Physics of Nucleon-Nucleus Scattering - Chapter VII

Chapter Seven

NEUTRON DENSITY DISTRIBUTION

Introduction

Neutron distributions in nuclei are not well established yet. In contrast to the proton root-mean-square (rms) radii, that are known to within an accuracy of ~ 0.02 fm [192], the neutron rms radii at best may be known to an accuracy of 0.2 fm [193]. Interest in the matter distributions of ²⁰⁸Pb, and its neutron density profile particularly, is quite topical   [194]. There is a proposal to determine its neutron rms radius at the Jefferson Laboratory (Jefferson Laboratory Experiment E-00-003, spokespersons R. Michaels, P. A. Souder, and G. M. Urciuoli) from analysis of parity-violating electron scattering data. However, the neutron rms radius in ²⁰⁸Pb was assessed in terms of modern Skyrme-Hartree-Fock (SHF) models [194]. With the Friedman-Pandharipande neutron equation of state as a constraint, the neutron rms radius in ²⁰⁸Pb was found to be 0.16 ± 0.02 fm larger than the proton rms radius. Empirical confirmation of that awaits.

           

In this chapter by using the g-folding approach to define optical potentials, I have analyzed data from the elastic scattering of protons from ⁹Be, ¹¹⁸Sn, and ²⁰⁸Pb to address the question of whether such establish a measure of the neutron density distributions by distinguishing between various model structures that have been proposed.

Results and discussions

Structure and angular distributions of ⁹Be

As I will show in the next chapter predictions of the total reaction cross sections from proton scattering are sensitive to the details of the model structures chosen for the ground states of nuclei. In particular, the results for p-⁹Be scattering reflect the appreciable deformation of that nucleus; a deformation when represented by a complete  shell model calculation suffices to explain the proton total  reaction cross sections. Such a model reproduces the r.m.s. radii but as yet fails to give the correct value for the quadrupole moment Q.

           

The neutron distributions for ⁹Be associated with two structure models entertained are given in Fig. 7.1. The solid curve shows the neutron density distribution with the  description of the nucleus while the dashed curve is the neutron density distribution when the structure model is used. The propriety of the larger space description of this nucleus (over that of the much simpler   shell model) is confirmed by predictions of the angular dependent measureables, the differential cross section and analyzing power in particular.

 

Figure 7.1: The neutron density profiles for ⁹Be used in calculations of the p-⁹Be total reaction cross sections. The solid curve shows the neutron density distribution assumed with the  description of the nucleus and the dashed curve is that given by the  structure model.

In Fig. 7.2, the 30.3 MeV differential cross sections and analyzing power data [141] are compared with the g-folding model  predictions. Those found using the basic   and the    model OBDME are portrayed by the solid and dashed curves respectively. With themodel a HO  length of 1.44 fm was used while in the folding with  the  shell model OBDME I  have used an oscillator length of 1.57 fm. The change in oscillator length is consistent with values used to specify details of the shell model in each case. The   results give slightly better agreement with the data. That is also the case with the analyzing power, although improvement in detail can be sought.

           

The differential cross section and analyzing power for 200 MeV proton scattering from ⁹Be are shown in Fig.7.3. The notation is as used in Fig. 7.2. In this case there is little distinction between the results that give reasonable agreement with the data [195] at least to 30^o.

Figure 7.2: The differential cross sections (top) and the analyzing powers (bottom)  for 30 MeV proton-⁹Be scattering.

Figure 7.3: As for Fig. 7.2, but for the differential cross sections (top) and analyzing power (bottom)  for 200 MeV proton-⁹Be scattering.

Structure and angular distributions of ¹¹⁸Sn

For ¹¹⁸Sn two neutron distributions are given in Fig. 7.4. The basic model result is depicted by the solid curve. The second result, portrayed in the figure by dashed curve, was obtained from a g-folding optical potential formed by varying the surface neutron orbit (h_11/2) to be that for an oscillator energy reduced by 20%. The effect of altered oscillator energy for the h_11/2 orbit is to reduce the neutron probability in the region of 4 fm to enhance the probabilities from 6.5 fm out so ensuring the correct neutron number. Note that the scale is semilogarithmic.

Figure 7.4: The neutron density profiles for ¹¹⁸Sn used in the  calculations. The different curves are identified in the text.

  

These diverse neutron profiles lead to differing angular distributions. The differential cross section and analyzing power for the scattering of 40 MeV protons from ¹¹⁸Sn are shown in Fig.7.5. Therein the results obtained with optical potentials formed with the OBDME and SP functions of the basic and of the extended   shell model are shown by the solid and dashed  curves respectively. Both g-folding calculations give results in reasonable agreement with the data [196] but the extended model results are slightly  the better.

Figure 7.5: The differential cross sections (top) and analyzing powers (bottom) for 40 MeV proton-¹¹⁸Sn scattering.

Structure and angular distributions of ²⁰⁸Pb

For this nucleus there are 3 model prescriptions considered. I have used two simple packing   shell model descriptions of ²⁰⁸Pb. With both shell models, an oscillator length b = 2.325 fm was chosen   for the proton SP wave functions.  The neutron SP wave functions were chosen differently. The first model, designated the   model hereafter, has the neutron oscillator length set as that of the protons.  In the second model, designated the extended  model hereafter, I adopted that same neutron    oscillator length for all neutrons except for the surface orbit. In the extended model b = 2.740 fm (15% greater) was used for the outer most shell (i_13/2) neutrons. The third  structure was obtained from an  SHF model [194].

           

Each model gives distinctive density distributions, as is evident in Fig.7.6. The normalization is such that their volume integrals equate to the proton and neutron numbers, 82 and 126 respectively.

 

Figure 7.6: Nucleon densities in ²⁰⁸Pb.

In the top segment of Fig. 7.6, the proton density, r_proton(r), of both the  and extended-  models are displayed by the solid curve while that from the SHF model is  shown  by the dot-dashed curve. These quite distinct shapes nevertheless give the same proton rms radius. They will, however, differ in the longitudinal electron scattering form factor which is essentially the Fourier transform of the proton density. The three model neutron densities, r_neutron(r), are shown in the bottom segment of Fig.7.6. As with the proton densities, both of the models have enhanced neutron probabilities in the nuclear interior over the SHF values. But these densities also have increased neutron probability at very large radii compared to the SHF prescription.

           

I have analyzed data from proton elastic scattering from ²⁰⁸Pb  for energies ranging from 30 to 800 MeV using the three  models  discussed above specifying the ²⁰⁸Pb ground state densities. Results are compared with the data taken at 12 energies in that range.  Specially I have considered energies of  30, 40, 65, 80, 120, 160, 200, 300, 400, 500, 650, and 800 MeV. Not only  at those energies do the most complete set of data exist including proton   integral observables and differential cross sections,  but also for   those energies the method of analysis has been used with great success for analyses of NA scattering from many stable nuclei [7, 76]. As a first test of the sensitivity of proton  scattering to the matter distribution of ²⁰⁸Pb, I  show in Table 7.1, the total  reaction cross sections at the energies 10 to 300~MeV for which data [197-202] are known.

           

In comparison with the available proton data, there is a preference for the SHF and extended- models of the ground state density, and so  to probe  these two models further, I have studied  the differential cross sections, analyzing powers and spin rotation (Q) at diverse energies. Proton scattering should be sensitive to the neutron distribution in nuclei given dominance of the isoscalar ³S₁ channel in the effective NN interactions [7]. In all figures I show,   the solid, dashed, and dot-dashed lines portray the predictions obtained from the calculations of ,  extended-  and SHF   models, respectively.

Table 7.1: Total reaction cross sections (in millibarn) of proton scattering from ²⁰⁸Pb. The models are as defined in the text. The energy units are MeV.

Model				Experiment
Energy		Extended-	SHF	sR	Energy	Refs.
10	98	145	106	216 ±148	9.92	[199]
20	1164	1356	1180	1511±64	21.1	[202]
25	1477	1665	1490	1706±52	24.2	[202]
30	1670	1849	1685	1862±41	30.3	[202]
40	1867	2026	1895	2023±100	40.0	[200]
50	1927	2070	1964	1842±93	49.5	[200]
65	1919	2045	1963	1993±95	60.8	[200]
80	1881	1995	1931	1665±60	77.0	[201]
100	1853	1945	1905	1831±51	99.2	[197]
120	1785	1869	1836	1716±56	113	[201]
200	1593	1664	1644	1550±160	185	[198]
300	1408	1476	1454	1480±150	305	[198]

The differential cross sections and analyzing powers for the scattering of 30 MeV proton from ²⁰⁸Pb are presented in Fig. 7.7. The shape of the differential cross section data are well reproduced by all the three model calculations. But as   seen before [78], the result from using the model considerably overestimates  the data at and above 50^o scattering angle, as do the cross sections found using the SHF model in this case.  Although the extended-   model result is not in perfect agreement with the data, it gives the best fit to data of the three model calculations. For the analyzing powers also, the trend of the three model calculations are similar to that with the differential cross sections. All three calculations underestimate structure seen in the data, but the shape (peaks and valleys) is well replicated. The extended- result is the best fit to the data.

Figure 7.7: Differential cross sections (top) and analyzing powers (bottom) from the elastic scattering of 30 MeV protons from ²⁰⁸Pb. The solid,  dashed, and dot-dashed lines portray the predictions obtained from the calculations made using the , extended- and SHF  models, respectively.The analyzing power  data were measured at 29 MeV [139], and the differential cross section  data were measured at 30 MeV [146].

In Fig. 7.8, the differential cross sections and analyzing powers obtained from the three model calculations for the scattering of 40 MeV protons from ²⁰⁸Pb are compared with data. The shape of the differential cross sections data are well reproduced by all the three model calculations. But now the structure is overaccentuated from what is evident in the data. Again, as the surface nucleons have a major effect at these energies of scattering, the extended- model yields the best result of those found using the three models. The analyzing power data also is well reproduced, although the calculated results all are slightly out of phase with the data at 40 MeV.

Figure 7.8: As for Fig. 7.7 but for 40 MeV. Experimental data were measured at 40 MeV [97].

The differential cross sections and analyzing powers for the scattering of 65 MeVprotons from ²⁰⁸Pb are presented in Fig.7.9. All three model calculations give similar results that are in good agreement with this data, indicating again that scattering at this energy largely is a surface phenomenon.  The SHF model now gives best agreement with the data. Recall also that the integral observables for 65 MeV proton scattering gives a preference for the SHF model.

 

Figure 7.9: As for Fig. 7.7 but for 65 MeV. Experimental data were measured at 65 MeV [203].

The differential cross sections for the scattering of 80, 120 and 160~MeV protons are presented in Fig. 7.10. The differential cross sections obtained from calculations made with the SHF model of structure are in very good agreement with all of these   data. The predictions made by both of the  models on the other hand underestimate the data particularly for scattering angles beyond  about 30 - 35^o.

Figure 7.10: Differential cross sections from the elastic scattering of 80 (top), 120 (middle), and 160 (bottom) MeV protons from ²⁰⁸Pb. The solid, dashed, and dot-dashed lines portray the predictions obtained from the calculations of ,  extended-  and SHF  models, respectively. Experimental data were measured at 79.9, 121.2, and 159.9 MeV [204].

The differential cross sections, analyzing powers and spin rotation parameter Q for 200 MeV proton scattering from ²⁰⁸Pb are compared with data in Fig. 7.11. The differential cross section data are well reproduced by the SHF model calculation, while both model structures give predictions that underestimate the data noticeably from 20^o onward. At this energy and higher, the optical potentials are such that scattering is most influenced by the bulk nuclear medium. Such is evident in the results and in the analyzing powers particularly. These obtained with the SHF model calculations are clearly the best, giving   very good agreement with the data  to 30^o scattering angle.That is also the case with  the spin rotation observable Q.

Figure 7.11: Differential cross sections (top), analyzing powers (middle) and spin rotation parameter, Q (bottom) from the elastic scattering of 200 MeV protons from ²⁰⁸Pb. The solid, dashed, and dot-dashed lines portray the predictions obtained from the calculations of , extended and SHF models, respectively. The cross section and analyzing power   data were taken from Ref. [205], and the Q data were taken from Ref. [206].

Figure 7.12: Same as Fig. 7.11 but for 300 MeV. The cross section and analyzing power data were taken from Ref. [205], and the Q data were taken from Ref. [207].

The predictions of differential cross sections, analyzing powers and spin rotation parameter Q from the elastic scattering of 300 MeV protons from ²⁰⁸Pb obtained using the three model structures are compared with the data in Fig.7.12. Differential cross section data are well reproduced to 15^o scattering by all  three  calculations. At larger momentum transfer the data are underestimated by both the models, and to some extent, overestimated by the SHF model results. Nevertheless the result of the SHF model calculation best replicates the data. Likewise the analyzing power data are best reproduced with the SHF model calculations and in the region 14^o ~ 40^o particularly. At forward angles, all these results  overestimate the data; a phenomenon noticed in other analyses [76] and which I attribute to inadequacies in the specification of the effective interactions. Again the spin rotation data are well described, and best by the SHF prescription.

Figure 7.13: As for Fig. 7.7 but for 400 MeV. Experimental data were taken from Ref. [205].

 

Figure 7.14: Same as Fig. 7.11 but for 500 MeV. The cross section and analyzing power data were measured at 500 MeV [205], and the Q data were measured at 497 MeV [208].

The results of my calculations for proton energies well above pion threshold are shown in Figs. 7.13, 7.14, 7.15 and 7.16 for the energies 400, 500, 650 and 800 MeV, respectively. As for p-¹²C scattering [76] analyses, at these energies  the effective NN interactions have been formed by supplementing the bare BCC3 interaction with complex short ranged Gaussian NN optical potentials with strengths set to ensure a match with the NN phase shifts at each relevant energy.

    

The differential cross sections and analyzing powers from the elastic scattering of 400 MeV protons from ²⁰⁸Pb are compared with the data in Fig. 7.13. The differential cross sections data are well reproduced by the calculation made with the SHF spectroscopy. Using either model again underestimates the data at larger momentum transfer values. The analyzing powers resulting by use of the SHF model densities are  in very good agreement with the data at and above 18^o scattering angles, but again the  forward angle analyzing power data are overestimated by all  three model calculations.

Figure 7.15: As for Fig. 7.7 but for 650 MeV. Experimental data were taken from Ref. [209].

Figure 7.16: Same as Fig. 7.11 but for 800 MeV. The cross section, analyzing power and Q data were taken from Refs. [115, 210]  and [211] respectively.

The predicted differential cross sections, analyzing powers and spin rotation parameter Q from the elastic scattering of 500 MeV protons from ²⁰⁸Pb obtained using the three models of density  are compared with the experimental data in Fig. 7.14. The differential cross section data are reasonably well reproduced by all the three model calculations, although there are now clear mismatches that put into question the propriety of the effective interaction used. Such a concern is heightened by the result I have for the spin observables. Similar quality results are found at 650 MeV (Fig.7.15).

           

Predictions of the differential cross section, analyzing power and the spin rotation parameter Q from the elastic scattering of 800 MeV protons from ²⁰⁸Pb are compared with the data in 7.16. In the case of the differential cross section the three model calculations give similar results. The data are well described to 14^o scattering but beyond that angle the predictions all overestimate the data significantly. The situation with the spin observables, the analyzing power and spin rotation Q is much worse. That data are not reproduced wll at all, neither in shape nor in magnitude. It is evident that at high energies (> 300 MeV) a better specification of effective NN interactions at least is needed.

Conclusions

The distinctions between the predictions of scattering found using different model neutron distributions, and particularly with the three models for the structure of ²⁰⁸Pb, demonstrate that the sensitivity of the procedure of analysis to the neutron density suffices to pin down the neutron matter distributions better than has been possible in the past. Such I predicted on the basis of the results for scattering below 300 MeV for which I am confident that the g-folding approach and the NN effective interactions are reliable. Below 65 MeV, the data analyses indicate strongly the properties of the nuclear surface while above that energy, it is the bulk density variation that plays the dominant role in scattering analyses. All in all the SHF specification of the nucleon distributions in ²⁰⁸Pb seem the most realistic although improvements are suggested.

Chapter - VIII