1 crystallographic point groups elizabeth mojarro senior colloquium april 8, 2010
TRANSCRIPT
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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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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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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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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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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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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.