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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 Alpha Modules

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/Iaa = (q/(1+q))[U(snna)/U(saa)]

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 an alpha module with a neutron spin pair to the interaction of two alpha modules is then given by

R = 4q(1+q)/[4(1+q)²] = q/(1+q)
therefore
q = R + Rq
and hence
q − Rq = R
and
q = R/(1−R)

The Computational Results

The incremental binding energies of alpha modules(first differences) were first computed The second differences in the binding energies of alpha modules and the cross differences of the binding energies of neutron spin pairs and alpha modules were computed and used as estimates of IaaT and Inna, respectively. Their ratios were computed and taken as estimates of q/(1+q). From these values of q were computed. There were 540 separate values for q computed. The average of these 540 values is −0.567,

The cases in which there are changes in the shell were not purged from the results but whose effect should be minor in the average over 540 cases. One graph below shows the averages over the number of alpha modules for the different numbers of neutron spin pairs. This gives some notion of the variability of the estimates. The other graph shows the averages over the number of neutron spin pairs for the different numbers of alpha modules.

The Role of the Electrostatic Repulsion Between Two Protons

The force between two protons is

F = Hf(s)/s² + Ke²/s²

where K is the force constant for the electrostatic force and e is the electrostatic charge of the proton.

The force between two protons can be represented as

F = H[1 + (Ke²)/f(s)]f(s)/s²

Thus the charge involved in the interaction between to protons is effectively (1+d)½. This means the interaction between two alpha modules is 2[(1+d)½+q]. Then the ratio R of the interaction of an alpha module with a neutron spin pair to the interaction of two alpha modules is given by

R = q/( (1+d)½+q)
and therefore
q = R (1+d)½ + Rq
and hence
q − Rq = R (1+d)½
and
q = R(1+d)½)/(1−R)
or, equivalently
q =[R/(1−R)] (1+d)½

From a separate study the value of (1+d)½ is approximately 1.2. Therefore the estimate of q corrected for the electrostatic repulsion between protons is

q = (−0.567)(1.2) = −0.680

This is quite consistent with previous estimates of q being −2/3.


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