two-dimensional powder diffraction - s. n. bose national centre...

Two-dimensional Powder Diffraction

Bob He, Bruker AXS

XRD2: Comparison with Conventional XRD (1)

The powder diffraction pattern in 3D space (blue) and the conventional diffractometer plane.

Conventional X-ray Diffractometer

Divergence slit

Detector-slit

Tube

Antiscatterslit

Sample

Mono-chromator

Bragg-Brentano Geometry.

Scanning over 2 range to collect XRD pattern.

Corundum Powder Diffraction

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

20 25 30 35 40 45 50

Two Theta

Inte

nsit

y

Göbel mirror

Tube

Soller Slit

Detector

Sample

Parallel beam Geometry.

Scanning over 2 range to collect XRD pattern.

Not sensitive to the sample height error and rough surface

Parallel beam Geometry

Bragg-Brentano Geometry

Axial

Equatorial

15.12.2011 Bruker Confidential 6

Soller slits are used to control axial divergence

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.

large flat sample

Most part of the diffraction ring is clipped off by:

XRD2: Two-dimensional X-ray Diffraction

XRD2: Choice of Detectors Sensitivity vs. Count Rate

MiKroGap

MWPC

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).

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.

2

)/(

)/(

in

out

NS

NSDQE

15.12.2011 Bruker Confidential 9

XRD2: Point Spread Function and Resolution

Consider a very small diffraction spot (blue line-delta 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.

RMS2.3548FWHM

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 200m

High maximum count rate: Global count rate: 1.5Mcps Local count rate: 250kcps/reflection

Low background noise: <5 cps/global

Maintenance-free: no re-gassing

VÅNTEC-500 – Tapered front for high 2θ

D

lrange arctan22

D

hm max2

15.12.2011 12 Bruker Confidential

XRD2: 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.

Debye Cone

Sample

Incident Beam

XRD2: Diffraction pattern with both g and 2 information

g

g

cos2sin

sin2sin

12cos10ss

H

Diffraction vector with g: Expressed in sample space:

z

y

x

h

h

h

aaa

aaa

aaa

h

h

h

333231

232221

131211

3

2

1

XRD2: Geometry Convention - Diffraction Space

Diffraction rings (blue) in the laboratory axes (red).

g

g

coscos

sincos

sin

L

z

y

x

h

h

h

h

XRD2: 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

cos2sin

sin2sin

12cos11

0

0

0

0

zz

yy

xx

ss

ss

ssss

H

g

g

coscos

sincos

sin

L

z

y

x

h

h

h

H

Hh

XRD2: Geometry Convention - Detector Space

Detector position in the laboratory coordinates is determined by the detector distance D and swing angle .

XRD2: From Pixel to 2θ and g in Diffraction Space

Conversion of pixel intensity into 2 and g intensity based on the detector position in the laboratory coordinates D and :

)20(,cossin

cos2222

1

yxD

Dx

)(,)sincos(

cossincos

sincos

22

1 g

g

Dxy

y

Dx

Dx

XRD2: Phase ID Measurement Geometry

XRD2: Single Frame Covering All

2 coverage: 70 at 8 cm detector distance

Sample with strong texture and large grain

ZrO2 12 cm sample-detector distance time 2x50s

37-1484 (* ) - B addeleyite, syn - ZrO2 - Y : 160.00 % - d x by: 0.998 - W L: 1.54056

Operations: Import

x - File: DUP12a.raw - Type: 2Th alone - S tart: 46.997 ° - E nd: 80.374 ° - S tep: 0.010 ° - S tep time: 50.0 s - Temp.: 25.0 °C (Room) - Time S tarted: 0 s - 2-Theta: 46.997 ° - Theta: 29.875 ° - Chi: 89.590 ° - P hi: -17.661 ° - X : 0.712 mm - Y : -5.722 mm - Z: 11.300 mm - A ux1: 3.500 - A ux2: 0.000 - Aux3: 0.000 - Display plane: 0 - A node: Cu - W L1: 1.54056 - WL2: 1.54439 - Int. Ratio: 0.50000 - S lit Meas.= n.a. - S lit S imul.= n.a. - X -Offset: 0.000 ° - Displ.: 0.000 mm - Company: BRUK ER A NALY TICA L - Operator: BRUK ER A NALY TICA L - - Creation: 02/19/99 14:01:29

Operations: Import

New Frame - File: dup12b.raw - Type: 2Th alone - S tart: 15.420 ° - End: 47.000 ° - S tep: 0.010 ° - S tep time: 50.0 s - Temp.: 25.0 °C (Room) - Time S tarted: 0 s - 2-Theta: 15.420 ° - Theta: 17.525 ° - Chi: 88.802 ° - P hi: 64.028 ° - X : 0.712 mm - Y : -5.722 mm - Z: 11.300 mm - A ux1: 3.500 - A ux2: 0.000 - A ux3: 0.000 - Display plane: 0 - A node: Cu - W L1: 1.54056 - W L2: 1.54439 - Int. Ratio: 0.50000 - S lit Meas.= n.a. - S lit S imul.= n.a. - X -Offset: 0.000 ° - Displ.: 0.000 mm - Company: BRUK ER A NALY TICA L - Operator: BRUK ER A NALY TICA L - - Creation: 02/19/99 11:20:42

Lin

(Cou

nts)

0

1

2

3

4

5

6

7

8

9

1 0

1 1

2-Theta - Scale

2 1 3 0 4 0 5 0 6 0 7 0 8 0

Fast Phase Identification for time resolved Studies

Corundum data 0.03 to 1 sec

High Speed Analysis

Y + 75.0 mm - 1 Second Data Collection

Y + 50.0 mm - 0.5 Second Data Collection

Y + 25.0 mm - 0.1 Second Data Collection

0.03 Second Data Collection

Rel (c

ps)

1

2

3

4

5

6

7

8

9

10

11

2-Theta - Scale

32 40 50 60

XRD2: Frame Merge and Integration

4 frames at 20 cm

Merged frames 2 coverage: 100

Integrated profile for phase ID search/match

15.12.2011 23 Bruker Confidential

XRD2: reflection vs. transmission

Reflection mode frame from corundum at 5° incident angle.

Transmission mode frame with perpendicular incident beam.

15.12.2011 Bruker

Confidential 24

XRD2 : Defocusing at low incident angle in reflection

Lower resolution when 2 or (2-) 90°

sin

)2sin(

sin

sin

1

2

b

B

15.12.2011 Bruker Confidential 25

Cylinder detector with 5°incident angle for 5~80°2

Flat detector with several (5, 15, 25, 35) incident angles for 5~80°2

XRD2: Defocusing effect in reflection mode

depends on detector and data collection strategy

15.12.2011 Bruker Confidential 26

XRD2: Defocusing effect with reflection sample

depends on detector and data collection strategy

Defocusing vs. Detectors

0

2

4

6

8

10

12

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80Two Theta

De

foc

us

ing

Fa

cto

r

Flat

Cylinder

BB

Defocus effect can be minimized with data collection strategy

Cylinder detector may collect large 2 range, but with large defocusing effect at high 2 angle

15.12.2011 27 Bruker Confidential

XRD2: Relative Intensity of Powder Pattern

The integrated intensity diffracted from random

polycrystalline materials is given by:

where:

kI - instrument constant;

phkl - the multiplicity of the planes; v - the volume of the unit cell;

(LPA) - the Lorentz-polarization and absorption factors;

- the structure factor of the crystal plane (hkl) and

exp(-2Mt-2Ms) - the attenuation factor due to lattice thermal vibrations and weak static displacements.

Denotes the factors which are different between Bragg-Brentano geometry and XRD2 geometry. kI is determined by the source, optics and detector

(LPA) will be given in this presentation.

Corundum Powder Diffraction

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

20 25 30 35 40 45 50

Two Theta

Inte

nsit

y

sthklhkl

Ihkl MMFLPAv

pk 22exp)( 23

2 I

2

hklF

XRD2: Polarization Correction

The polarization factor for

Bragg-Brentano geometry

with incident beam

monochromator is:

where 2θM is the Bragg

angle of the

monochromator.

The general polarization

factor for the diffracted

beam to point P is:

Geometric relationship between the monochromator

and detector in laboratory coordinates.

M

MIP

2cos1

2cos2cos12

22

M

MGP

2cos1

cossin2cos2cos)sincos2(cos2

2222222

XRD2: Polarization Correction

The unit vector of the diffraction vector HP and its projection on YL-

ZL plane, H'P, in the laboratory system are given respectively as:

The unit vector of YL is yL=[0,1,0], then:

Therefore, and

The polarization factor for XRD2 can then be given as a function of

both and g:

g

g

coscos

sincos

sin

L

z

y

x

h

h

h

h

g

g

cos

sin

00

L

z

y

h

hh

g sin),cos(cos LL yy LL hh

g 22 sincos g 22 cossin

M

MMP

ggg

2cos1

cos)2cos2(cossin)2cos2cos1(),(

2

222222

XRD2: Sample Absorption Correction

The absorption can

be measured by the

transmission

coefficient:

where is the total

beam path and A is

the average over all

the element dV. For

Bragg-Brentano

geometry, we have:

ABB=1/(2µ)

dVeV

AV

1

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: AAAT BB 2/

XRD2: Sample Absorption Correction

For reflection mode diffraction

with a thick plate:

and

The normalized transmission

coefficient:

with

0

0

1

os

g

g

cos2sin

sin2sin

2cos

s

sin

coscos

cossin

n

coscos

cos2

T

cossincos nso

cossin2coscos nsgg sincos2sincoscossin2sin

XRD2: Sample Absorption Correction

For transmission mode:

and

The normalized transmission

coefficient :

0

0

1

os

g

g

cos2sin

sin2sin

2cos

s

sincos

cossinsinsincos

coscossinsinsin

n

secsec

secexpsecexpsec2

ttT

coscossinsinsincos nso

g

g

cos2sinsincos

sin2sin)cossinsinsin(cos

2cos)coscossinsin(sincos

ns

XRD2: Texture Effect and Correction

The integrated intensity with texture is: where g() is the normalized pole density function. For the BB geometry, The texture effect for XRD2: Correct to the B-B equivalent with a texture effect:

sthklhklhkl

Ihkl MMFLPAv

pk 22exp),()( 23

2 gI

)0,(2

hklg

)( g

hkl

m

hklc

hklg

II

)(

)0,(2

g

hkl

m

hklhklBB

hklg

g II

12

2

1

)]([)(

gg

gggg

g

g

dhkl

hkl

gg

For fiber texture: and 3

1cos h

SCD, December 2008

XRD2: GADDS Microdiffraction

Horizontal th-2th, XYZ stage

Fatal Bicycle Accident Collection of Evidence

traces of car paint found on the bicycle

Dr. W. Kugler Landeskriminalamt Baden-Württemberg Stuttgart, Germany

Fatal Bicycle Accident Mapping of Car Paint with GADDS

video image (for documentation)

2-dimensional diffraction pattern

integration of data: diffractogram for phase identification

Fatal Bicycle Accident Phase Identification of Car Paint

36-0426 (*) - Dolomite - CaMg(CO3)2

05-0586 (*) - Calcite, syn - CaCO3

21-1276 (*) - Rutile, syn - TiO2

New Frame - File: 2708_07.raw - Start: 11.905 ° - End: 78.926 ° - Step: 0.040 °

Lin

(C

ounts

)

0

500

1000

1500

2000

2-Theta - Scale

12 20 30 40 50 60 70

Fatal Bicycle Accident The Car‘s Identification

New Frame - File: 2708_07.raw - Start: 12.000 ° - End: 79.030 ° - Step: 0.010 °

New Frame - File: 2708_05.raw - Start: 12.000 ° - End: 79.030 ° - Step: 0.010 °

New Frame - File: 2708_04.raw - Start: 12.000 ° - End: 79.030 ° - Step: 0.010 °

New Frame - File: 2708_06.raw - Start: 12.000 ° - End: 79.030 ° - Step: 0.010 °

Lin

(C

ou

nts

)

0

5000

2-Theta - Scale

13 20 30 40 50 60 70

sequence of coatings is characteristic of car type

Car Paint Analysis Five Layers of Car Paint

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

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

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

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

Phase ID Mapping

Rockmapping

Operations: Import

X + 24.0 mm - Y + 60.0 mm - Rockmapping

Operations: Import

X + 22.0 mm - Y + 55.0 mm - Rockmapping

Operations: Import

X + 20.0 mm - Y + 50.0 mm - Rockmapping

Operations: Import

X + 18.0 mm - Y + 45.0 mm - Rockmapping

Operations: Import

X + 16.0 mm - Y + 40.0 mm - Rockmapping

Operations: Import

X + 14.0 mm - Y + 35.0 mm - Rockmapping

Operations: Import

X + 12.0 mm - Y + 30.0 mm - Rockmapping

Operations: Import

X + 10.0 mm - Y + 25.0 mm - Rockmapping

Operations: Import

X + 8.0 mm - Y + 20.0 mm - Rockmapping -

Operations: Import

X + 6.0 mm - Y + 15.0 mm - Rockmapping -

Operations: Import

X + 4.0 mm - Y + 10.0 mm - Microverify - Fil

Operations: Import

X + 2.0 mm - Y + 5.0 mm - Microverify - File:

Operations: Import

Rockmapping - File: Spot-N_01.raw - Type:

Lin

(C

ou

nts

)

0

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30

40

50

60

70

80

90

100

110

120

130

140

150

160

170

2-Theta - Scale

6 10 20 30 40 50 60 70

High-throughput Screening (HTS)

Vertical theta-theta, Reflection/Transmission

Easy and accurate sample positioning without touching the sample surface

Video image of each material library spot can be automatically stored during data scan

XRD2: High throughput screening Laser/video sample alignment

PolySNAP for Combined Analysis: Correlation among XRD, Raman and other probes

Cell display

Dendrogram

3D plots

XRD2: Particle Size Analysis

1 nm 10 nm 100 nm 1 m 10 m 100 m 1 mm

2 profile analysis g profile analysis

particle size range in pharmaceutical systems

2 profile analysis, including measurement from peak FWHM by Scherrer equation, or profile 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.

XRD2: Particle size measurement by g profile analysis:

(111) (110) (100)

gprofile

Peak broadening - gold Nanoparticles

Particle size calculation:

Scherrer equation:

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.

cosB

Ct

15.12.2011 52 Bruker Confidential

XRD2: 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.

15.12.2011 53 Bruker Confidential

XRD2: Particle size measurement by g profile analysis:

For XRD2, the instrumental window is given by

so

s s- o

g

1

2

s

)]2/sin(arcsin[cos221 g

XRD2: Particle size measurement by g profile analysis:

For XRD2 in reflection mode, the crystallite size is given by

where is the linear absorption coefficient

For transmission mode with the incident beam perpendicular to the sample surface, the crystallite size is given by

where t is the sample thickness.

k is the instrument calibration factor or can be calculated from: if the instrument broadening in 2 direction is known.

31

2

2

)]2/sin(arcsin[cos

s

hkl

N

bpkd

g

31

2 )]2/sin(arcsin[cos

s

ihkl

N

tbpkd

g

31

4

3

k

More About XRD2

1. Introduction. 2. 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. 12. Quantitative Analysis. 13. Innovation and Future Development.

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 2011

56

Nanowire Film Orientation Analysis

Jim Britten, Weiguang Guan, Victoria Jarvis

McMaster University

Hamilton, Ontario, Canada

Jim Britten, Bruker-NYU XRD Workshop June 2011

57

XRD3 and Ewald’s Sphere

Jim Britten, Bruker-NYU XRD Workshop June 2011

58

Concentric spheres of Intensity at radii 1/d in Reciprocal Space

Cones of diffraction in real space

The Extremes of 3D Diffraction

Jim Britten, Bruker-NYU XRD Workshop June 2011

59

What can we see in between?

Residual Stress

Jim Britten, Bruker-NYU XRD Workshop June 2011

60

Looking for subtle changes in 2θ position of line/arc/shell to indicate orientation dependant residual stresses. Hard to see visually – need mathematical 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/d0

1D Ordering – Fibre Diffraction

Jim Britten, Bruker-NYU XRD Workshop June 2011

61

Extruded, distorted polypropylene. Elnagmi / Jain

C∞ -axis in diffraction pattern

Example 1 – Random Orientation GaAs NW on Carbon nanotube ‘fabric’

Jim Britten, Bruker-NYU XRD Workshop June 2011

62

Why bother with XRD3? Sometimes there are surprises!

Example 2 – Multiple (8) Orientation GaAs NW’s on Si Substrate

Jim Britten, Bruker-NYU XRD Workshop June 2011

63

2D scan sequence

Full scan in MAX3D

December 8, 2010 64 15. Dezember 2011 64 © Copyright Bruker Corporation. All rights reserved