DIFFRACTION METHODS IN MATERIAL SCIENCE. PD Dr. Nikolay Zotov Lecture 6

Transcription

1 DIFFRACTION METHODS IN MATERIAL SCIENCE PD Dr. Nikolay Zotov Lecture 6

2 OUTLINE OF THE COURSE 0. Introduction 1. Classification of Materials 2. Defects in Solids 3+4. Basics of X-ray and neutron scattering 5. Diffraction studies of Polycrystalline Materials 6. Measuring Powder Diffraction Patterns; 7. Microstructural Analysis by Diffraction 8. Diffraction studies of Thin Films 9. Diffraction studies of Nanomaterials 10. Diffraction studies of Amorphous and Composite Materials 2

3 OUTLINE OF TODAY S LECTURE Scattering from Polycrystalline Materials (Repetition) Powder Diffractometers Measuring Powder Diffraction Patterns Diffraction Line Broadening Deconvolution Profile Fitting 3

4 Scattering from Polycrystalline Samples Q = ng hkl Q k f k i For random orientation of the crystallites, the scattered X-rays (neutrons) lay on cone(s) with opening angle = 4Q. The reciprocal space of a polycrystal represents a system of concentric spheres. 4

5 Powder Diffraction Intensities I hkl = ( I o /R SD2 ) m hkl F hkl 2 Pol Lor Primary beam Intensity Distance Sample- Detector Polarization Factor Absorption Factor Multiplicity Factor Lorentz factor Structure Factor 5

6 Scattering conditions h + k + l = 2n bcc type lattice 6

7 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! 7

8 Q - Q Diffractometers Liquid Samples Molten Samples Low-Temperature/High-Temperature Furnaces Deformation Rigs Detector Tube Scattering Plane (vertical) Q k i k f Counter weights Sample Holder 8

9 Q - 2Q Diffractometers 1 X-ray Tube 2 Prinary-beam optics 3 Sample Holder 4 Micrometer for adjustment of the height 5 ¼ Eulerian cradle 6 Scattered-beam optics 7 Monochromator 8 Point Detector Scattering Plane (horisontal) k f k i Q 9

10 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 10

11 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 ~ (Intergration over the recieving slit) 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 11

12 Diffraction Pattern - So now what? 5000 Intensity (counts) / Peak Fitting List of d-spacings and Intensities l = nm 2Q (degrees) 12

13 Unknown (new) phase Diffraction Pattern Existing 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) 13

14 Chemical Constraints: Ni,Ti,O 14

15 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 Strong Overlap of Diffraction Lines Coincidential Overlap of Diffraction lines 15

16 Simple Example of Profile Fitting Intensity (counts) Peak Position Peak Intensity Integral Intensity FWHM Q (degrees) 16

17 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 or Rietveld 17

18 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 % 18

19 Measuring (Powder) Diffraction Patterns I Meas = I hkl * + I BKG + I Noise Structural BKG (diffuse Scat.) + Instrumental BKG (Furnace; Vacuum Chamber) Background Reduction Noise Reduction BKG Noise 19

20 2D diffraction pattern Be window Air scattering of the primary beam Artefacts of the CCD 20

21 Background Reduction Sources Air Scattering Auxilliary Scattering Substrate Scattering X-ray Fluorescence Solutions Vacuum (if possible) Proper Alignment ß Filter Use of secondary-beam monochromator* Narrow Slits Proper selection of substrate materials Glass, Thin Kapton Foil, Specially-cut Si-wafers Proper choice of wavelength E.g. Co radiation for Fe-containing materials, because Cu radiation leads to strong fluorescence * Q M = arcsin(l/2d) 21

22 Effect of Substrate Si(400) Specimen Si Substrate Intensity T 22

23 Graphite Monochromator d 002 = Å Tube Sample Effect of Monochromator Eliminates K ß Reduces background intensity Loss of Intensity of K a1 (and K a2 ) 23

24 Noise Reduction Detection Counting of the number of X-ray photons or neutrons in a certain direction for a given amount of time Noise - Variation of the intensity at a given scattering angle with time Types Time-independent Noise Detector-specific Noise Cosmic rays Low Gas detectors Scintilator detectors Solid State detectors Higher Films (foggy images) Image Plates CCD detectors (dark-current noise) Statistical Noise Arrises from the stohastic nature of the counting (detection) prosses 24

25 Statistical Noise The counting process of X-ray photons (or neutrons) is a random process. Intensity Most detectors have # constant counting rate Ř ( I meas = Ř t) # low counting rate (efficiency) 2Q fixed Time <I> = 1/N S I a ; s 2 = 1/(N-1) S (I a - <I>) 2 ; a = 1,2,... N What is the probability to measure given intensity I, if a large number N of scattered photons/neutrons impinge on the detector (at fixed 2Q)? <I> = i P(i) di 25

26 Statistical Noise The statistics of large number of events (photons impinging on the detector) with low probability of success (counting rate) is describes by the so-called Poisson distribution. Poisson Statistic X random variable (Intensity) P(X=k) = l k /k! exp(-l) k a specific numerical value <X> = l (Average value) s 2 = Var(X) = l (Variance) In scattering <X> = I Relative statistical error e = s(i)/i = 1/sqrt(I) (Noise/signal ratio) s = (s 2 ) ½ = (I) ½ ; (Noise) 26

27 Measuring Strategies Fixed time per step I = Řt; t I e Pre-defined number of counts per step t=1, e 1 = 1/sqrt(R) t=100, e 100 = 0.1 e 1 I = 100 counts, s(i) = 10, e = s/i = 10% Increases the time for measurements between the Bragg peaks Time-per-step increases with 2Q Compensates for the decrease of the intensities due to thermal vibrations 1500 Intensity (counts) Q (degrees) 27

28 Diffraction Line Broadening Scattering Condition Bragg Law: sin(q) hkl = l/2d hkl Delta function at 2Q hkl Origins of Line Broadening Instrumental Broadening Structural Broadening 28

29 Diffraction Line Broadening I hkl * = I Real * I Inst Warren, X-ray Diffraction 29

30 Properties of Convolution G Gausian function ~ exp[-ln2(x-x 0 ) 2 /w 2 ] L Lorentzian Function ~ 1/[1 + (x-x 0 ) 2 /w 2 ] G = G 1 * G 2 s 2 (G) = s 2 (G 1 ) + s 2 (G 2 ) Gaus L L = L 1 * L 2 s(l) = s(l 1 ) + s(l 2 ) Intensity X o = x 30

31 Instrumental Broadening Finite width of the wavelength distribution of the primary beam, instead of perfectly monochromatic radiation (Larger for laboratory X-ray tubes, much smaller for synchrotron sources) Finite slits width Imperfect scattering geometry (not perfectly parallel beam, monochromator) Natural Element Line Width (ev) Co Ka Ka Cu Ka Ka2 3.0 Mo Ka M.O. Krause and J.H. Oliver J. Phys. Chem. Ref. Data (1979) 31

32 DECONVOLUTION GOALS Full profile reconstruction Fourier (Stokes) method F(I S,Meas ) = F(I Real ).F(I Inst ) ; I Real = F -1 [F(I S,Meas )/F(I Inst )] Very sensitive to noise!!! Determination of the structural Full Widths at Half maximum (FWHM) - Profile Fitting Fourier methods

33 Profile Fitting Most precise way to determine peak positions, peak intensities and FWHMs 1/ Selection of profile functions Gauss (G) thermal neutrons Lorentz (L) intrinsic peak shape for many physical processes pseudo-voigt pv = hl + (1-h)G X-rays Voigt V = G*L X-rays Pearson VII PVII = 1/[1 + w 2 (x-x 0 ) 2 /m] m X-rays 2/Selection of starting parameters (graphical software) 3/ Iterative Least-squares refinement Minimization of the sum S[I O (2Q i ) I C (2Q i )] 2 33

34 Comparison of Gauss, Lorents and pseudo-voigt Functions 1.0 Gaus Lorentz h =.5 Intensity The mixing parameter is very sensitive to the noise and the BKG <x> = x 34

35 Pearson VII Functions m = 1 Lorentz m Gauss 35

36 350 Example of Profile Fitting 300 Ag(111) Intensity (counts) Ag 3 Sn(020) Ag 3 Sn(211) Points exp data Red line fitted profile Blue line residual Pink Lines Individual peaks Ag 4 Sn Q (degrees) Pseudo-Voigt Functions + Constant Background Each Line is the sum of K a1 + K a2 36

37 Whole Pattern Fitting Y 2 O 3 powder Cu K a1 radiation Pearson VII functions Different m and FWHM for the different peaks Langford & Louer (1996) 37

38 Fitting Programs FIT (Petkov) WinFIT (Krumm) MAUD MWP-Fit (Ungar et al) WinPlotR (ESRF) X Pert (Philips) Topas (Bruker) MAUD (Lutteroti) 38

39 Profile fitting with WinPlot (ESRF) # Select left/right background points # Select, for each peak, the left/right points of the FWHM and the Peak intensity # Select fitting parameters: Peak positions (3); peak intensities (3); 1 FWHM for all peaks and 1 shape parameter h for all peaks 39

40 Profile fitting Fitting of individual FWHMs for each reflection 40

41 Profile fitting 41

42 Fitting results Position Sigma Intensity Sigma FWHM Sigma Et Sigma Peak positions most accurate; Mixing parameters most sensitive to errors 42