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notes06-07-page237

notes06-07-page237 - 3 = x 1 y 1 x 2 y 2 x 3 y 3(32.3 For...

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The above considerations leave four vectors. Infact ,therew i l la lwaysbethesamenumbero fvectorsassymmetrye lements . Altogether, the vectors represent what is call an irreducible representation of the group. These vectors make up the : C 2 v ˆ E ˆ C 2 ˆ σ v ˆ σ 0 v A 1 1111 A 2 11 1 1 B 1 1 11 1 B 2 1 1 11 ∗∗∗ See Handout on Character Tables ∗∗∗ 32.3.1. Direct Products The direct product of a two vectors is de ned as ( x 1 ,x 2 ,x 3 ,... ) ( y 1 ,y
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Unformatted text preview: 3 , . . . ) = ( x 1 y 1 , x 2 y 2 , x 3 y 3 , . . . ) (32.3) For the example of the C 2 v group consider B 1 ⊗ B 2 = (1 , − 1 , 1 , − 1) ⊗ (1 , − 1 , − 1 , 1) = (1 , 1 , − 1 , − 1) = A 2 (32.4) 32.4. Symmetry Breaking and Crystal Field Splitting We shall investigate how degeneracies of energy levels are broken as one reduces the overall symmetry of the system. 225...
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