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: http://www.chem.tamu.edu/rgroup/ hughbanks/courses/634/chem634.html
1 Tim Hughbanks
Office: Chemistry Building, Room 330 Office phone: 845-0215 Office Hrs: Tues. 2:00 - 4:00 PM Other times are OK too! e-mail: [email protected]
2 Required Books, etc.
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 Two Covers
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 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 Good Undergraduate Background Text
• 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 Electron Paramagnetic Resonance of Transition Ions Canet Nuclear Magnetic Resonance Concepts and Methods Carrington & McLachlan Introduction to Magnetic Resonance Cheetham & Day Solid-State Chemistry Techniques Clegg Crystal Structure Determination Cotton Chemical Applications of Group Theory, 3rd Edition Drago Physical Methods for Chemists, 2nd Edition Ebsworth, Rankin & Cradock Structural Methods in Inorganic Chemistry, 2nd Ed. Friebolin Basic One-and Two-Dimensional NMR Spectroscopy Keeler Understanding NMR Spectroscopy Housecroft & Sharpe Inorganic Chemistry, 4th Ed. Orton Electron Paramagnetic Resonance Giacovazzo, et. all Fundamentals of Crystallography Solomon & Lever, eds. Inorganic Electronic Structure and Spectroscopy, Volume II
Some Chapters of Drago are available (zipped, password protected): http://www.chem.tamu.edu/rgroup/dunbar/chem634.htm
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7 Radiation & Energies
8 Plane and Space Groups
Symmetry of Crystals
9 References for this Topic
✿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: http://www.chem.tamu.edu/rgroup/ dunbar/Chem634/chem634.htm ✿ Wikipedia article on wallpaper groups is excellent: http://en.wikipedia.org/wiki/ Wallpaper_group
10 Burns & Glazer 3rd Edition, 2013
11 1-D Symmetry
• Seven types 1. unit translation 2. parallel (or longitudinal) reflection 3. perpendicular (or transverse) reflection
4. 2-fold rotational axis (C2)
5. a combination of C2 and one mirror 6. glide reflection 7. combination of: 3, 4, 6
12 1-D Symmetry Operations
unit translation longitudinal reflection
transverse reflection 2-fold rotational axis (C2)
13 1-D Symmetry Operations combination of C2 and glide reflection both types of mirrors
combination of C2, transverse mirror, and glide
14 Distortions in Mo4O6 Chains
15 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) = b⋅(c × a) = c⋅(a × b)
16 Plane Lattices • Lattices are generated by translational symmetry only. (“Lattice” is a highly misused term!) R = ta + ub ; t,u integers • Five Plane Lattices – Oblique, a, b, γ – Rectangular - primitive and centered a ≠ b, γ = 90˚ – Square a = b, γ = 90˚ – Hexagonal a = b, γ = 120˚ • Square and hexagonal lattices can’t be centered
17 Plane Lattices
Also called a rhombic lattice.
• What happens if you try to center square or hexagonal lattices?
18 Unit Cells NOT!
19 Rotation Operations
C2, 2 No other rotations are compatible with translational symmetry. C3, 3
C4, 4
C6, 6
20 Glide Planes
b-glide plane illustrated perpendicular the plane “of the paper”. The square of the operation is a unit translation along b.
21 Plane Groups
Rectangular Oblique Square Hexagonal primitive centered
p1 pm cm p4 p3
p2 pmm cmm p4m p6
pg p4g p31m
pgg p3m1
pmg p6m
22 Plane Groups (Full Symbols)
Rectangular Oblique Square Hexagonal primitive centered
p1 p1m1 c1m1 p4 p3
p211 p2mm c2mm p4mm p6
p1g1 p4gm p31m
p2gg p3m1
p2mg p6mm
23 Notation for Full Symbols 1) 1st symbol, p or c: primitive or a face-centered. 2) 2nd symbol, n: highest rotational symmetry: 1-fold (none), 2-fold, 3-fold, 4-fold, or 6-fold. 3) 3rd 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.
24 Symmetry not shown in Notation & Short Symbols
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.
25 Plane Groups
(see Cotton, Chapter 10)
26 Plane Groups
27 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:
C2v = mm ; D2h = mmm ; C4v = 4mm ; 4/mmm = ? ; 422 = ? ; 42m = ? ; 3m = ?
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 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
29 p1
p1
30 pm
pm
31 p2
32 Oblique lattices can’t have mirrors
33 pg
34 pmm
35 p4m
36 Plane Group symmetries from Symbol: pgg
37 Identifying plane groups: Examples
Unit Cell?
Symmetry Operations? p4g
38 Identifying plane groups: Examples
Unit Cell?
Symmetry Operations?
cmm
39 Identifying plane groups: Examples
Unit Cell?
Symmetry Operations?
p4g
40 Identifying plane groups: Examples
Unit Cell?
Symmetry Operations?
pgg
41 2– Identifying plane groups: [NiF4] (flattened)
Symmetry Operations?
Unit Cell?
42 Identifying plane groups: PtS2 (flattened)
Symmetry Operations?
Unit Cell?
43 Identifying plane groups: CaB2C2 (B-C nets)
Unit Cell?
Symmetry Operations?
44 Identifying plane groups: one graphite layer
Symmetry Operations?
Unit Cell?
45 Plane Group Flowchart
46 Identifying plane groups: Examples
Unit Cell?
Symmetry Operations? (not) ignoring colors
47 Identifying plane groups: Examples
Unit Cell?
Symmetry Operations? (not) ignoring colors
48 p3m1 vs. p31m
p3m1 p31m
p3m1 has mirror planes through all the 3-fold axes, p31m has mirror planes through one kind of the 3-fold axes.
49 Identifying plane groups: Examples
Unit Cell?
Symmetry Operations? (not) ignoring colors
50 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.
51 (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.
52 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): R = ta + ub + vc
t, u, v : integers • V = volume of unit cell
V = a ⋅(b × c) = b⋅(c × a) = c⋅(a × b)
53 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)
54 (b-axis unique) Bravais Lattices
(c-axis vertical)
http://isis.ku.dk/kurser/index.aspx? kursusid=27416&xslt=simple6¶m1=205806¶m8=false
55 Bravais Lattices, continued
56 32 Crystallographic Point Groups
Hermann-Mauguin Schoenflies 1,1,2,m,2 m,mm,222, C1,Ci ,C2 ,Cs ,C2h ,C2v , D2 , mmm,4,4,4 m,4mm,42m, D2h ,C4 ,S4 ,C4h ,C4v , D2d , 422,4 mmm,3,3,3m,32,3m, D , D ,C ,S ,C , D , D , 6,6,6 m,6m2,6mm,622, 4 4h 3 6 3v 3 3d C ,C ,C , D ,C , D , 6 mmm,23,m3,43m,432,m3m 6 3h 6h 3h 6v 6
D6h ,T,Th ,Td ,O,Oh
• These are all the point symmetries compatible with translational symmetry of crystals.
57 Structures vs. Lattices
• What is the distinction between a structure and a lattice? • Consider the two-dimensional “honeycomb net” of a graphite layer
Q: Do the carbon atom positions constitute a “lattice”?
58 Graphite - 3D
59 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 trans- lation are possible (with a component ⊥ to the plane).
60 41 Screw Axis
61 Screw Operations
62 Screw Operations
63 Space Group Operations
64 Space Group Symbols & Notation
65 Space Group Symbols
66 List of Space Groups with High Resolution Diagrams
http://img.chem.ucl.ac.uk/sgp/large/sgp.htm
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67 PtS - example
68 P4/m; All the info in the symbol
PerformApplyRecognize 4-fold translations 2-fold rotation axes andand m additionalreflection; recognizeinversion inversion centers
69 Two Symmorphic,
70 Triclinic Space Groups: P1 & P1
71 Orthorhombic Space Groups
72 Orthorhombic Space
73 β-Sn (White Tin)
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