First detection of the 2+ → 0+ ground-state transition in the β decay of
20
F
O. S. Kirseboma,∗, M. Hukkanenb , A. Kankainenb , W. H. Trzaskab , K. Andersena , L. Caneteb , J. Cederkällc ,
T. Enqvistd , T. Eronenb , H. O. U. Fynboa , S. Geldhofb , R. de Grooteb , D. G. Jenkinse , A. Jokinenb , P. Joshie ,
R. Julinb , A. Khanamb,∗∗, J. Kostensalob , P. Kuusiniemid , I. Mooreb , M. Muncha , D. Nesterenkob , S. V. i Onsèsf ,
J. D. Ovejazf , H. Penttiläb , I. Pohjalainenb , M. Reponenb , S. Rinta-Antilab , K. Riisagera , A. de Roubinb ,
P. C. Srivastavag , J. Suhonenb , J. A. Swartza , O. Tengbladf , M. Vilenb , J. Äystöb
a Department
of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark
of Jyvaskyla, Department of Physics, FIN-40014 University of Jyvaskyla, Finland
c Department of Physics, Lund University, SE-22100 Lund, Sweden
d University of Oulu, Oulu Southern Institute, FIN-90014, Finland
e Department of Physics, University of York, York YO10 5DD, United Kingdom
f Instituto de Estructura de la Materia, CSIC, E-28006 Madrid, Spain
g Department of Physics, Indian Institute of Technology, Roorkee 247667, India
b University
Abstract
We report the first detection of the second-forbidden, non-unique, 2+ → 0+ , ground-state transition in the β decay
of F. A low-energy, mass-separated 20 F beam produced at the IGISOL facility in Jyväskylä, Finland, was implanted
in a thin carbon foil and the β spectrum measured using a magnetic transporter and a plastic-scintillator detector. The
+0.00
branching ratio inferred from the observed β yield is [1.10 ± 0.21(stat) ± 0.17(sys)−0.11 (theo)] × 10−5 corresponding to
log f t = 10.47(11), making this the strongest known second-forbidden, non-unique transition. The experimental result
is supported by shell-model calculations and has important astrophysical implications.
20
Keywords:
20
F, β decay, second-forbidden, non-unique
1. Introduction
5
10
15
20
Recent studies have highlighted the importance of the
second-forbidden, non-unique, electron-capture transition 25
from the 0+ ground state of 20 Ne to the 2+ ground state of
20
F for the final evolution of Super-AGB stars [1, 2, 3]. The
strength of the transition is, however, not well constrained,
neither experimentally nor theoretically, making an experimental determination highly desirable. The strength
may be determined from the branching ratio of the inverse 2+ → 0+ transition in the β decay of 20 F (Fig. 1),
but this transition is not easily detected as it is masked by
the much faster, allowed, 2+ → 2+ transition to the firstexcited state in 20 Ne. Indeed, previous attempts to detect
the 2+ → 0+ transition have been unsuccessful [6, 7, 8]
yielding a rough upper limit of ∼ 10−5 on the branching
ratio [8]. The β-decay endpoint energies for the groundstate and first-excited state transitions are 7.025 MeV and
5.391 MeV, respectively [4], leaving a rather narrow energy
window for the detection of the ground-state transition.1
In this Letter, we report the first successful measurement
of the second-forbidden, non-unique, 2+ → 0+ transition
∗ Corresponding
author
address: Aalto University, P.O. Box 11000, FIN-00076
Aalto, Finland
Email address: oliskir@phys.au.dk (O. S. Kirsebom)
1 The endpoint energies are known to sub-keV precision [4, 9].
∗∗ Present
Preprint submitted to Physics Letters B
in the β decay of 20 F and present the results of shell-model
calculations which provide additional support for our result. The astrophysical implications for the evolution of
Super-AGB stars are dealt with elsewhere [10].
2+
7.025
20 F
19
t1/2 = 11.163(8)
s
5.788
1−
−6
5.621
3−
8 .2 (6 ) × 10 −5
4.967
2−
0 .999907
1.634
2+
0.0
0+
< 7 × 10 −7
< 1 .5 × 10
β
1 .1 (3 ) × 10 −5
20 Ne
10
Figure 1: 20 F β-decay scheme [4, 5] including the newly observed
ground-state transition. Energies are in MeV relative to the 20 Ne
ground state.
April 9, 2018
2. Experimental setup
35
40
Below, we briefly describe the experimental setup and
procedures; a detailed account will be published separately
[11]. The experiment was performed at the IGISOL-4 facility of the JYFL Accelerator Laboratory in Jyväskylä, 65
Finland [12]. Radioactive ions of 12 B and 20 F were produced via (d, p) reactions on targets of B and BaF2 at
energies of 9 MeV and 6 MeV, respectively. The ions were
accelerated to 30 keV and separated based on their massto-charge ratio, before being guided to the experimental 70
station where they were stopped in a thin (50 µg/cm2 )
carbon foil.
The detection system, shown in Fig. 2, consisted of
a Siegbahn-Slätis type intermediate-image magnetic electron transporter combined with an energy-dispersive detector. Such an arrangement is well suited for the measure-
1)
2)
3)
4)
Plasti s intillator
LaBr3 (Ce) rystal
Shield
Anti-positron bae
Plasti
Al
Fe
Brass
4
1
,
3
Coils
0
75
3. Data analysis and results
80
By only allowing electrons within a relatively narrow
energy band to reach the detector, the magnetic transporter effectively “carves out” a slice of the β spectrum.
This is clearly seen in Fig. 3 which shows energy spectra
measured at three different magnetic-field strengths. The
central energy selected by the magnetic transporter is apeβ ' 7.72I˜ + 3.01I˜2 MeV, where
proximately given by E
˜
I is the electrical current expressed as a fraction of the
maximum current provided by the power supply (700 A).
10 m
2
γ
20
F
β
Figure 2: Schematic diagram of the setup.
45
50
55
three detectors (veto, front and main) are plastic scintillators read out with silicon photomultipliers (SiPM). The
detector dimensions represent a compromise between the
requirement to fully stop a significant fraction of the most
energetic electrons (the nominal range of 7-MeV electrons
in plastic is 35 mm) and the requirement to minimize the
cosmic-ray exposure. A small LaBr3 (Ce) crystal placed inside the shield on the centre axis is used to detect the 1.63MeV γ ray from the decay 20 F, thereby providing absolute
normalisation of the β spectrum. Finally, a baffle placed
at the centre of the magnet prevents positrons, which spiral in the opposite direction of electrons, from reaching the
detector thereby eliminating positron emitters as a potential source of background, while only reducing the electron
flux by 11%.
ment of rare ground-state transitions in nuclear β decays
as the effective solid angle of the detector, and hence the
count rate, is greatly increased by the focussing action of
the magnetic field. Furthermore, and equally important,
the shield on the centre axis prevents γ rays and electrons
from excited-state transitions from reaching the detector.
This essentially eliminates βγ summing and ββ pile-up as
sources of background and leads to an improved sensitivity
towards the ground-state transition.
The magnetic transporter was constructed at the JYFL
Accelerator Laboratory in 1980s [13], but has been fully refurbished for the present experiment. The β detector has 85
the shape of a cylinder and consists of a 5-mm thick outer
detector, used as a veto against cosmic rays, and a 45 mm
× 45 mm inner detector, used to measure the full energy
of the electrons. The inner detector is further subdivided
into a 5-mm thick front detector and a 40-mm thick main 90
detector to provide additional discriminatory power. All
2
Counts / 100 keV / 105 decays
30
60
103
20
F
~
I = 35.3%
Exp.
Sim.
102
12
B
~
I = 67.7%
12
B
~
I = 79.0%
10
1
10−1
10−2
0
1
2
3
4
5
6
7
8
9
10
E β (MeV)
Figure 3: Comparison of experimental and simulated energy spectra obtained at 35.3% (20 F), and 67.7% and 79.0% (12 B) of the
maximum electrical current. The 20 F data have been subject to
both veto and front cut, while the 12 B data have been subject only
to the veto cut.
The spectrum obtained at I˜ = 35.3% shows a central
slice of the allowed β spectrum of 20 F, while the spectra
obtained at 67.0% and 79.0% show slices of the β spectrum
of 12 B which has an end-point energy of 13.37 MeV. The
spectrum obtained at 35.3% has been cleaned by requiring
that no coincident signal is recorded in the veto detector
(veto cut) and that the energy deposited in the front detector is between 650–1600 keV (front cut). The spectra
100
105
110
115
120
125
130
135
140
145
3
Counts / 10 keV
3
×10
105
20
104
10
103
1.4
10
1.6
1.8
2
10
0
1
2
3
4
5
6
7
8
9
E γ (MeV)
Figure 4: γ spectrum obtained at 67.7% of the maximum electrical
current. The arrow indicates the 1.63-MeV line from the decay of 20 F
and the inset shows a Gaussian fit to the peak.
Counts / 100 keV
95
obtained at 67.0% and 79.0% have also been subject to the
veto cut, but the front cut could not be applied because
the 12 B measurements were performed with an earlier version of the β detector, identical to the one described in
Sec. 2 except that the inner detector is not divided into
a front and a main section. The spectral shape is seen to
be well reproduced by the GEANT4 simulation, especially
at the lower current setting. Larger deviations are found
at the higher current settings, but the accuracy remains
sufficient for our needs.
The 20 F and 12 B data (only a subset of which is shown
in Fig. 3) and data obtained with an absolutely calibrated
207
Bi source, have been used to validate the absolute accuracy of the GEANT4 simulations all the way up to
eβ = 8.0 MeV. For the raw data (i.e. before application
E
of the veto and front cuts) we find good agreement between the experimental and simulated peak efficiency (the
number of counts in the full-energy peak normalized to
the number of decays) across the full energy range. However, the simulation tends to over-estimate the peak efficiency for the cleaned data, partly due to the presence of
what appears to be electronic cross-talk between the inner and the outer detectors, but also due to inaccuracies
in the modelling of the stopping process in the detector
eβ < 3.0,
volumes. The required correction is small for E
but grows with increasing energy and amounts to a factor of 1.25(7) at 6.0 MeV. Additionally, we observe small
temporal variations in the peak efficiency throughout the
experiment. These variations occur in response to slight
changes in beam optics which affect the source geometry,
and amount to a 13%-spread in peak efficiency which we
include as a systematic uncertainty on the final result.
Long-duration measurements were performed at the
current settings I˜ = 67.7% (67 h) and 70.5% (38 h) to
search for a signal in the energy region 5.4–7.0 MeV, and
at 79.0% (37 h) to demonstrate that any signal detected at
the two lower settings did not persist above 7.0 MeV. The
average 20 F implantation rate for these measurements was
11 kHz. Additionally, background data were collected at150
67.7% and 70.5% (183 h). The γ and β spectra obtained
at 67.7% are shown in Fig. 4 and 5, respectively. In the
γ spectrum we clearly see the well-resolved 1.63-MeV line
from the decay of 20 F which is used for absolute normalisation. The cosmic-ray background dominates the raw β155
spectrum above 5.4 MeV while electrons from the allowed
2+ → 2+ transition produce the the bump centered at
5.0 MeV and the continuum below it. In the signal region
the cosmic-ray background is reduced by a factor of 105
by the veto cut. The front cut removes another factor of160
3.35 resulting in an overall reduction factor of 352. On the
other hand, about 2/3 of the β particles survive the cuts,
a figure which would have been even larger in the absence
of cross-talk between the inner and outer detectors.
Fig. 6 shows a zoom-in on the cleaned spectrum in165
the region of interest. The spectra obtained at 67.7% and
70.5% reveal a clear excess of counts in the signal region
when compared to the background spectrum. Equally im-
No cuts
Veto and front cut
Fit region
104
103
102
10
1
0
1
2
3
4
5
6
7
8
9
10
E β (MeV)
Figure 5: β spectrum with cuts (gray) and without (yellow) obtained at 67.7% of the maximum electrical current. The dashed box
indicates the region used for the likelihood fits shown in Fig. 6.
portant, no excess of counts is observed above the signal
region in the data obtained at 79.0%. For example, the
spectrum at 67.7% has 55 ± 7 counts between 5.8–6.8 MeV
while the background spectrum, downscaled to account
for the longer measurement time, only has 30 ± 3 counts in
the same region. Furthermore, an analysis of the temporal distribution of the counts between 5.8–6.8 MeV reveals
a clear correlation with the 20 F implantation rate, which
varied by more than a factor of two during the experiment, while the temporal distribution of the counts above
7.0 MeV shows no such correlation. Thus, the observed
signal is consistent with being due to the hitherto unobserved, second-forbidden, ground-state transition in the β
decay of 20 F.
To quantify this statement and obtain an accurate determination of the branching ratio, we perform a likelihood
fit to the data between 5.0–8.0 MeV, in which we allow the
normalisations of the simulated energy spectra of the al-
Exp.
2++ → 0++
2 →2
Backgr.
Fit
2
10
(a)
~
I = 67.7%
20
F (2.4 × 109)
67 h
185
10
1
103
(b)
~
I = 70.5%
20
F (1.5 × 109)
38 h
102
goodness of fit worsens to χ2 /N = 171.8/113 = 1.52 corresponding to a significance level of only Pχ2 >171.8 = 0.0003,
providing clear evidence for a positive observation.
In Fig. 7 we show the dependence of the goodness of fit
on the assumed end-point energy of the forbidden transition. The best fit is clearly obtained by adopting the actual
end-point energy of 7.025 MeV. If we require a significance
χ 2/ N
Counts / 100 keV
103
10
1.45
N = 112
1.4
1.35
1%
1
1.3
103
(c)
~
I = 79.0%
20
F (1.8 × 109)
37 h
102
1.25
5%
1.2
10
1.15
6
7
8
103
(d)
~
I = 67.7%, 70.5%
Backgr.
183 h
102
5
5.5
6
6.5
7
7.5
8
E β (MeV) 195
Figure 6: Likelihood fits to the experimental data in the energy
region 5.0–8.0 MeV obtained with beam (a–c) and without (d), at
current settings focused on the region of interest (67.7% and 70.5%)
and immediately above it (79.0%). The contribution of the secondforbidden, ground-state transition in the β decay of 20 F is shown by200
the thin, solid (blue) curve.
180
11
12
190
1
175
10
Figure 7: Dependence of the goodness of fit on the assumed endpoint energy of the forbidden transition. The 5% and 1% significance
levels are shown by the dashed horizontal lines. The actual end-point
energy of 7.025 MeV is indicated by a star.
10
170
9
End-point energy (MeV)
1
lowed and forbidden transitions to vary freely while the
background is modelled by a simple exponential function
205
with two free parameters. While the normalisation of the
allowed transition is, in principle, fixed, in practice it is
necessary to allow the normalisation to vary because the
GEANT4 simulation becomes inaccurate in the low-energy
tail of the trasmission window. We also allow for a small
(< 50 keV) constant energy shift to account for inaccuracies in the energy calibration and allow for small varia-210
tions in the intrinsic detector resolution which is around
5% (FWHM). Assuming an allowed shape for the forbidden transition, a simultaneous fit to the four spectra shown
in Fig. 6 yields a branching ratio of 1.10(21) × 10−5 and a
goodness of fit of χ2 /N = 133.9/112 = 1.20 corresponding
to an acceptable significance level of Pχ2 >133.9 = 0.076. If,215
on the other hand, we fix the branching ratio to zero, the
4
level of 1% or above, end-point energies above 10 MeV and
below 6.2 MeV are excluded by the fit; if the cut is made
at 5%, the range of acceptable end-point energies shrinks
to 6.7–8.0 MeV. Thus, it seems unlikely that an unknown
β − -unstable beam contaminant should be the cause of the
observed signal. This is well in line with expectations as
20
F is the only β emitter with mass 20 produced by the
19
F(d, p) reaction at 6 MeV. Measurements performed on
neighboring masses and on mass 40 were used to rule out
the possibility that the signal was due to a β emitter with
a mass different from 20, transmitted to the setup through
the tails of the acceptance window of the dipole magnet,
as a molecule or as a doubly-charged ion.
Taking into account the uncertainties related to the
normalisation of the β spectrum discussed above, our result for the branching ratio of the forbidden transition is
[1.10 ± 0.21(stat) ± 0.17(sys)]×10−5 , which translates into
log f t = 10.47(11) when statistical and systematical uncertainties are added in quadrature.
4. Theoretical calculations
In order to validate our experimental result, we have
performed shell-model calculations using two ab initio approaches known as the in-medium similarity renormalization group (IM-SRG) [14] and the coupled-cluster effective
interaction (CCEI) [15]. For comparison, we have also used
225
Branching ratio × 105
220
the phenomenological USDB interaction [16]. The calculations were done with the code NUSHELLX [17]. The
formalism used for the calculation of the forbidden, nonunique β-decay transitions is that of Ref. [18]. Details on
the implementation of the formalism, including the use of
the next-to-leading-order nuclear-matrix elements, can be
found in Ref. [19].
To check the accuracy of the wave functions, we first
calculate the electromagnetic properties 20 F and 20 Ne and
compare to experimental data. As seen in Tables 1 and 2,
all three interactions reproduce the magnetic dipole moments reasonably well, and two of the interactions (CCEI
and USDB) also do a fairly good job at reproducing the
electric quadrupole moments. The accuracy of the wave
Level
µexp a
20 F
2+
1
2+
1
4+
1
+2.09335(9)
+1.08(8)
+1.5(3)
20 Ne
a Refs.
b Using
IM-SRG
µtheo b
CCEI
USDB
+2.171
+1.036
+2.086
+2.183
+1.037
+2.095
+2.092
+1.020
+2.052
Qexp a
20 F
2+
1
2+
1
0.0547(18)
−0.23(3)
20 Ne
a Refs.
b Using
230
235
240
245
−0.0515
+0.0359
Qtheo
CCEI
USDB
+0.0729
−0.1578
+0.0679
−0.1576
1.2
1
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
gA
Figure 8: Dependence of the calculated branching ratio on the value
adopted for the effective axial-vector coupling constant, gA . The dots
mark the values required to reproduce the experimental half-life of
20 F. The experimental branching ratio is shown by the horizontal
band (statistical and systematic uncertainties added in quadrature).
The dashed lines show the effect of using the shape predicted by the
shell-model calculations instead of the allowed shape.
b
IM-SRG
1.4
0.2
0.6
255
Level
1.6
0.4
Table 2: Comparison of experimental and theoretical quadropole
moments in units of eb.
Nucleus
1.8
0.6
250
[20, 21].
the free g factors for sd shell nuclei.
USDB
CCEI
IM-SRG
Exp. (allowed shape)
Exp. (theo. shape)
2
0.8
Table 1: Comparison of experimental and theoretical magnetic
dipole moments in units of µN .
Nucleus
2.2
260
[22, 23].
ep = 1.5e and en = 0.5e.
functions can also be tested by comparing the predicted
half life of 20 F to the experimental value of 11.163(8) s [4].
For IM-SRG and USDB, the experimental half life is repro-265
duced using an effective axial-vector coupling constant of
gA = 1.06, while gA = 1.12 must be used for CCEI. This is
well in line with previous studies in the sd shell, which have
found that an effective gA value close to unity is needed to
reproduce the experimental half-lives [24, 25, 26].
Since our calculations predict several key observables270
in 20 F and 20 Ne, we expect to obtain reasonably accurate estimates also for the second-forbidden, non-unique
2+ → 0+ transition. The results for the branching ratio
are shown in Fig. 8 where it is seen that the branching ratio
depends both on the interaction and the assumed value for275
gA . Overall, there is good agreement between theory and
experiment with theory underestimating the experimental
value somewhat. Using the values for gA that give the correct half life for the allowed transition, the branching ratio
comes out as 5.3 × 10−6 (USDB), 5.4 × 10−6 (CCEI) and280
7.7 × 10−6 (IM-SRG).
5
We have also calculated the half life of the 2+ → 0+
transition in the β decay of 36 Cl which was the only secondforbidden, non-unique transition measured in the sd shell
prior to this work. For this case, the choice of interaction
has a large impact on the calculated strength, resulting
in partial half-lives ranging from (6–7) × 104 yr (USDB)
to ∼ 107 yr (CCEI), which may be compared to the experimental half life of 3.07(2) × 105 yr. The very poor
performance of the CCEI interaction in this case is not
surprising as the interaction has not been designed for the
mass region near A = 36. On the other hand, the poor
performance of the USDB interaction (a factor of five deviation for 36 Cl compared to a factor of two for 20 F) is unexpected and poses the question whether the good agreement
for 20 F is in part coincidental.
5. Discussion
The experimental branching ratio given in Sec. 3 was
obtained assuming that the 2+ → 0+ transition has an
allowed shape even though significant deviations from the
allowed shape can occur. An assumption about the shape
had to be made in order to convert the observed β yield
between 5.4–7.0 MeV to a total β yield, as the experimental data have very limited sensitivity towards the spectral
shape. To quantify the model dependency introduced by
this assumption, we have repeated our analysis using the
shapes predicted by our shell-model calculations and find
that the branching ratio is reduced by 10%, improving the
agreement with the theoretical predictions somewhat. We
include this as a theoretical uncertainty on our final result.
With a log f t value of 10.5, the 2+ → 0+ transition
in the β decay of 20 F is, to our knowledge, the strongest
second-forbidden, non-unique transition ever measured, as
285
290
295
300
305
310
the 27 previously known cases have log f t values ranging
from 10.6 to 14.2 [27].
Shell-model calculations are known to reproduce the
strengths of second-forbidden, unique transitions in the
sd shell within a factor of two or better [28, 29]. However, little appears to be known about the accuracy for330
second-forbidden, non-unique transitions in the sd shell,
such as the 2+ → 0+ transition in the β decay of 20 F,
though we should expect the accuracy to be worse or, at
the best, comparable. The astrophysical importance of
the 2+ → 0+ transition was first pointed out by Ref. [1]335
and a shell-model calculation based on the USDB interaction was published soon after by the same team [30]. The
branching ratio was erroneously given as 1.3 × 10−6 while
the result was in fact 1.3 × 10−5 using Wood-Saxon radial wave functions. Moreover, when Harmonic-Oscillator340
wave functions were used, the value obtained was in agreement with the value of 5.3 × 10−6 obtained in this Letter. It would be of great interest to clarify the origin of
the factor of five discrepancy between theory and experiment in the case of 36 Cl (see Sec. 4), but this requires a345
careful theoretical analysis which is beyond the scope of
this Letter and is left for future work. Notwithstanding
these issues, it is encouraging that the calculations give
correct order-of-magnitude estimates for two transitions
with markedly different strengths that span almost three
orders of magnitude (log f t = 10.5 in the case of 20 F com-350
pared to log f t = 13.3 in the case 36 Cl).
6. Conclusion
355
+
+
The second-forbidden, non-unique, 2 → 0 groundstate transition in the β decay of 20 F has been observed
for the first time. The detection was made possible by the
development of a dedicated experimental setup consisting360
of Siegbahn-Slätis type intermediate-image magnetic electron transporter combined with a plastic-scintillator telescope. The branching ratio was determined to be
365
h
i
+0.00
1.10 ± 0.21(stat) ± 0.17(sys)−0.11 (theo) × 10−5 ,
315
320
325
in agreement with upper limits established in previous ex370
periments and implying log f t = 10.47(11), which makes
this the strongest second-forbidden, non-unique transition
ever measured. This remarkable result is supported by
shell-model calculations which yield correct order of mag-375
nitude estimates not only for this transition, but also for
the much weaker 2+ → 0+ transition in the β decay of
36
Cl (log f t = 13.3). The result has important astrophysical implications which are discussed elsewhere [10]. While380
the inferred branching ratio was found to be fairly insensitive to the assumed spectra shape, the impact on the
astrophysical electron-capture rate could be substantial (a
factor of 4–10 according to Ref. [1]), and hence future ex-385
periments should aim to provide improved constraints on
the shape.
6
Acknowledgements
We thank G. Martínez-Pinedo and A. Idini for enlightening discussions on the topic of shell-model calculations of
forbidden β-decay transitions and for informing us about
the typo in Ref. [30]. We are greatly indebted to the technical staff at the JYFL laboratory and Aarhus University
who contributed with their time and expertise to the refurbishment of the spectrometer. We would also like to thank
P. Schotanus and E. Bodewits at SCIONIX Holland B.V.
for a fruitful collaboration on the design of the β detector. We thank F. Lyckegaard for making the BaF2 targets,
H. Kettunen for preparing the catcher foil and E. Nacher
for providing technical support with the GEANT4 simulations. Finally, we thank T. Kibedi for valuable advice and
encouragement in the early phases of the project. This
work has been supported by the Academy of Finland under the Finnish Centre of Excellence Programme 2012–
2017 (Nuclear and Accelerator Based Physics Research at
JYFL) and the Academy of Finland grants No. 275389,
284516, 312544 and XXXXX. OSK acknowledges support
from the Villum Foundation.
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