Correlation between Nucleon-Nucleon Interaction, Pairing Energy Gap and Phase Shift for Identical Nucleons in Nuclear Systems ()
1. 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
between the two particles;
2) The distance between the nearest-approach, called impact parameter or the scattering length a;
3) Phase shift,
, 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 1s0 pairing gap
. Thus there must exist, a definite relationship between the interaction potential
. The scattering length a, the phase shift
and the pairing gap
. Thus the quantitative features of 1s0 pairing in nuclear matter and neutron matter can be obtained directly from the 1s0 phase shifts. The 1s0 neutron matter superfluid is relevant for the phenomena that occur in the inner crust of neutron stars [1] . The 1s0 pairing gap values and its density dependence show a peak value of about 3 MeV at a Fermi momentum close to
[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 1s0 state. The pairing gap energy for neutrons [4] in the 1s0 state rose with increasing density and reached a maximum of roughly 3 MeV at the density of about
,where
is the saturation density of symmetric nuclear matter (containing equal numbers of neutrons and protons), a density typical of the interiors of heavy nuclei. With further increase in density, the neutron pairing energy gap dropped and vanished at a density just below
(i.e. n has less than
). At such density, it becomes favorable for neutrons to pair in the 3P2 state
in which the pairs have unit orbital angular momentum
, unit spin
and total angular momentum
. It should be emphasized that the 3P2 state has more attractive interaction than the other 3P 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 3P2, the pairing energy gap is increased to about 0.5 MeV at a density of around
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
), which corresponds to low Fermi momentum and low density. The most attractive is channel is 1S0, which at higher densities the interaction in 3P2 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 3p2 state and 1S0 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 1S0 state, the gap is independent of the direction of k and the relation between gap
and the interaction potential
is given by [7]
(1)
where
, is the matrix element of the potential averaged over the angle between k and
given by
(2)
We can use two values for
one is the Yukawa potential, i.e.
(3)
where
and
(4)
Another is the simple Gaussian potential of the form,
(5)
where
(6)
This potential has been used in the phase shift analysis of the 2S1/2 scattering phase. Substituting the values of
from Equation (3) and Equation (5) in Equation (2) and then substituting the values of
in Equation (1) we can get the relation between
and
clearly emphasizing of the there exists a close relationship between the pairing energy gap and the interaction potential. A rough calculations assuming
leads a constant-value for
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
in Equation (5) and obtain the value of
. Hence there exists a definite correlation between the pairing energy gap
and obtain the interaction potential. Without making any approximations, exact value of
can also be calculated [8] .
2. 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 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
(7)
Another potential is a simple Gaussian potential of the form [11] [12] .
(8)
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
with the phase shift
the scattering length a and the energy gap
; and the dependence of
on the Fermi momentum
has also been studied. Calculations have been done using Yukawa potential only.
Using Yukawa potential calculations are done to relate interaction potential
to the phase shift
and energy gap
Yukawa potential
where
the interaction is potential, r is the inter-particle distance,
is the range of nucleon-nucleon force and
is potential well depth.
The Born approximation [13] is a relationship between phase shifts,
and interaction potential
(9)
Using Bessel function in form
, for
, ground state, where
(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
, varies with the Fermi momentum
. This potential is substituted in the Born-approximation phase shifts,
, for scattering from a spherical potential,
, in 3-D, and to find the values of phase shifts.
The pairing gap for small values of
is [14]
(11)
where
the scattering length in the ISO channel is
,
is a constant ≈ 1, e = 2.718, here
is related to the interaction potential between a pair of nucleons. However at saturation density
,
.
For low energy scattering especially in nuclear force which corresponds to the size of the potential and
, due to scattering is given by relation
here
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
therefore Equation (9) becomes,
(12)
Here
is the spherical Bessel function of Zeroth order i.e.
such that,
(13)
It is valid to represent the interaction energy of a particle with momentum
, with all the particles within the Fermi surface. The values for
[15] could be 0.1 fm−1, 0.2 fm−1, 0.3 fm−1 … and therefore for
hence
therefore Equation (13) reduces to
(14)
Substituting the value of the potential,
in Equation (12) we get
(15)
Integrating Equation (15) by parts we get,
(16)
Now for Yukawa potential, the value of the well-depth parameters S is
(17)
where M is the average mass of the two interacting nucleons
for the bound state of the nuclear matter and hence
is given by
(18)
Substituting Equation (17) in Equation (16) gives,
(19)
For 1S0 scattering and the reduced mass
of the two interacting nucleons is given by
(20)
where
the proton is mass equal to 938.27 meV, and
is the neutron mass equal to 939.57 meV. Substituting the values of the masses in Equation (20) reduced mass
. The average mass M of the proton and neutron is given by the equation,
(21)
Substituting for the value of the masses
, then Equation (19) becomes
(22)
Calculating the values of phase shift,
, for different values of
ranging from
. And plot a graph of phase shift
, against Fermi momentum
.
For scattering length in the 1S0 channel
and
Equation (11) for energy gap [14] ,
, becomes
(23)
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
and energy gap
vary with Fermi momentum
. For low energy scattering the phase shift is given by
for different values of the Fermi momentum
.
Singlet scattering length Equation [16] is given by
(24)
where critical constant
and
is the vacuum strength.
The Pairing Gap equation is given by
1) At very low density [17]
for
(25)
where
,
and
.
2) At low density [18]
, for
(26)
where: m is the free nucleon mass.
3. 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
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
measured from
and on the coordinate-space technique which involves an interaction [19] . At very low densities
(24)
where critical constant
and
is the vacuum strength,
, for
(27)
where
,
and
.
These results agree with the general analysis of the gap equation at low densities for
. It is valid only in the weak coupling regime which corresponds to negative scattering length (Figure 1).
At
from Equation (1) it shows that at low densities the pairing gap is small and it comes to zero.
Relation between the phase shift
, the nucleon-nucleon interaction
and the energy gap using Yukawa potential:
Using Equation (22) to calculate the values of phase shift
, for different values of
ranging from 0.1 - 1.6 fm−1 and plot a graph of phase shift
, against Fermi momentum
.
When the Fermi momentum
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
leads to an increase in phase shift
.
Using Equation (23) to compute the values of energy gap
against
![]()
Figure 1. A graph of pairing gap against scattering length for
.
![]()
Figure 2. A graph of pairing gap against Fermi momentum (fm−1).
changes in the Fermi momentum
and this variation is done and the data tabulated (Figure 3).
The energy gap
increase steadily and faster for low Fermi momentum
up to around 1.0 fm−1 and it is roughly constant with the value ≅0.8 MeV for
. In literature the values of the energy gap
are 0.4 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
increase steadily and faster for low Fermi momentum
up to around 0.4 fm−1 and it is roughly constant with the value ≅1.0 MeV for
. In literature the values of the energy gap
are 0.2 MeV, 0.4 MeV, 0.6 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 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
![]()
Figure 3. A graph of energy gap against Fermi momentum (fm−1).
center-of-mass. For higher Fermi momenta the energy gap tends to reach saturation since there is considerable model dependence also because inelastic channels start to open up in nucleon-nucleon scattering. The energy gap
increases steadily and faster for low values of Fermi momentum
and becomes roughly constant for values of
.
4. Conclusion
From the results obtained, it can be concluded that scattering length influences 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 forces. The singlet scattering length which is
agrees with the theoretical value which is equivalent to
. Numerical values of scattering length are negative for
which means attractive force and positive for
indicating repulsive force. 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
with nucleon-nucleon interaction assuming charge dependence. Interactions between nucleons may be characterized by a single parameter, the S-wave scattering length
which indicates the strength of interactions. The sign determines whether the interactions are effectively attractive or repulsive. When the scattering length is
the interaction is attractive and repulsive for
. The behavior of pairing gap (∆) at very low densities agrees well with calculations based on realistic NN forces. The singlet scattering length which is
at
agrees with the theoretical value which is equivalent to
. Numerical values of scattering length are negative for
which means attractive force and positive for
indicating repulsive force. When the Fermi momentum
of the interacting nucleons is at zero the value of phase shift
, is equal to zero in the ground state as seen from Figure 2. This show that, increase in Fermi momentum
leads to an increase in phase shift
.
5. 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. 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
,
and
, and there is a different relation with the nuclear force. These calculations can be done in the future.
Acknowledgements
We are grateful to our colleagues at the Department of Physics, University of Eldoret for their helpful discussions, contributions and suggestions during the preparation and writing of this research paper. We also appreciate the international Centre of theoretical physics for their eJDS platform for providing us with several articles and useful information in line with this publication.