Magic Numbers in Nuclei (English)

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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