DIFFRACTION METHODS IN MATERIAL SCIENCE. PD Dr. Nikolay Zotov Tel Room 3N16.
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1 DIFFRACTION METHODS IN MATERIAL SCIENCE PD Dr. Nikolay Zotov Tel Room 3N16 Lecture 5
2 OUTLINE OF THE COURSE 0. Introduction 1. Classification of Materials 2. Defects in Solids 3. Basics of X-ray and neutron scattering 4. Diffraction studies of Polycrystalline Materials 5. Microstructural Analysis by Diffraction 6. Diffraction studies of Thin Films 7. Diffraction studies of Nanomaterials 8. Diffraction studies of Amorphous and Composite Materials 2
3 OUTLINE OF TODAY S LECTURE Scattering from Single Crystals (Short Repetition) Scattering from Polycrystalline Materials Powder Diffraction Intensities Selection Rules Types of Detectors Powder Diffractometers Diffraction Pattern measured What to do now? 3
4 SCATTERING EXPERIMENT Source Detector Sample Sources Types; How to select them Sample Polycrystalline Sample Detectors Types; How to select them 4
5 Conditions for Scattering from Single Crystal Q = ng hkl The diffraction from a single crystal is located only at the reciprocal lattice points k' Q k 0 Ewald construction Every spot is a given reflection hkl with intensity (darkness) ~ F hkl 2. Some reciprocal lattice points are missing because of the presence of selection rules. 5
6 Scattering Amplitude/ Scattering Intensity A(Q) = = {Σ f j (Q)exp(-½u j Q 2 )exp(iq.x j )} {Σ exp (i.q.t mnp )} = F hkl {Σ Σ Σ exp (i.q.t mnp )} I(Q) ~ A(Q) 2 = F hkl 2 L(Q) ; L(Q) is the so-called Laue function 6
7 Laue Function L(Q) = {Σ Σ Σ exp (i.q.t mnp )} {Σ Σ Σ exp (i.q.t mnp )}* = T mnp = ma+nb+pc = [sin 2 (MQ.a/2)/sin 2 (Q.a/2)] [sin 2 (NQ.b/2)/sin 2 (Q.b/2)] [sin 2 (PQ.c/2)/sin 2 (Q.c/2)] M,N,P number of unit cells along x,y,z directions Sinc function Maximum of the Sinc Function Q.a = 2hp Q.b = 2kp Q.c = 2lp Another derivation of the Laue conditions 7
8 Scattering from a Polycrystalline Sample Polycrystalline sample = Aggregate of crystallites (small single crystals) with different shape, size and orientation. k i Q k f For random orientation of the crystallites, the scattered X-rays (neutrons or electrons) lay on cone(s) with opening angle = 4Q; r = (2p/l) sin(q) k=2 p/l Q r The reciprocal space of a polycrystal represents a system of concentric spheres with radii d hkl * = 1/d hkl 8
9 Effect of Preffered Orientation Random Orientations Preffered Orientations Debey-Scherrer rings r Ring For random orientation, the intensities depend only on the diffraction angle 2Q! For a sample with a preffered orientation individual diffraction spots on the DS rings with inhomogeneous intensity distribution. 9
10 Powder Diffraction Intensities I hkl = ( I o /R SD2 ) m hkl F hkl 2 Pol Lor Abs Primary beam Intensity Distance Sample- Detector Polarization Factor Absorption Factor Multiplicity Factor Lorentz factor Structure Factor 10
11 Multiplicity Factors The multiplicity of the reflections reflects the point symmetry of the crystal (100) Tetragonal Crystal {100} = (100); (010); (-100); (0-1 0) {001} = (001); (00-1) c (001) 11
12 Multiplicity Factors for general reflections (hkl) 12
13 Multiplicity Factors for general reflections (hkl) Point group 2/m generates 4-symmetry equivalent objects m = 4 13
14 Multiplicity Factors for general reflections (hkl) m = 8 14
15 Multiplicity Factors for general reflections (hkl) m = 8 15
16 Polarization Factor Z Electromagnetic wave is a transverse wave I= E sc 2 = E o 2 Pol(Q) k i k f 2Q I = E 2 Y X Vertical plane Z ~ 1 Incident wave polarized in the vertical scattering plane Pol(Q) = cos(2q) Incident wave polarized in the horisontal scattering plane ½[1 + cos 2 (2Q)] Unpolarized incident wave Laboratoty X-ray tubes give unpolarized X-rays 16
17 Lorentz Factor I Geometric factor, relating the scattered intensity to the density of scattered X-ray (neutrons). Diffraction Cone 2p/l 4Q Peremeter of the circular base of the cone: C = 2pr = (4p 2 /l)sin(2q) Density of diffraction spots along the cone (homogeneous distribution) : ~ 1/ sin(2q) 17
18 Lorentz Factor II Radius of Reciprocal Sphere (RS) Q = G hkl r RS = d hkl * = 1/d hkl = (2/l)sin(Q) Density of diffraction spots on the RS ~ 1/sin(Q) Bragg equation Polycrystalline materials Lor(Q) = 1/sin(Q)sin(2Q) 18
19 19
20 Beers Law I = I o exp(-µx) The thickness x varies with 2Q! Absorption Factor Reflection Geometry A(Q) = 1 exp(-2µd/sin(q)) Transmission Geometry (Cylindrical samples) Numerical Methods 20
21 A = (1/pR 2 ) 0 R rdr 0 2p df exp[-µl(r,f,2q)] µr Collaso et al. (1998) 21
22 Structure Factor/ General Selection Rules F Q =Σ f j (Q) exp(iq.x j ) F hkl = fexp[i2p(h.0+k.0+l.0)] + Q = G hkl BCC metals fexp[i2p(h.1/2+k.1/2 + l.1/2)] = f {1 + exp[ip(h+k+l)]} h+k+l = 2n (even) F hkl = f h+k+l = 2n+1 (odd) F hkl = 0 absent!!! 1 2 Atom 1 (0,0,0) Atom 2 (1/2,1/2,1/2) 22
23 Structure Factor/ General Selection Rules F Q =Σ f j (Q) exp(i.q.x j ) Hexagonal-closed packed (HCP) Metals Atoms at (0,0,0) and (1/3, 2/3,1/2) F hkl = f{1 + exp[2pi(h/3+2k/3+l/2)]} F hkl2 = 4f 2 cos 2 [p(h/3+2k/3+l/2]] 0 h+2k = 3n and l = odd absent f 2 h+2k = 3n±1 and l = even I hkl ~ 3f 2 h+2k = 3n±1 and l = odd present 4f 2 h+2k = 3n and l = even 23
24 Structure Factor/ General Selection Rules 4 Atome in the unit cell: (0,0,0) (1/2,1/2,0); (1/2,0,1/2); (0,1/2,1/2) Fcc metals F hkl = f Cu {exp[i2p(h0+k0+l0)] + exp[i2p(h1/2+k1/2 + l0)] + exp[i2p(h1/2+k0 + l1/2)] + exp[i2p(h0+k1/2 + l1/2)]} = f Cu {1 + (-1) (h+k) + (-1) (h+l) + (-1) (k+l) } F hkl = 4f Cu F hkl = 0 if all h,k,l are even or all are odd if mixed parity
25 General Selection Rules 25
26 Systematic Absences for Screw Axes 26
27 Systematic Absences for Glide Planes 27
28 Selection of Detectors 2D detector 1D detector Dimensionality 0D Detector ( Point detector) 1D Detector (PSD detector) 2D Detector High-resolution, lower cost Quick measurement of large 2Q ranges Quick measurement of large portions of rec. space Quick investigation of preffered orientation Expensive 28
29 Point Detectors Gas-proportional Detectors X-ray photon e - Ar/Xe + Scintilator Detectors photon NaI(Tl) X-ray photon Solid State Detectors Si(Li) High Energy resolution!!! (Cooling necessary) 29
30 POSITION SENSITIVE DETECTORS 2Q range ~ 2-4 deg 2Q range ~ deg 30
31 2D Detectors Debey-Scherrer Cameras Gas Detectors CCD Cameras 31
32 Debey Scherrer Cameras Debey-Scherrer (DS) rings The DS rings are cross-section of the diffraction cones with the cylindrical surface of the film. Wet photographic processes, Densitometer (scanner) for reading of intensities 32
33 2D Gaseous Dtectors Vantec 500 (Bruker) e - X-ray photon 2D network of wires The parts of the Debey-Sherrer rings are cross-sections of the diffraction cones with the flat 2D surface of the detector o 2Q range measured simultaneously 33
34 Sn Thin Film Large crystallites 34
35 Diffraction Patterns from 2D images Strong preffered orientation of some reflections 35
36 CCD DETECTORS Gd 2 O 2 S:(Eu,Tb) 36
37 CCD DETECTORS 37
38 Powder Diffractometers Historically, first were used Point-Detectors. In order to measure many diffraction lines, movement of tube/sample/detector are necessary: Q - Q diffractometers (scans) (sample fixed) Q - 2Q diffractometers (scans) (tube fixed) Scattering condition: angle between k i and k f = 2Q 38
39 Q - Q Diffractometers Liquid Samples Molten Samples Low-Temperature/High-Temperature Furnaces Deformation Rigs Detector Tube Sample Holder Counter weights 39
40 Q 2Q Diffractometers X-ray tubes Divergent Beam Divergent Slit(s)/ Colimator(s) Smaller Colimator: Smaller Illuminated Sample Area less Divergence Receiving Slit(s) Smaller Slit: Higher Resolution Lower Intensity Diffractometer Circle Larger Radius: Less Intensity (1/R 2 ) Higher resolution 40
41 Integral Intensity Geometrical Divergence of X-ray (neutrons) Dispersion of wavelengths Distribution of I(2Q) around the ideal Brag positions 2Q B. Misorientation of crystallites Integral Intensity ~ I o m F 2 LPA L(Q B)dQ B 2Q B L(Q B)dQ B ~ MNP = V/V uc I(2Q B ) ~ Vm F 2 LPA Given phase (structure) gives a unique set of diffraction lines at specific 2Q positions and with specific integral intensities proportional to the volume of the sample (untextured samples) 41
42 Diffraction Pattern - So now what? 5000 Intensity (counts) / Peak Fitting List of d-spacings and Intensities l = nm 2Q (degrees) 42
43 Unknown (new) phase Diffraction Pattern Known (exsisting) phase Indexing as a triclinic crystal Indexing Search Match Lattice Parameters Determination Lattice Parameters Refinement Phase Indentification (using the ICDD Data Base) Space Group Determination (selection rules) Structure Determination (Patterson methods and/or Direct methods) Structure Refinement (Rietveld method) 43
44 Chemical Constraints: Ni,Ti,O 44
45 Ag(200) Ag F m-3m (fcc) Sn (200) Ag (111) Sn I 4 1 /amd Ag 3 Sn P mna Ag Sn (101) Ag3Sn (201) Ag3Sn (020) Ag3Sn (012) (211) X Sn (211) Ag 3 Sn Sn 45
46 Simple Example of Profile Fitting Intensity (counts) Peak Position Peak Intensity Integral Intensity FWHM Q (degrees) 46
47 Lattice Parameter Calculations - Example Ag 3 Sn Orthorhombic [PDF Card (a=5.968, b = , c= Å)] 1/d hkl2 = h 2 /a 2 + k 2 /b 2 + l 2 /c 2 2Q (hkl) d hkl (201) (020) b = 2d 020 = Å (211) Accurate Determination of Lattice Parameters = Multiple Reflexions + Least-Squares Rietveld 47
48 Rietveld refinement Shape memory alloy NiTi, RT Martensite 89 wt% P 2 1 /m a = 4.639(2) Å b = 4.119(2) Å c = 2.898(1) Å ß = (2)o Austenite 11 wt% P m -3 m a = 11.31(1) Å Pseudo-Voigt R wp ~ 6 % 48
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