVat302.8keV 1250eV level becomes slightly higher below each line and the back- ground is described as a series of step functions, with the step height being proportional to the line intensity. With this approach,thebackgroundinbothanalysescanbereasonably therawdataofbothmeasurementsarecompatiblewitheach welldescribed. other. This prompted an exchange of the data and a mutual The measurement at PTB also offers a different approach analysis at both institutes to identify any systematic differ- toevaluatethebackground,whichisusedinthePTBanalysis. ences introduced by the data processing code or analysis Thesecondpixelofthedetectorisequippedwithanabsorber approach. The data processing and analysis were conducted withoutanyenclosedradioactivematerial,whichisalsoillu- as usual for local data sets and allowed for the comparison minatedbythecalibrationsource.Therefore,thespectrumof betweenthefourspectraintotalresultingfromtwomeasure- thesecondpixelshouldbeaverygoodapproximationforthe mentswithtwoanalyseseach. background of the primary pixel. Since the detector perfor- For both data sets, the two separate analyses yield sur- mance,e.g.,inrisetimeandenergyresolution,isnotexactly prisingly different numbers of counts, as can be seen in the same, these need to be adjusted and the amplitude of the TableIV.ThePTBmeasurementinvoluntarilystoppedduring measuredbackgroundspectrumscaledtomatchtheintensity the campaign and was restarted. This caused inconsistencies of the primary pixel, before the background is subtracted. inthedataand,fortheanalysisatPTB,successfulcorrections Since the primary pixel showed the better energy resolution wereimplemented.Incontrast,theLNHBanalysisonlyused ((cid:4)E = 72eV compared to (cid:4)E = 92eV), the pri- the largest continuous data set, which contained about three- FWHM FWHM mary spectrum was convolved with a normalized Gaussian quarters of the total data. The remaining difference, also for with(cid:4)E =28eVtomatchthesecondaryspectrumand theLNHBmeasurement,ismostlikelycausedbydifferences FWHM thesecondaryspectrumwasscaledbyafactorof0.73because intheimplementationandthesettingsforthesoftwaretrigger, ofitslargerintensity. andpossiblybycutsforeventselection.Thetriggerthreshold To account for energy losses in the absorber, mainly via shouldonlyinfluencetheenergythreshold,butothersettings, unstopped photons generated by excitation of Au atoms and suchastriggerhold-offorextending/nonextendingdeadtime, bremsstrahlung, an unfolding correction was applied to each etc.,canhaveanimpactoverthewholeenergyrange.Specifi- spectrum before determining the maximum β energy. The cally,fortheLNHB-LNHB(set3)analysis,averyshortevent unfolding procedure is based on an algorithm that does not time window was chosen, which allows one to reduce the requireanyaprioriknowledgeofthetrueβspectrum[29,41]. impact of pile-up, with the trade-off of a slightly diminished Itsbasisisthefollowingdiscreteunfoldingproblem: energyresolutionandhigherenergythreshold. Itwasagreedupontousethesamecalibrationlinesforthe h N meas =R N×N ·h N true, (3) energy calibration in the spectra and the corresponding liter- aturevaluesfortheγ [39,40]andx-ray[33]energies.These where hmeas is the measured histogram spectrum having N N lineswereveryeasytoidentifyandtofitinthespectrum.The energybins,R N×N denotestheresponsematrixoftheabsorber tabulatedenergiesanddifferencestothemeasurementdataare for the N energy intervals, and htrue is the unknown true N summarizedinTableIII. histogram spectrum. After a measurement, one only obtains a value for the left-hand side of (3) and thus the problem is typically high dimensional and greatly underdetermined. 4. Spectrumcorrections However, an excellent approximation of the response matrix ves B ti e g f a o te re d, th tw e o ac c t o u r a re l c s t h io ap n e sn o e f e t d he to 99 b T e c ap β pl s i p e e d c t t o ru t m he c m an ea b s e ur i e n d - R N sim ×N ≈R N×N can be acquired via large-scale Monte Carlo (MC) simulations of the corresponding absorber geometries. data.First,thecontributionsfromthecalibrationsourcesand Sinceitcanbeshownthattheapproximatematrixisinvertible additionalbackgroundneedtoberemoved.Second,thespec- withprobability1[41],thealgorithmprovidesanapproximate trumneedstobecorrectedforenergylossesoftheβelectrons, solutiontoEq.(3), mostlycausedbyx-rayfluorescenceoftheabsorbermaterial (cid:3) (cid:4) andescapingbremsstrahlung. htrue ≈halgo =! Rsim −1·hmeas. (4) Thesimplestwaytoaccountforbackgroundistoassumea N N N×N N constantbackground,withitslevelbeingdeterminedfromthe The adopted absorber geometries are, notably, just an ap- background above the endpoint of the β spectrum. This ap- proximationsincetheexactdistributionsoftheactivitywithin proachcould,e.g.,besufficientforthedatatakenatPTBand the rather complex sources are not precisely known. In ad- is used in its analysis at LNHB, since no major background dition, the imperfect knowledge of the cross sections (e.g., sourcesorγ linesareexpectedintheendpointregionoftheβ for bremsstrahlung) contribute to the related uncertainty 055503-6

HIGHPRECISIONMEASUREMENTOFTHE99Tc … PHYSICALREVIEWC110,055503(2024)

					energy range	from 150keV		to 250keV	as the	normalization
γ
					factor because	there	are no	lines	in that	range in	either
---	---	---	---	---	----------------	----------------	----------	--------------	-----------	------------	-------
					measurement.	Furthermore,		the difference		of the	normal-
					ized counts	was divided	by	the sum	of absolute	differences.
					Thus, the	resulting	normalized	residuals	sum	to unity,	i.e.,
					100%. Normalized		residual	plots of	the various	MMC	sets
					(see Table	IV) are depicted		in Fig. 6.	For the	same data	sets,
theanalysesyieldverysmallresidualsclosetothemeanand
					below 5%	(proportional	contribution		to the	residual	sum) at
---	---	---	---	---	--------	-------------	------------	---	------	--------	-------
1.5keV.
					energies	below	These	residuals	are	due to	different
---	---	---	---	---	--------	-----	-----	---------	---	------	---------
χ-squareandenergycuts,whicharesetmanually.TheLNHB
					data were	not background		cleaned	with experimental		data
---	---	---	---	---	---------	--------------	---	-------	-----------------	---	----
concerningthecalibrationpeaksandthesecanbeseenwhen
comparedtothePTBdata,showingaverysimilarpatternfor
thedifferentanalyses.
							B. Silicondetectors
---	---	---	---	---	---------------	---------	-------------------	-----------	--------	--------------	--------
					The MMC	results	were	compared	to	an independent
					measurement	performed	with	a detection		system	recently
					developed	at LNHB	[43,44].	The system	was	designed	for
					the measurement	of	β spectra	and	is based	on two	PIPS
detectorswiththinentrancewindows.Suchdetectorsarecom-
monlyusedtodetectchargedparticlesinnuclearandparticle
physics.

1. Experimentalsetup

   					The geometry	of	the overall	system	is designed	such	that
   					the PIPS	detectors face	each	other.	An ultrathin	radioactive
   					source developed	specifically		for this	application	[45,46]	is
   					placed in	their center.	The	experimental	configuration		cov-
   					ers more	than 98%	of the	solid angle	and	minimizes	the
   self-absorptionwithinthesource.Consequently,experimental
   					distortions	of the spectrum	are	significantly		reduced.	Source
   ---	---	---	---	---	----------------	---------------	----------	-------------	------------	----------	------
   					and detectors	are placed	in	an ultrahigh	vacuum	chamber,
   					while the	detection	system	is cooled	down	to 100K	with
   					liquid nitrogen,	in	order to	improve	the energy	resolution
   					and lower	the detection	threshold.		The detector	output	is
   preamplifiedinthechamberbeforebeingshapedandrecorded
   					using a LABZY	module	[47].	A complete	description		of the
   ---	---	---	---	---	-------------	------	-----	----------	-----------	---	------
   detectordesign,acquisitionsystem,andsourcefabricationcan
   befoundin[48].
   The99TcspectrameasuredbyMMCs,withtheirseparate
   FIG.5.

2. Analysisandcorrections analysesinlinearscale(top)andlogarithmicscale(bottom).Back- ground and energy losses have been corrected as described in the TheanalysisofthedatawasperformedinC ++usingthe

   text.					ROOTsoftware[49].Thesystemwascalibratedwith109Cdand
   					207Bi sources	whose	emissions	cover	an energy	range	from
   component.TheMCsimulationswererealizedwiththecode
   about22to1063keV,asdescribedin[48].Theenergyreso-
   egs_phdwithintheEGSNRCsoftware[42]andwereperformed
   lutionofthedetectionsystemwasdeterminedtobe9keVat
   separately for the PTB and LNHB detector geometries. The 65.52keV,whereanenergypeakduetoAgKshellelectrons,
   resulting	four spectra with	the background	removed	and en-
   ---------	-----------------	--------------	-------	-------	---------	----------	--------	----------	---	------------	---
   					which are	emitted by	internal	conversion	in	109Cd decay,	is
   ergylosscorrectionsappliedareshowninFig.5.
   situated.Thecalibrationsourceswerealsousedtodetermine
   					the detection	energy	threshold,	which	was	estimated	to be
   ---	---	---	---	---	-------------	------	----------	-----	---	---------	-----

3. Intercomparison about10keV. Afterapplyingthecorrectionstothespectraasdescribed, Theactivityofthe99Tcsourcewasabout800Bq inorder allfourspectrashould,inprinciple,showthesamespectrum to limit the dead-time ratio of the acquisition to less than shape. Before these can be directly compared, they need to 0.5%.Themeasurementwasperformedoverfiveconsecutive be normalized. We used the total number of events in the days. To estimate the contribution of the background to the 055503-7

M.PAULSENetal.					PHYSICALREVIEWC110,055503(2024)
FIG.6. AnalysiscomparisonforthetwoMMCmeasurementsofthe99Tcspectrum(data,analysis):normalizedresidualpointplots(blue)
andmeanofresiduals(orange).Rows1and2showthelow-energyrangefrom0to10keVandrows3and4depicttheremainingenergy
rangefrom10to400keV.Plotsareshownforsets3and2(rows1and3,left),sets2and4(rows1and3,center),sets3and4(rows1and3,
right),sets1and2(rows2and4,left),sets1and3(rows2and4,center),andsets1and4(rows2and4,right).
initialenergydistributionoftheβ
spectrum, a ten-day measurement was performed with a electrons.Moredetailson
source produced with a nonradioactive solution. The back- the algorithm and the simulation can be found in [48]. The
ground was found to have little effect on the measured response function of the detector is considered to be under
spectrum, as 80% of the background events was detected good control from 850keV down to at least 25keV. This
below15keV.Forthedataanalysis,thebackgroundspectrum is evidenced by the good agreement between the analyzed
wassubtractedfromthe99Tcspectrumafternormalizationof datafromseveralmeasuredsourcesandthesimulationofthe
thelifetimeratiosofthemeasurements.				detectionsystem.
-----------------------------------	-------	---------------------	-------------------	----------------	---	---	---
To account	for the	remaining distortions	in the experi-
mental spectrum,	mainly	due to energy	loss in dead layers
3. Comparison
and escape	of particles,	an unfolding algorithm	was adapted
----------	-------------	----------------------	-----------	---	---	---	---
fromtheprinciplepresentedin[41].BasedondetailedPENE- The corrected spectrum obtained from the PIPS measure-
LOPE [50] Monte Carlo simulations of the source-detector ment is compared in Fig. 7 to the two MMC measurements.
geometry,	the algorithm	provides a way	to reconstruct the
---------	-------------	--------------	------------------	-----------------	-----------------	--------------	---
				The three spectra	show an excellent	agreement from	the
response matrix of the detection system. From this knowl- minimum reconstructed energy of the PIPS system up to
edge,theexperimentalspectrumcanbeunfoldedtoobtainthe the endpoint of the spectra. The PIPS measurement is
055503-8

HIGHPRECISIONMEASUREMENTOFTHE99Tc … PHYSICALREVIEWC110,055503(2024) where N(W) corresponds to the measured data, F(Z,W) is the Fermi function with Z the daughter atomic number, p= (W2−1)1/2,W =1+E /m ,wherem istheelectronrest 0 max e e mass, and W =1+E/m . The constant G2 is the squared e β product of the weak interaction coupling constant g and the cosineoftheCabibboangle,cos(cid:11) ,andX(W)standsforthe C correctionoftheatomicscreeningandexchangeeffects.The atomicoverlapcorrectionisgivenby 1 ∂2 r(Z,W)=1− B(G). (6) W −W ∂Z2 0 For99Tc,theconstantB(cid:5)(cid:5) = ∂2 B(G)anditsuncertaintywas calculatedtobeB(cid:5)(cid:5)=0.211(1 ∂ 1 Z ) 2 keVwhenusingparametriza- tionsfromtheliterature[52,53]. The shape-factor function of the second forbidden FIG.7. ComparisonofthemeasuredPTBMMC,LNHBMMC, nonuniqueβ transitionof99Tcisoftenparameterizedasfirst andPIPSspectraafterremovingthebackground,inred,green,and black, respectively. Please note the corresponding coloring of the forbiddenuniqueusing first,second,andthirdyaxis.WhilethecountstatisticsofthePIPS C(W)=q2+λp2 = (W −W)2+λ(W2−1), (7) 0 measurement is approximately 50 times larger than those of the MMCs, its energy resolution is two orders of magnitude smaller with λ being a constant parameter (see, e.g., [15]). Hence, (9keVvs(cid:2)90eVat65.52keV).Showninblueistherecommended Eq.(5)canberearrangedtoget literaturespectrumof99Tc[15],whichwasplottedovertheexper- (cid:5) imental measurement interval (55 to 250keV) using the reported N(W) =K{[(W −W)2−B (cid:5)(cid:5) (W −W)] shapefactors.Notably,theliteraturespectrumshapedivergesbelow pWF(Z,W)X(W) 0 0 100keV. ×[(W −W)2+λ(W2−1)]}1/2, 0 (8) independent of the MMC measurements, both in terms of detectionmethodandsourcepreparation,andtheconsistency which is then used for the fit procedures. In contrast to stan- between the three spectra underlines the reliability of the dardKuriefits[54],thefitfunction[rightsideofEq.(8)isnot β spectra presented in this article. Compared to the recom- linear,butthethreeparameters(K,W ,andλ)canbedirectly 0 mendedβ spectrumof99Tcintheliterature[15],whichwas determinedinasinglefitprocess. reported by Reich and Schüpferling in 1974, there is rather TheanalysiswascarriedoutwithallfourMMCdatasets good agreement above 100keV, but divergence with a clear (twomeasurements×twoanalyses).Tothisend,spectrawith trend at lower energies. Very likely, the divergence is due to the background removed and corrected for bremsstrahlung experimental issues in the absorber geometry of [15]. The lossesandwithabinwidthof160eVwereusedasastarting solid angle was realistically less than 4π and, to the best of point.Iftheusedbackgroundmodelleftthephotonpeaksof ourknowledge,theresultingenergylosseswerenotcorrected, the external sources (57Co and 133Ba, respectively) in place, e.g.,usinganunfoldingprocedure. these were removed. The uncertainty assigned to the back- groundwasconservativelyestimated.Tothisend,theanalysis wasrepeatedwithoutanybackgroundsubtraction.Thediffer- III. COMBINEDANALYSIS ence to the previous result with background subtraction was The results of Sec. IIIA, concerning the maximum β en- then used to evaluate the corresponding uncertainty compo- nent assuming a rectangular distribution. It should be noted ergy, were carried out at the PTB with theory insight from thattheinfluenceofbackgroundtothedeterminedmaximum the LNHB, and the spectrum-shape method calculations of β energyalsodependsontheenergyrangethatisconsidered Sec.IIIBweredoneattheLNHB. forthefits. For all cases (sets 1–4), the spectrum unfolding shifted A. 99TcQ-value themaximumenergybyapproximately+200eV.Theimpact on the spectrum shape, however, is very subtle as the energy For the determination of the maximum β energy E of max lossesoftheabsorberswereverysmall(<1%)intheseexper- 99Tc, the methodology described in a previous article [51] iments. If the energy losses are larger, the unfolding affects was adopted to allow for an energy-dependent shape-factor thespectrumshapemoresignificantly[29,41,44]. functionC(W).Whenneglectingtheantineutrinomass,theβ Results of the fit procedure are shown in Table V and an spectrumisdescribedby uncertaintybudgetisshowninTableVI.Thefitrangeswere chosen in order to study several potential effects such as the G2 N(W)dW = 2π β 3 F(Z,W)pW(W 0 −W)2dW i r n an te g r e pla (1 y 2 b 5 e t t o we 2 e 9 n 3. s 5 h k ap e e V - ) fa w ct a o s r i f n u c n l c u t d io e n d a to nd de E m ma o x n . s T tr h a e te th th ir a d t ×X(W)C(W)r(Z,W), (5) consistentresultsareobtained,evenifahighupperlimitclose 055503-9

M.PAULSENetal.										PHYSICALREVIEWC110,055503(2024)
TABLE V. Results for the maximum β energy E obtained current (CVC) hypothesis derives from the gauge invariance
max
using spectra from the two analysis codes and two measurements. oftheweakinteractionandleadstog =1.Accordingtothe
V
Theresultscorrespondtomeanvaluesthatwereobtainedfromthree partially conserved axial-vector current (PCAC) hypothesis,
individual results using different energy ranges for the fits (70 to onecanadoptthefree-nucleonvaluegfree=1.2754(13)[55].
A
290keV, 140 to 290keV, and 125 to 293.5keV, respectively). In An ideal description of the nuclear structure would allow
all cases, the measurement uncertainties were taken into account inβ
								theuseofthesevalues			decay	calculations.However,			g V
---------	----------	----------	---	-----------	---	-------------	-------	-------------------	---	------------	-----	---------------------	------	-------------	---
(weighted	fits), the	background	was	subtracted,		spectrum	unfold-
								and g can	be	renormalized		because	of the	imperfections	of
						functionC(W)=		A
ing was taken into account, and the shape-factor thenuclearmodelthatisemployed.Thishasbeenwellknown
q2+λp2
	wasassumed.ThecurrentlyrecommendedE						max valuesin
---	-----------------------------------	---	---	---	---	---	------------	----------	----	----	---------	----------	----	-----------	-----
								for a long	time	from	the study	of partial	half	lives (see,	e.g.,
theliteratureare293.8keV[15]and297.5(9)keV[17],respectively.
thereviewofSuhonen[56]).Effectivevaluesofthecoupling
								constants	then	help to	compensate	for	some	approximations
---	-----------	---	--------	---	---	-----	---	---------	----	-------	----------	---	----	--------------	---
Set	Measurement		Analysis		E	(keV)	λ
max such as nonexistent or partial core excitations, or simplified
1 PTB LNHB 295.809 0.615 many-nucleoncorrelations.However,onehastonotethatthis
2 PTB PTB 295.786 0.652 renormalization could not have a physical meaning, but is
3 LNHB LNHB 295.845 0.651 moreanartifactduetothenuclearmodel.Thelastdecadehas
4	LNHB		PTB		295.862		0.659
---	----	---	---	---	-------	---	-----	---	---	---	---	---	---	---	---
seenimportantprogressthankstothedevelopmentofeffective
								field theories.		It was	recently	demonstrated		that for allowed
---	---	---	---	---	---	---	---	---------------	---	------------	--------	-------------	---	----------------	----
								Gamow-Teller		transitions,	the	0.8 quenching		factor	on g
A
								obtained	with	the nuclear	shell	model	becomes	unnecessary
---	---	---	---	---	---	---	---	--------	----	-----------	-----	-----	-------	-----------	---
totheendpointenergyischosen.Thismaybedifferentforhis-
torical studies, which can suffer from low-energy resolution to reproduce the decay rates when advanced ab initio com-
andtherelatedresolutiondistortioneffect. putations, including two-body currents, are employed [57].
								Inthepresentanalysis,wehaveconsiderednuclear						structure
---	---	---	---	---	---	---	---	--------------------------------------------	---	---	---	---	---	---------	---
Theevaluationoffurtheruncertaintycomponentswascar-
riedoutinasimilarmannerasdescribedinRef.[51].Forthe informationfromthenuclearshellmodelonly.
uncertainty analysis, fit ranges were varied and unweighted Recent theoretical studies suggested that geff can also
A
fits were compared with fits that take statistical uncertainties have asignificantinfluence onthespectrumshapeofforbid-
into account. In order to evaluate a model uncertainty, the den nonunique transitions [58,59]. Indeed, the calculation of
analyses were repeated, ignoring the correction for screen- forbiddennonuniquetransitionsinvolvesnonrelativisticaxial-
ing and the atomic exchange effect and by using a modified vectormatrixelementsthatareassociatedtotheg constant.
A
shape-factorfunctionC(W)=1+aW +b/W +cW2.Inthis In particular, 99Tc decay was predicted to be very sensitive
geff,
case, the mean result agrees to within 74eV. However, a to making this spectrum a good candidate for a precise
A
somewhatlargerspreadofresultsforthemaximumenergyis determination [8]. This approach, called the spectrum-shape
observedwhenusingthisshape-factorparametrization,which method and introduced in [58], was applied in the present
might be due to the larger number of adjustable parameters. work.Theanalysiswascarriedoutonthespectrummeasured
The uncertainty budgets for both measurements are listed in and analyzed at PTB, chosen for its lower background cor-
TableVI.TheindividualresultsE =295.798(156)keVfor rection. Because full computation of the β spectrum is time
max
=295.854(199)keV consuming, the energy binning was increased to 1keV. The
thePTBmeasurementandE			max				forthe
---------------------	---	---	---	---	---	---	------	---	---	---	---	---	---	---	---
LNHBmeasurementareusedtocalculateaweightedmeanas maximum energy used as the input parameter was from this
ourfinalresult:								work,asdescribedabove.
---------------	---	---	---	---	---	---	---	----------------------	---	------	-----	-------------	--------	--------	---
								In Suhonen’s		review	[56],	the effective	coupling	constant
=295.82(16)keV.
		E					(9)
---	---	---	---	---	---	---	---	-------------------------------------------------	---	---	---	---	---	---	---
		max						wasmostlypredictedtobequenched,andaformulaisgiven
toestimatethepartofthequenchingfactorrelatedtonuclear
The uncertainty		of the	final result	corresponds		to	the uncer-
---------------------------------------------------	----------------	-------	------------	---------------	------------	---	----------	---------------	-------	------------	-----------	---------------	--------	------------	-------
								medium	effects	from	a quenching	factor	in	infinite	nuclear
tainty of	the PTB	result	and	is more	conservative		than the
								matter.	Applied	to 99Tc	decay,	we	obtained	ginf =	1.120.
inner(123eV)andouter(3eV)uncertaintiesoftheweighted														A
								In addition,	a	conflicting	result	was	stressed	for the	forbid-
mean and	we can	exclude	an	underestimation		of	the uncer-
								den nonunique		transitions.	In	first forbidden		transitions,	the
tainty due	to correlations.		The	analysis	described		above also
								single-particle		estimates	of	the effective	coupling	constants
providesinformationontheparameterλ.However,theanaly-
arefoundtobegeff≈0.3–0.7andgeff≈0.46–0.56.Forhigher
sisisdedicatedtothedeterminationofthemaximumenergy
										V		A
---	---	---	---	---	---	---	---	---	---	---	---	---	---	---	---
anditdoesnotconsiderthelow-energypartoftheβspectrum. forbidden transitions, only two fourth forbidden nonunique
transitionswerestudied,namely,113Cdand115Indecays,and
Thus,thestatedλvaluesarenotnecessarilysuitedtodescribe
thespectrumshapeinawideenergyrange. thebestadjustmentwiththeexistingmeasurementsledtoan
≈
								unquenched	value	of	g and	geff	0.9. The	high precision
---	---	-----------------------	---	---	---	---	---	------------------------	-----	--------	--------	----	--------	--------------	---
											V	A
								measurement	of	the 99Tc	spectrum	from	this	work brings	a
	B.	Thespectrum-shapemethod						new,importantconstraint.
β
In usual	descriptions		of nuclear		decay,	the Hamiltonian
----------	------------	---	----------	---------	------	---------------	---------	---	---	---	---	---	---	---	---
density is	expressed	in	terms	of lepton	and	hadron	currents.
βspectrummodeling
Assuming	a pure	(V −A)	weak	interaction,		transition	ma-			1.
--------	------	------	----	------------	---	----------	---	---	---	---	---	---	---	---	---
β
trix elements are of the vector type, associated to the vector The method to calculate the spectrum has already been
coupling constant g , or of axial-vector type, associated to described in [51,60] and follows the formalism of Behrens
V
the axial-vector coupling constant g . The conserved vector and Bühring [61]. The theoretical spectrum is described by
A
055503-10

HIGHPRECISIONMEASUREMENTOFTHE99Tc … PHYSICALREVIEWC110,055503(2024) TABLE VI. Uncertainty budgets for the maximum β energy E for the two measurements. All uncertainties are stated as standard max uncertainties(k=1).

		LNHBmeas.			PTBmeas.
Uncertainty			u	Relative	u	Relative
component			(eV)	uncertainty	(eV) uncertainty					Comment
Energycalibration 45 0.015% 50 0.017% Severalwell-knownpeaksusedforthecalibration,high
reproducibility,differentcalibrationsourcesinthetwo
measurements(57Coand133Ba)
Resolution 10 0.003% 10 0.003% Binsathighenergiesavoided;theseeffectsarefurther
distortioneffect,								describedinKossertetal.[51].
-----------------	---	---	---	---	---	---	---	----------------------------	---	---	---	---
finiteenergy
resolution
Background			141	0.048%	69	0.023%	Backgroundtakenintoaccount
----------	---	---	---	------	---	------	--------------------------	---	---	---	---	---
Fitmethod 93 0.031% 89 0.030% Variationofenergyrange;weightedvsunweightedfit,
rebinning
Theoreticalmodel 75 0.025% 75 0.025% Analysiswithothershape-factorparametrization,
differencewith/withoutscreeningandexchange
Spectrumunfolding 57 0.019% 57 0.019% Dependenceoncrosssections(e.g.,bremsstrahlung)and
geometry
Analysis 10 0.003% 10 0.003% Deviationwhenusinginputspectraobtainedfromthetwo
software/pile-up								analysiscodes
----------------	---	---	---	------	---	------	---	-------------	---	---	---	---
Combined			199	0.067%	156	0.053%
Eq. (5) on which the radiative corrections given in [52] are (m R),(WR),and(αZ)andisherereferredtoasthetruncated
e
also applied. Two terms, however, are calculated differently: leptoncurrent.Inthecaseofthesecondforbiddennonunique
99Tcdecay,wefollowedBehrens’andBühring’srecommen-
theatomicexchangecorrectionandtheshapefactor.
Themodelingoftheatomicexchangeeffectwasdeveloped dation,keepingonlythedominantterms[61].Itisnoteworthy
decadesagofortheallowedtransitions[62],andwasrecently thatin[56],theauthorsalsoconsiderednext-to-leading-order
extended to the forbidden unique transitions [63]. Applying terms as described in [64]. This procedure is possible only
Eq. (3) in [63] allowed us to calculate the exchange effect withasimplifiedCoulombpotential.Asinourpreviousstudy
specifically for the second forbidden nonunique transition in of 151Sm decay [51], we also considered a full numerical
99Tc
decay. To this purpose, it is necessary to introduce nu- lepton current based on lepton wave functions determined
clearstructureinformation,whichisalsoneededfortheshape by solving the Dirac equation with a Coulomb potential that
factorandwhichwedetailnow. includes atomic screening. All the terms of the lepton wave-
TheshapefactorC(W)inEqs.(5)and(7)isexperimental, functionexpansionarethereforevirtuallyaccountedfor.
i.e.,itsglobalenergydependencyisbasedonsometheoretical For calculating the shape factor, an input from a realistic
arguments,butisadjustedonthemeasuredspectrum.Toex- nuclearstructuremodelisrequired.TheNUSHELLXcode[65]
tractthecouplingconstants,C(W)mustbepurelytheoretical. was used in this work to determine the list of nucleon-
99Tc
The shape factor is a convolution of the nuclear structure nucleon transitions that contribute to the decay. These
andtheleptondynamicsandisusuallyexpandedindifferent single-particletransitionsareweightedbytheircorresponding
multipolesofboththenuclearandleptoncurrents, one-body transition densities (OBTDs). Following [8], we
(cid:7) firstconsideredtheeffectiveinteractionfromGloeckner[66]
(cid:6)
C(W)= λ M 2(k ,kν)+m 2(k ,kν) w i th t h e G L v a l e n c e s p a c e s p a n n i n g t h e p ro to n o rb i ta l s 2 p /
		ke	K e	K e								1 2
---	----	---	---	---	-------	---	-------------	--------------------	-----------	-------------	---------------	---------------------
						a n	d 1 g	, an d t h e n e u	tr o n o r	b i t a l s 3	s a n d 2 d	. W i t h s u c h
	K,ke	,kν					9 / 2				1 / 2 5 /	2
					(cid:8)	a	d e s c r i p	t io n , t h e 9 9 T	c d e c a y	i s d r iv	e n b y a s i n	g l e n u c l e o n -
2μ γ
			ke	keM
---	---	---	---	---	---	---	---	---	---	---	---	---
− (k ,kν)m (k ,kν) . (10) nucleontransition,fromaneutronin2d 5/2 toaprotonin1g 9/2 .
			k W	K e K	e
----------	-------	---	----------	---------------	--------	----------------------------------------------------	---	-------------	---	-----------	------------	------
			e			Next,theeffectiveinteractionfromMach[67]withthewider
						GLEKPN		valence space	was	considered.	This valence	space
Quantities	labeled	by	the lepton	quantum numbers	k and kν
					e	spanstheprotonorbitals1f				,1f	,2p ,2p	,and1g
dependontherelativisticwavefunctions,andK istheprinci- 7/2 5/2 3/2 1/2 9/2 ,
						and	the neutron	orbitals	1g 9/2	, 1g 7/2	, 2d 5/2 , 2d 3/2	, and 3s 1/2 .
---	---	---	---	---	---	---	-----------	--------	------	--------	-----------------	--------------
palnuclearmultipoleorder.Theleptoncurrentwassimplified
						To	limit	the computational	burden,		the 1f proton	and the
---	---	---	---	---	---	---	-----	-----------------	-------	---	-------------	-------
by Behrens and Bühring in order to decouple the calculation 7/2
						1g	neutron	orbitals	were	constrained	to be full,	and four
-----------------------------------------------------	---	---	---	---	---	---	-------	--------	----	-----------	-----------	--------
ofleptonandnuclearmatrixelements.Theprocedureconsists							9/2
orbital.Asaresult,the99Tc
in expanding the lepton radial wave function in powers1 of protonswereblockedinthe1f 5/2
						decay	is	still dominantly	driven	by	the same single-particle
---	---	---	---	---	---	-----	---	----------------	------	---	------------------------	---
transition,butasmalladmixtureofatransitionfromaneutron
						in	1g 7/2	to a proton in	1g 9/2	appears	(see Table	VII for the
---	---	---	---	---	---	---	------	--------------	------	-------	----------	-----------
=2).Finally,wealsoconsidered
1R is the daughter nucleus radius and α is the fine-structure con- dominantmultipoleorderK
stant. the jj45pnb interaction [68] with the valence space spanning
055503-11

M.PAULSENetal.											PHYSICALREVIEWC110,055503(2024)
	TABLEVII.		One-bodytransitiondensities(OBTDs)ofthedom-
inant	multipole		order (K	=2), as	given	by NUSHELLX.		Coulomb
displacement		energies	for	each nucleon-nucleon			transition	are also
given(MeV).
One-particle
					(cid:4)E(3)	(cid:4)E(4)		(cid:4)E(4)
----------	---	---	-----	---	-----------	-----------	----	-----------	---	---	---	---	---	---	---
transition			OBTDs				C1,2	C2,1
C
1g	→	1g	0.00994		11.696	12.051		12.045
---	---	---	-------	---	------	------	---	------	---	---	---	---	---	---	---
	7/2	9/2
→
2d	5/2	1g 9/2	0.47752		10.560	10.536		10.537
-------	--------	---------	-------	-------------	------------	------	---------------	--------	---	---	---	---	---	---	---
1f	, 2p	, 2p	, and	1g	for protons,	and	1g	, 2d ,
	5/2	3/2	1/2	9/2			7/2	5/2
2d	3/2 , 3s	1/2 , and	1h 11/2	for neutrons.		The	only constraint
applied	was	a minimum		of four	protons	in	the 1f 5/2	orbital.
OBTDs	are	quite	close	to those	obtained	with	GLEKPN,	but
noticeablyofoppositesignforthedominantmultipoleorder:
0.01107for1g			→1g	and−0.43985for2d			→1g
------------	---	---	---	----------------	---	---	---	-----	---	---	---	---	---	---	---
			7/2	9/2			5/2	9/2 .
Thetransitionprobabilityofaforbiddennonuniquedecay
first	depends	on	a relativistic	vector	matrix	element.		While a
---------------	-----------	---	---------------	------	--------------	---------	----------	-------	---	---	---	---	---	---	---
nonrelativistic			matrix element		couples	the large	components
of	the nucleon		wave functions,		a relativistic		matrix	element
couplestheirsmallandlargecomponentstogether.However,
mostofthenuclearmodels,asNUSHELLX,arenonrelativistic.
Comparisonofthemeasured99Tcspectrumwithdiffer-
Anaccurateestimateofthisrelativisticvectormatrixelement									FIG.8.
-------------------------------------------------------	---	---	---	---	---	---	---	---	------	---	---	---	---	---	---
can be obtained from a nonrelativistic vector matrix element enttheoreticalcurves.CVCiseitherignoredorincludedfordifferent
employingtheCVChypothesis(see,e.g.,36Cldecayin[69]). estimatesoftheCoulombdisplacementenergy(cid:4)E .Inset:Thelow-
C
In99Tcdecay,therelationshipis									energypartofthespectrum.
-----------------------------	---	---	---	---	---	---	---	---	------------------------	---	---	---	---	---	---
R
		(cid:7)−√		−(m	−m )+(cid:4)E			,
---	---	---------	---	---	-------------	---	---	---	---	---	---	---	---	---	---
VF [W ]VF (11) whereg i andg f aretheradiallargecomponentsoftheinitial
	221		0	n	p	C	220
---	---	---	---	---	---	---	---	---	-----------------------------------------------------	---	---	---	---	---	---
			10						andfinalnucleonwavefunctions,respectively.ThisCoulomb
displacementenergy(cid:4)E(3)isthusdifferentforeachnucleon-
with	m	and m	the neutron		and proton		rest masses	and				C
--------	------	------	--------------	---	----------	---	-----------	-------	-------	-----------	---	--------	-----------	------	----------
	n		p												β-particle
(cid:4)E									nucleon	transition,	but	is still	independent	of the
	C th e	Coulom	b displacement		energy.	The	small	Q-value
9Tcdecaymakesacriticalgoodestimateof(cid:4)E energy. Finally, one can also introduce a dependency on the
of9								because
---	---	---	-----------------	---	---	---	---	-------	----------	------	---	-----------	---	---------------	----
							C		β-particle	energy	by	considering	the	complete lepton	cur-
[W	−(m	−m	)]=−0.487MeVonly.
0 n p rent, instead of the leading order (r/R)K. The corresponding
TheCoulombdisplacementenergycanbeestimatedfrom
Coulombdisplacementenergyisdenoted(cid:4)E(4)
andalsode-
different		methods.	Usually,	a	uniformly	charged	sphere	is						C
----------	---	--------	--------------	---	------------	-------	------	--------	--------	-----------------	-------	--------	---------	---------	-----------
									pends on	the lepton	quantum		numbers	k and kν.	This latter
considered		and	the expression		only depends		on the	daughter						e
									method	is, in principle,		the most	accurate.	In Table	VII, we
nucleusthroughitsatomicnumberanditsnuclearradius,
give(cid:4)E(3)andtheaveragevalueof(cid:4)E(4)
, forthedominant
										C			C	ke k ν
---	---	---	---	---	---	---	---	---	---	---	---	---	---	------	---
6αZ nucleon- nucleon transitions. In practic e, t h e dependency of
		(cid:4)E	(1) =	f	=13.476MeV.
---	---	--------	-----	---	-----------	---	---	----	--------------------------------------------	------	----------	------	---	-----------	---------
								(12)	(cid:4)E (4 )	in the	β-particle	energy	was	found to be	less than
			C	5 R					C k ,kν
				f					1keVovertheentireenergyrangeofthespectrum. e
Asimilarexpressioncanbeestablishedthatdependsonboth In Fig. 8, we present the spectrum calculated without the
the parent and daughter atomic numbers and their nuclear CVC hypothesis, and the spectra considering CVC for the
									different	(cid:4)E	estimates.	The	free-neutron	value	of g was
------	---	---	---	---	---	---	---	---	---------	--------	----------	---	------------	-----	--------
radii,										C					A
assumed.CVChasaninfluencebutdoesnotleadtoachange
		3 α		3	α
---	---	---	---	---	---	---	---	---	---	---	---	---	---	---	---
(cid:4)E(2) = −1)− −1)=12.814MeV. of the spectrum shape as spectacular as in 36Cl decay [69].
			Z (Z		Z(Z
---	---	---	----	---	---	---	---	---	---	---	---	---	---	---	---
C 5R f f 5R i i Thespectrummostlyexhibitsadependencyatlowenergyand
		f			i
---	---	---	---	---	---	---	---	----	-------------------------	---	---	--------------	---	---	---
								(13)	thespectrawith(cid:4)E(3)			and(cid:4)E(4)
arehardlydistinguished.In
		(cid:4)E									C	(cid:4)E	C
---	---	--------	---	---	---	---	---	---	---	---	---	--------	---	---	---
However, C is known to possibly be sensitive to the addition, we tried to adjust in order to be as close as
C
mismatch between the initial and final nucleon wave func- possibletothemeasuredspectrum.Thebestvaluewasfound
tions [70]. Behrens and Bühring approximated the single- tobe(cid:4)E =3.57MeVwithapoorreduced-χ2 of5.018,and
C
particle	potential		difference	by	the average		of the Coulomb
--------	---------	---	----------	---	-----------	---	--------------	---	---	---	---	---	---	---	---
anonlineartendencywasfoundintheresiduals.Mostofall,it
potential, keeping only the leading order of the lepton cur- isclearthattheadjustedCoulombdisplacementenergyisnot
rent[61],									realistic.
---------	---	---	---	---	---	---	---	---	----------	---	---	---	---	---	---
(cid:9)
				∞					This	result	supports	an adjustment		of the coupling	con-
---	---	---	---	---	---	---	---	---	----	------	--------	-------------	---	---------------	----
g V(r)g(r/R)Kr2dr
(cid:4)E(3) = 0(cid:9) f i , stants to retrieve the measured spectrum. We chose as a
				∞				(14)
---	---	---	---	---	---	---	---	----	---	---	---	---	---	---	---
C g g(r/R)Kr2dr reference for the fitting procedure the spectrum calculated
				0 f	i
---	---	---	---	---	---	---	---	---	---	---	---	---	---	---	---
055503-12

HIGHPRECISIONMEASUREMENTOFTHE99Tc … PHYSICALREVIEWC110,055503(2024) 2 1.5 40 1 0.5 30 FIG.9. Influence of the effective g value on the theoretical β 0 A 20 spectrumof99Tcdecay,comparedtothehighprecisionmeasurement 0.5 fromthiswork. 1 10 1.5 2 0 considering a full numerical treatment of the lepton current, 0 100 200 300 2 1 0 1 2 the OBTDs from the GLEKPN valence space with Mach in- teraction, and the CVC hypothesis with the (cid:4)E(4) Coulomb FIG.10. Comparison of the measured 99Tc spectrum with the C displacementenergy.Allotherpossibilitieswereusedtoesti- theoretical curve with the best adjusted geff value. Inset: The low- A mateuncertainties. energypartofthespectrumandthedistributionsoftheresiduals. 2. Effectivecouplingconstants such agreement would not have been possible with an erro- Based on the modeling described above, we followed the neousendpointenergy,whichvalidatestheextractedQ-value approachin[8,71]varyingonlyg .Oneshouldunderlinethat previously determined. Below 6keV, the atomic exchange A theCVChypothesis wasnotusedin[8],butwasin[71].As correctiondoesnotseemtobesufficientlyhightoaccountfor illustratedinFig.9,weindeedobservedahighsensitivityof the distortion observed in the measured spectrum. We found thespectrumshapeontheeffectiveg value.Quickinspection thesamediscrepancyintherecentstudyof151Smdecay[51]. A clearlyshowsthatageff valuebetween1.4and1.6couldgive A possible explanation could be an inaccurate modeling of A goodagreementwiththemeasuredspectrumfromthiswork. the atomic wave functions, as recently pointed out in [72]. Thefitproceduresimplyconsistedinredoingthecalculations These authors provide an analytical fit of their correction, fordifferentgeffvaluesuntilfindingtheminimumreduced-χ2. but only for allowed transitions. We applied it only to the A A reasonable energy range of 20 to 275keV was considered Fermifunction,whichshouldgivethemaincontribution.The todeterminethecentralvalueinordernottobeinfluencedby verylow-energypartofthespectrumisquitecomparableand thefirstandthelast20keVofthespectrum,wheredistortions doesnotresolvethediscrepancywiththemeasuredspectrum. can be significant. The reduced-χ2 distribution is very close The best adjusted value was found to be geff = 1.520 with A to a quadratic shape. The best adjusted value is geff = 1.526 reduced-χ2 =1.018,andwasusedtoestimateanuncertainty A withreduced-χ2 =1.010. component,asexplainedbelow. Thefinalspectrumiscomparedtothehighprecisionmea- The uncertainty budget is detailed in Table VIII. Several surementfromthisworkinFig.10.Theagreementisexcellent components were studied and estimated with the minimum- down to 6keV. The distribution of the residuals does not maximum method. Different probability distributions were showanyenergydependencyandfollowsanarrowGaussian considereddependingontheuncertaintycomponent,withthe distribution, perfectly centered on zero. It is noteworthy that objectiveofbeingrealisticandconservative. 055503-13

M.PAULSENetal.											PHYSICALREVIEWC110,055503(2024)
Uncertaintybudgetforthegeffvalueextractedfromthe99Tcβ
TABLEVIII. A spectrummeasuredinthiswork.Alluncertaintiesarestated
asstandarduncertainties(k=1).
Relative
Uncertaintycomponent					Valuein10−2		uncertainty					Comment
--------------------	---	---	---	---	-----------	---	-----------	---	---	---	---	-------	---	---
Estimateatχ2±1;includesbinstatisticscomponent
Fitmethod					7.45		4.88%
---------	---	---	---	---	----	---	-----	---	---	---	---	---	---	---
Fitrange 0.61 0.40% Largestdeviationobservedwithextremeenergyranges
Maximumenergy					0.38		0.25%		Q-valueanduncertaintyfromthiswork
-------------	---	---	---	---	----	---	-----	---	---------------------------------	---	---	---	---	---
Deviationwithjj45pnbinteraction
Nuclearmodel					2.94		1.93%
------------	---	---	---	---	----	---	-----	---	---	---	---	---	---	---
Leptoncurrent 1.27 0.83% Deviationwithsimplifiedleptoncurrentand(cid:4)E(3)
C
Largestdeviationobservedwith(cid:4)E(1)
Coulombdisplacementenergy					4.25		2.78%
-------------------------	---	---	---	---	----	---	-----	---	---	---	---	---	---	---
C
Atomicexchange					0.29		0.19%		Deviationwithcorrectionfrom[72]
--------------	---	---	---	---	----	---	-----	---	-------------------------------	---	---	---	---	---
Radiativecorrections 0.33 0.21% Conservativeestimate,withorwithoutincludingthem
(cid:14)2-norm,i.e.,thesquarerootofthequadraticsum
Combined					9.19		6.02%
--------	---	---	---	---	----	---	-----	---	---	---	---	---	---	---
The main contribution comes from the fit method, which extractedfrom99Tcspectrumiseventually
also includes	the	spectrum	statistics.		The quadratic		shape of
--------------	-----------	------------	-----------	-----------	-------------	-------------	--------	---	---	---	-----------------	---	---	----
											ge ff =1.526(92).			(15)
the reduced-χ2		distribution		allows	one to	estimate	the fit				A
method	uncertainty	from	the	geff values	at	reduced-χ2+1.
A It can be surprising that our g A value is not quenched but
≈
The chosen energy range for the fitting procedure also has enhanced, far from Suhonen’s review value of geff 0.9.
A
an influence. Two extreme cases were considered, namely, 0 However, our calculated half life obtained from spectrum
to 296keV and 100 to 200keV. The largest deviation was integrationis29.98(85)×103a,oneorderofmagnitudelower
usedwithatriangularprobabilitydistribution.Themaximum 211.5(11)×103a
								than	the evaluated		half life		[11].	We can
---	---	---	---	---	---	---	---	----	-------------	---	---------	---	-----	------
energy of the spectrum determined in this work was used very accurately describe the global 99Tc decay, i.e., both the
for the calculation and its uncertainty was propagated with spectrumshapeandthehalflife,bysimplyrenormalizingthe
calculations	at	±1σ	considering	a	rectangular		probability
-------------	---	---	-----------	---	-----------	---	-----------	--------------	---------	---	--------	--------------	---------------	---
								coupling	constants		with the	ratio of these	two half lives.	We
distribution.								thendetermined
Fortheothercomponents,spectrumcalculationswereper-
									ge ff	=0.376(5)		and ge ff =0.574(36).		(16)
---	---	---	---	---	---	---	---	---	-----	---------	---	---------------------	---	----
formed with different hypotheses, geff was extracted, and V A
A
the largest	deviation	from	ge	ff = 1.526	was	considered	with
-----------	---------	----	---	----------	---	----------	----	---	--------	---------	---	--------------	-----------	----
			A					The	coupling	constants	are	thus quenched,	with values	that
a triangular probability distribution. Uncertainty due to the arenoticeablyconsistentwithSuhonen’sreviewvaluesforthe
nuclear model		was estimated		from the	largest	obtained	devi-
-------------	---	-------------	---	--------	-------	--------	-----	---	---	---	---	---	---	---
firstforbiddennonuniquetransitions.Ourresultthusseemsto
ation,consideringeithertheGLmodelspacewithGloeckner
solvetheinconsistencyhighlightedbySuhonenin[56].
interaction[66]orthejj45pnbinteraction[68].Theusualsim- Consideringthefullenergyrange,wedeterminedtheaver-
plified lepton current was employed to assess an uncertainty ageenergyoftheβ spectrumfromourcalculation,
duetoleptoncurrenttreatment.Thedifferentmethodsforde-
=98.51(23)keV,
terminingtheCoulombdisplacementenergyweretestedand											Eβ			(17)
--------------------------------------------------	----	----------	---	--------	----------	---------	---------	----------------------	---	---	---	--------	---	----
themaximumdeviationwasfoundwith(cid:4)E						(1).Thege
							ff value	andthecorrespondinglog				f value,
						C	A
determined	with	the atomic		exchange	correction		from [72]
=−0.47660(22).
was considered to estimate an uncertainty due to this log f (18)
correction.								Withthepartialhalflifefrom[11],weobtained
-----------	---	----------	---	-----------	---------	---	------	-----------------------------------------	---	---	---	---	---	---
Finally,	we	considered	an	uncertainty	component		due to
=12.3478(23)
the radiative corrections. The latter include the emission of log ft (19)
real soft photons from the internal bremsstrahlung process. fortheground-statetoground-statetransitionin99Tcdecay.
β
This part	of the	correction	on	the	spectrum	assumes	that
---------	------	----------	---	---	--------	-------	----	--------	---	----------------	---	------------	---------	-----
								Notably,		the experimental		shape factor	from [15]	leads
thesephotonsarelostandthusnotdetected.However,thereis = 95.91(5)keV, =−0.8785(11), =
								to Eβ			log	f	and	log ft
--------	---------	---	---------	----------	---	------	---------	-----	---	---	---	---	---	------
no doubt	that they	are	partially	reabsorbed		by the	detection
11.9458(25),usingtheQ-valuefromthiswork.
system,	especially	the	low-energy		photons.	A conservative
-------	----------	---	----------	---	--------	--------------	---	---	---	---	---	---	---	---
estimatewasobtainedbysimplyignoringtheradiativecorrec-
IV. CONCLUSION
tions in	the spectrum	calculation.		It	is more	than	an extreme
--------	------------	------------	---	---	-------	----	----------	---	---	---	---	---	---	---
case of possible photon reabsorption because additional cor- In the work described here, it is once again demonstrated
rections that do not come from internal bremsstrahlung are thatMMCmeasurementsareexcellentlysuitedfordetermin-
alsoignored. ingboththeshapeandthemaximumenergyofβspectrawith
ThetotalcombineduncertaintyisgiveninTableVIII.The
Qβ-valuesbelow1MeV.Themeasurementsarecharacterized
main component comes from the statistics of the measure- not only by the high linearity and high-energy resolution,
ment.Thevalueoftheeffectiveaxial-vectorcouplingconstant but also by the fact that very low detection thresholds can
055503-14

HIGHPRECISIONMEASUREMENTOFTHE99Tc … PHYSICALREVIEWC110,055503(2024) be reached. The good agreement of two almost independent consistent the ones derived from first forbidden nonunique measurementsalsointhecrossanalysisofthedataincreases transitionsin[56]. confidenceintheobtainedβ spectra.Sincethebackgroundis Our Q-value is five times more precise than the recom- the major uncertainty contribution of the obtained maximum mended one [17] and shifted by −1.8keV, which is about β energy, as shown in Table VI, we conclude that experi- two standard deviations of the recommended value. The un- mental design and optimized calibration sources are crucial certaintyiscompetitivewithPenningtrapmeasurementsand toimproveuponsuchmeasurementsusingMMCs. we call for a confirmation of our Q-value using this method. High-quality PIPS detector measurements confirm the BeyondMMCs,theactiveArrayofCryogenicCalorimetersto shape of the spectrum above 25keV. We do not make a de- EvaluateSpectralShapes(ACCESS)projectaimstomeasure β tailed comparison with previous determinations of the shape forbidden decays such as 99Tc using an array of neutron of the β spectrum, but point out that parametrizations of the transmutation-doped germanium (Ge-NTD or NTD) detec- β spectrum found in the literature must now be considered tors [73]. Our results could then be confirmed in the near obsolete.Itshouldbenotedthatallpreviousmeasurementsof futurewithanotherindependenttechnique.

the 99Tc β	spectrum	had	significantly	higher-energy		thresh-
olds, so	that a	significant	part	of the	spectrum	at low
energies had to be considered as being unknown. Combin- ACKNOWLEDGMENTS
ing our measurements		with	detailed	theoretical		calculations
--------------------	---	----	--------	-----------	---	------------	---	---	---	---
based on the nuclear shell model, we extracted new decay This work is part of the project 20FUN04 PrimA-LTD
dataofinterest:Qβ =295.82(16)keV,Eβ =98.51(23)keV, that has received funding from the EMPIR programme co-
=−0.47660(22),andlog =12.3478(23).Thespec- financed by the Participating States and from the European
log f			ft
----------	---	--------	-------	---------	------	---------	---------------	-------------	--------------	----------
							Union’s Horizon	2020 research	and innovation	programme.
trum shape	was	found to	be very	sensitive	to the	effective
value of the axial-vector coupling constant, and the best The Linux-Compute-Cluster at the PTB Berlin was used to
runEGSNRC.WethankRolfBehrens,GerdLindner,Andreas
description	of the	decay	led to the	quenched	coupling	con-
-----------	------	-----	----------	--------	--------	----	---	---	---	---
=0.376(5)andgeff =0.574(36).Thesevaluesare Lübbert,andHayoZutzfortheirhelpfulcooperation.
stantsgeff
V			A
---	---	---	---	---	---	---	---	---	---	---
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