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    Introduction to X-Ray Powder Diffractometry
    Introduction to X-Ray Powder Diffractometry
    Introduction to X-Ray Powder Diffractometry
    Ebook659 pages6 hours

    Introduction to X-Ray Powder Diffractometry

    By Ron Jenkins and Robert Snyder

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    When bombarded with X-rays, solid materials produce distinct scattering patterns similar to fingerprints. X-ray powder diffraction is a technique used to fingerprint solid samples, which are then identified and cataloged for future use-much the way the FBI keeps fingerprints on file. The current database of some 70,000 material prints has been put to a broad range of uses, from the analysis of moon rocks to testing drugs for purity.

    Introduction to X-ray Powder Diffractometry fully updates the achievements in the field over the past fifteen years and provides a much-needed explanation of the state-of-the-art techniques involved in characterizing materials. It covers the latest instruments and methods, with an emphasis on the fundamentals of the diffractometer, its components, alignment, calibration, and automation.

    The first three chapters outline diffraction theory in clear language, accessible to both students and professionals in chemistry, physics, geology, and materials science. The book's middle chapters describe the instrumentation and procedures used in X-ray diffraction, including X-ray sources, X-ray detection, and production of monochromatic radiation. The chapter devoted to instrument design and calibration is followed by an examination of specimen preparation methods, data collection, and reduction. The final two chapters provide in-depth discussions of qualitative and quantitative analysis.

    While the material is presented in an orderly progression, beginning with basic concepts and moving on to more complex material, each chapter stands on its own and can be studied independently or used as a professional reference. More than 230 illustrations and tables demonstrate techniques and clarify complex material.

    Self-contained, timely, and user-friendly, Introduction to X-ray Powder Diffractometry is an enormously useful text and professional reference for analytical chemists, physicists, geologists and materials scientists, and upper-level undergraduate and graduate students in materials science and analytical chemistry.

    X-ray powder diffraction-a technique that has matured significantly in recent years-is used to identify solid samples and determine their composition by analyzing the so-called "fingerprints" they generate when X-rayed. This unique volume fulfills two major roles: it is the first textbook devoted solely to X-ray powder diffractometry, and the first up-to-date treatment of the subject in 20 years.

    This timely, authoritative volume features:
    * Clear, concise descriptions of both theory and practice-including fundamentals of diffraction theory and all aspects of the diffractometer
    * A treatment that reflects current trends toward automation, covering the newest instrumentation and automation techniques
    * Coverage of all the most common applications, with special emphasis on qualitative and quantitative analysis
    * An accessible presentation appropriate for both students and professionals
    * More than 230 tables and illustrations

    Introduction to X-ray Powder Diffractometry, a collaboration between two internationally known and respected experts in the field, provides invaluable guidance to anyone using X-ray powder diffractometers and diffractometry in materials science, ceramics, the pharmaceutical industry, and elsewhere.
    LanguageEnglish
    PublisherWiley-Interscience
    Release dateAug 29, 2012
    ISBN9781118520925
    Introduction to X-Ray Powder Diffractometry

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      Introduction to X-Ray Powder Diffractometry - Ron Jenkins

      CHAPTER 1

      CHARACTERISTICS OF X-RADIATION

      1.1. EARLY DEVELOPMENT OF X-RAY DIFFRACTION

      Following the discovery of X-rays by W. C. Röntgen in 1895, three major branches of science have developed from the use of this radiation. The first and oldest of these is X-ray radiography, which makes use of the fact that the relative absorption of X-rays by matter is a function of the average atomic number and density of the matter concerned. From this has developed the whole range of diagnostic methods for medical and industrial use. Early attempts to confirm the dual nature of X-rays, i.e., their particle and wave character, were frustrated by experimental difficulties involved with the handling of the very short wavelengths in question. Not until the classic work of Max von Laue in 1912 was the wave character confirmed by diffraction experiments from a single crystal. From this single experiment has developed the field of X-ray crystallography, of which X-ray powder diffractometry is one important member. X-ray crystallography, using single crystals or powder, is mainly concerned with structure analysis. The third technique, X-ray spectrometry, also has its fundamental roots in the early part of this century, but routine application of X-ray fluorescence spectrometry has only developed over the last 20 to 30 years.

      The purpose of this work is to discuss X-ray powder diffractometry. Powder diffractometry is mainly used for the identification of compounds by their diffraction patterns. The first X-ray powder diffractometer was developed in 1935 by Le Galley [1], but, due mainly to the lack of parafocusing conditions, the instrument gave relatively poor intensities. In 1945 Parrish and Gordon [2] developed a Geiger-counter spectrometer¹ for the precision cutting of quartz oscillator plates used in frequency control for military communication equipment. At the same time, Friedman [3] was working on X-ray spectrometer techniques at the U.S. Naval Research Laboratory in Washington, DC. The modern parafocusing X-ray powder diffractometer was based on these ideas, and the first commercial equipment was introduced by North American Philips in 1947. The latest versions of the powder diffractometer differ little in their construction and geometry, but considerable advances have been made in detection and counting systems, automation, and in the X-ray tubes themselves.

      1.2. ORIGIN OF X-RADIATION

      X-rays are relatively short-wavelength, high-energy beams of electromagnetic radiation. When an X-ray beam is viewed as a wave, one can think of it as a sinusoidal oscillating electric field with, at right angles to it, a similarly varying magnetic field changing with time. Another description of X-rays is as particles of energy called photons. All electromagnetic radiation is characterized either by its wave character using its wavelength λ (i.e., the distance between peaks) or its frequency v (the number of peaks that pass a point in unit time) or by means of its photon energy E. The following equations represent the relationships between these quantities:

      (1.1) equation

      (1.2) equation

      where c is the speed of light and h is Planck’s constant. The X-ray region is normally considered to be that part of the electromagnetic spectrum lying between 0.1 and 100 Å(1Å = 10-10 m), being bounded by the γ-ray region to the short-wavelength side and the vacuum ultraviolet region to the long-wavelength side. In terms of energy, the X-ray region covers the range from about 0.1 to 100 keV. From a combination of Equations 1.1 and 1.2, it follows that the energy equivalent of an X-ray photon is

      (1.3) equation

      Insertion of the appropriate values for the fundamental constants gives

      (1.4) equation

      or

      (1.5) equation

      where E is in keV and λ in angstroms. As an example the Cu Kα1, Kα2 doublet has an energy of about 8.05 keV, corresponding to a wavelength of 12.398/8.046= 1.541 Å.

      In the early days of crystallography there was no standard value for—or way to determine—the wavelength of any particular X-ray photon. A practical definition was made defining wavelength in terms of the cubic lattice parameter of calcite. These units are referred to as kX units and were used in the literature into the 1950s. The angstrom (Å) unit has always been the preferred measure of wavelength and is related to kX (the crystallographic unit) by 1 Å = 1.00025 kX units. Even though the latest recommendation from the International Union of Pure and Applied Chemistry (IUPAC) discourages use of the angstrom and encourages use of the nanometer (nm; 1 × 10-9 m), the powder diffraction community has fought for retention of the angstrom and this remains the common unit in use in the field today. For this reason, in this book we will use the angstrom unit. The common electron-volt energy unit is also not IUPAC approved in that the standard energy unit is the joule (J), which may be converted by 1 eV = 1.602 × 10-19 J.

      1.3. CONTINUOUS RADIATION

      X-radiation arises when matter is irradiated with a beam of high-energy charged particles or photons. When an element is bombarded with electrons the spectrum obtained is similar to that shown in Figure 1.1. The figure illustrates the main features of the spectrum that would be obtained from a copper anode (target) X-ray tube, operated at 8.5, 25, and 50 kV, respectively. It will be seen that the spectrum consists of a broad band of continuous radiation (bremsstrahlung, or white radiation) superimposed on which are discrete wavelengths of varying intensity. The continuous radiation is produced as the impinging high-energy electrons are decelerated by the atomic electrons of the target element. The continuum is typified by a minimum wavelength, λmin, which is related to the maximum accelerating potential V of the electrons. Thus, as follows from Equation 1.5,

      Figure 1.1. Continuous and characteristic radiation for copper.

      (1.6) equation

      Note from Figure 1.1 that as the operating voltage is increased from 8.5 to 25 to 50 kV, the λmin value shifts to shorter wavelengths and the intensity of the continuum increases. The intensity distribution of the continuum reaches a maximum intensity at a wavelength of about 1.5 to 2 times λmin. The wavelength distribution of the continuum can be expressed quantitatively in terms of the excitation conditions by means of Kramers’ formula [4]:

      (1.7) equation

      Kramers’ formula relates the intensity I(λ) from an infinitely thick target of atomic number Z with the applied current i where K is a constant. This expression does not correct for self-absorption by the target, which in practice leads to some modification of the intensity distribution.

      It will also be seen from Figure 1.1 that somewhere between X-ray tube potentials of 8.5 and 25 kV sharp lines appear, superimposed on the continuum. These lines were shown by Moseley [5] to be characteristic wavelengths since their values differ for each unique target element. These characteristic lines will only appear when their equivalent excitation potential value V is exceeded. While the wavelengths of these characteristic lines are completely independent of the X-ray tube conditions, the intensities of the lines are very much dependent on the X-ray tube current i and voltage V; see Section 4.3.

      1.4. CHARACTERISTIC RADIATION

      1.4.1. The Photoelectric Effect

      The processes whereby characteristic radiation is produced in an X-ray tube are based on interactions between the atomic electrons of the target and the incident particles. In the case described in Figure 1.2, the incident particles are high-voltage electrons. The incident particle can also be an X-ray photon, a γ-ray, or a proton. Each will produce similar effects if the energy of the particle is greater than the energy binding the electron to the nucleus. The atomic electron may be removed from its original atomic position leaving the atom in an ionized state. The free electron, called a photoelectron, will leave the atom with a kinetic energy E — ϕe, i.e., equal to the difference between the energy E of the incident photon and the binding energy ϕe of the electron.

      Figure 1.2. The Auger and photoelectric effects.

      From R. Jenkins, R. W. Gould, and D. Gedcke, Quantitative X-Ray Spectrometry, p. 16, Fig. 2-9. Dekker, New York, 1981. Reprinted by courtesy of Marcel Dekker Inc.

      Figure 1.2 shows the basic processes involved in a photoelectric interaction. Figure 1.2a shows an atom with its various energy levels ϕK, ϕL, ϕM, etc., and incident upon it is a photon of energy E. Figure 1.2b shows the ejected photoelectron leaving the atom with an energy equal to E — ϕK. Note that this process creates a vacancy in the atom, in this instance, with an equivalent energy of ϕK. One of the processes by which this vacancy can be filled is by transferring an outer orbital electron to fill its place. Such a transference is shown in Figure 1.2c, where an electron from the L level is transferred to the K vacancy. Associated with this electron transfer (and subsequent lowering of the ionized energy of the atom) will be the production of a fluorescent X-ray photon with an energy Ex-ray equal to ϕK — ϕL. As will be shown later, this photon is called a Kα photon.

      1.4.2. The Auger Effect

      An alternative deexcitation process, called the Auger effect, can also occur, and this effect is illustrated in Figure 1.2d. It may happen that the ionization of an inner shell electron produces a photon that in turn gets absorbed by an outer shell electron. Thus, the incident X-ray is absorbed by, for example, a K shell electron that leaves the atom. Next, an electron falls into the K shell, producing a Kα photon. The Kα photon, in turn, may be absorbed by an M electron, causing its ionization as an Auger electron. The kinetic energy of the emitted Auger electron is not just dependent on the energy of the initial X-ray photon (or particle) that ionized the K electron. Any incident particle with sufficient energy to create the initial vacancy can be responsible for the subsequent production of an Auger electron of unique energy. Study of the energy and intensities of Auger electrons, called Auger spectroscopy, allows measurement of the precise energy of the chemical bonds that involve the valence electrons.

      1.4.3. Fluorescent Yield

      It is apparent from the foregoing that there are two competing processes by which an ionized atom can regain its initial, or ground, state, these processes being the fluorescence of an X-ray photon and the Auger effect. The efficiency of the production of characteristic X-rays will be dependent upon the relative effectiveness of these two processes in a given atom. As the atomic number decreases, the production of Auger electrons becomes more probable and thus the production of K radiation falls off at low Z. The efficiency of a particular element producing fluorescent X-rays is quantified by the fluorescent yield ω. The fluorescent yield is the ratio of the number of photons produced from a given atomic shell to the number of equivalent shell vacancies created. For the production of K radiation from a specimen,

      (1.8) equation

      The probability of the production of an Auger electron is 1 — ω. Fluorescent yield values vary as the fourth power of atomic number and range from almost unity, for high atomic number elements, to 0.001, for low atomic numbers. For the wavelengths typically employed in powder diffraction, the K fluorescent yield values are about 0.5.

      1.4.4. Selection Rules

      Characteristic radiation arises from the rearrangement of the orbital electrons of the target element following the ejection of one or more electrons in the excitation process. The final resting place of the transferred electron determines the type of radiation, i.e., K, L, M, etc. Thus, ejection of a K electron leaves the atom in the highly energetic K+ state. Transference of an electron from the L shell reduces the energy state from K+ → L+ and the excess energy (K+ — L+) is emitted as K X-radiation, in this instance Kα radiation. Since every energy level has a unique binding energy, every element will have a unique set of binding energies and the energy state differences will also be unique (see, e.g., Table 1.1). There are obviously a great number of possibilities for electron transitions, particularly when one considers the various quantized states that each electron may have. However, in practice, X-ray spectra are far simpler than might appear at first sight, and just three selection rules cover the allowed transitions. These are Δn ≥ 1, Δl = 1, and ΔJ = 0 or 1, where n is the principal or group quantum number, l the angular quantum number, and J the vector sum of the angular and spin(s) quantum numbers. It is common practice to refer to transition groups K, LI, LII, etc. (Table 1.2), which are simply built up by combining l and J quantum numbers. The construction of the transition groups is given in Table 1.3, which also shows the number of electrons allowed in each group. The familiar spectroscopic names of the various electron states derive from the values of l such that when l=0, the state is called s; when l = 1, it is called p; when l=2, it is called d; when l=3, it is called f; and so on. The value of the J quantum number, which is the sum of l and s, determines the number of degenerate electron states for each energy level. For example, when l is 1 and J = 1/2, the two resulting energy states are called p¹/²; when l is 1 and J = 3/2, the four resulting states are referred to as p³/².

      Table 1.1. Binding Energies for the K, L, and M Levels of Copper

      Table 1.2. Siegbahn and IUPAC Notation for the K Series

      Table 1.3. Construction of Transition Groups and Number of Electrons Allowed in Each State (Multiplicity)

      Figure 1.3 shows the usual transitions for the K spectrum of copper, giving both transition groups and quantum numbers. It will be seen that the copper K spectrum is quite simple, consisting of two α lines (called a doublet) from 2p¹/² → 1s and 2p³/² → 1s transitions, and two β lines from 3p¹/² → 1s and 3p³/² → 1s transitions. Since in the case of both doublets the energy difference between the lines in each pair is simply that due to the spin quantum number, the relative energy difference is very small. As will be seen in later sections, the energy difference between α1 and α2 being very small causes only partial separation of the two wavelengths to generally occur in diffraction. In practice the β1, β3 doublet is never resolved, but the α1, α2 is resolved at moderate-to-high diffraction angles. It should be noted that since copper does not have electrons in the 4p level the 4p → 1s transition (the β3 doublet) is absent. In the case of higher atomic numbers such as silver (47) and molybdenum (42) the β2 doublet is observed as an unresolved line. The relative intensities of the characteristic wavelengths are determined by the appropriate quantum mechanical transition probabilities. It is evident that a K shell with a missing electron represents a higher energy state than a similar hole in the corresponding L shell. The transition probability is a rather complex function of the difference in energy of the two levels concerned.

      Figure 1.3. The copper Kα spectrum.

      The relative intensity ratio of possible lines for an element is constant but may differ from one element to another. The greater the energy difference, the less probable the transition becomes and the less intense is the resulting line. Consequently the intensity of Kα1, Kα2 > Kβ1, β3 > Kβ2. For a copper anode the ratio is about 5:1:0, and for molybdenum about 3:1:0.3. The relative intensity of α1, α2 (and also Kβ1, β3) is much simpler to predict since for these line pairs the Burger-Dorgelo rule [6] holds, stating that the intensity ratio is equal to the number of electrons that may make the transition. In the case of the Kα1:Kα2 ratio, there are four p³/² electrons (as shown in Table 1.3) giving rise to the Kα1 line and two p¹/² electrons giving rise to the Kα2 line. Thus, the intensity ratio is 4:2 or 2:1.

      The original nomenclature system for X-ray wavelengths was proposed by K. M. G. Siegbahn in the 1920s and is properly called the Siegbahn notation. Since the introduction of the Siegbahn notation a number of lines have been observed that have not been classified within the Siegbahn nomenclature, particularly for the M and N series. A further complication is that the Siegbahn notation is unsystematic and consequently rather difficult to learn. In recent years this problem has been addressed by the IUPAC, with the result that a new IUPAC notation has been recommended [7]. At the time of the publication of this book, the acceptance of the new IUPAC nomenclature among the X-ray community is still uncertain. However, for information, Tables 1.2 and 1.4 list typical Siegbahn and IUPAC equivalents.

      Table 1.4. Nomenclature for the Copper K Series Wavelengths

      1.4.5. Nondiagram Lines

      Not all observed characteristic X-ray lines can be satisfactorily described by the selection rules just outlined. Other lines occur following special conditions of ionization that generally fit into one of two categories—forbidden transitions and satellites. The origin of forbidden transitions is rather complex, but, in simple terms, forbidden transitions arise because outer orbital electrons are typically not distributed in absolutely unique and well-separated orbitals. For example, there is much hybridization of outer orbitals, meaning that an 5 electron may tend to show the character of a p electron and so on. Thus, transitions may occur that are close to obeying the selection rules, but in which-the electron being transferred acts as if it had a different angular quantum number than expected. As an example, in the Cu K spectrum a weak Kβ5 occurs from a 3d → Is transition. Since such a line corresponds to a Δl of 2 it is forbidden by the selection rules for normal lines.

      Satellite lines occur from transitions involving removal of more than one electron from a target atom (dual ionization). Although the excitation/deexcitation process is fast ( 10-12s), it is finite, and there is a probability that a second electron may be removed before the first vacancy is filled. Figure 1.4 shows the origin of the Kα satellites in the Cu K spectrum. The left-hand diagram shows the usual situation with a K vacancy being filled by a 2p³/² → 1s and a 2p¹/² → 1s transition, giving the Kα1 and Kα2 lines, respectively. The center, diagram shows similar transitions, except that now there are two atomic vacancies, one in the K shell and one in the LIII level. Removal of the LIII electron decreases the total electron charge of the atom and the attraction of the charge by the nucleus of the atom. There is a consequent widening of the energy gap between the K and L levels. Hence, the two transitions give a pair of lines similar to the Kα1 and Kα2, but of shorter wavelength. These lines are called the α3 and α4. The right-hand diagram shows a similar circumstance in which the second vacancy is in the L1 level, which gives rise to the α5 and α6 lines. Thus, each of the Kα1 and Kα2 lines is actually a triplet, and there are actually six lines (i.e., two triplets) in what is usually called the Kα1, Kα2 doublet. A major difference between the satellites and the forbidden transitions is that the satellites occur close to the α1, α2 doublet, and even though they are not resolved by the normal monochromatization devices, they do play some part [8] in the profile-fitting process.

      Figure 1.4. Origin of the copper Kα. satellite lines from doubly ionized atoms.

      1.4.6. Practical Form of the Copper K Spectrum

      From the foregoing discussion it will be clear that the characteristic α radiation emission from copper is much more complex than the simple α-doublet and β-doublet model generally employed in classical powder diffractometry. The relative intensities of the satellite lines in each line within each triplet differ somewhat in the α1 and α2 sets, which probably accounts for the higher degree of asymmetry of the α2 relative to the α1 typically observed. The largest energy gap within any of the triplets is only about 2.5 eV. Since the absolute energy resolution of the powder diffractometer using Cu Kα radiation ranges from about 200 eV at 10° 2θ to about 2.5 eV at 140° 2θ, the fine structure of the triplets is not resolved. However, as indicated in Figure 1.5, asymmetry is introduced in the α2, which starts to become apparent at very high 2θ values. Even more important, where profile-fitting methods are employed, the effective fitting resolution is probably on the order of a few electronvolts, and here allowance must be made for wavelengths other than the α1 and α2 if accurate (<2% or so) fitting is required. For most practical purposes, however, in powder diffractometry, the copper K spectrum is considered to consist simply of two pairs of lines, the Kα1, Kα2 doublet occurring from 2p → 1s transitions; and the Kβ1, β3 doublet from 3p → 1s transitions. In most experimental work the β doublet intensity is typically reduced to less than a few percent of the α-doublet intensity by use of filtration or is removed by use of a crystal monochromator or an Si(Li) energy-resolving detector. In each case, what remains is essentially bichromatic radiation.

      Figure 1.5. Full spectrum for copper K radiation.

      The most commonly used values for the wavelengths of Cu Kα1 and Cu Kα2 are 1.54056 and 1.54439 Å, respectively. These values were reported by Bearden [9] and have been recommended in the International Union of Crystallography (IUCr) publication International Critical Tables [10]. The values are generally given in terms of a unit length in angstrom units, based on the energy of the W Kα line of 59.31821 keV. There is a degree of uncertainty that arises because of the conversion of × units (Xu) to angstrom units. Bearden uses a value of 1.002056 for this conversion factor. Because of this uncertainty, minor differences will be found in other tables. As an example, Cauchois and Senemaud [11] list a value of 1537.400 Xu with a conversion factor of 1.0020802 to give 1.540598 Å. This value has also been used by the National Bureau of Standards (NBS; now the National Institute of Standards and Technology, NIST) and has been widely used in the powder diffraction community for the last 15 years. The NBS value is based on the techniques used by Deslattes et al. [12]. There is clearly some inconsistency between these values, and Bearden et al. [13] have suggested a new value of 1537.370 Xu for Cu Kα. Most recently Härtwig et al. [14] have suggested 1.54059292 Å. These minor variations will not affect most X-ray powder diffraction measurements, and we recommend the use of 1.54060 Å for the Cu Kα1 line.

      1.5. SCATTERING OF X-RAYS

      Electromagnetic radiation is a form of energy that can be described as an oscillating electric field E with an oscillating magnetic field H at right angles to it, as shown in Figure 1.6. The magnetic field will only interact with other magnetic fields and is therefore not generally important in considering the interactions of X-rays with matter. However, the oscillating electric field will couple to the charged electrons surrounding the atoms and cause them to accelerate and decelerate. Since an electron bound to an atom has an amount of energy fixed by the laws of quantum mechanics, the extra energy imparted to it from the acceleration must be reradiated or absorbed by perhaps stimulating a vibrational mode of the lattice. The phenomenon known as scattering occurs when any of the incident energy is reradiated.

      Figure 1.6. Electromagnetic radiation.

      1.5.1. Coherent Scatter

      Coherent scatter, or elastic scatter, can be thought of as a perfectly elastic collision between a photon and an electron. The photon changes direction after colliding with the electron but transfers none of its energy to the electron. The result is that the scattered photon leaves in a new direction but with the same phase and energy as that of the incident photon.

      From the wave perspective one thinks of the incoming wave being instantaneously absorbed by an electron and reemitted (i.e., in a time interval short enough to fall within the uncertainty principle) as spherical waves. Thus, each electron on a scattering atom in a material will act as a Huygens scattering center. If the scattering atoms are arranged in an orderly manner, with a distance between each on the order of the wavelength of the radiation, then the phase relationships between scatterers will become periodic and interference diffraction effects will be observed at various angles of view (see Section 3.1).

      1.5.2. Compton Scatter

      It can also happen that the X-ray photon loses part of its energy in the collision process, especially where the electron is only loosely bound. In this case the scatter is said to be incoherent (Compton scatter) and the wavelength of the incoherently scattered photons will be longer than the coherently scattered wavelength. Compton scatter amounts to an inelastic collision between a photon and an electron. Part of the energy of the incident photon is absorbed by an electron, and the electron is ionized. However, instead of all of the remaining energy of the original photon converting to kinetic energy of the ionized photoelectron, some of it is reemitted as an X-ray photon of lower energy. Not only has the energy of this Compton photon been lowered, but it loses any phase relationship to the incident photon. For this reason the process is often called incoherent scatter. Since the Compton (phase) modified photons are emitted in arbitrary directions very few of them will reach the detector and, therefore, this is also a source of absorption. Compton scatter decreases in importance as the atomic number of the scatterer increases.

      The total scatter σ is made up of both coherent and incoherent terms:

      (1.9) equation

      The first term is the coherent term, and the second is the incoherent term; f is called the atomic scattering factor and will be discussed in Section 3.6.2.

      1.6. ABSORPTION OF X-RAYS

      When a beam of X-radiation falls onto an absorber, a number of different processes may occur. The more important of these are illustrated in Figure 1.7. In this example, a monochromatic beam of radiation of wavelength λ and intensity I0 is incident on an absorber of thickness t (with differential thickness dt) and density ρ. A certain portion, I, of the radiation may pass through the absorber. Where this happens the wavelength of the transmitted beam is unchanged and the intensity of this transmitted beam I(λ) is given by

      (1.10) equation

      Figure 1.7. Absorption processes for X-rays.

      where μ/ρ is the mass attenuation coefficient of the absorber for the wavelength λ and the density ρ. Equation 1.10 is very general and is called the mass-absorption law. In other parts of the spectrum the same equation will be called the Lambert-Beer law. The value of the X-ray mass attenuation coefficient μ/ρ in Equation 1.10 is a function both of the photoelectric absorption τ and the scatter σ:

      (1.11) equation

      The scatter term contains contributions from coherent and incoherent scatter. However, τ is generally large in comparison with σ and generally μ/ρ f(τ). For this reason, the mass attenuation coefficient is often referred to as the mass absorption coefficient. The mass attenuation coefficient is independent of the physical state of a material (i.e., solid, liquid or gas) and depends only on the wavelength of the incident radiation. The wavelength dependence is roughly proportional to the cube of λ. However, since no one has found an exact theoretical relationship for the wavelength dependence, we must resort to measuring μ/ρ for all of the commonly used wavelengths and tabulating them. An empirical relationship,

      (1.12) equation

      known as the Bragg-Pierce law, has been established, where Z is the atomic number and K is an empirical constant that is different on each side of each of the absorption edges shown in Figure 1.8.

      Figure 1.8. The absorption curve for barium.

      The difference between I and I0 for a fixed wavelength is dependent on the thickness of the absorber and on the linear absorption coefficient μ, which is a constant related to the absorbing material. Since all of the absorption processes shown in Figure 1.7 ultimately depend on the presence of electrons, clearly the ability of a material to absorb electromagnetic radiation is related to the density of electrons. In turn, the electron density of a material is determined by the types of atoms composing the material and the closeness of their packing. The linear absorption coefficient of a material, therefore, depends on the types of atoms present and the density of the material. However, with the elimination of the functional dependence on density, which is determined by the type and strength of the chemical bonds in a material, a true constant for each type of element is obtained. Thus, μ/ρ is characteristic of each element at any specified wavelength. The absorption coefficients of the elements are listed in Appendix B.

      Photoelectric absorption occurs at each of the energy levels of the atom, and the total photoelectric absorption τ(total) is determined by the sum of each of the individual absorptions. Thus,

      (1.13)

      equation

      where τ(n) represents the outermost level of the atom containing electrons. It is apparent that all radiation produced as a result of electron transitions following ejection of orbital electrons must have a wavelength longer than that of the source which stimulated the excited state. Also, not all of the radiation produced is X-radiation; hence the photoelectric effect must be giving rise to X-radiation (λ) from the absorber and other photons.

      The various contributions to the total absorption from the different energy levels is illustrated in Figure 1.9, which shows the absorption curve for barium. As is seen in the figure, the value of the mass attenuation coefficient increases steadily with wavelength in accord with the Bragg-Pierce law (Equation 1.12); also the curve has very sharp discontinuities, called absorption edges, indicated as K, LI, LII, LIII, etc. These absorption edges correspond to the binding energy of electrons in the appropriate levels. Where the absorbed wavelength is shorter than the wavelength of one of the edges, an electron from the corresponding level can be excited. For instance, in Figure 1.9 an absorbed wavelength of 0.3 Å is shorter than the K absorption edge of barium (0.332 Å) and hence photoelectric absorption in the K level can occur. For an absorbed wavelength of 0.4 Å, however, photoelectric absorption in the K level certainly cannot occur. Thus, in general terms, each time the wavelength increases to a value in excess of a certain absorption edge, one of the terms in Equation 1.13 drops out with a corresponding decrease in the value of the total absorption term. Mass attenuation coefficients are well documented for most of the X-ray region, and the data are readily available in tabular form [15, 16]. Where the specimen is made up of n elements, the total mass attenuation coefficient of the specimen μs is given by

      Figure 1.9. The absorption curve for barium, plotted on a log/log scale showing the contributions of the different levels.

      From R. Jenkins, X-Ray Fluorescence Spectrometry, p. 10, Fig. 1-4.

      Copyright © 1988, John Wiley & Sons, Inc. Reprinted by permission of the publisher.

      (1.14) equation

      where wi is the weight fraction of element i.

      1.7. SAFETY CONSIDERATIONS

      It has been shown that X-rays are beams of energetic electromagnetic radiation that ionize matter with which they interact by ejecting electrons from their atoms. The extent of the ionization, absorption, and even molecular change of the material depends on the quantity (radiation flux and intensity) and the quality (the spectral distribution of the photon energy) of the radiation. Living organisms that are exposed to various doses of X-radiation can be injured by such exposures, and death may result if the exposure is particularly severe (see Figure 1.11). The amount of damage done to the body by the radiation is often difficult to estimate, but Figure 1.10 gives typical effects of various levels of exposure to human beings. The extent of the damage depends on the energy of the radiation, the total dose received, the type of tissue exposed, the dose rate, and the volume of the body exposed. It is thus vitally imperative that all operators of X-ray instruments be knowledgeable in their use to protect

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