Bob He, Bruker AXS April 15, 2014 Harvard University

Basics and Recent Advances in Two-dimensional X-ray Diffraction

Basic Concept – from XRD to XRD2

12/4/2017 2

Conventional X-ray Diffractometer

Divergence slit

Detector-slit

Tube

Antiscatterslit

Sample

Mono-chromator

Bragg-Brentano Geometry.

Point (0D) detector.

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

Diffraction Patterns vs. Atomic Arrangement

XRD2: Two-dimensional X-ray Diffraction

XRD2: From 2D pattern to -2 image

Fig. A. 2D diffraction pattern. Fig. B. 2D pattern in rectangular -2 image.

2

bob.he@bruker-axs.com 8

XRD2: 2D pattern in I on -2 coordinates Powder Texture

bob.he@bruker-axs.com 9

XRD2: 2D pattern in I on -2 coordinates Stress Large grains

bob.he@bruker-axs.com 10

Geometry Convention and Diffraction Vector Approach

12/4/2017 11

Single Crystals

l

k

h

)(

)(

)(

0

0

0

ssc

ssb

ssaLaue equation

S0

S

XRD2: Debye cones from powders

Bragg law

sin2dn

Debye Cone

Sample

Incident Beam

XRD2: Diffraction pattern with both and 2 information

cos2sin

sin2sin

12cos10ssH

Diffraction vector with

XRD2: Geometry Convention - Diffraction Space

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

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:

cos2sin

sin2sin

12cos11

0

0

0

0

zz

yy

xx

ss

ss

ssss

H

coscos

sincos

sin

L

z

y

x

h

h

h

H

Hh

XRD2: Sample Space and Eulerian Geometry

The angular relationships

between XLYLZL and

S1S2S3 are:

The transformation matrix

from the diffraction space

to the sample

space is:

X Y Z

S a a a

S a a a

S a a a

L L L

1 11 12 13

2 21 22 23

3 31 32 33

sincoscoscossin

coscossinsin

cossincos

sincos

cossinsin

sincoscossin

sinsincos

coscos

sinsinsin

333231

232221

131211

aaa

aaa

aaa

XRD2: Sample Space and Unit Diffraction Vector

The components of the unit

vector hS in the sample

coordinates S1S2S3 is then

given by

Or

in expanded form:

z

y

x

h

h

h

aaa

aaa

aaa

h

h

h

333231

232221

131211

3

2

1

)sincoscossin(sinsincos

cossincoscos)coscossinsin(sinsin1

h

)sinsincossin(cossincos

coscoscoscos)cossinsinsin(cossin2

h

h3 sin cos sin cos sin cos cos cos cos sin

Sources

12/4/2017 20

Characteristic X-ray generation

• All present monochromatic home laboratory sources are based on characteristic radiation from a material anode

• The efficiency of this process is very low

• Approximately 99% of the incident electron power is converted to heat, not X -rays

• Dissipation of this waste heat fundamentally limits the brightness of the source

12/4/2017 21

How to make brighter source I: Microfocus sources

Large spot

Quasi-one dimensional heat flow limits power loading

Small spot

Two dimensional heat flow (more efficient cooling)

Relative performance improved by 10 times

)2ln(

2 0max

r

TTp m

Brightness (B) is proportional to power loading (p) Power loading is higher for smaller spot focus

12/4/2017 23

ImS microfocus source

• Intensity 3x1010 X-rays/mm2-sec (Cu Ka)

• 8 times higher than conventional 5.4 kW rotating anode

• Typical lifetime >5 years

• High reliability

• 3 year warranty

• >300 installed

• Air-cooled

• Available in Cr, Cu, Mo, Ag

12/4/2017 24

ImS & VÅNTEC-2000 vs. Classic Set-up Corundum Comparison

Sealed Tube with Göbel Mirror

45kV, 40mA, (1800 W)

0.3mm collimator

total counts: 78K

Microsource (ImS)TM

45kV, 0.650mA, (30 W)

0.3mm collimator

total counts: 1235K

Intensity: 15.8x; Efficiency: 948x !

How to make a brighter source II: Rotating anode sources

• Power loading can be increased by over an order of magnitude by rotating the anode surface to spread out the heat load

• Power load is also (modestly) increased by smaller spot

• In latest generation rotating anodes angular velocity is 10,000 rpm

• This improves performance by 50 times

w

vTTp m 0max

w=beam width v=anode velocity

12/4/2017 26

TXS HB High brilliance rotating anode

• Highest intensity rotating anode

• 2x1011 X-rays/mm2-sec Cu Ka

• 50 times the intensity of a 5.4 kW classic RAG with multilayer optics

• Cu, Mo, Ag anodes • Low maintenance • Easy to align, highly stable mount

• No alignment base • Precrystallized, prealigned filaments

• No realignment required after filament changes (2X per year)

• Precision aligned anode

• No realignment required after anode exchange (1X per year)

12/4/2017 27

What are the limits of rotating anode performance?

• Faster rotation () allows higher power loading

• However, faster rotation also increases mechanical stress

• State-of-the-art rotating anodes are operated close to mechanical failure limits

• Little room for further improvement

2232

h Rt2

R

t

PR

wRp max

12/4/2017 28

NEW: Liquid metal sources

• High-speed liquid-metal-jet anode

• Anode is regenerative

• No longer limited by melting

• >500 kW/mm2 e-beam power density

• Rotating anode limited to maximum 50 kW/mm2

12/4/2017 29

E-beam

Interaction

point

Metal jet

Spot Size

20 µm 10 µm 5 µm

5 mm

12/4/2017 30

Spot size

[µm, FWHM]

Voltage

[kV]

Power

[W]

Ga Kα Brightness

[Photons/(s× mm2× mrad2× line]

5 60 50 1.5× 1011

10 60 100 7.6× 1010

20 60 200 3.8× 1010

• X-ray photon energy: Ga Ka 9.25 keV (=1.34 Å)

• Suitable replacement for Cu Ka 8.06 keV (=1.54 Å)

So, is it possible to put a synchrotron beamline on a table top?

• Yes, at least the equivalent of a typical present generation bending magnet beamline

12/4/2017 32

Detector

12/4/2017 33

2D Workshop

Characteristics of area detectors: Sensitivity vs. Counting 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

04.12.2017 Bruker Confidential 35

XRD2 : Choice of Detectors: Active Area and Pixel Size

The angular coverage of 2D detectors with various active areas (D=150mm)

04.12.2017 Bruker Confidential 36

XRD2 : Choice of Detectors: Active Area and Pixel Size

At D=150 mm, the angular resolution per pixel is 0.026 for large detector, but 0.21 and 0.11 for two smaller 2D detectors.

Large active area: 140 mm in dia.

Frame size: 2048 x 2048 pixels 1024 x 1024 pixels 512 x 512 pixels

Pixel size: 68 mm x 68 µm 136 mm x 136 µm 272 mm x 272 µm

High sensitivity: 80% DQE for Cu

High max linear count rate: 0.9 Mcps – global; 160 kcps/reflection -local

Low background noise:

Phase ID

12/4/2017 43

XRD2: Phase ID Measurement Geometry

XRD2: Single Frame Covering All

Multilayer battery anode.

2 coverage: 70 at 8 cm detector distance

A single frame showing information on phase, stress, texture and grain size

2D detector is essential for In-situ measurement

XRD2: Frame Merge and Integration

4 frames at 20 cm

Merged frames 2 coverage: 100

Integrated profile for phase ID search/match

Stress

12/4/2017 47

cos

sinsin

sincos

3

2

1

h

h

h

h

XRD: Sample Space & Unit Diffraction Vector

The components of the unit vector

hS in the sample coordinates S1S2S3

is then given by

which is a single value at each

sample orientation.

The sin2 - Method for Stress Measurements

The sin2 - Method for Stress Measurements

Effects of Texture and Large Grain on e- sin

2 – Functions

XRD & XRD2: Fundamental Equations for Strain

As a second order tensor, the relationship between the measured

strains and the strain tensor is given by:

0D/1D: 2D:

Both equations are in the same form except the difference in hs

Introducing =90 or 270 (diffractometer plane),

the above equation can be derived from the bottom equation

ee jiij hh

e e e e

e e e

11

2 2

12

2

22

2 2

13 23 33

2

2

2 2

cos sin sin sin sin sin

cos sin sin sin cos

jiij hh ee ),,,(

33

2

32332133122

2

2122111

2

1

}{

),,,( 222 eeeeeee hhhhhhhhhhkl

XRD2: Fundamental Equation for Stress Measurement

Introducing the

elasticity constants:

E & or

the macroscopic

elastic constants:

&

the fundamental

equation for stress measurement in XRD2:

sin

sinln)222

()(

0322331132112

2

333

2

222

2

111221

3322111

hhhhhh

hhhSS

12 2

1S E ( ) /

S E1 /

XRD2: 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 (211) with Cr

radiation.

The D8 DISCOVER with DAVINCI VÅNTEC-500 for stress measurement

Almen strip (spring

steel)

(211) & (200) rings

Ring distortion when

sample rotates with

and

Parameter setting

Results reporting: 11= -1068 18 MPa 22= -1085 18 MPa

XRD2: Stress Measurement –Example

Evaluate the data quality

Principal stress and stress ellipse –equibiaxial

Comparison of the sin2 and 2D method for stress measurement

04/12/2017 Bruker Confidential 59

6238 33

26 23

With increasing data points, the new 2D method can measure stress with higher accuracy than the conventional sin2 method for the same amount of data collection

XRD2: Stress Measurement with 2D method: Effect of Texture

low intensity

and poor profile

due to texture

tends to give

larger 2 shift

error.

Elastic

anisotropy

results in error

in stress

calculation

XRD2: Stress Measurement with 2D method: Effect of Large Grains

Poor profile due

to large grain

size gives larger

2 shift error.

Diffraction

profile of

extreme large

grain or substrate

at a rocking

angle brings

error in 2 .

XRD2: Innovations to 2D method: Intensity Weighted Least Squares Regression

low intensity and poor

profile due to texture and

large grain tends to give

larger 2 shift error. Ref: B. He, Two-dimensional X-ray Diffraction,

John Wiley & Sons, 2009, pp 299-303.

n

i

iii

n

i

ii yywrwS1

2

1

2 )(

2

i

ii

Iw

maximum (or integrated) intensity

the standard error of profile fitting

of the ith data point

XRD2: Innovations to 2D method: Stress Measurement from Multiple (hkl ) Rings Regression

Improve the accuracy and

reduce the effect of

anisotropy and texture

(420) (331)

Cu Film Stress vs. Loading

300

350

400

450

500

550

600

650

700

750

800

0 1 2 3 4 5 6 7 8Loading Strain (X 0.001)

Str

es

s (

MP

a)

30

32

34

36

38

40

42

44

46

48

50

52

54

56

58

60

Sta

nd

ard

err

or

(MP

a)

(331)+(420) Stress

(331) only stress

(420) only stress

Linear

(331)+(420)

Std.err(331) only std.err

(420) only std.err

Texture, Orientation and Fiber

12/4/2017 64

XRD2: Fundamental Equation for Texture Analysis

The pole figure angles

(a,) can be calculated

from the unit vector

components by the pole

mapping equations:

2

2

2

1

1

3

1 cossin hhh a

00

00cos

2

2

2

2

2

1

11

hif

hif

hh

h

The D8 DISCOVER with DAVINCI VÅNTEC-500 for texture measurement

Steel can

(200) & (110) rings

Intensity variation during

scan

XRD2: Data Collection Speed vs. 2D or 0D

2D: 108 exposures (frames); 0D: 973 exposures (points) multiple pole-figures; single pole-figures 2D detector is 1~2 orders of magnitude faster than 0D detector.

5

Texture Measurement Using VANTEC-500: Magnetron sputter-deposited Cu films

Cu (311) (220) (200) (111)

Si (311)

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.

Arts from XRD2: Pole-figure from Cu film and Si

12/4/2017 70 Beijing Application Lab

A

B C

D

A’ B’ C’ D’

S1 S2

O

XRD2: Miscut Angle

• Two spots are produced by sample rotates during data collection

• 2theta integration shows the gamma angle between two spots

in degrees

12/4/2017 71 Beijing Application Lab

XRD2: Miscut Angle

• The angle between two diffraction vector is given by the virtual oscillation angle

• The miscut angle a is from half of the

)]2/sin(arcsin[cos2

)]4/sin(arcsin[cos2 a

72 72

D8 Discover with VÅNTEC-500 PE Fiber: Orientation and Crystallinity

72

XRD2: Fiber with VÅNTEC-500

Crystal Size

12/4/2017 74

04.12.2017

XRD2: Crystal Size by profile analysis: Organic glass for food & drugs

*Courtesy of Prof. Lian Yu, U. of Wisconsin - Madison 75

04.12.2017 76 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 (-range), multiplicity of the crystal plane, and effective diffraction volume.

The size of jelly beans and candy bin determines how many you can fill.

04.12.2017 77 Bruker Confidential

XRD2: Particle size measurement by profile analysis:

so

s s- o

1

2

s

W

For XRD2, the instrumental window W is given by

)]2/sin(arcsin[cos221 W

XRD2: Particle Size Analysis

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

2 profile analysis profile analysis

particle size range in pharmaceutical systems

Scherrer equation:

cosB

Ct

Gold nano-particle

Thin film applications contributed by Jon Giencke jon.giencke@bruker-axs.com

12/4/2017 79

December 4, 2017 80

Instrument Setup

• ImS microfocus Cu Tube

• Montel Optic

• 0.5 mm Collimator

• Knife Edge

• VÅNTEC 500

= XRD coverage

a

Specular Reflection

Ewald Sphere

Innovative XRD Techniques Rapid X-ray Reflectometry with XRD

9.748 nm

18.095 nm

42.348 nm

116.199 nm

December 4, 2017 81

Superlattice with 4nm period

Scan Mode

Step Mode

One Frame

Innovative XRD Techniques Rapid X-ray Reflectometry with XRD

Layers of Various Thicknesses

December 4, 2017 83

Standard GID (Polycrystalline Film)

In Plane GID (Polycrystalline Film)

In Plane GID (Epitaxial Film)

Innovative XRD Techniques In Plane Grazing Incidence Diffraction with XRD

December 4, 2017 84

Innovative XRD Techniques In Plane Grazing Incidence Diffraction with XRD 10 nm Polycrystalline Film on Si – Conventional GID

GID

In Plane GID

December 4, 2017 85

Innovative XRD Techniques In Plane Grazing Incidence Diffraction with XRD Comparison of Conventional and XRD

December 4, 2017 86

IμS with Montel VÅNTEC 500

Sample

Ewald Sphere Construction

Ray Tracing Diagram

Ψ

Innovative XRD Techniques In Plane Grazing Incidence Diffraction with XRD

Raw Data

Gam

ma

2 Theta

0.05 Phi Step per frame

Innovative XRD Techniques

In Plane Grazing Incidence Diffraction with XRD Crystal Truncation Rod Measurement

What is a Crystal Truncation Rod?

25 nm Si / 50 nm SiGe / Si 20 nm STO on Si G

am

ma

Gam

ma

Phi Phi

Innovative XRD Techniques

In Plane Grazing Incidence Diffraction with XRD Crystal Truncation Rod Measurement

December 4, 2017 91

Innovative XRD Techniques In Plane Grazing Transmission Diffraction with XRD

8 nm

16 nm

40 nm

100 nm Si 220 STO 200

Z is set slightly low, so only the edge is tilted into the beam

Total Time for XRD2 RSM (27 peaks): 15 minutes Total Time for 1D RSMs (4 peaks): 1.5 hours

Innovative XRD Techniques Reciprocal Space Mapping with XRD Comparison of RSM with XRD and 1D LynxEye

1D Detector

December 4, 2017 92

1D Detector

December 4, 2017 93

Innovative XRD Techniques Lattice Parameter Refinement with DIFFRAC.TOPAS

3D Display Visualization of 3D Reciprocal Space with MAX3D

12/4/2017 94

Contributed by

Jim Britten, Weiguang Guan, Victoria Jarvis

McMaster University, Hamilton , Ontario, Canada

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

Jim Britten, Bruker-NYU XRD Workshop June 2011 96

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 97

2D scan sequence

Full scan in MAX3D

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.

More About XRD2

Volume H with a chapter on two-dimensional X-ray diffraction will be published in 2014 and available at

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=C-D7iu5nWTtGNM&tbnid=pG_ePcN_HqimhM:&ved=0CAUQjRw&url=http://www.amazon.com/International-Tables-Crystallography-Vol-Crystallographic/dp/1402031386&ei=OaGNUoulCeef2QXl8oDoAg&bvm=bv.56988011,d.b2I&psig=AFQjCNG5NeJS0MITRjpUT4-zHc8bJxnt0g&ust=1385099636988705

XRD2 Publications ICDD Most Downloaded AXA Papers

XRD2 Publications Most Downloaded AXA Papers #2

XRD2 Publications Most Downloaded AXA Papers #1

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 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

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

December 8, 2010 109 4. Dezember 2017 109 © Copyright Bruker Corporation. All rights reserved

Bob.He@Bruker-AXS.com