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2-5: The Magic Numbers, the Periodic Law

As learned on the last page, it has been clarified that the general feature about the nuclear binding energy is well reproduced by the Weizsaecker-Bethe Mass Formula which was established on the basis of the idea of the liquid drop model. (See the Figure at the bottom of the last page.)
However, the very detailed data of the binding energies are not completely reproduced. There are some points where the experimental data deviate from the values of the Weizsaecker-Bethe Mass Formula.
[Deviations of the Experimental Data from the Mass Formula]
The following figure shows the details of the deviations of the experimental binding energies from the values of the Weizsaecker-Bethe Mass Formula. The ordinate denotes the absolute values of the deviations (not per nucleon) :

[The Magic Numbers]
As seen in the above figure, the nuclei whose neutron number is 28, 50, 82 or 126 have especially large binding energies.
For the proton number, a similar analysis is also possible. According to detailed analyses, we can conclude that the nuclei of which the proton number Z or neutron number N is
2, 8, 20, 28, 50, 82, 126
have particularly large binding energies. (Since the nucleus with Z = 126 does not occur, the number 126 is only for neutrons.) These numbers are called the magic numbers in nuclei.
Evidence of the magic numbers is not restricted to the binding energy. We can find that the excitation energies of the first excited states are particularly large in the nuclei whose proton number or neutron number is one of the magic numbers; these nuclei are difficult to be excited and then especially stable. The following figure shows one of these examples.

[Doubly Magic Nuclei]
There are some nuclei in which both proton number Z and neutron number N are magic number as follows:

Their binding energy are large and they are very stable. They are often called doublly magic nuclei.
[The Periodic Law in Nuclei]
The periodic law of elements or periodic table is explained on the page in the present seminar: "Microscopic World -2-: 3-3: The Periodic Law of Elements".
Among atoms, the numbers Z = 2,10,18,36,54,86 have a special meaning. The elements with these atomic numbers, i.e. He,Ne,Ar,Kr,Xe,and Rn, are quite stable and called the Noble Gases.
It is known well that the properties of chemical elements are periodic functions of their atomic weights (or atomic number) and the noble gases are at the boundaries of the periodicity. This periodic law of elements has beatifully been derived from the shell structure of atoms.
Then, does a similar periodic law hold also among nuclei?
Comparing the magic numbers in atoms and in nuclei, we suppose that there might be a periodic law also in the nuclear world.
Surely, there is a periodic law in the Nuclear World. Let us explain it in detail on the next page.
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