Enhanced Abundance of Magic Number Nuclei

Part of the motivation for the shell model of nuclear structure is the existance of "magic numbers" of neutrons and protons at which the nuclei have exceptional stability, implying some kind of "closed shell". One indication of this stability is the enhanced abundance of isotopes which have a magic number of neutrons or protons.

The illustration examines the abundance of elements around iron and above. Our model of heavy element formation involves extraordinary processes in supernovae. Since these nuclei are born in the maelstrom of neutrons and neutrinos in the violent outer reaches of the supernovae, one would expect a statistical advantage for those isotopes which are most stable and therefore have the smallest cross-section for the kind of scattering which would disrupt them.

While the peaks in abundance for the magic number isotopes do not appear to be particularly prominent, keep in mind that the vertical scale is logaritmic.

The iron-56 is a particularly unique case, as shown by its extraordinary abundance. Iron-56 is an even-even nucleus and therefore expected to be particularly stable because of the Pauli contribution in the liquid drop model, but does not have magic numbers of either N or Z. It is exceeded in binding energy only by nickel-62 (the most stable nuclide) and iron-58. It is near the peak of the binding energy curve, and therefore can be considered to be one of the end points of both nuclear fusion and fission sequences, so perhaps that is the explanation for its extraordinary abundance.

"Magic Number" Nuclei at End of Radioactive Series

Part of the motivation for the shell model of nuclear structure is the existance of "magic numbers" of neutrons and protons at which the nuclei have exceptional stability, implying some kind of "closed shell". Further evidence of the uniqueness of these numbers is the fact that the end points of all four of the natural radioactive series are nuclei which have magic numbers of either N or Z.

The lead end products have 82 protons, a magic number, and the bismuth has 126 neutrons, also a magic number. The lead-208 is doubly magic with Z=82, N=126.

Neutron Absorption Cross-sections

Part of the motivation for the shell model of nuclear structure is the existance of "magic numbers" of neutrons and protons at which the nuclei have exceptional stability, implying some kind of "closed shell". Part of this evidence comes from absorption cross-sections for neutrons.

The stability of those nuclei with magic numbers of neutrons makes them less likely to be excited by neutron bombardment. The probability for absorption of an incident is expressed as an effective cross-section which is presented by the target nucleus to those incoming neutrons. The common unit for cross-sections is the barn, and the vertical axis on the illustration is labeled in millibarns.

Binding Energy for the Last Neutron as Evidence of Shell Structure

Part of the motivation for the shell model of nuclear structure is the existance of "magic numbers" of neutrons and protons at which the nuclei have exceptional stability, implying some kind of "closed shell". Part of this evidence comes from measuring the energy required to remove a neutron from the nucleus.

This dependence of the energy to remove the last neutron is strong evidence for a kind of shell structure. At the magic numbers, the shell is "closed" and it is hard to remove a neutron. Just above the closed shell, the added neutrons are less tightly bound, reminiscent of the alkali metals in the chemical shell structure. The zero in energy above is the expected binding energy from the Weizsaeker formula.