Two-dimensional Powder Diffraction. Bob He, Bruker AXS

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1 Two-dimensional Powder Diffraction Bob He, Bruker AXS

2 XRD : Comparison with Conventional XRD (1) The powder diffraction pattern in 3D space (blue) and the conventional diffractometer plane.

$Intensity Conventional X-ray Diffractometer Divergence slit Antiscatter slit Monochromator Bragg-Brentano Geometry.$

3 Intensity Conventional X-ray Diffractometer Divergence slit Antiscatter slit Monochromator Bragg-Brentano Geometry. Scanning over range to collect XRD pattern. Tube Detectorslit Corundum Powder Diffraction Sample Two Theta

4 Parallel beam Geometry Tube Göbel mirror Soller Slit Detector Parallel beam Geometry. Scanning over range to collect XRD pattern. Not sensitive to the sample height error and rough surface Sample

5 Bragg-Brentano Geometry Equatorial Axial

6 Soller slits are used to control axial divergence large flat sample Most part of the diffraction ring is clipped off by: 6 In Bragg-Brentano geometry, the line focus beam can be considered as a superposition of point beams. All in parallel with the diffractometer plane and the same geometry condition separated by soller slit foils. Bruker Confidential

7 XRD : Two-dimensional X-ray Diffraction

8 XRD : Choice of Detectors Sensitivity vs. Count Rate MWPC MiKroGap CCD Image Plate Detective Quantum Efficiency (DQE): The DQE is a parameter defined as the square of the ratio of the output and input signal-to-nose ratios (SNR). ( S / N) DQE ( S / N) out The DQE of a real detector is less than 100% because not every incident x-ray photon is detected, and because there is always some detector noise. MiKroGap has the best overall performance. in

9 XRD : Point Spread Function and Resolution Consider a very small diffraction spot (blue linedelta function) An adjacent spot red line RMS (root-mean-square) is another parameter for PSF: A perfect detector - dashed blue line. A real detector - intensity in a spread distribution. Can be measured if the separation is larger than FWHM. FWHM.3548 RMS Bruker Confidential 9

10 VÅNTEC-500 Outperform all previous gaseous detectors. High sensitivity: 80% DQE for Cu (detection quantum efficiency) High spatial resolution: The FWHM of the PSF is 00m High maximum count rate: Global count rate: 1.5Mcps Local count rate: 50kcps/reflection Low background noise: <5 cps/global Maintenance-free: no re-gassing

11 VÅNTEC-500 Tapered front for high θ range max arctan m h D l D

12 XRD : Diffraction vector approach Applications Vector approaches Phase ID: Polarization and absorption correction Texture Analysis: Orientation mapping angles; Data collection strategy (scheme) Stress Analysis: Fundamental equation derived by second order tensor transformation; Data collection strategy (scheme) Crystal Size: Equations for the effective volume calculation at both reflection and transmission modes Bruker Confidential 1

$XRD : Diffraction pattern with both g and information Debye Cone Incident Beam Sample Diffraction vector with$

13 XRD : Diffraction pattern with both g and information Debye Cone Incident Beam Sample Diffraction vector with g: cos 1 s s 1 H 0 sin sing sin cosg Expressed in sample space: h1 a11 a1 a13hx h a 1a a3 h y h3 a31a3 a33 hz

14 XRD : Geometry Convention - Diffraction Space Diffraction rings (blue) in the laboratory axes (red). g g cos cos sin cos sin L z y x h h h h

$XRD : Diffraction Vector & Unit Diffraction Vector The diffraction vector is$ can be represented by its unit vector given by: g g cos sin sin sin 1 cos 1 1

can be represented by its unit vector given by: g g cos sin sin sin 1 cos 1 1

15 XRD : Diffraction Vector & Unit Diffraction Vector The diffraction vector is given in laboratory coordinates by The direction of each diffraction vector can be represented by its unit vector given by: g g cos sin sin sin 1 cos z z y y x x s s s s s s s s H g g cos cos sin cos sin L z y x h h h H H h

16 XRD : Geometry Convention - Detector Space Detector position in the laboratory coordinates is determined by the detector distance D and swing angle.

$XRD : From Pixel to θ and g in Diffraction Space Conversion of pixel intensity into and g intensity based on the detector$

17 XRD : From Pixel to θ and g in Diffraction Space Conversion of pixel intensity into and g intensity based on the detector position in the laboratory coordinates D and : cos 1 x sin D cos, D x y (0 ) g x cos x cos Dsin cos Dsin 1 y y ( x cos Dsin ), ( g )

18 XRD : Phase ID Measurement Geometry

19 XRD : Single Frame Covering All coverage: 70 at 8 cm detector distance Sample with strong texture and large grain

Fast Phase Identification for time resolved Studies ZrO 1 cm sample-detector distance time x50s 11 10 9 8 7 Lin (Counts) 6 5 4 3 1 0 1 30 40 50 60 70 80 -Theta - Scale New Frame - File: dup1b.

20 Fast Phase Identification for time resolved Studies ZrO 1 cm sample-detector distance time x50s Lin (Counts) Theta - Scale New Frame - File: dup1b.raw - Type: Th alone - S tart: E nd: S tep: S tep time: 50.0 s - Temp.: 5.0 C (Room) - Time S tarted: 0 s - -Theta: Theta: Chi: Operations: Import x - File: DUP 1a.raw - Type: Th alone - S tart: E nd: S tep: S tep time: 50.0 s - Temp.: 5.0 C (Room) - Time S tarted: 0 s - -Theta: Theta: Chi: Operations: Import (* ) - B addeleyite, syn - ZrO - Y : % - d x by: W L:

21 Corundum data 0.03 to 1 sec High Speed Analysis Rel (cps) Theta - Scale 0.03 Second Data Collection Y mm Second Data Collection Y mm Second Data Collection Y mm - 1 Second Data Collection

22 XRD : Frame Merge and Integration 4 frames at 0 cm Merged frames coverage: 100 Integrated profile for phase ID search/mat ch

23 XRD : reflection vs. transmission Reflection mode frame from corundum at 5 incident angle. Transmission mode frame with perpendicular incident beam Bruker Confidential 3

24 XRD : Defocusing at low incident angle in reflection Lower resolution when or (-) 90 B b sin sin 1 sin( ) sin Bruker Confidential

25 XRD : Defocusing effect in reflection mode depends on detector and data collection strategy Cylinder detector with 5 incident angle for 5~80 Flat detector with several (5, 15, 5, 35) incident angles for 5~80 5 Bruker Confidential

26 Defocusing Factor XRD : Defocusing effect with reflection sample depends on detector and data collection strategy Defocusing vs. Detectors Two Theta Cylinder detector may collect large range, but with large defocusing effect at high angle Flat Cylinder BB Defocus effect can be minimized with data collection strategy 6 Bruker Confidential

27 Intensity XRD : Relative Intensity of Powder Pattern The integrated intensity diffracted from random polycrystalline materials is given by: I hkl where: k I p hkl v phkl 3 ki ( LPA) Fhkl exp Mt M v - instrument constant; - the multiplicity of the planes; - the volume of the unit cell; (LPA) - the Lorentz-polarization and absorption factors; F hkl - the structure factor of the crystal plane (hkl) and exp(-m t -M s ) - the attenuation factor due to lattice thermal vibrations and weak static displacements. Denotes the factors which are different between Bragg-Brentano geometry and XRD geometry. k I is determined by the source, optics and detector (LPA) will be given in this presentation. s Corundum Powder Diffraction Bruker Confidential Two Theta

28 XRD : Polarization Correction The polarization factor for Bragg-Brentano geometry with incident beam monochromator is: 1 cos M cos P I 1 cos M where θ M is the Bragg angle of the monochromator. The general polarization factor for the diffracted beam to point P is: P G (cos Geometric relationship between the monochromator and detector in laboratory coordinates. cos sin )cos M cos 1 cos M sin cos

29 XRD : Polarization Correction The unit vector of the diffraction vector H P and its projection on Y L - Z L plane, H' P, in the laboratory system are given respectively as: The unit vector of Y L is y L =[0,1,0], then: Therefore, and The polarization factor for XRD can then be given as a function of both and g: g g cos cos sin cos sin L z y x h h h h g g cos sin 0 0 L z y h h h sing ), cos( cos L L y y L h L h g sin cos g sin cos M M M P g g g cos 1 )cos cos (cos )sin cos cos (1 ), (

30 XRD : Sample Absorption Correction The absorption can be measured by the transmission coefficient: A 1 V e V dv where is the total beam path and A is the average over all the element dv. For Bragg-Brentano geometry, we have: A BB =1/(µ) Absorption correction of flat slab: (a) reflection (b) transmission. To make the relative intensity comparable to Bragg-Brentano geometry, we introduce a normalized transmission coefficient T: T A/ ABB A

$XRD : Sample Absorption Correction For reflection mode diffraction with a thick plate: 1 cos s o 0 s sin sing 0 sin cosg and sin cos n$

31 XRD : Sample Absorption Correction For reflection mode diffraction with a thick plate: 1 cos s o 0 s sin sing 0 sin cosg and sin cos n cos cos sin The normalized transmission coefficient: cos cos s o n sin cos T cos cos with cos sn cos sin cos sin sing cos cos sin cos g sin

XRD : Sample Absorption Correction For transmission mode: 1 cos s o 0 s sin 0 sin and sing cosg sin sin sin cos cos n cos sin sin sin cos cos sin The normalized

32 XRD : Sample Absorption Correction For transmission mode: 1 cos s o 0 s sin 0 sin and sing cosg sin sin sin cos cos n cos sin sin sin cos cos sin The normalized transmission coefficient : sec exp t sec exp t sec T sec sec cos s o n sinsin sin cos cos cos s n (sin sin sin cos cos)cos (cos sin sin sin cos)sin sing cos sin sin cosg

XRD : Texture Effect and Correction The integrated intensity with texture is: phkl 3 Ihkl ki ( LPA) Fhkl g hkl(, )exp Mt M v where g() is the normalized pole density function.

33 XRD : Texture Effect and Correction The integrated intensity with texture is: phkl 3 Ihkl ki ( LPA) Fhkl g hkl(, )exp Mt M v where g() is the normalized pole density function. For the BB geometry,,0) ( g hkl The texture effect for XRD : m c Ihkl Ihkl g ( g ) hkl Correct to the B-B equivalent with a texture effect: I BB hkl g hkl(,0) I g ( g ) hkl s For fiber texture: m hkl g hkl ( g ) g g 1 g cos hkl g [ ( g )] gdg 1 g h and 3 1

$Microdiffraction$ Horizontal th-th,

34 XRD : GADDS Microdiffraction Horizontal th-th, XYZ stage SCD, December 008

35 Fatal Bicycle Accident Collection of Evidence traces of car paint found on the bicycle Dr. W. Kugler Landeskriminalamt Baden-Württemberg Stuttgart, Germany

$-dimensional diffraction pattern integration$

36 Fatal Bicycle Accident Mapping of Car Paint with GADDS video image (for documentation) -dimensional diffraction pattern integration of data: diffractogram for phase identification

37 Fatal Bicycle Accident Phase Identification of Car Paint 000 Lin (Counts) Theta - Scale New Frame - File: 708_07.raw - Start: End: Step: (*) - Rutile, syn - TiO (*) - Calcite, syn - CaCO (*) - Dolomite - CaMg(CO3)

Fatal Bicycle Accident The Car s Identification Lin (Counts) 5000 sequence of coatings is characteristic of car type 0 13 0 30 40 50 60 70 -Theta - Scale New Frame - File: 708_06.raw - Start: 1.

38 Fatal Bicycle Accident The Car s Identification Lin (Counts) 5000 sequence of coatings is characteristic of car type Theta - Scale New Frame - File: 708_06.raw - Start: End: Step: New Frame - File: 708_04.raw - Start: End: Step: New Frame - File: 708_05.raw - Start: End: Step: New Frame - File: 708_07.raw - Start: End: Step: 0.010

39 Car Paint Analysis Five Layers of Car Paint

40 Police Officer s Pistol Exccessive Force? Death of a Bank Rober

41 Police Officer s Pistol Exccessive Force? Contact Trace on the Barrel

42 Police Officer s Pistol Exccessive Force? Contact Trace on the Projectile

43 Police Officer s Pistol Exccessive Force? Negative. Contact Trace and Control Specimen

Phase ID Mapping Rockmapping 170 160 150 140 130 10 Lin (Counts) 110 100 90 80 70 60 50 40 30 0 10 0 6 10 0 30 40 50 60 70 -Theta - Scale Rockmapping -

0 mm - Rockmapping Operations: Import Operations: Import Operations: Import Operations: Import X +.0 mm - Y + 5.0 mm - Microverify - File: X + 10.

0 mm - Microverify - Fil X + 1.0 mm - Y + 30.0 mm - Rockmapping X + 0.0 mm - Y + 50.

44 Phase ID Mapping Rockmapping Lin (Counts) Theta - Scale Rockmapping - File: Spot-N_01.raw - Type: X mm - Y mm - Rockmapping - X mm - Y mm - Rockmapping X mm - Y mm - Rockmapping Operations: Import Operations: Import Operations: Import Operations: Import X +.0 mm - Y mm - Microverify - File: X mm - Y mm - Rockmapping X mm - Y mm - Rockmapping Operations: Import Operations: Import Operations: Import X mm - Y mm - Microverify - Fil X mm - Y mm - Rockmapping X mm - Y mm - Rockmapping Operations: Import Operations: Import Operations: Import X mm - Y mm - Rockmapping - X mm - Y mm - Rockmapping X +.0 mm - Y mm - Rockmapping Operations: Import Operations: Import Operations: Import

45 High-throughput Screening (HTS) Vertical theta-theta, Reflection/Transmission

46 XRD : High throughput screening Laser/video sample alignment Easy and accurate sample positioning without touching the sample surface Video image of each material library spot can be automatically stored during data scan

47 PolySNAP for Combined Analysis: Correlation among XRD, Raman and other probes Cell display 3D plots Dendrogram

48 XRD : Particle Size Analysis profile analysis g profile analysis 1 nm 10 nm 100 nm 1 m 10 m 100 m 1 mm profile analysis, including measurement from peak FWHM by Scherrer particle equation, size range in pharmaceutical or profile systems analysis by Stokes and Wilson, is suitable for particle size below 100 nm. g profile analysis is suitable for particle size from sub-micrometer to a few millimeters. The size range of g profile analysis can be further extended by instrumentation and data collection strategy.

49 XRD : Particle size measurement by g profile analysis: (111) (110) (100) gprofile

50 Peak broadening - gold Nanoparticles

51 Particle size calculation: Scherrer equation: t C Bcos where is wavelength (Å), B is FWHM (radians) corrected for instrument broadening, is Bragg angle, C is a crystal shape factor from 0.9~1.

$XRD : Data Collection: Acetaminophen powder The spotty diffraction ring is due to the large crystallites$ $multiplicity of the crystal plane, and effective diffraction volume.$

52 XRD : Data Collection: Acetaminophen powder The spotty diffraction ring is due to the large crystallites compared to the sampling volume (beam size). The number of spots on the ring is determined by crystallite size, instrumental window (g-range), multiplicity of the crystal plane, and effective diffraction volume. The size of jelly beans and candy bin determines how many you can fill Bruker Confidential 5

53 XRD : Particle size measurement by g profile analysis: ss - o s o s 1 g For XRD, the instrumental window is given by 1 arcsin[cos sin( g / )] Bruker Confidential 53

54 XRD : Particle size measurement by g profile analysis: For XRD in reflection mode, the crystallite size is given by d k p hkl b arcsin[cos sin( g / )] N s where is the linear absorption coefficient 1 3 For transmission mode with the incident beam perpendicular to the sample surface, the crystallite size is given by d k p hkli where t is the sample thickness. b t arcsin[cos sin( g / )] Ns 1 3 k is the instrument calibration factor or can be calculated from: 3 k if the instrument broadening in direction is known.

More About XRD 1. Introduction.. Geometry Conventions. 3. X-Ray Source and Optics. 4. X-Ray Detectors. 5. Goniometer and Sample Stages. 6. Data Treatment. 7.

55 More About XRD 1. Introduction.. Geometry Conventions. 3. X-Ray Source and Optics. 4. X-Ray Detectors. 5. Goniometer and Sample Stages. 6. Data Treatment. 7. Phase Identification. 8. Texture Analysis. 9. Stress Measurement. 10. Small-Angle X-Ray Scattering. 11. Combinatorial Screening. 1. Quantitative Analysis. 13. Innovation and Future Development.

56 Visualization of 3D Reciprocal Space with MAX3D Jim Britten, Weiguang Guan, Victoria Jarvis McMaster University Hamilton, Ontario, Canada Jim Britten, Bruker-NYU XRD Workshop June

57 Nanowire Film Orientation Analysis Jim Britten, Weiguang Guan, Victoria Jarvis McMaster University Hamilton, Ontario, Canada Jim Britten, Bruker-NYU XRD Workshop June

$XRD 3 and Ewald s Sphere Cones of diffraction in real space Concentric spheres of$

58 XRD 3 and Ewald s Sphere Cones of diffraction in real space Concentric spheres of Intensity at radii 1/d in Reciprocal Space Jim Britten, Bruker-NYU XRD Workshop June

$Diffraction What can we$ see in between?

59 The Extremes of 3D Diffraction What can we see in between? Jim Britten, Bruker-NYU XRD Workshop June

60 Residual Stress Looking for subtle changes in θ position of line/arc/shell to indicate orientation dependant residual stresses. Hard to see visually need mathematica l analysis. High angle snapshots of diffraction shell segments in two series of φ steps at two different ω (incident) angles. Looking for elliptical deviation from spheres where r = 1/d 0 Jim Britten, Bruker-NYU XRD Workshop June

$1D Ordering Fibre Diffraction Extruded,$ $Elnagmi / Jain C -axis in diffraction$

61 1D Ordering Fibre Diffraction Extruded, distorted polypropylene. Elnagmi / Jain C -axis in diffraction pattern Jim Britten, Bruker-NYU XRD Workshop June

62 Example 1 Random Orientation GaAs NW on Carbon nanotube fabric Why bother with XRD 3? Sometimes there are surprises! Jim Britten, Bruker-NYU XRD Workshop June 011 6

63 Example Multiple (8) Orientation GaAs NW s on Si Substrate D scan sequence Jim Britten, Bruker-NYU XRD Workshop June 011 Full scan in MAX3D 63

Basics and Recent Advances in Two-dimensional X-ray Diffraction. Bob He, Bruker AXS April 15, 2014 Harvard University

$Basics and Recent Advances in Two-dimensional X-ray Diffraction. Bob He, Bruker AXS April 15, 2014 Harvard University$ Basics and Recent Advances in Two-dimensional X-ray Diffraction Bob He, Bruker AXS April 15, 2014 Harvard University Basic Concept from XRD to XRD 2 12/4/2017 2 Intensity Conventional X-ray Diffractometer