Crystal-field theory for magnetic crystals

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 

The splitting of the 2P level in G = 4mm due to the CF and weak spin-orbit coupling is 

In a weak m3 field the 2D multiplet is split by spin-orbit coupling into its 2Dy2(4) and 2Ds/2(6) components. The 2D3/2 state is 6(4) while the 2Ds/2 state sphts into s(2) and 6(4) components, which correlate with the intermediate field case in eq. (21). 

5 Verify that the 5 matrices in Table 14.4 satisfy the multiplication rules for co-representations. It is not necessary to examine all possible products. Select, as a typical example, R = crx and B = cra in eq. (14.2.12). [Hint Recall 02 = —1 for spinor representations.] 

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