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Spintronics in Nanostructures Assignment 2

SS 2007 Handing in on Monday 09.04.07

Assistant: Mircea Trif ­ Office 4.4 ­ Tel. 061 267 36 56 ­ Mircea.Trif@unibas.ch

Exercise 1*. Point Group Td . Consider the full point a) Find all elements of b) Determine the order c) Find the classes of T group of a tetrahedron (Td ). the group and determine the order of the group. of the elements of Td . d.

Exercise 2*. Great Orthogonality theorem. Consider irreducible representations of the point group C3v ( : C3v D GL2 (R)). Using matrix representation of C3v , calculate: D(gi ) D(gi )11 , 11
gi C
3v

D(gi ) D(gi )22 , 22
gi C
3v

D(gi ) D(gi )12 , 12
gi C
3v

D(gi ) D(gi )22 . 11
gi C
3v

Using the great orthogonality theorem, find the result for the previous relations. Compare the results. Exercise 3. Characters I Proof the following theorem: Theorem 1. The character for each element in a class is the same. Exercise 4. Characters I I Proof the following theorem: Theorem 2. The number of irreducible representations is equal to the number of classes.

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