Bilbao Crystallographic Server IR Raman Hyper-Raman Modes

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Information of the Point Group C₆ (6)

Character Table¹

C6(6)	#	1	6+	3+	2	3-	6-	functions
A	Γ1	1	1	1	1	1	1	z,x2+y2,z2,Jz
B	Γ4	1	-1	1	-1	1	-1	·
1E2 2E2	Γ3 Γ2	1 1	w w2	w2 w	1 1	w w2	w2 w	(x2-y2,xy)
2E1 1E1	Γ5 Γ6	1 1	-w2 -w	w w2	-1 -1	w2 w	-w -w2	(x,y),(xz,yz),(Jx,Jy)


w = exp(2iπ/3)

To get the list of the irreducible representations in their matrix form please click here.



Raman and Hyper-Raman Tensors

Raman Tensors	Hyper-Raman Tensors
A(z) 1E1(x) 2E1(y) 1E2 2E2 a · · · · c · · -d e f · f -e · · a · · · d · · c f -e · -e -f · · · b c d · -d c · · · · · · ·	A(z) B 1E1(x) 2E1(y) E2 · · · · b · c -c · · · -d 3e e f · · g -3g -g -h · · e · · · k -i · · · · b · · -d d · · · -c g 3g h · · e e 3e f · · -g · · · i k · b b a · · · · · · · · · · · · h f · · · · f -h · -i i · · · k


To transform the tensors please click here.

Information of the Space Group P6₅ (No. 170)

Selection Rules

In the following tables one can find which are the allowed representations for each one of the selected Wyckoff positions for IR, Raman and Hyper-Raman processes.


• IR Active Modes: (Γ_acoustic = A + ¹E₁ + ²E₁ )



WP	A	B	2E2	1E1	1E2	2E1
6a	3	·	·	3	·	3




• Raman Active Modes



WP	A	B	2E2	1E1	1E2	2E1
6a	3	·	3	3	3	3




• Hyper-Raman Active Modes



WP	A	B	2E2	1E1	1E2	2E1
6a	3	3	3	3	3	3



The polarization selection rules for Raman and Hyper-Raman processes.



Modes Activity

This table is a summary of the activity of the different modes of the space group.



-	A	B	2E2	1E1	1E2	2E1
Infrared	x	·	·	x	·	x
Raman	x	·	x	x	x	x
Hyper-Raman	x	x	x	x	x	x

Note: x ≡ represents the modes which can be detected.


More information:



Permutational Representation

The permutational representation represents the number of atoms that are invariant under the symmetry operations of the group. In this table one can find the representation for each one of the Wyckoff positions.



WP	A	B	2E2	1E1	1E2	2E1
6a	1	1	1	1	1	1




Mechanical Representation

The mechanical representation is defined as the direct product of the permutational representation with the vectorial representation. The mechanical representation for each one of the chosen Wyckoff positions is given in the following table.



WP	A	B	2E2	1E1	1E2	2E1	Modes
6a	3	3	3	3	3	3	Show

Note: Click in the Show option to obtain the symmetry adapted modes for a given orbit.


Irreps decompositions

In the following table one can find the decomposition of the vectorial, pseudo vectorial and so on representations.



C6 (6)	A	B	2E2	1E1	1E2	2E1
V	1	·	·	1	·	1
A	1	·	·	1	·	1
[V2]	2	·	1	1	1	1
[V2] x V	4	2	2	4	2	4

Information about second order processes:



Second Order IR Activity

In the next table one can find the information the second order IR active modes:



⊗ A B 2E2 1E1 1E2 2E1 A A B 2E2 1E1 1E2 2E1 B · A 1E1 2E2 2E1 1E2 2E2 · · 1E2 2E1 A B 1E1 · · · 1E2 B A 1E2 · · · · 2E2 1E1 2E1 · · · · · 2E2

Active               Inactive       bold   IR active modes





Second Order Raman Activity

In the next table one can find the information the second order Raman active modes:



⊗ A B 2E2 1E1 1E2 2E1 A A B 2E2 1E1 1E2 2E1 B · A 1E1 2E2 2E1 1E2 2E2 · · 1E2 2E1 A B 1E1 · · · 1E2 B A 1E2 · · · · 2E2 1E1 2E1 · · · · · 2E2

Active               Inactive       bold   Raman active modes


For more information about the second order Raman go the the program RAMAN