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Wednesday, April 2 Last Time: - The solid state - atomic arrangement in solids - why do solids form: energetics - Lattices, translations, rotation, & other symmetry operations Today: Continue with lattices, translations, rotation, & other symmetry operations Describing solid structures Crystal lattice types Crystal structures & properties Locations, directions, & planes Readings: Chang & Thoman: Chapter 8 Solid state reading (posted on Blackboard) Reminders: Spectroscopy assignment due Friday Chemistry Senior Symposium: Tuesday, April 25, CC MR5, 0:30 2 and 4

3-D Structure

20 Nobel Prize in Chemistry "for the discovery of quasicrystals Daniel Shechtman Technion Israel Institute of Technology Haifa, Israel Images from nobelprize.org Figure 7. Pentagonal Penrose tiling. Note the local 5-fold symmetry that even in the absence of any tran

Rectangular 2mm p2mm Patterson symmetry p2mm p 2 mm No. 6 p3m 3m Hexagonal No. 5 p3m Patterson symmetry p6mm Origin at 2mm Asymmetric unit 0 x 2; 0 y 2 Symmetry operations () (2) 2 0,0 (3) m 0,y (4) m x,0 Generators selected (); t(,0); t(0,); (2); (3) Positions Multiplicity, Wyckoff letter, Site symmetry Coordinates Reflection conditions General: 4 i () x,y (2) x,ȳ (3) x,y (4) x,ȳ no conditions 2 h. m. 2,y 2,ȳ 2 g. m. 0,y 0,ȳ 2 f..m x, 2 x, 2 2 e..m x,0 x,0 d 2 mm 2, 2 c 2 mm 2,0 b 2 mm 0, 2 a 2 mm 0,0 Maximal non-isomorphic subgroups I [2] pm(pm,3) ; 3 [2] pm (pm,3) ; 4 [2] p2(p2, 2) ; 2 IIa none IIb [2] p2mg(a = 2a)(7);[2]p2gm(b = 2b)(p2mg,7);[2]c2mm(a = 2a,b = 2b)(9) Maximal isomorphic subgroups of lowest index IIc [2] p2mm(a = 2a or b = 2b)(6) Minimal non-isomorphic supergroups I [2] p4mm() II [2] c2mm(9) Special: Origin at 3m Asymmetric unit 0 x 2 3; 0 y 2; x ( + y)/2; y min( x,x) Vertices 0,0 2,0 2 3, 3 2, 2 Symmetry operations () (2) 3 + 0,0 (3) 3 0,0 (4) m x,x (5) m x,0 (6) m 0,y Generators selected (); t(,0); t(0,); (2); (4) Positions Multiplicity, Wyckoff letter, Site symmetry Coordinates 6 d () x,y (2) ȳ,x y (3) x + y, x (4) y,x (5) x y,ȳ (6) x, x + y 3 c..m x,0 0,x x, x 2 b 3.. 3, 2 3 a 3. m 0,0 2 3, 3 Maximal non-isomorphic subgroups I [2] p3(p3, 3) ; 2; 3 { [3] pm (cm,5) ; 4 [3] pm (cm,5) ; 5 [3] pm (cm,5) ; 6 IIa none IIb [3] h3m (a = 3a,b = 3b)(p3m, 4) Maximal isomorphic subgroups of lowest index IIc [4] p3m (a = 2a,b = 2b)(5) Minimal non-isomorphic supergroups I [2] p6mm(7) II [3] h3m (p3m, 4) Reflection conditions General: no conditions Special:

M.C. Escher p3m Reflection Crystallography (2006). Vol. A, Plane group 5, p. 06. 3m p3m Patterson sym x 2 3; 0 y 2; x ( + y)/2; y min( x,x) 2, 3 2, 2

4 Bravais Lattices P Primitive F Face-centered I Body-centered A, B, or C Side-centered R Rhombohedral

32 Crystallographic Point Groups

Pm 3m O h m 3m Cubic No. 22 P 4/m 3 2/m Patterson symmetry Pm 3m CONTINUED No. 22 Pm 3m Generators selected (); t(,0,0); t(0,,0); t(0,0,); (2); (3); (5); (3); (25) Positions Multiplicity, Wyckoff letter, Site symmetry Coordinates 48 n () x,y,z (2) x,ȳ,z (3) x,y, z (4) x,ȳ, z (5) z,x,y (6) z, x,ȳ (7) z, x,y (8) z,x,ȳ (9) y,z,x (0) ȳ,z, x () y, z, x (2) ȳ, z,x (3) y,x, z (4) ȳ, x, z (5) y, x,z (6) ȳ,x,z (7) x,z,ȳ (8) x,z,y (9) x, z,ȳ (20) x, z,y (2) z,y, x (22) z,ȳ,x (23) z,y,x (24) z,ȳ, x (25) x,ȳ, z (26) x,y, z (27) x,ȳ,z (28) x,y,z (29) z, x,ȳ (30) z,x,y (3) z,x,ȳ (32) z, x,y (33) ȳ, z, x (34) y, z,x (35) ȳ,z,x (36) y,z, x (37) ȳ, x,z (38) y,x,z (39) ȳ,x, z (40) y, x, z (4) x, z,y (42) x, z,ȳ (43) x,z,y (44) x,z,ȳ (45) z,ȳ,x (46) z,y, x (47) z,ȳ, x (48) z,y,x 24 m..m x,x,z x, x,z x,x, z x, x, z z,x,x z, x, x z, x,x z,x, x x,z,x x,z, x x, z, x x, z,x x,x, z x, x, z x, x,z x,x,z x,z, x x,z,x x, z, x x, z,x z,x, x z, x,x z,x,x z, x, x Reflection conditions h,k,l permutable General: no conditions Special: 24 l m.. 2,y,z 2,ȳ,z 2,y, z 2,ȳ, z z, 2,y z, 2,ȳ z, 2,y z, 2,ȳ y,z, 2 ȳ,z, 2 y, z, 2 ȳ, z, 2 y, 2, z ȳ, 2, z y, 2,z ȳ, 2,z 2,z,ȳ 2,z,y 2, z,ȳ 2, z,y z,y, 2 z,ȳ, 2 z,y, 2 z,ȳ, 2 24 k m.. 0,y,z 0,ȳ,z 0,y, z 0,ȳ, z z,0,y z,0,ȳ z,0,y z,0,ȳ y,z,0 ȳ,z,0 y, z,0 ȳ, z,0 y,0, z ȳ,0, z y,0,z ȳ,0,z 0,z,ȳ 0,z,y 0, z,ȳ 0, z,y z,y,0 z,ȳ,0 z,y,0 z,ȳ,0 2 j m. m2 2,y,y 2,ȳ,y 2,y,ȳ 2,ȳ,ȳ y, 2,y y, 2,ȳ ȳ, 2,y ȳ, 2,ȳ y,y, 2 ȳ,y, 2 y,ȳ, 2 ȳ,ȳ, 2 2 i m. m2 0,y,y 0,ȳ,y 0,y,ȳ 0,ȳ,ȳ y,0,y y,0,ȳ ȳ,0,y ȳ,0,ȳ y,y,0 ȳ,y,0 y,ȳ,0 ȳ,ȳ,0 2 h mm2.. x, 2,0 x, 2,0 0,x, 2 0, x, 2 2,0,x 2,0, x 2,x,0 2, x,0 x,0, 2 x,0, 2 0, 2, x 0, 2,x 8 g. 3 m x,x,x x, x,x x,x, x x, x, x x,x, x x, x, x x, x,x x,x,x 6 f 4 m.m x, 2, 2 x, 2, 2 2,x, 2 2, x, 2 2, 2,x 2, 2, x 6 e 4 m.m x,0,0 x,0,0 0,x,0 0, x,0 0,0,x 0,0, x 3 d 4/mm.m 2,0,0 0, 2,0 0,0, 2 3 c 4/mm.m 0, 2, 2 2,0, 2 2, 2,0 Origin at centre (m 3m) Asymmetric unit 0 x 2; 0 y 2; 0 z 2; y x; z y Vertices 0,0,0 2,0,0 2, 2,0 2, 2, 2 Symmetry operations (given on page 674) b m 3 m 2, 2, 2 a m 3 m 0,0,0 Symmetry of special projections Along [00] p4mm Along [] p6mm a = a b = b a = 3(2a b c) b = 3( a + 2b c) Origin at 0,0,z Origin at x,x,x Along [0] p2mm a = 2( a + b) b = c Origin at x,x,0

Im 3m O 9 h m 3m Cubic CONTINUED No. 229 Im 3m No. 229 I 4/m 3 2/m Patterson symmetry Im 3m Generators selected (); t(,0,0); t(0,,0); t(0,0,); t( 2, 2, 2); (2); (3); (5); (3); (25) Positions Multiplicity, Wyckoff letter, Site symmetry Coordinates (0,0,0)+ ( 2, 2, 2)+ 96 l () x,y,z (2) x,ȳ,z (3) x,y, z (4) x,ȳ, z (5) z,x,y (6) z, x,ȳ (7) z, x,y (8) z,x,ȳ (9) y,z,x (0) ȳ,z, x () y, z, x (2) ȳ, z,x (3) y,x, z (4) ȳ, x, z (5) y, x,z (6) ȳ,x,z (7) x,z,ȳ (8) x,z,y (9) x, z,ȳ (20) x, z,y (2) z,y, x (22) z,ȳ,x (23) z,y,x (24) z,ȳ, x (25) x,ȳ, z (26) x,y, z (27) x,ȳ,z (28) x,y,z (29) z, x,ȳ (30) z,x,y (3) z,x,ȳ (32) z, x,y (33) ȳ, z, x (34) y, z,x (35) ȳ,z,x (36) y,z, x (37) ȳ, x,z (38) y,x,z (39) ȳ,x, z (40) y, x, z (4) x, z,y (42) x, z,ȳ (43) x,z,y (44) x,z,ȳ (45) z,ȳ,x (46) z,y, x (47) z,ȳ, x (48) z,y,x 48 k..m x,x,z x, x,z x,x, z x, x, z z,x,x z, x, x z, x,x z,x, x x,z,x x,z, x x, z, x x, z,x x,x, z x, x, z x, x,z x,x,z x,z, x x,z,x x, z, x x, z,x z,x, x z, x,x z,x,x z, x, x 48 j m.. 0,y,z 0,ȳ,z 0,y, z 0,ȳ, z z,0,y z,0,ȳ z,0,y z,0,ȳ y,z,0 ȳ,z,0 y, z,0 ȳ, z,0 y,0, z ȳ,0, z y,0,z ȳ,0,z 0,z,ȳ 0,z,y 0, z,ȳ 0, z,y z,y,0 z,ȳ,0 z,y,0 z,ȳ,0 48 i..2 4,y,ȳ + 2 3 4,ȳ,ȳ + 2 3 4,y,y + 2 4,ȳ,y + 2 ȳ + 2, 4,y ȳ + 2, 3 4,ȳ y+ 2, 3 4,y y+ 2, 4,ȳ Reflection conditions h,k,l permutable General: hkl : h + k + l = 2n 0kl : k + l = 2n hhl : l = 2n h00 : h = 2n Special: as above, plus y,ȳ + 2, 4 ȳ,ȳ + 2, 3 4 y,y + 2, 3 4 ȳ,y + 2, 4 3 4,ȳ,y + 2 4,y,y + 2 4,ȳ,ȳ + 2 3 4,y,ȳ + 2 y + 2, 3 4,ȳ y+ 2, 4,y ȳ + 2, 4,ȳ ȳ + 2, 3 4,y ȳ,y + 2, 3 4 y,y + 2, 4 ȳ,ȳ + 2, 4 y,ȳ + 2, 3 4 24 h m. m2 0,y,y 0,ȳ,y 0,y,ȳ 0,ȳ,ȳ y,0,y y,0,ȳ ȳ,0,y ȳ,0,ȳ y,y,0 ȳ,y,0 y,ȳ,0 ȳ,ȳ,0 24 g mm2.. x,0, 2 x,0, 2 2,x,0 2, x,0 0, 2,x 0, 2, x 0,x, 2 0, x, 2 x, 2,0 x, 2,0 2,0, x 2,0,x 6 f. 3 m x,x,x x, x,x x,x, x x, x, x x,x, x x, x, x x, x,x x,x,x 2 e 4 m.m x,0,0 x,0,0 0,x,0 0, x,0 0,0,x 0,0, x 2 d 4 m.2 4,0, 2 8 c. 3 m 4, 4, 4 6 b 4/mm.m 0, 2, 2 3 4,0, 2 3 4, 3 4, 4 2,0, 2 2, 4,0 2, 3 4,0 0, 2, 4 0, 2, 3 4 3 4, 4, 3 4 4, 3 4, 3 4 hkl : k,l = 2n 2, 2,0 noextraconditions 2 a m 3 m 0,0,0 Symmetry of special projections Along [00] p4mm a = 2(a b) b = 2(a + b) Origin at 0,0,z Along [] p6mm a = 3(2a b c) b = 3( a + 2b c) Origin at x,x,x Along [0] p2mm a = 2( a + b) b = 2c Origin at x,x,0 Origin at centre (m 3m) 73 Asymmetric unit 0 x 2; 0 y 2; 0 z 4; y x; z min( 2 x,y) Vertices 0,0,0 2,0,0 2, 2,0 4, 4, 4 Symmetry operations (given on page 74)

Fm 3m O 5 h m 3m Cubic No. 225 F 4/m 3 2/m Patterson symmetry Fm 3m CONTINUED No. 225 Fm 3m Generators selected (); t(,0,0); t(0,,0); t(0,0,); t(0, 2, 2); t( 2,0, 2); (2); (3); (5); (3); (25) Positions Multiplicity, Wyckoff letter, Site symmetry Coordinates (0,0,0)+ (0, 2, 2)+ ( 2,0, 2)+ ( 2, 2,0)+ 92 l () x,y,z (2) x,ȳ,z (3) x,y, z (4) x,ȳ, z (5) z,x,y (6) z, x,ȳ (7) z, x,y (8) z,x,ȳ (9) y,z,x (0) ȳ,z, x () y, z, x (2) ȳ, z,x (3) y,x, z (4) ȳ, x, z (5) y, x,z (6) ȳ,x,z (7) x,z,ȳ (8) x,z,y (9) x, z,ȳ (20) x, z,y (2) z,y, x (22) z,ȳ,x (23) z,y,x (24) z,ȳ, x (25) x,ȳ, z (26) x,y, z (27) x,ȳ,z (28) x,y,z (29) z, x,ȳ (30) z,x,y (3) z,x,ȳ (32) z, x,y (33) ȳ, z, x (34) y, z,x (35) ȳ,z,x (36) y,z, x (37) ȳ, x,z (38) y,x,z (39) ȳ,x, z (40) y, x, z (4) x, z,y (42) x, z,ȳ (43) x,z,y (44) x,z,ȳ (45) z,ȳ,x (46) z,y, x (47) z,ȳ, x (48) z,y,x 96 k..m x,x,z x, x,z x,x, z x, x, z z,x,x z, x, x z, x,x z,x, x x,z,x x,z, x x, z, x x, z,x x,x, z x, x, z x, x,z x,x,z x,z, x x,z,x x, z, x x, z,x z,x, x z, x,x z,x,x z, x, x 96 j m.. 0,y,z 0,ȳ,z 0,y, z 0,ȳ, z z,0,y z,0,ȳ z,0,y z,0,ȳ y,z,0 ȳ,z,0 y, z,0 ȳ, z,0 y,0, z ȳ,0, z y,0,z ȳ,0,z 0,z,ȳ 0,z,y 0, z,ȳ 0, z,y z,y,0 z,ȳ,0 z,y,0 z,ȳ,0 48 i m. m2 2,y,y 2,ȳ,y 2,y,ȳ 2,ȳ,ȳ y, 2,y y, 2,ȳ ȳ, 2,y ȳ, 2,ȳ y,y, 2 ȳ,y, 2 y,ȳ, 2 ȳ,ȳ, 2 48 h m. m2 0,y,y 0,ȳ,y 0,y,ȳ 0,ȳ,ȳ y,0,y y,0,ȳ ȳ,0,y ȳ,0,ȳ y,y,0 ȳ,y,0 y,ȳ,0 ȳ,ȳ,0 Reflection conditions h,k,l permutable General: hkl : h+k,h+l,k+l = 2n 0kl : k,l = 2n hhl : h + l = 2n h00 : h = 2n Special: as above, plus 48 g 2. mm x, 4, 4 x, 3 4, 4 4,x, 4 4, x, 3 4 4,x, 3 4 3 4, x, 3 4 x, 4, 3 4 x, 4, 4 32 f. 3 m x,x,x x, x,x x,x, x x, x, x x,x, x x, x, x x, x,x x,x,x 4, 3 4,x 4, 4, x 4, 4, x 4, 3 4,x hkl : h = 2n 24 e 4 m.m x,0,0 x,0,0 0,x,0 0, x,0 0,0,x 0,0, x 24 d m. mm 0, 4, 4 0, 3 4, 4 8 c 43m 4, 4, 4 4,0, 4 4,0, 3 4 4, 3 4,0 4, 4,0 hkl : h = 2n 4, 4, 3 4 hkl : h = 2n 4 b m 3 m 2, 2, 2 4 a m 3 m 0,0,0 Symmetry of special projections Along [00] p4mm Along [] p6mm a = 2a b = 2b a = 6(2a b c) b = 6( a + 2b c) Origin at 0,0,z Origin at x,x,x Along [0] c2mm a = 2( a + b) b = c Origin at x,x,0 Origin at centre (m 3m) Asymmetric unit 0 x 2; 0 y 4; 0 z 4; y min(x, 2 x); z y Vertices 0,0,0 2,0,0 4, 4,0 4, 4, 4 Symmetry operations (given on page 69)

Body-centered cubic

Face-centered cubic

Mechanical Behavior Material Crystal Structure Modulus (GPa) Yield Strength (MPa) Tensile Strength (MPa) Percent Elongation (%) Aluminum FCC 68 80 250 34 Gold FCC - - 20 30 Nickel FCC 207 59 37 30 Magnesium HCP 45 83 50 2 Fe-C (Steel) BCC 200 600 750 7

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