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Nuclear Rotations of
Alpha Module Rings

A neutron can form a spin pair with another neutron and with a proton. The same applies for a proton. This means that chains of nucleons can be formed involving a neutron pair being linked to a proton pair which in turn is linked to another neutron pair. Thus a chain contain sections of the form -n-p-p-n- or equivalently -p-n-n-p-. Such an arrangement is depicted below with the red dots representing protons and the black ones neutrons. The lines between the dots represent spin pair bonds.

This is not an exact description of the spatial arrangement of the nucleons in such a chain. The depiction of an alpha particle in style of the above would be the figure shown on the left below, whereas a more proper representation would be the tetrahedral arrangement shown on the right.

Here is a better visual depiction of an alpha particle.

The chains may be closed forming a ring.

Such chains are made up of modules involving two neutrons and two protons. In a module each neutron is involved in two spin pairs and three interactions and likewise for each proton just as in an alpha particle. Here is what is meant by the term alpha module.

Thus the potential and kinetic energies and binding energy is the same as in an alpha particle. An alpha particle is, in effect, a chain of length one alpha module.

Nuclear Rotation

Aage Bohr and Ben Mottleson found that nuclear rotations obey the I(I+1) rule; i.e., have energy levels EI such that

EI = [h²/(2J)]I(I+1)

where I is an integer, J is the moment of interia of the rotating nuclear shell and h is Planck's constant divided by 2π. Note what this implies about the quantification of angular momentum. If ω is the rate of rotation then

EI = ½Jω² = [h²/(2J)]I(I+1)
and hence
ω² = [h²/J²]I(I+1)
and therefore
ω = {h/J][I(I+1)]½

Thus angular momentum L is given by

L = Jω = h[I(I+1)]½

This is in contrast to the Old Quantum Physics of Niels Bohr in which L would be hI.

Note that

ω = h[I(I+1)]½/J

and thus the smaller the moment of inertia the faster a structure rotates. However the angular momentum is always h²[I(I+1)]]½. Thus there is an equipartition of angular momenta among the various modes of rotation.

The number of complete rotations per second, ν, is given by

ν = ω/(2π)

Modes of Rotation
of an Alpha Module Ring

The Number of Alpha
Modules in a Ring

If the nuclear magic numbers are (2, 6, 14, 28, 50, 82, 126, 184) then the shell occupancies in terms of numbers of nucleons are (2, 4, 8, 14, 22, 32, 44, 62). In terms of nuceon pairs the occupancies are (1, 2, 4, 7, 11, 16, 22, 31) and also for the number of alpha modules.

The Ring Radii

The radius of a nucleus in fermi is given by the empirical formula

R' = 1.2×A

where A is the number of nucleons in the nucleus, the sum of the numbers of neutrons and protons. Only exactly filled shells are being considered so the numbers of neutrons and protons are equal. The values that A can take on are (4, 12, 28, 56, 100, 164, 252).

An alpha module consists of two protons and two neutrons as -p-n-n-p- or, equivalently, -n-p-p-n-. The minimal ring of alpha modules is just an alpha particle. A larger ring of alpha modules rotates like a vortex ring to keep the neutrons and protons separate. The width of the vortex ring is the diameter of an alpha particle (3.6 fermi). The radius of the ring is then the radius of the nucleus less half the diameter of an alpha particle.

Order of Magnitude Estimates of the Rates
of Rotation of Alpha Module Rings

Computation of Rotation Rates of Nuclear Alpha Module Rings
1 2 3 4 5 6 7
A: Number
of Nucleons
4 12 28 56 100 164 252
Radius of
Nucleus (fermi)
1.904 2.745 3.640 4.585 5.561 6.557 7.566
R: Ring Radius
0.104 0.945 1.840 2.7849 3.761 4.7572 5.766
R² (fermi)² 0.011 0.893 3.385 7.756 14.1489 22.632 33.243
N: Number
of Alpha
1 3 7 14 25 41 63
Jr: Moment
of Inertia
×10-57 kg m²
0.0719 17.803 157.449 721.462 2350.192 6165.602 13915.970
ωr: Rotation
Rate (rad/sec)
1.304E+20 5.264E+17 5.952E+16 1.230E+16 3.987E+15 1.51983E+15 6.734E+14
ωf: Rotation
Rate (rad/sec)
2.075E+19 8.378E+16 9.47E+15 2.067E+15 6.346E+14 2.419E+14 1.072E+14
νr (times/sec) 2.075E+19 8.378E+16 9.47218E+15 2.067E+15 6.346E+14 2.419E+14 1.072E+14
νf (times/sec) 3.303E+18 1.333E+16 1.508E+15 3.290E+14 1.010E+14 3.850E+13 1.706E+13

The notation E+n means 10n.

Thus the order of magnitude of the rotations of an alpha module ring is upwards of 17 billion times per second. Its dynamic appearance would be that of a spherical shell. .

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