Chapter One Crystal Structure

Transcription

1 Chpter One Crystl Structure Drusy Qurtz in Geode Tbulr Orthoclse Feldspr Encrusting Smithsonite Peruvin Pyrite 1

2 Snow crystls the Beltsville Agriculturl Reserch Center

3 Solid : Crystl vs. Amorphous (glssy) Ordered rry of toms Disordered rrngement Competition between ttrctive (binding) force nd repulsive force. Regulr rry lowers system energy. Complicted! -- difficult to predict the structure of mterils 3

4 Importnce : structure plys mjor role in determining physicl properties of solids Determintion : X-ry nd neutron scttering re key tools for determining crystl structures. (bulk) Also microscopic techniques such s SEM, TEM, STM, AFM (surfce) Devitions : There is no perfect crystl. Mny key properties depend on devition more. Defects imperfection in crystl Phonons lttice vibrtions 4

5 Clcite(CCO 3 ) crystl is mde from sphericl prticles. Christin Huygen, Leiden 1690 A crystl is mde from sphericl prticles. Robert Hooke, London 1745 depicted by René Hüy, Pris, 18 5

6 Crystl periodic rry of toms : point lttice + bsis Point lttice mthemticl points in spce r = u 1 1,u, r +, u, u u integer = + + u 3 lttice vectors 3 r r 1 6

7 Primitive lttice cell A cell will fill ll spce by the repetition of suitble crystl trnsltion opertions A minimum volume cell. A.One lttice point per primitive cell..the bsis ssocited w/. primitive cell -- primitive bsis.not unique. cell r = r = 1 1 r r sinφ Sme for ll primitive cells Uniform mss density Dimensions r r 1 Wigner-Seitz Primitive cell -- lttice point is t its center the highest symmetry cell possible 7

8 Wigner-Seitz Primitive cell in D (or 3D) Drw lines to connect given lttice point to ll nerby lttice points.. Drw bisecting lines (or plnes) to the previous lines.. The smllest re (or volume) enclosed. D Oblique Lttice 8

9 Fundmentl types of Brvis lttices Bsed on symmetries : 1) Trnsltionl sme if trnslte by vector T = u u u3 3 ) Rottionl sme if lttice is rotted by n ngle bout point -fold by 180 o 4-fold by 90 o 3-fold by 10 o 6-fold by 60 o 3) Mirror symmetry sme if reflected bout plne Auguste Brvis 4) Inversion symmetry sme if reflected through point (equivlent to rottion 180 o nd mirror rottionl xis r -r 9

10 Five Brvis lttices in two dimensions Squre lttice 1 =, φ=90 o Unit cell Symmetry element r φ r 1 Rectngulr lttice 1, φ=90 o Symmetry Unit cell element 10

11 Oblique lttice 1, φ 60 o, 90 o Symmetry Unit cell element Centered Rectngulr lttice r r 1 φ 1 = cosφ Unit cell Symmetry element 11

12 Hexgonl lttice 1 =, φ=60 o Symmetry element Unit cell twofold xis (di) threefold xis (trid) fourfold xis (tetrd) sixfold xis (hexd) mirror line 1

13 A Brvis lttice is lttice in which every lttice point hs exctly the sme environment. How bout honeycomb lttice? A honeycomb crystl = A hexgonl lttice + two-pint bsis 13

14 System Triclinic Monoclinic The seven crystl systems divided into fourteen Brvis lttices : Number of lttices 1: Simple Simple, Bse-Centered Unit cell chrcteristics 1 3 α β γ 1 3 α = β = 90 o γ Chrcteristic symmetry elements None One -fold rottion xis Orthorhombic 4: BCC, FCC Simple, Bse-Centered 1 3 α = β = γ = 90 o Three mutully orthogonl -fold rottion xes Tetrgonl : Simple, BCC 1 = 3 α = β = γ = 90 o One 4-fold rottion xis Cubic 3: Simple, BCC, FCC 1 = = 3 α = β = γ = 90 o Four 3-fold rottion xes Trigonl 1: Simple 1 = = 3 α = β = γ<10 o 90 o One 3-fold rottion xis Hexgonl 1: Simple 1 = 3 α = β = 90 o, γ =10 o One 3-fold rottion xis 14

15 TRICLINIC (α β γ) MONOCLINIC (β=γ=90 o α) 1 β γ α 3 Simple 1 Simple Bse-centered 15

16 Orthorhombic (α=β=γ=90 o ) 1 3 β 1 3 α γ Simple Bse-centered BCC FCC 16

17 Tetrgonl (α=β=γ=90 o ) 1 β α γ 1 Simple BCC 17

18 Cubic (α=β=γ=90 o ) β α γ Simple BCC FCC 18

19 Trigonl (α=β=γ<10 o, 90 o ) Hexgonl (α=β=90 o, γ=10 o ) 3 3 γ=10 o Simple Simple 19

20 Chrcteristics of cubic lttices Simple Body-centered Fce-centered Lttice points/cell 1 4 Number of nerest neighbors Nerest- neighbor distnce Pcking frction π 6 = π = π = Mximum, sme s hexgonl 0

21 Crystl Periodic rrngement of toms Brvis lttice + Bsis of tom (Arry of point in spce) (Arrngement of toms within unit cell) Dimensions 5 types 3 Dimensions 14 types By symmetry Crystl my hve sme or less symmetry thn originl Brvis lttice. Bsis Symmetry 4mm 1

22 Introducing new symmetries for bsis with multiple toms : Point group symmetries : combintions of rottion, inversion, reflection tht hold one point fixed nd return originl structure. Dimensions, 10 3 Dimensions, 3 Spce group symmetries : point group opertions + trnsltions tht return originl structure. Dimensions, 17 3 Dimensions, 30 There re mny possible types by symmetry but most re never observed. Rel crystls form few types due to energies of crystl formtion.

23 3

24 Directions nd plnes in crystls Useful to develop nottion for describing directions nd identifying plnes of toms in crystls r = u + Consider lttice defined by 1,, u u3 3 Vector direction is described s [u 1 u u 3 ] Note : where u 1, u, nd u 3 re the lowest reduced integers. [8 6 0] sme s [4 3 0] use ū insted of -u [11] [u 1 u u 3 ] not s the sme s Crtesin coordinte direction except for simple cubic crystl 4

25 Cubic hs highest symmetric directions [ ] [ ] [ ] [ ] [ ] By symmetry, [ ], [ ], [ ] re equivlent [ ],[ ],[ ] Denote { } set of equivlent directions 5

26 Crystl plnes toms fll on plnes lbeled by Miller indices Note : Find the intercepts on the xes in terms of 1,, nd 3 h k l Tke reciprocls nd mke integers ( ) h' k ' l' h' k' l' Reduce to smllest three integers If the plne does not intersect one of the crystl xes, tht intercept is tken to be infinitely fr from the origin nd the corresponding index is zero. ( h k l ) is clled Miller index of the plne Miller indices specify vector norml to the plne, nd not specific plne : ll prllel plnes hve the sme indices 6

27 Schemtic illustrtions of lttice plnes Lines in two dimensionl crystls r r 1 (11) (01) (5) Low index plne : more dense nd more widely spced High index plne : Less dense nd more closely spced 7

28 Most common crystl structures : Simple Cubic lttice : Po conventionl cell : 1 tom/cube Body Centered Cubic lttice : Conventionl cell : toms/ cube Not primitive lttice (1/,1/,1/) 8 nerest neighbors (0,0,0) Alkli metls : Li, N, K, Rb, Cs Ferromgnetic metls : Cr, Fe Trnsition metls : Nb, V, T, Mo, W BCC lttice + single tom bsis SC lttice + bsis of toms t (0,0,0) nd (1/,1/,1/) 8

29 Body Centered Cubic lttice x 3 z 1 y 1 3 = (xˆ + ŷ ẑ) = ( xˆ + ŷ + ẑ) = (xˆ ŷ + ẑ) 9

30 Body-centered Cubic lttice Primitive cell : Rhombohedron w/ edge. the ngle between two djcent edges is 109 o 8 30

31 Fce Centered Cubic lttice : Conventionl cell : 4 toms/ cube Not primitive lttice (1/,0,1/) (0,1/,1/) 1 nerest neighbors (0,0,0) (1/,1/,0) Noble metls : Cu, Ag, Au Trnsition metls : Ni, Pd, Pt, Inert gs solids : Ne, Ar, Kr, Xe FCC lttice + single tom bsis SC lttice + bsis of 4 toms t (0,0,0), (1/,1/,0) (1/,0,1/), nd (0,1/,1/) 31

32 Fce Centered Cubic lttice x 3 z 1 y 1 3 = = = (xˆ + (ŷ + (xˆ + ŷ) ẑ) ẑ) Rhombohedrl Primitive cell The ngle between two djcent edges : 60 o Edge 3

33 Hexgonl Close-Pcked lttice c 3 c 1 =, = with n included ngle10 1 1,, nd c do not construct primitive lttice 1,, nd 3 construct primitive lttice 1 nerest neighbors 1 = = 3 o 1 Trnsition metls : Sc, Y, Ti, Zr, Co IIA metls : Be, Mg Bsl Plne Hexgonl lttice + bsis of toms t (0,0,0) nd (/3,1/3,1/) 33

34 HCP lttice A B A B A B z y x FCC lttice A B C A B C A B C 34

35 Dimond structure : two FCC displced from ech other by ¼ of body digonl FCC lttice + bsis of toms t (0,0,0) nd (1/4,1/4,1/4) 0 1/ 0 3/4 1/4 1/ 0 1/ 1/4 3/4 0 1/ 0 Tetrhedrl bonding : 4 nerest neighbors 1 next nerest neighbors The mximum pcking frction = 0.34 Si, Ge, Sn, C, ZnS, GAs, 35

36 Some toms form multiple stble structures: for exmple, C dimond or grphite (hexgonl) An STM imge of grphite surfce clerly shows the interconnected 6-membered rings of grphite 36

37 grphite dimond 37

38 Mny crystls undergo structurl chnges with T, P: for exmple, Fe δ-ferrite α-ferrite BCC FCC BCC Liquid 910 o C 1400 o C 100 o C Temperture N HCP 36K FCC 371K Liquid Temperture 38

39 Compounds NCl structures NCl, NF, KCl, AgCl, MgO, MnO, FCC lttice + bsis of two toms Cl (0,0,0), N (1/,1/,1/) SC w/. lternting toms CsCl structures CsCl, BeCu, ZnCu(brss), AlNi, AgMg, SC lttice + bsis of two toms Cs (0,0,0), Cl (1/,1/,1/) BCC w/. lternting toms 39

40 ZnS structures Znicblende ZnS, CuF, CuCl, CdS GAs, GP, InSb, compounds Photoconductor III-V semiconducting compounds FCC lttice + bsis of two toms Zn (0,0,0), S (1/4,1/4,1/4) G(0,0,0), As(1/4,1/4,1/4) dimond w/. lternting toms 40

41 41

42 High Tc superconductors c b O(4) Pr Pr Pr Cu() O() Cu() O() Cu() O() O(1) B/Pr O(1) B/Pr O(1) B/Pr Cu(1) O(3) O(5) Cu(1) O(3) Cu(1) O(3) Orthorhombic O(I) (Pmmm) Tetrgonl T (P4/mmm) Orthorhombic O(II) (Cmmm) 4

43 C-60 Bucky blls BuckminsterFullurene 43