Nuclear symmetry energy and neutron-proton pair correlations in microscopic mean field theory

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Nuclear Symmetry Energy and Neutron-Proton Pair

Correlations in Microscopic Mean Field Theory

SHUFANG BAN

Doctoral Thesis in Physics

Stockholm, Sweden 2007

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TRITA-FYS-2007:70 ISSN 0280-316x

ISRN KTH/FYS/−07:70−SE ISBN 978-91-7178-738-5

KTH, AlbaNova, University Center SE-10691, Stockholm SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till oﬀentlig granskning för avläggande av Teknologie Doktorsexamen fredagen den 24 August 2007 kl 11.00 i FA31, AlbaNova, Universitetscenturm, Kungl Tekniska Högskolan, Stockholm.

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Shufang Ban, July, 2007 Tryck: Universitetsservice US AB

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Abstract

A major task of nuclear structure theory is to develop an effective interaction that is capable to predict properties of nuclei under extreme conditions. In this thesis three different areas of effective interactions have been studied: i) the density dependence, which is of importance for the understanding of neutron stars and neutron rich nuclei; ii) the symmetry energy which governs the evolution of the binding energy with changing N/Z ratio; and iii) the neutron proton pairing energy. The symmetry energy governs the isospin dependent part of the nuclear interaction and determines basic properties of neutron stars. The isospin dependence of short range correlations are in turn important for properties of N≈Z nuclei as well as symmetric nuclear matter.

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List of Publications

This thesis is based on the ﬁrst three papers (attached at the end) and to the development of the HFB code in coordinate space with generalized pairing interac-tion using the pairing interacinterac-tion described in Appendix C and chapter 6.

[1] Nuetron-proton pairing correlations in Hartree-Fock Bogliubov theory S. F. Ban, R. A. Wyss, P. H. Heenen, and J. Meng, manuscript, 2007 [2] Nuclear symmetry energy in relativistic mean ﬁeld theory

S. F. Ban, J. Meng, W. Satula, and R. A. Wyss, Phys. Lett. B633 (2006) 231-236. [3] Density dependencies of interaction strengths and their inﬂuences on nuclear matter and neutron stars in relativistic mean ﬁeld theory

S. F. Ban, J. Li, S. Q. Zhang, H. Y. Jia, J. P. Sang, and J. Meng, Phys. Rev. C69 (2004) 045805.

[4] Recent progress in relativistic many-body approach

S. F. Ban, L. S. Geng, L. Liu, W. H. Long, J. Meng, J. M. Yao, S. Q. Zhang, and S. G. Zhou. Int. J. Mod. Phys. E15(7) (2006) 1447-1464.

[5] Response of the neutron star properties to the equation of state at low den-sity

Z. H. Zhu, S. F. Ban, J. Li, and J. Meng, High energy physics and nuclear physics, 29(6) (2005) 565-569.

[6] Description of the nuclear matter and neutron stars in relativistic mean ﬁeld theory with density-dependent interactions

J. Li. S. F. Ban, H. Y. Jia, J. P. Sang, and J. Meng, High energy physics and nuclear physics, 28(2) (2004) 140-147.

[7] Nuclear symmetry energy for A=48 isobars in relativistic mean ﬁeld theory S. F. Ban, J. Meng, and R. A. Wyss, High energy physics and nuclear physics, 28(supp) (2004) 66-68.

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[8] Description of the nuclear matter and neutron stars in density-dependent rela-tivistic mean ﬁeld theory

S. F. Ban, J. Li, S. Q. Zhang, H. Y. Jia, J. P. Sang, and J. Meng, Nucl. Phys. Rev. of China, 22(1) (2005) 29.

Published proceeding of the 12th national physics conference, Beijing, 2004. [9] Relativistic description of exotic nuclei and nuclear matter at extreme con-ditions

J. Meng, S. F. Ban, J. Li, W. H. Long, H. F. Lü, S. Q. Zhang, W. Zhang, and S. Q. Zhou, Physics of Atomic Nuclei, 67(9) (2004) 1619-1626.

Published proceeding of Conference in Dubna of Russia, 2003.

[10] New eﬀective interactions, new symmetry and new states in atomic nuclei J. Meng, S. F. Ban, J. Li, L. S. Geng, W. H. Long, H. F. Lü, S. Q. Zhang, W. Zhang, and S. Q. Zhou, High energy physics and nuclear physics, 18(12) (2004) 1291-1296.

Published Conference proceeding of 2003 Dalian International symposium on Nu-clear Physics.

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Introduction

This thesis is based on three papers [1–3] attached at the end and the develop-ment of a computer programm employing the generalized HFB transformation for neutron-proton pairing. The thesis work has been done under joint supervision of Ramon Wyss in KTH and Jie Meng in PKU.

A widely used and successful approach for nuclear matter and finite nuclei is the microscopic self-consistent mean field theory employing effective interactions. The mean field theory includes relativistic mean field (RMF) theory [4] starting from effective Lagrangian density and non-relativistic Hartree-Fock (HF) theory [5, 6] with effective interactions, such as Skyrme [7] or Gogny [8]. They have been used not only for describing the properties of nuclei near the valley of stability success-fully [9–13], but also for predicting the properties of nuclei with large neutron or proton excess [14–21] and neutron stars [22, 23]. As discussed in Ref. [24], mean field models have reached a level of accuracy allowing for detailed comparison to experiments. Still, many open questions remain with respect to the accuracy of present days models.

The physics of neutron stars has offered an intriguing interplay between nuclear processes and astrophysical observation, and has become a hot topic in nuclear physics and astrophysics [22,23]. In recent years, a number of effective interactions of meson-baryon couplings of RMF theory have been developed, including nonlinear effective interactions NL1, NL2 [25], NL3 [26], NLSH [27], TM1, TM2 [28], PK1, PK1r [29] and density-dependent effective interactions TW-99 [30], DD-ME1 [31], and PKDD [29]. To study the neutron stars, the equation of state (EOS) is essential to understand the structure and properties [22,23]. Usually, the EOS is obtained by extrapolating the theory, which is developed mainly for normal symmetric nuclear matter, to nuclear matter with extreme high isospin and high densities. Because the nonlinear interactions have problems of stability at high densities [25], one has to develop a different effective interactions from those used to finite nuclei, e.g.,

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10 CHAPTER 1. INTRODUCTION 97 [22] in order to study the properties of neutron stars in RMF theory. Density dependence in the coupling constants is a more natural alternative and the density-dependent effective interactions should be reasonably extrapolated to extreme con-ditions of isospin and/or density [30]. In this thesis, the density dependencies of various effective interactions including both the nonlinear and density-dependent versions in RMF theory are studied and the properties of nuclear matter and neu-tron stars are investigate by using density-dependent effective interactions of RMF theory [1].

The isospin properties are very important for such extrapolation of EOS. After almost all the eﬀective interactions in RMF theory are studied and carefully com-pared in nuclear matter and neutron stars, we pay our attention on their isospin limit by investigating the nuclear symmetry energy, which is not only important for the study of neutron stars, but also important for the study of exotic nuclei. Note that the uncertainty of the density dependence of the nuclear symmetry en-ergy accounts for 50% of the uncertainty of the radii of neutron stars [32], revealing how intimately properties of nuclear interactions are linked to the evolution and properties of astrophysical objects.

The study of exotic nuclei [33–35] with radioactive nuclear beams [36] is one of the most active and important areas in contemporary nuclear physics. Exotic nuclei have extreme isospin and mass and they may lie near the neutron/proton drip line. Detailed investigations of such systems have revealed a number of new interesting phenomena, such as neutron/proton halos and skins [37–40]. The size of neutron skins in turn are determined by the surface to volume ratio of the symmetry en-ergy. In theoretical calculations, to determine the nuclear drip line, the role of the continuum in loosely bound system and in particular, its impact on the treatment of neutron-neutron and proton-proton pair correlations has been discussed to great extent in recent times. However, the proper understanding and correct reproduc-tion of the nuclear symmetry energy may have even greater bearing for masses of loosely bound nuclei near the neutron drip line and certainly is a key issue in the study of exotic nuclei. In this thesis, the nuclear symmetry energy is studied within the framework of RMF theory and compared with those from Skyrme HF calcula-tions [2].

The nuclear symmetry energy (NSE) is the quantity which describes the ten-dency towards stability of N = Z line. The N = Z nuclei are close to the proton drip and oﬀer the ideal condition for ﬁnding neutron-proton (np) pair correla-tions [41–43]. In these nuclei, np pairing might create additional binding energy and provide stability to some nuclei which would otherwise be unstable. The

ra-dioactive nuclear beams allows studies of N = Z line up to 100_{Sn [44]. Therefore,}

a great interest in the study of np pairing in medium-mass nuclei has surfaced re-cently. The additional binding energy that may originate from T=0 np pairing has been associated with the ’Wigner’ energy [45] which in term inﬂuences the nuclear symmetry energy. In this thesis, the fully self-consistent microscopic calculations

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11 for np pairing without symmetry restricted except parity are performed and the np pair correlations are studied within the framework of the cranked Hartree-Fock-Bogoliubov theory [3].

This thesis is organized as follows: a brief description of the RMF theory is presented in Chapter 2; the density dependencies of various eﬀective interactions of the RMF theory and their descriptions of nuclear matter and neutron stars are given in Chapter 3. Chapter 4 contains the discussions of the nuclear symmetry energy in the framework of the RMF theory; The cranked HFB theory is presented in Chapter 5. In chapter 6, the isospin generalized HFB theory and primary results about np pairing are given. The conclusions are given in Chapter 7 and at last, we give brief summaries of the attached papers.

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Chapter 2

Relativistic Mean Field Theory

The Relativistic mean field (RMF) theory [4] has received wide attention because of its successful description of many nuclear phenomena. Recent reviews of the RMF theory can be found in Refs. [11–14]. The RMF theory is a quantum field theory, starting from a relativistic Lagrangian density containing mesonic and baryonic degrees of freedom. It can not be treated perturbably because of the large coupling constants. The lowest order of the full quantum field theory is the mean field approximation. Most of the applications of the RMF theory to the study of nuclear systems have been performed at the Hartree level. The relativistic Lagrangian is an effective functional of density and describes the many-body system exactly. In this approach, independent nucleons are treated as point-like particles, move in the mean field potential generated by all nucleons and obey the Dirac equations. Their interactions are described by exchange of mesons. The iso-scalar scalar σ meson offers medium-range attractive nuclear interaction, the iso-scalar vector ω meson offers short-range repulsive interactions, and the iso-vector vector ρ meson provides the necessary isospin asymmetry. One should note that the pseudoscalar π-meson is not taken into account at the Hartree level since the pion contributions are absent in the mean field treatment of a parity-symmetric system [11]. However, it will appear when exchange diagrams are included. There are also some studies where the pions are included [11, 46], but we will not present them in detail here. In addition, the Lagrange density can also include other masons, which have been found to be less important [11].

The relativistic models based only on one-meson exchange could not reproduce essential nuclear properties, e.g., they gave too large incompressibility. Boguta and Bodmer introduced the non-linear self-coupling of σ-meson [47], which has been used in almost all the recent applications. The meson self-coupling introduces a new density dependence into the Lagrangian and consequently, the nuclear mat-ter incompressibility can be lowered to reasonable values. As an extension, the nonlinear self-coupling of ω meson was introduced by Sugahara and Toki [48] and the nonlinear self-coupling of ρ, by Long etal. [29]. A number of nonlinear

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14 CHAPTER 2. RELATIVISTIC MEAN FIELD THEORY fective interactions have also been developed, including NL1, NL2 [25], NL3 [26], NLSH [27], TM1, TM2 [28], PK1, PK1r [29]. They have proved to be very success-ful in describing the properties of nuclear matter and ﬁnite nuclei near the valley of stability [11–13], as well as exotic nuclei with large neutron or proton excess [17–21], magnetic rotation [49], super-deformed bands and high spin state [50, 51], etc.

However the nonlinear effective interactions have problems of stability at high densities and their physical foundation has not been clarified [25]. A more natural alternative is to introduce the density dependence in the couplings [30]. Based on the Dirac-Brueckner calculations, Typel and Wolter proposed a density-dependent effective interaction TW-99 and expected that the model could be reasonably ex-trapolated to extreme conditions of isospin and/or density [30]. Along this line, Nikšić etal. developed DD-ME1 [31] and Long etal. developed PKDD [29]. Based on the RMF theory, there are also many studies on neutron stars and strange nuclear matter by using nonlinear and density-dependent effective interactions [1,22,52–55], which we will discuss in next chapter.

2.1 Lagrangian density

The basic ansatz of the RMF theory is an eﬀective Lagrangian density with baryons, mesons (σ, ω and ρ) and photons as degrees of freedom (¯h = c = 1) [13]:

L = X B ¯ ψB iγµ_∂ µ− mB− gσBσ − gωBγµωµ− gρBγµτB· ρµ− eγµAµ1 − τ3B 2 ψB +1 2∂µσ∂ µ_{σ −}1 2m 2 σσ2− U(σ) − 1 4ωµνω µν₊1 2m 2 ωωµωµ+ U (ω) −1 4ρµν· ρ µν₊1 2m 2 ρρµ· ρµ+ U (ρ) − 1 4AµνA µν_, _(2.1)

where the Dirac spinor ψB denotes the baryon B with mass mB and isospin

τB. The sum over B includes protons, neutrons and hyperons (Λ, Σ±, Σ0, Ξ−, Ξ0,

etc.). The masses of mesons are denoted by mσ, mω and mρ respectively. The

corresponding meson-baryon coupling constants are gσB, gωBand gρB respectively.

τ3B is the 3-component of τB. The ﬁeld tensors are,

   ωµν = ∂µων− ∂νωµ, ρµν = ∂µρν− ∂νρµ+ gρρ× ρ, Aµν = ∂µAν− ∂νAµ, (2.2) and the nonlinear self-couplings for mesons are,

   U (σ) = g2σ3/3 + g3σ4/4, U (ω) = c3(ωµωµ)2/4, U (ρ) = d3(ρµ· ρµ)2/4, (2.3) with the self-coupling constants g2, g3,c3, and d3. For the density-dependent

inter-actions, g2, g3, c3, and d3 do not appear and the meson-baryon coupling constants

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2.2. EFFECTIVE INTERACTIONS 15

2.2 Effective interactions

In the above Lagrange density (Eq. (2.1)), there are 11 parameters for nonlinear eﬀective interactions,

mB, gσB, gωB, gρB, mσ, mω, mρ, g2, g3, c3, d3, (2.4)

which are adjusted by ﬁtting the bulk nuclear properties. Thus the theory is phe-nomenological and the parameters are diﬀerent from the values in free space.

Gen-erally, the baryon mass m and ρ-meson mass mρ(also mωin some eﬀective

interac-tions) are set to their free values and the remaining nine (or eight) parameters are determined by ﬁtting the saturation properties of nuclear matter and experimental ground state properties of selected ﬁnite nuclei [29].

Instead of the self-coupling of meson ﬁelds, Typel and Wolter proposed the

density dependencies of coupling constants in the RMF theory [30], i.e., g2= g3=

c3= d3= 0 and the coupling constant gσ(ω)Bof the σ (ω) meson is replaced by:

gσ(ω)B(ρ) = gσ(ω)B(ρ0)fσ(ω)(x), (2.5) where fσ(ω)(x) = aσ(ω) 1 + bσ(ω)(x + dσ(ω))2 1 + cσ(ω)(x + dσ(ω))2 (2.6)

is a function of x = ρ/ρ0(where ρ is the vector density and ρ0is the saturation

den-sity of nuclear matter). The denden-sity-dependent ρ-meson coupling constant gρB(ρ)

is introduced as,

gρB(ρ) = gρB(ρ0) exp[−aρ(x − 1)]. (2.7)

Then the parameters for density-dependent eﬀective interactions are

mB, gσB(ρ0), gωB(ρ0), gρB(ρ0), mσ, mω, mρ, aσ(ω), bσ(ω), cσ(ω), dσ(ω), aρ,

(2.8)

where the eight real parameters aσ(ω), bσ(ω), cσ(ω), dσ(ω) in Eq. (2.6) are not

independent. The ﬁve constraints fσ(ω)(1) = 1, fσ(ω)′′ (0) = 0 and fσ′′(1) = fω′′(1)

reduce the number of independent parameters to three. Then the number of to-tal free parameters ﬁtted to the experimento-tal data for density-dependent eﬀective interactions is as above nine (or eight).

The effective interactions NL1, NL2 [25], NL3 [26], NLSH [27], and GL-97 [22] include the non-linear σ term, while TM1, TM2 [28], and PK1 [29] consider both the non-linear σ and ω terms. The effect interaction PK1r is with the non-linear σ, ω and ρ terms and TW-99 [30], DD-ME1 [31] and PKDD [29] are density-dependent. As an example, the effective interactions NL3 [26], TM1 [28], TW-99 [30],

DD-ME1 [31], PK1, PK1r and PKDD [29] are presented in Table 2.1. Where, mn and

mpare the masses of neutron and proton respectively and gσN, gωN, gρN denote the

meson-nucleon coupling constants, which are the same for neutrons and protons and

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16 CHAPTER 2. RELATIVISTIC MEAN FIELD THEORY Table 2.1: Eﬀective interactions in RMF theory NL3 [26], TM1 [28], TW-99 [30], DD-ME1 [31], PK1, PK1r and PKDD [29] NL3 TM1 TW-99 DD-ME1 PK1 PK1r PKDD mn 939.000 938.000 939.000 938.500 939.573 939.573 939.573 mp 939.000 938.000 939.000 938.500 938.280 938.280 938.280 mσ 508.194 511.198 550.000 549.526 514.089 514.087 555.511 mω 782.501 783.000 783.000 783.000 784.254 784.222 783.000 mρ 763.000 770.000 763.000 763.000 763.000 763.000 763.000 gσN 10.2169 10.0289 10.7285 10.4434 10.3222 10.3219 10.7385 gωN 12.8675 12.6139 13.2902 12.8939 13.0131 13.0134 13.1476 gρN 4.4744 4.6322 3.6610 3.8053 4.5297 4.5500 4.2998 g2 -10.4307 -7.2325 0 0 -8.1688 -8.1562 0 g3 -28.8851 0.6183 0 0 -9.9976 -10.1984 0 c3 0 71.3075 0 0 55.6360 54.4459 0 d3 0 0 0 0 0 350 0 aσ bσ cσ dσ aω bω cω dω aρ TW-99 1.366 0.226 0.410 0.902 1.403 0.177 0.344 0.984 0.515 DD-ME1 1.385 0.978 1.534 0.466 1.388 0.853 1.357 0.496 0.501 PKDD 1.327 0.435 0.692 0.694 1.342 0.371 0.611 0.738 0.183

the inclusion of hyperons in neutron stars in RMF theory are given in Refs. [1, 22]. Usually, one introduces the ratios of the meson-hyperons coupling strengths to those of nucleons as: xσh = gσh gσN, xωh= gωh gωN, xρh = gρh gρN. (2.9)

In Ref. [1], the ratios xσh = xωh = xρh =

p

2/3 are chosen. For the

density-dependent eﬀective interactions, gσN, gωN and gρN in Table 2.1 correspond to the

values at saturation density ρ0. gσh, gωh, and gρhhave the same density dependence

as gσN, gωN and gρN respectively in the RMF calculations [1].

2.3 Equation of motion

From the variational principle, δS = δ

Z

d4xL(x) = 0, with x = (x, t), (2.10)

the Euler−Lagrange Equation ∂L ∂φ(x)− ∂µ ∂L ∂(∂µφ) = 0 (2.11)

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2.3. EQUATION OF MOTION 17

can be obtained, where φ can be ψ, σ, ω or A. In a continued medium, the energy-momentum tensor is deﬁned by:

Tµν= ∂L ∂(∂µφ) ∂νφ − ηµνL (2.12) where, ηµν = ηµν =     1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 −1     . (2.13)

The Euler-Lagrange equation (2.11) ensures that the energy-momentum tensor is conserved,

∂µTµν = 0, (2.14)

which implies that the four-momentum

Pµ= (H, P) = ηµνPν, with Pν=

Z

d3xT0ν (2.15)

is a constant of the motion. H is the Hamiltonian with the value energy E,

P0= E =

Z

d3xH (2.16)

and the 00-component of energy-momentum tensor is the Hamiltonian density H:

H = T00₌ ∂L

∂(∂0φ)

∂0_{φ − L,} _(2.17)

The equations of motion for baryons and mesons can be derived from the La-grangian density Eq. (2.1) and the Euler−Lagrange Equation (2.11). For baryons, the equations of motion are the Dirac Equations:

γµ(i∂µ− gωBωµ− gρBτ· ρ_µ− e1 − τ3B 2 Aµ− Σ R µB) − m∗ ψB = 0, (2.18)

where the eﬀective mass m∗_{= m}

B+ gσBσ and ΣRµB is a “rearrangement” term due

to the density dependence of the couplings:

ΣR µB = jµ ρ( ∂gωB ∂ρ ψ¯Bγ ν_ψ Bων+ ∂gρB ∂ρ ψ¯Bγ ν τBψBρ_ν+∂gσB ∂ρ ψ¯BψBσ). (2.19)

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18 CHAPTER 2. RELATIVISTIC MEAN FIELD THEORY (∂µ_∂ µ+ m2σ)σ = − X B gσBψ¯BψB− U′(σ), (2.20) ∂µωµν+ m2ωων = X B gωBψ¯BγνψB− U′(ω), (2.21) ∂µρµν+ m2ρρν = X B gρBψ¯BγντBψB+ gρBρµ× ρµν− U′(ρ), (2.22) ∂µAµν = e ¯ψBγν1 − τ3B 2 ψB. (2.23)

Eqs. (2.18) and (2.20)-(2.22) are given in general for both kinds of eﬀective interactions. For the nonlinear eﬀective interactions, the “rearrangement” term

ΣR

µB is zero in Eq. (2.18) and for the density-dependent eﬀective interactions,

U′_{(σ) = U}′_{(ω) = U}′_{(ρ) = 0 in Eqs. (2.20)-(2.22), i.e., g}

2= g3= c3= d3= 0.

The above five coupled, nonlinear differential equations (2.18) and (2.20)-(2.22) are very complicated to solve. In RMF theory, two approximations of the effective Lagrangian density (2.1) are introduced: the mean-field approximation and no-sea approximation. The mean-field approximation corresponds to a classical treatment of the field theory that the meson fields are replaced by their mean values,

σ, ωµ ρ→< σ >, < ωµ>, < ρ > . (2.24)

Thus, the nucleons interact only via the mean field and move as independent par-ticles in this field. The RMF theory is a relativistic Hartree approximation. The spectrum of the Dirac equation (2.18) contains the negative energy continuum, negative energy bound state, positive energy bound state, and the positive energy continuum. In principle, all the solutions should be taken into account for a fully relativistic description. Most of the applications of the RMF theory are done in the no-sea approximation, i.e., the contribution of negative energy states are neglected. By using these two approximations and imposing certain symmetries, the equa-tions of motion can be solved. The details for the specific examples, nuclear matter and neutron stars, are discussed in next chapter. In Chapter 4, the RMF theory for axially deformed finite nuclei is presented.

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Chapter 3

Nuclear matter and neutron star in

RMF theory

In this chapter, the density dependencies of various eﬀective interaction strengths in RMF theory are studied and carefully compared for nuclear matter and neutron stars, mainly based on Ref. [1].

The physics of neutron stars has offered an intriguing interplay between nuclear processes and astrophysical observation, and has become a hot topic in nuclear physics and astrophysics. The existence of neutron stars was predicted following the discovery of neutron. In 1934, Baade and Zwicky suggested that neutron stars could be formed in “supernovae" [56]. The radio pulsars discovered by Bell and Hewith in 1967 [57] were identified as rotating neutron stars by Pacini [58] and Gold [59]. The first theoretical calculation of neutron stars was performed by Oppenheimer and Volkoff [60], and independently by Tolman [61] in 1939. In their calculation, the neutron stars were assumed as gravitationally bound states of neutron Fermi gas. Later, there are many studies as shown in Refs. [22, 62–67] and references therein.

3.1 Standard Model of neutron stars

The neutron star models including various so-called realistic equations of state have resulted in the following general picture of neutron stars [67], which can be divided into ﬁve regions: the surface, the outer crust, the inner crust, the neutron liquid region, and the core. The surface contains a negligible amount of mass and usually we don’t take it account. For the interior of the neutron star, as shown in Fig. 3.1, the crust of neutron star is a solid of thickness about 1 km. The outer crust (density

< 4 × 1011_g·cm−3_{) primarily contains a Coulomb lattice of nuclei and a relativistic}

electrons gas. In the inner crust (density 4 × 1011 _g·cm−3 _{∼ 2 × 10}14 _g·cm−3_),

neutrons begin to leak out of nuclei when density is above the neutron drip density and the nuclei are immersed in a sea of free neutrons, as well as the protons and electrons in beta-equilibrium. Inside the crust, higher pressure makes the nuclei

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CHAPTER 3. NUCLEAR MATTER AND NEUTRON STAR IN RMF THEORY merge together and form a uniform neutron liquid with a small concentration of protons and electrons in equal number. As the density continues to increase, proton

and electron densities increase as well, and the µ−_{, π, K mesons and other baryons}

( e.g., hyperons ) as well as a phase transition from baryon degrees of freedom to quark matter will appear.

Figure 3.1: Standard model of neutron stars.

To study the neutron stars, the equation of state (EOS) is essential to under-stand the structure and properties. The EOS determines properties such as the mass range, the mass-radius relationship, the crust thickness, the cooling rate and even the energy released in a supernova explosion. Usually, the EOS is obtained by extrapolating the theory, which is developed mainly for normal nuclear matter, to nuclear matter with extreme high isospin and high densities. In RMF theory, the neutron star is treated as a kind of uniform neutron matter with a small ad-mixture of protons, electrons, muons, and at high density hyperons can also be included [1, 23]. Clearly, the outer crust of the neutron stars is omitted and we considered the EOS of crust and studied it’s inﬂuence in Ref. [53].

3.2 RMF Formula for nuclear matter and neutron stars

A. Nuclear matter

Inﬁnite nuclear matter is an uniform static system including protons and neutrons with a given ratio N/Z. Baryon B in Lagrange Eq. (2.1) denotes neutron and

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3.2. RMF FORMULA FOR NUCLEAR MATTER AND NEUTRON STARS 21

the spatial coordinate x, and the spatial vector components of ω and ρ vanish. Charge conservation guarantees that only the 3-component of the isovector ρ

sur-vives, which we denote in the following by ρ0,3. Neglecting the coulomb ﬁeld and

introducing the mean-ﬁeld approximation, i.e., the meson ﬁelds are replaced by their mean values, the Klein-Gordon equations (2.20-2.22) reduce to the simpler form: m2σσ = − X B gσB < ¯ψBψB > −U′(σ), (3.1) m2ωω0 = X B gωB< ¯ψBγ0ψB > −U′(ω0), (3.2) m2ρρ0,3 = X B gρBτ3B< ¯ψBγ0ψB > −U′(ρ0,3), (3.3)

where σ, ω0 and ρ0,3 are mean values, not ﬁelds again.

The baryon ﬁelds are eigenstates of momentum k,

ψB(x) = ψB(k)e−ik·x, with k · x = kµxµ= k0t − k · x. (3.4)

Then the Dirac Equation (2.18) can be simpliﬁed as

γµ(kµ− gωBωµ− gρBτ· ρ_µ− ΣR_µB) − m∗ψB(k) = 0. (3.5)

The eigenvalues of the Dirac equation for baryons in Eq. (3.5) are obtained as: e±B(k) = gωBω0+ gρB· τ3B· ρ0,3+ ΣR0B±

q

k2_{+ m}∗2

B, (3.6)

where ΣR

0 is the time component of rearrangement term, "±" denote the

eigenval-ues for particle and antiparticle. Using no-sea approximation, i.e., neglecting the contribution of the antiparticle, the Dirac eigenvalues can be written as

eB(k) = gωBω0+ gρB· τ3B· ρ0,3+ ΣR0B+

q

k2_{+ m}∗2

B. (3.7)

The baryon vector density ρ and scalar density ρsare respectively:

ρ = X B < ¯ψBγ0ψB >= X B ρB = X B k3 B 3π2, (3.8) ρs = X B < ¯ψBψB >= X B ρsB= 1 π2 X B Z kB 0 k2dk m ∗ B p k2_{+ m}∗2 B , (3.9)

where kB denotes the Fermi momentum and Fermi energy is given by eB(k).

From Eq. (??) and the mean ﬁeld approximation, the energy density and pres-sure for nuclear matter can be obtained respectively,

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22

CHAPTER 3. NUCLEAR MATTER AND NEUTRON STAR IN RMF THEORY ε = 1 2m 2 σσ2+ 1 2m 2 ωω20+ 1 2m 2 ρρ20,3+ 1 3g2σ 3₊1 4g3σ 4₊3 4c3ω 4 0+ 3 4d3ρ 4 0,3 + 1 π2 X n,p Z k 0 k2dkpk2_{+ m}∗2_, _(3.10) P = −1₂m2σσ2+ 1 2m 2 ωω02+ 1 2m 2 ρρ20,3− 1 3g2σ 3 −1₄g3σ4+ 1 4c3ω 4 0+ 1 4d3ρ 4 0,3 + ρ · ΣR0 + 1 3π2 X n,p Z k 0 k4 √ k2_{+ m}∗2dk. (3.11)

B. Neutron Stars

The charge-neutral neutron star is treated as a kind of uniform neutron matter with a small admixture of protons, electrons, muons, and at high density hyperons can also appear. Baryon B in Lagrange Eq. (2.1) denotes neutron, proton and other

baryons, e.g., hyperon, and an additional term for leptons(λ = e−_{, µ}−_):

Lλ=

X

λ=e−_,µ−

¯

ψλ(iγµ∂µ− mλ)ψλ, (3.12)

is also need to added in Eq. (2.1) in order to get the total RMF lagrange density for neutron stars.

Introducing the mean-ﬁeld and no-sea approximation, the equations of motion for baryons and mesons can be derived, and the corresponding energy eigenvalues, baryon density and scalar density can be obtained for neutron stars, similar to

those for nuclear matter. The equations of motion for electron and µ− _{are free}

Dirac equations:

(iγµ∂µ− mλ)ψλ= 0, (3.13)

and their densities can be expressed in terms of their corresponding Fermi momenta as ρλ= k3λ/(3π2).

The chemical potentials µB for the baryons B are the energy eigenvalues of the

Dirac equation:

µB= eB(k) = gωBω0+ gρB· τ3B· ρ0,3+ ΣR0B+

q

k2_{+ m}∗2

B. (3.14)

The chemical potentials µλ for the leptons are the solutions of their equations of

motion:

µλ=

q k2

λ+ m2λ. (3.15)

In neutron stars, the chemical equilibrium conditions is required:

µB = bBµn− qBµe, µµ = µe, (3.16)

where bB and qB denote baryon charge and electronic charge of baryon B, µn, µe

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3.3. DENSITY DEPENDENCE OF EFFECTIVE INTERACTIONS 23

The baryon number conservation and charge-neutral conditions are given by:

ρ = X B ρB = X B bBk3B 3π2 , (3.17) Q = X B QB+ X λ Qλ= X B qBkB3 3π2 − X λ k3 λ 3π2 = 0. (3.18)

Similar as the case for nuclear matter, the energy density and pressure for neutron stars are respectively:

ε = 1 2m 2 σσ2+ 1 2m 2 ωω02+ 1 2m 2 ρρ20,3+ 1 3g2σ 3₊1 4g3σ 4₊3 4c3ω 4 0+ 3 4d3ρ 4 0,3 + 1 π2 X B Z kB 0 k2dkqk2_{+ m}∗2 B + 1 π2 X λ=e−,µ− Z kλ 0 k2dkqk2_{+ m}2 λ, (3.19) P = −1₂m2σσ2+ 1 2m 2 ωω20+ 1 2m 2 ρρ20,3− 1 3g2σ 3 −1₄g3σ4+ 1 4c3ω 4 0+ 1 4d3ρ 4 0,3 + X B ρB· ΣR0B+ 1 3π2 X B Z kB 0 k4 p k2_{+ m}∗2 B dk + 1 3π2 X λ=e−,µ− Z kλ 0 dk k 4 p k2_{+ m}2 λ . (3.20) The numerical details about how to solve above equations for nuclear matter and neutron stars can be found in Paper I (i.e., Ref. [1]).

3.3 Density dependence of effective interactions

The density dependencies of various eﬀective interaction strengths are studied and carefully compared in nuclear matter and neutron stars, including the nonlinear RMF interactions NL1, NL2 [25], NL3 [26], NLSH [27], TM1, TM2 [28], PK1, PK1r [29], and GL-97 [22], and the density-dependent interactions TW-99 [30], DD-ME1 [31], and PKDD [29], as shown in Figs. 3.2 and 3.3.

For nonlinear effective interactions, the “equivalent" density dependencies of the effective interaction strengths for σ, ω and ρ mesons are extracted from the meson field equations Eqs. (3.1) - (3.3) according to:

   gσB(ρ) = gσB+ U′(σ)/ρs= gσB+ (g2σ2+ g3σ3)/ρs, gωB(ρ) = gωB− U′(ω0)/ρ = gωB− (c3ω03)/ρ, gρB(ρ) = gρB− U′(ρ0,3)/PBτ3BρB = gρB− (d3ρ30,3)/(ρn− ρp). (3.21)

In Fig. 3.2, the density dependencies of the eﬀective interaction strengths for σ (top), ω (middle) and ρ (bottom) mesons in symmetric nuclear matter as functions of the baryon density are shown. The density dependencies of the interaction strengths for TW-99 and DD-ME1 are very similar, as noted in Ref. [31]. The

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24

CHAPTER 3. NUCLEAR MATTER AND NEUTRON STAR IN RMF THEORY 0 0.05 0.1 0.15 0.2 0.25 Baryon density ρ[ fm−3 ] 2 5 gρ 8 12 16 gω 4 9 14 gσ PK1 PK1r PKDD TW−99 DD−ME1 NLSH TW−99 NL1 NL3 TM1 GL−97 DD−ME1 NL1 NLSH NL3 TM1 GL−97 DD−ME1 TW−99 NL1 TM1 NL3 NLSH GL−97 NL2 TM2 NL2 TM2 NL2 TM2

Figure 3.2: The eﬀective interaction strengths for σ (top), ω (middle) and ρ (bot-tom) in symmetric nuclear matter as functions of the baryon density. The shadowed area corresponds to the empirical value of the saturation density in nuclear matter

(Fermi momentum kF = 1.35 ± 0.05 fm−1 or density ρ = 0.166 ± 0.018 fm−3).

σ and ω strengths of PKDD are very similar as TW-99 and DD-ME1, and those for PK1 and PK1r are almost same and have the similar slopes as that of TM1. Let’s note that the strengths for σ and ω mesons for GL-97 is obviously weaker than the others.

For the σ-meson, all the interaction strengths can be presented as density-dependent. The interaction strengths of the σ-meson given by TW-99, DD-ME1 and PKDD are quite different from the others in either magnitudes or slopes. The differences can also be seen even in the region of the empirical nuclear matter densities. For the ω-meson, except TW-99, DD-ME1, PKDD, PK1, PK1r, TM1 and TM2, all the other strengths are density-independent. All the strengths are more similar to each other in the region of the empirical saturation densities comparing with the case of σ-meson except GL-97, although large differences can also be seen at low densities. For the ρ-meson which describes the isospin asymmetry, the strengths for TW-99, DD-ME1 and PKDD show strong density dependencies in contrast with the constants in the other interactions, except PK1r, which shows a slight density dependence. TW-99 and DD-ME1 cross the nonlinear interactions at

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3.3. DENSITY DEPENDENCE OF EFFECTIVE INTERACTIONS 25

a density much lower than the empirical saturation density.

0 0.3 0.6 0.9 Baryon density ρ[ fm−3 ] 0 2 4 6 gρ (ρ) PK1 PK1r PKDD 6 9 12 15 gω (ρ) 2 6 10 14 gσ (ρ) NLSH NL1 NL3 TW−99 DD−−ME1 TM1 GL−97 NL1 NLSH NL3 DD−ME1 TW−99 GL−97 TM1 NL1 TM1 NL3 NLSH GL−97 DD−ME1 TW−99

Figure 3.3: Similar as Fig. 3.2, but for neutron stars without hyperons.

The results for neutron stars are shown in Fig. 3.3. As the eﬀective interactions NL2 and TM2 are mainly used in light nuclei, we don’t discuss them here. At densities ρ < 0.2 fm−3_{, Fig. 3.3 is similar to Fig. 3.2, i.e., the density dependencies}

are almost not related to the isospin symmetry.

The σ and ω strengths of TW-99 and DD-ME1 are diﬀerent at high density. At the same time, the behavior of PKDD is similar as that of DD-ME1. For the scalar σ-meson, the interaction strengths of TW-99, DD-ME1, PKDD, TM1 and GL-97 decrease with the baryon density in similar slopes, while those of NL1, NL3 and

NLSH decrease with baryon density for the densities ρ < 0.2 fm−3_{, then increase}

afterwards. This is due to the very big positive g3σ3 in Eq. (3.21) for NL1, NL3

and NLSH, in contrast with the negative ones for TM1 and GL-97. For the vector ω-meson, the interaction strengths of TW-99, DD-ME1, PKDD, Pk1, PK1r, and

TM1 decrease with the baryon density. At densities 0.2 < ρ < 0.55 fm−3_{, the}

strengths of DD-ME1, TW-99, PKDD, Pk1, PK1r, and TM1 are between those of NL3 and GL-97. The curve given by TM1 crosses with the line of GL-97 at density

ρ ≈ 0.57 fm−3_{, and then gives the weakest interaction strength. For isospin-vector}

ρ meson, the interaction strengths of TW-99 and DD-ME1 decrease with baryon density and trend to vanish at high densities, while the nonlinear interactions are

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26

CHAPTER 3. NUCLEAR MATTER AND NEUTRON STAR IN RMF THEORY constants except Pk1r decreasing a little. PKDD is in the between.

The properties of nuclear matter and neutron stars are studied and carefully compared by using diﬀerent eﬀective interactions of RMF theory. The detailed discussions can be found in Paper I (attached at the end, i.e., Ref. [1]).

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Chapter 4

Nuclear symmetry energy in RMF

theory

In this chapter, the nuclear symmetry energy is studied in the RMF theory with the eﬀective interactions NLSH, NL3, TM1, and PK1 and compared with those from Skyrme HF calculations. The result are published in the papers [2, 68] and paper [2] is attached at the end.

The semi-empirical mass formula contains a term called the nuclear symmetry energy(NSE) [69, 70], Esym= 1 2bsym (N − Z)2 A = asymT 2 _(4.1)

with T = |Tz| = |N − Z|/2, which is explained to originate from the kinetic energy

and the interaction itself, i.e., asym= akin+ aint in standard textbooks. The NSE

shows quadratic isospin dependence in semi-empirical mass formula and it may also contains a term linear in T , which is associated with the neutron-proton exchange and involves more detailed features of nuclear structure [69]. Conventionally, the NSE can be parameterized as

Esym= asym(A) T (T + λ) (4.2)

The linear term λT is found to be strongly model dependent and there is a common belief that the mean-ﬁled models in the Hartree approximation yield essentially only a quadratic term (λ ≈ 0). On the other hand, the nuclear shell model [71, 72] or models restoring isospin symmetry [73] suggest that λ ≈ 1. Although there is certain preference for λ ≈ 1, no consensus is reached so far concerning the value of λ, e.g., in the Winger super multiples model, the linear term gives value of λ = 4 [74].

Experiment evidence from masses of nuclei with small values of T supports the existence of the linear term [75]. Based on the experimental binding energies for

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28 CHAPTER 4. NUCLEAR SYMMETRY ENERGY IN RMF THEORY A < 80 nuclei, Jänecke determined the symmetry energy as [76]:

Esym= asymT (T + 1), with asym= a(A)

A (4.3)

and the shell structure was shown in the symmetry energy coeﬃcient a(A). The

coeﬃcient of the NSE admits typical volume and surface components, asym =

av/A + as/A4/3. One of the most accurate mass formulae, the so-called FRDM [77],

employs a value of λ ≈ 1 but inconsistently admits only a volume-like linear term. Assuming a T (T + 1) dependence, Duﬂo and Zuker performed a global ﬁt to nuclear mass data of 1751 nuclei and obtained the empirical NSE formula [78, 79]:

asym= ( av A + as A4/3) = ( 134.4 A − 203.6 A4/3) [MeV]. (4.4)

with volume term av and surface term as.

As mentioned above, the physical origin of NSE is traditionally explained in

terms of an interplay between the kinetic energy and the interactions, asym =

akin+ aint. The Fermi gas model has been used to calculate the kinetic part akin.

Based on the iso-cranking model [80, 81], a diﬀerent view on the origin of the NSE was presented recently by Satuła and Wyss that:

Esym=

1 2εT

2₊1

2κT (T + 1) (4.5)

where ε is the mean level spacing with the empirical value [82, 83]

εemp= e

A, with 53 MeV < e < 66 MeV, (4.6)

and the term εT2_{/2 is obtained from the iso-cranking model. κ is the average}

eﬀective strength of the isovector potential 1

2κ ˆT · ˆT , (4.7)

and its eigenvalue corresponds to the term κT (T + 1)/2. This concept has been veriﬁed in Skyrme HF calculation [82]. One can see that the total NSE does contain a linear term in T in Eq. (4.5), but with λ < 1.

Since the RMF theory is based on a very different concept from the Skyrme HF theory, it is very interesting to investigate the physical origin of the NSE in the framework of the RMF theory. In the RMF theory, the effective interaction between nucleons is described by the exchange of various mesons (the isoscalar scalar sigma (σ), isoscalar vector omega (ω) and isovector vector rho (ρ)) and the photon. The σ and ω mesons provide the attractive and repulsive parts of the nucleon-nucleon force, respectively. The necessary isospin asymmetry is provided by the isovector ρ meson. Hence, by switching on and off the coupling to the ρ meson, one can easily study the mean level spacing ε and effective isovector potential strength κ. Before turning to the RMF theory, let’s briefly explain the iso-cranking model first.

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4.1. ISO-CRANKING MODEL 29

4.1 Iso-cranking Model

The iso-cranking model was put forward by Satuła and Wyss in order to understand the neutron-proton superﬂuidity in N = Z nuclei [80, 81].

Figure 4.1: The single-particle Routhians (upper panel) versus the iso-cranking

frequency for the equidistant level model. Solid (dashed) lines depict τx = +1/2

(τx= +1/2) sp state. At each crossing frequency (indicated by arrows) the

conﬁg-uration changes, and hence excitation energy and iso-alignment (lower panel). This ﬁgure is taken from Ref. [80].

First, considering a single-particle Routhian ˆ

Hω_{= ˆ}_H

sp− ¯hωˆtx (4.8)

where the spectrum of ˆHsp is iso-symmetric and equidistant with spacing ε, i.e.

{ei= iε}. Therefore, at ¯hω = 0, each eigenstate of ˆHωis fourfold degenerate. The

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degener-30 CHAPTER 4. NUCLEAR SYMMETRY ENERGY IN RMF THEORY acy. The following doublets are obtained:

|− >= √1

2(|n > −|p >), |+ >= 1 √

2(|n > +|p >), (4.9)

with < ±|ˆtx|± >= ±1/2. As shown in Fig. 4.1, the ground state conﬁguration

changes stepwise at the crossing frequencies: ¯

hωc(n)= ε, 3ε, 5ε, ... (2n − 1)ε (4.10)

At each frequency two iso-scalar pairs (|+ >, |− >) are emptied and two iso-vector (|+ >, |+ >) become occupied. Hence, the total isoalignment, < ˆtx>≡ Tx, changes

in step of ∆Tx=2. This follows the fact that Tz = Ty = 0, which yields Tx = T

and ∆Tx = ∆T . The ground state band consists of only even isospin states, T =

0, 2, 4, .... Once the crossing frequencies are calculated, it is straightforward to compute the excitation energy (with respect to the ground state) as,

Eeven−T − ET =0= Eω+ ¯hωTx=1

2εT

2_, _(4.11)

Note that this result is completely general and dose not depend on any assumptions concerning the character of the nucleonic interactions. Odd-T excitations in even-even N = Z nuclei are discussed in Ref. [80]. It is important to underline that odd-T states can be reached only by the proper particle-hole excitation at ¯hω = 0, which leads to the property that the odd-T band is shifted in energy by ε/2. The excitation energy is,

Eodd−T − ET =0= 1

2ε +

1 2εT

2_, _(4.12)

When the cranking axis is changed from tx to tz, i.e., Tx= Ty= 0, T = Tz,

ne-glecting the inﬂuences of the Coulomb ﬁeld, the results are expected to be the same because of the isospin invariance. One may understand the iso-cranking around

the tz-axis within the simple equidistant level model, as illustrated in Fig. 4.2.

This model assumes that the single-particle spectrum is equidistant with spacing

ε and fourfold degeneracy. When Tz cranking is applying to the even-even

nu-clei, there are four particles in the highest level for T = 0 state. For T = 1, one proton state is emptied and one neutron state is occupied. The total energy diﬀerence between T = 0 and T = 1 thus is equal to ε. Similar remarks hold for the T = 2, 3, 4, ... states and one can get that for the evT states, the

en-ergy diﬀerence Eeven−T − ET =0 =

1 2εT

2_{, which is the same as for T}

x cranking

(see Eq. (4.11)), as expected due to isospin invariance. For the odd-T states, Eodd−T − ET =0 =

1

2ε +

1 2εT

2_{, identical to Eq. (4.12). Our calculations are}

per-formed for even-T states of Tz cranking starting from N = Z even-even nuclei.

Thus, Eq.(4.11) shows the contribution of mean level spacing ε to the NSE, i.e., εT2_/2.

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4.2. RMF THEORY FOR AXIALLY DEFORMED FINITE NUCLEI 31

Figure 4.2: Iso-cranking around tz-axis in the equidistant level model starting from

N = Z even-even nuclei.

4.2 RMF theory for axially deformed finite nuclei

In this thesis, the binding energies of nuclei from T = 0 to neutron drip line for several isobars are calculated in RMF theory and then from these calculated binding energies, the isospin and particle number dependencies of NES are studied. Most of the nuclei calculated here are deformed. Thus in our calculation, the axially deformation is included. Therefore, we start by studying the corresponding axially deformed formulae for ﬁnite nuclei in RMF theory.

For ﬁnite nuclei, baryon B in the Lagrangian (Eq. (2.1)) denotes neutron and proton. For simplicity, we suppress the symbol B in the following equations. The equations of motion (Eqs. (2.18) and (2.20-2.23)) can be rewritten as:

{−iα · ∇ + V (r) + β [m + S(r)]} ψi= ǫiψi, (4.13) and _       −∆ + m2σ σ(r) = −gσρs(r) − g2σ2(r) − g3σ3(r), −∆ + m2ω ωµ(r) = gωjµ(r) − c3(ωνων)ωµ(r), −∆ + m2 ρ ρµ(r) = gρ~jµ(r) − d3(ρνρν)ρµ(r), −∆Aµ_(r) _{= ej}µ p(r), (4.14) where    V (r) = β gωγµωµ(r) + gργµτ ρµ(r) + eγµ1 − τ 3 2 Aµ(r) , S(r) = gσσ(r), (4.15)

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32 CHAPTER 4. NUCLEAR SYMMETRY ENERGY IN RMF THEORY are respectively the vector and scalar potentials, and the source terms for the mesons and the photons are

                       ρs(r) = A P i=1 ¯ ψiψi, jµ_(r) ₌ PA i=1 ¯ ψiγµψi, ~jµ_(r) ₌ PA i=1 ¯ ψiγµτψi, jµ p(r) = A P i=1 ¯ ψiγµ1 − τ 3 2 ψi. (4.16)

The above equations hold for nonlinear effective interactions only. We will not present the density-dependent case here, since in our calculation, only nonlinear effective interactions NLSH [27], NL3 [26], TM1 [28], and PK1 [29] are used. In Eqs. (4.16), the summations are taken over the valence nucleons only, i.e., no-sea approximation is adopted. For Eqs. (4.13) and (4.14), the mean field approximation is generally used, i.e., the meson field operators in Eq. (4.14) are replaced by their corresponding expectation values, and the nucleons are considered to move independently in the classical meson fields. The coupled equations (4.13) and (4.14) can be solved self-consistently with certain symmetries in coordinate space [17, 18], harmonic oscillator basis [84] or Woods-Saxon basis [85].

The symmetries of the system simplify the calculations considerably. In the sys-tem considered here, as time reversal symmetry is preserved, there are no currents in the nucleus and the spatial vector components of ω, ρ and A vanish. This leaves

only the time-like components ω0, ρ0and A0. Charge conservation guarantees that

only the 3-component of the iso-vector ρ survives, which we denote in the following

by ρ0,3, For axially deformed nuclei (i.e., the systems which have rotational

sym-metry around a symmetrical axis, assumed to be the z-axis), the potentials in the

Dirac equation and the sources of meson ﬁelds depend only on the coordinates r⊥

and z and the equations (4.13) and (4.14) can be solved in polar coordinates, where we have

x = r⊥cosϕ, y = r⊥sinϕ, z. (4.17)

The Dirac spinor ψi for the nucleon with the index i can be characterized by the

quantum numbers Ωi, πi and ti, where Ωi = mli+ msi is the eigenvalue of the

angular momentum operator Jz, πi is the parity and ti is the isospin. The wave

function ψi(r, t) can be written as:

ψi(r, t) = fi(r) igi(r) = √1 2π     f_i+(z, r⊥)ei(Ωi−1/2)ϕ fi−(z, r⊥)ei(Ωi+1/2)ϕ ig_i+(z, r⊥)ei(Ωi−1/2)ϕ ig_i−(z, r⊥)ei(Ωi+1/2)ϕ     χi(t), (4.18)

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4.2. RMF THEORY FOR AXIALLY DEFORMED FINITE NUCLEI 33

where, fi±, g±i are four components of the Dirac spinor ψi. Thus, the Dirac equation

for nucleons can be written as:          (m + S + V )fi++ ∂zg+i (∂r_⊥+Ω+1/2₂ )gi− = εifi+, (m + S + V )f− i + ∂zgi−(∂r⊥− Ω−1/2 2 )g + i = εifi−, (m + S − V )gi++ ∂zfi+(∂r_⊥+Ω+1/2₂ )fi− = −εigi+, (m + S − V )g− i + ∂zfi−(∂r⊥− Ω−1/2 2 )f + i = −εigi−. (4.19)

For each solution with positive Ωi,

ψi ≡ {fi+, fi−, g +

i , gi−, Ωi}, (4.20)

there is the time-reversed solution with the same energy, ψ_¯i= T ψi≡ {−fi−, f

+

i , gi−, −g +

i , −Ωi}, (4.21)

where the time reversal operator is T = −iσyK and K is the complex conjugation.

For the system with time-reversal symmetry, the contributions of state i to nuclear densities are the same as those of its time-reversal state ¯i. Thus the nuclear scalar density can be written as

ρs= 2 X i>0 ni((|fi+| 2 + |f− i | 2 ) − (|g+i | 2 + |g− i | 2_)), _(4.22)

where ni is the occupation probability of state i, i > 0 means sum over all the

positive Ωistate. In a similar fashion, the densities ρv, ρ0,3and ρpcan be obtained.

These densities are source terms of meson and photon ﬁelds. In polar coordinates, the Klein-Gordon equations are

(−_r1 ⊥ ∂r⊥r⊥∂r⊥+ m 2 φ)φ(z, r⊥) = sφ(z, r⊥), (4.23) with sφ(z, r⊥) =        −gσρs(z, r⊥) − g2σ2(z, r⊥) − g3σ3(z, r⊥), gωρv(z, r⊥) − c3ω03(z, r⊥), gρρ0,3(z, r⊥) − d3ρ30,3(z, r⊥), eρp(z, r⊥). (4.24)

The source terms for the mesons and the photons are          ρs(z, r⊥) = PAi=1ψ¯iψi, ρv(z, r⊥) = PAi=1ψ + i ψi, ρ0,3(z, r⊥) = PZp=1ψ+pψp−PN_n=1ψ+nψn, ρp(z, r⊥) = PZp=1ψ+pψp. (4.25)

To solve the Eqs. (4.19) and (4.23), the Dirac spinor ψi as well as the meson ﬁelds

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34 CHAPTER 4. NUCLEAR SYMMETRY ENERGY IN RMF THEORY oscillator potential or Woods-Saxon potential, and the solution of the problem is transformed into a diagonalization of a Hermitian matrix. Details can be found in Ref. [84, 85].

The total energy of the system is given by

ERMF = Enucleon+ Eσ+ Eω+ Eρ+ Ec+ ECM, (4.26) with                                    Enucleon = P i ǫi, Eσ = − 1 2 Z d3r gσρs(r)σ(r) + 1 3g2σ(r) 3₊1 2g3σ(r) 4 , Eω = −1 2 Z d3r gωρ(r)ω0(r) −1 2c3ω0(r) 4 , Eρ = −1 2 Z d3r gρτ3ρ3(r)ρ0,3(r) −1 2d3ρ0,3(r) 4 , Ec = − e2 8π Z d3rρc(r)A0(r), ECM = −3 441A −1/3_, (4.27)

where Enucleon is the sum of the single particle energies ǫi, Eσ, Eω, Eρ, and Ec are

the contributions of the meson ﬁelds and the Coulomb ﬁeld, and ECM is the center

of mass correction. Note that the only iso-vector contribution to the total energy is Eρ.

4.3 Numerical details

The RMF equations (4.19) and (4.23) are solved by expansion in the harmonic oscil-lator basis with 14 osciloscil-lator shells for both the fermion ﬁelds and the boson ﬁelds.

The oscillator frequency of the harmonic oscillator basis is ﬁxed as ¯hω0= 41A−1/3

MeV and the deformation of the harmonic oscillator basis β0 is set to obtain the

lowest total energy. Generally speaking, the RMF calculation reproduce the exper-imental binding energy very well. For the present study we are mainly interested in the symmetry energy in RMF theory. Therefore the Coulomb potentials and the pairing correlations will be neglected in the following calculations. From the Dirac equation (4.13), the full RMF potential is given by,

Vtot= V (r) + βS(r) = gωω0(r) + gρρ0,3(r) + βgσσ(r), (4.28)

which can be readily separated into isovector and isoscalar components:

Visos(r) = gωω0(r) + βgσσ(r),

Visov(r) = gρρ0,3(r). (4.29)

By switching on or oﬀ the iso-vector potential, we evaluate the binding energies of nuclei for several isobaric chains and will study the mean level spacing ε, the

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4.3. NUMERICAL DETAILS 35

iso-vector potential eﬀective strength κ, and the NSE in detail, as discussed in the following.

Hints from the NSE of

A=48 isobars

We start by calculating the mean level spacing ε and iso-vector potential eﬀective strength κ for A = 48 isobars in RMF theory with the eﬀective interactions NLSH [27], NL3 [26], TM1 [28], and PK1 [29] as shown in Ref. [68]. The process is as followings:

(1) The full RMF potential is separated into iso-scalar and iso-vector parts and the Coulomb ﬁeld is neglected, as shown in Eqs. (4.28) and (4.29).

(2) By switching oﬀ the iso-vector part, i.e., setting gρ = 0, we can calculate

the binding energies ˜ET by using the iso-scalar potential Visos for diﬀerent nuclei

in one isobaric chain, then the mean level spacing ε can be obtained from ˜ ET− ˜ET =0= 1 2εT 2_. _(4.30) 0 4 8 12 16 T=(N−Z)/2 0.8 1.2 1.6 2.0 κ [MeV] NL3 NLSH PK1 TM1 0 4 8 12 16 T=(N−Z)/2 0.0 1.0 2.0 3.0 ε [MeV] NL3 NLSH PK1 TM1 0 4 8 12 16 20 0 4 8 12 16 20 (m*/m) ε T(T+1) T2

Figure 4.3: The mean level spacing ε (upper left) and its counterpart (upper right)

scaled by m∗_{/m and the average eﬀective strength κ of iso-vector potential ﬁtted}

along T2_{(lower left) and T (T + 1) (lower right) for A = 48 isobars in RMF theory.}

The shadowed areas correspond to the empirical mean level spacing, Eq. (4.6).

The results are shown in the upper part of Fig. 4.3. It is very clear that the behavior of ε is similar to that obtained from Skyrme HF calculations [82]. Strong variations in ε are seen for small values of T , which are associated with shell

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36 CHAPTER 4. NUCLEAR SYMMETRY ENERGY IN RMF THEORY

closures (e.g., T =4 corresponding to double magic nucleus 48_{Ca), and for larger}

values of T , ε become less sensitive to the shell structure and after re-scaled by

corresponding eﬀective mass i.e., ε∗_{= (m}∗_{/m)ε, it lies within the empirical values}

(shaded areas), as shown in the corresponding right panels. One need to point that

the eﬀective mass m∗_{in RMF theory we used in this thesis and Ref. [2] is the scalar}

eﬀective mass m∗_{= m − g}

σσ, which is diﬀerent from the eﬀective mass in Skyrme

HF calculation (m∗_{/m = 1 − dV (E)/dE, where V is potential and E is enenrgy).}

We also can calculate the same eﬀective mass in RMF theory and the value is

m∗_{/m=0.62∼0.68, which is similar to the scalar mass ((m − g}

σσ)/m=0.60∼0.64).

Thus, the above conclusions are kept.

(3) Switching on the iso-vector potential, binding energies ET can be obtained

by using the full RMF potential, Vtot. Thus, the average eﬀective strength of

iso-vector potential κ can be calculated from ET − ˜ET =

1 2κT

2 _or 1

2κT (T + 1). (4.31)

Note that ET =0 = eET =0, i.e., the iso-vector potential has no contribution to the

binding energy of T = 0 nuclei. The results are shown in the lower part of Fig. 4.3.

2 4 6 8 10 12 14 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 T(T+x) A=48, RMF, PK1 k a p p a [ M e V ] T=(N-Z)/2 x=0 x=1 x=1.25 x=1.5 x=2 x=3 x=4 x=5

Figure 4.4: The average effective strength κ of iso-vector potential fitted along T (T + x) with different x as marked in the figure for A=48 isobars calculated in RMF theory.

The κ calculated from the T (T + 1) dependence in Skyrme HF calculations are almost constant along T . However κ calculated from T (T + 1) dependence in RMF

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theory still decreases along T and towards isospin dependence like T (T + x) ET − ˜ET =

1

2κT (T + x) (4.32)

with x > 1. I.e., in RMF theory, κ shows a stronger linear term in T . The eﬀective strength κ from T (T + x) dependence with diﬀerent x are calculated, as shown in Fig 4.4. When x is larger, the slope is smaller and when x = 4 or 5, κ is a constant at large T . One can thus make the following ansatz for the NSE

Esym= ET − ET =0= 1 2εT 2₊1 2κT (T + x) ⇒ 1 2(ε + κ)T (T + 1), (4.33)

for x > 1. The NSE tends to T (T + 1) dependence, which is the same as the empirical formula in Refs. [76, 78].

(4) Next, we extract the NSE coeﬃcient asym for the A = 48 isobaric chain

from the diﬀerences of the binding energies ET:

Esym= ET − ET =0= asymT (T + 1). (4.34)

The result is shown in Fig. 4.5. Since the results for diﬀerent eﬀective interactions are very close to each other, we just present the results for TM1 and PK1. The

straight line corresponds to the empirical value of the NSE coeﬃcient asym in

Ref. [78]. The asymfrom RMF calculation is smaller than the empirical values but

it is almost constant when T = 6, 8, 10, 12 (i.e., it shows the T (T + 1) dependence very well). 0 4 8 12 16 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 A=48 1.6334 a _ s y m [ M e V ] T=(N-Z)/2 TM1 PK1

Figure 4.5: The NSE coeﬃcient asym extracted from Eq. (4.34) for the A = 48

isobaric chain calculated in RMF theory. The straight line corresponds to the number 1.6334, which is the empirical value from Eq. (4.4) [78] for A=48.

(5) The contribution of mean level spacing ε to NSE has a T2 _{dependence, as}

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38 CHAPTER 4. NUCLEAR SYMMETRY ENERGY IN RMF THEORY Fig. 4.5. Thus from Eqs. (4.30), (4.33) and (4.34), the value x = 1 + ε/κ can be obtained, i.e., the contribution of κ to the NSE is

ET− ˜ET =

1

2κT (T + 1 + ε/κ). (4.35)

It is also known that the empirical asymin Ref. [76] exhibits shell structure with

maxima at A = 16, 28, 40, 56. Thus, binding energies for isobaric chains, including

A = 40, 56, 88, 100, 120, 140, 160, 164, 180 are calculated, and their ε, κ, and asym

values are studied in detail as shown below.

0 4 8 12 16 20 0 1 2 3 4 5 6 7 A=40 e p s i l o n [ M e V ] T=(N-Z)/2 PK1 0 4 8 12 16 20 0 1 2 3 4 5 6 7 A=40 ( m */ m ) e p s i l o n [ M e V ] T=(N-Z)/2 PK1 0 4 8 12 16 20 0 1 2 3 4 5 A=56 e p s i l o n [ M e V ] T=(N-Z)/2 PK1 0 4 8 12 16 20 0 1 2 3 A=56 ( m */ m ) e p s i l o n [ M e V ] T=(N-Z)/2 PK1 0 4 8 12 16 20 0.0 0.4 0.8 1.2 1.6 2.0 A=88 m e a n l e v e l s p a c i n g : e [ M e V ] T=(N-Z)/2 TM1 0 4 8 12 16 20 0.0 0.4 0.8 1.2 1.6 2.0 A=88 ( m * / m ) e [ M e V ] T=(N-Z)/2 TM1

Figure 4.6: The mean level spacing ε extracted from Eq. (4.30) (left) and its

counterpart (right) scaled by m∗_{/m for A=40, 56, and 88 isobaric chains calculated}

in RMF theory. The two straight lines correspond to the empirical mean level spacing, Eq. (4.6).

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4.3. NUMERICAL DETAILS 39 0 5 10 15 20 25 0 1 2 3 4 5 A=100 e p s i l o n [ M e V ] T=(N-Z)/2 TM1 0 4 8 12 16 20 24 0.0 0.5 1.0 1.5 2.0 2.5 3.0 A=100 ( m */ m ) e p s i l o n [ M e V ] T=(N-Z)/2 TM1 0 4 8 12 16 20 24 28 0.0 0.2 0.4 0.6 0.8 1.0 A=120 ( m */ m ) e p s i l o n [ M e V ] T=(N-Z)/2 TM1 0 5 10 15 20 25 30 0.0 0.3 0.6 0.9 1.2 1.5 A=120 e p s i l o n [ M e V ] T=(N-Z)/2 TM1 0 4 8 12 16 20 24 28 32 36 0.0 0.5 1.0 1.5 A=140 e p s i l o n [ M e V ] T=(N-Z)/2 TM1 0 4 8 12 16 20 24 28 32 36 0.0 0.5 1.0 1.5 A=140 ( m */ m ) e p s i l o n [ M e V ] T=(N-Z)/2 TM1

Figure 4.7: Similar as Fig. 4.6, but for isobars A=100, 120, and 140.

A. The contribution of mean level spacing

ε to the NSE

The mean level spacing ε extracted from Eq. (4.30) in the RMF theory for A=40, 56, 88, 100, 120, 140, 160, 164, and 180 isobaric chains are shown in Figs. 4.6, 4.7 and 4.8.

All of them show similar behavior as the A =48 isobar. When the mean level

spacing ε is re-scaled by their corresponding eﬀective masses m∗_{/m, i.e., ε}∗ ₌

(m∗_{/m)ε, the eﬀective mean level spacing ε}∗ _{lies within the empirical values, as}

shown in the corresponding right panels. For small values of T , strong variations in ε are seen, which are associated with the shell closures (That is, in the isobaric chains A=40, 58, 100, and 164, T = 0 nuclei are double magic, hence there are very large mean level spacing for the corresponding T = 2 nuclei. T = 6 in A=88,

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40 CHAPTER 4. NUCLEAR SYMMETRY ENERGY IN RMF THEORY 0 4 8 12 16 20 24 28 32 36 40 0.0 0.5 1.0 1.5 2.0 2.5 A=160 e p s i l o n [ M e V ] T=(N-Z)/2 TM1 0 4 8 12 16 20 24 28 32 36 40 0.0 0.5 1.0 1.5 2.0 2.5 A=160 ( m */ m ) e p s i l o n [ M e V ] T=(N-Z)/2 TM1 0 4 8 12 16 20 24 28 32 36 40 44 0 1 2 3 4 5 A=164 e p s i l o n [ M e V ] T=(N-Z)/2 TM1 0 4 8 12 16 20 24 28 32 36 40 44 0 1 2 3 A=164 ( m */ m ) e p s i l o n [ M e V ] T=(N-Z)/2 TM1 0 4 8 12 16 20 24 28 32 36 40 44 0.0 0.5 1.0 1.5 A=180 e p s i l o n [ M e V ] T=(N-Z)/2 TM1 0 4 8 12 16 20 24 28 32 36 40 44 0.0 0.5 1.0 1.5 A=180 ( m */ m ) e p s i l o n [ M e V ] T=(N-Z)/2 TM1

Figure 4.8: Similar as Fig. 4.6, but for isobars A=160, 164, and 180.

T = 12 in A=140, T = 2 in A=160, and T = 8 in A=180 correspond respectively to neutron magic number N = 50, N = 82, N = 82, and Z = 82. Thus these nuclei give smaller mean level spacing than their neighbors). For larger values of T , ε become less sensitive to the shell structure and its value is stabilized for an isobaric chain after averaging over a certain interval. One should also note that with increasing A (i.e., from A = 40 to A = 180), all curves turn towards the upper limit of the empirical data, reflecting the decreasing role of surface effects with increasing A, similar to the Skyrme HF results [82]. Note that we just present the results of one effective interaction, PK1 or TM1, because the results of different effective interactions are very close to each other, as shown in Fig. 1 of Paper II (attached at the end of the thesis, i.e., Ref. [2]), where we chose three effective

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interactions, NL3, TM1, and PK1 and present the results for three isobaric chains A=48, 88, and 160.

When comparing to the results of the Skyrme HF calculations [82], several con-clusions can be drawn for the mean level spacing: i) even though the mean level spacing ε from the RMF calculation is much larger than that from Skyrme HF calculations, after the effective mass scaling both models are within the empirical values. ii) the results from the RMF calculations clearly confirm the general out-come of Ref. [82] that the iso-scalar field generates a symmetry energy of the form

εT2_{, which can also be obtained independently from iso-cranking model.}

B. The contribution of iso-vector potential to the NSE

After obtaining the contribution of the mean level density ε to the NSE, we now proceed to study the contribution of iso-vector potential. Taking into account the

iso-vector ρ meson, the binding energy ET can be calculated with the full RMF

potential. From Eqs. (4.32) and (4.35), the average eﬀective strength κ of the iso-vector potential can be calculated. Fig. 4.4 has shown that the eﬀective strength of the iso-vector potential κ has a T (T + x) dependence with x > 1, and in Fig. 4.9, we present the results of κ estimated from Eq. (4.32) with x = 0, 1, 2, for isobaric chains A=40, 56, 100, 120, 140, and 164.

Similar to the result for A=48 in Fig. 4.4, κ calculated from a T (T + 1) depen-dence decreases along T . When x = 2, κ is almost constant for isobars A=56 and 120, but the slope still can be found for A=40, 100, 140, and 164. Therefore, x is A dependent. Thus, κ are calculated from Eq. (4.35), as shown in Fig. 2 of Paper II (attached at the end, i.e., Ref. [2]). More detailed discussions can be found in the paper. One need to point that the isospin dependence of κ, T (T + x), x being clear larger than 1 in RMF theory. As analtz of the form T (T + 1 + ε/κ) yields a rather constant ﬁt, in variance to Skyrme HF calculations. Thus the total NSE in RMF can be given by,

Esym= 1 2εT 2₊1 2κT (T + 1 + ε/κ) = 1 2(ε + κ)T (T + 1). (4.36)

C. Global behavior of the NSE

We proceed further by investigating the A dependence of the NSE. As shown in Eq. (4.36), the total NSE in RMF theory can be presented by a T (T + 1) depen-dence. Based on the calculation with the full RMF potential, we extract the nuclear

symmetry energy coeﬃcient asym from the diﬀerence of the binding energies Eq.

(4.34). The NSE coeﬃcients asymfor the isobars A=40, 56, 88, 100, 120, 140, 160,

164, and 180 are shown in Fig. 4.10. The shell structure also can be see in asymat

small T , similar to the case of ε. The T =0 nuclei of isobars A=40, 56, 100 and 164 are double magic. They are more bound and result in an increase of the symmetry energies for nuclei with T > 0. At large T , from A=48 to A=180, the values of NSE decrease, but turn to the empirical values of Ref. [78].

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42 CHAPTER 4. NUCLEAR SYMMETRY ENERGY IN RMF THEORY 0 4 8 12 16 20 0.0 0.5 1.0 1.5 2.0 2.5 3.0 PK1 T(T+x) A=40 k a p p a [ M e V ] T=(N-Z)/2 x=0 x=1 x=2 0 4 8 12 16 20 0.0 0.4 0.8 1.2 1.6 2.0 PK1 T(T+x) A=56 k a p p a [ M e V ] T=(N-Z)/2 x=0 x=1 x=2 0 4 8 12 16 20 24 0.4 0.6 0.8 1.0 1.2 1.4 A=100 k a p p a [ M e V ] T=(N-Z)/2 x=0 x=1 x=2 0 4 8 12 16 20 24 28 0.0 0.3 0.6 0.9 1.2 1.5 A=120 k a p p a [ M e V ] T=(N-Z)/2 x=0 x=1 x=2 0 4 8 12 16 20 24 28 32 36 0.0 0.2 0.4 0.6 0.8 1.0 1.2 TM1 T(T+x) A=140 k a p p a [ M e V ] T=(N-Z)/2 x=0 x=1 x=2 0 4 8 12 16 20 24 28 32 36 40 44 0.0 0.2 0.4 0.6 0.8 1.0 1.2 T(T+x) A=164 k a p p a [ M e V ] T=(N-Z)/2 x=0 x=1 x=2

Figure 4.9: The eﬀective strength of the iso-vector potential κ estimated from Eq. (4.32) for isobaric chains A=40, 56, 100, 120, 140, and 164.

In order to compare with the empirical data of Ref. [76](where Esym= (a(A)/A)

T (T + 1) and a(A) exhibits shell structure with maxima at A = 16, 28, 40, 56), we

depict A ∗ asym as a function of A for all the 10 isobaric chains (A=40, 48, 56, 88,

100, 120, 140, 160, 164, and 180) calculated with the eﬀective interaction PK1. To

avoid the inﬂuence of shell structure, the value of the NSE coeﬃcients asymof two

nuclei at larger values of T in the vicinity of the drip line for each isobaric chain are

selected in Fig. 3 of paper II. From this Fig., one found that i) A ∗ asymcalculated

from RMF theory are in good agrement with empirical data in Ref. [76], especially the shell structure with maxima at A = 40 and 56; ii) all the calculated results are

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4.3. NUMERICAL DETAILS 43 0 4 8 12 16 20 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 1.8717 A=40 a _ s y m [ M e V ] T=(N-Z)/2 PK1 0 4 8 12 16 20 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 1.4497 A=56 a _ s y m [ M e V ] T=(N-Z)/2 PK1 0 4 8 12 16 20 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 1.007 A=88 a _ s y m [ M e V ] T=(N-Z)/2 TM1 0 4 8 12 16 20 24 0.0 0.4 0.8 1.2 1.6 2.0 2.4 0.9054 A=100 a _ s y m [ M e V ] T=(N-Z)/2 TM1 0 4 8 12 16 20 24 28 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.77599 A=120 a _ s y m [ M e V ] T=(N-Z)/2 TM1 0 4 8 12 16 20 24 28 32 36 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.67995 A=140 a _ s y m [ M e V ] T=(N-Z)/2 TM1 0 4 8 12 16 20 24 28 32 36 40 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 A=160 0.6056 a _ s y m [ M e V ] T=(N-Z)/2 TM1 0 4 8 12 16 20 24 28 32 36 40 44 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.59272 A=164 a _ s y m [ M e V ] T=(N-Z)/2 TM1 0 4 8 12 16 20 24 28 32 36 40 44 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.54633 A=180 a _ s y m [ M e V ] T=(N-Z)/2 TM1

Figure 4.10: The NSE coeﬃcients asym from Eq. (4.34) for isobars A=40, 56, 88,

100, 120, 140, 160, 164, and 180. The lines are empirical values of asymfrom Eq.

(4.4).

close to the line drawn from the average experiment formula Eq. (4.4).

Restricting the analysis to volume and surface terms only, the NSE in RMF leads to the following asymptotic formula, obtained by a least square ﬁt to the chosen NSE, asym= av A − as A4/3 = 133.20 A − 220.27 A4/3 [M eV ], (4.37)

which is very close to the empirical values, Eq. (4.4). In a similar manner, the volume and surface terms for the the mean level spacing ε and the average eﬀective strength κ are determined from the calculations RMF by using least squares ﬁtting as shown in Fig. 4 of Paper II. It should be noted that we have chosen the same

Nuclear symmetry energy and neutron-proton pair correlations in microscopic mean field theory