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Line groups

Line groups. Review To enumerate the 230 3-D space groups: pt. grps. taken in turn & extended with conforming 3-D lattice translation groups. Further, translations associated with glide planes & screw axes added by using homomorphism betwn isogonal pt. grps. & translation groups. Line groups.

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Line groups

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  1. Line groups Review To enumerate the 230 3-D space groups: pt. grps. taken in turn & extended with conforming 3-D lattice translation groups. Further, translations associated with glide planes & screw axes added by using homomorphism betwn isogonal pt. grps. & translation groups.

  2. Line groups Review To enumerate the 230 3-D space groups: pt. grps. taken in turn & extended with conforming 3-D lattice translation groups. Further, translations associated with glide planes & screw axes added by using homomorphism betwn isogonal pt. grps. & translation groups. The 17 2-D plane groups can be developed in a similar manner. Removal of a dimension severely restricts no. of symmetry combinations.

  3. Line groups Review To enumerate the 230 3-D space groups: pt. grps. taken in turn & extended with conforming 3-D lattice translation groups. Further, translations associated with glide planes & screw axes added by using homomorphism betwn isogonal pt. grps. & translation groups. The 17 2-D plane groups can be developed in a similar manner. Removal of a dimension severely restricts no. of symmetry combinations. What about 1-D line groups? Some surprises(?)

  4. Line groups What about 1-D line groups? Consider a 3-D object of arbitrary pt. symmetry repeated along 1-D lattice

  5. Line groups What about 1-D line groups? Consider a 3-D object of arbitrary pt. symmetry repeated along 1-D lattice Only one type of translation - no possibility for centering translations Translations simply described by single lattice parameter

  6. Line groups What about 1-D line groups? Consider a 3-D object of arbitrary pt. symmetry repeated along 1-D lattice Only one type of translation - no possibility for centering translations Translations simply described by single lattice parameter But lattice parameter not enough to describe structure of array

  7. Line groups Need to also describe symmetry of object Can use pt. grps. for this All objects can be described by one of the 32 pt. grps.

  8. Line groups Need to also describe symmetry of object Can use pt. grps. for this All objects can be described by one of the 32 pt. grps. Doesn't quite work Consider:

  9. Line groups Note glide in this planar structure repeat unit

  10. Line groups Note glide in this planar structure c-glides allowed repeat unit

  11. Line groups Note glide in this planar structure repeat unit 2/m 2/m 2/m m C2 m' C2" m" C2'

  12. Line groups Note glide in this planar structure repeat unit 2/m 2/m 2/m m,zt C2,1t m'.xt C2",3t m",yt C2',2t Rules: 1t 2t = 3t 1t zt = it 1t xt = yt (as before) 1t it = zt 1t yt = xt

  13. Line groups Note glide in this planar structure repeat unit 2/m 2/m 2/m m,zt C2,1t m'.xt C2",3t m",yt C2',2t Rules: 1t 2t = 3t 1t zt = it 1t xt = yt Choose: 1t = c/2, 3t = 0 (as before) 1t it = zt 1t yt = xt it = 0, yt = 0 Then: 2t = zt = xt = c/2

  14. Line groups Note glide in this planar structure C2,1t m'.xt C2",3t L 21/sm s2/c 2/m or L 21/mcm m,zt repeat unit m",yt C2',2t 2/m 2/m 2/m m,zt C2,1t m'.xt C2",3t m",yt C2',2t Rules: 1t 2t = 3t 1t zt = it 1t xt = yt Choose: 1t = c/2, 3t = 0 (as before) 1t it = zt 1t yt = xt it = 0, yt = 0 Then: 2t = zt = xt = c/2

  15. Line groups Note glide in this planar structure Also consider: repeat unit

  16. Line groups Note glide in this planar structure Also consider: "Non-crystallographic" rotations can be crystallographic in 1-D lattices repeat unit

  17. Line groups Rotation axes of any order, up to C∞, proper or improper, allowed C∞

  18. Line groups Rotation axes of any order, up to C∞, proper or improper, allowed Multiple rotation axes (n/n) allowed (n even perpendicular mirror) C∞

  19. Line groups Rotation axes of any order, up to C∞, proper or improper, allowed Multiple rotation axes (n/n) allowed (n even perpendicular mirror) Screw axes of any kind (21, 53, 149) allowed C∞

  20. Line groups Summary: In line groups, pt. symmetry extended considerably Translations along lattice direction allowed No translations in any other direction

  21. Line groups • Procedure • Combine lattice translations w/ 32 pt. grps. As in 2-D & 3-D, lattice group must be invariant under all pt. grp. operations Thus, while infinite no. of types of rotation axes allowed along lattice direction, only 2 & 2 axes allowed otherwise & must be perpendicular to lattice direction

  22. Line groups • Procedure • 1. Combine lattice translations w/ 32 pt. grps. • 2. Add rotation axes not allowed in 2-D & 3-D (use Euler construction) • Or remember: • New axis appears at angle of 1/2 throw of • main rotation axis new axis

  23. Line groups Procedure 1. Combine lattice translations w/ 32 pt. grps. 2. Add rotation axes not allowed in 2-D & 3-D (use Euler construction) 3. Determine line groups isogonal w/ all allowed point groups Infinite no. of line groups possible

  24. Line groups Procedure 1. Combine lattice translations w/ 32 pt. grps. 2. Add rotation axes not allowed in 2-D & 3-D (use Euler construction) 3. Determine line groups isogonal w/ all allowed point groups Infinite no. of line groups possible Examples for 1: L1, L4/m, L622, L2/m 2/m 2/m

  25. Line groups Procedure 1. Combine lattice translations w/ 32 pt. grps. 2. Add rotation axes not allowed in 2-D & 3-D (use Euler construction) 3. Determine line groups isogonal w/ all allowed point groups Infinite no. of line groups possible Examples for 1: L1, L4/m, L622, L2/m 2/m 2/m Examples for 2: L5, L14/m, L8/m 2/m 2/m

  26. Line groups Line groups (from Vujicic, Bozovic, & Herbut, J. Phys. A 10, 1271 (1977) Pt. grp. n odd n even (w/example) Cn Ln Lnp Cnv (4mm)Lnm Lnmm Lnc Lncc L(2q)qmc Cnh (4/m) Ln/m L(2q)q/m S2n (3) Ln L(2n) Dn (422) Ln2 Ln22 Lnp2 Lnp22 Dnd (42m) Lnm L(2n)2m Lnc L(2n)2c Dnh (4/m 2/m 2/m) L(2n)2m Ln/mmm L(2n)2c Ln/mcm L(2q)q/mcm n = 1, 2, …. p = 1, 2, …., n-1

  27. Line groups Line groups - example - nanotubes 84 + 1/2 (4,0)

  28. Line groups Line groups - example - nanotubes 84 + 1/2 84/m (4,0) ± 1/2

  29. Line groups Line groups - example - nanotubes 84 + 1/2 84/m_m (4,0) ± 1/2

  30. Line groups Line groups - example - nanotubes 84 + 1/2 84/mcm (4,0) 1/4 ± 1/2

  31. Line groups Line groups - example - nanotubes 0° Ch = na1 + ma2 (n,m)

  32. Line groups Line groups - example - nanotubes

  33. Line groups Line groups - example - nanotubes armchair zigzag chiral

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