International
Tables for Crystallography Volume H Powder diffraction Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk © International Union of Crystallography 2018 
International Tables for Crystallography (2018). Vol. H, ch. 1.1, pp. 911
Section 1.1.3. The peak intensity^{a}MaxPlanckInstitute for Solid State Research, Heisenbergstrasse 1, D70569 Stuttgart, Germany,^{b}Department of Applied Physics and Applied Mathematics, Columbia University, 500 West 120th Street, Room 200 Mudd, MC 4701, New York, NY 10027, USA, and ^{c}Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, PO Box 5000, Upton, NY 11973–5000, USA 
Bragg's law gives the positions at which diffraction by a crystal will lead to sharp peaks (known as Bragg peaks) in diffracted intensity. We now want to investigate the factors that determine the intensities of these peaks.
Xrays are electromagnetic (EM) waves with a much shorter wavelength than visible light, typically of the order of 1 Å (= 10^{−10} m). The physics of EM waves is well understood and excellent introductions to the subject are found in every textbook on optics. Here we briefly review the results that are most important in understanding the intensities of Bragg peaks.
Classical EM waves can be described by a sine wave of wavelength that repeats every 2π radians. If two identical waves are not coincident, they are said to have a phase shift, which is either measured as a shift, Δ, on a length scale in units of the wavelength, or equivalently as a shift in the phase, , on an angular scale, such thatThis is shown in Fig. 1.1.14.

Graphical illustration of the phase shift between two sine waves of equal amplitude. [Reproduced from Dinnebier & Billinge (2008) with permission from the Royal Society of Chemistry.] 
The detected intensity, I, is proportional to the square of the amplitude, A, of the sine wave. With two waves present that are coherent and can interfere, the amplitude of the resultant wave is not just the sum of the individual amplitudes, but depends on the phase shift . The two extremes occur when (constructive interference), where , and (destructive interference), where . In general, . When more than two waves are present, this equation becomeswhere the sum is over all the sine waves present and the phases, , are measured with respect to some origin.
Measuring Xray diffraction involves the measurement of the intensity of Xrays scattered from electrons bound to atoms. Waves scattered by atoms at different positions arrive at the detector with a relative phase shift. Therefore, the measured intensities yield information about the relative atomic positions.
In the case of Xray diffraction, the Fraunhofer approximation is valid. This is a farfield approximation, where the distances L_{1} from the source to the place where scattering occurs (the sample) and L_{2} from the sample to the detector are much larger than the separation, D, of the scatterers. This is an excellent approximation, since in this case . The Fraunhofer approximation greatly simplifies the mathematics. The incident Xrays come from a distant source and form a wavefront of constant phase that is a plane wave. Xrays scattered by single electrons are outgoing spherical waves, which again appear as plane waves in the far field. This allows us to express the intensity of the diffracted Xrays using equations (1.1.51) and (1.1.39).
This is the origin of equation (1.1.39), which gives the amplitude of the scattered radiation in terms of the scattering vector, h = s_{0} − s, and the atomic positions, r_{j}. In fact, the amplitude of the scattered radiation is only proportional to this expression. The actual intensity depends on the amplitude of the incident wave and also on the absolute scattering power of the scatterers. If we neglect for now the incident intensity and assume that our measured intensities are normalized to the incident beam intensity, we getwhere is the atomic form factor and is the magnitude of the scattering vector, and is described in more detail in International Tables for Crystallography, Volume C, Part 6 . This is a measure of the strength of scattering from the jth atom. At , scattering is in the forward direction with all electrons scattering in phase. As a result, f_{j}(0) equals the number of electrons bound to the atom (in units of the Thomson scattering cross section for an electron), usually taken to be the atomic number of the atomic species at the jth site. An additional hdependent reduction of the amplitude comes from positional disorder of the atoms. A Gaussian blurring is used with a width that is often falsely called the `temperature factor', but is more correctly known as the atomic displacement parameter (ADP). The Gaussian is known as the Debye–Waller factor, which is discussed below. More information can be found in Chapter 4.7 .
The crystal structure consists of periodic arrangements of atoms. The simplest structures have one atom in a periodically repeated unit cell. However, in general, there is a well defined group of atoms that forms a structural motif that is periodically repeated. This motif can range from one atom to thousands of atoms in complex protein structures. Solving the crystal structure consists of finding the unitcell parameters and determining the positions in the unit cell of the atoms in the structural motif. In this sense, the structure of the infinite crystal can be thought of mathematically as a convolution of the periodic lattice that we discussed above with the structural motif. This results in a perfect, orientationally ordered copy of the structural motif in every unit cell translated in threedimensional space.
As we discussed above, the directspace lattice has a reciprocal lattice associated with it which determines the positions of the Bragg peaks, or allowed delta functions of scattered intensity. The reciprocal lattice is actually a Fourier transform of the periodic lattice in direct space. The convolution theorem of Fourier transforms tells us that a convolution of two functions in direct space will result in a product of the Fourier transforms of those functions in the Fourier space. Since the structure is a convolution of the directspace lattice with the structural motif, the reciprocal lattice will be multiplied by the Fourier transform of the structural motif. This Fourier transform of the structural motif is called the crystallographic structure factor, F_{hkl}.
This result can be readily derived from equation (1.1.52). In this equation is the vector from the (arbitrary but fixed) origin to the jth atom in the material. If we now think of the crystal as consisting of n identical cells, each containing an identical structural motif consisting of m atoms, we can write as a sum of two vectors: a vector that goes from the origin to the corner of the sth unit cell that contains the jth atom, and a second vector that goes from the corner of the sth cell to the position of the jth atom. This is illustrated in Fig. 1.1.15.
Equation (1.1.52) can then be written aswhere it is readily seen that the first sum is taken over all the cells in the crystal and the second sum is taken over the m atoms in the structural motif. The equation is readily factored as follows:Taking n to infinity, we immediately recognise the first sum as the lattice sum of equation (1.1.43), and we can therefore rewrite equation (1.1.54) asThe delta functions determine the positions of the reciprocallattice points (directions of the Bragg peaks), and their intensities are multiplied by a factor, the crystallographic structure factor,If we write each term as a complex number denoted f_{t}, we can represent this complex sum as a vector sum in the complex plane, as illustrated in Fig. 1.1.16, where the . The intensity of the Bragg peak depends only on the length of the F_{hkl}, not its direction. However, its length depends on both the lengths and the phases of each contribution, which in turn depend on the positions of the atoms within the unit cell. This is the phase information that is `lost' in a diffraction experiment. Given a structure, we can directly calculate all the Braggpeak intensities (the `forward problem'). However, given all the Braggpeak intensities, we cannot directly calculate the structure (the `inverse problem'). Structure determination uses the measured intensities and reconstructs the lost phase information using various iterative methods and algorithms.

Graphical illustration of the summation of scattered wave amplitudes f_{t} in the complex plane, accounting for the phase shifts coming from the different positions of the atoms in the unit cell. 
In fact, the intensity of a Bragg reflection hkl is given by the squared absolute value of the structurefactor amplitude F_{hkl},where * indicates the complex conjugate. This analysis shows that the positions of the Bragg peaks determine the geometry of the periodic lattice (the size and shape of the unit cell, for example), but the intensities of the Bragg peaks are determined by the relative positions of atoms within the unit cell, scaled by their respective scattering power. To solve the internal structure of the structural motif within the unit cell, it is necessary to measure quantitatively the intensities of many Bragg peaks and use some kind of iterative procedure to move the atoms within the cell until the calculated structure factors selfconsistently reproduce the intensities of all the measured Bragg peaks.
The situation is not fundamentally different in a powder diffraction experiment from the singlecrystal case, except that the Bragg peaks in threedimensional reciprocal space are projected into one dimension, as shown in Fig. 1.1.17.

Schematic illustration of the projection of the reciprocal a*c* plane (representing the threedimensional reciprocallattice space) into the onedimensional powder pattern. 
`Indexing' is the term used for deriving the lattice parameters from the positions of the Bragg peaks (see Chapter 3.4 ). Once the size and shape of the reciprocal lattice is determined, Miller indices can be assigned to each of the Bragg peaks in a onedimensional powder pattern. If it is possible to extract the intensities of those peaks from the pattern, diffraction data from a powder can be used to reconstruct the threedimensional structure in exactly the same way as is done with data from a single crystal. This process is known as structure solution from powder diffraction, and is often successful, although it is less well automated than structure solution from data from single crystals. As mentioned above, the main problem with powder data is a loss of information due to systematic and accidental peak overlap, but this can often be overcome.
There are various methods for extracting quantitative peak intensities from indexed powder patterns by computer fitting of profiles to the Bragg peaks at their known positions. Two of the most common are Pawley refinement (Pawley, 1981) and Le Bail refinement (Le Bail et al., 1988), as discussed in Chapter 3.5 .
In general, the intensities of the Bragg reflections must be corrected by the product of various correction factors. Some common correction factors are given bywhere M_{hkl} is the multiplicity, Abs_{hkl} is an absorption correction, Ext_{hkl} is an extinction correction, LP_{hkl} is the geometrical Lorentz–polarization correction and PO_{hkl} is a correction for preferred orientation (see Chapter 4.7 ).
If there is more than one crystalline phase present in the sample, and the structures of all the crystalline phases are known, then we can find a scale factor for each phase in the mixture which reproduces the data. This is then a way of determining the proportion of each phase in the sample. This is called quantitative phase analysis (see Chapter 3.9 ).
References
Le Bail, A., Duroy, H. & Fourquet, J. L. (1988). Abinitio structure determination of LiSbWO_{6} by Xray powder diffraction. Mater. Res. Bull. 23, 447–452.Google ScholarPawley, G. S. (1981). Unitcell refinement from powder diffraction scans. J. Appl. Cryst. 14, 357–361.Google Scholar