ABSTRACT

We have determined a value for the ¹S₀ neutron-neutron scattering length (a_nn). The scattering length result is presented for the extended-soft-core (ESC04) interaction. The value obtained in the present work is a_nn = −18.6249 fm. The method of solution of the radial Schrödinger equation with nonlocal potential for nucleonnucleon pairs is described and the result is consistent with previous determinations of a_nn = −18.63 ( 0.10 (statistical) ( 0.44 (systematic) ( 0.30 (theoretical) fm. The nonlocal potentials are of the central, spin-spin, spin-orbital, and tensor type. The analysis from the ESC04 interaction is done at energies 0 ( T_lab ( 350 MeV. We compare the present result with experimental S-wave phase shifts analysis and agreement is found.

Keywords:Nucleon-Induced Reactions; S-Matrix Theory; Scattering Theory

1. Introduction

In nuclear physics, important information can be obtained from the scattering length associated with lowenergy nucleon-nucleon scattering. At these energies, the nucleon-nucleon interaction can be treated non-relativistically and the scattering was studied by means of a single particle Schrödinger equation which involves a nonlocal effective potential, derived from [1-]">4] using an extended soft-core model (ESC interaction). In the present manuscript, we consider a potential that involves a central part, a spin-spin interaction, a spin-orbital interaction and a tensor part and perform a numerical study of the associated Schrödinger equation. Also, we determine a numerical value for proton-proton and neutron-proton scattering lengths.

The present work is realized by considering energies in the range of 0 £ T_lab £ 350 MeV. For nucleon-nucleon scattering, it has been demonstrated that the interaction from the ESC model gives a description that is in good agreement with the nucleon-nucleon data. The extended soft-core model, also known as ESC, is used for nucleonnucleon (NN), hyperon-nucleon (YN), and hyperonhyperon (YY) scatterings. The particular version of the model ESC, called ESC04 [T. A. Rijken, Phys. Rev. C 73, 04007 (2006)], describes NN and YN interaction in an unified way using broken SU (3) symmetry.

A good fit with the experimental data is obtained by using the ESC04 model. The manuscript is organized as follows: in Section II, we give a theoretical review of the model; in Section III, we present our numerical results and in Section IV, we draw our conclusions.

2. Theory

2.1. The Schroedinger Equation with Non-Local Potential

The model we are going to study numerically involves a radial Schrödinger equation with ESC04 potential; namely

, (1)

where is the reduced mass of the nucleons whose individual masses are m₁ and m₂, and have spins and; r is the distance between the nucleons. The potential is parameterized as

where is a second rank tensor operator.

For an S-state we introduce, where

.

For a given value of the quantum number J,

, (2)

where we introduce

, (3)

where the symbol denotes a ClebschGordan coefficient, and Y_LML are the spherical harmonics, and

;

;

.

The subscript on c refers to the magnetic projection quantum number M_S of the spin-1 state, while a and b represent spin up and spin down for the particular spin-½ nucleon indicated by the subscript.

The Equation (2) forms an orthonormal set spanning the space of spin-1 functions and functions of the direction r. The normalization of requires that the radial functions satisfy,

. (4)

The Schrödinger equation [Equation (1)] is processed by the method of separation of variables, we obtain as its radial component,

. (5)

We use the parametrized potential

and

for an S-state to obtain,

, (6)

where [5], and S₁₂ may be written as an operator of the form

with l = 2 and j₁ = j₂ = 1. Here is the Clebsch-Gordan coefficient.

Using Racha algebra (see appendix A of [6]) we can show that

. (7)

2.2. Numerical Solution of the Schrödinger Equation

Considering the single state for the ¹S₀ wave, Equation (6) for the neutron-neutron system has the form (S = J = L = 0, L’ = −1, 0, 1),

(8)

where S_00-1 = S₀₀₁ = 0, S₀₀₀ = 2 are calculated from Equation (7).

For the proton-proton system we add the Coulomb effect to Equation (8),.

The numerical techniques necessary to solve equation (8) with this ESC04 potential are explained in chapter 3, Equation (3.28) of [7]. The solutions of u₀ from Equation (8) are introduced in the S matrix (Equation (10.58) of [7], which is,

, (9)

where the S matrix is evaluated in the last two points on a mesh of size e (). U_l are the solutions to Equation (8) with the ESC04 potential previously calculated and h_l are the spherical Hankel functions defined in Equation (10.52) of [7].

We insert the numerical solution of the S matrix in the solution of the S matrix for a real potential

, (10)

where d_l is real and is known as the phase shift.

Once the d₀ phase shift is found the a_nn scattering length and the effective range r_nn are calculated. For l = 0 the expression for can be parameterized in the following form,

. (11)

The quantity a is called the scattering length and r₀ is known as the effective range.

In the limit of low energies the scattering length is given in terms of the s-wave phase shift (see appendix B of [8]),

, (12)

where is the center-of-mass momentum (the wave number) and  indicates the real part.

2.3. Extended Soft-Core Potential (ESC04)

An Extended Soft-core potential is calculated consisting of a central, spin-spin, spin-orbital, and a tensor part. The potential of the ESC04 model is generated by one-bosonexchange (OBE), two-meson-exchange (TME) and meson-pair-exchange (MPE); this potential is calculated and explained in [1-4]. In Figure 1 the total ESC04 potential is plotted as a function of the r distance. In Figure 2 we show the central, spin-spin, spin-orbital, and tensor part of this total potential.

The algoritms for the YN potential are found in [9].

3. Results

The a_nn Scattering Length The a_nn scattering length is calculated obtaining a numerical value a_nn = −18.62497 fm and an effective range of r_nn = 2.746615 fm. We use an ESC04 potential below 350 MeV. In Figures 3 and 4 the phase shift is plotted for the proton-proton and neutron-proton case.

Table 1 shows the results for the low-energy parameters from the scattering lengths and the effective ranges for neutron-proton, proton-proton and neutron-neutron system using the ESC04 interaction.

4. Conclusions

In the present work, we have numerically solved the Schrödinger equation with an ESC04 potential and obtained the nucleon-nucleon scattering lengths. Summarizing our main conclusions:

1) Recent calculations using the ESC04 interaction for nucleon-nucleon dispersion have been realized [4], and reproduced with the Schrödinger equation.

2) The numerical solution of the radial Schrödinger equation has been realized and has been demonstrated to give a good fit to the nucleon-nucleon data.

3) The scattering lengths a_pp, a_np and a_nn have been calculated and are consistent with the experimental re-

sults. The final value for a_nn from this study is a_nn = −18.625 fm. Results from previous studies are

[12],

[13]and

[14]The presented ESC model is thus successful in describing the NN data.

Acknowledgements

This work was partially supported by PIEIC-UNACH 2012 and SIINV UNACH 2012.

REFERENCES

NOTES

^*Corresponding author.