Correlation between Nucleon-Nucleon Interaction , Pairing Energy Gap and Phase Shift for Identical Nucleons in Nuclear Systems

Assuming some known nucleon-nucleon interactions, and using the relations between phase shift δ and nucleon-nucleon interaction potential ( ) V r ; the relation between nucleon-nucleon interaction and scattering length a; the relation between energy gap ∆ , and scattering length a; an equation is obtained between energy gap ∆ and Fermi momentum F k via the phase shift ( ) F k δ . Assuming s0 (singlet) pairing between the nucleons, the energy gap ∆ has been calculated and it is found that 3.0 MeV ∆ = at Fermi momentum 1 0.8 fm F k − = .


Introduction
When one particle approaches another particle, and they are in the field of force of each other; they interact and scatter.In this process the following three parameters are involved.
1) The interaction potential ( ) V r between the two particles; 2) The distance between the nearest-approach, called impact parameter or the scattering length a; 3) Phase shift, δ , due to scattering.
However, when dealing with large finite-nuclei, or infinite nuclear matter and neutron matter (in stars), another important parameter gets involved, and this is the so called 1 s 0 pairing gap ∆ .Thus there must exist, a definite relationship between the interaction potential ( ) V r .The scattering length a, the phase shift δ and the pairing gap ∆ .Thus the quantitative features of 1 s 0 pairing in nuc- lear matter and neutron matter can be obtained directly from the 1 s 0 phase shifts.
The 1 s 0 neutron matter superfluid is relevant for the phenomena that occur in the inner crust of neutron stars [1].The 1 s 0 pairing gap values and its density dependence show a peak value of about 3 MeV at a Fermi momentum close to 1 0.8 fem F k − ≅ [2]; most of the calculations adopt the bare nucleon-nucleon interaction as the pairing force and it has been pointed out that the screening by the medium of interaction could strongly reduce the pairing strength in this channel [3].
After the discovery of neutron stars calculations were done for pairing gap for neutrons [4] in the 1 s 0 state.The pairing gap energy for neutrons [4]  ).At such density, it becomes favorable for neu- trons to pair in the 3 P 2 state ( ) the pairs have unit orbital angular momentum ( ) total angular momentum ( ) It should be emphasized that the 3 P 2 state has more attractive interaction than the other 3 P sates due to the fact that the spin-orbit interaction is attractive for nucleons.However in atomic physics the spin-orbit interaction is repulsive.For the neutron pair in the state 3 P 2 , the pairing energy gap is increased to about 0.5 MeV at a density of around 2 s n and dropped at higher densities.The qualitative behavior of the pairing energy gaps is understood in terms of the measured phase shifts for nucleon-nucleon interactions.A positive phase shifts corresponds to an attractive interaction between neutrons and therefore at low k (k is the propagation vector p  ), which corresponds to low Fermi momentum and low density.The most attractive is channel is 1 S 0 , which at higher densities the interaction in 3 P 2 channel is more attractive and hence the phase shift will be positive.
In BCS approximation, the pairing energy gap is calculated by solving the BCS equation with the free-space nucleon-nucleon interaction and free particles in intermediate states in the scattering process; the effects of the neutron medium on the normal state excitations and the pairing interaction are neglected [5] [6].
Many different techniques have been developed to include effects beyond the BCS approximation in the calculations of neutron pairing energy and most of them that the pairing gap reduces by a factor of 2 or more.However microscopic calculations have been for pairing energy gap for nuclear and neutron matter in the 1S0 state 3 p 2 state and 1 S 0 state for protons.The pairing gap ∆ has also been related to the nucleon-nucleon interaction and phase shift.
Going by BCS approximation and considering pairing in the 1 S 0 state, the gap Journal of High Energy Physics, Gravitation and Cosmology is independent of the direction of k and the relation between gap ( ) k ∆ and the interaction potential ( ) v r is given by [7]   ( ) ( ) ( ) ( ) where ( ) , V k k′ , is the matrix element of the potential averaged over the angle between k and k′ given by ( ) ( ) ( ) ( ) We can use two values for ( ) V r one is the Yukawa potential, i.e.
( ) where Another is the simple Gaussian potential of the form, ( ) where 2 0 5037.0MeV and 12.0 fm This potential has been used in the phase shift analysis of the 2 S 1/2 scattering phase.Substituting the values of ( ) V r from Equation (3) and Equation (5) in Equation ( 2) and then substituting the values of ( ) , V k k′ in Equation (1) we can get the relation between ( ) k ∆ and 0 V clearly emphasizing of the there exists a close relationship between the pairing energy gap and the interaction potential.A rough calculations assuming ( ) ( ) and substituting this in Equation (1) will give a finite value of ( ) assuming some finite limits for integral in Equation ( 1), similarly we can use the value of ( ) V r in Equation ( 5) and obtain the value of ( ) Hence there exists a definite correlation between the pairing energy gap ( ) k ∆ and obtain the interaction potential.Without making any approximations, exact value of ( ) can also be calculated [8].

Theoretical Derivations
The nuclear force has been at the heart of nuclear physics since the discovery of the neutrons by Chadwick [9].The interaction between two nucleons is basic for all of nuclear physics.The main aim of nuclear physics is to understand the properties of atomic nuclei in terms of the "bare" interaction between a pair of nucleons.Scattering of nucleons is due to the neutrons neutrons-interaction between the nucleons and hence the resulting phase shifts will have a definite correlation with the interaction potential ( ) From time to time, a nuclear of nucleon-nucleon interaction potentials has Journal of High Energy Physics, Gravitation and Cosmology been proposed.For instance, Yukawa potential [10] is the oldest attempt to explain the nature of the nuclear forces.According to Yukawa massive bosons (mesons) mediate the interaction between two nucleons it is given by ( ) ( ) Another potential is a simple Gaussian potential of the form [11] [12].
( ) where In the last few decades the major issues concerning the (nucleon-nucleon) interaction have been: 1) Charge-dependence; 2) The precise value of the π nucleon-nucleon compiling constant; 3) Improved phase shift analysis; 4) High precision nucleon-nucleon data; 5) High-precision nucleon-nucleon potentials; 6) Quantum-chromo-Dynamics (QCD) and the nuclear force; 7) Nuclear-nuclear scattering is at intermediate and high energies.
However, in this manuscript we are interested in some simple calculations that will correlate the well known nucleon-nucleon interaction potential ( ) v r with the phase shift δ the scattering length a and the energy gap ∆ ; and the dependence of ∆ on the Fermi momentum F k has also been studied.Calcu- lations have been done using Yukawa potential only.
Using Yukawa potential calculations are done to relate interaction potential ( ) V r the interaction is potential, r is the inter-particle distance, β is the range of nucleon-nucleon force and 0 V is potential well depth.
The Born approximation [13] is a relationship between phase shifts, ( ) , for 0 l = , ground state, where 1 The pairing gap for small values of 0 F k a is [14]   ( ) where 0 a the scattering length in the ISO channel is ( ) 718, here 0 a is related to the interaction potential between a pair of nucleons.However at saturation density ( ) , due to scattering is given by relation ( ) r is the effective range of the nuclear force which roughly corresponds to the size of the potential and is the S-wave scattering phase-shift.
At the ground state 0 =  therefore Equation ( 9) becomes, ( ) ( ) ( ) Here ( ) It is valid to represent the interaction energy of a particle with momentum i F k k < , with all the particles within the Fermi surface.The values for i k [15]   could be 0.1 fm −1 , 0.2 fm −1 , 0.3 fm −1 … and therefore for 1 therefore Equation (13) reduces to ( ) Substituting the value of the potential, ( ) in Equation (12) we get ( ) Integrating Equation ( 15) by parts we get, ( ) Now for Yukawa potential, the value of the well-depth parameters S is where M is the average mass of the two interacting nucleons 1 S ≅ for the bound state of the nuclear matter and hence 0 V is given by Equation ( 23) will give the values of the energy gap.
It will be interesting to see that if other potentials are used for instance the potential used Hassan and Ramadan study [15] how phase shift Singlet scattering length Equation [16] is given by where critical constant The Pairing Gap equation is given by 1) At very low density [17] π exp 2 where 2) At low density [18] where: m is the free nucleon mass.

Results and Discussion
Recent studies have shown the nuclear isotope shifts, the differential observables such as the odd-even mass differences and odd-even effects in charge radii along isotope chains can be reproduced with an effective density-dependent contact pairing interaction.The self-consistent LEDF calculations with density gradient term f ε α ∇ in pairing force provide desirable size of isotopic shifts.Using Lead isotopes some sets of parameters are deduced for the pairing force.Calculations are done based on the general variation-principle applied to local effective density-dependent function with a fixed energy cutoff 40 MeV c =  measured from F  and on the coordinate-space technique which involves an interaction [19].At very low densities where critical constant 0 2 1.912 where These results agree with the general analysis of the gap equation at low densities for 1 nn a  .It is valid only in the weak coupling regime which corresponds to negative scattering length (Figure 1). 1) it shows that at low densities the pairing gap is small and it comes to zero.
Relation between the phase shift l δ , the nucleon-nucleon interaction ( ) V r and the energy gap using Yukawa potential: Using Equation ( 22) to calculate the values of phase shift ( ) , for different values of F k ranging from 0.1 -1.6 fm −1 and plot a graph of phase shift ( ) When the Fermi momentum F k of the interacting nucleons is zero, the value of phase shift ( ) , is equal to zero in the ground state as seen from Figure 2, this show that an increase in Fermi momentum F k leads to an increase in phase shift Using Equation (23) to compute the values of energy gap ( )  changes in the Fermi momentum F k and this variation is done and the data tabulated (Figure 3).
The energy gap ( ) increase steadily and faster for low Fermi momentum F k up to around 1.0 fm −1 and it is roughly constant with the value ≅0.8 MeV for  [15] which are in agreement with the known values.The reason for the good agreement in that, for calculating energy gaps the quantity that matters is the scattering length at energies of order of the Fermi momentum and this is strongly constrained by nucleon-nucleon scattering data for nucleon momentum in the

Conclusion
From the results obtained, it can be concluded that scattering length influences

Suggestions
In future this problem can be done using nucleon interactions that may involve elementary particles.But such calculations will be quite complicated and will require the use of many-body techniques involving Greens functions can be done.
r to the phase shift

f k r < ( 10 ),
Using derived equations and the values of the constants available, data was then generated and tabulated.Graphs have been drawn to show how the phase shifts ( )t F k δ, varies with the Fermi momentum F k .This potential is substi- tuted in the Born-approximation phase shifts, for scattering from a spherical potential, ( ) Journal of High Energy Physics, Gravitation and Cosmology

=
For low energy scattering especially in nuclear force which corresponds to the size of the potential and of the Fermi momentum F k .

Figure 1 .
Figure 1.A graph of pairing gap against scattering length for 0 nn a < .

. 4 .
In literature the values of the energy gap ( ) MeV, 0.6 MeV, 0.7 MeV … for 0.2 fm −1 , 0.3 fm −1 , 0.4 fm −1 … respectively[15] which are in agreement with the known values.The energy gap ( ) F k ∆ increase steadily and faster for low Fermi momentum F k up to around 0.4 fm −1 and it is roughly constant with the value ≅1.0In literature the values of the energy gap ( ) , 0.4 MeV, 0.6 MeV … for 0.2 fm −1 , 0.3 fm −1 , 0.4 fm −1 … respectively

1 0,
the energy gap with different values of Fermi momentum.The behavior of pairing gap (∆) at very low densities agrees well with calculations based on realistic nucleon-nucleon The singlet scattering length which is 19force.At low densities the pairing effect is affected by strong repulsive and short-range component interaction; therefore, the predictions are set to go higher for pairing gap (∆) reaching a maximum of 1.69 MeV at nucleon interaction assuming charge dependence.Interactions between nucleons may be characterized by a single parameter, the S-wave scattering length nn a which in- dicates the strength of interactions.The sign determines whether the interactions are effectively attractive or repulsive.When the scattering length is 0 nn a < the interaction is attractive and repulsive for 0 nn a > .The behavior of pairing gap (∆) at very low densities agrees well with calculations based on realistic NN forces.The singlet scattering length which is 20.When theFermi momentum F k of the interacting nucleons is at zero the value of phase shift is equal to zero in the ground state as seen from Figure2.This show that, increase in Fermi momentum F k leads to an increase in phase shift Recent discoveries on elementary particles have led to the suggestions for different types of nuclear interactions involving many parameters.Elementary particles inside the nucleus were the neutrons and protons.But within the theory of quantum chromo-dynamics (QCD), neutrons and protons are no longer the elementary particles.Nuclear forces between neutrons (neutron-neutron force).Protons (proton-proton force) and neutron-proton (neutron-proton force) were treated as charge independent.Within the Theory of QCD, there is what is called Charge Symmetry Breaking (CSB) and consequently the neutron-neutron-scattering length, the proton-proton-scattering length and neutron-proton scattering length change.Consequently there is a corresponding phase shift variation with nn a , pp a and np a , and there is a different relation with the nuclear force.These cal- culations can be done in the future.