912 o C 1400 o C 1539 o C α iron γ iron δ iron. liquid iron BCC FCC BCC

Transcription

1 Polymorphism or Allotropy Mny elements or ompounds exist in more thn one rystlline form under different onditions of temperture nd pressure. This phenomenon is termed polymorphism nd if the mteril is n elementl solid is lled llotropy. Exmple: Iron (Fe Z = 26) liquid bove 1539 C. δ-iron (BCC) between 1394 nd 1539 C. γ-iron (FCC) between 912 nd 1394 C. α-iron (BCC) between -273 nd 912 C. 912 o C 1400 o C 1539 o C α iron γ iron δ iron liquid iron BCC FCC BCC

2 Another exmple of llotropy is rbon. Pure, solid rbon ours in three rystlline forms dimond, grphite; nd lrge, hollow fullerenes. Two kinds of fullerenes re shown here: bukminsterfullerene (bukybll) nd rbon nnotube.

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4 Crystllogrphi Plnes nd Diretions Atom Positions in Cubi Unit Cells A ube of lttie prmeter is onsidered to hve side equl to unity. Only the toms with oordintes x, y nd z greter thn or equl to zero nd less thn unity belong to tht speifi ell. z 0,0,1 0,1,1 0 x, y, z < 1 1,0,1 1,0,0 1,1,1 ½, ½, ½ 0,0,0 1,1,0 0,1,0 y x

5 Diretions in The Unit Cell For ubi rystls the rystllogrphi diretions indies re the vetor omponents of the diretion resolved long eh of the oordinte xes nd redued to the smllest integer. z Exmple diretion A 1,0,1 1,0,0 0,0,1 0,1,1 0,0,0 ½, ½, ½ A 1,1,1 1,1,0 0,1,0 y ) Two points origin oordintes 0,0,0 nd finl position oordintes 1,1,0 b) 1,1,0-0,0,0 = 1,1,0 ) No frtions to ler d) Diretion [110] x

6 z 0,0,1 Exmple diretion B ) Two points origin oordintes 1,1,1 nd finl position oordintes 0,0,0 b) 0,0,0-1,1,1 = -1,-1,-1 ) No frtions to ler _ d) Diretion [111] x C B 0,0,0 1,1,1 ½, 1, 0 y Exmple diretion C ) Two points origin oordintes ½,1,0 nd finl position oordintes 0,0,1 b) 0,0,1 - ½,1,0 = -½,-1,1 ) There re frtions to ler. Multiply times 2. 2( -½,-1,1) = -1,-2,2 d) Diretion [ 12 2]

7 Notes About the Use of Miller Indies for Diretions A diretion nd its negtive re not identil; [100] is not equl to [br100]. They represent the sme line but opposite diretions.. diretion nd its multiple re identil: [100] is the sme diretion s [200]. We just forgot to redue to lowest integers. Certin groups of diretions re equivlent; they hve their prtiulr indies primrily beuse of the wy we onstrut the oordintes. For exmple, [100] diretion is equivlent to the [010] diretion if we re-define the o-ordintes system. We my refer to groups of equivlent diretions s diretions of the sme fmily. The speil brkets < > re used to indite this olletion of diretions. Exmple: The fmily of diretions <100> onsists of six equivlent diretions < 100 > [100],[010],[001],[010],[001],[100]

8 Miller Indies for Crystllogrphi plnes in Cubi Cells Plnes in unit ells re lso defined by three integer numbers, lled the Miller indies nd written (hkl). Miller s indies n be used s shorthnd nottion to identify rystllogrphi diretions (erlier) AND plnes. Proedure for determining Miller Indies lote the origin identify the points t whih the plne interepts the x, y nd z oordintes s frtions of unit ell length. If the plne psses through the origin, the origin of the o-ordinte system must be moved! tke reiprols of these interepts ler frtions but do not redue to lowest integers enlose the resulting numbers in prentheses (h k l). Agin, the negtive numbers should be written with br over the number.

9 z A y Exmple: Miller indies for plne A ) Lote the origin of oordinte. b) Find the interepts x = 1, y = 1, z = 1 ) Find the inverse 1/x=1, 1/y=1, 1/z=1 d) No frtions to ler e) (1 1 1) x

10 More Miller Indies - Exmples 1/ b 2/3 b b b b b Notes About the Use of Miller Indies for Plnes A plne nd its negtive re prllel nd identil. Plnes nd its multiple re prllel plnes: (100) is prllel to the plne (200) nd the distne between (200) plnes is hlf of the distne between (100) plnes.

11 Certin groups of plnes re equivlent (sme tom distribution); they hve their prtiulr indies primrily beuse of the wy we onstrut the o-ordintes. For exmple, (100) plnes is equivlent to the (010) plnes. We my refer to groups of equivlent plnes s plnes of the sme fmily. The speil brkets { } re used to indite this olletion of plnes. In ubi systems the diretion of miller indies [h k l] is norml o perpendiulr to the (h k l) plne. in ubi systems, the distne d between plnes (h k l ) is given by the formul onstnt. Exmple: d = h 2 + k 2 + l where is the lttie The fmily of plnes {100} onsists of three equivlent plnes (100), (010) nd (001) 2

12 A fmily of rystl plnes ontins ll those plnes re rystllogrphilly equivlent. Plnes hve the sme tomi pking density fmily is designted by indies tht re enlosed by bres. - {111}: (111), (111), (111), (111), (111), (111), (111), (111)

13 Single Crystl Polyrystlline mterils Anisotropy nd isotropy

14 Two Types of Indies in the Hexgonl System , 2,nd re independent, 3 is not! 3 = - ( ) Miller: (hkl) (sme s before) Miller-Brvis: (hkil) i = - (h+k) (001) = (0001) - - (110) = (1100) (110) = (1100) (100) = (1010) -

15 Struture of Cermis Cermis kermikos - burnt stuff in Greek - desirble properties of ermis re normlly hieved through high temperture het tretment proess (firing). Usully ompound between metlli nd nonmetlli elements Alwys omposed of more thn one element (e.g., Al 2 O 3, NCl, SiC, SiO 2 ) Bonds re prtilly or totlly ioni, n hve ombintion of ioni nd ovlent bonding (eletronegtivity) Generlly hrd, brittle nd eletril nd therml insultors Cn be optilly opque, semi-trnsprent, or trnsprent Trditionl ermis bsed on ly (hin, briks, tiles, porelin), glsses. New ermis for eletroni, omputer, erospe industries.

16 Crystl Strutures in Cermis with predominntly ioni bonding Crystl struture is defined by The eletri hrge: The rystl must remin eletrilly neutrl. Chrge blne dittes hemil formul (C 2+ nd F - form CF 2 ). Reltive size of the tion nd nion. The rtio of the tomi rdii (r tion /r nion ) dittes the tomi rrngement. Stble strutures hve tion/nion ontt.

17 Coordintion Number: the number of nions nerest neighbors for tion. As the rtio gets lrger (tht is s r tion /r nion ~ 1) the oordintion number gets lrger nd lrger. Holes in sphere pking Tringulr Tetrhedrl Othedrl

18 Clulting minimum rdius rtio for tringle: B O O B 1 2 AO = 1 AB 2 AO r r r A AB r + r = r r + r = os30 = = C = osα ( α = 30 ) AO = 1 AB 2 AO r r r AB r + r = r r A + r = os45 = = 2 2 C = osα ( α = 45 ) o for n othedrl hole

19 C.N. = 2 r C /r A < C.N. = < r C /r A < C.N. = < r C /r A < C.N. = < r C /r A < C.N. = < r C /r A < 1.0

20 Ioni (nd other) strutures my be derived from the ouption of interstitil sites in lose-pked rrngements.

21 Comprison between strutures with filled othedrl nd tetrhedrl holes o/t f(p) hp ll ot. NCl NiAs ll tetr. CF 2 (ReB 2 ) o/t (ll) (Li 3 Bi) (N 3 As) ½ t sphlerite (ZnS) wurtzite (ZnS) (½ o CdCl 2 CdI 2 ) Lotion nd number of tetrhedrl holes in f (p) unit ell - Z = 4 (number of toms in the unit ell) - N = 8 (number of tetrhedrl holes in the unit ell)

22 Crystls hving filled Interstitil Sites Othedrl, Oh, Sites Ioni Crystls prefer the NCl Struture: NCl struture hs N+ ions t ll 4 othedrl sites Lrge intertomi distne LiH, MgO, MnO, AgBr, PbS, KCl, KBr N+ ions Cl- ions

23 Crystls hving filled Interstitil Sites Tetrhedrl, Th, Sites Both the dimond ubi struture And the Zin sulfide strutures hve 4 tetrhedrl sites oupied nd 4 tetrhedrl sited empty. Zn toms Covlently Bonded Crystls Prefer this Struture Shorter Intertomi Distnes thn ioni Group IV Crystls (C, Si, Ge, Sn) Group III--Group V Crystls (AlP, GP, GAs, AlAs, InSb) Zn, Cd Group VI Crystls (ZnS, ZnSe, CdS) Cu, Ag Group VII Crystls (AgI, CuCl, CuF) S toms

24 AX Type Crystl Strutures Rok Slt Struture (NCl) NCl struture: r C = r N = nm, r A = r Cl = nm r C /r A = 0.56 Coordintion = 6 Cl NCl, MgO, LiF, FeO, CoO N Cesium Chloride Struture (CsCl) CsCl Struture: r C = r Cs = nm, r A = r Cl = nm r C /r A = 0.94 Coordintion = 8 Cl Cs Is this body entered ubi struture?

25 Zin Blende Struture (ZnS) rdius rtio = Coordintion = 4 S Zn ZnS, ZnTe, SiC hve this rystl struture A m X p -Type Crystl Strutures If the hrges on the tions nd nions re not the sme, ompound n exist with the hemil formul A m X p, where m nd/or p 1. An exmple would be AX 2, for whih ommon rystl struture is found in fluorite (CF 2 ).

26 Fluorite CF 2 Fluorite (CF 2 ): r C = r C = nm, r A = r F = nm r C /r A = 0.75 From the tble for stble geometries we see tht C.N. = 8 Other ompounds tht hve this rystl struture inlude UO 2, PuO 2, nd ThO 2. A m B n X p -Type Crystl Strutures It is lso possible for ermi ompounds to hve more thn one type of tion; for two types of tions (represented by A nd B), their hemil formul my be designted s A m B n X p. Brium titnte (BTiO 3 ), hving both B 2 + nd Ti 4+ tions, flls into this lssifition. This mteril hs perovskite rystl struture nd rther interesting eletromehnil properties

27 CTiO 3 - dieletri BTiO 3 - ferroeletri Pb(Mg 1/3 Nb 2/3 )O 3 - relxor ferroeletri Pb(Zr 1-x Ti x )O 3 - piezoeletri (B 1-x L x )TiO 3 semiondutor Perovskite - n Inorgni Chmeleon ABX 3 - three ompositionl vribles, A, B nd X (Y 1/3 B 2/3 )CuO 3 -x - superondutor N x WO 3 - mixed ondutor; eletrohromi SrCeO 3 - H - protoni ondutor RECoO 3-x - mixed ondutor (Li 0.5-3x L 0.5+x )TiO 3 - lithium ion ondutor LMnO 3-x - Gint mgnetoresistne

28 The perovskite struture CTiO 3 - TiO 6 othedr - CO 12 ubothedr (C 2+ nd O 2- form ubi lose pking) preferred bsis struture of piezoeletri, ferroeletri nd superonduting mterils