Macroscopic symmetry in space-time

Abstract

Magnetic and non-magnetic crystallographic point groups may be distinguished by the different relations in which they stand to the time-inversion or anti-identity operator R. The 32 point groups of classical crystallography G may be used to generate both magnetic and non-magnetic groups. A non-magnetic group G prime is generated by the generators of any G together with the operator R. There are therefore 32 groups G prime, each with twice as many elements as the corresponding group G. The 90 magnetic groups, M prime, comprise the 32 classical crystallographic groups, G, and 58 additional groups, M, which contain R only in combination with elements other than the identity operator. After a brief introductory section, the classical point groups, G, are considered in detail in § 2, where the crystallographic symmetry operators are discussed and tabulated for the various crystal classes. The connection between the symmetry operators and the symmetry of macroscopic property tensors is examined and this connection is formalized by the use of generating matrices. By way of example, the evaluation of the form of a particular tensor in one of the hexagonal crystal classes is discussed in detail in § 2.4. The extension to the general case is considered in § 2.5 where a table is presented which displays the equalities between the forms of property tensors in the various crystal classes. This is augmented by another table showing the specific forms of tensors of ranks 0, 1, 2, 3 and 4 for the various point groups G. Section 2.6 deals with the additional restrictions imposed on property tensors by particularization and otherwise, and § 2 is concluded by a consideration of null property tensors and `forbidden' effects. Section 3 is concerned, primarily, with the magnetic point groups M prime. After an introductory discussion of time-inversion, the concepts of elementary group theory are used to show, in detail, how the magnetic groups may be derived from the classical crystallographic groups G. The symmetry operators and generating matrices of each group M prime are tabulated and the influence of the new symmetry operators on the symmetry of property tensors is examined. The results of this examination are embodied in a table which permits the form of any property tensor in any magnetic group (M prime) to be obtained from the corresponding result for a classical group (G). The tensor components are tabulated explicitly for tensors of ranks 0, 1, 2, 3 and 4, so that the form of any such tensor may be obtained immediately both for magnetic and non-magnetic classes. The use of these tables is exemplified, in § 3.5, by considering static physical effects which are peculiar to magnetic crystal classes. Pyromagnetism and the magneto-electric effect are taken as examples of tensors of the first and second rank respectively. Piezomagnetism provides an example of a third-rank tensor and also illustrates the simplifying effect of particularization. The section is concluded with a discussion of the application of symmetry considerations to dynamic properties such as transport phenomena. Experimental data on `forbidden' effects in antiferromagnetic materials are presented in § 4.1, where the experimental observations are correlated with the predictions made in § 3. The magnetic properties of ferromagnetic and ferrimagnetic crystals merit separate consideration and these are discussed in § 4.2. Finally, an alternative method of obtaining the forms of property tensors is mentioned.