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The Question of the Magicality of the Numbers
42, 78 and 132 in the Physics of Nuclei

In the late 1940's the German physicists, Maria Goeppert-Mayer and Hans Jensen, independently formulated what came to be known as the shell model of nuclei. In 1948 they were jointly awarded the Noble Prize in Physics for their work. Primarily they utilized the information on the number of stable isotopes and isotones to identify which numbers of protons and numbers of neutrons represented filled shells. (Isotopes are nuclides with the same number of protons and isotones are nuclides with the same number of neutrons.)

For example, consider the data in the following table.

Proton
Number
41424344454647484950
Number of
Stable Isotopes
16071626110

Proton
Number
51525354555657585960
Number of
Stable Isotopes
2419161215

The case of 50 protons (the element Tin) stands out with 10 stable isotopes, but not extraordinarily so.

The data for the number of stable isotones shows a similar pattern.

Neutron
Number
41424344454647484950
Number of
Stable Isotones
1534131425

Neutron
Number
51525354555657585960
Number of
Stable Isotones
2434232313

The case for the special nature of number 50 is less striking in the case of the number of neutrons than for the number of protons. Nevertheless the number 50 is taken to represent a case of the shells being filled and such numbers have become known as

magic numbers.

Goeppert-Mayer and Jensen desinated 2, 8, 20, 28, 50, 82 and 126 as magic numbers.

There is a more precise way of identifying magic numbers. It is in terms of the increments in binding energy resulting from additional nucleons in nuclides. For example, consider the case of Tin once again.

The data for other elements, such as Selenium, also show the same phenomenon.

The incremental binding energy drops sharply after a shell is filled and additional neutrons go into a higher shell. (The sawtooth pattern is due to the formation of neutron spin pairs.)

The incremental binding energy method identifies as magic numbers the set designated by Jensen and Goeppert-Mayer. However it also indentifies 6 and 14 as magic numbers.

In this case there is not only a sharp drop at 14 neutrons but also a lesser sharp drop at 20 neutrons.

In this case there is not only a sharp drop at 6 neutrons but also a lesser drop at 8 neutrons and another at 14 neutrons.

So 6 and 14 are nuclear magic numbers.

There exists an algorithm for explaining the sequence of magic numbers {2, 6, 14, 28, 50, 82, 126}. This is the magic numbers with 8 and 20 left out. Eight and 20 are definitely magic numbers but they appear to be ones in a different category from the others.

Both 8 and 20 can be expressed as the sum of the previous two magic numbers in the above sequence; i.e., 8=6+2 and 20=14+6. If there is anything to this pattern then 42=28+14, 78=50+28 and 132=82+50 should be in the nature of magic numbers.

The Number 42

Note that in the data for the number of stable isotopes and isotones the numbers for 42 were notably larger than for 41 and 43.

Proton
Number
41424344454647484950
Number of
Stable Isotopes
16071626110

Neutron
Number
41424344454647484950
Number of
Stable Isotones
1534131425

For neutrons the stable isotones for 42 is the same as for 50. There is some indication of the magicality of the number 42 in terms of incremental binding energy, as shown below, but in displays for other elements it does not show up.

While 42 cannot be said to be a magic number like 8 and 20 there is something special about it.

The Number 78

The data for 78 nucleons shows the number of stable isotopes or isotones not to be much different than the values for the other even numbers in its vicinities.

Proton
Number
71727374757677787980
Number of
Stable Isotopes
1524152516

Neutron
Number
71727374757677787980
Number of
Stable Isotones
4515242433

There is just the barest hint of something different happening to the slope of the incremental binding energy relation at 78 neutrons, as indicated below.

The Number 132

Again for 132 there is just the barest hint of something different happening to the slope of the incremental binding energy relation at 132neutrons.

There are no stable nuclides with either 132 protons or neutrons.

Conclusion

The pattern that exists for the magic numbers of 8 and 20 being equal to the sum of the previous two magic numbers in the sequence {2, 6, 14, 28, 50, 82, 126} does prevail for higher numbers. There is something special about the number 42 but little to no indication of this for the numbers 78 and 132.


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