Comparison with data

Results found for the elastic scattering of 65 MeV protons from ¹¹⁸Sn and of 200 MeV protons from ¹²⁰Sn have been found from optical potentials formed by folding the effective interactions for those energies that were used previously with much success for data analysis of many nuclei [16], with the densities for the Sn isotopes discussed above and also with the densities of those nuclei specified by a simple $0\hbar\omega$ oscillator model. The oscillator length was taken as A^1/6; i.e. 2.215 and 2.221 fm, respectively, for the two isotopes. Those results found using the SLy4 and SkP model densities are shown in the following figures by the solid and dashed curves respectively. Those obtained using the HO model densities are portrayed by the dot-dashed curves.

The differential cross sections and analyzing powers for the elastic scattering of 65 MeV protons from ¹¹⁸Sn, measured at the RCNP, Japan [26], are compared with our predictions in Fig. 13. In the case of the differential cross section, there is little difference ($\sim 1\%$) between the SLy4 (solid) and SkP (dashed) results.

  

Figure 13: The differential cross sections (top) and analyzing powers (bottom) for the elastic scattering of 65 MeV protons from ¹¹⁸Sn. The data [26] are compared to the results of the calculations made using the SLy4 (solid line), SkP (dashed line) and HO (dot-dashed line) models.

The calculated results are in good agreement with the structure and magnitude of the data; the SLy4 and SkP model results particularly so. The preference for the SLy4 (SkP) model also is evident with the total reaction cross section. They predict 1.48 b while the HO calculation yields 1.41 b. The measured value [27] is $1.535 \pm 0.047$ b. But it is with the analyzing power at 65 MeV that the SLy4 (SkP) densities give a significantly better result in comparison to the HO model one. While the HO result is a fair reproduction of the data structure, the SLy4 (SkP) model result gives not only the correct location of the maxima and minima but also shows the overall trend seen in the data of increasing positive in magnitude with increasing angle. They also depict best the marked asymmetry of each peak structure of the data.

The 200 MeV scattering results from ¹²⁰Sn are also quite good, although with the exception of the differential cross section produced using the HO model the reproductions of data obtained with the mean-field model densities are not as good as found at 65 MeV for ¹¹⁸Sn.

  

Figure 14: The differential cross sections (top) and analyzing powers (bottom) for the elastic scattering of 200 MeV protons from ¹²⁰Sn. The data [28] are compared to the results of the calculations made as given in the text. The curves are as for Fig. 13. The SkP result is not given for the cross section as it is indistinguishable from the SLy4 result.

The differential cross sections and analyzing powers are shown in Fig. 14 from which it is clearly evident that the HO model tracks the measured data [28] well for all angles shown. The SLy4 model result is not as good, though it agrees with the forward angle ( $< 30^\circ$) scattering data well enough. The SkP model result is indistinguishable from the SLy4 one on this scale. The actual differences in cross section are less than a percent at most scattering angles. Both the SLy4 and SkP model calculations thus overestimate the cross section and predict sharper structure than seen in the data for larger scattering angles. Such discrepancies have been noted in other circumstances, notably in a comparative study [15] of model structures for ²⁰⁸Pb and in identifying ⁶He and ¹¹Li as nuclei with extended neutron distributions (neutron halos) [29,30]. For this energy the SLy4, SkP and HO calculations gave total reaction cross sections of 1.18, 1.17 and 1.14 b respectively. To compare there is a value measured at 221 MeV [31] (from ¹¹⁸Sn however) of $1.18\pm 0.03$ b. As with the differential cross section results, our predictions for the analyzing powers (shown in the bottom half of Fig. 14) are good but there is room for improvement. The asymmetry seen with the 65 MeV data is less severe at this higher energy but, of course, there are more peaks within the scattering angle range shown. Both features are evident in the HO and SLy4 calculations without the exact angular structures in the data being reproduced. It is notable that the SLy4 and SkP results do match the observed peak magnitudes and valley depths very well and the two densities yield slight but noticeable differences in the spin measurable.