Chapter 2.3. Powder and related techniques: X-ray techniques

The reflection specimen with θ–2θ scanning in the focusing arrangement shown in Fig. 2.3.1.3 is the most widely used powder diffraction method. It is estimated that about 10 000 to 15 000 of these diffractometers have been sold since they were introduced in 1948, which makes it the most widely used X-ray crystallographic instrument. Some authors have called it the Bragg–Brentano parafocusing method (Bragg, 1921; Brentano, 1946), but the X-ray optics (described below) are significantly different from the methods and instruments described by these authors.

Figure 2.3.1.3| top | pdf | X-ray optics in the focusing plane of a `conventional' diffractometer with reflection specimen, diffracted-beam monochromator, and θ–2θ scanning: ψ take-off angle, DC diffractometer circle, MFC monochromator focusing circle, αES and αRS entrance- and receiving-slit apertures, θ Bragg angle, 2θ reflection angle, O diffractometer and specimen rotation axis; other symbols listed in Fig. 2.3.1.1.

The X-ray tube spot focus was first used as the source and gave broad reflections. A narrow entrance slit improved the resolution but caused a large loss of intensity. Early diffractometers were described by LeGalley (1935), Lindemann & Trost (1940), and Bleeksma, Kloos & DiGiovanni (1948); see Parrish (1983). The use of the line focus with parallel slits to limit axial divergence was developed in the late 1940's and gave much higher resolution. A collection of papers by Parrish and co-workers (Parrish, 1965) and Klug & Alexander (1974) describe details of the instrumentation and method.

2.3.1.1.1. Geometrical instrument parameters

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The powder diffractometer is basically a single-axis goniometer with a large-diameter precision gear and worm drive. The detector and receiving-slit assembly are mounted on an arm attached to the gear in a radial position. The specimen is mounted in a holder carried by a shaft precisely positioned at the centre of the gear. 2/1 reduction gears drive the specimen post at one-half the speed of the detector. Some diffractometers have two large gears, making it possible to drive only the detector with the specimen fixed or vice versa, or to use 2/1 scanning. Synchronous motors have been used for continuous scanning for rate-meter recording and stepping motors for step-scanning with computer control.

The geometry of the method requires that the axis of rotation of the diffractometer be parallel to the X-ray tube focal line to obtain maximum intensity and resolution. The target is normal to the long axis of the tube; vertically mounted tubes require a diffractometer that scans in the vertical plane while a horizontal tube requires a horizontal diffractometer. The X-ray optics are the same for both. The incident angle θ and the reflection angle 2θ are defined with respect to the central ray that passes through the diffractometer axis of rotation O.

The axis of rotation of the specimen is the central axis of the main gear of the diffractometer, as shown in Fig. 2.3.1.3. The centre of the specimen is equidistant from the source F and receiving slit RS. The instrument radius R_DC = F − O = O − RS. The radius of commercial instruments is in the range 150 to 250 mm, with 185 mm most common. Changing the radius affects the instrument parameters and a number of the aberrations. Larger radii have been used to obtain higher resolution and better profile shapes. For example, the asymmetric broadening caused by axial divergence is decreased because the chord of the diffraction cone intercepted by the receiving slit has less curvature. However, if the same entrance slit is used, moving the specimen further from the source proportionately increases the length of specimen irradiated and decreases the intensity.

The imaginary specimen focusing circle SFC passes through F, O and the middle of RS and its radius varies with θ: The specimen holder is set parallel to the central ray at 0° and the gears drive the RS-detector arm at twice the speed of the specimen to maintain the θ–2θ relation at all angles. The source F is the line focus of the X-ray tube viewed at a take-off angle ψ. The actual width, , is foreshortened to In a typical case, = 0.4 mm and, at ψ = 5°, = 0.03 mm and the projected angular width is 0.025° for R = 185 mm. The angular aperture α_ES of the incident beam in the equatorial (focusing) plane is determined by the entrance slit width ES_w (also called the `divergence slit' since it limits the divergence of the beam) and its distance D₁ from F: Because the beam is divergent, the length of specimen irradiated S_l in the direction of the incident beam normal to O varies with θ: where α is in radians and is the distance from F to the crossover point before ES and is given by . The approximate relation is close enough for practical purposes (Parrish, Mack & Taylor, 1966). The intensity is nearly proportional to ES_α but the maximum aperture that can be used is determined by S_l and the smallest angle to be scanned 2θ_min, as shown in Fig. 2.3.1.4 . The entrance-slit width may be increased to obtain higher intensity at the upper angular range; for example, ES = 1° for the forward-reflection region and 4° for back-reflection.

Figure 2.3.1.4| top | pdf | Length of specimen irradiated, Sl, as a function of 2θ for various angular apertures. Sl = αR/sin θ, R = 185 mm.

Some slit designs are shown in Fig. 2.3.1.5 . The base is machined with a pair of rectangular shoulders whose separation A is the sum of the diameters of the two rods (a) or bar widths (b) and the central spacers on both ends that determine the slit opening. The distance P between the centre of the slit opening and the edge of the slit frame is kept constant for all slits to avoid angular errors when changing slits. The rods may be molyb­denum or other highly absorbing metal and are cemented in place. It may also be made in one piece (c) using machinable tungsten (Parrish & Vajda, 1966).

Figure 2.3.1.5| top | pdf | Slit designs made with (a) rods, (b) bars, and (c) machined from single piece. (d) Parallel (Soller) slits made with spacers or slots cut into the two side pieces (not shown) to position the foils.

A variable ES whose width increases with 2θ so that the irradiated length is about the same at all angles has been described by Jenkins & Paolini (1974). It is called a θ-compensating slit in which a pair of semicircular cylinders with a fixed opening is rotated around the axis of the opening by a linkage attached to the specimen shaft of the diffractometer to vary the aperture continuously with θ. The observed intensities must be corrected to obtain the relative intensities and the angular dependence of the aberrations is different from the fixed aperture slit.

Another way to irradiate constantly the entire specimen length is to use a self-centring slit which acts as an entrance and antiscatter slit (de Wolff, 1957). A 1 mm thick brass plate with rounded edge is mounted above the centre of the specimen and is moved in a plane normal to the specimen surface so that the aperture is proportional to sin θ. It can only be used for forward reflections.

Owing to the beam divergence, the geometric centre of the irradiated specimen length shifts a small amount during the scan (see also §2.3.5.1.5). It is generally advisable to centre the beam at the smallest 2θ to be scanned. Below about 20°, the irradiated length increases rapidly and it is essential to use small apertures and to align the entrance and antiscatter slits carefully. Failure to do this correctly could cause (a) errors in the relative intensities owing to the primary beam exceeding the specimen area, (b) cutoff by the walls of the specimen holder for low-absorbing thick specimens, and (c) increased background from scattering by the specimen holder or the primary beam entering the detector. The transmission specimen method (Subsection 2.3.1.2) has advantages in measuring large d's.

The beam converges after reflection on the receiving slit RS, whose width defines the reflection and profile width. Only those rays that are within the θ–2θ setting are in sharp convergence, i.e. `in focus'. The reflections become broader with increasing distance from the RS, and, therefore, this method is not suited for position-sensitive detectors. The RS aperture is the dominant factor in determining the intensity and resolution. For RS_w = 0.1 mm and R = 185 mm, α_RS = 0.031°.

Antiscatter slits AS are slightly wider than the beam and are essential in this and other geometries to make certain the detector can receive X-rays only from the specimen area. They must be carefully aligned to avoid touching the beam.

The use of the long X-ray source makes it necessary to reduce the axial divergence, which would cause very large asymmetry. This is done with two sets of thin (25 to 50 μm) parallel metallic foils PS (`Soller slits'; Soller, 1924) placed before and after the specimen. If a monochromator is used, the set on the side of the monochromator is not essential because the crystal reduces the divergence. The angular aperture of a set of slits is The overall width of the set and δ determine the width of the specimen irradiated in the axial direction, which remains constant at all 2θ's. The construction is illustrated in Fig. 2.3.1.5(d). The aperture δ is usually 2 to 5°. Each set of parallel slits reduces the intensity; for example, with 12.5 mm long foils with 1 mm spacings, the intensity is about one-half of that without the parallel slits. The aperture can be selected with any combination of spacings and lengths but the greater the length, the fewer foils are needed, and the less is the intensity loss due to thickness of the metal foils (usually 0.025 mm). These slits can be made as interchangeable units of different apertures.

2.3.1.1.3. Alignment and angular calibration

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It is essential to align and calibrate the diffractometer properly. Failure to do so degrades the performance of the instrument, leading to a loss of intensity and resolution, increased background, incorrect profile shapes, and errors that cannot be readily diagnosed. Procedures and devices for this purpose are often provided by the manufacturer. The principles and mechanical devices to aid in making a proper alignment have been described by Parrish & Lowitzsch (1959) and the general procedure by Klug & Alexander (1974, p. 280).

The alignment requires setting the diffractometer axis of rotation to the selected X-ray tube take-off angle at a distance equal to the radius of the diffractometer. The long axes of the X-ray tube focal line, entrance, receiving, and antiscatter slits must be centred, be parallel to the axis of rotation, and lie in the same plane when the instrument is at 0°. The slits are made parallel to the axis of rotation in the manufacture of the diffractometer, and these steps require positioning of the instrument with respect to the line focus. The parallel-slit foils must also be normal to the rotation axis. A flat fluorescent screen made as a specimen to fit into the diffractometer specimen post is used to centre the primary beam by small movements of the ES and/or diffractometer. The diffracted beam can be centred on the curved monochromator with a narrow slit placed at the centre of the monochromator position (with the monochromator removed). The detector arm is then moved to the highest intensity. The procedure is repeated with the receiving slit in position. This is very close to the 0° position described below.

The angular calibration of the diffractometer is usually made by accurately measuring the 0° position to establish a fiducial point. It assumes that the gear system is accurate and that the receiving-slit arm moves exactly to the angle indicated on the scale at all 2θ positions. The determination of the angular precision of the gear train requires special equipment and methods; see, for example, Jenkins & Schreiner (1986). It is essential that the setting of the worm against the gear wheel be adjusted for smooth operation. In practice, this is a compromise between minimum backlash and jerky movement. The backlash can be avoided by scanning in the same direction and running the diffractometer beyond the starting angle before beginning the data collection. Incremental angle encoders have been used when very high precision is required.

The 0° position of the diffractometer scale, 2θ₀, can be determined with a pinhole placed in the specimen post as shown in Fig. 2.3.1.6(a) (Parrish & Lowitzsch, 1959). The receiving slit is step scanned in 0.01° or smaller increments and the midpoints of chords at various heights are used to determine the angle. To avoid mis-centring errors of the pinhole, two measurements are made with the specimen post rotated 180° between measurements. The median angle of the two plots is the 0° position as shown in Fig. 2.3.1.6(b) and the diffractometer scale is then reset to this position. The shape of the curves is determined by the relative sizes of the pinhole and the receiving-slit width. With care, the position can be located to about 0.001°. The 2θ₀ position can be corrected by using it as a variable in the least-squares refinement of the lattice parameter of a standard specimen.

Figure 2.3.1.6| top | pdf | Zero-angle calibration. (a) XRT X-ray tube anode, ψ take-off angle, O axis of rotation, PH pinhole, RS receiving slit. Intensity distribution at right. (b) 0° position is median of two curves recorded with 180° rotation of PH.

Another method measures the peak angles of a number of reflections on both sides of 0°, which is equivalent to measuring 4θ. This method may be mechanically impossible with some diffractometers.

The θ–2θ setting of the specimen post is made with the diffractometer locked in the predetermined 0° position and manually (or with a stepping motor) rotating the post to the maximum intensity. A flat plate can be used as illustrated in Fig. 2.3.1.7(a) . The setting can be made to a small fraction of a degree. Fig. 2.3.1.7(b) shows the effect of incorrect θ–2θ setting, which combines with the flat-specimen aberration to cause a marked broadening and decrease of peak height but no apparent shift in peak position (Parrish, 1958). The effect increases with decreasing θ and could cause systematic errors in the peak intensities as well as incorrect profile broadening.

Figure 2.3.1.7| top | pdf | (a) θ–2θ setting at 0°. Flat plate or long narrow slit is rotated to position of highest intensity. (b) and (c) Profiles obtained with correct θ–2θ setting (solid profile) and 1 and 2° mis-settings (dashed profiles) at (b) 21° and (c) 60° (2θ).

2.3.1.1.4. Instrument broadening and aberrations

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The asymmetric form, broadening and angular shifts of the recorded profiles arise from the Kα doublet and geometrical aberrations inherent in the imperfect focusing of the particular diffractometer method used. There are additional causes of distortions such as the time constant and scanning speed in rate-meter strip-chart recording, small crystallite size, strain, disorder stacking and similar properties of the specimen, and very small effects due to refractive index and related physical aberrations.

Perfect focusing in the sense of reflection from a mirror is never realized in powder diffractometry. The focusing is approximate (sometimes called `parafocusing') and the practical selection of the instrument geometry and slit sizes is a compromise between intensity, resolution, and profile shape. Increasing the resolution causes a loss of intensity. When setting up a diffractometer, the effects of the various instrument and specimen factors should be taken into account as well as the required precision of the results so that they can be matched. There is no advantage in using high resolution, which increases the recording time (because of the lower intensity and smaller step increments), if the analysis does not require it. A set of runs to determine the best experimental conditions using the following descriptions as a guide should be helpful in obtaining the most useful results.

In the symmetrical geometries where the incident and reflected beams make the same angle with the specimen surface, the effect of absorption on the intensity is independent of the θ angle. This is an important advantage since the relative intensities can be compared directly without corrections. The actual intensities depend on the type of specimen. For a solid block of the material, or a compacted powder specimen, the intensity is proportional to , where μ is the linear absorption coefficient of the material. The transparency aberration [equation (2.3.1.13)], however, depends on the effective absorption coefficient of the composite specimen.

The need to correct the experimental data for the various aberrations depends on the nature of the required analysis. For example, simple phase identifications can often be made using data in which the uncertainty of the lattice spacing Δd/d is of the order of 1/1000, corresponding to about 0.025 to 0.05° precision in the useful identification range. This is readily attainable in routine practice if care is taken to minimize specimen displacement and the zero-angle calibration is properly carried out. The experimental data can then be used directly for peak search (Subsection 2.3.3.7) to determine the peak angles and intensities (Subsection 2.3.3.5) and the data entered in the search/match program for phase identification.

However, in many of the more advanced aspects of powder diffraction, as in crystal-structure determination and the characterization of materials for solid-state studies, much more detailed and more precise data are required, and this involves attention to the profile shapes. The following sections describe the origin of the instrumental factors that contribute to the shapes and shift the peaks from their correct positions. Many of these factors can be handled individually. With the use of computer programs, they can be determined collectively by using a standard sample without profile broadening and profile-fitting methods to determine the shapes (Subsection 2.3.3.6). The resulting instrument function can then be stored and used to determine the contribution of the specimen to the observed profiles.

A series of papers describing the geometrical and physical aberrations occurring in powder diffractometry has been published by Wilson (1963, 1974). His work provides the mathematical foundation for understanding the origin and treatment of the various sources of errors. The major aberrations are described in the following and are illustrated with experimental profiles and plots of computed data for better visualization and interpretation of the effects. The information can be used to correct the experimental data, interpret the profile broadening and shifts, and evaluate the precision of the analysis. Chapter 5.2 contains tables listing the centroid displacements and variances of the various aberrations.

The magnitudes of the aberrations and their effects are illustrated in Figs. 2.3.1.8 (a) and (b), which show the Cu spectrum inside the experimental profile. At high 2θ's, the shape of the experimental profile is largely determined by the spectral distribution, but at low 2θ's the aberrations are the principal contributors. The basic experimental high-resolution profile shapes from specimens without appreciable broadening effects (NIST silicon powder standard) are shown in Figs. 2.3.1.8(c)–(f). The solid-line profiles were obtained with a reflection specimen (Fig. 2.3.1.3). The differences in the doublet separations are explained in Subsection 2.3.1.2. These profiles are the basic instrument functions which show the profile shapes contained in all reflections recorded with these methods. The shapes are modified by changing slit sizes.

Figure 2.3.1.8| top | pdf | Diffractometer profiles. (a) and (b) Spectral profiles λ of Cu Kα doublet (dashed-line profiles) inside experimental profiles R (solid line). (c)–(f) Experimental profiles with reflection specimen (R) geometry (Fig. 2.3.1.3) with αES 1° and αRS 0.046° (solid line profiles), and transmission specimen (T) (Fig. 2.3.1.12) with αES 2° and receiving axial divergence parallel slits (dotted profiles). Cu Kα radiation. (a) Si(531), (b) quartz(100), (c) Si(111), (d) Si(220), (e) Si(311), and (f) Si(422).

2.3.1.1.5. Focal line and receiving-slit widths

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The projected source width and receiving-slit width RS_w each add a symmetrical broadening to the profiles that is constant for all angles. Both the profile width and the intensity increase with increasing take-off angle (Section 2.3.5). However, the contribution of is small when the line focus is used, Fig. 2.3.1.9(a) . The receiving slit can easily be changed and it is one of the most important elements in controlling the profile width, intensity, and peak-to-background ratio, as is shown in Figs. 2.3.1.9(a) and (c). Because of the contributions of other broadening factors, α_RS can be about twice α_F (line focus) without significant loss of resolution.

Figure 2.3.1.9| top | pdf | (a) Effect of source size on profile shape, Cu Kα, αES 1°, αRS 0.05°, Si(111). Effect of receiving-slit aperture αRS on profiles of quartz (b) (100) and (c) (121); peak intensities normalized, Cu Kα, αES 1°.

The projected width of the X-ray tube focus is given in equation (2.3.1.2). The aperture is For a line focus with actual width = 1 mm, ψ = 5°, and R = 185 mm, α_F = 0.011°. The receiving-slit aperture is For RS_w = 0.2 mm and R = 185 mm. α_RS is 0.062°. The FWHM of the profiles is always greater than the receiving-slit aperture because of the other broadening factors.

2.3.1.1.6. Aberrations related to the specimen

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The major displacement errors arising from the specimen are (1) displacement of the specimen surface from the axis of rotation, (2) use of a flat rather than a curved specimen, and (3) specimen transparency. These are illustrated schematically for the focusing plane in Fig. 2.3.1.10(a) . The rays from a highly absorbing or very thin specimen with the same curvature as the focusing circle converge at A without broadening and at the correct 2θ. The rays from the flat surface cause an asymmetric profile shifted to B. Penetration of the beam below the surface combined with the flat specimen causes additional broadening and a shift to C.

Figure 2.3.1.10| top | pdf | (a) Origin of specimen-related aberrations in focusing plane of conventional reflection specimen diffractometer (Fig. 2.3.1.3). A no aberration from curved specimen; B flat specimen; C specimen displacement from 0. (b) Computed angular shifts caused by specimen displacement, R = 185 mm. (c) Flat-specimen asymmetric aberration, Si(422), Cu Kα1, Kα2 peak intensities normalized. (d) Computed flat-specimen centroid shifts for various apertures; parabola for constant irradiated 2 cm specimen length. (e) Transparency asymmetric aberration, LiF(200) powder reflection, Cu Kα, peak intensities normalized, thin specimen (solid-line profile) 0.1 mm thick; thick specimen (dotted-line profile) 1.0 mm, αES 1°, αRS 0.046°. (f) Computed transparency centroid shifts for various values of linear absorption coefficient.

The most frequent and usually the largest source of angular errors arises from displacement of the specimen surface from the diffractometer axis of rotation. It is not easy to avoid and may arise from several sources. It is advisable to check the reproducibility of inserting the specimen in the diffractometer by recording an isolated peak at low 2θ for each insertion. If only a radial displacement s occurs, the reflection is shifted where R is the diffractometer radius. A plot of equation (2.3.1.10) is shown in Fig. 2.3.1.10(b). The shift is to larger or smaller angles depending on the direction of the displacement and there is no broadening if the displacement is only radial and relatively small. Even a small displacement causes a relatively large shift; for example, if s = +0.1 mm and R = 185 mm, = + 0.06° at 20°2θ. This gives rise to a systematic error in the recorded reflection angles, which increases with decreasing 2. It could be handled with a plot, providing it was the only source of error. There are other possible sources of displacement such as (a) if the bearing surface of the specimen post was not machined to lie exactly on the axis of rotation, (b) improper specimen preparation or insertion in which the surface was not exactly coincident with the bearing surface or (c) nonplanar specimen surface, irregularities, large particle sizes, and specimen transparency. Source (a) leads to a constant error in all measurements, and errors due to (b) and (c) vary with each specimen.

Ideally, the specimen should be in the form of a focusing torus because of the beam divergence in the equatorial and axial planes. The curvatures would have to vary continuously and differently during the scan and it is impracticable to make specimens in such forms. An approximation is to make the specimen in a flexible cylindrical form with the radius of curvature increasing with decreasing 2θ (Ogilvie, 1963). This requires a very thin specimen (thus reducing the intensity) to avoid cracking and surface irregularities, and also introduces background from the substrate. A compromise uses rigid curved specimens, which match the SFC (Fig. 2.3.1.3) at the smallest 2θ angle to be scanned, and this eliminates most of the aberration (Parrish, 1968). A major disadvantage of the curvature is that it is not possible to spin the specimen.

In practice, a flat specimen is almost always used. The specimen surface departs from the focusing circle by an amount h at a distance l/2 from the specimen centre: This causes a broadening of the low-2θ side of the profile and shifts the centroid to lower 2θ: For α = 1° and = 20°, = −0.016°. The peak shift is about one-third as large as the centroid shift in the forward-reflection region. This aberration can be interpreted as a continuous series of specimen-surface displacements, which increase from at the centre of the specimen to a maximum value at the ends. The effect increases with α and decreasing 2θ. The profile distortion is magnified in the small 2θ-angle region where the axial divergence also increases and causes similar effects. Typical flat-specimen profiles are shown in Fig. 2.3.1.10(c) and computed centroid shifts in Fig. 2.3.1.10(d).

The specimen-transparency aberration is caused by diffraction from below the surface of the specimen which asymmetrically broadens the profile (Langford & Wilson, 1962). The peak and centroid are shifted to smaller 2θ as shown in Fig. 2.3.1.10(e). For the case of a thick absorbing specimen, the centroid is shifted and for a thin low-absorbing specimen where μ is the effective linear absorption coefficient of the specimen used, t the thickness in cm, and R the diffractometer radius in cm. The intermediate absorption case is described by Wilson (1963). A plot of equation (2.3.1.13) for various values of μ is given in Fig. 2.3.1.10(f). The effect varies with sin2θ and is maximum at 90° and zero at 0° and 180°. For example, if μ = 50 cm⁻¹, the centroid shift is −0.033° at 90° and falls to −0.012° at 20°2θ.

The observed intensity is reduced by absorption of the incident and diffracted beams in the specimen. The intensity loss is , where μ is the linear absorption coefficient of the powder sample (it is almost always smaller than the solid material) and x_s is the distance below the surface, which may be equal to the thickness in the case of a thin film or low-absorbing material specimen. The thick (1 mm) specimen of LiF in Fig. 2.3.1.10(e) had twice the peak intensity of the thin (0.1 mm) specimen.

The aberration can be avoided by making the sample thin. However, the amount of incident-beam intensity contributing to the reflections could then vary with θ because different amounts are transmitted through the sample and this may require corrections of the experimental data. Because the effective reflecting volume of low-absorbing specimens lies below the surface, care must be taken to avoid blocking part of the diffracted beam with the antiscatter slits or the specimen holder, particularly at small 2θ.

There are additional problems related to the specimen such as preferred orientation, particle size, and other factors; these are discussed in Section 2.3.3.

2.3.1.1.7. Axial divergence

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Divergence in the axial direction (formerly also called `vertical divergence') causes asymmetric broadening and shifts the reflections. The aberration is illustrated in Fig. 2.3.1.11 for a low-2θ reflection in the transmission-specimen mode (Subsection 2.3.1.2). The narrow profile was obtained with δ = 4.4° parallel slits placed between the monochromator and detector, and the broad profile with the slits removed. The slits caused a 33% reduction in peak intensity. This problem was recognized in the first design of the diffractometer using the X-ray tube line focus when parallel slits were used in the incident and diffracted beams to limit the effect (Parrish, 1949). Increasing the radius reduces the effect if the slit length is kept constant. The intensity is also reduced because the chord length intercepted is a smaller fraction of the longer radius diffraction cone. The construction of parallel (Soller) slits (Soller, 1924) is shown in Fig. 2.3.1.5(d).

Figure 2.3.1.11| top | pdf | Effect of axial divergence on profile shape. Narrow profile recorded with parallel slits (PS), δ = 4.4° between monochromator and detector (Fig. 2.3.1.12), and broad profile with these parallel slits removed. Faujasite, Cu Kα, αES 2°.

The calculation of the aberration and the present status is summarized by Wilson (1963, pp. 40–45). The results depend on the aperture of the parallel slits, the length of the entrance and receiving slits, and 2θ. In the limit of small s, the shift of the centroid is where s is the spacing and l the length of the foils. The shift becomes very large at small 2θ's but not infinite as equation (2.3.1.15) implies. The shift is to smaller 2θ's in the forward-reflection region and to larger 2θ's in back-reflection. However, the mathematical formulation is difficult to quantify because in the forward-reflection region the axial divergence convolves with the flat-specimen aberration to increase the asymmetry. In the back-reflection region, the effect is not so obvious because the distortion is smaller and the Lorentz and dispersion factors also stretch the profiles to higher angles.

Transmission-specimen methods are not as widely used as reflection methods but they provide important supplemental data and have advantages in a number of applications. Reflections occur from lattice planes oriented normal to the specimen surface rather than parallel. Reflection and transmission patterns can be compared to determine texture and preferred-orientation effects. The transmission method is better suited to the measurement of large d's. Smaller specimen volumes are required. The surface `roughness' which may cause large intensity errors due to the microabsorption in reflection specimens is largely reduced.

The same basic diffractometer is used for both methods but the geometry is different because the diffracted beam continues to diverge after it passes through the specimen and the monochromator is required to refocus the beam, on the detector as shown in Fig. 2.3.1.12 (de Wolff, 1968b; Parrish, 1958). The monochromator can be placed before or after the specimen and the position has different effects on the pattern. Using the monochromator in the diffracted beam, the intensity and width of the profiles are determined by the X-ray focal line width and the quality of the bent monochromator rather than the receiving slit which serves as an antiscatter slit. This geometrical arrangement places the virtual image VI of the focal line at the intersection of the focusing circles. After reflection from the specimen, the divergent beam is again reflected by the focusing crystal M and converges on the detector. The pattern is recorded with θ–2θ scanning with the monochromator and detector both mounted on a rigid arm rotating around the diffractometer axis. A beam stop MS can be translated and moved in and out near the crossover point to prevent the primary beam from entering the detector at small 2θ's. To avoid long radii, the crystal surface is cut at an angle σ (about 3°) to the reflecting lattice plane. The distances are related by where is the Bragg angle of the monochromator for the selected wavelength and the l's are shown in Fig. 2.3.1.12(a).

Figure 2.3.1.12| top | pdf | X-ray optics of the transmission specimen with asymmetric focusing monochromator and θ–2θ scanning. (a) Monochromator in diffracted beam. θM Bragg angle of monochromator with surface cut at angle σ to reflecting plane, MS adjustable beam stop, I1, I2, and I3 defined in text and other symbols listed in Fig. 2.3.1.3. (b) Monochromator in incident beam, equivalent to Guinier focusing camera.

Because the profile shape and the intensity are determined by the monochromator, the crystal quality and the accuracy of the bending are crucial factors in determining the quality of the pattern. A flat thin quartz (101) wafer bent with a special device to approximate a section of a logarithmic spiral has been successfully used (de Wolff, 1968b). The curvature can be varied to obtain the sharpest focus. Thin silicon crystals that can be bent are now available, and Johann and Johannsen asymmetric crystals may be used. Pyrolytic graphite monochromators are not applicable; the radii would be longer because graphite is too soft to be cut at an angle, and a receiving slit would be necessary to define the diffracted beam because the monochromator produces a broad reflection.

A polarization factor is introduced by the monochromator, where for mosaic crystals and for perfect crystals. The value of k is strongly dependent on the surface finish of the crystal and the crystal should be measured to determine the effect. A specimen with accurately known structure factors such as silicon can be used to calibrate the intensities.

The Kα-doublet separation is zero at the 2θ angle at which the dispersion of the specimen compensates that of the monochromator, i.e. the 2θ at which the monochromator is aligned and also depends on the distances. The Kα₁ and Kα₂ peaks are superposed and appear as a single peak over a small range of 2θ's. The Kα₂ peak gradually separates with increasing 2θ but the separation is less than calculated from the wavelengths and the intensity ratio may not be 2:1 until higher angles are reached as shown in Fig. 2.3.1.8.

A larger angular aperture α_T can be used for transmission than for reflection α_R because the specimen is more nearly normal to than parallel to the primary beam: where the diffractometer radius R_D = l₁. For R_D = 170 mm, specimen length = 20 mm and l₂ = 65 mm; α_T could be 4.7 times larger than α_R but the monochromator length usually limits it to about 3°. The smallest reflection angle that can be measured is Using α_T = 0.5°, 2θ_min = 1.75° and d = 50 Å for Cu Kα radiation.

Specimen preparation is not difficult and the preparation can be easily tested and changed. The specimen must be X-ray transparent and can be a free-standing film or foil, or a powder cemented to a thin substrate. The substrate selection is important because its pattern is included. If both transmission and reflection patterns are to be compared, the substrate should be selected to have a minimal contribution to both. For example, Mylar is a good substrate for transmission but has a strong reflection pattern, and although rolled Be foil has a few reflections it is often satisfactory for both.

The absorption factor is where t is the powder thickness and s is the sum of the products of the absorption coefficients and thicknesses of the powder and the substrate. The optimum specimen thickness to give the highest intensity is μt = 1, i.e. the specimen should transmit about 38% of the incident Kα intensity. The transmission can be easily measured with a standard specimen set to reflect the Kα and the specimen to be measured inserted normal to the diffracted beam in front of the detector. It is not critical to achieve the exact value and a range of ±15–20% of the transmission can be tolerated. This minimizes the effect of the absorption change with 2θ, and corrections of the relative intensities are required only when accurate values are required.

The intensity of the incident beam can be measured at 0° in the same geometry and used to scale the relative intensities to `absolute' values. The flat specimen, transparency, and specimen surface displacement aberrations are similar to those in reflection specimen geometry except that they vary as rather than . This is an important factor in the measurement of large-d-spacing reflections. The flat-specimen effect is smaller because the irradiated specimen length is usually smaller. The transparency error is also usually smaller because thin specimens are used.

An important advantage of the method is that the specimen displacement can be directly determined by measuring the peak in the normal position and again after rotating the specimen holder 180°. The correct peak position is at one-half the angle between the two values. The axial divergence has the same effect as in reflection. The limitations are that only the forward-reflection region is accessible, and the intensity is about one-half of the reflection method (except at small angles) because smaller specimen volumes are used.

An alternative arrangement for the transmission specimen mode is to use an incident-beam monochromator as shown in Fig. 2.3.1.12(b). This is similar to the geometry used in the Guinier powder camera with the detector replacing the film. A high-quality focusing crystal is required. Wölfel (1981) used a symmetrical focusing monochromator with 260 mm focal length for quantitative analysis. Göbel (1982) used an asymmetric monochromator with a position-sensitive detector for high-speed scanning, see §2.3.5.4.1. By proper selection of the source size and distances, the Kα₂ can be eliminated and the pattern contains only the Kα₁ peaks (Guinier & Sébilleau, 1952). This geometry can have high resolution with the FWHM typically about 0.05 to 0.07°. The profile widths are narrower for the subtractive setting of the monochromator than for the additive setting.

The pattern is recorded with θ–2θ scanning. The 0° position can be determined by measuring 4θ, i.e. peaks above and below 0°, or calibration can be made with a standard specimen. A slit after the monochromator limits the size of the beam striking the specimen. The width and intensity of the powder reflections are limited by the receiving-slit width. A parallel slit is used between the specimen and detector to limit axial divergence.

The full spectrum from the X-ray tube strikes the monochromator and only the monochromatic beam reaches the specimen, so that it is preferred for radiation-sensitive materials. On the other hand, the radiation reaching the specimen may cause fluorescence (though considerably less than the full spectrum) which adds to the background.

The Seemann–Bohlin (S–B) diffractometer has the specimen mounted on a radial arm instead of the axis of rotation and a linkage or servomechanism moves the detector around the circumference of a fixed-radius focusing circle while keeping it pointed to the stationary specimen. All reflections occur simultaneously focused on the focusing circle as shown in Fig. 2.3.1.13(a) . The method was originally developed for powder cameras by Seemann (1919) and Bohlin (1920) but was not widely used because of the limited angular range and the broad reflections caused by inclination of the rays to the film. The diffractometer eliminates the broadening and extends the angular range. Diffractometers designed for this geometry have been described by Wassermann & Wiewiorosky (1953), Segmüller (1957), Kunze (1964a , b, c), Parrish, Mack & Vajda (1967), King, Gillham & Huggins (1970), Feder & Berry (1970), and others.

Figure 2.3.1.13| top | pdf | Seemann–Bohlin method. (a) X-ray optics using incident-beam monochromator. (b) X-ray tube line-focus source showing geometrical relations: γ mean angle of incident beam, βH inclination of reflecting plane H to specimen surface, θH Bragg angle of H plane, t tangent to focusing circle at O. (c) Diffractometer settings for various angular ranges.

The geometry is shown in Fig. 2.3.1.13(b) (Parrish & Mack, 1967). Reflections occur from lattice planes with varying inclinations β_H to the specimen surface. The reflecting position of a plane H is , where γ is the incidence angle and the reflection angle. The maximum value of β_H is about 45°. It is essential to align the specimen tangent to FC. This is a critical adjustment because even a small misalignment causes profile broadening and loss of peak intensity.

The source may be the line focus of the X-ray tube [F in Fig. 2.3.1.13(b)] or at the focus of a monochromator [ES in Fig. 2.3.1.13(a)]; in the latter case, the entrance slit at F′ limits the divergent beam reaching the specimen. The source, specimen centre O, and receiving slit RS lie on the specimen focusing circle SFC, which has a fixed radius r. The incidence angle γ is given by where b is the distance from F or F ′ to O, or 2r sin γ. The γ angle determines the angular range that can be recorded with a given r, decreasing γ decreases 2θ_min. The relationships of specimen position on the focusing circle and the recording range are illustrated in Fig. 2.3.1.13(c). To change the range requires rotation of the X-ray tube axis or the diffractometer around F. The detector must also be repositioned. For forward-reflection measurements, γ is usually . Extreme care must be used in the specimen preparation to avoid errors due to microabsorption (particle-shadowing) effects, which increase with decreasing γ. The 0° position cannot be measured directly and a standard is used for calibration. The range from 0° to about 15°2θ is inaccessible because of mechanical dimensions. At γ = 90°, only the back-reflection region can be scanned.

The aperture of the beam striking the specimen is where ES_w is the entrance slit width and a the distance between F or F′ and the slit. The irradiated specimen length l is constant at all angles, l = 2αr. A large aperture can be used to increase intensity since the specimen is close to F. However, the selection of α is limited if γ is small, and also because of the large flat-specimen aberration.

The receiving-slit aperture varies with the distance of the slit to the specimen Consequently, the resolution and relative intensity gradually change across the pattern. The S–B has greater widths at the smaller 2θ's and nearly the same widths at the higher angles compared with the θ–2θ diffractometer. The aperture can be kept constant by using a special slit with offset sides (to avoid shadowing) and pointing the opening to C while the detector remains pointed to O (Parrish et al., 1967). The slit opening is tangent to FC and inclined to the beam and rotates while scanning. The constant aperture slit has

The axial divergence is limited by parallel slits as in conventional diffractometry and the effects are about the same. The equatorial aberrations are also similar but larger in magnitude. The specimen-aberration errors are listed in Table 5.2.7.1 . The flat specimen causes asymmetric broadening; the shift is proportional to and increases with decreasing θ. It can be eliminated by making the specimen with the same curvature as r = FC. In this case, one curvature satisfies the entire angular range because the focusing circle has a fixed radius. However, the curvature precludes rotating the specimen. The specimen transparency also causes asymmetric broadening and a peak shift that increases with decreasing θ. For , the geometric term is the same as for specimen displacement (Mack & Parrish, 1967).

The diffracted intensity is proportional to I₀A(μ h)TB, where I₀ is the incident intensity determined by α, δ, and the axial length L of the incident-beam assembly, A(μh) is the specimen absorption factor, T the transmission of the air path, and B the length L_RS of the diffracted ring intercepted by the slit. The X-rays reflected at a depth x below the specimen surface are attenuated by where μ is the linear absorption coefficient. The asymmetric geometry causes the absorption to vary with the reflection angle. The air absorption path varies with the distance O to RS and reaches a maximum at 180° + 2γ. The expression for air transmission includes the radius of the X-ray tube R_T, which is needed only for the case where the X-ray tube focal line is used as F. In a typical instrument with X-ray tube source F and r = 174 mm, the transmission of Cu Kα decreases from 90% at 40°4θ to 62% at 210°, and Cr Kα from 73 to 23% at the same angles.

Some of the advantages of the method include the following: (a) the fixed specimen makes it possible to simplify the design of specimen environment devices; (b) a large aperture can be used and the intensities are higher than for conventional diffractometers; (c) the flat-specimen aberration can be eliminated by a single-curvature specimen; (d) a small γ angle can be used to increase the path length l in the specimen, and hence the intensity of low-absorbing thin-film samples ( and for γ = 5°, l = 11.5t); (e) the method is useful in thin-film and preferred-orientation studies because about a 45° range of lattice-plane orientations can be measured and compared with conventional patterns. The limitations include (a) the more complicated diffractometer and its alignment, (b) limited angular range of about 10 to 110°2θ for the forward-reflection setting, (c) extreme care required in specimen preparation, and (d) larger aberration errors.

In this geometry, the specimen is fixed in the horizontal plane and the X-ray tube and detector are synchronously scanned in the vertical plane in opposite directions above the centre of the specimen as shown in Fig. 2.3.1.14 . The distances source to S and S to RS are equal to that the angles of incidence and diffraction and a constant dθ/dt are maintained over the entire angular range. A focusing monochromator can be used in the incident or diffracted beam. High- and low-temperature chambers are simplified because the specimen does not move. The arms carrying the X-ray tube and detector must be counterbalanced because of the unequal weights. The method has advantages in certain applications such as the measurement of liquid scattering without a covering window, high-temperature molten samples, and other applications requiring a stationary horizontal sample (Kaplow & Averbach, 1963; Wagner, 1969).

Figure 2.3.1.14| top | pdf | Optics of θ–θ scanning diffractometer. X-ray tube and detector move synchronously in opposite directions (arrows) around fixed horizontal specimen. A focusing monochromator can be used after the receiving slit.

There are two types of microdiffraction: (a) only a very small amount of powder is available, and (b) information is required from very small areas of a conventional-size specimen. Small-volume samples have been analysed with a conventional diffractometer by concentrating the powder over a small spot centred on a single-crystal plate such as silicon (510) or an AT-cut quartz plate, or on Mylar for transmission. It is essential to rotate the specimen and increase the count time. A Gandolfi camera has also been used for very small specimens (see Section 2.3.4). A high-brilliance microfocus X-ray source has been used with a collimator made of 10 to 100 μm internal-diameter capillary tube. An X–Y stage is used with an optical microscope to locate selected areas of the specimen.

A microdiffractometer has been designed for microanalysis, Fig. 2.3.1.15 (Rigaku Corporation, 1990). It has been used to determine phases and stress in areas  μm² (Goldsmith & Walker, 1984). The key to the method is the use of an annular-ring receiving slit, which transmits the entire diffraction cone to the detector instead of a small chord as in conventional diffractometry, thereby utilizing all the available intensity. The pattern is scanned by translating the ring and detector along the direct-beam path so that where is the radius of the ring slit and L the distance from the fixed specimen. For R_RS = 15 mm, L varies from 171 to 9 mm in the transmission range 5 to 60°2θ; a 50 mm diameter scintillation counter is used. A doughnut-shaped proportional counter (3/4 of a full circle) is used for the 30 to 150° reflection specimen mode. The slit width is 0.2 mm and the aperture varies with 2θ. The intensities fall off at the higher 2θ's because of the small incidence angles to the slit. An alternative method uses a position-sensitive proportional counter. Steinmeyer (1986) has described applications of microdiffractometry.

Figure 2.3.1.15| top | pdf | Rigaku microdiffractometer for microanalysis. C collimator, PC ring proportional counter, RS ring slit with radius r, S specimen, SC scintillation counter, PBS primary beam stop, PH pinhole for alignment, L specimen-to-receiving-slit distance.

By using synchrotron radiation (Section 2.3.2), single-crystal data for structure determination can now be obtained from a microcrystal about 5–10 μm in size; see Andrews et al. (1988), Bachmann, Kohler, Schultz & Weber (1985), Harding (1988), Newsam, King & Liang (1989), Cheetham, Harding, Mingos & Powell (1993), Harding & Kariuki (1994), and Harding, Kariuki, Cernik & Cressey (1994).

The X-ray optics of a plane-wave parallel-beam diffractometer is shown schematically in Fig. 2.3.2.4(a) . The primary white beam is limited by slits at C1. A channel monochromator CM is used because it has the important property of maintaining the same direction and position for a wide range of wavelengths. It may be used in the dispersive setting with respect to the specimen or in the parallel setting [Fig. 2.3.2.4(b)]. The monochromatic beam is larger than the entrance slit ES and it is unnecessary to realign the powder diffractometer when changing wavelengths. The monochromator can be mounted on a stripped diffractometer for easy alignment and step scanning.

Figure 2.3.2.4| top | pdf | (a) Optics of dispersive parallel-beam method for synchrotron X-rays. C1 primary-beam collimator, D1 diffractometer for channel monochromator CM, C2 antiscatter shield, Be beryllium foil for monitor, SC1 and SC2 scintillation counters, ES entrance slit on powder diffractometer D2, VPS vertical parallel slits to limit axial divergence, HPS horizontal parallel slits, which determine the resolution. (b) CM in nondispersive setting and crystal analyser A used as a narrow receiving slit. (c) Fibre specimen FS with receiving slit RS or with position-sensitive detector (not shown) with RS removed.

There are no characteristic spectral lines and the wavelength calibration of the monochromator is made by step scanning the monochromator across absorption edges of elements in a specimen or pure element foils placed in the beam. The wavelength accuracy is limited by the uncertainty as to what feature of the edge should be measured and which one was used for the wavelength tables. A standard powder sample such as NIST silicon 640 b whose lattice parameter is known with moderately high precision can also be used. An alternative method is to measure the reflection angle of a single-crystal plate of float-zones oxygen-free silicon whose lattice parameter is known to 1 part in 10⁻⁷ and to determine the wavelength from the Bragg equation (Hart, 1981). The accuracy is then limited by the angular accuracy of the diffractometer and the orientation setting.

It is necessary to monitor the monochromatic beam intensity , which changes during the recording due to decreasing storage-ring current, orbital shifts or other factors. This can be done by inserting a low-absorbing ionization chamber in the beam or by using a scintillation counter to measure scattering from an inclined thin beryllium foil, kapton or other low-absorbing material. The data are recorded and used to correct the observed intensities. The monitored counts can also be used as a timer for step scanning if a sufficient number are recorded for good counting statistics.

The entrance slit ES determines the irradiated specimen length, which is equal to . Vertical parallel slits VPS with ° are used to limit the axial divergence. The longer the distance between the specimen and detector, the smaller the asymmetry, and a vacuum path should be used to avoid air-absorption losses. The specimen may be used in either reflection or transmission simply by rotating it 90° around the diffractometer axis from its previous position.

The diffracted beam can be defined by a receiving slit (Parrish, Hart & Huang, 1986), horizontal parallel slits HPS [Fig. 2.3.2.4(a)] (Parrish & Hart, 1985) or a high-quality single-crystal plate which acts as a very narrow receiving slit [Fig. 2.3.2.4(b)] (Cox, Hastings, Thomlinson & Prewitt, 1983; Hastings et al., 1984). If a receiving slit is used, the intensity, profile width and shape are determined by the widths of both ES and RS. If either one is much wider than the other, the profile has a flat top. Increasing the RS width and keeping ES constant causes symmetrical profile broadening and increases the intensity as in conventional focusing diffractometry. There are disadvantages in using a receiving slit because the intensities are low and it causes the same specimen-surface-displacement and transparency errors as the focusing geometries.

A set of horizontal parallel (Soller) slits is advantageous because of the much higher intensity and it eliminates the displacement errors. The profiles of specimens without broadening effects have the same FWHM as the aperture of the slits, equation (2.3.1.7). The FWHM increases as due to wavelength dispersion. By increasing the length of the foils and keeping the same spacing, the aperture can be reduced to increase the resolution without large loss of intensity. A set of 365 mm long slits with 0.05° aperture has been used and even smaller apertures are feasible. Longer slits decrease the fluorescence intensity (if any) reaching the detector. They must be carefully made and aligned to avoid loss of intensity and should be evacuated or filled with He to avoid air-absorption losses.

The use of a crystal analyser eliminates fluorescence and gives the highest resolution powder profiles with FWHM = 0.02 to 0.05°2θ, depending on the quality of the crystal (Hastings et al., 1984). The alignment of the crystal is critical and must be done with remote automated control every time the wavelength is changed. Displacement aberrations are eliminated but the intensity is much lower than the HPS because of the small rocking angle and low integrated reflectivity of the crystal.

The correct orientation of crystalline powder particles for reflection is far more restrictive for the parallel beam than the X-ray tube divergent beam. A much smaller number of particles will have the exact orientation for reflection, and thus the recorded intensity will be lower and relative intensities less accurate. If the specimen is stationary, the standard deviations of the intensities due to particle size are six to nine times higher than in focusing methods (Parrish, Hart & Huang, 1986). It also becomes more difficult to achieve the completely randomly oriented specimens required for structure determination and quantitative analysis and, as in X-ray tube data, a preferred-orientation term is included in the structure refinement. It is, therefore, essential to use small particles  μm and to rotate the specimen. Some investigators prefer to oscillate the specimen over a small angle but this is not as effective as rotation.

The profiles are virtually symmetrical except at small angles where axial divergence causes asymmetry. The profiles in Fig. 2.3.2.5 show the differences in the shape and resolution obtained with conventional focusing (a) and parallel-beam synchrotron methods (b). The effect of the higher resolution on a mixture of nearly equal volumes of quartz, orthoclase, and feldspar recorded with X-ray tube focusing methods is shown in Fig. 2.3.2.5(c) and with synchrotron radiation in Fig. 2.3.2.5(d). The symmetry and nearly constant simple instrument function make it easier to separate overlapping reflections and simplify the profile-fitting procedures and the interpretation of specimen-broadening effects.

Figure 2.3.2.5| top | pdf | Comparison of patterns obtained with a conventional focusing diffractometer (a) and (c), and synchrotron parallel-beam method (b) and (d). (a) and (b) quartz powder profiles; (c) and (d) mixture of equal amounts of quartz, orthoclase, and feldspar.

The early crystal-structure studies using Rietveld refinement were not as successful with X-ray tube focusing methods as they were with neutron diffraction because the complicated instrument function made profile fitting difficult and inaccurate. The development of synchrotron powder methods with simple symmetrical instrument function, high resolution, and the use of longer wavelengths to increase the dispersion have made structural studies as successful as with neutrons, and have the advantage of orders-of-magnitude higher intensity. Some examples are described by Attfield, Cheetham, Cox & Sleight (1988), Lehmann, Christensen, Fjellvåg, Feidenhans'l & Nielsen (1987), and ab initio structure determinations by McCusker (1988), Cernik et al. (1991), Morris, Harrison, Nicol, Wilkinson & Cheetham (1992), and others.

Structures have also been solved using a two-stage method in which the integrated intensities are determined by profile fitting the individual reflections and used in a powder least-squares refinement method (POWLS) (Will, Bellotto, Parrish & Hart, 1988). The method was tested with silicon, which gave R(Bragg) 0.7%, and quartz, which gave 1.6%, which is a good test of the high quality of the experimental data and the profile-fitting procedure. Fig. 2.3.2.6 shows Fourier maps of orthorhombic Mg₂GeO₄ calculated using Fourier coefficients taken directly from the profile-fitting intensities.

Figure 2.3.2.6| top | pdf | (a) and (c) Fourier maps of orthorhombic Mg2GeO4 calculated directly from profile-fitted synchrotron powder data. (b) Fourier section of isostructural Mg2SiO4 calculated from single-crystal data for comparison with (a).

Other types of powder studies have been carried out successfully. For example, these have been used in anomalous-scattering studies (Will, Masciocchi, Hart & Parrish, 1987; Will, Masciocchi, Parrish & Hart, 1987), Warren–Averbach profile-broadening analysis (Huang, Hart, Parrish & Masciocchi 1987), studies of texture in thin films (Hart, Parrish & Masciocchi, 1987), and precision lattice-parameter determination (Hart, Cernik, Parrish & Toraya, 1990).

The flat specimen can be replaced by a thin cylindrical [Fig. 2.3.2.4(c)] specimen as used in powder cameras. The powder can be coated on a thin fibre or reactive materials can be forced into a capillary to avoid contact with air. The intensity is lower than for flat specimens because of the smaller beam, and less powder is required. Thompson, Cox & Hastings (1987) used the method to determine the structure of Al₂O₃ by Rietveld refinement. They used a two-crystal incident-beam Si(111) monochromator; the first crystal was flat and the second a cylindrically bent triangular plate for sagittal focusing to form a  mm beam with spectral bandwidth .

The method can also be used with a receiving slit or position-sensitive detectors (Lehmann et al., 1987; Shishiguchi, Minato & Hashizume, 1986). The latter can be a short straight detector, which can be scanned to increase the data-collection speed (Göbel, 1982), or a longer curved detector.

In conventional focusing geometry, the specimen and detector are coupled in θ–2θ relation at all 2θ's to avoid defocusing and profile broadening. In Seemann–Bohlin geometry, changing the specimen position necessitates realigning the diffractometer and very small incidence angles are inaccessible. In parallel-beam geometry, the specimen and detector positions can be uncoupled without loss of resolution. This freedom makes possible the use of different geometries for new applications. The specimen can be set at any angle from grazing incidence to slightly less than 2θ, and the detector scanned. Because the incident and exit angles are unequal, the relative intensities may differ by small amounts from those of the θ–2θ scan due to specimen absorption. The reflections occur from differently oriented crystallites whose planes are inclined (rather than parallel) to the specimen surface so that particle statistics becomes an important factor. The method is thus similar to Seemann–Bohlin but without focusing.

The method can be used for depth-profiling analysis of polycrystalline thin films using grazing-incidence diffraction (GID) (Lim, Parrish, Ortiz, Bellotto & Hart, 1987). If the angle of incidence is less than the critical angle of total reflection , diffraction occurs only from the top 35 to 60 Å of the film. Comparison of the GID pattern with a conventional θ–2θ pattern in which the penetration is much greater gives structural information for phase identification as a function of film depth. The intrinsic profile shapes are the same in the two patterns and broadening may indicate smaller particle sizes. However, if the film is epitaxic or highly oriented, it may not be possible to obtain a GID pattern.

For , the penetration depth is (Vineyard, 1982) and, for , where μ is the linear absorption coefficient. The thinnest top layer of the film that can be sampled is determined by the film density, which may be less than the bulk value. As approaches , the penetration depth increases rapidly and fine control becomes more difficult. Fig. 2.3.2.7 shows this relation and the advantage of using longer wavelengths for a wider range of penetration control. For example, for a film with μ = 200 cm⁻¹, λ = 1.75 Å, and , only the top 45 Å contribute, and increasing to 0.35° increases the depth to 130 Å. The patterns have much lower intensity than a θ–2θ scan because of the smaller diffracting volume.

Figure 2.3.2.7| top | pdf | Penetration depth t' as a function of grazing-incidence angle α for γ-Fe2O3 thin film. The critical angle of total reflection αc is shown by the vertical arrows for different wavelengths.

Fig. 2.3.2.8 shows patterns of a 5000 Å polycrystalline film of iron oxide deposited on a glass substrate and recorded with (a) θ–2θ scanning and (b) 0.25° GID. The film has preferred orientation as shown by the numbers above the peaks in (a), which are the relative intensities of a random powder sample. The relative intensities are different because in (a) they come from planes oriented parallel to the surface and in (b) the planes are inclined. The glass scattering that is prominent in (a) is absent in (b) because the beam does not penetrate to the substrate.

Figure 2.3.2.8| top | pdf | Synchrotron diffraction patterns of annealed 5000 Å iron oxide film, λ = 1.75 Å, (a) θ–2θ scan; relative intensities of random powder sample shown above each reflection. (b) Grazing incidence pattern of same film with α = 0.25° showing only reflections from top 60 Å of film, superstructure peak S.S. and α-Fe2O3 peaks not seen in (a). Absolute intensity is an order of magnitude lower than (a).

By step scanning the channel monochromator instead of the specimen, a different wavelength reaches the specimen at each step and the pattern is a plot of intensity versus wavelength or energy (Parrish & Hart, 1985, 1987). The X-ray optics can be the same as described in Subsection 2.3.2.1 and determines the resolution. A scintillation counter with conventional electronic circuits can be used. As in the conventional white-beam energy-dispersive diffraction (EDD) described in Section 2.5.1 , the specimen and detector remain fixed at selected angles during the recording. This makes it possible to design special experiments that would not be possible with specimen-scanning methods. It also simplifies the design of specimen-environment chambers for high and low temperatures. The advantages of the method over conventional EDD are the order-of-magnitude higher resolution that can be controlled by the X-ray optics, the ability to handle high peak count rates with a high-speed scintillation counter and conventional circuits, and much lower count times for good statistical accuracy.

The accessible range of d's that can be recorded using a selected wavelength range is determined by the 2θ setting of the detector. Changing 2θ causes the separation of the peaks to expand or compress in a manner similar to a variation of λ in conventional diffractometry. This is illustrated in Figs. 2.3.2.9 (a)–(d) for a quartz powder specimen using an Si(111) channel monochromator and = 19 to 5° (2.04 to 0.55 Å, 6.1 to 22.7 keV) and four detector 2θ settings. At small 2θ settings, only the large d's are recorded and the peak separation is large. Increasing the 2θ setting decreases the d range and the separation of the peaks as shown in Fig. 2.3.2.9(e). These patterns were recorded with the pulse-height analyser set to discriminate only against scintillation counter noise.

Figure 2.3.2.9| top | pdf | (a)–(d) High-resolution energy-dispersive diffraction patterns of quartz powder sample obtained with 2θ settings shown in upper left corners. (e) d range as a function of detector 2θ setting for λ = 0.4 to 2 Å. (f) Effect of 2θ setting and E on profile widths of quartz. Right: 121 reflection, 20°2θ, Ep 10.45 keV; left: 100 reflection, 45°2θ, ep 8.35; both reflections broadened by X-ray optics and peak intensity of 100 twice that of 121.

For given X-ray optics, the profiles symmetrically broaden with decreasing X-ray photon energy and with θ. This type of broadening remains symmetrical if E is increased and 2θ decreased, or vice versa, Fig. 2.3.2.9(f). The two profiles shown have been broadened by the X-ray optics but the intrinsic resolution is far better. The number of points recorded per profile thus decreases with decreasing profile width since is constant. At the higher energies, it may be desirable to use smaller steps to increase the number of points to define better the profile. Alternatively, increments in steps rather than θ steps would eliminate this variation.

Many electronic solid-state devices use thin films that are purposely prepared to have single-crystal structure (e.g. epitaxic growth), or with a selected lattice plane oriented parallel or normal to the film surface to enhance certain properties. The properties vary with the degree of orientation and textural characterization is essential to make the correct film preparation. Preferred orientation can be detected by comparing the relative intensities of the thin-film pattern with those of a random powder. The pattern can be recorded with conventional θ–2θ scanning (λ fixed) or by EDD. However, this only gives information on the planes oriented parallel to the surface. To study inclined planes requires uncoupling the specimen surface and detector angles. This can be done with the EDD method described above without distorting the profiles (Hart et al., 1987).

The principle of the method is illustrated in Fig. 2.3.2.10 . The set of lattice planes (hkl) oriented parallel to the surface has its highest intensity in the symmetric θ–2θ position. Rotating the specimen by an angle while keeping 2θ fixed reduces the intensity of (hkl) and brings another set of planes (pqr), which are inclined to the surface, to its symmetrical reflecting position. The required rotation is determined by the interplanar angle between (hkl) and (pqr). The angular distribution of any plane can be measured with respect to the film surface by step scanning at small steps. The specimen is rotated clockwise with the limitation 2θ. A computer automation program is desirable for large numbers of measurements.

Figure 2.3.2.10| top | pdf | Specimen orientation for symmetric reflection (a) from (hkl) planes and (b) specimen rotated θr for symmetric reflection from (pqr) planes.

Fig. 2.3.2.3(c) shows the appearance of a pattern of a specimen containing elements with absorption edges in the recording range and using electronic discrimination only against electronic noise. Starting at the incident high-energy side, the Zn and Ni K fluorescence increases as the energy approaches the edges (λ³ law), decreases abruptly when the energy crosses each edge, and disappears beyond the Ni K edge. Long-wavelength fluorescence is absorbed in the windows and air path.

The method is of doubtful use for structure determination or quantitative analysis. The wide range of wavelengths, continually varying absorption and profile widths, and other factors create a major difficulty in deriving accurate values of the relative intensities.

Conventional energy-dispersive diffraction methods using white X-rays and a solid-state detector are described in Chapter 2.5 and Section 5.2.7 .

Ideally, the specimen should contain a large number of small equal-sized randomly oriented particles. The surface must be flat and smooth to avoid microabsorption effects, i.e. particle interferences which reduce the intensities of the incident and reflected beams and can lead to significant errors (Cline & Snyder, 1983). The specimen should be homogeneous, particularly if it is a mixture or if a standard has been added. Low packing density and specimen-surface displacement (§2.3.1.1.6) may cause significant errors. It is recommended that the powder and the prepared specimen be examined with a low-power binocular optical microscope. Smith & Barrett (1979), Jenkins, Fawcett, Smith, Visser, Morris & Frevel (1986), and Bish & Reynolds (1989) have surveyed methods of specimen preparation and they include bibliographies on special handling problems. Powder diffraction standards for angle and intensity calibration are described in Section 5.2.9 .

2.3.3.1.1. Preferred orientation

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Preferred orientation changes the relative intensities from those obtained with a randomly oriented powder sample. It occurs in materials that have good cleavage or a morphology that is platy, acicular or any special shape in which the particles tend to orient themselves in specimen preparation. The micas and clay minerals are examples of materials that exhibit very strong preferred orientation. When they are prepared as reflection specimens, the basal reflections dominate the pattern. It is common in prepared thin films where preferred orientation occurs frequently or may be purposely induced to enhance certain optical, electrical, or magnetic properties for electronic devices. By comparison of the relative intensities with the random powder pattern, the degree of preferred orientation can be observed.

Powder reflections take place from crystallites oriented in different ways in the instrument geometries as shown in Fig. 2.3.1.2. In reflection specimen geometry with θ–2θ scanning, reflections can occur only from lattice planes parallel to the surface and in the transmission mode they must be normal to the surface. In the Seemann–Bohlin and fixed specimen with 2θ scanning methods, the orientation varies from parallel to about 45° inclination to the surface. The effect of preferred orientation can be seen in diffraction patterns obtained by using the same specimen in the different geometries.

The effect is illustrated in Fig. 2.3.3.1 for m-chlorobenzoic acid, C₇H₅ClO₂, with reflection and transmission patterns and the pattern calculated from the crystal structure. The degree of preferred orientation is shown by comparing the peak intensities of four reflections in the three patterns:

Figure 2.3.3.1| top | pdf | Differences in relative intensities due to preferred orientation as seen in synchrotron X-ray patterns of m-chlorobenzoic acid obtained with a specimen in reflection and transmission compared with calculated pattern. Peaks marked × are impurities, O absent in experimental patterns.

Care is required to make certain the differences are not caused by a few fortuitously oriented large particles.

Various methods have been used to minimize preferred orientation in the specimen preparation (Calvert, Sirianni, Gainsford & Hubbard, 1983; Smith & Barrett, 1979; Jenkins et al., 1986; Bish & Reynolds, 1989). These include using small particles, loading the powder from the back or side of the specimen holder, and cutting shallow grooves to roughen the surface. The powder has also been sifted directly on the surface of a microscope slide or single-crystal plate that has been wetted with the binder or petroleum jelly. Another method is to mix the powder with an inert amorphous powder such as Lindemann glass or rice starch, or add gum arabic, which after setting can be reground to obtain irregular particles. Any additive reduces the intensity and the peak-to-background ratio of the pattern. A promising method that requires a considerable amount of powder is to mix it with a binder and to use spray drying to encapsulate the particles into small spheres which are then used to prepare the specimen (Smith, Snyder & Brownell, 1979).

Preferred orientation would not cause a serious problem in routine identification providing the reference standard had a similar preferred orientation and both patterns were obtained with the same diffractometer geometry. However, when accurate values of the relative intensities are required, as in crystal-structure refinement and quantitative analysis, it may be the major factor limiting the precision.

In practice, it is very difficult to prepare specimens that have a completely random orientation. Even materials that do not have good cleavage or special morphological forms, such as quartz and silicon, show small deviations from a completely random orientation. These show up as errors in the structure refinement and a correction factor is required.

An empirical correction factor determined by the acute angle between the preferred-orientation plane and the diffracting plane (hkl) can be used (Will et al., 1988). Three functions have been used to represent and the term GP is the variable refined: (Rietveld, 1969) for transmission specimens; for reflection specimens; and (Dollase, 1986).

These functions are quite similar for small amounts of non-randomness. The preferred-orientation plane is selected by trial and error. For example, a modified fast routine of the powder least-squares refinement program with only seven cycles of refinement on each plane for the first dozen allowed Miller indices can be used to find the plane that gives the lowest R(Bragg) value as shown in Table 2.3.3.1. All three functions improve the R(Bragg) value as shown in Table 2.3.3.2 but the evidence is not conclusive as to which is the best. More research is required in this area. Several specimens made of the same material may show different preferred-orientation planes, and in some cases the preferred-orientation plane never occurred in the crystal morphology. A more complicated method examines the polar-axis density distribution using a cubic harmonic expansion to describe the crystallite orientation of a rotating sample (Järvinen, Merisalo, Pesonen & Inkinen 1970; Ahtee, Nurmela, Suortti & Järvinen, 1989; Järvinen, 1993).

Table 2.3.3.1| top | pdf | Preferred-orientation data for silicon h k l R(Bragg) (%) GP 1 1 1 1.86 −0.11 2 2 0 2.02 −0.11 3 1 1 2.01 0.17 4 0 0† 0.86 −0.15 3 3 1 1.73 −0.19 4 2 2 2.43 0.04 5 1 1 1.36 0.19 5 3 1 2.44 −0.08 4 4 2 1.69 −0.19 6 2 0 1.25 0.29 5 3 3 2.40 −0.04 †Selected preferred orientation plane.

Table 2.3.3.2| top | pdf | R(Bragg) values obtained with different preferred-orientation formulae   R(Bragg) Si SiO2 Mg2GeO4 No corrections 3.50 2.57 12.5 Gaussian 1.65 1.60 5.71 Exponential 0.75 1.83 5.30 March/Dollase 0.75 1.64 4.87 Preferred-orientation plane 100 211 100

2.3.3.1.2. Crystallite-size effects

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In addition to profile broadening, which begins to appear when the crystallite sizes are 1–2 μm, the sizes have a strong effect on the absolute and relative intensities (de Wolff, Taylor & Parrish, 1959; Parrish & Huang, 1983). The particle sizes have to be less than about 5 μm to achieve 1% reproducible relative intensities from a stationary specimen in conventional diffractometer geometry (Klug & Alexander, 1974). The statistical errors arising from the number of particles irradiated can be greatly reduced by using smaller particles and rotating the specimen around the diffraction vector. This brings many more particles into reflecting orientations.

The particle-size effect is illustrated in Fig. 2.3.3.2 for specimens of NIST silicon standard powder 640 sifted to different size fractions. The powders were packed in a 1 mm deep cavity in a 25.4 mm diameter Al holder using 5% collodion/amyl acetate binder. They were rotated by a synchronous motor (a stepper motor can also be used) around the axis normal to the centre of the specimen surface with the detector arm fixed at the peak position and the intensity recorded with a strip-chart. Rapid rotation, ∼60 r min⁻¹, gives the average peak intensity for all azimuths of the specimen and the small variations result only from the counting statistics. Scaling the intensities to (111) = 100% for the 5–10 μm fraction, the 10–20 μm fraction is 94%, 20–30 μm 88% and  μm 59%. The decrease is probably due to lower particle-packing density and increasing interparticle microabsorption. The  μm fraction = 95% may be due to the larger ratio of oxide coating around the particles to the mass of the particles.

Figure 2.3.3.2| top | pdf | Effect of specimen rotation and particle size on Si powder intensity using a conventional diffractometer (Fig. 2.3.1.3) and Cu Kα. Numbers below fast rotation are the average intensities.

Slow rotation, ∼1/7 r min⁻¹, shows the variation of the peak intensity with azimuth angle . The pattern repeats after 360° rotation and the magnitude of the fluctuations increases with increasing particle sizes and resolution. There is no correlation between the fluctuations of different reflections, as can be seen by comparing the 111, 220 and 311 reflections of the 10–20 μm specimen (lower left side) for which the incident-beam intensity was adjusted to give the same average amplitude. The horizontal lines are ±10% of the average. This shows the magnitude of errors that could occur using stationary specimens. Similar particle-size effects were found using the integrated intensities derived from profile fitting. The above discussion and Fig. 2.3.3.2 refer to a continuous scan. If the step-scan mode is used to collect data, it is clearly not necessary to rotate the specimen through more than one revolution at each step.

The rotating specimen also averages the in-plane preferred orientation but has virtually no effect on the planes oriented parallel to the specimen surface. The slow rotation method is useful in testing the grinding and sifting stages in specimen preparation. When calibrated with known size fractions, it can be used as a rough qualitative measure of the particle sizes.

A common source of error arises from the Kα doublet which produces a pair of peaks for each reflection. The separation of the Cu Kα₁, Kα₂ peaks increases from 0.05° at 20°2θ to 1.08° at 150°2θ. The overlapping is also dependent on the instrument resolution and may cause errors in the peak angles and intensities when strip-chart recording or peak-search methods (described below) are used. The Kα₁ wavelength is generally used to calculate all the d's even when the low-angle peaks are unresolved. In the region where the doublet is only slightly resolved, the apparent Kα₁ peak angle is shifted to higher angles because of the overlapping Kα₂ tail and similarly the peak intensities will be in error. The relative peak intensities of a reflection with superposed doublet compared to a resolved doublet could have an error as large as 50%. Relative peak intensities are used in the ICDD standards file and cause no problem because the unknowns are measured in the same way. The integrated intensity avoids this difficulty but is impractical to use in routine identification.

Rachinger (1948) described a simple graphical procedure for removing Kα₂ peaks. The method causes errors because it makes the incorrect assumption that Kα₂ is the exact half-scale version of Kα₁. Ladell, Zagofsky & Pearlman (1975) developed an exact algorithm using the actual mathematical shapes observed with the user's diffractometer but, with line-profile-fitting programs now available, the Kα₂ component can be modelled precisely along with the Kα₁.

It is possible to isolate the Kα₁ line when using a high-quality incident-beam focusing monochromator as described in Subsection 2.3.1.2, Fig. 2.3.1.12(b), but there may be a loss of intensity. The source size must be narrow and the focal length long enough to separate the components.

The most obvious and commonly used measure of the reflection angle of a profile is the position of maximum intensities (Fig. 2.3.3.3 ). The midpoints of chords at various heights have often been used but their values vary with the profile asymmetry. Another method is to connect the midpoints of chords near the top of the profile and extrapolate to the peak. The computer methods using derivatives are the most accurate and fastest as described in Subsection 2.3.3.7.

Figure 2.3.3.3| top | pdf | Various measures of profile.

A more fundamental measure that uses the entire intensity distribution is the centre of gravity (or centroid) defined as The variance (mean-square deviation of the mean) is defined as

The use of the centroid and variance has two important advantages: (1) most of the aberrations (§2.3.1.1.6) were derived in terms of the centroid and variance; and (2) they are additive, making it easy to determine the composite effect of a number of aberrations. Mathematically, the integration extends from to but the aberrations have a finite range. However, the practical use of these measures causes some difficulty. If the profile shapes are Lorentzian, the tails decay slowly. A very wide range would be required to reach points where the signal could no longer be separated from the background and the profiles must be truncated for the calculation. Truncation limits that have been used are 90% ordinate heights of Kα₁ (Ladell, Parrish & Taylor, 1959), and equal 2θ or limits from the centroid (Taylor, Mack & Parrish, 1964; Langford, 1982). The limits such as and in Fig. 2.3.3.3 must be carefully chosen to avoid errors and this involves the correct determination of the background level. It is not practical to use centroids for overlapping peak clusters unless the profile fitting can accurately resolve the individuals with their correct positions and intensities. Their use has, therefore, been confined to simple patterns with small unit cells in which the profiles were well separated.

The difference between the angle derived from the peak and the centroid depends on the asymmetry of the profile, which in turn varies with the Kα-doublet separation and the aberration broadening. Tournarie (1958) found that the centre of a horizontal chord at 60.6% of the Kα₁ peak height corresponds well to the centroid of that line in fairly well resolved doublets. The number, of course, depends on the profile shape. There is also the basic problem that most of the X-ray wavelengths were probably determined from the spectral peaks and, if the centroids are measured for the powder pattern, the Bragg equation becomes nonlinear in the sense that the 1:1 correspondence between λ and is lost.

Formerly, the most common method of obtaining diffractometer data was by using a rate-meter and strip-chart recorder with the paper moving synchronously with the constant angular velocity of the scan. This simple analogue method is still used and a large fraction of the JCPDS (ICDD) file prior to about 1982 was obtained in this way.

The method has several limitations: the data are not in the digital form required for computers, and are distorted; manual measurement of the chart takes a long time and has low accuracy. The output of the strip chart lags behind the input by an amount determined by the product of the scanning speed and the time constant of the rate-meter, including the speed of the recorder pen. The peak height is decreased and shifted in the direction of the scan causing asymmetric broadening with loss of resolution. The profile shape, Kα-doublet separation, and scan direction also contribute to distortion. When the product of the scan speed and time constant have the same value, the profile shapes are the same even though the total count is determined by the scan speed, Figs. 2.3.3.4 (a) and (b). If the product is large, the distortion is severe (c), and very weak peaks may be lost.

Figure 2.3.3.4| top | pdf | Rate-meter strip-chart recordings. REV: scan direction reversed. Scan speed and time constant shown at top.

Most diffractometers are now sold with computer automation. Older instruments can be easily upgraded by adding a stepping motor to the gear-drive shaft. A large variety of computers and programs is available, and it is not easy to make the best selection. Continuing improvements in computer technology have been made to handle expanded programs with increased speed and storage capabilities. The collected data are displayed on a VDU screen and/or computer printer and stored on hard disk or diskette for later use and analysis. Microprocessors are often used to select the X-ray-generator operating conditions, shutter control, specimen change, and similar tasks that were formerly performed manually. Aside from the elimination of much of the manual labour, automation provides far better control of the data-collection and data-reduction procedures. However, computers do not preclude the necessity of precise alignment and calibration. Smith (1989) has written a detailed description of computer analysis for phase identification and also includes related programs and their sources.

Personal computers are widely used for powder-diffraction automation and a typical arrangement is shown in Fig. 2.3.3.5(a) . The automation may provide for step scanning, continuous scanning with read-out on the fly, or slewing to selected angles to read particular points. Step scanning is the method most frequently used. It is essential that absolute registration and step tracking be reliably maintained for all experimental conditions.

Figure 2.3.3.5| top | pdf | Fig. 2.3.3.5. (a) Block diagram of typical computer-controlled diffractometer and electronic circuits. The monitor circuit enclosed by the dashed line is optional. HPIB is the interface bus. (b) A full-screen menu with some typical entries.

The step size or angular increment Δ2θ and count time t at each step, and the beginning and ending angles are selectable. For a given total time available for the experiment, it usually makes no difference in the counting statistical accuracy if a combination of small or large Δ2θ and t (within reasonable limits) is used. A minimal number of steps of the order of to 0.2 FWHM is required for profile fitting isolated peaks. It is clear that the greater the number of steps, the better the definition of the profile shape. The step size becomes important when using profile fitting to resolve patterns containing overlapped reflections and to detect closely spaced overlaps from the width and small changes in slopes of the profiles. A preliminary fast run to determine the nature of the pattern may be made to select the best run conditions for the final pattern. Will et al. (1988) recorded a quartz pattern with 1.28 Å synchrotron X-rays and 0.01° steps to test the step-size role. The profile fitting was done using all points and repeated with the omission of every second, third, and fourth point corresponding to Δ2θ = 0.02, 0.03 and 0.04°. The R(Bragg) values were virtually the same (except for 0.04° where it increased), indicating the experimental time could have been reduced by a factor of three with little loss of precision; see also Hill & Madsen (1984). Patterns with more overlapping would require smaller steps. Ideally, the steps could be larger in the background but this also requires a prior knowledge of the pattern and special programming.

A typical VDU screen menu for diffractometer-operation control is shown in Fig. 2.3.3.5(b). A number of runs can be defined with the same or different experimental parameters to run consecutively. The run log number, date, and time are usually automatically entered and together with the comment and parameters are carried forward and recorded on the print-outs and graphics to make certain the runs are completely identified. The menu is designed to prompt the operator to enter all the required information before a run can be started. Error messages appear if omissions or entry mistakes are made. There are, of course, many variations to the one shown.

X-ray quanta arrive at the detector at random and varying rates and hence the rules of statistics govern the accuracy of the intensity measurements. The general problems in achieving maximum accuracy in minimum time and in assessing the accuracy are described in books on mathematical statistics. Chapter 7.5 reviews the pertinent theory; see also Wilson (1980). In this section, only the fixed-time method is described because the fixed-count method takes too long for most practical applications.

Let be the average of N, the number of counts in a given time t, over a very large number of determinations. The spread is given by a Poisson probability distribution (if is large) with standard deviation Any individual determination of N or the corresponding counting rate n (= N/t) will be subject to a proportionate error which is also a function of the confidence level, i.e. the probability that the result deviates less than a certain percentage from the true value. If Q is the constant determined by the confidence level, then where Q = 0.67 for the probable relative error (50% confidence level) and Q = 1.64 and 2.58 for the 90 and 99% confidence levels respectively. For a 1% error, N = 4500, 27 000, 67 000 for , , , respectively. Fig. 2.3.3.6 shows various percentage errors as a function of N for several confidence levels.

Figure 2.3.3.6| top | pdf | Percentage error as a function of the total number of counts N for several confidence levels.

In practice, there is usually a background count . The net peak count is dependent on the P/B ratio as well as on and separately. The relative error of the net peak count is which shows that is similarly influenced by both absolute errors and . The absolute standard deviation of the net peak height is and expressed as the per cent standard deviation is The accuracy of the net peak measurement decreases rapidly as the peak-to-background ratio falls below 1. For example, with N_B = 50, the dependence of on P/B is It is obviously desirable to minimize the background using the best possible experimental methods.

The accurate location of the 2θ angle corresponding to the peak of the profile has been discussed in many papers (see, for example, Wilson, 1965). Computers are now widely used for data reduction, thereby greatly decreasing the labour, improving the accuracy, and making possible the use of specially designed algorithms. It is not possible to present a description of the large number of private and commercial programs. The peak-search and profile-fitting methods described below have been successfully used for a number of years and are representative of the results that can now be obtained. They have greatly improved the results in phase identification, integrated intensity measurement, and analyses requiring precise profile-shape determination. It is likely that even better programs and methods will be developed in this rapidly changing field.

There are two levels of the types of data reduction that may be done. The easiest and most frequently used method is usually called `peak search'. It computes the 2θ angles and intensities of the peaks. The results have good precision for isolated peaks but give the values of the composite overlapping reflections as they appear, for example, on a strip-chart recording. The calculation is virtually instantaneous and is often all that is needed for phase identification, lattice-parameter determination, and similar analyses. The second, profile fitting, described below, is a more advanced procedure that can resolve overlapping peaks into individual reflections and determines the profile shape, width, peak and integrated intensities, and reflection angle of each resolved peak. This method requires a prior knowledge of the profile-fitting function. It is used to determine the integrated intensities for analyses requiring higher precision such as crystal-structure refinement and quantitative analysis, and profile-shape parameters for small crystallite size, microstrain and similar studies.

To measure weak peaks, the counting statistical accuracy must be sufficient to delineate the peak from the background. When the intensity and peak-to-background ratio are low, the computing time is much increased. Since powder patterns often contain a number of weak peaks that may not be required for the analysis, computer programs often permit the user to select a minimum peak height (MPH) and a standard deviation (SD) that the peak must exceed to be included in the data reduction. For example, MPH = 1 would reject peaks less than 1% of the highest peak in the recorded pattern, and SD = 4 requires the intensity to exceed the background adjacent to the peak by . The number of peaks rejected depends on the intensity and peak-to-background ratio as illustrated in Fig. 2.3.3.7 , where the cut-off level was set at for two recordings of the same pattern with about a 40 times difference in intensities. All visible peaks are included in the high-intensity recording and several are rejected by the cut-off level selected in the lower-intensity pattern.

Figure 2.3.3.7| top | pdf | Effect of 4σ maximum peak height (horizontal line) on dropping weak peaks from inclusion in computer calculation. Step scan with (a) t = 5 s and (b) t = 0.1 s. Five-compound mixture, Cu Kα.

Before carrying out the computer calculations, it may be desirable to subtract unusual background such as is caused by a glass substrate in a thin-film pattern.

The following method was developed using computer-generated profiles having the same shapes as conventional diffractometer (Fig. 2.3.1.3) profiles and adding random counting statistical noise (Huang & Parrish, 1984; Huang, 1988). The best results were obtained using the first derivative (dx/dy = 0) of a least-squares-fitted cubic polynomial to locate the peaks, combined with the second derivative (d²y/dx² = minimum) of a quadratic/cubic polynomial to resolve overlapped reflections (Fig. 2.3.3.8 ). Overlaps with a separate 0.5 FWHM can be resolved and measured and the accuracy of the peak position is 0.001° for noise-free profiles. Real profiles with statistical noise have a precision of ±0.003 to 0.02° depending on the noise level. The Savitzky & Golay (1964) method (see also Ateiner, Termonia & Deltour, 1974; Edwards & Willson, 1974) was used for smoothing and differentiation of the data by least squares in which the values of the derivatives can be calculated using a set of tabulated integers. The convolution range CR expressed as a multiple of the FWHM of the peak can be selected. A minimum of five points is required. For asymmetric peaks, such as occur at small 2θ's, a CR 0.5 FWHM gives the best precision. The larger the CR the larger the intrinsic error but the smaller the random error, and the smaller the number of peaks identified in overlapping patterns. The larger CR also avoids false peaks in patterns with poor counting statistics. Fig. 2.3.3.8(c) shows the dependence of the accuracy of the peak determination on P/ σ. The computer results list the 2θ's, d's, absolute and relative intensities (scaled to 100) of the identified peaks. The calculation is made with a selected wavelength such as and the possible peaks are flagged.

Figure 2.3.3.8| top | pdf | (a) Si(220) Cu Kα reflection. (b) First (circles), second (crosses), and third (triangles) derivatives of a seven-point polynomial of data in (a). (c) Average angular deviations as a function of P/σ for various derivatives.

Profile fitting has greatly advanced powder diffractometry by making it possible to calculate the intensities, peak positions, widths, and shapes of the reflections with a far greater precision than had been possible with manual measurements or visual inspection of the experimental data. The method has better resolution than the original data and the entire scattering distribution is used instead of only a few features such as the peak and width. Individual profiles and clusters of reflections can be fitted, or the entire pattern as in the Rietveld method (Chapter 8.6 ).

The procedure is based on the least-squares fitting of theoretical profile intensities to the digitized powder pattern. The profile intensity at the ith step is calculated by where is the background intensity, is the integrated intensity of the jth reflection, is the peak-maximum position, is the profile function to represent the profile shape, and is taken over j, in which the has a finite value at . Unlike the Rietveld method, a structure model is not used. In the least-squares fitting, and are refined together with background and profile shape parameters in . Smoothing the experimental data is not required because it underestimates the estimated standard deviations for the least-squares parameters, which are based on the counting statistics.

The experimental profiles are a convolution of the X-ray line spectrum λ and all the combined instrumental and geometrical aberrations G with the true diffraction effects of the specimen S (Parrish, Huang & Ayers, 1976), i.e.

The profile shapes and resolution differ in the various diffractometer geometries and there is no universal profile-fitting function. In conventional X-ray tube focusing methods, the profiles are asymmetric and the shapes change continually across the scattering-angle range owing to the aberrations and the Kα doublet. To avoid problems caused by the Kα doublet, a few authors used the Kβ line, but it has only about 1/7 the intensity. The profiles obtained with synchrotron radiation are symmetrical and narrower, and the widths increase with increasing 2θ. The different shapes and rates of decay of the tails make it necessary to find an analytical function that best fits the particular experimental profile. Langford (1987) and Young & Wiles (1982) have compiled and reviewed various profile-fitting functions and several are described below. Howard & Preston (1989) give details of the computations in their review of the method.

Early profile analyses used Gaussian or Lorentzian (Cauchy) curves. Fig. 2.3.3.9(a) shows that the most obvious difference is the rate of decay of the tails. X-ray synchrotron profiles lie between the two as shown in Figs. 2.3.3.9(c)–(f). The function must fit the tails as well as the main body and single-element functions are generally unsatisfactory. The Voigt function is a convolution of Lorentzian (L) and Gaussian (G) functions of different widths: where x corresponds to (Langford, 1978). It has been used for profile fitting and also to determine certain physical properties such as crystallite size and strain broadening from the constituent L and G profiles (see, for example, Langford, 1978; Suortti, Ahtee & Unonius, 1979; de Keijser, Langford, Mittemeijer & Vogels, 1982; Langford, Delhez, de Keijser & Mittemeijer, 1988). Evaluation of a symmetrical Voigt function involves the real part of the complex error function and an algorithm for calculating this has been given by Langford (1992).

Figure 2.3.3.9| top | pdf | (a) Computer-generated symmetrical Lorentzian profile L and Gaussian G with equal peak heights, 2θ and FWHM. (b) Double Gaussian GG shown as the sum of two Gaussians in which I and FWHM of G1 are twice those of G2 and 2θ is constant. (c)–(f) Profile fitting with different functions. Differences between experimental points and fitted profile shown at one-half height. Synchrotron radiation, Si(111).

The two most frequently used functions are at present the pseudo-Voigt (Wertheim, Butler, West & Buchanan, 1974) and the Pearson VII (Hall, Veeraraghavan, Rubin & Winchell, 1977). They cannot be easily deconvoluted analytically and have no direct physical interpretation but the equivalent Voigt parameters can be derived. Both can be split into symmetric and asymmetric portions to adjust better to the profile asymmetries and tails. The shapes can also be varied systematically by changing the L/G ratio (de Keijser, Langford, Mittemeijer & Vogels, 1982).

The pseudo-Voigt function is similar to the Voigt except that an addition is used in place of the convolution. It is easier to use and Wertheim et al. (1974) found there was only a small difference in the intensities and widths obtained with this approximation. It is defined as where η is the ratio of Lorentz to Gauss and they have the same widths. The refined η and width of the full fitted profile can be related by a polynomial expansion (Hastings, Thomlinson & Cox, 1984; David, 1986; Cox, Toby & Eddy, 1988) to the widths of the L and G components of the original Voigt function. It is frequently used to fit synchrotron-radiation profiles. In the particular case shown in Figs. 2.3.3.9(c)–(f), the pseudo-Voigt has the best fit as shown by the difference curve at half-height and the lowest R_p [= R(PF)] value.

The Pearson VII function is defined as where m is a refinable parameter based on the G/L content (for m = 1, the curve is 100% Lorentzian and for m = ∞ it is 100% Gaussian), and 1/b = 2[2^1/m− 1]^1/2/W, where W is the FWHM.

The peak asymmetry can be incorporated into the profile function in several ways. One is to multiply (or add) the symmetrical profile function with an asymmetric function (Rietveld, 1969). Another is to dispose two or three functions asymmetrically (Parrish, Huang & Ayers, 1976). A third is to use a split-type function, consisting of two profile functions, each of which defines one-half the total peak, i.e. the low- or high-angle sides of the peak and each has different profile widths and shapes but the same height (Toraya, Yoshimura & Somiya, 1983; Howard & Snyder, 1983).

Some other functions that have been used include the double Gaussian [Fig. 2.3.3.9(b)] for low-resolution synchrotron data (Will, Masciocchi, Parrish & Hart, 1987), a Gaussian with shifted Lorentzian component to account for the asymmetry on the low-2θ side of the tail (Will, Masciocchi, Parrish & Lutz, 1990), profile modelling of single isolated peaks with a rational function, e.g. the ratio of two polynomials (Pyrros & Hubbard, 1983). In contrast to these analytical-type functions, some empirical functions have been developed. They are the `learned' (experimental) peak-shape function (Hepp & Baerlocher, 1988) and the direct fitting of experimental data represented by Fourier series (Mortier & Constenoble, 1973).

The sum of Lorentzians has been used for X-ray tube focusing profiles (Parrish & Huang, 1980; Taupin, 1973). The instrument function is determined (see below) by a sum of Lorentzian curves, three each for Kα₁ and Kα₂ and one for the weak Kα₃ satellite. Three Lorentzians were used to match the asymmetry although a greater or lesser number could be used depending on the profile symmetry. Each curve has three parameters (intensity, half-width at half-height, and peak position) and the 21 parameters are adjusted by the computer program to give the best fit to the experimental data, which may contain 150 to 300 points. This is done only once for each particular instrument set-up. After is determined, the profile fitting is easy and fast because only the specimen contribution S must be convoluted with . If the specimen has no asymmetric broadening other than , S can be approximated by a single symmetrical Lorentzian for each reflection; a split Lorentzian can be used if there is asymmetric broadening.

A function can be tested using isolated profiles of a standard specimen such as silicon, tungsten, quartz, and others which have 10 µm particles and no specimen broadening (Fawcett et al., 1988). It is necessary to do this test carefully and whenever the instrument parameters are changed.

The instrument function can be measured using isolated profiles of standard specimens as stated above. The measurements should be made with small ° steps and count times long enough to accumulate about 25 000 to 50 000 counts on the Kα₁ peaks for good counting statistics. By measuring a number of profiles separated by no more than about 5 to 10°, the instrument function can be established for the angular range of interest. A linear interpolation of the profile-fitting parameter between adjacent profiles gives a continuous function for use at any 2θ in the range. The intensities of the derived profile parameters are normalized and stored in the computer for later use. Note that any change in the X-ray spectrum or instrument geometry requires another set of measurements. The instrument function is also an important aid in computer graphics as described in Subsection 2.3.3.9.

The fitting of conventional diffractometer profiles was considerably improved by the use of a convolution function, in which the Pearson VII function is convoluted with the observed instrument function (Toraya et al., 1983; Toraya, 1988). Enzo, Fagherazzi, Benedetti & Polizzi (1988; Benedetti, Fagherazzi, Enzo & Battagliarin, 1988) used the convolution of a pseudo-Voigt function as the true data function and the convolution of exponential and pseudo-Voigt functions as the instrumental function for crystallite size and strain analysis. These functions have advantages in analysing the crystallite size and strain, although they require longer computation time for calculating the convolution.

Background intensity is usually included in the refinement. A first- or second-order polynomial is used to represent the background function B(x) in equation (2.3.3.12) in a small 2θ range, and the polynomial coefficients are adjusted during the least-squares refinement. In some cases, the background is subtracted from the pattern before the refinement by using the lowest intensities between the reflections. The background in the vicinity of high-intensity peaks and peak clusters is usually higher and should be avoided.

In the least-squares refinement, the following quantity is minimized: where N is the number of observations, is the weight assigned to the ith observation, and is the observed profile intensity. A statistical weighting factor such as , where , is frequently used. The quality of the fitting procedure is generally expressed by R factors such as , the weighted R factor for profile intensity, which includes the entire scattering range and the background. The definitions of these factors are summarized by Young, Prince & Sparks (1982). The and factors are given as If the selected function is inappropriate, it will show up on the difference curve (experimental – calculated; see Fig. 2.3.3.9), and high and factors.

A whole-powder-pattern fitting technique without using the structural model was proposed for analysing neutron powder data (Pawley, 1981) and then extended to X-ray data (Toraya, 1986). The method executes the whole pattern decomposition (i.e. fitting all the profiles) in one step. In this technique, the peak position T_j in equation (2.3.3.12) is a function of unit-cell parameters, and the unit-cell parameters are refined instead of individual peak positions. Furthermore, the angular dependence of the profile width can be expressed approximately as where , , and are adjustable parameters (Caglioti, Paoletti & Ricci, 1958); but see also Louër & Langford (1988). The profile-shape dependency on 2θ is ignored when fitting a small 2θ range, but it must be taken into account in the whole-powder-pattern fitting in both focusing and parallel-beam geometries. The least-squares ill conditioning is handled by imposing the constraints on the peak positions in the profile-fitting procedure.

Approximate unit-cell parameters are required to start the refinement. Advantages of this technique are: (1) the unit-cell parameters are refined to high precision; (2) the analysis is rapid and straightforward; (3) it is also powerful in analysing complex powder patterns. The output of indices and integrated intensities of all reflections can be used to calculate Patterson and Fourier diagrams, and thus used for ab initio structure determination (McCusker, 1988) and the structure refinement based on the integrated intensities such as used in the POWLS program (Will, 1979).

The convolution equation is used in place of P(x) in equation (2.3.3.12), in which the true data function represented by a pseudo-Voigt or Pearson VII has adjustable parameters of crystallite size and strain (Toraya, 1989). The anisotropic crystallite size assuming cylindrical shape has been determined by whole-powder-pattern fitting for complex powder patterns.

The advantage of profile fitting is illustrated in Fig. 2.3.3.10 for the quartz cluster at 68° with Cu Kα where the doublet separation is 0.19° and the FWHM is 0.14°. The relative intensities of the 122, 203, and 301 Kα₁ peaks are 81:97:100, which differ from the profile-fitted peaks, 90:100:67, due to the overlapping. The sum of the fitted curves is the solid line which passes through the experimental points. The peak-search (or strip-chart) intensities that are not corrected for overlaps are more likely to correspond to the ICDD powder file than the profile-fitted values. Profile fitting is capable of about ±0.0004°2θ and 0.2% intensity for good experimental data (Parrish & Huang, 1980). Even in data with poor counting statistical accuracy, it is possible to identify very weak peaks with low P/B as shown in Fig. 2.3.3.11 .

Figure 2.3.3.10| top | pdf | Profile fitting with sum-of-Lorentzians method. Individual reflections shown as dashed-line curves and sum as solid line passing through experimental points. Quartz peak cluster, Cu Kα1, Kα2, conventional diffractometer.

Figure 2.3.3.11| top | pdf | Profile fitting of poor statistical data.

An interactive graphics display program is a very important asset for interpreting and analysing powder diffraction data. If a colour graphics station is used, the display can be enhanced by using various colours. The simplest form is the VDU display of experimental points connected with straight lines, which appears similar to a strip-chart recording but has no time-constant error and is printed on page-size paper. It avoids storing large numbers of charts because hundreds of patterns can be stored on a diskette and displayed and printed at any time.

The basic parameters required in one of the published methods (Parrish, Huang & Ayers, 1984) are the d's and I's of the reflections, the wavelength and profile shapes instrument function). This makes it possible to produce a pattern exactly as it would appear on the user's diffractometer, aside from contributions arising from sample microstructure. The step size can be included if experimental patterns are to be reproduced or if patterns are to be subtracted. A section of the pattern can be enlarged to the full screen size by entering the desired angular range and highest peak intensity. A linear background can be added by entries at the low- and high-2θ points. Nonlinear background, e.g. from an amorphous substrate, can be transferred from a stored file. Counting statistical noise can be added to a simulated pattern by using a normally distributed random number with a standard deviation scaled to the calculated . Noise in an experimental pattern can be smoothed. A profile broadening factor can be added to the function. Quantitative synthesis of a mixture can be simulated by entering the relative weight percentage and reference intensity, the ratio of the intensities of the strongest lines of each pattern in a 50–50 mixture, or the ICDD values compared to α-Al₂O₃ (de Wolff & Visser, 1988; Davis & Smith, 1988). The program has access to the ICDD file stored on disk so that any card can be reproduced as a pattern using any wavelength. Some examples are shown in Fig. 2.3.3.12 .

Figure 2.3.3.12| top | pdf | Some examples of computer graphics of powder patterns. (a) Overlay of three patterns with ICDD card numbers. (b) Effect of adding 0.15° to FWHM. (c) Synchrotron 0.6888 Å radiation pattern of Si powder. (d) Low-intensity section enlarged and 11-point smoothing.

The design of cylindrical powder cameras with Straumanis film mounting was described by Buerger (1945) and the collimators by Parrish & Cisney (1948). Straumanis developed the method to an art and used it to measure lattice parameters, thermal expansion, and other properties of many materials; see, for example, Straumanis (1959), which contains references to many of his papers. In the USA, the camera diameter was usually made 57.3 or 114.6 mm to simplify measuring the film with a millimetre scale, 1 mm = 1° or 2°2θ. One of the major advantages of the method is that the full reflection range is recorded simultaneously on the film. Other advantages are that the effects of preferred orientation are immediately apparent on a film, lines can have non-uniform intensity (`spottiness') owing to size effects or there can be broadening owing to structural imperfections. These visual effects, which are less evident with diffractometer data, can be valuable aids in identifying a mixture of substances.

The camera is basically a cylindrical light-tight metal body with removable cover, and the film is pressed around the inside circumference. The beam is defined by an entrance collimator and the undiffracted portion is conducted out by an exit tube; both are mounted on the central plane of the camera and extend inside nearly to the specimen. The specimen is centred and rotated continuously during the exposure; translation may be added to bring more particles into the beam. Evacuating the camera or filling it with helium removes the air scattering which darkens the film in the vicinity of the 0° hole.

If the specimen is too thick or has high absorption, the forward reflection lines split because the beam penetrates only the top and bottom of the rod. The diameter of the rod determines the widths of the lines. The line widths are about twice the diameter of the rod at small 2θ's and decrease with increasing 2θ. The absorption causes a systematic error in the positions of the lines, which can be handled with a cos² θ or Nelson–Riley plot (Section 5.2.8 ). The sample may be small – only about 0.1 mg is required. Axial divergence causes the well known `umbrella' or `broom' broadening illustrated in Fig. 2.3.4.1(b). It is essential to measure the film along the equator where the lines are narrowest and shifts the smallest. The specimens should be less than 0.5 mm diameter and may be coated on a fine wire or glass fibre (silica or Lindemann glass), or packed into a capillary (commercially available).

Read & Hensler (1972) modified a Debye–Scherrer camera to use flat specimens for thin-film analysis (Tao & Hewett, 1987).

The Guinier camera (Guinier, 1937, 1946; Guinier & Dexter, 1963) uses a high-quality asymmetric focusing monochromator and cylindrical camera with a thin transmission specimen, Fig. 2.3.4.1(c). The film must be placed at the focal point of the monochromator, which can be adjusted to reflect only the Kα₁ line. When the camera is in the position shown, the angular range is larger on one side of the film than the other (asymmetric setting). If the camera is placed so that the rays from the monochromator are along the camera diameter, the angular range is the same on both sides of the 0° point (symmetric setting) and the usable range is about 60°2θ. The sharpest lines are obtained when the rays are nearly normal to the film. The lines are broadened by inclination of the rays to the film, axial divergence, and specimen thickness. The camera can also be used with the specimen in reflection so that it becomes a Seemann–Bohlin camera with only the back reflections accessible [Fig. 2.3.4.1(d)]. Hofmann & Jagodzinski (1955) designed a double camera in a single body that can record transmission and reflection patterns on separate films.

de Wolff (1948) described a novel Guinier-type camera that can simultaneously record up to four patterns of different specimens on one film with a single monochromator and long fine-focus X-ray tube. The patterns are separated by horizontal partitions. There are some differences in the line widths in the top and bottom patterns. Malmros & Werner (1973) developed an automated film-measuring densitometer to improve the precision in measuring the Guinier films; see also Sonneveld & Visser (1975).

The symmetrical back-reflection camera, Fig. 2.3.4.1(e), is mainly used for lattice-parameter and solid-solution studies because the high reflection angles can be recorded. The specimen can be mounted on a curved holder matching the film curvature to obtain sharp lines and is oscillated during exposure.

The flat-plate camera, Fig. 2.3.4.1(f), can be used for forward- or back-reflection. The angular range is small and varies inversely with the specimen-to-film distance. Polaroid film is frequently used. The same method is used for Laue photographs, usually in back-reflection with a goniometer to orient the crystal. The method is often used for fibre and polymer specimens because the entire cone can be recorded (Alexander, 1969).

The Gandolfi (1967) camera produces a powder-like pattern from a tiny single crystal by simultaneous rotation of the crystal around two inclined axes. It is often made as a modification to the cylindrical camera. The crystal may be very small but the pattern is greatly improved by using several crystals. The smoothness of the lines depends on the chance orientation of the crystal with respect to the rotation axes, and the multiplicity of the reflection. The centring of the specimen and the rotation axes must be done precisely. Anderson, Zolensky, Smith, Freeborn & Scheetz (1981) obtained patterns routinely from 5 μm particles in 2–4 d exposure at 40 keV, 20 mA in an evacuated camera; see also Sussieck-Fornefeld & Schmetzer (1987) and Rendle (1983). A high-brilliance microfocus X-ray tube can greatly increase the intensity.

Another type of camera for the same purpose was developed by Parrish & Vajda (1971). The small crystal is mounted on a glass fibre at the end of a vertical shaft that rotates continuously and simultaneously scans about 90°. The film is mounted in a half-cylinder with about 20 mm radius. A microscope is used for precise alignment and centring.

A camera with a wide film cassette has been used for high-temperature diffraction patterns. The cassette can be translated synchronously with the change in temperature, or held in fixed positions during exposure at selected temperatures. The advantage is that all the patterns are recorded on a single film showing the phase changes and thermal expansion as a function of temperature. A Weissenberg camera can be adapted for this purpose.

Vacuum-sealed water-cooled X-ray tubes of the type shown in Fig. 2.3.5.1 are almost exclusively used for powder diffraction. They are installed in either a vertical or a horizontal shield (sometimes called a tower) mounted on the generator, or remotely operated with a long high-voltage cable. The shield is designed to seat the tube cap in the correct position, which allows tube replacement without realigning the instruments. Rotating-anode tubes are becoming more popular. They may be operated at higher currents and, although they require continual pumping, recent designs incorporating a ferromagnetic seal and turbomolecular pump make their use virtually as simple as sealed tubes. For additional background information see Phillips (1985) and Yoshimatsu & Kozaki (1977). End-window tubes with large focal spot have been used mainly for X-ray-fluorescence spectroscopy (Arai, Shoji & Omote, 1986), and fine-point-focus tubes for Kossel diagrams.

Figure 2.3.5.1| top | pdf | Sealed X-ray diffraction tube (Philips), dimensions are given in mm. a = `short' focus, b = `long' focus.

The maximum permissible power ratings for sealed water-cooled diffraction tubes are about 60 kV, 60 mA and 3 kW. The rating varies with the focal-spot size, anode element, and the particular manufacturer's specifications. Table 2.3.5.1 lists some typical maximum ratings of sealed and rotating-anode tubes. The brightness or specific loading, expressed as watts per square mm, increases with decreasing focal-spot size. There is a very large increase in brightness in the small microfocus sources that operate at lower total power. X-ray tubes normally have a life of several thousand hours. It varies with power, anode-cooling efficiency, on–off cycles, and similar factors.

Most X-ray generators are now designed for constant-potential operation using solid-state rectifiers and capacitors in the high-voltage transformer tank. They produce higher intensity at the same voltage than self-rectified or full-wave-rectified operation because the characteristic line spectrum is produced only in the portion of the cycle in which the voltage exceeds the critical excitation voltage of the target element. The gain thus increases with decreasing wavelength. The operation of modern X-ray generators is very simple and requires little attention. Safety interlocks provide electrical protection, and window-shutter interlocks aid in radiation safety. Large ray-proof plastic enclosures are available to surround the X-ray tube tower and diffraction instrumentation and are recommended for safety. Some legal requirements are outlined in Part 10 .

Air-cooled tubes can operate at only a fraction of the power of water-cooled tubes and are used for special applications where low intensities can be tolerated. Small portable air-cooled X-ray tubes have recently become available in a variety of forms (see, for example, Kevex Corporation, 1990). The tube, high-voltage generator and control electronics are packaged in compact units with approximate dimensions 27 by 10 cm weighing about 3 kg. They have a single 0.13 mm Be window, a focal-spot size 0.25 to 0.50 mm, and are available with a number of target elements. They can be AC or battery operated. Some tubes are rated at 70 kV, 7 W, and others at 30 kV, 200 W, depending on the model.

2.3.5.1.3. Source intensity distribution and size

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The intensity distribution of the focal line is usually not uniform. This has no apparent effect on the shapes of powder reflections but may cause difficulties with single crystals (Parrish, Mack & Taylor, 1966). The distribution can be measured with a small pinhole placed between the X-ray tube focal line and a dental or Polaroid film. The ratio of the distances between line-to-pinhole and pinhole-to-film determines the magnification of the image. The pinhole diameter should be small for good resolution. About 0.1 mm diameter is satisfactory and can be made with a special microdrill, spark erosion or other methods. The thickness of the metal must be minimal to avoid having the aperture formed by the length and diameter of the pinhole limit the length of focus photographed. Avoid overexposure which broadens the image. Also, the Polaroid film should be exposed outside the cassette to avoid broadening caused by the intensifying screen.

A more accurate method is to scan a slit and detector (mounted on the same arm) normal to the central ray from the focus as shown in Fig. 2.3.5.2(b) (Parrish, 1967). The slits are a pair of molybdenum rods (or other high-absorbing metal) with opening normal to the scan direction, and the slit width determines the resolution. This method gives a direct measurement of the intensity distribution from which the projected size can be determined.

Figure 2.3.5.2| top | pdf | (a) Transmission of Be, Al and air as a function of wavelength. (b) Method for measuring X-ray tube focus by scanning slit S2 and detector D. Slit S1 is fixed and the ratio of the distances d2/d1 gives the magnification. (c) Intensity of a copper target tube as a function of kV for various take-off angles. (d) Intensity and brightness as a function of take-off angle of a copper target tube operated at 50 kV. The intensity distributions for 1 and 4° entrance-slit apertures are shown at the top, and terms used to define ψ and αES are shown in the lower insert.

The actual size of the focus is foreshortened to by the small take-off angle ψ, . A typical 0.5 10 mm focus viewed at 6° appears to be a line 0.05 10 mm or a spot 0.05 1 mm [Fig. 2.3.1.9(a)]. The line focus is generally used for powder diffractometry and focusing cameras and the spot focus for powder cameras and single-crystal diffractometry.

X-rays emerge from three or four Be windows spaced 90° apart around the circumference. Their diameter and position with respect to the plane of the target determine the usable ψ-angle range. The length of line focus that can pass through the window can be seen with a flat fluorescent screen in the specimen holder using the largest entrance slit. The Be window thickness often used is 300 μm and the transmission as a function of wavelength is shown in Fig. 2.3.5.2(a).

The X-ray tube spectrum consists of sharp characteristic lines superposed on broad continuous radiation as shown in Fig. 2.3.5.3 . The continuous spectrum begins at a wavelength determined by the voltage on the X-ray tube, λmin 12.4/kV. It reaches a maximum at about 1.5 to 2λmin and gradually falls off with increasing λ [Fig. 2.3.5.4(a) ]. The intensity increases with voltage and current, and also with the atomic number of the target element. The integrated intensity is greater than that of the spectral lines. It is used for Laue patterns, fluorescence analysis, and energy-dispersive diffraction. It is troublesome in powder diffraction because it contributes to the background by scattering and by causing specimen fluorescence.

Figure 2.3.5.3| top | pdf | X-ray spectrum of copper target tube with Be window, 50 kV constant potential, 12° take-off angle. (a) Unfiltered, (b) with Ni filter, (c) unfiltered with pulse-height discrimination (PHD), (d) Ni filter + PHD. (1) λmin = 0.246 Å (4.5°2θ), (2) I K-absorption edge (from NaI scintillation crystal), (3) peak of continuous radiation (about 19% of Cu Kα peak), (4) W Lγ contaminant, (5) W Lβ, (6) Cu K-absorption edge, (7) Cu Kβ, (8) W Lα, (9) Cu Kα1 + Kα2, (10) Co Kα, (11) Fe Kα, (12) Mn Kα, (13) Ni K-absorption edge, (14) escape peak. Experimental conditions: Si(111) single-crystal analyser, vacuum path, Ni filter 0.18 mm, scintillation counter with 45% resolution for Cu Kα, lower-level discrimination only against circuit noise. ES 0.25 × 1.5 mm, AS 1.4 mm, no RS, Δ2θ 0.05°, FWHM 0.3°2θ.

Figure 2.3.5.4| top | pdf | (a) Continuous X-ray spectrum of tungsten target X-ray tube as a function of voltage and constant current. Full-wave rectification, silicon (111) crystal analyser, scintillation counter. (b) Plot of Moseley's law for four characteristic X-ray spectral lines.

The wavelengths of the spectral lines decrease with increasing atomic number Z of the target element [Moseley's law, Fig. 2.3.5.4(b)]. All the lines in a series appear when the critical excitation voltage is exceeded. For a Cu target, this is 9 kV and the approximate relative intensities are Cu Kα₂ 50, Kα₁ 100 and Kβ 20. The peak intensities of Cu Kα₁ and Cu Kα₂ in diffractometer patterns may not be exactly 2:1 but closer to 2.1:1 in resolved doublets because of the different profile widths. The profile widths of the spectral lines vary among the different elements used for X-ray tube targets (Compton & Allison, 1935), as does the Kβ/Kα ratio (Smith, Reed & Ware, 1974). The observed ratio varies with the degree of overlap. The rate of increase with voltage and other factors is described above.

A broad weak group of satellite peaks, Kα₃, occurs near the bottom of the short-wavelength tail of the Kα₁ peak (see Fig. 2.3.3.3). The intensity varies with the target element and is about 0.5% for the Cu K spectrum. The satellites appear as a small, broad, ill defined peak in powder diffraction patterns (Parratt, 1936; Parrish, Mack & Taylor, 1963; Edwards & Langford, 1971).

The spectral lines have an approximately Lorentzian shape when measured with a two-crystal diffractometer. They usually have a small asymmetry and their widths vary among the elements and also in the same series of lines. Bearden (1964) defined the wavelength as the peak determined by extrapolation of the centres of chords near the top of the peak. The corresponding energy levels have been compiled by Bearden & Burr (1965). The centroid of the , peaks of Cu and Fe has been calculated from the Bearden experimental two-crystal data (Mack, Parrish & Taylor, 1964). X-ray wavelengths are discussed in Chapter 4.2 . The standard targets provide the K spectra of Ag, Mo, Cu, Co, Fe and Cr, and the W L spectrum. Other targets may be obtained on special order. The K spectra of the elements of high atomic number require a radiographic tube and power supply that can operate continuously at about 150 kV or higher. (Caution: The radiation-shielding problems multiply exponentially at high voltages.)

2.3.5.2.1. Wavelength selection

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The selection of the X-ray tube anode is determined by several factors such as intensity, specimen fluorescence, and dispersion. The intensity of the characteristic line radiation varies among the target elements depending on the voltage and if a vacuum or He path is used. The recorded intensities also change abruptly at the absorption edges of the elements in the specimen. If a diffracted-beam monochromator or solid-state detector with narrow window centred on the characteristic line energy is used, the specimen fluorescence is eliminated (except for the element that is the same as the anode), and one tube can be used for all compositions. If the pattern has severe overlapping, the separation of the peaks can be increased with longer wavelengths, which increase the dispersion expressed as °θ Å⁻¹ of d. Fig. 2.3.5.5 shows portions of diffractometer patterns of topaz in which the same d ranges were recorded with Cu Kα (a) and Cr Kα (b). The greater separation of the peaks is clearly advantageous in analysing the patterns.

Figure 2.3.5.5| top | pdf | Portion of diffractometer pattern of topaz showing effect of increasing dispersion on separation of peaks. (a) Cu Kα, (b) Cr Kα.

Copper-anode tubes are most frequently used for powder work because of their high intensity and good dispersion. Chromium tubes are often used for specimens containing iron and other transition elements to avoid fluorescence, and for larger dispersion, but require a vacuum or helium path and the intensity is usually one-half or less than that of copper. Molybdenum tubes are often used for single-crystal analysis, but not often for powders because of the low dispersion.

The remarkable properties of synchrotron-radiation sources, which produce very high intensity parallel beams of continuous `white' radiation, are described in Subsection 4.2.1.5 , and their use in powder diffraction in Section 2.3.2.

Fluorescent sources produced by primary X-ray tube excitation of a selected element have the advantage of a wide range of wavelengths but have too low brightness to be useful for powder diffraction. The intensity is 2–3 orders of magnitude lower than an X-ray tube source (Parrish, Lowitzsch & Spielberg, 1958).

Radionuclides that decay by K-electron capture and produce X-rays (e.g. Mn Kα from ⁵⁵Fe) have too low brightness for use in powder diffraction. They are often used to calibrate detectors and to measure the stability of a counting system (Dyson, 1973).

The powder method is based on approximately monochromatic radiation and requires the isolation of a spectral line and/or reduction of the white radiation, except of course for energy-dispersive diffraction. This is done with one or more of the following techniques:

Special methods such as total reflection from a highly polished surface are rarely used in powder diffraction.

2.3.5.4.1. Crystal monochromators

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Reflection from a single-crystal plate is the most common way to obtain monochromatic X-rays. Although the reflected beam is not strictly monochromatic because of the natural width of the spectral line and the rocking angle of the crystal, it is sufficient for practical powder diffraction. The crystal reflects λ and may also reflect subharmonic wavelengths λ/2, λ/3, etc., and higher-order hkl's depending on its crystal structure. Crystals can be selected to avoid the subharmonics, for example, Si and Ge cut parallel to (111); they have a negligible 222 reflection and λ/3 can be easily rejected with pulse-amplitude discrimination. Crystals are selected with relatively small Bragg angles to minimize polarization effects. Virtually all monochromators are of the reflection type. Transmission monochromators such as thin mica have been used occasionally in X-ray spectroscopy but not in powder diffraction.

When a crystal monochromator is placed in the direct beam from an X-ray tube or synchrotron-radiation source, the crystal also reflects other wavelengths from the continuous radiation. It is necessary to take a photograph of the reflected beam to see if Laue spots may be close to the spectral line and might pass through. If Laue spots are a problem and a flat crystal is used, a small rotation will move the spots. The entrance and exit slits should be made as narrow as possible for the experiment and a narrow pulse-height analyser window (see Section 7.1.2 ) may be helpful. In any case, a simple powder pattern will show if the unwanted wavelengths are reaching the specimen.

To achieve maximum performance in terms of intensity and resolution, it is essential to design the X-ray optics so that the properties of the monochromator match the characteristics of the source, specimen, and instrument geometry. A flat crystal is used for parallel beams and a curved crystal for focusing geometries. The curved crystal can accept a much larger divergent primary beam and has the property of converting the incident divergent beam to a convergent beam after reflection. The quality of the crystal and its surface preparation by fine lapping and etching are crucial.

The crystal materials most commonly used are silicon, germanium, and quartz, which have small rocking angles, and graphite and LiF which have large mosaic spreads. A large variety of crystals is available with large and small d spacings for use in X-ray fluorescence spectroscopy. The crystal must be chemically stable and not deteriorate with X-ray exposure. Synthetic multilayer microstructures have recently been developed for longer-wavelength X-rays. A lower atomic number element avoids fluorescence from the crystal.

The common types of monochromators are illustrated in Fig. 2.3.5.6 . The beam reflected from a flat crystal (a) is nearly parallel. If the incident beam is divergent and the crystal is rotated, the reflection will broaden as the rays that make the correct Bragg angle `walk' across the surface. If the crystal is cut at an angle γ to the reflecting plane, the beam is broadened as shown in (b) (or narrowed if reversed) (Fankuchen, 1937; Evans, Hirsch & Kellar, 1948).

Figure 2.3.5.6| top | pdf | Crystal monochromators most frequently used in powder diffraction. (a)–(c) Non-focusing parallel beam, (d)–(f) focusing bent crystals. All may be cut parallel to the reflecting lattice plane (symmetric cut) or inclined (asymmetric cut). The latter are used to expand or condense beam depending on the direction of inclination, and to change focal lengths. (a) Flat symmetric plate. (b) Flat asymmetric plate in orientation to expand beam and increase intensity (Fankuchen, 1937). (c) Channel monochromator cut from highly perfect ingot (Bonse & Hart, 1965). (d) Focusing crystal bent to radius 2R (Johann, 1931). (e) Crystal bent to 2R and surface ground to R (DuMond & Kirkpatrick, 1930; Johannson, 1933). (f) Crystal bent to section of logarithmic spiral (Barraud, 1949; de Wolff, 1968b).

A channel-cut monochromator [Fig. 2.3.5.6(c)] is cut from a single-crystal ingot and both plates, therefore, have exactly the same orientation (Bonse & Hart, 1965, 1966). They are usually made from a high-quality dislocation-free silicon ingot. They can also be designed to give more than two reflections per channel, and can be cut at an angle to the reflecting plane (Deutsch, 1980). Originally designed for small-angle scattering, they are now also used for parallel-beam diffractometry, interferometry, and spectroscopy. They have the important property that the position and direction of the monochromatic beam remain nearly the same for a wide range of wavelengths. This avoids realignment and recalibration of the diffractometer when changing wavelengths in synchrotron diffractometry. The reflections are narrow with minimal tails. The resolution is determined by the energy spread of the perfect-crystal bandpass [which is for Si(111)] and the wavelength dispersion, which is small at small 2θ's and increases with (Beaumont & Hart, 1974; Hart, Rodrigues & Siddons, 1984).

Thin crystals can be bent to form a section of a cylinder for focusing, Fig. 2.3.5.6(d) (Johann, 1931). The safe bending radius is of the order of 1000 to 2000 times the thickness of the crystal plate. The bending radius 2R forms a surface tangent to the focusing circle of radius R. The cylindrical form allows the line focus of the X-ray tube to be used. Because the lattice planes are not always tangent to the focusing circle, as would be required for perfect focusing, the aberrations broaden the focus, but this may not be a serious problem in powder diffraction. If the crystal is also ground so that its surface radius R matches the focusing circle, the aberrations are removed, Fig. 2.3.5.6(e) (DuMond & Kirkpatrick, 1930; Johannson, 1933). The crystal may be initially cut at an angle γ to the surface to change the focal length FL of the incident and reflected beams. Here, and .

Another type of focusing monochromator requires a plane-parallel thin single-crystal plate bent into a section of a logarithmic spiral, Fig. 2.3.5.6(f) (Barraud, 1949). de Wolff (1968b) developed a method of applying unequal forces to the ends of the plate in adjusting the curvature to give a sharp focus (Subsection 2.3.1.2). It has the important advantage that the curvature can be changed while set on a reflection to obtain the best results in setting up the diffractometer.

The most widely used monochromator is highly oriented pyrolytic graphite in the form of a cylindrically curved plate. It is generally used in the diffracted beam after the receiving slit. The basal reflection d(002) = 3.35 Å. Because of its softness, it cannot be ground or cut at an angle to the plane. It is not a true single crystal and has a broad rocking angle of 0.3 to 0.6°, but this is not a problem when the receiving slit determines the profile. Its greatest advantage is the extraordinarily high reflectivity of about 50% for Cu Kα, which is far higher than any other crystal (Renninger, 1956). In practice, some graphite plates may have a reflectivity as low as about 25–30%.

The advantage of placing the monochromator in the diffracted beam is that it eliminates specimen fluorescence except for the wavelength to which it is tuned. In conventional focusing geometry, the receiving slit controls the resolution and intensity. The set of parallel slits that limits the axial divergence in the diffracted beam can be eliminated because the crystal has a smaller effective aperture. By eliminating the slits and the Kβ filter, each of which reduces the intensity by about one half, there is about a twofold gain of intensity. The results are the same using the parallel or antiparallel position of the graphite with respect to the specimen. The dispersive setting makes it easier to use shielding for radioactive samples.

There is no advantage in using a perfect crystal such as Si after the receiving slit because it does not improve the resolution or profile shape, and the intensity is much lower. However, if the monochromator is to be used in the incident beam, it is advisable to use a high-quality crystal because the incident-beam aperture and profile shape are determined by the focusing properties of the monochromator. A narrow slit would be needed to reduce the reflected width of a graphite monochromator and would cause a large loss of intensity.

The use of a small solid-state detector in place of the monochromator should be considered if the count rates are not too high (see Subsection 7.1.5.1 ).

2.3.5.4.2. Single and balanced filters

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Single filters to remove the Kβ lines are also used, but better results are generally obtained with a crystal monochromator. The following description provides the basic information on the use of filters if monochromators are not used. A single thin filter made of, or containing, an element that has an absorption edge of wavelength just less than that of the , doublet will absorb part of that doublet but much more of the Kβ line and part of the white radiation, as shown in Fig. 2.3.5.3. The relative transmission throughout the spectrum depends on the filter element and its thickness.

A filter may be used to modify the X-ray spectral distribution by suppressing certain radiations for any of several reasons:

(1) β lines. β-line intensity need be reduced only enough to avoid overlaps and difficulties in identification in powder work. In single-crystal work, the large peak intensities may require a larger reduction of the β lines, which may be virtually eliminated if so desired. The Kα intensity is also reduced by the filter. For example, a 0.015 mm thick Ni filter reduces Cu Kβ by 99% but also reduces Cu Kα₁ by 60%.

(2) Continuum. The continuum is reduced by the filter but by no means eliminated (see Fig. 2.3.5.3). The greatest reduction occurs for those wavelengths just below the K-absorption edge of the filter. The reduction of the continuum appears greater for Mo than for Cu and lower atomic number targets because the Mo K lines occur near the peak of the continuum. Care must be taken in measuring integrated line intensities when using filters because the K-absorption edge of the filter may cause an abrupt change in the background level on the short-wavelength side of the line.

(3) Contaminant lines. Lines arising from an element other than the pure target element may be absorbed. For example, an Ni filter is an ideal absorber for the W L spectrum.

The filter thickness required to obtain a certain : peak or integrated-intensity ratio at the detector requires the unfiltered peak or integrated-intensity ratio under the same experimental conditions. Then, where the thickness t is in cm and μ is the linear absorption coefficient of the filter for the given wavelength. Table 2.3.5.2 lists the calculated thicknesses of β filters required to reduce the : integrated-intensity ratio to 1/100 and 1/500 for seven common targets. A brass filter has been used to isolate W . The L-absorption edges of high atomic number elements have been used for filtering purposes, but the high absorption of these filters causes a large reduction of the Kα intensity.

Table 2.3.5.2| top | pdf | β filters for common target elements Target element β filter % loss % loss (mm) g cm −2 (mm) g cm −2 Ag Pd 0.62 0.074 60 0.092 0.110 74 Rh 0.062 0.077 59 0.092 0.114 73 Mo Zr 0.081 0.053 57 0.120 0.078 71 Cu Ni 0.015 0.013 45 0.023 0.020 60 Ni Co 0.013 0.011 42 0.020 0.017 57 Co Fe 0.012 0.009 39 0.019 0.015 54 Fe Mn 0.011 0.008 38 0.018 0.013 53 Mn2O3 0.027 0.012 43 0.042 0.019 59 MnO2 0.026 0.013 45 0.042 0.021 61 Cr V 0.011 0.007 37 0.017 0.010 51 V2O5 0.036 0.012 48 0.056 0.019 64

The object of filtering is to obtain an optimum effect at the measuring device (photographic film, counter, etc.), and the distribution of intensity before and after diffraction by the crystalline specimen has to be taken into account in deciding the best position of the filter. The continuum, line spectrum or both cause all specimens to fluoresce, that is, to produce K, L, and M line spectra characteristic of the elements in the specimen. The longer-wavelength fluorescence spectra (λ 2.5 Å) are usually absorbed in the air path or counter-tube window and, hence, are not observed. When using vacuum or helium-path instruments and low-absorbing detector windows, the longer-wavelength fluorescence spectra may appear.

When specimen fluorescence is present, the position of the β filter may have a marked effect on the background. If placed between the X-ray tube and specimen, the filter attenuates a portion of the primary spectrum just below the absorption edges of the elements in the specimen, thereby reducing the intensity of the fluorescence. When placed between the specimen and counter tube, the filter absorbs some of the fluorescence from the specimen. The choice of position will depend on the elements of the X-ray tube target and specimen. If the filter is placed after the specimen, it is advisable to place it close to the specimen to minimize the amount of fluorescence from the filter that reaches the detector. The fluorescence intensity decreases by the inverse-square law. Maximizing the distance between the specimen and detector also reduces the specimen fluorescence intensity detected for the same reason. If the filter is to be placed between the X-ray tube and specimen, the filter should be close to the tube to avoid fluorescence from the filter that might be recorded. It is sometimes useful to place the filter over only a portion of the film in powder cameras to facilitate the identification of the β lines.

If possible, the X-ray tube target element should be chosen so that its β filter also has a high absorption for the specimen X-ray fluorescence. For example, with a Cu target and Cu specimen, the continuum causes a large Cu K fluorescence that is transmitted by an Ni filter; if a Co target is used instead, the Cu K fluorescence is greatly decreased by an Fe Kβ filter. A second filter may be useful in reducing the fluorescence background. For example, with a Ge specimen, the continuum from a Cu target causes strong Ge K fluorescence, which an Ni filter transmits. Addition of a thin Zn filter improves the peak/background ratio (P/B) of the Cu Kα with only a small reduction of peak intensity (Ge Kα, λ = 1.25 Å; Zn K-absorption edge, λ =1.28 Å).

X-ray background is also caused by scattering of the entire primary spectrum with varying efficiency by the specimen. The filter reduces the background by an amount dependent on its absorption characteristics. When using pulse-amplitude discrimination and specimens whose X-ray fluorescence is weak, the remaining observed background is largely due to characteristic line radiation. The β filter then usually reduces the background and the Kα radiation by roughly the same amount and P/B is not changed markedly regardless of the position of the filter.

The β filter is sometimes used instead of black paper or Al foil to screen out visible and ultraviolet light. Filters in the form of pure thin metal foils are available from a number of metal and chemical companies. They should be checked with a bright light source to make certain they are free of pinholes.

The balanced-filter technique uses two filters that have absorption edges just above and just below the Kα₁, Kα₂ wavelengths (Ross, 1928; Young, 1963). The difference between intensities of X-ray diffractometer or film recordings made with each filter arises from the band of wavelengths between the absorption edges, which is essential that of the Kα₁, Kα₂ wavelengths. The thicknesses of the two filters should be selected so that both have the same absorption for the Kβ wavelength. Table 2.3.5.3 lists the calculated thicknesses of filter pairs for the common target elements. The (A) filter was chosen for a 67% transmission of the incident Kα intensity, and only pure metal foils are used. Adjustment of the thickness is facilitated if the foil is mounted in a rotatable holder so that the ray-path thickness can be varied by changing the inclination of the foil to the beam.

Table 2.3.5.3| top | pdf | Calculated thickness of balanced filters for common target elements Target material Filter pair (A) (B) Thickness Thickness (A) (B) mm g cm−3 mm g cm−2 Ag Pd Mo 0.0275 0.033 0.039 0.040 Mo Zr Sr 0.0392 0.026 0.104 0.027 Mo Zr Y 0.0392 0.026 0.063 0.028 Cu Ni Co 0.0100 0.0089 0.0108 0.0095 Ni Co Fe 0.0094 0.0083 0.0113 0.0089 Co Fe Mn 0.0098 0.0077 0.0111 0.0083 Fe Mn Cr 0.0095 0.0071 0.0107 0.0077 Cr V Ti 0.0097 0.0059 0.0146 0.0066

Although the two filters can be experimentally adjusted to give the same Kβ intensities, they are not exactly balanced at other wavelengths. The use of pulse-amplitude discrimination to remove most of the continuous radiation is desirable to reduce this effect. The limitations of the method are (a) the difficulties in adjusting the balance of the filters, (b) the band-pass is much wider that that of a crystal monochromator, and (c) it requires two sets of data, one of which has low intensity and consequently poor counting statistics.