Chapter 6. Isospin dependence of the Microscopic Optical potential for Neutron rich isotopes of Ni, Sn and Zr.

Transcription

1 Chapter 6 Isospin dependence of the Microscopic Optical potential for Neutron rich isotopes of Ni, Sn and Zr.

2 (6.1) Introduction: Experiments with radioactive nuclear beams provide an opportunity to study nuclei with large neutron excess close to the neutron drip line. Moreover the long isotopic chains with increasing neutron excess provide an excellent laboratory for investigating the isospin dependence of various components of the optical potential. For these exotic nuclei, nuclear shell effects are non-negligible and hence the spin-orbit part of the nuclear potential is expected to play an important role and may lead to the discovery of new magic numbers. Further, the spin-orbit term plays an important role in the calculation of the spin observables such as analyzing power and spin rotation function in the polarized proton-nucleus scattering. Theoretical studies [1,2] based on the mean-field models have suggested that an increasing excess of neutrons may lead to a steady decrease in the strength of the spin-orbit interaction and this may contributes to a radical change in the ordering of the single particle energy levels. Schiffer et al. [3] have reported the results that reveal the onset of this effect in Sn isotopes suggesting the decrease of nuclear spin-orbit interaction with the addition of neutrons. In view of these results Hemalatha et al. [4] studied the effect of increasing neutron excess on the calculated spin-orbit potential for Zr and Sn isotopes. It was found that the effect of decreasing spin-orbit potential leads to a decrease in the first maxima of the analyzing power and hence an experimentally verifiable fact. In this chapter we present the results of our study concerning the isospin dependence of the microscopic proton nucleus optical potential. We have chosen Ni(A=52-112) as targets. However differential cross section and polarization data is available only for p Ni at 39.6 and 65MeV. In order to calculate the microscopic optical potential in first order Brueckner theory we require realistic inter-nucleon interaction and point proton and neutron densities. The method of calculation is discussed in detail in Chapter 4. In Section 2, we present an optical model analysis of the differential cross section and analyzing power data (wherever available) for the scattering of protons from Ni isotopes ( 58 Ni, 60 Ni, 62 Ni and 64 Ni) using three Hamiltonians,Argonne v18[5],reid93 and NijmII [6]. The experimental data has been taken from Ref. [7]. For comparison we also present results 109

3 from an empirical analysis of the same data using a global potential by A.J.Koning and J.P.Delaroche [8]. In Section 3 we have presented our detailed results concerning the iso-spin dependence of the spin orbit potential for the scattering of protons from Ni (A=52-114) isotopes. We have used the microscopically calculated p-nucleus optical potentials in the framework of first order Brueckner-Hartree-Fock (BHF) theory employing more recent Argonne v18 [5],Reid and NijmII[6] soft-core inter nucleon potentials along with the relativistic mean field (RMF) densities [9]both for protons and neutrons in the targets. Ref. [4] have used only Urbana V14 [10] interaction in BHF for Sn (96-136) and Zr (76-110) isotopes. Our results show that the calculated spin-orbit potential for Ni isotopes decreases with increasing neutron excess as found earlier for Zn and Sn isotopes thus reconfirming this tendency. It is shown that the decrease of spin orbit potential leads to a corresponding decrease in the magnitude of first maxima in the analyzing power. In Section 4 we present our findings concerning the (N-Z)/A dependence of the volume integrals of the calculated central real part of the potential. We outline a simple approximate procedure to evaluate the Lane term of the nucleon optical potential. Section 5 describes our conclusions. (6.2) Analysis of p- 58,60,62,64 Ni scattering data at 39.6 and 65MeV. It is well known that calculated optical potential in BHF is sensitive to the densities of the target nuclei [11]. Hence we show in Figure 1 the RMF density distribution [9] of protons and neutrons of all even isotopes of Ni ranging from A= 52 to 114 considered by us. We note that although the charge number (28) is constant the proton density shows noticeable changes as neutron number increases in the isotopic chain. Further as more neutrons are added the neutron densities extend beyond the proton distribution, as expected giving rise to neutron skin as shown in Fig. 2. Figure 2 also shows (red circles) results from Ref.[12] using Bonn A potential in density dependent relativistic hadron DDRH field theory. It is satisfying to note that the neutron skin of the RMF densities used in the present work are 110

4 very close to those of Ref.[12]. The kinks appearing in the neutron skin are at A=68, 78, 100 corresponding to magic numbers as also discussed in detail in Ref. [12]. Using the method as described in Chapter 4, we calculate the proton optical potential for the required isotopes of Ni at 39.6 and 65MeV from the three Hamiltonians mentioned in earlier. The calculated optical potential were used to obtain differential crossection and polarization for all the isotopes at both energies for which data were available. We find that we are able to get good agreement with the differential cross-section (Fig. 3) and a reasonable agreement with the analyzing-power data (Fig. 4) for all 58,60,62,64 Ni isotopes using only three scaling parameters. The normalization parameters at 39.6 (65) MeV are : λ R = 0.96(0.98), λ I = 0.760(0.72), and λ SO R = 1.2(1.19), whereas λ SO I = 1 for all targets. Thus the agreement with experimental data required only three parameters at each energy for all isotopes. We note that λ R is close to unity while λ I is much less than 1, implying that the calculated imaginary potential is larger than that required by the experimental data. We note that the agreement obtained using our microscopic optical potential with all three Hamiltonians is almost as good as that from the empirical optical potential by Koning and Delaroche [8]. (6.3) Spin orbit potential for p-ni (A=52-114) isotopes. In this section we present our results concerning the calculated spin orbit potential for the isotopic chain of Ni (A=52-114) in BHF using only Av18. Use of other NN interactions give similar results. Fig.5 shows the calculated real spin-orbit potential for the scattering of protons from Ni isotopes at 39.6 and 65 MeV. We note that at both energies the peak value of the real spin orbit potential decreases as more neutrons are added in the Ni target. This is a general result and is independent of incident energy. Further as the neutron number increases the position of the peak shifts outward as expected. To further clarify this result we have shown in Fig.6 the magnitude of the first maxima in the spin orbit potential at 65 MeV. We note that when the neutron number is less than proton number the spin-orbit 111

5 potential increases and then it decreases regularly for all neutron rich isotopes with kinks at the magic numbers. In Fig.7 we show the volume integral of the calculated proton-ni spin orbit potential at both energies considered in the present work. We note that at both energies the volume integral decreases linearly with the addition of neutrons. A straight line fit to the calculated volume integral is quite close to the results from the empirical potential of Koning and Deloroche[8]. Fig. 8 shows the variation of first maxima of the predicted analyzing power using our calculated potential for some representative isotopes of Ni considered in the present work. We observe a decrease in the magnitude of the first maxima with increasing neutrons in the target which is a reflection of a significant weakening of the calculated spin orbit potential, thus an experimentally verifiable result. In order to test this one would require analyzing power data from a larger number of isotopes. (6.4) Isospin dependence of Real microscopic optical potential for Ni, Zr and Sn isotopes. In this section we briefly describe our procedure to extract (N-Z)/ A dependence of real optical potential. We have considered proton scattering from the even isotopic chains of Ni,Sn and Zr. Since Lane [13] has suggested the isotopic spin dependence of the proton optical potential and has shown that it gives rise to a (N-Z)/ A dependence of the depth of the Optical Potential. Theoretically such a dependence is expected because there is a shifting of the energy levels in the nuclei as (N-Z)/A changes [14]. The depth of the real nucleon-nucleus optical potential can be written as V=V 0 ±V 1 (N-Z) /A+Vc V 1 is the strength of isospin potential. + (-) sign applies to proton (neutron) Optical potential respectively. As Lane pointed out, knowledge of V 1 is important for several distinct classes of phenomena including nuclear symmetry energy, nuclear scattering data and exchange scattering to isobaric analogue 112

6 states. Assuming the linear energy dependence of the real part of the optical potential V 0 (E)=V 0 (0)+ βe where E is the incident kinetic energy of the proton and β=-0.3, Perey [15] estimated the magnitude of the Coulomb correction term Vc to be 0.4Z/A 1/3. Hodgson [16], Rook [17], Sood [18], Koning and Delaroche [8], Nadasen [19] and J.Ball [20] et al. independently studied the dependence of real nuclear potential well depth on nuclear symmetry parameter (N-Z)/A using empirical analysis. In this section, we have followed the procedure outlined below, to investigate the isospin term microscopically. We have calculated the volume integral per nucleon J v /A from our microscopic potential for a range of isotopes of Ni, Zr and Sn each at two different energies. Since the radial form of the calculated potential is not necessarily Saxon-Wood we would have to study (N-Z)/ A dependence of the volume integrals. From the calculated volume integrals we would have to subtract the contribution coming from volume integrals of the isoscalar term V 0 and the Coulomb term Vc. To estimate the contribution in the volume integrals of Coulomb correction term we use the approximation of a Saxon-Woods form factors for only V c. We define Volume integral J v as Jv = 4 π ( ) 0 2 V r r dr V (r) is the real part of Optical Model Potential. If we assume a Saxon-Woods radial form, the Real Volume Integral Jv can be evaluated (approximately to order a/r 2 ) as: If we take Jv = Aπr0V ( πa0 / R0) ( N Z ) V ( r) = V Z / A + V1 f ( r) A where f () r is the Saxon-Woods form factor. 1 2 ( N Z) π Vrrdr ( ) = V Z/ A + V1 4 π f( rrdr ) A ( N Z) 3 Jv 0.4 Z/ A f() r d r= V0+ V1 f() r d r A 113

7 The left hand side of this equation is known from our microscopic calculation of the optical potential; hence no approximation regarding its shape is used. For estimating the coulomb correction we approximate the form factor to be Saxon-Woods. For a Saxon Woods form factor we can write: f() r d r = πr0 A 1 + ( πa0 / R0) 3 J v Z 4 4 ( N Z) r 1 ( a / R ) V r 1 ( a / R ) V r 1 ( a / R ) A π + π = π + π + π + π 1 A A 3 { } { } { } This equation can be written as: J v A N Z J J J A c = 0 + 1, Z where J c is 0.4 πr 1 0 { 1 + ( πa0 / R0) 3 } (volume integral of the coulomb correction A 3 term). We calculate this term for each isotope of Ni, Sn and Zr. The left hand side of the above equation is known for each isotope as J v /A has been calculated microscopically using first order Brueckner theory. The right hand side of this equation can be treated as a straight line equation with (N-Z)/A as the independent variable. The straight line fit gives the valve of J 0 and J 1 and thus we get the estimate of potentials V 0 and V 1. The variation of Jv J A with (N-Z)/A is shown in the Fig 9. c In estimating V 0 and V 1 from their Volume integrals J 0 and J 1 we have assumed the Saxon- Woods behavior of their form factors. The radius and diffuseness parameters r 0, a 0 from the recently obtained global optical potential of Koning & Deloroche [8] have been used. We also compare our results with the values obtained by different people using empirical analysis. Empirical analysis by Perry and Buck [21] concluded that the average value of V 1 is 20 MeV. Hodgson and Rook [22] obtained 35MeV.Another independent determination of magnitude of V 1 by Sood [18] led to the value 28.5MeV. Analysis by J.B.Ball et al [20] yielded a value of 27 ± 6.0MeV for V

8 Our results show that the average values for all the isotopes of Ni, Zr and Sn are: For Zr at 22.5MeV V 0 =42.65Me V 1 =32.05MeV Zr at 50MeV V 0 =32.93MeV V 1 =19.21MeV Sn at 20.4MeV V 0 =44.69MeV V 1 =24.58MeV Sn at 50 MeV V 0 =33.25MeV V 1 =12.95MeV Ni at 39.6MeV V 0 =37.577MeV V 1 =18.37MeV Ni at 65MeV V 0 =30.10MeV V 1 =12.33MeV We observe that both V 0 as well as V 1 are energy dependent. Value of β comes out to be for Zr, for Sn, for Ni which is close to the value calculated by Perey [15]. We observe that our results for the calculation of both V 0 and V 1 are in agreement with the results from above approximations and both decrease with energy as also found in [22].The values of V 1 are smaller than empirical results as found by Hodgson [23] also. (6.5) Conclusion: We have shown that the results of our calculated microscopic optical potential are in reasonable agreement with experimental data of proton elastic scattering from Ni isotopes. Argonne v18,reid 93 and Nijm II inter nucleon-interactions have been successfully used in BHF. Further, the calculated optical potential depends sensitively on the proton and neutron density distribution in the target. The volume integral per nucleon, of the real part of spinorbit interaction, is found to decrease with an increase in neutron number. The magnitude of the first maxima in the calculated analyzing power decreases with the addition of neutrons for Ni isotopes reflecting the reduction of the spin-orbit interaction.we conclude from our calculations that the p-nucleus spin-orbit potential decreases as (N-Z)/A increases. Lastly we have been able to obtain isospin dependence of real central optical potential for Ni,Zr and Sn isotopes. Our results are in satisfactory agreement with the earlier studies. 115

9 References: [1] J. Dobaczewski, I. Hamamoto,W. Nazarewicz, and J. A. Sheikh, Phys. Rev. Lett. 72, 981 (1994). [2] G. A. Lalazissis, D. Vretenar,W. P oschl, and P. Ring, Phys. Lett. B418, 7(1998). [3] J. P. Schiffer et al., Phys. Rev. Lett. 92, (2004). [4] M. Hemalatha,Y. K. Gambhir,W. Haider, and S. Kailas,Phys. Rev. C79, (2009). [5] R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, Phys. Rev. C51, 38 (1995). [6] V. G. J. Stoks, R. A. M. Klomp, C. P. F. Terheggen, and J. J. deswart, Phys. Rev. C 49, 2950 (1994). [7] H.Sakaguchi et al. Phys. Rev. 26, 944 (1982). [8] A. J. Koning and J. P. Delaroche, Nucl. Phys. A 713 (2003) 231. [9] Y. K. Gambhir, P. Ring, and A. Thimet, Ann. Phys. (NY) 198, 132 (1990), and references cited therein. [10] I. E. Lagris and V. R. Pandharipande, Nucl. Phys. A 359, 331(1981). [11] S. M. Saliem and W.Haider; J. Phys. G: Nucl. Part. Phys. 28 (2002) [12] F.Hofmann,C.M.Keil, and H.Lenske. Phys. Rev. 64, (2001). [13] A.M.Lane, Nucl. Phys.35 (1962) 676. [14] Schiffer, Lee and Zeidman, Phys. Rev. 115 (1959) 427. [15] F.G.Perey,Phys Rev.131 (1963)745. [16] P.E.Hodgson,Nucl.Phys.21,21 (1960). [17] P. E. Hodgson and J. R. Rook, Nucl.Phys.37,632(1962). [18] P.C.Sood,Nucl.Phys,37,624(1962). [19] A. Nadasen,Phys.Rev.C 23(1981) [20] J.B.Ball,C.B.Fulmer,etal Phys.Rev.135,3B(1964)B706. [21] F.G.Perey and B.Buck, Nucl. Phys.32(1962)353. [22] K.Kwiatkowski and N.S.wall Nucl.Phys.A 301(1978)349. [23] P.E.Hodgson,Phys.Letters,3(1963)

10 Fig. 1. The neutron and proton distribution in Ni isotopes used in our analysis. 117

11 Fig 2. Difference of the root mean square radii of neutrons and protons in Ni isotopes obtained using RMF calculations compared with those obtained from DBHF calculations[12] 118

12 Fig.3. Differential cross section for scattering of protons from Ni isotopes using Av 18,NijmII,Reid93 and global internucleon potentials in BHF framework (A) at 39.6MeV and (B) at 65MeV. 119

13 Fig.4. Same as Fig. 1 but for proton analyzing power data (a) 39.6 MeV and (b) 65MeV. 120

14 Fig.5. The calculated proton-ni (isotopes) spin orbit potential at 39.6 and 65MeV. 121

15 Fig. 6. The magnitudes of the first maxima of the calculated Real Spin orbit potential,with mass number (A) for 65MeV proton incident on Ni isotopes. 122

16 Fig.7. The volume integral per nucleon of the real part of spin-orbit potential with (N-Z)/A for Ni isotopes. Stars represent the results at 65MeV and circles at 39.6MeV.Solid lines show the straight line fits. 123

17 Fig.8. The magnitudes of the first maxima of the calculated analyzing power,with mass number (A) for 65MeV proton incident on Ni isotopes. 124

18 Fig. 9. J v J A with (N-Z)/A for p-ni isotopes at 39.6 and 65MeV, -Zr at 22.5 and 50MeV c and -Sn at 20.4 and 50MeV. 125