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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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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.

Groups Theory Definitions…

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Group Theory Definitions…

DEFINITION: Let X be a nonempty set. Then a bijection f: XX 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 BB. 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:  

Moreover the composition of two isometries will preserve distance.

BBBB ))(())(()( 111

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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. 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

|r|

A’

Together the two rotations yield:

B’

-α α

A B

|r’|

|r| |r|

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Possible rotations:

|r| |r| |r|

Case 1: |r'|=0 Case 2: |r'| = |r|  

Case 3 : |r'| = 2|r| Case 4: |r'| = 3|r|  

α= π/3 = 2π/6 α= π/2 = 2π/4

α= 2π/3

α= π = 2π/2

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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 3:

Case 1: Case 2:

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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)

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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The Crystalline Restriction Theorem in R3 yields

14 BRAVAIS LATTICES in

7 CRYSTAL SYSTEMS

Described by “centerings” on different “facings” of the unit cell

Bravais Lattices in R3

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The Seven Crystal Systems Yielding 14 Bravais Latttices 

Triclinic: Monoclinic: Orthorhombic:

Tetragonal: Trigonal:

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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/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

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