Measurement of the Nucleon Nucleon Scattering Length with the ESC04 Interaction ()

Roberto Arceoro, Gerardo Jesús Escalera Santos, Orlando Díaz-Hernández

Centro de Estudios en Física y Matemáticas Básicas y Aplicadas, Universidad Autónoma de Chiapas, Tuxtla Gutiérrez, México.

**DOI: **10.4236/wjnst.2014.41003
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Centro de Estudios en Física y Matemáticas Básicas y Aplicadas, Universidad Autónoma de Chiapas, Tuxtla Gutiérrez, México.

**We have determined a value for the ^{1}S_{0} 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 Schr?dinger equation with nonlocal potential for nucleonnucleon pairs is described and the result is consistent with previous determinations of a_{nn} = -18.63 **

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R. Arceoro, G. Santos and O. Díaz-Hernández, "Measurement of the Nucleon Nucleon Scattering Length with the ESC04 Interaction," *World Journal of Nuclear Science and Technology*, Vol. 4 No. 1, 2014, pp. 16-20. doi: 10.4236/wjnst.2014.41003.

**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 m_{1} and m_{2}, 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 S_{12} may be written as an operator of the form

with l = 2 and j_{1} = j_{2} = 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 ^{1}S_{0} wave, Equation (6) for the neutron-neutron system has the form (S = J = L = 0, L’ = −1, 0, 1),

(8)

where S_{00-1} = S_{001} = 0, S_{000} = 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 u_{0} 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 d_{0} 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 r_{0} 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 Â 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-

Figure 1. Total potential in the partial wave ^{1}S_{0}, for I = ½.

Table 1. ESC04 low-energy parameters: S-wave scattering lengths and effective ranges.

Figure 2. Central (a), spin-spin (b), spin-orbital (c), and tensor (d) part of the YN potential.

Figure 3. Solid curve, proton-proton I = 1 phase shifts (degrees), as a function of T_{lab} (MeV), numerical solution for the ESC04 model. Dots, phases of the Rijken analysis [4]. Circles, s.e. phases of the Nijmegen93 PW analysis. Triangles, the m.e. phases of the Nijmegen93 PW analysis [10].

Figure 4. Solid curve, neutron-proton I = 0 phase shifts (degrees), as a function of T_{lab} (MeV), numerical solution for the ESC04 model. Dots, phases of the Rijken analysis [4]. Circles, s.e. phases of the Nijmegen93 PW analysis. Triangles, the m.e. phases of the Nijmegen93 PW analysis [10]. Diamonds, Bugg s.e. [11].

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.

NOTES

Conflicts of Interest

The authors declare no conflicts of interest.

[1] |
Th. A. Rijken and V. G. J. Stocks, “Soft two-Meson-Exchange Nucleon-Nucleon Potentials. I. Planar and CrossedBox Diagrams”, Physical Review C, Vol. 54, No. 6, 1996, pp. 2851-2868. http://dx.doi.org/10.1103/PhysRevC.54.2851 |

[2] |
Th. A. Rijken and V. G. J. Stocks, “Soft Two-MesonExchange Nucleon-Nucleon Potentials. II. One-Pair and Two-Pair Diagrams”, Physical Review C, Vol. 54, No. 6, 1996, pp. 2869-2882. http://dx.doi.org/10.1103/PhysRevC.54.2869 |

[3] |
Th. A. Rijken, H. Polinder and J. Nagata, “ExtendedSoft-Core NN Potentials in Momentum Space. I. Pseudoscalar-Pseudoscalar Exchange Potentials”, Physical Review C, Vol. 66, No. 4, 2002, pp. 044008-1-044008-19. http://dx.doi.org/10.1103/PhysRevC.66.044008 |

[4] |
Th. A. Rijken, “Extended-Soft-Core Baryon-Baryon Model. I. Nucleon-Nucleon Scattering with the ESC04 Interaction,” Physical Review C, Vol. 73, No. 4, 2006, pp. 044007-1-044007-16. http://dx.doi.org/10.1103/PhysRevC.73.044007 |

[5] | J. M. Eisenberg and W. Greiner, “Microscopy Theory of the Nucleus,” North-Holland Publishing Company, Amsterdam, 1972. |

[6] | J. M. Eisenberg and W. Greiner, “Excitation Mechanisms of the Nucleus,” North-Holland Publishing Company, Amsterdam, 1972. |

[7] | W. R. Gibbs, “Computation in Modern Physics,” 3rd Edition, World Scientific Publishing, Singapore, 2006. |

[8] | S. S. M. Wong, “Introductory Nuclear Physics,” 2nd Edition, Wiley-VCH Verlag GmbH & Co. KGaA, New York, 2004. |

[9] | ESC04 YN Potentials. (2006) http://nn-online.org |

[10] |
V. G. J. Stocks, R. A. M. Klomp M. C. M. Rentmeester and J. J. de Swart, “Partial-Wave Analysis of All Nucleon-Nucleon Scattering Data Below 350 MeV,” Physical Review C, Vol. 48, No. 2, 1993, pp. 792-815. http://dx.doi.org/10.1103/PhysRevC.48.792 |

[11] |
D. V. Bugg and R. A. Bryan, “Comments on np Elastic Scattering, 142-800 MeV,” Nuclear Physics A, Vol. 540, No. 3-4, 1992, pp. 449-460. http://dx.doi.org/10.1016/0375-9474(92)90168-J |

[12] |
B. Gabioud, J. C. Alder, C. Joseph, J.-F. Loude, N. Morel, A. Perrenoud, J. P. Perroud, M. T. Tran, E. Winkelmann, W. Dahme, H. Panke, D. Renker, C. Zupancic, G. Strassner and P. Truol, “n-n Scattering Length from the Photon Spectra of the Reactions π-d→γnn and π-p→γn,” Physical Review Letters, Vol. 42, No. 23, 1979, pp. 1508-1511; http://dx.doi.org/10.1103/PhysRevLett.42.1508 |

[13] | O. Schori, B. Gabioud, C. Joseph, J. P. Perroud, D. Rüegger, M. T. Tran, P. Truol, E. Winkelmann and W. Dahme, “Measurement of the Neutron-Neutron Scattering Length Ann with the Reaction π-d → nnγ in Complete Kinematics,” Physical Review C, Vol. 35, No. 6, 1987, pp. 2252-2257. http://dx.doi.org/10.1103/PhysRevC.35.2252 |

[14] |
Q. Chen, C. R. Howell, T. S. Carman, W. R. Gibbs, B. F. Gibson, A. Hussein, M. R. Kiser, G. Mertens, C. F. Moore, C. Morris, A. Obst, E. Pasyuk, C. D. Roper, F. Salinas, H. R. Setze, I. Slaus, S. Sterbenz, W. Tornow, R. L. Walter, C. R. Whiteley and M. Whitton, “Measurement of the Neutron-Neutron Scattering Length Using the π-d Capture Reaction,” Physical Review C, Vol. 77, No. 5, 2008, pp. 054002-1-054002-19. http://dx.doi.org/10.1103/PhysRevC.77.054002 |

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