Some Groups of Mathematical Crystallography Part Deux

Quick Review Crystals are regular arrangements of atoms/molecules in solids Model symmetry using group theory We are considering rotations and reflections-we left off with the discussion of the dihedral group: groups of 90° rotations and reflections across diagonals and axes

Overview Chemistry and Physics Viewpoint Lattices as crystal models The groups E(2) and O(2) Crystallographic space groups and their point groups Concept of Equivalence Point Group Classification

Chemistry Viewpoint “Most solid substances are crystalline in nature” “Every crystal consists of atoms arranged in a three-dimension pattern that repeats itself regularly.” “It is the regularity of arrangement of the atoms in a crystal that gives to the crystal its characteristic properties,…”

Chemistry Viewpoint cont’d “The principal classification of crystals is on the basis of their symmetry.” “Chemists often make use of the observed shapes of crystals to help them in the identification of substances.” -Linus Pauling, General Chemistry

Physics Viewpoint In solids, the atoms will arrange themselves in a configuration that has the lowest energy possible. This arrangement is infinitely repetitive in three dimensions. “The arrangement of the atoms in a crystal- the crystal lattice-can take on many geometric forms.”

Physics Viewpoint cont’d “…[I]ron has a body-centered cubic lattice at low temperatures, but a face-centered cubic lattice at higher temperatures. The physical properties are quite different in the two crystalline forms.” Richard Feynman, Lectures on Physics

Examples Source: Face-Centered Cubic Body-Centered Cubic

Techniques to Study Molecular Structure X-ray diffraction Neutron diffraction Electron diffraction

Space Groups and their corresponding Lattices

Space Groups Wishing to examine symmetry groups of crystals – namely, those symmetries which map a crystal to itself, we look to Space Groups.

Space Groups Since crystals are repetitive formations of atoms, it can be said that there is some lattice T with basis t 1, t 2 such that any translation is of the form T a, where a = m t 1 + n t 2 (m,n ε Z) Note: Any crystal which arises from these translations, is a map onto itself from the lattice T.

Space Groups Def: A crystallographic space group is a subgroup of E(2) whose translations are a set of the form {(I, t) | t ε T} where T is a lattice. Remark: The set of translations {(I,t) | t εT} forms an abelian subgroup of G (the translation group of G). Clearly there is a 1-1 correspondence (I, t) tbetween G and the elements of T.

Space Groups * All translations come in some sense from a fixed lattice * Ex. Let T be the lattice with basis (1,0) and (0,1). The matrices and vectors below are written with respect to this lattice basis. 1) G = T is a space group 2) Let G be the set consisting of the translation subgroup T along with all elements of the form (A,t), t εT where A =

Space Groups and their Lattices 1) 2) Note that the lattice itself does not identify the crystal by symmetry type – again think of the crystal as having identical patterns of atoms at each lattice point – the type of atom pattern determines the full symmetry group.

Lattices as Crystal Models Given that the crystal lattice is the arrangement of atoms in a crystal, we can model crystals using lattices. We’ll do this by defining space groups, point groups, and their relationships.

The Groups E(2) and O(2) O(2) – the orthogonal group in the plane R 2 E(2) – the Euclidean isometry group on R 2 ; the group (under function composition) of all symmetries of R 2

Crystallographic Space Groups and Their Point Groups The symmetries of a crystal are modeled by a group called the crystallographic space group (G  E(2) ). –The translations for this group can be identified with a lattice T  G. –G 0 = { A | (A, a)  G }; (A, a) represents Ax + a where A is an orthogonal matrix –We can associate a point group (G 0 ) with a space group: G/T  O(2) where G/T is isomorphic to G 0

Concept of Equivalence Two point groups G 0 and G 0 ` are equivalent if they are conjugate as subgroups of all 2 x 2 unimodular matrices. –A unimodular matrix is one with determinant ±1 with integer entries. Two space groups are equivalent if they are isomorphic and their lattice structure is preserved.

Point Group Classification Finiteness of point groups Crystallographic restriction The 10 Crystal Classes

Finiteness of Point Groups

The Point Group G 0 THM: The point Group G 0 of a space group MUST be a finite group. Proof: First consider a circle about the origin containing a lattice basis {t 1, t 2 } of T. N: # of lattice points in the circle

G 0 Proof (Cont.)

There are only finitely many lattice points inside this circle, say n (Note: n  4) f = mt 1 + nt 2 (m,n   )  Thus finitely many

G 0 Proof (Cont.) Matrix A  G 0 is distance preserving if a lattice is moved (A maps lattice points to lattice points in the circle) A permutes the N lattice points in the circle –N! permutations of N lattice points  N! A matrices Thus, G 0 is must be finite

G 0 Proof (Cont.) Observations: (If A  G 0 ) A(T) = T t  T At  T

Finite Subgroups of O(2) Finite subgroups of O(2) are either cyclic or dihedral Proof : (next slides) Note: R in the next slides is the rotation in the plane around the origin

Finite Subgroups of O(2) G is a subgroup of O(2), G - finite Proof: Cyclic

Finite Subgroups of O(2) Set = least non-zero Why is this true??? Proof: Cyclic (cont)

Finite Subgroups of O(2) Union the set when Proof: Cyclic (cont)

Finite Subgroups of O(2) Proof: Cyclic (cont)

Finite Subgroups of O(2) Proof: Cyclic (cont)

Finite Subgroups of O(2) Proof: Cyclic Conclusion

Finite Subgroups of O(2) F is a subgroup of G - finite Proof: Dihedral

Finite Subgroups of O(2) Proof: Dihedral (cont)

Finite Subgroups of O(2) Proof: Dihedral (cont) H is the cyclic group of rotation matrices HF is the reflection coset

Finite Subgroups of O(2) Proof: Dihedral (cont) Every group can be written as the union of distinct cosets

Finite Subgroups of O(2) Proof: Dihedral Conclusion

Crystallographic Restriction Theorem (CRT)

Definition: – Let R be a rotation in a point group through an angle 2  /n. Then n is 1, 2, 3, 4, or 6.

CRT Proof Let R be an element of O(2) with the matrix: cos  -sin  sin  cos  The trace of the matrix is 2cos 

CRT Proof cont’d The matrix R with respect to a lattice basis has Z entries, because the matrix is unimodular. Thus it has an Z trace. –Note: matrices with the same linear transformations with respect to different basis have the same trace. Since cos  ±  or ±1/2, the corresponding n values are: 1, 2, 3, 4, 6

CRT Proof (Continued) C n and D n both contain rotations through 2  /n, the implications assert that: –Any point group must be associated with the 10 crystal classes within C n and D n, where C n and D n are the cyclic group of order n and dihedral group of order 2n respectively.

The 10 Crystal Classes C1 C2 C3 C4 C6 D1 D2 D3 D4 D6

Examples of Patterns Formed Example of orthogonal group of D6

Examples cont’d Example of C6 orthogonal group.

References “Modern Geometries, 5 th Ed.” by James R. Smart, Brooks/Cole Publishing Company 1998 “Symmetry Groups and their Applications” by Willard Miller Jr., Academic Press 1972 “General Chemistry” by Linus Pauling, Dover 1970 “The Feynman Lectures on Physics”, Feynman, et al, Addison-Wesley 1963 “Applications of Abstract Algebra”, by George Mackiw, Wiley 1985

Questions?