Mathematical Foundations of Crystallography

Bill SteinBill Stein Adam ClouesAdam Cloues David BauerDavid Bauer Britt WooldridgeBritt Wooldridge Mang YangMang Yang Chris BouzekChris Bouzek

What are Crystals?

““Regular Arrangements” of Regular Arrangements” of Atoms/Molecules in Solids.Atoms/Molecules in Solids.

Question: What kinds of arrangements are Question: What kinds of arrangements are possible? possible?

Answer: Use Group Theory to describe the Answer: Use Group Theory to describe the possibilities. possibilities.

Applied Group Theory

““Groups” model concepts of symmetry.Groups” model concepts of symmetry. Set of objects w/ a binary operation such Set of objects w/ a binary operation such

that:that: An identity element e exists: ge = eg = gAn identity element e exists: ge = eg = g There exist inverses: g(gThere exist inverses: g(g-1-1) = (g) = (g-1-1)g = e)g = e Operation is associative: (gh)k = g(hk)Operation is associative: (gh)k = g(hk)

Group Examples

Z = “the integers” Z = “the integers”

= {…, -3, -2, -1, 0, 1, 2, 3, …..}= {…, -3, -2, -1, 0, 1, 2, 3, …..} Orthogonal Group O(n)Orthogonal Group O(n) Matrix Groups:Matrix Groups:

GL(n,C) = {A a complex invertible matrix}GL(n,C) = {A a complex invertible matrix} SL(n,C) = {A SL(n,C) = {A GL(n,C) | det(A) = 1} GL(n,C) | det(A) = 1}

Orthogonal Group

Definition:Definition: A 2x2 real matrix A is orthogonal if:A 2x2 real matrix A is orthogonal if:

AAAATT = A = ATTA = I (i.e. if AA = I (i.e. if ATT = A = A-1-1).). Let ||x|| be the length of vector x. A Let ||x|| be the length of vector x. A

distance preservingdistance preserving linear transformation linear transformation T is said to be T is said to be orthogonalorthogonal. Such . Such transformations form the transformations form the orthogonal orthogonal groupgroup O(n). O(n).

Subgroups of the Orthogonal Group

An example of a subgroup of the orthogonal An example of a subgroup of the orthogonal group is rotation about the origin on the group is rotation about the origin on the plane:plane:

cosθ sinθ

sinθ- cosθ

Subgroups

Non-empty subsets H are subgroups if:Non-empty subsets H are subgroups if: h,k h,k H H hk hk H (closure) H (closure) h h H H (h (h-1-1) ) H (inverses) H (inverses)

Examples:Examples: ZZ33 = {0,1,2} is a subgroup of the integers = {0,1,2} is a subgroup of the integers SL(n,R) = {A SL(n,R) = {A GL(n,R) | det(A) = 1} is GL(n,R) | det(A) = 1} is

a subgroup of the General Linear group a subgroup of the General Linear group GL(n,R).GL(n,R).

Equivalence Classes

““~” is an equivalence relation if it is:~” is an equivalence relation if it is: Reflexive: h Reflexive: h G, h ~ h G, h ~ h Symmetric: h, k Symmetric: h, k G, G, h ~ k h ~ k k ~ h k ~ h Transitive: h ~ k, k ~ g Transitive: h ~ k, k ~ g h ~ g h ~ g

Equivalence Classes cont’d.

An equivalence class is defined as a An equivalence class is defined as a subset subset of the formof the form {x {x G | x~y}, where y is an G | x~y}, where y is an element of element of GG and the and the notationnotation "x~y" is used "x~y" is used to mean that there is an to mean that there is an equivalence relationequivalence relation between x and y. between x and y.

Equivalence classes are maximal sets of Equivalence classes are maximal sets of equivalent elements.equivalent elements.

Example of Equivalence Class

ZZ33 (Z “mod” 3) (Z “mod” 3)

In this world, two elements are equivalent if In this world, two elements are equivalent if their difference is divisible by three.their difference is divisible by three.

Coset Partitioning

The concept of partitioning a group G The concept of partitioning a group G allows a better understanding of the allows a better understanding of the structure of G by breaking the group into structure of G by breaking the group into parts (e.g. left cosets).parts (e.g. left cosets).

A A subsetsubset of G of the form gH for some of G of the form gH for some g g G is said to be a G is said to be a left cosetleft coset of H. of H. gH = {gh | h gH = {gh | h H} H}

Example of Coset Partitioning

Example: G can be partitioned by cosets, Example: G can be partitioned by cosets, also by conjugacy classes.also by conjugacy classes.

Conjugacy Classes

Given G (any group), we say that for Given G (any group), we say that for h, k, g h, k, g G, G, h~k ( “h is conjugate to k” ) if h~k ( “h is conjugate to k” ) if k = ghgk = ghg-1.-1. This is known as This is known as conjugation.conjugation.

A conjugacy class is an equivalence class.A conjugacy class is an equivalence class. In this case, equivalence classes are called In this case, equivalence classes are called

conjugacy classes.conjugacy classes.

Conjugacy Classes cont’d.

Each element in a group belongs to exactly Each element in a group belongs to exactly one class, and the identity element ‘e’ is one class, and the identity element ‘e’ is always its own conjugacy class.always its own conjugacy class.

The Full Symmetric Group SX

Let X be a set.Let X be a set. SSXX is the set of all 1:1 and onto maps of the is the set of all 1:1 and onto maps of the

set X onto itself. set X onto itself. The permutation groups are subgroups The permutation groups are subgroups

under Sunder SXX..

Permutation Groups

Definition: A permutation group is a Definition: A permutation group is a subgroup of the group Ssubgroup of the group SXX..

Permutation groups are otherwise known as Permutation groups are otherwise known as transformation groups.transformation groups.

Examples of Permutation Groups

The Symmetric Group for the natural The Symmetric Group for the natural numbers (Snumbers (SNN).).

The Dihedral Group (DThe Dihedral Group (Dnn).).

The Alternating Group (AThe Alternating Group (Ann).).

Symmetric Group SN

SSNN is the full symmetric group for the is the full symmetric group for the

natural numbers. natural numbers. Example: SExample: S3 3 is the set of all self-bijections is the set of all self-bijections

of X = {1,2,3}.of X = {1,2,3}.

Permutation

A permutation is a rearrangement of an A permutation is a rearrangement of an ordered list, let’s say S, into a 1:1 ordered list, let’s say S, into a 1:1 correspondence with S itself. correspondence with S itself.

The number of different permutations in a The number of different permutations in a set of order n is n factorial (written n!).set of order n is n factorial (written n!).

              

An Example of Permutation Notation

Permutation Example

Let’s look at the set S = {A, B, C}.Let’s look at the set S = {A, B, C}. The order of its permutation is: The order of its permutation is:

n! = 3 * 2 * 1 = 6.n! = 3 * 2 * 1 = 6.

Permutation Example cont’d.

The elements of S’s permutation are: The elements of S’s permutation are: ABCABC ACBACB BACBAC BCABCA CABCAB CBACBA

Cycle Notation

First introduced by the great French First introduced by the great French mathematician Cauchy in 1815.mathematician Cauchy in 1815.

Has theoretical advantages in that certain Has theoretical advantages in that certain important properties of the permutation can important properties of the permutation can be readily determined when cycle notation be readily determined when cycle notation is used.is used.

Permutation

The act of changing the linear order of The act of changing the linear order of objects in a group.objects in a group.

ABC

BCACAB

Examples of Elements of S6

Permutation Notation

( )1 2 3 4 5 6

2 3 4 5 6 1

1-Cycle Notation

(1 2 3 4 5 6)

Permutation Notation

( )1 2 3 4 5 6

4 3 2 5 6 1

2-Cycle Notation

(1 4 5 6) (2 3)

Concepts of Cycle Notation

Don’t need to write cycles that have only Don’t need to write cycles that have only one entry. The missing element is mapped one entry. The missing element is mapped to itself.to itself.

In a sense, the “order” does not matter.In a sense, the “order” does not matter.

G-Equivalence

Permutation groups induce G-Equivalence.Permutation groups induce G-Equivalence. Suppose we have a permutation group G, Suppose we have a permutation group G,

where G where G S SXX..

If x, y If x, y X, we say that x and y are X, we say that x and y are “G-Equivalent” (x~y) if gx = y for some “G-Equivalent” (x~y) if gx = y for some g g G. G.

G-Equivalence cont’d.

Fact: “G-Equivalence” is an equivalence Fact: “G-Equivalence” is an equivalence relation.relation.

Definition: The equivalence classes of X Definition: The equivalence classes of X under the equivalence relation “~” are under the equivalence relation “~” are called G-orbits (or orbits).called G-orbits (or orbits).

Orbits

Definition of orbit: Definition of orbit: Gx = {y Gx = {y X | y = gx for some g X | y = gx for some g G}. G}.

This is also called G-equivalence.This is also called G-equivalence. All elements in X are G-equivalent if there All elements in X are G-equivalent if there

is only one orbit. In this case, we say that is only one orbit. In this case, we say that the action of G is transitive.the action of G is transitive.

Examples of Orbits

Example of an element of the full Example of an element of the full symmetric group on the plane: symmetric group on the plane: = R = R22 R R22 (1:1, onto). (1:1, onto).

Rotation group about the origin.Rotation group about the origin.

Rotation Group About the Origin

(0,0)

*

**

cos sin

sin cos

x

y

The orbits are the concentric circles.

Invariance and Symmetry Groups

Groups as Models of Geometric Symmetry

Crystals are formed through repetition or Crystals are formed through repetition or clustering of crystal atoms. clustering of crystal atoms.

Symmetry groups of invariant permutations Symmetry groups of invariant permutations are the foundation of crystallography are the foundation of crystallography because it models symmetry of objects because it models symmetry of objects (specifically crystal atoms and molecules) (specifically crystal atoms and molecules) using group theory. using group theory.

What is Invariance of Objects Under Permutation?

Definition:Definition: Recall that SRecall that SXX is the set of all is the set of all

permutations on X. permutations on X. Suppose we have a set Y Suppose we have a set Y X and a X and a

group G group G S SXX. Then the subset Y is . Then the subset Y is “G-invariant” if gY “G-invariant” if gY Y, where Y, where gY = g(Y), the image of Y under G, gY = g(Y), the image of Y under G, g g G G..

Invariance of objects under permutation

What does that mean? What does that mean? For any object on a plane, symmetric For any object on a plane, symmetric

permutations (rotation, translation, permutations (rotation, translation, and reflection) leave an object and reflection) leave an object invariant if it preserves the motif of invariant if it preserves the motif of that object.that object.

Invariance of objects under permutation

Here y is acted upon by g(y) under reflection. As Here y is acted upon by g(y) under reflection. As we can see although the position of y has changed, we can see although the position of y has changed, its pattern remains unchanged. its pattern remains unchanged.

Specifically what are symmetry groups of objects?

Symmetry groups are the consequence of an Symmetry groups are the consequence of an invariant permutation on a plane.invariant permutation on a plane.

e.g. Triangle permutations (De.g. Triangle permutations (D33))

RR00, R, R120120, R, R240240

Reflection: RReflection: R11, R, R22, R, R33

Example of a permutation that is Example of a permutation that is notnot invariant on Dinvariant on D33 is R is R9090

Symmetry Group of a Square

Or, the dihedral group of a squareOr, the dihedral group of a square

Motivation

Knowing conjugacy classes and valid Knowing conjugacy classes and valid geometric transforms allows easier geometric transforms allows easier modeling for computer applications by modeling for computer applications by reducing the number of transformations that reducing the number of transformations that the computer needs to deal with.the computer needs to deal with.

Why is the Dihedral Group a Group?

The dihedral group is a subset of the full The dihedral group is a subset of the full symmetric group.symmetric group.

Show each element has an inverse.Show each element has an inverse. Show that the group is closed under the Show that the group is closed under the

operation.operation.

Elements of the dihedral group Group elements: {e, r, rGroup elements: {e, r, r22, r, r33, v, h, m, n}, v, h, m, n} e = identity (no transformation)e = identity (no transformation) r = 90r = 90ºº CCW rotation CCW rotation rr22 = 180 = 180ºº CCW rotation CCW rotation rr33 = 270 = 270ºº CCW rotation CCW rotation h = horizontal reflectionh = horizontal reflection v = vertical reflectionv = vertical reflection m = reflection across main diagonalm = reflection across main diagonal n = reflection across minor diagonaln = reflection across minor diagonal

Cayley Table for the Dihedral Group

e r r2 r3 h v m n

r r2 r3 e n m h v

r2 r3 e r v h n m

r3 e r r2 m n v h

h m v n e r2 r r3

v n h m r2 e r3 r

m v n h r3 r e r2

n h m v r r3 r2 e

Inverses of the Elements of the Dihedral Group

rr-1-1 = r = r33

rr-2 -2 = r= r22

rr-3-3 = r = r11

hh-1-1 = h = h vv-1-1 = v = v mm-1 -1 = m= m nn-1-1 = n = n ee-1-1 = e = e

Conjugacy Classes

Recall that an equivalence class is a subset Recall that an equivalence class is a subset of a group whose members are equivalent of a group whose members are equivalent under some operation.under some operation.

A conjugacy class is a set whose members A conjugacy class is a set whose members are equivalent under conjugation (ghgare equivalent under conjugation (ghg -1-1)) for for g, h g, h G. G.

Thus, a conjugacy class is a way to partition Thus, a conjugacy class is a way to partition a group into equivalence classes.a group into equivalence classes.

Determining Conjugacy Classes

Technique 1: Pick h, k Technique 1: Pick h, k G. h ~ k if G. h ~ k if k = ghgk = ghg-1-1 for some g for some g G. G.

Technique 2: Consider the conjugacy class Technique 2: Consider the conjugacy class as an orbit. Pick h as an orbit. Pick h G, then find the G, then find the conjugation conjugation g g G. This will give the G. This will give the conjugacy class for h.conjugacy class for h.

Dihedral Group Conjugacy Classes

Conjugacy classes of the dihedral group: Conjugacy classes of the dihedral group:

{ { e }, { r, r{ { e }, { r, r33 }, { r }, { r22 }, { h, v }, { m, n } } }, { h, v }, { m, n } } To find these, we will use Technique 2: To find these, we will use Technique 2:

select an h select an h G, then find the conjugation. G, then find the conjugation.

Finding the Conjugacy Classes Pick r. I claim that { r ~ rPick r. I claim that { r ~ r33 }. }. rr33rrrr-3-3 = r = r rr22rrrr-2-2 = r = r hrhhrh-1-1 = r = r33 vrvvrv-1-1 = r = r33 mrmmrm-1-1 = r = r33 nrnnrn-1-1 = r = r33 Thus { r, rThus { r, r33 } is one conjugacy class. } is one conjugacy class. A similar procedure gives us the remainder of the A similar procedure gives us the remainder of the

conjugacy classes.conjugacy classes.

Impact on Crystallography

Crystallographers study the geometric Crystallographers study the geometric structure of crystals in many materials in structure of crystals in many materials in order to to determine a material’s physical order to to determine a material’s physical properties, which benefits society as a properties, which benefits society as a whole.whole.

Knowing these conjugacy classes (i.e. Knowing these conjugacy classes (i.e. knowing that certain transformations are knowing that certain transformations are equivalent) can speed up their computer equivalent) can speed up their computer analyses of the material.analyses of the material.

References ““Modern Geometries, 5Modern Geometries, 5thth Ed.” by James R. Ed.” by James R.

Smart, Brooks/Cole Publishing Company Smart, Brooks/Cole Publishing Company 19981998

““Symmetry Groups and their Applications” Symmetry Groups and their Applications” by Willard Miller Jr., Academic Press 1972by Willard Miller Jr., Academic Press 1972

““General Chemistry” by Linus Pauling, General Chemistry” by Linus Pauling, Dover 1970Dover 1970

““The Fenyman Lectures on Physics”, The Fenyman Lectures on Physics”, Fenyman, et al, Addison-Wesley 1963Fenyman, et al, Addison-Wesley 1963

Questions?