1 / 16

Point Symmetry

Learn about point symmetry groups, including improper operations and the relation between 1st and 2nd sort operations. Discover how groups can be formed, decomposed, and transformed while exploring rotoinversion axes and proper point groups.

michaelcampos
Download Presentation

Point Symmetry

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Point Symmetry Groups w/ improper operations Any group containing 2nd sort operations also contains 1st sort operations (Aa i) (Ba i) = Aa Ba i i = Aa Ba

  2. Point Symmetry Groups w/ improper operations Any group containing 2nd sort operations also contains 1st sort operations (Aa i) (Ba i) = Aa Ba i i = Aa Ba 1st sort operations in such a group G form a subgroup of index 2 g = 1 a2 …. an Bg = B Ba2 …. Ban B is any 2nd sort operation in G

  3. Point Symmetry Groups w/ improper operations Any group containing 2nd sort operations also contains 1st sort operations (Aa i) (Ba i) = Aa Ba i i = Aa Ba 1st sort operations in such a group G form a subgroup of index 2 g = 1 a2 …. an Bg = B Ba2 …. Ban B is any 2nd sort operation in G (Ba i) ar = (Ba ar) i all 2nd sort operations in Bg 1st sort

  4. Point Symmetry Groups w/ improper operations 1. Any group of 2nd sort contains equal nos. of operations of 1st & 2nd sort 2. In any group of the 2nd sort, 1st sort operations form a subgroup of index 2

  5. Point Symmetry Groups w/ improper operations 1. Any group of 2nd sort contains equal nos. of operations of 1st & 2nd sort 2. In any group of the 2nd sort, 1st sort operations form a subgroup of index 2 3. A group of the 2nd sort can be formed by adding any second sort operation (and its products) to a group such that it transforms that group into itself

  6. Point Symmetry Groups w/ improper operations 1. Any group of 2nd sort contains equal nos. of operations of 1st & 2nd sort 2. In any group of the 2nd sort, 1st sort operations form a subgroup of index 2 3. A group of the 2nd sort can be formed by adding any second sort operation (and its products) to a group such that it transforms that group into itself 4. All groups of the 2nd sort can be formed by starting with all groups of the 1st sort & adding each possible 2nd sort operation (i or m) & products which transforms the group into itself

  7. Point Symmetry Groups w/ improper operations Any rotoinversion axis, except those for which n = 4N, can be decomposed into a combination of rotation axis & inversion or reflection

  8. Point Symmetry Groups not containing i G = g, ng = g, gn (g of index 2…cosets =…n transforms g into itself) For every pt grp G of 1st sort which can be expressed as product of subgroup g of index 2 and operation of 1st sort n (n even), can get a corresponding group G from same subgrp & 2nd sort operation n

  9. Point Symmetry Groups not containing i G = g, ng = g, gn (g of index 2…cosets =…n transforms g into itself) For every pt grp G of 1st sort which can be expressed as product of subgroup g of index 2 and operation of 1st sort n (n even), can get a corresponding group G from same subgrp & 2nd sort operation n 2 (=m) 4 6 (If n odd, order of n = 2x that of n)

  10. Point Symmetry

  11. Point Symmetry C4 C4 C4 C2 C2 C4 C2 C4 C2 C4 2 3 2 3 C4 C4 C4 m m C4 m C4 m C4 2 3 3 2

  12. Point Symmetry Groups containing i wrt i, any proper point group is of index 2 … i transforms any group into itself So, get new groups by G i

  13. Point Symmetry G G i 1 1 2 2/m 3 3 4 4/m 6 6/m 222 2/m 2/m 2/m 32 3 2/m 422 4/m 2/m 2/m 622 6/m 2/m 2/m 23 2/m 3 432 4/m 3 2/m

  14. Point Symmetry 4 i = 4/m C4 C2 C4 i i C4 i C2 i C4 G G i 1 1 2 2/m 3 3 4 4/m 6 6/m 222 2/m 2/m 2/m 32 3 2/m 422 4/m 2/m 2/m 622 6/m 2/m 2/m 23 2/m 3 432 4/m 3 2/m -1 -1

  15. Point Symmetry 4 i = 4/m C4 C2 C4 i i C4 i C2 i C4 2 i = 2/m C2 i i C2 G G i 1 1 2 2/m 3 3 4 4/m 6 6/m 222 2/m 2/m 2/m 32 3 2/m 422 4/m 2/m 2/m 622 6/m 2/m 2/m 23 2/m 3 432 4/m 3 2/m -1 -1

  16. Point Symmetry 4 i = 4/m C4 C2 C4 i i C4 i C2 i C4 2 i = 2/m C2 i i C2 2' i = 2'/m' C2 i i C2 G G i 1 1 2 2/m 3 3 4 4/m 6 6/m 222 2/m 2/m 2/m 32 3 2/m 422 4/m 2/m 2/m 622 6/m 2/m 2/m 23 2/m 3 432 4/m 3 2/m -1 -1 l l

More Related