We generally think of nuclei as spherical -- this would be the equilibrium shape for an uncharged liquid drop bound by short range forces. However, most nuclei are deformed into an ellipsoidal shape. The reasons for this lie in the shell structure of nuclei, and the interaction of outer (`valence') nucleons with an inner `closed shell' core.
This ellipsoidal deformation can be quantified in terms of a quadrupole moment Q (of the mass (or charge) distribution). This can be obtained by an expansion of the mass or charge distribution in spherical harmonics and a truncation to the lowest order terms (the monopole, which gives the mass, and the quadrupole; the dipole term vanishes by symmetry). Let us rather see how this arises in a simpler argument.
Suppose that the charge of the nucleus is distributed according to some function . Then the charge M of the nucleus is
The mean square radius of the distribution is
Now in cartesian coordinates,
For a spherically symmetric distribution there is nothing to distinguish the three axes, so .
Now deform the spherical distribution into an ellipsoidal distribution with the z axis as the axis of symmetry. Then , so that .
Thus a measure of the deviation of the distribution from spherical symmetry is given by
Q is known as the quadrupole moment. With the above definition it is positive for prolate deformations and negative for oblate deformations. Similar definitions can be given for the charge distribution of the nucleus (which might differ in shape from the mass distribution).
Non-spherical nuclei are able to rotate about axes other than the axis of symmetry; this rotation gives rise to characteristic spectral features which permit the quadrupole moment to be measured.