A Global Elastic Nucleon-Nucleus Optical Potential

By: S. Weppner
Applet by: G. Diffendale and G. Vittorini
Phys. Rev. C 80, 034608 (2009)
correction in red to equation 1. of the paper below. Also a wrong sign in one of the numbers in the chart, also in red.

This is a Non Relativistic Distorted Wave Born Approximation
with a Relativistic Correction↑ using a modified Numerov Routine
Please make suggestions by emailing S. Weppner

Java Applet that calculates the Global Optical Potential

If the Calculate button is depressed then the program is working, it may take a few moments.
The applet will error unless the energy ranges are between 30 and 160

WP stands for Weppner-Penney, KD stands for Koning-Delaroche and is Ref. [5]. MD stands for Madland and is Ref. [4] The figures below were generated with the help of TALYS. Some TALYS input files for the WP and MD potentials are located here. This directory has source code (*.rb) to generate more files, potential files (*.pot), and example TALYS input files for Ca40.

$\displaystyle f_{WS}(r,{\cal R}_i,{\cal A}_i)=(1+\exp\left ((r-{\cal R}_i)/{\cal A}_i)\right)^{-1},$ (0.1)CORRECTION TO PAPER IN RED

Our complex phenemological optical model potential contains the traditional volume ($ V$ ), surface ($ S$ ), and spin-orbit ($ SO$ ) nuclear terms which are delineated using the standard Woods-Saxon form factors

$\displaystyle f_{WS}(r,{\cal R}_i,{\cal A}_i)=(1+\exp\left ((r-{\cal R}_i)/{\cal A}_i)\right)^{-1},$ (0.1)

where $ {\cal R}_i$ is the radius parameter and $ {\cal A}_i$ is the diffusive parameter. The $ i$ is a placeholder for the $ V$ , $ S$ , or $ SO$ designation. The phenemological optical model potential takes a standard form
    $\displaystyle {\cal U}(r,E,A,N,Z,MN)=$  
    $\displaystyle \Big(-{\cal V}_V(E,A,N,Z,MN)-i{\cal W}_V(E,A,N,Z,MN)\Big)
f_{WS}(r,{\cal R}_V,{\cal A}_V)$  
    $\displaystyle +4{\cal A}_S\Big({\cal V}_D(E,A)+i{\cal W}_D(E,A,N,Z)\Big)
\frac{d}{dr}f_{WS}(r,{\cal R}_S,{\cal A}_S)$  
    $\displaystyle +\frac{2}{r}\Big({\cal V}_{SO}(E,A,N,Z)+i{\cal W}_{SO}(E,A,N,Z)\Big)
\frac{d}{dr}f_{WS}(r,{\cal R}_{SO},{\cal A}_{SO})({\bf l\cdot\sigma})$  
    $\displaystyle +f_{coul}(r,{\cal R}_C,A,N,Z)$ (0.2)

where the $ {\cal V}_i$ and $ {\cal W}_i$ are the real and imaginary potential amplitudes respectively and $ f_{coul}(r,{\cal R}_C,A,N,Z)$ is the coulomb term which has the following traditional format with a proton projectile:
$\displaystyle f_{coul}(r,{\cal R}_C,A,N,Z) = \frac{Ze^2}{r},\;\;\;\;\;r\ge {\cal R}_C,$      
$\displaystyle f_{coul}(r,{\cal R}_C,A,N,Z) = \frac{Ze^2}{2{\cal R}_C}
\Big(3-\frac{r^2}{{{\cal R}_C}^2}\Big ), r\le {\cal R}_C.$     (0.3)

For a neutron projectile it is set to zero. For this potential $ {{\cal R}_C}$ takes on a functional form and is detailed below. The amplitudes, radii, and diffusive parameters have the following dependent variables:
$ \bullet$ E - projectile nucleon laboratory energy in MeV
$ \bullet$ A - Atomic number of the target nucleus
$ \bullet$ N - Number of neutrons in the target nucleus
$ \bullet$ Z - Number of protons in the target nucleus
$ \bullet$ $ \cal P$ - +1 if projectile is a proton, -1 if a neutron
$ \bullet$ MN - set to 1 if the target is traditionally singly magic
- set to 2 if the target is tradittionally doubly magic
- otherwise set to 0.

Explicity the amplitudes are given in the following polynomial formats:
The volume amplitudes:

    $\displaystyle {\cal V}_V=V_{V_0}+V_{V_1}A+V_{V_2}A^2+V_{V_3}A^3
$\displaystyle +$   $\displaystyle {\cal P} (N-Z)\Big(V_{V_{i0}}+V_{V_{i1}}A+V_{V_{i2}}A^2+V_{V_{i3}}A^3+V_{V_{i4}}A^4
$\displaystyle +$   $\displaystyle MN\Big(V_{V_{m0}}+V_{V_{m1}}A+V_{V_{m2}}A^2+V_{V_{m3}}A^3

    $\displaystyle {\cal W}_V=W_{V_0}+W_{V_1}A+W_{V_2}A^2+W_{V_3}A^3
$\displaystyle +$   $\displaystyle {\cal P} (N-Z)\Big(W_{V_{i0}}+W_{V_{i1}}A+W_{V_{i2}}A^2+W_{V_{i3}}A^3+W_{V_{i4}}A^4
$\displaystyle +$   $\displaystyle MN\Big(W_{V_{m0}}+W_{V_{m1}}A+W_{V_{m2}}A^2+W_{V_{m3}}A^3

The surface amplitudes:
    $\displaystyle {\cal V}_S=V_{S_0}+V_{S_1}A+V_{S_2}A^2+V_{S_3}A^3

    $\displaystyle {\cal W}_S=W_{S_0}+W_{S_1}A+W_{S_2}A^2+W_{S_3}A^3
$\displaystyle +$   $\displaystyle {\cal P} (N-Z)\Big(W_{S_{i0}}+W_{S_{i1}}A+W_{S_{i2}}A^2+W_{S_{i3}}A^3+W_{S_{i4}}A^4

The spin orbit amplitudes:
    $\displaystyle {\cal V}_{SO}=V_{{SO}_0}+V_{{SO}_1}A+V_{{SO}_2}A^2+V_{{SO}_3}A^3
$\displaystyle +$   $\displaystyle {\cal P} (N-Z)\Big(V_{{SO}_{i0}}+V_{{SO}_{i1}}A+V_{{SO}_{i2}}A^2+V_{{SO}_{i3}}A^3+V_{{SO}_{i4}}A^4

    $\displaystyle {\cal W}_{SO}=W_{{SO}_0}+W_{{SO}_1}A+W_{{SO}_2}A^2+W_{{SO}_3}A^3
$\displaystyle +$   $\displaystyle {\cal P} (N-Z)\Big(W_{{SO}_{i0}}+W_{{SO}_{i1}}A+W_{{SO}_{i2}}A^2+W_{{SO}_{i3}}A^3+W_{{SO}_{i4}}A^4

The volume radius and diffusive parameters:
    $\displaystyle {\cal R}_V=R_{V_0}+R_{V_1}A+R_{V_2}A^2+R_{V_3}A^3

    $\displaystyle {\cal A}_V=A_{V_0}+A_{V_1}A+A_{V_2}A^2+A_{V_3}A^3
$\displaystyle +$   $\displaystyle {\cal P}(N-Z)\Big(A_{V_{i0}}+A_{V_{i1}}A+A_{V_{i2}}A^2+A_{V_{i3}}A^3

The surface radius and diffusive parameters:
    $\displaystyle {\cal R}_S=R_{S_0}+R_{S_1}A+R_{S_2}A^2+R_{S_3}A^3

    $\displaystyle {\cal A}_S=A_{S_0}+A_{S_1}A+A_{S_2}A^2+A_{S_3}A^3

the spin-orbit radius and diffusive parameters:
    $\displaystyle {\cal R}_{SO}=R_{{SO}_0}+R_{{SO}_1}A+R_{{SO}_2}A^2+R_{{SO}_3}A^3
$\displaystyle +$   $\displaystyle {\cal P}(N-Z)\Big(R_{{SO}_{i0}}+R_{{SO}_{i1}}A+R_{{SO}_{i2}}A^2+R_{{SO}_{i3}}A^3

    $\displaystyle {\cal A}_{SO}=A_{{SO}_0}+A_{{SO}_1}A+A_{{SO}_2}A^2+A_{{SO}_3}A^3

and finaly the the coulumb radius parameter:
    $\displaystyle {\cal R}_{C}=R_{{C}_0}+R_{{C}_1}A+R_{{C}_2}A^2+R_{{C}_3}A^3
$\displaystyle +$   $\displaystyle {\cal P}(N-Z)\Big (\frac{2Z}{A}\Big )^{\frac{1}{3}}
\Big (R_{{C}_...
+R_{{C}_{i5}}E+R_{{C}_{i6}}E^2+R_{{C}_{i7}}E^3\Big )$  

These parameters are listed in table 1.

Table 1: These are the polynomial parameters which are used in the formula above.
Model 0 $ 1$ $ 2$ $ 3$ $ 4$ $ 5$ $ 6$ $ 7$
$ V$ +5.703E1 +4.099E-1 -8.656E-3 +5.793E-5 -- -5.881E-1 +1.822E-3 --
$ V_i$ -7.810E0 +1.054E0 -4.616E-2 +8.384E-4 -5.416E-6 -6.729E-3 +3.684E-5 --
$ V_m$ -3.723E-1 +6.563E-3 -5.308E-4 +7.987E-6 -- +2.515E-3 -5.607E-6 --
$ W$ -1.897E0 -1.843E-1 +5.034E-3 -3.814E-5 -- +2.367E-1 -1.423E-3 2.556E-6
$ W_i$ +8.216E0 -8.359E-1 +3.221E-2 -5.426E-4 +3.320E-6 +8.446E-3 -2.644E-5 --
$ W_m$ -3.781E0 +1.818E-1 -4.772E-3 +3.374E-5 -- +4.157E-2 -2.149E-4 --
$ V_S$ -4.612E-1 -1.178E-2 +9.658E-4 -1.270E-5 -- +7.906E-3 -4.230E-5 --
$ W_S$ +6.189E0 +1.740E-1 -4.790E-3 +3.670E-5 -- -6.423E-2 -3.753E-4 +3.096E-6
$ W_{S_i}$ +3.471E0 -4.265E-1 +1.670E-2 -2.828E-4 +1.744E-6 +1.449E-2 -8.093E-5 --
$ V_{SO}$ +1.562E1 -1.202E-1 +1.765E-3 -- -- -1.923E-1 +1.168E-3 -2.400E-6
$ V_{SO_i}$ -3.666E0 +7.228E-1 -3.524E-2 +6.493E-4 -4.151E-6 +2.472E-3 -3.317E-6 --
$ W_{SO}$ +3.929E-1 +1.660E-1 -5.369E-3 +4.646E-5 -- -3.702E-2 +9.223E-5 --
$ W_{SO_i}$ +5.399E0 -4.639E-1 +1.718E-2 -2.809E-4 +1.696E-6 -1.720E-2 +1.234E-4 --
$ R_V$ +1.491E0 -1.971E-2 +5.447E-4 -4.561E-6 -- -6.255E-3 +9.064E-5 -3.187E-7
$ A_V$ +1.933E-1 +3.484E-2 -9.172E-4 +6.999E-6 -- +5.762E-3 -6.097E-5 +1.929E-7
$ A_{V_i}$ +2.207E-3 +5.253E-3 -1.970E-4 +2.043E-6 -- -5.014E-4 +1.898E-6 --
$ R_S$ +8.599E-1 -5.657E-3 +8.884E-5 +7.253E-7 -- +1.024E-2 -4.166E-5 --
$ A_S$ +9.477E-1 +5.097E-3 +1.201E-4 -2.824E-6 -- -1.255E-2 +4.597E-5 --
$ R_{SO}$ +8.293E-1 +3.098E-2 -7.747E-4 +6.035E-6 -- -3.894E-3 +1.799E-5 --
$ R_{SO_i}$ -1.132E-1 -5.916E-4 +3.596E-6 -- -- +4.458E-3 -4.652E-5 +1.521E-7
$ A_{SO}$ +9.239E-1 +3.091E-2 -7.702E-4 +5.982E-6 -- -1.874E-2 +1.576E-4 -4.161E-7
$ R_C$ +3.604E0 -2.103E-1 +7.753E-3 -8.155E-5 -- +1.074E-1 -6.348E-4 --
$ R_{C_i}$ +3.404E-1 -1.038E-1 +1.294E-3 -- -- +4.501E-2 -3.729E-4 +9.467E-7

↑ A. Nadasen et al. "Elastic Scattering of 80-180 MeV Protons and the Proton-Nucleus Optical Potential."
Physical Review C 23 (1981): 1023-1043.

If you have any questions or problems please contact S. Weppner

Stephen Weppner 2009-06-24