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If number of non-degenerate irreducible representation is eight for D4th point group. The number of double degerate is irreducible representation is
- a)two
- b)four
- c)six
- d)eight
Correct answer is option 'A'. Can you explain this answer?
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Number of Non-Degenerate Irreducible Representations in D4 Point Group
The D4 point group has 8 symmetry operations- identity, C4, C2, C4^-1, σv, σd, σv', and σd'. Using character table, the number of non-degenerate irreducible representations for D4 point group can be calculated. It is given by:
Number of non-degenerate irreducible representations = (1/8) [h(Γ1) + h(Γ2) + h(Γ3) + 2h(Γ4)]
where h(Γi) denotes the number of elements in the i-th irreducible representation. Substituting the values in the above formula, we get:
Number of non-degenerate irreducible representations = (1/8) [1 + 1 + 2 + 2] = 1
Number of Double Degenerate Irreducible Representations in D4 Point Group
Since the total number of irreducible representations for D4 point group is 8 and the number of non-degenerate irreducible representations is 1, the remaining 7 irreducible representations must be degenerate. The degenerate irreducible representations are of two types- double degenerate and triple degenerate. Using the formula below, we can calculate the number of double degenerate irreducible representations:
Number of double degenerate irreducible representations = (1/2) [h(Γ5) + h(Γ6) + h(Γ7)]
where h(Γi) denotes the number of elements in the i-th irreducible representation. Substituting the values in the above formula, we get:
Number of double degenerate irreducible representations = (1/2) [2 + 2 + 2] = 2
Therefore, the correct option is A: Two.
Number of Non-Degenerate Irreducible Representations in D4 Point Group
The D4 point group has 8 symmetry operations- identity, C4, C2, C4^-1, σv, σd, σv', and σd'. Using character table, the number of non-degenerate irreducible representations for D4 point group can be calculated. It is given by:
Number of non-degenerate irreducible representations = (1/8) [h(Γ1) + h(Γ2) + h(Γ3) + 2h(Γ4)]
where h(Γi) denotes the number of elements in the i-th irreducible representation. Substituting the values in the above formula, we get:
Number of non-degenerate irreducible representations = (1/8) [1 + 1 + 2 + 2] = 1
Number of Double Degenerate Irreducible Representations in D4 Point Group
Since the total number of irreducible representations for D4 point group is 8 and the number of non-degenerate irreducible representations is 1, the remaining 7 irreducible representations must be degenerate. The degenerate irreducible representations are of two types- double degenerate and triple degenerate. Using the formula below, we can calculate the number of double degenerate irreducible representations:
Number of double degenerate irreducible representations = (1/2) [h(Γ5) + h(Γ6) + h(Γ7)]
where h(Γi) denotes the number of elements in the i-th irreducible representation. Substituting the values in the above formula, we get:
Number of double degenerate irreducible representations = (1/2) [2 + 2 + 2] = 2
Therefore, the correct option is A: Two.
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If number of non-degenerate irreducible representation is eight for D4th point group. The number of double degerate is irreducible representation isa)twob)fourc)sixd)eightCorrect answer is option 'A'. Can you explain this answer?
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