Crystallographic Point Groups
Elizabeth Mojarro
Senior Colloquium
April 8, 2010
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Outline
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Group Theory
– Definitions
– Examples
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Isometries
Lattices
Crystalline Restriction Theorem
Bravais Lattices
Point Groups
– Hexagonal Lattice Examples
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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 <g>={gn | n  Z} is a
subgroup of G. G is called a cyclic group if g  G with G=<g>. 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 Cn={1,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 n=2,3,…) is the group of symmetries
of a regular polygon of n-sides including both rotations and reflections.
n=3
n=4
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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 2π/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 α,β  L.
αβ(B)= α(β(B))
= α(B) Since β(B)=B
=B
Identity: The identity isometry I satisfies I(B)=B and Iα= αI= α for α L.
Inverse:  1 ( B)   1 ( ( B))  ( 1 )(B)  B
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
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Two non-collinear vectors a, b of minimal length form a unit cell.
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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.
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Unit Cell:
Lattice :
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Lattice + Unit Cell
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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. 2π/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 α , yielding A’
Rotating B by - α yields
Together the two rotations yield:
A’
B’
|r ’|
|r|
|r|
α
-α
A
|r|
B
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Possible rotations:
Case 1: |r'|=0
Case 2: |r'| = |r|
α= π/3 = 2π/6
Case 3 : |r'| = 2|r|
α= π/2 = 2π/4
Case 4: |r'| = 3|r|
|r|
|r|
|r|
α= π = 2π/2
α= 2π/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, C4 , D4
Point groups:
C2, D3 , D6, C3 , C6 , V
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C3
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Isometries in R3 (see handout)
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Rotations
Reflections
Improper Rotations
Inverse Operations
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Lattices in R3
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Three non-coplanar vectors a, b, c of minimal length form a unit cell.
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DEFINITION: The integral combinations of three non-zero, non-coplanar
vectors (points) is called a space lattice.
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Unit Cell:
Lattice:
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Bravais Lattices in R3
The Crystalline Restriction Theorem in R3 yields
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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:
Tetragonal:
Monoclinic:
Orthorhombic:
Trigonal:
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Hexagonal:
Cubic:
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Crystallography Groups and Point Groups in R3

Crystallography group (space group)
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(Crystallographic) point group
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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/Crystal_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
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