Field dependent tensors in solid state physics

Abstract

The concept of field-dependent tensors (tensors the components of which depend on the direction of an applied field) is generalized to define quantities which have different tenserial character in different subspaces (field subspace and geometrical or physical subspace) of the whole space. The transformation laws for these field-dependent tensors are worked out and the weighted and relative field dependent tensors are defined. The calculus of field dependent tensors is established through which the operations of addition, subtraction, inner and outer multiplication, contraction and differentiation of field dependent tensors are defined and the conditions under which each operation can be performed are discussed, and the quotient law for field dependent tensors is also worked out. The effect of magnetic crystal symmetry on the forms of field-dependent tensors is considered and a generalized Neumann's principle is defined. Furthermore, it is proved that in working out the magnetic point group of symmetry operations, identification of the magnetic moment inversion operator with the time-inversion operator is not correct. The effect of the symmetry operations of the magnetic point groups on the field dependent tensors representing the transport properties of a magnetic crystal is considered. Different prescriptions (A, B and C) given by different workers for finding the magnetic symmetry restricted forms of the magnetoconductivity tensor σ(_ij)(B) are discussed, the objections to each prescription are pointed out and then a new prescription (D) is given. Following prescription D, the restriction on the forms of the magnetoconductivity tensor σ(_ij)(B) the magnetoresistlvlty tensor σ(_ij)(B), the magnetothermoelectric power σ(_ij)(B) the magnetothermal conductivity K(_ij)(B) and the magneto-Peltier effect π(_ij)(B) imposed by the symmetry of crystals belonging to each magnetic point group are found. Finally the way in which the permittivity of a crystal belonging to a magnetic point group can be represented by a field dependent tensor is discussed.