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The Dimensionality of Nucleon Shells

There is overwhelming evidence that the neutrons and protons of nuclei are separately organized in shells. The spatial nature of those shells has never been specified. It has been presumed that the shells are concentric, but they could be concentric rings, cylinders or spheres. The material below examines this question of dimensionality.

There are two notions of how the spatial arrangement of the shells. The scale of a shell, such as a radius, may be determined by quantum mechanical considerations, as in the orbits of electrons in atoms. Alternatively the spacing may be determined by the distance at which wave function overlap invokes the Pauli Exclusion Principle and keeps the structures separated. It is this latter possibility that is being investigated here. Let r be the separation distance between the substructures of a nucleus. The substructures of a shell could be individual nucleons, nucleon pairs, or perhaps alpha particles. The exact nature of the substructures does not have to be specified at this point.

Shells of Rings

The first ring could be a single substructure. The second ring would then have a circumference of 2πr. The number of substructures in the second ring would be [2πr/r]=[2π]=6, where [z] denotes the largest integer less than or equal to z. The third ring would have a circumference of 2π(2r)=4πr and hence the number in a third ring would be [4π]=12. A fourth ring would have 18 and a fifth 25. Basically there is a linear relationship between the shell order number and the occupancy of the shell.

Shells of Spheres

The first shell would be a single substructure. A second spherical shell would have a surface area of 4πr². Each substructure would occupy an area proportional to πr². Thus the occupancy would be proportional to [4πr²/(πr²)]=4. The third shell would have a radius of 2r and an area of 4π(2r)²=16πr². The occupancy of the third shell would be proportional to 16. The occupancy of the fourth shell would be proportional to 36. Thus the occupancies of spherical shells beyond the first would be proportional to {4, 16, 36, 64, …} or equivalently {1, 4, 9, 16, … }; i.e., the square of the shell number less one.

The Actual Occupancy Levels

The conventional magic numbers of nuclear structure are {2, 8, 20, 28, 50, 82, 126} based upon isotope stabilities. An examination of incremental binding energies reveals that 6 and 14 are also magic numbers. An algorithm based upon four quantum numbers generates the sequence {2, 6, 14, 28, 50, 82, 126} indicating that the numbers 8 and 20 are special cases different from the other magic numbers. The shell occupancy levels based upon the sequence {2, 6, 14, 28, 50, 82, 126} are {2, 4, 8, 14, 22, 32, 44}. The relationship of between the logarithm of occupancy and the logarithm of shell number should reveal the dimensionality of the shells. The graph of this relationship is shown below.

Except for the first shell the data falls nearly perfectly on a straight line. A regression leaving out the first shell gives the following equation.

ln(shell occupancy) = 0.00643 + 1.92456ln(shell number)

The coefficient of determination (R²) for this equation is 0.9984.

For a ring arrangement the coefficient would be 1.0 and for a spherical arrangement it would be 2.0. The value of 1.92456 is clearly closer to the spherical arrangement.

Let n denote the shell number. The occupancy for the spherical arrangement was shown to be proportional to (n-1)² rather than n². However the numbering of the shells was to a degree arbitrary. The first shell could have be designated the zeroeth shell. A replotting of the previous graph using the logarithm of (n-1) rather than n is shown below. The plotting has to start with n=2 rather than n=1.

Again the first data point does not fit into the pattern of the other data points. A regression of the last five data points gives

ln(shell occupancy) = 0.96975 + 1.55195ln(shell number - 1)

The coefficient of determination (R²) for this equation is 0.9975.

This result is squarely midway between the case of the ring arrangement and the spherical arrangement.

Cylindrical Arrangements

A cylindrical arrangement in which the height of the cylinder is proportional to its radius would result in occupancy being proportional to the square of the shell number less one, as in the case of a spherical arrangement of shells.

The Numbering of the Shells

Since the numbering of the shells is at least to a degree arbitrary some other numbering can be explored. Above the effect of using (n-1) instead of n was investigated. Below is the case of using (n+1) instead of n.

The regression equation for this data is

ln(shell occupancy) = −1.31318 +2.45422ln(shell number + 1)

The coefficient of determination (R²) for this equation is an impressive 0.99995.

The data seems to resist giving a near integral value for the slope of the relationship.

The shell numbering corresponds to a quantum number for the system. c coefficient is 2.19 and the R² is 0.99966, not quite as good as for n+1. coefficient is 2.71 and the R² is 0.99976, also not quite as good as for n+1.


From the above analysis the dimensionality of the nuclear shells seems to be clearly three dimensional, such as concentric spheres or cylinders, rather than two dimensional as in the case of concentric rings. If the scale of the shells is determined by some quantum mechanical consideration and the spacing between shells is not constant then the conclusion may be different.

(To be continued.)

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