San José State University
Thayer Watkins
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The Functional Form and Estimates for the
Parameter Values of the Nuclear Strong Force


The so-called nuclear strong force is alleged to be what holds the nucleons in a nucleus together. in reality it is but one of four forces maintaining a nucleus' structure.

  1. Forces associated with the formation of spin pairs of the three types, neutron-neutron, proton-proton and neutron-proton.

    These are effectively forces of attraction. The binding energy associated with the formation of a spin pair consists of that due to formation itself and the change in potential energy due to any rearrangement or adjustment of the structure of the rest of the nucleus.

    The forces associated with spin pair formation are exclusive and this precludes them from being identified as the nuclear strong force. Spin pairing is exclusive in the sense that one nucleon can pair with one nucleon of the same type and one nucleon of the opposite type and no more.

  2. A force usually called the nuclear strong force which is distance-dependent and drops off faster than inverse distance-squared.

    This name is inappropriate because it is not all that strong at relevant distances compared with the forces involved in spin pair formation. A more appropriate name would be nucleonic force, the force between nucleons.

    Under this force like nucleons are repelled from each other and unlike ones attracted. However, when nucleons are smeared over concentric spherical shells, a repelling force between them that drops off faster than inverse distance squared promotes structural stability.

  3. The electrostatic (Coulomb) repulsion between protons, which is inversely proportional to distance squared.

    This force only affects interactions between protons. Neutrons have no net electrostatic charge but have a radial distribution of electrostatic involving an internal positive charge and a negative outer charge.

  4. In principle gravity is also involved but the magnitude of the gravity forces is so small in comparison to the other forces that gravity is ignored.

The Functional Form of the Nuclear Strong Force
Must Be Justified by a Physical Prosess

An inverse distance squared law for force is justified by the force-carrying particles being spread over a sphere of area 4πR² where R is the distance from the source of the field. There are not many functional forms that arise from physical processes.

If the force-carrying particles decay over time then their intensity at a distance R from the source is exp(−R/R0), where the parameter R0 depends upon the rate of decay of the force-carrying particles and their velocity. This means that the force between two particles of nucleonic charges q1 and q2 separated by a distance R is

F = Hq1q2exp(−R/R0)/r²

where H and R0 are parameters.

A negative value of a force represents an attraction, something that reduces the separation distance. A positive value is associated with increasing the separation distance; i.e., a repulsion. The sign of the force is determined by the sign of the product of the charges. Thus charges of the same sign repel each other and charges of opposite sign attract.

The dependence of force on the product of the charges stems from each bit of one charge interacting with all of the bits of the other charge.

The potential energy for a separation distance of S is

V = Hq1q2S(exp(−R/R0)/R²)dR

If R/R0 is denoted as z, so R=R0z and dR=R0dz then the above formula is

V = [Hq1q2/R0]∫Z(exp(−z)/z²)dz

From Yukawa we have an estimate of R0 as 1.522 fermi (1.522×10-15 m).

The separation distance between the centers of the neutron and proton in a deuteron is 2.29 fermi. Thus Z is equal to 1.5043. The value of ∫1.5043[exp(-z)/z²]dz is approximately 0.1.

Binding energy is based upon the mass deficits of nuclides, but it behaves like the loss of potential energy, For the value of V we will take the estimated interaction binding energy of a neutron and proton in the same shell. That is 0.621286749 MeV which is 9.954×10-14 joules. If the nucleonic charge of a proton is taken to be +1 then the nucleonic charge of a neutron is −2/3.

Thus we have

9.954×10-14 = H(−2/3)(0.1)/1.522×10-15
and hence
H = 2.2725×10-27 joule-meters.

A joule-meter is the same as kg-m³/sec².


Almost nothing is known about the so-called nuclear strong force. What is thought to be known is mostly incorrect. Here is a specification of it

F = −Hq1q2exp(−R/R0)/r²

where H = 2.2725×10-27 kg-m³/sec². and R0 = 1.522×10-15 m.

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