Exact calculation of a microscopic nucleon spin-orbit potential: Reexamination of Brieva-Rook localization

Brieva-Rook approximations for calculating the equivalent local microscopic nucleon-nucleus spin-orbit potential has been examined in the energy region of $65<E<500\mathrm{MeV}$. We present an exact method for calculating both the direct and the exchange parts of the spin-orbit potential. It has been shown that the series expansion used earlier overestimates both the real and the imaginary parts by about $50\%$. Our results show that the exchange contribution to the total spin-orbit potential at low energies is only $15\%$ (much smaller than earlier estimates) and becomes negligible around 200-MeV nucleon incident energy. Predictions of the new spin-orbit potential with minor scaling has been found to be in fair agreement with $p\text{−}{}^{208}\mathrm{Pb}$ scattering observables from 65 to 500 MeV.

Figure 1

Total real spin-orbit potential for $p\text{−}{}^{208}\mathrm{Pb}$ at (a) 65 MeV and at (b) 200 MeV. The solid, dashed, and dotted continuous lines represent the BHF results using only Av-18, Av-18 + UV1X, and Av-18 + TNI interactions, respectively. See the text for details.

Figure 2

The direct part of the calculated real and imaginary proton-nucleus spin-orbit potential at 65 MeV. The solid (dotted) curves represent results using Eq. (3.7) ((3.10)).

Figure 3

The same as for Fig. 2 but only for $p\text{−}{}^{16}\mathrm{O}$ at four incident energies.

Figure 4

The calculated exchange part of the proton real spin-orbit potential for (a) four targets at 65 MeV and (b) at six energies for ${}^{16}\mathrm{O}$. The solid (dashed) curves represent results using Eq. (4.10) ((4.11)).

Figure 5

The total proton spin-orbit potential for four targets, (a) real part and (b) imaginary part at 65 MeV.

Figure 6

The energy dependence of the calculated peak value of the total proton spin-orbit potential for four targets, (a) real part and (b) imaginary part. Continuous lines have been drawn to guide the eye.

Figure 7

The calculated total proton spin-orbit potential compared with exchange part at four energies, (a) real part for ${}^{16}\mathrm{O}$ and ${}^{208}\mathrm{Pb}$ and (b) imaginary part for ${}^{16}\mathrm{O}$.

Figure 8

(a) Differential cross section, (b) analyzing power, and (c) spin rotation for the scattering of 65-MeV protons from ${}^{208}\mathrm{Pb}$. The solid curves represent our predictions with scaling factor of ${\lambda }^{\mathrm{I}}=0.75$, and the dashed curves are the results using the BR spin-orbit potential. Similar scaling is used for the BR results. (See the text for details.)

Figure 9

The same as for Fig. 8 but at 200 MeV. The solid curves represent our predictions with scaling factor ${\lambda }^{\mathrm{I}}=0.7$, the dotted curves are our results with ${\lambda }^{\mathrm{I}}=0.7$ and ${\lambda }^{\mathrm{R}}=0.5$, and the dashed curves are the results using the BR spin-orbit potential. Similar scaling $\left({\lambda }^{\mathrm{I}}=0.7\right)$ is used for the BR results. (See the text for details.)

Figure 10

The same as for Fig. 8 but at 500 MeV. The solid curves represent our predictions with scaling factor ${\lambda }^{\mathrm{I}}=0.8$, the dotted curves are our results with ${\lambda }^{\mathrm{I}}=0.8$ and ${\lambda }^{\mathrm{R}}=0.5$, and the dashed curves are the results using the BR spin-orbit potential. Similar scaling is used for the BR results. (See the text for details.)