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

BỘ KHOA HỌC VÀ CÔNG NGHỆ BỘ GIÁO DỤC VÀ ĐÀO TẠO VIỆN NĂNG LƯỢNG NGUYÊN TỬ VIỆT NAM _____________________ ​BÙI MINH LỘC ​Chuyên ngành: Vật lý Nguyên tử Mã số: ​62 44 01 06 ​TÓM TẮT ​LUẬN ÁN TIẾN SĨ VẬT LÝ NGUYÊN TỬ ​Hà Nội - 2017 HẠN QUA PHẢN ỨNG TRAO ĐỔI ĐIỆN TÍCH NGHIÊN CỨU NĂNG LƯỢNG ĐỐI XỨNG CỦA CHẤT HẠT NHÂN VÀ LỚP DA NEUTRON CỦA HẠT NHÂN HỮU Công trình được hoàn thành tại Viện Khoa học và Kỹ thuật Hạt nhân, Viện Năng lượng Nguyên tử Việt Nam, Bộ Khoa học và Công nghệ Việt Nam, 179 Hoàng Quốc Việt, Nghĩa Đô, 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 được bảo vệ trước Hội đồng cấp viện chấm luận án tiến sĩ họp tại Trung tâm Đào tạo Hạt nhân, Viện Năng lượng Nguyên tử Việt Nam, 140 Nguyễn Tuân, Thanh Xuân, Hà Nội, Việt Nam, vào hồi 9 giờ ngày 14 tháng 6 năm 2017. Có thể tìm hiểu luận án tại: - Thư viện Quốc gia Việt Nam - Thư viện Trung tâm Đào tạo Hạt nhân Chapter 1 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 (3He,t)IAS reaction. The charge- exchange (p, n)IAS or (3He,t)IAS reaction to the IAS can be approx- imately considered as an “elastic” scattering process, with the isospin of the incident proton or 3He 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, 1 the charge-exchange , isospin-flip scattering to the IAS is naturally caused by the isovector part (IV) of the optical potential (OP), ex- pressed 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 3 for nucleon and 3He, 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 charge- exchange scattering to the IAS [1]. The IV term of the empirical proton-nucleus or 3He-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 (3He,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 Tz = (N − Z)/2 and T˜z = Tz − 1, respectively. We denote the state formed by adding proton or 3He to A as |aA〉 and that formed by adding a neutron or triton to A˜ as |a˜A˜〉, so that the DWBA charge-exchange FF for the (p, n)IAS or (3He,t)IAS scattering to the IAS can be obtained [4] from the transition matrix element of the OP (1.1) as Fcx(R) = 〈a˜A˜|4U1(R) t.TaA |aA〉 = 2 aA √ 2TzU1(R). (1.2) 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 scat- tering to the IAS [5]. However, a direct connection of the OP to the nuclear density can be revealed only when the OP is obtained micro- scopically from the folding model calculation. In this case, the FF of the (3He,t)IAS scattering to the IAS is given by the double-folding model (DFM) [7, 6] compactly in the following form Fcx(R) = √ 2 Tz ∫∫ [∆ρ1(r1)∆ρ2(r2)vD01(E, s) + ∆ρ1(r1, r1 + s) × ×∆ρ2(r2, r2 − s)vEX01 (E, s) j0(k(E,R)s/M)]d3r1d3r2,(1.3) where vD01 and v EX 01 are the direct and exchange parts of the isospin- dependent part of the central nucleon-nucleon (NN) force; ∆ρi(r, r′) = ρ(i)n (r, r′)−ρ(i)p (r, r′) is the IV density matrix of the i-th nucleus, which gives the local IV density when r = r′; s = r2 − r1 + R, and M = aA/(a + A). The relative-motion momentum k(E,R) is obtained self- consistently from the real OP at the distance R (see details in Ref. [7, 6]). In the limit a → 1 and ∆ρ1 → 1, the integration over r1 disap- pears and Eq. (1.3) is reduced to a single-folded expression for the FF of the (p, n)IAS scattering to the IAS [6]. 3 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]. There- fore, 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 (3He,t)IAS scattering to the IAS could be sensitive to higher nu- clear densities (ρ & ρ0) formed in the spatial overlap of 3He pro- jectile with the target. This is an essential feature of the folding model analysis of elastic nucleus-nucleus scattering that the nucleus- nucleus OP at small internuclear radii is determined by the effective NN interaction at high nuclear medium densities [11]. In fact, the (3He,t)IAS scattering to the IAS has been studied in the DWBA us- ing the FF obtained from a single-folding calculation with the effec- tive (isospin-dependent) 3He-nucleon interaction [12, 13], and a fully 4 double-folding formalism for the charge-exchange (3He,t)IAS scat- tering 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 = 〈r2n〉1/2 − 〈r2p〉1/2, (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 determi- nation 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 − ρp, and can be probed, therefore, in the folding model analysis of the charge-exchange (p, n)IAS or (3He,t)IAS scattering to the IAS. 5 Chapter 2 Results and discussion An extensive folding model analysis of the (p, n)IAS scattering to the IAS of 48Ca, 90Zr, 120Sn, and 208Pb targets at 35 MeV and 45 MeV, and the (3He,t)IAS scattering to the IAS of 14C at 72 MeV, and 48Ca 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 (3He,t)IAS data under study. To show a direct link of this folding analysis to the study of the nuclear symmetry energy, 6 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 20 40 60 80 100 120 Symmetry energy (fm -3 ) S ( M e V ) N V1 =1.5 N V1 =1.3 N V1 =1.0 CDM3Y6s 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 7 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 in- teraction, 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 renormal- ization factor agrees well with that given by the folding model anal- ysis of the (p, n)IAS and (3He,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 be- havior 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 8 0 20 40 60 80 100 120 140 160 10 -2 10 -1 10 0 10 1 0 20 40 60 80 100 120 140 160 180 E p = 35 MeV 48 Ca(p,n) 48 Sc IAS d / d ( m b / s r ) c.m. (deg.) N V1 =1.5 N V1 =1.3 N V1 =1.0 CDM3Y6s E p = 45 MeV c.m. (deg.) Figure 2.2: The DWBA description of the (p, n)IAS scattering to the IAS of 48Ca target at Ep = 35 and 45 MeV given by the folded charge- exchange FF obtained with the different inputs for the IV part of the CDM3Y6 interaction. The data were taken from Ref. [19]. zero angle cannot be accounted for by the FF obtained with the soft CDM3Y6s interaction. The EOS of the uniform core of neutron star of the npeµ composition in the β-equilibrium at zero temperature has also been calculated using the stiff and soft versions of the IV inter- 9 0.0 0.5 1.0 1.5 2.0 2.5 10 -2 10 -1 10 0 10 1 10 2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 c.m. (deg) Best-fit, R np = 0.09 ± 0.03 fm HFB density, R np = 0.07 fm d / d ( m b / s r ) 90 Zr( 3 He,t) 90 Nb IAS E lab = 420 MeV 208 Pb( 3 He,t) 208 Bi IAS E lab = 420 MeV Best-fit, R np = 0.16 ± 0.04 fm HFB density, R np = 0.17 fm Figure 2.3: DWBA description of the (3He,t)IAS scattering to the IAS of the 90Zr and 208Pb 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 abso- lute 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 Tolman- Oppenheimer-Volkov equations to determine the gravitational mass and radius of neutron star. The most obvious effect caused by chang- ing 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 re- cently. The results of our DWBA analysis of the (3He,t)IAS scatter- ing to the IAS of 90Zr and 208Pb targets are shown in Fig. 2.3. For the incident energies of 100 ∼ 200 MeV/nucleon, the impulse ap- proximation is reasonable and the t-matrix interaction suggested by Franey and Love [20, 21] has been used as the folding model in- put. The radial parameter of the empirical neutron density by Ray 11 et al. [22] was slightly adjusted to obtain the best fit to the charge- exchange data. Such a simple linear fit resulted on the neutron skin ∆Rnp ≈ 0.09 ± 0.03 fm for 90Zr and ∆Rnp ≈ 0.16 ± 0.04 fm for 208Pb. The uncertainty of the best-fit ∆Rnp is associated with the experimen- tal 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-Fock- Bogoliubov (HFB) calculation [23]. Although the PREX data seem to provide an accurate, model- independent determination of the neutron skin of 208Pb [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 208Pb con- structed 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 (3He,t)IAS data. Conse- quently, we still cannot rule out the large neutron skin of 208Pb 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 re- solve the uncertainty shown in Fig. 2.5. 12 0.0 0.5 1.0 1.5 2.0 2.5 5 10 15 0.0 0.5 1.0 1.5 2.0 STIFF CDM3Y3 CDM3Y4 CDM3Y6 CDM3Y3s CDM3Y4s CDM3Y6s SOFT R (km) M / M S o l a r 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 val- ues calculated at the maximum central densities. The thick solid (red) line is the limit allowed by the General Relativity [29]. 13 0.0 0.5 1.0 1.5 2.0 2.5 3.0 10 -2 10 -1 10 0 10 1 10 2 c.m. (deg) 208 Pb( 3 He,t) 208 Bi IAS , E lab =420 MeV Neutron density ~ PREX data d / d ( m b / s r ) Figure 2.5: DWBA description of the charge-exchange scattering to the IAS of 208Pb given by the charge-exchange FF based on the em- pirical 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 3 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 208Pb and 90Zr. The method of calculation based on a consistent fold- ing model study of the charge-exchange (p, n) and (3He,t) scattering to the IAS. The consistency was archived by using the same density- and 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 calcu- lation for NM study. Consequently, it would be of great interest to perform experiments for the nuclear charge-exchange reaction pur- sued at the modern rare isotope beam facilities. The CDM3Y interaction is a semi-microscopic effective NN in- teraction. 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 labo- ratories and the NM in the universe is the modern theories of the in- medium 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 de- veloped 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 princi- ple, 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 perspec- tive. The nuclear charge-exchange reactions can excite the target not 16 only into the IAS by the Fermi transition (∆Jpi = 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 prop- erties 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 sta- bility 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. 8 (1962) 171. [4] G.R. Satchler, Direct Nuclear Reactions (Clarendon Press, Ox- ford, 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 Oertzen, H.G. Bohlen, and S. Ohkubo: J. Phys. G 34 (2007) R111. [12] S.Y. van Der Werf, S. Brandenburg, P. Grasduk, W.A. Sterren- burg, M.N. Harakeh, M.B. Greenfield, B.A. Brown, and M. Fu- jiwara: Nucl. Phys. A496 (1989) 305. [13] J. Ja¨necke et al.: Nucl. Phys. A526 (1991) 1. [14] M.B. Tsang et al.: Phys. Rev. Lett. 102 (2009) 122701; M.B. Tsang et al.: Prog. Part. Nucl. Phys. 66 (2011) 400. 19 [15] A. Ono, P. Danielewicz, W.A. Friedman, W.G. Lynch, and M.B. Tsang: Phys. Rev. C 68 (2003) 051601(R). [16] L. Trippa, G. Colo`, and E. Vigezzi: Phys. Rev. C 77 (2008) 061304(R). [17] R.J. Furnstahl: Nucl. Phys. A706 (2002) 85. [18] D.T. Loan, N.H. Tan, D.T. Khoa, and J. Margueron: Phys. Rev. C 83 (2011) 065809. [19] R.R. Doering, D.M. Patterson, and A. Galonsky: Phys. Rev. C 12 (1975) 378. [20] W.G. Love and M.A. Franey: Phys. Rev. C 24 (1981) 1073. [21] M.A. Franey and W.G. Love: Phys. Rev. C 31 (1985) 488. [22] L. Ray, G.W. Hoffmann, G.S. Blanpied, W.R. Coker, and R.P. Liljestrand: Phys. Rev. C 18 (1978) 1756. [23] M. Grasso, N. Sandulescu, N.V. Giai, and R.J. Liotta: Phys. 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. Colo`, M. Centelles, X. Vin˜as, 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 1. 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). 2. 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 Re- view C 89, 024317 (2014). 3. 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 Euro- pean Physical Journal A 50: 34 (2014). 22 List of Publications 1. Nuclear mean field and double-folding model of the nucleus- nucleus optical potential. Dao T. Khoa, Nguyen Hoang Phuc, Doan Thi Loan, and Bui Minh Loc, Physical Review C 94, 034612 (2016). 2. Extended Hartree-Fock study of the single-particle potential: The nuclear symmetry energy, nucleon effective mass, and fold- ing model of the nucleon optical potential. Doan Thi Loan, Bui Minh Loc, Dao T. Khoa, Physical Review C 92, 034304 (2015). 3. Low-energy nucleon-nucleus scattering within the energy den- sity functional approach. TV Nhan Hao, Bui Minh Loc, Nguyen Hoang Phuc, Physical Review C 92, 014605 (2015). 4. 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). 5. 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 Re- view C 89, 024317 (2014). 6. 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 Euro- pean Physical Journal A 50: 34 (2014). 7. Neutron scattering from 208Pb 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