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Elliott's model of a straight‐fronted cleavage crack is used to conduct an analytical Peierls‐Nabarro‐type calculation appropriate for a two‐dimensional lattice where the atomic spacings are respectively a and b normal and parallel to the cleavage crack. The analysis enables the effect of the discreteness of the atomic structure on brittle cleavage crack extension to be assessed and circumvents some of the limitations in a previous analysis due to Thomson, Hsieh, and Rana. The general conclusion emerging from the present work is that the magnitude of the discreteness effect, reflected in the ratio between the crack extension stress and that predicted by the continuum‐based Griffith relation, is greater the smaller the crack front width which thereby substantiates Thomson, Hsieh, and Rana's inferences. Proceeding from this conclusion, the wider problem of relating the magnitude of the discreteness effect, crack front width, and the type of atomic bonding is discussed from an essentially physical viewpoint. As a result of such considerations it is suggested that, in general, the discreteness effect will be small and the crack front will be wide with a material that is more resistant to elastic shear than tension, as in a covalently bonded solid where the bonding is directional. Since dislocation mobility is difficult in such a material, it would therefore seem that existing arguments regarding brittle cleavage crack extension should not be significantly affected by the introduction of discreteness considerations and that very special circumstances are likely to be required to give a large discreteness effect, as would be reflected in a brittle crack extension stress markedly in excess of that predicted by the Griffith relation.