MAGNETIC GROUPS AND THEIR COREPRESENTATIONS
CRACKNELL ARTHUR P. (ARTHUR PHILIP)
CRACKNELL ARTHUR P. (ARTHUR PHILIP)
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Abstract
A brief outline of the history of the development of the study of point groups and space groups is given, leading up to the introduction by Shubnikov of the idea of antisymmetry and the enumeration of the "black and white'' or "magnetic" point groups and space groups, If the operation of antisymmetry is identified with the operation of time-inversion or the operation of the reversal of a magnetic moment these groups are non--unitary groups, It is well known that the wave function of an electron or of a normal mode of vibration belongs to one or other of the irreducible representations of the point group or space group of the system in which it finds itself, The modification of representation theory and the development of the theory of corepresentations was done by Wigner and some preliminary work on the application of this theory to the crystallographic magnetic groups was done by Dimmock and Wheeler. The present work completes the derivation of the irreducible corepresentations ( "coreps") of the 58 magnetic point groups, Complete tables of the characters of the single-valued and double-valued representations of the ordinary point groups, together with the matrix representatives for the degenerate representations, are given. The corepresentations of the grey point groups are considered and it is shown that the theory is equivalent to that of Herring considering the consequences of invariance under the operation of time-inversion. The unitary subgroups of the 58 black and white or magnetic point groups are identified in terms of the ordinary point groups and tables are then presented which give the irreducible corepresentations of each of the magnetic point groups in terms of' the irreducible representations of the unitary subgroup, An example of the use of the tables is given for the magnetic point group m'3. The problem of determining the irreducible corepresentations of the magnetic space groups is considered with particular attention being given to the cubic magnetic space groups. The little group of the wave vector is given at each special point of the Brillouin zone of every cubic space group and the unitary subgroup is identified in all of the magnetic groups derived from each of these. As an example the irreducible coreprescntations of the little group at each of the points of symmetry are derived in the case of the magnetic space groups related to T7h (Ia3). Finally the application of the theory of the irreducible corepresentations of the magnetic point groups is considered in relation to crystal field theory, in the absence of spin-orbit coupling. Both the splitting of the energy levels and the symmetry properties of the wave function are considered in magnetic point groups. Examples of 4' mm' and 4m' m' are studied.
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Date
1967
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