1. Line groups
Review
To enumerate the 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 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 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 1 C2 C2' C2" m m' m" i
repeat unit 1 C2
C2'
C2" m m'
m" i
2/m 2/m 2/m m m' C2
C2"
m"
C2'

12. Line groups G Note glide in this planar structure 1t 2t 3t zt xt yt it
repeat unit G 1t 2t 3t zt xt yt it
2/m 2/m 2/m
m,zt
m'.xt
C2,1t
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 G Note glide in this planar structure 1t 2t 3t zt xt yt it
repeat unit G 1t 2t 3t zt xt yt it
2/m 2/m 2/m
m,zt
m'.xt
C2,1t
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 G Note glide in this planar structure 1t 2t 3t zt xt yt it
m'.xt
C2",3t
L 21/sm s2/c 2/m
or L 21/mcm
m,zt
repeat unit
m",yt
C2',2t G 1t 2t 3t zt xt yt it
2/m 2/m 2/m
m,zt
m'.xt
C2,1t
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 84/m + 1/2 (4,0)
± 1/2

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

30. Line groups Line groups - example - nanotubes 84 84/mcm + 1/2 (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