Nuclear Shell Model (English)

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2-6: The Nuclear Shell Model

As learned on the preceding page,

the nuclei whose

proton number Z

neutron number N

is one of the

magic numbers

2, 8, 20, 28, 50, 82, 126

are especially stable

and

they correspond to

the noble gases

in the atomic world.

The periodic law

and the periodic table

are valid in the Atomic World.

Similarly,

the idea of the periodic table

in the nuclear world

was also proposed

in a rather early time.

[Why the Periodic Law]

As discussed on the page

in the present seminar,

"Microscopic World -2-: 3-3: The Periodic Law of Elements",

the periodic law of elements

could be interpreted

with the shell structure of the atoms.

Let us review the

atomic shell structure

again.

The fundamental properties

of atoms are determined by

the structure of electron distributions

surrounding the nucleus

at the center of atom.

Namely, they are determined by

the electron number

and their arrangements

(i.e. configurations).

The electrons in an atom

make a layer structure

starting from the central nucleus

and going outward.

Each of these layers is called

a shell, and

this is none other than

the shell structure.

An electron in atoms

is bound by the attractive

average potential

whose main part is

the Coulomb force between

the positive charge of the nucleus

and the negative charge of the electron.

The electrons bound in the atom

by the average potential

are staying in

eigenstates with

discrete energy eigenvalues.

These eigenstates form

the shell structure

of which mechanism was explained

in detail

on the page,

"Microscopic World -2-: 3-3: The Periodic Law of Elements".

In this shell structure,

each shell is occupied by

a certain number of electrons

according to the

Pauli principle.

When one of the shells is

filled up, a noble gas

is formed.

This is the reason why the

periodic law appears.

[The Nuclear Shell Model]

The idea of the nuclear

shell model

had been proposed

at a rather early time

on the analogy of

the atomic shell structure.

However, there was some doubt

on this idea because

there were differences

between the cases

of atoms and nuclei.

The differences are as follows:

(1)

In the case of an atom,

there is a heavy nucleus

at the central position of the system,

and it attracts

light electrons around

by the Coulomb interaction.

In contrast to this,

there is nothing special

at the nuclear center.

(2)

In the case of a nucleus,

it is clear that the whole system

can be described

in the image of

a "raindrop".

Namely, the liquid drop model

is valid for nuclei.

However, such an image is not

valid for atoms.

Moreover, the biggest problem

is the difference

in the magic numbers

of both the systems,

atoms and nuclei.

Although

the magic numbers in atoms are

Z =

2, 10, 18, 36, 54, 86,

the magic numbers in nuclei are

or N =

2, 8, 20, 28, 50, 82, 126.

There is a big difference

between these two kinds of systems,

atoms and nuclei.

Even if the shell structure

is held in both the systems,

the average potentials

are presumed to be

considerably different.

In the case of atoms,

if we assume an average potential

consisting of the Coulomb

attractive potential

acting from the +Ze

charge of the nucleus

(and the averaged Coulomb repulsive

potential between electrons),

it was rather easy

to reproduce the atomic

magic numbers,

(Z = 2, 10, 18, 36, 54, 86).

(See the previous page of the present seminar,

"Microscopic World -2-: 3-3: The Periodic Law of Elements").

It was, however, very difficult

to find out the average potential

which can reproduce

the nuclear magic numbers,

or N

= 2, 8, 20, 28, 50, 82, 126.

[Mayer-Jensen's shell Model]

In 1949, M. G. Mayer

(USA, 1906 - 72)

and H. D. Jensen

(Germany, 1907 - 73) et al.

independently proposed

an average potential

which can reproduce the

nuclear magic numbers.

Thereby the shell structure in nuclei

was theoretically

established

and it became

a firm foundation of

the Nuclear Structure Theory.

The average potential

proposed by Mayer, Jensen et al.

includes a very special term

called LS-potential.

Because of this,

Mayer-Jensen's shell model

is sometimes called

the jj-coupling shell model.

The details on the LS-potential

are somewhat expertized discussions,

so that we omit them here.

In the following figure,

the single-particle levels

in both the models,

the nuclear shell model

and the atomic shell model,

are compared.

[The Energy Levels in the Nuclear and Atomic Shell Models]

The red levels

show the single-particle levels

in the Mayer-Jensen shell model

for nuclei and

the blue levels

are those in the atomic

shell model.

In each of the red levels,

specified numbers of

protons and neutrons

are filled one by one

from the bottom,

and in blue levels

electrons are filled.

The circled numbers

indicate the particle number

which can be filled up

to the levels.

They are just the

magic numbers.

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