San José State University
Thayer Watkins
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The Spatial Structure of Nuclei

Without a doubt there is some shell structure to nuclei. Remarkably protons and neutrons apparently have the same shell structure. The data on nuclear binding energy and also on nuclear magnetic moments indicate that protons and neutrons (nucleons) form spin pairs; i.e., proton-proton, proton-neutron, and neutron-neutron. These nucleon spin pairs are overwhelmingly responsible for holding the nucleons together in a nucleus. But this spin pairing is exclusive in the sense that a nucleon can pair with one nucleon of the same type and one of the opposite type and no more. This results in chains made up of modules of the form -n-p-p-n- or equivalently -p-n-n-p-. The ends of these chains can link together forming a ring. The smallest such ring is the alpha particle (He4) so the modules can appropriately be called alpha modules and the rings alpha module rings.

An alpha module ring can rotate in four different modes: 1. As a vortex ring 2. As a wheel 3. and 4. Like a flipped coin. Here is an animation that shows all of the modes except as a vortex ring.

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.)

The overall structure of a nucleus of filled shells is then of the form

That is to say, concentric spherical shells.

For a nucleus consisting of filled shells plus extra neutrons (called halo neutrons) the dynamic appearance is a spherical core of filled shells with pairs of halo neutrons in orbits about the core.

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 nuclear rates of rotation are found to be many, 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. .

In addition to the spin pairing in nuclei there is a nonexclusive force due to the interaction of nucleons. However the analysis of binding energies indicates this interaction force is such that like nucleons repel each other and unlike ones attrack. This can be explained by protons and neutrons having a nucleonic charge of different signs. Statistical analysis indicates that if the nucleonic charge of a proton is taken to be 1 the nucleonic charge of a neutron is −2/3.

The conventional theory of all nucleons attracting each other equally, the so-called nuclear strong force, is simply wrong empirically. When it is used in an attempt to statistically explain binding energies all but one of the regression coefficients are of the wrong sign and relative magnitude. The nuclear strong force theory conflates the disparate phenomena of spin pairing and nucleonic interaction.

An examination of nuclear densities indicates that there is close packing of the nucleons. The shell capacities and the close packing of nucleons indicates that the apparent spatial arrangement of a shell is three dimensional, like a sphere or cylinder, rather than two dimensional like a ring.

At rates of rotation of many 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.

(To be continued.)

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