Elastic Cross Section for the Reactions 235 U ( n, n ) and 238 U ( n, n ) at Low Energies Using Woods-Saxon and Coulomb Potentials

The elastic cross section for neutrons by Uranium atoms is calculated for the reactions 235 U ( n, n ) and 238 U ( n, n ) in the regime of low energies using Coulomb and Woods-Saxon potentials. The parameters used in the calculations were chosen by comparison with experimental data previously published by reducing chi-square. Using the same parameters in the equations the ratio 238 U / 235 U is calculated. The results are shown next to the experimental data to illustrate the agreement of the model in the range of energies from 1.5 keV to 1.0 MeV .


Introduction
Since the neutron discovery by Chadwick in 1932 (Chadwick, 1932), neutrons have been used in scattering experiments for several decades (Carpenter & Lander, 2010).Given that neutrons lack electric charge, they weakly interact with matter (Ederth, 2018) and penetrate deeper into the nucleus of different isotopes.For energies of a few MeV, the reduced wavelength of a neutron-nucleus system is approximately of the same order of magnitude as the nucleus (Block et. al., 2010).Therefore, for these energies, neutrons are able to interact with atomic nucleus and neutron scattering is well suited for estimating cross sections at nuclear level.In the present work, we have used the data from GMAP database (Carlson et. al., 2018) of the 238 U (n, n) and 235 U (n, n); then, we have found solutions to the Schroedinger equation with an optical Woods-Saxon potential in the low energy regime (Carlson et. al., 2018;Arceo et. al., 2020).A Woods-Saxon potential can be considered as a smoothed version of a barrier potential (Rojas & Villalba, 2005).In order to find the best fitting parameters to the experimental data (Carlson et. al., 2018) from the Coulomb barrier, we use a model that reduced the chi-squared uncertainty for the choice of the parameters such as the depth of the potential well, the length surface thickness of the nucleus, and the nuclear radius.There are few theoretical models and experimental data carried out for the reactions 238 U (n, n) and 235 U (n, n) (Poenitz, 1981;Poenitz & Aumeier, 1997;Chadwick et. al., 2006Chadwick et. al., , 2011;;Brown et. al., 2018;Otuka et. al., 2014).In section II, we present the theoretical framework of the dispersion S matrix and the Woods-Saxon potential used in the estimation of the total elastic cross section.

Theory
The radial Schroedinger equation was solved to calculate the scattering of neutrons by Uranium atoms using an optical Woods-Saxon potential.The calculations described in (Arceo et. al., 2020) are briefly summarized as it follows.
The Schroedinger equation is, where µ = m1m2 m1+m2 is the reduced mass for a system of two single particles, E is the energy and V (r) is the radial effective potential.
We introduce U (r) as, and using separation of variables in the Schroedinger equation ( 1), we obtain its radial component, with where r is the radius given in f m, V (r) is the potential given in M eV and Z 1,2 is the atomic number of each nucleus.
V C (r) is de Coulomb potential, and V W S (r) is the Woods-Saxon potential which will be described in the section 2.2.
The radial equation (equation ( 3)) takes the final form,

The Scattering S Matrix
The S matrix takes several mathematical forms (Gibbs, 2006).We used the form in which the solutions from the radial Schroedinger equation U l (equation ( 6)) are introduced to include the scattering process for the reactions 238 U (n, n) and 235 U (n, n).
The solutions of U l from equation ( 6) are introduced in the S matrix (Eq.10.58 of (Gibbs, 2006)), which is, where the S matrix is evaluated in the last two points on a mesh of size δ ( r = 0, δ, 2δ, ..., nδ).U l are the solutions to the Schroedinger equation with the potential previously calculated and h l are the spherical Hankel functions defined in Eq. 10.52 of (Gibbs, 2006).
The scattering amplitude for a partial wave decomposition in terms of the S matrix is, For states with well defined spin and isospin, the total cross section for neutrons via Uranium atoms scattering into a solid angle element dΩ is given by the scattering amplitude f (θ) of the reaction where k is the center-of-mass momentum and f (0 o ) is the forward amplitude.
The results from the dispersion are shown in the next section.

The Woods-Saxon Potential
The Woods-Saxon potential (Woods & Saxon, 1954) is used and explained in section 2.1 of (Arceo et. al., 2020), where V 0 is the potential well depth, a is the length surface, R is the nuclear radius, and r * is the distance from the center of the nucleus.
In the Figure 1 the Woods-Saxon and the Coulomb potential are plotted as a function of r * .We have used the parameters obtained in Table 1 and the equation ( 5).
Figure 1.The solid line is for the Woods-Saxon potential, the dashed line is for the Coulomb potential, and the dashed-dot line is for the Woods-Saxon and Coulomb potential using the parameters from the Table 1 In Figure 2 is shown the results of the chi-squared calculations obtained with the variations of the parameters in the Woods-Saxon potential.In this case, we compare the experimental data from (Carlson et. al., 2018) with the proposed optical model.The parameters shown in Table 1 are in the range from 0 to 50 in the chi-squared distribution, this range corresponds to the larger count number.In the Table 1 are showed the parameters that minimize the chi-squared for the reactions 238 U (n, n) and 235 U (n, n) using the experimental data from (Carlson et. al., 2018).

Elastic Cross Section
The analysis is performed for the reactions 238 U (n, n) and 235 U (n, n) in the energy range from 1.5 keV to 1.0 M eV .We observe that the cross section decreases as the incident neutron energy increases.
Figure 3 shows the total cross section for the interaction of neutrons by Uranium isotopes.We evaluated the cross sections at energies up to E= 1.0 M eV by considering the radial effective Woods-Saxon and Coulomb potentials, and setting the parameters calculated by reducing the chi-squared in fitting the experimental data in (Carlson et. al., 2018) and shown in Table 1.
Figure 3.The elastic scattering cross sections for 238 U (n, n) and 235 U (n, n) are plotted as a function of the incident neutron energy.The experimental data were taken from (Carlson et. al., 2018) In Figure 4 the total cross section ratio from 238 U (n, n) to 10 times 235 U (n, n) is plotted as a function of the incident neutron energy.The experimental data were taken from (Carlson et. al., 2018).
Figure 4.The elastic cross section ratio from 238 U (n, n) to 10 times 235 U (n, n) is plotted as a function of the incident neutron energy.The experimental data were taken from (Carlson et. al., 2018) χ 2 histogram for 235 U (n, n) (left plot) and the χ 2 histogram for 238 U (n, n) (right plot)

Table 1 .
Parameters obtained for the reactions 238 U (n, n) and 235 U (n, n)