MINISTRY OF EDUCATION AND TRAINING MINISTRY OF SCIENCE AND TECHNOLOGY VIETNAM ATOMIC ENERGY INSTITUTE ABSTRACT OF PHD THESIS CONSISTENT MEAN-FIELD APPROACH TO EQUATION OF STATE OF NUCLEAR MATTER AND NUCLEON OPTICAL POTENTIAL PhD student: Doan Thi Loan Supervisor: Prof Dao Tien Khoa Major: Atomic physics Code: 9.44.01.06 Hanoi - 2019 Chapter - Introduction The concept of nuclear mean field (NMF) was proposed first by Hans Bethe 80 years ago to describe the single-particle (SP) potential generated by strong interactions between nucleons bound in the nucleus NMF is the basis of the nuclear mean-field potential used in shell model to describe nuclear structure, as well as the optical potential (OP) used to describe nucleon-nucleus scattering at low and medium energies With NMF determined microscopically by many-body theories, using the different versions of nucleon-nucleon (NN) interactions, we can describe equation of state (EOS) of nuclear matter (NM) EOS of NM is an important topic of nuclear physics and nuclear astrophysics Especially, the research results of EOS of neutron rich NM are necessary for the studies of neutron star formation from supernovas [1–5], as well as the structure studies of exotic nuclei lying close to neutron dripline Among many microscopic studies of NMF, Hartree-Fock (HF) method has been widely used in the structure studies and the EOS of NM studies with the input chosen between different versions of effective NN interactions [6–12] These effective NN interactions are constructed explicitly depending on the nucleon density in the nuclear medium surrounding the two interacting nucleons Higher-order interaction in NN scattering in NM environment are considered as the physics origin of density dependence of NN effective interactions Among them, quite popular are CDM3Yn interactions, the density dependent versions of the M3Y interaction (originally constructed to reproduce the G-matrix elements of the Reid [13] and Paris [14] NN potentials in an oscillator basis) These interactions have been successfully used in the HF studies of NM [6–8, 15–17], as well as in the folding model studies of the nucleon-nucleus and nucleus-nucleus scattering [9, 18–23] by the theory group at Institute for Nuclear Science and Technology (INST) Recently, we have successfully built a NMF approach based on the Hartree-Fock (HF) method to study EOS of neutron-rich NM at different densities and temperatures in proto-neutron stars and neutron stars, using the latest versions of NN density-dependent interactions EOS of NM is described by energy per particle, pressure, compress- ibility, and nuclear symmetry energy In present HF approach, total energy E of homogeneous NM, which has consists of kinetic and potential energies of all nucleons bound in NM, using nucleon SP wave function - a plane wave |kστ - and NN interaction vNN , E = Ekin + nτ (k)nτ (k )A kστ, k σ τ |vNN |kστ, k σ τ , (1) kστk σ τ with k, σ and τ are momentum, spin, and isospin respectively of nucleon, nτ (k) is momentum distribution function of nucleons and A is antisymmetry operator acting on potential matrix elements according to Pauli principle Energy per paticle of NM (E/A) denpends on desity ρ and neutron/proton (NP) asymmetry parameter δ = (ρn − ρp )/ρ as E E E ≡ = ε(ρ, δ) = (ρ, δ = 0) + S(ρ)δ + O(δ ) + (2) A N A The quantity S(ρ) in (2) is called nuclear symmetry energy of NM The understanding of symmetry energy S(ρ) is essential to nuclear astrophysics studies on the crust and inner core structure of neutron stars [25], as well as cooling process of hot protoneutron stars toward cold neutron stars with T ≈ In addition, symmetry energy is also an important quantity in the microscopic models of the structure studies which is directly related to the description of neutron-skin of medium and heavy nuclei [23] The HF method also determines the NMF of the bound nucleon in NM which is called the single particle (SP) potential This is an important quantity in the study of asymmetric NM that helps to determine the EOS of NM in the core of neutron stars [4], as well as the structure of finite nuclear [28, 29] The NMF model presented in this thesis can be used consistently to describe nucleon SP potential as a continuous function of momentum, from the bound nucleon in NM to the scattered nucleon on NM, similar to continuous approximation applied in the BHF microscopic theory [28, 29] According to the Landau theory for an infinite fermion system [30], the SP nucleon energy eτ can be evaluated [17] from average energy of (2) of NM at density ρ as eτ (ρ, k) = k2 ∂ε(ρ) = + Uτ (ρ, k), với τ = n, p ∂nτ (k) 2mτ (3) eτ is the change of the NM energy when a nucleon with the momentum k removed or added Following the Landau theory, the nucleon SP potential Uτ from (3) consists of both the HF and rearrangement term (RT) RT appears because of higher-order correction in NN scattering or the density dependence of NN effective interactions However, it is impossible to determine explicitly RT in HF scheme BHF calculation has proved that higher-order correlations in NN interaction or threebody interaction contribute mainly in RT [26–29] In this thesis, we have successfully built an effective method to determine the contribution of RT in the extended HF model according to the Hugenholtz-Van Hove (HvH) theorem [31], using CDM3Yn density dependent NN interaction including RT ∆vNN For scattering nucleons on NM (E > 0), the SP potential of nucleons is also called optical potential (OP) After RT ∆vNN is determined explicitly for CDM3Yn interaction [17], the contribution of RT to SP potential and OP in NM need to be evaluated On the other hand, according to the HvH theorem [? ], the SP nucleon energy at the Fermi level is directly related to the symmetric energy S(ρ) of NM [11] Therefore, the effect of RT on the symmetric energy in the EOS study of neutron-rich NM should also be investigated in the HF model using the CDM3Yn interaction Thus, it is easy to see the central role of the SP nucleon potential in the EOS description of the neutron-rich NM These results are presented in detail in chapter and also were published in Physical Review C [41] In addition, an important content directly related to SP potential is the effective mass of nucleons in NM medium Understanding the nucleon effective mass is essential for the studies of nuclear physics and astrophysics [32, 33] The nucleon effective mass is directly related to symmetric energy S(ρ) of NM [25, 27] as well as the thermodynamic properties of hot NM at different densities [24] Moreover, the difference between the effective mass of neutron and proton is called the effective mass splitting, which is also a important topic of nuclear physics and nuclear astrophysics This quantity directly relates to the neutron/proton ratio in stellar evolution or proto-neutron star cooling toward neutron stars [32, 33] The results of effective mass and effective mass splitting from SP potential and OP are detailed in chapter and chapter 3 The folding model which is used to build microscopically nucleonnucleus OP, with inputs consisting of the effective NN interaction and the wave function of the nucleons bound in the nucleus, is a tool to evaluate structure models as well as versions of NN interactions With the important role of RT from the many-bodies correlation with ∆vNN term included in CDM3Yn interaction, the folding model can de extended to calculate the exact OP in the the microscopic description of nucleon-nucleus scattering Similar to SP by HF method, the OP potential determined by folding model also includes direct and exchange terms because of antisymmetrization of nucleon system [34] In particular, the exchange potential has a nonlocal form depending on two coordinate variables This extended folding model is presented in chapter of the thesis, with results confirming the important contribution of RT in nonlocal nucleon OP The Schrăodinger equation for nucleon-nucleus scattering using nonlocal OP is an integro-differential equation, more complicated than normal differential equation using local OP The nonlocal exchange term of nucleon OP is usually reduced to the local form in order to simplify the solution of the scattering equation [35] This approximation has been widely applied to calculate nucleon OP [18, 19, 21, 22] In addition, the nonlocal effect of OP nucleons is investigated by comparing the results of the cross-section of nucleon scattering on different targets such as 40,48 Ca, 90 Zr and 208 Pb, using local OP from the traditional folding model [18, 19, 21] and nonlocal OP from the extended folding model In the present work, we use R-matrix method [36, 37] to solve exactly the Schrăodinger equation with a nonlocal potential [41] Chapter - Nuclear mean field and EOS of asymmetric nuclear matter In chapter 2, NMF potential in asymmetric NM is calculated by HF method, using CDM3Yn NN effective interactions including RT term [9, 17] The contribution of RT to NMF is investigated in detail based on the relation of nucleon SP potential to symmetry energy and effective nucleon mass 2.1 CDM3Yn interaction and saturation property of symmetric NM Within HF scheme, the total energy (1) of NM is contributed mainly by the potential given by the central component vc of the effective NN interaction The total potential of NM determined by HF method also consists of direct and exchange terms Epot = nτ (k)nτ (k )[ kστ, k σ τ |vcD |kστ, k σ τ kστ k σ τ + kστ, k σ τ |vcEX |k στ, kσ τ ] (4) The nucleon momentum distribution nτ (k) in cold NM (T=0) is a step function determined at the Fermi momentum kF τ = (3π ρtau )1/2 as k kFτ nτ (k) = (5) k > kFτ For spin-saturated NM, the contribution of the spin-dependent component of vc to the potential (4) vanishes and the HF calculation is implemented by using only the isoscalar (IS) part and isovector (IV) part of the CDM3Yn interaction as D(EX) vcD(EX) (ρ, s) = F0 (ρ)v00 D(EX) (s)τ τ , với s = |r − r | (6) The density fuctions F0(1) (ρ) are parametrized [9, 42] in following form (s) + F1 (ρ)v01 F0(1) (ρ) = C0(1) [1 + α0(1) exp(−β0(1) ρ) + γ0(1) ρ] D(EX) (7) The radial functions v00(01) (s) are kept unchanged in terms of three Yukawa functions as the original density-independent M3Y interaction [14], exp(−Rν s) D(EX) D(EX) v00(01) (s) = Y00(01) (ν) (8) Rν s ν=1 120 100 CDM3Y6 80 60 40 =0.0 20 =0.3 =0.6 E/A (MeV) =1.0 -20 120 100 CDM3Y3 80 60 40 20 -20 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -3 (fm ) Figure 2.1 Total NM energy per particle E/A (2) at the different neutronproton asymmetries δ given by the HF calculation, using the CDM3Y3 (lower panel) and CDM3Y6 (upper panel) The solid circles are the saturation densities of the NM at the different asymmetries One of the most basic characters of NM is the appearance of a saturation point ρsat , with energy E < satisfying the minimum condition ∂ E = (9) ∂ρ A ρ = ρsat In figure 2.1, one can see that the saturation density ρsat rapidly decreases with the increasing asymmetry δ , and the pure neutron matter δ → is unbound by the (in-medium) NN interaction The result in Figure 2.1 confirms the reliability of the CDM3Yn interaction [9, 17] in the HF calculations for EOS of NM 2.2 Consistent NMF approach to the single-particle potential and optical potential The NMF model presented in this thesis can be used to describe consistently the nucleon SP as a continuous function of momentum, from k < kF for nucleons bound in NM to k > kF for incident nucleons on NM The nucleon SP potential is determined based on the HF method by summing all two-body interaction potential of the considered nucleon with other bound nucleons in NM as Uτ(HF) (ρ, k) = nτ (k )A kστ, k σ τ |vc |kστ, k σ τ kστ (10) (HF) Because of antisymmetrization A, the nucleon SP Uτ (10) also consists of direct term (Hartree) and exchange term (Fock term) The nucleon SP energy, which is determined from average energy of NM (2) according to Landau theory [30], also contains RT beside HF term (10), Uτ (ρ, k) = Uτ(HF) (ρ, k) + Uτ(RT) (ρ, k), với Uτ(RT) (ρ, k) = nτ1 (k1 )nτ2 (k2 ) (11) k1 σ1 τ1 k2 σ2 τ2 × A k1 σ1 τ1 , k2 σ2 τ2 ∂vc k1 σ1 τ1 , k2 σ2 τ2 (12) ∂nτ (k) Using the CDM3Yn interactions (6)-(8), we obtain the HF term of the SP potential including IS and IV parts (M3Y) (M3Y) Uτ(HF) (ρ, δ, k) = F0 (ρ)UIS (ρ, k) ± F1 (ρ)UIV (ρ, δ, k), (M3Y) với UIS (M3Y) UIV (ρ, k) = ρJ0D + EX A0 (r)v00 (r)j0 (kr)d3 r, (ρ, δ, k) = ρJ1D δ + (13) EX A1 (r)v01 (r)j0 (kr)d3 r where D A0(1) (r) = ρn jˆ1 (kF n r) ± ρp jˆ1 (kF p r), J0(1) = D v00(01) (r)d3 r, j0 (x) = sin x/x, jˆ1 (x) = 3j1 (x)/x = 3(sin x − x cos x)/x3 The (-) sign on pertains to the single-proton (τ = p) and (+) sign to the single-neutron (τ = n) potentials Because the original M3Y interaction is momentum independent, the momentum dependence of the nucleon SP potential is entirely determined by the exchange terms Applying the HVH theorem, the RT of the SP potential is also obtained explicitly in terms of the IS and IV parts, but at the Fermi momentum kFτ only, (RT) Uτ(RT) (ρ, δ, kF τ ) = UIS (RT) (ρ, kF τ ) + UIV (ρ, δ, kF τ ) (14) We adopt phenomenologically a momentum dependent RT of the SP potential in the similar form as HF term (13) by adding a density dependent function ∆F0(1) into the CDM3Yn interaction (M3Y) Uτ(RT) (ρ, δ, k) = ∆F0 (ρ)UIS (M3Y) (ρ, k)+∆F1 (ρ, δ)UIV (ρ, δ, k) (15) where RT contributions to the IS and IV density dependencies of the CDM3Yn interaction are determined consistently from the exact expression (14) and phenomenological expression (15) at the Fermi momentum (RT) ∆F0 (ρ) = UIS (ρ, kF τ ) (M3Y) UIS (ρ, kF τ ) (RT) and ∆F1 (ρ, δ) = UIV (ρ, δ, kF τ ) (M3Y) UIV (16) (ρ, δ, kF τ ) Consequently, the total SP potential is determined in the present HF approach with the CDM3Yn interaction including RT (16) as Uτ (ρ, δ, k) = UIS (ρ, k) ± UIV (ρ, δ, k) (M3Y) = [F0 (ρ) + ∆F0 (ρ)]UIS ± [F1 (ρ) ± (ρ, k) (M3Y) ∆F1 (ρ, δ)]UIV (ρ, δ, k) (17) The nucleon optical potential in nuclear matter In the NM limit, the nucleon OP is determined as the interaction potential between the incident nucleon on the NM at a given energy E > and the nucleons bound in the Fermi sea [43] Applying a continuous choice for the nucleon SP potential [28, 29] at the positive energies E, we obtain the nucleon OP in the NM similar to the nucleon SP potential (17) of bound nucleons In general, the nucleon OP contains both the IS and IV parts like the total SP potential (17) One needs first to explore the IS term predicted by the HF calculation of the symmetric NM EX U0 (ρ, E) = [F0 (ρ)+∆F0 (ρ)] J0D+ ˆj1 (kF r)j0 k(E)r v00 (r)d3 r (18) Here k(E) is the (energy dependent) momentum of the incident nucleon propagating in the mean field of the nucleons bound in the NM k(E) = 2m [E − U0 (ρ, E)] (19) Following Fig 2.2, the inclusion of the RT significantly improved the agreement of the calculated U0 with the data at the lowest energies, it remains somewhat more attractive at the higher energies in comparison with the empirical trend Thus, we scale in the present work the CDM3Yn interaction (6) by a momentum-energy dependent function g k(E) , and iteratively adjust its strength to the best agreement of 0.84 k F -1 2.1 1.6 k (fm ) 2.8 2.5 CDM3Y6 = -40 U (MeV) -20 -60 U U HF HF +U RT g(k)*(U HF +U RT ) -80 -50 50 100 150 200 E (MeV) Figure 2.2 Energy dependence of the nucleon OP in the symmetric NM (evaluated at the saturation density ρ0 with and without the RT using the CDM3Y6 interaction) in comparison with the empirical data taken from Refs.[44] (circles), [45] (squares), and [46] (triangles) The momentum dependent factor g(k) has been iteratively adjusted to the best agreement of the calculated nucleon OP (18) with the empirical data (solid line) 40 40 Symmetric nuclear matter Pure neutron matter 20 20 0 -20 no g k ( ) no -20 g k ( ) -40 U U U (RT) (HF) U + -60 (RT) 20 U U (MeV) -80 -40 (HF) (MeV) U U U -60 (HF) (RT) (HF) U + (RT) -80 20 0 -20 with g k ( ) with -20 g k ( ) -40 -40 -60 -60 -80 -100 -80 k k -1 (fm ) Figure 2.5 Momentum dependence of the total SP potential in the symmetric NM at ρ0 with the explicit contributions from the RT and HF parts -1 (fm ) Figure 2.6 The same as Fig 2.5 but for the total SP potential in the pure neutron matter δ = the (HF+RT) nucleon OP obtained at the saturation density ρ0 with the empirical data, as shown in Fig 2.2 The total SP potentials obtained at the saturation density ρ0 from where s = |R − r| is the distance between the incident nucleon and the bound nucleon in the target The local and nonlocal densities of proton and neutron are calculated in terms of the SP wave functions (τ )∗ ϕj (r, σ) of bound nucleons in the target (τ )∗ ρτ (r, r ) = ϕj (τ ) (r, σ)ϕj (r , σ ), (32) j∈A with ρτ (r) ≡ ρτ (r, r) and τ = n, p Multiplying both sides of Eq (29) with the spherical harmonics and integrating out the angular variables ˆ , we obtain the radial equation for partial wave χLJ (k, R), ˆ and k R − d2 L(L + 1) − χLJ (k, R) + VD (E, R) + VC (R) 2µ dR R2 + ALJ Vso (R) χLJ (k, R)+ (33) (LJ) KEX (E, R, r)χLJ (k, r)dr = EχLJ (k, R), where ALJ = L if J = L + 12 and ALJ = −L − if J = L − 21 In this thesis, we apply R-matrix method [36, 37] to solve exactly Schrăodinger equation (33) with the nonlocal OP [41] 3.3 Local approximation for the optical potential The integro-differential equation (33) can be converted to a differential form containing only local OP based on some approximations proposed by Brieva and Rook (BR) [35] Using the plane-wave approximation the scatter wave function is writen as χ(k, r) = χ(k, R + s) χ(k, R) exp(ik(E, R)s) (34) Then the nonlocal exchange term of the OP can be converted to the local form KEX (E, R, r)χ(k, r)dr with (loc) UEX (E, R)= (loc) UEX (E, R)χ(k, R) ρτ (R, r)vcEX (s, ρ) exp(ik(E, R)s)d3 r (35) (36) τ where s = r−R and k(E, R) is relative motion momentum of nucleonnucleus system 2µ (loc) (37) k(R, E) = [E − UD (R) − UEX (R) − VC (R)] 15 Multiplying both sides of the equation with the spherical harmonics and integrating out the angular variables, we obtain the radial equation for partial wave χLJ (k, R) This deferential equation has been solved by the traditional Numerov method, using ECIS06 code written by Raynal [52] 40,48 3.4 Elastic nucleon scattering on Ca,90 Zr and 208 Pb targets The OP based on the extended folding model taking into account the contribution of RT is used to describe the elastic nucleon scattering on stable targets 40,48 Ca, 90 Zr, 208 Pb at low and medium energy to evaluate the contribution of the RT term as well as the effect of the nonlocality The CDM3Yn interaction used in the HF calculation for NM was applied to the finite nuclei via local density approximation (LDA) The single-nucleon wave function and the density of the target were determined from the calculation of the nuclear structure by HF [53, 54] using NN effective interaction D1S-Gogny [55] 10 10 Pb( n,n) 26 MeV 30.4 MeV d /d (mb/sr) 10 208 x10 10 -2 -2 40 MeV x10 10 -4 -4 Nonlocal without RT Nonlocal with RT Local with RT 10 -6 20 40 60 80 c.m 100 120 140 160 180 (deg) Figure 3.1 OM description of the elastic n+208 Pb scattering data measured at 26, 30, and 40 MeV [56, 57] given by the complex nonlocal folded OP obtained with or without the inclusion of RT, in comparison with that given by the local folded OP with the RT included 3.4.1 Contribution of RT and the role of nonlocality From the OM results obtained with the CDM3Y6 interaction shown in Fig 3.1 for elastic n+208 Pb scattering, one can see that the inclusion 16 10 10 Nonlocal with RT Ca 10 Nonlocal without RT Nonlocal with RT (mb) Ca 16.8 MeV -1 -2 10 35 MeV 10 10 -1 x10 45 MeV Zr 24 MeV 10 x10 -3 -2 -3 -4 x10 10 10 -2 d /d (mb/sr) d /d 10 Local with RT 30.3 MeV 90 10 p,p) x10 10 Pb( Local with RT 16.9 MeV 48 10 208 10 10 Nonlocal without RT 40 10 10 Neutron elastic scattering -4 -5 -4 10 -5 20 40 60 80 100 c.m 120 140 160 -7 180 (deg) 20 40 60 80 c.m 100 120 140 160 180 (deg) Figure 3.6 The same as Fig 3.1 but for the elastic p+208 Pb scattering data measured at 30, 35, and 45 MeV [60] Figure 3.5 The same as Fig 3.1 but for the data of the elastic neutron scattering on 40 Ca, 48 Ca and 90 Zr targets measured [58, 59] at 17 and 24 MeV, respectively of the RT into the folding model calculation of the nonlocal neutron OP is essential for a proper OM description of the data over the whole angular range The same important role of the RT and good accuracy of the local folding approach can be seen in the OM results given by the folded OP for the elastic neutron scattering on the mediummass 40 Ca, 48 Ca and 90 Zr targets (see Fig 3.5) The elastic p+208 Pb 40 10 Ca( p,p) Nonlocal without RT Nonlocal with RT Local with RT 10 d /d (mb/sr) 30 MeV 10 35 MeV 10 x10 -2 -2 45 MeV 10 -4 x10 10 -4 -6 20 40 60 80 c.m 100 120 140 160 180 (deg) Figure 3.7 The same as Fig 3.1 but for the elastic p+40 Ca scattering data measured at 30, 35, and 45 MeV [61] Figure 3.8 The same as Fig 3.1 but for the elastic p+90 Zr scattering data measured at 30 and 40 MeV [62] 17 scattering data measured at 30.4, 35, and 45 MeV [60]] are compared in Fig 3.6 with the OM results given by the same three versions of the folded OP as those discussed in Fig 3.1 At forward angles, the effect of the RT is somewhat suppressed by the interference between the Coulomb and nuclear scatterings, and not as strong as found in the elastic neutron scattering However, the inclusion of the RT is still vital for a proper OM description of the data over the whole angular range as shown in Fig 3.6 Both the nonlocal and local versions of the folded OP give about the same OM results for the elastic p+208 Pb cross section at the forward and medium angles, while the data at the most backward angles are clearly better described by the nonlocal folded OP, especially, at 45 MeV The same picture but with a slightly more pronounced difference between the results given by three versions of the folded OP as discussed above can be seen in the OM results for the elastic scattering p+40 Ca (Fig 3.7) and p+90 Zr (Fig 3.8) 3.4.2 Microscopic nonlocal folded OP versus the global parametrization We note here the early work by Perey and Buck (PB) [38] and the recent revision of the PB parametrization by Tian, Pang, and Ma (TPM) [39], where the nonlocal nucleon OP is built up from a Woods-Saxon form factor multiplied by a nonlocal Gaussian While the PB parameters were adjusted to the best OM fit of the two data sets (elastic n+208 Pb scattering at 7.0 and 14.5 MeV), those of the TPM potential were fitted to reproduce the data of elastic nucleon scattering on medium and heavy targets at energies of to 30 MeV More recently, an energy dependence has been introduced explicitly into the imaginary parts of the PB and TPM potentials, dubbed as PBE and TPME potentials, with the parameters adjusted to achieve the overall good OM description of neutron elastic scattering on 40 Ca, 90 Zr and 208 Pb targets at energies E ∼ − 40 MeV [40] The OM results for the elastic n+208 Pb scattering at 26, 30.4, and 40 MeV given by the nonlocal folded OP obtained with the CDM3Y6 interaction are compared with the OM results given by the PB parametrization of the nonlocal neutron OP [38] and the recent energy dependent version PBE [40] in Fig 3.9 One can see that the microscopic folded OP performs quite well, with the predicted elastic cross section agreeing closely with the 18 10 208 Pb( n,n) 10 208 PB Pb( n,n) TPM PBE TPME Nonlocal folded 10 30.4 MeV 10 Nonlocal folded 26 MeV 10 d /d 10 (mb/sr) 26 MeV d /d (mb/sr) 10 -1 x10 30.4 MeV -2 10 -1 x10 -2 40 MeV 40 MeV 10 -3 x10 10 -4 10 -3 x10 -4 -5 20 40 60 80 c.m 100 120 140 160 10 180 -5 20 40 60 (deg) Figure 3.9 OM description of the elastic n+208 Pb scattering data measured at 26, 30, and 40 MeV [56, 57] given by the nonlocal folded OP including the RT, in comparison with that given by the original PB OP [38] and the energy dependent version PBE [40] 80 c.m 100 120 140 160 180 (deg) Figure 3.10 The same as Fig 3.9 but in comparison with the OM results given by the nonlocal neutron OP by Tian, Pang, and Ma (TPM) [39] and its energy dependent version (TPME) [40] data like that given by the global PBE potential.It can be seen in Fig 3.10 that the elastic n+ 208 Pb cross section predicted by the nonlocal folded OP agrees with the data slightly better than that predicted by the global TPM and TPME 3.4.3 Nucleon effective mass from the local OP The effective mass of incident nucleon (E > and k kFτ ) can be determined from the momentum dependence of the local OP (22) Using the explicit form of the real folded potential Re Uτ (E, R, kτ ) = g(kτ )Vτ (E, R, kτ ), (38) The nucleon effective mass is calculated as −1 m ∂g(kτ ) m∗τ (E, R, kτ ) ∂Vτ = 1+ Vτ (E, R, kτ )+g(kτ ) (39) m kτ ∂kτ ∂kτ It is noteworthy that the momentum kτ of incident nucleon is medium dependent, and determined self-consistently from the real folded OP at the energy E by relation (37) From the radial shape of the real folded potential for 208 Pb target at different energies shown in Fig 3.12 and 3.13, it becomes obvious from relation (37) that the momentum of incident nucleon depends explicitly on the nucleon-nucleus 19 0.20 208 Pb 2.0 45 MeV 0.15 ) 1.5 kFn -1 ) (fm -3 k (fm n 0.10 1.0 MeV 0.05 0.5 0.0 0.00 n+ 0.95 E=1 MeV - 45 MeV m */ m -20 Re n 208 Pb E = MeV - 45 MeV 0.90 Un (MeV) -10 0.85 -30 45 MeV 0.80 -40 45 MeV 0.75 MeV MeV 0.70 -50 10 12 R (fm) Figure 3.11 Radial shape of the real folded neutron OP obtained at energies E ≈ − 45 for 208 Pb target (lower panel), and the neutronand total g.s densities of 208 Pb (upper panel) given by the HF calculation [54] using the finite-range D1S Gogny 10 12 R (fm) Figure 3.12 Radial dependence of the neutron effective mass obtained from the real folded OP at energies E ≈ − 45 for 208 Pb target (lower panel), the momentum of incident neutron and Fermi momentum kF n extracted from the neutron g.s density of 208 Pb (upper panel) distance R at energies E = ∼ 45 MeV One can see that at each energy the kn momentum changes gradually from its maximum of about 1.6 ∼ fm−1 in the nuclear center to 0.2 ∼ 1.5 fm−1 at the surface, lying above the corresponding Fermi momentum Over the same radial range, the neutron effective mass m∗n is changing from about 0.75 ∼ 0.78 to unity at the surface Such a radial dependence of m∗ is similar to that found in the nuclear structure studies [63–65] One can see in upper panel of Fig 3.11 that the average total density in the center of 208 P b target is ρ¯ ≈ ρ0 , and the NP asymmetry is δ¯ = (¯ ρn − ρ¯p )/¯ ρ ≈ 0.185 In local density approximation (LDA), the difference between m∗n and m∗p in the center of the targets can be compared with that obtained from HF calculation for NM at Fermi momentum kF Following Fig 3.14, m∗ /m is linear dependent on the NP asymmetry δ with opposite tendency for neutron and pro- 20 Table 3.1 The effective mass of proton and neutron (39) at the average density ρ¯ ≈ ρ0 , based on the real folded OP in the radii R fm in the targets 48 Ca, 90 Zr, and 208 Pb Target ρ¯ δ¯ m∗n /m m∗p /m m∗n−p /δ¯ 48 Ca 90 Zr 208 Pb 0.159 ± 0.003 0.0966 ± 0.0069 0.7495 ± 0.0015 0.7332 ± 0.0008 0.1687 ± 0.0021 0.160 ± 0.002 0.0691 ± 0.0021 0.7441 ± 0.0003 0.7326 ± 0.0001 0.1664 ± 0.0010 0.160 ± 0.001 0.1853 ± 0.0060 0.7557 ± 0.0003 0.7245 ± 0.0003 0.1684 ± 0.0019 0.80 2.0 45 MeV 0.78 208 1.5 0.76 48 ) -1 40 m m (fm MeV */ k kFp 1.0 90 Ca Zr Pb Ca 0.74 0.72 0.5 proton 0.70 neutron 0.0 p+ 0.95 208 0.68 Pb 0.00 E = MeV - 45 MeV p m */ m 0.90 0.85 0.80 45 MeV 0.75 MeV 0.70 10 12 R (fm) Figure 3.13 The same as Fig 3.12 but for proton effective mass 0.05 0.10 0.15 0.20 0.25 0.30 Figure 3.14 Nucleon effective masses m∗τ obtained from the HF results (dashed and dotted lines, respectively) for nucleons bound in asymmetric NM at ρ = ρ0 , kτ = kF τ , and different NP asymmetries δ The symbols are those obtained from the folded OP of finite nuclei at ρ ≈ ρ0 , and E = 0.05 MeV ton, similar to the NM (Fig 2.6) In Table 3.1, the average value of the effective mass splitting m∗n−p (ρ0 , δ) ≈ (0.167 ± 0.018)δ is still in the range m∗n−p (ρ0 , δ) ≈ (0.27 ± 0.25)δ determined from experimental experiments and astronomical observations [51] 21 Conclusion The NMF derived from nucleon free degrees has been developed and applied to study the characteristics of NM such as saturation point, SP nucleon potential, symmetry energy and nucleon effective mass using CDM3Yn NN effective interaction The nucleon-nucleus OP has been also built microscopically from folding model to describe the data of nucleon-nucleus elastic scattering at low and medium energies The NMF was used to describe consistently the nucleon SP potential as a continuous function of nucleon momentum, from k < kF for nucleons bound in NM to k > kF for nucleons scattered on NM, with important contribution of RT which occurs naturally in the SP due to the density-dependent NN interaction Based on the HvH theorem, a simple method is effectively used to determine the momentum dependence of RT of the SP potential in HF scheme by adding a density dependent function ∆F0(1) (ρ) corresponding to RT The nucleon (HF+RT) SP potentials obtained at different densities and asymmetry are quite consistent with those obtained from the microscopic BHF calculation for NM in which RT appears because of higher-level correlation of the NN interaction and the contribution of three-body force In parabolic approximation, the symmetry energy of NM can be directly determined from neutron and proton SP energy at Fermi level At each density ρ and asymmetry δ of NM, the contribution of RT to neutron SP potential and proton SP potential is the same Consequently, the contribution of RT to symmetry energy S(ρ) vanishes The nucleon effective mass was investigated when there was an important contribution of RT as well as the correction of the momentum dependence of SP potential by the function g(k) The contribution of RT makes the effective mass m∗τ the splitting m∗n−p close to the semi-empirical data and astronomical observations [51] For the first time in the folding calculation, we have successfully built an extended folding model for nucleon-nucleus OP including the contribution of the RT The nonlocal exchange term of the nucleon OP is converted to local form via Brieva-Rook approximation This local OP is used to calculate cross-section of the elastic nucleon scattering on 40 Ca, 90 Zr and 208 Pb targets The obtained results are consistent with experimental data in the low and medium range of incident energies 22 RT takes a very important role in the nucleon OP to describe better the nucleon-nucleus elastic scattering, especially for neutron scattering The similar results obtained by using local and nonlocal show that the local approximation for the OP is still effective in analyzing the nucleon-nucleus elastic scattering Thus, the NMF method 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rearrangement term and nucleon effective mass", Submitted to J Phys G: Nucl Part Phys (2019) Conferences "Nonlocal nucleon optical potential and nucleon effective mass", Vietnam Conference on Nuclear Science and Technology–VINANST13, Ha Long (2019) "Mean-field description of nonlocal nucleon optical potential and R-matrix method", International Symposium on Physics of Unstable Nuclei, September 25–30, Ha Long, Vietnam(2017) "The nonlocal nucleon optical potential: Folding model and Rmatrix method", Vietnam Conference on Nuclear Science and Technology–VINANST12, Nha Trang (2017) "Consistent mean-field description of nuclear matter and nucleonnucleus scattering", Hội nghị chuyên ngành Vật lý hạt nhân, Vật lý lượng cao vấn đề liên quan (NHEP), Hanoi (2016) 29 ... differential scattering cross section for the elastic nucleon scattering can be obtained as dσ(θ) = |fC (θ)δms ms + fms ms (θ)|2 , (28) dΩ 2mm χscatt,ms ms (k, R) ∼ s s where fC (θ) is scattering amplitude... (2017) "Consistent mean-field description of nuclear matter and nucleonnucleus scattering", Hội nghị chuyên ngành Vật lý hạt nhân, Vật lý lượng cao vấn đề liên quan (NHEP), Hanoi (2016) 29 ... elastic p+40 Ca scattering data measured at 30, 35, and 45 MeV [61] Figure 3.8 The same as Fig 3.1 but for the elastic p+90 Zr scattering data measured at 30 and 40 MeV [62] 17 scattering data measured