c 2003, Michael Marder

Transcription

1 Expeimental Detemination of Cystal Stuctues 1 8th Januay 003 c 003, Michael Made

2 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. Willingness to disobey supeviso, and belief that expeiment was safe than theoy. X-ay tubes, photogaphic plates, and expeience with thei use. Pesistence. Coheent expeiments dagging incoheent theoy along behind. 8th Januay 003 c 003, Michael Made

3 Tems 3 Bagg scatteing, elastic and inelastic Bagg angle Bagg peak Bagg planes Atomic fom facto Recipocal lattice Mille indices Stuctue facto Extinctions Ewald constuction Laue method Debye-Schee method, powde diffaction 8th Januay 003 c 003, Michael Made

4 Scatteing Theoy 4 k k 0 k 0 Incoming adiation 8th Januay 003 c 003, Michael Made

5 Scatteing Theoy 5 3 a k 0 K k 0 a K 1 0 a d Plane wave tavels towad solid, scattes off atoms. Coheent scatteing patten eveals cystalline patten. 8th Januay 003 c 003, Michael Made

6 Scatteing fom a paticle at the oigin 6 Schiff page 115 o Jackson Eq. 9.8 Ae i t e i k 0 f e ik 0 (L1) I atom d d atom f (L) f is atomic fom facto. 8th Januay 003 c 003, Michael Made

7 Scatteing fom paticle at R 7 Ae i t ei k 0 R e i k 0 R f e ik 0 R R (L3) Fo sufficiently lage, k 0 R k0 k 0 R (L4) Using Eq. (L4) and defining and k q k0 k 0 k (L5) (L6) gives Note that Ae q i t e i k 0 f e ik 0 iqr (L7) k0 sin (L8) 8th Januay 003 c 003, Michael Made

8 Many scatteing paticles 8 Assume multiple scatteing and inelastic scatteing can be ignoed Ae i t e i k 0 l f l e ik 0 iq R l (L9) Look away fom incoming beam Ae i t l f l e ik 0 i q R l (L10) Intensity pe unit solid angle I f l f l e i q R l R l (L11) l l Eq. (L11) is tue no matte how atoms ae aanged. 8th Januay 003 c 003, Michael Made

9 Scatteing fom cystal 9 I Iatom l e i q R l (L1) Laue condition: find q so that fo all atom locations R l exp iq R l 1(L13) 8th Januay 003 c 003, Michael Made

10 One-Dimensional Sum 10 q N 1 l0 e ilaq (L14) q N 1 l0 e ilaq (L15) eiaq q N 1 e i l 1 aq (L16) l0 q N 1 l0 e ilaq 1 einaq(l17) 1 q einaq(l18) 1 (L19) 1 e inaq e iaq e inaq e iaq sin Naq sinaq (L0) 8th Januay 003 c 003, Michael Made

11 One-Dimensional Sum 11 q q sin Naq sin (L1) aq q e inaq e iaq 1 1 (L) sin Naq sin (L3) aq 8th Januay 003 c 003, Michael Made

12 One-Dimensional Sum q a 4 6 a a Wave numbe q 8 a 8th Januay 003 c 003, Michael Made

13 One-Dimensional Sum 13 Peaks when aq l q l a (L4) View as sum of delta functions: N 1 l0 e ilaq l N L q l a (L5) 8th Januay 003 c 003, Michael Made

14 Scatteing in thee dimensions 14 Main esult: when q k 0 k K satisfies exp i K R 1 o K R l(l6) thee is a stong scatteing peak. The scatteing sum can be ewitten e i R q N 3 q K (L7) R K When the vectos R lie in a Bavais lattice, then vectos a lattice the ecipocal lattice. K satisfying Eq. (L6) also lie in 8th Januay 003 c 003, Michael Made

15 Scatteing in thee dimensions 15 Fist conside K R 0(L8) Once the diection of K is chosen, the R satisfying this condition lie in a plane passing though the oigin 0 K 0 K a 1 a a 3 1 a 1 a a 3 1 8th Januay 003 c 003, Michael Made

16 Scatteing in thee dimensions 16 The magnitude of K is esticted by the need to satisfy Eq. (L8) fo all Bagg planes. In the plane, l 1 a 1 l a (L9) Fo any a 3 in an adjacent plane, suppose a 3 K (L30) Then K R K l 1 a 1 l a l3 a 3 l 3 (L31) Explicit constuction: b 1 a a 1 a a 3 a 3 b a 3 a a 3 a 1 a 1 b 3 a 1 a 3 a 1 a a (L3a) K 3 m l b l (L3b) l1 8th Januay 003 c 003, Michael Made

17 Scatteing in thee dimensions 17 Recipocal lattice of a simple cubic lattice of lattice spacing a is anothe simple cubic lattice, of spacing a. The ecipocal lattice of an fcc lattice of spacing a is, howeve, a bcc lattice of spacing 4 a The ecipocal lattice of a bcc lattice of spacing a is an fcc lattice of spacing 4 a. 8th Januay 003 c 003, Michael Made

18 Mille Indices 18 i jk efes to a diection ix jy kz (L33) in the lattice specified by the thee integes i, j, and k. i jk efes to a lattice plane pependicula to i jk i jk efes to the family of lattice planes pependicula to i jk and elated by symmety. 8th Januay 003 c 003, Michael Made

19 Lattice with a Basis 19 Regouping of basic sum fist caied out by Laue R u l l (L34) e i q R e i q u l l (L35) R ll e i q u l e i q l (L36) l l I e iq u j u j e i q l l (L37) j j ll Stuctue facto fo the unit cell is F q l e i q l (L38) When Fq vanishes, have an extinction: Laue ovelooked this possibility, leading to yeas of confusion intepeting pattens. 8th Januay 003 c 003, Michael Made

20 Lattice with a Basis 0 l l l l Example: Diamond Theefoe, K 4 l 1 a l a a l 3 a (L39) (L40) FK is K l 1 l l 3 (L41) F K 1 e i l 1 l 3 (L4) 4 if l 1 if l 1 l l 3 is odd 1 (L43) 0 if l 1 l th Januay 003 c 003, Michael Made

21 Expeimental Methods 1 Ewald constuction k q k 0 Shining geneic monochomatic X-ay upon cystal gives no scatteing peaks 8th Januay 003 c 003, Michael Made

22 Laue Method Intensity (Abitay Units) Wavelength (Å) k min 0 k max 0 8th Januay 003 c 003, Michael Made

23 Rotating Cystal Method 3 Film Cylindical Film Incoming Beam Sample k 0 8th Januay 003 c 003, Michael Made

24 Rotating Cystal Method 4 k D sin k 0 D sin D 8th Januay 003 c 003, Michael Made

25 Powde Method 5 1 sin K and the adius on film of the scatteing ing due to ecipocal lattice vecto K is Dtan k 0 (L44) (L45) 8th Januay 003 c 003, Michael Made

26 Moe on Scatteing Expeiments 6 Chage 0 0 Mass X-ays Neutons Electons e 7 kg Typical enegy 10 kev 0.03 ev 100 kev Typical wavelength 1Å 1 Å 0.05Å Typical attenuation length 100 Typical atomic fom facto, f 10 Inteactions of X ays with matte m 5 cm 1 3 Å 10 m 4 Å 10 Å 31 kg I atom f e 4 m c 4 e mc 1 P cos e 4 m c 4 P P (L46) m(l47) 8th Januay 003 c 003, Michael Made

27 Moe on Scatteing Expeiments Lase Synchoton Radiation Photons [s mad 0.1% Bandwidth] Cu K Mo K X-Ray Tubes Bemsstahlung Al K Cu L C K He I Sun Bulb Wavelength (Å) Neutons ae almost completely isotopic. Elastic scatteing (neutons lose no enegy) 8th Januay 003 c 003, Michael Made

28 Moe on Scatteing Expeiments 8 gives vey pecise infomation about static stuctue. Inelastic scatteing gives vey pecise infomation about mechanical excitations. Neutons ae sensitive to the spins of the nuclei fom which they scatte. Intensity (Abitay units) 80K 93K (111) () (311) (400) (331) (511) (440) (111) (6) (311) (00) (0) Infomation on a neuton detecto Scatteing angle (degees) 8th Januay 003 c 003, Michael Made

29 75 yeas of futhe advances... 9 Insetion of heavy atoms allows extemely complex cystals to be decipheed. Cystallogaphy Online Stuctue of Hemoglobin asmol viewe fo molecules Computes do most of the wok now (fo bette o wose) 8th Januay 003 c 003, Michael Made