A study of the band structure of a massless particle in a cosine potential is made via the Dirac equation. It is shown that every alternate band gap disappears as a consequence of a periodicity of the potential combined with a peculiar symmetry of the Dirac equation. This basic potential is then used to study a simple one-dimensional model of the nucleus from which it is ascertained that modelling the mean field of the quarks in the nucleus via a pure scalar potential is unsatisfactory. A simple extension involving a combined scalar and vector potential is then proposed as a possible solution to this problem. The effect of the addition of this vector component to the band structure is also investigated. 32 refs.