A local-density-approximation description of high-momentum tails in isospin asymmetric nuclei Xiao-Hua Fan1Zu-Xing Yang2Peng Yin3 4yPeng-Hui

2025-04-27 0 0 562.26KB 8 页 10玖币

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A local-density-approximation description of high-momentum tails in isospin

asymmetric nuclei

Xiao-Hua Fan,1Zu-Xing Yang,2, ∗Peng Yin,3, 4, †Peng-Hui

Chen,5Jian-Min Dong,4Zhi-Pan Li,1and Haozhao Liang6, 7

1School of Physical Science and Technology, Southwest University, Chongqing 400715, China

2RIKEN Nishina Center, Wako, Saitama 351-0198, Japan

3Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA

4Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China

5College of Physics Science and Technology, Yangzhou University, Yangzhou, Jiangsu 225002, China

6Department of Physics, Graduate School of Science,

The University of Tokyo, Tokyo 113-0033, Japan

7RIKEN iTHEMS, Wako 351-0198, Japan

We adapt the local density approximation to add the high-momentum tails (HMTs) to ﬁnite

nuclei’s Slater-determinant momentum distributions. The HMTs are extracted by the extended

Brueckner-Hartree-Fock (EBHF) method or by the lowest order cluster approximation. With a

correction factor being added to EBHF, it is suﬃciently in agreement with the experimental bench-

mark, i.e., the high-momentum N/Z ratios approximately equal to 1, and the low-momentum N/Z

ratios approximately equal to N/Z of the systems. It is also found that the tensor force makes the

nucleon-nucleon correlations appear more easily on the nuclear surface region and the percentage of

high-momentum (p > 300 MeV/c) nucleons, around 17%–18%, independent of isospin asymmetry.

I. INTRODUCTION

Recently, worldwide experiments have revealed many

exciting results on the dynamical correlations of nu-

cleons [1–8]. It was surprising to observe that the

proton-neutron (pn) short-range correlations (SRCs) in

nuclei are much stronger than the proton-proton (pp)

and neutron-neutron (nn) correlations by a factor of

about 20, for the internal momenta of 250–600 MeV/c,

where the tensor forces dominate the nucleon-nucleon in-

teractions [4–6]. The two-nucleon knockout experiment

demonstrated that these pn-dominated correlated pairs

are formed with large relative momenta and small center-

of-mass momenta [7]. Recently, the (e, e0p) and (e, e0n)

quasi-elastic knockout event-sampling experiments have

further displayed that some nucleons in nuclei form close-

proximity neutron-proton pairs with high nucleon mo-

mentum at diﬀerent isospin-asymmetries [8]. The SRC

quenches the neutron superﬂuidity and neutrino emissiv-

ity of neutron stars, and hence visibly aﬀects the neu-

tron star cooling [9, 10]. Moreover, the quasi-free αclus-

ter–knockout reactions showed a direct experimental evi-

dence for forming αclusters at the surface of neutron-rich

Sn isotopes [11].

For these new experimental discoveries, theorists are

trying to provide a self-consistent and reliable expla-

nation. Various theoretical methods have been em-

ployed to calculate the nucleon-nucleon correlations in

nuclear matter, such as the correlated basis functions

[12–14], the quantum Monte Carlo method [15], the

self-consistent Green’s function (SCGF) [16–20], the in-

∗zuxing.yang@riken.jp

†yinpeng@impcas.ac.cn

medium T-matrix method [21–23], and the Brueckner-

Hartree-Fock (BHF) method [24–31]. In particular, in

Ref. [32] the isospin- and density-dependent momentum

distribution calculated by extended Brueckner-Hartree-

Fock (EBHF) has been parameterized. For ﬁnite nuclei,

the local density approximation (LDA) based on the re-

sults of the lowest order cluster (LOC) approximation

[12, 33–35] and the light-front dynamics method [35] have

been utilized to describe the momentum distributions

with initial success. However, these methods cannot ade-

quately explain the existence of the high-momentum pn-

dominated close-proximity correlated pairs [8]. There-

fore, a phenomenological (i.e., experiment-based) pn-

dominance model [6, 36], which uses a mean-ﬁeld mo-

mentum distribution at low momentum (k < kf) and a

scaled deuteron-like high-momentum tail, has been de-

veloped.

In this work, we employ the LDA method to include

the high momentum tails (HMTs) in ﬁnite nuclei as a sig-

niﬁcant correction to the Slater determinant momentum

distributions. This paper is organized as follows. The

theoretical approaches, including the EBHF theory and

the LOC approximation, are brieﬂy reviewed in Sec. II. In

Sec. III, the momentum properties with the two methods

are compared, and a modiﬁcation to the EBHF is pro-

posed as a new scheme. In Sec. IV, we employ the mod-

iﬁed model to study the SRC eﬀects on selected nuclei

and compare the results with the available experimental

data. Finally, a summary is given in Sec. V.

II. THEORETICAL FRAMEWORK

In nuclear matter, dynamical correlations modify the

occupation probability of nucleon from that in the Fermi

gas model. At zero temperature, this process can be

arXiv:2210.05957v1 [nucl-th] 12 Oct 2022

characterized as [33]

nτ

NM(k;ρ, δ) = Θ(kτ

f−k) + δnτ

NM(k;ρ, δ).(1)

Here, the nuclear matter is characterized by its total

density ρ=ρn+ρpand isospin asymmetry δ= (ρn−

ρp)/(ρn+ρp). The corresponding Fermi momenta read

kτ

f= (3π2ρτ)1/3for neutrons and protons (τ=n, p),

respectively. The nτ

NM(k), which is dimensionless and

satisﬁes 0 ≤nτ

NM(k)≤1, is the correlated momentum

distribution in nuclear matter, Θ(kτ

f−k) denotes the oc-

cupation probability of the independent-particle model.

And δnτ

NM(k) is the correction caused by the dynamical

correlations. By deﬁnition, Rδnτ

NM(k)d3k= 0 to ensure

the conservation of particle numbers.

For a ﬁnite nucleus, the momentum distribution can

also be written as a sum of the single-particle contribu-

tion and the correlation eﬀect, i.e.,

nτ

A(k) = nτ

SD(k) + δnτ

A(k),(2)

where

nτ

SD(k) = X

α∈τ

αφ∗

α(k)φα(k) (3)

is the Slater-determinant momentum distribution gener-

ated by the single-particle wave functions φα(k) written

in the momentum-space representation, with the corre-

sponding occupation probabilities v2

α. The subscript A

labels the nuclide and αlabels the single-particle quan-

tum numbers. The δnτ

A(k) corresponds to the dynamical

correlation, satisfying Rδnτ

A(k)d3k= 0 due to the parti-

cle number conservation. Note that, in the whole paper,

the momentum distributions in ﬁnite nuclei are normal-

ized to the particle numbers, i.e.,

Znτ

A(k)d3k=Nτ,(4)

with Nτthe number of protons (Z) or neutrons (N).

The unit of nτ

A(k) is fm3.

In this paper, nSD(k) is calculated by the self-

consistent Skyrme Hartree-Fock (SHF) model with

Bardeen-Cooper-Schrieﬀer (BCS) pairing, by using the

SkM* interaction [37] and adopting the spherical sym-

metry.

Based on LDA, one can obtain the δnτ

A(k) from the

superposition of δnτ

NM(k) at diﬀerent densities,

δnτ

A(k) = Rλτ(r)δnτ

NM(k;ρ(r), δ(r))ρτ(r)d3r

Nτ,(5)

where ρτ(r) is the Slater-determinant density distri-

bution of proton or neutron, ρ(r) and δ(r) are the

corresponding local total density and isospin asymme-

try, respectively. The normalization factor λτ(r) =

Nτ/[4π3ρτ(r)] here takes care of the diﬀerences in the

units and normalization conditions between nτ

NM(k) and

nτ

A(k). Combining Eqs. (1), (2), and (5), one can ob-

tain the correlated momentum distributions of a ﬁnite

nucleus. To this end, the EBHF method or the LOC ap-

proximation is adopted to determine the only unknown

quantity δnτ

NM(k).

A. Extended Brueckner-Hartree-Fock Method

The Brueckner-Hartree-Fock method is one of the

widely used ab initio approaches for inverstigating the

properties of nuclear matter. In Refs. [38, 39], the

BHF model with the realistic Argonne V18 [40] two-

body interaction was extended to include the microscopic

three-body force, and it is called the EBHF method.

For the details of the EBHF method, one can refer to

Refs. [38, 39, 41, 42].

In this scheme, the realistic nuclear force is converted

into the eﬀective interaction G-matrix of the Bethe-

Brueckner-Goldstone theory by a self-consistent solution

of the Bethe-Goldstone equation. This G-matrix, which

includes all ladder diagrams of nucleon-nucleon interac-

tions and embodies the tensor correlations and SRCs,

can be used to compute the mass operator M(k, ω) [32].

The mass operator M(k, ω) allows us to write down the

Green’s function in the energy-momentum representa-

tion,

G(k, ω) = 1

ω−k2

2m−M(k, ω).(6)

Futhermore, the spectral function S(k, ω), which de-

scribes the probability density of removing a particle with

momentum kfrom a target nuclear system and leaving a

ﬁnal system with excitation energy ω, is thus given by

S(k, ω) = i

2π[G(k, ω)− G(k, ω)∗],(7)

with the sum rule R∞

−∞ S(k, ω)dω = 1. Finally, one can

obtain the momentum distributions using the spectral

function by

nNM(k) = Zεf

−∞

S(k, ω)dω, (8)

where the Fermi energy εfsatisﬁes the on-shell condition

εf=k2

f/2m+ Re M(k, εf).

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Alocal-density-approximationdescriptionofhigh-momentumtailsinisospinasymmetricnucleiXiao-HuaFan,1Zu-XingYang,2,PengYin,3,4,yPeng-HuiChen,5Jian-MinDong,4Zhi-PanLi,1andHaozhaoLiang6,71SchoolofPhysicalScienceandTechnology,SouthwestUniversity,Chongqing400715,China2RIKENNishinaCenter,Wako,Saitama351-019...

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A local-density-approximation description of high-momentum tails in isospin asymmetric nuclei Xiao-Hua Fan1Zu-Xing Yang2Peng Yin3 4yPeng-Hui

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