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unzhakov 876e1e3ab8 Add massive neutrino support and improve CSV/CLI output 
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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
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| | | | | 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 | | | | | | | |
| ----------------- | ------ | ----------- | -------- | ------------- | ------- | ------------------ | ------ | ---------------- | ------- | ----------------- | ----------- | ---------------- | ------------ | ----------- |
| | | | | | | | | 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 | | | | | | | |
| --------------- | --- | --- | ------- | ----------- | --- | --- | ----------- | --- | --- | --- | --- | --- | --- | --- |
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- | | | | | | | | |
| ---------- | ----- | ------ | ----- | ------------- | --- | ---- | ---------- | --- | --------- | ---------------------------------------------- | --- | --- | --- | --- | --- |
| | | | | | | | | | 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 | | | | | | | |
| ---------- | ------- | ------ | ----------- | --- | ------- | ------------ | --- | ------ | --- | --- | --- | ---------- | --- | --- | --------- |
| 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 |
| --- | --- | --- | --- | --- | ------------- | ------ | ---------- | ----- | --- | --------- | ----- |
5. 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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