Two-dimensional X-ray Diffraction

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This edition first published 2018

© 2018 John Wiley & Sons, Inc.

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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

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

X–rays–Diffraction–Industrial applications.

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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 1

Introduction

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].

X‐ray diffraction (XRD), by definition, covers single crystal diffraction and powder diffraction as well as many X‐ray diffraction techniques. However, it has been accepted as convention that SCD is distinguished from XRD. By this practice, XRD is commonly used to represent various X‐ray diffraction applications other than SCD. These applications include phase‐identification, texture analysis, stress measurement, percentage crystallinity, particle (grain) size, and thin film analysis. An analogous method to X‐ray diffraction is the small angle X‐ray scattering (SAXS) technique. SAXS measures scattering intensity at scattering angles within a few degrees from the incident angle. SAXS pattern reveals the material structures, typically particle size and shape, in the nanometer to micrometer range. In contrast to SAXS, other X‐ray diffraction techniques are also referred to as wide angle X‐ray scattering (WAXS).

1.2 Geometry of Crystals

Solids can be divided into two categories: amorphous and crystalline. In an amorphous solid, glass for example, atoms are not arranged with long range order. Thus amorphous solids are also referred to as glassy solids. In contrast, a crystal is a solid formed by atoms, molecules, or ions stacking in three‐dimensional space with a regular and repeating arrangement. The geometry and structure of a crystalline solid determines the X‐ray diffraction pattern. Comprehensive knowledge of crystallography has been covered by many books [2, 5–9]. This section gives only some basics to help further discussion on X‐ray diffraction.

1.2.1 Crystal Lattice and Symmetry

A crystal structure can be simply expressed by a point lattice, as shown in Figure 1.1(a). The point lattice represents the three‐dimensional arrangement of the atoms in the crystal structure. It can be imagined as being comprised of three sets of planes, each set containing parallel crystal planes with equal interplane distance. Each intersection of three planes is called a lattice point and may represent the location of an atom, ion, or molecule in the crystal. A point lattice can be minimally represented by a unit cell, highlighted in bold in the bottom left corner. A complete point lattice can be formed by the translation of the unit cell in three‐dimensional space. This feature is also referred to as translation symmetry. The shape and size of a unit cell can be defined by three vectors a, b, and c, all starting from any single lattice point as shown in Figure 1.1(b). The three vectors are called the crystallographic axes of the cell. As each vector can be defined by its length and direction, a unit cell can also be defined by the three lengths of the vectors (a, b, and c) as well as the angles between them (α, β, and γ). The six parameters (a, b, c, α, β, and γ) are referred to as the lattice constants or lattice parameters of the unit cell.

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

One important feature of crystals is their symmetry. In addition to the translation symmetry in point lattices, there are also four basic point symmetries: reflection, rotation, inversion, and rotation‐inversion. Figure 1.2 shows all four basic point symmetries on a cubic unit cell. The reflection plane is like a mirror. The reflection plane divides the crystal into two sides. Each side of the crystal matches the mirrored position of the other side. The cubic structure has several reflection planes. The rotation axes include 2‐, 3‐, 4‐, and 6‐fold axes. A rotation of a crystal about an n‐fold axis by 360°/n will bring it into self‐coincidence. A cubic unit cell has several 2‐, 3‐, and 4‐fold axes. The inversion center is like a pinhole camera, the crystal will maintain self‐coincidence if every point of the crystal is inverted through the inversion center. Any straight line passing through the inversion center intersects with the same lattice point at the same distance at both sides of the inversion center. A cubic unit cell has an inversion center in its body center. The rotation‐inversion center can be considered as a combined symmetry of rotation and inversion.

Figure 1.2 Symmetry elements of a cubic unit cell.

The various relationships among the six lattice parameters (a, b, c, α, β, and γ) result in various crystal systems. The simplest crystal system is cubic system in which all three crystallographic vectors are equal in length and perpendicular to each other ( c01-i0001.png and c01-i0002.png ). Seven crystal systems are sufficient to cover all possible point lattices. The French crystallographer Bravais found that there are a total of 14 possible point lattices. Seven point lattices are given by the seven crystal systems for the case that only one lattice point is in each unit cell and that the lattice point is located in the corner of the unit cell. These seven types of unit cells are called primitive cells and labeled by P or R. By adding one or more lattice points within a unit cell one can create non‐primitive cells depending on the location of the additional lattice points. The location of a lattice point in the unit cell can be specified by fractional coordinates within a unit cell (u, v, w). For example, the lattice point in a primitive cell is (0, 0, 0). Therefore, we can define three types of non‐primitive cells. The label I represents the body centered point lattice, which has one additional lattice point at the center of the unit cell, or can be defined by the fraction (½, ½, ½). The label F represents the face centered point lattice with additional lattice points at the center of unit cell face, or (0, ½, ½), (½, 0, ½), and (½, ½, 0). The label C represents the base centered point lattice with an additional lattice point at the center of the base face (½, ½, 0). All seven crystal systems and 14 Bravais lattices are summarized in Table 1.1. The unit cells of the 14 Bravais lattices are shown in Figure 1.3.

Table 1.1 Crystal Systems and Bravais Lattices

Figure 1.3 Unit cells of the 14 Bravais lattices.

1.2.2 Lattice Directions and Planes

The direction of any line in a crystal lattice can be specified by drawing a line starting from the unit cell origin parallel to the given line and then taking the coordinates (u′, v′, w′) of any point on the line. The coordinates (u′, v′, w′) are not necessarily integers. However, by conventional, (u′, v′, w′) are multiplied by the smallest number that produces integers u, v, and w. The crystal direction is described by putting the three integers in square brackets as [uvw]. The [uvw] are the indices of a specific crystal direction, and each of the indices can take the value of a positive or negative integer. All directions in a crystal that are symmetry equivalent to [uvw] are represented by a notation with the integers in angular brackets as <uvw>. For example, in a cubic crystal all diagonals of the unit cell are symmetry equivalent. So all the directions [111], [ c01-i0019.png 11], [1 c01-i0020.png 1], [11 c01-i0021.png ], [ c01-i0022.png 1], [ c01-i0023.png 1 c01-i0024.png ], [1 c01-i0025.png ], and [ c01-i0026.png ] can be represented by <111>. The bar over the number is for negative indices. Figure 1.4(a) shows some lattice directions and their indices in a unit cell.

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

The orientation of lattice planes can be described by using a set of three integers referred to as Miller indices. Miller indices are the reciprocal intercepts of the plane on the unit cell axes. If the crystal plane makes fractional intercepts of 1/h, 1/k, 1/l with the three crystal axes respectively, the Miller indices are (hkl). If the plane runs parallel to an axis, the intercept is at ∞, so the Miller index is 0. Miller indices describe the orientation and spacing of a family of planes. Figure 1.4(b) shows some lattice planes and their Miller indices in a unit cell. The spacing between adjacent planes in a family is referred to as the d‐spacing. The symbol {hkl} refers to all planes that are symmetry equivalent to (hkl). This group of equivalent planes is referred to as planes of a form. For the cubic system all the planes (100), (010), (001), ( c01-i0027.png 00), (0 c01-i0028.png 0), and (00 c01-i0029.png ) belong to the form {100}. For a tetragonal crystal, c01-i0030.png , only the first two indices imply the same interception distance on the crystal axes, so the form {100} would only include (100), (010), ( c01-i0031.png 00), and (0 c01-i0032.png 0).

Figure 1.5(a) shows the hexagonal unit cell and indices of some directions. It follows the same definition as other lattice types. However, lattice planes are often described by a different system of plane indexing, called Miller–Bravais indices. In hexagonal unit cells it is common to use four axis coordinates, a1, a2, a3, and c, in which a1, a2, a3 are lying in the basal plane and c is perpendicular to all three axes. The indices of a plane in the hexagonal system are written as (hkil). Figure 1.5(b) shows some lattice planes in a hexagonal lattice described by Miller–Bravais indices. Since a1, a2, a3 are symmetry equivalent and 120° apart each other, there are only two independent axes among them. So that the first three values in the Miller–Bravais indices maintain the relation

(1.1)

Since all cyclic permutations of h, k, and i are symmetry equivalent, (10 c01-i0033.png 0), ( c01-i0034.png 100), (0 c01-i0035.png 10) are equivalent.

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.

A zone is defined as a set of non‐parallel planes that are all parallel to one axis. This axis is called the zone axis. Miller indices for all planes in a zone obey the relationship

(1.2)

where [uvw] defines the zone axis and (hkl) are the Miller indices of each plane in the zone. Figure 1.6 shows some of the crystal planes in the cubic lattice that belong to the [001] zone.

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

The distance between two adjacent planes with the same indices is called the interplanar spacing or d‐spacing, which is an important parameter in Bragg's law. The interplanar spacing dhkl is a function of both the plane indices (hkl) and the lattice parameters (a, b, c, α, β, γ). The equations of d‐spacings for all seven crystal systems are listed in Table 1.2. More equations on the unit cell volume and interplanar angles can be found in appendix 3 of ref. [2].

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

1.2.3 Atomic Arrangement in Crystal Structures

Actual crystal structures can be described by the Bravais lattice filled with the same or different kind of atoms. The atoms take either the exact lattice points and/or points with fixed offset to the lattice points. The three most common crystal structures of metals are body‐centered cubic (BCC), face‐centered cubic (FCC), and hexagonal close‐packed (HCP) structures, as shown in Figure 1.7. BCC has two atoms per unit cell located at the coordinates 0 0 0 and ½ ½ ½ respectively. Many metals, such as α‐iron, niobium, chromium, vanadium, and tungsten have BCC structure. FCC has four atoms per unit cell at the coordinates 0 0 0, 0 ½ ½, ½ 0 ½, and ½ ½ 0 respectively. Metals with FCC structure include γ‐iron, aluminum, copper, silver, nickel, and gold. HCP contains three equivalent hexagonal unit cells, each has two atoms at the coordinates 0 0 0 and ⅔ ⅓ ½ (or at equivalent position ⅓ ⅔ ½). Metals with HCP structure include beryllium, magnesium, zinc and α‐titanium. Both FCC and HCP are close‐packed arrangements. Both FCC (111) plane and HCP (0002) have the same atomic arrangement within the plane, but have different stacking sequences.

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

Structures of crystals formed by unlike atoms are built by a Bravais lattice with certain conditions. One is that the translation of a Bravais lattice must begin and end on the atoms of the same kind. The other is that the space arrangement of each kind of atom poses the same symmetry elements as the whole crystal. The structure of NaCl (rock salt) is shown in Figure 1.8. The unit cell of NaCl contains eight ions, located at the following coordinates:

four Na+ ions at 0 0 0, ½ ½ 0, ½ 0 ½, and 0 ½ ½,

four Cl− ions at ½ ½ ½, 0 0 ½, 0 ½ 0, and ½ 0 0.

It can be seen that Na+ ions formed a FCC structure and 4 Cl− ions form an FCC with ½ ½ ½ translation from the Na+ lattice. Therefore, the Bravais lattice of NaCl crystal is face‐centered cubic.

Figure 1.8 The structure of NaCl. Na+ is FCC and Cl− is FCC with ½, ½, ½ translation.

1.2.4 Imperfections in Crystal Structure

Thus far we have assumed that crystals have a very regular atomic arrangement following the crystal structure. However, most crystalline materials are not perfect. The regular pattern of atomic arrangement may be interrupted by crystal defects. There are various types of crystal defects, such as point defects, line defects, planar defects, and bulk defects.

Point defects are defects that involve randomly distributed extra or missing atoms. There is no strict definition of the size of a point defect, but generally a point defect is not extended in space in any dimension, but within a region of one or a few atoms. Vacancies are sites that should be occupied by an atom in a perfect crystal but that are unoccupied. Interstitials are extra atoms inserted between the normal atomic sites. Typically interstitials are significantly smaller atoms compared to the matrix atoms in the crystal, for example, hydrogen, carbon, boron, or nitrogen atoms in metal crystals. Crystals with interstitials are also referred to as interstitial solid solutions. A substitutional solid solution contains another type of point defects – substitutional defects. In a substitutional solid solution of B in A, B atoms replace the sites normally occupied by A atoms. In a typical substitutional solution, B atoms are randomly distributed in the crystal. Under certain conditions, B atoms may replace A atoms in a regular pattern, called long range order. The solution is then called ordered or superlattice structure. Point defects may change the lattice parameters in proportion to the concentration of the defects. Point defects play an important role in semiconductors.

Line defects are defects that extend in one dimension within a region of one or a few atoms in the other two dimensions. Crystal dislocations are line defects. There are two basic types of dislocations, the edge dislocation and the screw dislocation. An edge dislocation is caused by the termination of a plane of atoms in the middle of a crystal, or it can be thought of as the result of adding or subtracting a half crystal plane between two adjacent full crystal planes. A screw dislocation is a line defect along which the atom arrangement is distorted like a screw thread, or it can be thought of as the result of cutting partway through the crystal and displacing it parallel to the edge of the cut. Dislocations can dramatically reduce the energy barrier to shearing a crystal along a crystal plane, so that the density of dislocations in a crystal can change the resistance of the crystal to plastic deformation.

Plane defects are crystal defects that extend in two dimensions and within a region of one or a few atoms in the third dimension. Grain boundaries are interfaces between contacting crystals that have different orientations. Depending on the degree of misorientation between the two contacting crystals, grain boundaries are categorized as low angle grain boundaries and high angle grain boundaries. The difference between low angle grain boundary misorientation and high angle grain boundary misorientation varies in the range of 10–15° depending on the material. The structure and property of low angle grain boundaries have a strong dependence on the misorientation angle, while high angle grain boundaries are not dependent on the misorientation. Antiphase boundaries are another type of plane defect existing in ordered alloys. The crystals on both sides of the boundary have the same structure and orientation with the interruption of the order by removing or adding a layer of atoms. For example, if the ordering is in the sequence of ABABABAB an antiphase boundary takes the form of ABABBABA or BABAABAB. Stacking faults are another type of plane defect. Stacking faults commonly occur in close‐packed structures. The {111} planes of FCC and the {0002} planes of HCP have the same close‐packed atomic planes with six‐fold symmetry. Any two adjacent close‐packed crystal planes in FCC and HCP are stacked in an identical sequence and labeled as AB. Each atom in the B plane is directly on top of the center of triangles formed by three atoms in A plane. In an HCP structure, the atomic location in the third plane is directly above those of the first plane, so the stacking sequence continues as ABABABAB. In an FCC structure, the atoms in the third layer fall on a location not directly above either A or B, but at a third location C. The atoms in the fourth plane are directly above those of the A plane, so the sequence continues as ABCABCABC. A stacking fault is a one or two plane deviation from the above perfect sequence. For example, ABCABCBCABCABC in FCC is a stacking fault and ABABABCABAB in HCP is a stacking fault. All plane defects disrupt the motion of dislocations through a material; so introducing the plane defects can change the mechanical properties of a material.

Bulk defects, also known as volume defects, are either clusters of the above defects or small regions of a different phase. The latter are often called precipitates. Bulk defects are obstacles to dislocation motion, so they are one of the mechanisms for strengthening materials. A crystal may contain many small regions or blocks with identical lattice structure, but separated by faults and dislocation clusters, as shown in Figure 1.9. The adjacent blocks are slightly disoriented so that the perfect crystal lattice extends only within each block. This kind of structure is referred to as a mosaic structure. The extent of the mosaic structure is also described as mosaicity. To say that a crystal has low mosaicity means that it has larger perfect crystal blocks or smaller misorientation between blocks.

Figure 1.9 Illustration of crystal mosaicity.

1.3 Principles of X‐Ray Diffraction

X‐rays are electromagnetic radiation with a wavelength in the range of 0.01 to 100 angstroms (Å). X‐rays belong to a portion of the electromagnetic spectrum overlapping with gamma rays in the shorter wavelengths and with ultraviolet in the longer wavelengths. The wavelength of typical X‐rays used in X‐ray diffraction is in the region of 1 Å, which is comparable to the range of interatomic spacing in crystals. When a monochromatic X‐ray beam hits a sample, in addition to absorption and other phenomena, it generates scattered X‐rays with the same wavelength as the incident beam. This type of scattering is also known as elastic scatter or coherent scattering. The scattered X‐rays from a sample are not evenly distributed in space, but are a function of the electron distribution in the sample. The atomic arrangement in the sample can be ordered as with a single crystal or disordered for glass or liquid. As such, the intensities and spatial distributions of the scattered X‐rays form a specific diffraction pattern that is uniquely determined by the structure of the sample.

1.3.1 Bragg's Law

There are many theories and equations about the relationship between the diffraction pattern and the material structure. Bragg's law is a simple way to describe the diffraction of X‐rays by a crystal. In Figure 1.10(a), the incident X‐rays hit the crystal planes with an incident angle θ and reflection angle θ. The diffraction peak is observed when the Bragg condition is satisfied:

(1.3)

where λ is the wavelength, d is the distance between each adjacent crystal planes (d‐spacing), θ is the Bragg angle at which one observes a diffraction peak, and n is an integer, called the order of reflection. That means that the Bragg condition with the same d‐spacing and 2θ angle can be satisfied by various X‐ray wavelengths (energies). Or for the same wavelength and d‐spacing, the Bragg condition may be satisfied by several 2θ angles. In X‐ray diffraction using a single wavelength, the Bragg equation is typically expressed with c01-i0045.png for the first order of diffraction because the higher order reflections can be considered as being from different lattice planes. For instance, the second order reflection from (hkl) planes is equivalent to the first order reflection from (2h, 2k, 2l) planes. The diffraction peak is displayed as diffracted intensities at a range of 2θ angles. For perfect crystals with perfect instrumentation, the peak is a delta functions (the black straight vertical line) as shown in Figure 1.10(b). The intensity is denoted by I.

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.

The delta function is an oversimplified model, which requires a perfect crystal without mosaic structure and a perfectly collimated monochromatic X‐ray beam. A typical diffraction peak is a broadened peak displayed by the blue curved line in Figure 1.10(b). The peak broadening can be due to many different effects, including imperfect crystal conditions, such as strain, mosaic structure, and finite size; ambient conditions, such as atomic thermal vibration; and instrumental conditions, such as X‐ray beam size, beam divergence, beam spectrum distribution, and detector resolution. The curved line gives a peak profile which is the diffracted intensity distribution in the vicinity of the Bragg angle. The highest point on the curve gives the maximum intensity of the peak, Imax. The width of a peak is typically measured by its full width at half maximum (FWHM). The total diffracted energy of a diffracted beam for a peak can be measured by the area under the curve, which is referred to as integrated intensity. The integrated intensity is a more consistent value for measuring the diffracted intensity of a reflection since it is less affected by all the peak broadening factors. Causes of peak broadening, while increasing FWHM, typically also reduce the maximum intensity at the same time. Therefore, overall variation of the integrated intensity is less significant compared to the variations of FWHM and Imax.

1.3.2 Diffraction Patterns

This diffraction condition is based on the existence of the long periodicity of crystalline materials. In general, X‐ray diffraction can provide information on the atomic arrangement in materials with long range order, short range order, or no order at all, like gases, liquids, and amorphous solids. A material may have one of the above atomic arrangement types, or a mixture of the above types. Figure 1.11 gives a schematic comparison of diffraction patterns for crystalline solids, liquid, amorphous solids, and monatomic gases as well as their mixtures. The diffraction pattern from crystals has many sharp peaks corresponding to various crystal planes based on Bragg's law. The peaks at low 2θ angles are from crystal planes of large d‐spacing and vice versa at high 2θ angles. In order to satisfy the Bragg condition at all crystal planes, the crystal diffraction pattern is actually generated from polycrystalline materials or powder materials. Therefore, the diffraction pattern is also called the powder diffraction pattern. A similar diffraction pattern can be collected with a single crystal if the crystal has been rotated at various angles during data collection so that Bragg's law can be satisfied when the crystal is at the right orientation. The technique has been used in the Gandolfi camera in which the crystal is rotated above an axis tilted 45° from the camera axis. The powder‐like pattern generated by rotating a single crystal sample with other types of diffractometers is also referred as a Gandolfi pattern.

Both amorphous solid and liquid materials do not have the long range order as a crystal does, but the atomic distance has a narrow distribution due to the atoms being tightly packed. In this case, the intensity of the scattered X‐rays forms one or two maxima with a very broad distribution in the 2θ range. The intensity vs. 2θ distribution reflects the distribution of the atomic distances. In principle, a pattern like this should be called a scattering pattern since there is no diffraction as we have defined earlier, but we may call it a diffraction pattern for convenience. A monatomic gas has no order at all. The atoms are distributed randomly in space. The scattering curve shows no features at all except that the scattered intensity drops continuously with the increase of the 2θ angle. The scattering curve for air or gas shows a similar feature although the molecules have a preferred distance between atoms within each molecule. The diffraction pattern from a material containing both amorphous and crystalline solids has a broad background from the amorphous phase and sharp peaks from crystalline phase. For example, many polymer materials have an amorphous matrix with crystallized regions. The diffraction pattern may contain air‐scattering background in addition to sharp diffraction peaks. The air‐scattering can be generated from the incident beam or diffracted beam. If the air‐scattering is not removed by the diffractometer, the diffraction pattern contains a high background at low 2θ angle and the background gradually decreases with increasing 2θ angle.

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

1.4 Reciprocal Space and Diffraction

Bragg's law gives a simple relationship between the diffraction pattern and the crystal structure. Many X‐ray diffraction applications can be easily explained by Bragg's law. X‐ray diffraction phenomena can also be explained in reciprocal space by the reciprocal lattice and the Ewald sphere. X‐ray diffraction analysis with concepts in reciprocal space is a powerful way of understanding and solving many diffraction problems [2–5].

1.4.1 Reciprocal Lattice

The reciprocal lattice is a transformation of the crystal lattice in real space to reciprocal space. The shape and size of a unit cell in real space can be defined by three vectors a, b, and c, all starting from any single lattice point. The unit cell of the corresponding reciprocal lattice is then give by three vectors a*, b*, and c* (also referred to as reciprocal lattice axes, and

(1.4)

where V is the volume of the crystal unit cell in the real space and

(1.5)

Since each reciprocal lattice axis is the vector product of two lattice axes in real space, it is perpendicular to the planes defined by the two lattice axes. The original lattice axes and reciprocal lattice axes maintain the following relations:

(1.6)

and

(1.7)

Figure 1.12 illustrates the relationship between the original lattice in real space and reciprocal lattice. The unit cell of the original lattice is drawn as green lines. The three reciprocal lattice axes define a unit cell of the reciprocal lattice (blue lines). The origin of the reciprocal lattice axes, denoted by O, is the origin of the reciprocal lattice. The repeat translation of the reciprocal lattice unit cell in three dimensions forms the complete reciprocal lattice. Except for the origin, each lattice point is denoted by a set of three integers (hkl), which are the number of translations of the three reciprocal lattice axes respectively to reach the lattice point. In other words, the vector drawn from the origin to the lattice point (hkl) is given by

(1.8)

and the direction of the vector c01-i0046.png is normal to the lattice planes (hkl) in real space. The magnitude of the vector c01-i0047.png is given by the d‐spacing of the (hkl) planes by

(1.9)

Therefore, each point (hkl) in the reciprocal lattice represents a set of lattice planes (hkl) in the real space lattice. The position of the point in the reciprocal lattice defines the orientation and d‐spacing of the lattice planes in the real space lattice. The further away a reciprocal lattice point is from the origin, the smaller is the d‐spacing of the corresponding lattice planes. For example, the reciprocal lattice point (111) represents the (111) lattice planes in the real space lattice, and the lattice vector is given by

and

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.

1.4.2 The Ewald Sphere

The relationship between the Bragg condition and the reciprocal lattice can be explained visually by the Ewald sphere, also referred to as the reflection sphere. Ewald came up with a geometrical construction to help visualize which Bragg planes are in the correct orientation to diffract. In Figure 1.13, the diffracting crystal is located in the center of the Ewald sphere, C. The radius of the Ewald sphere is defined as 1/λ. The incident beam can be visualized as the vector from I to C, and the diffracted beam is the vector from C to P. Both the incident beam and the diffracted beam are at an angle θ from a set of crystal planes (hkl). The d‐spacing of the crystal planes is c01-i0048.png . In Ewald sphere, both the incident beam vector s0/λ and the diffracted beam vector s/λ start at the point C and end at points O and P respectively. The vector from O to P is the reciprocal lattice vector c01-i0049.png and is perpendicular to the crystal planes. The three vectors have the relationship

(1.10)

and the magnitude of the vectors has the relationship based on Bragg's law:

(1.11)

The point O is the origin of the reciprocal lattice and the point P is the reciprocal point (hkl). The Bragg condition is satisfied only when the reciprocal lattice point falls on the Ewald sphere. For a single crystal, the chance of having a reciprocal lattice point on the Ewald sphere is very small if the crystal orientation is fixed. Multiplying both ends of equation (1.10) by the three lattice axes in real space respectively, we obtain the Laue equations

(1.12)

The Laue equations establish that a periodic three‐dimensional lattice produces diffraction maxima at specific angles depending on the incident beam direction and the wavelength. The Laue equations are suitable to describe the diffraction geometry of a single crystal. Bragg's law is more conveniently used for powder diffraction. Both the Laue equations and Bragg's law define the diffraction condition in different formats.

The distance between the origin of the reciprocal lattice (O) and the lattice point (P) is reciprocal to the d‐spacing. The largest possible magnitude of the reciprocal lattice vector is given by 2/λ. This means that the smallest d‐spacing satisfying the Bragg condition is λ/2. In powder X‐ray diffraction, the random orientation of all crystallites can take all possible orientations assuming an infinite number of crystallites in diffraction. The trace of the reciprocal lattice points from all crystallites can be considered as a series of spherical surfaces with the origin O as the center. Therefore, the condition for satisfying Bragg's law is only if the d‐spacing is greater than half of the wavelength. In other words, the Bragg condition can be satisfied if a reciprocal lattice point falls in a sphere of 2/λ from the origin O. This sphere is called the limiting sphere for powder diffraction. Figure 1.14 illustrates the limiting sphere for powder diffraction in a two‐dimensional cut through the origin. Only two of the three reciprocal lattice vectors a* and b* are shown in the 2D illustration. All the reciprocal lattice points within the limiting sphere are denoted by blue dots. For powder samples, all the reciprocal lattice points having the same distance from the origin form a sphere shown by a circle with a blue line. For example, the reciprocal lattice point P(hkl) would not fall on the Ewald sphere for a single crystal with fixed orientation. But for powder samples, the equivalent reciprocal lattice point from some crystallites would fall on the Ewald sphere at point P′. The same explanation can also be given for a rotating single crystal. In this case, the reciprocal lattice point P(hkl) can cross with the Ewald sphere by a proper rotation. The Gandolfi camera works on this principle.

There is another set of vectors, usually in physics and related fields, which are used to describe the above diffraction condition with the following relations:

(1.13)

and

(1.14)

where, k and k0 are called the wave vectors for the incident and diffracted beams, and both have a magnitude of 2π/λ. The scattering vector Q has a similar physical meaning of Hhkl, except with a factor of 2π in magnitudes. Overall, we have the following relations between the two sets of vectors:

(1.15)

in which, s and s0 can be considered as the unit vectors of k and k0 respectively. H is a general expression of the diffraction vector by removing the indices (hkl) from the subscript. For consistency, the vector set s, s0, and H will be used in the following chapters except when the vector set k, k0, and Q are specified.

Figure 1.14 Limiting sphere for the powder diffraction.

1.4.3 Diffraction Cone and Diffraction Vector Cone

In powder diffraction, for a fixed incident X‐ray vector s0/λ, the diffracted beam vector s/λ takes all directions at a 2θ angle from the incident beam direction, as shown in Figure 1.15. The end of the s/λ vector forms a circle on the Ewald sphere passing through the reciprocal lattice point P(hkl), P′(hkl) and all equivalent reciprocal lattice points. The diffracted beams form a cone with the incident beam on the rotation axis. This cone is referred to as diffraction cone. The 2θ angle can take values from 0° to 180°, corresponding to all the directions of the diffracted beams. The diffraction vector c01-i0050.png , starting from the origin of the reciprocal lattice (O) to the trace circle of the lattice point P(hkl) and equivalents, also forms a cone, called the diffraction vector cone. The angle between the diffraction vector and the incident beam is 90° + θ. In the illustration of the Ewald sphere, the diffraction cone and the diffraction vector cone start from different points. In real space geometry, both the diffraction cone and the diffraction vector cone are considered as starting from the same point (the sample location or the instrument center).

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

1.5 Two‐Dimensional X‐Ray Diffraction

1.5.1 Diffraction Pattern Measured by Area Detector

The diffraction patterns shown in Figure 1.11 are displayed as diffracted intensity vs. 2θ angle assuming that the diffracted intensity is a unique function of diffraction angle. The actual diffraction pattern is distributed in the 3D space around the diffracting sample. Figure 1.16 illustrates the diffraction patterns from a single crystal and from polycrystalline samples. The diffracted beams from a single crystal point to discrete directions each correspond to a family of diffracting planes, as shown in Figure 1.16(a). Each diffracted beam is a direct reflection of the incident X‐ray beam based on Bragg's law. The diffracted beams are intercepted by an area detector and the X‐ray intensity distribution on the sensing area is converted to an image‐like diffraction pattern, also referred to as a frame. The region representing each diffracted beam in the frame is called a diffraction spot. Figure 1.16(b) is a diffraction frame from a single crystal of the sweet protein thaumatin. Due to the large and complex unit cell of this protein crystal, there are many diffraction spots in the frame. Today, in the area of single crystal diffraction, two‐dimensional detectors are required to collect enough diffraction data to solve the structure of a complex crystal. Single crystal X‐ray diffraction (SCD) has been covered by much literature [9, 10]. This book will mainly cover diffraction from polycrystalline materials or other non‐single crystal materials in the following chapters.

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.

Polycrystalline materials consist of many crystalline domains, ranging from a few to more than a million in the incident beam. In single‐phase polycrystalline materials, all these domains have the same crystal structure but various orientations. Polycrystalline materials can also be multiphase materials with more than one kind of crystal structure blended together. Polycrystalline materials can also be mixed or bonded to different materials such as thin films or coatings on single‐crystal substrates. The crystalline domains can be embedded in an amorphous matrix. Most often, the sample undergoing X‐ray analysis is not a randomly oriented polycrystalline material, but a combination of polycrystalline, amorphous, and single crystal contents, polycrystalline with preferred orientation or deformed due to residual stresses. The diffracted beams from a polycrystalline (powder) sample forms a series of diffraction cones in 3D space since large numbers of crystals oriented randomly in the space are covered by the incident X‐ray beam, as shown in Figure 1.16(c). Each diffraction cone corresponds to the diffraction from the same family of crystalline planes in all the participating grains. The diffraction frame from a polycrystalline sample is a cross‐section of the detecting surface and the diffraction cones. Figure 1.16(d) is a diffraction frame collected from corundum powder by an area detector with flat detecting surface. The diffraction pattern collected with an area detector is typically given as a two‐dimensional image frame, so the X‐ray diffraction with an area detector is called two‐dimensional X‐ray diffraction [11]. In this book and a lot of literature, XRD² or 2D‐XRD may be used alternatively as an abbreviation for two‐dimensional X‐ray diffraction.

1.5.2 Materials Characterization with 2D Diffraction Pattern

A 2D diffraction pattern can be considered as the intensity distribution of the scattered X‐rays as a function of γ and 2θ angles. Figure 1.17(a) shows such a 2D pattern collected from corundum powder with a flat 2D detector. It contains several diffraction rings with various Bragg angles, denoted by 2θ. The diffraction ring closest to the trajectory of the incident beam has the smallest 2θ value. The angle along each diffraction ring can be described by γ, which defines the dimension orthogonal to the 2θ direction. In order to explain the basic concept of materials characterization with 2D diffraction patterns, the frame can be converted to an image with rectangular γ‐2θ coordinates, as shown in Figure 1.17(b). This image can be referred to as a γ‐2θ plot or simply a gamma plot.

A 2D diffraction pattern can be analyzed directly or by data reduction to the intensity distribution along γ or 2θ. The γ‐integration can reduce the 2D pattern into a diffraction profile analogous to the conventional diffraction pattern which is the diffraction intensity distribution as a function of 2θ angles. This kind of diffraction pattern can be evaluated by most exiting software and algorithms for conventional applications, such as phase identification, structure refinement, and 2θ‐profile analysis. However, the materials structure information associated with the intensity distribution along the γ direction is lost through γ‐integration. In order to evaluate the material's structure associated with the intensity distribution along the γ angle, either the 2D diffraction pattern should be directly analyzed or the γ‐profile generated by 2θ‐integration should be used.

Figure 1.17 (a) 2D diffraction pattern from corundum powder, (b) 2D pattern displayed in rectangular γ‐2θ coordinates.

The relationship between the diffraction ring profile and materials characterization can be explained by the following four types of γ‐2θ plots. Figure 1.18 illustrates four γ‐2θ plots of a single diffraction ring in 3D view. Figure 1.18(a) shows a straight wall with a constant 2θ and constant intensity along γ. Due to the large amount of randomly oriented fine powder irradiated by the incident X‐ray, the number of crystallites satisfying the Bragg condition is statistically the same at all γ angles. Figure 1.18(b) shows the intensity variation along γ due to the preferred orientation. The texture in the sample will change the diffraction intensity measured at a different angle, but not the 2θ value. Figure 1.18(c) illustrates the intensity distribution when the sample contains residual stress or applied stress loading. The d‐spacing variation at different orientation due to stress will result in the 2θ value variation measured at different γ angles. Figure 1.18(d) shows a spotty intensity distribution along γ due to only a limited number of large crystals satisfying the Bragg condition. Apparently, the spottiness is related to the crystal size and its size distribution.

A material may contain texture, stress, and large crystal size simultaneously so the 2D diffraction pattern can be a mix of the above four models. Figure 1.19(a) is a diffraction frame collected from a proprietary multilayer battery anode with Bruker VÅNTEC‐500 2D detector. Figure 1.19(b) is the same diffraction pattern displayed as a 3D plot with intensity, denoted as I, in the vertical direction. The material characterization can be observed directly from the 2D frame. For example, some diffraction rings show very strong intensity variation but smooth transition from high intensity to low intensity. These diffraction rings are produced from materials with strong preferred orientation and fine crystal grains. Some other diffraction rings are very spotty with sharp transition from high intensity to low intensity. These diffractions are produced from materials with large crystal grain size. By comparing two types of the diffraction rings, it can also be concluded that the sample contains at least two phases, because it is impossible for a sample with homogenous single phase to produce diffraction rings so dramatically different.

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.

1.5.3