X-Ray powder diffraction

X-ray powder diffraction is a technique applied to finely powdered crystals or mixtures of crystals to identify and determine the relative amounts of the crystal phase or phases present.

Structure of Crystals

X-ray powder diffraction is a technique used in the analysis of fine powders. It can be used to distinguish glasses from crystals, to identify crystal and mineral phases, and to determine the relative amounts of crystal phases in mixtures. It can also be used to determine the composition of crystals that have a range of ionic substitution, and the size and shape of the unit cell of a crystal substance.

The technique is based on the structure of crystals—that is, on their orderly, periodically repeating system of atoms and molecules. For example, if the minerals quartz (silicon dioxide) and calcite (calcium carbonate) are present in a rock, then X-ray powder techniques can be used to determine their presence and relative amounts. It is not a chemical technique for determining the presence and amount of particular elements, except for those minerals with limited ionic substitution. It does not specifically determine the presence of the silicon or calcium in quartz or calcite. The procedure depends on the fact that the wavelength of X-rays and the spacing between layers of atoms that make up the periodic structure of crystalline substances (d-spacing) are similar: 0.5 to 2.5 angstroms. As a consequence, when X-rays are swept over a crystal lattice and geometric conditions are correct, an energy peak will be emitted that represents each lattice plane of the crystal. Every mineral has a unique set of peaks whose position and size are characteristic of its crystal structure and chemical composition.

A crystal is a homogeneous solid with an orderly, periodically repeating atomic structure. This structure is responsible for the flat faces on large crystals; the orientation of these faces relative to one another is a consequence of the internal structure of the crystal. A two-dimensional analogy to the periodic structure in a crystal lattice is the repeating pattern in wallpaper. Each design unit in the wallpaper can be envisioned as representing an atom or molecule (a cluster of atoms). The pattern in a wallpaper design obeys the same mathematical laws that pervade nature, including the structure of crystals. For simplicity, the structure of crystals is commonly envisioned as a series of points periodically repeating in a three-dimensional space. The atoms form sets of parallel planes called lattice planes, and the distance between each lattice plane is symbolized by the letter d. Even in a simple rectangular array of two dimensions, there are many possible lattice d-spacings (d1, d2, d3, d4, and so on). Intersecting sets of lattice planes delimit a minimal group of atoms that forms the unit cell, the fundamental building block of each crystal. There are strict mathematical laws that govern the way atoms can repeat in space, and there are only six possible crystal systems: cubic, tetragonal, orthorhombic, hexagonal, monoclinic, and triclinic. These six systems together have 230 non-identical space groups, or arrangements of points in space. When combined with the variation supplied by the ninety-two natural elements, this diversity means that no two crystals have identical structures.

Generation of X-rays

One of the ways X-rays can be generated is when high-energy photons bombard a metal. Copper is the metal most used in X-ray powder diffraction procedures, but many other metals can be used, including molybdenum, nickel, cobalt, iron, and chromium. An X-ray tube consists of a tungsten filament and a copper (or other metal) target in a vacuum. The tungsten filament supplies electrons that are accelerated into the copper target by voltages of 30 to 50 kilovolts and currents of 10 to 50 milliamperes. The photons interact with the copper target in two ways. First, many photons are absorbed by a variety of processes that give rise to a broad and continuous spectrum of X-ray energy. Second in powder diffraction, X-rays of a very precise wavelength will be emitted by electrons changing orbits within the copper atoms when these electrons absorb and then release a fixed amount of energy. Thus, the wavelength of X-rays emitted by a copper tube consists of a broad “hump” with “spikes” at highly specific wavelengths. A typical commercial X-ray diffractometer with a copper tube is designed to allow only the X-rays of wavelength 1.5418 angstroms to hit the target powder.

Bragg's Law

When X-rays impinge on a crystal lattice, many interactions take place. The one of importance here is diffraction. The diffraction relation between X-ray wavelength and the d-spacing between lattice planes of a crystal is expressed by Bragg's law. Bragg's law is so named because it was developed by British chemist William Henry Bragg and his son. Bragg's law is nλ = 2d sinθ. In this equation, n is a whole number (usually taken as 1), λ is the wavelength of the X-rays (a known quantity), d is the distance between successive parallel planes in a crystal, and θ is the angle between the direction of incoming X-rays and the lattice plane of interest. (The angle is measured by a goniometer.) When the angle θ is such that the difference in the X-ray paths from adjacent crystal planes is not a multiple of a whole wavelength, the diffracted energy will be low, because the emitted wave will be out of phase. When the angle ABC is equal to whole multiples of the wavelength, however, then the emitted waves will be in phase; they will reinforce each other, and an energy peak will be emitted. Bragg's equation can be readily solved, because the X-ray wavelength is known, the angle θ can be read on the goniometer when a peak appears, and one can assume that n is 1. Thus, the only unknown in the equation is the lattice d-spacing (d).

Single-Crystal Diffraction

There are two X-ray diffraction techniques: single crystal and powder. Single-crystal diffraction requires one crystal of the substance of interest. The crystal must be oriented in such a way that the relation between the crystal lattice, the X-ray beam, and the detector is precisely defined. The technique is difficult, requiring precise orientation, refined analysis, and considerable mathematical skill. Single-crystal techniques give a “Laue spot pattern” of energy peaks and are used for determination of the details of crystal structure.

X-ray powder diffraction is a comparatively simple and routine analytical procedure, producing a series of rings of peak energy. It depends on statistics, with the crystallites having uniformly random orientations. The fine powder is packed in its holder so that the millions of crystallites, or minute crystals, are randomly oriented relative to the X-ray beam. Usually, a powder with particles measuring 45 micrometers or less is required. The effect is as though there were one average-sized crystal present. If not already fine grained, the sample must be ground into a powder.

Powder Diffraction

There are two powder techniques: camera and diffractometer. In the camera method, a special cylindrical Debye-Scherrer camera is used. The sample powder is placed in a thin, glass tube, and the tube is placed on the axis of rotation on the centerline of the cylinder. Photographic film is placed along the inner circumference of the cylinder. X-rays enter along a hole in the side of the camera, hit the rotating powder, are diffracted, and then hit the film. Peaks are recorded as a series of circles (rings) on the film. Careful measurement of the line position relative to the hole where the X-rays entered and knowledge of the geometry involved yield the location of the peaks, which in turn leads to the solution of Bragg's equation. The camera technique was the first to be developed. It is still useful when the amount of sample is very small, but it is tedious and time-consuming. A typical run may take four or five hours.

The diffractometer method is the fastest, easiest, most quantitative, and most widely used powder technique. This technique uses a goniometer to correctly position the X-ray beam, the surface of the sample powder, and the X-ray detector so that proper geometry for the solution of Bragg's equation is maintained. First, the proper operating conditions for the X-ray generation are set and the goniometer positioned at the desired start angle. Then, the powder is pressed into a holder, so that it has a smooth surface, and placed in the sample chamber. The goniometer scans from the starting angle to the ending angle. In modern machines, that procedure is controlled by a computer program. The output signal is received by a scintillation counter, electronically enhanced, and sent to an output device, typically a strip chart recorder. The output data are called the X-ray pattern, presented as a graph of peak position and size versus angle degrees. A typical run takes thirty minutes.

Each diffraction peak corresponds to an interplaner d-spacing. The size of the peak is a function of the electron density along the “surface” of the lattice. In general, the heavier the element, the larger the peak. For identification of a single-phase powder, the three largest peaks in the low-angle range are normally adequate. Data on their position and relative size are compared to the X-ray powder diffraction card index file published by the American Society for Testing and Materials. When a match is found for the three main peaks, the identification is verified using the balance of the peaks.

If there are two or more phases in the powder, identification becomes a matter of experience, and guesses are used until a phase is identified. Elimination of the peaks of an identified phase is followed by repetitions of the procedure until all peaks are accounted for. Computers then search for likely combinations of peaks. Where precise measurement of the d-spacings is desired, such as in the determination of the unit cell, peaks in the high-angle region are used. The d-spacings are partially controlled by composition variation. For example, when small ions substitute for larger ions, the d-spacing decreases. This decrease results in a shift of the peak toward higher angles.

Limitations and Benefits

X-ray powder diffraction has several limitations. It is difficult to detect components that form less than 1 percent of a mixture. Furthermore, estimates of the amounts of specific minerals present in a mineral mixture are seldom more accurate than plus or minus 1 percent. Nevertheless, the technique does not alter the character of the sample, and after X-ray analysis, the powder can be used for further analyses.

In fact, X-ray diffraction is widely used in the study of rocks, sediments, meteorites, and any crystalline solid in which the particles are too fine-grained for analysis by standard optical techniques. Because of its simplicity, it is also used to identify large crystals after they have been ground into a suitable powder.

One area in which X-ray diffraction is used is the classification of silicate minerals. Attempts to classify this large and diverse family of minerals on chemical grounds resulted in contradictions and confusion. X-ray diffraction is used to divide silicates into structural groups—such as orthosilicates, phyllosilicates (which include clay minerals), and isosilicates—giving rise to a logical and meaningful classification scheme.

Role in Technological Developments

X-ray diffraction procedures are vital to the technological developments that one tends to take for granted in geology, oceanography, meteoritics, ceramics, electronics, and cements. The search for a diminishing body of finite earth resources means that geologists must look more and more closely at rocks that contain the needed materials. Many of these rocks are very fine grained, and X-ray diffraction is the only suitable analytical tool. In oceanography, the X-ray analysis of the very fine sediments and rocks on the sea floor is essential to understanding the origin and history of the ocean basins. Meteorites—rocky and metallic fragments from beyond the atmosphere—are typically fine grained. Nondestructive analysis by X-ray diffraction is essential to their classification. In the ceramics industry, the development of tougher and more durable pottery, grinding compounds, and insulators requires this analytical technique; in electronics, the development of new transistors, thermisters, and superconductors requires X-ray-based, single-crystal analysis. Cements form a complex paste of reactants that, upon curing, are best studied by X-ray powder diffraction.

In summary, whenever the samples are in the form of very fine crystallites, X-ray powder diffraction is the most powerful and easy-to-use system for mineral or crystal analysis.

Principal Terms

Bragg's law: the fundamental equation that relates X-ray wavelength, interatomic distances, and the angle between the X-ray beam and the lattice plane of crystals

ceramic: a human-made mineral, crystal, or aggregate thereof, excluding metals

crystal: a solid consisting of a regular periodic arrangement of atoms; its external form and physical properties express the repeated units of the structure

diffractometer: an instrument used for X-ray powder diffraction analysis

d-spacing: the distance between successive parallel layers of atoms in a crystal

glass: a solid with no regular periodic arrangement of atoms; an amorphous solid

goniometer: the mechanism that maintains the correct arrangement among the sample powder, the X-ray beam, and the X-ray detector in a diffractometer

mineral: a natural substance of fixed or narrowly limited chemical and physical properties; most minerals are also crystals

phase: a homogeneous, physically distinct, mechanically separable portion of matter present in a nonhomogeneous chemical system

X-ray: a photon with a much higher energy and shorter wavelength than those of visible light; its wavelength is of the same order of magnitude as the spaces between atoms in a crystal

Bibliography

Azaroff, L. V., and M. J. Buerger. The Powder Method in X-ray Crystallography. New York: McGraw-Hill, 1958. This text is best suited to a second college course in X-ray diffraction analysis. Its focus is the use of powder cameras. As with all books written on this topic, the authors expect the reader to have a background in elementary physics and crystallography. They discuss the design and alignment of cameras, how to take photographs, how to interpret powder photographs in terms of unit cell size and geometry, the causes of errors, and how to overcome them.

Bowen, David Keith, and Brian K. Tanner. High-Resolution X-ray Diffractometry and Topography. London: Taylor and Francis, 1998. This book examines the procedures involved with and the equipment required within the field of crystallography. Bowen and Tanner lay the foundation for a thorough look at the processes and applications of X-ray diffraction and X-ray crystallography. A somewhat technical book intended for the specialist.

Buhrke, Victor E., Ron Jenkins, and Deane K. Smith, eds. A Practical Guide for the Preparation of Specimens for X-ray Fluorescence and X-ray Diffraction Analysis. New York: John Wiley & Sons, 1998. Provides the best techniques for issues with XRF and XRD analysis. Covers material usually left for manuals along with theoretical discussion.

Bunn, C. W. Chemical Crystallography. Oxford, England: Clarendon Press, 1958. A readable text for a graduate-level course in X-ray diffraction procedures. Its emphasis is on basic principles of crystallography, and it provides comparatively little information on the source and interaction of X-rays. It is a valuable resource for camera techniques, both powder and single crystal. There is a minimum of math and chemistry; the author relies instead on photographs and diagrams. Includes a chapter with examples of successful solutions of crystallographic structures.

Bush, Laura. “The Dynamic World of X-ray Fluorescence.” Spectroscopy 26 (2011): 40-44. Provides current information on X-ray fluorescence technology and applications. Written in a nontechnical manner accessible to the layperson, but contains enough detail to be relevant to professionals in the field of X-ray fluorescence.

Cullity, B. D., and S. R. Stock. Elements of X-ray Diffraction. 3rd ed. Addison-Wesley, 2001. Some more recent information in addition to that from the classic second edition. This text uses Bragg's law, so the reader does not need knowledge of reciprocal lattice. Covers fundamentals, experimentation, and applications of XRD.

Hammond, Christopher. The Basics of Crystallography and Diffraction. 3rd ed. London: Oxford University Press, 2009. Hammond offers a clear understanding of the principles and practices of crystallography and X-ray crystallography. Index and bibliography.

Jenkins, Ron. Introduction to X-ray Powder Diffractometry. 2d ed. New York: John Wiley, 1996. This classic text is intended for an introductory college course in X-ray crystallography. It is a basic source for information on the principles and practice of X-ray powder diffraction as applied to inorganic materials. Jenkins discusses crystallography, X-ray production, the interaction of X-rays and crystals, and the details of X-ray diffractometer design.

Jones, Christopher, et al., eds. Crsytallographic Methods and Protocol. Totowa, N.J.: Humana Press, 1996. Part of the Methods in Molecular Biology series, this volume examines the use of X-ray diffraction to determine the structure of compounds such as nucleic acids and proteins. A large portion of the book is dedicated to discussing the practices and protocols surrounding X-ray diffraction and X-ray crystallography.

Klein, Cornelis, and Barbara Dutrow. Manual of Mineral Science. 23rd ed. New York: John Wiley & Sons, 2008. A classic college-level introduction to mineralogy, updated numerous times since its original publication in 1912. Contains a thorough discussion of crystal systems and concise descriptions of all common minerals, including essential optical data. Chapters 13 and 14 contain a summary of optical microscopy, X-ray and electron imaging methods, and mass spectrometry. Well illustrated and indexed, with key references after each chapter.

Klug, H. P., and L. E. Alexander. X-ray Diffraction Procedures for Polycrystalline and Amorphous Materials. New York: John Wiley & Sons, 1954. This classic text is intended for an introductory college course in X-ray crystallography. It is a basic source for information on the principles and practice of X-ray powder diffraction as applied to inorganic materials. It discusses crystallography, X-ray production, the interaction of X-rays and crystals, and the details of X-ray diffractometer design. The specific diffractometers discussed are dated, but the principles remain the same.

Nuffield, E. W. X-ray Diffraction Methods. New York: John Wiley & Sons, 1966. This relatively brief book combines information on powder and single-crystal techniques. It is intended as a laboratory aid for students with limited mathematical backgrounds. Discussions of elementary crystallography and X-ray generation are followed by chapters devoted to specific methods, techniques, and concepts. A good introduction to how single-crystal and powder techniques are related.

Potts, Philip J., and Margaret West, eds. Portable X-ray Fluorescence Spectrometry: Capabilities for In Situ Analysis. Royal Society of Chemistry, 2008. This text provides an overview of the limitations and capabilities of the new instruments available in X-ray fluorescence spectrometry. Written for the undergraduate student.