two-dimensional powder diffraction - s. n. bose national centre...
TRANSCRIPT
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
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5
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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
)
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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