13.3 Size of the Nucleus

As we have seen in Chapter \(12,\) Rutherford was the pioneer who postulated and established the existence of the atomic nucleus. At Rutherford's suggestion, Geiger and Marsden performed their classic experiment: on the scattering of \(\alpha\) -particles from thin gold foils. Their experiments revealed that the distance of closest approach to a gold nucleus of an \(\alpha\) -particle of kinetic energy \(5.5 \mathrm{MeV}\) is about \(4.0 \times 10^{-14} \mathrm{~m}\) The scattering of \(\alpha\) -particle by the gold sheet could be understood by Rutherford by assuming that the coulomb repulsive force was solely responsible for scattering. Since the positive charge is confined to the nucleus, the actual size of the nucleus has to be less than \(4.0 \times 10^{-14} \mathrm{~m}\).

If we use \(\alpha\) -particles of higher energies than \(5.5 \mathrm{MeV}\), the distance of closest approach to the gold nucleus will be smaller and at some poin the scattering will begin to be affected by the short range nuclear forces, and differ from Rutherford's calculations. Rutherford's calculations ar based on pure coulomb repulsion between the positive charges of the \(\alpha\) particle and the gold nucleus. From the distance at which deviations set in, nuclear sizes can be inferred.

By performing scattering experiments in which fast electrons, instead of α -particles, are projectiles that bombard targets made up of various elements, the sizes of nuclei of various elements have been accurately measured.

It has been found that a nucleus of mass number A has a radius \[ R=R_{0} A^{1 / 3} \]

where \(R_{0}=1.2 \times 10^{-15} \mathrm{~m}\left(=1.2 \mathrm{fm} ; 1 \mathrm{fm}=10^{-15} \mathrm{~m}\right) .\) This means the volume of the nucleus, which is proportional to \(R^{3}\) is proportional to \(A\). Thus the density of nucleus is a constant, independent of \(A\), for all nuclei. Different nuclei are like a drop of liquid of constant density. The density of nuclear matter is approximately \(2.3 \times 10^{17} \mathrm{~kg} \mathrm{~m}^{-3}\). This density is very large compared to ordinary matter, say water, which is \(10^{3} \mathrm{~kg} \mathrm{~m}^{-3}\). This is understandable, as we have already seen that most of the atom is empty. Ordinary matter consisting of atoms has a large amount of empty space.