X-ray analysis methods Mauro Sardela, Ph.D. FS-MRL, UIUC Platinum - PowerPoint PPT Presentation

X-ray analysis methods Mauro Sardela, Ph.D. FS-MRL, UIUC Platinum Sponsors Sound test

X-ray interactions with matter Coherent scattering Incoherent scattering E 1 <E 0 E 0 (3) (Diffraction, Thompson or (3) (Compton scattering) Rayleigh scattering) E 0 E 0 (1) (2) (1) (2) Electron Electron Photon Photon Nucleus Nucleus (1) Incoming photon (2) Oscillating electron (1) Incoming photon (3) Scattered photon (2) Energy is partially transferred to electron No loss of energy. (3) Scattered photon (energy loss). Auger electron E 1 =E L -E K Fluorescence (3) (4) E 0 (1) (3) (4) E 0 (1) (5) (2) (6) Photon Nucleus Nucleus (2) K K L Electron shell Electron (1) Incoming photon L (1-4) Hole created (3) after photoelectron emission (2) (2) Expelled electron (photoelectron) shell is occupied by outer electron (4). (3) Hole is created in the shell (5) Excitation energy is transferred to electron (4) Outer shell electron moves to the inner shell hole (6) Electron ejected from atom (Auger electron) (5) Energy excess emitted as characteristic photon.

Fundaments of diffraction F “Real” space Reciprocal space Point Set of planes (h k l) d h k l 2 p /d origin M. von Laue 1879-1960 X-rays from crystals, 1912. 3

Fundaments of diffraction F “Real” space Reciprocal space d 2 p /d origin 4

Fundaments of diffraction F “Real” space Reciprocal space origin 5

Bragg’s law and Ewald’s sphere F “Real” space Reciprocal space Bragg’s law Ewald’s sphere w 2q k 1 q 2q k 0 (= radius) 2 d sin q = n l q = k 1 – k 0 Elastic (Thompson’s) scattering q: scattering vector q = (4 p / l) sin q Paul P. Ewald 6 1888-1985 1862 – 1942 1890-1971

Single crystal Point detector scan X-ray Diffraction plane Poly crystal (random) Point detector scan X-ray Diffraction plane Poly crystal (texture) X-ray Diffraction plane

Coupled 2theta-omega scans q k 1 k 0 w 2q X-ray source d 2q Detector w Sample k 1 2 q-w scan: q 2q Probes d -spacing variation Along q k 0 (= radius) Phases id, composition, lattice constants Grain sizes, texture, strain/stress

Rocking curve omega scans q k 1 k 0 w 2q Detector X-ray source d w Sample k 1 w scan: q 2q Probes in-plane variations Normal to q k 0 (= radius) Mosaicity, texture and texture strength

Information contents in the XRD pattern Detector X-ray source 2q w Sample Peak position: identification, structure, lattice parameter Peak width: crystallite size, strain, defects Peak area or height ratio: preferred orientation Peak tails: Diffuse scattering, point defects Background: amorphous contents

Powder diffraction methods Crystalline? Amorphous? What elements, compounds, phases are present? Structure? Lattice constants? Strain? Grain sizes? Grain orientations? Is there a mixture? What % ? Powders, bulk materials, thin films, nanoparticles, soft materials. 11

Bragg-Brentano focusing configuration Single crystal monochromator Receiving ( l ) Focus slit X-ray source Detector Divergence slit Secondary Optics: Scatter and soller slits Focusing circle Detector (variable during specimen rotation measurement) ( 2 q ) Divergent beam Angle of Sample height not good for incidence positioning is critical ( w ) grazing incidence Diffractometer circle 12 analysis (fixed during measurement)

Added Ground to fine amorphous powder + phases (glass) (random grain to complicate orientations) things… 

XRD powder analysis walkthrough Crystalline phases Intensity (sqrt counts) Amorphous (zero background holder) 20 30 40 50 60 70 80 90 100 110 2-theta (degrees), Cu K-alpha radiation

XRD powder pattern ~ 20 w% amorphous added Intensity (sqrt counts) No amorphous added 20 30 40 50 60 70 80 90 100 110 2-theta (degrees), Cu K-alpha radiation

Peak fit and shape analysis Peak fit: Data + Peak Instrument + shape resolution FWHM = f (2 q ) function Intensity (sqrt counts) Σ Peak areas Crystallinity = = 81.7 % Total area Amorphous = 1 – (crystallinity) = 18.3 % contents 20 30 40 50 60 70 80 90 100 110 2-theta (degrees), Cu K-alpha radiation

Search / match Peak position Match! Fingerprinting ICDD PDF4 database Search against… + intensity ratio identification of phases ICSD, etc. Hits Formula FOM PDF RIR Space group Calcite CaCO 3 1.1 04-012-0489 3.45 R-3c(167) Dolomite Ca 1.07 Mg 0.93 (CO 3 ) 2 15.0 04-011-9830 2.51 R-3(148)

Search / match Peak position Match! Fingerprinting ICDD PDF4 database Search against… + intensity ratio identification of phases ICSD, etc. Hits Formula FOM PDF RIR Space group √ Calcite CaCO 3 1.1 04-012-0489 3.45 R-3c(167) Dolomite Ca 1.07 Mg 0.93 (CO 3 ) 2 15.0 04-011-9830 2.51 R-3(148)

Search / match Focus on Second unmatched peaks round Hits Formula FOM PDF RIR Space group √ Calcite CaCO 3 1.1 04-012-0489 3.45 R-3c(167) Dolomite Ca 1.07 Mg 0.93 (CO 3 ) 2 15.0 04-011-9830 2.51 R-3(148)

Search / match Identify additional Focus on Second Search / Match phases (~ > 1 w%) unmatched peaks round Hits Formula FOM PDF RIR Space group √ Calcite CaCO 3 1.1 04-012-0489 3.45 R-3c(167) √ Dolomite Ca 1.07 Mg 0.93 (CO 3 ) 2 15.0 04-011-9830 2.51 R-3(148)

Quant: RIR reference intensity ratio Ratio of ( ( ) ) Ratio of peak areas corrected crystalline ~ RIR ~ I / I corundum by RIR of each phase phases 1 Ratio of crystalline phases: 2 Calcite: 79.2 w% Intensity (sqrt counts) Dolomite: 20.8 w% (no amorphous included) 1 1 1 1 1 1 1 2 2 1 2 2 45 50 40 30 35 2-theta (degrees), Cu K-alpha radiation

Rietveld refinement Non-linear least square minimization H. Rietveld (1932-) For each data point i : Minimize this function: Sum over n data points n data points m Bragg reflections for each data i p phases w i , b i , K l , Y l,j weight, background, scale factor and peak shape function Refinement parameters: Background Data + Sample displacement, transparency and zero-shift correction preliminary structure: Peak shape function Unit cell dimensions Minimize and converge Preferred orientation figures of merit/quality: Scale factors R Atom positions in the structure Atomic displacement parameters

Rietveld refinement Peak fit: Data + Peak Instrument + shape resolution Calcite: 80.7 w% FWHM = f (2 q ) function Dolomite: 22.2 w% Amorphous: 17.1 w% Intensity (sqrt counts) Σ Peak areas Crystallite size: Crystallinity = = 81.7 % Calcite: 56.8 nm Total area Dolomite: 35.6 nm Amorphous = 1 – (crystallinity) = 18.3 % contents 20 30 40 50 60 70 80 90 100 110 2-theta (degrees), Cu K-alpha radiation

Rietveld refinement Peak fit: Data + Peak Instrument + shape resolution FWHM = f (2 q ) function Intensity (sqrt counts) Σ Peak areas Crystallinity = = 81.7 % Total area _ Amorphous = 1 – (crystallinity) = 18.3 % Calcite, CaCO 3 , hexagonal, R3c (167) contents 0.499 nm/ 0.499 nm / 1.705 nm <90.0/90.0/120.0> 20 30 40 50 60 70 80 90 100 110 2-theta (degrees), Cu K-alpha radiation

Rietveld refinement Peak fit: Data + Peak Instrument + shape resolution FWHM = f (2 q ) function Intensity (sqrt counts) Σ Peak areas Crystallinity = = 81.7 % Total area Amorphous _ = 1 – (crystallinity) = 18.3 % contents Dolomite, Ca 1.07 Mg 0.93 (CO 3 ) 2 , hexagonal, R3 (148) 0.481 nm/ 0.4819 nm / 1.602 nm <90.0/90.0/120.0> 20 30 40 50 60 70 80 90 100 110 2-theta (degrees), Cu K-alpha radiation

Crystallite size analysis Scherrer’s equation: Simplistic approximation! k * l Not accounting for peak Size = broadening from strain and defects cos ( q ) * ( FWHM ) k : shape factor (0.8-1.2) Directional measurement! l : x-ray wavelength Measured along the FWHM : full width at specific direction normal to half maximum (in radians) the (hkl) lattice plane given by the 2 q peak position Peak width (FWHM or integral breadth) ___ Measurement ___ Fit Peak position 2 q 27

Crystallite size analysis Intensity(Counts) Intensity(Counts) 20 nm 6.5 o (111) grains 0.5 o in Cu foil 2 nm Fe 3 O 4 nanoparticle 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 Two-Theta (deg) 41.5 42.0 42.5 43.0 43.5 44.0 44.5 45.0 45.5 Two-Theta (deg) 145 nm Intensity(Counts) Si powder 0.17 o 113.5 114.0 114.5 115.0 115.5 Two-Theta (deg) 28

Peak shape analysis Peak fit functions: Gaussian Lorentzian Pearson-VII (sharp peaks) Pseudo-Voigt (round peaks) Measurement Information from fit: Fit • Position • Width (FWHM) • Area • Deconvolution • Skewness ___ Measurement ___ Fit 29

Correction for instrument resolution Use FWHM curve as a function of 2 q from standard sample (NIST LaB 6 ): FWHM: b specific for each diffractometer b D = ( b meas ) D – ( b instr ) D D : deconvolution parameter Measurement D : 1 (~ Lorentzian) b = ( b meas ) – ( b instr ) Instrument function D : 2 (~ Gaussian) b 2 = ( b meas ) 2 – ( b instr ) 2 D : 1.5 b 1.5 = ( b meas ) 1.5 – ( b instr ) 1.5 30