Tim Hughbanks CHEMISTRY 634. Two Covers. Required Books, etc.

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$edu 1 2 Required Books, etc. Two Covers C. Hammond, The Basics of Crystallography and Diffraction ; Oxford, 4th Edition.$

1 CHEMISTRY 634 This course is for 3 credits. Lecture: 2 75 min/week; TTh 11:10-12:25, Room 2122 Grades will be based on the homework (roughly 25%), term paper (15%), midterm and final exams Web site: hughbanks/courses/634/chem634.html Tim Hughbanks Office: Chemistry Building, Room 330 Office phone: Office Hrs: Tues. 2:00-4:00 PM Other times are OK too! trh@mail.chem.tamu.edu 1 2 Required Books, etc. Two Covers C. Hammond, The Basics of Crystallography and Diffraction ; Oxford, 4th Edition. Harris & Bertolucci, Symmetry and Spectroscopy, An Introduction to Vibrational and Electronic Spectroscopy, Dover. A. F. Orchard, Magnetochemistry,Oxford. J. Iggo, NMR Spectroscopy in Inorganic Chemistry, Oxford. Handouts, posted lecture outlines, and reference materials will be necessary (see reading list in syllabus) - download all you see there! 3 4

Prerequisites Undergraduate chemistry courses, especially inorganic and physical chemistry Chem 673 important in several places In the handout section of the web page there are short introductions to

2 Prerequisites Undergraduate chemistry courses, especially inorganic and physical chemistry Chem 673 important in several places In the handout section of the web page there are short introductions to vectors, matrices, and quantum mechanical techniques (perturbation theory, spin operators, etc.) I will often include background physical derivations that you needn t memorize, but try to understand the physical content! 5 Electron Paramagnetic Resonance of Transition Ions Nuclear Magnetic Resonance Concepts and Methods Introduction to Magnetic Resonance Solid-State Chemistry Techniques Crystal Structure Determination Cotton Drago Ebsworth, Rankin & Cradock Friebolin Keeler Chemical Applications of Group Theory, 3rd Edition Physical Methods for Chemists, 2nd Edition Structural Methods in Inorganic Chemistry, 2nd Ed. Basic One-and Two-Dimensional NMR Spectroscopy Understanding NMR Spectroscopy Housecroft & Sharpe Orton Giacovazzo, et. all Solomon & Lever, eds. Inorganic Chemistry, 4th Ed. Electron Paramagnetic Resonance Fundamentals of Crystallography Inorganic Electronic Structure and Spectroscopy, Volume II Housecroft & Sharpe, Inorganic Chemistry, 4th Edition, Chapters 19 and 20. We cover much more, but these chapters fill in almost everything I haven t done in detail. 6 Other Notable books Abragam & Bleaney Canet Carrington & McLachlan Cheetham & Day Clegg Good Undergraduate Background Text Radiation & Energies Some Chapters of Drago are available (zipped, password protected):

3 References for this Topic Plane and Space Groups Symmetry of Crystals Cotton, Chemical Applications of Group Theory, Chapter 10. Burns & Glazer, Space Groups for Solid State Scientists, 2nd Edition Dunbar lecture notes are especially complete for this topic: dunbar/chem634/chem634.htm Wikipedia article on wallpaper groups is excellent: Wallpaper_group D Symmetry Burns & Glazer 3rd Edition, 2013 Seven types 1. unit translation 2. parallel (or longitudinal) reflection 3. perpendicular (or transverse) reflection 4. 2-fold rotational axis (C 2 ) 5. a combination of C 2 and one mirror 6. glide reflection 7. combination of: 3, 4,

1-D Symmetry Operations unit translation longitudinal reflection 1-D Symmetry Operations combination of C 2 and both types of mirrors glide reflection transverse reflection 2-fold rotational axis (C

4 1-D Symmetry Operations unit translation longitudinal reflection 1-D Symmetry Operations combination of C 2 and both types of mirrors glide reflection transverse reflection 2-fold rotational axis (C 2 ) combination of C 2, transverse mirror, and glide Distortions in Mo 4 O 6 Chains Bravais Lattices Direct Lattice: A regular, periodic array of points with a spacing commensurate with the unit cell dimensions. The environment around each points in a lattice is identical. In 3-D, the set of direct lattice points can be written as (vectors): R = ta + ub + vc t,u,v integers V = volume of unit cell V = a ( b c) = ( ) = c ( a b) b c a 15 16

Oblique, a, b, γ Rectangular - primitive and centered a b, γ = 90 Square a = b, γ = 90 Hexagonal a = b, γ = 120 What happens if you try to

5 Plane Lattices Plane Lattices Lattices are generated by translational symmetry only. ( Lattice is a highly misused term!) R = ta + ub ; t,u integers Five Plane Lattices Also called a rhombic lattice. Oblique, a, b, γ Rectangular - primitive and centered a b, γ = 90 Square a = b, γ = 90 Hexagonal a = b, γ = 120 What happens if you try to center square or hexagonal lattices? Square and hexagonal lattices can t be centered 17 Unit Cells 18 NOT! Rotation Operations C 2, 2 C 3, 3 No other rotations are compatible with translational symmetry. C 4, 4 C 6,

6 Glide Planes Plane Groups Oblique Rectangular primitive centered Square Hexagonal p1 pm cm p4 p3 b-glide plane illustrated perpendicular the plane of the paper. The square of the operation is a unit translation along b. p2 pmm cmm p4m p6 pg p4g p31m pgg p3m1 pmg p6m Plane Groups (Full Symbols) Rectangular Oblique primitive centered Square Hexagonal p1 p1m1 c1m1 p4 p3 p211 p2mm c2mm p4mm p6 p1g1 p4gm p31m p2gg p3m1 p2mg p6mm Notation for Full Symbols 1) 1 st symbol, p or c: primitive or a face-centered. 2) 2 nd symbol, n: highest rotational symmetry: 1-fold (none), 2-fold, 3-fold, 4-fold, or 6-fold. 3) 3 rd symbol, m, g, or 1, for mirror, glide reflection, or none: These indicate symmetries relative to one translation axis of the pattern, referred to as the "main" one. If there is a mirror perpendicular to a translation axis we choose that axis as the main one (or if there are two, one of them). The axis of the mirror or glide reflection is perpendicular to the main axis for the first letter, and either parallel or tilted 180 /n (when n > 2) for the second letter

7 Symmetry not shown in Notation & Short Symbols Plane Groups 4) Many groups include other symmetries implied by the given ones. The short notation drops digits or an m that can be deduced, so long as that leaves no confusion with another group. (see Cotton, Chapter 10) Plane Groups Building up Groups from Generators The philosophy of the Hermann-Mauguin (international) notation is to supply the minimal number of group operations in the group symbol (and those operations generate the rest ). Point group examples: C 2v = mm ; D 2h = mmm ; C 4v = 4mm ; 4/mmm =? ; 422 =? ; 42m =? ; 3m =? 27 28

Plane and Space Groups add translations as additional Generators In plane (and space) groups, the symbols convey information about the whether the lattice is primitive or centered plus point group

8 Plane and Space Groups add translations as additional Generators In plane (and space) groups, the symbols convey information about the whether the lattice is primitive or centered plus point group operation information. Translations and operations that are products of point group operations with each other or with translations are assumed to be generated. Examples: p1 ; pm ; p2 ; pg ; pmm ; p4m p1 p p2 pm pm 31 32

9 Oblique lattices can t have mirrors pg pmm p4m 35 36

10 Plane Group symmetries from Symbol: pgg Identifying plane groups: Examples p4g Identifying plane groups: Examples Identifying plane groups: Examples p4g cmm 39 40

Identifying plane groups: Examples Identifying plane groups: [NiF4]2 (flattened) pgg 41 42

11 Identifying plane groups: Examples Identifying plane groups: [NiF4]2 (flattened) pgg Identifying plane groups: PtS2 (flattened) Identifying plane groups: CaB2C2 (B-C nets) 43 44

12 Identifying plane groups: one graphite layer Plane Group Flowchart Identifying plane groups: Examples Identifying plane groups: Examples (not) ignoring colors 47 (not) ignoring colors 48

p3m1 vs. p31m Identifying plane groups: Examples p3m1 p31m (not) ignoring colors p3m1 has mirror planes through all the 3-fold axes, p31m has mirror planes through one kind of the 3-fold axes.

13 p3m1 vs. p31m Identifying plane groups: Examples p3m1 p31m (not) ignoring colors p3m1 has mirror planes through all the 3-fold axes, p31m has mirror planes through one kind of the 3-fold axes Group Theory: Comments Space groups are groups: they have all the properties you learned about point groups, but are of infinite order. We won t worry about the irreducible representations of space groups in this class, but the group theoretic machinery when applied to crystals is very powerful! Just as for point groups, the more symmetry that is present, the more symmetry can be used to simplify physical problems. Lattices are associated with the infinite-order translational subgroup of space groups. (Non)symmorphic Groups Symmorphic space groups are direct product groups of one of the 32 crystallographic point groups and the translation group (i.e., every operation is a product of a translation and a simple point group operation). Nonsymmorphic space groups contain elements (glide planes and or screw axes) that are not simply generated as a product of a lattice translation with a point group operation. Symmorphic space (and plane) groups normally do not have screw axis or glide plane symbols appear in their symbols (though this rule can be relaxed when the setting chosen for the unit-cell axes is changed). A symmorphic space group must have at least one point in the cell with a site symmetry which is the same as the point group of the space group

14 Bravais Lattices Direct Lattice: A regular, periodic array of points with a spacing commensurate with the unit cell dimensions. The environment around all points in a lattice is identical. The set of direct lattice points can be written as (vectors): V = volume of unit cell R = ta + ub + vc t, u, v : integers V = a ( b c) = b ( c a) = c ( a b) 3-D Lattices 14 Bravais Lattices - fall under 7 crystal systems Triclinic a b c,α β γ Monoclinic a b c, α = γ = 90, β no condition Primitive and body centered (I-centered) Orthorhombic a b c, α = β = γ = 90 One-faced centered (A-, B, C- centered) Body centered (I-centered) Face centered (F-centered) Bravais Lattices (b-axis unique) Bravais Lattices, continued (c-axis vertical) kursusid=27416&xslt=simple6&param1=205806&param8=false 55 56

32 Crystallographic Point Groups Hermann-Mauguin 1,1,2,m,2 m,mm,222, mmm,4, 4,4 m,4mm, 42m, 422,4 m mm,3, 3,3m,32, 3m, 6, 6,6 m, 6m2,6mm,622, 6 mmm,23,m3,43m,432,m3m Schoenflies C 1,C i,c 2,C s,c

15 32 Crystallographic Point Groups Hermann-Mauguin 1,1,2,m,2 m,mm,222, mmm,4, 4,4 m,4mm, 42m, 422,4 m mm,3, 3,3m,32, 3m, 6, 6,6 m, 6m2,6mm,622, 6 mmm,23,m3,43m,432,m3m Schoenflies C 1,C i,c 2,C s,c 2h,C 2v, D 2, D 2h,C 4,S 4,C 4h,C 4v, D 2d, D 4, D 4h,C 3,S 6,C 3v, D 3, D 3d, C 6,C 3h,C 6h, D 3h,C 6v, D 6, Structures vs. Lattices What is the distinction between a structure and a lattice? Consider the two-dimensional honeycomb net of a graphite layer D 6h,T,T h,t d,o,o h These are all the point symmetries compatible with translational symmetry of crystals. Q: Do the carbon atom positions constitute a lattice? Graphite - 3D Glide Planes b-glide plane illustrated perpendicular the plane of the paper. Square of the operation is a unit translation along b. n-glide (diagonal glide) plane illustrated in the plane of the paper. Comma indicates that the object is flipped over. perpendicular the plane of the paper. Square of the operation is a unit translation along a + b. in tetragonal and cubic, n-glides involving a (a + b + c)/2 translation are possible (with a component to the plane)

16 4 1 Screw Axis Screw Operations 61 Screw Operations 62 Space Group Operations 63 64

17 Space Group Symbols & Notation 65 Space Group Symbols 66 List of Space Groups with High Resolution Diagrams PtS - example

18 P4/m; All the info in the symbol Two Symmorphic, Apply Perform Recognize 4-fold translations 2-fold rotation axes and and m additional reflection; recognize inversion inversion centers Triclinic Space Groups: P1 & P1 Orthorhombic Space Groups 71 72

19 β-sn (White Tin) Orthorhombic Space 73 74

Crystallographic Point Groups and Space Groups

Crystallographic Point Groups and Space Groups Physics 251 Spring 2011 Matt Wittmann University of California Santa Cruz June 8, 2011 Mathematical description of a crystal Definition A Bravais lattice