Solid state physics. Lecture 2: X-ray diffraction. Prof. Dr. U. Pietsch

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1 Solid state physics Lecture : X-ray diffraction Prof. Dr. U. Pietsch

2 Die Idee von Max von Laue (9) X-Strahlen sind elektromagnetische Wellen mit einer Wellenlänge viele kürzer als die des sichtbaren Lichtes. X-Strahlen werden an periodisch angeordneten Atomen/Molekülen im Kristall gebeugt 94 Nobelpreis für Physik

3 Originalexperiment von Laue, Friedrich und Knipping Photographic film 0.April 93 X-ray tube Collimator Crystal Ausgestellt im Deutschen Museum in München

4 Erstes Laue Experiment Photographic film Erster Kristall Cu SO 4 5 H O

5 9: Begin der moderen Kristallographie X-Strahlen sind elektromagnetische Wellen mit sehr kurzer Wellenlänge ( ~ Å = 0-0 m). Kristalle sind periodische Strukturen: die innerkristallinen Abstände sind von gleicher Größenordnung wie die Wellenlänge der Röntgenstrahlen. Röntgenbeugung ist eine Methode zur Untersuchung der Struktur von Festkörpern!

6 Erklärung durch Interferenz am 3D Gitter

7 Explanation of Laue pattern

10 recip. lattice vector

11 a = b = c h² k² l² a a h² k² l² d sin 4 sin sin

12 Alternative description of Laue-pattern by W.H.Bragg und W.L.Bragg Bragg equation N=dsinQ Interference at dense backed lattice planes

13 Reciprocal space, Ewald construction 0,... * 0, * * * * c b b a c c b b a a ) ( * ) ( * ) ( * c b a b a c c b a a c b c b a c b a *) * * ( lc kb ha G K ) ( hkl d G k k f i

14 Setup of modern diffraction experiment Storage ring Monochromator Slits Analysator Q f Detec tor pre-mirror Slits Q i Absorberbox Sample

15 X-ray tube and tube spectrum E = h v = e U min = h c / e U a min =.4/U a Bremsstrahlung Characteristic radiation

16 ESRF

17 Generation of Synchrotron radiation Electron circulates with relatvististic energy in orbit of storage ring with orbit frequency w= 0 6 Hz E e = 5 GeV m0c² m0c² E e v² ß² c² E e is much larger compared to rest mass energy mc², g = 5GeV/ 0.5 GeV 0 4 g Ee m c² ß² o value /g is the vertical open angle Using g one can estimate the electron velocity ß [ ] g ² / g ² 6*0 9 v 6*0 9 c

18 Generation of Synchrotron radiation Supposing the circulating electron emits a photon at point A of orbit. M Elektron proceeds to point C via B on orbit. At point C it emits a second photon. For the electron the length For the photon emitted in A reaches this distance is AC (electron) = v* dt. AC(photon = c *dt The spatial distance between both waves : (c-v) dt An observer at point C the light pulse has a length t ( c v) dt c ( ²) dt

19 Generation of Synchrotron radiation Considering the opening angle a = /g due to electron orbit t a² ( ag)² ( cosa) dt [ ( )( )] dt dt g ² g ² For a =0 t dt 0 8 dt g ² Dilatation of time a consequences for spectral distribution of photon emission Orbit time of electron along AC differs from time of photon t ( electron) t( photon) T 0 g gw T g ² gw 0 s 0 g ³ w 8 s

20 Generation of Synchrotron radiation Inverse pulse length determines the spectral with of light emission w c 3 g ³ w 8 0 Hz The emission spectrum has a critical wave length given by 8 c c This is x-ray range. In energy : 0 m * E e w c wg ³ ( GeV ) B( T )

21 Generation of Synchrotron radiation Bending radius R of electron orbit is gmv Re B R mc g eb Ee mc Ee( GeV) 3.3* mc² eb B( T) Emission power is P( kw).66* Ee ( GeV) B²( T) Ra( m) I( ma) For ESRF, E=6GeV, B=0.8T R= 4.8m P( kw).66*6e ( GeV)0.8²( T)4.8m 0.05mrad *00mA 7. 3W

22 w c Bend magnet spectrum Well predictable ESRF

23 Kinematic X-ray scattering Dipole scattering of x-ray wave by an electron: Electron excites dipole radiation I E² E e² mc² R² cos ² o Superposition of dipole radiation generated by two electrons at positions X and X e² E E0 exp( i[ wt ( X X )]) mc² X For n electrons e² R E E exp( i[ t ] exp( [ S ]) mc² R k 0 S 0 n f k i ( S S 0 ) Stotal scattering amplitude of one atom : atomic form factor

24 Kinematic X-ray scattering Scattering by all electrons of one atom : atomic formfactor e² R E E0 exp( i[ t ] exp( [ S S 0 ]) mc² R n s s0 e * exp( i[ r( S S0 )]) exp(i[ r ] dv n Form factor f[(s-s 0 )/] result from quantun mechanical calculations f(0) = Z ; f[(s-s 0 )/] decays continuosly as function of (s-s 0 )/

25 Kinematic X-ray scattering Scattering by two atoms : E f R R ( S S0 ) R R R exp( i[ t ] f exp( i[ t ]) exp( i[ t ]( f f exp( i[ ( S 0 )]) S Scattering by many disordered atoms : * I E * E f f f f cos[ ( S S0 )] i sin( krnm) exp [ S S0 ] rnm] k kr nm I n, m Debye sche Streugleichung Debye equation f m f n sin( kr kr nm nm )

R i E fn exp( iwt)exp( i )exp( ( S S0)[ ma ma m3a3] E I Summation over all unit cells N a, N a, N 3

26 Kinematic X-ray scattering Scattering by small crystal: r nm = R nm +r n distances between unit cells shape factor, distances betwen atoms within one unit cell structure factor R nm m a m a m 3a3 R i E fn exp( iwt)exp( i )exp( ( S S0)[ ma ma m3a3] E I Summation over all unit cells N a, N a, N 3 a 3 K k f k i K sin( NKa) sin( NKa) sin( F² sin( Ka ) sin( Ka ) sin( EE* F² S² N 3 Ka Ka 3 3 ) ) FWHM

27 Kinematic X-ray scattering I <> 0 only in near vicinity of the peak maxima Ka Ka Ka 3 h k l h,k,l integer FWHM of a Bragg peak can be approximated sin ²( Nx) I( x) N² exp( ( xn)² / ) sin ²( x) x ka k B cos Q N ² a² B N ² exp( ( ( )² cos ² Q) ² FWHM B(Q) is inversely proportional to crystal size L B(Q) (ln s) / NacosQ 0.94 NacosQ 0.94 LcosQ

28 Structure factor F F n i f i exp( i( kr i )) k ( ha* kb* lc*) r i x a i y b i z c i F n i f i exp( i( ha * kb* lc*)( x a i y b i z c)) i F n i f i exp( i( hx i ky i lz i )) Examples

29 bcc- structure: xyz = 000, ½ ½ ½ F f ( exp( i( h k l)) F= f if h+k+l = even; F= 0 if h+k+l odd fcc- structure: xyz = 000, ½ ½ 0, ½ 0 ½, 0 ½ ½ F f ( exp( i( h k) exp( i( h l) exp( i( k l)) F= 4f if h,k,l all are even or odd; F= 0 if h,k,l are mixed hcp- structure: xyz = 000, /3 /3 ½ F f ( exp( i(( h k) / 3 l / )) F= f if h+k =3n, l even; F=.73 if h+k= 3n+, l odd F= 0 if h+k= 3n+ l even

30 Extinktion rules Not all combinations of h,k,l are allowed, Bragg peak intensity can be vanishing due to crystal (point group ) symmetry In fcc system, only reflections appear with h²+k²+l² =,4,8,,. =, 00, 0, 3,,.. In NaCl In ZnSe, 3.. Are weak reflectons 0, Are strong reflections 0, 400.strong, 3.middle 00,.weka Structure (point group) identification considering extinktion rules and Bragg peak intensities

$diffraction$ pattern

31 Powder diffraction pattern from CdSe Nanowires

32 Debye-Waller Factor Due to thermal atom oscillations exp( ik( r r)) exp( ikr) exp( ikr) exp( ix) i x x²... exp( x² ) exp( ikr) exp( k² r² ) exp( M ) M 6 ² u² sin ² ² F( T) F exp( M ) X-ray intensity becomes damped, Bragg peaks exit

35 Data evaluation

36 Electron charge density distribution from x-ray diffraction data ( r ) F( k)exp( ikr) dk h k l i ( r ) Fi ( H ) exp( Hr); H ( hkl ) V a h k l Electronic charge density of silicon Valence charge density difference charge density (bonding charges) U.Pietsch phys.stat.sol.(b) 37, 44,(986)

37 Electron density distribution of GaAs from x-ray diffraction data Ga As As Ga Valence charge density difference charge density (bonding charges)

Crystal Structure SOLID STATE PHYSICS. Lecture 5. A.H. Harker. thelecture thenextlecture. Physics and Astronomy UCL

Crystal Structure SOLID STATE PHYSICS. Lecture 5. A.H. Harker. thelecture thenextlecture. Physics and Astronomy UCL Crystal Structure thelecture thenextlecture SOLID STATE PHYSICS Lecture 5 A.H. Harker Physics and Astronomy UCL Structure & Diffraction Crystal Diffraction (continued) 2.4 Experimental Methods Notes: examples