In this work, the elastic cross section is calculated at energies above the Coulomb barrier for 3He + 58Ni using a Woods-Saxon potential. The solutions of the radial Schr ödinger equations are calculated numerically and they are introduced in the S matrix, after which the cross section is obtained. The parameters in the potential are adjusted to satisfy known experimental data.
We have studied the scattering of nuclei by helium and nickel atoms using the Schrödinger equation at energies up to 35 MeV for the reaction 3He + 58Ni using a radial Woods-Saxon potential. We treated the Schrödinger equation numerically, for the case of the Woods-Saxon potential [
The value of chi-squared was minimized by using a theoretical model and the experimental data from Fujisawa et al. [
Recently we have results at low energies for the reaction 3,4,6He + 58Ni [
This paper is divided into four sections as follows. In Section 2, we briefly describe the setup; Section 2.1 is dedicated to obtaining the Woods-Saxon potential. In Section 2.2, we discuss the elastic cross section for the scattering of helium by nickel atoms. In Section 3, the obtained results are shown. Finally, in Section 4 we focus on the discussion of our results.
In this section we describe the procedure used to compute the Woods-Saxon potential produced by a point particle. We then calculate the cross section for the collision of two particles of mass m 1,2 and atomic number Z 1,2 . Our approach to this problem is numerical, and we make the assumption that the interaction of the incident particle with the rest atom can be accounted for by the effective Woods-Saxon potential which we calculate below. We minimize the value of chi squared from the experimental 3He + 58Ni data [
The Woods-Saxon potential is a mean field potential for the nucleons (protons and neutrons) inside the atomic nucleus, which is used to describe approximately the forces applied on each nucleon, in the nuclear shell model for the structure of the nucleus.
The standard Woods-Saxon potential [
V ′ ( r * ) = − V 0 1 + exp ( r * − R a ) , a ≪ R (1)
| E | l | V 0 | R 1 | a 1 | W 0 | R 2 | a 2 | σ R | σ T | χ 2 / N |
|---|---|---|---|---|---|---|---|---|---|---|
| (MeV) | up to | (MeV) | (fm) | (fm) | (MeV) | (fm) | (fm) | (mb) | (mb) | |
| 24.15 | 11 | 174.400 | 1.30 | 0.750 | 17.1 | 1.41 | 0.71 | 1476.904 | 2871.663 | 11.95 |
| 27.64 | 15 | 174.275 | 0.93 | 0.601 | 17.1 | 1.41 | 0.71 | 1526.071 | 3542.422 | 5.85 |
| 34.14 | 13 | 174.500 | 0.94 | 0.75 | 18.6 | 1.41 | 0.73 | 1620.872 | 2695.711 | 2.21 |
| E | l | V 0 | R 1 | a 1 | W 0 | R 2 | a 2 | σ T | χ 2 / N |
|---|---|---|---|---|---|---|---|---|---|
| (MeV) | up to | (MeV) | (fm) | (fm) | (MeV) | (fm) | (fm) | (mb) | |
| 24.15 | 11 | 175.0 | 1.3 | 0.75 | 18.2 | 1.41 | 0.72 | 3277.768 | 17.903 |
| 27.64 | 15 | 173.6 | 1.3 | 0.75 | 17.1 | 1.41 | 0.71 | 3046.658 | 5.047 |
| 34.14 | 13 | 173.9 | 0.94 | 0.75 | 18.9 | 1.41 | 0.73 | 2804.289 | 1.832 |
where V 0 (with dimensions of energy, MeV) represents the potential well depth, a is a length representing the “surface thickness” of the nucleus, and R = r 0 A 1 / 3 is the nuclear radius where r 0 = 1.25 fm and A is the atomic mass number.
It is interesting to examine the consequences of the radial effective Woods-Saxon potential, V W S ( r * ) , by using both real and imaginary terms in experiments such as scattering events. We do so in the following section, where we include the Coulomb interaction potential V C ( r ) . The total radial effective potential used is
V ( r ) = V C ( r ) + V W S ( r * ) , (2)
V ( r ) = 1.44 Z 1 , 2 r − V 0 1 + exp ( r * − R 1 a 1 ) − i W 0 1 + exp ( r * − R 2 a 2 ) . (3)
In
In this section we solve the radial Schrödinger equation using the radial potential (Equation (3)). The Schrödinger equation is,
[ − ℏ 2 2 μ ∇ 2 + V ( r ) ] Ψ ( r ) = E Ψ ( r ) , (4)
where μ = m 1 m 2 m 1 + m 2 is the reduced mass for a two-particle system, E is the
energy and V ( r ) is the radial effective potential calculated in the previous section.
We introduce U ( r ) , where
Ψ ( r ) = Ψ ( r ) = U ( r ) r , (5)
and the Schrödinger Equation (4) is solved by the method of separation of variables. For the radial component we obtain
U ″ l ( r ) + 2 μ ℏ 2 [ E − V ( r ) ] U l ( r ) − l ( l + 1 ) r 2 U l ( r ) = 0. (6)
The radial equation takes the final form,
U ″ l ( r ) + 2 μ ℏ 2 [ E − 1.44 Z 1,2 r + V 0 1 + exp ( r * − R 1 a 1 ) + i W 0 1 + exp ( r * − R 2 a 2 ) ] U l ( r ) − l ( l + 1 ) r 2 U l ( r ) = 0. (7)
The next step is to determine the set of the parameters for the Woods-Saxon potential. In the
χ 2 = ∑ i = 1 N [ σ t h ( θ i ) − σ e x p ( θ i ) Δ σ e x p ( θ i ) ] 2 . (8)
The calculations for this analysis were done using the experimental data from Fujisawa et al. [
In
The numerical techniques necessary to solve the Schrödinger equation with a radial potential are explained in chapter 3, Equation (3.28) of Ref. [
S l = U l ( r n − 1 ) r n h l − ( k r n ) − U l ( r n ) r n − 1 h l − ( k r n − 1 ) U l ( r n ) r n − 1 h l + ( k r n − 1 ) − U l ( r n − 1 ) r n h l + ( k r n ) , (9)
where the S matrix is evaluated in the last two points on a mesh of size δ ( r = 0 , δ , 2 δ , ⋯ , n δ ). U l are the solutions to the Schrödinger equation with the potential previously calculated and h l are the spherical Hankel functions defined in Eq. 10.52 of Ref. [
f ( θ ) = 1 2 i k ∑ l = 0 ∞ ( 2 l + 1 ) P l ( cos θ ) ( S l − 1 ) . (10)
For states with well defined spin and isospin the elastic and total cross section of nucleon-nucleon scattering into a solid angle element d Ω is given by the scattering amplitude f ( θ ) of the reaction
d σ d Ω = | f ( θ ) | 2 , (11)
σ T = 4 π k I m [ f ( 0 ∘ ) ] , (12)
where k is the center-of-mass momentum and f ( 0 ∘ ) is the forward amplitude.
The reaction cross section is defined as the subtraction from the integral of the elastic cross section from the total cross section,
σ R ≡ 2 π k 2 ∑ l = 0 ∞ ( 2 l + 1 ) R e ( 1 − S l ) − ∫ σ ( θ ) d Ω . (13)
Doing the integration gives
σ R = 2 π k 2 ∑ l = 0 ∞ ( 2 l + 1 ) R e ( 1 − S l ) − π k 2 ∑ l = 0 ∞ ( 2 l + 1 ) [ | S l | 2 + 1 − 2 R e S l ] , (14)
σ R = π k 2 ∑ l = 0 ∞ ( 2 l + 1 ) ( 1 − | S l | 2 ) . (15)
The results from the calculations are shown in the next section.
With the analysis performed, we proceed to evaluate numerically the Equations (11)-(12) and (15). We compare the theoretical results with experimental data for elastic cross sections for elastic scattering of helium by nickel atoms [
In Figures 2-4 the elastic cross section is analyzed for helium by nickel atoms. We evaluate the elastic cross section at energies from TLab = 24.15, 27.64 and 34.14 MeV considering the radial effective Woods-Saxon potential and setting the parameters to adjust the experimental points (see
In
In this work, we present a numerical solution of the radial Schrödinger equation using a Woods-Saxon potential. We have examined the scattering of helium atoms via nickel. The scattered from alpha particles via nickel atoms was performed and the use of an imaginary term in the Woods-Saxon potential gives a better fit to the experimental data. The parameters for the Woods-Saxon potential were varied until χ 2 was minimized and they are shown in
Finally, the total cross section and integrated elastic scattering cross section are calculated and compared with experimental data. We obtain excellent agreement with the experimental data of Fujisawa et al. (Ref. [
This work was partially supported by FCFM-UNACH. R. Arceo acknowledges the support of the PFCE 2018.
The authors declare no conflicts of interest regarding the publication of this paper.
Arceo, R., Pedraza, O., Sandoval, L.M., López, L.A., Álvarez, C., Hueyotl-Zahuantitla, F., Flores-Rosas, A., Raya, G.L. and Martínez-Castro, J. (2020) Elastic Cross Sections for 3He + 58Ni above the Coulomb Barrier. World Journal of Nu- clear Science and Technology, 10, 1-8. https://doi.org/10.4236/wjnst.2020.101001