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Crystal Lattice Defects and Differential Geometry

Forty years ago K. Kondo discovered the close connexion between the crystal lattice defect dislocation and Cartan's notion of torsion of a geometrical space. This discovery of Kondo opened the way to a new understanding of lattice defects in terms of modern field theory. However, in a closed theory point defects too, must play a role because these and the dislocations mutually convert, e.g. in dislocation climb. It is shown that the elementary point defects, namely vacancies, self-interstitials and shear defects, can be included by an extension of the differential geometry of dislocations. The point defects disturb the metricity of the crystal, but also can lead to torsion. When elementary shear defects (point stacking faults) are neglected, the geometry becomes of type Weyl. Our results have a bearing on the statics, too, if these are developed in the dual stress space where the stress function tensor (tensor potential) has the meaning of a metric tensor.