# Statistical properties of the well deformed 153,155Sm nuclei and the
scissors resonance

K. L. Malatji<EMAIL_ADDRESS>Department of Subatomic Physics, iThemba
LABS, P.O. Box 722, Somerset West 7129, South Africa  Physics Department,
Stellenbosch University, Matieland 7602, South Africa  K. S. Beckmann
<EMAIL_ADDRESS>Department of Physics, University of Oslo, N-0316, Oslo,
Norway
M. Wiedeking<EMAIL_ADDRESS>Department of Subatomic Physics, iThemba
LABS, P.O. Box 722, Somerset West 7129, South Africa  School of Physics,
University of the Witwatersrand, Johannesburg 2050, South Africa  S. Siem
Department of Physics, University of Oslo, N-0316, Oslo, Norway
S. Goriely  Institut d’Astronomie et d’Astrophysique, Université Libre de
Bruxelles, CP 226, B-1050 Brussels, Belgium
A. C. Larsen  Department of Physics, University of Oslo, N-0316, Oslo, Norway
K. O. Ay  Department of Physics, Faculty of Science and Letters, Eskisehir
Osmangazi University, TR-26040 Eskisehir, Turkey
F. L. Bello Garrote  L. Crespo Campo  A. Görgen  Department of Physics,
University of Oslo, N-0316, Oslo, Norway
M. Guttormsen  V. W. Ingeberg  Department of Physics, University of Oslo,
N-0316, Oslo, Norway
P. Jones  Department of Subatomic Physics, iThemba LABS, P.O. Box 722,
Somerset West 7129, South Africa  B. V. Kheswa  Department of Subatomic
Physics, iThemba LABS, P.O. Box 722, Somerset West 7129, South Africa
Department of Applied Physics and Engineering Mathematics, University of
Johannesburg, Doornfontein 2028, South Africa
P. von Neumann-Cosel  Institut für Kernphysik, Technische Universität
Darmstadt, D-64289 Darmstadt, Germany
M. Ozgur  Department of Physics, Faculty of Science and Letters, Eskisehir
Osmangazi University, TR-26040 Eskisehir, Turkey
G. Potel  Lawrence Livermore National Laboratory, Livermore, California 94551,
USA  L. Pellegri  Department of Subatomic Physics, iThemba LABS, P.O. Box 722,
Somerset West 7129, South Africa  School of Physics, University of the
Witwatersrand, Johannesburg 2050, South Africa  T. Renstrøm  G. M. Tveten
Department of Physics, University of Oslo, N-0316, Oslo, Norway
F. Zeiser  Department of Physics, University of Oslo, N-0316, Oslo, Norway

###### Abstract

The Nuclear Level Densities (NLDs) and the $\gamma$-ray Strength Functions
($\gamma$SFs) of 153,155Sm have been extracted from (d,p$\gamma$) coincidences
using the Oslo method. The experimental NLD of 153Sm is higher than the NLD of
155Sm, in accordance with microscopic calculations. The $\gamma$SFs of
153,155Sm are in fair agreement with QRPA calculations based on the D1M Gogny
interaction. An enhancement is observed in the $\gamma$SF for both 153,155Sm
nuclei around 3 MeV in excitation energy and is attributed to the M1 Scissors
Resonance (SR). Their integrated strengths were found to be in the range 1.3 —
2.1 and 4.4 — 6.4 $\mu^{2}_{N}$ for 153Sm and 155Sm, respectively. The
strength of the SR for 155Sm is comparable to those for deformed even-even Sm
isotopes from nuclear resonance fluorescence measurements, while that of 153Sm
is lower than expected.

##  I Introduction

The stable samarium isotopic chain provides an excellent opportunity to
systematically investigate the evolution of nuclear structure effects, from
the semi-magic and near spherical 144Sm to the highly-deformed 154Sm isotope.
As the nuclear shape changes, statistical quantities such as the Nuclear Level
Density (NLD) and $\gamma$-ray Strength Function ($\gamma$SF) are expected to
change and provide evolutionary information across the isotopic chain.
Furthermore, the behavior of resonance modes, such as the $M1$ Scissors
Resonance (SR), $E1$ Pygmy Dipole Resonance (PDR), and the Low-Energy
Enhancement (LEE), can be tracked.

The strength of the SR is sensitive to the ground state deformation Richter
(1990); Margraf  _et al._ (1993); Goriely  _et al._ (2016); Goriely and Plujko
(2019). The SR was first predicted in 1978 by Lo Iudice and Palumbo Iudice and
Palumbo (1978) before it was observed experimentally a few years later Bohle
_et al._ (1984). Even-even nuclei were initially considered to be the best
experimental candidates to exhibit strong SR modes. However, it soon became
apparent that this mode should also present in odd-even and odd-odd systems,
although its intensity may be fragmented significantly, making it more
difficult to detect Enders  _et al._ (1997); Heyde  _et al._ (2010). Since
then, many heavy deformed even-even and odd-mass rare-earth nuclei have been
systematically investigated. So far, the SR mode has been experimentally
observed in vibrational and rotational Heyde  _et al._ (2010), as well as in
$\gamma$-soft nuclei Linnemann  _et al._ (2003); von Garrel  _et al._ (2006)
and has also been observed in the actinide region Margraf  _et al._ (1990);
Guttormsen  _et al._ (2012, 2013, 2014); Tornyi  _et al._ (2014); Laplace  _et
al._ (2016); Zeiser  _et al._ (2019), as well as in the rare-earth mass region
Voinov  _et al._ (2001); Melby  _et al._ (2001); Siem  _et al._ (2002);
Guttormsen  _et al._ (2003); Krtička  _et al._ (2004); Agvaanluvsan  _et al._
(2004); Nyhus  _et al._ (2010); Baramsai  _et al._ (2015); Simon  _et al._
(2016); Renstrøm  _et al._ (2018); Goriely and Plujko (2019).

A range of different experimental techniques have been used to investigate the
low-energy SR. These include ground state absorption experiments such as
inelastic electron scattering Bohle  _et al._ (1984), nuclear resonance
fluorescence (NRF) Kneissl  _et al._ (1996) and average resonance capture
(ARC) Kopecky  _et al._ (2017), as well as $\gamma$-decay experiments such as
radiative neutron capture Krtička  _et al._ (2004); Baramsai  _et al._ (2015)
and the Oslo method Schiller  _et al._ (2000); Larsen  _et al._ (2011), which
extracts information from ion scattering or transfer reactions. In general,
$\gamma$-decay experiments have yielded larger SR strengths than the ground
state absorption experiments. Summaries of experimental techniques to measure
$\gamma$SFs can be found in Goriely et al. Goriely  _et al._ (2019).

Recent measurements of the $\gamma$SF in the actinides Guttormsen  _et al._
(2012); Laplace  _et al._ (2016); Tornyi  _et al._ (2014) have uncovered that
the SR exhibits a pronounced double-hump structure, seemingly independent of
whether the nucleus has an even or odd number of neutrons. The splitting has
also been observed, albeit weaker, in the transitional nucleus 181Ta Angell
_et al._ (2016); Brits  _et al._ (2019). It is suggested that the splitting of
the SR may be due to the isovector spin-scissors mode Balbutsev  _et al._
(2018), or due to triaxiality Iachello (1981). The splitting in the SR has not
been reported for any of the rare-earth nuclei studied with the NRF technique
Pietralla  _et al._ (1998), with the Oslo method Voinov  _et al._ (2001);
Melby  _et al._ (2001); Siem  _et al._ (2002); Guttormsen  _et al._ (2003);
Agvaanluvsan  _et al._ (2004); Nyhus  _et al._ (2010); Simon  _et al._ (2016);
Renstrøm  _et al._ (2018) or other techniques Krtička  _et al._ (2004);
Baramsai  _et al._ (2015); Goriely and Plujko (2019).

The $\gamma$SFs of the isotopes 148,149Sm were measured, and analyzed with the
Oslo method, almost two decades ago and a weak structure was identified to
possibly be due to the SR, called a pygmy at the time Siem  _et al._ (2002).
With data already available on these two weakly deformed isotopes, together
with the recent measurements of 151,153Sm Simon  _et al._ (2016) (also
analyzed with the Oslo method), it is interesting to extend the investigation
towards more deformed Sm nuclei where the SR is expected to be more prominent.

In this paper, the measurement of the NLDs and $\gamma$SFs for the odd-even
153,155Sm extracted with the Oslo method from the 152,154Sm(d,p$\gamma$)
reactions are reported. The integrated strength of the SR in both isotopes are
extracted and compared to previous experimental data. In Sec. II the
experimental setup is presented and Sec. III provides a brief overview of the
Oslo method and the normalization procedures used, as well as the measured
NLDs and $\gamma$SFs. Section IV investigates the presence of the SR in Sm
isotopes and its integrated strength. Discussion follows in Sec. V and a brief
summary in Sec. VI.

##  II Experimental Setup

Two experiments were performed at the Oslo Cyclotron Laboratory (OCL) at the
University of Oslo using self-supporting 152Sm (enriched to 98.3%) and 154Sm
(enriched to 98.7$\%$) targets with thicknesses of 2.9 and 3.2 mg/cm2,
respectively. Deuteron beams of 13.5 MeV and 13 MeV were used to populate
excited states in 153,155Sm. The SiRi particle telescope Guttormsen  _et al._
(2011) and CACTUS scintillator Guttormsen  _et al._ (1990) arrays were used to
detect charged particles and $\gamma$-rays in coincidence.

The $\Delta E$-$E$ SiRi particle-telescope consisted of eight 130 $\mu$m thin,
segmented silicon $\Delta E$ detectors and eight 1550 $\mu$m thick $E$ silicon
detectors. These were placed at backward angles to reduce detection of
elastically scattered events and covered a polar angular range of
$\theta_{lab}=126^{\circ}-140^{\circ}$ with respect to the beam direction. The
energy resolution, as determined from the elastic peaks, is $\sim$ 130 keV.
The CACTUS array consisted of 26 and 24 NaI(Tl) detectors for the 153Sm and
155Sm measurements, respectively. The $5"\times 5"$ crystals were positioned
22 cm from the target, covering solid angles of 17$\%$ and 15$\%$ of $4\pi$
sr, respectively. CACTUS has an energy resolution of 7$\%$ FWHM for a 1.332
MeV $\gamma$-ray transition.

The $E$ detectors provided a start signal and the NaI(Tl) detectors a stop
signal for the time-to-digital converters, enabling event-by-event sorting for
the $\gamma$-particle coincidence data. Calibration of the SiRi and CACTUS
detectors was achieved using distinct $\gamma$-ray transitions of 29Si
obtained from 28Si(d,p$\gamma$) calibration runs which provided well-resolved
particle and $\gamma$-ray peaks. During offline analysis, charged-
particle-$\gamma$ coincidence events were extracted within a prompt time gate
of 20 ns. Equivalently wide time gates were used to remove the majority of
randomly correlated events from the prompt particle-$\gamma$ events. The
excitation energy ($E_{x}$) versus $\gamma$-ray energy ($E_{\gamma}$) matrices
were constructed from the particle-$\gamma$ coincidence events and are shown
in Fig. 1 (a) and (d). The Oslo method Schiller  _et al._ (2000); Larsen  _et
al._ (2011) is applied on these matrices to extract simultaneously the NLDs
and the $\gamma$SFs up to the neutron-binding energies ($B_{n}$) through
several iterative methods, discussed in the next section. The analysis in this
work was performed using the Oslo method software version 1.1.2 Guttormsen
_et al._ (2018).

Figure 1: (Color online) The raw (panel a, d), unfolded (panel b, e) and first
generation (primary) $\gamma$-ray matrix (panel c, f) of 153,155Sm. The dashed
lines indicate the energy regions from which the NLDs and $\gamma$SFs were
extracted.

##  III Data Analysis

###  III.1 The Oslo Method

The $\gamma$-ray spectra, extracted for each $E_{x}$ bin, were unfolded with
the iterative procedure of Ref. Guttormsen  _et al._ (1996) and then corrected
for the known NaI(Tl) response functions remeasured in 2012 Campo  _et al._
(2016), in order to obtain the full-energy $\gamma$-ray spectra. At this
point, the first-generation $\gamma$-ray method Guttormsen  _et al._ (1987) is
used to extract the primary $\gamma$-rays from the decay cascades in each
$E_{x}$ bin. The raw matrices (a) and (d), the unfolded matrices (b) and (e),
and the resulting first generation $\gamma$-ray matrices (c) and (f),
P($E_{\gamma}$, $E_{x}$), are summarized in Fig. 1. The diagonals where
$E_{\gamma}$=$E_{x}$ represent all direct decays to the ground state.

For 153Sm, the region used for extraction of the NLD and $\gamma$SF was from
$E_{\gamma}$ = 0.96 MeV to and including the $E_{x}$ = $E_{\gamma}$ diagonal,
and from $E_{x}$ = 2.88 MeV up to $E_{x}$ = 5.69 MeV. For 155Sm, the limits
were from $E_{\gamma}$ = 1.65 MeV to and including the $E_{x}$ = $E_{\gamma}$
diagonal, and from $E_{x}$ = 2.49 MeV up to $E_{x}$ = 5.73 MeV. They were
chosen to exclude regions characterized by discrete transitions at low
excitation energies. The regions in Fig. 1 (c) and (f) that correspond to
$E_{\gamma}<$ 1 MeV have low statistics due to over-subtraction of discrete
and strong $\gamma$-ray transitions. This energy region is therefore also
excluded from further analysis.

The NLDs and $\gamma$SFs of 153,155Sm were extracted simultaneously from the
$P$($E_{\gamma}$,$E_{x}$) matrix, using the ansatz Tveter  _et al._ (1996);
Midtbø  _et al._ :

$\centering
P(E_{x},E_{\gamma})\propto\rho(E_{x}-E_{\gamma})\mathcal{T}(E_{\gamma}),\@add@centering$
(1)

where $\rho(E_{x}-E_{\gamma})$ is the level density at the final $E_{x}$ to
which the nucleus decays. The parameter $\mathcal{T}(E_{\gamma})$ is the
$\gamma$-ray transmission coefficient, and assuming the generalized Brink-Axel
Hypothesis Brink (1957); Axel (1962), it is only dependent on the $\gamma$-ray
energy. It is also assumed in Eq. (1) that the $\gamma$-decay pattern from any
initial excitation energy is independent of whether the nucleus was populated
into this excitation energy directly from a nuclear reaction or by
$\gamma$-ray decays from higher-lying states Bohr and Mottelson (1969); Henden
_et al._ (1995). A $\chi^{2}$ minimization is performed, between the
experimental $P$($E_{\gamma}$,$E_{x}$) and a theoretical
$P_{theo}$($E_{\gamma}$,$E_{x}$) in which $\rho(E_{x}-E_{\gamma})$ and
$\mathcal{T}(E_{\gamma})$ are treated as free parameters Schiller  _et al._
(2000):

$\centering\begin{split}\tilde{\rho}(E_{x}-E_{\gamma})&=A\rho(E_{x}-E_{\gamma})\exp[{\alpha(E_{x}-E_{\gamma}})],\\\
\tilde{\mathcal{T}}(E_{\gamma})&=B\mathcal{T}(E_{\gamma})\exp(\alpha
E_{\gamma}).\end{split}\@add@centering$ (2)

The transformation parameters $\alpha$, $A$ and $B$ correspond to physical
solutions and are deduced from external experimental data to get the solution
to Eq. (1). At this point, the features in the NLDs and $\gamma$-ray
transmission coefficients are fixed, except for the slopes and absolute
values. Note that the resulting $\rho$ and $\mathcal{T}$ functions do not
depend on the initial parameters used in the iterative procedure.

###  III.2 Normalization of the NLDs

Figure 2: (Color online) Spin distributions at $B_{n}$ estimated from HFB+comb
(blue solid line) and RMI (red dashed line) models for 153Sm.

The extracted NLD is normalized to the known experimental discrete states NND
(2019) at $E_{x}\lesssim$ 1 MeV and extrapolated to the NLD, $\rho(B_{n})$, at
the neutron-binding energy ($B_{n}$), determining its slope and absolute
value. The level density $\rho(B_{n})$ is determined from the average $s-$wave
neutron-resonance spacing $D_{0}$ Capote  _et al._ (2009), using Eq. (28) of
Ref. Schiller  _et al._ (2000). The parameters used for the normalization are
listed in Tab. 1. Due to the unavailability of experimental data on the spin
and parity $J^{\pi}$ distribution at $B_{n}$, the Rigid Moment of Inertia
formula (RMI) which assumes equiparity von Egidy and Bucurescu (2006) and the
Hartree-Fock-Bogoliubov plus Combinatorial (HFB+comb) Goriely  _et al._ (2008)
models were utilized to model the distributions. The RMI and HFB+comb spin
distributions at ${B_{n}}$ are shown in Fig. 2 for 153Sm.

Figure 3: (Color online) Experimental NLDs of (a) 153Sm and (b) 155Sm from the
present (d,p$\gamma$) experiments, normalized using the RMI and HFB+comb
models and extrapolated with the CT and Fermi Gas models to match the
estimated $\rho$(Bn). The error bars on the NLD data points represent only
statistical uncertainties, whereas the error band represent both statistical
uncertainties as well as systematic uncertainties (see text for details). The
error bands beyond the last data point (grey shaded) indicate the range of the
extrapolation models and the solid line represent the average of the limits.
The NLDs of 153,155Sm are compared to the NLDs of (p,d$\gamma)^{151,153}$Sm
from Ref. Simon  _et al._ (2016) in panel (c).

The HFB+comb model is a global microscopic approach to calculate the energy-
and $J^{\pi}$\- dependent NLD Goriely  _et al._ (2008). The HFB+comb model can
be renormalized to match the known experimental discrete states and the
average $s-$wave neutron-resonance spacing $D_{0}$, as detailed in Ref.
Goriely  _et al._ (2008).

Figures 3 (a) and (b) present the extracted NLDs for 153,155Sm. At low $E_{x}$
the NLDs follow closely the experimental discrete states NND (2019). The NLD-
bands represent the standard deviation of the level densities as a result of
the statistical errors stemming from the Oslo method Schiller  _et al._
(2000); Larsen  _et al._ (2011), as well as systematic errors. The systematic
errors take into account the variations of slope and absolute value due to
normalizing using the HFB+comb (upper limit) and RMI (lower limit) as well as
from varying $D_{0}$ within its uncertainties and varying a reduction factor
which is used to scale the width of the spin distribution at $B_{n}$. The data
points in Fig. 3 (a) and (b) represent the average of the upper and lower
limits with the statistical error bars. It is important to distinguish between
statistical and systematic errors because the statistical errors are small and
they limit the possibility of fluctuations between neighboring points. As can
be seen in Fig. 3, the extracted experimental NLD is not available up to
$B_{n}$ due to the exclusion of transitions below the Eγ-cutoff discussed in
Sec. III.1. In order to bridge the gap between the highest $E_{x}$ data point
and $\rho(B_{n})$, an extrapolation is performed using the Fermi Gas (upper
limit) Egidy  _et al._ (1988); von Egidy and Bucurescu (2006) and Constant
Temperature (lower limit) Ericson (1959); Gilbert and Cameron (1965) models.

In Fig. 3 (c) the NLDs of 153,155Sm are compared to the NLDs of
(p,d$\gamma)^{151,153}$Sm from Ref. Simon  _et al._ (2016) and are shown to be
in reasonable agreement. Note that the level density is decreasing
significantly from 153Sm to 155Sm. This will be discussed in Sec. V.

Table 1: Parameters used for extraction and normalization of $\rho(E_{x})$ and
$\mathcal{T}(E_{\gamma})$ in 153,155Sm. The temperature parameter $T_{CT}$ is
used for the CT model extrapolation to the $B_{n}$.

Isotope | $J_{\pi}$ | $B_{n}$ | $a$ | $E1$ | $T_{CT}$ | $\sigma(B_{n})_{RMI}$ | $D_{0}$ | $\rho_{RMI}(B_{n})$ | $\rho_{HFB}(B_{n})$ | $\langle\Gamma_{\gamma}(B_{n})\rangle$
---|---|---|---|---|---|---|---|---|---|---
|  | (MeV) | (MeV-1) | (MeV) | (MeV) |  | (eV) | ($10^{6}$MeV-1) | ($10^{6}$MeV-1) | (meV)
153Sm | $3/2^{+}$ | 5.868 | 18.5 | -0.66 | 0.57 | 6.0(6)a | 41.0(28)b | 1.77(36)a | 2.52(49) | 60.0c(37)d(110)e
155Sm | $3/2^{-}$ | 5.807 | 18.0 | -0.56 | 0.55 | 6.1(6)a | 112(15)b | 0.66(16)a | 0.80(17) | 74c(11)d(13)e

aCalculated with the rigid moment of inertia formula of von Egidy and
Bucurescu von Egidy and Bucurescu (2006).

bTaken from Ref. Mughabghab (2018).

cWeighted average, $\bar{x}$, calculated from the $x_{i}\pm\sigma_{i}$
resonances listed in Ref. Mughabghab (2018), using
$\bar{x}=\sum_{i=1}^{N}w_{i}x_{i}/\sum_{i=1}^{N}w_{i}$, where
$w_{i}={1}/{\sigma_{i}^{2}}$.

dUncertainty propagation calculated using
$\sigma_{\bar{x}}=\sqrt{1/\sum_{i=1}^{N}{w_{i}}}$.

eStandard deviation of the weighted average calculated using
$\sigma_{\bar{x},std}=\sqrt{N^{\prime}\sum_{i=1}^{N}w_{i}(x_{i}-\bar{x})^{2}/(N^{\prime}-1)\sum_{i=1}^{N}w_{i}}$,
where $N^{\prime}$ is the number of non-zero weights.

The (d,p$\gamma$) reaction may populate a limited spin range due to its low-
angular momentum transfer. This again influences the primary $\gamma$-ray
spectra $P(E_{x},E_{\gamma})$. A slope correction of the $\gamma$SF might
therefore be necessary in particular for sub-Coulomb barrier reactions
Guttormsen  _et al._ (2012, 2013, 2014); Tornyi  _et al._ (2014); Laplace  _et
al._ (2016); Zeiser  _et al._ (2019); Ingeberg  _et al._ (2020). To verify
whether such a correction is necessary, the $J^{\pi}$ distribution populated
by the (d,p$\gamma$) reaction has been investigated with the statistical
nuclear reaction code TALYS (v1.95) Koning and Rochman (2012) for the deuteron
absorption compound reaction formation assuming isotropic emission. The non-
elastic breakup, in which the neutron is absorbed by the target in a two-step
(direct deuteron breakup + neutron absorption) mechanism has been
investigated, as a function of the excitation energy of the residual nucleus
following the Green’s function transfer formalism of Refs. Potel  _et al._
(2015, 2017). The results indicate low non-elastic breakup cross sections
($\approx$ 2 - 3 mb/sr MeV) for $\theta_{lab}=126^{\circ}-140^{\circ}$, and
the deuteron-fusion proton-evaporation dominates, leading to a broad spin
distribution. Therefore, the discrepancy in the populated spin range is
considered small and no slope correction was performed.

Figure 4: (Color online) The experimental $\gamma$SF of 153Sm (a) and 155Sm
(b) compared to the 152,154Sm photo-nuclear reaction GEDR measurements of
Filipescu  _et al._ (2014), and $E$1 and $M$1 (n,$\gamma$)155Sm data measured
in ARC experiments Kopecky  _et al._ (2017). The $\gamma$SFs of 153Sm and
155Sm, which only include the systematic errors, are compared in panel (c).
The $\gamma$SFs of 151,153Sm from Ref. Simon  _et al._ (2016) are included for
comparison.

###  III.3 Normalization of the $\gamma$SFs

The $\gamma$-ray transmission coefficient, $\mathcal{T}_{XL}(E_{\gamma})$, of
multipolarity $L$ and electromagnetic character $X$ (electric, $E$, or
magnetic, $M$) is transformed to the total experimental $\gamma$SF through the
relationship Capote  _et al._ (2009):

$\centering\begin{split}f(E_{\gamma})&\approx
f_{E1}(E_{\gamma})+f_{M1}(E_{\gamma})\\\ &\approx\frac{1}{2\pi
E_{\gamma}^{3}}B[\mathcal{T}_{E1}(E_{\gamma})+\mathcal{T}_{M1}(E_{\gamma})],\end{split}\@add@centering$
(3)

assuming dominance of dipole transitions for statistical $\gamma$-ray decays.
This assumption is strongly supported by data, see e.g. Ref. Kopecky and
Chrien (1987). The absolute normalization parameter $B$ in Eq. (3) is
determined using the experimental average total radiative width
$\langle\varGamma_{\gamma}\rangle_{\ell=0}$ at $B_{n}$. The corresponding
parameters obtained and used for the normalization are summarized in Tab. 1.
The value of $\langle\varGamma_{\gamma}(B_{n})\rangle_{\ell=0}$ was obtained
by calculating the weighted average of the resonance widths listed in Ref.
Mughabghab (2018).

Using the total average radiative width, the parameter $B$ is determined by
Kopecky and Uhl (1990); Schiller  _et al._ (2000):

$\centering\begin{split}\langle{\Gamma_{\gamma}(B_{n})}\rangle=\frac{D_{0}B}{2\pi}&\int_{0}^{B_{n}}dE_{\gamma}\mathcal{T}(E_{\gamma})\times\rho(B_{n}-E_{\gamma})\cdot\\\
&\sum_{J=-1}^{J=1}g(B_{n}-E_{\gamma},J_{t}+J\pm
1/2),\end{split}\@add@centering$ (4)

where $J_{t}$ is the target spin (0 for 152,154Sm) and the function
$g(E_{x},J)$ is the relative probability of a given spin at excitation energy
$E_{x}$. In order to calculate the integral in Eq. (4), a log-linear function
for the $\gamma$SF is fitted for $\gamma$-energies between 0 and
$E_{\gamma,l}$ and from $E_{\gamma,h}\rightarrow B_{n}$, where $E_{\gamma,l}$
and $E_{\gamma,h}$ represent $\gamma$-ray energies for the lowest and highest
data point, respectively. For the RMI, a function approximating the spin
distribution is implemented Bethe (1936); Ericson (1959):

$\displaystyle
g(\sigma(E_{x}),J)=\frac{1}{2\sigma(E_{x})^{2}}(2J+1)\text{exp}\left[\frac{-(J+\frac{1}{2})^{2}}{2\sigma(E_{x})^{2}}\right],$
(5)

with

$\displaystyle\sigma^{2}(E_{x})=0.0146A^{5/3}\frac{1+\sqrt{1+4a(E_{x}-E_{1})}}{2a},$
(6)

where $a$ is the level density parameter, $E_{1}$ is the total back-shift
parameter and the spin-cutoff parameter, $\sigma(E_{x})$, is a modeled
variable related to the width of the distribution. However, in the HFB+comb
model the explicit probability for each spin is given, as illustrated in Fig.
2. Therefore, the relevant spin probabilities required for Eq. (4) are
directly obtained from the tables Koning and Rochman (2012), while correcting
for the excitation-energy shift used to normalize the NLDs.

The extracted experimental $\gamma$SFs of 153,155Sm are shown in Fig. 4. As in
the case of the NLDs, the $\gamma$SF-bands include both statistical and
systematic errors. Here, the systematic errors also take into account the
uncertainty of the $\langle\varGamma_{\gamma}\rangle$ parameter. The average
of the limits and the statistical errors are shown as data points within the
error bands. In Fig. 4, the 153,155Sm experimental $\gamma$SFs are compared to
the experimental Giant Electric Dipole Resonance (GEDR) data from
152,154Sm($\gamma$,n) photo-nuclear ($\gamma$-absorption) data Filipescu  _et
al._ (2014), as there is no GEDR data on 153,155Sm. The $\gamma$SFs from this
work appear steep in comparison to the available GEDR data, possible
explanations are discussed in Sec. V. The present data are further compared to
$E1$ and $M1$ (n,$\gamma$)155Sm data measured in average resonance capture
(ARC) experiments Kopecky  _et al._ (2017). The cross sections
$\sigma_{\gamma}(E_{\gamma})$ are transformed to $\gamma$SFs using the
relation Axel (1962):

$\centering
f(E_{\gamma})=\frac{1}{3\pi^{2}\hbar^{2}c^{2}}{\frac{\sigma_{\gamma}(E_{\gamma})}{E_{\gamma}}},\@add@centering$
(7)

where the factor $1/{3\pi^{2}\hbar^{2}c^{2}}$ = 8.674$\times 10^{-8}~{}{\rm
mb^{-1}MeV^{-2}}$.

Finally, in Fig. 4 (c), the $\gamma$SFs of 153Sm and 155Sm are compared to
each other as well as to the $\gamma$SFs of 151,153Sm from Ref. Simon  _et
al._ (2016). Several points of interest emerge from this comparison, such as
the absolute value difference between the 153Sm and 155Sm from the current
experiment. Previous measurements on close lying nuclei give consistently
similar absolute values for the $\gamma$SFs. This apparent deviation is
discussed in Sec. V. Comparing the results for 153Sm from Ref. Simon  _et al._
(2016) and from the present work there are clear differences. Given that both
data sets were analyzed using the Oslo method it is important to understand
the differing features. Firstly, there seems to be a large discrepancy at
$\gamma$-energies below 2 MeV, where the (p,d$\gamma)^{153}$Sm data indicate a
strong LEE, while the (d,p$\gamma)^{153}$Sm keeps trending downward. Secondly,
the SR appears to be significantly more pronounced in the
(p,d$\gamma)^{153}$Sm data. This will be explored below after the $B_{SR}(M1)$
strengths are extracted and compared.

##  IV The Scissors Resonance

Experiments using the Oslo method can only extract the SR built on excited
states in the quasi-continuum, whereas NRF measurements can only extract SR
built on the ground state. From the $\gamma$SF, the integrated reduced
transition strength for magnetic dipole transitions, $B_{SR}(M1)$, is obtained
by determining the shape of the resonance and numerically integrating over the
distribution using the Standard Lorentzian Function (SLo) Brink (1957); Axel
(1962); Capote  _et al._ (2009), in the energy range relevant to the SR:

$B_{SR}=\frac{27(\hbar c)^{3}}{16\pi}\int
f_{SR}^{SLo}(E_{\gamma})dE_{\gamma},$ (8)

where the factor $27(\hbar c)^{3}/{16\pi}=2.5980\times
10^{8}~{}{\mu_{N}^{2}}$MeV2.

Figure 5: (Color online) The extracted SR of 153Sm (a) and 155Sm (b), fitted
with the exponential background and SLo (see text for details). Table 2:
Experimental integrated $B_{SR}(M1)$ strength for samarium isotopes extracted
in the given energy ranges. The quadrupole deformation of the nuclear ground
state $\delta^{2}$ is taken from the FRDM12 database Möller  _et al._ (2016).
The quantity $\omega_{SR}$ denotes the energy centroid of the SR.

Isotope | Deformation | Energy range | $\omega_{SR}$ | $B_{SR}(M1)$ | Ref. | Technique
---|---|---|---|---|---|---
AX | $\delta^{2}$ | [MeV] | [MeV] | [$\mu^{2}_{N}$] |  |
153Sm | 0.26 | Full Rangea | 2.89(0) | 1.27 — 2.13 | Present Exp. | Oslo method
153Sm | 0.26 | 2.0 — 4.0 | 2.89(0) | 1.09 — 1.85 | Present Exp. | Oslo method
155Sm | 0.27 | Full Rangea | 2.98(8) | 4.40 — 6.44 | Present Exp. | Oslo method
155Sm | 0.27 | 2.0 — 4.0 | 2.98(8) | 3.58 — 5.30 | Present Exp. | Oslo method
144Sm | 0.08 | 2.0 — 4.0 | 3.97(4) | 0.28(0) | Ziegler et al. Ziegler  _et al._ (1990) | NRF
148Sm | 0.18 | 2.0 — 4.0 | 3.07(3) | 0.51(1) | Ziegler et al. Ziegler  _et al._ (1990) | NRF
150Sm | 0.21 | 2.0 — 4.0 | 3.13(3) | 0.97(10) | Ziegler et al. Ziegler  _et al._ (1990) | NRF
151Sm | 0.22 | 0.0 — 5.0 | 3.00(2) | 7.80(340) | Simon et al. Simon  _et al._ (2016) | Oslo method
152Sm | 0.24 | 2.0 — 4.0 | 2.99(3) | 2.35(20) | Ziegler et al. Ziegler  _et al._ (1990) | NRF
153Sm | 0.26 | 0.0 — 5.0 | 3.00(2) | 7.80(200) | Simon et al. Simon  _et al._ (2016) | Oslo method
154Sm | 0.27 | 2.0 — 4.0 | 3.20(3) | 2.65(30) | Ziegler et al. Ziegler  _et al._ (1990) | NRF

aEquation 8 was integrated from 0 to 20 MeV. This equates to an unrestricted
range, as including higher energies did not alter the tabulated value to a
significant digit.

Several fitting methods were explored, such as including the GEDR data and
fitting a collection of SLo peaks, or making a model of an exponential
background plus one SLo-peak. However, none of the methods converged
successfully and required many of the parameters to be fixed. Therefore a less
sophisticated, but transparent method was chosen. To extract the strength of
the SR an exponential function was fitted by two points in the experimental
$\gamma$SFs, to approximate the background in the vicinity of the resonance,
as shown in Fig. 5. It is deemed reasonable that the background is of
exponential shape, as all conventional empirical models for the GEDR are of
Lorentzian type, where the tail can be approximated by an exponential for a
small energy interval. This background was then extracted from the data. An
SLo was subsequently fitted to the extracted points, with statistical errors
as weights in the fit. A fit was performed on the upper and lower limit of the
$\gamma$SF, as well as for the average as shown in Fig. 5. The integrated
$B_{SR}(M1)$ was found numerically from Eq. (8), and the resulting strengths
are listed in Tab. 2. Due to the extraction method, a range of the SR strength
is given instead of a recommended value with uncertainties. The energy
centroid is consistent across the different fits and is also given with its
small uncertainties in Tab. 2.

##  V Discussion

Figure 6: (Color online) Comparison between 153Sm (solid lines) and 155Sm
(dashed lines) NLDs obtained within the combinatorial (red curves) Goriely
_et al._ (2008) and statistical, $\rho_{stat}$, (dark blue curves) Demetriou
and Goriely (2001) models. Also shown are the energy-dependent level density
parameters $a_{stat}$ (light blue curves) obtained within the microscopic
statistical approach Demetriou and Goriely (2001).

In the current experiments, the NLD and $\gamma$SF of the deformed even-odd
155Sm below $B_{n}$, were measured for the first time, as well as that of
153Sm in the energy range 4 $\leq E_{\gamma}\leq B_{n}$ MeV.

It is interesting to note in Fig. 3 (c) that even though the 153,155Sm NLDs
have similar slopes, the 153Sm level density is higher than that of 155Sm,
which is counter-intuitive. As more neutrons are added and deformation
increases, it may be expected that there should be more levels in 155Sm
compared to 153Sm. The same behavior has also been observed in neodymium
isotopes Ay  _et al._ (2016); Guttormsen  _et al._ (2020) where the lighter,
spherical or less deformed isotopes exhibit higher NLDs than the more
deformed, heavier isotopes. This phenomenon is consistent with the microscopic
description of the NLDs obtained within the combinatorial (HFB+combinatorial)
Goriely  _et al._ (2008) and statistical (HFBCS+statistical) Demetriou and
Goriely (2001) models which are compared to the experimental data for the
even-odd 153,155Sm isotopes, in Fig. 6.

The same feature of a lower NLD for 155Sm is seen to be predicted by both
microscopic models. Also shown in Fig. 6 are the energy-dependent level
density parameters $a_{stat}$ obtained within the microscopic statistical
approach Demetriou and Goriely (2001). The $a$ parameter is a measure of the
single-particle level density at the Fermi surface and is consequently
sensitive to shell and pairing effects. The difference between 153Sm and 155Sm
NLDs is essentially due to stronger shell plus pairing effects in 155Sm in
comparison with 153Sm leading to a smaller single-particle level density in
155Sm at the Fermi energy. This structure effect is also found in the ground-
state microscopic energy predicted by most macroscopic-microscopic mass models
Wender (2000); Möller  _et al._ (2016) which give a maximum microscopic energy
within the Sm neighboring isotopes for $A=148-150$ and a lower value for 155Sm
compared to 153Sm.

The 153,155Sm $\gamma$SFs show pronounced strength, which is localized at mean
excitation energies of about $\omega_{SR}\approx 3$ MeV, a feature observed in
most deformed rare-earth nuclei. The analytical technique used to extract the
$\gamma$SFs, the Oslo method Schiller  _et al._ (2000); Larsen  _et al._
(2011), cannot reveal fine structures in the $\gamma$-spectra. However, the
observed resonances at $\sim$ 3 MeV are believed to be due to the $M$1 SR,
which is consistent with other observations in this mass region. High
resolution measurements (see Ref. Heyde  _et al._ (2010) and references
therein) have shown that these resonances are due to $M$1 transitions between
high-$j$ orbitals and dominated by $J^{\pi}=1^{+}$ states when excited
directly from a $J^{\pi}=0^{+}$ ground state Richter (1990). The increase in
the integrated SR strength from 153Sm to 155Sm is not entirely consistent with
previous experimental findings and theoretical descriptions Heyde  _et al._
(2010). Considering that the deformation is comparable for the two nuclei the
strengths should be comparable as well. A possible explanation might be that
the reported calculated deformation for 155Sm is underestimated and therefore
an experiment to measure both 153,155Sm deformation is highly desirable.

In contrast to what was observed for the actinides Guttormsen  _et al._
(2012); Tornyi  _et al._ (2014); Laplace  _et al._ (2016), the SR in the odd-
even rare-earth 153,155Sm $\gamma$SF does not exhibit any double-hump
structure.

Fig. 7 compares the present integrated $B_{SR}(M1)$ with the experimental NRF
values and those of Ref. Simon  _et al._ (2016) extracted with the Oslo method
analysis. Given the comparable deformation parameters in Tab. 2, the
$B_{SR}(M1)$ for (d,p$\gamma)^{155}$Sm is in reasonable agreement with the
measurements of Ziegler et al. Ziegler  _et al._ (1990) on 152,154Sm when
extracted over the same energy region, while the $B_{SR}(M1)$ for
(d,p$\gamma)^{153}$Sm is lower than that of the neighbouring nuclei. Also the
$B_{SR}(M1)$ for (d,p$\gamma)^{155}$Sm is in excellent agreement with
measurements of the other rare-earth nuclei extracted with the Oslo Method
Voinov  _et al._ (2001); Melby  _et al._ (2001); Siem  _et al._ (2002);
Guttormsen  _et al._ (2003); Agvaanluvsan  _et al._ (2004); Nyhus  _et al._
(2010); Simon  _et al._ (2016); Renstrøm  _et al._ (2018). The results for
151,153Sm Simon  _et al._ (2016) (also listed in Tab. 2) are significantly
higher than those from both the 150,152,154Sm NRF values and the present
measurements.

Figure 7: (Color online) The experimental low-lying $M1$ strength,
$B_{SR}(M1)$ plotted against mass number $A$. The present measurements (lines)
extracted over a full energy range are compared to 151,153Sm data from Ref.
Simon  _et al._ (2016) (blue squares) and 150,152,154Sm from NRF measurements
of Ref. Ziegler  _et al._ (1990) (green stars).

This follows predictably from the discussion in Sec. (III.3) where the SR
appears much more pronounced in the $\gamma$SFs of (p,d$\gamma)^{151,153}$Sm
data. To explain this, and the LEE discrepancy, it is important to highlight
the differences in the analyses performed. One difference between how the two
data sets were analyzed is the region of extraction of the NLDs and
$\gamma$SFs from the primary $\gamma$-ray matrices, as the
(p,d$\gamma)^{153}$Sm data were extracted for $E_{\gamma}$ as low as 0.645 MeV
and from $E_{x}$ = 2.525 MeV to $E_{x}$ = 4.045 MeV. Depending on the
statistical nature of the nucleus at these energies, the resulting $\gamma$SFs
might be inconsistent. There might also be some residual transitions at low
$\gamma$-energies in the primary $\gamma$-ray matrices for the
(p,d$\gamma$)153Sm data that lead to the differences in the $\gamma$SFs below
2 MeV. Importantly, the approaches for extracting the SR are different, for
(p,d$\gamma)^{151,153}$Sm a fit for the energy region Eγ = 0 - 11 MeV is
performed. Lastly, the highest data points for the $\gamma$SFs of
(p,d$\gamma)^{153}$Sm have high uncertainties and are located at
$\gamma$-energies which could give the appearance of a large SR and depending
on the reliability of the last data points, the slopes could be in sharp
contrast.

In previous measurements, the $B_{SR}(M1)$ has proven to be less fragmented
and stronger in even-even nuclei than in even-odd nuclei Heyde  _et al._
(2010); Pietralla  _et al._ (1998). The strength seen in ($\gamma$,$\gamma$’)
experiment for odd-even cases can differ to a great extent. However, the
unobserved strength can be estimated from a fluctuation analysis of the data,
see Ref. Enders  _et al._ (1997). If this is taken into account, the same
accumulated strengths as in the even-even cases and the same dependence on
deformation is found. Different types of experiments and theoretical
calculations that extract the $B_{SR}(M1)$ over the same energy region, yield
similar strengths as shown for Dy isotopes Renstrøm  _et al._ (2018). The
present 155Sm measurement is in reasonable agreement whereas 153Sm appears to
contract these findings. When the extraction performed here is limited by the
same integration limits in Eq. (8), the 153Sm $B(M1)$ value is found to be
significantly lower (higher limit of 1.9 $\mu^{2}_{N}$) than measurements of
the other rare-earth nuclei extracted with the Oslo Method Renstrøm  _et al._
(2018) and that of the neighboring even-even deformed 154Sm isotope of Ref.
Ziegler  _et al._ (1990).

A steep increase in the $\gamma$SF of 155Sm above $E_{\gamma}>4$ MeV might be
an indication of a PDR. This excessive strength is also observed in the
$\gamma$SF of 153Sm, as shown in Fig. 5. However, it is not possible to
determine the electromagnetic nature of the resonance from Oslo method-type
experiments and therefore information from other experiments such as NRF
Kneissl  _et al._ (1996) or inelastic proton scattering measurements Tamii
_et al._ (2009); Neveling  _et al._ (2011) is crucial. The LEE is not seen in
the $\gamma$SFs of 153,155Sm for the energy range under investigation. This
may be due to the current experimental conditions, which limit the extraction
of useful data below $\sim$ 1 and 1.6 MeV, respectively. This is in contrast
to the (d,p$\gamma$)151,153Sm data shown in Fig. 4.

Figure 8: (Color online) Comparison between the D1M+QRPA+0lim Goriely  _et
al._ (2018, 2019) $E1$ (blue solid lines), $M1$ (red solid lines) and $E1+M1$
(black solid lines) with experimental data, a) for 153Sm and b) for 155Sm. The
present Oslo data including systematic uncertainties correspond to gray
squares and the ARC data to triangles (blue for $E1$ and red for $M1$). To
give a fair approximation of the $\gamma$SF in the GDR region, the $\gamma$SF
extracted from photoneutron cross section of the neighboring 154Sm is shown by
solid Filipescu  _et al._ (2014) and open triangles Carlos  _et al._ (1974).

Finally, we compare in Fig. 8 the D1M+QRPA+0lim $E1$, $M1$ and $E1+M1$
$\gamma$SF with available experimental data, i.e. the present Oslo data, ARC
data known separately for $E1$ and $M1$ strengths in 155Sm, and the $\gamma$SF
extracted from photoneutron cross section of the neighboring 154Sm isotope.
The D1M+QRPA+0lim model corresponds to axially deformed Gogny-HFB plus quasi-
particle random phase approximation (QRPA) predictions obtained with the D1M
interaction. The model has been complemented by phenomenological shell-model-
inspired $E1$ and $M1$ LEE contributions to describe the de-excitation
strength function Goriely  _et al._ (2018, 2019). As seen in Fig. 8, the
D1M+QRPA+0lim predictions are in rather good agreement with data, especially
for 153Sm. In the 155Sm case, the total calculated $\gamma$SF remains somewhat
lower than the one found with the Oslo method though the agreement with ARC
data is good. For both Sm isotopes, the large $\gamma$SF around 5 MeV cannot
be explained by the D1M+QRPA+0lim model. Due to the phenomenological inclusion
of an $M1$ LEE, the D1M+QRPA+0lim model can reproduce rather well the low-
energy points found by the Oslo method below and around 2 MeV.

##  VI Summary

The NLD and the $\gamma$SF of the deformed even-odd 153,155Sm have been
measured with the reaction (d,p$\gamma$) below $B_{n}$ at the Oslo Cyclotron
Laboratory (OCL) using the Oslo method. A pronounced resonance, the SR, was
observed for both nuclei. The SR integrated strengths, in 153,155Sm were
determined to be in the range 1.3 — 2.1 and 4.4 — 6.4 $\mu^{2}_{N}$,
respectively. These values are comparable to those of neighboring nuclei for
155Sm and somewhat smaller in 153Sm.

The experimental NLD of 153Sm is found by the Oslo method to be larger than
that of 155Sm, a counter-intuitive pattern that is confirmed by microscopic
models and explained by stronger pairing plus shell effects in 155Sm. QRPA
calculations based on the D1M Gogny interaction are also found to predict the
153,155Sm $\gamma$SF in fairly good agreement with the Oslo data, though the
large strength around 5 MeV is not be described by the model.

## Acknowledgments

The authors thank the cyclotron team at the University of Oslo for providing
high-quality experimental conditions. This work is based on the research
supported in part by the National Research Foundation of South Africa (Grant
No. 118846, 92600, 90741 and 92789) and by the IAEA under Research Contract
20454. This work was partially supported by the Fonds de la Recherche
Scientifique - FNRS, the Fonds Wetenschappelijk Onderzoek - Vlaanderen (FWO)
under the EOS Project No O022818F and the U.S. Department of Energy under
Contract DE-AC52-07NA27344. The authors gratefully acknowledge funding from
the Research Council of Norway (NFR) project grant No. 263030 (A.G., S.S.,
F.Z. and V.W.I.), project grant No. 262952 (G.M.T.) and by the Deutsche
Forschungsgemeinschaft (DFG) under Grant No. SFB 1245, project ID 279384907
(P.v.N.-C.). A.C.L. gratefully acknowledges support by the European Research
Council through ERC-STG-2014 under grant agreement No. 637686.

## References

## References

  * Richter (1990) A. Richter, Nuclear Physics A 507, 99 (1990).
  * Margraf  _et al._ (1993) J. Margraf, R. D. Heil, U. Kneissl, U. Maier, H. H. Pitz, H. Friedrichs, S. Lindenstruth, B. Schlitt, C. Wesselborg, P. von Brentano, R.-D. Herzberg, and A. Zilges, Physical Review C 47, 1474 (1993).
  * Goriely  _et al._ (2016) S. Goriely, S. Hilaire, S. Péru, M. Martini, I. Deloncle, and F. Lechaftois, Physical Review C 94 (2016), 10.1103/physrevc.94.044306.
  * Goriely and Plujko (2019) S. Goriely and V. Plujko, Physical Review C 99 (2019), 10.1103/physrevc.99.014303.
  * Iudice and Palumbo (1978) N. L. Iudice and F. Palumbo, Physical Review Letters 41, 1532 (1978).
  * Bohle  _et al._ (1984) D. Bohle, A. Richter, W. Steffen, A. Dieperink, N. L. Iudice, F. Palumbo, and O. Scholten, Physics Letters B 137, 27 (1984).
  * Enders  _et al._ (1997) J. Enders, N. Huxel, P. von Neumann-Cosel, and A. Richter, Physical Review Letters 79, 2010 (1997).
  * Heyde  _et al._ (2010) K. Heyde, P. von Neumann-Cosel, and A. Richter, Reviews of Modern Physics 82, 2365 (2010).
  * Linnemann  _et al._ (2003) A. Linnemann, P. von Brentano, J. Eberth, J. Enders, A. Fitzler, C. Fransen, E. Guliyev, R.-D. Herzberg, L. Käubler, A. Kuliev, P. von Neumann-Cosel, N. Pietralla, H. Prade, A. Richter, R. Schwengner, H. Thomas, D. Weisshaar, and I. Wiedenhöver, Physics Letters B 554, 15 (2003).
  * von Garrel  _et al._ (2006) H. von Garrel, P. von Brentano, C. Fransen, G. Friessner, N. Hollmann, J. Jolie, F. Käppeler, L. Käubler, U. Kneissl, C. Kohstall, L. Kostov, A. Linnemann, D. Mücher, N. Pietralla, H. H. Pitz, G. Rusev, M. Scheck, K. D. Schilling, C. Scholl, R. Schwengner, F. Stedile, S. Walter, V. Werner, and K. Wisshak, Physical Review C 73 (2006), 10.1103/physrevc.73.054315.
  * Margraf  _et al._ (1990) J. Margraf, A. Degener, H. Friedrichs, R. D. Heil, A. Jung, U. Kneissl, S. Lindenstruth, H. H. Pitz, H. Schacht, U. Seemann, R. Stock, C. Wesselborg, P. von Brentano, and A. Zilges, Physical Review C 42, 771 (1990).
  * Guttormsen  _et al._ (2012) M. Guttormsen, L. A. Bernstein, A. Bürger, A. Görgen, F. Gunsing, T. W. Hagen, A. C. Larsen, T. Renstrøm, S. Siem, M. Wiedeking, and J. N. Wilson, Physical Review Letters 109 (2012), 10.1103/physrevlett.109.162503.
  * Guttormsen  _et al._ (2013) M. Guttormsen, B. Jurado, J. N. Wilson, M. Aiche, L. A. Bernstein, Q. Ducasse, F. Giacoppo, A. Görgen, F. Gunsing, T. W. Hagen, A. C. Larsen, M. Lebois, B. Leniau, T. Renstrøm, S. J. Rose, S. Siem, T. Tornyi, G. M. Tveten, and M. Wiedeking, Physical Review C 88 (2013), 10.1103/physrevc.88.024307.
  * Guttormsen  _et al._ (2014) M. Guttormsen, L. A. Bernstein, A. Görgen, B. Jurado, S. Siem, M. Aiche, Q. Ducasse, F. Giacoppo, F. Gunsing, T. W. Hagen, A. C. Larsen, M. Lebois, B. Leniau, T. Renstrøm, S. J. Rose, T. G. Tornyi, G. M. Tveten, M. Wiedeking, and J. N. Wilson, Physical Review C 89 (2014), 10.1103/physrevc.89.014302.
  * Tornyi  _et al._ (2014) T. G. Tornyi, M. Guttormsen, T. K. Eriksen, A. Görgen, F. Giacoppo, T. W. Hagen, A. Krasznahorkay, A. C. Larsen, T. Renstrøm, S. J. Rose, S. Siem, and G. M. Tveten, Physical Review C 89 (2014), 10.1103/physrevc.89.044323.
  * Laplace  _et al._ (2016) T. A. Laplace, F. Zeiser, M. Guttormsen, A. C. Larsen, D. L. Bleuel, L. A. Bernstein, B. L. Goldblum, S. Siem, F. L. B. Garotte, J. A. Brown, L. C. Campo, T. K. Eriksen, F. Giacoppo, A. Görgen, K. Hadynska-Klek, R. A. Henderson, M. Klintefjord, M. Lebois, T. Renstrøm, S. J. Rose, E. Sahin, T. G. Tornyi, G. M. Tveten, A. Voinov, M. Wiedeking, J. N. Wilson, and W. Younes, Physical Review C 93 (2016), 10.1103/physrevc.93.014323.
  * Zeiser  _et al._ (2019) F. Zeiser, G. M. Tveten, G. Potel, A. C. Larsen, M. Guttormsen, T. A. Laplace, S. Siem, D. L. Bleuel, B. L. Goldblum, L. A. Bernstein, F. L. B. Garrote, L. C. Campo, T. K. Eriksen, A. Görgen, K. Hadynska-Klek, V. W. Ingeberg, J. E. Midtbø, E. Sahin, T. Tornyi, A. Voinov, M. Wiedeking, and J. Wilson, Physical Review C 100 (2019), 10.1103/physrevc.100.024305.
  * Voinov  _et al._ (2001) A. Voinov, M. Guttormsen, E. Melby, J. Rekstad, A. Schiller, and S. Siem, Physical Review C 63 (2001), 10.1103/physrevc.63.044313.
  * Melby  _et al._ (2001) E. Melby, M. Guttormsen, J. Rekstad, A. Schiller, S. Siem, and A. Voinov, Physical Review C 63 (2001), 10.1103/physrevc.63.044309.
  * Siem  _et al._ (2002) S. Siem, M. Guttormsen, K. Ingeberg, E. Melby, J. Rekstad, A. Schiller, and A. Voinov, Physical Review C 65 (2002), 10.1103/physrevc.65.044318.
  * Guttormsen  _et al._ (2003) M. Guttormsen, A. Bagheri, R. Chankova, J. Rekstad, S. Siem, A. Schiller, and A. Voinov, Physical Review C 68 (2003), 10.1103/physrevc.68.064306.
  * Krtička  _et al._ (2004) M. Krtička, F. Bečvář, J. Honzátko, I. Tomandl, M. Heil, F. Käppeler, R. Reifarth, F. Voss, and K. Wisshak, Physical Review Letters 92 (2004), 10.1103/physrevlett.92.172501.
  * Agvaanluvsan  _et al._ (2004) U. Agvaanluvsan, A. Schiller, J. A. Becker, L. A. Bernstein, P. E. Garrett, M. Guttormsen, G. E. Mitchell, J. Rekstad, S. Siem, A. Voinov, and W. Younes, Physical Review C 70 (2004), 10.1103/physrevc.70.054611.
  * Nyhus  _et al._ (2010) H. T. Nyhus, S. Siem, M. Guttormsen, A. C. Larsen, A. Bürger, N. U. H. Syed, G. M. Tveten, and A. Voinov, Physical Review C 81 (2010), 10.1103/physrevc.81.024325.
  * Baramsai  _et al._ (2015) B. Baramsai, F. Bečvář, T. Bredeweg, R. Haight, M. Jandel, J. Kroll, M. Krtička, G. Mitchell, J. O’Donnell, R. Rundberg, J. Ullmann, S. Valenta, and J. B. Wilhelmy, EPJ Web of Conferences 93, 01037 (2015).
  * Simon  _et al._ (2016) A. Simon, M. Guttormsen, A. C. Larsen, C. W. Beausang, P. Humby, J. T. Burke, R. J. Casperson, R. O. Hughes, T. J. Ross, J. M. Allmond, R. Chyzh, M. Dag, J. Koglin, E. McCleskey, M. McCleskey, S. Ota, and A. Saastamoinen, Physical Review C 93 (2016), 10.1103/physrevc.93.034303.
  * Renstrøm  _et al._ (2018) T. Renstrøm, H. Utsunomiya, H. T. Nyhus, A. C. Larsen, M. Guttormsen, G. M. Tveten, D. M. Filipescu, I. Gheorghe, S. Goriely, S. Hilaire, Y.-W. Lui, J. E. Midtbø, S. Péru, T. Shima, S. Siem, and O. Tesileanu, Physical Review C 98 (2018), 10.1103/physrevc.98.054310.
  * Kneissl  _et al._ (1996) U. Kneissl, H. Pitz, and A. Zilges, Progress in Particle and Nuclear Physics 37, 349 (1996).
  * Kopecky  _et al._ (2017) J. Kopecky, S. Goriely, S. Péru, S. Hilaire, and M. Martini, Physical Review C 95 (2017), 10.1103/physrevc.95.054317.
  * Schiller  _et al._ (2000) A. Schiller, L. Bergholt, M. Guttormsen, E. Melby, J. Rekstad, and S. Siem, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 447, 498 (2000).
  * Larsen  _et al._ (2011) A. C. Larsen, M. Guttormsen, M. Krtička, E. Běták, A. Bürger, A. Görgen, H. T. Nyhus, J. Rekstad, A. Schiller, S. Siem, H. K. Toft, G. M. Tveten, A. V. Voinov, and K. Wikan, Physical Review C 83 (2011), 10.1103/physrevc.83.034315.
  * Goriely  _et al._ (2019) S. Goriely, P. Dimitriou, M. Wiedeking, T. Belgya, R. Firestone, J. Kopecky, M. Krtička, V. Plujko, R. Schwengner, S. Siem, H. Utsunomiya, S. Hilaire, S. Péru, Y. S. Cho, D. M. Filipescu, N. Iwamoto, T. Kawano, V. Varlamov, and R. Xu, The European Physical Journal A 55 (2019), 10.1140/epja/i2019-12840-1.
  * Angell  _et al._ (2016) C. Angell, R. Hajima, T. Shizuma, B. Ludewigt, and B. Quiter, Physical Review Letters 117 (2016), 10.1103/physrevlett.117.142501.
  * Brits  _et al._ (2019) C. P. Brits, K. L. Malatji, M. Wiedeking, B. V. Kheswa, S. Goriely, F. L. B. Garrote, D. L. Bleuel, F. Giacoppo, A. Görgen, M. Guttormsen, K. Hadynska-Klek, T. W. Hagen, S. Hilaire, V. W. Ingeberg, H. Jia, M. Klintefjord, A. C. Larsen, S. N. T. Majola, P. Papka, S. Péru, B. Qi, T. Renstrøm, S. J. Rose, E. Sahin, S. Siem, G. M. Tveten, and F. Zeiser, Physical Review C 99 (2019), 10.1103/physrevc.99.054330.
  * Balbutsev  _et al._ (2018) E. B. Balbutsev, I. V. Molodtsova, and P. Schuck, Physical Review C 97 (2018), 10.1103/physrevc.97.044316.
  * Iachello (1981) F. Iachello, Nuclear Physics A 358, 89 (1981).
  * Pietralla  _et al._ (1998) N. Pietralla, P. von Brentano, R.-D. Herzberg, U. Kneissl, N. L. Iudice, H. Maser, H. H. Pitz, and A. Zilges, Physical Review C 58, 184 (1998).
  * Guttormsen  _et al._ (2011) M. Guttormsen, A. Bürger, T. Hansen, and N. Lietaer, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 648, 168 (2011).
  * Guttormsen  _et al._ (1990) M. Guttormsen, A. Atac, G. Løvhøiden, S. Messelt, T. Ramsøy, J. Rekstad, T. F. Thorsteinsen, T. S. Tveter, and Z. Zelazny, Physica Scripta T32, 54 (1990).
  * Guttormsen  _et al._ (2018) M. Guttormsen, F. Zeiser, J. E. Midtbø, Vetle W. Ingeberg, and Ann-Cecilie Larsen, “oslocyclotronlab/oslo-method-software: Oslo method v1.1.2,”  (2018).
  * Guttormsen  _et al._ (1996) M. Guttormsen, T. Tveter, L. Bergholt, F. Ingebretsen, and J. Rekstad, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 374, 371 (1996).
  * Campo  _et al._ (2016) L. C. Campo, F. L. B. Garrote, T. K. Eriksen, A. Görgen, M. Guttormsen, K. Hadynska-Klek, M. Klintefjord, A. C. Larsen, T. Renstrøm, E. Sahin, S. Siem, A. Springer, T. G. Tornyi, and G. M. Tveten, Physical Review C 94 (2016), 10.1103/physrevc.94.044321.
  * Guttormsen  _et al._ (1987) M. Guttormsen, T. Ramsøy, and J. Rekstad, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 255, 518 (1987).
  * Tveter  _et al._ (1996) T. S. Tveter, L. Bergholt, M. Guttormsen, E. Melby, and J. Rekstad, Physical Review Letters 77, 2404 (1996).
  * (45) J. E. Midtbø, F. Zeiser, E. Lima, A.-C. Larsen, G. M. Tveten, M. Guttormsen, F. L. B. Garrote, A. Kvellestad, and T. Renstrøm, 1904.13248 .
  * Brink (1957) D. Brink, Nuclear Physics 4, 215 (1957).
  * Axel (1962) P. Axel, Physical Review 126, 671 (1962).
  * Bohr and Mottelson (1969) A. Bohr and B. Mottelson,  _Nuclear Structure_ , Vol. 1 (New York, 1969).
  * Henden  _et al._ (1995) L. Henden, L. Bergholt, M. Guttormsen, J. Rekstad, and T. Tveter, Nuclear Physics A 589, 249 (1995).
  * NND (2019) _Data extracted from NuDat database on the National Nuclear Data Center_, Brookhaven National Laboratory, USA, as of September 2019 (2019).
  * Capote  _et al._ (2009) R. Capote, M. Herman, P. Obložinský, P. Young, S. Goriely, T. Belgya, A. Ignatyuk, A. Koning, S. Hilaire, V. Plujko, M. Avrigeanu, O. Bersillon, M. Chadwick, T. Fukahori, Z. Ge, Y. Han, S. Kailas, J. Kopecky, V. Maslov, G. Reffo, M. Sin, E. Soukhovitskii, and P. Talou, Nuclear Data Sheets 110, 3107 (2009).
  * von Egidy and Bucurescu (2006) T. von Egidy and D. Bucurescu, Physical Review C 73 (2006), 10.1103/physrevc.73.049901.
  * Goriely  _et al._ (2008) S. Goriely, S. Hilaire, and A. J. Koning, Physical Review C 78 (2008), 10.1103/physrevc.78.064307.
  * Egidy  _et al._ (1988) T. V. Egidy, H. Schmidt, and A. Behkami, Nuclear Physics A 481, 189 (1988).
  * Ericson (1959) T. Ericson, Nuclear Physics 11, 481 (1959).
  * Gilbert and Cameron (1965) A. Gilbert and A. G. W. Cameron, Canadian Journal of Physics 43, 1446 (1965).
  * Mughabghab (2018) S. F. Mughabghab, in  _Atlas of Neutron Resonances_ (Elsevier, 2018) pp. 89–822.
  * Ingeberg  _et al._ (2020) V. W. Ingeberg, S. Siem, M. Wiedeking, K. Sieja, D. L. Bleuel, C. P. Brits, T. D. Bucher, T. S. Dinoko, J. L. Easton, A. Görgen, M. Guttormsen, P. Jones, B. V. Kheswa, N. A. Khumalo, A. C. Larsen, E. A. Lawrie, J. J. Lawrie, S. N. T. Majola, K. L. Malatji, L. Makhathini, B. Maqabuka, D. Negi, S. P. Noncolela, P. Papka, E. Sahin, R. Schwengner, G. M. Tveten, F. Zeiser, and B. R. Zikhali, The European Physical Journal A 56 (2020), 10.1140/epja/s10050-020-00070-7.
  * Koning and Rochman (2012) A. Koning and D. Rochman, Nuclear Data Sheets 113, 2841 (2012).
  * Potel  _et al._ (2015) G. Potel, F. M. Nunes, and I. J. Thompson, Physical Review C 92 (2015), 10.1103/physrevc.92.034611.
  * Potel  _et al._ (2017) G. Potel, G. Perdikakis, B. V. Carlson, M. C. Atkinson, W. H. Dickhoff, J. E. Escher, M. S. Hussein, J. Lei, W. Li, A. O. Macchiavelli, A. M. Moro, F. M. Nunes, S. D. Pain, and J. Rotureau, The European Physical Journal A 53 (2017), 10.1140/epja/i2017-12371-9.
  * Filipescu  _et al._ (2014) D. M. Filipescu, I. Gheorghe, H. Utsunomiya, S. Goriely, T. Renstrøm, H.-T. Nyhus, O. Tesileanu, T. Glodariu, T. Shima, K. Takahisa, S. Miyamoto, Y.-W. Lui, S. Hilaire, S. Péru, M. Martini, and A. J. Koning, Physical Review C 90 (2014), 10.1103/physrevc.90.064616.
  * Kopecky and Chrien (1987) J. Kopecky and R. Chrien, Nuclear Physics A 468, 285 (1987).
  * Kopecky and Uhl (1990) J. Kopecky and M. Uhl, Physical Review C 41, 1941 (1990).
  * Bethe (1936) H. A. Bethe, Physical Review 50, 332 (1936).
  * Möller  _et al._ (2016) P. Möller, A. Sierk, T. Ichikawa, and H. Sagawa, At. Data Nucl. Data Tables 109-110, 1 (2016).
  * Ziegler  _et al._ (1990) W. Ziegler, C. Rangacharyulu, A. Richter, and C. Spieler, Physical Review Letters 65, 2515 (1990).
  * Demetriou and Goriely (2001) P. Demetriou and S. Goriely, Nuclear Physics A 695, 95 (2001).
  * Ay  _et al._ (2016) K. O. Ay, M. Ozgur, E. Algin, M. Guttormsen, F. L. B. Garrote, L. C. Campo, A. Görgen, T. W. Hagen, V. W. Ingeberg, B. V. Kheswa, M. Klintefjord, A. C. Larsen, J. E. Midtbo, V. Modamio, T. Renstrom, S. J. Rose, E. Sahin, S. Siem, G. M. Tveten, and F. Zeiser, Journal of Physics: Conference Series 766, 012027 (2016).
  * Guttormsen  _et al._ (2020) M. Guttormsen, Y. Alhassid, W. Ryssens, K. O. Ay, M. Ozgur, E. Algin, A. C. Larsen, F. L. B. Garrote, L. C. Campo, T. Dahl-Jacobsen, A. Görgen, T. W. Hagen, V. W. Ingeberg, B. V. Kheswa, M. Klintefjord, J. E. Midtbø, V. Modamio, T. Renstrøm, E. Sahin, S. Siem, G. M. Tveten, and F. Zeiser, “Strong enhancement of level densities in the crossover from spherical to deformed neodymium isotopes,” (2020), arXiv:2012.01902 [nucl-ex] .
  * Wender (2000) S. Wender, ed.,  _Proceedings of the 10th International Symposium on Capture Gamma-Ray Spectroscopy and Related Topics_ , Vol. 529 (AIP Conference Proceedings, 2000).
  * Tamii  _et al._ (2009) A. Tamii, Y. Fujita, H. Matsubara, T. Adachi, J. Carter, M. Dozono, H. Fujita, K. Fujita, H. Hashimoto, K. Hatanaka, T. Itahashi, M. Itoh, T. Kawabata, K. Nakanishi, S. Ninomiya, A. Perez-Cerdan, L. Popescu, B. Rubio, T. Saito, H. Sakaguchi, Y. Sakemi, Y. Sasamoto, Y. Shimbara, Y. Shimizu, F. Smit, Y. Tameshige, M. Yosoi, and J. Zenhiro, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 605, 326 (2009).
  * Neveling  _et al._ (2011) R. Neveling, H. Fujita, F. Smit, T. Adachi, G. Berg, E. Buthelezi, J. Carter, J. Conradie, M. Couder, R. Fearick, S. Förtsch, D. Fourie, Y. Fujita, J. Görres, K. Hatanaka, M. Jingo, A. Krumbholz, C. Kureba, J. Mira, S. Murray, P. von Neumann-Cosel, S. O'Brien, P. Papka, I. Poltoratska, A. Richter, E. Sideras-Haddad, J. Swartz, A. Tamii, I. Usman, and J. van Zyl, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 654, 29 (2011).
  * Goriely  _et al._ (2018) S. Goriely, S. Hilaire, S. Péru, and K. Sieja, Physical Review C 98 (2018), 10.1103/physrevc.98.014327.
  * Carlos  _et al._ (1974) P. Carlos, H. Beil, R. Bergère, A. Leprêtre, A. D. Miniac, and A. Veyssière, Nuclear Physics A 225, 171 (1974).