San José State University

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Thayer Watkins
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The Dynamics of a Deuteronic System
with an Exponentially Weighted Central Force

Particles with Empty Centers

In atomic and subatomic systems the lowest energy state is usually one of zero angular momentum. This means from a classical physics perspective that a particle appears to pass through another particle. This is not impossible even on a macroscopic level. All that is necessary is for at least one of the particles to have an empty center of mass, such as when there are subparticles distributed about that center of mass.

Below is an illustration of one such possible arrangement.

Although the other particle would not have to have a smaller scale for it to pass through the first particle such a case is easy to visualize, as for example below.

The reason this case was illustrated with three subparticles is of course because the nucleons are believed to be composed of three subparticles called quarks.

The square of the separation distance of the subparticles of one particle to the center of mass of the other particle is then (r²+b²).

The radial component of the force is the force multiplied by the cosine of the angle between the vector to the subparticles and the radial vector to the other particle's center of mass. This cosine is equal to r/(r²+b²)½.

The Functional Form of the Nuclear Strong Force

A force which is carried by particles, such as gravitation or the electrostatic (coulomb) force, must have an inverse distance squared dependence. When the force-carrying particles decay then the force also has a negative exponential dependece to take into the survival of those particles. The nuclear force is carried by the π mesons and they decay over time and hence over distance. Thus the form of the nuclear strong force is

F = −H*exp(−s/s0)/s²

where H and s0 are constants.

The distance s for the empty-center particles is equal to (r²+b²)½. The radial component of the force is then given by

Fcos(θ) = −H*exp(−(r²+b²)½/s0)r/(r²+b²)3/2

The Dynamics of the System

The center of mass of the two particles remains fixed so this can be used as the origin of the coordinate system. Consider first the simple case of two particles of equal mass m. Let ρ be distance from the center of mass of the system to a particle. The dynamics of the particle are given by

m(d²ρ/dt²) = Fcos(θ)

Since ρ=r/2 this equation can be expressed as

½m(d²ρ/dt²) = −H*exp(−(r²+b²)½/s0)r/(r²+b²)3/2

Thus the acceleration of the separation distance r is given by

a(t) = d²r/dt² = −(2H/m)exp(−(r²+b²)½/s0)r/(r²+b²)3/2

The time path for the separation distance in such a model is given below along with the velocity and acceleration for the case (2H/m)=1, b=1 and s0=100.

Quantization of Radial Momentum

According to the Wilson-Sommerfeld condition the radial momentum pr=m(dr/dt) for the system must be such that

prdr = nh

where n is an integer and h is Planck's constant.

In the model reported on above the maximum separation distance determines the total energy of the system. The time paths for a variety of maximum separation distances were computed and the values of prdr determined. The graph of the result is shown below.

The relationship is essentially linear. For a given value of the principal quantum number n for the system a value of maximum separation distance can be determined that generates a radial momentum equal to nh. That value of maximum separation distance then determines the total energy of the system. Thus the energy would be quantized as a function of n and likewise the maximum separation distance. Since the maximum separation distance is measurable in some fashion a value can be determined for the force constant H. The scale factor s0 is usually presumed to be 1.522 fermi (10-15 m) based upon the Yukawa relation and the mass of the force-carrying particles, the π mesons.

(To be continued.)


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