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 Crystallographic Point Groups

•  Elizabeth Mojarro
• Senior Colloquium
• April 8, 2010

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Outline

• Group Theory
• Definitions
• Examples
• Isometries
• Lattices
• Crystalline Restriction Theorem
• Bravais Lattices
• Point Groups
• Hexagonal Lattice Examples
• We will be considering all of the above in R2 and
  R3

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Groups Theory Definitions

• DEFINITION  Let G denote a non-empty set and let
  denote a binary operation closed on G. Then
  (G,) forms a group if
• (1) is associative (2) An identity element e
  exists in G (3) Every element g has an inverse
  in G
• Example 1 The integers under addition. The
  identity element is 0 and the (additive) inverse
  of x is x.
• Example 2 R-0 under multiplication.
• Example 3 Integers mod n. Zn 0,1,2,,n-1.
• If H is a subset of G, and a group in its own
  right, call H a subgroup of G.

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Group Theory Definitions

• DEFINITION Let X be a nonempty set. Then a
  bijection f X?X is called a permutation. The set
  of all permutations forms a group under
  composition called SX. These permutations are
  also called symmetries, and the group is called
  the Symmetric Group on X.
• DEFINITION Let G be a group. If g ? G, then
  ltggtgn n ? Z is a subgroup of G. G is called
  a cyclic group if ?g ? G with Gltggt. The element
  g is called a generator of G. 
• Example Integers mod n generated by 1. Zn
  0,1,2,,n-1.
• All cyclic finite groups of n elements are the
  same (isomorphic) and are often denoted by
  Cn1,g,g2,,gn-1 , of n elements.
•  

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Other Groups

• Example The Klein Group (denoted V) is a
  4-element group, which classifies the symmetries
  of a rectangle.

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More Groups

• DEFINITION A dihedral group (Dn for n2,3,) is
  the group of symmetries of a regular polygon of
  n-sides including both rotations and reflections.
  
• n3
  n4

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• The general dihedral group for a n-sided regular
  polygon is
• Dn e,f, f2,, fn-1,g,fg, f2g,,fn-1g, where
  gfi f-i g, ?i. Dn is generated by the two
  elements f and g , such that f is a rotation of
  2p/n and g is the flip (reflection) for a total
  of 2n elements.

f
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Isometries in R2

• DEFINITION An isometry is a permutation ? R2
  ? R2 which preserves Euclidean distance the
  distance between the points of u and v equals the
  distance between of ?(u) and ?(v). Points that
  are close together remain close together after ?.

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Isometries in R2

• The isometries in are Reflections, Rotations,
  Translations, and Glide Reflections.
•  

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Invariance

• Lemma The set of all isometries that leave an
  object invariant form a group under composition.
•  Proof Let L denote a set of all isometries
  that map an object B?B.
• The composition of two bijections is a bijection
  and composition is associative.
• Let a,ß ? L.
  
• aß(B) a(ß(B))
• a(B) Since ß(B)B
• B
• Identity The identity isometry I satisfies
  I(B)B and Ia aI a for ?a? L.
• Inverse  
• Moreover the composition of two isometries will
  preserve distance.

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Crystal Groups in R2

• DEFINITION A crystallography group (or space
  group) is a group of isometries that map R2 to
  itself.
• DEFINITION If an isometry leaves at least one
  point fixed then it is a point isometry.
• DEFINITION A crystallographic group G whose
  isometries leave a common point fixed is called a
  crystallographic point group.
• Example D4

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Lattices in R2

• Two non-collinear vectors a, b of minimal length
  form a unit cell.
• DEFINITION If vectors a, b is a set of two
  non-collinear nonzero vectors in R2, then the
  integral linear combinations of these vectors
  (points) is called a lattice.
• Unit Cell Lattice

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Lattice Unit Cell

• Crystal in R2 superimposed on a lattice.

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Crystalline Restriction Theorem in R2

• What are the possible rotations around a fixed
  point?
• THEOREM The only possible rotational symmetries
  of a lattice are 2-fold, 3-fold, 4-fold, and
  6-fold rotations (i.e. 2p/n where n 1,2,3,4 or
  6).

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Crystalline Restriction Theorem in R2

• Proof Let A and B be two distinct points at
  minimal distance.
• Rotate A by an angle a , yielding A

Rotating B by - a yields
Together the two rotations yield
B
r
r
r
-a
a
A
B
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Possible rotations

• Case 1 r'0

• Case 2 r' r  

a p/3 2p/6
a p/2 2p/4
Case 3 r' 2r
Case 4 r' 3r  
a p 2p/2
a 2p/3
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Bravais Lattices in R2

• Given the Crystalline Restriction Theorem,
  Bravais Lattices are the only lattices preserved
  by translations, and the allowable rotational
  symmetry.

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Bravais Lattices in R2 (two vectors of equal
length)
Case 1 Case 2
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Bravais Lattices in R2 (two vectors of unequal
length)
Case 1
Case 2
Case 3
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Point Groups in R2 Some Examples

• Three examples

Point groups C2, D3 , D6, C3 , C6 , V
Point groups C2, C4 , D4
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C3
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Isometries in R3 (see handout)

• Rotations
• Reflections
• Improper Rotations
• Inverse Operations

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Lattices in R3

• Three non-coplanar vectors a, b, c of minimal
  length form a unit cell.
• DEFINITION The integral combinations of three
  non-zero, non-coplanar vectors (points) is called
  a space lattice.
• Unit Cell Lattice

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Bravais Lattices in R3

• The Crystalline Restriction Theorem in R3 yields
  
• 14 BRAVAIS LATTICES in
• 7 CRYSTAL SYSTEMS
• Described by centerings on different facings
  of the unit cell

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The Seven Crystal Systems Yielding 14 Bravais
Latttices 
Triclinic Monoclinic Orthorhombic
Tetragonal Trigonal Orthorhombic
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Hexagonal Cubic
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Crystallography Groups and Point Groups in R3

• Crystallography group (space group)
• (Crystallographic) point group
• 32 Total Point Groups in R3 for the 7 Crystal
  Systems

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Table of Point Groups in R3
Crystal system/Lattice system Point Groups (3-D)
Triclinic C1, (Ci )
Monoclinic C2, Cs, C2h
Orthorhombic D2 , C2v, D2h
Tetragonal C4, S4, C4h, D4 C4v, D2d, D4h
Trigonal C3, S6 (C3i), D3 C3v, D3d
Hexagonal C6, C3h, C6h, D6 C6v, D3h, D6h
Cubic T, Th ,O ,Td ,Oh
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The Hexagonal Lattice
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1,6?6,5
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1,6?5,4 5,4?12,11
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1,6?6,5 6,5?13,12
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1,6?6,5 6,5?13,8
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1,6?5,4 5,4?8,9 8,9?1,2
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1,6?6,5 6,5?8,13 8,13?6,1
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1,6? 6,5 6,5?2,3
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Boron Nitride (BN)
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Main References

• Boisen, M.B. Jr., Gibbs, G.V., (1985).
  Mathematical Crystallography An Introduction to
  the Mathematical Foundations of Crystallography.
  Washington, D.C. Bookcrafters, Inc.
• Crystal System. Wikipedia. Retrieved (2009
  November 25) from http//en.wikipedia.org/wiki/Cry
  stal_system
• Evans, J. W., Davies, G. M. (1924). Elementary
  Crystallography. London The Woodbridge Press,
  LTD.
• Rousseau, J.-J. (1998). Basic Crystallography.
  New York John Wiley Sons, Inc.
• Sands, D. E (1993). Introduction to
  Crystallography. New York Dover Publication,
  Inc.
• Saracino, D. (1992). Abstract Algebra A First
  Course. Prospect Heights, IL Waverland Press,
  Inc.

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Special Thank You

• Prof. Tinberg
• Prof. Buckmire
• Prof. Sundberg
• Prof. Tollisen
• Math Department
• Family and Friends