6. Contraction energy

In the case of the hydrogen atom, we have seen that a reduction in the radius of the unit increased the electrical energy. Whether or not this is true in general must be considered. If we write equation (1.4) with the assumption that Co is an absolute constant, a reduction in the radius of circulation requires a compensating frequency increase. We write

(6.1) 2pr/N Nf₀ = C₀

Then for a group of N particles, each of which is reduced in radius by a factor of N , the focal energy is given by

(6.2) E = Nh (N f₀) = N²h f_o

The decay of neutron into a proton requires the emission of an electron. The present philosophy that the electron was created at the time of emission is not in accord with the facts. However, the physical dimensions of the electron in the neutron is of interest.

The matter wave energy radius of the electron at rest is given by (1.7) in the form

(6.3) r =h/2pM₀c₀ = 3.84 x 10^-11

in the unite of centimeters. The spin radius is 1.92 x 10^-11 cm as a result. This latter value must be considered the distance from the center to the axis of the torus ring. However, there is a third radius given by (1.8) in the form

(6.4) r = e²/M₀ = 2.81 x 10^-13

This can be considered the radius of the inner vortex of the toroid, but even in this case, we find the radius of the electron to be about twice the size of the neutron radius.

In analogy with equations (4.1) ant (4.2), we write the total electrical energy of the group in the form

(6.5) N²eC₀²/r [1-v²/C₀²] + N²eV²/r = N²eC₀²/r

Then in analogy with (6.1), we write

(6.6) N²eC₀²/r₀ = N²eC₀²/r₀/N

In this case, we have

(6.7) r = r₀/N = 2.81 x 10^-13/1860 = 1.51 x 10^-16

The fully contracted form in analogy with (2.4) is then

(6.8)

where N is the number of particles of primary matter forming the group and V is the velocity of translation of the group as a unit.

It is necessary to conclude that the energy represented by (6.6) is real.

The energy of N separate particles widely dispersed is given by

(6.9) E_N = NM₀c₀² = N e²/r₀ c₀²

The energy of the same N particles in a group is

(6.10) N² e²/r₀ c₀² = N² M_o c₀²

where the energy is increased by a factor of N with respect to the dispersed state. However, no corresponding mass increase is found by measurement. We conclude that energy can increase without a corresponding mass increase. In the absence of any source to provide the added energy, we must assume that it was created by mutual proximity of particles of primary matter.

As a final note, we observe that the radius of the electron as it exists within the neutron structure is given by (6.7). Neglect of the velocity contraction factor used in (6.8) does not change the fact that the circumference of the neutron ring is more than adequate to contain all of the primary particles required by the calculation. We conclude that in the case of neutron decay, the emitted electron existed within the structure prior to the time of emission.