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A Notable Characteristic of Composite Stuctures
Consisting of Particle Pairs of Different Types

This material has to do with the potential energy of a structure consisting of four particles, two of one type and two of another type. An alpha particle is of this nature, but a hydrogen molecule might also be considered to fit this description as well.

The alpha particle will be considered as a neutron-neutron spin pair and a proton-proton spin pair revolving about their center of mass.

The force between two particles, 1 and 2, is assumed to be given by this general formula

F = q1q2H·f(z/z0)/z²

where q1 and q2 are the charges of particle 1 and particle 2, respectively. Their values may be positive or negative. The symbol H denotes a positive constant, z is the separation distance of the centers of the two particles and z0 is a positive constant.

A negative value for the force between two particles indicates that there is an attraction between them and a positive values indicates a repulsion.

The nature of the force between two particles is taken care of by the product of the charges. There is a repulsion between like particles and an attraction between unlike particles if the charges of the two types of particles are of opposite in sign. For convenience in explanation one type of particle will be called a neutron and the other type proton. If the charge of the proton is taken to be +1 then the charge of the neutron can be denoted as −q. H denotes a positive constant, z is the separation distance of the centers of the two particles and z0 is a positive constant. The dependence on the inverse distance squared indicates there are particles like photons which carry the force. Since the function f( ) is unspecified the force formula is still a completely general central force.

For convenience the four particles are labeled 1, 2, 3, 4 with particles 1 and 2 being the neutrons and 3 and 4 the protons. Then si,j is the separation distance between the center of particle i and the center of particle j.

Potential Energy

The potential energy due to the force between two particle is given by

V(s) = ∫sF(z)dz
which reduces to
V(s) = qiqjH∫s(f(z/z0)/z²)dz
which can be represented as
V(s) = qiqjW(s)

where W(s)=H·∫s(f(z/z0)/z²)dz.

The potential energies due to the six interactions are

V1,2 = q²W(s1,2)
V1,3 = −q·W(s1,3)
V1,4 = −qW(s1,4)
V2,3 = −q·W(s2,3)
V2,4 = −qW(s2,4)
V3,4 = 1·W(s3,4)

Now suppose that all of the separation distances are equal; i.e., si,j=s. This would be the case if the four particle centers are located at the vertices of a regular tetrahedron. Under this condition the total potential energy due to the strong force, VTSF, is equal to

VTSF = (1−4q+q²)W(s)

The shape of the factor (1−4q+q²) is shown below.

Thus for q<(2−√3)=0.27 and q>((2+√3)=3.73=1/0.27 the potential energy is positive but between those two extremes it is negative. In particular for q=2/3 the total is −(11/9)W(s). For any q outside of the range (0.27, 1/0.27) the potential energy due to the strong force would be positive. This would be an indication of instability in as much as the disintergration of the arrangement would release energy.

It is to be emphasized that this condition is valid only if the separation distances are equal. For any value of q if the separation distances between the like particles is sufficiently greater than the separation distances between unlike particles the potential energy of the arrangement would be negative.

If the energies involved in the formation of spin pairs are taken into account the total potential energy would be mmore negative.

(To be continued.)


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