Microscopic Description of 295 MeV Polarized Proton Incident on Sn-Isotopes

Transcription

1 Chapter 5 Microscopic escription of 295 MeV Polarized Proton Incident on Sn-Isotopes 5.1 Introduction The scattering of intermediate energy protons provide an excellent means of extracting information about certain aspects of nuclear structure as well as the two nucleon interaction inside nuclear matter. The dependence of scattering observables on neutron and proton density distribution in nuclei is expected to provide information that may be compared with results using other probes and also from nuclear structure calculations. The relative transparency of the nucleus at intermediate energies increases the sensitivity of the models used and in particular to the components depending on spin. It has been observed that the differential cross-section data shows greater sensitivity to the spin-orbit potential at intermediate energies [1]. Thus the intermediate energy nucleon scattering can be used as a test of the effective nucleon-nucleon interaction and the density distribution of nucleons. Electron scattering has proved to be helpful in obtaining reliable charge distribution and thus proton densities in nuclei. It has been shown [2,3] that the neutron skin is closely related to the symmetry term of the equation of state. Thus the determination of neutron density distribution has become increasingly important. Karataglidis et al.[4], using reaction matrix approach, have shown that the proton differential cross-section is sensitive to the assumed neutron distribution in target nuclei. In view of this Amos et al. have studied [5] the proton and neutron matter distribution in tin isotopes in an extensive analysis of proton scattering data up to 200 MeV using matter densities obtained from spherical mean field Hartree-Fock- Bogoliubov (HFB) model with Skyrme type interactions. irac phenomenology [6,7], involving several parameters, has proved to be quite successful in explaining proton scattering data, specially spin observables, in the intermediate energy region. Huthcheon et al. [8] analyzed the MeV proton scattering data from 40 Ca and 208 Pb using two phenomenological and two microscopic models. They found that the reaction matrix approach [9] gives agreement only in the extreme forward angles and overestimate the differential cross-sections at intermediate angles. They conclude that it is perhaps due to inadequate treatment of self-consistency in obtaining the reaction matrices. Further their results 106

2 indicate that even at these energies the scattering observables are sensitive to only the surface region of the potentials used. Kaki and Toki [10] successfully analyzed the MeV proton elastic scattering data from 58 Ni and 120 Sn using relativistic impulse approach [10-12] and MF densities with the TMA parameter set [13]. ecently Terashima et al [14] have measured differential cross-section and analyzing power data for the scattering of 295 MeV protons from five tin isotopes with an aim of studying systematic changes in neutron distribution. They were able to obtain satisfactory agreement with the data using relativistic impulse approximation [11]. However as a first step they rescaled [14,15] masses and coupling constants of some of the mesons to fit the proton scattering data from 58 Ni at 295 MeV, assuming same neutron and proton density distribution in this target. This rescaling is identified as medium effect [14-16]. Using the rescaled interaction they parameterized the neutron distribution in tin isotopes as a sum of twelve Gaussians for each target. The parameters of the Gaussians were adjusted to obtain agreement with the proton differential elastic scattering and analyzing power data at 295 MeV. Neutron skin thus obtained is in satisfactory agreement with those obtained from other sources (see Fig. 12. in ef. [14]). In this chapter we present an analysis of this new data [14] for tin isotopes ( Sn) at 295 MeV using first order Brueckner theory with soft-core Urbana v14 (UV14) [24]internucleon potential. The required nucleon density distributions in the relativistic mean field (MF) approach were provided by Prof. Y. K. Gambhir (I.I.T Mumbai, India) [17-19]. The MF is now known to give exceptionally good account of the ground state properties of nuclei spread over the entire periodic table. In view of the failure of the earlier reaction matrix approach [9] and the importance of relativity [10] at intermediate energies we use relativistic kinematics for calculating the reaction matrices. Further we have made a careful calculation of the spin-orbit potential avoiding the short range approximation for the dominant direct part of the spin-orbit potential. Thus the present approach differs in two important respects. Firstly we use relativistic kinematics which necessitates a recalculation of self-consistent nucleon matter optical potential and to see its effect on observables we compare our results with those obtained using nonrelativistic kinematics. Secondly the folding integral for the direct part of the spin-orbit potential has been calculated exactly and we show that the effect of approximation used by Brieva and ook (to be denoted as B) [23] on the calculated spin-orbit potential is non- 107

3 negligible. Further we note that the approximation affects the cross-section and analyzing power data even at intermediate angles. For the exchange parts we use the equivalent local approximation as described in chapter 4 (section A). A similar approximation has also been used in ef. [8] and for p- 40 Ca [25] at 200 MeV. We are able to obtain satisfactory agreement using Urbana v14 interaction with the data using only three scaling parameters for all five isotopes of tin. The method of calculating optical potential and of the dominant direct part of spin-orbit potential without any approximation is described in section 5.2. Our results of the proton differential cross-section and analyzing power data from 116,118,120,122,124 Sn are discussed in section 5.3. In the same section we also show that the predictions of our model give fairly good agreement with proton total reaction cross-section data at two energies(22.8 MeV and 65 MeV) from all even isotopes of tin (for which the data is available). Further we have been able to get good agreement with reaction cross section data for p- 118 Sn from 65MeV to 225 MeV. Section 5. 4 describes our conclusions. 5.2 Method of Calculation In order to calculate the microscopic optical potential in Brueckner theory one requires only two inputs namely the basic nucleon nucleon interaction to calculate the reaction matrices and point nucleon densities to be used for folding. The method of calculating the p- nucleus optical potential is described in detail in chapter 4 (section A). In view of the importance of the spin at intermediate energies we avoid the normally used short range approximation and calculate the folding integral for the direct part without any approximation, as described below. The nucleon-nucleus spin-orbit part of the potential has been calculated by several authors [1,5-7] as a series expansion. The earliest formula, for only the direct part of the spin-orbit potential was given by Blin-Stoyle [5]: V so = const. 1 dρ( r)., (5.1) r dr where ρ ( r ) is the density distribution of nucleons in the nucleus. The right hand side of this formula is the first term in a series expansion for V so ; the full series has been given by Greenlees et al. [7]. The expansion parameter in this series is the ratio of two distances. The first is 108

4 the range of the spin-orbit part of the effective interaction and the second is the distance in which ρ changes appreciably. Scheerbaum [6] in his investigation of the spin-orbit potential gave a different series and then used various approximations to sum the series. We shall show that this series and that of Greenlees et.al. [7] are formally equivalent. An important result of ef. [6] and confirmed by other authors [1-4], is that the exchange terms contribute substantially to the total spin-orbit potential. The direct part of the nucleon-nucleus spin-orbit potential in the folding model approach [23] is: V (r 1,E) = N + φ N ( r 2 ) G L r. S r φ N ( r 2 ) d r 2, (5.2) where G is the direct part of the nucleon-nucleon spin-orbit t-matrix as defined in [1], L r and S r are respectively the total orbital and spin angular momentum for the nucleon pair. φ N ( r 2 ) is the bound single particle wave function in the target nucleus and the label N represents the appropriate quantum numbers of the bound nucleon. The summation in eq.(5.2) is over all the occupied states in the target nucleus. We use label 1 for the incident nucleon and label 2 for a typical nucleon in the target nucleus. For L r. S r we take L r. S r 1 = ( r r 1 ) x ( 2 p r r 1 p ). ( s r r s ), (5.3) 2 where p r, p r and s r,s r refer to the momenta and spins of the respective particles. Changing the integration variable in eq.(5.2) to x = 2 2 r r r 1, we obtain 1 r r V (r 1,E) = - 2 ρ ( 1 + x ) t r r where ρ ( + + r r 1 x ) = φ ( + x N N r r r r d x r, (5.4) x r x ( p1 p2 )(. s1 + s2 ) φ ( 1 ) N r r 1 + x ) is the density distribution in the target nucleus. We consider only spin zero nuclei, and hence the sum over s r 2 is zero. Further since no direction of nucleons in the target nucleus is specified the integration over p r 2 vanishes. This gives us: 109

5 1 r r V (r 1,E) = - 2 ρ ( + 1 x ) G x r x p r.s r 1 1d x r. (5.5) This is the expression obtained by Brieva and ook [23]. The integration in eq. (5.5) must be in the direction of rr 1 to contribute to the spin-orbit potential. We finally obtain V (r 1,E) = - 1 A( r ). /r l s1 1, (5.6) r r r where l1 = 1xp1 is the orbital angular momentum of the incident nucleon, and A( r1 ) rr 1 /r 1 = ρ ( r r 1 + x ) G x r dx r. (5.7) Eq.(5.6) is our basic equation to calculate spin-orbit part of the nucleon-nucleus optical potential. Once G is calculated using first order Brueckner theory, eq. (5.6) along with Eq. (5.7) can be easily used to calculate V (r 1,E) without any further approximation. This equation has also been discussed by Breiva and ook [23]. B makes a series expansion of ρ( r 1 + x ) and uses only the lowest order term of this infinite series. However, we calculate the integral numerically in Eq. (5.5) without making any approximation. We find that the differences have substantial effects on both differential cross-section and analyzing power data at 295 MeV. The exchange part of the spin-orbit interaction is calculated as in Chapter 4 (Section A). The required nucleon density distributions are obtained in the relativistic mean field (MF) approach [17]. Using the MF densities and the reaction matrices we calculate the optical potential ( V(E, r), W(E, r), V (E, r) and W (E, r) ) using folding procedure as described in chapter 4(section A) for all five tin isotopes at 295 MeV. In order to obtain agreement with the experimental data we multiply each component of the calculated potential by scaling parameters λ. The potential used by us in a spherical optical model code is 110

6 I U ( E, r) = λ V ( E, r) + iλ W ( E, r) + λ V ( E, r) + iλ W ( E, r) (5.8) I Thus we have four adjustable parameters ( λ s) to obtain best fit to data by minimizing 2 χ /F (F stands for degrees of freedom). 5.3 esults and iscussion The calculated neutron and proton densities are shown in Fig. 1. The figure shows that the MF density distributions both for protons and neutrons resemble closely with the corresponding HFB distributions with Skyrme type interactions as reported by Amos et. al. [5]. Notice that as we add more neutrons (going from 112 Sn Sn) the shape of the proton density distribution remains almost unchanged while the depression at the centre changes to a hump in the neutron distribution due to shell filling and is consistent with the results of ef.[5]. The rms charge radius, shown in Fig. 2(a), increasing with neutron excess is in close agreement with electron scattering [26], muonic X-ray data [27] and combined results of electron scattering and muonic X-ray data [28]. The root mean square radii of both the proton and neutron densities slowly increase with the addition of neutrons. The neutron skin (shown in Fig. 2 (b)) increases with neutron excess from 0.115fm for 112 Sn to 0.280fm for 124 Sn, a value slightly higher than those found in ef.[14]. Fig. 2(b) shows that the MF results for neutron thicknesses though higher are still within the uncertainties of results from 800MeV proton scattering [29], giant-dipole resonance [30, spin-dipole resonance [31], anti-protonic x-ray data [32] and results from Terashima et al. [14]. Thus various probes are able to give a range of skin thicknesses consistent with the data [33]. Using the method described in section 2 we have calculated the optical potential for the scattering of protons from tin isotopes. In Fig. 3 we show the calculated direct and exchange parts of the real central (Fig. 3(a)) and the spin-orbit part (Fig. 3(b)) of the optical potential for p- 120 Sn at 295 MeV. We note that the exchange part at this energy is quite small and hence the equivalent local approximation would not cause big error. Fig. 3(b) shows our results (denoted as NEW) for the calculated real spin-orbit potential using Eq.(6) and the B 111

7 approximation (denoted as OL) (Eq.(9) of ef. [23]). We find that besides reduction of strength near origin, use of equation (6) leads to non-negligible changes around peak value. The calculated optical potential is used in a spherical optical model code to calculate the differential cross-sections and analyzing power data for the scattering of 295 MeV protons from 116,118,120,122,124 Sn. The scaling parametersλ s are adjusted to obtain χ /F minimum. A value of λ <1 (>1) implies that the calculated potential is larger (smaller) than that required by the data. We find that we are able to get good agreement with the differential cross-section (Fig. 4) and a reasonable agreement with the analyzing power data (Fig. 5) for all isotopes 116,118,120,122,124 Sn using only three scaling parameters in comparison with twelve Gaussians for each target [14]. Our values are: λ =.722, λ and λ while was 0 I = = kept unity for all targets. Thus we have only three parameters for all five targets. Table I shows our results for the total proton reaction cross section from even isotopes of tin and 208 Pb at 22.8 MeV and 65.5 MeV. It is satisfying to note that the predictions of our model for almost all targets are within experimental uncertainties of the measurements [34,35] even at low energies. Further to compare our predictions over a wider energy region we have also calculated reaction cross-section for p- 118 Sn up to 220 MeV. Fig. 6 shows that we are able to get satisfactory agreement using relativistic kinematics with the data [34,36] up to 220MeV. For the low energy region: 10 E<65.5 MeV we find that the predictions of our model (i.e. all λ s kept unity) are in good agreement with the data, while for energies E: 65.5 E<220 MeV we have 2 I λ used the same normalizations ( λ =.722, λ 0.760, λ and =1.0) as for 0 I = = I λ Figs. 4 and 5. In order to see the effect of relativistic kinematics we compare our results with those obtained by using non-relativistic kinematics. Although we do not show figures, the calculated central potentials are slightly deeper while the effect on spin-orbit potential is marginal with the use of relativistic kinematics hence the normalizations obtained for p- 120 Sn with the nonrelativistic potentials are different ( λ = 0.80, λ = 0.9 and λ = 0.8). The results for other targets are similar. Fig. 7 shows our results for the differential cross-section and analyzing power due to the use of relativistic (solid curve) and non-relativistic (dotted curve) kinematics in our model. I 112

8 We see that the differential cross-section show noticeable changes only for angles greater than 55 o while analyzing power data starts showing the effect of relativity for smaller angles (>35 o ) also. Since the proton scattering data at this energy is available only up to 50 o the analyzing power data exhibits greater sensitivity to the relativistic kinematics. Note that there is negligible effect of relativistic kinematics for angles < 35 o. In view of the above we conclude that the relativistic kinematic effects are non-negligible only for angles > 35 o for proton scattering at 295MeV. In Fig. 7 we also show the effect on differential cross-section and analyzing power data of using Eq.(6) instead of B approximation for calculating the spin orbit part of the potential. To see the effect of spin-orbit potential we keep the same central potential as used for Fig. 4 and Fig. 5. Solid line show our results while dashed line show the results using B approximation for the direct part of the spin orbit potential. We note that the use of the approximate spin orbit potential overestimate the differential cross section at all angles except the extreme forward angles. Further the analyzing power predictions are also severely affected specially for angles 40 o. A similar result with the use of Brueckner theory [8] was obtained for the proton scattering data from 40 Ca and 208 Pb in the MeV region. Thus we find that the proton differential cross-section data is quite sensitive even at intermediate angles to the spin-orbit potential at intermediate energies. Further since Eq.(6) can be easily evaluated there is no need to use the approximation suggested by Brieva and ook [23] for calculating the direct part of the spin-orbit potential. In conclusion we find that the relativistic kinematic effects are important at large angles while the spin-orbit potential influences the observables even at small angles. In order to see the sensitivity of our predictions on neutron distribution in the target we have repeated our calculations with neutron densities reduced by 10%. This reduction of density would not change the rms radius of neutron distribution. Our results are shown in Fig. 8. We o note that there are only minor changes in the differential cross-section for angles 60 while non-negligible effects on analyzing power even at small angles. 5.4 Conclusions We have been able to extend the application of BHF to 300 MeV proton scattering and obtain satisfactory agreement with the new 295 MeV proton scattering data [14] from tin isotopes using non-relativistic g-matrix approach with relativistic kinematics. We have further im- 113

9 proved the calculation of spin orbit potential. The MF nucleon density distributions used here yields a satisfactory description of tin isotopes and predicts a regular increase of neutron skin as neutron number increases. The skin thicknesses predicted by MF calculations are slightly larger than the IA results in [14]. In conclusion we find that the approach used here gives a satisfactory description of proton scattering from tin isotopes at 295 MeV. eferences [1] A. Nadasen, P. Schwandt, P. P. Singh, A.. Bacher, P. T. ebevee, M.. Kaichuck and J. T. Meck, Phys. ev. C 53, 1023 (1981); P. Schwandt, H. O. Meyer, W. W. Jacobs, A.. Bacher, S. E. Vigdor, M.. Kaitchuck and T.. onoghue, Phys. ev. C 26, 55 (1982). [2] S. Typel and B. A. Brown, Phys. ev. C 64, (2001). [3] P. anielewicz, Nucl. Phys. A 727, 233 (2003). [4] S. Karataglidis, K. Amos, B. A. Brown and P. K. eb, Phys. ev. C 65, (2002). [5] K.Amos, S. Karataglidis and J. obaczewski, Phys. ev. C 70, (2004). [6] L.G.Arnold, B. C. Clark et al., Phys. ev. C 19, 917 (1979); Phys. ev.c 25, 936 (1982). [7] A. M. Kobos, E.. Coopper, J. I. Johansson and H. S. Sherif, Nucl. Phys. A 445, 605 (1985) and references therein. [8]. A. Hutcheon et al., Nucl. Phys. A 483, 429 (1988). [9] H. V. von Geramb, in Interaction between medium energy nucleons in nuclei, Proc. of Bloomington Conf. ed. H. O. Mayer (AIP Conf. Proc. No. 97, N. Y. 1982). [10] K. Kaki and H. Toki, Nucl. Phys. A 696, 453 (2001). [11]. P. Murdock and C. J. Horowitz, Phys. ev. C 35, 1442 (1987), C. J. Horowitz, Phys. ev. C 31, 1340 (1985) [12] N. Ottenstein, S. J. Wallace and J. A. Tjon, Phys. ev. C 38, 2289 (1988). [13] Y. Sugahara and H. Toki, Nucl. Phys. A 579, 557 (1994); Y. Sugahara, octoral Thesis, [14] S. Terashima et al., Phys. ev. C 77, (2008). [15] H. Sakaguchi et al., Phys. ev. C 57, 1749 (1998). 114

10 [16] G. E. Brown, A. Sethi and N. M. Hintz, Phys. ev. C 44, 2653 (1991). [17] Y. K. Gambhir, P. ing and A. Thimet, Ann. Phys. (NY) 198, 132(1990). [18] P. ing, Prog. Part. Nucl. Phys. 37, 193 (1996) and references cited therein. [19] Y. K. Gambhir, A. Bhagwat and M. Gupta, Ann. Phys. (NY) 320, 429 (2005) and references cited therein. [20] G. A. Lalazissis, J. Konig and P. ing, Phys. ev. C 55, 540 (1997). [21] J. F. Berger, M. Girod and. Gogny, Nucl. Phys. A 428, 32 (1984). [22] T. Gonzalez-Llarena et al. Phys. Lett. B 379, 13 (1996). [23] F. A. Brieva and J.. ook, Nucl. Phys. A 297, 206 (1978), Nucl. Phys. A 291(1977) 317 [24] L. E. Lagaris and V.. Pandharipande, Nucl. Phys. A 359, 331 (1981). [25] M. Hemalatha, Y. K. Gambhir, S. Kailas and W. Haider, Phys. ev. C 75, (2007). [26] H. de Vries, C. W. de Jager and C. de Vries, At. ata Nucl. ata Tables 36, 495 (1987). [27] C. Piller et al., Phys. ev. C 42, 182 (1990). [28] I. Angeli, At. ata Nucl. ata Tables 87, 185 (2004). [29] L. ay, Phys. ev. C 19, 1855 (1979). [30] A. Krasznahorkay et al., Nucl.Phys A 567,521(1994) [31] A. Krasznahorkay et al., Phys. ev. Lett. 82, 3216(1999). [32] A. Trzcinska, J. Jastrzebski, P. Lubinski, F. J. Hartmann,. Schmidt, T. von Egidy, and B. Klos, Phys. ev. Lett. 87, (2001). [33] J. Piekarewicz, S. P. Weppner, Nucl.Phys A 778,10(2006). [34]. F. Carlson, Atomic ata and Nuclear data Tables 63, (1996). [35] A. Auce, A. Ingemarsson,. Johansson, M. Lantz, and G. Tibell, Phys. ev. C 71, (2005). [36] A. V. Sannikov and E. N. Savitskaya, adiation Protection osimetry 110 (2004)

11 theo σ TABLE I. Calculated reaction cross section (mb) for 22.8 MeV and 65.5 MeV protons, compared with experimental reaction cross-section Exp σ (mb). E p = 22.8 MeV E p = 65.5 MeV theo exp Target σ (mb) σ (mb) ref.[38] σ (mb) σ (mb) ref.[39] theo 112 Sn ± ± Sn ± Sn ± ± Sn ± ± Sn ± ± Sn ± Sn ± ± Pb ± 54.3 exp 116

12 Figure 1. The neutron and proton distribution in tin isotopes as obtained by our MF calculations. 117

13 Figure 2. (a) Existing experimental data of rms charge radii of the isotopes along with the present MF results (solid triangles). Solid squares, stars, solid circles and open triangles are respectively the results from electron scattering using different shapes of the charge distribution [26], muonic atom x- ray [27] and combined electron scattering and muonic x-ray data [28]. (b). Shows neutron skin thicknesses. Solid triangles are the results from present work (MF). Solid squares, open triangles, crossed circles, stars and shaded circles are respectively the results from Terashima et al. [14], 800 MeV proton scattering [29], giant dipole resonances [30], spin dipole resonances [31] and anti-protonic x-ray data [32]. 118

14 Figure. 3. The alculated direct, exchange and total (a) real central and (b) the spin-orbit part of the calculated optical potential for p- 120 Sn at 295 MeV. NEW denotes the results obtained using Eq.(5.6) while OL denotes the use of B approximation [23] (see text for details). 119

15 Figure 4. Solid lines show the best fit obtained in the present work to the differential elastic cross-section data for the scattering of protons from tin isotopes. Sold circles are the data from ref.[14]. 120

16 Figure 5. Same as for Fig. 4 but for analyzing power data. 121

17 Figure 6. Calculated eaction cross section for p- 118 Sn. Solid line shows our result while open circles are the experimental data taken from refs.[34,36]. Figure 7. Our predictions for the differential cross-section and analyzing power for p- 120 Sn at 295 MeV. Solid and dotted curves show the results due to relativistic (el) and non-relativistic (Nel) kinematics. The dashed curve shows the effect of B approximation [23]. OL and NEW symbols are same as for Fig

18 Figure 8. Our results for the differential cross-section and analyzing power for p- 120 Sn. Solid curves are same as in Fig. 4, while dashed curves show our predictions after 10% reduction of neutron densityρ n. 123