﻿ Nuclear Rotations of Alpha Module Rings
San José State University

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Thayer Watkins
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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

## Modes of Rotation of an Alpha Module Ring

• The neutrons and protons in the modules are attracted to each other and there must be rotation to keep them separate. This is like the rotation involved in a vortex ring.
• The ring can rotate around the axis that runs through the center of the ring and perpendicular to the plane of the ring. The moment of inertia of the ring for this mode of rotation is Jr=NMαR², where N is the number of alpha modules in the ring, Mα is the mass of an alpha module and R is the radius of the midline of the ring.
• The ring can rotate about two diameters in its plane which are perpendicular to each other. This is rotation in the nature of a flipped coin. The moment of inertia of a ring of mass NMα and radius R is

#### Jf = NMαR²/(2π) and therefore Jf = Jr/(2π)

These alpha module rings rotate in four modes. They rotate as a vortex ring to keep the neutrons and protons (which are attracted to each other) separate. The vortex ring rotates like a wheel about an axis through its center and perpendicular to its plane. The vortex ring also rotates like a flipped coin about two different diameters perpendicular to each other.

The above animation shows the different modes of rotation occurring sequentially but physically they occur simultaneously. (The pattern on the torus ring is just to allow the wheel-like rotation to be observed.)

Aage Bohr and Dan Mottleson found that the angular momentum of a nucleus (moment of inertia times the rate of rotation) is quantized to h(I(I+1))½, where h is Planck's constant divided by 2π and I is a positive integer. Using this result the rates of rotation are found to be many billions of times per second. Because of the complexity of the four modes of rotation each nucleon is effectively smeared throughout a spherical shell. So, although the static structure of a nuclear shell is that of a ring, its dynamic structure is that of a spherical shell.

At rates of rotation of billions of times per second all that can ever be observed concerning the structure of nuclei is their dynamic appearances. This accounts for all the empirical evidence concerning the shape of nuclei being spherical or near-spherical. The Alpha Module Model thus gives an explanation for the observed spherical shapes of nuclei.

## 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×A⅓

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 moodules 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
Ring
Number
1 2 3 4 5 6 7
A: Number
of Nucleons
4 12 28 56 100 164 252
Nucleus (fermi)
1.904 2.745 3.640 4.585 5.561 6.557 7.566
(fermi)
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
Modules
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