Some Groups of Mathematical Crystallography

1. Some Groups of Mathematical Crystallography Part Deux

2. 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

3. 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

4. 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,…”

5. 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

6. 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.”

7. 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

8. Examples Face-Centered Cubic Body-Centered Cubic Source: http://cst-www.nrl.navy.mil/lattice/index.html

9. Techniques to Study Molecular Structure • X-ray diffraction • Neutron diffraction • Electron diffraction

10. Space Groups and their corresponding Lattices

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

12. Space Groups • Since crystals are repetitive formations of atoms, it can be said that there is some lattice T with basis t1, t2 such that any translation is of the form Ta, where a = m t1 + n t2 (m,n ε Z) Note: Any crystal which arises from these translations, is a map onto itself from the lattice T.

13. 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) t between G and the elements of T.

14. 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 =

15. 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.

16. 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.

17. The Groups E(2) and O(2) • O(2) – the orthogonal group in the plane R2 • E(2) – the Euclidean isometry group on R2; the group (under function composition) of all symmetries of R2

18. 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. • G0 = { A | (A, a)  G }; (A, a) represents Ax + a where A is an orthogonal matrix • We can associate a point group (G0) with a space group: G/T  O(2) where G/T is isomorphic to G0

19. Concept of Equivalence • Two point groups G0 and G0` 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.

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

21. Finiteness of Point Groups

22. The Point Group G0 • THM: The point Group G0 of a space group MUST be a finite group. • Proof: First consider a circle about the origin containing a lattice basis {t1, t2} of T. • N: # of lattice points in the circle

23. G0 Proof (Cont.)

24. G0 Proof (Cont.) • There are only finitely many lattice points inside this circle, say n (Note: n  4) • f = mt1 + nt2 (m,n  )  Thus finitely many

25. G0 Proof (Cont.) • Matrix A  G0 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, G0 is must be finite

26. G0 Proof (Cont.) • Observations: (If A  G0) A(T) = T t  T At  T

27. 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

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

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

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

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

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

33. Finite Subgroups of O(2) Proof: Cyclic Conclusion

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

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

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

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

38. Finite Subgroups of O(2) Proof: Dihedral Conclusion

39. Crystallographic Restriction Theorem(CRT)

40. 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.

41. CRT Proof • Let R be an element of O(2) with the matrix: cosq -sinq sinq cosq • The trace of the matrix is 2cosq

42. 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 cosq = 0, ±1, or ±1/2, the corresponding n values are: 1, 2, 3, 4, 6

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

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

45. Examples of Patterns Formed Example of orthogonal group of D6

46. Examples cont’d • Example of C6 orthogonal group.

47. References • “Modern Geometries, 5th 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

48. Questions?