Draw the plot of potential energy of a pair of nucleons as a function of their separation. Write two important conclusions that can be drawn from this plot.
Show Hint
Remember the three important regions of the nucleon potential energy curve:
\[
r>2\ \text{fm}
\Rightarrow
\text{Force is negligible}
\]
\[
0.8\ \text{fm}<r<2\ \text{fm}
\Rightarrow
\text{Force is attractive}
\]
\[
r<0.5\ \text{fm}
\Rightarrow
\text{Force is strongly repulsive}
\]
This behaviour explains why nuclei are stable and why nucleons remain confined inside the nucleus.
Concept:
The interaction between two nucleons (proton-proton, neutron-neutron, or proton-neutron) is governed by the nuclear force. The nature of this force can be understood by studying the variation of the potential energy of a pair of nucleons with their separation.
The potential energy curve shows that the nuclear force is:
• Attractive at intermediate distances.
• Repulsive at extremely small separations.
• Negligible at large separations.
The typical graph of potential energy \(U\) versus separation \(r\) of two nucleons is shown below.
A schematic representation of the graph can also be drawn as
The graph indicates a deep negative potential well at intermediate separations and rises sharply to positive values at very small separations.
Step 1: Interpret the behaviour of the curve at large distances.
As the separation between the nucleons increases beyond approximately
\[
2\ \text{fm}
\quad
(1\ \text{fm}=10^{-15}\ \text{m}),
\]
the potential energy approaches zero.
This means that the nuclear force becomes practically negligible beyond a few femtometres.
Hence, nuclear forces are short-ranged forces.
\[
\boxed{\text{Nuclear force acts only over a very short range of about }2-3\ \text{fm}.}
\]
Step 2: Interpret the behaviour of the curve at intermediate distances.
For separations around
\[
0.8\ \text{fm}
\text{ to }
2\ \text{fm},
\]
the potential energy is negative.
Negative potential energy indicates that the force between the nucleons is attractive.
Therefore, nucleons remain bound together inside the nucleus due to this attractive nuclear force.
\[
\boxed{\text{Nuclear force is strongly attractive at intermediate separations.}}
\]
Step 3: Interpret the behaviour of the curve at extremely small separations.
As the separation becomes very small,
\[
r\lesssim 0.5\ \text{fm},
\]
the potential energy rises sharply and becomes positive.
This indicates that the nuclear force becomes strongly repulsive at extremely short distances.
This repulsive nature prevents the nucleons from collapsing into one another.
\[
\boxed{\text{Nuclear force becomes strongly repulsive at very small separations.}}
\]
Important Conclusions from the Graph: • Nuclear force is a short-range force; it becomes negligible for separations greater than about \(2-3\ \text{fm}\).
• Nuclear force is attractive at intermediate distances but becomes strongly repulsive at very small separations, thereby ensuring the stability of the nucleus.
