_{1}

This article proposes the potential energy function of nucleon in nucleus, derives the expression equation of nuclear force, shows that nucleus has the shell structure by the solving the Schr?dinger equation of nucleon, obtains the magic numbers, and interprets the past experimental results in theory; for example the radius of nucleus is proportional to the cubic root of nucleon number, the nuclear force is repulsive in the depths of nucleus and attractive in the surface layer, and the variation of average binding energy of nucleons with the nucleon number.

The study on nuclear force and nuclear structure is the center subject of nuclear physics. Although the liquid drop model, the shell model and other models of nucleus had been put forward on the basis of experiments long ago [

Since the nuclear force is independent of charge, the nucleons (protons and neutrons) could be considered as identical point particles, and the mass of free nucleon is

where

and the interaction mass of the nucleus within the radius r

the introduction of factor 3 is due to the strong interaction with the three colors [

Substituting N for

where

is both the average mass density of nucleus and the mass density on its border. So

this shows the radius of nucleus is proportional to the cubic root of nucleon number, this is consistent with the experiments [

The expression equation of nuclear force can be obtained from Equation (1):

Obviously, the nuclear force is short-range, and

So there is

There is

we can draw the function curve of the potential energy of nucleon and the nuclear structure diagram, as shown in

In the following we show that the nucleus has the shell structure and explain why the nuclear force changes from the repulsion to the attraction with the increase of the ra-

dius. Take

where

Write

the Equation (13) can be divided into the two equations as follows:

where

where

There is

the Equation (18) becomes

Substituting the form solution

Write

According to the theory of ordinary differential equations the above equation has the series solution as follows:

where

is the associated Le Gail polynomial,

Notice

, (25)

the radial wave function of nucleon is

where

where

Since the nuclear force is mainly in the radial direction we introduce the radial quantum number

So the angular momentum quantum number is

each subshell is

level corresponding to

The first shell:

The second shell:

The third shell:

The fourth shell:

The fifth shell:

The sixth shell:

The seventh shell:

…,

So the total number of nucleons fulfilled the first shell equals 2, the total number of nucleons fulfilled the first two shells equals 8, the total number of nucleons fulfilled the first three shells equals 20, and so on, and these numbers are the called magic numbers:

We know that nucleons are fermions, there is the repulsion between them according to the Pauli Exclusion Principle, at the same time there is the attraction due to the exchange of meson

Since the distribution probability

quantum number m, the shape of fulfilled nucleus is spherical, and the shape of unfulfilled nucleus is ellipsoid [

For simplicity, suppose every energy level is fulfilled by nucleons, the Equation (24) gives the average binding energy of nucleon as follows:

where k is the number of energy levels fulfilled by nucleons. From the equation above we see that the average binding energy of nucleons of the medium nucleus with major nucleons is greater than that of the light nucleus with fewer nucleons, this is consistent with the experiments. But, in the heavy nucleus whose nucleon number more than 100, some nucleons are located in the repulsive region of

The above analysis shows that a series of results consistent with the experiments can be

derived from the potential energy function of nucleon as shown in the Equation (1), therefore, the potential energy function of nucleon given by this article is reasonable.

Liang, B. (2016) The Potential Energy of Nucleon and Nuclear Structure. Journal of Modern Physics, 7, 1866-1873. http://dx.doi.org/10.4236/jmp.2016.714165