San José State University

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Thayer Watkins
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 A Rough Approximation of the SpatialStructure of the Nucleonic Shells of Nuclei

One of the unsolved puzzles of nuclear physics is the spatial arrangement of nucleons in the nucleus. It is known that the neutrons of nuclei are arranged in a shell structure and the capacities of these shells are known, except for some minor quibbles. Likewise protons are arranged in shells and the structure of those shells is the same as for neutrons.

Scattering experiments have given some notion of the size of nuclei. A simple formula that applies for the larger nuclei gives the radius r of a nucleus having A nucleons as

#### r = r0A1/3

where r0 is equal to about 1.25 fermi (10-15 meters).

If the value of A for a nucleus in which both the N-th neutron and proton shell are filled is substituted into the formular the result gives the approximate radius of the N-th shells.

## Nuclear Shells

As mentioned above protons and neutrons are arranged separately in shells. The numbers corresponding to the shells filled to full capacity are known as the nuclear magic numbers. Conventionally the magic numbers are {2, 8, 20, 28, 50, 82, 126}, but a case can be made for the magic numbers being instead {2, 6, 14, 28, 50, 82, 126} with 8 and 20 being in a different category of magic numbers. For more on this see Magic Numbers.

The structure of the nuclear shells, both for neutrons and protons, is given in the following table.

 ShellNumber 1 2 3 4 5 6 7 8 Capacity 2 4 8 14 22 32 44 58 Range 1 to 2 3 to 6 7 to 14 15 to 28 29 to 50 51 to 82 83 to 126 127 to 184

There is reason to expect the proton shells to have a larger radius than the corresponding neutron shell. This is because protons are subject to an electrostatic repulsion that neutrons are not subject. Thus the radius of a nucleus that has the first N of both the neutron and proton shells would be the radius of the N-th proton shell. The radius of the N-th neutron shell would be that of a nucleus in which the first N neutron shells are filled and the first (N-1) proton shells are filled. Using the above formula and the scheme just mentioned the radii of the two types of shells are as follows.

Radii of the Neutron Shells of Nuclei
Shell
Number
Total
Neutrons
Total
Protons
Total
Nucleons
(fermi)
Difference
(fermi)
26282.500
3146203.3930.893
42814424.3450.952
55028785.3410.996
682501326.3641.024
7126822087.4061.042

Radii of the Proton Shells of Nuclei
Shell
Number
Total
Protons
Total
Neutrons
Total
Nucleons
(fermi)
Difference
(fermi)
12241.984
266122.8620.877
31414283.7960.934
42828564.7820.987
550501005.8021.020
682821646.8421.040

The differences in the radii of the corresponding shells are of interest.

 ShellNumber Radius ofProton Shell(fermi) Radius ofNeutron Shell(fermi) Difference(fermi) 2 2.862 2.500 0.362 3 3.796 3.393 0.403 4 4.782 4.345 0.437 5 5.802 5.341 0.461 6 6.842 6.364 0.478

(To be continued.)