# Isoscalar giant monopole resonance in 24Mg and 28Si: Effect of coupling
between the isoscalar monopole and quadrupole strength

A. Bahini<EMAIL_ADDRESS>School of Physics, University of the
Witwatersrand, Johannesburg 2050, South Africa  iThemba Laboratory for
Accelerator Based Sciences, Somerset West 7129, South Africa  V. O. Nesterenko
<EMAIL_ADDRESS>Laboratory of Theoretical Physics, Joint Institute for
Nuclear Research, Dubna, Moscow Region 141980, Russia  State University
“Dubna”, Dubna, Moscow Region 141980, Russia  Moscow Institute of Physics and
Technology, Dolgoprudny, Moscow Region 141701, Russia  I. T. Usman  School of
Physics, University of the Witwatersrand, Johannesburg 2050, South Africa  P.
von Neumann-Cosel  Institut für Kernphysik, Technische Universität Darmstadt,
D-64289 Darmstadt, Germany  R. Neveling  iThemba Laboratory for Accelerator
Based Sciences, Somerset West 7129, South Africa  J. Carter  School of
Physics, University of the Witwatersrand, Johannesburg 2050, South Africa  J.
Kvasil  Institute of Particle and Nuclear Physics, Charles University,
CZ-18000, Prague 8, Czech Republic  A. Repko  Institute of Particle and
Nuclear Physics, Charles University, CZ-18000, Prague 8, Czech Republic
Institute of Physics, Slovak Academy of Sciences, 84511, Bratislava, Slovakia
P. Adsley  School of Physics, University of the Witwatersrand, Johannesburg
2050, South Africa  iThemba Laboratory for Accelerator Based Sciences,
Somerset West 7129, South Africa  Department of Physics, Stellenbosch
University, 7602 Matieland, Stellenbosch, South Africa Irene Joliot Curie Lab,
UMR8608, IN2P3-CNRS, Université Paris Sud 11, 91406 Orsay, France N. Botha
School of Physics, University of the Witwatersrand, Johannesburg 2050, South
Africa  J. W. Brummer  iThemba Laboratory for Accelerator Based Sciences,
Somerset West 7129, South Africa  Department of Physics, Stellenbosch
University, 7602 Matieland, Stellenbosch, South Africa L. M. Donaldson
iThemba Laboratory for Accelerator Based Sciences, Somerset West 7129, South
Africa  S. Jongile  iThemba Laboratory for Accelerator Based Sciences,
Somerset West 7129, South Africa  Department of Physics, Stellenbosch
University, 7602 Matieland, Stellenbosch, South Africa T. C. Khumalo  School
of Physics, University of the Witwatersrand, Johannesburg 2050, South Africa
iThemba Laboratory for Accelerator Based Sciences, Somerset West 7129, South
Africa  Department of Physics, University of Zululand, Richards Bay, 3900,
South Africa M. B. Latif  School of Physics, University of the Witwatersrand,
Johannesburg 2050, South Africa  iThemba Laboratory for Accelerator Based
Sciences, Somerset West 7129, South Africa  K. C. W. Li  iThemba Laboratory
for Accelerator Based Sciences, Somerset West 7129, South Africa  Department
of Physics, Stellenbosch University, 7602 Matieland, Stellenbosch, South
Africa P. Z. Mabika  Department of Physics and Astronomy, University of the
Western Cape, Bellville 7535, South Africa  P. T. Molema  School of Physics,
University of the Witwatersrand, Johannesburg 2050, South Africa  iThemba
Laboratory for Accelerator Based Sciences, Somerset West 7129, South Africa
C. S. Moodley  School of Physics, University of the Witwatersrand,
Johannesburg 2050, South Africa  iThemba Laboratory for Accelerator Based
Sciences, Somerset West 7129, South Africa  S. D. Olorunfunmi  School of
Physics, University of the Witwatersrand, Johannesburg 2050, South Africa
iThemba Laboratory for Accelerator Based Sciences, Somerset West 7129, South
Africa  P. Papka  iThemba Laboratory for Accelerator Based Sciences, Somerset
West 7129, South Africa  Department of Physics, Stellenbosch University, 7602
Matieland, Stellenbosch, South Africa L. Pellegri  School of Physics,
University of the Witwatersrand, Johannesburg 2050, South Africa  iThemba
Laboratory for Accelerator Based Sciences, Somerset West 7129, South Africa
B. Rebeiro  Department of Physics and Astronomy, University of the Western
Cape, Bellville 7535, South Africa  E. Sideras-Haddad  School of Physics,
University of the Witwatersrand, Johannesburg 2050, South Africa  F. D. Smit
iThemba Laboratory for Accelerator Based Sciences, Somerset West 7129, South
Africa  S. Triambak  Department of Physics and Astronomy, University of the
Western Cape, Bellville 7535, South Africa  J. J. van Zyl  Department of
Physics, Stellenbosch University, 7602 Matieland, Stellenbosch, South Africa

###### Abstract

Background: In highly deformed nuclei, there is a noticeable coupling of the
Isoscalar Giant Monopole Resonance (ISGMR) and the $K=0$ component of the
Isoscalar Giant Quadrupole Resonance (ISGQR), which results in a double peak
structure of the isoscalar monopole (IS0) strength (a narrow low-energy
deformation-induced peak and a main broad ISGMR part). The energy of the
narrow low-lying IS0 peak is sensitive to both the incompressibility modulus
$K_{\infty}$ and the coupling between IS0 and isoscalar quadrupole (IS2)
strength.

Objective: This study aims to investigate the two-peaked structure of the
ISGMR in the prolate 24Mg and oblate 28Si nuclei and identify among a variety
of energy density functionals based on Skyrme parameterisations the one which
best describes the experimental data. This will allow for conclusions
regarding the nuclear incompressibility. Because of the strong IS0/IS2
coupling, the deformation splitting of the ISGQR will also be analysed.

Methods: The ISGMR was excited in 24Mg and 28Si using $\alpha$-particle
inelastic scattering measurements acquired with an $E_{\alpha}=196$ MeV beam
at scattering angles $\theta_{\text{Lab}}=0^{\circ}$ and $4^{\circ}$. The
K$600$ magnetic spectrometer at iThemba LABS was used to detect and momentum
analyse the inelastically scattered $\alpha$ particles. An experimental energy
resolution of $\approx 70$ keV (FWHM) was attained, revealing fine structure
in the excitation-energy region of the ISGMR. The IS0 strength distributions
in the nuclei studied were obtained with the Difference-of-Spectrum (DoS)
technique. The theoretical comparison is based on the quasiparticle random-
phase approximation (QRPA) with a representative set of Skyrme forces.

Results: IS0 strength distributions for 24Mg and 28Si are extracted and
compared to previously published results from experiments with a lower energy
resolution. With some exceptions, a reasonable agreement is obtained. The IS0
strength is found to be separated into a narrow structure at about $13-14$ MeV
in 24Mg, $17-19$ MeV in 28Si and a broad structure at $19-26$ MeV in both
nuclei. The data are compared with QRPA results. The results of the calculated
characteristics of IS0 states demonstrate the strong IS0/IS2 coupling in
strongly prolate 24Mg and oblate 28Si. The narrow IS0 peaks are shown to arise
due to the deformation-induced IS0/IS2 coupling and strong collective effects.
The cluster features of the narrow IS0 peak at $13.87$ MeV in 24Mg are also
discussed. The best description of the IS0 data is obtained using the Skyrme
force SkPδ with an associated low nuclear incompressibility $K_{\infty}=202$
MeV allowing for both the energy of the peak and integral IS0 strength in 24Mg
and 28Si to be reproduced. The features of the ISGQR in these nuclei are also
investigated. An anomalous deformation splitting of the ISGQR in oblate 28Si
is found.

Conclusions: The ISGMR and ISGQR in light deformed nuclei are coupled and thus
need to be described simultaneously. Only such a description is relevant and
consistent. The deformation-induced narrow IS0 peaks can serve as an
additional sensitive measure of the nuclear incompressibility.

###### pacs:

21.60.Jz, 27.30.+t, 21.10.-k

##  I Introduction

For many years, the Isoscalar Giant Monopole Resonance (ISGMR) served as an
important source of information on the nuclear incompressibility Blaizot ;
Harakeh ; GC_review2018 . In deformed nuclei, this resonance demonstrates a
remarkable coupling of the isoscalar monopole and quadrupole $(K=0)$ modes,
which leads to a double peak structure of the IS0 strength abgrall1980 ;
jang1983 ; nishizaki1985 ; buenerd1980 ; garg1984 ; E0_Kvasil_PRC2016 ;
E0_Colo_PLB2020  (here, $K$ stands for the projection of the total angular
momentum on the symmetry axis of an axially deformed nucleus bohr1998nuclear
). Strongly deformed light nuclei, like prolate 24Mg and oblate 28Si, are
especially attractive for an exploration of the nuclear incompressibility
$K_{\infty}$ and the role of the IS0/IS2 coupling. Indeed, in these nuclei,
one may expect a deformation-induced strong narrow peak in the IS0 strength
located below the main ISGMR region 24Mg_GG_PLB2015 ; 24Mg_GG_PRC2016 ;
24Mg_PPJC2016 . The energy of this peak should be sensitive to both the
incompressibility $K_{\infty}$ and the IS0/IS2 coupling strength.

The ISGMR in 24Mg has been explored in $(\alpha,\alpha^{\prime})$ and
(6Li,6Li′) experiments since the $1980$s (see Ref. Harakeh  for the extensive
review and discussion). More recent $(\alpha,\alpha^{\prime})$ experiments
were performed at the Research Center for Nuclear Physics (RCNP) for 24Mg
kaw2013 ; 24Mg_GG_PLB2015 ; 24Mg_GG_PRC2016  and 28Si 28Si_PGG_PRC2016 . In
$2021$, the results of an RCNP (6Li,6Li′) experiment on IS0 strength in 24Mg
were published Zabora2021 . The IS0/IS2 coupling was mentioned in Refs.
24Mg_GG_PLB2015 ; Zabora2021  in order to explain the existence of the IS0
structure located around $16-19$ MeV excitation in 24Mg. The existence of a
prominent IS0 peak at about $18$ MeV in oblate 28Si was noted but not yet
explained 28Si_PGG_PRC2016 . These results generally confirm previous findings
from $(\alpha,\alpha^{\prime})$ experiments performed by the Texas A&M
University (TAMU) group on 24Mg youngblood2009isoscalar  (for the dataset
where the out-of-plane angle was measured) and 28Si youngblood2007isoscalar .
For 28Si, discrepancies between RCNP 28Si_PGG_PRC2016  and TAMU
youngblood2007isoscalar  results in the high excitation-energy region above
$20$ MeV were noted and attributed to the phenomenological background
subtraction method employed by TAMU GC_review2018 .

The IS0/IS2 coupling was earlier considered within the quasiparticle random
phase approximation (QRPA) in 24Mg 24Mg_GG_PLB2015 ; 24Mg_GG_PRC2016 ;
24Mg_PPJC2016 ; Zabora2021 ; yoshida2010 ; adsley2021isoscalar  and 28Si
adsley2021isoscalar . However, in the study adsley2021isoscalar , the narrow
IS0 peaks in 24Mg and 28Si were not specifically discussed. Furthermore, one
can dispute whether the low-energy narrow IS0 peak in 24Mg has been correctly
assigned (and used in the QRPA analysis) in the RCNP studies 24Mg_GG_PLB2015 ;
24Mg_GG_PRC2016 ; Zabora2021  (see discussion in Sect. IV.2).

In this paper, the ISGMR in prolate 24Mg and oblate 28Si is explored using
$\alpha$-particle inelastic scattering measurements with an $E_{\alpha}=196$
MeV beam measured at scattering angles $\theta_{\text{Lab}}=0^{\circ}$ and
$4^{\circ}$ using the high energy-resolution K$600$ magnetic spectrometer at
the iThemba Laboratory for Accelerator Based Science (LABS), South Africa. The
study focuses on the deformation-induced splitting of the ISGMR into two
parts: a narrow peak arising due to IS0/IS2 coupling, and a wide structure
representing the main ISGMR. The former exists only in deformed nuclei while
the latter appears in both spherical and deformed nuclei.

By comparing the experimental data obtained with QRPA calculations, we will
identify the narrow IS0 peak in 24Mg, explain the origin of the narrow IS0
peaks in both prolate 24Mg and oblate 28Si, and suggest a Skyrme force
suitable for the description of IS0 strength in these nuclei. Moreover, we
will analyse in detail the IS2 strength, which determines the energy of the
narrow IS0 peak, and show that, in strongly deformed nuclei, only a
simultaneous description of IS0 and IS2 strengths can be considered as
consistent. The simultaneous availability of data for a prolate (24Mg) and an
oblate (28Si) case allows for the generalisation of the conclusions.

The paper is organised as follows: in Sec. II, the experimental details are
given. The method of extraction and the resulting IS0 strength distributions
are presented in Sec. III while, in Sec. IV, details of the QRPA calculations
are given and the comparison with the data is shown and discussed. In Sec. V,
conclusions are drawn and finally in Appendix A, the predicted deformation
splitting of the ISGQR in 24Mg and 28Si is presented.

Figure 1: Double-differential cross sections for the
24Mg($\alpha,\alpha^{\prime}$) reaction at $E_{\alpha}=196$ MeV with
$\theta_{\text{Lab}}=0^{\circ}-1.91^{\circ}$ (top) and
$\theta_{\text{Lab}}=2^{\circ}-6^{\circ}$ (bottom). Figure 2: Same as Fig. 1,
but for 28Si.

##  II Experimental details

The experimental procedure followed in this study is similar to that described
elsewhere adsley2021isoscalar ; adsley2017alpha . As such, only salient
details are provided here. The experiment was performed at the Separated
Sector Cyclotron (SSC) facility of iThemba LABS using a beam of $196$ MeV
$\alpha$ particles. Inelastically scattered $\alpha$ particles were momentum
analysed by the K$600$ magnetic spectrometer after interacting with either a
$0.23$ mg/cm2 thick 24Mg or a $0.23$ mg/cm2 thick 28Si foil. The horizontal
and vertical positions of the scattered $\alpha$ particles in the focal plane
of the spectrometer were measured using two multiwire drift chambers. Energy
deposition in plastic scintillators in the focal plane as well as time-of-
flight measurements relative to the cyclotron radio frequency were used for
particle identification.

Spectra were acquired with the spectrometer set at scattering angles
$0^{\circ}$ and $4^{\circ}$. In the former, scattering angles
$\theta_{\text{Lab}}=0^{\circ}\pm 1.91^{\circ}$ and in the latter, scattering
angles from $\theta_{\text{Lab}}=2^{\circ}$ to $6^{\circ}$ were covered by a
circular spectrometer aperture, respectively. The trajectory of the scattered
particles through the focal plane was used to construct the scattering angle.
A multihole collimator was used to calibrate the measured focal-plane angles
to scattering trajectories into the spectrometer.

In the zero-degree mode, both the unscattered beam and the inelastically
scattered particles are transported through the K$600$ magnetic spectrometer
with the beam exiting the spectrometer only a few centimeters away from the
drift chamber position in the high-dispersion focal plane mode of the K$600$.
For such measurements, suppression of beam-induced background is critical. In
order to subtract the instrumental background, a standard method used at the
iThemba LABS K$600$ neveling2011high  and the RCNP magnetic spectrometers
tamii2009measurement  that exploits differences in the vertical distribution
of real and background events was employed. This method establishes background
spectra from the regions of the focal plane above and below the vertically
focussed band of true events and allows a direct subtraction of these
components from the central region of interest. In contrast to the zero-degree
mode, improved instrumental background conditions in the small-angle mode
allows one to operate the spectrometer in vertical off-focus mode and also use
the medium-dispersion focal plane. In the off-focus mode, the vertical
position on the focal plane relates to the vertical component of the
scattering angle into the spectrometer aperture and allows for its
reconstruction tamii2009measurement .

The same techniques as employed in Ref. adsley2017alpha  were adopted for the
analysis of the data. This includes software corrections of kinematic effects
and optical aberrations in the horizontal focal-plane position that depend on
the vertical focal-plane position and the scattering angle into the
spectrometer. The energy calibration was based on well-known states in 24Mg
kaw2013 ; bor1981 . An energy resolution of $\approx 70$ keV (FWHM) was
attained. Figures 1 and 2 show the inelastic scattering cross sections
extracted at $0^{\circ}-1.91^{\circ}$ and $2^{\circ}-6^{\circ}$ for 24Mg and
28Si, respectively. The broad structure seen in Fig. 2 below $10$ MeV is due
to hydrogen contaminants in the targets. However, it does not affect the
excitation-energy ranges relevant for the extraction of the IS0 strength as
demonstrated in the next section.

##  III Experimental Results

The Multipole Decomposition Analysis (MDA) technique was employed in numerous
studies to extract the IS0 strength distribution in nuclei Zabora2021 ;
GC_review2018 ; gupta2018isoscalar ; 24Mg_GG_PRC2016 ; 28Si_PGG_PRC2016 ;
24Mg_GG_PLB2015 ; patel2014excitation ; youngblood2009isoscalar ;
youngblood2007isoscalar . However, due to the limited number of angular data
points in the present study, the IS0 strength distributions were rather
determined via the Difference-of-Spectra (DoS) method DoSpaper . In the DoS
method, an excitation-energy spectrum for the angular region associated with
the first minimum of the $L=0$ angular distribution is subtracted from the
excitation-energy spectrum taken at $0^{\circ}$. This method, therefore,
requires the determination of the suitable angle cut from the measurement at
$\theta_{\text{Lab}}=2^{\circ}-6^{\circ}$, which is obtained from DWBA
calculations.

In the present study, the DWBA calculations were performed according to the
method described in Ref. satchler1997missing . A density-dependent single-
folding model for the real part of the potential $U(r)$, obtained with a
Gaussian $\alpha$-nucleon potential, and a phenomenological Woods-Saxon
potential for the imaginary term of $U(r)$ were used, so that the
$\alpha$-nucleus potential can be written as

$U(r)=V_{\text{fold}}(r)+i\dfrac{W}{\left\\{1+\exp\left[\left(r-R_{\text{I}}\right)/a_{\text{I}}\right]\right\\}}~{},$
(1)

with radius
$R_{\text{I}}=r_{\text{0I}}(A_{\text{p}}^{1/3}+A_{\text{t}}^{1/3})$ and
diffuseness $a_{\text{I}}$. The subscripts p and t refer to projectile and
target, respectively, and $A$ denotes the mass number. The potential
$V_{\text{fold}}(r)$ is obtained by folding the ground-state density with a
density-dependent $\alpha$-nucleon interaction

$V_{\text{fold}}(r)=-V\int
d^{3}r^{\prime}\rho(r^{\prime})\left[1-\beta\rho(r^{\prime})^{2/3}\right]\exp(-\mathit{z}^{2}/t^{2})~{},$
(2)

where $\mathit{z}=|r-r^{\prime}|$ is the distance between the centre of mass
of the $\alpha$ particle and a target nucleon, and $\rho(r^{\prime})$ is the
ground-state density of the target nucleus at the position $r^{\prime}$ of the
target nucleon. The parameters $\beta=1.9$ fm2 and range $t=1.88$ fm were
taken from Ref. satchler1997missing . The ground-state density $\rho(r)$ of
the target nucleus at the position $r$ is given by

$\rho(r)=\dfrac{\rho_{0}}{1+\exp\left(\frac{r-c}{a}\right)}~{},$ (3)

where the Fermi-distribution parameters $c$ and $a$ describe the half-density
radius and the diffuseness, respectively. Numerical values for 24Mg and 28Si
were taken from Ref. fricke1995nuclear .

DWBA calculations were carried out using the computer code PTOLEMY
rhoades1980techniques ; rhoades1980techniques2 . The Optical Model Parameters
(OMP) used in the DWBA calculations, viz. $r_{\text{0I}},a_{\text{I}},V$ and
$W$, for each nucleus are usually obtained by fitting elastic scattering cross
sections. The procedure is fully described in Ref. nolte1987global . However,
due to the lack of elastic scattering data in this study, the parameters were
taken from studies of the TAMU group on the same nuclei. This is justified by
the similarity of their beam energy ($240$ MeV) compared to this study. The
OMP parameters are summarised in Table 1.

Table 1: Optical model parameters taken from the TAMU group publications. Nucleus |  $V$ (MeV) |  $W$ (MeV) |  $r_{\text{0I}}$ (fm) |  $a_{\text{I}}$ (fm) | Refs.
---|---|---|---|---|---
24Mg | $41.02$ | $35.39$ | $0.934$ | $0.614\par$ | youngblood1999giant
28Si | $44$ | $32.5$ | $0.9306$ | $0.687\par$ | youngblood2002isoscalar

The measured cross sections can be converted to fractions of the EWSR
$(a_{0})$ by comparing with DWBA calculations assuming $100\%$ EWSR, which are
shown in Fig. 3. The strength is then calculated using the $a_{0}$ values and
is expressed as

$S_{0}(E_{\text{x}})=\dfrac{2\hbar^{2}A\langle
r^{2}\rangle}{mE_{\text{x}}}a_{0}(E_{\text{x}})~{},$ (4)

where $m$ is the nucleon mass, $E_{\text{x}}$ is the excitation energy
corresponding to a given state or energy bin, and $\langle r^{2}\rangle$ is
the second moment of the ground-state density. We use $\langle
r^{2}\rangle=9.945$ fm2 for 24Mg and $9.753$ fm2 for 28Si.

The DoS technique relies on the subtraction of the contributions of all
multipoles except $L=0$ from the $0^{\circ}$ spectrum using data from an angle
range defined by the first minimum of the $L=0$ contribution. It was shown
that the sum of angular distributions of all multipoles $L>0$ is similar at
the maximum and first minimum of the $L=0$ distribution GC_review2018 ;
DoSpaper . Hence, the subtraction of the inelastic spectrum associated with
the flat angular distribution (where $L=0$ is at a minimum from the
$0^{\circ}$ spectrum) is assumed to represent essentially the IS0 component
excited in $\alpha$-inelastic scattering close to $0^{\circ}$.

The direct subtraction of the angle cut spectrum from the spectrum at
$0^{\circ}\leq\theta_{\text{Lab}}\leq 1.91^{\circ}$ (blue) yields the black
difference spectra shown in the upper and lower panels of Fig. 4 for 24Mg and
28Si, respectively. The angle cut for each nucleus is indicated by the
highlighted yellow area in Fig. 3 and listed in Table 2. The angle
corresponding to the minimum of the $L=0$ distribution is also indicated.

Figure 3: DWBA calculations of the differential cross sections for the
24Mg($\alpha$, $\alpha^{\prime}$) (top panel) and 28Si($\alpha$,
$\alpha^{\prime}$) (bottom panel) reaction at $E_{\alpha}=196$ MeV for various
isoscalar electric multipoles. The calculations have been normalised to
$100\%$ of the appropriate EWSR at an excitation energy of $16.9$ MeV in 24Mg
and $17.8$ MeV in 28Si.

Figure 4: Double-differential cross sections for 24Mg($\alpha$,
$\alpha^{\prime}$) (top panel) and 28Si($\alpha$, $\alpha^{\prime}$) (bottom
panel) at $E_{\alpha}=196$ MeV. The blue and red spectra represent the data
acquired at $0^{\circ}\leq\theta_{\text{Lab}}\leq 1.91^{\circ}$ and at
$3.39^{\circ}\leq\theta_{\text{Lab}}\leq 4.4^{\circ}$ for 24Mg and
$3.03^{\circ}\leq\theta_{\text{Lab}}\leq 4.07^{\circ}$ for 28Si, respectively.
For clarity, these spectra are shifted by $40$ mb/sr MeV. The black spectra
represent the difference calculated using the DoS technique (see text). Table
2: Angle cuts implemented in the $4^{\circ}$ dataset to define the angular
region around the first $L=0$ minimum.

Nucleus |  24Mg |  28Si
---|---|---
Angle cut ($\theta_{\text{Lab}}$) | $3.39^{\circ}-4.4^{\circ}$ | $3.03^{\circ}-4.07^{\circ}$
Angle cut ($\theta_{\text{c.m.}}$) | $4^{\circ}-5.2^{\circ}$ | $3.5^{\circ}-4.7^{\circ}$
$L=0$ minimum ($\theta_{\text{c.m.}}$) | $4.6^{\circ}$ | $4.1^{\circ}$

IS0 strength distributions in 24Mg and 28Si were determined from the
difference spectra. The fraction of IS0 strength in EWSR per MeV was obtained
by dividing the extracted experimental IS0 component by the corresponding
integral value of $L=0$, $100\%$ EWSR obtained at the average angles
$\theta_{\text{c.m.}}^{\text{av.}}=1.13^{\circ}$ (24Mg) and $1.12^{\circ}$
(28Si) in the $0^{\circ}$ spectra. The results are shown in Figs. 5 and 6 for
$0.5$ MeV wide energy bins. The uncertainties shown include both systematical
and statistical errors. For comparison purposes, the IS0 strengths from the
RCNP experiments (kaw2013 ; 24Mg_GG_PLB2015 ; 24Mg_GG_PRC2016  for 24Mg and
28Si_PGG_PRC2016  for 28Si) and the TAMU experiments (youngblood2009isoscalar
for 24Mg and youngblood2007isoscalar  for 28Si) are also shown. The results
from Ref. youngblood2009isoscalar  are for the dataset where the out-of-plane
angle was measured. For better visibility the comparison of IS0 strength
distributions with the present results in Fig. 5 for 24Mg is split into two
parts. The upper part shows results from Refs. 24Mg_GG_PLB2015 ;
24Mg_GG_PRC2016 , Zabora2021  and the lower those from Refs. kaw2013  and
youngblood2009isoscalar .

Figure 5: IS0 strength distributions in 24Mg. The present iThemba LABS data
are shown as black filled circles. Also shown are the ($\alpha$,
$\alpha^{\prime}$) 24Mg_GG_PLB2015 ; 24Mg_GG_PRC2016  and (6Li, 6Li′) data
Zabora2021  from the RCNP shown as blue and red histograms, respectively (top
panel). The bottom panel displays results from TAMU youngblood2009isoscalar
(magenta histogram) and a different measurement at RCNP kaw2013  shown by dark
cyan histogram. Figure 6: IS0 strength distribution in 28Si. The present
iThemba data are shown as black filled circles. Data shown as blue and magenta
histograms with vertical error bars are from RCNP 28Si_PGG_PRC2016  and TAMU
youngblood2007isoscalar .

Data for 24Mg from the present study generally agree within error bars with
the RCNP and TAMU data. However, above $17$ MeV the IS0 strength from the TAMU
group is consistently higher than both the iThemba LABS and RCNP strengths.
These discrepancies might be attributed to the background-subtraction method
employed by the TAMU group. At the same time, Fig. 5 shows that the RCNP
24Mg_GG_PLB2015 ; 24Mg_GG_PRC2016 , TAMU youngblood2009isoscalar  and present
iThemba LABS data for 24Mg give somewhat different strength distributions
below the main ISGMR region. In particular, the RCNP experiments
24Mg_GG_PLB2015 ; 24Mg_GG_PRC2016  suggest the highest IS0 structures at
$16-20$ MeV and a two times lower peak at $13-16$ MeV. A similar structure was
observed in the recent (6Li, 6Li′) experiment Zabora2021 . Instead, the TAMU
and iThemba LABS data (bottom panel of Fig. 5) give the highest IS0 peak at
about $13.8$ MeV and a broader peak with a maximum at about $18$ MeV. It is
remarkable that, unlike the RCNP data 24Mg_GG_PLB2015 ; 24Mg_GG_PRC2016 ,
another ($\alpha$, $\alpha^{\prime}$) experiment at RCNP with superior energy
resolution kaw2013  does show the strong narrow peak at $13.8$ MeV (bottom
panel of Fig. 5). Moreover, this peak was clearly observed in the early
($\alpha$, $\alpha^{\prime}$) Lu86  and (6Li, 6Li′) Denn95  experiments (see
the discussion in Sec. $4.2.3$ of Ref. Harakeh ). So most of the experimental
datasets confirm the existence of the strong narrow IS0 peak at $13.8-13.9$
MeV in 24Mg. As can be seen in the next section, this peak is of a crucial
importance for our present study. The differences between the various
datasets, especially for strong narrow peaks of IS0 strength, can most likely
be ascribed to differences in the experimental energy resolution and a subtle
interplay between binning effects and differences in the calibration of the
excitation energy.

As for 28Si (Fig. 6), the iThemba LABS results are mainly consistent with the
previous findings by the TAMU group, except for a slightly lower value of the
IS0 strength at the peak of the distribution. The results from RCNP, on the
other hand, exhibit almost a factor of two larger IS0 strength above $19$ MeV
than both the iThemba LABS and the TAMU results. One should also note that
some of the IS0 strength at lower excitation energy might not belong to the
ISGMR. It was suggested that, in 28Si, the $0^{+}$ states at $9.71$ MeV,
$10.81$ MeV, $11.14$ MeV, and $12.99$ MeV, i.e. most of the IS0 strength below
$15$ MeV, can be considered as potential band heads for superdeformed bands
adsley2017alpha .

Table 3: Excitation of prominent $J^{\pi}=0^{+}$ states in 24Mg. $S$(IS0) is the experimental IS0 strength exhausted by the state. $E_{\text{x}}$ (MeV)111Present experiment. |  $E_{\text{x}}$ (MeV)222From Ref. nndc . |  $E_{\text{x}}$ (MeV)333From Ref. adsley2021isoscalar . | % EWSR1 | % EWSR3 |  $S$(IS0) (fm4)1
---|---|---|---|---|---
$9.31(1)$ | $9.30539(24)$ |  | $1.0(1)$ | $1.4(3)$ | $19.9(20)$
$10.68(1)$ | $10.6797(4)$ |  | $0.31(5)$ | $0.29(6)$ | $5.32(61)$
$11.73(1)$ | $11.7281(10)$ |  | $0.77(11)$ | $1.0(2)$ | $12.1(16)$
$13.36(2)$ |  | $13.37(1)$ | $0.40(5)$ | $0.5(1)$ | $4.32(71)$
|  | $13.79(1)$ |  | $1.7(3)$ |
$13.87(2)$444The integration region for the $13.87(2)$ state from the present experiment includes both the $13.79$ MeV and $13.89$ MeV states from Ref. adsley2021isoscalar . | $13.884(1)$ | $13.89(1)$ | $2.8(3)\ref{footnote5}$ | $2.6(5)$ | $37.7(38)\ref{footnote5}$
$15.32(2)$ |  | $15.33(3)$ | $1.7(2)$ | $1.9(4)$ | $20.7(25)$
$15.78(2)$ |  | $15.79(3)$ | $0.40(5)$ | $1.1(2)$ | $4.6(10)$
Table 4: Same as in Table 3 but for excitation of $J^{\pi}=0^{+}$ states in
28Si.

$E_{\text{x}}$ (MeV)555Present experiment. $E_{\text{x}}$ (MeV)666From Ref.
adsley2017alpha . % EWSR5 % EWSR6 $S$(IS0) (fm4)5 $9.70(2)$ $9.71(2)$
$0.22(4)$ $0.38(8)$ $5.2(6)$ $10.81(2)$ $10.81(3)$ $0.27(4)$ $0.35(7)$
$5.7(6)$ $11.14(2)$ $11.142(1)$ $0.8(1)$ $0.9(2)$ $15.3(17)$ $13.00(2)$
$12.99(2)$ $0.95(12)$ $0.8(2)$ $16.9(18)$ $15.03(3)$ $15.02(3)$ $0.40(9)$
$0.8(2)$ $6.0(15)$ $15.77(3)$ $0.7(1)$ $9.7(16)$

With the good energy resolution of the present experiment, $J^{\pi}=0^{+}$
states could be resolved up to $16$ MeV in 24Mg and 15 MeV in 28Si. The
corresponding strengths $S$(IS0) and % EWSR exhausted by the strongest
discrete states evident in Fig. 4 are summarised in Tables 3 and 4,
respectively. The values obtained in this study are in good agreement with
results from Ref. adsley2021isoscalar  which analysed angular distributions of
individual states, employing a different DWBA code as well as OMPs.

##  IV QRPA calculations and comparison with experiment

###  IV.1 Details of the calculations

The calculations were performed within the QRPA model Rep_arxiv ; Rep_EPJA17 ;
Rep_sePRC19 ; Kva_seEPJA19  based on the Skyrme functional Ben_RMP03 ;
Stone_PPNP07 . The model is fully self-consistent. Both the mean field and the
residual interaction are derived from the initial Skyrme functional. The
residual interaction takes into account all the terms following from the
Skyrme functional and Coulomb (direct and exchange) parts. Both particle-hole
and particle-particle channels are included Rep_EPJA17 . Pairing is treated at
the BCS level Rep_EPJA17 . Spurious admixtures caused by pairing-induced
violation of the particle number are removed using the method from Ref.
Kva_seEPJA19 .

A representative set of Skyrme forces is used, see Table 5. For our aims, the
most important characteristics of the forces are the nuclear incompressibility
$K_{\infty}$ and the isoscalar effective mass $m^{*}_{0}/m$ affecting ISGMR
and ISGQR energies, respectively. The corresponding values are listed in Table
5. We employ the standard force SkM* SkM*  and the most recent force SV-bas
SVbas . Both forces were previously used in the analysis presented in
24Mg_GG_PLB2015 ; 24Mg_GG_PRC2016 ; 28Si_PGG_PRC2016 ; 24Mg_PPJC2016 .
Further, we use the earlier force SkT6 SkT6  and more recent force SV-mas10
SVbas  which both have a large effective mass, $m^{*}_{0}/m=1$, and so are
favourable for the description of the ISGQR E0_Colo_PLB2020 ; Ne_odd ; Kureba
. Finally, we exploit the force SkPδ SkP  which has a very low
incompressibility $K_{\infty}=202$ MeV. In a recent study of the ISGMR in Mo
isotopes, this force gave the best results E0_Colo_PLB2020 .

The nuclear mean field and pairing are computed by the code SKYAX Skyax  using
a two-dimensional grid in cylindrical coordinates. The calculation box extends
up to three nuclear radii and the grid step size is $0.4$ fm. The axial
quadrupole equilibrium deformation is obtained by minimisation of the energy
of the system. As shown in Fig. 7, both 24Mg and 28Si have oblate and prolate
minima. However, in the ground state, 24Mg is prolate and 28Si is oblate. As
shown in Table 6, the calculated deformation parameters $\beta$ somewhat
underestimate the experimental values nndc . This is typical for light
deformed nuclei, see e.g. Ref. Ne_PRL18 . For all the applied Skyrme forces,
the pairing in 24Mg and 28Si was found to be very weak. This is explained by
the sparse single-particle spectrum in light nuclei.

Table 5: Incompressibility $K_{\infty}$ and isoscalar effective mass $m^{*}_{0}/m$ for the Skyrme forces SV-bas, SkM*, SkPδ, SkT6 and SV-mas10 used in the present analysis. | SV-bas | SkM* | SkPδ | SkT6 | SV-mas10
---|---|---|---|---|---
$K_{\infty}$ (MeV) | $234$ | $217$ | $202$ | $236$ | $234$
$m^{*}_{0}/m$ | $0.9$ | $0.79$ | $1$ | $1$ | $1$

The calculations use a large configuration space. The single-particle spectrum
extends from the bottom of the potential well up to $30$ MeV. For example,
SkPδ calculations use $404$ proton and $404$ neutron single-particle levels
for 24Mg and $422$ proton and $422$ neutron single-particle levels for 28Si.
The two-quasiparticle ($2$qp) basis in QRPA calculations with SkPδ extends up
to $70$ MeV. It includes $1056$ proton and $956$ neutron $K^{\pi}=0^{+}$
states in 24Mg and $1142$ proton and $1002$ neutron states in 28Si. The
isoscalar monopole energy-weighted sum rule EWSR(IS0)$=2\hbar^{2}/m\cdot
A\langle r^{2}\rangle_{0}$ is exhausted by $97-99$%.

QRPA strength functions for isoscalar monopole $(L=0)$ and quadrupole $(L=2)$
transitions have the form

$S_{L}(E)=\sum_{K=0}^{L}(2-\delta_{K,0})\sum_{\nu\in
K}|\langle\nu|{\hat{O}}_{LK}|0\rangle|^{2}\xi_{\Delta}(E-E_{\nu}),$ (5)

where $\nu$ labels QRPA eigenvalues $|\nu\rangle$ with energies $E_{\nu}$, and
$|0\rangle$ is the QRPA ground state. The monopole and quadrupole isoscalar
transition operators are $\hat{O}_{00}=\sum_{i}^{A}r^{2}_{i}$ and
$\hat{O}_{20}=\sum_{i}^{A}r^{2}_{i}Y_{20}(\hat{r}_{i})$, respectively. For
convenient comparison with the experimental data, the strength is smoothed by
a Lorentz function
$\xi_{\Delta}(E-E_{\nu})=\Delta/(2\pi[(E-E_{\nu})^{2}-\Delta^{2}/4])$ with an
averaging parameter $\Delta=1$ MeV. The dimension of the strength functions is
fm4/MeV. We also present 2qp strength functions calculated without the
residual interaction. In this case, the number $\nu$ in Eq. (5) labels 2qp
states.

Figure 7: Total energy of 24Mg (left) and 28Si (right), calculated in the framework of QRPA with the forces SkPδ, SkM* and SVbas as a function of the deformation parameter $\beta$. Table 6: Experimental and calculated deformation parameters $\beta$ in 24Mg and 28Si. | Exp.777From Ref. nndc . | SV-bas | SkM* | SkPδ | SkT$6$ | SV-mas$10$
---|---|---|---|---|---|---
24Mg | $0.613$ | $0.527$ | $0.522$ | $0.545$ | $0.506$ | $0.518$
28Si | $-0.412$ | $-0.308$ | $-0.291$ | $-0.322$ | $-0.311$ | $-0.295$
Figure 8: Upper panels: QRPA (solid black line) and 2qp (short-dash red line)
IS0 strength functions calculated with the force SVbas ($K_{\infty}=234$ MeV)
in 24Mg (left) and 28Si (right) and comparison with the present iThemba LABS
experimental data shown as black filled circles. Middle panels: comparison
with RCNP 24Mg_GG_PRC2016 ; 28Si_PGG_PRC2016  (black empty squares) and TAMU
youngblood2009isoscalar ; youngblood2007isoscalar  (red filled diamonds)
experimental data. Bottom panels: QRPA (solid black line) and 2qp (short-dash
red line) IS20 strengths.

###  IV.2 Strength distribution and IS0/IS2 coupling

The QRPA IS0 strength functions from Eq. (5) are compared with the
experimental data in Figs. 8 \- 11. The iThemba LABS experimental data are
shown by black filled circles where each data point accumulates the IS0
strength in an energy interval of $0.5$ MeV.

In Fig. 8, the strength functions calculated with the force SV-bas are shown.
In our set of Skyrme forces, SV-bas has a large (though rather typical for
Skyrme forces) incompressibility $K_{\infty}=234$ MeV. The calculated QRPA IS0
strengths (blue solid line) in 24Mg and 28Si are separated into a narrow
structure at $\sim 14$ MeV in 24Mg and $\sim 18$ MeV in 28Si and a broad
structure at higher energy, which is a typical picture for deformed nuclei
E0_Kvasil_PRC2016 . It is instructive to compare QRPA IS0 strength with the
2qp IS0 one (red dashed line) obtained without the residual interaction. We
see a strong collective effect: the residual interaction essentially shifts
down the IS0 strength and creates collective peaks at $\sim 14$ MeV in 24Mg
and $\sim 18$ MeV in 28Si.

The bottom panels of Fig. 8 show the $LK=20$ branches of the ISGQR (see
Appendix A for discussion of the deformation splitting of the ISGQR in prolate
and oblate nuclei) in the QRPA and 2qp cases. Here, we also observe a strong
collective effect. Additionally, one observes that the main IS20 QRPA peak
lies at the same energy as the narrow IS0 QRPA peak. Thus, both peaks are
related, i.e., we observe a clear deformation-induced coupling of the IS0 and
IS20 modes. The larger the nuclear deformation, the more strength is
concentrated in the narrow IS0 peak E0_Kvasil_PRC2016 . Altogether, we see
that the narrow IS0 peaks in 24Mg and 28Si have a dual (collective effect +
deformation-induced IS0/IS20 coupling) origin.

Now the question arises which structures in the experimental data should be
associated with the calculated IS0 narrow peak. To inspect this point, let us
consider in Fig. 8 the experimental data from iThemba LABS (upper panel), RCNP
24Mg_GG_PRC2016 ; 28Si_PGG_PRC2016  and TAMU youngblood2009isoscalar ;
youngblood2007isoscalar  (middle panel). In 28Si, all the data give a
distinctive strong narrow peak at $17-19$ MeV. In 24Mg, the issue is more
complicated. Here, the iThemba LABS and TAMU data give a distinctive narrow
peak at $13-14$ MeV but the RCNP data 24Mg_GG_PRC2016  do not. At the same
time, a hump at $13-14$ MeV also appears in the (6Li, 6Li′) data Zabora2021 .
Moreover, it is clearly observed at $13.8$ MeV in the RCNP ($\alpha$,
$\alpha^{\prime}$) data of Ref. kaw2013 , which have an energy resolution
comparable to the present experiment. Finally, the narrow peak at $13-14$ MeV
was observed in early ($\alpha$, $\alpha^{\prime}$) Lu86  and (6Li, 6Li′)
Denn95  experiments. So, in 24Mg, the peak at $13-14$ MeV is a good candidate
for the comparison with the narrow IS0 peak in our QRPA calculations and the
analysis of IS0/IS2 coupling.

Note that the proper choice of the experimental IS0 peak for the comparison
with theory is of crucial importance in the present study. As argued above,
this peak arises due to IS0/IS2 coupling in deformed nuclei and its energy
location can be used to find the optimal Skyrme force and thereby determine
the relevant incompressibility $K_{\infty}$ and isoscalar effective mass
$m^{*}_{0}/m$. In Ref. 24Mg_GG_PRC2016 , the local maximum at $16-17$ MeV was
assumed to be the narrow peak, which resulted in the preference of the force
SkM* ($K_{\infty}=217$ MeV, $m^{*}_{0}/m=0.79$). Below we will show that in
24Mg, the experimental peak at $13-14$ MeV in the iThemba LABS and TAMU data
is a better suited candidate. This will change the predicted values of
$K_{\infty}$ and $m^{*}_{0}/m$.

To justify the latter, let us compare the calculated and experimental IS0
strengths in Fig. 8. It is seen that the calculated narrow IS0 peaks lie
higher by $1.5-2$ MeV than the experimental peaks at $13.8$ MeV (24Mg) and
about $18$ MeV (28Si). If we additionally upshift the calculated IS0 strength
in both nuclei, we could match the RCNP structure at $16-17$ MeV in 24Mg but,
simultaneously, would essentially worsen the description of the strength in
28Si. Instead, if we assume the $13.8$ MeV peak as the relevant narrow
structure in 24Mg, then it remains possible to describe IS0 results in 24Mg
and 28Si consistently by an equal downshift of the QRPA narrow peaks.

Figure 9: Same as upper and lower panels of Fig. 8 but for the force SkM* with
$K_{\infty}=217$ MeV.

Figure 8 shows that SV-bas generally well describes the main ISGMR located at
$16-25$ MeV in 24Mg and $20-25$ MeV in 28Si. The only noticeable discrepancy
is that the theory noticeably underestimates IS0 strength at $16-20$ MeV in
24Mg where the experiment shows a sequence of prominent peaks. This
discrepancy can be caused by the disregard in our calculations of the Coupling
with Complex Configurations (CCC). The CCC could redistribute the strength,
reducing the height of the narrow IS0 peak and shifting a part of its strength
to the range $16-20$ MeV.

Figure 10: Same as in Fig. 9 but for the force SkPδ with $K_{\infty}=202$ MeV.

Previous work has shown that the lower $K_{\infty}$, the smaller the ISGMR
centroid energy GC_review2018 ; E0_Kvasil_PRC2016 . In Fig. 8, SV-bas gives
IS0 narrow peaks at too high excitation energy. So it is worth to try a force
with a smaller $K_{\infty}$. In Fig. 9, results for the SkM* force with
$K_{\infty}=217$ MeV are shown. As in Fig. 8, we see a strong collective
effect. The energy of the narrow IS0 peak again coincides with the energy of
the quadrupole $LK=20$ branch, which means that this peak is produced by the
IS0/IS2 coupling. However, SkM* does not improve the results: the energy of
the IS0 narrow peak is still noticeably overestimated. The downshift of the
IS0 strength due to the smaller $K_{\infty}=217$ MeV is countered by an
upshift due to the smaller value of the effective mass $m^{*}_{0}/m=0.79$ (see
the discussion of the impact of the effective mass, in connection with Fig.
12).

Next, in Fig. 10, we present results for the force SkPδ with a low
incompressibility $K_{\infty}=202$ MeV and high effective mass
$m^{*}_{0}/m=1$. Now both factors downshift the IS0 strength as compared with
the SV-bas case. An improved agreement with the experimental IS0 strength
distributions is observed: both in 24Mg and 28Si the calculated IS0 narrow
peaks now match the experimental structures. Note that SkPδ also provides the
best description of the ISGMR in deformed Mo isotopes E0_Colo_PLB2020 . In
Fig. 10, the narrow peaks are again produced by IS0/IS2 coupling and a strong
collective effect takes place.

Figure 11: QRPA IS0 strength functions (solid lines) calculated with the
Skyrme forces SkT6 (upper panels) and SV-mas10 (bottom panels).

In Figs. 8 \- 10, the quadrupole IS20 peak in 24Mg is stronger and has a lower
energy than in 28Si. As a result, the narrow IS0 peak in 24Mg is more
separated from the main ISGMR than in 28Si. This is explained by the larger
absolute quadrupole deformation of 24Mg as compared with 28Si, which results
in a stronger downshift of the IS20 peak in 24Mg. Besides, Figs. 8 \- 10 show
that IS0/IS2 coupling is relevant also for lower-energy states.

Due to the IS0/IS2 coupling, the energy of the narrow peak is sensitive to the
isoscalar effective mass $m^{*}_{0}/m$ which affects the ISGQR energy
E0_Colo_PLB2020 ; Ne_odd . In general, the larger $m^{*}_{0}/m$, the smaller
the ISGQR centroid energy Ne_odd . So, in principle, one can try to downshift
the energy of the narrow IS0 peak solely using a large value of $m^{*}_{0}/m$.
In Fig. 11, the forces SkT6 ($K_{\infty}=236$ MeV) and SV-mas10
($K_{\infty}=234$ MeV), which both have an effective mass $m^{*}_{0}/m=1$, are
applied for the calculation of the IS0 strength. It is seen that these two
forces still overestimate the energies of the narrow IS0 peaks like in the
cases of SV-bas and SkM*. Thus, the optimum Skyrme force also needs a lower
value of $K_{\infty}$.

We note that a QRPA calculation of the ISGMR strength in 24Mg with a Gogny
force is discussed in Ref. Zabora2021 . There, a reasonable agreement with the
data could only be achieved after an ad-hoc upshift of the IS0 strength
distribution by $2$ MeV.

###  IV.3 IS2 strength

Since the energy of the narrow monopole peak is determined by the $K=0$ branch
of the ISGQR, it is important to consider the ISGMR and ISGQR simultaneously.
In particular, we should see how well we can describe the ISGQR and its
deformation splitting in 24Mg and 28Si. This is investigated in Appendix A for
various values of the deformation parameter $\beta$. For 24Mg, only prolate
deformations are inspected. For 28Si, where oblate and prolate deformations
for excited states were predicted adsley2017alpha ; Taniguchi_PRC2009 ;
Taniguchi_PLB2020 ; Afan+Ray , both types of deformations are considered. In
24Mg, the calculations show the behaviour expected for prolate deformation
splitting where the energies of the branches $LK=20,21,$ and $22$ follow the
relation $E_{20}<E_{21}<E_{22}$. The branch $LK=20$ is strong. It is
downshifted by $4-5$ MeV when increasing the deformation from $\beta=0.1$ to
$0.522$. The same splitting features are seen in prolate 28Si. However, in
oblate 28Si, the deformation splitting of ISGQR looks surprising. The branch
$LK=20$ is not upshifted, as would be expected, but downshifted (by $3-4$ MeV
from $\beta=-0.1$ to $-0.412$) and becomes rather weak.

Figure 12: ISGQR in 24Mg (left) and 28Si (right), calculated with the forces
SkM* (upper panels), SV-bas (middle panels) and SkPδ (bottom panels). Energy-
weighted strength functions $S(\text{IS2})$ for the branches $LK=20$ (thin
solid black line), $21$ (dashed red line) and $22$ (dotted blue line) as well
as for the summed $20$+$21$+$22$ strength (bold green line) are shown. The
experimental data are taken from Ref. Zabora2021  for 24Mg and Ref.
youngblood2007isoscalar  for 28Si. For better visibility, the latter are
multiplied by a factor of 1$0$.

These features of the deformation splitting are illustrated in Fig. 12, where
SkM*, SV-bas and SkPδ QRPA results are compared with ISGQR data obtained in
the reactions (6Li, 6Li′) for 24Mg Zabora2021  and ($\alpha$,
$\alpha^{\prime}$) for 28Si youngblood2007isoscalar . For better visibility,
the experimental values for 28Si are multiplied by a factor of $10$. All three
forces reasonably reproduce the IS2 experimental data. In 24Mg, the best
description is obtained with SV-bas. In 28Si, SkM* and SV-bas give reasonable
results. It is seen that the humps of the experimental strength at $14$ and
$19$ MeV in 24Mg are produced by the $LK=20$ and $21$ branches, respectively.
In 28Si, the main peak at $19$ MeV is produced by the $LK=22$ branch. Figure
12 demonstrates a regular dependence of the ISGQR features on increasing the
isoscalar effective mass $m^{*}_{0}/m$ from $0.79$ (SkM*) to $0.9$ (SV-bas) to
$1.0$ (SkPδ). The larger the value of $m^{*}_{0}/m$, the stronger the
deformation (and the deformation splitting) and the larger the energy
downshift of the ISGQR. The force SkPδ, being best in description of the
ISGMR, turns out to be not very good for the ISGQR. A consistent description
of the ISGMR and ISGQR would need a Skyrme force with $m^{*}_{0}/m\approx 0.9$
and $K_{\infty}<217$ MeV (the same conclusion was drawn earlier in the study
of ISGMR/ISGQR in Mo isotopes E0_Colo_PLB2020 ). The forces used above do not
fulfil these requirements.

Note that our QRPA IS2 strength functions are computed for band heads
$I^{\pi}K=0^{+}0,1^{+}1,2^{+}2$ while IS2 experimental data are obtained for
the states $I^{\pi}K=2^{+}0,2^{+}1,2^{+}2$, i.e. for rotational states in
$K=0$ and $1$ cases. So, for a more accurate comparison with the experiment,
the calculated IS2 $(K=0,1)$ strengths should be shifted to a higher energy by
the rotational correction. It can be estimated roughly from the energy of the
$2^{+}0$ state in the ground-state rotational band, which is $1.368$ MeV for
24Mg and $1.779$ MeV for 28Si nndc . Though the corrections are significant,
they do not affect the conclusions drawn from Fig. 12.

Table 7: Peak energy $E_{\text{p}}$ of the narrow IS0 resonance and summed
strength $\sum_{\nu}B_{\nu}(\text{IS0})$ in the energy interval $9-25$ MeV
from various Skyrme forces compared with the present experimental data.

24Mg 28Si $K_{\infty}$ $E_{\text{p}}$ $\sum B(\text{IS0})$ $E_{\text{p}}$
$\sum B(\text{IS0})$ (MeV) (MeV) (fm4) (MeV) (fm4) Exp. 13.75(2) 728(41)
17.75(3) 895(40) SkPδ 202 14.3 796 17.0 908 SkM* 217 15.6 706 18.0 780 SVbas
234 15.4 634 19.4 685

Table 8: Energy-weighted strengths $\sum_{\nu}E_{\nu}B_{\nu}(\text{IS0})$ for 24Mg and 28Si, summed in the energy intervals $0-16$ MeV, $16-25$ MeV, $25-70$ MeV, and $0-70$ MeV. The strengths are given in $\%$ of the monopole EWSR (see text for more detail). In the interval $0-16$ MeV, the experimental strength is measured only in the interval $9-16$ MeV. |  24Mg |  28Si
---|---|---
|  $0-16$ |  $16-25$ |  $25-70$ |  $0-70$ |  $0-16$ |  $16-25$ |  $25-70$ |  $0-70$
Exp. | 15(1) | 55(4) | 0 | 70(4) | 7(1) | 68(4) | 0 | 75(4)
SkPδ | 38 | 35 | 25 | 98 | 11 | 63 | 25 | 97
SkM* | 31 | 37 | 29 | 97 | 5 | 64 | 29 | 98
SV-bas | 35 | 25 | 37 | 97 | 6 | 55 | 38 | 99

###  IV.4 Integral IS0 strength

For further comparison, it is worth considering the integrated IS0 strengths
$\sum_{\nu}B_{\nu}(\text{IS0})$ in the energy range $9-25$ MeV covered by the
present experiment. Here,
$B_{\nu}(\text{IS0})=|\langle\nu|{\hat{O}}_{2K}|0\rangle|^{2}$ is the reduced
probability of the IS0 transition from the ground to $\nu$-th QRPA state. The
results for the main three forces used in the present study (SkPδ, SkM* and
SV-bas) are shown in Table 7. One observes an anticorrelation between
$K_{\infty}$ and $\sum_{\nu}B_{\nu}(\text{IS0})$: the calculated
$\sum_{\nu}B_{\nu}(IS\text{0})$ steadily decreases from SkPδ to SV-bas, i.e.
with increasing $K_{\infty}$ from $202$ MeV to $234$ MeV. As demonstrated for
the strength distributions in Figs. 8 \- 11, the best agreement with
experiment is obtained for the force SkPδ. Indeed, this force gives a
reasonable reproduction of the summed strength in 24Mg (together with the
force SkM*) and the best reproduction of the summed strength in 28Si. Besides,
SkPδ describes well the peak energy $E_{\text{p}}$ of the narrow IS0 resonance
in 24Mg and reasonably in 28Si. So, with this force, we simultaneously
describe the energy of the narrow IS0 peak and integral strength
$\sum_{\nu}B_{\nu}(\text{IS0})$. To the best of our knowledge, this is
achieved for the first time.

It is also interesting to inspect contributions of IS0 strength from different
energy intervals to the energy-weighted sum rule EWSR=$2\hbar^{2}/m\cdot
A\langle r^{2}\rangle_{0}$ summarised in Table 8. It is seen that the
experimental strengths exhaust about $70\%$ of the EWSR in 24Mg and $77\%$ in
28Si. The forces SkPδ, SkM* and SV-bas give rather similar results. All
calculations exhaust close to $100\%$ EWSR in the model space $0-70$ MeV. In
24Mg, the calculations overestimate the experimental strength at $0-16$ MeV
(low-energy IS0 strength and the narrow peak) and underestimate it at $16-25$
MeV (main part of ISGMR). It seems that the Skyrme forces redistribute too
much strength from the main ISGMR to the narrow IS0 peak. A large part of the
strength, $25-37\%$, is located above $25$ MeV, i.e. beyond the scope of the
iThemba LABS experiment. For 28Si, the agreement of the experiment and theory
is much better. The calculations reproduce rather well a small strength at
$0-25$ MeV and its major part at $16-25$ MeV.

Table 9: Characteristics of low-energy $K^{\pi}=0^{+}$ states with excitation energy $E_{\nu}>5$ MeV and reduced transition probability $B(\text{IS0})>40~{}\rm{fm}^{4}$ in 24Mg and 28Si, calculated within QRPA with the force SkPδ. For each state, we show the excitation energy, the reduced transition probabilities $B(\text{IS0})$ and $B(\text{IS20})$, and the main $2$qp components (contribution to the state norm in $\%$, and structure in terms of Nilsson asymptotic quantum numbers). Nucleus | $E_{\nu}$ | $B(\text{IS0})$ | $B(\text{IS20})$ | main $2$qp components
---|---|---|---|---
| (MeV) | (fm4) | (fm4) |  $\%$ |  $[N,n_{z},\Lambda]$
| 5.8 | 51 | 56.5 | 49 | pp $[220\uparrow-211\downarrow]$
| 12.9 | 75 | 11.7 | 96 | pp $[211\uparrow-431\uparrow]$
24Mg | 13.5 | 77 | 26.5 | 92 | pp $[220\uparrow-440\uparrow]$
| 14.3 | 166 | 77.1 | 40 | pp $[211\uparrow-411\uparrow]$
| 14.5 | 81 | 40.0 | 48 | pp $[211\uparrow-411\uparrow]$
| 15.3 | 42 | 3.0 | 91 | pp $[200\uparrow-431\downarrow]$
| 16.6 | 59 | 7.1 | 90 | pp $[202\uparrow-413\downarrow]$
28Si | 16.9 | 91 | 10.5 | 68 | nn $[200\uparrow-411\downarrow]$
| 17.3 | 47 | 4.3 | 39 | nn $[200\uparrow-420\uparrow]$
| 17.6 | 62 | 6.0 | 75 | pp $[202\downarrow-431\uparrow]$
| 17.8 | 82 | 18.8 | 22 | pp $[200\uparrow-600\uparrow]$

###  IV.5 Features of individual IS0 transitions

The energy range $9-25$ MeV embraces many QRPA $0^{+}$ states. For example, in
the SkPδ case, we have $631$ states in 24Mg and $537$ states in 28Si. It is
instructive to consider in more detail QRPA states which form the narrow IS0
peak and strong IS0 excitations at a lower energy. In Table 9, the
characteristics of IS0 states in the energy range $5-18$ MeV with a strength
$B(\text{IS0})>40$ fm4 are presented. In 24Mg, there are two transitions with
a large monopole strength below $13$ MeV. The three transitions at energies
between $13.5$ MeV and $14.5$ MeV form the strong narrow IS0 peak shown in
Fig. 10. Following our calculations, two of these states, at $14.3$ and $14.5$
MeV, are very collective (composed by many 2qp components with the largest 2qp
contributions to the state norm as $40\%$ and $48\%$, respectively), which
justifies the collective effect mentioned above. Most probably just these two
states with close energies correspond to the $13.87$ MeV level from the
iThemba LABS data (see Table 3). It is difficult to establish one-to-one
correspondence between the measured transitions in Table 3 and calculated
transitions in Table 9 since our Skyrme QRPA calculations neglect some
important effects (coupling with complex configurations and shape coexistence)
and thus have not enough accuracy for such a detailed comparison. For 24Mg and
28Si, there are attempts of one-to-one comparison of the experimentally
observed discrete $0^{+}$-levels using Gogny QRPA Zabora2021  and
Antisymmetrised Molecular Dynamics (AMD) Chiba_PRC2015  calculations. However,
these calculations achieve a reasonable agreement with the data using only an
ad-hoc upshift of the IS0 strength distribution by $2$ MeV.

All the states in Table 9 have large values for $0^{+}_{\text{gs}}\to
2^{+}0_{\nu}$ isoscalar quadrupole transitions with $\Delta K=0$. This once
more confirms the strong IS0/IS2 coupling in strongly prolate 24Mg. The main
IS0 resonance at $18-25$ MeV (not considered in Table 9) is formed by a large
number of mildly collective states with $B(\text{IS0})<1$ fm4. The results for
strongly oblate 28Si generally agree with those for 24Mg. However, as seen
from Table 9, in 28Si our calculations do not predict monopole states with a
high $B(\text{IS0})$ and $B(\text{IS2},K=0)$ below $15$ MeV.

###  IV.6 Cluster features of the narrow IS0 peak in 24Mg

The strong narrow peak at $13.87$ MeV in 24Mg lies close to the thresholds of
cluster configurations ${}^{12}\text{C}$+${}^{12}\text{C}$ and
${}^{16}\text{O}+2\alpha$ and, following the coincidence experiment kaw2013 ,
should exhibit cluster features. To check this point, we calculated the
density distribution of the QRPA $14.3$ MeV $0^{+}$ state (assuming
correspondence to the observed $13.87$ MeV excitation) and compared it with
the density of the ground state. In Fig. 13, both densities are exhibited in
the $x-z$ plane where $z$ is the symmetry axis. Another relevant collective
$0^{+}$ state at $14.5$ MeV has a similar density contribution and so is not
considered here.

Figure 13: Density profiles in $x-z$ plane, calculated with the force SkPδ in
24Mg for the ground state (left) and excited QRPA $K^{\pi}=0^{+}$ state at
$14.3$ MeV (right).

In the ground state, prolate 24Mg has two spots of the enhanced density in the
pole regions. These spots persist, though being somewhat shrunk, in the $14.3$
MeV $0^{+}$ state. The density spots can be considered as precursors of
clustering. They can be treated as predecessors of two ${}^{12}\text{C}$
clusters or, to a lesser extent, of two $\alpha$-particles with 16O. Note that
the plot for $14.3$ MeV state agrees well with the
${}^{12}\text{C}$+${}^{12}\text{C}$ density plots in Ref. Chiba_PRC2015  where
the cluster features of 24Mg are analysed within the AMD method. The
similarity of the density plots for the ground and excited states can be
explained by a dual nature (mean field + clustering) of the ground state of
light nuclei, see the discussion and further citation in Ref. Chiba_PRC2015 .
Altogether we see that the strong narrow IS0 peak in 24Mg might combine both
mean-field (IS0/IS2 coupling) and cluster features. The nucleus 28Si also
demonstrates cluster properties (see e.g. discussions in Refs.
adsley2021isoscalar ; Taniguchi_PRC2009 ). However analysis of these
properties is complicated by the shape coexistence in this nucleus and so is
skipped here.

##  V Conclusions

The isoscalar monopole strength in the energy interval $9\leq E_{\text{x}}\leq
25$ MeV in 24Mg and 28Si has been investigated using $\alpha$-particle
inelastic scattering with a $196$ MeV beam at scattering angles
$\theta_{\text{Lab}}=0^{\circ}$ and $4^{\circ}$. The K$600$ magnetic
spectrometer and the Separated Sector Cyclotron (SSC) at iThemba LABS were
used.

The DoS technique was applied in order to extract the IS0 strength
distributions. The difference spectrum obtained for each nucleus was converted
to the fraction of the IS0 EWSR per bin and compared with MDA analysis results
obtained by the TAMU youngblood2009isoscalar ; youngblood2007isoscalar  and
RCNP 24Mg_GG_PLB2015 ; 24Mg_GG_PRC2016 ; 28Si_PGG_PRC2016 ; Zabora2021 ;
kaw2013  groups. Overall, the strength distributions obtained in this study
show a reasonable agreement with results from both groups, with exception of
the RCNP data 24Mg_GG_PLB2015 ; 24Mg_GG_PRC2016  for IS0 structures in the
energy range $13-18$ MeV in 24Mg. At the same time, we find good agreement for
this energy range with another $(\alpha,\alpha^{\prime})$ experiment performed
at RCNP kaw2013 .

The extracted IS0 strength distributions were compared to calculations
performed in the framework of QRPA. A representative set of Skyrme forces
(SkM*, SV-bas, SkPδ, SkT6 and SV-mas10) with different incompressibility
$K_{\infty}$ values and isoscalar effective masses $m^{*}_{0}/m$ was used.
Similar to a recent study for Mo isotopes E0_Colo_PLB2020 , the present
iThemba LABS experimental data for the ISGMR in 24Mg and 28Si are best
described by the force SkPδ with a low incompressibility $K_{\infty}=202$ MeV.
This force allows the reproduction of both i) the energy of the narrow IS0
peaks at $13.8$ MeV in 24Mg and $18$ MeV in 28Si and ii) the integral IS0
strengths $\sum B(\text{IS0})$.

Figure 14: ISGQR branches $LK=20$ (solid black line), $21$ (dashed red line)
and $22$ (dotted blue line) in 24Mg and 28Si, calculated with the force SkM*
for different quadrupole deformations. For 28Si, both oblate (middle panels)
and prolate (right panels) deformations are considered.

In the theoretical analysis, the main attention was paid to the narrow IS0
peak constituting a convenient and sensitive test case to determine the
nuclear incompressibility in deformed nuclei. The comparison of IS0 and IS2
strength distributions justifies that the narrow IS0 peak appears due to the
deformation-induced coupling between the ISGMR and the $K=0$ branch of the
ISGQR. The important role of a large collective downshift of the IS0 strength
(due to the isoscalar residual interaction) in the production of the narrow
IS0 peak is demonstrated. In 24Mg, the narrow IS0 peak lies close to the
${}^{12}\text{C}$+${}^{12}\text{C}$ threshold kaw2013  and so, in accordance
with our analysis and observations kaw2013 , can demonstrate cluster features.

In connection with the strong IS0/IS2 coupling effect, we also performed a
detailed analysis of the ISGQR in 24Mg and 28Si. An unusual deformation
splitting in oblate 28Si was found. The calculations show that the quadrupole
$K=0$ branch is strongly downshifted in prolate 24Mg and slightly downshifted
in oblate 28Si. This explains why the narrow IS0 peak is well separated from
the main ISGMR in 24Mg but superimposed on the broad structure of the main
ISGMR in 28Si.

Our analysis demonstrates that the ISGMR in deformed nuclei must be described
simultaneously with the ISGQR. Only then can such a description be considered
as relevant and consistent. In our calculations, the Skyrme force SkPδ gives
the best results for a simultaneous reproduction of ISGMR and ISGQR. However,
even this force has shortcomings for the description of the ISGQR. There is a
need for new Skyrme parameterisations with $K_{\infty}<217$ MeV and
$m^{*}_{0}/m\approx 0.9$.

## ACKNOWLEDGEMENTS

The authors thank the Accelerator Group at iThemba LABS for the high-quality
dispersion-matched beam provided for this experiment. A.B. acknowledges
financial support through iThemba LABS, NRF South Africa. R.N. acknowledges
support from the NRF through Grant No. $85509$. The authors thank Prof P.-G.
Reinhard for the code SKYAX. This work was supported by the Deutsche
Forschungsgemeinschaft under contract SFB $1245$ (Project ID No. $79384907$)
and NRF-JINR grant JINR200401510986. V.O.N. and J.K. acknowledge the Votruba -
Blokhintsev (Czech Republic - BLTP JINR) grant. J.K. appreciates the support
by a grant of the Czech Science Agency, Project No. 19-14048S. V.O.N. and
P.v.N.-C. appreciate the support by a Heisenberg-Landau grant (Germany - BLTP
JINR). A.R. acknowledges support by the Slovak Research and Development Agency
under Contract No. APVV-20-0532 and by the Slovak grant agency VEGA (Contract
No. 2/0067/21). P. A. acknowledges support from the Claude Leon Foundation in
the form of a postdoctoral fellowship.

##  Appendix A ISGQR in 24Mg and 28Si

As shown in the main text, the $K=0$ branch of the ISGQR determines the energy
of the narrow IS0 peak. The energy of this branch is governed by the
deformation splitting of the ISGQR into $K=0,1$ and $2$ components. So, it is
worthwhile to investigate the deformation splitting of the ISGQR in 24Mg and
28Si in more detail. Figure 14 shows the strength distributions of the
branches for different axial quadrupole deformations. The calculations are
performed for the parametrisation SkM* as an example but other forces lead to
the same qualitative conclusions.

In the plots a)-d), the ISGQR strength distribution in prolate 24Mg is
exhibited at $\beta=0.1$, $0.3$, $0.522$ (calculated equilibrium value) and
0.613 (experimental value). We see a splitting picture typical for prolate
nuclei: the $K=0$ branch is downshifted from $21$ MeV ($\beta=0.1$) to $16$
MeV ($\beta=0.613$), the $K=1$ branch shows less downshift from $21$ to $19$
MeV, and the $K=2$ branch is upshifted from $22$ to $23$ MeV. The energy of
the $K=0$ branch is most sensitive to the deformation. However, even at
$\beta=0.613$, this branch does not approach an excitation energy $13-14$ MeV
where experiment gives the narrow monopole peak. So, to describe the
experimental data we need a force like SkPδ with a smaller incompressibility.

Table 10: EWSR exhaustion for the $LK=20$, $21$ and $22$ branches of the
ISGQR, calculated with the force SkM*.

24Mg, $\beta=0.522$ 28Si, $\beta=-0.291$ ($\rm{fm}^{4}$ MeV) (%)
($\rm{fm}^{4}$ MeV) (%) 20 4468 26.5 3650 17.5 21 7396 42.0 6998 33.7 22 5534
31.5 10144 48.8

In the plots e)-h), the dependence of the ISGQR in 28Si on an oblate
quadrupole deformation is shown. The deformations are $\beta=-~{}0.1$,
$-~{}0.291$ (calculated equilibrium values), $-~{}0.412$ (experimental value)
and $-~{}0.7$. In the oblate case, the deformation splitting demonstrates some
surprising features. The $K=0$ branch is not upshifted, as would be expected
from analytic estimates E0_Kvasil_PRC2016 , but downshifted from $21$ MeV to
$\sim 13$ MeV. The $K=0$ strength is redistributed to a lower-energy region
and significantly suppressed (see also Table 10). The $K=1$ branch keeps
almost the same energy. The $K=2$ branch is strongly downshifted from $21$ MeV
to $8-15$ MeV and its strength is significantly enhanced (see also Table 10).
At the calculated ($\beta_{\text{th}}=-~{}0.291$) and experimental
($\beta_{\text{ex}}=-~{}0.412$) values of the quadrupole deformation, the
$K=0$ peak approaches the energy $17-18$ MeV of the narrow monopole peak. So
the force SkM* can in principle describe the narrow IS0 peak in 28Si once we
take a proper value of oblate quadrupole deformation.

In the plots i)-l), the dependence of ISGQR in 28Si on a prolate quadrupole
deformation is inspected. This case is interesting since some studies predict
at $9-13$ MeV in 28Si superdeformed rotational bands based on prolate $0^{+}$
band heads adsley2017alpha ; Taniguchi_PRC2009 ; Taniguchi_PLB2020 ; Afan+Ray
. As seen from Fig. 7, the total energy of 28Si indeed has a local prolate
minimum. The plots i)-l) show that the deformation splitting in prolate 28Si
would be rather similar to that in prolate 24Mg. The $K=0$ branch in prolate
28Si indeed covers the relevant energy interval $8-16$ MeV. Our calculations,
therefore, do not contradict the suggestion adsley2017alpha ;
Taniguchi_PRC2009 ; Taniguchi_PLB2020 ; Afan+Ray  that 28Si can possess highly
prolate states at excitation energies $9-13$ MeV. Moreover, at $\beta=0.3$,
the $K=0$ strength covers the energy interval $16-19$ MeV where the
experimental narrow IS0 peak is located. So, our SkM* calculations do not
contradict that 28Si, being oblate in the ground state, can have a prolate
shape for $0^{+}$ excitations.

Altogether, the present analysis allows for the conclusion that the
description of the experimental narrow IS0 peaks: i) in the case of 24Mg,
needs a Skyrme force like SkPδ ($K_{\infty}=202$ MeV) with a low nuclear
incompressibility and ii) in the case of 28Si, can be achieved with both SkPδ
and SkM* ($K_{\infty}=217$ MeV) forces once we allow for oblate and prolate
quadrupole deformation.

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