]Present address: Univeristy of Jyväskylä, 40014 Jyväskylä, Finland

# Production of neptunium and plutonium nuclides from uranium carbide using
1.4-GeV protons

M. Au  Corresponding author<EMAIL_ADDRESS>European Organization for Nuclear
Research (CERN), Meyrin, 1211 Geneva, Switzerland  Johannes Gutenberg-
Universität Mainz, 55099 Mainz, Germany  M. Athanasakis-Kaklamanakis  European
Organization for Nuclear Research (CERN), Meyrin, 1211 Geneva, Switzerland
Katholieke Universiteit Leuven, B-3001 Leuven, Belgium  L. Nies  European
Organization for Nuclear Research (CERN), Meyrin, 1211 Geneva, Switzerland
University of Greifswald, 17489 Greifswald, Germany  R. Heinke  European
Organization for Nuclear Research (CERN), Meyrin, 1211 Geneva, Switzerland  K.
Chrysalidis  European Organization for Nuclear Research (CERN), Meyrin, 1211
Geneva, Switzerland  U. Köster  European Organization for Nuclear Research
(CERN), Meyrin, 1211 Geneva, Switzerland  Institut Laue-Langevin, 38000
Grenoble, France  P. Kunz  TRIUMF, V6T 2A3 Vancouver, Canada  B. Marsh
European Organization for Nuclear Research (CERN), Meyrin, 1211 Geneva,
Switzerland  M. Mougeot  [  European Organization for Nuclear Research (CERN),
Meyrin, 1211 Geneva, Switzerland  Max Planck Institut für Kernphysik, 69117
Heidelberg, Germany  L. Schweikhard  University of Greifswald, 17489
Greifswald, Germany  S. Stegemann  European Organization for Nuclear Research
(CERN), Meyrin, 1211 Geneva, Switzerland  Y. Vila Gracia  European
Organization for Nuclear Research (CERN), Meyrin, 1211 Geneva, Switzerland
Ch. E. Düllmann  Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany
GSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, Germany
Helmholtz Institute Mainz, 55099 Mainz, Germany  S. Rothe  European
Organization for Nuclear Research (CERN), Meyrin, 1211 Geneva, Switzerland

###### Abstract

Accelerator-based techniques are one of the leading ways to produce
radioactive nuclei. In this work, the Isotope Separation On-Line method was
employed at the CERN-ISOLDE facility to produce neptunium and plutonium from a
uranium carbide target material using 1.4-GeV protons. Neptunium and plutonium
were laser-ionized and extracted as 30-keV ion beams. A Multi-Reflection Time-
of-Flight mass spectrometer was used for ion identification by means of time-
of-flight measurements as well as for isobaric separation. Isotope shifts were
investigated for the 395.6-nm ground state transition in 236,237,239Np and the
413.4-nm ground state transition in 236,239,240Pu. Rates of 235-241Np and
234-241Pu ions were measured and compared with predictions of in-target
production mechanisms simulated with Geant4 and FLUKA to elucidate the
processes by which these nuclei, which contain more protons than the target
nucleus, are formed. 241Pu is the heaviest nuclide produced and identified at
a proton-accelerator-driven facility to date. We report the availability of
neptunium and plutonium as two additional elements at CERN-ISOLDE and discuss
the limit of accelerator-based isotope production at high-energy proton
accelerator facilities for nuclides in the actinide region.

##  I Introduction

The actinide region of the nuclear chart is a focus of research for topics
including the r-process and astrophysical isotopic abundances [1], nuclear
fission [2, 3], nuclear medicine [4, 5], environmental monitoring [6], and
energy production [7]. Experimental measurements of nuclear masses,
$\beta$-decay, neutron capture, and fission properties are required to
benchmark theoretical nuclear structure models, but are missing for many
nuclei [8, 9]. Laser spectroscopy has the capability to reveal both nuclear
and atomic information through techniques such as high-resolution resonance
ionization and collinear laser spectroscopy [10, 11, 12]. Resonance ionization
laser schemes have been developed for many actinide elements [13, 14],
facilitating the use of laser ionization as a spectroscopic technique, a tool
for efficient and element-selective production of radioactive ion beams, as
well as for trace element detection [15].

All actinide nuclides are radioactive. 232Th and 235,238U are available
naturally in macroscopic quantities, but the majority of the other actinide
isotopes must be artificially produced, with production often being one of the
most limiting factors in their study and use. Research reactors are the
leading method of production for the study of some actinide isotopes,
producing a selection of species [16]. Some specific isotopes can also be
obtained through generators made from decaying parent nuclides, for example
227Th (and 227Ra) produced from 227Ac [17]. Alternatively, actinides can be
produced at accelerator facilities using a projectile on a thin target to
induce fusion-evaporation reactions or multi-nucleon transfer. The reaction
products can then be separated in-flight as ion beams; long-lived isotopes can
be collected for off-line isolation and use [18]. The Isotope Separation On-
Line (ISOL) method uses an energetic driver beam that interacts with a thick
target, generating reaction products that diffuse through the target material
and effuse to the ion source where they are ionized and extracted as an ion
beam. The ISOLDE facility at CERN uses 1.4-GeV protons from the CERN Proton
Synchrotron Booster to produce and deliver more than 1000 different nuclide
species [19, 20]. Uranium carbide (UC${}_{\textrm{x}}$) from depleted uranium
is one of the most commonly used target materials [21, 22]. The ISOL method is
typically able to provide isotopes lighter than the target nucleus through
fission, fragmentation, and spallation reactions. The production of above-
target elements (higher proton number $Z$ than the target nucleus) has barely
been studied. In the case of the 238U target nucleus, above-target production
pathways would reach into the elusive transuranium region.

Figure 1: Schematic of the experimental setup. Isotopes are generated in the
UC${}_{\textrm{x}}$ target by 1.4-GeV protons (up to 2 $\mu$A) from the CERN
Proton Synchrotron Booster. Reaction products diffuse and effuse into the hot
cavity ion source where they can be resonantly ionized by lasers from RILIS
[23]. Ions are extracted as a beam via the 30-kV extraction electrode and
separated by their mass-to-charge ratio in the separator magnet. The mass-
separated beam is either sent to a MagneToF detector [24] or through the
central beamline to ISOLTRAP, where the ions are cooled and bunched in the
RFQ-cb and sent to the MR-ToF MS for isobaric mass separation.

Beams of 238Np (2019) and 239Pu (2013) were produced at TRIUMF, Canada from
the ISOL method with 500-MeV protons from two separate UC${}_{\textrm{x}}$
targets using resonance laser ionization in rhenium hot cavity surface ion
sources, and isotopic yields were reported [25, 26]. The extraction behaviour
of actinide beams from the UC${}_{\textrm{x}}$ target matrix still remains
largely uncharacterized. Experimental data on availability, intensity and
purity of Np and Pu ion beams will enable experiments on these elements at
ISOL facilities. Identifying the high-$Z$ limits of the accelerator-based
production technique at proton accelerator facilities gives essential
information for the field of actinide research. Theoretical estimates can be
provided by models describing the processes occurring in collisions of 1.4-GeV
protons with target nuclei like the Monte Carlo code FLUKA [27, 28], which is
used to describe nuclide production at ISOLDE [20].

##  II Methods

The depleted UC${}_{\textrm{x}}$ target (UC2-C2 with 0.25 wt% 235U, 82.95 wt%
238U, 16.8 wt% C, density 3.67 g/cm3) used for the experiment was irradiated
using 5($10^{18}$) protons over a time of 313 h ($\approx$2 weeks) while
actively used for isotope production and extraction during an experiment
ending on the 12${}^{\textrm{th}}$ of November 2018. After a 929(1) day
($\approx$2.5 year) cooling period, first investigations were conducted using
the remaining long-lived inventory. At 934.0(5) days of cooling time, another
irradiation of 8.5($10^{17}$) 1.4-GeV protons was performed on the target
unit. Subsequent measurements were taken both with and without direct
irradiation over an experimental period ending at 942.4(5) days after the end
of the initial irradiation. In this study, measurements were taken under three
conditions:

  1. “before, off”:

Before any additional irradiation (after the cooling period).

  2. “after, off”:

After re-irradiation, without the proton beam actively hitting the target.

  3. “after, on”:

After re-irradiation, with the proton beam actively hitting the target.

The FLUKA model of in-target production was used to predict production of
elements up to plutonium for a beam of 1.4-GeV protons, Gaussian full-width at
half-maximum (FWHM) 6.35 mm, on a depleted UC${}_{\textrm{x}}$ target [20].
Particle fluence spectra caused by the 1.4-GeV proton beam are modeled along
with the inventory of radionuclides produced during irradiation. The simulated
fluence was additionally used as input for the ActiWiz software [29] to
calculate in-target inventory of long-lived actinides after 734 days
($\approx$2 years) of cooling time [30]. A Geant4 [31, 32] package-based model
developed at TRIUMF [33] was adapted to the ISOLDE proton beam energy and
geometry to model reaction pathways. This was used to model $10^{9}$ 1.4-GeV
primary protons irradiating a UC${}_{\textrm{x}}$ target (83.2 wt% U and 16.8
wt% C) of natural uranium, consisting of 234U (0.005 %), 235U (0.72 %), and
238U (99.275 %). The difference caused by the 235U content is expected to be
negligible. Excessive decay time was included in the Geant4 model to evaluate
the largest feasible effect of previous irradiation and cooling time, i.e.,
all inventory has been allowed to decay, considering a decay time extending to
timescales relevant to the scheduling of future experiments [9792 days
($\approx$27 years) after primary impact]. The FLUKA and Geant4 models were
used to compare modeled rates with experimental rates achievable by using a
previously irradiated target. Further details of the models can be found in
Appendix A.

The experimental setup is shown schematically in Fig. 1. Ions were formed
using the Resonance Ionization Laser Ion Source (RILIS) [23, 34] and a rhenium
hot cavity surface ion source. Ions were extracted as 30-keV ion beams, which
were mass-separated using the General Purpose Separator (GPS) dipole magnet.
The two-step ionization schemes [14] shown in Fig. 1 were chosen, using intra-
cavity second harmonic generation in titanium:sapphire (Ti:Sa) lasers [35, 36]
for the first excitation step (FES) and second excitation step (SES) of Np and
Pu laser ionization schemes. A grating Ti:Sa laser [37] was also used to
perform scans of the FES wavelength, with typical laser FWHM of 7 GHz.
Simultaneous application of a uranium laser ionization scheme [38] during beam
composition studies enabled the identification of potential contaminants such
as 238U from the target material. Further details of the ion source are
described in Appendix B.

The target and ion source are heated separately by resistive (Joule) heating,
such that the ion source could be at 2000 ∘C for surface ionization while
keeping the target at temperatures estimated to be 500(200) ∘C by conduction
only. With the ion source heated to facilitate surface ionization of the
released species, the target temperature was increased stepwise in 100-300 ∘C
increments from 1000 ∘C to a maximum of 2100 ∘C in the “before, off”
condition. At nominal target temperature (2000 ∘C), additional heat deposition
from the proton beam is expected to contribute less than 10$\%$ to the total
target heating power. Further details regarding the target temperature are
described in Appendix B.

Mass scans with the laser ionization switched on/off were conducted using the
ISOLDE GPS separator magnet in the “after, off” condition. Because of the
potential increase in surface ionization caused by additional heating from
laser power deposition, mass scans with lasers “off” were done with the FES
blocked or detuned from resonance, and the SES on resonance. Isotope shifts
are on the order of 1-2 GHz per neutron number and were not corrected for
during the mass scans. In this work, “MagneToF detector” refers to the
detector immediately after the mass-separator magnet, unless “ISOLTRAP
MagneToF detector” is explicitly indicated. Rates on the MagneToF detector are
reported as the Faraday cup (FC) equivalent absolute ion intensity calculated
from the manufacturer-provided gain curve [24].

For analysis of the beam composition using mass spectrometry, the beam from
the ISOLDE GPS was cooled and bunched in the ISOLTRAP Radio-Frequency
Quadrupole Cooler-Buncher (RFQ-cb) [39] using helium buffer gas. Ion bunches
were then injected into the ISOLTRAP Multi-Reflection Time-of-Flight Mass
Spectrometer (MR-ToF MS) [40] and captured between the electrostatic mirror
potentials using the in-trap lift method [41]. Further details can be found in
Appendix C.

Table 1: Production mechanisms in % of total events for selected nuclides of
interest calculated using Geant4 QGSP_INCLXX+ABLA with $10^{9}$ 1.4-GeV proton
primaries. Inelastic and Decay columns give the sums over inelastic and
radioactive decay processes, respectively. The larger contribution is
indicated in bold. Various capture reactions are included in the model but not
shown in the table. Some processes with event fractions below 1 % are not
shown and only the dominant parent nucleus and its corresponding percentage
fraction of total events are given. The number of total events is scaled from
$10^{9}$ protons to obtain the nuclides per $\mu$C equivalent

isotope |  Inelastic: | p | n | d | t |  3He | $\alpha$ | ions |  Decay: | parent |  | nuclides$/\mu$C
---|---|---|---|---|---|---|---|---|---|---|---|---
234U | 36.4 | 28.4 | 7.2 | 0.4 | - | - | - | - | 63.6 |  234Pa | 59.4 | 6.7$\times 10^{9}$
236U | 65.5 | 52.2 | 11.8 | 0.8 | - | - | - | - | 34.3 |  236Pa | 34.3 | 1.3$\times 10^{10}$
237U | 70.5 | 61.7 | 7.4 | 0.9 | 0.1 | - | - | - | 29.0 |  237Pa | 29.0 | 1.9$\times 10^{10}$
239Ua | 1.9 | - | - | 1.3 | 0.5 | - | 0.1 | - | - | - | - | 2.2$\times 10^{9}$
240U | 100.0 | - | - | - | 85.7 | - | 11.9 | 2.4 | - | - | - | 2.1$\times 10^{6}$
231Np | 99.0 | 93.1 | - | 3.5 | 0.5 | - | - | - | 1.0 |  231Pu | 1.0 | 1.3$\times 10^{6}$
232Np | 100.0 | 92.3 | 0.3 | 5.4 | 0.3 | 0.1 | - | - | - | - | - | 1.3$\times 10^{7}$
233Np | 99.8 | 91.1 | 0.2 | 6.1 | 0.5 | 0.1 | - | - | 0.2 |  233Pu | 0.2 | 3.6$\times 10^{7}$
234Np | 100 | 89.9 | 0.2 | 7.7 | 0.7 | 0.1 | - | - | - | - | - | 2.0$\times 10^{8}$
235Np | 99.8 | 88.4 | 0.2 | 8.9 | 1.0 | 0.2 | - | - | 0.2 |  235Pu | 0.2 | 3.8$\times 10^{8}$
236Np | 100.0 | 86.2 | 0.1 | 11.1 | 1.4 | 0.2 | - | - | - | - | - | 8.7$\times 10^{8}$
237Np | 3.7 | 3.0 | - | 0.6 | 0.1 | - | - | - | 96.3 |  237U | 96.3 | 2.0$\times 10^{10}$
238Np | 100.0 | 61.0 | - | 27.4 | 8.8 | 1.1 | 0.4 | - | - | - | - | 3.9$\times 10^{8}$
239Np | 3.0 | - | - | 2.2 | 0.6 | 0.2 | 0.1 | - | 97.0 |  239U | 97.0 | 2.2$\times 10^{9}$
240Np | 73.0 | - | - | - | 47.8 | 5.2 | 18.8 | 1.3 | 27.0 |  240U | 27.0 | 7.8$\times 10^{6}$
241Np | 100.0 | - | - | - | - | - | 96.3 | 3.7 | - | - | - | 6.8$\times 10^{5}$
235Pu | 99.2 | 62.5 | - | - | - | 18.8 | 10.9 | - | 0.8 |  235Am | 0.8 | 8.0$\times 10^{5}$
236Pu | 97.1 | 67.8 | - | - | - | 13.5 | 12.5 | - | 2.9 |  236Np | 1.9 | 1.3$\times 10^{6}$
237Pu | 99.5 | 39.7 | - | - | - | 15.5 | 39.7 | 0.3 | 0.5 | - | - | 2.3$\times 10^{6}$
238Pu | 0.9 | 0.1 | - | - | - | 0.1 | 0.7 | - | 99.1 |  238Np | 99.1 | 4.0$\times 10^{8}$
239Pu | 0.3 | - | - | - | - | - | 0.3 | - | 99.7 |  239Np | 99.7 | 2.2$\times 10^{9}$
240Pu | 24.0 | - | - | - | - | 2.5 | 20.9 | 0.6 | 76.0 |  240Np | 75.5 | 1.0$\times 10^{7}$
241Pu | 53.9 | - | - | - | - | - | 51.7 | 2.2 | 46.1 |  241Np | 46.1 | 1.4$\times 10^{6}$

a 239U is 98.1 % produced by neutron capture reactions.

##  III Results

The production mechanisms extracted from the Geant4 model predict that some
species (239,240U, 236,238,241Np, 234-237Pu) are entirely or primarily
generated by inelastic reactions. Others (237,239Np, 238,239Pu) form primarily
through decay of a parent nucleus. The reaction pathways summarized in Table 1
predict different possibilities for in-target inventory of these species using
pre-irradiated targets compared to the situation during direct irradiation.

The isotopes of Np, Pu and Am observed with at least one event in the Geant4
model with $10^{9}$ primary proton projectiles are: 228-244Np, 234-244Pu and
235-244Am. Several of these isotopes (228-229,242-244Np, 242-244Pu, and
235-240,242-244Am) were predicted with event rates less than 100 per $10^{9}$
primaries and are therefore not included in Table 1. The Geant4 model suggests
that the production of some isotopes features a large contribution from decay.
For example, the dominant mechanism to form 239Pu from the target material is
238U undergoing neutron capture to 239U, which decays with a half-life of
23.45 min through 239Np (2.3 days) into 239Pu, with 98.2 % of the 239U
produced from neutron capture reactions, 97.1 % of the 239Np produced by 239U
decay, and 99.7 % of the 239Pu events produced by 239Np decay. This makes
239Pu, and other long-lived isotopes with significant decay feeding, available
in measurable quantities without continuous proton irradiation. Other
isotopes, such as 236,237Pu, are predicted to form primarily through reactions
rather than decay, particularly in the case of 237Pu since there is no
contribution from in-target decay of the long-lived 237Np. With the exceptions
of 241Am (162 events, 97.5 % from decay of 241Pu), and 243Am, (7 events, 5
from decay of 243Pu), all Am isotopes were predicted to come entirely from
inelastic reactions with protons and heavy ions. These isotopes produced
dominantly through direct reactions are thus not expected to be available in
off-line operation.

Pu ion beams were detectable at the lowest target temperatures among the
studied actinide beams, with resonant laser response visible on the MagneToF
detector at a target temperature of at least 1150 ∘C in the “before, off”
condition. Np ion beams were identified only at target temperatures above 2100
∘C. The laser ionization schemes presented in Fig. 1 for Np and Pu showed
resonant enhancement on nominal mass 237 (Fig. 2) and 239 (Fig. 3) beams,
respectively, corresponding to the isotopes 237Np and 239Pu, which were
predicted to have the largest in-target inventory for the elements Np and Pu
(Tab. 1).

Figure 2: Np laser ionization resonance on mass 237 at a target temperature of
2110 ∘C and the “after, off” condition. Rates from the GLM MagneToF detector.
Left: FES scan using 25645 cm-1 for the SES. The Gaussian fit shown gives a
peak width of 6.3 GHz. Right: SES scan using 25277.8 cm-1 for the FES. The
Gaussian fit shown gives a peak width of 27.57 GHz which is typical for auto-
ionizing states. Blue line: a shoulder at 25646.25 cm-1 which could come from
another atomic or molecular ion with the same mass-to-charge ratio.  Figure 3:
Pu laser ionization resonance on mass 239 at a target temperature of 1990 ∘C
and the “after, off” condition. Rates from the MagneToF detector. Left: FES
scan using 24690.4 cm-1 for the SES. Gaussian fit gives a peak width of 5.7
GHz. Right: SES scan using 24188.4 cm-1 for the FES. Gaussian fit gives a peak
width of 33.8 GHz.  Figure 4: a) Signal on the MagneToF detector during a mass
scan of the ISOLDE GPS in the “after, off” condition. The mass spectrum taken
with the laser scheme for resonant ionization of Pu (Fig. 1) is shown in red.
The mass spectrum with the FES laser off resonance is shown in black. b) The
difference between “lasers on” and “lasers off” mass scans in a) fitted with a
model of combined Gaussians. c) Difference between the experimental data and
the fit shown in b).

After irradiating the target with additional proton beam and heating to 1990
∘C, a mass scan of the GPS separator magnet on the MagneToF detector showed a
laser effect for Pu on the masses 236, 239 and 240 (Fig. 4). The laser
wavelengths were optimized on 239Pu. Fitting the difference between the mass
spectra with lasers on and lasers off with a sum model of seven Gaussian peaks
fixed at the masses of 234-240Pu (Fig. 4) gives peaks corresponding to the
laser response on each mass. For 237Pu, only an upper limit can be extracted.

For Np, a very small laser enhancement effect was visible in a mass spectrum
at a target temperature of 2110 ∘C (Fig. 5). 237Np, the longest-lived Np
isotope, can be seen as a shoulder on the large 238U peak. The difference was
modeled with Gaussian peaks fixed at the masses of 235,236-239Np.

Figure 5: a) Signal on the MagneToF detector during a mass scan of the ISOLDE
GPS in the “after, off” condition. The mass spectrum taken with the laser
scheme for resonant ionization of Np (Fig. 1) is shown in red. The mass
spectrum with the FES laser off resonance is shown in black. b) The difference
between “lasers on” and “lasers off” mass scans in a) fitted with a model of
combined Gaussians. c) Difference between the experimental data and the fit
shown in b).

ToF spectra from the MR-ToF MS were used for beam composition identification.
Non-isobaric contamination in an isobaric ToF spectrum can be matched with the
corresponding mass on a given number of revolutions by using the ToF
calibration (e.g. Fig. 6). In cases where a laser scheme was available for the
contaminant (e.g. 235,238U from the target material), the laser effect on the
contaminant was used for additional identification (see Fig. 7).

On nominal mass 236, 236Pu was identified in the MR-ToF MS after trapping for
1000 revolutions using laser response. It was present in the ToF spectrum
along with a non-isobaric contaminant that showed no change in intensity with
the Pu FES. The contaminant arrived 3.7 $\mu$s earlier than the 236Pu. This
corresponds within 60 ns to the expected time-of-flight of 235U16O2 on 940
revolutions (Fig. 6). Since the mass resolving power of the GPS would have
prevented simultaneous injection of Pu and UO2 into the ISOLTRAP RFQ-cb, some
surface-ionized 235U appears to have formed 235U16O2 in the RFQ-cb. 235U is
also known to be present in the target material and is an expected contaminant
that would not be fully suppressed by the separator magnet.

Figure 6: ToF spectrum of mass 236 separated by the GPS with the Pu ionization
scheme in the ion source. Trapping the ions for 1000 revs in the ISOLTRAP MR-
ToF MS gives two visible ion ToF traces in the ToF range of interest for 236Pu
(red). The Pu FES was blocked around 1000 shots, which led to a significant
reduction in intensity of the trace around 37018.5 $\mu$s.

On nominal mass 239, 239Pu was identified with the ISOLTRAP MR-ToF MS along
with contaminants. At 1000 revolutions, the ToF centroid of the contaminant
was earlier than the 239Pu as shown in Fig. 7. The contaminant at 1000
revolutions responded to the Pu laser scheme and matched the predicted time-
of-flight for 239Pu16O at a lower number of revolutions. Laser-ionized 239Pu
appears to have formed 239Pu16O in the RFQ-cb. With trapping time
corresponding to 992 and 990 revolutions of 239Pu, the 239PuO was not visible,
confirming that it is non-isobaric. 238U from the target material was
identified as another contaminant. The ToF trace matched the expected ToF of
238U and responded to the U laser ionization scheme, but not the Pu laser
scheme. On 800 revolutions another contaminant was identified, matching the
expected ToF of 238U16O.

Figure 7: ToF spectrum of mass 239 separated by the GPS, trapped for 1000,
992, 990 and 800 revolutions in the ISOLTRAP MR-ToF MS. Vertical axes show the
summed counts and counts per 10 cycles (shots) for each number of revolutions.
The color scale indicates the number of counts in a 10 ns time bin. Horizontal
axes show ToF since bunch ejection from the RFQ-cb with respect to the
predicted ToF for 239Pu. Horizontal dotted lines indicate a change in the
laser configuration; laser scheme(s) applied (U, Pu) are indicated on the
left.  Figure 8: ToF spectrum of mass 240 separated by the GPS and trapping
for 990 revs in the ISOLTRAP MR-ToF MS. Red and blue show the expected ToFs
for 240Pu and 240Pu16O2 at 990 and 930 revolutions, respectively. Horizontal
dotted lines indicate a change in the laser configuration; laser scheme(s)
applied (U, Pu) are indicated.

On nominal mass 240, 240Pu was identified in the MR-ToF MS with only one trace
visible at 1000 revolutions. On 990 revolutions, a contaminant was present
4.86 $\mu$s later than 240Pu, which matched with the expected ToF of 240Pu16O2
on 930 revolutions (Fig. 8). Both traces dropped noticeably in count rate when
the Pu FES was blocked. No count rates above background were observed on this
mass with the U laser in the ion source and the Pu FES blocked.

Figure 9: Intensity on the MagneToF detector vs. the laser wavenumber of the
Np FES for mass-separated ion beams: a) A=236, b) A=237, c) A=239 from the
GPS. Wavenumbers are with respect to the centroid of the FES resonance of
237Np, shown as the dotted blue line in b). d) Plot of isotope shift with
respect to 237Np and linear best fit (black dashed line). The fit’s large 95%
confidence interval indicated in grey extends beyond the plot range. e)
residuals shown as the difference between the measured isotope shift and the
best fit shown in d).  Figure 10: Intensity on the GLM MagneToF detector vs.
the wavenumber of doubled Ti:Sa light of the Pu FES for mass-separated ion
beams: a) A=236, b) A=239, c) A=240 from the GPS. Wavenumbers in a,b,c) are
with respect to the centroid of the FES resonance of 239Pu, shown as the
dotted blue line in b). d) Plot of isotope shift with respect to 239Pu and
linear best fit (black dashed line) with its 95% confidence interval indicated
in grey. e) residuals shown as the difference between the measured isotope
shift and the best fit shown in d).

The 395.6-nm transition in Np and the 413.4-nm transition in Pu were scanned
for 3 isotopes each (Figs. 9,10) using the grating Ti:Sa laser. The Gaussian
fit gave a FWHM of approximately 5 GHz. From the three isotopes, the linear
best fits found for the isotope shifts with respect to nuclide mass were:
$y_{\textrm{Np}}=0.320(63)m-76(15)$ cm-1 and
$y_{\textrm{Pu}}=-0.0514(7)m+12.29(0.16)$ cm-1. For Np, deviations from
linearity are larger than experimental uncertainty, suggesting that the
isotope shifts are not linear. 236Np, with an odd neutron number, appears to
exhibit a smaller isotope shift than the extrapolation of the data points
237,239Np, which both have even numbers of neutrons. For Pu, deviations from
linearity are within the absolute uncertainty of the wavemeter (0.03 cm-1) and
the uncertainty given by the linear best fit.

The time structure of isotope release from thick targets exhibits an element-
dependent behavior that limits extraction of short-lived isotopes using the
ISOL technique. For 239Pu, the release time was evaluated by recording the
response to stopping direct proton irradiation (Fig. 11) after irradiation
with a proton intensity of 1 $\mu$A with a constant ion current indicating
steady-state conditions of production and release. For Pu, steady-state
conditions were achieved after irradiation times on the order of 600 s. The
signal on an FC for 239Pu was recorded immediately after stopping irradiation.
An exponential of the form $A\exp{\frac{-t}{\tau}}+C$ was used to fit the
data. Here, $C$ represents the constant background on the FC. The amplitude
$A$ of the exponentially falling signal indicates the rate of isotopes
generated by protons on target. The exponential decrease captures the combined
release following many individual pulses, and therefore sets an upper limit
for the time required to extract a Pu isotope from the UCx target. At a target
temperature of 1990(50) ∘C, this upper limit of the extraction time constant
$\tau$ was evaluated to be 430(34) s. For the 24110-year half-life of the
239Pu used in this experiment, the contribution of radioactive decay to the
decrease of signal during the 500 s measurement is 6.6$\times 10^{-10}$ of the
initial amount and is therefore negligible in the determination of the release
time.

Figure 11: Release response of 239Pu at 1990 ∘C target temperature after the
target was irradiated with an integrated current of 1 $\mu$A of protons. Data
is shown in blue and the best fit using an exponential decay is shown in
orange. Time 0 indicates the stop of irradiation. The grey vertical line
indicates blocking the Pu FES from the ion source and the resulting surface
ionized background.

The modeled inventory generated by 1 $\mu$C of proton primaries using Geant4
and FLUKA is compared against the measured values in Fig. 12. Resonance signal
height on each mass and peak heights from the sum of the Gaussian fits gives a
rate of the laser-ionized species without considering the surface-ionized
species or surface-ionized contaminants. ToF compositions indicate negligible
contributions from surface ionization of Pu compared to the laser-ionized Pu
ions. From evaluation of the isotope shifts and the laser linewidth of
approximately 5 GHz, the effect in intensity of the different isotopes can be
estimated (Figs. 9,10).

Figure 12: Combined rates of laser-ionized Np (top) and Pu (bottom) beams from laser resonances (circles), mass scans (left arrows), protons on/off measurements (right arrows), and FC measurements (up arrows) shown with predicted total in-target production rates from the FLUKA and Geant4 models as bars. The rates from direct reactions are shown in light blue and pink for FLUKA and Geant4 respectively. The fractions from decay reactions are shown using orange and purple bars for ActiWiz and Geant4 respectively. Marker colors indicate the target temperature in ∘C during the measurement (blue: below nominal, black: nominal, red: above nominal) and the marker fill indicates the irradiation state during the measurement (empty: offline measurement before re-irradiation, half-filled: measurement taken after re-irradiation with the proton beam “off”, filled: online measurement during irradiation).  Table 2: Maximum rates reported for Np and Pu ions at target temperatures between 1850 and 2150 ∘C. Measurements where uncertainties are larger than the measured value are not included.  mass (u) | Np (ions/s) | Pu (ions/s)
---|---|---
235 | 2(1)E+4 | 2.3(2)E+5
236 | 1.76(14)E+3 | 1.38(2)E+5
237 | 1.56(51)E+5 |  $<$ 2.8E+4
238 | 1.3(1)E+6 | 8.5(2.6)E+6
239 | 1.55(4)E+5 | 6.82(9)E+6
240 | 1.7(14)E+2 | 2.14(4)E+4
241 | 7.1(4)E+1 | 2.6(7)E+4

##  IV Discussion

Isotope shifts are connected to the changes in nuclear charge radii and are
therefore not necessarily expected to be linear with neutron number [42],
especially in the actinide region of the nuclear chart [12]. The 236,239,240Pu
isotope shifts agree with linearity, but the 236,237,239Np isotope shifts are
not well-described by a linear fit. The offsets could suggest odd-even
staggering or the presence of a kink in the nuclear charge radius. These
preliminary measurements motivate future campaigns to extract further
information about the nuclear charge radii of Np.

From laser resonances, laser on/off mass scans and proton on/off measurements,
ion beam rates are reported for 235-241Pu (Fig. 12). These rates are
achievable with the use of pre-irradiated uranium carbide targets combined
with production of isotopes through inelastic reactions during irradiation and
can be achieved at nominal (1850-2150 ∘C) UC${}_{\textrm{x}}$ target
temperatures. In contrast, rates of Np isotopes are below $10^{6}$ ions s-1
even with protons on target and target temperatures above 2100 ∘C, indicating
poor release efficiency as follows from a comparison with simulated production
rates. Maximum rates observed experimentally in this work are reported in Tab.
2. Further developments may be required to enhance extraction from the target
or improve the efficiency of resonance ionization laser schemes.

Mass scans and ToF measurements of ion beams mass-separated by the separator
magnets show considerable rates of surface-ionized 235,238U on the nearby
masses of interest. The mass-separated actinide beams from ISOLDE formed
molecules in the ISOLTRAP RFQ-cb, creating contaminants several mass units
away from the species of interest, particularly for U and Pu. These
contaminants account for non-isobaric contaminants in the ToF spectra. In many
cases the ToF could be matched with a laser effect and corresponding number of
revolutions to identify a likely contaminant (Fig. 7). The molecular formation
decreased the effective efficiency of the species of interest, therefore ion
rates from the MR-ToF MS measurements are not reported.

The Geant4 model (Table 1) predicts trends in production mechanisms for these
above-target isotopes. The light Np isotopes 228-236Np are produced to more
than 99% through inelastic reactions, with direct irradiation contributing
more than 85% to the total production for all of these isotopes. 237,239,240Np
show large production fractions through decay, with 96.3%, 97.0% and 27.0%,
respectively. 240,241Np were mostly produced through inelastic reactions with
240Np mainly from tritons (47.8%) and $\alpha$-particles (18.8%), and 241Np
dominantly from $\alpha$-particles (96.3%) and heavier ions (3.7%). Heavier
isotopes 242-244Np were modeled with less than 5 events each, entirely from
heavy ions. For Pu, inelastic reactions with protons and $\alpha$-particles
contributed dominantly to the modeled production of the lighter isotopes
234-237Pu, while the heavier isotopes 238-244Pu result largely from the
$\beta$-decay of Np isotopes and, for isotopes above 240Pu, also from heavy
ion inelastic reactions. For 237,239Np and 238,239Pu, decay contributions are
described in both models. The FLUKA model and the direct reaction contribution
from the Geant4 model agree well for Np but show slightly different behaviour
in the case of Pu. Differences in the predictions of decay contributions given
by the ActiWiz model and the Geant4 model could be attributed to the
difference in the modeled cooling time.

Events of 235-240,242-244Am were modeled to be between 1 and 100 events per
109 primary protons. 162 events of 241Am were modeled dominantly from 241Pu
decay. Two of the Am isotopes seen in the model, 242,244Am, have short half
lives (16 h and 10 h respectively) and $\beta$-decay into 242,244Cm. Neither
242Cm nor 244Cm decay into the next element, Bk, marking the $Z$-limit of
elements available for experiments at high-energy proton-accelerator-based
thick target facilities. To reach usable rates of nuclides with higher proton
numbers, these facilities could employ heavier target nuclei, improve
extraction efficiency by several orders of magnitude, or improve experimental
sensitivity and increase duration. There are limited practical options for
targets of bulk material with nuclei heavier than 238U, where decreasing half-
lives and available quantities of actinide targets come with increased costs
and radioactivity handling considerations. These results demonstrate the
availability of two new transuranium beams from UC${}_{\textrm{x}}$ targets at
the ISOLDE facility with rates typically required for current experimental
sensitivity. Additionally, comparisons of production pathways with measured
rates suggest the potential availability of a select few Am isotopes for
experiments with state-of-the art sensitivity.

##  V Conclusions

From 1.4-GeV proton irradiations of a UCx target, above-target nuclides of
neptunium and plutonium have been produced, released from the hot target,
resonantly laser-ionized and identified as ion beams at the ISOLDE facility.
Production mechanisms modeled by Geant4 predict production-pathway-specific
yields. On the neutron-deficient side of stability, Np and Pu isotopes are
formed through inelastic reactions, mostly induced by protons. Decay
contributions from $\beta$-decaying U precursors are significant for
237,239,240Np and 238-244Pu, with the $\beta$-decay of Np contributing to the
Pu inventory. The use of previously irradiated targets with long cooling times
thus contributes to the in-target inventory especially of Pu isotopes, while
neutron-deficient isotopes are produced dominantly from direct irradiation.

Two-step ionization schemes using intra-cavity doubled Ti:Sa lasers were
applied to resonantly ionize both Np and Pu. Isotope shifts were investigated
for the 395.6-nm ground state transition in 236,237,239Np and the 413.4-nm
ground state transition in 236,239,240Pu. To our knowledge, there is no data
available in literature for the charge radius of Np nuclei. With two odd-odd
nuclei and odd-even 236Np, Np isotope shifts did not follow a linear trend.
Further measurements are required to extract information about the nuclear
charge radii. In the case of Pu, isotope shifts are linear with nucleon number
in the investigated range and can be extrapolated for use over a wider range
of isotopes.

Ion production rates were measured for 235-240Np and 234-240Pu (Fig. 12). ToF
MS spectra show surface-ionized contamination on the masses of interest,
originating from the uranium carbide target material. Oxide formation is
observed in a gas-filled RFQ-cb. Comparing the observed rates with predicted
in-target inventory gives an estimation for the extraction efficiency of Np on
the order of 0.001% and Pu on the order of 0.1%. Target temperatures required
for observable rates of Np were above 2100 ∘C, while target temperatures
required for Pu were above 1150 ∘C. Pu is released within 400-500 s at nominal
target temperatures. The demonstrated availability of Np and Pu beams at
ISOLDE brings two new actinide elements into reach for experiments at ISOL
facilities with useful intensities of specific isotopes as functions of
different experimental parameters such as decay time. Information on laser
ionization schemes, magnitudes of isotope shifts, and release temperatures
informs future experiments requesting beams of these elements.

241Pu is the heaviest nuclide identified at a proton-accelerator-driven ISOL
facility to date. While extremely sensitive experiments may be capable of
extending studies one element higher to specific isotopes of Am discussed
here, the investigation of production mechanisms suggests that plutonium is
the high-$Z$ limit of heavy nuclide production from 1.4-GeV protons on a 238U
target with typical rates required for current experimental sensitivity.

###### Acknowledgements.

The authors gratefully acknowledge technical support from E. Barbero, M.
Bovigny, B. Crepieux, J. Cruikshank and the ISOLDE operations staff. This
project has received funding from the European Union’s Horizon 2020 Research
and Innovation Program (grant No. 861198 project ‘LISA’ MSCA ITN). The authors
acknowledge support from the German Federal Ministry of Education and Research
(BMBF) for ISOLTRAP (grant No. 05P18HGCIA and 05P21HGCI1). L.N. acknowledges
support from the Wolfgang Gentner Program (grant No. 13E18CHA).

##  Appendix A In-target production models

The FLUKA Monte-Carlo model of in-target production is a standard tool
available for users of the CERN-ISOLDE facility, with a database of modeled
values [20]. This model uses the ABLA description of nuclear de-excitation
through particle evaporation and fission for the decay pathways of nuclear
systems [43]. The FLUKA model incorporates no cooling time for radioactive
decay. Instead, the FLUKA model of particle fluence spectra in the target was
used as input to the ActiWiz Creator software version 3.4 [29]. ActiWiz is
based on 100 CPU years of FLUKA calculations modeling interactions of protons,
charged pions, photons, and neutrons above 20 MeV with the evaluated libraries
JEFF 3.3, ENDF VIII.0 and EAF2010 for neutrons below 20 MeV. The ActiWiz model
was used to extend the FLUKA model to include the inventory of radionuclides
produced during the cooling period and is detailed in Ref. [30].

Geant4 (version geant4-11) was used with physics list QGSP_INCLXX_HP_ABLA, an
experimental physics list using the Liège intranuclear cascade model for
nucleon-nucleus interactions below 3 GeV, with neutronHP (high precision
neutron package) for neutron cross-sections below 20 MeV and de-excitation
modeled using ABLA with datasets for neutron, proton and pion cross-sections
for elastic, inelastic, capture and fission reactions as described in [32].

Predictions from the INCLXX and ABLA models show good agreement for a
selection of lighter isotopic chains measured at ISOLDE [44].

##  Appendix B Target and resonance ionization laser ion source

The measured temperature was calibrated against the target and ion source
heating current and required power using an optical pyrometer up to 2000 ∘C.
The calibration data were extrapolated to estimate the target and ion source
temperatures during the experiment with an uncertainty estimated to be 100 ∘C.
The calibration was performed before the loading of the target material and
the first irradiation of the target.

Three diode pumped solid-state (DPSS) Nd:YAG lasers were used to pump a total
of four titanium:sapphire (Ti:Sa) lasers based on wavelength selection by
grating tuning [35] or a combination of a Lyot filter and Fabry-Perot etalon
[36] with a repetition rate of 10 kHz. One intra-cavity—second-harmonic
generation (IC-doubled) grating Ti:Sa laser [37] was used to provide the first
excitation step (FES) for either Pu or Np, with typical output power up to 400
mW. The other three IC-doubled Ti:Sa lasers were used to provide the second
excitation step (SES) of Pu, the identical wavelength of the FES and SES of U,
and the SES of Np, each with up to 2.5 W of output power. The Pu resonance
ionization scheme was developed by Kneip et al [14] and uses a FES from the
ground state to the $5f^{5}6d^{2}7s$ J=1 state followed by a SES to an auto-
ionizing state located above the ionization potential. The U scheme was
developed by Savina et al [38] and features a two-step, single wavelength
ionization from the ground state to an auto-ionizing state. Several schemes
were investigated for Np using a FES of 25277.6 cm-1 from the ground state to
a J=9/2 state.

To switch between ionizing Pu or Np, the grating Ti:Sa was changed between the
Pu FES and the Np FES. Wavelength scans of the FES were done by changing the
diffraction grating angle. Small-range wavelength scans of the second step
were done manually by changing the etalon angle. The frequency of the Ti:Sa
lasers was measured using a High Finesse WS7 wavemeter (uncertainty 20% of the
laser linewidth) to sample the fundamental, non-frequency-doubled mode. Atomic
transitions in the ion source are expected to be Doppler-broadened, with the
FWHM given by $\Delta\nu_{D}$ where:

$\Delta\nu_{D}=\nu_{0}\sqrt{\frac{8k_{B}T\ln 2}{mc^{2}}}$ (1)

where $T$ is the absolute temperature, $k_{B}$ is the Boltzmann constant, $m$
is the mass of the atom, $\nu_{0}$ is the nominal laser frequency, and $c$ is
the speed of light in vacuum. For the employed transitions around 400 nm, the
Doppler broadening at 2000 ∘C thus corresponds to Gaussian peak shapes with
FWHM above 1.6 GHz for a mass of 238 u. Laser linewidth and power-broadening
may increase the experimentally observed peak widths further [23].

##  Appendix C Ion beam analysis

The ion beams mass-separated by the GPS separator magnet were injected into
the ISOLTRAP RFQ-cb, where they were collisionally cooled with the He buffer
gas at pressures up to 10-5 mbar measured within 1 m of the injection. After
ejection from the RFQ-cb, the ion bunch energy was reduced to 3.2 keV using a
pulsed drift tube. Ion bunches were then injected into the ISOLTRAP MR-ToF MS
[40] and trapped between the electrostatic mirror potentials using the in-trap
lift method [41] for typically 1000 revolutions before release and detection
of the ions’ time of flight (ToF). The difference in ToF allows components
with different mass-over-charge ratios $m/q$ to be separated in time $t$ with
a mass resolving power $R$ given by:

$R=\frac{m}{\Delta m}=\frac{t}{2\Delta t}$ (2)

where $t$ is the absolute time-of-flight and $\Delta t$ is the FWHM of the ToF
distribution. After trapping times of up to 40 ms, the ions were ejected onto
the ISOLTRAP MagneToF detector. Ion arrival times were measured from the time
of ejection from the RFQ-cb and recorded with 100 ps resolution. Mass
resolving powers $R$ in excess of 105 were achieved for the investigated mass
range.

Additionally, non-isobaric beam components formed in the trapping and cooling
process from the mass-separated beam can be identified. These components can
appear in the same ToF spectrum when travelling through the in-trap lift
cavity of the MR-ToF MS with the isobaric beam component on a different number
of revolutions during bunch ejection. Non-isobaric components were identified
by varying the trapping time of the isobaric beam bunch in the MR-ToF MS. This
leads to different absolute ejection times, which increases the chance that
the non-isobaric component is separated from the isobaric beam bunch. The
ISOLTRAP MR-ToF MS was calibrated using 85,87Rb and 133Cs from the ISOLTRAP
offline ion source and using 238U from the ISOLDE GPS target and ion source.
The reference measurements were used to create a calibration to relate known
masses to an expected ToF in the form:

$t_{\mathrm{calibration}}(m)=a\sqrt{m/q}+b$ (3)

where $a$ and $b$ are the calibrated parameters.

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From Uniform Continuity to Absolute Continuity

Kai Yang, Chenhong Zhu

The notion of uniform continuity emerged slowly in the lectures of
Dirichlet(1854) and of Weierstrass(1861) [2]. Then in 1905, Vitali established
the absolute continuity for a class of functions in the paper “Sulle funzioni
integrali” [3]. Sooner or later, several equivalent and sufficient conditions
for a function to be absolutely continuous were derived (see [4]), which
depend on several results of measure theory and integration. For example,
Banach-Zarecki Criterion[5] states that, “$f$ is absolutely continuous on
$[a,b]$ if and only if $f$ is continuous and of bounded variation on $[a,b]$
and maps null sets to null sets. ”, and in particular, every Lipschitz
continuous function is absolutely continuous.

Absolute continuity implies uniform continuity, but generally not vice versa.
However, under certain conditions (piecewise convexity), uniform continuity
will also imply absolute continuity. In this short note, we will present a
sufficient condition for a uniformly continuous function to be absolutely
continuous.

###### Definition 1

(piecewise convex function) A function $f$ defined on an interval $I_{a,b}$ of
the real line is piecewise convex, if there exits a finite partition
$P=\\{a_{i}\\}_{i=0}^{N}$ such that on each subinterval $[a_{i},a_{i+1}]$
$(i=0,\dots,$ $N-1)$, $f$ is concave or convex.

Remark: Here $I_{a,b}$ is an interval of $\mathbb{R}$ with $a,b$ as the
endpoints and it can be open, closed or half-open; moreover, $a,b$ can also
take the value $\pm\infty$ when $a$ or $b$ is not contained in $I_{a,b}$ (in
this case, we use $(a_{0},a_{1}]$ or $[a_{N-1},a_{N})$ instead of the closed
subinterval).

###### Theorem 1

For a uniformly continuous function $f$ defined on $I_{a,b}$ of $\mathbb{R}$,
if it is piecewise convex, then it is also absolutely continuous on $I_{a,b}$.

Remark: We can verify that if $I_{a,b}=[a,b]$, then the conditions in Theorem
1 satisfy the Banach-Zarecki Criterion. However, our attempt is based on some
elementary properties of uniform continuity and convexity. In particular, one
simple example is that $f(x)=\sqrt{x}$ is absolutely continuous (not Lipschitz
continuous) on $[0,c]$. Moreover, the converse statement of this theorem is
false. One proper counterexample is $f(x)=x^{2}\sin(1/x)$ on $[0,1]$ (define
$f(0)=0$). In addition, the cantor function[7] is uniformly continuous but not
absolutely continuous.

Figure 1:  graph of $f(x)=\sqrt{x}$ when $\sigma=0.5$

This picture gives us the idea of Lemma 1 that the slope of an increasing
concave function of two points with the same distance of the $x$ coordinate
will decrease. In addition, the monotonicity of the change of slopes will
still be valid for other monotone convex or concave functions.

###### Lemma 1

If $f$ is monotone and concave or convex on an interval $I$ of $\mathbb{R}$,
then $G_{\sigma}(x)=|f(x+\sigma)-f(x)|$ is monotone with respect to $x$, where
$\sigma$ is any positive real number.

Proof. We only demonstrate the case that $f(x)$ is monotone increasing and
concave, other situations can be proved similarly. Obviously, in this case,
$G_{\sigma}(x)=f(x+\sigma)-f(x)$ for $\sigma>0$. We will show that
$G_{\sigma}(x)$ is decreasing on $I$ with respect to $x$.

Suppose $x,y+\sigma\in I$, and $x<y$. Since $f(x)$ is concave, for any
$\theta\in(0,1)$, we get

$f(\theta a+(1-\theta)b)\geq\theta f(a)+(1-\theta)f(b),\,\,\,\,\forall a,b\in
I.$ (1)

1) Set $a=x$, $b=y+\sigma$, $\theta=\frac{y-x}{y-x+\sigma}$: By (1) , we have

$f(x+\sigma)\geq\frac{y-x}{y-x+\sigma}f(x)+\frac{\sigma}{y-x+\sigma}f(y+\sigma),$

which is equivalent to

$\frac{f(y+\sigma)-f(x)}{y-x+\sigma}\leq\frac{f(x+\sigma)-f(x)}{\sigma}.$ (2)

2) Set $a=x$, $b=y+\sigma$, $\theta=\frac{\sigma}{y-x+\sigma}$: Similarly, we
can get

$\frac{f(y+\sigma)-f(y)}{\sigma}\leq\frac{f(y+\sigma)-f(x)}{y-x+\sigma}.$ (3)

By (2) and (3) , we obtain

$\frac{f(y+\sigma)-f(y)}{\sigma}\leq\frac{f(x+\sigma)-f(x)}{\sigma}.$

Thus, $G_{\sigma}(y)\leq G_{\sigma}(x)$, which implies that $G_{\sigma}(x)$ is
a monotone decreasing function.

Remark: Lemma 1 is one special case of a classical property of convex or
concave functions on an interval of $\mathbb{R}$, see [6].

###### Lemma 2

For a monotone concave or convex function $f(x)$ defined on an interval $I$ of
$\mathbb{R}$, if $f$ is uniformly continuous, then $f$ is absolutely
continuous.

Proof. Since $f$ is uniformly continuous on $I$, for any $\epsilon>0$, there
exists $\delta>0$ such that for any $x,y\in I$ and $|x-y|<\delta$, we have

$|f(x)-f(y)|<\epsilon.$

Now, we consider every finite collection $\\{(x_{i},y_{i})\\}_{i=1}^{n}$ of
nonoverlapping subintervals of $I$ with $x_{i}<x_{i+1}$, and

$\sum_{i=1}^{n}(y_{i}-x_{i})<\delta.$

Define $\sigma_{i}=y_{i}-x_{i}>0$, then

$|f(y_{i})-f(x_{i})|=G_{\sigma_{i}}(x_{i}).$

By Lemma 1, we know that $G_{\sigma}(x)$ is monotone.

1) $G_{\sigma}(x)$ is monotone deceasing. Since $(x_{2},y_{2})$ and
$(x_{1},y_{1})$ are nonoverlapping, we get

$G_{\sigma_{2}}(x_{2})\leq
G_{\sigma_{2}}(y_{1})=G_{\sigma_{2}}(x_{1}+\sigma_{1}).$

Inductively, for $i\geq 2$, we will obtain

$G_{\sigma_{i}}(x_{i})\leq G_{\sigma_{i}}(x_{1}+\sum_{j=1}^{i-1}\sigma_{j}).$

Therefore, we have

$\displaystyle\sum_{i=1}^{n}|f(y_{i})-f(x_{i})|$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}G_{\sigma_{i}}(x_{i})\leq
G_{\sigma_{1}}(x_{1})+\sum_{i=2}^{n}G_{\sigma_{i}}(x_{1}+\sum_{j=1}^{i-1}\sigma_{j}).$

Define $z_{1}=x_{1}$, and $z_{i}=x_{1}+\sum_{j=1}^{i-1}\sigma_{j}$ $(2\leq
i\leq n+1)$. Since $f$ is monotone, the above inequality is equivalent to

$\sum_{i=1}^{n}|f(y_{i})-f(x_{i})|\leq\sum_{i=1}^{n}|f(z_{i+1})-f(z_{i})|=|f(z_{n+1})-f(z_{1})|.$

In addition,
$|z_{n+1}-z_{1}|=\sum_{i=1}^{n}\sigma_{i}=\sum_{i=1}^{n}(y_{i}-x_{i})<\delta$,
since $f$ is uniformly continuous, we obtain

$\sum_{i=1}^{n}|f(y_{i})-f(x_{i})|\leq|f(z_{n+1})-f(z_{1})|<\epsilon.$

Hence, $f$ is also absolutely continuous on $I$.

2) $G_{\sigma}(x)$ is monotone increasing. The strategy is quite similar to
the previous one; however, we just fix $(x_{n},y_{n})$ first and define
$z_{n+1}=y_{n}$, $z_{i}=y_{n}-\sum_{j=i}^{n}\sigma_{j}$ for $1\leq i\leq n$.
Similarly, we have

$\sum_{i=1}^{n}|f(y_{i})-f(x_{i})|\leq\sum_{i=1}^{n}|f(z_{i+1})-f(z_{i})|=|f(z_{n+1})-f(z_{1})|<\epsilon.$

Therefore, $f$ is absolutely continuous on $I$.

Remark: The idea of this lemma is to glue disjoint subintervals
$(x_{i},y_{i})$ together as one subinterval then apply the property of uniform
continuity. By Lemma 2, it is clear that $f(x)=\sqrt{x}$ is absolutely
continuous on $[0,c]$. In addition, we can consider more general functions
that oscillate finite times, and on each monotone subinterval, they also admit
convexity. If these functions are uniformly continuous, then they are also
absolutely continuous by utilizing the same strategy in the proof of Lemma 2
on each subinterval.

Proof of Theorem 1. If $f$ is not monotone on $[a_{i},a_{i+1}]$, then we can
split $[a_{i},a_{i+1}]$ into two subintervals such that on each of them, $f$
is monotone, since $f$ assumes convexity. If we relabel them, then on each
$[a_{i},a_{i+1}]$, $f$ is monotone and concave or convex. The following proof
is based on this situation.

Since $f$ is uniformly continuous on $I$, for any $\epsilon/N>0$, there exists
$\delta>0$ such that for any $x,y\in I$ and $|x-y|<\delta$, we have

$|f(x)-f(y)|<\epsilon/N.$

Choose $\delta_{1}<\min\\{a_{1}-a_{0},\dots,a_{N}-a_{N-1},\delta\\}$, then for
every finite collection $\\{(x_{i},y_{i})\\}_{i=1}^{n}$ of nonoverlapping
subintervals of $I$ with $x_{i}<x_{i+1}$ and
$\sum_{i=1}^{n}(y_{i}-x_{i})<\delta_{1},$ each subinterval $(x_{i},y_{i})$ can
contain at most one $a_{j}$. For such interval $(x_{i},y_{i})$ containing
$a_{j}$, by triangle inequality, we have

$|f(y_{i})-f(x_{i})|\leq|f(y_{i})-f(a_{j})|+|f(a_{j})-f(x_{i})|.$ (4)

Now, we still treat $(x_{i},a_{j})$ and $(a_{j},y_{i})$ as two “nonoverlapping
subintervals”. Then, we relabel all the subintervals as
$\\{(x_{j},y_{j})\\}_{j=1}^{m}$ ($n\leq m\leq 2n$). Through the above
strategy, each new $(x_{j},y_{j})$ will lie exactly in one $[a_{i},a_{i+1}]$
and

$\sum_{j=1}^{m}(y_{j}-x_{j})=\sum_{i=1}^{n}(y_{i}-x_{i})<\delta_{1}.$

For all the new subintervals that lie in $[a_{i},a_{i+1}]$ $(i=0,\dots,N-1)$,
since $f$ is also monotone and concave or convex and

$\sum_{(x_{j},y_{j})\subset[a_{i},a_{i+1}]}(y_{j}-x_{j})<\delta_{1}<\delta,$

by the method in Lemma 2, we obtain that

$\sum_{(x_{j},y_{j})\subset[a_{i},a_{i+1}]}|f(y_{j})-f(x_{j})|<\epsilon/N.$

Therefore,

$\sum_{j=1}^{m}|f(y_{j})-f(x_{j})|=\sum_{i=0}^{N-1}{\sum_{(x_{j},y_{j})\subset[a_{i},a_{i+1}]}|f(y_{j})-f(x_{j})|}<\epsilon.$
(5)

By (4) and (5) , we get

$\sum_{i=1}^{n}|f(y_{i})-f(x_{i})|\leq\sum_{j=1}^{m}|f(y_{j})-f(x_{j})|<\epsilon.$

Hence, $f$ is also absolutely continuous on $I_{a,b}$.

Question: We consider the case on the real line; however, one can think about
the situation for the multidimensional Euclidean space.

## References

  * [1]
  * [2] E. Hairer, G. Wanner, Analysis by Its History, Springer-Verlag, New York, 1996.
  * [3] M. B. Porter, Concerning Absolutely Continuous Functions, Bulletin of the American Mathematical Sociaty, Vol.22, No.3, p109-111, 1915.
  * [4] E. V. Dale, On Absolutely Continuous Functions, American Mathematical Monthly, Vol.72, No.8, p831-841, 1965\.
  * [5] J. Yeh, Lectures on Real Analysis, World Scientific, Singapore, 2000.
  * [6] H. L. Royden, Real Analysis, 3rd ed., Macmillan Publishing Company, New York, 1988.
  * [7] D. R. Chalice, A Characterization of the Cantor Function, Amererican Mathematical Monthly, Vol.98, No.3, p255-258, 1991.

Department of Mathematics and Statistics, Memorial University of Newfoundland,
NL, Canada.

<EMAIL_ADDRESS>
Applied Mathematical and Computational Sciences, University of Iowa, IA, USA.

<EMAIL_ADDRESS>


# Improving Baselines in the Wild

Kazuki Irie1 Imanol Schlag1 Róbert Csordás1 Jürgen Schmidhuber1,2
1The Swiss AI Lab, IDSIA, University of Lugano (USI) & SUPSI, Lugano,
Switzerland
2King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi
Arabia
{kazuki, imanol, robert<EMAIL_ADDRESS>

###### Abstract

We share our experience with the recently released WILDS benchmark, a
collection of ten datasets dedicated to developing models and training
strategies which are robust to domain shifts. Several experiments yield a
couple of critical observations which we believe are of general interest for
any future work on WILDS. Our study focuses on two datasets: iWildCam and
FMoW. We show that (1) Conducting separate cross-validation for each
evaluation metric is crucial for both datasets, (2) A weak correlation between
validation and test performance might make model development difficult for
iWildCam, (3) Minor changes in the training of hyper-parameters improve the
baseline by a relatively large margin (mainly on FMoW), (4) There is a strong
correlation between certain domains and certain target labels (mainly on
iWildCam). To the best of our knowledge, no prior work on these datasets has
reported these observations despite their obvious importance. Our code is
public.111https://github.com/kazuki-irie/fork--wilds-public

##  1 Introduction

It goes without saying that common benchmark datasets and solid baselines are
essential ingredients for developing new models and correctly measuring
progress in machine learning. Key datasets such as ImageNet [1] in image
recognition, Switchboard [2] in automatic speech recognition, or Penn Treebank
[3, 4] in language modelling, to name only a few, have played a crucial role
in demonstrating the impressive performance of neural networks (NNs)
especially in the 2010s. They have contributed to gradually marking the
progress obtained by new techniques and models. Equally importantly, strong
baselines are crucial to properly measure progress. In some cases, by
revisiting some old or standard baseline configurations, we end up with
surprisingly good results, including results of LSTMs for language modelling
[5], ResNets for vision [6], or Transformers for systematic generalization
[7].

Nowadays, few researchers are surprised by NNs performing very well on some
in-domain data distribution, and there has been increasing interest in
developing models that are robust to domain shifts [8, 9, 10, 11]. Here we
focus on the recently proposed benchmark for evaluating domain robust systems,
WILDS [11], and we share our empirical experience with two datasets of WILDS,
iWildCam and FMoW, as well as their baseline models. A handful of experiments
exhibit important, previously unreported facets of the data and the baselines.
Our main findings can be summarised as follows: (1) Conducting separate cross-
validation for each evaluation metric is crucial for both datasets, (2) A weak
correlation between validation and test performance might make model
development difficult (iWildCam), (3) Minor changes in the training hyper-
parameters improve the baseline by a relatively large margin (mainly on FMoW),
(4) There is a strong correlation between certain domains and certain target
labels (mainly on iWildCam). Any practitioner should be aware of these aspects
when developing new ideas based on these two datasets, as well as on other
datasets of WILDS.

##  2 Datasets and Experimental Settings

Here we briefly summarize essential properties of the datasets used in our
experiments. For further information, we refer to the original paper by Koh et
al. [11].

### iWildCam2020-WILDS.

iWildCam2020-WILDS (iWildCam for short) is a variant of the iWildCam 2020
Competition Dataset [12]. The task is classification of photos of 182
different animal species from various camera traps. The camera traps define
various domains of this dataset. The training set contains about 130 K photos
taken by 243 camera traps. The out-of-distribution validation and test sets
consist of photos from 32 and 48 different camera traps which are not part of
the training domains. Following Koh et al. [11], we report the classification
accuracy, as well as the macro F1 score which Koh et al. [11] set as the most
relevant evaluation metric.

### FMoW-WILDS.

FMoW-WILDS (FMoW for short) is a variant of the functional map of the world
dataset [13]. This is also an image classification task using satellite images
containing one of 62 building or land classes. There are two attributes for
grouping these images: years and geographical regions (Africa, the Americas,
Oceania, Asia, or Europe). The training data consist of images taken before
2013, the validation from 2013 to 2015, and the test set from 2016. Koh et al.
[11] set the worst region accuracy (WR) as the main evaluation metric, and
also report the total accuracy (TA).

### Experimental settings.

Unless otherwise stated, we use all default settings provided by the publicly
available official codebase for WILDS [11].

Table 1: Performance on iWildCam. “CV” denotes the cross-validation criterion
(on the OOD validation set) used to select the best checkpoint. * denotes
information which is not available in the original paper. Following Koh et al.
[11], we report mean and std over 3 training runs.

Model CV Accuracy (AC) Macro F1 (F1) Valid Test Valid Test ID OOD ID OOD ID
OOD ID OOD Koh et al. [11] * 82.5 (0.8) 62.7 (2.4) 75.7 (0.3) 71.6 (2.5) 48.8
(2.5) 37.4 (1.7) 47.0 (1.4) 31.0 (1.3) Our run F1 82.7 (1.1) 61.7 (0.4) 74.9
(0.2) 69.9 (0.1) 47.9 (0.6) 36.2 (0.4) 45.3 (0.4) 30.2 (1.2) AC 82.7 (0.1)
64.7 (0.4) 75.6 (0.3) 71.9 (1.8) 45.0 (4.2) 33.6 (3.3) 41.1 (5.3) 27.2 (3.4)
\+ freq ckpt F1 82.5 (0.8) 64.1 (1.7) 76.2 (0.1) 69.0 (0.3) 46.7 (1.0) 38.3
(0.9) 47.9 (2.1) 32.1 (1.2) AC 82.6 (0.7) 66.6 (0.4) 75.8 (0.4) 68.6 (0.3)
46.2 (0.9) 36.6 (2.1) 44.9 (0.4) 28.7 (2.0)

##  3 Core findings

In this section, we highlight our findings on iWildCam2020-wilds (iWildCam for
short) and FMoW-wilds (FMoW for short). We note that all our results are
produced using the official baseline code provided by Koh et al. [11] with
small modifications specified below.

### Metric-wise cross-validation is crucial.

In both iWildCam and FMoW, the performance of models is measured on two or
more evaluation metrics (one of them is considered to be the “main” metric).
E.g., for iWildCam, the classification accuracy (AC) and macro F1 (F1) are
reported on in-domain (ID) and out-of-domain (OOD) subsets of validation and
test sets. Assuming that we mainly care about the OOD performance, this
results in two metrics to monitor during the model development: OOD validation
AC and F1. Technically, this implies a necessity for conducting cross-
validation and checkpoint tracking separately for each metric. We note that
such a metric-wise cross-validation is not part of the official setting [11].
Furthermore, we observe that in the official setting, the cross-validation is
carried out only at the end of each epoch. We increase the cross-validation
frequency (to every 1000 steps) such that we do not miss good checkpoints
between epochs. Table 1 shows the performance of different checkpoints
selected using two different CV criteria within the same training runs on
iWildCam. The first thing to notice here is that simply introducing checkpoint
tracking between epochs can improve the baseline performance by a non-
negligible margin. In Table 1, if we compare “Our run” which uses the default
setting by Koh et al. [11] and “+ freq ckpt” which does cross validation every
1000 training steps: the OOD validation accuracy improves from 64.7% to 66.6%
and the F1 from 36.2% to 38.3%.

The impact of metric-wise CV is also relatively large: In our best
configuration (“+ freq ckpt” in Table 1), the AC-CV checkpoint (i.e. the best
checkpoint found by cross validation based on OOD validation accuracy)
achieves an OOD validation accuracy of 66.6% vs. 64.1% for the F1-CV one.
Similarly, the F1-CV model achieves an OOD validation F1 score of 38.3%
compared to 36.6% for the AC-CV one.

We stress that these performance gaps are obtained without any technical
changes on the algorithmic level, revealing that very careful tuning is needed
to compare models on this dataset. Strictly speaking, we should also do
separate cross-validations on the ID validation metrics to report and compare
the ID performance. We omit this here as our results on the OOD metrics
already clearly exhibit the issue. While our goal is not to develop separate
models for different metrics, we observe how crucial it is to report the
cross-validation setting for fair comparisons between different approaches.

### Correlation between validation and test performance is weak.

Another important observation we drawn from Table 1 is that despite a
relatively large performance gap between the F1-CV and AC-CV models in terms
of OOD validation accuracy (64.1 vs. 66.6%), their OOD test accuracy is
similar (69.0 vs. 68.6%). This trend is less visible for FMoW (Table 2, batch
size 64), but still: our WR-CV model outperforms the official baseline on the
WR validation accuracy (50.2 vs. 48.9%) while it underperforms on the
corresponding WR test accuracy (31.4 vs. 32.3%). This is actually not
surprising as the validation and test sets are also OOD to each other, hence
improvements on the validation set do not necessarily transfer to the test
set. However, this is problematic since we only have access to the OOD
validation performance while developing models. The common practice of
selecting the checkpoint based on a validation set (also recommended by the
original work [11]; in the paragraph on avoiding overfitting to the test sets)
might not work here: we thus have no reliable reference metric for developing
new models. A similar problem regarding the lack of useful validation sets has
been recently pointed out [7] in the context of systematic generalization
[14]. This problem seems even harder for general domain shifts: unlike in the
case of systematic generalization, there is no a priori way of controlling the
degree of "domain shifts" systematically. This calls for discussing the
construction of useful validation sets when studying OOD generalization.

Table 2: Performance on FMoW. “Batch” column indicates the batch size. “CV”
denotes the Cross-validation Criterion (on the OOD validation set). ∗not
mentioned in the corresponding paper. Following Koh et al. [11], we report
mean and std dev over 3 training runs.

Batch CV Total Accuracy (TA) Worst Region Accuracy (WR) Valid Test Valid Test
IID OOD IID OOD IID OOD IID OOD 64 [11] * 61.2 (0.5) 59.5 (0.4) 59.7 (0.7)
53.0 (0.6) 59.2 (0.7) 48.9 (0.6) 58.3 (0.9) 32.3 (1.3) 64 TA 60.8 (0.2) 59.3
(0.3) 59.7 (0.1) 52.6 (0.4) 58.7 (0.3) 47.6 (1.9) 58.2 (0.3) 30.9 (2.6) WR
59.8 (0.8) 57.9 (0.5) 58.7 (0.5) 51.9 (0.4) 57.8 (1.0) 50.2 (1.2) 57.0 (0.5)
31.4 (0.9) 32 TA 61.2 (0.1) 59.8 (0.1) 60.1 (0.1) 53.3 (0.2) 59.2 (0.3) 48.1
(1.1) 58.5 (0.5) 33.4 (0.1) WR 60.4 (0.9) 58.8 (0.6) 59.3 (0.6) 52.5 (0.5)
58.3 (0.9) 51.0 (0.9) 57.7 (0.5) 33.6 (1.6) 20 TA 62.1 (0.1) 60.3 (0.2) 60.7
(0.1) 53.8 (0.3) 60.4 (0.2) 49.6 (0.1) 59.0 (0.1) 33.8 (0.5) WR 61.3 (0.3)
58.9 (0.6) 59.9 (0.4) 52.5 (0.2) 59.0 (0.3) 52.4 (0.8) 58.5 (0.5) 34.4 (0.3)
\+ higher lr TA 64.0 (0.1) 62.1 (0.0) 62.3 (0.4) 55.6 (0.2) 62.3 (0.4) 51.4
(1.3) 61.1 (0.6) 34.2 (1.2) WR 63.9 (0.2) 62.1 (0.0) 62.3 (0.2) 55.6 (0.2)
62.2 (0.5) 52.5 (1.0) 60.9 (0.6) 34.8 (1.5) Fish [9] * * 57.8 (0.2) * 51.8
(0.3) * 49.5 (2.3) * 34.6 (0.2)

### There is sub-optimality in the baseline.

Another crucial observation regarding FMoW is the sub-optimality of the
baseline setting. While keeping all configurations equal to the original
configuration [11], we modify the training batch size in the FMoW baseline. As
is shown in Table 2, simply by reducing the batch size from 64 to 20, the
best/worst region accuracy (WR; the main evaluation metric for FMoW) improves
from 50.2% to 52.4% on the OOD validation set, and from 31.4% to 34.4% on the
OOD test set. By further modifying the learning rate from 1e-4 to 3e-4 and
increasing the checkpoint frequency to be every 200 steps instead of 1000, we
finally obtain a worst region accuracy of 34.8% (rows “+ higher lr”). We note
that these final OOD test accuracies we obtain are competitive compared to the
number reported222These numbers were taken from
https://wilds.stanford.edu/leaderboard/ as recommended by Shi et al. [9]
(personal communication). by Shi et al. [9], 34.6%, achieved using a technique
specifically designed for the domain shift problem (a method called Fish; last
row in Table 2). Such “trivial” improvements over the baseline call for
further tuning of the baseline settings before developing and evaluating new
models on this dataset.

### Correlation between domains and target labels is rather strong.

Our last observation to be shared here is that certain domains have a bias
towards certain target labels. Each image in iWildCam is labeled with a target
class label (one of 182 animal species) and a domain label (one of 323 camera
traps). For each domain, we count the number of unique class labels covered
within the domain i.e., if this number is 1 for a certain domain, all images
belonging to that domain have the same target class label. Figure 1 shows the
cumulative histogram of the corresponding statistics. As can be seen, the
number of unique class labels (x-axis) vary only from 1 to 24 (while the
maximum is 182) and the 50% of the domains contain less than 5 distinct
classes (indicated by the yellow line). This number improves to 10 distinct
classes only when we take 75% of the domains (red line). Overall, the
diversity of target class labels is very limited within each domain. This bias
can potentially be problematic when developing training strategies which make
use of some grouping of datapoints by the domain label. A number of techniques
has been proposed to exploit domain information [15, 16, 17] for domain robust
learning. Interestingly, none of these techniques have resulted in consistent
improvements over the basic empirical risk minimisation baseline [11]. We
point out these previously unreported important statistical properties
although further analysis is required to draw conclusions regarding the causal
relationship between these two observations.

Figure 1:  Cumulative histogram over the number of domains with the
corresponding number of unique target class labels within the domain. The
yellow line indicates the coverage of 50% of the total number of domains,
while the red one indicates 75%.

##  4 Conclusion

To properly evaluate new machine learning methods and measure progress, it is
crucial to start from strong and well-established baselines. We reported a
couple of important observations from our experiments with the baseline
settings of WILDS. The latter can be improved by simply tweaking a few
configurations (frequent metric-wise cross-validation) and one hyper-parameter
for FMoW (batch size). Our observations seem to indicate that a systematic
study of baseline configurations is necessary before starting the development
of new models based on WILDS. We hope that practitioners will find our
observations useful, and take them into account for future work on WILDS.

## Acknowledgments and Disclosure of Funding

This research was partially funded by ERC Advanced grant no: 742870, project
AlgoRNN, and by Swiss National Science Foundation grant no: 200021_192356,
project NEUSYM. This work was partially supported by computational resources
at the CSCS Swiss National Supercomputing Centre, project d115. We thank
NVIDIA Corporation for donating several DGX machines, and IBM for donating a
Minsky machine.

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