Directions Within The Unit Cell NCSU

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1 Smmetr In Crstalline Materials SiO 2 1) Draw the unit cell, and label the internal rotational/ mirror/inversion smmetries. C 2 2) Show all positions of the molecule generated b smmetr (out to 4 unit cells). C 4 The chemical structure is given b the unit cell and position of the atoms inside. Directions Within The Unit Cell [010] Heagonal Unit Cell [100] [001] z [111] [110] [010] [hkl] lattice vector (hkl) lattice plane (perpendicular to the vector) z [001] [100] Cubic Unit Cell 1

2 Solid State Chemistr: Smmetr Smmetr elements in space groups are alwas given with reference to special aes in the crstal sstem (H-M notation). Crstal Sstem Triclinic Smmetr Direction [PST] Primar Secondar Tertiar None Monoclinic [010] Orthorhombic [100] [010] [001] Tetragonal [001] [100]/[010] [110] Heagonal/ [001] [100]/[010] [120]/[1 1 0] Trigonal Cubic [100]/[010]/ [001] [111] [110] Crstal Sstem Triclinic Monoclinic Orthorhombic Trigonal Solid State Chemistr: Smmetr Unit Cell Metrics a b c α β γ Herman-Mauguin Point Group Schoenflies Point Group 1, 1 C 1,C i a b c α=γ=90 o β 2, m, 2/m C 2, C s, C 2h a b c 222, mm2, mmm D 2, C 2v, D 2h α= β=γ=90 o a=b=c,, 2, C, S 6, D, α=β=γ 90 o m, m C v, D d Heagonal a=b c 6, 6, 6/m, 622, α=β=90 o ; γ=120 o 6mm, 62m, 6mm Tetragonal a=b c 4, 4, 4/m, 422, α= β=γ=90 o 4mm, 42m, 4/mmm Cubic a=b=c 2, m, 42, α= β=γ=90 o 42, m m 2 Point Groups + 14 Bravais Lattice = 20 Space Groups C 6, C h, C 4h, D 6, C 6v, D h, D 6h C 4, S 4, C 4h, D 4, C 4v, D 2d, D 4h T, T h, O, T d, O h 2

Solid State Chemistr: Smmetr Eample: YBa 2 Cu O 7

SQUID magnetometers, for efficient power

Solid State Chemistr: Smmetr Eample: SiO 2 z

3 Solid State Chemistr: Smmetr Eample: YBa 2 Cu O 7 (9K superconductor) Superconductors are used in SQUID magnetometers, for efficient power transmission, etc. Solid State Chemistr: Smmetr Eample: SiO 2 z Commonl used in filters, timing, and frequenc control devices. Found in: electronic hand calculators, watches, clocks, timers, color TV s, RF converters, telephones, cop machines, video recorders, etc. z

4 Eamples: C 60 (fullerene) Solid State Chemistr: Smmetr Solid State Chemistr: Smmetr Self Assembl of Two Distinct Supramolecular Motifs In A Single Crstalline Framework Cohen et al., Angew. Chem. Int. Ed. 2004, 4, 285. molecular heagon double heli Supramolecular chemistr = construction of molecular assemblies held together b noncovalent interactions, hdrogen bonding, pi-pi interactions, etc. 4

Building block : cali[4]resorcinarene (H 2 O) 8 cluster superimposed on a fragment of ice

5 Solid State Chemistr: Smmetr A Well-Resolved Ice-Like (H 2 O) 8 Cluster In An Organic Supramolecular Comple Atwood et al., J. Am. Chem. Soc. 2001, 12, Building block : cali[4]resorcinarene (H 2 O) 8 cluster superimposed on a fragment of ice structure-tpe Ic. Space group : P4 2 /nmc Solid State Chemistr: Smmetr Schoenflies S.G. Crstal Sstem H-M Space Group Smbols (Short & Long) b Point Group c a a b c 5

In Class Problem Set Starting from the below unit cells, a) generate their full 2D patterns out to unit cells, b) label all rotational, mirror and inversion smmetries in one unit cell, and c)

6 In Class Problem Set Starting from the below unit cells, a) generate their full 2D patterns out to unit cells, b) label all rotational, mirror and inversion smmetries in one unit cell, and c) determine how man smmetr-unique atoms there are. A B Smmetr: Fractional Coordinates B/C net present in LaB 2 C 2 Atomic positions within the unit cell are given as fractional coordinates of the unit cell lengths Positions: (, ) C = 0.500, , B = 0.226, , Plane group: P2mm Positions outside unit cell are related to atoms inside b 1, 2, etc. translations. e.g , 0.17 translated to 1.500,

Smmetr: Fractional Coordinates 2D Projection of U Si 2 Plane Group: P4gm Positions: (, ) U = 0, 0 0.5, 0.5 0.182, 0.682 0.818, 0.18 0.18, 0.182 0.682, 0.818 Si = 0.84, 0.884 0.616, 0.116 0.116, 0.

7 Smmetr: Fractional Coordinates 2D Projection of U Si 2 Plane Group: P4gm Positions: (, ) U = 0, 0 0.5, , , , , Si = 0.84, , , , Which are smmetr unique? (not interchangeable b a smmetr operation) Calculating Interatomic Distances Distance between two atoms, in fractional coordinates ( 1, 1, z 1 ) and ( 2, 2, z 2 ), is given b the law of cosines in three dimensions. L = {( a) 2 + ( b) 2 + ( zc) 2 2ab cosγ 2ac zcosβ 2bc zcosα}½ When all angles are 90 o, this reduces to: L = {( a) 2 + ( b) 2 + ( zc) 2 }½ Give = =.822Å, what is the distance between the two nearest carbon atoms: 0.5, and 0.5, 1.17 Distance (C-C) = {(( )*.822) 2 + (( )*.822) 2 }½ = {( )}½ = 1.22Å 7

8 Solid State Chemistr: Smmetr Wckoff Sites (1) = 1; (2) = 2 1 down z; () = 2 1 down ; (4) = 2 1 down (5) = -1; (6) = a-glide z; (7) = mirror ; (8) = n-glide Solid State Chemistr: Smmetr Compact crstallographic descriptions: TiO 2 (rutile) P4 2 /nmn ; a = 4.594, c =2.958Å. Ti in 2a; O in 4f with = 0.05 Calculate positions of all atoms in unit cell? Z = 2 MgAl 2 O 4 (spinel) Fdm ; a = 8.080Å. Mg in 8a; Al in 16d; O in 2e with = atoms per unit cell! (econom of description) Z = 8 Eamples in ATOMS drawing program 8

Roger Johnson Structure and Dynamics: The 230 space groups Lecture 3

Roger Johnson Structure and Dynamics: The 230 space groups Lecture 3 Roger Johnson Structure and Dnamics: The 23 space groups Lecture 3 3.1. Summar In the first two lectures we considered the structure and dnamics of single molecules. In this lecture we turn our attention