Origin of the Powder Diffraction Pattern

8.1 Origin of the Powder Diffraction Pattern

As has been established in Sect. 7.2.3, the primary monochromatic beam is scattered in a particular direction, which is easily predicted using Ewlad’s representation (see Figs. 7.10–7.12). A similar, yet fundamentally different situation is observed in the case of diffraction from powders or from polycrystalline specimens, that is, when multiple single crystals (crystallites or grains) are irradiated simultaneously by a monochromatic incident beam. When the number of grains in the irradiated volume is large and their orientations are completely random, the same is true for the recip- rocal lattices associated with each crystallite. Thus, the ends of the identical recip- rocal lattice vectors, d ∗ hkl , become arranged on the surface of the Ewald’s sphere in

a circle perpendicular to the incident wavevector, k 0 . The corresponding scattered wavevectors, k 1 , are aligned along the surface of the cone, as shown in Fig. 8.1. The apex of the cone coincides with the center of the Ewald’s sphere, the cone axis is parallel to k 0 , and the solid cone angle is 4 θ. Assuming that the number of crystallites approaches infinity (the randomness of their orientations has been postulated in the previous paragraph), the density of the scattered wavevectors, k 1 , becomes constant on the surface of the cone. The diffracted intensity will therefore, be constant around the circumference of the cone base or, when measured by a planar area detector as shown in Fig. 8.1, around the ring, which the cone base forms with the plane of the detector. Similar rings but with different intensities and diameters are formed by other independent reciprocal

lattice vectors, and these are commonly known as the Debye 2 rings. The appearance of eight diffraction cones when polycrystalline copper powder is irradiated by the monochromatic Cu K α 1 radiation is shown in Fig. 8.2. All Bragg peaks, possible in the range 0 ◦ < 2θ < 180 ◦ , are also listed with the corresponding Miller indices and relative intensities in Table 8.1.

Assuming that the diffracted intensity is distributed evenly around the base of each cone (see the postulations made earlier), there is usually no need to measure the intensity of the entire Debye ring. Hence, in a conventional powder diffraction experiment, the measurements are performed only along a narrow rectangle centered at the circumference of the equatorial plane of the Ewald’s sphere, as shown in

2 Petrus (Peter) Josephus Wilhelmus Debye (1884–1966). Dutch physical chemist credited with numerous discoveries in physics and chemistry. Relevant to the subject of this book is his cal-

culation of the effect of temperature on the scattered X-ray intensity (the Debye-Waller fac- tor) and his work together with Paul Scherrer on the development of the powder diffraction method. In 1936 Peter Debye was awarded the Nobel Prize in Chemistry for “his contributions to our knowledge of molecular structure through his investigations on dipole moments and on the diffraction of X-rays and electrons in gases.” For more information see http://nobelprize.org/ nobel prizes/chemistry/laureates/1936/debye-bio.html.

8.1 Origin of the Powder Diffraction Pattern 153

Fig. 8.1 The origin of the powder diffraction cone as the result of the infinite number of the com- pletely randomly oriented identical reciprocal lattice vectors, d ∗ hkl , forming a circle with their ends placed on the surface of the Ewald’s sphere, thus producing the powder diffraction cone and the

corresponding Debye ring on the flat screen (film or area detector). 3 The detector is perpendicular to both the direction of the incident beam and cone axis, and the radius of the Debye ring in this geometry is proportional to tan 2 θ.

Fig. 8.2 and indicated by the arc with an arrow marked as 2 θ. Because of this, only one variable axis (2 θ) is fundamentally required in powder diffractometry, yet the majority of instruments have two independently or jointly controlled axes. The latter is done due to a variety of reasons, such as the limits imposed by the geometry, more favorable focusing, particular application, etc. More details about the geometry of modern powder diffractometers is given in Chap. 11.

In powder diffraction, the scattered intensity is customarily represented as a func- tion of a single independent variable – Bragg angle – 2 θ, as modeled in Fig. 8.3 for

a polycrystalline copper. This type of the plot is standard and it is called the pow- der diffraction pattern or the histogram. In some instances, the diffracted intensity may be plotted versus the interplanar distance, d, the q–value (q = 1/d 2 =d ∗2 ), or sin θ/λ (or the Q-value, which is different from sin θ/λ by a factor of 4π, see Footnote 9 on page 146).

3 In this geometry (flat detector perpendicular to the incident beam placed behind the sample) it is fundamentally impossible to measure intensity scattered at 2 θ ≥ 90 ◦ . One alternative is to place

the detector between the focal point of the X-ray tube and the sample; this enables to measure in- tensity scattered at 2 θ > 90 ◦ . In either case, when diffraction occurs at 2 θ = 90 ◦ , the measurement is impossible (tan 90 ◦ = ∞). Furthermore, when 2θ ∼ = 90 ◦ , the size of a flat detector becomes pro- hibitively large. For practical measurements, a flat detector may be tilted at any angle with respect to the propagation vector of the incident beam. Instead of a flat detector, a flexible image plate detector may be arranged as a cylinder with its axis perpendicular to the incident wavevector and

154 8 The Powder Diffraction Pattern Ewald’s

Fig. 8.2 The schematic of the powder diffraction cones produced by a polycrystalline copper sam- ple using Cu K α 1 radiation. The differences in the relative intensities of various Bragg peaks (diffraction cones) are not discriminated, and they may be found in Table 8.1. Each cone is marked with the corresponding triplet of Miller indices.

Table 8.1 Bragg peaks observed from a polycrystalline copper using Cu K α 1 radiation. 4 hkl

The scattered intensity is usually represented as the total number of the accumu- lated counts, counting rate (counts per second – cps) or in arbitrary units. Regardless of which units are chosen to plot the intensity, the patterns are visually identical be- cause the intensity scale remains linear, and because the intensity measurements are normally relative, not absolute. In rare instances, the intensity is plotted as a com- mon or a natural logarithm, or a square root of the total number of the accumulated counts, in order to better visualize both the strong and weak Bragg peaks on the same

traversing the location of the specimen, which facilitates simultaneous measurement of the entire powder diffraction pattern.

4 The data are taken from the ICDD powder diffraction file, record No. 4-836: H.E. Swanson, E. Tatge, National Bureau of Standards (US), Circular 359, 1 (1953).

8.1 Origin of the Powder Diffraction Pattern 155

111 Cu Kα 1 /Kα 2 radiation

Relative intensity 113 022

Bragg angle, 2θ (deg.)

Fig. 8.3 The simulated powder diffraction pattern of copper (space group Fm¯3m, a = 3.615 ˚ A, Cu K α 1 /Kα 2 radiation, Cu atom in 4(a) position with x = 0, y = 0, z = 0).

plot. The use of these two nonlinear intensity scales, however, always increases the visibility of the noise (i.e., highlights the presence of statistical counting errors). A few examples of the nonconventional representation of powder diffraction patterns are found in Sect. 8.2.

In the Chap. 7, we assumed that the diffracted intensity is observed as infinitely narrow diffraction maxima (delta functions). In reality, the Ewald’s sphere has finite thickness due to wavelength aberrations, and reciprocal lattice points are far from infinitesimal shapeless points – they may be reasonably imagined as small diffuse spheres (do not forget that the reciprocal lattice itself is not real and it is nothing else than a useful mathematical concept). Therefore, Bragg peaks always have nonzero widths as functions of 2 θ, which is illustrated quite well in Figs. 8.4 and 8.5 by

the powder diffraction pattern of LaB 6 . 5 The data were collected using Mo K α radiation on a Bruker SmartApex diffractometer equipped with a flat CCD detector placed perpendicular to the primary beam. This figure also serves as an excellent experimental confirmation of our conclusions made at the beginning of this section (e.g., see Figs. 8.1 and 8.2).

As shown in Fig. 8.4, when diffraction cones, produced by the LaB 6 powder, intersect with the flat detector placed perpendicularly to the incident wavevector, they create a set of concentric Debye rings. As in a typical powder diffractometer, only a narrow band has been scanned, and the result of the integration is also shown in Fig. 8.4 as the scattered intensity versus tan 2 θ (note that the radial coordinate of the detector is tan 2 θ, and not 2θ). The resultant diffraction pattern is shown in the standard format as relative intensity versus 2 θ in Fig. 8.5, where each Bragg peak is labeled with the corresponding Miller indices. It is worth noting that the diffractometer used in this experiment is a single crystal diffractometer, which was

5 NIST standard reference material, SRM 660 (see http://ts.nist.gov/measurementservices/ referencematerials/index.cfm).

156 8 The Powder Diffraction Pattern

Intensity

tan2 θ

Fig. 8.4 Left – the X-ray diffraction pattern of a polycrystalline LaB 6 obtained using Mo K α radiation and recorded using a flat CCD area detector placed perpendicular to the incident beam wavevector (compare with Figs. 8.1 and 8.2). Measured intensity is proportional to the degree of darkening. The diffuse white line extending from the center of the image to the top left corner is the projection of the wire holding the beam stop needed to protect the detector from being damaged by the high intensity incident beam. The white box delineates the area in which the scattered intensity was integrated from the center of the image toward its edge. Right – the resultant intensity as a function of tan2 θ shown together with the area over which the integration has been carried out.

LaB 6 , Mo Kα, CCD detector 2500

(arb. units) Y

211 Relative intensity,

Bragg angle, 2θ (deg.)

Fig. 8.5 The powder diffraction pattern of the polycrystalline LaB 6 as intensity versus 2 θ obtained by the integration of the rectangular area from the two-dimensional diffraction pattern shown in Fig. 8.4.

8.2 Representation of Powder Diffraction Patterns 157 not designed to take full advantage of focusing of the scattered beam. As a result, the

Bragg peaks shown in Fig. 8.5 are quite broad and the K α 1 /α 2 doublet is unresolved even when 2 θ approaches 30 ◦ . As we will see in Chap. 12 (e.g., see Fig. 12.21) a much better resolution is possible in high resolution powder diffractometers, where the doublet becomes resolved at much lower Bragg angles.