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BỘ KHOA HỌC VÀ CÔNG NGHỆ BỘ GIÁO DỤC VÀ ĐÀO TẠO _ VIỆN NĂNG LƯỢNG NGUYÊN TỬ VIỆT NAM B ÙI MINH LỘC NGHIÊN CỨU NĂNG LƯỢNG ĐỐI XỨNG CỦA CHẤT HẠT NHÂN VÀ LỚP DA NEUTRON CỦA HẠT NHÂN HỮU HẠN QUA PHẢN ỨNG TRAO ĐỔI ĐIỆN TÍCH Chuyên ngành: Vật lý Nguyên tử Mã số: 62 44 01 06 T UẬN ÁN TIẾN SĨ VẬT LÝ NGUYÊN TỬ ÓM TẮT L H Nội - 2017 Cơng trình hồn thành Viện Khoa học Kỹ thuật Hạt nhân, Viện Năng lượng Nguyên tử Việt Nam, Bộ Khoa học Công nghệ Việt Nam, 179 Hồng Quốc Việt, Nghĩa Đơ, Cầu Giấy, Hà Nội, Việt Nam Người hướng dẫn khoa học: GS TS Đào Tiến Khoa Phản biện: PGS TS Nguyễn Quang Hưng Phản biện: PGS TS Nguyễn Tuấn Khải Phản biện: PGS TS Phạm Đức Khuê Luận án bảo vệ trước Hội đồng cấp viện chấm luận án tiến sĩ họp Trung tâm Đào tạo Hạt nhân, Viện Năng lượng Nguyên tử Việt Nam, 140 Nguyễn Tuân, Thanh Xuân, Hà Nội, Việt Nam, vào hồi ngày 14 tháng năm 2017 Có thể tìm hiểu luận án tại: - Thư viện Quốc gia Việt Nam - Thư viện Trung tâm Đào tạo Hạt nhân Chapter Introduction In the structure of isobaric nuclei, there are the analog states called the Isobaric Analog States (IAS) They form a group of states related by a rotation in the isospin space These states are strongly excited by the charge-exchange (p, n)IAS or (3 He,t)IAS reaction The chargeexchange (p, n)IAS or (3 He,t)IAS reaction to the IAS can be approximately considered as an “elastic” scattering process, with the isospin of the incident proton or He being flipped, because the two IAS’s are members of an isospin multiplet which have similar structures and differ only in the orientation of the isospin T [1] In this picture, the charge-exchange , isospin-flip scattering to the IAS is naturally caused by the isovector part (IV) of the optical potential (OP), expressed in the following Lane form [3] U(R) = U0 (R) + 4U1 (R) t.T , aA (1.1) where t is the isospin of the projectile and T is that of the target with mass number A, a=1 and for nucleon and He, respectively The second term is the symmetry term of the OP, and U1 is known as the Lane potential that contributes to both the elastic and chargeexchange scattering to the IAS [1] The IV term of the empirical proton-nucleus or He-nucleus OP in the Woods-Saxon form has been used some 40 years ago [2] as the charge-exchange form factor (FF) to describe the (p, n)IAS or (3 He,t)IAS scattering to the IAS within the distorted wave Born approximation (DWBA) In the isospin representation, the target nucleus A and its isobaric analog A˜ can be considered as the isospin states with T z = (N − Z)/2 and T˜z = T z − 1, respectively We denote the state formed by adding proton or He to A as |aAi and that formed by adding a neutron or ˜ so that the DWBA charge-exchange FF for the triton to A˜ as |˜aAi, (p, n)IAS or (3 He,t)IAS scattering to the IAS can be obtained [4] from the transition matrix element of the OP (1.1) as p ˜ (R) t.T |aAi = 2T z U1 (R) Fcx (R) = h˜aA|4U aA aA (1.2) Only in a few cases has the Lane potential U1 been deduced from the DWBA studies of (p, n)IAS scattering to the IAS With the Coulomb correction properly taken into account, the phenomenological Lane potential has been shown to account quite well for the (p, n)IAS scattering to the IAS [5] However, a direct connection of the OP to the nuclear density can be revealed only when the OP is obtained microscopically from the folding model calculation In this case, the FF of the (3 He,t)IAS scattering to the IAS is given by the double-folding model (DFM) [7, 6] compactly in the following form s ZZ Fcx (R) = [∆ρ1 (r1 )∆ρ2 (r2 )vD 01 (E, s) + ∆ρ1 (r1 , r1 + s) × Tz 3 ×∆ρ2 (r2 , r2 − s)vEX 01 (E, s) j0 (k(E, R)s/M)]d r1 d r2 ,(1.3) EX where vD 01 and v01 are the direct and exchange parts of the isospin- dependent part of the central nucleon-nucleon (NN) force; ∆ρi (r, r0 ) = (i) 0 ρ(i) n (r, r ) − ρ p (r, r ) is the IV density matrix of the i-th nucleus, which gives the local IV density when r = r0 ; s = r2 − r1 + R, and M = aA/(a + A) The relative-motion momentum k(E, R) is obtained selfconsistently from the real OP at the distance R (see details in Ref [7, 6]) In the limit a → and ∆ρ1 → 1, the integration over r1 disappears and Eq (1.3) is reduced to a single-folded expression for the FF of the (p, n)IAS scattering to the IAS [6] It is obvious that the folded (Lane consistent) OP serves as a direct link between the isospin dependence of the in-medium NN interaction and the charge-exchange scattering to the IAS On the other hand, within a Hartree-Fock (HF) calculation of nuclear matter (NM), the symmetry energy S (ρ) of the NM depends entirely on the density- and isospin dependence of the in-medium NN interaction [7, 8] Therefore, our recent folding model studies of the (p, n)IAS scattering to the IAS [9, 10] were aimed to gain some information on the nuclear symmetry energy The isospin dependence of the chosen in-medium NN interaction, fine-tuned to the best fit of the (p,n) data, has been shown [10] to give the symmetry energy at the saturation NM density ρ0 very close to the empirical values deduced from other studies In difference from the (p, n)IAS data, the high-precision data of the (3 He,t)IAS scattering to the IAS could be sensitive to higher nuclear densities (ρ & ρ0 ) formed in the spatial overlap of He projectile with the target This is an essential feature of the folding model analysis of elastic nucleus-nucleus scattering that the nucleusnucleus OP at small internuclear radii is determined by the effective NN interaction at high nuclear medium densities [11] In fact, the (3 He,t)IAS scattering to the IAS has been studied in the DWBA using the FF obtained from a single-folding calculation with the effective (isospin-dependent) He-nucleon interaction [12, 13], and a fully double-folding formalism for the charge-exchange (3 He,t)IAS scattering to the IAS has also been suggested [6] The neutron skin thickness determined as the difference between the neutron and proton (root mean square) radii, ∆Rnp = hrn2 i1/2 − hr2p i1/2 , (1.4) was found by structure studies to be strongly correlated with the slope of the nuclear symmetry energy S (ρ), a key quantity for the determination of the equation of state (EOS) of the neutron-rich NM On the other hand, ∆Rnp is directly linked to the difference between the neutron and proton densities, ρn − ρ p , and can be probed, therefore, in the folding model analysis of the charge-exchange (p, n)IAS or (3 He,t)IAS scattering to the IAS Chapter Results and discussion An extensive folding model analysis of the (p, n)IAS scattering to the IAS of 48 Ca, 90 Zr, 120 Sn, and 208 Pb targets at 35 MeV and 45 MeV, and the ( He,t)IAS scattering to the IAS of 14 C at 72 MeV, and 48 Ca at 82 MeV has been done in Ref [6] These results show that the real IV part of the CDM3Y6 interaction (that was constructed based on the JLM results) needs to be enhanced by about 30 ∼ 40% to give a consistently good coupled-channel or DWBA description of the (p, n)IAS and (3 He,t)IAS data under study To show a direct link of this folding analysis to the study of the nuclear symmetry energy, 120 Symmetry energy 100 N N 80 N =1.5 V1 =1.3 V1 =1.0 V1 S (MeV) CDM3Y6s 60 40 20 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -3 (fm ) Figure 2.1: HF results for the nuclear symmetry energy S (ρ) given by the CDM3Y6 interaction we show in Fig 2.1 the results of the Hartree-Fock calculation of the nuclear symmetry energy S (ρ) given by different IV strengths of the CDM3Y6 interaction In Fig 2.1, the shaded (magenta) region marks the empirical boundaries implied by the isospin diffusion data and double ratio of neutron and proton spectra of heavy-ion collisions [14, 15] The circle is the prediction by the many-body calculations [7, 8] The square and triangle are the constraints deduced from the structure studies of the giant dipole resonance [16] and neutron skin [17], respectively CDM3Y6s is a “soft” version of the CDM3Y6 interaction, with the same density dependence assumed for both the IS and IV parts [18] The use of the unrenormalized strength of the IV interaction clearly underestimates S (ρ0 ) compared to the empirical values At the saturation density, S (ρ0 ) is approaching the empirical value of around 30 − 31 MeV only if the IV strength of the CDM3Y6 interaction is rescaled by a factor NV1 ≈ 1.3 − 1.5 Such a renormalization factor agrees well with that given by the folding model analysis of the (p, n)IAS and (3 He,t)IAS data [6], like example shown in Fig 2.2 The difference in the slope of the NM symmetry energy (stiff or soft) at low NM densities shown in Fig 2.1 can also be traced in the calculated (p, n)IAS cross section From Fig 2.2 one can see that the best DWBA fit to the charge-exchange data prefers the stiff behavior of S (ρ), especially at large angles Although nearly the same description of the data at forward angles is given by both stiff and soft choices of the IV interaction, the 35 MeV data points approaching the 10 Rnp= 0.09 ± 0.03 fm Rnp= 0.07 fm Best-fit, Best-fit, Rnp= 0.16 ± 0.04 fm Rnp= 0.17 fm HFB density, HFB density, 10 d /d (mb/sr) 10 -1 10 90 t 208 90 Zr( He, ) E t 208 Pb( He, ) Nb IAS E = 420 MeV lab Bi IAS = 420 MeV lab -2 10 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 (deg) c.m Figure 2.3: DWBA description of the (3 He,t)IAS scattering to the IAS of the 90 Zr and 208 Pb targets, given by the charge-exchange FF based on the empirical neutron density by Ray et al [22] adjusted to fit the data The error of the best-fit neutron skin was determined to account for the experimental uncertainty around 10% of the absolute cross section measured at the most forward angles (the hatched area) The dash curve is the prediction given by the microscopic HFB densities [23] 10 action [18] These EOS’s were then used as input for the TolmanOppenheimer-Volkov equations to determine the gravitational mass and radius of neutron star The most obvious effect caused by changing slope of S (ρ) from stiff to soft is the reduction of the maximum gravitational mass M and radius R as illustrated in Fig 2.4, with a much worse description of the empirical mass-radius data [28] In Fig 2.4, the empirical data (shaded contours) deduced by Steiner et al [28] from the observation of the X-ray burster 4U 1608-52 The circles are values calculated at the maximum central densities The thick solid (red) line is the limit allowed by the General Relativity [29] Together with the results of the folding model analysis of the charge-exchange scattering to the IAS, it is plausible the EOS with a soft behavior of the symmetry energy can be ruled out in the modeling of neutron star Given the indirect relation of the neutron skin to the behavior of the nuclear symmetry energy, it has become a hot research topic recently The results of our DWBA analysis of the (3 He,t)IAS scattering to the IAS of 90 Zr and 208 Pb targets are shown in Fig 2.3 For the incident energies of 100 ∼ 200 MeV/nucleon, the impulse approximation is reasonable and the t-matrix interaction suggested by Franey and Love [20, 21] has been used as the folding model input The radial parameter of the empirical neutron density by Ray 11 et al [22] was slightly adjusted to obtain the best fit to the chargeexchange data Such a simple linear fit resulted on the neutron skin ∆Rnp ≈ 0.09 ± 0.03 fm for 90 Zr and ∆Rnp ≈ 0.16 ± 0.04 fm for 208 Pb The uncertainty of the best-fit ∆Rnp is associated with the experimental uncertainty around 10% of the absolute cross section measured at the most forward angles The best-fit values of ∆Rnp turned out to be rather close to those of the densities given by the Hartree-FockBogoliubov (HFB) calculation [23] Although the PREX data seem to provide an accurate, modelindependent determination of the neutron skin of 208 Pb [24], the mean ∆Rnp value deduced from the PREX data is significantly higher than that given by other studies [25, 26] For a comparison, we have made a DFM + DWBA calculation using the neutron density of 208 Pb constructed to give the same neutron skin as that given by the PREX data [27] One can see from Fig 2.5 that the lower edge of the PREX values agrees nicely with the measured (3 He,t)IAS data Consequently, we still cannot rule out the large neutron skin of 208 Pb given by the PREX measurement, as was discussed recently by Fattoyev and Piekarewicz [26] The new PREX experiment planned to pin down the uncertainty of ∆Rnp to about 0.06 fm [30] would surely resolve the uncertainty shown in Fig 2.5 12 2.5 2.0 STIFF 1.5 CDM3Y3 1.0 CDM3Y4 0.0 / M M Solar CDM3Y6 0.5 2.0 SOFT 1.5 1.0 CDM3Y3s CDM3Y4s 0.5 CDM3Y6s 0.0 10 R 15 (km) Figure 2.4: The gravitational mass of neutron star versus its radius obtained with the EOS’s given by the stiff-type (upper panel) and soft-type (lower panel) CDM3Yn interactions, in comparison with the empirical data (shaded contours) deduced by Steiner et al [28] from the observation of the X-ray burster 4U 1608-52 The circles are values calculated at the maximum central densities The thick solid (red) line is the limit allowed by the General Relativity [29] 13 10 208 t Pb( He, ) 10 Bi IAS , E =420 MeV lab d /d (mb/sr) 10 208 10 -1 Neutron density ~ PREX data 10 -2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 (deg) c.m Figure 2.5: DWBA description of the charge-exchange scattering to the IAS of 208 Pb given by the charge-exchange FF based on the empirical neutron density by Ray et al [22], adjusted to reproduce the PREX data for the neutron skin, ∆Rnp ≈ 0.33+0.16 −0.18 fm [24] 14 Chapter Conclusions and Future Perspectives In conclusion, the results of the study showed the indirect relation of the nuclear charge-exchange exciting to the IAS with the symmetry energy of NM and the neutron-skin thickness of finite nuclei such as 208 Pb and 90 Zr The method of calculation based on a consistent fold- ing model study of the charge-exchange (p, n) and (3 He,t) scattering to the IAS The consistency was archived by using the same densityand isospin-dependent effective NN interaction, namely CDM3Y, to 15 calculate the OMPs and charge-exchange form factors required by the DWBA or the CC calculation in nuclear reactions and the HF calculation for NM study Consequently, it would be of great interest to perform experiments for the nuclear charge-exchange reaction pursued at the modern rare isotope beam facilities The CDM3Y interaction is a semi-microscopic effective NN interaction Although it is simple and flexible for the consistent study of the nuclear reaction and NM calculation, it is interested to construct a fully microscopic interaction Moreover, in the nuclear theoretical aspect, the key connection between the nuclear experiment in laboratories and the NM in the universe is the modern theories of the inmedium effective interaction in many-fermion systems A promising candidate is the new Brueckner’s g-matrix constructed with the latest models for NN interaction in vacuum together with the recently developed many-body forces effective in high-density NM Moreover, the g-matrix in recent years is only the symmetric g-matrix which means that the g-matrix was established in symmetric NM In principle, an asymmetric g-matrix is able to be constructed A description for the nuclear elastic and inelastic scattering using a new version of the Brueckner’s g-matrix is perfectly suitable for our future perspective The nuclear charge-exchange reactions can excite the target not 16 only into the IAS by the Fermi transition (∆J π = 0+ , and ∆T = 1), but also into many different states at many different high energies by the Gamow-Teller transition (∆J = 0, ±1, and ∆T = 1) These data contains plenty information about the nuclear structure and the properties of spin-isospin transitions in nuclei Nowadays, there are many new experimental methods to study charge-exchange experiments at intermediate energies in inverse kinematics These new experiments enable to extract the Gamow-Teller transition strengths over large excitation-energy ranges in unstable isotopes that are far from the stability line Therefore, our studies about studying nuclear structure and nuclear interaction by using nuclear scattering and charge exchange reaction have just been started 17 Bibliography [1] G.R Satchler, Isospin in Nuclear Physics (Edited by D.H Wilkinson, North-Holland Publishing Company, Amsterdam, 1969) p.390 [2] G.R Satchler, R.M Drisko, and R.H Bassel, Phys Rev 136 (1964) B637 [3] A.M Lane: Phys Rev Lett (1962) 171 [4] G.R Satchler, Direct Nuclear Reactions (Clarendon Press, Oxford, 1983) [5] R.P DeVito, D.T Khoa, S.M Austin, U.E.P Berg, and B.M Loc: Phys Rev C 85 (2012) 024619 18 [6] D.T Khoa, B.M Loc, and D.N Thang: Eur Phys J A 50 (2014) 34 [7] D.T Khoa, W von Oertzen, and A.A Ogloblin: Nucl Phys A602 (1996) 98 [8] W Zuo, I Bombaci, and U Lombardo: Phys Rev C 60 (1999) 024605 [9] D.T Khoa and H.S Than: Phys Rev C 71 (2005) 044601 [10] D.T Khoa, H.S Than, and D.C Cuong: Phys Rev C 76 (2007) 014603 [11] D.T Khoa, W von 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Rev C 64 (2001) 064321 [24] S Abrahamyan et al (PREX Collaboration): Phys Rev Lett 108 (2012) 112502 20 [25] X Roca-Maza, M Brenna, G Col`o, M Centelles, X Vi˜nas, B K Agrawal, N Paar, D Vretenar, and J Piekarewicz: Phys Rev C 88 (2013) 024316 [26] F.J Fattoyev and J Piekarewicz: Phys Rev Lett 111 (2013) 162501 [27] Bui Minh Loc, Dao T Khoa, R.G.T Zegers: Phys Rev C 89 (2014) 024317 [28] A.W Steiner, J.M Lattimer, and E.F Brown: Astrophys J 722 (2010) 33 [29] N.K Glendenning, Compact Stars: Nuclear Physics, Particle Physics and General Relativity (Springer: Springer-Verlag New York, Inc 2000) [30] C.J Horowitz, E.F Brown, Y Kim, W.G Lynch, R Michaels, A Ono, J Piekarewicz, M.B Tsang, and H.H Wolter: J Phys G, Topical Review 41 (2014) 093001 21 List of Related Publications Charge-Exchange Excitation of the Isobaric Analog State and Implication for the Nuclear Symmetry Energy and Neutron Skin Dao T Khoa, Bui Minh Loc, R G T Zegers, Proceedings of the Conference on Advances in Radioactive Isotope Science (ARIS2014) (2015) Charge-exchange scattering to the isobaric analog sate at medium energy as the probe of the neutron skin Bui Minh Loc, Dao T Khoa, and R G T Zegers, Physical Review C 89, 024317 (2014) Folding model study of the charge-exchange scattering to the isobaric analog state and implication for the nuclear symmetry energy Dao T Khoa, Bui Minh Loc and Dang Ngoc Thang, The European Physical Journal A 50: 34 (2014) 22 List of Publications Nuclear mean field and double-folding model of the nucleusnucleus optical potential Dao T Khoa, Nguyen Hoang Phuc, Doan Thi Loan, and Bui Minh Loc, Physical Review C 94, 034612 (2016) Extended Hartree-Fock study of the single-particle potential: The nuclear symmetry energy, nucleon effective mass, and folding model of the nucleon optical potential Doan Thi Loan, Bui Minh Loc, Dao T Khoa, Physical Review C 92, 034304 (2015) Low-energy nucleon-nucleus scattering within the energy density functional approach TV Nhan Hao, Bui Minh Loc, Nguyen Hoang Phuc, Physical Review C 92, 014605 (2015) Charge-Exchange Excitation of the Isobaric Analog State and Implication for the Nuclear Symmetry Energy and Neutron Skin Dao T Khoa, Bui Minh Loc, R G T Zegers, Proceedings of the Conference on Advances in Radioactive Isotope Science (ARIS2014) (2015) Charge-exchange scattering to the isobaric analog state at medium energies as a probe of the neutron skin 23 Bui Minh Loc, Dao T Khoa, and R G T Zegers, Physical Review C 89, 024317 (2014) Folding model study of the charge-exchange scattering to the isobaric analog state and implication for the nuclear symmetry energy Dao T Khoa, Bui Minh Loc and Dang Ngoc Thang, The European Physical Journal A 50: 34 (2014) Neutron scattering from 208 Pb at 30.4 and 40.0 MeV and isospin dependence of the nucleon optical potential R P DeVito, Dao T Khoa, Sam M Austin, U E P Berg, and Bui Minh Loc, Physical Review C 85, 024619 (2012) 24
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