﻿ The Estimation of the Nucleonic Charge of a Neutron Relative to that of a Proton Using Data Based on Nuclei made up of Alpha Modules and Neutron Spin Pairs and Starting with the Incremental Binding Energies of Neutron Spin Pairs
San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
USA

 The Estimation of the Nucleonic Charge of a Neutron Relative to that of a Proton Using Data Based on Nuclei made up of Alpha Modules and Neutron Spin Pairs and Starting with the Incremental Binding Energies of Neutron Spin Pairs

## Background

Nuclei are composed of nucleons (neutrons and protons) but whenever possible those nucleons form spin pairs. The spin pairing is exclusive in the sense that one neutron can form a spin pair with one other neutron and with a proton. The same applies to protons. This means that the nucleons in a nucleus are linked together in chains that consists of modules of the form -n-p-p-n-, or equivalently, -p-n-n-p-. These can be called alpha modules. They link together into rings occupying shells. The smallest such shell is an alpha particle.

A neucleus then can consists of rings of alpha modules, excess neutron spin pairs or excess proton spin pairs and possibly a singleton neutron or proton. The binding energy of a nuclide is largely determined by the number of alpha modules it contains.

## The Model

The purpose of this analysis is to examine the binding energies deriving from the interaction of alpha modules and neutron spin pairs and compare these with the binding energies of alpha modules with each other. The analyis is limited to those nuclides that contain nothing other than alpha modules and neutron spin pairs.

The relative values of these interactions depend upon the nucleonic charge of a neutron relative to that of a proton and therefore the numerical values may be used to estimate that relative value.

What is conventionally called the nuclear strong force is here called the nucleonic force. The reason for this change in terminology is that this force is not so strong in comparison with another type of force involved in the structure of nuclei. That other type of force is the force involved in the formation of spin pairs. That force, although strong, is exclusive whereas the nucleonic force is not exclusive and for nuclides containing a larger number of nucleons the larger number of interactions of smaller interactions outweighs the two interactions of spin pairing for a nucleon.

Each nucleon has a charge with respect to the nucleonic force. The force between two nucleons or aggregates of nucleons is of the form

#### F = Hq1q2f(s)/s²

where H is a constant, q1 and q2 are the nucleonic charges, s is the distance between the particles and f(s) is a declining function of distance. Since f(s) is a declining function this means the nucleonic force drops off with distance faster than inverse distance-squared.

For now the repulsion between two protons is ignored.

## The Interactions

The loss in potential energy involved in the formation of the system is then proportional q1q2U(s), where U(s) is an integral of f(z)/z² from ∞ down to s.

Let the nucleonic charge of the proton be taken to be 1 and denote the nucleonic charge of the neutron as q, where q can be a negative or positive number. The binding energy of a nucleus is closely related to the loss of potential energy involved. The three types of interactions of individual nucleons through the nucleonic force are then

#### neutron-neutron: Inn = Hq²U(snn) neutron-proton: Inp = HqU(snp) proton-proton: Ipp = HU(s)

The nucleonic charge of a neutron spin pair is 2q and that of an alpha module is equal to 2(1+q). The three types of interactions of alpha modules and neutron spin pair are then:

#### neutron pair-neutron pair: Inn = H(4q²)U(snn) neutron pair-alpha module: Inna = H(4q(1+q))U(snna) alpha module-alpha module: Iaa = H(4(1+q)²))U(s)

Thus, among other relationships, there is

#### Inna/Inn = (q/(1+q))[U(snna)/U(snn)]

If snn is equal to snna then the ratio R=Inn/Ina gives an estimate of q/(1+q).

Since the ratio R of the interaction of two neutron spin pairs to the interaction of an alpha module with a neutron spin pair is then given by

## The Computational Results

The incremental binding energies of neutron spin pairs.(first differences) were first computed The second differences in the binding energies of neutron spin pairs and the cross differences of the binding energies of neutron spin pairs and alpha modules were computed and used as estimates of Inn and Inna, respectively. Their ratios were computed and taken as estimates of q/(1+q). From these values of q were computed. There were 504 separate values for q computed. The average of these 504 values is −0.971,

The cases in which there are changes in the shell were purged from the results. This elimated the estimates that were beyond ±2. One graph below shows the averages over the number of neutron spin pairs for the different numbers of alpha modules. This gives some notion of the variability of the estimates. The other graph shows the averages over the number of alpha modules for the different numbers of neutron spin pairs.

Taking into account the electrostatic repulsion between two protons would give an estimate of q which is more negative than the value of −0.971,