1 Crystallographic Point Groups Elizabeth Mojarro Senior Colloquium April 8, 2010

2 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 R 2 and R 3

3 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. Z n = {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…

4 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 S X. 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 n | n  Z} is a subgroup of G. G is called a cyclic group if  g  G with G=. The element g is called a generator of G. Example: Integers mod n generated by 1. Z n = {0,1,2,…,n-1}. All cyclic finite groups of n elements are the same (“isomorphic”) and are often denoted by C n ={1,g,g 2,…,g n-1 }, of n elements.

5 Other Groups… Example: The Klein Group (denoted V) is a 4-element group, which classifies the symmetries of a rectangle.

6 More Groups… DEFINITION: A dihedral group (D n 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

7 The general dihedral group for a n-sided regular polygon is D n ={e,f, f 2,…, f n-1,g,fg, f 2 g,…,f n-1 g}, where gf i = f -i g,  i. D n 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

8 Isometries in R 2 DEFINITION: An isometry is a permutation  : R 2  R 2 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 .

9 Isometries in R 2 The isometries in are Reflections, Rotations, Translations, and Glide Reflections.

10 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: Moreover the composition of two isometries will preserve distance.

11 Crystal Groups in R 2 DEFINITION: A crystallography group (or space group) is a group of isometries that map R 2 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: D 4

12 Lattices in R 2 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 R 2, then the integral linear combinations of these vectors (points) is called a lattice. Unit Cell: Lattice :

13 Lattice + Unit Cell Crystal in R 2 superimposed on a lattice.

14 Crystalline Restriction Theorem in R 2 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).

15 Crystalline Restriction Theorem in R 2 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’A’ Together the two rotations yield: B’B’ -αα AB |r ’ | |r|

16 Possible rotations: |r| Case 1: |r'|=0Case 2: |r'| = |r| Case 3 : |r'| = 2|r|Case 4: |r'| = 3|r| α= π/3 = 2π/6α= π/2 = 2π/4 α= 2π/3 α= π = 2π/2

17 Bravais Lattices in R 2 Given the Crystalline Restriction Theorem, Bravais Lattices are the only lattices preserved by translations, and the allowable rotational symmetry.

18 Bravais Lattices in R 2 (two vectors of equal length) Case 1:Case 2:

19 Bravais Lattices in R 2 (two vectors of unequal length) Case 3: Case 1: Case 2:

20 Point Groups in R 2 – Some Examples Three examples Point groups: C 2, C 4, D 4 Point groups: C 2, D 3, D 6, C 3, C 6, V

21 C3C3

22 Isometries in R 3 (see handout) Rotations Reflections Improper Rotations Inverse Operations

23 Lattices in R 3 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:

24 The Crystalline Restriction Theorem in R 3 yields 14 BRAVAIS LATTICES in 7 CRYSTAL SYSTEMS Described by “centerings” on different “facings” of the unit cell Bravais Lattices in R 3

25 The Seven Crystal Systems Yielding 14 Bravais Latttices Triclinic: Monoclinic:Orthorhombic: Tetragonal:Trigonal:

26 Hexagonal:Cubic:

27 Crystallography Groups and Point Groups in R 3 Crystallography group (space group) (Crystallographic) point group 32 Total Point Groups in R 3 for the 7 Crystal Systems

28 Table of Point Groups in R 3 Crystal system/Lattice system Point Groups (3-D) TriclinicC 1, (C i ) MonoclinicC 2, C s, C 2h OrthorhombicD 2, C 2v, D 2h TetragonalC 4, S 4, C 4h, D 4 C 4v, D 2d, D 4h TrigonalC 3, S 6 (C 3i ), D 3 C 3v, D 3d HexagonalC 6, C 3h, C 6h, D 6 C 6v, D 3h, D 6h CubicT, T h,O,T d,O h

29 The Hexagonal Lattice

30 {1,6}  {6,5}

31 {1,6}  {5,4} {5,4}  {12,11}

32 {1,6}  {6,5} {6,5}  {13,12}

33 {1,6}  {6,5} {6,5}  {13,8}

34 {1,6}  {5,4} {5,4}  {8,9} {8,9}  {1,2}

35 {1,6}  {6,5} {6,5}  {8,13} {8,13}  {6,1}

36 {1,6}  {6,5} {6,5}  {2,3}

37 Boron Nitride (BN)

38 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 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.

39 Special Thank You Prof. Tinberg Prof. Buckmire Prof. Sundberg Prof. Tollisen Math Department Family and Friends