As the discrete lattice structure of crystal is considered, it is easy to see that slip can only take place between atoms, thus no atom is situated at the centre of dislocation exactly, the discrete elastic energy belongs to each atom near the centre of dislocation is limited everywhere. Under the assumption of rigid dislocation, using the analytical results of the elastic theory for dislocations directly, the inner cut radius of a singular crystalline dislocation and its periodic variation along with the position of dislocation centre are obtained. For the case of singular screw dislocation in simple orthogonal crystal, the first order approximation coincides with the result of Peierls model. The self energies and effective inner cut radii for two kinds of crystalline system (fcc and bcc) are calculated, and the effect of anisotropic elasticity is taken into account primarily.It is shown that the slip plane for a dislocation is not a geometric plane. The slip plane for the screw dislocation in bcc crystal looks like a bee nest.The possibility to further estimate some second ary effects by this discrete elasticity approach is pointed out.