It is found that nuclei with even numbers of protons and neutrons are more stable than those with odd numbers. In particular, there are "magic numbers" of neutrons and protons which seem to be particularly favored in terms of nuclear stability:2,8,20,28,50,82,126

Magic Numbers

Nuclei which have both neutron number and proton number equal to one of the magic numbers can be called "doubly magic", and are found to be particularly stable.

Calcium provides a good example of the exceptional stability of "doubly magic" nuclei since it has two of them. The existence of several stable isotopes of calcium may have to to with the fact that Z=20, a magic number. The two highlighted isotopes have neutron numbers 20 and 28, also magic numbers. Compared to the binding energy calculated from the Weizsaecker formula, they both have more than the expected binding energy.

The existence of these magic numbers suggests closed shell configurations, like the shells in atomic structure. They represent one line of reasoning which led to the development of a shell model of the nucleus. Other forms of evidence suggesting shell structure include the following.

Enhanced abundance of those elements for which Z or N is a magic number.

The stable elements at the end of the naturally occuring radioactive series all have a "magic number" of neutrons or protons.

The neutron absorption cross-sections for isotopes where N = magic number are much lower than surrounding isotopes.

The binding energy for the last neutron is a maximum for a magic neutron number and drops sharply for the next neutron added.

Electric quadrupole moments are near zero for magic number nuclei.

The excitation energy from the ground nuclear state to the first excited state is greater for closed shells.

## Tuesday, 8 December 2009

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