1. PUNEQE

Forms of the potential function

Most of nuclear materials is dependent upon the use of different forms of interatomic/internucleonic potential. These forms are used throughout the field. This material is present in ({Nastasi, et. al. 1996).

Hard Sphere

The hard sphere states that another nucleon has no probability of existings within a certain radius \(\Gamma_{r}\), and is given

\[\begin{split}V\left(r\right)=\begin{cases} \infty & r<\Gamma_{r}\\ 0 & r>\Gamma_{r} \end{cases}\end{split}\]
Figure 1: Shape of the hard sphere potential

Square Well Potential

The square well potential allows there to be some certain attractive force when the other nucleon is near enough to the nucleus, but still has the inifite potential at close range

\[\begin{split}V\left(r\right)=\begin{cases} \infty & r<\Gamma_{r}\\ -\varepsilon & \Gamma_{r}<r<R\\ 0 & r>R \end{cases}\end{split}\]
Figure 2: Shape of the square well potential

Inverse Power Potential

The potential is obviously not discrete as has been shown above, so one way to retain the infinite potential at zero but have it continuous is the inverse power potential

\[V\left(r\right)=\varepsilon\left(\frac{\Gamma_{r}}{r}\right)^{n}\]
Figure 3: Shape of the inverse power potential

Lennard-Jones Potential

Finally, combining everything, we have the most accurate description of the potential yet, the Lennard-Jones potential. This potential preserves the infinite potential at zero, and the attraction between a certain distance. It is given by

\[V\left(r\right)=\varepsilon\left[\left(\frac{\Gamma_{r}}{r}\right)^{m}-\left(\frac{\Gamma_{r}}{r}\right)^{n}\right]\]
Figure 4: Shape of the Lennard-Jones potential

Bibliography