# Electric Dipole Polarizability of 40Ca

R. W. Fearick  Department of Physics, University of Cape Town, Rondebosch
7700, South Africa  P. von Neumann-Cosel<EMAIL_ADDRESS>Institut für
Kernphysik, Technische Universität Darmstadt, 64289 Darmstadt, Germany  S.
Bacca  Institut für Kernphysik and PRISMA+ Cluster of Excellence, Johannes
Gutenberg-Universität Mainz, 55128 Mainz, Germany  Helmholtz-Institut Mainz,
Johannes Gutenberg Universität Mainz, 55099 Mainz, Germany  J. Birkhan
Institut für Kernphysik, Technische Universität Darmstadt, 64289 Darmstadt,
Germany  F. Bonaiti  Institut für Kernphysik and PRISMA+ Cluster of
Excellence, Johannes Gutenberg-Universität Mainz, 55128 Mainz, Germany  I.
Brandherm  Institut für Kernphysik, Technische Universität Darmstadt, 64289
Darmstadt, Germany  G. Hagen  Physics Division, Oak Ridge National Laboratory,
Oak Ridge, Tennessee 37831, USA  Department of Physics and Astronomy,
University of Tennessee, Knoxville, Tennessee 37996, USA  H. Matsubara
Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka
567-0047, Japan  Faculty of Radiological Technology, Fujita Health University,
Aichi 470-1192, Japan  W. Nazarewicz  Facility for Rare Isotope Beams and
Department of Physics and Astronomy, Michigan State University, East Lansing,
Michigan 48824, USA  N. Pietralla  Institut für Kernphysik, Technische
Universität Darmstadt, 64289 Darmstadt, Germany  V. Yu. Ponomarev  Institut
für Kernphysik, Technische Universität Darmstadt, 64289 Darmstadt, Germany
P.-G. Reinhard  Institut für Theoretische Physik II, Universität Erlangen,
D-91058 Erlangen, Germany  X. Roca-Maza  Dipartimento di Fisica, Università
degli Studi di Milano, 20133 Milano, Italy and INFN, Sezione di Milano, 20133
Milano, Italy  A. Richter  Institut für Kernphysik, Technische Universität
Darmstadt, 64289 Darmstadt, Germany  A. Schwenk  Institut für Kernphysik,
Technische Universität Darmstadt, 64289 Darmstadt, Germany  ExtreMe Matter
Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291
Darmstadt, Germany  Max-Planck-Institut für Kernphysik, Saupfercheckweg 1,
69117 Heidelberg, Germany  J. Simonis  Institut für Kernphysik and PRISMA+
Cluster of Excellence, Johannes Gutenberg-Universität Mainz, 55128 Mainz,
Germany  A. Tamii  Research Center for Nuclear Physics, Osaka University,
Ibaraki, Osaka 567-0047, Japan

###### Abstract

The electric dipole strength distribution in 40Ca between 5 and 25 MeV has
been determined at RCNP, Osaka, from proton inelastic scattering experiments
at very forward angles. Combined with total photoabsorption data at higher
excitation energy, this enables an extraction of the electric dipole
polarizability $\alpha_{\mathrm{D}}(^{40}{\rm Ca})=1.92(17)$ fm3. Together
with the measured $\alpha_{\rm D}$ in 48Ca, it provides a stringent test of
modern theoretical approaches, including coupled cluster calculations with
chiral effective field theory interactions and state-of-the art energy density
functionals. The emerging picture is that for this medium-mass region dipole
polarizabilities are well described theoretically, with important constraints
for the neutron skin in 48Ca and related equation of state quantities.

Introduction.– The nuclear equation of state (EOS) determines not only basic
properties of nuclei [1] but also plays a key role for the properties of
neutron stars and the dynamics of core-collapse supernovae and neutron star
mergers [2]. New observations from neutron stars and mergers provides
constraints for the EOS of neutron-rich matter that can be compared with those
derived from nuclear physics (see, e.g., Refs. [3, 4, 5]). However, while the
EOS of symmetric nuclear matter is well determined around saturation density,
the properties of neutron-rich matter are less explored experimentally. The
latter depends on the symmetry energy, whose properties are typically encoded
in an expansion around saturation density $n_{0}$, with the symmetry energy at
saturation density $J(n_{0})$ and its density dependence $L=3n_{0}\partial
J(n_{0})/\partial n$.

Theoretically, a model-dependent correlation between $L$ and the neutron-skin
thickness $r_{\rm skin}$ in nuclei with neutron excess has been established
[6, 7, 8, 9]. This correlation was also recently confirmed in ab initio
computations of the neutron skin in 208Pb [10]. Experimental attempts to
determine the neutron skin thickness have been performed with a variety of
probes (see, e.g., Ref. [11] and references therein), but many of them suffer
from systematic uncertainties entering in the description of the reaction
processes. Parity-violating elastic electron scattering (a weak process
mediated by the $Z^{0}$ boson) can be used for a model-dependent extraction of
the neutron distribution in nuclei and, by comparison with accurately measured
charge radii, the neutron skin thickness. Recently, results with this
technique have been reported by the CREX and PREX collaborations for 48Ca [12]
and 208Pb [13], respectively. The $r_{\rm skin}$ values inferred with selected
nuclear models favor a comparatively small neutron skin in the former and a
large skin in the latter case.

Alternatively, the electric dipole polarizability $\alpha_{\rm D}$ has been
established as a measure of the neutron skin, based on the strong correlation
with $r_{\rm skin}$ [8, 14]. Data for $\alpha_{\rm D}$ extracted from proton
inelastic scattering experiments at extreme forward angles have been presented
for both 48Ca [15] and 208Pb [16]. In these papers, two theoretical approaches
have been used to describe $\alpha_{\rm D}$: ab initio coupled-cluster (CC)
calculations [17, 18] starting from chiral two- and three-nucleon interactions
[19, 20] and energy density functional (EDF) theory [21].

Attempts to simultaneously describe $\alpha_{\rm D}$(208Pb) and the parity-
violating asymmetry from PREX and CREX with EDF models have shown limited
success [22, 23, 24, 25]. The values derived for $r_{\rm skin}$ [13] and $L$
[26] from PREX are in tension with EDFs capable of describing [27] the
presently available results on $\alpha_{\rm D}$ in 48Ca [15], 68Ni [28], 120Sn
[29, 30], and 208Pb [16]. While the CREX results is in excellent agreement
with ab initio predictions [18], the PREX result is in mild tension with the
recent ab initio computations of 208Pb [10].

Correlations between experimental observables and symmetry energy properties
are well explored in EDF theory [6, 7, 8, 14], but predictions for isovector
observables like $\alpha_{\rm D}$ are less well constrained. On the other
hand, ab initio calculations provide a direct link to the EOS, as nuclear
matter properties can be calculated based on the same chiral interactions [19,
18, 31, 10, 32]. Results presented here are based on the set of two- and
three-nucleon interactions from Refs. [19, 20] applied to study $\alpha_{\rm
D}$ in 48Ca [18, 15]. The calculations of the $E1$ response are based on
merging the Lorentz Integral Transform approach with CC theory, as described
in Refs. [33, 34]. Recent work has extended the original two-particle–two-hole
(2p-2h) CC truncation to include correlations up to three-particle–three-hole
(3p-3h), so-called triples corrections, in the computation of $\alpha_{\rm D}$
[35]. Their inclusion leads to a reduction of the predictions for $\alpha_{\rm
D}$(48Ca) of the order of 10%, allowing an improved simultaneous description
of the charge radius [35]. A similar improvement was achieved for 68Ni [36].

In this Letter, we present the measurement of the dipole polarizability for
40Ca and confront it with CC and EDF calculations. This tests the emerging
picture that nuclear theory can describe very well the neutron skin in medium-
mass nuclei and related observables.

Experiment.– Cross sections for the 40Ca$(p,p^{\prime})$ reaction have been
measured at RCNP, Osaka, at an incident proton energy of 295 MeV. Data were
taken with the Grand Raiden spectrometer [37] in a laboratory scattering angle
range $0.4^{\circ}-14.0^{\circ}$ and for excitation energies in the range
$5-25$ MeV. Dispersion matching techniques were applied to achieve an energy
resolution of about 30 keV (full width at half maximum). The experimental
techniques and the raw data analysis are described in Ref. [38].

In the top panel of Fig. 1 we show representative energy spectra measured at
laboratory scattering angles $\Theta_{\rm lab}=0.4^{\circ}$, $1.74^{\circ}$,
$3.18^{\circ}$, and $5.15^{\circ}$. The predominant cross sections lie in the
energy region above 10 MeV. $M1$ strength in 40Ca is known to be concentrated
in a single prominent transition at 10.32 MeV [39]. The cross sections above
10 MeV show a broad resonance structure peaking at about 19 MeV increasing
towards $0^{\circ}$. The angular dependence is consistent with relativistic
Coulomb excitation of $E1$ transitions. We identify this resonance structure
as the isovector giant dipole resonance.

Figure 1:  Top panel (a): Spectra of the 40Ca$(p,p^{\prime})$ reaction at
$E_{0}=295$ MeV and scattering angles $\Theta_{\rm lab}=0.4^{\circ}$,
$1.74^{\circ}$, $3.18^{\circ}$ and $5.15^{\circ}$. Bottom panel (b): Example
of the MDA of the spectrum at $\Theta_{\rm lab}=0.4^{\circ}$ in 200 keV bins
(blue) and decomposition into contributions of $\lambda\neq 1$ multipoles
(orange), continuum background (green), and $E1$ (red).

The various contributions to the spectra were separated using a multipole
decomposition analysis (MDA) as described in Ref. [40]. Results for the most
forward angle measured are presented in the bottom part of Fig. 1 as example,
where the spectra was rebinned to 200 keV. Theoretical angular distributions
for the relevant multipoles were obtained from Distorted Wave Born
Approximation calculations with transition amplitudes from quasiparticle-
phonon-model calculations similar to the analysis of 48Ca [15]. Additionally,
a background due to pre-equilibrium multistep scattering was considered. Its
angular dependence was taken from experimental systematics [41, 42] while the
amplitude was derived by two means. Initially, an unconstrained fit was done
at each energy bin of the set of spectra. The resulting cross sections could
be well approximated by a simple Fermi function but showed strong fluctuations
for certain excitation energy bins due to the similarity to some of the $E1$
theoretical angular distributions. Thus, in the final analysis, the continuum
contribution was determined by fitting a Fermi function to the unconstrained
excitation energy dependence.

Figure 2:  Top panel (a): Photoabsorption cross section derived at a
scattering angle of $0.40^{\circ}$ using the virtual photon method. Bottom
panel (b): Electric dipole polarizability $\alpha_{D}$ derived from the
photoabsorption cross sections. The blue curve shows the present data, while
the orange and green curves show the extrapolation to higher energies using
the data of Refs. [43, 44]. The open (full) black circles are the CC results
for the NNLOsat interaction including up to doubles (triples) contributions in
the cluster expansion.

Photoabsorption cross sections and dipole polarizability.– The $E1$ cross
sections resulting from the MDA were converted into equivalent photoabsorption
cross sections using the virtual photon method [45]. The virtual photon
spectrum was calculated in an eikonal approach [46] to Coulomb excitation,
integrated over the distribution of scattering angles covered in the solid
angle of each angular bin. The photoabsorption spectra derived from scattering
data at $0.40^{\circ}$ and $1.00^{\circ}$ were essentially identical, and that
at $1.74^{\circ}$ deviated only slightly, consistent with an estimate of the
grazing angle ($1.33^{\circ}$) at which Coulomb-nuclear interference becomes
relevant. The resulting photoabsorption cross section is displayed as blue
histogram in Fig. 2 (a).

The electric dipole polarizability $\alpha_{D}$ was obtained from the
photoabsorption cross section over the energy range $10-25$ MeV leading to a
contribution 1.60(14) fm3. The integration was extended to 60 MeV, where the
cumulative sum plotted in Fig. 2 (b) shows saturation. The data at higher
excitation energies were taken for $25-31$ MeV from Ref. [44] and for $31-60$
MeV from Ref. [43] to obtain the total $\alpha_{D}(^{40}{\rm Ca})=1.92(17)$
fm3. The uncertainty considers systematic errors of (i) the absolute cross
sections, (ii) the MDA (determined as described, e.g., in Ref. [47]), and
(iii) the parameterization of the continuum background (evaluated by varying
the amplitude of the Fermi function), added in quadrature. The latter,
dominating the total uncertainty budget, was estimated by the variation needed
to change the $\chi^{2}$ value of the MDA fit by one. Statistical errors
turned out to be negligible. A detailed breakdown of the error contributions
is given in Table 1.

Table 1: Budget of error contributions to $\alpha_{\rm D}$(40Ca). Source | Value (%)
---|---
Trigger efficiency | 0.1
Drift chamber efficiency | 0.8
Charge collection | 0.3
Target thickness | 1.0
Determination of solid angle | 3.0
MDA | 1.2
Background parameterization | 8.3
Total | 9.0

Comparison with coupled-cluster calculations.– The extracted value of
$\alpha_{\rm D}$ serves as a benchmark for CC theory [33, 18, 35, 34].
Coupled-cluster calculations were recently performed for the dipole
polarizability of 48Ca [15] and 68Ni [36], which led to an improved
understanding of the neutron and proton distributions in nuclei, as well as
their difference encoded in the neutron skin. We have performed CC
computations of $\alpha_{\rm D}$ in 40Ca starting from a Hartree-Fock
reference state considering a basis of 15 major harmonic oscillator shells. To
gauge the convergence of our results we varied the oscillator frequency in the
range $\hbar\omega=12-16$ MeV. Three-nucleon contributions had an additional
energy cut of $E_{\rm 3max}=16\hbar\omega$.

Figure 3 explores the correlation between $\alpha_{\rm D}$ for 40Ca and 48Ca
as predicted by theory. Panel (a) shows the CC results including triples
contributions. The theoretical uncertainties for the different Hamiltonians
stem from the truncation of the CC expansion and the residual dependence on CC
convergence parameters, calculated as described in Ref. [34]. Similarly to
48Ca, we find that the inclusion of 3p-3h correlations reduces the value of
$\alpha_{\rm D}$(40Ca) by an amount varying between 10% – 20% for different
interactions. While the EM and PWA interactions [19] are not simultaneously
compatible with both 40Ca and 48Ca experimental data, the set of employed
interactions shows an approximately linear trend between the two quantities
overlapping with both experimental results. A particular improvement in the
reproduction of both $\alpha_{\rm D}$(48Ca) and $\alpha_{\rm D}$(40Ca) is seen
for the NNLOsat interaction [20], which is capable of accurately describing
binding energies and radii of nuclei up to 40Ca as well the saturation point
of symmetric nuclear matter. The different interactions predict a range of
symmetry energy parameters $J=27-33$ MeV, $L=41-49$ MeV [18], with the NNLOsat
values at the lower end ($J=27$ MeV, $L=41$ MeV).

Figure 3:  Comparison of the experimental dipole polarizabilites of 40Ca
(present work) and 48Ca [15] shown as blue bands with (top panel, a) CC
calculations with different interactions, including triples contributions and
(bottom panel, b) EDF calculations with different energy density functionals
[22]. For details see text.

Comparison with EDF approaches.– Recently, it was investigated whether the
dipole polarizability and the parity-violating asymmetry $A_{\rm PV}$ for
208Pb and 48Ca can be simultaneously accounted for with modern EDFs [24]. We
use the four representative forms of functionals from that study: non-
relativistic Skyrme functionals SV [48] and RD [49], the latter with different
forms of density dependence, and relativistic functionals DD [50] with finite-
range meson-exchange coupling and PC [51] with point coupling. All four have
been calibrated to the same set of ground-state data to determine the model
parameters. With these sets, it was shown that PREX and CREX results for
$A_{\rm PV}$ (and $r_{\rm skin}$) cannot be consistently explained within the
model uncertainties while the $\alpha_{\mathrm{D}}$ were reproduced. Hence,
the present result in 40Ca provides an important test of the global predictive
power of these EDFs.

Figure 3 (b) displays the EDF results for $\alpha_{\rm D}$ with $1\sigma$
error ellipses (for their definition see Refs. [22, 24]). The parametrizations
as given from the ground-state fits are shown by filled ellipses. The DD
functional performs rather well. The other predictions tend to slightly
overestimate the experimental mean values of both 40Ca and 48Ca, while their
$1\sigma$ error ellipses do overlap with the experimental bands, except for
PC. In all cases, the two $\alpha_{\mathrm{D}}$ are highly correlated. The
dashed ellipses show results from a refit where additionally the experimental
$\alpha_{\mathrm{D}}$ value of 208Pb [16] corrected for the quasideuteron part
[27] was included yielding the functionals SV-alpha; RD-alpha, PC-alpha, and
DD-alpha [22, 24]. This improves the agreement with experiment, particularly
for the PC model, and shrinks most error ellipsoids. The uncertainty reduction
is especially large for the DD model because this functional has the least
isovector freedom. The linear trend shown by the different theoretical
approaches in Fig. 3 is similar although the CC calculations tend to
underestimate the $\alpha_{D}$ in 40Ca and perform nicely for 48Ca. The bulk
symmetry energies range from $J=30$ MeV for DD to 35 MeV for PC and
accordingly from 32 MeV to 82 MeV for $L$. The fits which include also
$\alpha_{\mathrm{D}}$ in 208Pb narrow the prediction to $J=30-32$ MeV and
$L=35-52$ MeV which correlates nicely to the narrower range of predictions for
$\alpha_{\mathrm{D}}$ in 40,48Ca.

Conclusions.– We have extracted the dipole polarizability of 40Ca from a
combination of relativistic Coulomb excitation measurement in inelastic proton
scattering under very forward angles with total photoabsorption data at high
excitation energies. Together with a similar analysis on 48Ca the new data
serve as a benchmark test of state-of-the art theoretical approaches. A
representative set of EDFs can describe these data. An improvement is obtained
when the EDFs are optimized by adding the dipole polarizability of 208Pb to
the calibration dataset. Coupled-cluster computations for the NNLOsat
interaction simultaneously describe well the dipole polarizability of 40Ca and
48Ca, as well as the corresponding charge radii and the neutron skin thickness
[34]. A nearly linear systematic trend is obtained for other interactions, as
in the case of EDF theory. This analysis supports the robustness of current
theoretical approaches in the description of $\alpha_{D}$ while at the same
time poses further doubts on our understanding of the recently measured
parity-violating asymmetries for 208Pb.

###### Acknowledgements.

This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) – Project-ID 279384907 –- SFB 1245, through the Cluster
of Excellence “Precision Physics, Fundamental Interactions, and Structure of
Matter” (PRISMA+ EXC 2118/1, Project ID 39083149), by the U.S. Department of
Energy, Office of Science, Office of Nuclear Physics under award numbers DE-
SC0013365 and DE-SC0023175 (NUCLEI SciDAC-5 collaboration), under the contract
DE-AC05-00OR22725 with UT-Battelle, LLC (Oak Ridge National Laboratory), and
by the University of Cape Town. Computer time was provided by the Innovative
and Novel Computational Impact on Theory and Experiment (INCITE) programme.
This research used resources of the Oak Ridge Leadership Computing Facility at
the Oak Ridge National Laboratory, which is supported by the Office of Science
of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725.

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