P9812a. Fall Lecture Crystal Lattices 1.2 The Reciprocal Lattice 1.3 Experimental Determination of Crystal Structure

Transcription

1 P981a Lectue Cystal Lattices 1. The Recipocal Lattice 1.3 Expeimental Detemination of Cystal Stuctue Cystal: a solid composed of atoms, ions, o molecules aanged in a patten that is epeated in thee dimensions A mateial in which atoms ae situated in a epeating o peiodic aay ove lage atomic distances Refeences: 1. Made, Chaptes 1-3. Kittel, Chapte 1 and 3. Ashcoft and Memin, Chapte Buns, Chaptes 1-5. Ziman, Chapte 1 Lectue 1 1 Cystalline mateials - atoms (ions o molecules) in epeating 3D patten (a lattice) - long-ange ode; ex.: NaCl, Amophous (noncystalline) mateials - Shot ange ode, not peiodic; ex.: liquid wate, glass Factals - long-ange ode, symmety, but not epeating Liquid cystals - long ange ode of one type; disode of anothe - nematic and smectic Lectue 1 Factals Liquid cystals Lectue 1 3 Fall 011 1

2 P981a 1.1 Cystal Lattices Atomic Stuctue Questions: What is the basic stuctue of matte? How do atoms spontaneously oganize? Basic Answes: Scaling theoy elates atom-scale units to macoscopic solids Atoms fom cystalline aays Idea comes fom special class of solids: mineals See vast numbes of mineals at Lectue Two-Dimensional Lattices Definitions: Bavais lattice Pimitive vecto Basis vecto Unit cell (pimitive o not) Wigne-Seitz cell (Voonoi polyhedon) Tanslation, space and point goups Lectue 1 5 Bavais Lattices Lectue 1 6 Fall 011

3 P981a Bavais Lattices Lectue 1 7 Questions Ae pimitive vectos unique? No Fo hexagonal lattice a = a(1 0) 1 a = a 1 ( 3 ) We can also choose a 1, alt a, alt 1 = a( = a 1 ( 3 ) 3 ) Lectue 1 8 Lattice with Basis Note pesence of glide plane, showing that space goup is not the same as the poduct of tanslation and point goup Some, but not all symmeties of tiangula lattice destoyed Lectue 1 9 Fall 011 3

4 P981a Symmeties and The Space Goup The complete set of igid body motions that takes a cystal into itself is called space goup G = a + R( nˆ, θ ) Two subgoups: tanslation and point goups Tanslation: tanslation though all lattice vectos defined by and it leaves the cystal unchanged (invaiant) Point goup consists of otations that leave the cystal invaiant plus cew axis and glide planes n a na +... Lectue 1 10 Summay: Classification of D peiodic Stuctues Unit cell: a convenient epeating unit of a cystal lattice; the axial lengths and axial angles ae the lattice constants of the unit cell Wigne Seitz cell Lage than needed Unit cell is not unique! Wigne Seitz Cell : place the symmety cente in the cente of the cell; daw the pependicula bisecto planes of the tanslation vectos fom the chosen cente to the neaest equivalent lattice site Lectue 1 11 Questions How many distinct Bavais lattices ae thee? Five How many distinct two-dimensional lattices ae thee? 17 Lectue 1 1 Fall 011 4

5 P981a 1.1. Thee Dimensional Cystals Distibution of stuctues among elements A small numbe of popula cystal stuctues Cystal symmeties: 7 cystal systems 14 Bavais lattices 3 point goups 30 space goups Lectue 1 13 Cystallization of Pue Elements Fom Made: Web of Elements: cystal_stuctue.html Lectue 1 14 Allotopy Allotopy the ability of element to exist in two o moe cystalline stuctues Fe: bcc fcc bcc In case of compound it is called polymophism Cabon allotopic foms:? diamond gaphite fulleene o buckyballs nanotubes o buckysheets Lectue 1 15 Fall 011 5

6 P981a Allotopy Many elements adopt multiple cystal stuctues between 0 K and thei melting tempeatues Plutonium has a ich phase diagam Lectue 1 16 Popula Lattices >90% of elemental metals cystallize upon solidification into 3 densely packed cystal stuctues: Body-centeed cubic (bcc) Face-centeed cubic (fcc) Hexagonal closepacked (hcp) ex.: Fe, W, C ex.: Cu, Ag, Au ex.: Z, Ti, Zn Lectue 1 17 Impotant to know: Distance between atoms (d) - in tems of a Numbe of atoms in the unit cell - each cone atoms shaed by 8 cells: 1/8 - each face atom shaed by cells: ½ - each edge atom shaed by 4 cells: ¼ Coodination numbe - Numbe of neaest neighbous (n.n.); fo metals all equivalent Atomic Packing Facto (APF) APF = Volume of atoms in unit cell / Volume of unit cell (a 3 ) Lectue 1 18 Fall 011 6

7 P981a Diamond and Silicon dioxide C (diamond) Si, Ge Cistobalite (SiO ) Si C.N. = 4 SiO 4-4 O C.N. = Lectue 1 19 Sodium Chloide NaCl (Rocksalt) Laye 1 Laye Lectue 1 Laye 3 = Laye 1 0 Cesium Chloide - CsCl Laye 1 Laye Cs (0, 0, 0) Cl (1/, 1/, 1/) Laye 3 = Laye 1 Lectue 1 1 Fall 011 7

8 P981a Intestitial Sites in fcc Cystal Lattice Octahedal sites Tetahedal sites Lectue 1 Zinc Blend (ZnS) cystal stuctue Laye 5=1 Zn (0, 0, 0) S (x+0.5, y+0.5, z+0.5) Laye 4 Laye 3 Laye Laye 1 Laye 1 Laye Laye 3 Laye 4 Lectue 1 3 Calcium Fluoide CaF Laye 5=1 Laye 4 Laye 3 Ca (0, 0, 0) F (+0.5, +0.5, +0.5) (-0.5, -0.5, -0.5) Laye Laye 1 Laye 1 Laye Laye 3 Laye 4 Lectue 1 4 Fall 011 8

9 P981a CaF - coodination CO, CdF, CeO, CoSi, ZO Lectue 1 5 Peovskite CaTiO 3 ABO 3 A: M + (Ca, S, Ba, La) B: M 4+ (Ti, Z, Mn) Laye : TiO Laye 1: CaO Ti octahedal coodination by O (CN=6) d(ti-o)= a / Ca cuboid coodination by O (CN = 1) O octahedal by Ti and Ca Lectue Bavais Lattices and 7 Cystal Systems Lectue 1 7 Fall 011 9

10 P981a Symmety Elements Symmety of Lattices Lectue 1 8 Schönflies and Intenational Notations Schönflies C = Cyclic; allows successive otation about main axis. D = Dihedal; contains two-fold axes pependicula to main axis. S = Spiegel; unchanged afte combination of eflection and otation. T = Tetagonal. O = Octahedal. A subscipt n=1 6 denotes the ode of a otational axis, and subscipts denote the thee types of mio plane on pevious slide Intenational Associates each goup with a list of its symmety axes. Notation such as 6m efes to a mio plane containing a six-fold axis, while 6/m efes to a mio plane pependicula to the six-fold axis Lectue Cystallogaphic Point Goups Lean moe hee: 3dSpaceGps/3DSPGRP.HTM Lectue 1 30 Fall

11 P981a Symmety Opeations Symmety opeation fo a molecule o cystal is an opeation that intechanges the positions of the vaious atoms in such a way that the molecule o cystal appea exactly as befoe the opeation Find of the symmety opeations: 3 otations aound P-F axis 3 mio planes Cl, P and diffeent F atoms 1 mio plane with P and F atoms otation by 10 o and 40 o aound c-axis identity opeation Note: the axis of highest symmety of a molecule o cystal is called the pinciple axis o c-axis o z-axis HW task #1: daw a steeogam fo PF Lectue 3 Cl 1 31 Bulk Tuncation Stuctue b z Ideal flat suface: tuncating the bulk stuctue of a pefect cystal z Mille Indices, evisited - Fo plane with intesections at b x, b y b b y z wite ecipocals: bx by bz x b x - If all quotients ae ational integes o 0, this is Mille index e.g., b x, b y, b z = 1, 1, 0.5 (11) z 4 b x, b y, b z = 1,, (100) - In geneal cd cd cd Mille index ( i, j, k) =, whee cd - common denom.of bx,by,b 3 z bx by b z x e.g., cd = 1 ; ( i, j, k) = = (643) 3 4 Lectue 1 3 y y Angles between the planes [ ijk ] = [ lmn] [ opq] Θ Coss poducts of two vectos in a plane defines diection pependicula to plane [lmn] and [opq] ae both vectos in plane (ijk) Angle between two planes (diections) cos Θ = i + j + k [ ijk] [ lmn] l + m + n e.g., fo [111], [11] : cos Θ = Θ = o Lectue 1 33 Fall

12 P981a Planes in hexagonal cystals 4 coodinate axes (a 1, a, a 3, and c) of the HCP stuctue (instead of 3) Mille-Bavais indices - (h k i l) based on 4 axes coodinate system a 1, a, and a 3 ae 10 o apat: h k i c axis is 90 o : l 3 indices (aely used): h + k = - I (h k i l) (h k l) Note: in hcp, (001) (100) Lectue 1 34 Basal and Pizm Planes Basal planes; a 1 = ; a = ; a 3 = ; c = 1 ( ) Pizm planes: ABCD a 1 = +1; a = ; a 3 = -1; c = ( ) Lectue 1 35 Compaison of cystal stuctues FCC and HCP metal cystal stuctues void a void b A B Bb (111) planes of fcc have the same aangement as (0001) plane of hcp cystal 3D stuctues ae not identical: stacking has to be consideed Lectue 1 36 Fall 011 1

13 P981a FCC and HCP cystal stuctues void a void b A B C A B fcc B plane placed in a voids of plane A Next plane placed in a voids of plane B, making a new C plane Stacking: ABCABC hcp B plane placed in a voids of plane A Next plane placed in a voids of plane B, making a new A plane Stacking: ABAB Lectue 1 37 Steeogaphic Pojections cystal Poject nomals onto plana suface Nomals to planes Lectue 1 38 fom K.Kolasinski Steeogam fo PF 3 Cl E C 3, C 3 3 C σ h 3 σ v S 3, S 3 5 Lectue 1 39 Fall

14 P981a 1. Recipocal Space Recipocal space is also called Fouie space, k- space, o momentum space in contast to eal space o diect space The ecipocal space lattice is a set of imaginay points constucted in such a way that the diection of a vecto fom one point to anothe coincides with the diection of a nomal to the eal space planes and the sepaation of those points (absolute value of the vecto) is equal to the ecipocal of the eal inteplana distance The things which ae lage in eal space ae smalle in ecipocal space by definition Lectue Recipocal Space Lattices - Given a unit cell with basis vectos ( a 1, a ) - Thee is a complementay ecipocal lattice ( a *, *) 1 a ai a j* = δ ij ( i, j = 1, ) a1 * a and a * a1 a Rectangula Lattice 1 a Aea of unit cell A * α a 1 α a A = a1 a sin α * A* = a1* a* sinα * 1 A* = A eal space Non-ectangula lattice a 1 * a * ecipocal space a 1 * a * ecipocal space eal space Lectue Expeimental Detemination of Cystal Stuctues Lectue 1 4 Fall

15 P981a Histoy Expeiments and theoy in 191 finally evealed locations of atoms in cystalline solids Essential ingedients: Theoy of diffaction gating Skiing, and physics table at Café Lutz X-ay tubes, photogaphic plates, and fist expeiments with thei use Pesistence Coheent expeiments with incoheent theoy along behind Incident paticles to conside: X-ays Neutons Electons? Atoms? Lectue 1 43 Tem Mille indices Recipocal lattice Stuctue Detemination Bagg scatteing, elastic and inelastic Bagg angle, Bagg peak and cystal planes Atomic fom facto Stuctue facto Extinctions Expeimental Methods: Ewald constuction Laue method Debye-Schee method, powde diffaction Lectue 1 44 Theoy of Scatteing fom Cystals Geomety of scatteing expeiment Elastic scatteing: fequency of outgoing adiation is the same as that of incoming Radiation of wave vecto k o aives at a sample, intoducing a cicula ing of adiation fom each atom If k o is chosen just ight, the scatteing adiation fom the atoms adds constuctively in cetain diections X-ay EM n, e - QM Lectue 1 45 Fall

16 P981a Scatteing fom a paticle At the oigin Schiff, page 115 o Jackson Eq. 9.8 iko iωt ik e o ψ Ae [ e + f ( ) ] dσ I atom = = f ( ) f is atomic fom facto dωatom Contains details of inteactions between the scatteing potential and the scatteed wave Let s assume we know f () At RR iko ( R) iωt ik ( ) e or iko R ψ ~ Ae e [ e + f ( ) ] R q = k Fo sufficient ly lage, 0 sinθ hq - momentum tansfe ko ( R) ~ ko ko R between incoming and Using Eq. aboveand defining k = ko and q = ko k outgoing waves iko+ iqr θ - Bagg angle iωt ik e o ψ ~ Ae [ e + f ( ) ] Lectue 1 46 Scatteing Theoy Coheent scatteing patten eveals cystalline patten Illustation of Bagg scatteing at angle θ = 6.56 fom the (1) planes of a squae lattice. The magnitudes of k o, k, and K ae detemined using Eqs. (3.38) and (3.39), Made. Lectue 1 47 Many scatteing paticles Assuming multiple scatteing and inelastic scatteing can be ignoed iko + iqr1 iωt e ψ ~ Ae [ fl ( ) ] l Intensity pe unit solid angle iko + iqr1 iωt ik e o ψ ~ Ae [ e + fl ( ) ] l In diection away fom incident beam * iq( R1 R1' ) I = fl fl' e l, l' Equation above is tue no matte how atoms ae aanged! Lectue 1 48 Fall

17 P981a Scatteing fom cystal If all of the scattes ae identical and aanged in a Bavais lattice: I = Iatom e Laue condition : find q iqr1 e = 1 l iqr1 so that fo all atom locations One-Dimensional Sum: lattice points must be of the fom la N 1 q = l= 0 e ilaq (follow Made, p. 48) inaq e = 1 ; iaq q e 1 q = Naq sin aq sin R Lectue Scatteing fo one-dimension Peak when aq πl = lπ q = a Assuming delta functions: N 1 e ilaq = N 1 π πl' N δ ( q ) L l= 0 l' = a Lectue 1 50 thee is a stong peak Scatteing in thee dimensions Main esult : when q = ko k = K satisfies ikr e = 1o K R = πl The scateing sum can be ewitten 3 irq (π ) e = N ( q K) δ ν R K When the vectos R lie in a Bavais lattice, then vectos K satisfying equation above also lie in a lattice the ecipocal lattice Lectue 1 51 Fall

18 P981a Lattice with a Basis Lectue 1 5 Expeimental Methods Ewald constuction Shining geneic monochomatic X-ay upon cystal gives no scatteing peaks!!! Lectue 1 53 Laue Method Lectue 1 54 Fall

19 P981a Rotating Cystal Method Lectue 1 55 Powde Diffaction K θ = sin 1 k o And the adius on film of the scatteing ing due to the ecipocal lattice vecto K is = D tan( θ ) Lectue 1 56 Powde diffaction Main method fo detemining cystal stuctue Conside an X-ay of wavelength λ hits a set of planes sepaated by d unde an angle Θ - some of the X-ays go staight though - some ae eflected (scatteed), but only if specific conditions met Conside a mateial to be a stack of planes at a constant sepaation - d Θ d d sin Θ Out-of-phase n λ n λ whole numbe In phase Lectue 1 57 Fall

20 P981a Bagg s law The diffaction (the coheent elastic scatteing of waves by the cystal lattice) condition n λ = d sin Θ Bagg s law (X-ays, neutons, electons) whee λ wavelength of X-ay beam, d spacing of eflecting planes, Θ angle of incidence and eflection, n ode of diffaction (fo most of the cases we discuss n=1) The lattice plane spacing d depends on the cystal stuctue and indices {hkl} of the planes a a dcubic _ st = dhexagonal_ st = h + k + l 4 l a ( h + k + hk) + 3 c d set by the cystal λ set by appaatus (constant fo a given setup) can change Θ (theta) o often Θ!!! Kittel, pp.9-30 Lectue 1 58 Constuctive and destuctive intefeence X-ay waves scatte in phase (constuctive intefeence): λ, λ, 3λ,, nλ (n whole numbe) Out of phase (destuctive intefeence): 1/λ, 3/λ, 5/λ, What about the othe planes? 1 - if in phase condition holds fo plane 1 and, it also holds fo the plane 3, 4, etc. - if plane 1 and ae out of phase, the 3 d will be in phase will the 1 st, but the 3 4 th will cancel it out 4 Othe planes ae also impotant: λ/8 λ/4 λ/ Unless constuctive intefeence condition met (n whole numbe), thee is vey little intensity at a Lectue 1 given angle 59 Additional ules Conside diffaction fom the (100) face of the fcc cystal Θ a If d sin Θ = λ (i.e., n=1) but thee is always anothe plane at (n=1/) no intensity Rules fo detemining the diffacting {hkl} planes in cubic cystals Lattice bcc Reflection pesent (h+k+l)=even Reflection absent (h+k+l)=odd fcc (h,k,l) all odd o even (h,k,l) not all odd o even Details of cystal unit cell ae impotant Diffeent ules fo diffeent unit cells Lectue 1 60 Fall 011 0

21 P981a Possible peaks fo cubic stuctues a d hkl = h + k + l d hkl a Family of planes {100} sc fcc bcc Lectue 1 61 Powde diffaction Use polycystalline sample All possible planes ae at angle Θ to beam Only ones satisfying Baggs condition povide diffaction Need to change angle Θ to detect all Bagg peaks Recod of the diffaction angles fo a W (tungsten) sample obtained by the use of a diffactomete with Cu adiation Lectue 1 6 Expeimental details (powde diffaction) Use polycystalline sample Souce Collimato (slits) Sample holde (need otation) Detecto (moves in ac aound sample; intensity vs Θ is ecoded) Lectue 1 63 Fall 011 1