Order Variables and the Adiabatic Potential

Abstract

Crystals constitute a representative group of condensed matter, composed of a large number of identical ions and molecules. Because of imprecise theories of cohesive forces available today, we considered their geometrical structure as granted [1]. Nevertheless, their correlation energy is essential for structural stability, for which we may consider permutable constituents quantum-mechanically. On the other hand, if symmetry in a chemically pure crystal pertains to the thermodynamic environment, the stability can be attributed to the invariance under symmetry operations like rotation, reflection, translation, etc.; the structural invariance determines the correlation energy in a stable crystal. However, the local symmetry can be disrupted below the critical temperature, if a displacement occurs in the constituent. Exhibiting local fluctuations, the strained lattice suffers instability, which is however relaxed thermally to a strain-free structure with decreasing temperature, as postulated by Born and Huang. Meanwhile, symmetry in stable crystals is well documented by X-ray crystallography, as classified by the group theory. In this chapter, leaving details of the symmetry group to references, we discuss the origin of internal variables at the critical temperature, which are known as order variables and responsible for structural transitions.