San José State University

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Thayer Watkins
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 The Special Significance of the Neutron Number 42 in Nuclear Structure

An investigation of the prevalence of stable nuclides by Maria Goeppert-Mayer and Hans Jensen revealed that certain values for the numbers protons and neutrons were associated with relatively large numbers of such stable nuclides. These numbers were called magic numbers. Goeppert-Mayer and Jensen were awarded the Nobel prize in physics in 1963 (along with Eugene Wigner) for their efforts.

The magic numbers found by Goeppert-Mayer and Jensen were {2, 8, 20, 28, 50, 82, 126}. These are believed to correspond to filled shells in nuclei. Their differences would represent the capacities of the shells.

The reasoning from the numbers of stable nuclides might be questioned as to its objectivity. However, an alternate approach in terms of the changes in the binding energy for incremental increases in the number of neutrons confirms the magicality of the Goeppert-Mayer and Jensen magic numbers, as illustrated below for tin (Sn) when the number of neutrons reaches 82 and for lead (Pb) when the neutron number reaches 126.

However this approach reveals that 6 and 14 are also magic numbers, as is illustrated below.

A simple algorithm was found for generating the sequence of magic numbers {2, 6, 14, 28, 50, 82, 126}, which is the magic numbers with 8 and 20 left out. This raises the possibility that there are two categories of magic numbers, the main sequence and a subsidiary sequence that includes 8 and 20. Both 8 and 20 are the sum of the two previous numbers in the main sequence; i.e., 8=2+6 and 20=6+14. Since the magic numbers are the sum of the shell capacities, numbers which involve a sum of magic numbers would represent duplication of shells; i.e., 8=2+(2+4) and 20=(2+4)+(2+4+8).

If there is any substance to the idea that numbers which are the sum of two consecutive magic numbers in the main sequence then 42 which is the sum of 14 and 28 should be of some significance.

Below are shown some of the incremental binding energy profiles for various elements.

For copper (Cu) there is clearly a change in the magnitude of the fluctuations for odd and even at 42 neutrons. For germanium (Ge) there is also a change in the magnitude of the fluctuations but it may occur at 40 rather than 42 neutrons.

Likewise the change in the magnitude of the fluctuations occurs in the vicinity of 42 neutrons.

For gallium and nickel there appears to be one shift at 38 and another at 42.

For bromine (Br) there is a downward shift at 42, whereas for selenium (Se) there is a change in the magnitude of the fluctuations at 42.

## Conclusion

While 42 does not have the special significance of a magic number there is something special about it and this reveals something about the structure of nuclides.

(To be continued.)

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