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- 1. Group symmetry and the 32 Point Groups Shyue Ping Ong Department of NanoEngineering University of California, San Diego
- 2. An excursion into group theory NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 2
- 3. Definition In mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element satisfying four conditions called the group axioms, namely closure, associativity, identity and invertibility. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 3
- 4. Group axioms ¡ Closure ¡ For all a, b in G, the result of the operation, a • b, is also in G. ¡ Associativity ¡ For all a, b and c in G, (a • b) • c = a • (b • c). ¡ Identity ¡ There exists an element e in G, such that for every element a in G, the equation e • a = a • e = a holds. ¡ Invertibility ¡ For each a in G, there exists an element b in G such that a • b = b • a = e, where e is the identity element. ¡ (optional) Commutativity. ¡ a • b = b • a . Groups satisfying this property are known as Abelian or commutative groups. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 4
- 5. A very simple symmetry example ¡ Let’s consider a simple 4-fold rotation axis. We can construct a full multiplication table (Cayley table) for this set of symmetry operations as follows: ¡ How do you identify the inverse of each member? ¡ Is this group Abelian? NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 5 a b e 4 42 43 e e 4 42 43 4 4 42 43 e 42 42 43 e 4 43 43 e 4 42
- 6. More complicated example ¡ Quartz has configuration 223, i.e., it has a 2-fold rotation axis and a 3-fold rotation axis that are mutually perpendicular. As a consequence of Euler’s theorem, the 2u and 2y rotations are automatically determined by the combination of 2x and 3. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 6 32 (D3) e 3 32 2x 2y 2u e e 3 32 2x 2y 2u 3 3 32 e 2u 2x 2y 32 32 e 3 2y 2u 2x 2x 2x 2y 2u e 32 3 2y 2y 2u 2x 3 e 32 2u 2u 2x 2y 32 3 e Important note: For Cayley tables, each cell is given by row.column, e.g., the element in the red box on the right implies that D(3)D(2x) = D(2u)
- 7. Properties of a group ¡ Order: # of elements in group ¡ Isomorphism: 1-1 mapping between two groups ¡ Homomorphous groups: Two groups are homomorphous if there exists a unidirectional correspondence between them. ¡ Cyclic groups: A group is cyclic if there is an element O such that successive powers of O generates all the elements in the group. O is then called the generating element. ¡ Group generators: The minimal set of elements from which all group elements can be constructed. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 7
- 8. Subgroups and supergroups ¡ If a subset of elements of a group form a group, this set is called a subgroup of , and is denoted as ¡ Note that the identity is always a subgroup of all groups. ¡ If the subgroup is not the identity or itself, it is known as a proper subgroup. ¡ Can you identify all the subgroups in the quartz example? NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 8 Gk ⊂ G Gk G G
- 9. Tips for Symmetry Table Construction ¡ You can basically quickly fill up the cyclic subgroups parts of the table because those are simply powers of a rotation matrix. ¡ The inversion operation can be treated as simply equal to -1 multiplied by any matrix, because D(i) = -E. ¡ All symmetry operations must appear once, and only once in each row and column (think of a Sudoku table) ¡ This means that once you get the table partially filled, you can already work out the rest of the table using the above constraints without doing matrix multiplications. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 9
- 10. Derivation of the 32 3D-Crystallographic Point Groups NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 10
- 11. Preliminaries ¡ Crystallographic Point Groups: ¡ Crystallographic – Only symmetries compatible with crystals, i.e., for rotations, only 1, 2, 3, 4 and 6-fold. ¡ Point: Symmetries intersect at a common origin, which is invariant under all symmetry operations. ¡ Group: Satisfy group axioms of closure, associativity, identity and invertibility. ¡ All point groups will be presented as: ¡ We just saw our first point group! ¡ C1or identity point group. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 11 31 more to go….
- 12. Notation and Principal Directions ¡ The International or Hermann-Mauguin notation for point groups comprise of at most three symbols, which corresponds to the symmetry observed in a particular principal direction. The principal directions for each of the Bravais crystal systems are given below: NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 12
- 13. Principal directions in a cube NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 13 Primary Secondary Tertiary
- 14. Proper rotations ¡ Notation: ¡ All cyclic groups of order n (hence the “C”). ¡ Principal directions given by directions of monoclinic, trigonal, tetragonal and hexagonal systems respectively. ¡ What is the generating element? NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 14 n Cn[ ]
- 15. Dihedral groups ¡ Notation: ¡ Earlier, we derived the possible combinations of rotation axes. One set of possible rotation combination contains a 2-fold rotation axis perpendicular to another rotation axis. ¡ How many unique 2-fold rotations are there in each group? NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 15 n2(2) Dn[ ]
- 16. Rotations + Inversion ¡ Notation: ¡ We have already derived these in the previous lecture on symmetry operations NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 16 =3/m n Sn[ ]
- 17. Rotation + Perpendicular Reflections ¡ Notation: ¡ m and 3/m are already derived in previous slide. ¡ The /m notation indicates that the mirror is perpendicular to the rotation axis. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 17 n / m Cnh[ ]
- 18. Rotations + Coinciding Reflection ¡ Notation: ¡ Note that the coincidence of a mirror with a n-fold rotation implies the existence of another mirror that is at angle π/n to the original mirror plane. ¡ Generating elements are and ¡ Which mirror planes are related by symmetry? NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 18 nm Cnv[ ] n m
- 19. Roto-inversions + Coinciding Reflection ¡Notation: ¡Can you show that ? ¡Generators: Inversion rotation + mirror plane NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 19 nm Dnd[ ] 1m ≡ 2 / m 2m ≡ mm2
- 20. Rotations with Coinciding and Perpendicular Reflections ¡ Notation: ¡ Only even rotations result in new groups. ¡ Exercise: What do the 1 and 3-fold rotations lead to when we add coinciding and perpendicular mirror planes? ¡ Full vs shorthand symbol: NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 20 n / m m (m) Dnh[ ] 2 m 2 m 2 m ≡ mmm 4 m 2 m 2 m ≡ 4 m mm 6 m 2 m 2 m ≡ 6 m mm n-fold rotation axes omitted if the rotation axis can be unambiguously obtained from the combination of symmetry elements presented in the symbol.
- 21. Combination of Proper Rotations (not at right angles) ¡ Notation: NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 21 n1n2 T[ ] or O[ ]
- 22. Adding reflection to n1n2 ¡ Notation: NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 22 n1n2 Td[ ], Th[ ]or Oh[ ] Adding mirror plane to 2-fold rotation axes of 23. Adding mirror plane to 3-fold rotation axes of 23. Adding inversion or mirror to 432.
- 23. Laue Classes ¡ Only 11 of the 32 point groups are centrosymmetric, i.e., contains an inversion center. All other non- centrosymmetric point groups are subgroups of these 11. Each row is called a Laue class. ¡ Polar point groups are groups that have at least one direction that has no symmetrically equivalent directions. Can only happen in non- centrosymmetric point groups in which there is at most a single rotation axis (1, 2, 3, 4, 6, m, mm2, 3m, 4mm, 6mm) – basically all single rotation axes + coinciding mirror planes. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 23
- 24. Interpreting the full Hermann- Mauguin symbols ¡O and Oh – Cubic System NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 24 Primary Secondary Tertiary 432 4 m 3 2 m
- 25. Interpreting the full Hermann- Mauguin symbols ¡6mm and 6/mmm – Hexagonal System NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 25 6mm 6 m mm Primary Secondary Tertiary Secondary Tertiary Top view
- 26. Group-subgroup relations NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 26
- 27. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 27Molecule Linear? Contains two or more unique C3 axes? Contains an inversion center? Contains two or more unique C5 axes? C∞v Yes No D ∞h No Yes Contains an inversion center? Ih Yes I Contains two or more unique C4 axes? Contains one or more reﬂection planes? No Contains an inversion center? Contains an inversion center? Yes Yes Oh T No No Th YesYes Yes No No O Yes Td No No NANO106 Handout 4 Flowchart for Point Group Determination Continued on next page Red: Non-crystallographic point groups Green: Crystallographic point groups
- 28. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 28 Yes No Contains a proper rotation axis (Cn)? Identify the highest Cn. Are there n ⟂ C2 axes? Contains a reﬂection plane? Contains an inversion center? C1 No No Contains a horizontal reﬂection plane ⟂ to Cn axis (σh)? Yes Contains n dihedral (between C2) reﬂection planes (σd)? Dnh Contains a horizontal reﬂection plane ⟂ to Cn axis (σh)? No Yes No Dnd Dn No Yes Contains a vertical reﬂection plane (σv)? Cnv No Yes Contains a 2n-fold improper rotation axis? Cn S2n Yes No No Cnh Yes Cs Ci Yes Yes From previous page
- 29. Molecular Point Group NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 29
- 30. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 30Ethylene (H2CCH2) Methane (CH4) SF5Cl CO2 BF3 PF6
- 31. Practicing point group determination ¡ Set of molecule xyz files with different point groups are provided at https://github.com/materialsvirtuallab/nano106/tree/master /lectures/molecules ¡ Other online resources ¡ https://www.staff.ncl.ac.uk/j.p.goss/symmetry/Molecules_pov.html ¡ http://csi.chemie.tu-darmstadt.de/ak/immel/misc/oc- scripts/symmetry.html NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 31
- 32. Matrix representations of Point Groups ¡ As we have seen earlier, all symmetry operations can be represented as matrices. ¡ As point symmetry operations do not have translation, we only need 3x3 matrices to represent these operations (homogenous coordinates are needed only to include translation operations). ¡ We have also seen how working in crystal reference frame simplifies the symmetry operation matrices considerably, and can be obtained simply by inspecting how the crystal basis vectors transform under the symmetry operation. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 32
- 33. Generator matrices ¡ From the group multiplication tables, we know that all symmetry elements in a group can be obtained as the product of other elements. ¡ The minimum set of symmetry operators that are needed to generate the complete set of symmetry operations in the point group are known as the generators. ¡ All point groups can be generated from a subset of the 14 fundamental generator matrices. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 33
- 34. The 14 generator matrices NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 34
- 35. Simple example: mmm ¡ Consider the mmm point group with order 8. Let’s choose the three mirror planes as the generators (note that these are not the same as the ones from the 14 generator matrices! I am choosing these to illustrate how you can derive these from first principles). What are the generator matrices? ¡ Using the generator matrices, we can now generate the 8 symmetry operations in this point group. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 35 Blackboard Blackboard
- 36. Simple example: mmm ¡ Using the generator matrices, we can now generate the 8 symmetry operations in this point group. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 36 1 0 0 0 1 0 0 0 −1 " # $ $ $ % & ' ' ' = m1 1 0 0 0 −1 0 0 0 1 " # $ $ $ % & ' ' ' = m2 −1 0 0 0 1 0 0 0 1 " # $ $ $ % & ' ' ' = m3 −1 0 0 0 1 0 0 0 −1 " # $ $ $ % & ' ' ' = m1 ⋅m3 = 2y −1 0 0 0 −1 0 0 0 1 " # $ $ $ % & ' ' ' = m1 ⋅m2 = 2z 1 0 0 0 −1 0 0 0 −1 " # $ $ $ % & ' ' ' = m2 ⋅m3 = 2x −1 0 0 0 −1 0 0 0 −1 " # $ $ $ % & ' ' ' = m1 ⋅m2 ⋅m3 = i 1 0 0 0 1 0 0 0 1 " # $ $ $ % & ' ' ' = m1 ⋅m2 ⋅m3 ⋅m1 ⋅m2 ⋅m3 = i⋅i = E E i m1 m2 m3 2x 2y 2z E E i m1 m2 m3 2x 2y 2z i i E 2z 2y 2x m3 m2 m1 m1 m1 2z E 2x 2y m2 m3 i m2 m2 2y 2x E 2z m1 i m2 m3 m3 2x 2y 2z E i m1 m3 2x 2x m3 m2 m1 i E 2z 2y 2y 2y m2 m3 i m1 2z E 2x 2z 2z m1 i m2 m3 2y 2x E http://nbviewer.ipython.org/github/materialsvirtuallab/nano106/blob/master/lectures/lecture_4_point_ group_symmetry/Symmetry%20Computations%20on%20mmm%20%28D_2h%29%20Point%20Gro up.ipynb
- 37. Procedure for constructing a symmetry multiplication table ¡Identify point group ¡Identify compatible crystal system (if not provided) ¡Align symmetry elements with crystal axes ¡Derive a set of minimal symmetry matrices ¡Iteratively multiply to get all the symmetry matrices NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 5 37

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