Two-dimensional X-ray Diffraction E-Book

Bob B. He

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 986

Veröffentlichungsjahr: 2018

Beschreibung

An indispensable resource for researchers and students in materials science, chemistry, physics, and pharmaceuticals

Written by one of the pioneers of 2D X-Ray Diffraction, this updated and expanded edition of the definitive text in the field provides comprehensive coverage of the fundamentals of that analytical method, as well as state-of-the art experimental methods and applications. Geometry convention, x-ray source and optics, two-dimensional detectors, diffraction data interpretation, and configurations for various applications, such as phase identification, texture, stress, microstructure analysis, crystallinity, thin film analysis, and combinatorial screening are all covered in detail. Numerous experimental examples in materials research, manufacture, and pharmaceuticals are provided throughout.

Two-dimensional x-ray diffraction is the ideal, non-destructive analytical method for examining samples of all kinds including metals, polymers, ceramics, semiconductors, thin films, coatings, paints, biomaterials, composites, and more. Two-Dimensional X-Ray Diffraction, Second Edition is an up-to-date resource for understanding how the latest 2D detectors are integrated into diffractometers, how to get the best data using the 2D detector for diffraction, and how to interpret this data. All those desirous of setting up a 2D diffraction in their own laboratories will find the author’s coverage of the physical principles, projection geometry, and mathematical derivations extremely helpful.

• Features new contents in all chapters with most figures in full color to reveal more details in illustrations and diffraction patterns
• Covers the recent advances in detector technology and 2D data collection strategies that have led to dramatic increases in the use of two-dimensional detectors for x-ray diffraction
• Provides in-depth coverage of new innovations in x-ray sources, optics, system configurations, applications and data evaluation algorithms
• Contains new methods and experimental examples in stress, texture, crystal size, crystal orientation and thin film analysis 

Two-Dimensional X-Ray Diffraction, Second Edition is an important working resource for industrial and academic researchers and developers in materials science, chemistry, physics, pharmaceuticals, and all those who use x-ray diffraction as a characterization method. Users of all levels, instrument technicians and X-ray laboratory managers, as well as instrument developers, will want to have it on hand.

Leseprobe

Table of Contents

Cover

Copyright

Preface

Chapter 1: Introduction

1.1 X‐Ray Technology, a Brief History

1.2 Geometry of Crystals

1.3 Principles of X‐Ray Diffraction

1.4 Reciprocal Space and Diffraction

1.5 Two‐Dimensional X‐Ray Diffraction

References

Chapter 2: Geometry and Fundamentals

2.1 Introduction

2.2 Diffraction Space and Laboratory Coordinates

2.3 Detector Space and Detector Geometry

2.4 Sample Space and Goniometer Geometry

2.5 Transformation from Diffraction Space to Sample Space

2.6 Reciprocal Space

2.7 Summary

References

Chapter 3: X‐Ray Source and Optics

3.1 X‐Ray Generation and Characteristics

3.2 X‐Ray Optics

References

Chapter 4: X‐Ray Detectors

4.1 History of X‐Ray Detection Technology

4.2 Point Detectors in Conventional Diffractometers

4.3 Characteristics of Point Detectors

4.4 Line Detectors

4.5 Characteristics of Area Detectors

4.6 Types of Area Detectors

References

Chapter 5: Goniometer and Sample Stages

5.1 Goniometer and Sample Position

5.2 Goniometer Accuracy

5.3 Sample Alignment and Visualization Systems

5.4 Environment Stages

References

Chapter 6: Data Treatment

6.1 Introduction

6.2 Non‐Uniform Response Correction

6.3 Spatial Correction

6.4 Detector Position Accuracy and Calibration

6.5 Frame Integration

6.6 Multiple Frame Merge

6.7 Scanning 2D Pattern

6.8 Lorentz, Polarization, and Absorption Corrections

References

Chapter 7: Phase Identification

7.1 Introduction

7.2 Relative Intensity

7.3 Geometry and Resolution

7.4 Sampling Statistics

7.5 Preferred Orientation Effect

References

Chapter 8: Texture Analysis

8.1 Introduction

8.2 Pole Density and Pole‐Figure

8.3 Fundamental Equations

8.4 Data Collection Strategy

8.5 Texture Data Process

8.6 Orientation Distribution Function

8.7 Fiber Texture

8.8 Polymer Texture

8.9 Other Advantages of XRD for Texture

References

Chapter 9: Stress Measurement

9.1 Introduction

9.2 Principle of X‐ray Stress Analysis

9.3 Theory of Stress Analysis with XRD

9.4 Process of Stress Measurement with XRD

9.5 Experimental Examples

Appendix 9.1 Calculation of Principal Stresses from the General Stress Tensor

Appendix 9.2 Parameters for Stress Measurement

References

Chapter 10: Small Angle X‐ray Scattering

10.1 Introduction

10.2 2D SAXS Systems

10.3 Applications Examples

10.4 Some Innovations in 2D SAXS

References

Chapter 11: Combinatorial Screening

11.1 Introduction

11.2 XRD Systems for High Throughput Screening

11.3 Combined Screening with XRD and Raman

References

Chapter 12: Miscellaneous Applications

12.1 Percent Crystallinity

12.2 Crystal Size

12.3 Retained Austenite

12.4 Crystal Orientation

12.5 Thin Film Analysis

References

Chapter 13: Innovation and Future Development

13.1 Introduction

13.2 Scanning Line Detector for XRD

13.3 Three‐Dimensional Detector

13.4 Pixel Direct Diffraction Analysis

13.5 High Resolution Two‐Dimensional X‐Ray Diffractometer

References

Appendix A: Values of Commonly Used Parameters

Appendix B: Symbols

Index

End User License Agreement

List of Tables

Chapter 01

Table 1.1 Crystal Systems and Bravais Lattices

Table 1.2 Equation of D‐Spacing for All Seven Crystal Systems

Chapter 02

Table 2.1 Simple Tests for Forward Diffraction

Table 2.2 The Direction of the

φ

axis in

X

L

Y

L

Z

L

at Special

ω

and

ψ

Chapter 03

Table 3.1 Wavelengths of Characteristic Lines of Common Anode Elements

Table 3.2 Focal Spot Size, Line Focus Size, and Spot Focus Size of Some Typical X‐ray Tubes

Table 3.3 Focal Spot Brilliance for Sealed Tubes, Rotating Anode Generators, and Microsources with Cu Anode and Metal Jet with Ga Anode

Table 3.4 The β‐Filter for Common Target Elements

Table 3.5 Bragg Angles of Graphite Crystal (002) Plane for Various Target Materials and the Knife‐Edge Gap for 10 mm Crystal and 0.4 mm Spot Focus

Table 3.6 X‐ray Beam Divergence Angle (

β

), Convergence Angle (

α

), and Beam Spot Size on Sample (

S

) for a 0.8 mm Point Focus Tube with Graphite Monochromator or Cross‐Coupled Göbel Mirrors

Table 3.7 Comparison between Single Pinhole Collimator and Double Pinhole Collimator in Terms of Intensity Gain, Beam Divergence Angle (

β

), and Beam Spot Size on Sample (

S

)

Table 3.8 Intensity Gain (Calculation and Experimental) and Beam Spot Size Containing 90% Beam Intensity for Monocapillaries Compared with Double Pinhole Collimators

Chapter 04

Table 4.1 Specifications of Typical MWPC, CCD, CPAD, Mikrogap and HPC Detectors (Values on the Similar Detector Technologies are Given in Parentheses)

Chapter 05

Table 5.1 Specifications and Applications of Sample Stages

Table 5.2 Substances for High Temperature Stage Calibration [19, 25, 26]

Chapter 06

Table 6.1 The Recommended Angle and Generator Power for Hi‐Star Calibration with Iron Foil

Table 6.2 Detector Position Tolerance and Reproducibility

Table 6.3 Correction Factors for Pixel to Pixel Intensity Correction

Chapter 07

Table 7.1 Multiplicity Factor

p

hk

l

for Various Systems and Miller Indices

Table 7.2 Coefficients

and

C

ij

(

χ

) to the Order of n = 16 for Cubic Structure

Chapter 08

Table 8.1 Order of the Series of Harmonics for the Number of Pole‐Figures

Chapter 09

Table 9.1 Values of

A

R

X

for the Some Common Cubic Materials

Table 9.2 The Geometric Conditions Equivalent to the Conventional Diffractometer in an XRD

2

System

Table 9.3 The Equations of Stress Coefficients

Table 9.4 The Simulation Conditions in Figure 9.13

Table 9.5 Goniometer Angles and Measurable Stress Components for the Six Schemes. The Scanning Angles are Listed as Scan Range and Step in Parentheses

Table 9.6 Unknown Stress Components to be Solved by Least Squares Regression

Table 9.7 Measured Stress with the Conventional sin

2

ψ

Method and the 2D Method

Table 9.8 Residual Stresses ( MPa) of TiO

2

Films and Ti Substrates

Table 9.9 Equations of Dependent 2

θ

0

for Stress Analysis with Multiple {

hkl

} Rings

Table 9.10 Gage R&R Test Results from Ten Sample by Three Operators for Three Measurements

Chapter 10

Table 10.1 The Resolution of Various SAXS Configurations

Chapter 12

Table 12.1 Instrumental Calibration for Crystallite Size with LaB

6

Table 12.2 Retained Austenite Measurement of a Steel Roller: %RA = 21.3

List of Illustrations

Chapter 01

Figure 1.1 A point lattice (a) and its unit cell (b).

Figure 1.2 Symmetry elements of a cubic unit cell.

Figure 1.3 Unit cells of the 14 Bravais lattices.

Figure 1.4 (a) Indices of lattice directions and (b) Miller indices of lattice planes.

Figure 1.5 (a) Hexagonal unit cell (heavy lines) and indices of some lattice directions and (b) Miller–Bravais indices of some lattice planes in a hexagonal lattice.

Figure 1.6 All shaded crystal planes belong to the [001] zone in the cubic lattice.

Figure 1.7 Atomic arrangements in three common crystal structures of metals.

Figure 1.8 The structure of NaCl. Na

+

 is FCC and Cl

−

is FCC with ½, ½, ½ translation.

Figure 1.9 Illustration of crystal mosaicity.

Figure 1.10 (a) The incident X‐rays and reflected X‐rays make an angle of

θ

symmetric to the normal of crystal plane. (b) The diffraction peak is observed at the Bragg angle

θ

.

Figure 1.11 Diffraction patterns from crystalline solids, liquid, amorphous solid and monatomic gases as well as their mixtures.

Figure 1.12 Relationship between the original crystal lattice in real space and reciprocal lattice.

Figure 1.13 Ewald sphere and Bragg condition in reciprocal space.

Figure 1.14 Limiting sphere for the powder diffraction.

Figure 1.15 Diffraction cone and diffraction vector cone illustrated on the Ewald sphere.

Figure 1.16 Patterns of diffracted X‐rays: (a) from a single crystal, (b) diffraction frame from single crystal of protein thaumatin, (c) diffraction cones from a polycrystalline sample. (d) diffraction frame from corundum powder.

Figure 1.17 (a) 2D diffraction pattern from corundum powder, (b) 2D pattern displayed in rectangular

γ

‐2

θ

coordinates.

Figure 1.18 Illustrations of gamma plots from samples with (a) random fine powder, (b) texture, (c) stress, (d) large crystal size.

Figure 1.19 (a) 2D diffraction pattern from a battery anode, (b) the 2D pattern displayed in 3D plot with the intensity in vertical direction.

Figure 1.20 Five basic components in an XRD

2

system: X‐ray source (sealed tube generator); X‐ray optics (monochromator and collimator); goniometer and sample stage; sample alignment and monitor (laser‐video); and area detector.

Chapter 02

Figure 2.1 Diffraction patterns in 3D space from a powder sample and the diffractometer plane.

Figure 2.2 Coverage comparison: point, line, and area detectors.

Figure 2.3 Geometric definition of diffraction rings in laboratory axes.

Figure 2.4 Diffraction vector satisfying Bragg's law.

Figure 2.5 Relation between the diffraction cone and the corresponding diffraction vector cone.

Figure 2.6 Schematics of an ideal detector covering 4

π

solid angle.

Figure 2.7 A diffraction cone and the conic section by a 2D detector plane.

Figure 2.8 Detector position in the laboratory system

X

L

Y

L

Z

L

:

D

is the sample‐to‐detector distance;

α

is the swing angle of the detector.

Figure 2.9 Relationship between a pixel P and detector position in the laboratory coordinates,

X

L

Y

L

Z

L

.

Figure 2.10 Detector motion out of diffractometer plane: (a) azimuthal rotation of the detector about the

X

L

axis, (b) detector swing angle (

η

) about the instrument center and towards the

Z

L

axis, (c) diffraction space coverage of various detector motions.

Figure 2.11 Cylinder‐shaped detector in vertical direction: (a) detector position in the laboratory coordinates,

X

L

Y

L

Z

L

, (b) pixel position in the flattened image.

Figure 2.12 Cylinder shaped detector in horizontal direction: (a) detector position in the laboratory coordinates,

X

L

Y

L

Z

L

, (b) pixel position in the flattened image, (c) two flat detector and a cylinder detector combined to cover the complete diffraction space.

Figure 2.13 Sample rotation and translation: (a) three rotation axes in

X

L

Y

L

Z

L

coordinates, (b) rotation axes (

ω

,

χ

g

,

ψ

,

ϕ

) and translation axes

XYZ

.

Figure 2.14 Sample translation axes

XYZ

and the sample coordinates

S

1

S

2

S

3

.

Figure 2.15 Typical XRD

2

diffractometer configurations: (a) horizontal

θ

‐2

θ

, (b) vertical

θ‐θ

, (c) and (d) diffractometers in horizontal and vertical configurations (Bruker D8 Discover™).

Figure 2.16 Unit vector of diffraction vector in (a) the laboratory coordinates

X

L

Y

L

Z

L

and (b) the sample coordinates

S

1

S

2

S

3

.

Figure 2.17 Two‐dimensional detector coverage on Ewald sphere: (a) single frame covers a surface area on Ewald sphere, (b) multiple frames at various sample orientation cover a volume in reciprocal space.

Figure 2.18 Relationship between the three spaces and the laboratory coordinates.

Figure 2.19 Relationship between the fundamental equations.

Chapter 03

Figure 3.1 X‐ray spectrum generated by a sealed X‐ray tube or rotating anode generator: (a) spectrum consisting of continuous (white) radiation and characteristic radiation lines K

α

and K

β

, (b) K

α

line as a combination of two lines K

α1

and K

α2

.

Figure 3.2 Schematic of a sealed X‐ray tube showing filament (cathode), anode, focal spot on anode, takeoff angle, line focus projection, and spot focus projection.

Figure 3.3 Schematics of X‐ray source, optics, and X‐ray image on the sample and their relationship expressed by Liouville's theorem.

Figure 3.4 Conventional diffractometer in the Bragg–Brentano geometry: (a) Components arrangement projected in the diffractometer plane, (b) Axial divergence controlled by the soller slits.

Figure 3.5 Conventional diffractometer in parallel beam geometry.

Figure 3.6 X‐ray optics in an XRD

2

system.

Figure 3.7 Typical X‐ray optics in standard GADDS (courtesy of Bruker AXS) includes X‐ray tube, monochromator, collimator, and beamstop.

Figure 3.8 Diffraction frame from corundum powder: (a) smearing effect from line beam, (b) diffraction rings from point beam.

Figure 3.9 Effect of β‐filter: (a) spectrum in the vicinity of the K

α

line and K

β

line, (b) spectrum after transmission from the β‐filter. The dashed line is the mass absorption coefficient of the β‐filter.

Figure 3.10 Illustration of a crystal monochromator. Monochromatic X‐rays are obtained by diffraction from a single crystal plate.

Figure 3.11 (a) Geometry of a parabolic mirror, (b) Geometry of an elliptical mirror.

Figure 3.12 (a) A single parabolic mirror produces a parallel beam, (b) cross‐coupled Göbel mirrors produce a parallel point beam, (c) side‐by‐side mirror produces a convergence beam.

Figure 3.13 Comparison of X‐ray intensity between cross‐coupled Göbel mirrors and monochromator with various collimator pinhole size.

Figure 3.14 Schematic of the X‐ray beam path in a pinhole collimator, showing the parallel, divergent and convergent X‐rays and beam spot on the sample surface.

Figure 3.15 Schematic of polycapillary optics: (a) focusing polycapillary, (b) parallel polycapillary.

Chapter 04

Figure 4.1 Debye–Scherrer camera (a) and the diffraction lines on the film (b).

Figure 4.2 Description of counting statistics: (a) Gaussian distribution and standard deviation, (b) the relative standard deviation vs. the counts.

Figure 4.3 Detector quantum efficiency (DQE) as a function of wavelength for three typical point detectors: A – scintillation counter, B – Si(Li) solid state detector and C – xenon filled proportional counter. (Schematic comparison based on data from references [2, 4, 5].)

Figure 4.4 Illustration of the detector counting curves from: A – an ideal detector with 100% DQE; B – a detector ideal linearity; C – a detector with non‐paralyzable dead time; D – a detector with paralyzable dead time; E – a detector with semi‐paralyzable dead time.

Figure 4.5 Comparison of detector energy resolution of scintillation counter, proportional counter, and solid state detector with the characteristic lines of copper.

Figure 4.6 Illustration of line detectors: (a) straight line, (b) curved line.

Figure 4.7 (a) Våntec‐1 detector mounted on a diffractometer (Courtesy of Bruker AXS). (b) In‐situ phase transition investigation of NH

4

NO

3

with a Våntec‐1 detector.

Figure 4.8 Angular coverage of 2D detectors of three different active areas.

Figure 4.9 The

γ

angular coverage of rectangular flat 2D detector and cylindrical detector.

Figure 4.10 The

γ

range as a function of 2

θ

at various detector distances and orientations.

Figure 4.11 Detector dimensions and maximum measurable 2

θ

.

Figure 4.12 Illustration of a flat area detector showing the relationship between the solid angle covered by each pixel and its location on the detector.

Figure 4.13 (a) Point spread function (PSF) from a parallel point beam, (b) line spread function (LSF) from a sharp line beam, (c) first derivative of the integrated intensity curve (broken line) from the image of a razor blade.

Figure 4.14 Total number of pixels (megapixel or MP) of 2D detectors with three different active areas.

Figure 4.15 Point spread function (PSF) and 2

θ

resolution: (a) FWHM of PSF, optics, and overall instrument broadening, (b) resolution triangle.

Figure 4.16 Effective diffraction volume for: (a) large beam size, PSF, detector size, (b) beam size, PSF, detector size, and detector distance reduced by half.

Figure 4.17 Cutaway view of a MWPC detector (Bruker Hi‐Star™).

Figure 4.18 Schematic showing the side section of three pixels and the direct detection of an X‐ray photon in a front illuminated CCD.

Figure 4.19 CCD detector: (a) schematic showing phosphor, fiberoptic taper, and CCD chip, (b) a picture of a Fairchild 486 CCD chip.

Figure 4.20 (a) Bruker APEX II™ CCD detector, (b) schematic showing phosphor screen, fiberoptic faceplate, and CCD chip.

Figure 4.21 Illustration of a PAD detector with detection layer and readout layer connected pixel by pixel with connecting bumps.

Figure 4.22 Relation between DQE and reflection size for a HPC detector with 75 µm pixel size.

Figure 4.23 The Eiger2 detector (a) and a diffractometer with the detector (b).

Figure 4.24 Bruker Photon II™ detector, 2D frame and integrated profile from Al foil.

Figure 4.25 Two‐dimensional mikrogap detector: (a) cross‐section of the detector showing the beryllium window, grid, anode, and delay line, (b) schematic of the parallel‐plate resistive‐anode chamber with the readout electrode separated from the anode, (c) Våntec‐2000™ and (d) Våntec‐500™ (Courtesy of Bruker AXS).

Figure 4.26 Diffraction frame of NIST 1976 standard corundum plate measured with Våntec‐2000 showing the K

α1

‐K

α2

split in a peak at a 2

θ

of 35.2°.

Figure 4.27 X‐ray image from a homogeneous X‐ray source: (a) MWPC (Hi‐Star™) image showing stripes in horizontal and vertical directions, (b) mikrogap detector (Våntec‐500™) image showing homogeneous intensity distribution.

Figure 4.28 Comparison of six types of detectors (MWPC, IP, CCD, Mikrogap, HCP and CPAD) in terms of their effective DQE as a function of incoming X‐ray fluence.

Chapter 05

Figure 5.1 D8™ goniometer with two main axes in vertical orientation (Bruker AXS).

Figure 5.2 Various sample stages (Bruker AXS): (a) Fixed‐chi, (b) two‐position chi, (c) XYZ, (d) quarter‐circle Eulerian cradle.

Figure 5.3 Sphere of confusion (SoC) from the contributions of

φ

axis and

ψ

axis in a quarter‐circle Eulerian cradle.

Figure 5.4 The sphere of confusion (SoC) from the contributions of

ψ

axis in a quarter‐circle Eulerian cradle has three components.

Figure 5.5 Angular accuracy and precision in Eulerian geometry.

Figure 5.6 Counterbalance for the inner and outer circles of a vertical

θ‐θ

goniometer (Bruker AXS).

Figure 5.7 Laser video sample alignment system: (a) the working principle of laser‐video alignment system, (b) image of laser spot and crosshair on a gold connection pad, (c) picture of the laser video system (Bruker AXS).

Figure 5.8 Domed hot stage (Anton Paar DHS 900™) mounted on the diffractometer (Bruker AXS D8 Discover GADDS™).

Figure 5.9 Hot stage calibration: (a) the hot stage (Huber™) mounted on the diffractometer (Bruker AXS D8 Discover GADDS™) with a thermocouple mounted in the sample position, (b) schematic showing the positions of the calibration thermocouple.

Figure 5.10 Temperature calibration: temperature plot and the difference.

Chapter 06

Figure 6.1 X‐ray image from isotropic source: (a) raw image used for flood‐field correction, (b) image collected with flood‐field correction (Bruker Hi‐Star™ MWPC).

Figure 6.2 Geometry of the spatial correction for a mikrogap detector (Bruker VånteC‐2000).

Figure 6.3 Process of spatial correction: (a) pinhole pattern in a fiducial plane (projected detector plane), (b) image with fiducial spots and spline curves showing the distortion, (c) interpolated curves defining the area of pixel projection, (d) subpixels in the contributing pixels.

Figure 6.4 Spatial correction of a MWPC (Bruker Hi‐Star™): (a) raw frame with distorted array of fiducial spots, (b) indexed fiducial spot, (c) correction table transforming pixel

x

and

y

coordinates from raw to corrected values, (d) correction table transforming pixel

x

and

y

coordinates from corrected to raw values, (e) fiducial spots in the corrected frame.

Figure 6.5 Spatial correction of a mikrogap area detector (Bruker Våntec‐2000™): (a) raw frame with distorted array of fiducial spots, (b) indexed fiducial spot, (c) correction table transforming pixel

x

and

y

coordinates from raw to corrected values, (d) correction table transforming pixel

x

and

y

coordinates from corrected to raw values, (e) fiducial spots in the corrected frame.

Figure 6.6 Detector position in the laboratory coordinates and the six parameters determining the tolerance of the detector position.

Figure 6.7 Detector calibration with diffraction pattern of corundum with the calculated rings (purple lines) on top of the measured diffraction rings.

Figure 6.8 At 2

θ

 = 90°, the diffraction cone becomes a plane perpendicular to the incident X‐ray beams.

Figure 6.9 The 2D frames collected from corundum at 2

θ

near 90°: (left) minor or no roll error, (right) visible roll error.

Figure 6.10 A single diffraction cone from one detector position (left), intersections between three diffraction cones with different swing angles (right).

Figure 6.11 (a) Diffraction pattern collected from silver behenate, (b) intersections from silver behenate rings at three swing angles, (c) single diffraction ring collected from a corundum in transmission mode, (d) intersections with the overlapping of patterns collected at many swing angles.

Figure 6.12 Measured diffraction rings from corundum at two swing angles (a) and (b) (middle) and the intersection grid by overlapping multiple frames at various swing angles (right).

Figure 6.13 Relationship between intersection points and detector orientation errors (roll, pitch, yaw).

Figure 6.14 A 2D frame from corundum powder: (a) the

γ

‐integration within a “wedge” region from 60° to 120°, (b) the

γ

‐integration with a slice region of 200 pixels.

Figure 6.15 Illustration of integration by Bresenham algorithm: (a) conic line approximated by a series of pixels labeled with a dark dot inside the circle, (b) modified algorithm calculated the arc length for each pixel.

Figure 6.16 Illustration of integration by bin method: (a) conic line and Δ2

θ

define the region to be integrated, (b) each contribution pixel is divided into two or three bins, (c) each pixel is divided into subpixels.

Figure 6.17 Merging of multiple frames collected with a flat 2D detector: (a) illustration of three detector swing angles, (b) merged 2D pattern from three flat frames.

Figure 6.18 Projection of multiple frames collected with a flat 2D detector onto a cylindrical surface.

Figure 6.19 Projection of single frame collected with a flat 2D detector onto a cylindrical surface: (a) projection geometry, (b) flattened image of the projection.

Figure 6.20 Geometry and algorithms to project flat 2D image onto the cylindrical surface.

Figure 6.21 Pixel‐to‐pixel projection from the flat 2D image onto the flattened cylindrical image.

Figure 6.22 Merging of overlapping region: (a) detector with round active area, (b) detector with rectangular active area, (c) top view of two detection planes, (d) weighting factor to combine overlapping pixels.

Figure 6.23 Merged cylindrical image from three frames collected with a Bruker Photon II™ detector from 1 µm Al

2

O

3

powder.

Figure 6.24 The 2D detector scans along the detection circle while collecting diffraction signals.

Figure 6.25 Smearing effect when the sequential 2D frames are simply superposed from the original rectangular frames.

Figure 6.26 Accurate diffraction rings without smearing effect when the cylindrical projections from sequential 2D frames are combined.

Figure 6.27 Exposure time of scanned 2D image and ramping up and ramping down regions.

Figure 6.28 The 2D diffraction pattern collected from corundum by scanning an Eiger2 detector in

γ

‐optimized mode within 2

θ

range of 20° to 80°.

Figure 6.29 Illustration of geometric relationship between the monochromator and the detector in the laboratory coordinates,

X

L

Y

L

Z

L

.

Figure 6.30 Geometry of the window absorption correction.

Figure 6.31 Absorption correction of flat slab: (a) reflection, (b) transmission.

Chapter 07

Figure 7.1 Diffraction pattern merged from three 2D frames collected from corundum and integrated diffraction profile.

Figure 7.2 Geometry aberration of XRD

2

in reflection mode.

Figure 7.3 Defocusing effects: (a) 5° incident angle with cylindrical detector, (b) various incident angles (5°, 15°, 25°, 35°) and detector swing angles (10°, 30°, 50°, 70°), (c) comparison of defocusing factors.

Figure 7.4 Geometry aberration of XRD

2

in transmission mode.

Figure 7.5 Diffraction pattern from corundum: (a) reflection mode diffraction with 5° incident angle, (b) transmission mode diffraction with perpendicular incident beam.

Figure 7.6 Angular window of instrument from the incident beam convergence.

Figure 7.7 Relation between the

γ

‐range, Δ

γ

, and the virtual oscillation angle, Δ

ψ

.

Figure 7.8 Diffraction frames from Si powder (NIST SRM 640c) collected on Bruker GADDS in transmission mode: (a) sample in still position, (b) sample oscillation in an area of ΔX = ΔY = 1mm

2

, (c) sample angular oscillation with Δ

ω

 = ±10°.

Figure 7.9 Relationship between the diffraction vector and pole density function for Bragg–Brentano geometry and XRD

2

geometry.

Figure 7.10 Fiber texture plot of the peak (111), (200), and (220) calculated and measured from a hot extruded rod of Cu‐Be alloy.

Chapter 08

Figure 8.1 The definition of (a) pole and (b) diffraction peak intensity change due to texture and peak shift due to stress.

Figure 8.2 Diffraction cone distortion due to stress and intensity variation along

γ

due to texture.

Figure 8.3 (a) Definition of pole direction angles

α

and

β

, (b) stereographic projection in pole‐figure.

Figure 8.4 Comparison between the pole‐figure measurement with the conventional X‐ray diffraction and two‐dimensional X‐ray diffraction.

Figure 8.5 Data collection strategy: (a) a scheme generated at 2

θ

 = 40°,

ω

 = 20°,

ψ

 = 35.26° has a hole in the center, (b) a modified scheme with

ω

 = 23°,

ψ

 = 30° covers the center of pole‐figure.

Figure 8.6 Texture analysis system and 2D frame: (a) Cu thin film sample mounted on a GADDS™ system (Bruker AXS), (b) each frame contains three Cu lines and a Si spot.

Figure 8.7 Data collection strategy for Cu thin films with two

φ

scans: (a) (111) pole‐figure with strategy A+B, (b) (200) with A+B, (c) (220) with A+B, (d) (220) with A+C.

Figure 8.8 Data collection strategy for Al with a combination of a

ϕ

scan and an

ω

scan.

Figure 8.9 Effect of wrong

ϕ

rotation direction on pole‐figure mapping.

Figure 8.10 Texture measurement with transmission mode diffraction.

Figure 8.11 Texture measurement in transmission mode on an aluminum plate: (a) 2D diffraction frame, (b) data collection scheme for (200) pole-figure, (c) (111) pole‐figure, (d) (200) pole‐figure, (e) (220) pole‐figure.

Figure 8.12 Comparison of data collection scheme: (a) strategy with area detector, (b) strategy with point detector.

Figure 8.13 Pole‐figure data process: (a) a frame from Al sample with the 2

θ

‐integration ranges for (220) ring, (b) 2

θ

profile showing the background and peak, (c) integrated intensity distribution as a function of

γ

.

Figure 8.14 Pole‐figure processes: (a) the 2

θ

‐integrated pole density mapped to the pole‐figure, (b) interpolation within five pixel half‐width box, (c) second interpolation with the same box size, (d) symmetry process with Laue symmetry

mmm

.

Figure 8.15 Eulerian angles and space: (a) Eulerian angles defining the orientation of the crystal coordinates in the sample coordinates, (b) Eulerian space given by the three Eulerian angles in Cartesian coordinates.

Figure 8.16 ODF of the aluminum plate calculated from the pole‐figures (111), (200), and (222).

Figure 8.17 Pole‐figure calculation from ODF: (a) Contour plots of three measured pole‐figures, (b) recalculated pole‐figures of the same planes from ODF, (c) calculated pole‐figures of unmeasured crystallographic planes.

Figure 8.18 Pole‐figures and ODF of fiber texture: (a) 3D surface plots of the three measured pole‐figures, (b) contour plots of the three measured pole‐figures, (c) ODF at four cross‐sections, (d) calculated pole‐figures from the ODF.

Figure 8.19 Normalized ODF of fiber texture of a Cu‐Be alloy sample: (a) in contour plot, (b) in 3D surface plot.

Figure 8.20 Pole‐figures of biaxially oriented BOPE films after various biaxial drawings.

Figure 8.21 Calculated (002) pole‐figures from ODF for various biaxial strains.

Figure 8.22 Combined pole‐figure of Cu film (111) and substrate Si (400): (a) pole‐figure in 2D projection, (b) pole‐figure as a 3D surface plot.

Figure 8.23 2D frames and (111) pole‐figures collected from two γ‐TiAl alloy samples with different microstructures: (a) and (c) larger grain and weak texture, (b) and (d) fine grain and strong texture.

Chapter 09

Figure 9.1 Stress and strain: (a) a force applied to an area

A

, (b) stress components on a volume element, (c) stress ellipsoid and principal stresses, (d) strain components on a volume element.

Figure 9.2 Illustration of the three kinds of residual stresses relative to the grain size.

Figure 9.3 Illustration of the strain measurement based on Bragg's law.

Figure 9.4 Schematic showing the strain measured by X‐ray diffraction in the sample coordinates.

Figure 9.5 Stress measurement from the

‐

plot: (a) linear when

, (b)

ψ

‐split due to shear

, (c) fluctuation due to texture, (d) curve due to stress or composition gradient.

Figure 9.6 Stress measurement from the measured

d

vs

relations: (a)

plot for

, (b)

plot for

, (c)

plot for

, (d)

plot for

.

Figure 9.7 The

ψ

‐tilt and depth of penetration: (a)

ψ

‐tilt iso‐inclination (

ω

‐rotation) or side‐inclination (

ψ

‐rotation), (b) depth of penetration at different inclination modes.

Figure 9.8 Experimental example of

method with area detector: (a) a spring with laser spot on inside surface, (b) alignment by laser‐video system, (c) 2D diffraction frame with shadow of the spring wire. (d) The

d

‐

curve.

Figure 9.9 Diffraction cone distortion due to stresses.

Figure 9.10 The diffraction cones and diffractometer plane.

Figure 9.11 Measured biaxial stress tensor and pseudo‐hydrostatic stress as a function of input

.

Figure 9.12 Simulated diffraction ring distortion due to stresses in radar chart (top) and 2

θ

vs.

γ

plot (bottom).

Figure 9.13 Simulated diffraction ring distortion in radar chart: (a) equibiaxial with

ψ

scans, (b) equibiaxial with

ω

scan, (c) uniaxial with

ψ

scans, (d) uniaxial with

ω

scan.

Figure 9.14 Effect of wrong

φ

rotation direction on stress results and principal tress orientation.

Figure 9.15 Two‐dimensional XRD system for stress analysis in vertical

θ‐θ

configuration (Bruker D8 Discover™).

Figure 9.16 Diffraction vector distribution in stress measurement with conventional method and 2D method.

Figure 9.17 Data collection strategy schemes: (a)

ω

scan, (b)

ψ+φ

(180°) scan, (c)

ω+φ

(90°) scan, (d)

ψ+φ

(45°)scan, (e)

ω+φ

(45°) scan, (d)

ψ+φ

(90°+135°) scan.

Figure 9.18 Data integration for stress measurement.

Figure 9.19 Pearson VII function with

, 2, or ∞.

Figure 9.20 Quality of diffraction profiles: (a) strong texture, (b) large grain sizes.

Figure 9.21 Stress‐free 2

θ

values measured at various directions.

Figure 9.22 Number of diffraction contributing crystallites: (a) point detector, (b) area detector.

Figure 9.23 Stress calculation with 2D method and

method: (a) data points taken from the diffraction ring, (b) measured stress and standard deviation by different methods and from various numbers of data points.

Figure 9.24 Virtual oscillation by

γ

‐integration over Δ

γ

 = 20° on a spotty diffraction ring taken from an SS304 stainless steel plate produces a smooth diffraction profile.

Figure 9.25 (a) Schematic illustration of the friction stir welding process, (b) specimen of friction stir welded aluminum alloy.

Figure 9.26 (a) Specimen loaded on the XYZ stage of Eulerian cradle and mapping spot is aligned with the laser‐video system, (b) magnified image of the mapping area with the laser spot pointing to the instrument center.

Figure 9.27 Diffraction frames collected at three typical regions: (a) original material, (b) friction‐stirred region; and (c) mixture of both.

Figure 9.28 Residual stress mapping on friction stir welded Al alloy plate: (a)

σ

22

on the top surface within 40 mm from the weld center line of both specimens, (b)

σ

22

on both the top surface and the bottom surface from the specimen edge to the weld center line.

Figure 9.29 Diffraction vector distribution in the sample coordinates for the sin

2

ψ

method (in purple) and for the XRD

2

method with low incident angles for high and low 2

θ

angle peaks (in red).

Figure 9.30 Anatase (101) and Ti (101) peaks collected at 15° incident angle used for stress measurement.

Figure 9.31 Measured average stress value

for each measurement depth

and the corresponding calculated stress value

at the depth

.

Figure 9.32 (a) Illustration of frame with multiple {

hkl

} rings, (b) diffraction frame of Cu film containing (331) and (420) rings.

Figure 9.33 Data integration region and stress calculation settings with LEPTOS software (Bruker AXS).

Figure 9.34 Stresses of Cu film measured at various loading strains.

Figure 9.35 Diffraction vector distribution for 0D and 2D detectors.

Figure 9.36 Diffraction vector distribution range Δ

ψ

as a function of 2

θ

: (a) for flat 2D detector and cylindrical detector, (b) for Eiger 2R 500k™ detector in

γ

‐optimized orientation.

Figure 9.37 Data collection strategy schemes with single tilt at

ψ

 = 22.5° and complete

φ

rotation of 45° steps: (a) PE polymer (020), 2

θ

 = 36.3° and

D

 = 20 cm, (b) Al

2

O

3

(116), 2

θ

 = 57.5° and

D

 = 15 cm.

Figure 9.38 Data evaluation setting with LEPTOS software for 1 µm thick Al

2

O

3

coating on cutting insert.

Figure 9.39 Data evaluation results with LEPTOS software for 1 µm thick Al

2

O

3

coating on cutting insert.

Figure 9.40 Data evaluation setting and fitting results with LEPTOS software for the HDPE pipe.

Figure 9.41 Stress values calculated by the conventional method and the 2D method from the ten samples, averaged over nine measurements for each sample.

Chapter 10

Figure 10.1 Pinhole collimation for SAXS: (a) conventional collimation with three pinholes, (b) collimation with two scatterless pinholes.

Figure 10.2 Attachments for SAXS: (a) beamstop attached to Hi‐Star detector, (b) helium beam path, (c) 2D frame with air scatter background (left) and with helium beam path (right), (d) integrated scattering intensities showing the air scattering background.

Figure 10.3 Two‐dimensional SAXS system (Bruker AXS Nanostar™): (a) side view, (b) top view showing complete beam path.

Figure 10.4 Diffraction pattern from silver behenate as a low‐angle calibration material and calibration ring.

Figure 10.5 The 2D SAXS data frame integration (a)

γ

‐integration of a frame from rat tail tendon, (b) 2

θ

‐integration of a frame from plate‐shaped iron‐oxide precipitates inside single crystal Cu (reproduced from reference [49] with permission).

Figure 10.6 SAXS from solution samples: (a) scattering plot of spherical gold colloidal particles in toluene, (b) size distribution of gold particle, (c) scattering plot of protein in water, (d) distance distribution function of the protein in water. (Reproduced from Bruker AXS Application Notes [51] with permission.)

Figure 10.7 Scanning SAXS: (a) SAXS system with XY stage, (b) radiography and SAXS patterns from selected spots (reproduced from Bruker Application Notes with permission [50]).

Figure 10.8 modified beamstop in which the transmitted direct beam from the sample passes the beamstop through an attenuator and the transmission size is limited by the pinhole.

Figure 10.9 Center portion of the frame collected with six layers of the oriented polyester films with magnification of 8×. The bright spot inside the shadow of the beamstop represents the transmission.

Figure 10.10 Transmission coefficient as a function of sample thickness (number of 0.2 mm polyester layers).

Figure 10.11 Schematics of the vertical small angle X‐ray scattering system.

Chapter 11

Figure 11.1 The XRD

2

system for combinatorial screening: (a) schematics, (b) Bruker D8 Discover GADDS™ CS.

Figure 11.2 Grid points are determined by the starting and ending points and the steps.

Figure 11.3 Screening parameters displayed against the cells in the material library plate.

Figure 11.4 Defocusing and cross‐cell contamination with XRD

2

screening in reflection mode.

Figure 11.5 Motorized retractable knife‐edge designed for the Bruker D8 Discover with GADDS for combinatorial screening.

Figure 11.6 Knife‐edge defining the irradiated area of diffraction.

Figure 11.7 Diffraction profiles from corundum power without knife‐edge (blue line) and with knife‐edge (red line).

Figure 11.8 Diffraction profiles from the combinatorial screening verification fixture (1 mm Cu wire with 5 mm separation): (a) cross contamination without the knife‐edge, (b) no cross contamination with knife‐edge.

Figure 11.9 XRD

2

system for combinatorial screening in transmission mode: (a) schematic, (b) Bruker D8 Discover GADDS™ CST.

Figure 11.10 Schemes of various material library plates with: (a) no transmission beam path, (b) transmission path for beam‐up configuration, (c) transmission path for beam‐down configuration, (d) transmission path for both beam‐up and beam‐down configurations.

Figure 11.11 One system with automated reflection (a) and transmission (b) conversion (Bruker AXS D8 Discover™ HTS).

Figure 11.12 Combinatorial screening system (Bruker AXS D8 ScreenLab™) and XRD pattern, Raman pattern, and optical image of Carbamazepine powder.

Chapter 12

Figure 12.1 (a) Amorphous scattering, (b) random polycrystalline scattering, (c) oriented polycrystalline and amorphous scattering.

Figure 12.2 (a) Integrated diffraction profile showing the sharp crystalline peaks and broad amorphous background, (b) separation of the sharp crystalline peaks and amorphous background.

Figure 12.3 2D diffraction pattern from an oriented polycrystalline polymer sample, (a) diffraction profile integrated from a horizontal region analogous to a profile collected with point detector, (b) diffraction profile integrated from all part of the 2D frame.

Figure 12.4 (a) Nylon frame with air scatter, (b) air scatter frame without sample, (c) nylon frame with air scatter subtracted.

Figure 12.5 (a) Internal method for

γ

‐nylon powder, (b) external method for

γ

‐nylon fiber.

Figure 12.6 Schematic of rolling ball background subtraction.

Figure 12.7 Percent crystallinity data process with a frame collected from a nylon fiber: (a) original frame with air scatter removed, (b) smoothed frame with air scatter removed, (c) amorphous background frame generated from frame (b), (d) crystalline scattering.

Figure 12.8 2D frames and

γ

‐integrated profile for crystallite size analysis: (a) NIST SRM 660 (LaB

6

), (b) Cu (111) peak from a semiconductor tab tape.

Figure 12.9 Atomic force microscope (AFM) images and XRD

2

patterns of samples crystallized from organic glass with large and small particle size. (Reproduced from [42] with permission.)

Figure 12.10 Instrumental window with two‐dimensional diffraction.

Figure 12.11 Relative variation of effective sampling volume as a function of 2

θ

and

γ

.

Figure 12.12 Three diffraction lines collected from SRM660a (LaB

6

) illustrate the

γ

‐profile analysis for crystallite size analysis.

Figure 12.13 Retained austenite measurement from a steel roller: (a) 2D frame collected by CCD detector, (b)

γ

‐integrated profile showing three peaks, (200), (220), and (311), from the retained austenite and two peaks, (200) and (211), from the martensite.

Figure 12.14 Orientation of the crystalline planes with respect to the sample orientation.

Figure 12.15 Measurement of miscut angle: (a) geometry, (b) 2D frame with two spots.

Figure 12.16 (a) GIXRD with a point detector, (b) GIXRD with a 2D detector.

Figure 12.17 Grazing incidence X‐ray diffraction with a 2D detector (GIXRD

2

): (a) standard geometry, (b) in‐plane geometry.

Figure 12.18 Diffraction data collected on 10 nm polycrystalline NiSi file on Si wafer with (a) OP‐GIXRD

2

, (b) IP‐GIXRD

2

, and (c) comparison of IP‐GIXRD

2

and OP‐GIXRD

1

(reproduced from [58] with permission).

Figure 12.19 Thin film thickness measurement with 2D detector. (a) 2D XRR pattern and (b) thickness evaluation from the integrated profiles. (Reproduced from [58] with permission.)

Figure 12.20 Reciprocal space mapping with a 2D detector.

Figure 12.21 Three‐dimensional RSM constructed by MAX3D with 2D frames collected at various sample rotation angles: (a) random corundum powder, (b) Au/Pt nanolayered sheet with rolled texture, (c) GaAs nanowires on Si substrate, (d) epitaxial thin film with texture on substrate, (e) extruded, distorted polypropylene, and (f) subsection of (e) with higher resolution. (Reproduced from [67, 68] with permission.)

Figure 12.22 Integrated RSM from 200 nm BiFeO

3

thin film on (001) SrTiO

3

substrate. (Reproduced from [69] with permission.)

Chapter 13

Figure 13.1 Geometrical features of two‐dimensional diffraction by scanning line detector.

Figure 13.2 Line detector position and a point of line detector in the diffraction space.

Figure 13.3 Illustration of a two‐dimensional diffraction image.

Figure 13.4 Air scatter from incident X‐ray beam and diffracted X‐rays is blocked by a scatter shield for the line detector.

Figure 13.5 A multilayer mirror used as a monochromator in front of the scanning line detector.

Figure 13.6 Geometry of a conceptual three‐dimensional detector.

Figure 13.7 Comparison of detectors with Ewald sphere: (a) point, line, and area detectors, (b) three‐dimensional detector.

Figure 13.8 High resolution XRD

2

in reciprocal space and the Ewald sphere.

Figure 13.9 Two‐dimensional X‐ray diffractometer with left‐hand goniometer in horizontal 2

θ‐β

configuration.

Figure 13.10 High resolution two‐dimensional X‐ray diffractometer in (a) horizontal 2

θ‐β

configuration at off‐axis position (top view), (b) horizontal 2

θ‐β

configuration at on‐axis position (top view), (c) vertical 2

θ‐β

configuration (side view).

Figure 13.11 (a) X‐ray transparent window for the incident X‐ray beam, (b) coupling of the collimator inside the sample chamber and X‐ray optics outside of the chamber.

Figure 13.12 Various designs for the beamstop: (a) fixed to the center, (b) strip beamstop, (c) motorized beamstop.

Guide

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TWO-DIMENSIONAL X-RAY DIFFRACTION

BOB BAOPING HE

Second Edition

Copyright

This edition first published 2018

© 2018 John Wiley & Sons, Inc.

Edition History

John Wiley & Sons, Inc. (1e, 2009)

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.

The right of Bob Baoping He to be identified as the author of this work has been asserted in accordance with law.

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Library of Congress Cataloging–in–Publication Data

Names: He, Bob B., 1954– author.

Title: Two–dimensional x–ray diffraction / by Bob Baoping He.

Description: Second edition. | Hoboken, NJ : John Wiley & Sons, 2018. |

Includes bibliographical references and index. |

Identifiers: LCCN 2018001011 (print) | LCCN 2018007081 (ebook) | ISBN

9781119356066 (pdf) | ISBN 9781119356097 (epub) | ISBN 9781119356103

(cloth)

Subjects: LCSH: X–rays–Diffraction. | X–rays–Diffraction–Experiments. |

X–rays–Diffraction–Industrial applications.

Classification: LCC QC482.D5 (ebook) | LCC QC482.D5 H4 2018 (print) | DDC

548/.83–dc23

LC record available at https://lccn.loc.gov/2018001011

Cover Design: Wiley

Cover Image: Image courtesy of Bob B. He

Preface

Two‐dimensional X‐ray diffraction is the ideal non‐destructive analytical method for characterizing many types of materials, such as metals, polymers, ceramics, semiconductors, thin films, coatings, paints, biomaterials, and composites. It has been widely used for material science and engineering, drug discovery and processing control, forensic analysis, archeological analysis, and many emerging applications. In the long history of powder X‐ray diffraction, data collection and analysis have been based mainly on one‐dimensional diffraction profiles measured with scanning point detectors or linear position‐sensitive detectors. Therefore, almost all X‐ray powder diffraction applications – such as phase identification, texture, residual stress, crystallite size, and percent crystallinity – are developed in accord with the one‐dimensional diffraction profiles collected by conventional diffractometers. In recent years, use of two‐dimensional detectors has dramatically increased due to the advances in detector technology. A two‐dimensional diffraction pattern contains abundant information about the atomic arrangement, microstructure, and defects of a solid or liquid material. Because of the unique nature of the data collected with a two‐dimensional detector, many algorithms and methods developed for conventional X‐ray diffraction are not sufficient or accurate for interpreting and analyzing the data. New concepts and approaches are necessary for designing a two‐dimensional diffractometer and for understanding and analyzing two‐dimensional diffraction data. In addition, the new theory should also be consistent with conventional theory because two‐dimensional X‐ray diffraction is also a natural extension of conventional X‐ray diffraction.

The purpose of this book is to give an introduction to two‐dimensional X‐ray diffraction. Chapter 1 gives a brief introduction to X‐ray diffraction and its extension to two‐dimensional X‐ray diffraction. Discussion of the general principles of crystallography and X‐ray diffraction is kept to a minimum since many books on the subjects are available. Chapter 2 describes the geometry conventions and diffraction vector analysis which establish the foundation for the subjects discussed in the following chapters. A diffraction vector approach is used for deriving many fundamental equations for various applications in the following chapters. The critical steps are included so that readers can use similar approach for any analysis not included in the book. Chapters 3 to 6 focus on the instrumentation technologies, including X‐ray source and optics, detectors, goniometers, system configurations and basic data collection, and evaluation algorithms. Chapters 7 to 12 cover the basic concepts, fundamental theory, diffractometer configurations, data collection strategy, data analysis algorithms, and experimental examples for various applications, such as phase identification, texture, stress, microstructure analysis, crystallinity, thin film analysis, and combinatorial screening. Chapter 13 presents some ideas on innovations and future development.

Since publication of the first edition of this book in 2009, there has been remarkable progress in the instrumentation and applications of two‐dimensional X‐ray diffraction. I have received many constructive reviews, comments, and suggestions from readers. In addition to the recent progress in instrumentation and applications, the most important improvement in the second edition is to have most figures in full color, so that different elements in illustrations and the details of diffraction patterns can be better revealed.

I would like to express my sincere appreciation to Professors Mingzhi Huang, Huijiu Zhou and Jiawen He, Charles Houska, Guoquan Lu, and Robert Hendricks for their guidance, assistance, and encouragement in my education and career development. I wish to acknowledge the support, suggestions, and contributions from my colleagues, especially from Kingsley Smith, Uwe Preckwinkel, Roger Durst, Yacouba Diawara, John Chambers, Gary Schmidt, Peter LaPuma, Lutz Brügemann, Frank Burgäzy, Martin Haase, Mark Depp, Hannes Jakob, Kurt Helmings, Arnt Kern, Geert Vanhoyland, Alexander Ulyanenkov, Jens Brechbuehl, Ekkehard Gerndt, Hitoshi Morioka, Keisuke Saito, Susan Byram, Michael Ruf, Charles Campana, Joerg Kaercher, Bruce Noll, Delaine Laski, Beth Beutler, Rob Hooft, Alexander Seyfarth, Joseph Formica, Richard Ortega, Brian Litteer, Bruce Becker, Detlef Bahr, Heiko Ress, Kurt Erlacher, Christian Maurer, Olaf Meding, Christoph Ollinger, Kai‐Uwe Mettendorf, Joachim Lange, Martin Zimmermann, Hugues Guerault, Ning Yang, Hao Jiang, Jon Giencke, and Brain Jones. I would also like to thank Robert Cernik, Shepton Steve, Christian Lehmann, George Kauffman, Gary Vardon, Werner Massa, Joseph Reibenspies, and Nattamai Bhuvanesh for spending their valuable time to publish book reviews of the first edition, which encouraged and helped me to make many corrections and improvements in the second edition.

I am grateful to those who have so generously contributed their ideas, inspiration, and insights through many thoughtful discussions and communications, particularly to Thomas Blanton, Davor Balzar, Camden Hubbard, James Britten, Joseph Reibenspies, Timothy Fawcett, Scott Misture, James Kaduk, Ralph Tissot, Mark Rodriguez, Matteo Leoni, Herbert Göbel, Scott Speakman, Thomas Watkins, Jian Lu, Xun‐Li Wang, John Anzelmo, Brian Toby, Ting Huang, Alejandro Navarro, Peter Zavalij, Mario Birkholz, Kewei Xu, Berthold Scholtes, Chang‐Beom Eom, Gregory Stephenson, Raj Suryanarayanan, Shawn Yin, Naveen Thakral, Lian Yu, Siddhartha Das, Chris Frampton, Chris Gilmore, Keisuke Tanaka, Wulf Pfeiffer, Dierk Raabe, Robert Snyder, Jose Miguel Delgado, Winnie Wong‐Ng, Xiaolong Chen, Chuanhai Jiang, Wenhai Ye, Weimin Mao, Leng Chen, Kun Tao, Erqiang Chen, Danmin Liu, Dulal Goldar, Vincent Ji, Peter Lee, Yan Gao, Lizhi Liu, Yujing Tang, Minqiao Ren, Ying Shi, Chunhua Tony Hu, Shaoliang Zheng, Ravi Ananth, Philip Conrad, Linda Sauer, Roberta Flemming, Chan Park, Dongying Ju, Milan Gembicky, Hui Zhang, Willard Schultz, Licai Jiang, Ning Gao, Fangling Needham, and John Faber. I am particularly indebted to my wife Judy for her patience, care, and understanding, and to my son Mike for his support.

Serving as a scientist and as director of R&D and business development for over 20 years for Bruker AXS, an industry leader in X‐ray diffraction instrumentation and solutions, gives me the opportunity to meet many scientists, engineers, professors, and students working in the field of X‐ray diffraction, and the necessary resources to put many ideas into practice. The many pictures and experimental data in this book are collected from diffractometers manufactured by Bruker AXS Inc. This should not be construed as an endorsement of a particular vendor, but rather a convenient way to illustrate the ideas contained in this book. The approaches and algorithms suggested in the book are not necessarily the best alternatives, and some errors may exist due to my mistakes. A list of references is included in each chapter, but I apologize for missing any original and important references due to my oversight. I welcome any and all comments, suggestions, and criticisms.

BOB BAOPING HE

Chapter 1Introduction

1.1 X‐Ray Technology, a Brief History

X‐ray technology has more than a hundred years of history and its discovery and development have revolutionized many areas of modern science and technology [1]. X‐rays were discovered by the German physicist Wilhelm Conrad Röntgen in 1895, who was honored with the Noble prize for physics in 1901. In many languages today X‐rays are still referred to as Röntgen rays or Röntgen radiation. This mysterious light was found to be invisible to human eyes, but capable of penetrating opaque object and expose photographic films. The density contrast of the object is revealed on the developed film as a radiograph. Since then X‐rays have been developed for medical imaging, such as for detection of bony structures and diseases in soft tissues like pneumonia and lung cancer. X‐rays have also been used to treat disease. Radiotherapy employs high energy X‐rays to generate a curative medical intervention to the cancer tissues. A recent technology, tomotherapy, combines the precision of a computerized tomography scan with the potency of radiation treatment to selectively destroy cancerous tumors while minimizing damage to surrounding tissue. Today, medical diagnoses and treatments are still the most common use of X‐ray technology.

The phenomenon of X‐ray diffraction by crystals was discovered in 1912 by Max von Laue. The diffraction condition in a simple mathematical form, which is now known as Bragg's law, was formulated by Lawrence Bragg in the same year. The Nobel Prize in Physics in consecutive two years (1914 and 1915) was awarded to von Laue and the elder and junior Braggs for the discovery and explanation of X‐ray diffraction. X‐ray diffraction techniques are based on elastic scattered X‐rays from matter. Due to the wave nature of X‐rays, the scattered X‐rays from a sample can interfere with each other, such that the intensity distribution is determined by the wavelength and the incident angle of the X‐rays and the atomic arrangement of the sample structure, particularly the long range order of crystalline structures. The expression of the space distribution of the scattered X‐rays is referred to as an X‐ray diffraction pattern. The atomic level structure of the material can then be determined by analyzing the diffraction pattern. Over its hundred year history of development, X‐ray diffraction techniques have evolved into many specialized areas. Each has its specialized instruments, samples of interest, theory, and practice. Single‐crystal X‐ray diffraction (SCD) is a technique used to solve the complete structure of crystalline materials, typically in the form of a single crystal. The technique started with simple inorganic solids and grew into complex macromolecules. Protein structures were first determined by X‐ray diffraction analysis by Max Perutz and Sir John Cowdery Kendrew in 1958, and both shared the 1962 Nobel Prize in Chemistry. Today, protein crystallography is the dominant application of SCD. X‐ray powder diffraction (XRPD), alternatively called powder X‐ray diffraction (PXRD), got its name from the technique of collecting X‐ray diffraction patterns from packed powder samples. Generally, X‐ray powder diffraction involves the characterization of the crystallographic structure, crystallite size, and orientation distribution in polycrystalline samples [2–5].