Stress and Texture by XRD Bob He, Bruker AXS

Transcription

1 Stress and Texture by XRD Bob He, Bruker AXS

2 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

3 Diffraction Patterns vs. Atomic Arrangement

4 XRD : Two-dimensional X-ray Diffraction

5 XRD : Innovation and Development The most dramatic development in XRD happens in detector and data evaluation algorithms: Detector: collect D pattern with correct intensity, position and angular coverage. Data evaluation algorithm: data collection strategy and data evaluation. (Diffraction Vector Approach)

6 VÅNTEC-500 Outperform all previous gaseous detectors. High sensitivity: 80% DQE for Cu High spatial resolution: FWHM of PSF is 00m High maximum count rate: 1.5Mcps global; 50kcps/reflection -local Low background noise: <5 cps/global Maintenance-free: no re-gassing

7 VÅNTEC-500 Outperforms all previous gaseous detectors Detector geometry: Be-window opening 140 mm in dia. Frame size: 048 x 048 pixels 104 x 104 pixels 51 x 51 pixels Pixel size: 68 m x 68 µm 136 m x 136 µm 7 m x 7 µm Detector working distance: 5~30 cm in D8 DISCOVER enclosure θ range in a single frame: 5 cm cm cm 4 0 cm 33 5 cm 7 30 cm 3

8 Stress/Texture/Microdiffraction Horizontal th-th, CEC

9 No barrier between 0D/1D/D Vertical theta-theta, CEC for microdiffraction/stress/texture

10 XRD : Diffraction vector approach Applications Vector approaches Phase Identification: 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 Analysis: Equations for the effective volume calculation at both reflection and transmission modes Bruker Confidential 10

11 Single Crystals S S 0 Laue equation a b c ( s ( s ( s s s s ) ) ) h k l

12 XRD : Debye cones from powders Bragg law n d sin

13 XRD : Diffraction pattern with both g and information Debye Cone Incident Beam Sampl e Diffraction vector with g H s s 0 cos 1 1 sin sing sin cosg Bruker Confidential 13

14 XRD : Sample Space & Eulerian Geometry The angular relationships between X L Y L Z L and S 1 S S 3 are: X Y Z L L L S a a a S a a a 1 3 S a a a The transformation matrix from the diffraction space a11 a1 a13 to the sample a space is: 1a a3 a 31 a 3 a 33 sin sin sin cos cos sin sin cos cos sin sin cos cos sin sin sin cos cos sin cos sin sin cos cos cos sin cos cos sin

15 XRD : Sample Space & Unit Diffraction Vector The components of the unit vector h S in the sample coordinates S 1 S S 3 is then given by h1 a11 a1 a13hx h a 1a a3 h y h3 a31a3 a33 hz In which each component is a function of sample orientation and g. Or in expanded form: sin (sin sin sin h h 1 cos cos) cos cosg sin cos cos sing (sin sin cos cos sin) sin (cos sin sin sin cos) cos cosg cos cos cos sing (cos sin cos sin sin) h 3 sin cos sin cos sing cos cos cos cosg sin

16 XRD: Sample Space & Unit Diffraction Vector The components of the unit vector h S in the sample coordinates S 1 S S 3 is then given by h h h h 1 3 cos sin sin sin cos h which is a single value at each sample orientation.

17 XRD & XRD : Fundamental Equations for Strain As a second order tensor, the relationship between the measured strains and the strain tensor is given by: 0D/1D: D: ij h h i j Both equations are in the same form except the difference in h s 11 cos sin 1 sin sin sin sin cos sin sin sin cos Introducing g=90 or 70 (diffractometer plane), the above equation can be derived from the bottom equation ( g,,, ) { hkl} ( g,,, ) h1 11 h1h 1 h h1h ij 3 h h h h h i 3 33 j

18 The sin - Method for Stress Measurements

19 The sin - Method for Stress Measurements

20 The sin - Method for Stress Measurements

21 Basic Types of - sin Functions

22 XRD : Fundamental Equation for Stress Measurement Introducing the elasticity constants: E & or the macroscopic elastic constants: & 1 S 1 the fundamental equation for stress measurement in XRD : S ( ) S ( h h 1 ( ) / S / 1 E 11 E h h h h 3 h 33 h 3 3 h 3 ) sin0 ln sin

23 XRD : Diffraction Rings Distortion Due to Stress The simulated diffraction ring distortion in radar chart: (a) equibiaxial stress with scan; (b) uniaxial stress with ω scan for Fe (11) with Cr radiation.

24 Comparison of the sin and D method for stress measurement Sin method measures one data point at each sample orientation. Data points follow the longitudinal line (constant ). D method measures many data points at each sample orientation and not limited to constant.

25 Comparison of the sin and D method for stress measurement With increasing data points, the new D method can measure stress with higher accuracy than the conventional sin method for the same amount of data collection 5 Bruker Confidential 15/1/011

26 The D8 DISCOVER with DAVINCI VÅNTEC-500 for stress measurement Almen strip (spring steel) (11) & (00) rings Ring distortion when sample rotates with and

27 XRD : Stress Measurement Example A standard reference sample (SN# B04) from Prof. Chuanhai Jiang distributed at the 15th CCRS, Chongqing, China, October 11-18, 009. The reference value of MPa measured with Cr radiation by the Sin method. Co tube, 40kV/30mA, a slightly focusing polycapillary lens and 1 mm pinhole collimator produces a beam size 0.3 mm on the sample. Total data collection time: 5 minutes for 3 frames (5s each+rotation). 1 or minutes are sufficient too.

28 XRD : Stress Measurement Example cracks?

29 XRD : Stress Measurement Example Parameter setting Evaluate the data quality Results reporting: 11 = MPa = MPa Principal stress and stress ellipse equibiaxial

30 Stress Measurement Using GADDS: A spring loaded on the XYZ stage of GADDS Micro-diffractometer A spring mounted on the XYZ stage. Size: coil diameter- 10mm wire diameter-1mm coil pitch-4mm The inside surface is aligned to the goniometer center. Both the incident and diffracted beams can pass through the gap Residual Stress: -864 (±48) MPa

31 D8 DISCOVER with GADDS: FSW Specimen Loaded on ¼-Circle Cradle Specimen is loaded on the XYZ stage of the Eulerian cradle Each mapping spot is aligned to the instrument center with the laser/video alignment system. Stress data is collected with and scans. Cr-K radiation Beam size: 0.8 mm Aluminium (311) planes Five frames, each 30 seconds, for each point.

32 Residual Stress (MPa) D8 DISCOVER with GADDS: Residual Stress Mapping on FSW: 150 Longitudinal Normal Stress Distance from Weld Center Line (mm) Residual stress ( ) distribution on top surface

33 XRD :Diffraction Pattern Across FSW Original material Boundary FSW weldment

34 Standard error (MPa) Stress (MPa) XRD : Innovations to D method: Stress Measurement from Multiple (hkl) Rings Regression Cu Film Stress vs. Loading (40) (331) 650 (331)+(40) Stress (331) only stress 48 Improve the accuracy and reduce the effect of anisotropy and texture (40) only stress Linear (331)+(40) Std.err (331) only std.err (40) only std.err Loading Strain (X 0.001)

35 XRD : Innovations to D method: Intensity Weighted Least Squares Regression low intensity and poor profile due to texture and large grain tends to give larger θ shift error. Ref: B. He, Two-dimensional X-ray Diffraction, John Wiley & Sons, 009, pp S w i n i1 I w r i i i i n i1 w ( y i i yˆ maximum (or integrated) intensity the standard error of profile fitting of the i th data point i )

36 XRD : Innovations to D method: Pixel Direct Diffraction (PDD) Analysis PDD analysis is based on diffraction vectors associated with each pixel in diffraction frames collected with area detectors. Conventionally, a D frame is treated as a continuous distribution of diffraction intensity, and D diffraction patterns are integrated to diffraction profiles as intensity distributions versus θ or g angles. In the DPP method, the diffraction meaning of each pixel is analyzed as an independent event and the final diffraction results are generated by a collective consideration of all pixel diffractions. (Each pixel is a point detector.) The final crystallographic results are based on the statistical analysis of all pixels. PDD results in a more accurate diffraction analysis with less human interference.. For stress analysis, the larger θ shift error due to texture and large grain can be eliminated. No profile analysis is required. Ref: B. He, Two-dimensional X-ray Diffraction, John Wiley & Sons, 009, pp S n i1 I i r i n i1 I ( y I i is the pixel intensity used as weighting factors, n is the number of total pixels in the selected region for all D frames i i yˆ i )

37 Texture: Crystal Orientation and Pole Figure The direction of poles is defined by and b angles in a spherical coordinates. The pole densities at all directions are mapped on the equatorial plane by Stereographic Projection. The D map is called pole figure

38 Texture: XRD vs. XRD(0D/1D) d XRD XRD

39 XRD : Sample Space & Unit Diffraction Vector The components of the unit vector h S in the sample coordinates S 1 S S 3 is then given by h1 a11 a1 a13hx h a 1a a3 h y h3 a31a3 a33 hz In which each component is a function of sample orientation and g. Or in expanded form: sin (sin sin sin h h 1 cos cos) cos cosg sin cos cos sing (sin sin cos cos sin) sin (cos sin sin sin cos) cos cosg cos cos cos sing (cos sin cos sin sin) h 3 sin cos sin cos sing cos cos cos cosg sin

40 XRD : Fundamental Equation for Texture Analysis The pole figure angles (,b) can be calculated from the unit vector components by the pole mapping equations: sin 1 b cos 1 h 3 h 1 h cos 1 h 1 h 1 b b h 0 0 if if h h 0 0

41 The D8 DISCOVER with DAVINCI VÅNTEC-500 for texture measurement Steel can (00) & (110) rings Intensity variation during scan

42 XRD : Data Collection Strategy (Scheme D=7cm) =40, =0, =35.6 =40, =3, =30 The pole figure coverage can be simulated from the diffraction θ angle, detector swing angle, detector distance, goniometer angles, and scanning steps.

43 XRD : Data Collection Strategy (Scheme) Data collection strategy for Cu thin films with two scans.

44 XRD : Data Collection Speed vs. D or 0D 5 D: 108 exposures (frames); 0D: 973 exposures (points) multiple pole-figures; single pole-figures D detector is 1~ orders of magnitude faster than 0D detector.

45 Texture Measurement Using VANTEC-500: Magnetron sputter-deposited Cu films Cu (311) (0) (00) (111) Si (311) 3m Cu film on Si substrate. Four diffraction rings are observed at 10 cm detector distance. Four pole figures can be measured simultaneously. Diffraction spots from Si wafer appear on some frames.

46 Arts from XRD : Pole-figure from Cu film and Si

47 Frames Collected from One Oriented Nylon Sheet for Two Orientations Phi-Axis Machine Direction Sample provided by Prof. Brisson of University of Laval

48 Debye Rings from textured aluminum sheet D Data collected in 1.5 s per frame coverage from =-40 to =+40 highly textured Aluminum good statistics from fast transmission measurement

49 Percent Crystallinity with preferred orientation: Stretching of a Rubber Band unstretched bright spots caused by large grains of filler stretched 00% elongation, 10 hours

50 Data Collection Strategy with Multex Area

51 Transparent Texture Evaluation

52 Texture: g-tial alloys (1): Comparison of D Patterns Full, spotty Debye Ring visible Large Grains Weak Texture Partial, smooth Debye Ring visible Fine Grains Sharp Texture Note the strong Differences in the dimensional Diffraction Pattern

53 Texture: g-tial alloys (): Comparison of Pole Figures (110) (111) Fine Grains Sharp Texture Large Grains Weak Texture

54 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.

55 15. December Copyright Dezember 8, Bruker Corporation. All rights reserved 55