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 = r₀A^1/3

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.

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.

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

(To be continued.)