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 (a, ß, and ?). The six parameters (a, b, c, a, ß, and ?) 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, a, ß, and ?) 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 ( and ). 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...