Gross Features of Binding Energies

Up to this point of the course we have been examining in detail the two-body interaction between nucleons. The hope has been that we could build up nuclei from by understanding pair-wise interactions. We have not given up on this hope and we will attempt to implement this approach at all stages of our analysis. However, we must introduce and analyse other important features of nuclei that have been known for many years in order to get a better idea of what the ``big-picture'' is.

One of the simplest measured quantities we can determine for a nucleus is its binding energy, $B(A,Z)$ which can depend on its global quantum numbers, electric charge and total nucleon number of the nucleus (baryon number). This quantity is defined in terms of the mass of the atom $M(A,Z)$, the mass of the neutron $M_n$ and the mass of a hydrogen atom $M_H$. By definition

 

$$B(A,Z) = Z M_H + (A-Z) M_n - M(A,Z) \ \ \ ,$$ (1)

which is a positive quantity for bound states.

A quantity that is directly related to $B(A,Z)$ that is found to be essentially indepndent of $A$ is the binding energy per nucleon $B/A$. The experimentally observed binding energy per nucleon is shown in the fig (1). One sees that over the range of nuclei that are stable (which we will get to later) the binding energy per nucleon is essentially constant, lying between $7.5$ and about $8.8$ MeV.

  

Figure: Binding energy per nucleon

This behaviour tells us a lot about the nuclear force. Imagine that the interaction between nucleons was entirely yukawa type mediated by pion exchange, that dominates the long-distance part. For distances inside of an inverse pion mass, the interaction looks coulombic. We know that for a coulomb system the particles want to sit on top of each other, this is the configuration that has the lowest potential energy. The actual configuration is a trade off between the kinetic and potential, that is why there is a finite radius of the hydrogen atom. However, when we put together more and more particle that interact via coulomb type interactions, which are infinite range, then the interaction energy depends on the number of pair-wise interactions that exist. In the case of nuclear interactions this would correspond to a term dependent upon $A^2$. Clearly, such dependence is inconsistent with the nearly constant $B/A$. One can understand this behaviour if in fact a given nucleon interacts with only a few of the other nucleons in the nucleus, such that the number of pair-wise interactions is not growing with $A^2$. A nucleon on one side of the nucleus is far outside its interaction range of a nucleon on the other side of the nucleus. This picture also leads to the idea that the density of nuclear matter is constant. This is indeed true and we will return to this later.