††thanks: Present address: National Superconducting Cyclotron Laboratory,
Michigan State University, East Lansing, Michigan 48824, USA

# Evolution of the $\gamma$-ray strength function in neodymium isotopes

M. Guttormsen<EMAIL_ADDRESS>Department of Physics, University
of Oslo, N-0316 Oslo, Norway  K. O. Ay  Department of Physics, Eskisehir
Osmangazi University, Faculty of Science and Letters, TR-26040 Eskisehir,
Turkey  M. Ozgur  Department of Physics, Eskisehir Osmangazi University,
Faculty of Science and Letters, TR-26040 Eskisehir, Turkey  E. Algin
Department of Physics, Eskisehir Osmangazi University, Faculty of Science and
Letters, TR-26040 Eskisehir, Turkey  Department of Metallurgical and Materials
Engineering, Pamukkale University, 20160 Denizli, Turkey  A. C. Larsen  F. L.
Bello Garrote  H. C. Berg  L. Crespo Campo  T. Dahl-Jacobsen  F. W. Furmyr  D.
Gjestvang  A. Görgen  T. W. Hagen  V. W. Ingeberg  Department of Physics,
University of Oslo, N-0316 Oslo, Norway  B. V. Kheswa  Department of Physics,
University of Oslo, N-0316 Oslo, Norway  Department of Physics, University of
Johannesburg, P.O. Box 524, Auckland Park 2006, South Africa  I. K. B.
Kullmann  Institut d’Astronomie et d’Astrophysique, CP-226, Université Libre
de Bruxelles, 1050 Brussels, Belgium  M. Klintefjord  Department of Physics,
University of Oslo, N-0316 Oslo, Norway  M. Markova  J. E. Midtbø  V. Modamio
Department of Physics, University of Oslo, N-0316 Oslo, Norway  W. Paulsen  L.
G. Pedersen  T. Renstrøm  E. Sahin  S. Siem  Department of Physics, University
of Oslo, N-0316 Oslo, Norway  G. M. Tveten  Department of Physics, University
of Oslo, N-0316 Oslo, Norway  M. Wiedeking  SSC Laboratory, iThemba LABS, P.O.
Box 722, Somerset West 7129, South Africa  School of Physics, University of
the Witwatersrand, Johannesburg 2050, South Africa

###### Abstract

The experimental $\gamma$-ray strength functions ($\gamma$SFs) of
142,144-151Nd have been studied for $\gamma$-ray energies up to the neutron
separation energy. The results represent a unique set of $\gamma$SFs for an
isotopic chain with increasing nuclear deformation. The data reveal how the
low-energy enhancement, the scissors mode and the pygmy dipole resonance
evolve with nuclear deformation and mass number. The data indicate that the
mechanisms behind the low-energy enhancement and the scissors mode are
decoupled from each other.

Gamma-ray strength function, resonances, spin distributions, Oslo method

††preprint: APS/123-QED

##  I Introduction

The $\gamma$-ray strength function ($\gamma$SF) is a fruitful concept in the
nuclear quasicontinuum region, describing the average reduced $\gamma$-ray
transition probabilities between groups of levels. The definition was
established by Bartholomew et al. Bartholomew1973 , connecting the $\gamma$SF
directly to the average partial $\gamma$-ray width of the initial levels. For
$\gamma$ decay (de-excitation), the $\gamma$SF is given by

$f_{\rm
XL}(E_{\gamma},E_{i},J_{i}^{\pi_{i}})=\frac{\langle\Gamma(E_{i},J_{i}^{\pi_{i}},E_{\gamma})\rangle}{E_{\gamma}^{2L+1}}\rho(E_{i},J_{i}^{\pi_{i}}),$
(1)

where the initial and final spin/parity obey the selection rules for
transitions of type $X$ and multipolarity $L$. The average $\gamma$ energy is
given by $E_{\gamma}=E_{i}-E_{f}$, and $\rho$ is the level density at the
initial excitation energy $E_{i}$ with spin/parity $J_{i}^{\pi_{i}}$.

There are many experimental methods to determine the $\gamma$SF. Recently,
Goriely et al. Goriely2019  have summarized the various techniques, and have
compiled experimental $\gamma$SF results in a database hosted at the IAEA
IAEA2020 .

Figure 1: (Color online) Proton-$\gamma$ coincidence matrices obtained in the
146Nd($d,p\gamma$)147Nd reaction with deuteron energies of 13.5 MeV. The (a)
raw, (b) unfolded, and (c) primary matrices are shown for initial excitation
energies $E_{i}$ between the ground level and the neutron binding energy
$S_{n}$. The primary $P(E_{\gamma},E_{i})$ matrix of panel (c) represents the
starting point for the Oslo method. We also show the $Q(E_{\gamma},E_{f})$
matrix of panel (d), which reveals the intensities of $\gamma$-ray transitions
populating final excitation energies $E_{f}$. All matrices have $(x,y)$-pixel
sizes of $(28,31)$ keV.

The $\gamma$SF is composed of several collective modes where the dominant
component is the giant electric dipole resonance (GDR) centered typically
around 15 MeV of $\gamma$-ray energy. The $\gamma$SF below the neutron
separation energy ($S_{n}\approx 5-10$ MeV) represents the low-energy part of
the GDR. As the GDR strength is $10-100$ times lower for these $\gamma$-ray
energies as compared to its maximum, other smaller structures appear, such as
the low-energy enhancement (LEE), the scissors mode (SM) and the pygmy dipole
resonance (PDR). The magnetic spin-flip resonance is also expected at these
energies, but with a negligible strength Kopecky2018 ; IAEA2020 .

Interpreting several of the $\gamma$SF structures has been a long-standing
problem. Of particular interest is the nature of the LEE and SM structures and
their dependence on deformation. It has been speculated schwengner2017 ;
naqvi2019 ; frauendorf2022  that these two (presumably) M1 structures are in
some way connected and add up to an integrated strength, which is relatively
independent on deformation and mass number. One of the experimental challenges
is to measure the LEE $\gamma$SF down to the lowest $\gamma$-ray energies.
Furthermore, the impact and interpretation of the PDR as due to neutron skin
oscillations outside the $N=Z$ core are of utmost interest savran2013 . The
resonance energy of the PDR is close to $S_{n}$, which is the upper energy
limit for the conventional Oslo method and where photonuclear ($\gamma,n)$
cross sections might be an unreliable probe of the $\gamma$SF.

The chain of stable neodymium isotopes covers nuclei from almost-spherical to
well-deformed shapes. In the present work, we have measured the $\gamma$SFs of
142,144-151Nd using particle-$\gamma$ coincidences from light-ion reactions on
stable neodymium targets. The data were analyzed within the framework of the
Oslo method Gut87 ; Gut96 ; Schiller00  and the recently developed Shape
method Wiedeking2021 . By also exploiting experimental results from other
measurements, the evolution of the $\gamma$SF in the energy region of
$E_{\gamma}\approx 0-17$ MeV and the interplay between its various components
are discussed as function of deformation and mass number.

Since both experimental and theoretical studies of the NLD in the neodymium
isotopes have been recently published Gut2021 , we focus mainly on the
$\gamma$SF in the present work.

The manuscript is outlined as follows. Section II describes the experiments at
the Oslo Cyclotron Laboratory and the Oslo method. In Sect. III the spin
distributions of the two applied reactions are investigated with the help of
$\gamma$-ray sidefeeding intensities into the rotational ground band of 150Nd
and by means of the Shape method. The experimental $\gamma$SF results are
presented in Sect. IV together with model fitting to the experimental data.
Summary and conclusions are given in Sect. V.

##  II Experiments and the Oslo method

The chain of 142,144-151Nd isotopes was studied with light-ion reactions at
the Oslo Cyclotron Laboratory. The targets were self-supporting metallic foils
of 142,144,146,148,150Nd with thicknesses of $\approx 2$ mg/cm2 and
enrichments of $\approx 97$%. The targets were bombarded with proton and
deuteron beams of energies 16.0 MeV and 13.5 MeV, respectively.

The SiRi particle-telescope system siri  was applied to determine the outgoing
particle type and energy. The 64 particle telescopes were located $\approx$ 5
cm from the target in eight angles between $126^{\circ}$ and $140^{\circ}$
with respect to the beam direction. The front ($\Delta E$) and back ($E$)
detectors had thicknesses of 130 and 1550 $\mu$m, respectively. The total
particle energy ($E+\Delta E$) resolution was $\approx$ 150 keV (FWHM).

During the neodymium experimental campaign, the $\gamma$-detector array CACTUS
CACTUS  was replaced by the OSCAR array Zeiser2021 ; Goergen2021 . The CACTUS
array is equipped with 26 NaI(Tl) 5” $\times$ 5” collimated scintillator
detectors at a distance of 22 cm from the target. This $\gamma$-detector array
was used for the 144,148,150Nd$(d,p)$ reactions111We also analyzed the
$(d,d^{\prime})$ reaction, but this channel had too low of a cross section at
high excitation energies.. The new OSCAR array consists of LaBr3(Ce)
scintillators characterized by high efficiency combined with excellent timing
and energy resolution. In the 142,146Nd$(p,p^{\prime})$ and 146Nd$(d,p)$
reactions a total of 15 LaBr3(Ce) detectors were placed 22 cm from the target,
and in the 144,148,150Nd$(p,p^{\prime})$ reactions, 30 detectors at 16 cm were
used.

The Oslo method allows a simultaneous extraction of NLD and $\gamma$SF from
the same particle-$\gamma$ coincidence data set. The first task is to sort the
coincidence events into $\gamma$ spectra for each initial excitation energy
$E_{i}$ of the residual nucleus. The $E_{i}$ value is calculated event-by-
event from the detected energy of the charged ejectile and the reaction
kinematics. Figure 1(a) shows this raw $(E_{\gamma},E_{i})$ matrix of 147Nd.
Each $\gamma$ spectrum is then unfolded using the known detector response
functions for CACTUS or OSCAR Gut96 ; om2022 . The matrix obtained from the
unfolding procedure Gut96  is displayed in Fig. 1(b), which presents all
$\gamma$ cascades from each $E_{i}$. The $\gamma$ multiplicity of these
statistical transitions is typically $\approx 1-4$, which can be calculated
from Gut87

$M_{\gamma}(E_{i})=\frac{E_{i}}{\langle E_{\gamma}(E_{i})\rangle},\\\ $ (2)

where $\langle E_{\gamma}\rangle$ is the average $\gamma$-ray energy of the
$\gamma$ spectrum for an initial excitation energy $E_{i}$.

Figure 2: (Color online) Probability densities of primary $\gamma$ spectra
(crosses) from various initial excitation energies $E_{i}$ in 147Nd. The
spectra are compared to the product
$\rho(E_{i}-E_{\gamma}){\mathcal{T}}(E_{\gamma})$ (blue histogram). The
statistical error bars are less than the data crosses. Both the $\gamma$ and
excitation energy dispersions are 124 keV/ch.

An important step of the Oslo method is to obtain a reliable first-generation
(primary) $\gamma$-ray matrix. The construction of this matrix is based on an
iterative subtraction technique Gut87  which separates the energy distribution
of the first emitted $\gamma$ rays from the distribution of higher-generation
$\gamma$ rays.

Let $u_{E_{i}}(E_{\gamma})$ be the unfolded $\gamma$ spectrum measured at the
initial excitation energy $E_{i}$, as shown in Fig. 1(b). Then, the primary
spectrum at initial excitation energy $E_{i}$ can be obtained by subtracting a
sum of weighted $u_{E_{i}^{\prime}}(E_{\gamma})$ spectra from lower excitation
energies

$p_{E_{i}}(E_{\gamma})=u_{E_{i}}(E_{\gamma})-\sum_{E_{i}^{\prime}<E_{i}}w_{E_{i}}(E_{i}^{\prime})u_{E_{i}^{\prime}}(E_{\gamma}),$
(3)

where the weighting coefficients $w$ and first-generation spectrum $p$ are
determined by an iterative procedure as described in Ref. Gut87 . After a few
iterations, the multiplicity of the primary spectrum is found to be
$M_{\gamma}\approx 1$, which is compared to the higher multiplicity of the $u$
spectrum of the average total cascades. The reason for the fast convergence is
the close relationship between $p$ and $w$. At a given initial excitation
energy $E_{i}$, the functional form of $p_{E_{i}}(E_{\gamma})$ should end up
by having the same functional form as $w_{E_{i}}(E_{i}^{\prime})$ when taken
as function of $E_{i}-E_{i}^{\prime}$, which represents the argument
$E_{\gamma}$ of $p$.

Figure 3: Level densities of 147Nd. The open and filled square data points
show the results of the Oslo method applied with a total ($\rho_{\rm tot}$)
and reduced ($\rho_{\rm exp}$) level density, respectively, having $\rho_{\rm
exp}(S_{n})=0.11\rho_{\rm tot}(S_{n})$. The data points of the two level
densities are connected to $\rho(S_{n})$ (shown as diamonds) with a constant
temperature (CT) model. The solid line shows the level density of known levels
NNDC .

The extracted primary $P(E_{\gamma},E_{i}$) matrix for the
146Nd($d,p\gamma$)147Nd reaction is shown in Fig. 1(c). Compared to the matrix
of cascades in panel (b), we see that the intensities of $\gamma$ rays with
energies at the initial excitation energy $E_{i}$ and slightly below, are
about the same since they represent primary transitions in both matrices. On
the other side, we see that the intensities of lower-energy $\gamma$ rays at
high $E_{i}$ are strongly suppressed in the primary matrix as they represent
higher-generation $\gamma$ rays. For illustration, in Fig. 1(d), we have also
constructed the intensities of $\gamma$ rays feeding final excitation energies
$E_{f}$ given by $Q(E_{\gamma},E_{f})=P(E_{\gamma},E_{i}-E_{\gamma})$. The $Q$
matrix shows that $\gamma$ rays below 3 MeV decay mainly directly to four to
five low-lying levels at $E_{f}<200$ keV. The decay to these levels are also
recognized as the lowest diagonal in the $P$ matrix.

In order to extract NLD and $\gamma$SF, we first normalize the primary spectra
to unity by $\sum_{E_{\gamma}}P(E_{\gamma},E_{i})=1$. We then factorize $P$ by
Schiller00

$P(E_{\gamma},E_{i})\propto\rho(E_{i}-E_{\gamma}){\cal{T}}(E_{\gamma}),\ $ (4)

where we assume that the decay probability is proportional to the NLD at the
final energy $\rho(E_{i}-E_{\gamma})$ according to Fermi’s golden rule dirac ;
fermi . The decay is also proportional to the $\gamma$-ray transmission
coefficient ${\cal{T}}$, which is assumed to be independent of excitation
energy according to the Brink hypothesis brink ; guttormsen2016 .

Provided that relation (4) holds, we extract the one-dimensional vectors
$\rho$ and ${\cal{T}}$ from the two-dimensional $P$ matrix using the least
$\chi^{2}$ fit iteration procedure of Schiller et al. Schiller00 . For a
successful fit we adopt a part of the $P$ matrix where the decay can be
considered statistical. A guide to this qualitative statement is to monitor at
what energy region the first-generation method works well, meaning that we
obtain the expected multiplicity of $M_{\gamma}\approx 1$. In the case of
147Nd in Fig. 2, the energy region of 3.0 MeV $<E_{i}<$ 5.4 MeV with
$E_{\gamma}>0.5$ MeV works well as confirmed by the agreement between
experimental data points and the results (blue histogram) using the fit
functions $\rho$ and ${\cal T}$. The figure includes six out of the 21
available $\gamma$ spectra that have been used to determine $\rho$ and
${\cal{T}}$ in the fitting procedure.

With the transmission coefficient ${\cal{T}}$ in hand, we obtain the
$\gamma$SF by kopecky1990

$f(E_{\gamma})=\frac{1}{2\pi}\frac{{\cal{T}}(E_{\gamma})}{E_{\gamma}^{3}},$
(5)

as dipole transitions dominate the decay in the quasicontinuum Capote09 .

The local variation of data points of $\rho$ and $\cal{T}$ are uniquely
determined through the fit, but the scale and slope of these functions are
still undetermined. It has been shown that transformations of the type
Schiller00

$\displaystyle\tilde{\rho}(E-E_{\gamma})$ $\displaystyle=$ $\displaystyle
A\exp[\alpha(E-E_{\gamma})]\,\rho(E-E_{\gamma}),$ (6)
$\displaystyle\tilde{{\mathcal{T}}}(E_{\gamma})$ $\displaystyle=$
$\displaystyle B\exp(\alpha E_{\gamma}){\mathcal{T}}(E_{\gamma}),$ (7)

give identical fits to the primary $\gamma$ spectra. Therefore, $A$, $\alpha$
and $B$ are parameters that have to be determined from other experimental data
or systematics.

Table 1: The quadrupole deformation $\beta_{2}$ and parameters for extracting
experimental NLD and $\gamma$SF.

Nucleus | $\beta_{2}$ | $T_{\rm CT}$ | $E_{d}$ | $\sigma_{d}$ | $S_{n}$ | $\sigma(S_{n})$ | $D_{0}$ | $\rho(S_{n})$ | $\langle\Gamma_{\gamma}\rangle$
---|---|---|---|---|---|---|---|---|---
|  | (MeV) | (MeV) |  | (MeV) | RMI | (eV) | (106MeV${}^{-1})$ | (meV)
142Nd | 0.092(2) | 0.65(5) | 2.5 | 3.0 | 9.828 | 6.6 | 19(4)a | 1.23(35)b | 77(20)c
144Nd | 0.125(2) | 0.63(3) | 2.5 | 2.8 | 7.817 | 6.3 | 37.6(21) | 0.32(5) | 74.2(18)
145Nd | 0.138(5)d | 0.59(3) | 1.3 | 2.9 | 5.755 | 5.9 | 450(50) | 0.16(4) | 51(4)
146Nd | 0.151(2) | 0.62(3) | 1.5 | 2.6 | 7.565 | 6.2 | 17.8(7) | 0.67(11) | 74(3)
147Nd | 0.176(5)d | 0.57(3) | 0.5 | 2.0 | 5.292 | 5.8 | 346(50) | 0.20(5) | 54(4)
148Nd | 0.200(2) | 0.59(3) | 1.4 | 2.5 | 7.333 | 6.1 | 5.9(11) | 2.4(6) | 68.8(60)
149Nd | 0.242(5)d | 0.54(3) | 0.5 | 2.3 | 5.039 | 5.8 | 165(14) | 0.42(9) | 45(3)
150Nd | 0.283(2) | 0.61(4) | 1.2 | 2.9 | 7.376 | 6.2 | 3.0(10)a | 4.8(18)b | 70(20)c
151Nd | 0.314(10)d | 0.54(3) | 0.4 | 2.6 | 5.335 | 6.0 | 169(11) | 0.43(9) | 67(25)

aAdjusted to reproduce $\rho(S_{n})$. bEstimated from systematics Gut2021 .
cEstimated from 144,146,148Nd. dInterpolated between even-mass neighbors.

At low excitation energies, we normalize the NLD to known discrete levels NNDC
. At high excitation energies, we use the measured average neutron $s$-wave
resonance spacing $D_{0}$ MugAtlas  at the neutron separation energy $S_{n}$.
To convert the measured $D_{0}$ to the total level density, we insert
$E=S_{n}$ into the spin distribution Ericson59

$g(E,J)\simeq\frac{2J+1}{2\sigma^{2}(E)}\exp\left[-(J+1/2)^{2}/2\sigma^{2}(E)\right],$
(8)

where $J$ is the spin quantum number. The function of the spin cutoff
parameter is given by Capote09

$\sigma^{2}(E)=\sigma_{d}^{2}+\frac{\sigma^{2}(S_{n})-\sigma_{d}^{2}}{S_{n}-E_{d}}\left(E-E_{d}\right),$
(9)

where $\sigma_{d}^{2}$ is determined from known discrete levels at low
excitation energy $E=E_{d}$ and $\sigma^{2}(S_{n})$ is determined from the
rigid-body moment of inertia (RMI) estimate, as shown in our previous work
Gut2021 .

Table 1 lists the quadrupole deformations and parameters needed for extracting
$\rho(S_{n})$. The $\beta_{2}$ values of the even-mass isotopes are taken from
the compilation of Pritychenko et al. pritychenko2016 . For the odd-mass
isotopes, we assume a deformation that is the average of their even-mass
neighbors. The table also includes the temperature $T_{\rm CT}$ extracted by a
$\chi^{2}$ fit of the constant-temperature (CT) formula Ericson59

$\rho_{\text{CT}}(E)=(1/T_{\rm CT})\exp{[(E-E_{0})/T_{\text{CT}}]}$ (10)

to the experimental high-energy data points and the predicted $\rho(S_{n})$
value of Table 1. Such a fit is shown in Fig. 3 where $\rho_{\text{CT}}$ was
fitted to the data points in the excitation region of $E=2.7-4.1$ MeV. The
energy shift parameter is given by
$E_{0}=S_{n}-T_{\text{CT}}\ln[T_{\text{CT}}\;\rho(S_{n})]$. Further details on
the extraction of NLDs in the neodymiums are given in Ref. Gut2021 .

The last column of Table 1 lists the average $\gamma$ widths for $\ell=0$
neutron capture reactions compiled in Ref. MugAtlas . These quantities are
exploited to determine the scaling of ${\cal{T}}$ (parameter $B$ of Eq. (7))
by reproducing the average, experimental $\gamma$-decay width Schiller00 ;
voin1

$\displaystyle\langle\Gamma_{\gamma}(S_{n})\rangle=\frac{1}{2\pi\rho(S_{n},J_{i},\pi)}\sum_{J_{f}}$
$\displaystyle\int_{0}^{S_{n}}{\mathrm{d}}E_{\gamma}{\cal{T}}(E_{\gamma})$
(11) $\displaystyle\times\rho(S_{n}-E_{\gamma},J_{f}),$

where the summation and integration run over all final levels with spin
$J_{f}$ that are accessible from initial spin $J_{i}$ by $E1$ or $M1$
transitions with energy $E_{\gamma}$. The Oslo method has been extensively
tested and discussed by Larsen et al. Lars11 . The Oslo method software is
available on the Oslo Cyclotron Laboratory GitHub om2022 .

##  III Spin distributions of the applied reactions

Before we can extract the $\gamma$SFs from the first-generation matrix, we
have to consider the spin distribution populated in these light-ion reactions
compared to the intrinsic spin distribution of the heavy nuclei studied. Table
1 shows typically $\sigma(S_{n})\approx 6$, which represents (see Eq. (8)) an
average spin of $\langle J\rangle\approx 7$ with a maximum of $J\approx
15-20$. This is a significantly larger spin distribution than expected from
the applied $(p,p^{\prime})$ and $(d,p)$ reactions at low beam energies222For
previous helium-induced reactions on lighter nuclei, the populated spin
distribution was much closer to the real intrinsic spin distribution, and
corrections were not necessary to perform..

In the following, we aim to estimate experimentally the ratio

$\eta=\frac{\rho_{\rm exp}(S_{n})}{\rho_{\rm tot}(S_{n})}$ (12)

between the level density populated in the reaction and the total, intrinsic
level density333By intrinsic level density we mean all available levels within
an excitation energy bin (independent on the nuclear reaction applied). given
in Table 1. With this reduction factor, we can estimate the experimental level
density $\rho_{\rm exp}(E)$ for excitation energies up to the neutron
separation energy $S_{n}$, as demonstrated in Fig. 3 for 147Nd by the filled
black data points.

It is obvious that the observed level density has a less steep slope compared
to that without a reduction of available spins. Since the observed
$P(E_{\gamma},E_{i})$ matrix represents the experimental spin range,
$\rho_{\rm exp}(E)$ must be adopted in Eq. (4). By replacing $\rho_{\rm
tot}(E)$ with $\rho_{\rm exp}(E)$, the slope of ${\cal{T}}(E_{\gamma})$ will
correspondingly change to fit the observed $P(E_{\gamma},E_{i})$ landscape. As
seen from Eqs. (6) and (7), a less steep slope of $\rho$ will induce a less
steep slope of ${\cal{T}}$ as well.

In the following, we adopt two techniques to estimate $\eta$, namely (i) the
sidefeeding into the rotational ground-state band for well-deformed nuclei and
(ii) the recently developed Shape method Wiedeking2021 . Only 150Nd works for
the sidefeeding method, whereas the Shape method may be used for all nuclei if
the final levels are known and experimentally separable.

Figure 4: (Color online) Spin distributions in 150Nd. Panel (a) shows the
intrinsic spin distribution (solid curve) with a spin cutoff parameter of
$\sigma=6.2$. The experimental data points are the $S(J)$ values obtained by
the evaluated sidefeeding, which are fitted to a spin distribution (dashed
curve) with $\sigma=2.9$. Panel (b) shows the experimental spin distribution
(dashed curve) with $\sigma=2.9$ normalized to the lowest spin of the
intrinsic spin distribution. The dashed-dotted curve (arbitrary units) shows
the ratio between the experimental and total spin distributions, and will be
referred to as the average probability $p^{\rm level}(E_{i},J_{i})$ of
populating individual levels of spin $J_{i}$.

###  III.1 Spin distribution of the $(p,p^{\prime})$ reaction

The 150Nd isotope is a well-behaving rotor with a quadrupole deformation of
$\beta_{2}=0.283$ pritychenko2016 . With an initial excitation energy gate of
$E_{i}=7.2-7.8$ MeV, we have evaluated the efficiency-corrected ground-state
band $\gamma$-ray intensities $I_{\gamma}(J\rightarrow J-2)$ and from these
values estimated the sidefeeding of spin $J$ from the quasicontinuum by

$S(J)=I_{\gamma}(J\rightarrow J-2)-I_{\gamma}(J+2\rightarrow J).$ (13)

The intensities in the ground-state band fade exponentially with spin, and the
highest transition found was the $10^{+}\rightarrow 8^{+}$ 468.9-keV
$\gamma$-ray line. Assuming that this $10^{+}$ level also collects the decay
from higher spins, we fit the sidefeeding spin distribution to Eq. (8) with
the spin cutoff parameter as a free parameter. The fit result for the
experimental data is shown in Fig. 4(a), with $\sigma_{\rm exp}=2.9(2)$. For
comparison, also the intrinsic spin distribution with $\sigma_{\rm tot}=6.2$
is displayed as a solid curve.

The two spin distributions $g(S_{n},J)_{\rm{tot}}$ and $g(S_{n},J)_{\rm{exp}}$
of Fig. 4(a) are normalized to unity. However, with the assumption that there
are no spin reduction for the lowest spins, we find the level density
reduction factor by

$\eta=\frac{g(S_{n},J=0)_{\rm{tot}}}{g(S_{n},J=0)_{\rm{exp}}}=0.22(2).$ (14)

The function $\eta g(S_{n},J)_{\rm{exp}}$ is shown in Fig. 4(b), which
coincides with the lowest spins of the $g(S_{n},J)_{\rm{tot}}$ distribution.

The average probability for populating individual levels of spin $J_{i}$ at
excitation energy $E_{i}$ is given by

$p^{\rm
level}(E_{i},J_{i})\propto\frac{g(E_{i},J_{i})_{\rm{exp}}}{g(E_{i},J_{i})_{\rm{tot}}},$
(15)

which is shown as the red dashed-dotted curve (arbitrary units) in Fig. 4(b).
The probability $p^{\rm level}(E_{i},J_{i})$ has to be taken into account in
the Shape method if $\gamma$-decay rates to final levels with different spins
are compared. The population probabilities for various initial spins are
listed in Table 2. It is interesting to note from Eqs. (8) and (15) that the
functional form of $p^{\rm level}$ for the present surface-induced light-ion
reactions follows the right part of a Gaussian centered at $J=-1/2$, see red
dashed-dotted curve in Fig. 4(b).

Table 2: Average probability $p^{\rm level}$ for populating individual levels in the $(p,p^{\prime})$ and $(d,p)$ reactions at $S_{n}$. The data are normalized to unity for $J_{i}=0$ or $1/2$. 150Nd$(p,p^{\prime})^{150}$Nd |  144Nd$(d,p)^{145}$Nd
---|---
$J_{i}$ | $p^{\rm level}$ | $J_{i}$ | $p^{\rm level}$
0 | 1.00 | 1/2 | 1.00
1 | 0.91 | 3/2 | 0.72
2 | 0.76 | 5/2 | 0.41
3 | 0.57 | 7/2 | 0.19
4 | 0.39 | 9/2 | 0.07
5 | 0.25 | 11/2 | 0.02
6 | 0.14 | 13/2 | 0.00
7 | 0.07 | 15/2 | 0.00
8 | 0.04 | 17/2 | 0.00
9 | 0.02 | 19/2 | 0.00
10 | 0.01 | 21/2 | 0.00
Figure 5: (Color online) (a) The primary $\gamma$ matrix $P(E_{\gamma},E_{i})$
of 150Nd showing the cuts for the two diagonals. (b) The resulting $\gamma$SF
from the Shape method (filled and open blue triangles) compared to the Oslo-
method using $\eta=1.0$ (solid grey squares) and $\eta=0.22$ (solid black
squares) from the side-feeding technique.

We will now test if the above findings are consistent with the Shape method
results and known $(n,\gamma)$ data. The Shape method Wiedeking2021  relies on
measuring the number of counts $N_{D}$ in a diagonal $D$ of the primary matrix
defined by $E_{i}=E_{\gamma}+E_{f}$ for a fixed value of $E_{f}$. Such a
diagonal is revealed in Fig. 1(c) for the final excitation energies of
$E_{f}\approx 0-200$ keV. The number of $\gamma$-ray transitions with energy
$E_{\gamma}$ from a given initial excitation energy $E_{i}$, is given by

$N_{D}\propto
f(E_{\gamma})E_{\gamma}^{3}\sum_{[J_{f}]}\sum_{J_{i}=J_{f}-1}^{J_{i}=J_{f}+1}p^{\rm
level}(E_{i},J_{i})\;g(E_{i},J_{i})_{\rm tot},$ (16)

where the fixed final excitation energy $E_{f}=E_{i}-E_{\gamma}$ defines the
diagonal $D$. All transitions are assumed to be dipole as the dipole strength
is known to be dominant within the quasi-continuum Capote09 . The notation
$[J_{f}]$ describes the spins of the final levels within the diagonal, e.g.,
if the diagonal contains four levels with $[J_{f}]$, then $\sum_{[J_{f}]}$ is
the sum over those corresponding four terms. The second sum is restricted to
the available spins $J$ populated by dipole transitions connecting initial and
final levels, which generally include three initial spins. However, in the
case of $J_{f}=0$, only the $J_{i}=1$ spin is included and for $J_{f}=1/2$,
only the $J_{i}=1/2$ and $J_{i}=3/2$ spins are included.

Figure 6: (Color online) (a) The primary $\gamma$ matrix $P(E_{\gamma},E_{i})$
of 144Nd showing the cuts for the two the diagonals. (b) The resulting
$\gamma$SF from the Shape method (filled and open blue triangles) compared to
the Oslo-method using $\eta=1.0$ and $0.22$. The filled red square data point
is taken from discrete resonance capture data (DRC) of Kopecky Kopecky2018 ;
IAEA2020 . Figure 7: (Color online) (a) The primary $\gamma$ matrix
$P(E_{\gamma},E_{i})$ of 145Nd showing the cuts for the two diagonals. (b) The
resulting $\gamma$SF from the Shape method (filled and open blue triangles)
compared to the Oslo-method using $\eta=1.0$ and $0.11$.

The primary $P(E_{\gamma},E_{i})$ matrix for 150Nd is shown in Fig. 5(a)
including two diagonals with their integration limits shown as black lines.
Diagonal $D_{1}$ includes the 0+ (0 keV), 2+ (130 keV) and 4+ (381 keV) final
levels and diagonal $D_{2}$ includes 12 levels in the final excitation energy
region $0.85-1.3$ MeV with average spin of $\langle J_{f}\rangle\approx 3.0$.

The Shape method implemented in the present work is based on the code diablo.c
available on the Oslo Cyclotron GitHub om2022 . The algorithm of the code
steps through one initial excitation energy $E_{i}$ of the primary
$P(E_{\gamma},E_{i})$ matrix and integrates the number of counts at the
diagonals $D_{1}$ and $D_{2}$ within the window $E_{i}\pm\Delta E_{i}/2$,
$\Delta E_{i}$ being the bin size. From the counts $N_{D1}$ and $N_{D2}$, a
pair of internally normalized values $f(E_{\gamma 1})$ and $f(E_{\gamma 2})$
is extracted by exploiting the proportionality of Eq. (16). These pairs are
then connected together by a sewing technique based on logarithmic
interpolation. The obtained strength function $f(E_{\gamma})$ has in principle
the correct functional form, but the absolute normalization is arbitrary and
must be determined by other means. More details of the Shape method are given
by Wiedeking et al. Wiedeking2021 .

Figure 5(b) shows the result of the Shape method giving a perfect overlap with
the Oslo method data using a level-density reduction factor $\eta=0.22$ at
$S_{n}$. Here, the Shape-method data points are multiplied by a common
absolute-normalization factor, which is found by a $\chi^{2}$ fit to the Oslo
data in the $E_{\gamma}=2.5-7.3$ MeV energy region. The $\gamma$SF data points
from populating the two diagonals (filled and open blue triangles) scatter
slightly indicating that the systematic uncertainties with the Shape method is
small in the case of 150Nd. The fact that the sidefeeding technique and the
Shape method give consistent results is very gratifying.

We also test the Shape method on 144Nd where known $(n,\gamma)$ data exist for
comparison. Figure 6(a) shows that another advantage with this almost
spherical nucleus ($\beta_{2}=0.125$) is that the diagonal to the 0+ (0 keV)
and 2+ (697 keV) are well separated and thus more accurate integrals for
$D_{1}$ and $D_{2}$ can be obtained. The 0+ ground level is reached by dipole
transitions from initial spin/parities 1±, whereas the 2+ level is populated
by decay from the 1±, 2± and 3±. It is therefore important to use reasonable
probabilities $p^{\rm level}$ (see Table 2) for the initial spins populated in
the reaction444If the two diagonals represent decay to levels with identical
spin/parities or with a broad range of spin/parities, the values of $p^{\rm
level}$ can be kept fixed for all spins..

The results of the Shape method are displayed in Fig. 6(b). Again we see a
good agreement between the $\gamma$SF from the Shape method and the Oslo
method using $\eta=0.22$. In addition, the two $\gamma$SFs agree well with the
discrete resonance capture data (DRC) of Kopecky Kopecky2018 ; IAEA2020 . The
Kopecky point gives additional support to our procedure described above.

We conclude that the two test cases 144,150Nd strongly suggest that a common
level density reduction factor of $\eta=0.22(2)$ at $S_{n}$ is reasonable for
the $(p,p^{\prime})$ reaction with 16-MeV protons on these neodymium isotopes.

Figure 8: (Color online) The $\gamma$SFs with spin reduction factors of
${\eta=0.22}$ and $0.11$ for 142,144,146,148,150Nd (red symbols) and
145,147,149,151Nd (blue symbols), respectively (see text). Except for 142Nd,
the $\gamma$SFs are separated by multiplying the next data set with a factor
of 3 for better visualization.

###  III.2 Spin distribution of the $(d,p)$ reaction

In the present work, $(d,p)$ reactions are used to study the odd-$A$ neodymium
isotopes. Here, the sidefeeding method cannot be applied due to many close-
lying $\gamma$-ray lines that are not separated due to the limited detector
resolution. Also, intraband transitions connecting close-lying rotational
bands complicate the extraction of sidefeeding from the quasicontinuum.
However, the Shape method is applicable provided that the two diagonals
include levels of known spin/parities.

The best case for the Shape method applied to the $(d,p)$ reaction is 145Nd,
where the lowest diagonal $D_{1}$ is well defined with the levels 7/2- (0
keV), 3/2- (67 keV) and 5/2- (73 keV). Diagonal $D_{2}$ is more problematic,
however, we have taken 10 levels in the final excitation region $0.66-1.09$
MeV with average spin of $\langle J_{f}\rangle\approx 2.9$.

Figure 7(a) shows the diagonals and integration limits for 145Nd, and the
Shape method results are displayed as filled and open blue triangles in Fig.
7(b). As shown, the Oslo method with intrinsic spin distribution (solid grey
squares) exhibits a $\gamma$SF too steep compared to the shape method. By
introducing a level density reduction factor at $S_{n}$ of $\eta=0.11(2)$, a
very good overlap between the Oslo and Shape method is obtained. As for the
$(p,p^{\prime})$ reaction, we assume that the experimental spin distribution
follows Eq. (8). Again we can estimate the probability $p^{\rm
level}(E_{i},J_{i})$ for populating individual levels of spin $J_{i}$ at
excitation energy $E_{i}$ using Eq. (15). Table 2 lists the ($d,p$) spin
population probabilities $p^{\rm level}$ at $S_{n}$, which are normalized to
unity for $J=1/2$.

Figure 8 summarizes the results obtained by introducing spin corrections in
the Oslo method. With reduction factors of $\eta=0.22(2)$ and $\eta=0.11(2)$
for the $(p,p^{\prime})$ and $(d,p)$ reactions, respectively, the $\gamma$SFs
follow a systematic trend from isotope to isotope. This feature is encouraging
and indicates that our corrections are sound.

##  IV Composition and evolution of the $\gamma$SF

The $\gamma$SF is composed of several structures which interplay and add up to
the total $\gamma$SF. Many of these structures depend strongly on the
quadrupole deformation $\beta_{2}$, which makes the chain of neodymiums
isotopes of particular interest. The giant dipole resonance (GDR) is known to
split into two components with deformation and the pygmy dipole resonance
(PDR) is expected to be stronger with increasing neutron excess. Furthermore,
the scissors mode (SM) strongly depends on deformation and finally the low-
energy enhancement (LEE) seems to be absent for deformed, heavy systems.

By introducing semi-empirical models Capote09 ; Goriely2019  for all these
structures, a total of 18 parameters have to be determined. This complicates a
simultaneous fit to the data, and the following fitting strategy is chosen.
Since the GDR is well separated from other structures around $\approx 15$ MeV,
we will first fit this part of the GDR. Using these parameters, the low-energy
E1 tail from the GDR is established, and we may introduce the weaker and
lower-lying structures on top of this. In cases where no experimental data
exist for certain $\gamma$-energy regions, we use the neighbouring isotopes as
a guidance. We apply the fit method implemented in ROOT, which is based on the
Minuit package james1981  with Hessian matrix error analysis.

We should point out that the Oslo method cannot separate the data into E1 or
M1 contributions. Furthermore, the technique is restricted to an excitation
energy of maximum $S_{n}$, which is typically around $5-6$ MeV and $7-8$ MeV
for the odd- and even-mass isotopes, respectively. Nevertheless, by also
exploiting other experimental data, we will obtain a reliable description of
the $\gamma$SF. Figure 9 presents our data together with other external data
for 142,144-151Nd. The various models are shown as curves with corresponding
model parameters listed in Tables 3 and 4.

Figure 9: (Color online) Experimental $\gamma$SFs (solid squares) with its
error bands (shaded blue) due to the uncertainty in $\sigma$, $D_{0}$,
$\langle\Gamma_{\gamma 0}\rangle$ and $\eta$ parameters. The photoneutron data
of Carlos carlos1971  and Nyhus nyhus2015  (purple circles and open diamonds)
are shown for the high-energy regions. Also the resonance capture data of
Kopecky Kopecky2018 ; IAEA2020  are shown as a filled red square data point in
144,146Nd. The various models applied are shown as curves, see assignments in
panel (f).

###  IV.1 Giant dipole resonance

In the macroscopic picture, the GDR describes the dipole oscillations of the
proton and neutron clouds against each other. The GDR is known to split into
two components (GDR1 and GDR2) for nuclei with an appreciable ground-state
deformation.

There exist photoneutron cross sections from the Saclay measurements on
142-146,148,150Nd by Carlos et al. carlos1971  and from inverse Compton
scattering on 143-146,148Nd by Nyhus et al. nyhus2015 . The $\gamma$SF, which
is calculated from the cross section by axel1962

$f(E_{\gamma})=\frac{1}{3(\pi\hbar c)^{2}}\frac{\sigma_{\gamma
n}(E_{\gamma})}{E_{\gamma}},$ (17)

where the constant reads $1/3(\pi\hbar c)^{2}=8.674\times 10^{-8}$mb-1MeV-2.
The GDR data of Fig. 9 show the expected spreading width rusev2008  of
$\approx 4$ MeV for the almost spherical 142Nd. For the heavier neodymiums,
the GDR gets broader with deformation and finally splits into two clear GDR
components for 150Nd. We also recognize for 144,145,148Nd that the high-energy
Oslo data match very well with the low-energy ($\gamma,n$) data of Nyhus et
al. nyhus2015 , which give further support to the spin restrictions introduced
in Sect. III.

To describe the GDR data, we use two generalized Lorentzians (GLOs), each with
the functional form of Capote09 :

$\displaystyle f_{\rm
E1}(E_{\gamma})=\frac{1}{3\pi^{2}\hbar^{2}c^{2}}\sigma_{E1}\Gamma_{E1}$
$\displaystyle\times\left[\frac{E_{\gamma}\Gamma(E_{\gamma},T_{f})}{(E_{\gamma}^{2}-E_{E1}^{2})^{2}+E_{\gamma}^{2}\Gamma^{2}(E_{\gamma},T_{f})}+0.7\frac{\Gamma(E_{\gamma}=0,T_{f})}{E_{E1}^{3}}\right]$
(18)

with

$\Gamma(E_{\gamma},T_{f})=\frac{\Gamma_{E1}}{E_{E1}^{2}}(E_{\gamma}^{2}+4\pi^{2}T_{f}^{2}).$
(19)

The resonance parameters are the energy centroid $E_{E1}$, the strength
$\sigma_{E1}$, and the width $\Gamma_{E1}$.

The $T_{f}$ parameter gives a non-zero tail for the lower $\gamma$ energies.
The parameter is extracted from 142,144Nd that are assumed to have no low-
lying structures. We find $T_{f}=0.46(1)$ MeV for both isotopes, which is
somewhat lower than the corresponding constant-temperature values of $T_{\rm
TC}=0.65(5)$ and $0.63(3)$ MeV. To reduce the number of free parameters in the
fit, we adopt the value of $T_{f}=0.50(5)$ MeV for all nine isotopes.

Figure 9 presents our data together with other external data for
142,144-151Nd. The E1 GLOs (long-dashed black curves) are fitted to the data
of Carlos and Nyhus carlos1971 ; nyhus2015 , using $T_{f}=0.50$ MeV. This part
of the $\gamma$SFs demonstrates the interesting broadening and splitting of
the GDR with deformation. For the three isotopes with no ($\gamma,n$) data,
the systematics from the neighbors are adopted. In the following, we exploit
the GDR data to obtain an estimate for the GDR E1 strength. The parameter sets
used for modelling the GDRs are listed in Table 3.

Table 3: Parameters for the GDR resonances with $T_{f}=0.50$ MeV. Parameters with uncertainties are from the fit. Nucleus | GDR1 | GDR2
---|---|---
| $E_{\rm GDR1}$ | $\sigma_{\rm GDR1}$ | $\Gamma_{\rm GDR1}$ | $E_{\rm GDR2}$ | $\sigma_{\rm GDR2}$ | $\Gamma_{\rm GDR2}$
| (MeV) | (mb) | (MeV) | (MeV) | (mb) | (MeV)
142Nd | 13.5(3) | 89(18) | 3.3(8) | 15.3(4) | 325(22) | 3.8(1)
144Nd | 14.4(7) | 179(93) | 4.4(11) | 15.9(2) | 184(103) | 4.3(7)
145Nd | 14.0(4) | 166(48) | 3.7(9) | 16.1(2) | 209(48) | 4.8(9)
146Nd | 13.1(7) | 89(19) | 3.9(19) | 15.7(5) | 435(30) | 4.6(2)
147Nd | 13.5 | 130 | 5.2 | 15.8 | 292 | 4.6
148Nd | 13.8(9) | 172(32) | 6.4(28) | 15.9(8) | 147(58) | 4.6(11)
149Nd | 13.6 | 193 | 5.0 | 16.2 | 146 | 4.3
150Nd | 13.4(2) | 213(5) | 6.7(5) | 16.5(8) | 145(9) | 4.0(4)
151Nd | 13.4 | 213 | 6.7 | 16.5 | 145 | 4.0

Table 4: Parameters for the PDR, SM and LEE structures and the integrated $B(M1)$ for LEE and SM. A common slope parameter of $\kappa_{\rm LEE}=1.9$ MeV-1 is set for all isotopes. Parameters with uncertainties are from the fit. Nucleus | PDR1 | PDR2 | SM | LEE
---|---|---|---|---
| $E_{\rm PDR1}$ | $\sigma_{\rm PDR1}$ | $\Gamma_{\rm PDR1}$ | $E_{\rm PDR2}$ | $\sigma_{\rm PDR2}$ | $\Gamma_{\rm PDR2}$ | $E_{\rm SM}$ | $\sigma_{\rm SM}$ | $\Gamma_{\rm SM}$ | $\sum B_{\rm SM}$ | $C_{\rm LEE}$ | $\sum B_{\rm LEE}$
| (MeV) | (mb) | (MeV) | (MeV) | (mb) | (MeV) | (MeV) | (mb) | (MeV) | ($\mu^{2}_{N}$) | ($10^{-8}$MeV-1) | ($\mu^{2}_{N}$)
142Nd | 6.90 | 3.0. | 0.6 | 11.1 | 25.0 | 1.6 | - | - | - | - | 6.3(1) | 7.7(22)
144Nd | 7.12(5) | 5.7(6) | 0.74(9) | 9.81(10) | 14.4(12) | 1.95(10) | - | - | - | - | 10.9(1) | 14.6(11)
145Nd | 7.5 | 9.0 | 0.9 | 9.3 | 16.0. | 1.6 | - | - | - | - | 12.8(2) | 16.6(19)
146Nd | 8.13(2) | 7.5(3) | 0.6 | 9.7 | 14 | 1.6 | - | - | - | - | 17.7(2) | 21.6(23)
147Nd | 7.1 | 4.7 | 0.59 | 9.2 | 13 | 1.64 | 2.27(3) | 0.134(5) | 0.97(6) | 1.7(15) | 17.1(1) | 20.9(17)
148Nd | 5.99(5) | 1.8(2) | 0.58(9) | 8.65(7) | 11.8(8) | 1.68(11) | 2.54(5) | 0.196(10) | 1.6(2) | 3.5(16) | 6.9(3) | 9.1((11)
149Nd | 6.03 | 3.9 | 0.85 | - | - | - | 2.66(7) | 0.284(10) | 2.4(2) | 6.2(18) | - | -
150Nd | 6.07(3) | 6.0(2) | 1.12(5) | - | - | - | 2.92(2) | 0.625(16) | 1.21(4) | 7.5(30) | - | -
151Nd | 6.07 | 6.0 | 1.12 | - | - | - | 2.89(4) | 0.637(39) | 1.07(6) | 6.7(38) | - | -

###  IV.2 Pygmy dipole resonance

The pygmy dipole resonance (PDR) is believed to originate from oscillations of
the neutron skin against the $N=Z=60$ core. The number of excess neutrons
$\Delta N=A-2Z$ changes from $\Delta N=22$ for 142Nd to $\Delta N=31$ for
151Nd, and the strength of the PDR resonance is expected to increase with this
neutron excess. An overview of experimental approaches to study the low-lying
electric dipole strength and experimental results are given by Savran et al.
savran2013 .

The resonance is described by the standard Lorentzian (SLO) model Capote09
given by

$f(E_{\gamma})=\frac{1}{3\pi^{2}\hbar^{2}c^{2}}\frac{\sigma
E_{\gamma}\Gamma^{2}}{(E_{\gamma}^{2}-E^{2})^{2}+E_{\gamma}^{2}\Gamma^{2}},$
(20)

with parameters $E$, $\sigma$, and $\Gamma$ appropriate for the specific
resonance.

By inspecting Fig. 9, we find resonance structures in the energy regions of
$6-8$ MeV and $9-11$ MeV that we for simplicity denote PDR1 and PDR2,
respectively. The lower resonance PDR1 is probably the one traditionally
called the pygmy dipole resonance. The high-lying resonance PDR2, which
accounts for the strength in the $\gamma$-energy region starting around 11.3
MeV for 142Nd and decreasing with mass number to 8.7 MeV for 148Nd, may as
well be due to a fragmentation of the GDR. Reinhard and Nazarewicz
reinhard2013  have used nuclear density functional theory to describe a weak
collective E1 structure in the region below the GDR. Thus, the proper
interpretation of the observed PDR2 is uncertain.

The most clear fingerprints for the PDR1 are found in the 145Nd data of Nyhus
et al. nyhus2015  and in the present 148,150Nd Oslo data. For 145Nd, the PDR1
is located at $E_{\gamma}\approx 7.5$ MeV and drops to $\approx 6.0$ MeV for
148,150Nd. However, there is no clear evidence in the compiled data sets that
the energy centroids decrease in a smooth way with increasing mass number.
Also, the evolution of the PDR1 strength seems complicated as it has a minimum
for the transitional 148Nd isotope.

For 147,149,151Nd there are data of neither the PDRs nor the GDRs, but we
include these structures in the over-all fit in order to get a consistent
description of the underlying strength of the low-energy LEE and SM
structures. For these isotopes, the resonance parameters are estimated from
the neighboring isotopes with values listed in Tables 3 and 4.

###  IV.3 Low-energy enhancement and scissors mode

The low-energy enhancement (LEE), often called upbend, was first observed in
56Fe voinov2004  and then in 93-98Mo guttormsen2005 . The phenomenon manifests
itself as an increase in $\gamma$ strength with decreasing energy below
$E_{\gamma}\approx 3$ MeV. The LEE, which is embedded in the nuclear
quasicontinuum, was completely unexpected at that time. Eight years later,
Wiedeking et al. wiedeking2012  confirmed the same structure in 95Mo using
other detectors and techniques. The LEE has been found for many lighter
nuclei, but has recently also been observed in heavier isotopes like 138,139La
kheswa2015  and 147,149,151,153Sm simon2016 . However, more recent experiments
on 153,155Sm malatji2021  did not reveal any LEE. This makes it questionable
if LEE exists also for heavy deformed systems. Midtbø et al. midtbo2018  have
given a review of the present status of the LEE. They report on a systematic
large-scale shell model study of 283 nuclei, which reveals in general a more
steep and pronounced LEE as the mass number increases.

The first modelling of the LEE structure was inspired by shell-model
calculations and experimental data for 90Zr and 94-96Mo. Here, Schwengner et
al. schwengner2013  suggest an exponential form of the LEE given by

$f(E_{\gamma})=C\exp(-\kappa E_{\gamma}),$ (21)

where $C$ and $\kappa$ are parameters.

The slope parameter $\kappa$ of Eq. (21) is determined by fits to the
pronounced LEE structures of 142,144-147Nd shown in Fig. 9, which give an
average value of $\kappa=1.9(2)$ MeV-1. The $\kappa$ parameter may be
interpreted as the inverse of temperature schwengner2013 . However,
$1/1.9\approx 0.53$ MeV do not coincide with the average value of $T_{\rm
CT}\approx 0.61$ MeV for 142,144-147Nd, see Table 1. Therefore, we use a fixed
value of $\kappa=1.9(2)$ MeV-1 in the fits for the 142,144-148Nd isotopes. The
fitted values of the remaining $C$ parameter are listed in Table 4. For
isotopes heavier than 148Nd, we see no clear LEE structure with the present
experimental lower $\gamma$-energy threshold.

The scissors mode (SM), which appears in deformed nuclei, was already
described in 1984 by Lo Iudice and Palumbo iudice1984 . They proposed a
geometrical picture where the deformed proton and neutron clouds oscillate
against each other like scissors blades. An alternative description was given
by Chen and Leander chen1982  that are based on strong M1 transitions between
$\Delta\Omega=1$ Nilsson orbitals555The Nilsson orbitals are assigned a
quantum number $\Omega$, which represents the projection of $j$ on the nuclear
symmetry axis. originating from the same spherical single-particle level
characterized by the angular momentum $j$. These predictions were confirmed by
the observation of a bump at $E_{\rm SM}\approx 2-3$ MeV in the $\gamma$-ray
spectra from various excitation energies in the quasicontinuum of 161Dy
guttormsen1984 . Later in the same year, the ($e,e^{\prime}$) reaction was
used to reveal M1 type resonant levels in 156,158Gd bohle1984 . It is
interesting that these authors also report on the absence of such levels in
146Nd, which is highly relevant for the present work. It has been shown that
the SM increases in strength with deformation peter1995 . A comprehensive
review of the magnetic dipole excitations build on the ground level is given
by Heyde et al. heyde2010 .

In the present work we study the SM embedded in the nuclear quasicontinuum
applying the Oslo method, see e.g. the study of 160-164Dy by Renstrøm et al.
renstrom2018  and references therein. In deformed rare earth nuclei, the SM
follows closely a standard Lorentzian at $E_{\rm SM}\approx 2-3$ MeV. In the
actinide region, the SM strength increases and is best modelled by two close-
lying Lorentzians tornyi2014 ; guttormsen2014 .

Figure 9(e) indicates a weak SM at $E_{\rm SM}\approx 2.3$ MeV in 147Nd, which
results in a rather flat $\gamma$SF located in-between the LEE structure and
the tail of the GDR. It is interesting to observe how the $\gamma$SF at these
energies grows in strength with deformation and finally reveals a clear and
strong SM structure in 150,151Nd at $E_{\rm SM}\approx 2.9$ MeV.

Figure 10: (Color online) The integrated LEE (blue diamonds) and SM (red
triangles) strengths $B(M1)$ together with the summed strength of the two
structures (orange squares) as function of quadrupole deformation $\beta_{2}$.
The strengths are integrated between $E_{\gamma}=0$ and $5$ MeV.

An intriguing idea was proposed by Schwengner et al. schwengner2017  based on
large-scale shell-model calculations for 60,64,68Fe. They find that the
strength of the SM increases by a factor of two when going from 60Fe to 68Fe.
At the same time, the LEE strength decreases correspondingly and thereby
conserves the total strength of $B(M1)\approx 9.8\mu_{N}^{2}$. The
conservation of the summed SM and LEE strengths has been experimentally tested
for 147,149,151,153Sm naqvi2019 , but large uncertainties prohibit a firm
conclusion.

The evolution of the LEE and SM is shown in Fig. 9. It is clear that the LEE
is present in 142,144-148Nd and the SM is present in 147-151Nd. Here, 148Nd is
a key nucleus since both the LEE and SM are significantly present. It is
interesting that no SM strength seems to be present for 146Nd, as also
confirmed in the ($e,e^{\prime}$) experiments bohle1984 .

In order to quantify the strength of the LEE and SM resonances as function of
deformation (or mass number), we integrate the corresponding Lorentzians for
the two structures. The $\gamma$-decay strength from an initial level with
spin $J_{i}$ is proportional with the number of available final spins $J_{f}$
that can be reached with a transition of electromagnetic character $XL$. If we
further assume an initial spin with $J_{i}\geq L$, we obtain Bartholomew1973 ;
Lone1986

$\displaystyle\frac{dB(XL)}{dE_{\gamma}}$
$\displaystyle=\frac{L\left[(2L+1)!!\right]^{2}{\left({\hbar
c}\right)^{2L+1}}}{8\pi(L+1)}f_{XL}(E_{\gamma})\sum_{J_{f}}1$
$\displaystyle=\frac{L\left[(2L+1)!!\right]^{2}{\left({\hbar
c}\right)^{2L+1}}}{8\pi(L+1)}f_{XL}(E_{\gamma})(2L+1).$ (22)

This expression is now used to evaluate the total strength in the
$\gamma$-energy region $0-5$ MeV. Assuming M1 electromagnetic character, the
integrated strength with $L=1$ is given by

$B(M1)=\frac{27(\hbar c)^{3}}{16\pi}\int_{0}^{\rm
5MeV}f(E_{\gamma})dE_{\gamma},$ (23)

where $f(E_{\gamma})$ is modelled by Eqs. (20) and (21) for SM and LEE,
respectively. The factor in front of the integral has the value $27(\hbar
c)^{3}/16\pi=2.598\times 10^{8}\mu^{2}_{N}$MeV2.

The LEE and SM strengths can now be evaluated from their respective fit
functions shown in Fig. 9 using Eq. (23). Figure 10 summarizes the total M1
strengths for the nine isotopes of the present experiment, which are also
included in Table 4. The error bars are mainly due to uncertainties in the
$T_{f}=0.50(5)$ parameter and the experimental $\langle\Gamma_{\gamma}\rangle$
value.

The blue curve of Fig. 10 shows the rise and fall of the LEE strength with a
maximum of $\approx 21\mu^{2}_{N}$ for 146,147Nd with $\beta_{2}\approx 0.16$.
For 148Nd the strength drops to $\approx 9\mu^{2}_{N}$ and then vanishes for
the heavier isotopes. The strength of the SM (red curve) starts increasing at
147Nd and reaches a plateau of $\approx 7\mu^{2}_{N}$ for the well-deformed
150,151Nd.

The summed strength of the LEE and SM structures is shown as an orange curve
in Fig. 10. It is clear that this curve is far from constant and contradicts
the picture of Schwengner et al. schwengner2017 . Firstly, we find that the SM
structure does not account for the missing strength of the LEE at 149-151Nd;
it reaches only one third of the maximum LEE strength. Secondly, there is a
strong increase of LEE strength from 142Nd to 146,147Nd where no SM strength
is present that could eventually account for this behavior.

The 148Nd seems to be a key nucleus where both the LEE and SM are coexisting.
It is noteworthy that the clear onset of the SM structure for this
transitional nucleus coincides with the onset of collectivity in the mean-
field solution for the shell-model interaction Gut2021 .

##  V Conclusions and outlook

The present study has shown that the limitation of transferred spin in the
present ($p,p^{\prime}$) and the ($d,p$) reactions have to be taken into
account when using the Oslo method. By including the spin reduction, the
$\gamma$SF is reduced with more than a factor of three at $E_{\gamma}\approx
S_{n}$, however, remaining the same around $E_{\gamma}\approx 3$ MeV. The
applied spin corrections are supported by the Shape-method and $\gamma$-ray
side-feeding into the rotational ground-state band. The corrected $\gamma$SF
also matches ($n,\gamma$) and ($\gamma,n$) data available from literature.

The ($\gamma,n$) data on the giant dipole resonances are modelled using the
GLO model with a fixed $T_{f}=0.50(5)$ MeV. The fitted resonance parameters
are exploited to obtain the E1 GDR tail, which is underlying the $\gamma$SF
structures located at lower $\gamma$ energies. The ($\gamma,n$) data confirm
the gradual splitting of the GDR with deformation.

The pygmy dipole resonance is partly obtained from the present Oslo data and
the ($\gamma,n$) data from literature. The data of 142,144-146,148Nd indicate
the presence of two resonances. However, we suggest that the lower in energy
is related to the pygmy dipole resonance built on neutron skin oscillations.

The present Oslo data bring new insight into the evolution of the low-energy
enhancement (LEE) and the scissors mode (SM). By increasing deformation, we
observe around 148Nd with $\beta_{2}\approx 0.2$ that the LEE strength
decreases and the SM strength increases. Apart from that, there seems not to
be any connection between the amount of $B(M1)$ strengths carried by the two
structures as function of deformation. This contradicts the idea that the LEE
and SM structures schwengner2017 ; naqvi2019 ; frauendorf2022  are connected
by exchanging strength with each other. Thus, the evaluated $B(M1)$ strengths
indicate that the two structures are due to different mechanisms.

In conclusion, the present work reveals the fascinating evolution of various
$\gamma$-ray structures in the cross over from spherical to deformed neodymium
isotopes. There is a clear need for more data and in particular, more robust
theoretical descriptions to understand the interplay between $\gamma$-decay
mechanisms in the nuclear quasicontinuum.

###### Acknowledgements.

We thank J. C. Müller, P. Sobas and J. Wikne for providing excellent
experimental conditions. We gratefully acknowledge valuable discussions with
Sean Liddick, Dennis Mücher, Artemis Spyrou and Fabio Zeiser. This work was
supported by The Scientific and Technological Research Council of Turkey
(TUBITAK) with project number 115F196. A. C. L. acknowledges funding of this
research by the European Research Council through ERC-STG-2014, grant
agreement no. 637686. This work was partially supported by projects 263030 and
262952 of the Norwegian Research Council. The OSCAR detector was funded by the
Norwegian Research Council project 245882. This work is based on the research
supported in part by the National Research Foundation of South Africa (Grant
Number: 118846).

## References

  * (1) G. A. Bartholomew, E. D. Earle, A. J. Fergusson, J. W. Knowles and M. A. Lone, Adv. Nucl. Phys. 7, 229 (1973).
  * (2) S. Goriely et al., Eur. Phys. J. A 55, 72 (2019).
  * (3) IAEA Nuclear Data Services, htpps://www-nds.iaea.org/PSFdatabase.
  * (4) J. Kopecky, private communication (2018).
  * (5) R. Schwengner, S. Frauendorf, and B.A. Brown, Phys. Rev. Lett. 118, 092502 (2017).
  * (6) F. Naqvi et al., Phys. Rev. C99, 054331 (2019).
  * (7) S. Frauendorf and R. Schwengner, Phys. Rev. C105, 034335 (2022).
  * (8) D. Savran, T. Aumann, and A. Zilges, Prog. Part. Nucl. Phys. 70, 210 (2013).
  * (9) M. Guttormsen, T. Ramsøy, and J. Rekstad, Nucl. Instrum. Methods Phys. Res. A 255, 518 (1987).
  * (10) M. Guttormsen, T. S. Tveter, L. Bergholt, F. Ingebretsen, and J. Rekstad, Nucl. Instrum. Methods Phys. Res. A 374, 371 (1996).
  * (11) A. Schiller, L. Bergholt, M. Guttormsen, E. Melby, J. Rekstad, and S. Siem, Nucl. Instrum. Methods Phys. Res. A 447 494 (2000).
  * (12) M. Wiedeking, M. Guttormsen, A. C. Larsen, F. Zeiser, A. Görgen, S. N. Liddick, D. Mücher, S. Siem, and A. Spyrou. Phys. Rev. C 104, 014311 (2021).
  * (13) M. Guttormsen et al., Phys. Lett. B 816, 136206 (2021) and supplemantary material.
  * (14) M. Guttormsen, A. Bürger, T. E. Hansen, and N. Lietaer, Nucl. Instrum. Methods Phys. Res. A 648, 168 (2011).
  * (15) M. Guttormsen, A. Atac, G. Løvhøiden, S. Messelt, T. Ramsøy, J. Rekstad, T. F. Thorsteinsen, T.S. Tveter, and Z. Zelazny, Phys. Scr. T 32, 54 (1990).
  * (16) F. Zeiser et al., Nucl. Instr. Methods A 985, 164678 (2021).
  * (17) A. Görgen, M. Guttormsen, A. C. Larsen, S. Siem, E. Adli, N. F. J. Edin, H. Gjerstad, G. Henriksen, E. Malinen, V. Modamio, B. Schoultz, P. A. Sobas, T. A. Theodossiou and J. C. Wikne, Eur. Phys. J. Plus 136, 181 (2021).
  * (18) M. Guttormsen, A. C. Larsen, F. Zeiser, J. E. Midtbø and V. W. Ingeberg, Oslo Method Software v1.1.6 (2022). Available on-line at https://github.com/oslocyclotronlab.
  * (19) P. A. M. Dirac, The Quantum Theory of Emission and Absorption of Radiation, Proc. R. Soc. Lond. A 1927 114, 243-265.
  * (20) E. Fermi, Nuclear Physics. University of Chicago Press (1950).
  * (21) D. M. Brink, doctorial thesis, Oxford University, 1955.
  * (22) M. Guttormsen, A. C. Larsen, A. Görgen, T. Renstrøm, S. Siem, T. G. Tornyi and G. M. Tveten, Phys. Rev. Lett. 116, 012502 (1990).
  * (23) J. Kopecky and M. Uhl, Phys. Rev. C 41, 1941 (1990).
  * (24) R. Capote, M. Herman, P. Oblozinsky, et al., Nuclear Data Sheets 110, 3107 (2009). Reference Input Library RIPL-3 available online at http://www-nds.iaea.org/RIPL-3/.
  * (25) Data from the NNDC On-Line Data Service database; available at http://www.nndc.bnl.gov/nudat2/.
  * (26) S. F. Mughabghab, Atlas of Neutron Resonances. (Elsevier Science, Amsterdam, 2018). 6th ed.
  * (27) T. Ericson, Nucl. Phys. 11, 481(1959).
  * (28) B. Pritychenko et al. Atomic Data and Nuclear Data Tables 107, 1–139 (2016).
  * (29) A. Voinov, M. Guttormsen, E. Melby, J. Rekstad, A. Schiller, and S. Siem, Phys. Rev. C 63, 044313 (2001).
  * (30) A. C. Larsen et al., Phys. Rev. C 83, 034315 (2011).
  * (31) F.James. Determining the statistical Significance of experimental Results. Technical Report DD/81/02 and CERN Report 81-03, CERN, 1981
  * (32) P. Carlos, H. Beil, R. Bergere, A. Lepretre and A. Veyssiere, Nucl. Phys. A 172, 437 (1971).
  * (33) H.-T. Nyhus et al., Phys. Rev. C 91, 015808 (2015).
  * (34) P. Axel, Phys. Rev. 126, 671 (1962).
  * (35) G. Rusev et al., Phys. Rev. C 77, 064321 (2008).
  * (36) P.-G. Reinhard and W. Nazarewicz, Phys. Rev. C 87, 014324 (2013).
  * (37) A. Voinov, E. Algin, U. Agvaanluvsan, T. Belgya, R. Chankova, M. Guttormsen, G.E. Mitchell, J. Rekstad, A. Schiller, and S. Siem, Phys. Rev. Lett. 93, 142504 (2004).
  * (38) M. Guttormsen et al., Phys. Rev. C 71, 044307 (2005).
  * (39) M. Wiedeking et al., Phys. Rev. Lett.108, 162503 (2012).
  * (40) B. V. Kheswa et al., Phys. Lett. B744, 268 (2015).
  * (41) A. Simon et al., Phys. Rev. C93, 034303 (2016).
  * (42) K. L. Malatji et al., Phys. Rev. C103, 014309 (2021).
  * (43) J.E. Midtbø, A.C. Larsen, T. Renstrøm, F.L. Bello Garrote, and E. Lima, Phys. Rev. C98, 064321 (2018).
  * (44) R. Schwengner, S. Frauendorf, and A. C. Larsen, Phys. Rev. Lett. 111, 232504 (2013).
  * (45) N. Lo Iudice and F. Palumbo, Phys. Lett. B 137, 27 (1984).
  * (46) Y. S. Chen and G. A. Leander, Phys. Rev. C 26, 2607 (1982).
  * (47) M. Guttormsen, J. Rekstad, A. Henriquez, F. Ingebretsen, and T.F. Thorsteinsen, Phys. Rev. Lett. 52, 102 (1984).
  * (48) D. Bohle, A. Richter, W. Steffen A.E.L.Dieperink, N. Lo Iudice, F. Palumbo, and O. Scholte, Phys. Lett. B 137 27 (1984).
  * (49) P. von Neumann-Cosel, J.N. Ginocchio, H. Bauer, and A. Richter, Phys. Rev. Lett. 75, 23 (1995).
  * (50) K. Heyde, P. von Neumann-Cosel, and A. Richter, Rev. Mod. Phys. 82, 2365 (2010).
  * (51) T. Renstrøm et al., Phys. Rev. C98, 054310 (2018).
  * (52) T. G. Tornyi et al., Phys. Rev. C89, 044323 (2014).
  * (53) M. Guttormsen et al., Phys. Rev. C89, 014302 (2014).
  * (54) M. A. Lone. Neutron Induced Reactions, eds. J. Kris̆tiak and E. Bĕták, Springer, Dordrecht (1986) p.238-252.