Scattering from Hydrogen - gedanken results

We have used the canonical single-particle wave functions for all of the Sn isotopes to fold an effective NN interaction and thus generate optical potentials for use in modified versions of the scattering program DWBA98 [18]. Those optical potentials are complex and non-local since they can be written in coordinate space with $\mbox{{\boldmath {$r$ }}}_{12} = \mbox{{\boldmath {$r$ }}}_1 - \mbox{{\boldmath {$r$ }}}_2$, as

$$U\left( \mbox{{\boldmath {$r$ }}}_1, \mbox{{\boldmath {$r$ }}... ...ldmath {$r$ }}}_1, \mbox{{\boldmath {$r$ }}}_2; E \right) \, ,$$ (1)

where the direct U_D and non-local exchange U_E terms are

$$\sum_n \zeta_n \left\{ \int \varphi^{\ast}_n( \mbox{{\boldmath {$... ...varphi_n( \mbox{{\boldmath {$s$ }}} ) \, d\mbox{{\boldmath {$s$ }}} \right\} \,$$ U_D =

$$\sum_n \zeta_n \left\{ \varphi^{\ast}_n( \mbox{{\boldmath {$r$ }}... ...boldmath {$r$ }}}_{12} ) \varphi_n( \mbox{{\boldmath {$r$ }}}_2 ) \right\} \, .$$ U_E = (2)

Here v_D and v_Ex are combinations of the components of the effective NN interactions, $\zeta_n$ are ground state one body density matrices (effectively bound state shell occupancies), and $\varphi_n(\mbox{{\boldmath {$s$ }}})$ are single nucleon bound states. All details and the prescription of solution of the associated nonlocal Schrödinger equations are given in the recent review [16]. The results to be discussed herein have been obtained by solving the actual nonlocal Schrödinger equations defined with potentials as given by Eq. (1). For the present calculations, the effective NNinteractions have been defined by their mapping to the BBG gmatrices of the BonnB NN interaction [25]. We consider the elastic scattering of 200 MeV protons from each of the even mass Sn isotopes (A = 100 to 176). By inverse kinematics the cross sections we determine are also those for the scattering of 200A MeV Sn ions from hydrogen; some of which can be, or may soon be, obtained in sufficient numbers to form a radioactive ion beam for experiment. The choice of 200 MeV was made, not only because the effective force at this energy has been used in many successful predictions of actual scattering data from stable nuclei [16] but also since that energy provided data from ²⁰⁸Pb in which a clear signature of the neutron density profile has been found [15].

In Fig. 10, the differential cross sections for the scattering from all of the even mass Sn isotopes are plotted to $45^{\circ}$ in the center of mass by which angle the cross section magnitude is O(0.1) mb/sr. The SLy4 model calculations were used to specify the OBDME and single particle wave functions by which the optical potentials were generated. Several trends are noticed in the differential cross section as the mass is increased. The first three minima tend to forward angles and the intervening (first) maximum becomes more pronounced as the neutron number increases. There are model signatures in these results for scattering angles $30^\circ$(and higher) that have magnitudes still possible to be detected in experiment. Since the p-n force (isoscalar ³S₁ channel) is the strongest, such effects primarily reflect attributes of the neutron density variation in the surface region of the nuclei. They are revealed also by considering the ratio to Rutherford values which are shown in Fig. 11.

  

Figure 10: The differential cross section for the elastic scattering of 200 MeV protons from the Sn isotopes. The SLy4 model was used in forming the optical potentials.

  

Figure 11: The results of Fig.10 shown as ratios to Rutherford scattering.

Therein the marked change in the magnitudes of results for the isotope in the mass range 110 to 120 as is the trend of peak and valley magnitudes to smaller momentum transfer as mass increases.

If ever exotic Sn nuclei can be formed as a radioactive beam for scattering from polarized hydrogen targets, spin measurables such as the spin rotation, Q, could be found. That quantity from the SLy4 model calculations is shown in Fig. 12.

  

Figure 12: The spin observable Q from 200 MeV proton scattering with the Sn isotopes.

The graphical view has been rotated from the aspect given with the cross section results to show more readily the model dependencies in our calculated results. For clarity, the mass variation now is shown across the page while the scattering angle variation occurs into the page. Again the prime feature of these results is the trend with mass for the peaks and valleys to smaller momentum transfer values, but there is also a strong model dependence of magnitudes, notably in the mass range 110 to 120.