Present Address: ]Lawrence Livermore National Laboratory, Livermore, CA 94550,
USA. Present Address: ]Accelerator Laboratory, Department of Physics,
University of Jyväskylä, FI-40014 Jyväskylä, Finland. Present Address: ]Oak
Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA. Present Address:
]Lawrence Livermore National Laboratory, Livermore, CA 94550, USA. Present
Address: ]Morgan State University, 1700 East Cold Spring Lane, Baltimore,
Maryland 21251, USA. Present Address: ]Florida State University, Tallahassee,
Florida 32306, USA Present Address: ]Brookhaven National Laboratory, National
Nuclear Data Center, Upton, New York 11973, USA. ]Deceased.

# Experimental Study of the 38S Excited Level Scheme

C. R. Hoffman<EMAIL_ADDRESS>Physics Division, Argonne National
Laboratory, 9700 S. Cass Ave., Argonne, IL 60439, USA  R. S. Lubna  TRIUMF,
4004 Wesbrook Mall, Vancouver, British Columbia V6T 2A3, Canada  Facility for
Rare Isotope Beams, Michigan State University, East Lansing, Michigan 48824,
USA  E. Rubino  [  Department of Physics, Florida State University,
Tallahassee, Florida 32306, USA  S. L. Tabor  Department of Physics, Florida
State University, Tallahassee, Florida 32306, USA  K. Auranen  [  Physics
Division, Argonne National Laboratory, 9700 S. Cass Ave., Argonne, IL 60439,
USA  P. C. Bender  Department of Physics and Applied Physics, University of
Massachusetts Lowell, Lowell, Massachusetts 01854, USA  C. M. Campbell
Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley,
California 94720, USA  M. P. Carpenter  Physics Division, Argonne National
Laboratory, 9700 S. Cass Ave., Argonne, IL 60439, USA  J. Chen  Physics
Division, Argonne National Laboratory, 9700 S. Cass Ave., Argonne, IL 60439,
USA  M. Gott  [  Physics Division, Argonne National Laboratory, 9700 S. Cass
Ave., Argonne, IL 60439, USA  J. P. Greene  Physics Division, Argonne National
Laboratory, 9700 S. Cass Ave., Argonne, IL 60439, USA  D. E. M. Hoff  [
Department of Physics and Applied Physics, University of Massachusetts Lowell,
Lowell, Massachusetts 01854, USA  T. Huang  Physics Division, Argonne National
Laboratory, 9700 S. Cass Ave., Argonne, IL 60439, USA  Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou 730000, China  H. Iwasaki
Facility for Rare Isotope Beams, Michigan State University, East Lansing,
Michigan 48824, USA  Department of Physics and Astronomy, Michigan State
University, East Lansing, MI 48824, USA  F. G. Kondev  Physics Division,
Argonne National Laboratory, 9700 S. Cass Ave., Argonne, IL 60439, USA  T.
Lauritsen  Physics Division, Argonne National Laboratory, 9700 S. Cass Ave.,
Argonne, IL 60439, USA  B. Longfellow  Lawrence Livermore National Laboratory,
Livermore, CA 94550, USA  C. Santamaria  [  Nuclear Science Division, Lawrence
Berkeley National Laboratory, Berkeley, California 94720, USA  D. Seweryniak
Physics Division, Argonne National Laboratory, 9700 S. Cass Ave., Argonne, IL
60439, USA  T. L. Tang  [  Physics Division, Argonne National Laboratory, 9700
S. Cass Ave., Argonne, IL 60439, USA  G. L. Wilson  Department of Physics and
Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA
Physics Division, Argonne National Laboratory, 9700 S. Cass Ave., Argonne, IL
60439, USA  J. Wu  [  Physics Division, Argonne National Laboratory, 9700 S.
Cass Ave., Argonne, IL 60439, USA  S. Zhu  [  Physics Division, Argonne
National Laboratory, 9700 S. Cass Ave., Argonne, IL 60439, USA

###### Abstract

Information on the 38S level scheme was expanded through experimental work
utilizing a fusion-evaporation reaction and in-beam $\gamma$-ray spectroscopy.
Prompt $\gamma$-ray transitions were detected by the Gamma-Ray Energy Tracking
Array (GRETINA) and recoiling 38S residues were selected by the Fragment Mass
Analayzer (FMA). Tools based on machine-learning techniques were developed and
deployed for the first time in order to enhance the unique selection of 38S
residues and identify any associated $\gamma$-ray transitions. The new level
information, including the extension of the even-spin yrast sequence through
$J^{\pi}=8^{(+)}$, was interpreted in terms of a basic single-particle picture
as well shell-model calculations which incorporated the empirically derived
FSU interaction. A comparison between the properties of the yrast states in
the even-$Z$ $N=22$ isotones from $Z=14$ to $20$, and for 36Si-38S in
particular, was also presented with an emphasis on the role and influence of
the neutron $1p_{3/2}$ orbital on the structure in the region.

##  I Introduction & Background

First discovered in 1958 [1], 38S is comprised of sixteen protons ($Z=16$),
twenty-two neutrons ($N=22$), and isospin $T=3$. Its positioning on the chart
of the nuclides is such that two valence neutrons reside outside of the
traditional $N=20$ shell closure defined by the $1s0d$-$0f1p$ shell gap while
valence protons fill two-thirds of the $1s0d$ shell including a large fraction
of the $\pi 0d_{5/2}$ orbital. The ground state and low-lying structure in 38S
lends itself to a competition between the single-particle and coherent
deformation-driven aspects of the $\pi 1s0d$ and $\nu 0f1p$ orbitals. In
particular, the proximity and occupancy of the proton
$0d_{5/2}-1s_{1/2}-0d_{3/2}$ orbitals and the absence of a robust $N=28$
shell-gap between the $\nu 0f_{7/2}-1p_{3/2}$ orbitals, provide scenarios for
strong proton-neutron quadrupole-based correlations to thrive. The higher-
lying excited levels in 38S are expected to be additionally influenced by
various particle-hole $N=20$ cross-shell excitations. Similar competitions and
effects are known to drive the deformation that has been observed in the low-
lying levels of the $N\approx 28$ Si and S isotopes (see Sec. 4.4 of Ref. [2]
and references therein).

The 38S ground state has a lifetime of T${}_{1/2}=170.3(7)$ min and
$\beta^{-}$ decays with a branch of 100% [3]. A number of excited levels have
been observed below 7 MeV based on data collected from $\beta$-delayed
$\gamma$-ray spectroscopy [4], two-particle transfer reactions - some that
incorporated $\gamma$-ray detection [5, 6, 7, 8, 9], and in-beam $\gamma$-ray
spectroscopy measurements utilizing deep inelastic reactions [10, 11, 12, 13]
or intermediate energy fragmentation [14]. A summary of the level energies and
spin-parity ($J^{\pi}$) values was presented in Fig. 6.46 of Ref. [13]. Firm
spin-parity assignments were established only for the yrast even-$J$ levels
through $J^{\pi}=4^{+}$ and the $2^{+}_{2}$ level at 2.806 MeV. A probable
candidate for the yrast $6^{+}_{1}$ level was first observed in the work of
Ref. [10]. Candidates for possible negative parity (intruder) states starting
at around 3.5 MeV up through $\sim$6 MeV were also identified, primarily in
the ($t$,$p$) and $\beta$ decay works [4, 7]. Additional spectroscopic
information with respect to the excited levels in 38S, such as lifetimes and
transitions stengths of various excited levels and the ratio of the multipole
matrix elements ($M_{n}/M_{p}$) plus the $g$ factor for the $2^{+}_{1}$
$\rightarrow$ $0^{+}_{1}$ transition, has been extracted from in-beam Coulomb
excitation, inelastic scattering, and deep-inelastic reaction data [15, 16,
17, 18, 19, 20, 21, 22].

The energies of the established yrast levels in 38S have been well described
by having valence protons contained within the $1s0d$ shell and valence
neutrons contained with the $0f1p$ shell. This was demonstrated by the weak-
coupling model prescription of Bansal and French [23] presented in Fig. 4 of
Ref. [7], for example. Unsurprisingly, a number of effective interactions
derived within this model space have also been successful in reproducing the
observed level energies and the even-$J$ yrast states in particular, for
instance, the results of the $SDPF$ interactions of Refs. [24, 25, 26] shown
in Fig. 4 of both Refs. [7] and [8]. More recently developed interactions such
as the SDPF-MU interaction [27, 28], the sdpf-nr interaction [29], and the
SDPF-U interaction [30], have also been successful in describing the low-lying
level energies (see Fig. 4 of Ref. [14] and Fig. 6.46 of Ref. [12]). Shell-
model interactions which have been sure to incorporate the $1p_{3/2}$ neutron
orbital within their model space, have shown promise in describing the degree
and nature of the deformation in 38S. This includes agreement with
experimental B($E2$) transition strengths [15, 21, 22] as well as the measured
$0^{+}\rightarrow 2^{+}_{1}$ $g$ factor [19, 18]. A common theme building from
these past works has been an emphasis on the key role played by the $\nu
1p_{3/2}$ orbital in generating the proper amount of coherent proton-neutron
(quadrupole) correlations in the low-lying levels.

The present work describes the first investigation of the 38S level scheme
through population in a fusion-evaporation reaction (Section II). In doing so,
a significant number of new excited levels and transitions have been
determined from the in-beam $\gamma$-ray data, uniquely selected following 38S
recoil identification (Section III). In particular, the yrast even-$J$ levels
have been extended to $8^{(+)}$ and a number of higher-$J$ candidates have
been found. Multipolarities have been deduced for some transitions based on
$\gamma$-ray yields and were used to assign or suggest $J$-values where
possible (Section IV). The resulting level scheme is discussed in terms of
simple symmetry arguments, the nearby nuclei in the region, the role of the
$\nu 1p_{3/2}$ orbital, and comparisons are made directly with shell-model
calculations using the FSU cross-shell interaction [31] (Section V).

##  II Experimental Details

Excited states in 38S were populated in the 18O(22Ne,2$p$) fusion-evaporation
reaction. A 48.5 MeV 22Ne6+ primary beam at an intensity of 30 particle-nano-
Amperes was provided by the Argonne Tandem Linear Accelerator System (ATLAS)
facility located at Argonne National Laboratory. The 18O targets were prepared
by electrodeposition. A 1 mL aliquot of 99% 18O H2O was added to the
deposition chamber seated on top of a $\sim$1.1 mg/cm2 Ta foil.
Electrodeposition was performed using a platinum cathode and the Ta foil as
the anode, which was held at a constant $+100$ V DC for approximately 5 hours.
As the 18O diffused into the Ta foil, the surface passivated resulting in the
current dropping from several hundred $\mu$A to tens of $\mu$A. Mass gain and
alpha loss measurements indicated a few hundred $\mu$g/cm2 of 18O were
diffused into one side of the Ta foil. The Ta foil was arranged so that the
18O material resided on the downstream target side allowing the best possible
release of heavy-ion recoils. The beam energy was chosen to take into account
the energy loss through $\sim 1/2$ of the Ta foil and assumed that the 18O
material was evenly distributed in the latter half of the foil. No degradation
in the amount or distribution of the 18O material was observed over the
duration of the run ($\sim$6 days).

Prompt $\gamma$-ray transitions emanating from nuclei following their
population in the fusion-evaporation reaction were detected in the Gamma-Ray
Energy Tracking Array (GRETINA) [32, 33]. GRETINA consisted of 12 HPGe modules
and was positioned to cover polar-angles between
$70^{\circ}<\theta<170^{\circ}$ relative to the beam direction. Relative
energy-dependent detection efficiencies and energy-response calibrations were
carried out with standard $\gamma$-ray sources of 152Eu and 56Co. The data
collected was processed in an add-back mode whereby $\gamma$-ray interactions
that occurred within 10 cm of each other were energy summed. Both the source
and in-beam $\gamma$-ray data was processed the same way. The position
information from the largest-energy interaction were used to define the
outgoing $\gamma$-ray angle for Doppler reconstruction. Similarly, the
timestamp from largest-energy interaction was used as the timestamp for that
event and used for relative-timing coincidences. An average recoil velocity of
$\beta=0.003375(15)$/c was uniquely determined for the 38S recoils from a fit
of the known 1293-keV and 1534-keV $\gamma$-ray energies as a function of
polar angle. An energy width of $\sim$0.5% FWHM was measured for the 1293-keV
$\gamma$-ray in 38S.

The Fragment Mass Analayzer (FMA) [34] provided selection capabilities for
recoiling 38S ions via energy-to-charge, $E/q$, and mass-to-charge, $A/q$,
ratios. The magnetic and electric elements of the FMA were optimized for the
transmission of 38S8+ at 18.7 MeV, i.e., $A/q=38/8=4.75$ and
$E/q=18.7/8\approx 2.338$. Movable slits in the dispersive (horizontal)
direction were used at the FMA focal plane to suppress recoils with similar
$A/q$ values and to reduce the intensity of scattered un-reacted primary beam.
Both dispersive and non-dispersive (vertical) position information was
collected on an event-by-event basis at the FMA focal plane by a parallel-grid
avalanche counter (PGAC) filled with isobutane gas to a pressure of $\sim$400
Pa (3 Torr). Timing signals to both trigger data collection and for use in the
event-by-event time-of-flight reconstruction originated from the PGAC anode
signal. An ionization chamber (IC) having a segmented-anode readout consisting
of two 5 cm long sections followed by a 20 cm long section, was positioned
directly behind the PGAC. The IC was operated with isobutane gas at a pressure
of $\sim$1067 Pa (8 Torr) which ensured the stopping of all recoils of
interest within its volume. The recoil-ion energy information was recorded
independently for each of the three sections, $E_{i}$ ($i=1-3$).

All data were collected by a digital data acquisition system which sampled the
input analog signals at a frequency of 100 MHz. The valid condition to
activate data collection was a relative time-coincidence of $<1\mu$s between
timestamps of the PGAC anode and GRETINA - when the $\gamma$-ray multiplicity
condition of $M_{\gamma}>0$ was met. Typical data-collection rates were on the
order of $\sim$5 kHz.

##  III Experimental & Analysis Methods

###  III.1 38S recoil selection

There are a number of example two-proton evaporation analyses prior to this
work which involved the selection and identification of isotopes of interest
by the FMA [35, 36, 37]. Each of those leveraged some combination of the
following quantities to extract the events of interest: i) the relative times
between prompt $\gamma$-rays and the PGAC ($T_{\gamma-PGAC}$) to provide
suppression of random coincidences, ii) the individual or summed ionization
chamber energy information ($E_{i},E_{ij},E_{ijk}$) to identify the element
($Z$) of interest, iii), the dispersive position of the recoil at the FMA
focal plane ($x$) to define $A/q$, and iv) the combined (total) ion chamber
energy, $E_{123}$, with the recoil time-of-flight ($T_{\gamma-PGAC}$) to
distinguish $A$ through the classical mass relation, $m\sim E_{123}T_{\gamma-
PGAC}^{2}$. In the present work, the beam energy and extra target thickness
due to the Ta foil matrix resulted in relatively low energies for the
recoiling 38S ions ($\lesssim 20$ MeV). The consequence was a less than
optimal identification of 38S recoils based on a series of manual selections
or gates applied to the quantities listed above. For instance, as viewed in
Figs. 1(a) - (d), the central region of the S recoils (indicated by the white
line) from the ionization chamber energies overlapped with the neighboring
isobar 38Cl. In addition, the dispersive position $x\propto A/q$ plot in Fig.
1(e) identifies an overlap between the $A/q=38/8$ recoils of interest with a
nearby $A/q=33/7$ ambiguity belonging to 33P7+.

Figure 1: The energy of the first ion chamber section ($E_{1}$) plotted
against the energy of the second section ($E_{2}$) (a)-(b), the third section
($E_{3}$) (c)-(d), and the dispersive position at the FMA focal plane ($x$) in
(e)-(f). The time-coincidence gate $|\Delta T_{\gamma-PGAC}|<200$ ns was
applied to the grey-scale histograms which are the same across each row.
Element ($Z$) labels in (a) and (c) identify the general regions of interest,
while the white line identifies the central region covered by 38S recoils. The
labels in (e) correspond to $A/q$ values. The overlaid spectra in (b), (d),
and (f) include data which define the manual cuts (grey points) and from
differing model output values, $k_{ml}>0.25$ (yellow), 0.60 (blue), and 0.81
(pink), as defined in Fig. 4 (see text for additional details). The white line
in (a) - (d) represents the central region of the S isotopes and is the same
for each row.

####  III.1.1 Application of a fully-connected feed-forward neural network
(NN) model

To improve upon the selection of 38S recoils over the application of manual
gating alone, a fully-connected feed-forward neural network (NN) model was
developed and implemented. This is the first application of such a model to be
used for assisting with recoil selection at the FMA. The power of trainable NN
models for various classifications scenarios is now wide-spread and impactful
across nuclear science research, see for example Ref. [38] and references
therein. In the present work, we developed and trained through supervision an
NN model with the ability to process our event data in a goal oriented One-Vs-
All mode. In this mode, the 38S recoil data is meant to be distinguished, or
classified, against all other types of recoil data. Types of data not
belonging to 38S recoils included other fusion-evaporation reaction channels
resulting in 38Cl and 33P recoils, random data originating from the 181Ta
lattice and scattered primary beam, and Compton-scattering background data.
The framework of the NN model was designed to output a single value, $k_{ml}$,
ranging from 0 – 1, for each data input into the model. In one sense, $k_{ml}$
is representative of the degree or likely-hood to which the model has found
the corresponding input data to be to classified as 38S (approaching a value
of 1) or not (approaching towards 0). An individual $\gamma$-ray energy, after
the add-back procedures described in Section II, was used to define a unique
set of input data to the NN model. For each $\gamma$ ray, eleven experimental
values were associated with it, defining the input size of the first NN model
layer. For example, an event with a $\gamma$-ray multiplicity of 5,
$M_{\gamma}=5$, was considered as five independent sets of data inputs into
the model. The input values included the individual and summing permutations
of the ion chamber energies
($E_{1}$,$E_{2}$,$E_{3}$,$E_{12}$,$E_{13}$,$E_{23}$,$E_{123}$), the FMA focal
plane dispersive plane position information ($x$), the relative recoil time-
of-flight from the target to the FMA focal plane ($T_{\gamma-PGAC}$), the
calculated mass ($m\sim E_{123}T^{2}_{\gamma-PGAC}$), and the $\gamma$-ray
multiplicity ($M_{\gamma}$) of the complete event to which the individual
$\gamma$-ray belonged. The number of input parameters could have been reduced
due to redundancies and correlations, for example, the parameters used to
generate $m$ were all included independently. However, the calculated values
were already readily available and the number of inputs were relatively small.
As is standard practice, each input value range was independently normalized
to span from 0 – 1 to reduce the possibility of any biases appearing
throughout the NN model training.

The framework of the NN model itself was standard and considered to be
shallow. It consisted of three total layers, an input layer with dimensions
(x11,x40), a hidden layer (x40,x20), and an output layer (x20,x1). Each
corresponding layer was linearly- and fully-connected with the previous one.
The rectified linear unit (Relu) and Parametric Relu (PRelu) functions [39]
were applied to the input and hidden layers, respectively. A Sigmoid
activation function was applied to the output layer. A dropout function [40]
was active between the input and hidden layers with a 20% probability for
zeroing any individual neuron within that layer. In total, there were 1322
trainable parameters: 480 for the input layer, 820+1 for the hidden layer plus
PRelu function, and 21 for the output layer. The NN model was trained under
supervision through gradient descent and back-propagation using the Binary
Cross Entropy Loss (BCELoss) function and Adam optimizer [41] with a learning
rate of $10^{-3}$ and exponential decay rate values of $\beta_{1}=0.9$ and
$\beta_{2}=0.999$ on the first and second moment estimates, respectively. In
addition to exploring the number and sizes of neutron layers, the drop-out
fraction, and learning rate parameters, other so-called hyper-parameters
explored such as the batch size and the epoch number. No hyper-parameter
showed significant impact on the quality of the training and outputs. The
largest sensitivity to the overall quality and performance of the NN model
came from the type, size, and scope (in terms of the manual gating) of the
training-data used for the supervised training.

####  III.1.2 Manual data reduction and defining the training data

The NN model was able to be trained, under supervision, by labeling a sub-set
of data based on $\gamma$-ray energy. This was possible because there are
known transitions for both the recoil of interest, 38S [10, 11, 12, 14], as
well as for 38Cl [42], 33P [43], and 181Ta [3]. Fig. 2 and Table 1 show and
list, respectively, the $\gamma$-ray energies used for labeling the NN
training data. Input values that were associated with a 38S $\gamma$-ray
energy $\pm 1-2$ keV were labeled with a value $\equiv$1\. Other types of data
with the requisite $\gamma$-ray energies $\pm 1-2$ keV were similarly labeled
as $\equiv$0\. Note that: i) not all $\gamma$-rays with the appropriate
energies were labeled and used for training (additional details below), and
ii) $\gamma$-ray spectra inherently contain backgrounds from various sources
as shown in Fig. 2. Due to the latter, generation of a “clean” data set for
training was not possible. As a result, there were large contributions to
mislabeled events, ranging anywhere from $\sim$10% to $>50$% for each of the
$\gamma$-ray energies included in the training data set.

Figure 2: The total spectrum of $\gamma$ rays encompassed within the manual gating regions (see text and Fig. 1). The labeled events used for the NN model training data set are distinguished by the colored regions. Note that for the labeling of the 38Cl and 181Ta $\gamma$-ray transitions, a down-sampling of x4 was used, hence, their reduced size relative to the total counts in spectrum. Table 1: Energies and numbers of known $\gamma$-ray transitions that were included in the manual gating, selected for labeling, and used as NN model training data.  |  $E_{\gamma}$ (keV) | # labeled | label | Refs.
---|---|---|---|---
38S | 384 | 8600 | $\equiv 1$ |  [10, 11, 12, 14]
1293 | 18460
1535 | 11940
38Cl | 171 | 20610 | $\equiv 0$ |  [42]
292 | 21110
638 | 10210
1190 | 7260
2044 | 4790
2275 | 5880
2972 | 6530
3142 | 5630
33P | 1432 | 14840 | $\equiv 0$ |  [43]
1848 | 21250
2379 | 12900
181Ta | 136 | 25910 | $\equiv 0$ |  [3]

A manual reduction of the data was used to pre-define a region of events for
inclusion into the NN model for training data selection as well as for the
analysis which followed. The manual selections aided in further reducing
random coincidences, background events, and hence, mislabeled training data,
over a timing-coincidence requirement alone. Also, the manual gates were aimed
at removing an initial overwhelming number of events from 38Ar and 38Cl that
were not interfering with the cleanliness of the 38S spectra. Therefore, in
addition to a requirement that events have a time relation of $|\Delta
T_{\gamma-PGAC}|<200$ ns, selections were also derived from the
$E_{1}$-$E_{2}$, $E_{1}$-$E_{3}$, and $E_{1}$-$x$ 2-dimensional spectra as
shown in Fig. 1. The energy selections represented by the grey data points in
Figs. 1(b) and (d) were able to remove all events linked to the 38Ar isobar,
as well as a large portion of the 38Cl recoil events. The gates also reduced
the number of random $\gamma$-ray events caused by scattered 22Ne primary
beam. The selection region shown in Fig. 1(f) on $E_{1}$-$x$ removed most of
the mass ambiguities. However, the $A/q\approx 33/7$ region was purposefully
not removed so as to ensure enough 33P events remained for labeling and
inclusion in the NN model training. In total, $\sim 5.5\times 10^{6}$ $\gamma$
rays remained after the manual selections (Fig. 2). Included within those data
were the $\sim$195k events labeled for training, $\sim$39k 38S, $\sim$82k
38Cl, $\sim$49k 33P, and $\sim$26k Ta (Table 1 and 2). In order to provide
better balance to the training data, the labeling of the 38Cl and 181Ta data
was down-sampled by a factor of 4 each.

Figure 3: (a - d) The evolution of the distribution of model output values,
$k_{ml}$, for the training data at snapshots throughout the NN model training
at completed epoch cycles (1, 5, 10, and 100). Purple lines correspond to the
$k_{ml}$ values of the labeled 38S training data and the black lines are the
corresponding values of the sum of the 38Cl, 33P, and 181Ta labeled training
data. (e - h) The fraction of the total number of integrated counts (given in
%) over the integral range starting from the $k_{ml}$ value given on the
x-axis through $k_{ml}=1$, i.e. $\int_{k_{ml}}^{1}dN$/$\int_{0}^{1}dN$. Figure
4: The distribution of the model output values, $k_{ml}$, for the fully-
trained NN model (epochs = 200). The black line represents the full complement
of data encompassed within the manual gating regions, both labeled and un-
labeled. The colored lines correspond to the final distributions of each of
the separately labeled training data only (purple - 38S, blue - 38Cl, grey -
33P, pink - 181Ta). The yellow ($>0.25$), blue ($>0.60$), and pink ($>0.81$)
vertical lines identify the lower-limit $k_{ml}$ integral values, where the
upper limit was defined $=1$, applied throughout the $\gamma$-ray analysis
(see text for additional details).

####  III.1.3 NN model training & output

The training of the NN model specifically refers to the adjustment and
eventual optimization of the 1322 model parameters described above. The
supervised training of the NN model, whereby we utilized our labeled data, was
completed by cycling through the labeled training data 200 times (200 epochs).
The loss calculations and backward propagation were calculated and carried out
in batches of 500 pieces of data. Changes in the epoch and batch size
parameters were explored but did not significantly alter the model
performance. Throughout the training procedure, the model output value
$k_{ml}$ was rounded to its nearest integer (0 or 1) prior to the loss
calculation. Various iterations into the type and scope of the manual data
selection, as described above, were explored in order to reach the adopted
parameters for the NN model. Metrics typically used in defining the goodness
of trained machine-learning models were not crucial in the determination of
the applicability of the model in this case. This is in part due to the known
mislabeling of a large fraction of training events which would lead to
ambiguities in most standard metrics.

To first order, the distribution and trends of the $k_{ml}$ values provided
straight-forward visual feedback of the NN model’s ability to separate 38S
data from the rest. This can be visualized through the evolution of the
$k_{ml}$ values for the training data throughout the actual training process
(Fig. 3). In Figs. 3(a - d) the distributions for $k_{ml}$ are shown at the
conclusion of different epoch numbers (1, 5, 10, and 100) for the labeled 38S
training data (purple lines) as well as the sum of all other labeled training
data (black lines). As expected, after a single pass through the data (epochs
= 1) both types of labeled data have similar distributions. As the epoch
number grows, the 38S data starts trending towards $\equiv 1$ faster than the
other data types, and even after only 10 epochs, a peak forms for the 38S data
above $k_{ml}>0.5$. By the completion of 100 epochs, the distributions are
different and a sub-set of data, peaking towards $k_{ml}=1$ identifies the 38S
data within the model. Fig. 4 also shows the breakdown of the individually
labeled components for the training data $k_{ml}$ distributions with the
fully-trained model (epochs = 200). One sees that little evolves between
epochs 100 - 200. Also, the behavior of each non-38S type is similar.

Beyond the distribution of the $k_{ml}$ values, the integrated number of
counts for a specific $\gamma$-ray energy between a starting value of
$k_{ml}=X$ up to $k_{ml}=1$, or $k_{ml}>X$, was also calculated throughout
each training. The behavior of the fractional counts remaining for the 38S
$\gamma$ rays, as well as the combined $\gamma$ rays of the contaminants, are
given relative to the total (in percent) in Figs. 3(e - h). The trends mimic
those of the $k_{ml}$ distributions as expected and highlight the unique
persistence of the 38S counts for increasing the lower limit on the
integration.

The entirety of the data that was encompassed within the manual-cut regions
(Figs. 1 and 2), not just the data that was labeled, was similarly evaluated
by the fully-trained NN model. The distribution of all of the output $k_{ml}$
values is shown by the black line in Fig. 4. The complete data set and the
labeled training data were used to define the $k_{ml}$ values applied in the
final analysis. Contaminant $\gamma$-ray yields showed the most dramatic
reduction in counts when $k_{ml}\gtrsim 0.25$, i.e. the inclusion of counts
between $0.25\leq k_{ml}\leq 1$. The 38S yields varied only slightly through
this transition point. In addition, the 38S recoils remained nearly constant
over $0.25\lesssim k_{ml}\lesssim 0.8$, while contaminant yields continued to
reduce. Above $k_{ml}\gtrsim 0.8$ all events showed diminishing yields.

Finally, the validity of each trained model was also evaluated by the quality
of the $\gamma$-ray singles spectra that they could generate. Specifically,
the labeled training data was used to check ratios between the summed areas of
known transitions in 38S [10, 11, 12, 14] compared to the relative summed
areas of the known transitions in 38Cl [42] and 33P [43]. For consistency, the
comparison between different trained models was made at a $k_{ml}$ value in
which the area of the 38S 1293-keV transition was $\equiv 5000$ counts. The
$\gamma$-ray spectra from each NN model training were also visually inspected
and overlaid with previously generated spectra in order to qualify background
suppression. The resulting spectra are presented in Section IV.

###  III.2 Additional prompt $\gamma$-ray information

In addition to a $\gamma$-ray singles spectrum for a specific $k_{ml}$ value,
a recoil-$\gamma$-$\gamma$ coincidence matrix was also generated from events
which had $\gamma$-ray multiplicities $M_{\gamma}>1$. To be included in the
matrix, a pair of $\gamma$ rays must have had a relative timing relation of
$<200$ ns and both have been within the accepted $k_{ml}$ cut range. The cut
range $k_{ml}>0.25$ was primarily used in the analysis due to its compromise
between cleanliness and statistics. In cases where contaminants may have been
present, including those from 38Cl in particular, coincidences were checked
with data from the more stringent $k_{ml}>0.60$ cut. Statistics limited the
observation of any higher-order $\gamma$-ray multiplicity studies.

The semi-alignment of magnetic sub-states provided by the near-symmetric
fusion evaporation reaction was leveraged to inform on some transition
multipolarities. The ratio of the $\gamma$-ray singles yields,
$R_{\theta_{2}/\theta_{1}}$, was extracted from data included within the
$k_{ml}>0.60$ cut. Although GRETINA has essentially a continuous angular
coverage, only two angular bins were used due to the limited statistics with
one centered around $\theta_{2}=149^{\circ}$ and the other centered around
$\theta_{1}=90^{\circ}$. An energy-dependent efficiency curve was determined
independently for each of the corresponding angular binning regions from the
$\gamma$-ray source data. Above $E_{\gamma}>500$ keV the ratio of the two
efficiency curves to one-another was uniform. A systematic uncertainty of 5%
was adopted for that energy region. Below this energy, a systematic
uncertainty of 10% was adopted due to an increase in the sensitivity of the
efficiency-curve fit parameters to the $\theta_{1}=90^{\circ}$ source data.
The final uncertainty on $R_{149^{\circ}/90^{\circ}}$ also included those from
statistics. Stretched-quadrupole ($\Delta J=2$, $L=2$) transitions are
expected to reside above $R_{149^{\circ}/90^{\circ}}\gtrsim 1.2$, while the
dipole $\Delta J\leq 1$, $L=1$ counterpart transitions are more probable for
$R_{149^{\circ}/90^{\circ}}\lesssim 1$ values. The known 1293-keV and 1535-keV
38S $\gamma$-rays, both having $\Delta J=2$, gave
$R_{149^{\circ}/90^{\circ}}~{}\gtrsim$ 1.2, in agreement with expectations.

##  IV Experimental Results

###  IV.1 Selection of 38S $\gamma$-ray transitions

The $\gamma$-ray transitions attributed to 38S based on the present analysis
are labeled by their observed energies in the $\gamma$-ray singles spectra of
Fig. 5 and they are also listed in Table 2. The $\gamma$-ray energies carry an
uncertainty of $\sim$0.5 - 1 keV. Transitions in 38S were identifiable by
showing little change in their yields between the spectra with the
$k_{ml}>0.25$ cut versus the $k_{ml}>0.60$ cut as shown in Figs. 5(a), (c),
and (e). Spectra generated from the $k_{ml}>0.60$ cut (blue) were far-removed
of the contaminant lines, in particular those from 38Cl, which appear
prominently in the $k_{ml}>0.25$ selection (yellow) but diminish largely for
the $k_{ml}>0.6$ selection. Note that a smooth background has been subtracted
from the singles spectra in Figs. 5(a), (c), and (e). The summed spectra,
including each individual recoil-$\gamma$-$\gamma$ spectra ($k_{ml}>0.6$),
shown in the lower half of each panel of Fig. 5, provide additional support of
the associated 38S transitions. The relative intensities of the 38S
transitions were determined from the singles spectra of the $k_{ml}>0.60$
data. They are given in Table 2, normalized to the intensity of the 1293-keV
line ($\equiv 1000$). Uncertainties on the relative intensities were dominated
by statistics over the systematic contributions from the peak fitting and
background subtraction ($<5$%) or the energy-dependent efficiency correction
($<5$%).

Figure 5: (a),(c),(e) Background-subtracted $\gamma$-ray singles spectra from
a model output cut of $k_{ml}>0.6$ (blue) overlaid on a cut of $k_{ml}>0.25$
(yellow). All transitions belonging to 38S are labeled by their energies while
those in bold have been newly determined in the present work. Parenthesis
indicate transitions that were unable to be placed in the level scheme (Fig.
6) and may be considered tentative. The recoil-$\gamma$-$\gamma$ spectrum
($k_{ml}>0.6$ cut) for each 38S transition labeled in (a), (c), and (e), were
combined to generate the inclusive summed spectra shown in (b), (d), and (f).
The blue vertical lines correspond to the labeled energies of the above
histogram. Figure 6: The level scheme of 38S based upon the present work and
other in-beam measurements. Dashed levels and transitions indicate tentative
placement. Transitions in blue ($L=2$) and purple ($L=1$) reflect
multipolarities based on their $R_{149^{\circ}/90^{\circ}}$ values (Fig. 7 and
Table 2). Figure 7: The ratio of the extracted $\gamma$-ray yields centered
around $\theta_{2}=149^{\circ}$ by that around $\theta_{1}=90^{\circ}$,
$R_{149^{\circ}/90^{\circ}}$, for identified transitions in 38S as a function
of their energy. The colored data points correspond to either fully-stretched
quadrupole transitions residing above $\approx 1.2$ (blue for multipolarity
$L=2$) or dipole transitions residing below $\approx 1$ (purple for
multipolarity $L=1$). Grey data points were not assigned multipolarities
primarily due to large uncertainties.

###  IV.2 38S level scheme

An updated 38S level scheme incorporating the present data is shown in Fig. 6.
Some details pertaining to the construction of specific levels or locations of
transitions are given below. In general, the placement of transitions into the
38S level scheme was done through the use of $\gamma$-ray and excited level
energy summations, the relative $\gamma$-ray intensities (Table 2), and where
possible, utilizing the recoil-$\gamma$-$\gamma$ coincidence data (Fig. 8).
Dashed transitions or levels in Fig. 6 identify cases where the coincidence
data was inconclusive or where only the energy-sums were used. Newly assigned
or speculative $J^{\pi}$ information was extracted from the
$R_{149^{\circ}/90^{\circ}}$ values (Fig. 7 and Table 2), as well as
transition selection rules and the propensity of the fusion evaporation
reaction mechanism to populate higher-$J$ or yrast states with increasing
excitation energy.

####  IV.2.1 The yrast even-$J$ levels

The energies and $J^{\pi}$ values for the even-$J$ yrast levels up to
$J^{\pi}=4^{+}$ have been firmly established (0.000 MeV – 0+, 1.293 MeV – 2+,
and 2.827 MeV – 4+). A tentative $J^{\pi}=(6^{+}_{1})$ assignment to the
3.677-MeV level was made previously based on speculative ($t$,$p$) angular
distributions [7], as well as observation of sizeable population and a
predominant decay branch to the 4${}^{+}_{1}$ level by various in-beam
reaction work [10, 11, 12, 13, 14, 22]. The $\gamma$-ray coincidence data and
the relative intensities extracted in the present work agree with the
established 1293-, 1535-, and 850-keV $\gamma$-ray cascade and arrangement
[Figs. 8(a) and (b)]. $R_{149^{\circ}/90^{\circ}}\gtrsim 1.2$ values were
extracted for each transition, including the 850-keV $\gamma$ ray, showing
consistent stretched quadrupole multipolarity throughout the cascade (Fig. 7
and Table 2). The assignment of the 3.677-MeV level has therefore been
modified to $J=6^{(+)}_{1}$, where positive parity is most likely.

A coincidence relationship was observed between the known yrast
$J^{\pi}=6^{(+)}\rightarrow 4^{+}\rightarrow 2^{+}\rightarrow 0^{+}$ sequence
and the 2668-keV $\gamma$ ray [Figs. 8(a) and (b)]. First identified in Refs.
[12, 13], the 2668-keV transition has now been placed to directly feed the
3.677-MeV level from a new level at 6.346 MeV. No additional transitions were
observed to decay from this new level suggesting a decay branch of near
$\sim$100% to the 6${}^{(+)}_{1}$ level. The efficiency corrected intensities
for the 1535- and 850-keV transitions relative to the 1293-keV [$\equiv
1.00(12)$] were 0.98(12), 0.85(11) in the 2668-keV gated $\gamma$-$\gamma$
coincidence spectrum. The extracted $R_{149^{\circ}/90^{\circ}}=1.74(19)$
value of the 2668-keV $\gamma$ ray shows a clear quadrupole multipolarity.
Hence, the new level at 6.346 MeV has been assigned as the yrast
$J=8^{(+)}_{1}$ level. Similar to the 3.677-MeV level, this level is likely
$\pi=+$.

Extending the even-$J$ yrast sequence beyond $J=8$ with any certainty proved
difficult. Exploration of the summed and individual coincidence spectra for
the yrast even-$J$ transitions revealed two possible candidate $\gamma$ rays
at 1617-keV and 2385-keV [Fig. 8(b)]. Any coincidence relationship between
these two lines was uncertain due to the limited statistics. Coincidence with
the 2668-keV transition in both cases led to their tentative placements
directly feeding into the $J^{\pi}=8^{(+)}$ 6.346-MeV level. The tentative
level at 8.730 MeV has no spin-parity suggestion as the multipolarity
information on the 2385-keV $\gamma$-ray is inconclusive
[$R_{149^{\circ}/90^{\circ}}=1.61(73)$] though $J\geq 8$ is most likely. The
placement of the 1617-keV transition generated a level with an energy sum of
$<$2 keV within that of the 7.963-MeV level established by the 1950-keV
transition (see sub-section IV.2.5 below). The 1617-keV $\gamma$-ray’s
$R_{149^{\circ}/90^{\circ}}<1.0$ value favored a dipole multipolarity and the
7.963-MeV level having $J=(8,9)$ is consistent with the 1950-keV cascade.

Figure 8: Projected recoil-$\gamma$-$\gamma$ coincidence spectra for
$k_{ml}>0.25$. The labels provide the $\gamma$-ray energy or energies selected
for the projection. The ”$+$” sign represents the summation of two separate
recoil-$\gamma$-$\gamma$ spectra. Only some of the transitions of interest
have been labeled in each spectra. No background subtractions have been
applied.

####  IV.2.2 The 2.806-MeV, 3.520-MeV, 3.615-MeV and 3.999-MeV levels

The energies of the 2.806-, 3.520-, 3.615-, and 3.99-MeV levels were
previously established with $J^{\pi}$ values of $2^{+}_{2}$, ($1-3$),
($2^{+},3^{+}$) and ($3^{+},4^{+}$), respectively (see Fig. 6.46 of Ref. [13]
for example). The doublet of states around 2.8 MeV was cleared up definitively
in Ref. [9] and a recent work confirmed the assignment of the 2.806-MeV level
as $J^{\pi}=2^{+}_{2}$ [21]. The 3.520-MeV level was first identified in the
$\beta$-decay of 38P [4] leading to a limit on $J$ from the ($2^{-}$) ground
state. The same level was also weakly populated in the deep-inelastic work of
Refs. [12, 13]. While the 2224-keV transition matches the energy of the
previous work within errors, it is unclear if the observed weakly-populated
3.522-keV transition is the ground state transition as it is outside the
expected energy uncertainty. The 3.615-MeV and 3.999-MeV levels were suggested
to be part of a multiplet of levels with $J^{\pi}=2^{+}-5^{+}$ (including the
4.437-MeV level discussed below) based on the $(\nu 0f_{7/2}\nu 1p_{3/2})$
configuration [11, 12].

All of the previously observed transitions from these levels have been
confirmed in the present work (Fig. 5). Two new linking transitions of 810-keV
and 788-keV were observed to decay from the 3.615-MeV level. The location of
810-keV line was established through the recoil-$\gamma$-$\gamma$ coincidence
data with the 1513-keV line. The weak 788-keV transition meets the energy
requirements to link the 2.827-MeV and 3.615-MeV levels. It also showed
marginal coincidence with the 384-keV line and the lower-lying 1293- and
1535-keV lines. The appearance of the 1535-keV transition within the
coincidence spectrum of the 384-keV line also indirectly supported its
placement.

No new information on the properties of the $J^{\pi}=2^{+}_{2}$ 2.806-MeV
level was determined in the present work due to the limited population of this
non-yrast state. The $R_{149^{\circ}/90^{\circ}}$ values for the 2323-keV and
384-keV transitions support the sequence of dipole transitions proposed in
Refs. [11, 12, 13] for the 3.615-MeV and 3.999-MeV levels. There is a
propensity for the higher of the two possible $J$-value assignments in each
case due to the fusion-evaporation reaction mechanism, however, no direct
empirical evidence is available to solidify this point. Therefore, the two
levels remain with $J^{\pi}=(2,3)$ and $(3,4)$.

####  IV.2.3 The 3.657-MeV level

A new level was established at an excitation energy of 3.657 MeV and assigned
with tentative spin values of $J=(3,4)$. Energy summations and
recoil-$\gamma$-$\gamma$ coincidence relations between the 830-keV, 780-keV,
and 2365-keV transitions with the known 4${}^{+}_{1}\rightarrow
2^{+}_{1}\rightarrow 0^{+}_{1}$ cascade were used [Fig. 8(d)]. There were no
previously recorded levels having energies consistent with 3.657 MeV, although
the 2365-keV was also observed in Refs. [12, 13]. The $\sim$830-keV data
observed in Ref. [14] originated from the 850-keV transition but was shifted
in energy due to its $>100$ ps lifetime, hence, it is not the same transition
as has been observed here. The tentative spin-parity range came from the
limits of the 2365-keV transition to the known $2^{+}_{1}$ level and the
830-keV transition to the known 4${}^{+}_{1}$ level ($2\leq J\leq 4$). The
$R_{149^{\circ}/90^{\circ}}=0.81(18)$ value of the 780-keV transition favors
$3\leq J\leq 5$ when coupled with the possible $J$ values of the 4.437-MeV
level (see subsection IV.2.4 below). The $R_{149^{\circ}/90^{\circ}}=1.88(44)$
of the 830-keV line is not obviously consistent with either $J=(3,4)$
$\rightarrow$ $4^{+}_{1}$ transition.

####  IV.2.4 The 4.437-MeV level

A key marketplace of the 38S level scheme has appeared at the 4.437-MeV level.
Similar levels were previously identified in the deep inelastic works of Refs.
[11, 12, 13] at both 4.436-MeV and 4.437-MeV. The former was tentatively
assigned $J=(4^{+},5^{+})$ based on theoretical arguments that the level was a
member of the ($\nu 0f_{7/2}\nu 1p_{3/2}$)J=2-5 multiplet. The latter was only
postulated based on the placement of an observed 1611-keV transition, though
the two levels agreed in energy within uncertainties. A broad peak around 4.43
MeV was also observed in the heavy-ion transfer work of Ref. [6] as well as a
level at 4.478(22) MeV which was given $J=(3^{-},4^{+})$ in the ($t,p$) work
of Ref. [7].

The $4.437\rightarrow 3.999\rightarrow 3.615$-MeV decay sequence suggested in
Refs. [11, 12] was confirmed by the recoil-$\gamma$-$\gamma$ coincidence
relations between the 384-keV and 438-keV transitions. The coincidence spectra
of the 780-keV and 1609-keV lines showed a number of transitions that were
also found to be in coincidence with the 438-keV transition, including the
1577-keV, 1950-keV, and 1020-keV $\gamma$ rays [Figs. 8(c) and (d)]. The
energy summations for each possible decay path stemming from the 4.437-MeV
level were in agreement to within $\sim 1$ keV. Therefore, within the $\sim$1
keV energy uncertainty a single level at 4.437 MeV has been placed in the
level scheme having four outgoing transitions (760-keV, 780-keV, 1609-keV, and
438-keV). A possible fifth transition through the 822-keV $\gamma$-ray was
also included based on energy arguments alone. The 1577-keV and 1020-keV gated
recoil-$\gamma$-$\gamma$ coincidence projections, both of which were found to
feed the 4.437-MeV level directly, showed coincidences with the 760-keV,
780-keV, 1609-keV and 438-keV $\gamma$ rays. Unfortunately, a check for
consistency between relative yields for the exiting transitions was not
possible due to the low statistics and background counts in the $\gamma$-ray
coincidence data. The 1950-keV and 1625-keV lines, and to a lesser extent the
559-keV line, each had a coincidence spectrum which supported a single level
at 4.437-MeV [Figs. 8(e) and (f)].

The placement of only a single level at 4.437-MeV is in contrast to the
proposed levels in Table 6.15 of Ref. [13] but supported by the coincidence
data and within the uncertainties of that work. The relative intensities
between the 438-keV and 1609-keV lines are consistent between the two works.
The 4.437-MeV level has been given a tentative range of $J=(4,5)$ with the
$J=(5)$ assignment slightly favored due to the reaction mechanism and
multipolarity information. From the linking transitions between the 4.437-MeV
level to levels with known $J^{\pi}$, spins of $J=(4-6)$ were possible. The
760-keV transition placement was critical as it set the lower limit on $J$.
The 1609-keV $R_{149^{\circ}/90^{\circ}}=0.74(13)$ ratio favors a dipole
multipolarity and therefore, $J<6$. The 760-keV transition has a suggestive
$R_{149^{\circ}/90^{\circ}}=0.72(35)$ ratio of a dipole transition and is
consistent with values of $(5)\rightarrow 6^{(+)}$ for the 4.437-MeV and
3.677-MeV levels. The observed dipole multipolarity of the 780-keV transition
was also consistent with either spin due to the uncertain final state spin of
the 3.657-MeV level, $J=(3,4)$. The 438-keV and 822-keV multipolarities were
inconclusive. It is unclear whether the 4.437-MeV level is the same as that
observed in the ($t$,$p$) work of Ref. [7]. If it is, then a $J=(4)$, $\pi=+$
assignment could be made as the ($t$,$p$) proton angular distributions rule
out $L=5$ and the 760-keV transition rules out a $J^{\pi}=3^{-}$ assignment.

####  IV.2.5 The 5.456-MeV, 6.014-MeV, 7.081-MeV, and 7.963-MeV levels

Levels with energies of 5.456 MeV, 6.014 MeV, 7.081 MeV, and 7.963 MeV were
identified in the present data. Only the 6.014-MeV level had parallels with
some previous work which found levels at 6.020(30), 6.000(30), and 6.006 MeV
[6, 7, 4]. It is likely that a common level having $J=(3^{-})$ was observed in
both the ($t,p$) and $\beta$ decay works at 6.006 MeV. However, this does not
appear to be the same level observed in the present work due to disagreements
in possible $J$ values. The strongest of the transitions comprising these new
levels, the 1577-keV $\gamma$ ray, was first observed in the deep inelastic
work of Ref. [12] but here placed higher in the level scheme than postulated
in their Table 6.15.

The combination of recoil-$\gamma$-$\gamma$ coincidence data with the
3.999-MeV and 4.437-MeV levels, as well as energy summations, led to the
placement of the eight transitions (one as tentative) to or from these new
levels. A lack of presence of the 438-keV $\gamma$ ray in the 1457-keV gated
$\gamma$-ray spectrum and the relation between the 4.437-MeV level and the
1020-keV transition, established the 5.456-MeV level. The appearance of the
1625-keV transition in both the 1457-keV and 1020-keV gated spectra, and vice-
versa, determined its location directly feeding the 5.456-MeV level. The
1950-keV to 1577-keV sequence was established through commensurate
coincidences with one another as well as their observed intensities from the
1609-keV gated coincidence spectrum [Fig. 8(c)]. The relatively weak 559-keV
transition linking the 6.014-MeV and 5.456-MeV levels was placed primarily
based on its energy agreement, though it did also show tentative evidence for
a coincidence relation with the 1950-keV [Fig. 8(f)] and 1020-keV transitions
. Finally, the 1067-keV $\gamma$ ray was observed to be at least a doublet.
One 1067-keV transition was in coincidence with the 1577-keV transition and
its energy was consistent with the difference between the 7.082-MeV to
6.014-MeV levels. However, the placement of any further 1067-keV $\gamma$ rays
was not able to be reconciled in the present work.

The 5.456-MeV level has been tentatively assigned $J=(5,6)$ based on the
extracted dipole nature of the 1020-keV transition
($R_{149^{\circ}/90^{\circ}}=0.86(11)$) and the quadrupole nature of the
1457-keV transition ($R_{149^{\circ}/90^{\circ}}=2.16(67)$) which feed the
$J=(4,5)$ 4.437-MeV and $J=(3,4)$ 3.999-MeV levels, respectively. The
extracted multipolarities of the 1577-keV (quadrupole) and 559-keV (dipole)
transitions similarly limit the $J$ range of the 6.014-MeV level to $J=(6,7)$.
The 1950-keV transitions $R_{149^{\circ}/90^{\circ}}=1.52(40)$ ratio is
consistent with a quadruple transition building upon the 6.014-MeV level
giving $J=(8,9)$ for the 7.963-MeV level. As noted above in sub-section
IV.2.1, the suggested $J$ values for the 7.963-MeV level are consistent with
the multipolarity of the tentatively placed 1617-keV transition which feeds
into the $J=8^{(+)}_{1}$ 6.346-MeV level. Finally, only a limit of $J=6-8$
could be surmised for the 7.081-MeV level due to the doublet nature of the
1067-keV transition and the non-distinct $R_{149^{\circ}/90^{\circ}}$ value of
the 1625-keV transition.

####  IV.2.6 Levels residing above 9.5 MeV in excitation energy

Two additional levels at 9.885 MeV and 10.996 MeV have been placed into the
38S level scheme based on the observed 2804-keV and 3033-keV $\gamma$ rays,
respectively. Due to the weak nature of these higher lying states, their
placement was most apparent in the $\gamma$-ray coincidence data provided by
the $k_{ml}>0.6$ recoil selection, though the 3033-keV line is visible in Fig.
8(e). The 2804-keV transition was within a few keV of the known 2.806-MeV
$2^{+}_{2}$ level. However, it showed clear coincidences with the 1293-keV and
1535-keV transitions amongst others, leading to its placing higher in the
level scheme. Furthermore, no evidence for a $\sim$2806-keV transition
appeared in the 810-keV gated coincidence data which clearly showed the
1513-keV line. The 2804-keV line did appear in the summed $\gamma$-ray
spectrum consisting of the combined 559-, 1020-, 1067-, and 1625-keV gated
transitions and the 2804-keV coincidence gated spectrum reciprocated the
stronger three of these transitions as well. The 2804-keV line also showed a
preference for a dipole multipolarity, however, no $J$ information could be
concluded for the 9.885-MeV level as only range of spins was placed on the
7.081-MeV level.

It is unlikely that the presently observed 3033-keV transition is the same as
the 3010-keV line observed in the deep inelastic work [12, 13] due to the
energy differences. A clear coincidence was observed between the 3033-keV line
and the 1293-keV transition which eliminated the possibility for direct
ground-state feeding. The summed $\gamma$-ray coincidence spectrum comprised
of both 1577-keV and 1950-keV gates demonstrated a clear coincidence with the
3033-keV line. A limit of $J\geq 8$ could be made for the 10.996-MeV level
based on the reaction mechanism and the tentative assignment of the 7.963-MeV
level.

####  IV.2.7 Unplaced $\gamma$-ray transitions

There were a sub-set of $\gamma$-ray transitions which were identified as
belonging to the 38S level scheme but were unable to be placed. They have been
labeled with parenthesis around their energies in Table 2. Of these, it is
probable that the 2057-keV, 2195-keV, 2486-keV, and 2572-keV, correspond to
transitions which have been previously listed in the deep inelastic work of
Ref. [12, 13]. It may be speculated that the 2057-keV transition feeds the
1.293-MeV level and that the 2195-keV level feeds the $J=4^{+}$ 2.827-MeV
state, corresponding to the 3.375(17)-MeV and 5.064(27)-MeV levels observed in
Ref. [7]. However, no coincidence data was available to weigh in on these
hypotheses.

Table 2: Identified $\gamma$-ray transitions in 38S based on the present work. Those which were unable to be placed in the level scheme are listed in parentheses. The $\gamma$-ray energies, Eγ, have uncertainties of $\sim$0.5-1 keV. The relative intensities have been normalized to the 1293-keV transitions ($\equiv 1000$). The ratio of the angular yields, $R_{149^{\circ}/90^{\circ}}$, deduced multipolarities ($L$), and the relative intensities, $I_{\gamma}$, were extracted from data based on the $k_{ml}>0.60$ cut and include statistical and systematic uncertainties. Ei [MeV] | $J^{\pi}_{i}$ | Eγ [keV] | Iγ | Ef [MeV] | $J^{\pi}_{f}$ | $R_{149^{\circ}/90^{\circ}}$ | Multipolarity ($L$)
---|---|---|---|---|---|---|---
0.000 | $0^{+}_{1}$ | – | – | – | – | – | –
1.293 | $2^{+}_{1}$ | 1292.7 | $\equiv 1000$ | 0.000 | $0^{+}_{1}$ | 1.29(5) | 2
2.806 | $2^{+}_{2}$ | 1513.1 | 38(6) | 1.293 | $2^{+}_{1}$ | 0.94(30) | –
2.827 | $4^{+}_{1}$ | 1534.5 | 598(35) | 1.293 | $2^{+}_{1}$ | 1.22(5) | 2
3.520 | (1-3) | 2224.4 | 19(5) | 1.293 | $2^{+}_{1}$ | – | –
|  | 3522.3111Placement is tentative. | 24(6) | 0.000 | $0^{+}_{1}$ | – | –
3.615 | (2,3) | 788.3${}^{\textrm{\ref{a}}}$ | 11(3) | 2.827 | $4^{+}_{1}$ | 0.76(48) | –
|  | 809.6 | 14(3) | 2.806 | $2^{+}_{2}$ | 0.64(29) | –
|  | 2322.8 | 140(12) | 1.293 | $2^{+}_{1}$ | 0.83(11) | 1
3.657 | (3,4) | 830.2 | 37(4) | 2.827 | $4^{+}_{1}$ | 1.88(44) | –
|  | 2364.8 | 89(9) | 1.293 | $2^{+}_{1}$ | 0.99(19) | –
3.677 | $6^{(+)}_{1}$ | 850.1 | 201(12) | 2.827 | $4^{+}_{1}$ | 1.38(9) | 2
3.999 | (3,4) | 383.6 | 39(5) | 3.615 | (2,3) | 0.75(13) | 1
4.437 | (4,5) | 438.3 | 18(3) | 3.999 | (3,4) | 1.10(36) | –
|  | 759.7 | 16(2) | 3.677 | $6^{(+)}_{1}$ | 0.72(35) | –
|  | 779.8 | 40(4) | 3.657 | (3,4) | 0.81(18) | 1
|  | 822.0${}^{\textrm{\ref{a}}}$ | 15(4) | 3.615 | (2,3) | – | –
|  | 1609.3 | 86(7) | 2.827 | $4^{+}_{1}$ | 0.74(13) | 1
5.456 | (5,6) | 1019.6 | 81(9) | 4.437 | (4,5) | 0.86(11) | 1
|  | 1457.0 | 28(6) | 3.999 | (3,4) | 2.16(67) | 2
6.014 | (6,7) | 558.9 | 18(4) | 5.456 | (5,6) | 0.64(21) | 1
|  | 1576.7 | 112(6) | 4.437 | (4,5) | 1.25(16) | 2
6.346 | $8^{(+)}_{1}$ | 2668.2 | 194(25) | 3.677 | $6^{(+)}_{1}$ | 1.74(19) | 2
7.081 | (6-8) | 1066.9222Identified as a doublet. | 69(6) | 6.014 | (6,7) | – | –
|  | 1625.4 | 45(5) | 5.456 | (5,6) | 1.24(41) | –
7.963 | (8,9) | 1617.3${}^{\textrm{\ref{a}}}$ | 40(5) | 6.346 | $8^{(+)}_{1}$ | 0.79(20) | 1
|  | 1950.3 | 48(6) | 6.014 | (6,7) | 1.52(40) | 2
8.730${}^{\textrm{\ref{a}}}$ | – | 2384.7${}^{\textrm{\ref{a}}}$ | 43(7) | 6.346 | $8^{(+)}_{1}$ | 1.61(73) | –
9.885 | – | 2804.0 | 40(8) | 7.081 | (6-8) | 0.70(28) | 1
10.996 | – | 3032.6 | 17(7) | 7.963 | (8,9) | – | –
– | – | (887) | 21(5) | – | – | – | –
– | – | (2015) | 17(5) | – | – | – | –
– | – | (2057) | 34(6) | – | – | – | –
– | – | (2195) | 26(6) | – | – | – | –
– | – | (2486) | 25(6) | – | – | – | –
– | – | (2572) | 34(10) | – | – | – | –
– | – | (3630) | 14(5) | – | – | – | –
– | – | (3764) | 13(4) | – | – | – | –

##  V Calculations & Discussion

The low-lying positive-parity energy levels of 38S may be described by the
coupling of a pair of protons occupying the $1s_{1/2}$ and $0d_{3/2}$ orbitals
(outside of a filled $0d_{5/2}$ orbital) and a pair of neutrons which occupy
the $0f_{7/2}$ and $1p_{3/2}$ orbitals. Within this space, $J$ values range
from $0^{+}$–$8^{+}$ through the combination of
$\pi(1s_{1/2}0d_{3/2})_{J=1,2}$, $\pi(1s_{1/2})^{2}_{J=0}$ or
$\pi(0d_{3/2})^{2}_{J=0,2}$ configurations with
$\nu(0f_{7/2}1p_{3/2})_{J=2-5}$, $\nu(0f_{7/2})^{2}_{J=0,2,4,6}$, or
$\nu(1p_{3/2})^{2}_{J=0,2}$ configurations. Excited levels reaching
$J^{\pi}>8^{+}$ are most readily accessible from either the promotion of a
$0d_{5/2}$ proton into the $1s_{1/2}0d_{3/2}$ orbitals or from the promotion
of pairs of $1s0d$ particles into the upper $0f1p$ shell, so-called $2p-2h$
excitations. Though contained within the valence space of previous
calculations, i.e. the SDPF-MU [27, 14] and the SDPF-U [30, 12] interactions,
there were no published predictions of the locations of $J^{\pi}>6^{+}$ levels
based on a $\pi(0d_{5/2})^{-1}$ configuration or high-$J$ states based on
$2p-2h$ configurations as they were not the focus of those works. However,
lower-$J$ values having $2p-2h$ character have been predicted to appear at
around 3–4 MeV in excitation energy [7, 8]. Experimentally, there are a few
possible candidates for such $2p-2h$ states, for instance levels with natural
parity that were weakly populated in previous $(t,p)$ or 2-$n$ transfer work
[5, 6, 7, 8, 9], but there are no concrete correspondences.

The lowest-lying negative parity states are expected to have single-particle
configurations built upon odd numbers of particle-hole excitations across the
$1s0d$-$0f1p$ shell gap, so-called $1p-1h$ excitations. Experimentally, a
number of candidates exist for levels with negative parity based on the
analysis of past $\beta$-decay and $(t,p)$ data [7, 8, 9, 4]. The lowest
energy candidate is around $\sim$3.5 MeV [$J^{\pi}=(1^{-}-3^{-})$], with other
candidates residing in the $\sim$4 - 5 MeV region. The lowest candidate is
well below both the $\sim 4.5$–5 MeV predictions from the
$\nu(0d_{3/2}^{4-n}0f_{7/2}^{n+2})^{6}$, $n=$ odd model-space constrained
shell-model calculations presented in Ref. [8] as well as the weak-coupling
prediction (35S$\otimes^{43}$Ca) in Ref. [7].

###  V.1 Shell-model calculations based on the FSU interaction

To further interpret the level scheme of 38S, shell-model calculations were
completed using an empirically-adjusted effective interaction, the so-called
FSU [31] interaction. This $0p$-$1s0d$-$0f1p$ shell interaction was developed
to prosper in the vicinity of 38S by including a valence space for particles
within both the $1s0d$ and $0f1p$ shells. There was an emphasis on describing
intruder states consisting of particle-hole excitations ($np-nh$) across
either the traditional $0p$-$1s0d$ shell closure or from within the $1s0d$
shell across the traditional $N=Z=20$ shell closure into the adjacent $0f1p$
sub-shells [31]. The FSU interaction has been successful in describing,
amongst other things, levels based on the $\pi 1s0d$\- and $\nu 0f1p$-shell
configurations [44, 45, 46] and in particular both normal and intruder levels
in 38Cl and 38,39Ar [42, 47].

Positive parity $0p-0h$ states in 38S were calculated using the FSU
interaction and requiring 8 valance protons to be confined within the $1s0d$
shell and the 2 valence neutrons to be confined within the full $0f1p$ shell.
The interaction was also used to provide predictions of negative parity
($1p-1h$) states and additional positive parity ($2p-2h$) states by allowing
fixed-numbers of excitations (protons or neutrons) to proceed across either
the $N=Z=8$ $0p$-$1s0d$ shell gap or the $N=Z=20$ $1s0d$-$0f1p$ gap. No mixing
was allowed between the positive parity $0p-0h$ and $2p-2h$ states. A subset
of the calculated levels are presented in Fig. 9 including the two lowest-
energy $0p-0h$ states for spins through $J^{\pi}=10^{+}$. The lowest energy
level for each spin up to $J^{-}=10^{-}$ is also shown for the $1p-1h$
calculations, and similarly, for even-$J^{+}$ states from the $2p-2h$
calculations.

Figure 9: A subset of the experimental excited-state levels in 38S are
presented along with $J^{\pi}$ assignments where available. Those in blue
emphasize the established yrast even-$J$ levels. Calculated levels based on
the FSU [31] interaction in which the protons were confined to the $1s0d$
shell and neutrons to the $1p0f$ shell are labeled as $0p-0h$. Those that
allowed for one or two particle-hole excitations across the $N=20$ shell gap,
from within the $1s0d$ shell to the $0f1p$ shell, are labelled by $1p-1h$ or
$2p-2h$, respectively. Only a subset of the calculated levels are also shown
(see text for additional details).

The calculations carried out with the FSU interaction give similar low-lying
level orderings and energies as the calculations presented in Refs. [7, 8, 14,
18, 21]. The yrast $8^{+}$ level is calculated to have a $0p-0h$ configuration
and reside at an energy of 6.32 MeV. The predicted lowest-energy positive
parity $2p-2h$ excitations appear at $\sim$3 MeV in excitation energy in Fig.
9. However, the $2p-2h$ even-$J$ states do not begin to compete in energy with
the $0p-0h$ levels so as to become yrast states up to at least
$J^{\pi}=10^{+}$. There the lowest-lying $0p-0h$ 10+ level is predicated at
around 11.1 MeV, while the partner $2p-2h$ 10+ energy is at 11.4 MeV, only 300
keV higher. The FSU interaction also predicts the lowest-energy negative
parity states of $1p-1h$ character ($J^{\pi}=3^{-}-5^{-}$) in the 4.2 – 4.9
MeV energy region. This is consistent with the previous model predictions
discussed above. The negative parity $1p-1h$ states are predicted to become
yrast in spin at around $J=9$ or 10, and near 8.5 MeV in excitation energy,
$\sim$1 MeV below the predicted $0p-0h$ 9+ and 10+ levels.

###  V.2 The even-$J$ yrast states

In addition to the excitation energies plotted in Fig. 9, the calculated
energies and occupancies for the even-$J^{+}$ yrast levels up to
$J^{\pi}=10^{+}$ ($0p-0h$) are also shown in Fig. 10. The calculations
reproduce the excitation energies for the $J^{\pi}=2^{+},4^{+},6^{+}$ and
$8^{+}$ levels, assuming positive parity for the latter two experimentally.
The states with $J\leq 6^{+}$ show mixed but near constant proton $1s_{1/2}$
and $0d_{3/2}$ occupancies while the neutron configurations evolve with
increasing $J$ from mixed $0f_{7/2}-1p_{3/2}$ occupancy to a pure
$(0f_{7/2})^{2}$ configuration. As discussed below and in Sec. V.3, the $\nu
1p_{3/2}$ occupancy deviates from the simple $(1f_{7/2})^{2}_{J=0,2,4,6}$
seniority $\nu=2$ picture for the low-lying even-$J$ multiplet for these two
levels. This deviation is a critical component in the microscopic description
of the collective nature of these low-lying levels. At larger spin, however,
the $\nu(0f_{7/2})^{2}$ configuration is restored out of the demand for
increased angular momentum. Namely, in order to generate spins of
$6^{+}-10^{+}$ within the $0p-0h$ space, a pure $\nu(0f_{7/2})^{2}$ neutron
configuration is needed as the inclusion of any $1p_{3/2}$ neutrons is limited
to $J_{\text{max}}=5\hbar$ as $(0f_{7/2}1p_{3/2})_{J=2-5}$.

Figure 10: The excitation energies and calculated occupancies ($0p-0h$) of the
even-$J^{+}$ yrast states in 38S through the $10^{+}$ level. The relative
occupancies for the ’last’ three $sd$-shell protons and the two neutrons
within the $0f1p$-shell are labelled by color and connected by lines to
highlight their evolution. The five remaining $sd$-shell protons fill the $\pi
0d_{5/2}$ orbital and the occupancy is only for the ‘6th’ proton, i.e. an
occupancy of one on the plot represents a filled $\pi 0d_{5/2}$ orbital
($0d_{5/2})^{6}$ and a value of zero gives $\pi(0d_{5/2})^{-1}$.

In accordance with Fig. 10 the predominant factor determining the relative
energy spacing between the $6^{+}$ – $8^{+}$ – $10^{+}$ levels is the
arrangement of the proton occupancies. Starting with $(0f_{7/2})^{2}_{J=6}$
from the neutrons, there is a migration in the proton occupancy from the
$1s_{1/2}$ into the $0d_{3/2}$, gaining 2$\hbar$ in angular momentum, which
extends $J^{\pi}=6^{+}$ up to $8^{+}$. The experimental spacing between the
$J^{\pi}=6^{+}-8^{+}$ levels, $\Delta E_{x}=2.67$ MeV, is well reproduced by
the FSU interaction, $\Delta E_{x}=2.56$ MeV. It can be concluded,
unsurprisingly, that the FSU interaction has a solid grasp on the description
of the proton $1s_{1/2}-0d_{3/2}$ single-particle energies and the associated
matrix elements.

One method to construct a $J^{\pi}=10^{+}$ level is through the excitation of
a proton from within the $\pi 0d_{5/2}$ orbital into the $\pi 1s_{1/2}$
orbital where the $(0d_{5/2})^{-1}$ proton hole provides the additional
angular momentum. The FSU interaction predicts this configuration to be yrast
at 11.122 MeV. A second option for generating $J^{\pi}>8^{+}$ levels is
through $2p-2h$ configurations. The addition of two neutrons into the
$0f_{7/2}$ and $1p_{3/2}$ orbitals increases the neutron $J_{\text{max}}$
contribution to $J_{\text{max}}>6$ via the $(0f_{7/2})^{4}_{J_{\text{max}}=8}$
and $(0f_{7/2}1p_{3/2})^{4}_{J_{\text{max}}=9}$ configurations. The lowest-
lying $2p-2h$ $10^{+}$ state predicted by the FSU interaction is only $\sim
300$ keV above its corresponding $0p-0h$ level. Hence, the calculations point
towards a possible mixing between these two states experimentally.
Unfortunately, no solid candidates were empirically determined for either
$10^{+}$ level though the state at 10.996 MeV is in the approximate energy
range of the predictions.

###  V.3 The role of the $\nu 1p_{3/2}$ occupancy

Appreciable occupancy of the neutron $1p_{3/2}$ orbital, and the implicit
weakening of the traditional $N=28$ shell closure, is known to play a role in
the low-lying structure of the neutron-rich S isotopes. Together with the
neutron $0f_{7/2}$ orbital, these orbitals provide the foundation for the
emergence of a region of low-lying deformation in the neutron-rich $N=28$
isotones [30, 28] via coherent proton-neutron correlations. The inclusion and
occupancy of the $\nu 1p_{3/2}$ orbital specifically, has been required for a
proper description of the experimental transition rates [14, 21] and the
$2^{+}_{1}$ excited-state $g$ factor [19, 18] in 38S. In the work of Ref. [18]
the role of the $\nu 1p_{3/2}$ orbital in dictating the proton occupancies of
the $1s_{1/2}$ and $0d_{5/2}$ via coherent quadrupole correlations was
emphasized. Such correlations and the migration of nucleon occupancy led to a
correct description of the observed $2^{+}_{1}$ $g$ factor while also
correctly increasing the magnitude of the calculated quadrupole transition
strength [B($E2$)] of this state.

One seemingly conflicting piece of data with the coherent proton-neutron
picture is the extracted ratio of the neutron-to-proton multi-pole transition
matrix elements, $M_{n}/M_{p}=(1.5\pm 0.3)N/Z$, for the $2^{+}_{1}$ state
[16]. The $\gtrsim 1$ value points towards a non-symmetric (isovector)
contribution of neutrons-to-protons to the $2^{+}_{1}$ excitation typically
indicative of a closed nucleon shell, in this case a closed proton shell. As
discussed in Ref. [16], similar behavior has been observed in 18O, and to a
less degree in 42Ca. One possible reconciliation is that while the $\nu
1p_{3/2}$ occupancy is crucial to driving coherent behavior there is still a
large fraction of the $\nu(0f_{7/2})^{2}$ configuration within the $2^{+}_{1}$
and ground state wave functions which manifests itself in the individual
muliti-pole transition matrix elements differently than in the $g$ factor
value.

The calculated excitation energies and neutron $1p_{3/2}$ occupancies from the
FSU interaction ($0p-0h$) are shown by the lines in Figs. 11(a) and (b),
respectively, for the Si, S, Ar, and Ca $N=22$ isotones. Both 36Si and 38S
show distinct increases in their $1p_{3/2}$ occupancy ($\gtrsim 0.4$ nucleons
for the $0^{+}_{1}$ and $2^{+}_{1}$ levels) relative to the heavier isotones
of 40Ar and 42Ca. As discussed in the work of Ref. [48], and shown in Fig. 11,
striking similarities exist experimentally between the low-lying even-$J$
levels of 36Si and 38S. The energies of their yrast levels through
$J^{\pi}=6^{+}$ and their ground state to $2^{+}_{1}$ dynamic quadrupole
transitions strengths, B($E2,0^{+}_{1}\rightarrow 2^{+}_{1}$) values, agree
within $\sim$10% of each other. The FSU interaction calculations reproduce the
excitation energies well [Fig. 11(a)] and slightly under predict the
B($E2,0^{+}_{1}\rightarrow 2^{+}_{1}$) values but reproduce the trend.
Additionaly, the $\nu 1p_{3/2}$ occupancies from the FSU interaction are
larger than those predicted by the SDPF-MU interaction for the same two levels
in 38S by $>0.2$ nucleons (see Fig. 20 of Ref. [14]). This may be an indicator
as to why there is an improper spacing between the 38S $0^{+}_{1},2^{+}_{1}$
energies relative to the $4^{+}_{1},6^{+}_{1}$ level energies by about $\sim
350-400$ keV (see Fig. 4 of Ref. [14]).

Interpretation of the above information and that in Fig. 11 suggests similar
magnitudes of low-lying deformation. Even still, the slight increase in the
$1p_{3/2}$ occupancy ($\sim$0.1 nucleons) in 38S is consistent with it having
a lower 2${}^{+}_{1}$ energy and a larger B($E,0^{+}_{1}\rightarrow
2^{+}_{1}$) value. It is perhaps initially surprising that the highest
$1p_{3/2}$ occupancy does not reside in the most neutron-rich system, 36Si,
considering its neutron $0f_{7/2}$-$1p_{3/2}$ energy spacing should be the
most reduced. However, it appears that the correlation energy gained by the
arrangement of the two additional protons in 38S is enough to overcome a
change in a slight increase in the neutron orbital energy spacing. It should
be noted, that while the magnitude of the deformations in 36Si and 38S are
similar, it has been postulated theoretically that 36Si has an axially-
symmetric oblate shape compared to a prolate shape in 38S [28]. The prolate
axially-symmetric shape in 38S has been effectively established by the
$g$-factor work [19, 18] while experimental information for 36Si is lacking.

###  V.4 The neutron $0f1p$ shell states

The coupling of a single neutron in each the $0f_{7/2}$ and $1p_{3/2}$
orbitals results in a multiplet of states ranging from $J=2^{+}-5^{+}$. While
the lower spins of this multiplet are susceptible to mixing, a correspondence
between states with this ideal single-particle configuration was found with
the calculations of the FSU interaction for the $5^{+}_{1}$ and $4^{+}_{2}$
levels at 5.146 MeV and 4.153 MeV, respectively. The calculated neutron
occupancies for these two levels, $0f_{7/2}^{\sim 1.2}$ and $1p_{3/2}^{\sim
0.7}$, close to the single-particle picture, and the calculated proton
occupancies were also similar, ($1s_{1/2}^{\sim 1.5}$ and $0d_{3/2}^{\sim
0.5}$). There is a predicted $3^{+}_{1}$ level by the FSU interaction at 4.146
MeV that is a possible member of the multiplet, though it shows variation in
the proton occupancy and an increase of 0.2 in the $\nu 0f_{7/2}$ occupancy
relative to the aforementioned $4^{+}_{2}$ and $5^{+}_{1}$ levels. Under the
assumption that the $5^{+}_{1}$ level specifically, has a reasonably pure
$(0f_{7/2}1p_{3/2})$ neutron configuration and the $6^{+}_{1}$ level has a
nearly pure $(\nu 0f_{7/2})^{2}$ configuration (Figs. 10 and 11), the energy
spacing between these levels is primarily dependent upon the description of
the $0f_{7/2}$-$1p_{3/2}$ interaction and their single-particle energies. The
calculated energy difference between the $5^{+}_{1}$ and $6^{+}_{1}$ states is
$\Delta E_{x}\approx 1.4$ MeV. Experimentally, there are a series of levels at
3.615 MeV, 3.999 MeV, and 4.437 MeV, that have been eluded to as belonging to
this $(0f_{7/2}1p_{3/2})_{J=2-5}$ multiplet [11, 12, 13]. Assuming this
sequence of levels are the $J^{\pi}=2-4^{+}$ multiplet members, and the newly
established level at 5.456 MeV is the $5^{+}$ candidate, a consistent (though
non-unique) picture between theory and experiment emerges. Under this
scenario, $\Delta E_{x}\approx 1.8$ MeV for the energy difference between the
experimental $5^{+}_{1}$ and $6^{+}_{1}$ levels, which suggests some
discrepancy with the calculations ($\approx 400$ keV). However, as mentioned,
there are a other plausible corresponding $J$ assignments between the observed
5.456 MeV level and calculation, including the $5^{+}_{2}$, $5^{-}_{1}$,
$6^{+}_{2}$, and $6^{-}_{1}$ calculated levels.

###  V.5 Other possible correspondences between the observed and calculated
levels

The measured state at 4.437 MeV has additional counterparts in the
calculations beyond the suggested $J^{\pi}=4^{+}$ assignment and its
membership in the $(0f_{7/2}^{1}1p_{3/2}^{1})_{J=2-5}$ multiplet discussed
above. In particular, the relatively large amount of fractional incoming and
outgoing yield through the level may suggest it is a low-lying negative parity
intruder state. The calculated $J^{\pi}=4^{-}$ and $5^{-}$ states are nearby
at $\sim 4.7-4.9$ MeV. Based on energy arguments alone, the subsequent
6.014-MeV and 7.963-MeV levels, each connected via quadrupole transitions
(Fig. 7 and Table 2) agree best with the $J^{\pi}=6^{-}$ and $8^{-}$ levels at
6.027 and 7.832 MeV. Though a lowering of the calculated $J^{-}$ states on the
order of $\sim$300-400 keV, would also create a viable scenario for the
$9^{-}\rightarrow 7^{-}\rightarrow 5^{-}$ sequence.

Figure 11: The even-$J^{+}$ yrast excitation energies ($E_{x}$) and $\nu$1p3/2
occupancies from the $0p-0h$ shell-model calculations using the FSU
interaction [31] (lines) for a selection of even $Z$ $N=22$ isotones. The
experimental energies for the $J^{\pi}=0^{+}_{1}$ (blue squares), $2^{+}_{1}$
(green circles), $4^{+}_{1}$ (pink triangles), and $6^{+}_{1}$ (purple
diamonds) are also given [3].

##  VI Summary & Conclusions

The level scheme of 38S has been extended in both excitation energy ($\sim$11
MeV) and maximum spin ($J\gtrsim 8$) based on new in-beam $\gamma$-ray data.
38S was populated through a fusion-evaporation reaction involving a beam of
22Ne interacting with an 18O target at the Argonne National Laboratory ATLAS
Facility. Prompt $\gamma$ rays emanating from the reaction were detected by
the GRETINA array and an event-by-event selection of the recoiling nuclei was
carried out by the Fragment Mass Analyzer. In order to improve upon the
traditional methods used in the unique recoil identification of 38S, machine-
learning techniques were employed for the first time. The supervised training
of a feed-forward neural network in a One-Vs-All mode was carried out using
labelled data based on known $\gamma$-ray energies and standard training
procedures. Clear signatures were observed in the distribution and behaviour
of the output values from the fully-trained model for $\gamma$-ray transitions
belonging to 38S relative to those belonging to other isotopes or backgrounds.
Hence, the selection of specific ranges of model output values provided a far-
improved determination of $\gamma$-ray transitions in 38S utilizing both
singles and coincidence $\gamma$-ray spectra.

The extension of the known even-$J$ yrast levels up through $J^{\pi}=8^{(+)}$
facilitated a discussion on their energy spacing and underlying single-
particle structures. In particular, the energy spacing between the
$6^{(+)}_{1}$ and $8^{(+)}_{1}$ levels was used to extract the amount of
energy required to promote a proton from the $1s_{1/2}$ orbital into the $\pi
0d_{3/2}$ orbital, $\Delta
E_{x}=(6.346~{}\mathrm{MeV}-3.677~{}\mathrm{MeV})\approx 2.7$ MeV.
Furthermore, candidates for the high-$J$ levels belonging to the near-pure
neutron $(0f_{7/2}1p_{3/2})_{J=2-5}$ configuration were identified, for
instance the level at 5.465 MeV was postulated as the $5^{+}$ member. The
energy difference between this $5^{+}_{1}$ level and that aforementioned
$6^{+}_{1}$ yrast state, $\Delta
E_{x}=(5.465~{}\mathrm{MeV}-3.677~{}\mathrm{MeV})\approx 1.8$ MeV, is
determined by the rearrangement of the $0f_{7/2}$-$1p_{3/2}$ occupancies,
their corresponding single-particle energies, and their interactions.

Shell-model calculations incorporating the FSU interaction [31] well
reproduced the energies of the even-$J$ yrast sequence and low-lying level
scheme in general. They also provided guidance on a number of possible spin-
parity scenarios for other corresponding states. The calculations reproduced
the measured energy spacing between the $6^{+}_{1}$ – $8^{+}_{1}$ levels
giving to the interactions accurate description of the proton single-particle
energies and interaction strengths within the $1s0d$ shell. A discrepancy of
$\sim$400 keV was noted between the theoretical spacing of the
$6^{+}_{1}-5^{+}_{1}$ levels and the experimental value, noting however that
the experimental $5^{+}_{1}$ assignment was speculative. Furthermore, the
calculations supported discussions pertaining to the spectroscopic
similarities between 36Si and 38S. In particular, the striking resemblance of
their neutron $1p_{3/2}$ occupancy across the even-$J^{+}$ yrast levels [48],
excitation energies, and B($E2,0^{+}_{1}\rightarrow 2^{+}_{1}$) values. The
role of the neutron $1p_{3/2}$ occupancy and its importance in developing
coherent proton-neutron correlations in the low-lying states of the $N=22$
systems was also reiterated, building upon the previous discussions in Refs.
[19, 18, 28, 30, 14, 21]. In closing, a few open questions about 38S stemming
from the present work include: i) why is the $(M_{n}/M_{p})$ value extracted
for the $2^{+}_{1}\rightarrow 0^{+}_{1}$ transition from inelastic proton
scattering data not consistent with the hydrodynamical, $N/Z$, limit? ii)
where is the location of the yrast $10^{+}$ level and how mixed is this state?
iii) where do the negative parity intruder levels appear for certain and where
do they become yrast?

##  VII Acknowledgments

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, Office of Nuclear Physics, under Contracts No. DE-
AC02-06CH11357 (Argonne), No. DE-FG02-94ER40848 (UML), No. DE-AC02-05CH11231
(LBNL), and No DE-SC0020451 (FRIB). This research used resources of ANL’s
ATLAS facility, which is a DOE Office of Science User Facility. We gratefully
acknowledge the computing resources provided on Bebop, a high-performance
computing cluster operated by the Laboratory Computing Resource Center at
Argonne National Laboratory. GRETINA was funded by the U.S. DOE, Office of
Science, Office of Nuclear Physics, and operated by the ANL and LBNL contract
numbers above. TRIUMF receives federal funding via a contribution agreement
through the National Research Council Canada (NRC). Work at LLNL was performed
under DOE Contract No. DE-AC52-07NA27344 and was supported by the LLNL-LDRD
Program under Project No. 23-LW-047. Shell model calculations used the
computational facilities of Florida State University supported by the Grant
No. DE-SC0009883 (FSU).

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