Microscopic nucleus-nucleus optical potentials from nuclear matter with uncertainty analysis from chiral forces T. R. Whitehead1

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Microscopic nucleus-nucleus optical potentials from nuclear matter

with uncertainty analysis from chiral forces

T. R. Whitehead1

1Facility for Rare Isotope Beams, Michigan State University, East Lansing, Michigan 48824, USA

Nucleus-nucleus optical potentials are constructed from an energy density functional approach

ﬁrst outlined by Brueckner et al. The interaction term of the energy density functional comes

from the complex nucleon self-energy computed in nuclear matter with two- and three-body chiral

nuclear forces. Nuclear density distributions are calculated from Skyrme functionals constrained to

the equations of state calculated from the same chiral forces used for the self-energy. Predictions for

elastic scattering cross sections and fusion cross sections are compared to experimental data. Very

good agreement is found with experiment for elastic scattering of heavier nucleus-nucleus systems at

energies in the range of 20 < E < 90 MeV/N, while accurate descriptions of lighter and lower-energy

systems may require the inclusion of collective excitations.

Introduction - As the ﬁeld of low-energy nuclear theory

increasingly turns its focus to exotic nuclei, theoretical

predictions with uncertainty quantiﬁcation will be essen-

tial for experimental eﬀorts at radioactive beam facilities.

In past studies of nuclei near stability, phenomenological

reaction models tuned to experimental data were consid-

ered suﬃcient. However, as modern experimental capa-

bilities allow for studies of rare isotopes and as advances

in nuclear many-body theory and our understanding of

the nuclear force enable predictive microscopic calcula-

tions of exotic nuclei, it is imperative that nuclear re-

action models be further developed to accommodate the

needs of experimental eﬀorts.

Optical potentials play a central role in the theoretical

modeling of nuclear reactions and the interpretation of

experimental data. There has been much recent develop-

ment in the area of microscopic nucleon-nucleus optical

potentials [1–5]. However, most reaction experiments in

the rare isotope era will involve nucleus-nucleus interac-

tions and therefore require nucleus-nucleus optical po-

tentials. The study of reaction channels such as knock-

out, transfer, charge-exchange, and fusion could bene-

ﬁt substantially from the implementation of microscopic

nucleus-nucleus optical potentials with quantiﬁed uncer-

tainties. In contrast to nucleon-nucleus optical poten-

tials, less attention has been paid to the development

of microscopic nucleus-nucleus optical potentials, with a

few notable exceptions that focus on light systems. These

works follow the double-folding approach [11] where ei-

ther a free-space NN interaction [6–8] or a g-matrix inter-

action [9, 10] is folded with nuclear density distributions.

When a free-space NN interaction is employed, the dou-

ble folding approach yields a real interaction which must

be supplemented by an imaginary term to account for

absorption. Typically this is done by assuming the real

and imaginary radial dependences to be equal and ad-

justing the strength of the imaginary term to reproduce

experimental data. Recently in Ref. [7], a more theo-

retically rigorous approach is taken by utilizing the dis-

persion relation to derive the strength of the imaginary

term from the real part. Another important distinction

of the work by Durant et al. [6–8] amongst the double

folding approaches is the estimation of the theoretical un-

certainty by varying the radial cutoﬀ of a local chiral NN

interaction. Aside from these works, the quantiﬁcation

of theoretical uncertainties in the calculation of nucleus-

nucleus optical potentials has been largely neglected. In

Refs. [9, 10] a double folding approach is carried out us-

ing the Melbourne g-matrix interaction modiﬁed to ap-

proximately account for eﬀects from three-body forces.

By utilizing the g-matrix instead of a free-space interac-

tion, in-medium eﬀects are accounted for. In Ref. [10] the

authors show the importance of including collective ex-

citations in a coupled-channels framework. They achieve

substantially better agreement with data in the coupled-

channels calculations, however discrepancies still persist

at large scattering angles.

An alternative method for constructing the nucleus-

nucleus optical potential is through an energy density

functional based on the nuclear matter single-nucleon po-

tential. Brueckner et al. ﬁrst outlined how energy density

functionals may be used to construct real nucleus-nucleus

interactions in Refs. [12, 13]. This energy density func-

tional approach may be generalized to include the imag-

inary term of the nucleus-nucleus optical potential on

equal footing to the real term [14–22] in contrast to the

double folding approach when carried out with an NN

interaction. Another advantage of utilizing many-body

calculations in nuclear matter is that Pauli blocking ef-

fects and correlations are included.

The present work constructs a nucleus-nucleus optical

potential starting from many-body perturbation theory

calculations of the self-energy in nuclear matter, which

are also the basis of the WLH global nucleon-nucleus op-

tical potential [1]. The results of the current work may

be combined with the WLH model for a consistent micro-

scopic treatment of the eﬀective interactions in few-body

reaction models. Two-body nuclear forces from chiral ef-

fective ﬁeld theory (EFT) calculated to N3LO with three-

body forces at N2LO [23–27] are used in the calculation

of the nuclear matter self-energy. To estimate the theo-

retical uncertainty from the nuclear force in predictions

of nuclear reaction observables, three chiral interactions

with momentum space cutoﬀs in the range of Λ= 414-

arXiv:2210.03031v1 [nucl-th] 6 Oct 2022

500 MeV are employed. Results are benchmarked by

constructing optical potentials for a variety of systems

where elastic scattering data are available for compari-

son. Fusion reactions are also used as a benchmark for

the nucleus-nucleus interactions, as they are important

in many contexts ranging from supernova nucleosynthe-

sis to experiments at rare isotope beam facilities. Fu-

sion cross sections are calculated for a set of systems and

compared to recent studies also based on Brueckner’s ap-

proach [28, 29].

Formalism - The present work follows Brueckner [12]

in constructing a nucleus-nucleus (A1+A2) interaction

from the nuclear matter self-energy through an energy

density functional. An imaginary term is also included

as in Ref. [14]. The A1+A2interaction is obtained by

integrating the energy density of the combined system

minus the energy densities of each isolated nucleus over

space for a given separation distance R.

U(R, E) = Z[H(ρT, E)−H(ρ1, E)−H(ρ2, E)]d3¯r

(1)

In the ﬁrst term of the integrand, the density ρand

isospin asymmetry δare taken to be the sum of each

nucleus where the total density is ρT=ρ1+ρ2and

total isospin asymmetry is δT=ρn

1+ρn

−ρp

ρ1+ρ2. This

so-called frozen density approximation assumes the two

nuclear densities simply add together, neglecting Pauli

blocking eﬀects between nucleons in diﬀerent nuclei. Im-

provements on this assumption will be the focus of future

works. The density and isospin asymmetry in the second

and third terms of the integrand in Eq. (1) are of each

isolated nucleus. The form of the energy density is given

Hρ¯r, ¯

R, E=3

10MN3π22/3ρ¯r, ¯

R5/3

72MN

∇ρ¯r, ¯

R2

ρ¯r, ¯

R

+ρ¯r, ¯

RUρ¯r, ¯

R, δ¯r, ¯

R, E.

(2)

In this work, the single-nucleon potential Uis the com-

plex and energy dependent nuclear matter self-energy

U(ρ, δ, E) = V(ρ, δ, E) + iW (ρ, δ, E). The self-energy

is taken to be the average between the proton and neu-

tron self-energies weighted by the isospin asymmetry at

a given point in the A1+A2system:

U=1 + δUn+1−δUp

2.(3)

Similarly, the energy Eis taken to be the average energy

of each nucleus in the lab frame E1, E2weighted by their

FIG. 1. The real (V) and imaginary (W) terms of a selection

of optical potentials as a function of the separation distance

between the nuclei. Results from N3LO chiral interactions

with momentum space cutoﬀs of Λ = 414, 450, 500 MeV are

shown in blue, green, and red. The light grey region repre-

sents where the frozen density approximation yields densities

beyond nuclear saturation. The dark grey region represents

the region of the potential that does not aﬀect elastic scatter-

ing cross sections for the given case.

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Microscopicnucleus-nucleusopticalpotentialsfromnuclearmatterwithuncertaintyanalysisfromchiralforcesT.R.Whitehead11FacilityforRareIsotopeBeams,MichiganStateUniversity,EastLansing,Michigan48824,USANucleus-nucleusopticalpotentialsareconstructedfromanenergydensityfunctionalapproachrstoutlinedbyBrueckne...

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Microscopic nucleus-nucleus optical potentials from nuclear matter with uncertainty analysis from chiral forces T. R. Whitehead1

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