On relations between the symmetries of elastic constants and stability conditions

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For a variety of crystal types (orthorhombic, hexagonal, and monoclinic) it is shown that the conditions which must be applied to force constants in rigid ion models to ensure the correct symmetry of the elastic constants can be related to derivatives of the Madelung constant. The derivatives are of two types: (i) with respect to internal lattice parameters; and (ii) with respect to combinations of unit cell parameters which leave the crystal type unchanged and which keep the volume constant. If the short-range interactions are assumed to be central then the symmetry conditions are identical to the constraints which stabilize the crystal energy with respect to the above deformations; but this identity only holds for central interactions.