Understanding of Powder Diffraction Patterns

8.3 Understanding of Powder Diffraction Patterns

The best way to appreciate and understand how structural information is encoded in

a powder diffraction pattern is to consider the latter as a set of discrete diffraction peaks (Bragg reflections) superimposed over a continuous background. Although the background may be used to extract information about the crystallinity of the specimen and few other parameters about the material, we are concerned with the Bragg peaks and not with the background. In the majority of powder diffraction applications, the background is an inconvenience which has to be dealt with, and generally every attempt is made to achieve its minimization during the experiment.

Disregarding the background, the structure of a typical powder diffraction pattern may be described by the following components: positions, intensities, and shapes of multiple Bragg reflections, for example, compare Figs. 8.6a and 8.7a. Each of the three components italicized here contains information about the crystal structure of the material, the properties of the specimen (sample), and the instrumental parame- ters, as shown in Table 8.2. Some of these parameters have a key role in defining a particular component of the powder diffraction pattern, while others result in vari- ous distortion(s), as also indicted in Table 8.2. It is worth noting that this table is not

160 8 The Powder Diffraction Pattern Table 8.2 Powder diffraction pattern as a function of various crystal structure, specimen and in-

strumental parameters. a Pattern

Instrumental component

Crystal structure

Specimen

parameter Peak

property

Unit cell parameters: Absorption Radiation (wavelength) position

(a , b, c, α, β, γ)

Porosity

Instrument/sample alignment Axial divergence of the beam

Preferred orientation

Peak Atomic parameters

Geometry and configuration intensity

Absorption

(x , y, z, B, etc.)

Radiation (Lorentz, polarization) Crystallinity

Porosity

Radiation (spectral purity) Peak shape Disorder

Grain size

Geometry Defects

Strain

Beam conditioning a Key parameters are shown in bold. Parameters that may have a significant influence are shown in italic.

Stress

comprehensive and additional parameters may affect the positions, intensities, and shapes of Bragg peaks.

In addition to the influence brought about by the instrumental parameters, there are two kinds of crystallographic (structural) parameters, which essentially define the makeup of every powder diffraction pattern. These are the unit cell dimensions and the atomic structure (both the unit cell content and spatial distributions of atoms in the unit cell). Thus, a powder diffraction pattern can be constructed (or simulated) as follows:

– Positions of Bragg peaks are established from the Braggs’ law as a function of the wavelength and the interplanar distances, that is, d-spacing. The latter can be easily calculated from the known unit cell dimensions (Sect. 8.4). For instance, in the case of the orthorhombic crystal system permissible Bragg angles are found from

λ 2 h k 2 l 2 −1/2

2 θ hkl = 2 arcsin

, where d hkl =

c 2 Since h, k , and l are integers, both the resultant d-values and Bragg angles form

hkl

2d b 2

arrays of discrete values for a given set of unit cell dimensions. Bragg angles are also dependent on the employed wavelength. The example of the discontinuous distribution of Bragg angles is shown using short vertical bars of equal length in Fig. 8.8a.

– As noted in Sect. 7.1, the intensity of diffraction maxima is a function of the periodicity of the scattering centers (unit cells) and therefore, the intensities can

be calculated for individual Bragg peaks from the structural model. The latter requires the knowledge of the coordinates of atoms in the unit cell together with other relevant atomic and geometrical parameters. The influence of the varying intensity on the formation of the powder diffraction pattern is illustrated using the varying lengths of the bars in Fig. 8.8b – the longer the bar, the higher the

8.3 Understanding of Powder Diffraction Patterns 161

2 θ Fig. 8.8 The appearance of the powder diffraction pattern: (a) only Bragg peak positions (e.g.,

see (8.1)) are represented by the vertical bars of equal length; (b) in addition to peak positions, their intensities are indicated by using the bars with variable lengths (the higher the intensity, the longer the bar); (c) peak shapes have been introduced by convoluting individual intensities with appropriate peak-shape functions, and a constant background has been indicated by the dash- double dotted line ; (d) the resultant powder diffraction pattern is the sum of all components shown separately in (c), i.e., discrete but partially overlapped peaks and continuous background.

intensity. Although not shown in Fig. 8.8, certain combinations of Miller indices may have zero or negligibly small intensity and, therefore, the corresponding Bragg reflections disappear or become unrecognizable in the diffraction pattern.

– The shape of Bragg peaks is usually represented by a bell-like function – the so- called peak-shape function. The latter is weakly dependent on the crystal struc- ture and is the convolution of various individual functions, established by the instrumental parameters and to some extent by the properties of the specimen, see Table 8.2. The shape of each peak can be modeled using instrumental and specimen characteristics, although in reality ab initio modeling is difficult and most often it is performed using various empirically selected peak-shape func- tions and parameters. If the radiation is not strictly monochromatic, that is, when

both K α 1 and K α 2 components are present in the diffracted beam, the resultant peak should include contributions from both components as shown in Fig. 8.9. Thus, vertical bars with different lengths are replaced by the corresponding peak shapes, as shown in Fig. 8.8c. It should be noted that although the relative inten- sities of different Bragg reflections may be adequately represented by the lengths of the bars, this is no longer correct for peak heights: the bars are one-dimensional and have zero area, but peak area is a function of the full width at half maximum, which varies with Bragg angle. Individual peaks should have their areas propor- tional to intensities of individual Bragg reflections (see Sect. 8.6.1).

162 8 The Powder Diffraction Pattern

2 θ 2 θ Fig. 8.9 The two individual peak-shape functions corresponding to monochromatic K α 1 and K α 2

wavelengths (left) and the resulting combined peak-shape function for a K α 1 /Kα 2 doublet as the sum of two peaks (right). Since both K α 1 and K α 2 peaks correspond to the same d ∗ hkl , their posi- tions, θ 1 and θ 2 , are related as sin θ 1 / λ Kα 1 = sin θ 2 / λ Kα 2 (see (7.9), while their areas (intensities) are related as approximately 2 to 1 (see Fig. 6.5).

– Finally, the resultant powder diffraction pattern is a sum of the individual peak- shape functions and a background function as illustrated in Fig. 8.8d, where the background function was assumed constant for clarity.

It is generally quite easy to simulate the powder diffraction pattern when the crystal structure of the material is known (the peak-shape parameters are empirical and the background, typical for a given instrument, may be measured). The inverse process, that is, the determination of the crystal structure from powder diffraction data is much more complex. First, individual Bragg peaks should be located on the pattern, and both their positions and intensities determined by fitting to a certain peak-shape function, including the background. Second, peak positions are used to establish the unit cell symmetry, parameters and content. Third, peak intensities are used to determine space-group symmetry and coordinates of atoms. Fourth, the en- tire diffraction pattern is used to refine all crystallographic and peak-shape function parameters, including the background. All these issues are discussed and illustrated beginning from Chap. 13.