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The sub-structure of Bi₂Sr₂Ca₁Cu₂O₈ can best be described on a non-centric orthorhombic A2aa cell, which permits the oxygen in the BiO plane to move off the center of the Bi square to approach to within 2.2 Å of a pair of Bi atoms. Each Bi then has two close oxygens within the BiO plane, and a third at 2.12 Å connecting to the CuO layer. The new structure permits the apparent Bi valence to approach 3+ with a more reasonable Bi-O co-ordination than for earlier approximate structural models. This model satisfies the high resolution neutron data of Bordet et al., while agreeing in part with the co-ordination proposed on chemical grounds by von Schnering et al.