_{1}

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The central part of the nuclear potential energy is shown to depend on the interacting masses of the nuclear matter. This mass dependent potential energy reduces to the usual Newtonian potential energy of the interacting masses when both the interacting masses are more than a certain limiting mass. This strong potential energy results when both the interacting masses are less than the limiting mass. The potential energy is applied to two more systems here and out of which one nucleus is in the middle of periodic table.

The gross properties of a nucleus are obtained by applying approximate potentials derived from quantum chromodynamics. In this note, a mass dependent potential energy is derived and it applies to any nucleus. It only depends how we choose the interacting masses of the given nucleus. Whenever the masses are quite large compared to the binding energy of the system, non-relativistic quantum mechanics can be used with any potential energy. For example, the electron and proton masses are very large compared to the binding energy of hydrogen atom and non-relativistic quantum mechanics provides a very good idea of the various properties of the hydrogen atom. The universal constant of gravitation is assumed to be universal for all values of the interacting masses. Once we relax this assumption for small masses like those of a nucleus, the results are stunningly accurate as shown in [

What are the gross properties of a nucleus? 1) The binding energy, 2) the ground state wave function, 3) the total spin and 4) the principal energy levels. These are some properties with which we can decide the acceptability of the given potential energy. The mass dependent potential energy is presented in Section 2. In Section 3, the mass dependent potential energy is applied to two more nuclei. These results are in addition to our results that are presented in [

In general, the main nuclear potential energy consists of two parts. 1). The central part and 2) the Yukawa exchange factor

where,

In order to account for various gross properties of nuclei such as Deuteron Helium-4 and many other nuclei, we were led to choose,

The expression in Equation (2.1) looks different from the usual expression of Newtonian potential energy for interacting masses

Equation (2.4) ensures that whenever even if one of the masses is greater than or equal to the cut-off mass

The same neutron experiences a different potential energy near a proton. It experiences the potential energy given by Equation (2.1) where for a proton-neutron there is no prior information as to the value of

Let the universal constant of gravitation G be a function of r where r is the distance between the interacting masses.

where k is a dimensionless parameter and a has the dimensions of inverse length. The central part of the nuclear potential energy is given by,

Suppose we choose k in the following way:

where, f(N) is a function of the variable,

The above condition ensures that the factor k becomes equal to and the exponential factor inside the brackets in Equation (2.6) goes to zero irrespective of the value of r. On the other hand,

If a system can be imagined to be made up of two interacting masses

Newtonian potential energy. From Equation (2.8), it is clear that k is dimensionless as the factor

The above factor is not a YUKAWA factor. In case of the Yukawa factor the mass appears as a single factor or as a sum but not as a product of masses. The factor a also indicates the dependence of

That

Assuming that f(N) = 0 for interacting nuclear masses or for those products of masses

From the above expression, it is quite clear why

Equation (2.14) we have,

From Equation (2.15) it must be very clear that

If the results of an experiment or observation match the theoretical prediction then there is good reason to accept the theory.

The potential energy in general should also include the YUKAWA factor

The factor

Boron nucleus contains 5 protons and 6 neutrons. The core for this nucleus consists of 5 neutrons and 5 protons. There is a neutron outside this core. The total spin of this nucleus in the ground state is

where,

The energy spectrum is given by,

Here n = 2, 3, 4 is the principal quantum number as in the case of hydrogen atom.

The ground state (n = 2) energy for this nucleus is given by,

The binding energy of this nucleus is 76.2046 MeV. The solution of the time independent Schrodinger equation led to the result given by Equation (3.5). By following

The Silver nucleus has many isotopes. One isotope of silver has a mass number of 95. This isotope is

There must be a proton outside this core of 48 neutrons and 46 protons with an orbital angular momentum quantum number of

where

The mass defect for this nucleus is given by

This is equivalent to a binding energy of 766.5079 MeV. With the potential energy given by Equation (3.8) the time independent Schrodinger Equation can be solved as in Refs. (1,2,3). The energy eigen-values are given by,

where,

In Equation (3.10),

When n = 5, the ground state energy is −766.508 Mev. The binding energy of this nucleus is 766.5 MeV. To decide whether a neutron or proton is outside the core we have to examine the magnetic moment of the silver-95 nucleus.

In this note, the Newtonian potential energy is modified through a limiting mass

In general, we should use Equation (3.1) for any nucleus. Here, we use it without the Yukawa factor to explain the gross properties of silver nucleus which is in the middle of the periodic table and has 95 nucleons. Is it possible to explain these results for this nucleus through QCD?

The author is very grateful to Prudhvi RChintalapati.

Cvavb ChandraRaju, (2015) Strong Interaction and Newtonian Potential Energy. Journal of Modern Physics,06,1814-1819. doi: 10.4236/jmp.2015.613185