San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

Confirmation of 6 and 14
as proper nuclear magic numbers

The neutrons and protons in a nucleus, whenever possible, form spin pairs; neutron-neutron, proton-proton and neutron-proton. But these spin pair formations are exclusive; one neutron can form a spin with one other neutron and with one proton and likewise for a proton. This leads to chains of the form -n-p-p-n-, or equivalently -p-n-n-p-. These may be appropriately called alpha modules because an alpha particle is the smallest such unit. These alpha module chains may link together to form rings.

The nuclides which are made up entirely of alpha modules will be calleld alpha nuclides. The binding energy of an alpha nuclide less that of a nuclide with one less neutron pair is the incremental binding energy of a neutron pair, IBEnn. The incremental binding energies of a proton pair and a neutron-proton pair are defined analogously.

Here is a graph of the data.

The abbreviation MeV stands for the energy unit of a million electron volts, the energy an electron gains from dropping through a potential difference of one million volts.

When a shell is filled the incremental binding energy for the next pair should drop significantly. For all three types of pairs the drops occur at 3, 7 and 14 alpha modules. But 3 alpha modules correspond to 6 neutrons and 6 protons. Likewise 7 alpha modules correspond to 14 neutrons and 14 protons. And 14 alpha modules correspond to 28 neutrons and 28 protons.

The conventional magic numbers for filled nucleon shells, as found by Maria Goeppert Mayer and Hans Jensen using the the numbers of stable isotopes and isotones are {2, 8, 20, 28, 50, 82, 126}. Incremental binding energies confirm {28, 50, 82, 126}. but, as shown above, reveal 6 and 14 as filled shell magic numbers. There is a degree of magicalness to 8 and 20 but as the values for fillled subshells rather than shells.

The fillled shell magicalness of 2, corresponding to 1 alpha module, is also confirmed by the data for the incremental binding energy of a neutron-proton pair. The incremental binding energy of a neutron-proton pair for 1 alpha module is 26.071101 MeV but drops to 24.50491 MeV for 2 alpha modules.

It has been found elsewhere that if the strong force charge of the proton is take to be +1 then the strong force charge of the neutron is −2/3. This means the net strong force charge of an alpha module is +2/3. Thus a neutron pair with a strong force charge of −4/3 is attracted to an alpha module. On the other hand a proton pair with a strong force charge of +2 is strongly repelled by an alpha module. Likewise a neutron-proton pair with a net strong force charge of +1/3 should be repelled by an alpha module.

Within a shell the IBEnn rises because, on balance neutron pairs are attracted to alpha modules. But within a shell the IBEpp declines reflecting the fact that proton pairs are repelled by an alpha module. The IBEnp should decline within a shell but at a rate of only 1/6 as much as does IBEpp. The graph shows IBEnp to be flat for the shells above 7 (14 neutorns and 14 protons) and rising in the shell that goes fron 4 to 7 alpha modu les.

The significance of 6 and 14 being the filled shell magic numbers rather than the conventional 8 and 20 is that a simple algorithm will generate the sequence {2, 6, 14, 28, 50, 82, 126}

Take the number sequence {0, 1, 2, 3, 4, 5, 6} and generate the cumulative sums; i.e., {0, 1, 3, 6, 10, 15, 21}. Now add 1 to each of these numbers to get {1, 2, 4, 7, 11, 16, 22}. Now take the cumulative sums of that sequence to get {1, 3, 7, 14, 25, 41, 63}. These are doubled because there are two spin orientations for each nucleon. The result is {2, 6, 14, 28, 50, 82, 126} which is just the magic numbers with 8 and 20 left out. If the sequence is extended the next magic number after 126 is 184.

HOME PAGE OF applet-magic
HOME PAGE OF Thayer Watkins