Copyright © 2009 by Phillip E. Fraley

Six models are basic: proton (already covered), neutron, deuteron, triton, helion and alpha. Larger nuclei contain mixtures of these.

Equation (3) will be the nuclear radius:

r = r_{0} A^{1/3} = 0.853 A^{1/3} (3)

This is a lower bound on nuclear radius calibrations, just as the proton radius was the lowest value that could reasonably fit the SOM idea.

Figure 4

Alpha ModelThe radius of the black circle is the average proton radius. Axis 3 is perpendicular as indicated by the small red bullseye.

In SOM, pairs of neutrons are replaced by a p p- pair. Therefore, the helium-4 nucleus, alpha, becomes 3 protons and one antiproton as shown in Fig. (4).

The radius (and p p- distance) is 0.853 4^{1/3} = 1.354 fm It is represented
by the vertical light blue line in the figure. The p p distance is √3 1.354 fm = 2.345 fm
and one (of three) is the darker blue horizontal line in the figure. The black circles were
scaled to give a 0.546 fm average proton radius.

p p-: -65.5/1.354

p p- overlap (Fig. (3)): +10.0

p p: +65.5/2.345

p p overlap: -2.4

There are 3 of each of these for a total of -38.38 MeV. A positive energy of 12.78 MeV is
needed to give the binding energy of -25.60 MeV. The BE in SOM is 2.6 MeV less than normally
calculated because 2 neutrons are replaced by p p-. The energies for
axes 1, 2 and 3 are 0.75 M v^{2}, M v^{2} and 1.5 M v^{2}. v/c for
axes 1, 2 and 3 are 0.1167c, 0.1348c and 0.09529c (equating to 12.78 MeV and dividing both
sides of equation by M c^{2} = 938.259 MeV). M is a proton mass.

The angular momentum for axes 1, 2 and 3 are: 1.5 M r v = 1.303ћ, √3 M r v = 1.303ћ
and 3 M r v = 1.843ћ. The average is 1.483ћ. r is 1.354 fm and M is a proton mass.
Multiply and divide by M r_{p} c = ћ. Example axis-1: ћ 1.5 1.354 0.1167/0.210
where r_{p} is 0.210 fm.

What fraction h' of ћ pointing and turning with the particles gives the same energy?
E = h' ћ v/r

Replace ћ by M r_{p} c and divide both side of equation
by M c^{2}. The totals of the two or three proton masses for axes 1,2 and 3
are 0.6515ћ, 0.6515ћ and 0.9215ћ. The average is 0.7415ћ.
Example axis-1: h'_{1} = 2/3 (12.78 1.354/(938.259 0.1348)) = 2/3 x 0.6515ћ
while the other 2 masses each give 1/6 x 0.6515ћ

Scale, 65.5 MeV, and r_{0} = 0.853 multiplying A^{1/3} were the parameters
selected. Secondarily, the selected radius of the proton, 0.546 fm, shifts the scale of the
overlap correction. The whole result is non-linear and I did not study it as function of
the proton radius.

Because of symmetry, the alpha is a keystone of SOM. The simplest model is given: overlapping bonds plus rotation of a semi-rigid structure. An angular momentum close to √2 ћ results (or, from an SOM perspective, √2/2 ћ total component in the direction of motion). The QM angular momentum, spin and orbital, are zero. One has to imagine the orbits as continuously and randomly changing. The cause is the randomness of the underlying partics and particles. Think of the outer layers of a ball of yarn - different colors for each particle. An analogy to the model here is the n=1 hydrogen electron orbit where the angular momentum is ћ (or a component of 1/2ћ in the direction of the velocity) even though the QM value is zero. The yarn’s outer layer can be thought of as looser in the hydrogen case since the radii are not precise by comparison to the alpha model. The average in each case is zero because the (instantaneous) vectors point in random directions.

Figure 4

Triton ModelThe radius of the black circle is the average proton radius. Axis 3 is perpendicular as indicated by the small red bullseye.

For the triton, 2 neutrons are replaced by p p- as shown in the figure.

The p p distance is allowed to vary, but the average radius is set by Eqn. (3) to about 1.230 fm.
The angular momentum is constrained to be about √2ћ. Since the formulas and methods
are similar to the alpha case, only the results are given.

p p: 65.5/1.7

p p overlap: -5.216 (Fig 3)

p p-: -65.5/2.3 (2 times)

p p- overlap:1.951 (2 times)

PE = -19.741 MeV

BE = -5.90 MeV (again, p p-, not 2 neutrons: so 2.6 MeV less negative)

KE: 13.842 MeV

Axis-3 energy is different: KE = M v_{p+}^{2} + 1.6989/2 M v_{p+}^{2}

(1.6989 is the square of the radius ratios: 1.4606/1.1206)

p p distance: 1.70 fm (horiz. darker blue line in Fig. 4)

p p- distance: 2.30 fm

r_{p+}: 1.1206 fm

r_{p-}: 1.4606 fm (vertical light blue line in Fig. 4)

r_{ave.}: 1.234 fm

angular momentum, axes 1, 2 and 3: 1.463ћ, 0.983ћ and 1.763ћ with average 1.403ћ

velocities, axes 1, 2 and 3: 0.1402c, 0.1215c and 0.08931c (p+)

(axis-3: multiply p+ velocity by the radius ratio above to get the p- velocity)

Figure 5

Neutron ModelThe radius of the black circle is the average proton radius.

The soe+ is modeled as a partic with random paths on a sphere (“outer layer of a ball of yarn”)
centered on the p- and whose radius is somewhere in the fringe of the Gaussian antiproton.
If n r_{p} is the radius, then c/n is the approximate velocity for a magnetic moment
of one nuclear magneton. r_{p} is 0.210 fm. The kinectic energy is about 938/n^{2}
MeV or 938 0.88/(r/r_{p})^{2} MeV when the correct magnetic moment value,
0.88 μ_{N} is used.. As for the p p- overlap case, the average 1/r component of
Eqn. (1) can be calculated from random geometric positions of the soe+ partic and the p-
antipartic. The average Δv factor is assumed to be a new scale factor Scale'.
However, since only one component has magnitude c and the other is c/n, Scale' should be about Scale/2.
Since Scale is about about 1/2 the “rigidly-passing” value of 131 MeV or 65.5 MeV, Scale' is
about 32.7 MeV, a value best checked (for self-consistency) in the deuteron model to follow.
A range of Scale' vs. soe+ radius is calculated by equating the negative potential Scale'
times (1/r)_{ave.} to the kinectic energy. (a small offset of 1.293 MeV for the positive
binding energy is also used). The soe+ radius for Scale' = 32.7 MeV is 1.215 fm which
is 5.79 r_{p}, 1.78 σ_{r} and 2.23 times the average p- radius. (The
average p- radius is about 0.8 σ_{r}.) Figure 4 shows the two neutron radii
alongside a p- σ-unit scale.

Figure 6

Deuteron ModelThe radius of the black circle is the average proton radius.

The deuteron is the only nucleon whose measured value is much greater than Eqn. (3) predicts.
Calculating the angular momentum for 0.853 2^{1/3} and the scale and soe+ radius values
just above, yields 1.12ћ. Adjusting the radius to get 1.41ћ yields 1.75 fm for the
radius, closer to the measured value of 1.6 to 1.7 fm. At 2 r = 3.5 fm, 0.2776 is the fraction
multiplying -65.5 MeV for the p p- potential energy and 0.2820 is the fraction multiplying +32.7
MeV for the soe+ p potential energy. Since 1/(2r) is 0.2857, there is overlap for both cases
and the difference for the p p- overlap of (0.2857 - 0.2776) times 65.5 can be seen (by
extrapolation) in Fig. 3. The overlap calculation for soe+ p is very similar, but only the
result is given. BE = 2.22 MeV was subtracted from the PE and KE = M v^{2}. 2 M v r
is the angular momentum. Figure 6 gives the geometry.

Figure 7

Helion ModelThe radius of the black circle is the average proton radius.

Figure 7 shows the model. A complication arises because the p- mass is centered in the
middle of the p p interaction. It is not obvious how to model this. Using 0.853 3^{1/3}
for the radius and 2^{1/2}ћ angular momentum gives an estimate that 10 MeV of 23.5
MeV of the p p interaction was blocked. So the helion is not useful for verifying any of the
models and scaling factors.