A theory has been given for the scattering of neutrons by anharmonic crystals, for which terms of the typeV⁽³⁾ (k_1j1; —k_1j1;o_j) which contribute to the sublattice displacements are not neglected. Using the standard perturbation theory in the interaction picture or Green’s function method, an expression has been derived for the differential scattering cross-section which brings in the shift and the width of the phonons in one-phonon energy exchange processes. It is shown that the sublattice displacements will modify the phase factor arising from the scattering by any atom in the unit cell, and the Debye-Waller factor also gets altered both by the sublattice displacements as well as by higher order terms arising from anharmonicity. It is shown that the differential scattering cross-section contains a term linearly depending on the third order anharmonicity coefficientV⁽³⁾ (k_1j1;k_2j2;k_3j3) and neutron scattering by crystals should provide a useful method for evaluating the third order anharmonicity coefficients.