Magnetic point groups

The thirty-two point groups without any form of the time reversal operator form a category of magnetic point groups. The representations of these are obtained by standard means and are presented in Section 12.6. 

The remaining fifty-eight magnetic point groups include the time reversal operator only in combination with rotation and rotation-reflection operators. The representations of these groups may be obtained from Eq. (12-27). 

Spin Warns.—In Application to Point Groups, above, we considered the irreducible representations of magnetic point groups. These would be useful in obtaining the symmetry properties of localized states in magnetic crystals, as impurity or single ion states in the tight... 

Magnetic ordering, 746 Magnetic point groups, 738, 739 international notation, 739 properties of, 740 Schonflies notation, 739 Shubnikov notation, 739 Magnetic point symmetry, determination of, 744... 

Scattering processes, 586 Schiff, L. J., 437,444 Schiffer, M.% 363 SchUchting, H., 24 Schmidt orthogonalization, 65 Schonflies notation for magnetic point groups, 739... 

Shannon, C. E., 190,195,219,220,242 Shapley, L. S316 Skirokovski, V. P., 768 Shortley, O. H., 404 Shot noise process, 169 Shubnikov, A. V., 726 Shubnikov groups, 726 Shubnikov notation for magnetic point groups, 739 Siebert, W. M., 170 Signum function, 313 Similar matrices, 68 Simon, A408 Simplex method, 292 Simulation, 317... 

Table 14.2. The fifty-eight type III magnetic point groups.

Example 14.2-2 Find the co-representations of the magnetic point group Amm or C4v(C2v). Take Q = rr.A, with a the unit vector along [110]. 

Exercise 14.3-1 Write down the non-zero CG coefficients for the inner DPs of the point group mm2 (C2V). [Hints See Table 14.4. Recall that %3 (R) means x(r3) and that for this group T3 = T4.] Using Table 14.6 derive expressions for the non-zero CG d coefficients of the magnetic point group 4mm in terms of the cijk and evaluate these. Hence write down the CG decomposition for the Kronecker products of the IRs T of 4mm. 

The splitting of atomic energy levels in a crystal field (CF) with the symmetry of one of the magnetic point groups has been considered in detail by Cracknell (1968). Consider an atomic 2P level (L = 1) in an intermediate field of 2mm or C2v symmetry and assume that Hs. L < Hcf. The degenerate 2P level is split into three components, T i T3 T4. But in a field of 4mm symmetry the two levels T3 and F4 stick together, that is, are degenerate, since 14 = r3 r4 (case (b)), while 14 is re-labeled as 14 (case (a)). (See Table 14.4 and... 

Properties of crystals with magnetic point groups... 

Table 15.9. The thirty-one magnetic point groups in which ferromagnetism is possible.

Example 15.5-2 Determine the structure of Q for the magnetic point groups 4mm and 4 mm. 

The piezomagnetic tensor Q for all magnetic point groups is given by Bhagavantam (1966), p. 173, and Nowick (1995) in his Table 8-3. 

Furthermore, there are many more "magnetic point groups" than 32 and there are other, more specialized point groups one can also go to more than three dimensions, and so on. 

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