Cooperative displacements

In some crystalline materials a phase transition on lowering the temperature may produce a modulated structure. This is characterized by the appearance of satellite or superstructure reflections that are adjacent reflections (called fundamental reflections) already observed for the high temperature phase. The superstructure reflections, usually much weaker than fundamental reflections, can in some cases be indexed by a unit cell that is a multiple of the high temperature cell. In such a case the term commensurate modulated structure is commonly used. However, the most general case arises when the additional reflections appear in incommensurate positions in reciprocal space. This diffraction effect is due to a distortion of the high temperature phase normally due to cooperative displacements of atoms, ordering of mixed occupied sites, or both. Let us consider the case of a displacive distortion. 

The rhombohedral R3c perovskite structure (Figure 39A) is formed via a minor cooperative displacement of the oxygen atoms from their ideal positions. As a... 

Cooperative displacements of the oxygen atoms from the middle of a (180° - 0) M—0—M bond may be superimposed on the structural modifications introduced by the mismatch of the mean equilibrium (A—O) and (M—0) bond lengths. This situation was first proposed [12] for Mn(III) t e ions in LaMnOs and is now well-established where cooperative oxygen displacements remove a ground-state orbital degeneracy at a localized-electron configuration. This situation represents a cooperative Jahn-Teller orbital-ordering distortion. 

There are more general problems of stability of materials and of phase transformations that are closely related to the tensile tests described above. Namely, the tensile test may be considered as a special case of so-called displacive phase transformation path. These paths are well known in studies of martensitic transformations. Such transformations play a major role in the theory of phase transitions. They proceed by means of cooperative displacements of atoms away from their lattice sites that alter crystal symmetry without changing the atomic order or composition. A microscopic understanding of the mechanisms of these transformations is vital since they occur prominently in many materials. 

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