Space Groups of Crystal Lattices

1
Space Groups of Crystal Lattices

• Ramsey Majzoub
• PHYS 215C Final

• Contents
• Group Theory Refresher
• Notation, Notation, Notation
• Specific Examples 2D
• Carbon Nanotubes and Line Groups

2
Where we left off the H2O molecule

• By considering symmetries of a molecule we can
  deduce information about the vibrational modes
  
• Extending Lecture Note Ideas
• Horizontal, Vertical and Diagonal
• Reflection Planes sh, sv, sd
• Improper Rotations Sn
• Algorithim
• (1)Determine Matrix Representations of Symmetry
  Operators
  
• (a) By assigining orientation vectors to atoms
  and graphically operating on them
  
• (2) Calculate the Character Table
• (3)Find out which irreducible representations the
  vibration modes contain and make sure you remove
  all translations and rotations!
  
• For our homework problem we did a 2D highly
  symmetric molecule but generalizing this should
  at least seem possible to you by now
  

3
Space Group Symmetry Operations

• Now lets consider an infinite lattice of these
  molecules.
  
• Space Group Symmetries will depend on point group
  symmetry of molecule and translational symmetry
  of lattice
  
• Side Note Symmetries not completely phase
  dependent! ie amorphous glass (solid) does not
  have a space group and liquid crystals (not
  solid) do have space groups

4
Hermann-Manguin Notation

• Space Group Operators must be able to represent
  translations and rotations

Pure Translation
Pure Rotation
Reflections, Rotations
Translations
-Form a subgroup which leave the Bravais
lattice intact -Limit Rotations to 1,2,3,4,6fol
d
Matrix Representation acting on a coordinate
system
Space Groups are more than just a direct product
of translations and point group operations
The two types of symmetry operations dont even
commute
5
A Proof Using Matrix Representation

• Consider two space group operations

Pure Translations or Rotations DO commute
Since space groups are more then direct products
between translations and point groups there
might be extra or missing symmetries
6
New Symmetries

• Glide Plane
• Translation Reflection
• Screw Axis
• Translation Rotation (3D)

7
Considering all the possibilities

• 230 Space Groups in 3D
• 73 are symmorphic
• 157 are nonsymmorphic
• 17 Wallpaper or 2D Space Groups
• Detailed information in X Ray Crystallography Book

8
Wallpaper Group Examples

• Notation
• p4gm
• p - unit cell (primitive)
• 4 - Highest Order Symmetry Preserving
  Rotation
  
• g - glide plane
• m - mirror plane
• p31m

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Your Turn!

• What are the rotational symmetries?
• Where are the mirror/glide planes?
• What would we classify this lattice as in HM
  notation?

p2mm
Joyce, David http//www.clarku.edu/djoyce/wallpap
er 1994, 1997
10
Carbon Nanotubes and Line Groups

• Nanotubes Chiral and Achiral

Circumference
All types of tubes can be described by (n, m)
Define
Which lets us define an area and number of carbon
atoms in unit cell
But any translation should be a symmetry
operation, lets write it in terms of Ch and T
This just a screw axis symmetry operation Ci
gT Now we can calculate electron and phonon pro
perties of nanotubes!
11
Concluding Remarks

• Space groups can yield information about phonon
  and electron energy levels
  
• Point Groups Translation Space Group
• Questions?

References Chatterjee, Sanat K. Crystallograph
y and the World of Symmetry Springer 2008
Jaswon, M.A. and Rose, M.A. Crystal Symmetry The
ory of Colour Crystallography Ellis Horwood
1983 Dresselhaus et al. Group Theory Applicati
ons to the Physics of Condensed Matter Springer
2008 M. Damnjanovic et al. Full Symmetry, optic
al activity and potentials of single wall and
multwall nanotubes Phys. Rev. B 60 2728 (1999)
Joyce, David http//www.clarku.edu/djoyce/wallp
aper 1994, 1997 http//en.wikipedia.org/wiki/Wa
llpaper_group