Physics of Nucleon-Nucleus Scattering - Chapter VI

Chapter Six

PROTON SCATTERING FROM HELIUM ISOTOPES

Introduction

While interest in  exotic, radioactive beam, systems exists for many and diverse reasons [178], a primary theoretical one is to find their matter distributions.  The stable nuclei have spatial distributions of protons and neutrons that are roughly the same.  In contrast, exotic systems such as neutron-rich  ¹¹Li [7, 179] seem to have a neutron distribution that is more extensive than the usual structure models predict. That ``halo'' is   formed by some of the valence neutrons, two for ¹¹Li, concomitant with the separation energies of those neutrons being small.  Conventional (shell)  models for such nuclei give  what has been termed  neutron skins [178]. Such seem appropriate for ⁸He and the Na isotopes in particular.  The mass variation of Na interaction cross sections indicate that the proton rms radii are fixed while the neutron ones vary with mass.  But the models used to make these reaction cross section calculations were very simple and give information on the matter profiles only indirectly, as do analyses of most other experiments involving the exotic nuclei [178].     

The light mass systems are of interest since their structures have been studied by  Navratil and  Barrett [170, 180]. They have made large-space shell model calculations for nuclei up to and including mass 11. In those calculations all nucleons were active and the NN potential energies were generated from a realistic NN interaction; the Nijmegen Reid93 interaction usually. These structure calculations give good low excitation spectra for the stable isotopes and for many radioactive ones as well. The calculations also give good results for the binding energies, ground state static moments, and rms radii.

It is of very current interest to investigate the structure of nuclei such as ^6,8He. And some studies in these particular isotopes have been published [179, 181-183]. In this chapter the results of analyses of elastic (and inelastic in one case) scattering of exotic helium isotopes from a hydrogen target are presented.

Determination of single particle wave functions

From many studies,  it has been found that to describe light nuclei in appropriate detail, large basis shell model calculations are required [75, 123, 184]. With those made with g-matrix elements described above, not only are the OBDME given but so also are the SP bound states. They are HO functions. However, as noted, such shell model specifications do not yield a neutron ``halo'' distribution. Therefore, to associate such extended distributions with any nucleus, one must change from the HO specification. Use of WS functions is one way  to control the nucleon distributions by the choice of state binding energies.

           

The WS single particle wave functions are used as determined from solution of the Schrodinger equation with

......................(6.1)

and

 ................................................(6.2)

where R = r₀ A^1/3.  With the parameter values are given in Table 6.1, the s- and p- state WS and HO orbit functions have a large overlap.

Table 6.1: Binding energies (in MeV) and r_o (in fm) for single particle wave functions. The remaining Woods--Saxon parameters are a_o = 0.65 fm, r_c = 1.05 fm, and  l = 7.0.

Orbital	4He	6He	8He
0s1/2 0p3/2 0p1/2 All others	-23.11 -6.44 -4.21 -0.50	-23.79 -7.76 -6.53 -0.50	-25.97 -8.77 -6.84 -0.50
r0	1.60	1.70	1.55

For ^6,8He, no electron scattering data exists to set the proton SP  wave functions for these nuclei. Therefore, I  assume WS wave functions with the same parameters and binding energies as the equivalent states in ^6,7Li [123] for ^6,8He. These choices represent the closest comparisons I have for systems with mass numbers of 6 and 8.

           

To ensure a halo structure with any of the nuclei, the SP bound state WS  potentials then were adjusted to reproduce the separation energies of the weakly bound nucleons in the given shell model state. A halo structure is enforced on ^6,8He by setting the 0p shell binding at 2 MeV and the sd shell and higher states at 0.5 MeV.  With this approach, ⁸He acts as a  control  since it  is believed to be an example of a neutron skin  given that the single neutron separation energy  for this nucleus  is 2.584 MeV [185].  Then, by artificially ascribing a halo to this nucleus, the question is to ascertain whether or not the optical potential calculations and existing data are sensitive enough to detect the enforced property. Such has been found to be the case in refs [179].

Results and discussions

The neutron density profiles for the exotic nuclei ^6,8He as given by large space shell model calculations described are shown in Fig. 6.1.

 

Figure 6.1: Shell model neutron density profiles for the nuclei ^4,6,8He. The dashed and solid curves represent, respectively, the profiles when  a halo is and is not contained in those structures. The dots represent the proton density for each nucleus.

Therein the dashed and solid lines portray respectively the profiles found with and without halo conditions being implemented. Density profiles for ⁴He are also given to show that the proton and neutron distributions for the stable nucleus are the same.

           

Note, however, that a primary role of halo conditions is to reduce the neutron densities for radii less than the rms  radius.  With the folding process to define the optical potentials, the short range details, r < r_rms, have significant effects and so influence calculations of differential cross sections notably at momentum transfers where scattering is influenced by the nuclear medium.  But the extended nature of the halo shapes also influence the optical potentials with their effects in scattering cross sections being evident in changes to small angle results, typically   for beam energies 60A to 70A MeV [7].

Elastic proton-⁴He scattering

The predicted differential cross sections for 25, 30 , 40 and 65 MeV protons from  ⁴He are presented in Fig. 6.2 compared with data measured at 25 [186], 31[187], 40 [188] and 65 [108] MeV respectively. At these energies the calculated differential cross sections are poor when compared with data, although the trend of the curves reasonably replicate that evident in the data. 

           

At 30, 40 and 65 MeV, the calculations reproduce the data up to   60^o scattering angle. But at 25 MeV the comparison is very poor. I note that the breakup threshold is near this excitation energy so making the scattering from this nucleus at this energy special. It does not detract from analyses of 25 MeV proton scattering from the other isotopes.

Figure 6.2: Predictions of the differential cross sections from the scattering of 25, 30, 40 and 65  MeV  protons from ⁴He compared with experimental data.

In Fig. 6.3, differential cross sections calculated for  the elastic scattering of 700A MeV ⁴He  ions from hydrogen   are compared with data that has been  measured at  699A MeV [189, 190]. The same results are depicted in both panels, with the right panel emphasizing those for small scattering angles. The data which are known only to 13^o in the center of mass are well reproduced.

Figure 6.3: Predictions of the differential cross sections from the scattering of 700A MeV ⁴He from hydrogen compared with experimental data (left panel). Right panel shows the same but for small angles only.

⁶He-hydrogen scattering

Results on the scattering of 24.2A MeV ⁶He ions both elastic and inelastic scattering, and for 717A MeV ⁶He ions elastic scattering are given  here.  Predictions made using the g-folding optical potential approach are compared with the experimental data in the Figs. 6.4 and 6.5.

           

In Fig. 6.4, predictions of both the elastic and inelastic scattering of 25A MeV ⁶He ions  from hydrogen are compared with the data in the left and right  panels respectively. The data were measured at 24.2A MeV [191] with the open and full dots being  the results obtained  from two experimental runs.  Given that  good results in comparison with 25 MeV proton scattering from diverse stable mass targets have been found [76, 77], these results are convincing evidence that ⁶He has a halo like structure.

           

Clearly the prediction made using a halo prescription for ⁶He is a better (and good) fit to the measured elastic scattering data; a result that is most evident for the scattering angles in the range 80^o to 140^o. Such  concurs with the findings of the analyses of 40A MeV data [121].

 

Figure 6.4: Predictions of the differential cross sections from the elastic scattering (left panel) and inelastic scattering (right panel) of 25A MeV ⁶He from hydrogen compared with experimental data. The results, assuming that the nucleus has a halo structure are portrayed by the dashed curves while the results assuming that the nucleus does not have a halo structure are portrayed by the solid curves.

The momentum transfer values associated with this angular range implies sensitivity to the potential properties within the nuclear volume. I associate this with the halo character of ⁶He then by the reduction of neutron matter density required within the nuclear volume (from that given by standard models of structure) to create the extended halo property.

 

Figure 6.5: Predictions of the differential cross sections from the scattering of 700A MeV ⁶He from hydrogen compared with experimental data (left panel). The   small angle values are shown in the right panel. The results, assuming that the nucleus has a halo structure are portrayed by the dashed curves and those assuming that the nucleus does not have a halo structure are portrayed by the solid curves.

The differential cross section from the inelastic scattering of 25A MeV ⁶He from hydrogen leading to the 1.8 MeV 2⁺ state in ⁶He  are compared with the data in the right panel of the  Fig. 6.4. In this case a DWA was used to give the predictions shown.  In that DWA, the distorted waves were generated from the microscopically defined nonlocal potentials (defined for both the initial and final channels by the relevant shell occupancies), by the transition OBDME given by the large space shell model study, by the single particle bound state wave functions as used to form the optical potentials, and with the same effective NN interaction as the transition operator.  Thus there is no adjustable parameter or undefined added term given the assumption that the shell model does not require core polarization corrections.  The analysis of the 40A MeV inelastic cross section data indicated that assumption to be reasonable [121].

           

The predicted differential cross sections for the scattering of 700A MeV ⁶He from hydrogen are presented in Fig. 6.5, wherein they are compared with data that were measured at 717A MeV [190]. The existing data range only to 12^o in the center of mass, but they are very well described by the predictions. Now, however, one should not discriminate between the results found using halo and nonhalo structures.

Elastic ⁸He-hydrogen scattering

The predicted differential cross sections for the scattering of 30A MeV ⁸He ions from hydrogen are compared with experimental data in Fig. 6.6. Data were measured at 32A  MeV [25]. At this particular energy both the results from halo and nonhalo structures are so similar that one cannot distinguish between the two possibilities.

Figure 6.6: Predictions of the differential cross sections from the scattering of 30A MeV ⁸He from hydrogen compared with experimental data. The results, assuming that the nucleus has a halo structure are portrayed by the dashed curve and the results assuming that the nucleus does not have a halo structure are portrayed by the solid curve.

The differential cross sections measured from the elastic scattering of 674A MeV ⁸He from hydrogen are compared with the predictions of 700A MeV ⁸He in Fig. 6.7. Data,   which exist only to 11^o, were taken from Ref. [190].

Figure 6.7: As for Fig. 6.5 but for 700A MeV ⁸He from hydrogen.

Again this data is not sufficient for one to discriminate between the halo and nonhalo structure prescriptions of the nucleus.

           

The differential cross sections for the scattering of 700A MeV ^4,6,8He from hydrogen are presented along with the comparison between the calculations and the measurements in one figure (Fig. 6.8) to provide  a  different comparative picture of the isotope variation. Clearly the g-folding models give optical potentials that have the same mass trend in this data, whether the nuclei have or do not have a neutron halo attribute in these matter distributions.

Figure 6.8: Predictions of the differential cross sections from the scattering of 700A MeV ^4,6,8He from hydrogen compared with experimental data. The results, assuming that the nucleus has a halo structure are portrayed by the dashed curves and the results assuming that the nucleus does not have a halo structure are portrayed by the solid curves.

Conclusions

The g-folding approach leads to nonlocal, complex and energy dependent optical potentials in coordinate space which can be sufficiently sensitive to extended nucleon distributions of exotic nuclei that signatures of those distributions are revealed in cross sections from the elastic scattering cross sections of those exotic nuclei from hydrogen. To be so, however, careful measurement of data to quite high momentum transfer values may be necessary. However, it seems more likely that data from the inelastic excitation of RIB nuclei is sensitive to specifics of the extended matter profiles. That may also be the case with total reaction cross sections as are discussed in a later chapter.

Chapter - VII