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. 2.5, pp. 132147
Section 2.5.4. Applications^{a}Bruker AXS Inc., 5465 E. Cheryl Parkway, Madison, WI 53711, USA 
In materials science, a phase is defined as a region that has uniform chemical composition and physical properties, including crystal structure. Therefore, every phase should give a unique diffraction pattern. A sample for Xray diffraction may contain a single phase or multiple phases. Analysis of the diffraction pattern can accurately and precisely determine the contents of the sample. This qualitative analysis is called phase identification (phase ID). One of the most efficient methods of phase identification is to compare the diffraction pattern from an unknown material to those in a database of a large number of standard diffraction patterns. The most comprehensive database is the Powder Diffraction File (PDF), updated annually by the International Centre for Diffraction Data (ICDD).
Twodimensional Xray diffraction has enhanced phase identification in many respects (Rudolf & Landes, 1994; Sulyanov et al., 1994; Hinrichsen, 2007). Because of its ability to collect diffracted Xrays in a large angular range in both the 2θ and γ directions, it can collect diffraction data with high speed and better sampling statistics than obtained by conventional diffraction. Owing to pointbeam illumination on the sample, a relatively small sample size is required for phase identification. The large 2D detector allows for a large 2θ range to be analysed without any movement of the sample and detector. This makes it possible to perform in situ phase investigation on samples during phase transformations, chemical reactions and deformations. The diffraction information in the γ direction allows accurate phase identification of samples with large grains and preferred orientation.
In the Bragg–Brentano geometry, the 2θ resolution is controlled by the selection of the divergence slit and receiving slit in the diffractometer plane, and the axial divergence is controlled by Soller slits, while in a diffractometer with a 2D detector, the 2θ resolution is mainly determined by the spatial resolution of the detector and the sampletodetector distance. The relative peak intensity in a diffraction pattern from a sample with texture measured with a 2D detector can be significantly different from the results measured with Bragg–Brentano geometry. It is imperative to study the nature of these discrepancies so that the diffraction patterns collected with 2D detectors can be used for phase ID with proper understanding and correction if necessary.
When twodimensional diffraction is used for phase identification, the first step is to integrate the 2D diffraction frame into a diffraction profile resembling the diffraction pattern collected with a conventional diffractometer (Cervellino et al., 2006; RodriguezNavarro, 2006; Boesecke, 2007; FuentesMontero et al., 2011; Hammersley, 2016). The integrated diffraction profiles can be analysed with all existing algorithms and methods, including profile fitting with conventional peak shapes and fundamental parameters, quantification of phases, and latticeparameter determination and refinement (Ning & Flemming, 2005; Flemming, 2007; Jabeen et al., 2011). The results can be used to search a powderdiffraction database to find possible matches. Since there is a great deal of literature covering these topics (Cullity, 1978; Jenkins & Snyder, 1996; Pecharsky & Zavalij, 2003), this section will focus on the special characteristics of twodimensional Xray diffraction as well as system geometry, datacollection strategies and data analysis in dealing with relative peak intensities, 2θ resolution, grain size and distribution, and preferred orientation. Many factors and correction algorithms described here can help in understanding the characteristics of twodimensional diffraction. In most applications, however, the γintegrated profile can be used for phase identification without these corrections.
The integrated intensity diffracted from polycrystalline materials with a random orientation distribution is given bywhere k_{I} is an instrument constant that is a scaling factor between the experimental observed intensities and the calculated intensity, p_{hkl} is the multiplicity factor of the crystal plane (hkl), v is the volume of the unit cell, (LPA) is the Lorentz–polarization and absorption factors, λ is the Xray wavelength, is the structure factor for the crystal plane (hkl), is the normalized poledensity distribution function and exp(−2M_{t} − 2M_{s}) is the attenuation factor due to lattice thermal vibrations and weak static displacements (Warren, 1990; He et al., 1994). Except for the texture effect, all the factors in the above equation are either discussed in the previous sections or have the same definitions and values as in conventional diffraction.
Phaseidentification studies by XRD are preferably carried out on powders or polycrystalline samples with a random orientation distribution of crystallites. Preferred orientation causes relative intensities to deviate from theoretical calculations or those reported in reference databases. In practice, a sample with a perfectly random orientation distribution of crystallites is very hard to fabricate and most polycrystalline samples have a preferred orientation to a certain extent. Discrepancies in the relative peak intensities between conventional diffraction and 2DXRD are largely due to texture effects. For BB geometry, the diffraction vector is always perpendicular to the sample surface. With a strong texture, it is possible that the pole density of certain reflections in the sample normal direction is very low or even approaches zero. In this case, the peak does not appear in the diffraction pattern collected in BB geometry. In 2DXRD, several diffraction rings may be measured with a single incident beam; the corresponding diffraction vectors are not necessarily in the sample normal direction. The diffraction profiles from 2D frames are produced by γ integration, therefore the texture factor should be replaced by the average normalized poledensity function within the γ integration range . The relation between (α, β) and (2θ, γ) is given in Chapter 5.4 . The chance of having zero pole density over the entire γintegration range is extremely small. Therefore, phase identification with 2DXRD is much more reliable than with conventional diffraction.
The 2θ resolution with BB geometry is controlled by the size of the slits. Smaller apertures of the divergence slit are used for higher 2θ resolution and larger apertures for fast data collection. With a twodimensional Xray diffraction system, the 2θ resolution is achieved with different approaches. A flat 2D detector has the flexibility to be used at different sampletodetector distances. The detector resolution is determined by the pixel size and pointspread function. For the same detector resolution and detector active area, a higher resolution can be achieved at larger distance, and higher angular coverage at shorter distance. The sampletodetector distance should be optimized depending on the 2θ measurement range and required resolution. In situations where the 2θ range of one frame is not enough, several frames at sequential 2θ ranges can be collected. The integrated profiles can then be merged to achieve a large 2θ range. Fig. 2.5.16 shows four 2D frames collected from a battery material with a microgap detector. The slice integration region is defined by two conic lines and two horizontal lines. The diffraction profile integrated from the merged frames is displayed below.
A 2D diffraction pattern over a range of 2θ is measured simultaneously with a single incident angle, so the incident angle has to be lower than the minimum 2θ angle. Since the reflected angle cannot always be the same as the incident angle, geometric aberrations are observed. The defocusing effect occurs when the incident angle is lower than the reflection angle. At low incident angles, the incident beam spreads over the sample surface into an area much larger than the size of the original Xray beam. The observed diffracted beam size is magnified by the defocusing effect if the diffracted beam makes an angle larger than the incident angle. The defocusing effect for reflectionmode diffraction can be expressed aswhere θ_{1} is the incident angle, b is the incident beam size and B is diffracted beam size. The ratio of B to b is a measurement of the geometric aberration and will be referred to as the defocusing factor. In principle, defocusing occurs only when B/b is larger than 1. The reflected beam is actually focused to the detector when θ_{2} < θ_{1}. The defocusing effect occurs when θ_{2} > θ_{1} and the defocusing factor increases with increasing θ_{2} or decreasing θ_{1}. The maximum defocusing appears at θ_{2} = 90°. For the θ–2θ configuration, the incident angle ω (= θ_{1}) is used in the equation.
For BB geometry with a divergent slit and receiving slit of the same size the defocusing factor is always 1. With a 2D detector the defocusing factor varies with the 2θ angle. If a large 2θ range is measured on a flat sample in reflection mode, it is always desirable to collect several frames at different incident angles for each 2θ range so as to improve the 2θ resolution. A cylindrical detector may collect a diffraction pattern over a large 2θ range (Gelfi et al., 2005). However, the defocusing effect prevents it from being used for a large 2θ range for a flat sample. Fig. 2.5.17 compares the effect for a flat detector and a cylindrical detector. Fig. 2.5.17(a) shows a cylindrical detector being used to collect a diffraction pattern from a flat sample for a 2θ range of 5 to 80°. The incident angle must be kept at 5° or lower. Fig. 2.5.17(b) shows a flat detector being used to collect the diffraction pattern over the same 2θ range. In order to minimize the defocusing effect, the data collection is done at four different incident angles (5, 15, 25 and 35°) with four corresponding detector swing angles (10, 30, 50 and 70°). Fig. 2.5.17(c) compares the defocusing factors of the two configurations. The horizontal dotdashed line with defocusing factor B/b = 1 represents the situation with BB geometry. The defocusing factor continues to increase with 2θ angle up to B/b = 11 for cylindrical detector. That means that the 2θ resolution would be 10 times worse than for the BB geometry. For the diffraction pattern collected with a flat detector in four steps, the defocusing factor fluctuates above 1, with the worst value being less than 3. Another approach to avoiding defocusing is to collect the diffraction pattern in transmission mode. There is no defocusing effect in transmission when the incident beam is perpendicular to the sample surface. Therefore, the transmission pattern has significantly better 2θ resolution. Transmissionmode diffraction also has other advantages. For instance, the air scattering from the primary beam may be blocked by a flat sample, therefore lowering the background from air scattering. However, transmissionmode diffraction data can only be collected from samples with limited thickness, and the maximum scattering intensity is achieved at low 2θ angles with a sample thickness of , where μ is the linear absorption coefficient. The scattering intensity drops dramatically when the thickness increases.
In powder Xray diffraction, the number of crystallites contributing to each reflection must be sufficiently large to generate reproducible integrated peak intensities (see Chapter 2.10 ). A larger number of contributing crystallites gives better precision or sampling statistics (also referred to as particle statistics). Sampling statistics are determined by both the structure of the sample and the instrumentation. For a powder sample in which the crystallites are perfectly randomly oriented, the number of contributing crystallites for a diffraction peak can be given aswhere p_{hkl} is the multiplicity of the diffracting planes, V is the effective sampling volume, f_{i} is the volume fraction of the measuring crystallites (f_{i} = 1 for singlephase materials), v_{i} is the volume of individual crystallites and Ω is the angular window of the instrument (given as a solid angle). The multiplicity term, p_{hkl}, effectively increases the number of crystallites contributing to the integrated intensity from a particular set of (hkl) planes. The volume of individual crystallites, v_{i}, is an average of various crystallite sizes. The combination of the effective sampling volume and the angular window makes up the instrumental window, which determines the total volume of polycrystalline material making a contribution to a Bragg reflection. For 2DXRD, the instrumental window is not only determined by the incident beam size and divergence, but also by the detective area and the sampletodetector distance (γ angular coverage).
In BB geometry, the effective irradiated volume is a constant,where A_{o} is the crosssection area of the incident beam measured on the sample surface, is the transmission coefficient for BB geometry, and μ is the linear absorption coefficient. For 2DXRD, the effective volume is given aswhere A is the transmission coefficient and T is the transmission coefficient with BB normalization for either transmission or reflection as given previously.
The angular window is given as a solid angle. The incident beam has a divergence angle of β_{1} within the diffraction plane and β_{2} in the perpendicular direction. The angular window corresponding to the incidentbeam divergence is given by
For 2DXRD, the angular window is not only determined by the incidentbeam divergence, but also significantly increased by γ integration. When γ integration is used to generate the diffraction profile, it actually integrates the data collected over a range of various diffraction vectors. Since the effect of γ integration on sampling statistics is equivalent to the angular oscillation on the ψ axis in a conventional diffractometer, the effect is referred to as virtual oscillation and Δψ is the virtual oscillation angle. In conventional oscillation, mechanical movement may result in some sampleposition error. Since there is no actual physical movement of the sample stage during data collection, virtual oscillation can avoid this error. This is crucial for microdiffraction. The angular window with the contributions of both the incidentbeam divergence and the virtual oscillation iswhere β is the divergence of the incident beam. While increasing the divergence angle β may introduce instrumental broadening which deteriorates the 2θ resolution, virtual oscillation improves sampling statistics without introducing instrumental broadening.
In the cases of materials with a large grain size or preferred orientation, or of microdiffraction with a small Xray beam size, it can be difficult to determine the 2θ position because of poor counting statistics. In these cases, some kind of sample oscillation, either by translation or rotation, can bring more crystallites into the diffraction condition. Angular oscillation is an enhancement to the angular window of the instrument. The effect is that the angular window scans over the oscillation angle. Any of the three rotation angles (ω, ψ, ϕ) or their combinations can be used as oscillation angles. Angular oscillation can effectively improve the sampling statistics for both large grain size and preferred orientation. As an extreme example, a powderdiffraction pattern can be generated from singlecrystal sample if a sufficient angular window can be achieved by sample rotation in such as way as to simulate a Gandolfi camera (Guggenheim, 2005). Sample oscillation is not always necessary if virtual oscillation can achieve sufficient sampling statistics.
Most natural or artificial solid materials are polycrystalline, consisting of many crystallites (also called grains) of various sizes, shapes and orientations. When the orientations of the crystallites in a material have a random distribution, it presents isotropic properties. The anisotropic orientation distribution of crystallites is referred to as preferred orientation or texture. Depending on the degree of the preferred orientation, a sample is referred to as having a weak, moderate or strong texture. Many electrical, optical or mechanical properties of materials are affected or determined by their texture. The determination and interpretation of textures are therefore of fundamental importance in materials science and technology (Bunge, 1983).
When a conventional Xray diffractometer with a point detector is used for texture measurement, the crystallite orientation distribution in one direction is measured at a time, and full texture information is measured by rotating the sample to all the desired orientations. When a twodimensional Xray diffraction system is used for texture measurement, the orientation distributions of several crystallographic planes over a range of angles can be measured simultaneously so as to get better measurement results in a shorter datacollection time (Smith & Ortega, 1993; Blanton, 1994; Bunge & Klein, 1996; Helming et al., 2003; Wenk & Grigull, 2003; He, 2009). The orientation relationships between different phases or between different layers of thin films and substrates can also be easily revealed. The texture effect may be observed and evaluated directly from the 2D diffraction frames without data processing.
XRD results from an `ideal' powder in which the crystallites are randomly oriented normally serve as a basis for determining the relative intensity of each diffraction peak. The deviation of the grain orientation distribution of a polycrystalline material from that of an ideal powder is measured as texture. The pole figure for a particular crystallographic plane is normally used to represent the texture of a sample. Assuming that all grains have the same volume, each `pole' represents a grain that satisfies the Bragg condition. The number of grains satisfying the Bragg condition at a particular sample orientation can be larger or smaller than the number of grains for an ideal sample, and likewise for the integrated intensity of that peak. The measured 2D diffraction pattern contains two very important parameters at each γ angle: the partially integrated intensity I and the Bragg angle 2θ. Fig. 2.5.18 shows a 2D frame for a Cu thin film on an Si wafer collected with a microgap 2D detector. It contains four Cu lines and one Si spot. The diffraction intensity varies along γ because of the anisotropic poledensity distribution. For each diffraction ring, the intensity is a function of γ and the sample orientation (ω, ψ, ϕ), i.e. I = I(γ, ω,ψ, ϕ).

Diffraction frame collected from a Cu film on an Si substrate showing intensity variation along γ due to texture. 
Plotting the intensity of each (hkl) line with respect to the sample coordinates in a stereographic projection gives a qualitative view of the orientation of the crystallites with respect to a sample direction. These stereographic projection plots are called pole figures. As is shown in Fig. 2.5.19(a), the sample orientation is defined by the sample coordinates S_{1}, S_{2} and S_{3}. For metals with rolling texture, the axes S_{1}, S_{2} and S_{3} correspond to the transverse direction (TD), rolling direction (RD) and normal direction (ND), respectively. Let us consider a sphere with unit radius and the origin at O. A unit vector representing an arbitrary pole direction starts from the origin O and ends at the point P on the sphere. The pole direction is defined by the radial angle α and azimuthal angle β. The pole density at the point P projects to the point P′ on the equatorial plane through a straight line from P to the point S. The pole densities at all directions are mapped onto the equatorial plane by stereographic projection as shown in Fig. 2.5.19(b). This twodimensional mapping of the pole density onto the equatorial plane is called a pole figure. The azimuthal angle β projects to the pole figure as a rotation angle about the centre of the pole figure from the sample direction S_{1}. When plotting the pole density into a pole figure of radius R, the location of the point P′ in the pole figure should be given by β and
For easy computer plotting and easy angular readout from the pole figure, the radial angle α may be plotted on an equally spaced angular scale, similar to a twodimensional polar coordinate system. Other polefigure mapping styles may be used, but must be properly noted to avoid confusion (Birkholz, 2006).
The α and β angles are functions of γ, ω, ψ, ϕ and 2θ. As shown in Fig. 2.5.19(a), a pole has three components h_{1}, h_{2} and h_{3}, parallel to the three sample coordinates S_{1}, S_{2} and S_{3}, respectively. The polefigure angles (α, β) can be calculated from the unitvector components by the following polemapping equations:where α takes a value between 0 and 90° () and β takes values in two ranges ( when and when ). The condition for reflectionmode diffraction is . For transmission diffraction it is possible that . In this case, the pole with mirror symmetry about the S_{1}S_{2} plane to the diffraction vector is used for the polefigure mapping. The absolute value of h_{3} is then used in the equation for the α angle. When in the above equation, β takes one of two values depending on the value of h_{1} ( when and when ). For Eulerian geometry, the unitvector components are given by equation (2.5.11).
The 2θ integrated intensity along the diffraction ring is then converted to the poledensity distribution along a curve on the pole figure. The α and β angles at each point of this curve are calculated from ω, ψ, ϕ, γ and 2θ. The sample orientation (ω, ψ, ϕ) and 2θ for a particular diffraction ring are constants; only γ takes a range of values depending on the detector size and distance.
For a textured sample, the 2θintegrated intensity of a diffraction ring from a family of (hkl) planes is a function of γ and the sample orientation (ω, ψ, ϕ), i.e. . From the polefigure anglemapping equations, we can obtain the integrated intensity in terms of polefigure angles asThe pole density at the polefigure angles (α, β) is proportional to the integrated intensity at the same angles:where is the 2θintegrated intensity of the (hkl) peak corresponding to the pole direction , is the scaling factor covering the absorption, polarization, background corrections and various instrument factors if these factors are included in the integrated intensities, and is the poledensity distribution function. Background correction can be done during the 2θ integration and will be discussed in Section 2.5.4.2.4. The pole figure is obtained by plotting the poledensity function based on the stereographic projection.
The poledensity function can be normalized such that it represents a fraction of the total diffracted intensity integrated over the pole sphere. The normalized poledensity distribution function is given byThe poledensity distribution function is a constant for a sample with a random orientation distribution. Assuming that the sample and instrument conditions are the same except for the poledensity distribution, we can obtain the normalized poledensity function by
The integrated intensity from the textured sample without any correction can be plotted according to the stereographic projection as an `uncorrected' pole figure. The same can be done for the sample with a random orientation distribution to form a `correction' pole figure that contains only the factors to be corrected. The normalized pole figure is then obtained by dividing the `uncorrected' pole figure by the `correction' pole figure. This experimental approach is feasible only if a similar sample with a random orientation distribution is available.
If the texture has a rotational symmetry with respect to an axis of the sample, the texture is referred to as a fibre texture and the axis is referred to as the fibre axis. The sample orientation containing the symmetry axis is referred to as the fibre axis. The fibre texture is mostly observed in two types of materials: metal wires or rods formed by drawing or extrusion, and thin films formed by physical or chemical deposition. The fibre axis is the wire axis for a wire and normal to the sample surface for thin films. Fibre texture can also be artificially formed by rotating a sample about its normal. If the fibre axis is aligned to the S_{3} direction, the poledensity distribution function becomes independent of the azimuthal angle β. For samples with fibre texture, or artificially formed fibre texture by rotating, the poledensity function is conveniently expressed as a function of a single variable, . Here, χ is the angle between the sample normal and pole direction.
The poledensity function for fibre texture can be expressed as a fibre plot. The fibre plot can be calculated from the relative intensity of several peaks (He, 1992; He et al., 1994) and artificial fibre texture can be achieved by sample spinning during data collection.
Since a onedimensional poledensity mapping is created from each 2D frame, it is important to lay out a datacollection strategy so as to have the optimum polefigure coverage and minimum redundancy in data collection. The polefigure coverage can be simulated from the diffraction 2θ angle, detector swing angle, detector distance, goniometer angles and scanning steps. When a large 2D detector is placed close to the sample, it is possible to collect a pole figure with a single ϕ scan. Fig. 2.5.20(a) shows an example of a scheme generated by a single ϕ scan of 5° steps with a detector 10.5 cm in diameter and D = 7 cm. The data collected with a single exposure at ϕ = 0° would generate a onedimensional pole figure as shown in the curve marked by A and B. The pole figure can be generated by a fullcircle rotation of 360°. The pole density at the centre represents the diffraction vector perpendicular to the sample surface. It is important to have the poledensity information in the centre region of the pole figure, especially for fibre texture. The polefigure angle at the centre is α = 90°, and the best strategy is to put point A at the centre of pole figure. That is

Datacollection strategy: (a) 2D detector with D = 7 cm; (b) 2D detector with D = 10 cm; (c) point detector. 
In some cases, a single ϕ scan is not enough to cover sufficient polefigure angles because of a large detector distance or limited detector area, so it is necessary to collect a set of data with ϕ scans at several different sample tilt angles. Fig. 2.5.20(b) illustrates the datacollection scheme with a detector that is 10.5 cm in diameter and D = 10 cm for the (111) plane of a Cu thin film. In this case, each pole figure requires two ϕ scans at different sample orientations. The datacollection strategy should also be optimized for several crystallographic planes if all can be covered in a frame. The step size of the datacollection scan depends highly on the strength of the texture and the purpose of the texture measurements. For a weak texture, or quality control for metal parts, ϕ (or ω, or ψ) scan steps of 5° may be sufficient. For strong textures, such as thin films with epitaxial structure, scan steps of 1° or smaller may be necessary.
The effectiveness of twodimensional data collection for a texture can be compared with that using a point detector with the datacollection strategy of the Cu thin film as an example. Fig. 2.5.20(c) shows the polefigure datacollection strategy with a point detector. For the same polefigure resolution, significantly more exposures are required with a point detector. Considering that several diffraction rings are measured simultaneously with a 2D detector, the polefigure measurement is typically 10 to 100 times faster than with a point detector. Therefore, quantitative highresolution polefigure measurements are only practical with a 2DXRD system (Bunge & Klein, 1996).
For a specific diffraction ring, 2θ is a constant or at least assumed to be constant for texture analysis, and the sampleorientation angles (ω, ψ, ϕ) for a frame are also constants. Therefore, the poledensity information is given by the diffractionintensity distribution as a function of γ only, or I = I(γ). Integration of the diffraction intensities in the 2θ direction converts 2D information into the function I(γ).
Fig. 2.5.21(a) shows a 2D diffraction ring for texture analysis. The low and high background and diffractionring 2θ–γ range are defined by three boxes, noted as B_{L}, B_{H} and I(γ), respectively. All three boxes have the same γ range from γ_{1} to γ_{2}. The 2θ ranges for the diffraction ring, low background and high background should be determined based on the width of the 2θ peak and available background between adjacent peaks. Assuming a normal distribution, a 2θ range of 2 times the FWHM covers 98% of the intensity peak, and 3 times the FWHM covers more than 99.9%. The 2θ range should also be broad enough to cover the possible 2θ shifts caused by residual stresses in the sample. Fig. 2.5.21(b) is the 2θ profile integrated over the section Δγ in Fig. 2.5.21(a). The background ranges on the low and high 2θ sides are given by 2θ_{L1}–2θ_{L2} and 2θ_{H1}–2θ_{H2}, respectively. The 2θintegrated diffraction intensities as a function of γ are plotted in Fig. 2.5.21(c). The background can be calculated and removed from the intensity values of the low and high backgrounds or ignored if the contribution of the background is very small.

Polefigure data processing: (a) a frame with the 2θ integration ranges for the (220) ring; (b) 2θ profile showing the background and peak; (c) integrated intensity distribution as a function of γ. 
2θ integration without a background correction can be expressed asA similar equation can be used for 2θ integration of the low and high backgrounds B_{L}(γ) and B_{H}(γ). Assuming a linear background change in the vicinity of the 2θ peak, the background under the peak, B(γ), is then given byThen the background B(γ) can be subtracted from the integrated intensity distribution I(γ).
The algorithms of γ integration given in Section 2.5.4.2.3 can be easily modified for 2θ integration by exchanging γ and 2θ in the equations. Algorithms with solidangle normalization should be used to get consistent integrated intensity over all areas of the detector. The 2θintegrated intensity distribution can then be mapped onto a pole figure based on the fundamental equations (2.5.53) and (2.5.54). When a polefigure pixel is overlapped by more than one data point from different scans, as shown in the region covered by two scans in Fig. 2.5.20(b), the average value should be mapped to that polefigure pixel. Fig. 2.5.22(a) shows poledensity mappings on the pole figure. There are big gaps between the measured poledensity data points due to the large ϕscan steps of 5°.

Polefigure processing: (a) I(γ) mapped to the pole figure; (b) Pole figure after interpolation and symmetry processing. 
All factors affecting relative intensities, such as Lorentz, polarization, air scattering, and Bewindow and sample absorption, will have an effect on the measured pole densities for the pole figures. Some or all these corrections may be applied to the diffraction frames before 2θ integration if the texture study demands high accuracy in the relative pole densities. Among these factors, the most important factor is sample absorption, since data sets for pole figures are typically collected at several different incident angles. A ridge between the poledensity regions covered by two different incident angles may be observed if sample absorption is not properly corrected.
The pole figure is stored and displayed as a bitmap image. The poledensity data from the data set may not fill up all the pixels of the polefigure image. In order to generate a smooth pole figure, the unmapped pixels are filled with values generated from the interpolation of the surrounding pixels. A linear interpolation within a defined box is sufficient to fill the unmapped pixels. The size of the box should be properly chosen. A box that is too small may not be able to fill all unmapped pixels and a box that is too big may have a smearing effect on the pole figure, especially if a sharp pole figure is processed. All the gaps between the measured poledensity points are filled after this interpolation. For a sample with sharp texture, smaller ϕscan steps should be used.
All pole figures possess symmetry as a consequence of the Laue symmetry of the crystallites in the sample. This symmetry can be used to fill in values for pixels in the pole figure for which data were not measured, or to smooth the pole figure. For example, orthorhombic materials exhibit mmm symmetry, thus one needs to collect only an octant or quadrant of the pole sphere to generate the entire pole figure. The pole figures of materials with higher symmetry may be treated by using lower symmetry in the processing. For instance, one can use 2/m or mmm symmetry for hexagonal materials and mmm for cubic materials. In symmetry processing, all the symmetryequivalent polefigure pixels are filled by the average value of the measured pixels. For the unmeasured polefigure pixels, this symmetry processing fills in a value from the average of all the equivalent pixels. For the measured pixels, this average processing serves as a smoothing function. Fig. 2.5.22(b) shows the results after both interpolation and use of symmetry.
A 2DXRD system can measure texture from a sample containing a single phase, multiple phases or single crystals. The orientation relationship between different phases, or thin films and substrates, can be revealed because data are collected from all phases of the sample simultaneously. One example is the measurement of pole figures for a magnetron sputterdeposited Cu film on an Si wafer (He et al., 2005). Fig. 2.5.23 shows the overlapped pole figures of the Cu (111) film and Si (400) substrate in a 2D pole figure (a) and 3D surface plot (b). The three sharp spots from the (400) spots of the Si wafer show the wafer cut orientation of (111). The Cu (111) pole density maximized in the centre of the pole figure shows a strong (111) fibre texture. The orientation relationship between the film fibre axis and the substrate is clearly described by the combined pole figures. For samples containing multiple thinfilm layers, the orientation relationships between the different layers of the films and substrate can be revealed by superimposing their pole figures.
When a solid material is elastically deformed by a force, each crystallite in it changes shape or size. Assuming that the stresses in each crystallite represent the stresses in the solid, the stresses can be measured by measuring the lattice dspacing changes in the crystallites. These dspacing changes can be measured by the changes in diffractionpeak positions based on Bragg's law. In this case, the dspacing serves as a gauge of the deformation. Stress measurement by Xray diffraction is typically done using a point detector or line detector (Walter, 1971; James & Cohen, 1980; Noyan & Cohen, 1987; Lu, 1996); this will be referred to as the conventional method. The stress or stress tensor is calculated from many strain measurements from diffractionpeak 2θ shifts of a specific latticeplane family. With a point or line detector, only a small cross section of the diffraction cone is measured at one sample orientation (ψ, ϕ). Compared to using a conventional detector, 2D detectors have many advantages in stress measurement (Borgonovi, 1984; Korhonen et al., 1989; Yoshioka & Ohya, 1992; Fujii & Kozaki, 1993; He & Smith, 1997; Kämpfe et al., 1999; Hanan et al., 2004). Since a 2D diffraction pattern covers the whole or a large portion of the diffraction rings, it can be used to measure stress with higher accuracy and can be collected in a shorter time than a conventional diffraction pattern, especially when dealing with highly textured materials, large grain sizes, small sample areas, weak diffraction, stress mapping and stresstensor measurement. The 2D method for stress measurement is based on the fundamental relationship between the stress tensor and the diffractioncone distortion (He & Smith, 1997; He, 2000; European Standard, 2008).
There are two kinds of stresses, which depend on the source of the loading forces that produce them. One kind is applied stress, caused by external forces acting on the solid object. Applied stress changes when the loading forces change and it disappears once the forces are removed. The stresses measured by Xray diffraction method are mostly residual stresses. Residual stress is caused by internal forces between different parts of a solid body. Residual stress exists without external forces or remains after the external forces have been removed. The net force and moment on a solid body in equilibrium must be zero, so the residual stresses in the body must be balanced within the body. This means that a compressive stress in one part of the body must come with a tensile stress in another part of the body. For example, the residual stress in a thin film is balanced by the stresses in the substrate. When residual stress in a solid body is mentioned it typically refers to a specific location.
Residual stresses are generally categorized as macroscopic or microscopic depending on the range over which the stresses are balanced. The macroscopic residual stress is the stress measured over a large number of grains. This kind of stress can be measured by Xray diffraction through the shift of the Bragg peaks. The microscopic stress is the stress measured over one or a few grains, or as small a range as micro or nanometres. This kind of stress alone will not cause a detectable shift of diffraction peaks, but is reflected in the peak profiles. In this chapter, we will focus on the Xray diffraction method for stress measurement at the macroscopic level.
Stress is a measure of the deforming force applied to a solid per unit area. The stress on an elemental volume in the sample coordinates S_{1}, S_{2}, S_{3} contains nine components, given by
A component is normal stress when the two indices are identical, or shear stress when the two indices differ. The group of the nine stress components is called the stress tensor. The stress tensor is a tensor of the second order. Under equilibrium conditions, the shear components must maintain the following relations:Therefore, only six independent components define the stress state in a solid. The following stress states are typically measured:
Strain is a measure of the resulting deformation of a solid body caused by stress. Strain is calculated from the change in the size and shape of the deformed solid due to stress. Analogous to normal stresses and shear stresses are normal strains and shear strains. The normal strain is calculated from the change in length of the solid body along the corresponding normal stress direction. Like the stress tensor, the strain tensor contains nine components:The directions of all strain components are defined in the same way as for the stress tensor. Similarly, there are six independent components in the strain tensor. Strictly speaking, Xray diffraction does not measure stresses directly, but strains. The stresses are calculated from the measured strains based on the elasticity of the materials. The stress–strain relations are given by the generalized form of Hooke's law:where are elastic stiffness coefficients. The stress–strain relations can also be expressed aswhere are the elastic compliances. For most polycrystalline materials without texture or with weak texture, it is practical and reasonable to consider the elastic behaviour to be isotropic and the structure to be homogeneous on a macroscopic scale. In these cases, the stress–strain relationship takes a much simpler form. Therefore, the Young's modulus E and Poisson's ratio ν are sufficient to describe the stress and strain relations for homogeneous isotropic materials:It is customary in the field of stress measurement by Xray diffraction to use another set of macroscopic elastic constants, S_{1} and ½S_{2}, which are given by
Although polycrystalline materials on a macroscopic level can be considered isotropic, residual stress measurement by Xray diffraction is done by measuring the strain in a specific crystal orientation of the crystallites that satisfies the Bragg condition. The stresses measured from diffracting crystallographic planes may have different values because of their elastic anisotropy. In such cases, the macroscopic elasticity constants should be replaced by a set of crystallographic planespecific elasticity constants, and , called Xray elastic constants (XECs). XECs for many materials can be found in the literature, measured or calculated from microscopic elasticity constants (Lu, 1996). In the case of materials with cubic crystal symmetry, the equations for calculating the XECs from the macroscopic elasticity constants ½S_{2} and S_{1} arewhereIn the equations for stress measurement hereafter, either the macroscopic elasticity constants ½S_{2} and S_{1} or the XECs and are used in the expression, but either set of elastic constants can be used depending on the requirements of the application. The factor of anisotropy (A_{RX}) is a measure of the elastic anisotropy of a material (He, 2009).
Fig. 2.5.24 illustrates two diffraction cones for backward diffraction. The regular diffraction cone (dashed lines) is from the powder sample with no stress, so the 2θ angles are constant at all γ angles. The diffraction ring shown as a solid line is the cross section of a diffraction cone that is distorted as a result of stresses. For a stressed sample, 2θ becomes a function of γ and the sample orientation (ω, ψ, ϕ), i.e. . This function is uniquely determined by the stress tensor. The strain measured by the 2θ shift at a point on the diffraction ring is , based on the true strain definitionwhere d_{o} and θ_{o} are the stressfree values and d and θ are measured values from a point on the diffraction ring corresponding to . The direction of in the sample coordinates S_{1}, S_{2}, S_{3} can be given by the unitvector components h_{1}, h_{2} and h_{3}. As a secondorder tensor, the relationship between the measured strain and the straintensor components is then given byThe scalar product of the strain tensor with the unit vector in the above equation is the sum of all components in the tensor multiplied by the components in the unit vector corresponding to the first and the second indices. The expansion of this equation for i and j values of 1, 2 and 3 results inOr, taking the true strain definition,where θ_{o} corresponds to the stressfree dspacing and θ are measured values from a point on the diffraction ring. Both θ and {h_{1}, h_{2}, h_{3}} are functions of . By taking γ values from 0 to 360°, equation (2.5.74) establishes the relationship between the diffractioncone distortion and the strain tensor. Therefore, equation (2.5.74) is the fundamental equation for strain measurement with twodimensional Xray diffraction.
Introducing the elasticity of materials, one obtainsor
It is convenient to express the fundamental equation in a clear linear form:where p_{ij} are stress coefficients given byIn the equations for the stress measurement above and hereafter, the macroscopic elastic constants ½S_{2} and S_{1} are used for simplicity, but they can always be replaced by the XECs for the specific lattice plane {hkl}, and , if the anisotropic nature of the crystallites should be considered. For instance, equation (2.5.76) can be expressed with the XECs as
The fundamental equation (2.5.74) may be used to derive many other equations based on the stress–strain relationship, stress state and special conditions. The fundamental equation and the derived equations are referred to as 2D equations hereafter to distinguish them from the conventional equations. These equations can be used in two ways. One is to calculate the stress or stresstensor components from the measured strain (2θshift) values in various directions. The fundamental equation for stress measurement with 2DXRD is a linear function of the stresstensor components. The stress tensor can be obtained by solving the linear equations if six independent strains are measured or by linear leastsquares regression if more than six independent measured strains are available. In order to get a reliable solution from the linear equations or leastsquares analysis, the independent strain should be measured at significantly different orientations. Another function of the fundamental equation is to calculate the diffractionring distortion for a given stress tensor at a particular sample orientation (He & Smith, 1998). The fundamental equation for stress measurement by the conventional Xray diffraction method can also be derived from the 2D fundamental equation (He, 2009).
The general triaxial stress state is not typically measured by Xray diffraction because of low penetration. For most applications, the stresses in a very thin layer of material on the surface are measured by Xray diffraction. It is reasonable to assume that the average normal stress in the surfacenormal direction is zero within such a thin layer. Therefore, , and the stress tensor has five nonzero components. In some of the literature this stress state is denoted as triaxial. In order to distinguish this from the general triaxial stress state, here we name this stress state as the `biaxial stress state with shear'. In this case, we can obtain the linear equation for the biaxial stress state with shear:where the coefficient and σ_{ph} is the pseudohydrostatic stress component introduced by the error in the stressfree dspacing. In this case, the stresses can be measured without the accurate stressfree dspacing, since this error is included in σ_{ph}. The value of σ_{ph} is considered as one of the unknowns to be determined by the linear system. With the measured stresstensor components, the general normal stress (σ_{ϕ}) and shear stress (τ_{ϕ}) at any arbitrary angle ϕ can be given by
Equation (2.5.81) can also be used for other stress states by removing the terms for stress components that are zero. For instance, in the biaxial stress state , so we have
In the 2D stress equations for any stress state with σ_{33} = 0, we can calculate stress with an approximation of d_{o} (or 2θ_{o}). Any error in d_{o} (or 2θ_{o}) will contribute only to a pseudohydrostatic term σ_{ph}. The measured stresses are independent of the input d_{o} (or 2θ_{o}) values (He, 2003). If we use to represent the initial input, then the true d_{o} (or 2θ_{o}) can be calculated from σ_{ph} with
Care must be taken that the σ_{ph} value also includes the measurement error. If the purpose of the experiment is to determine the stressfree dspacing d_{o}, the instrument should be first calibrated with a stressfree standard of a similar material.
The practice of stress analysis with 2DXRD involves the selection of the diffractionsystem configuration and the datacollection strategy, frame correction and integration, and stress calculation from the processed data points. Most concepts and strategies developed for a conventional diffractometer are still valid for 2DXRD. We will focus on the new concepts and practices due to the nature of the 2D detectors.
The diffraction vector is in the normal direction to the measured crystalline planes. It is not always possible to have the diffraction vector in the desired measurement direction. In reflection mode, it is easy to have the diffraction vector normal to the sample surface, or tilted away from the normal, but impossible to have the vector on the surface plane. The stress on the surface plane, or biaxial stress, is calculated by elasticity theory from the measured strain in other directions. The final stressmeasurement results can be considered as an extrapolation from the measured values. In the conventional sin^{2} ψ method, several ψtilt angles are required, typically at 15° steps from −45° to +45°. The same is true with a 2DXRD system. The diffraction vectors corresponding to the data scan can be projected onto a 2D plot in the same way as the poledensity distribution in a pole figure. The 2D plot is called a datacollection strategy scheme.
By evaluating the scheme, one can generate a datacollection strategy suitable for the measurement of the intended stress components. Fig. 2.5.25 illustrates two schemes for data collection. In the bisecting condition ( or and ), the trace of the diffraction vector falls in the vicinity of the scheme centre. Either an ω tilt or a ψ tilt can move the vectors away from the centre. The circles on the scheme are labelled with the tilt angle of 15°, 30° and 45°. Scheme (a) is for an ω tilt of 0°, ±15°, ±30° and ±45° with the ϕ angle at 0° and 90°. It is obvious that this set of data would be suitable for calculating the biaxialstress tensor. The data set with , as shown within the box enclosed by the dashed lines, would be sufficient on its own to calculate . Since the diffractionring distortion at or is not sensitive to the stress component , strategy (a) is suitable for the equibiaxial stress state, but is not able to determine accurately. In scheme (b), the ψ scan covers 0° to 45° with 15° steps at eight ϕ angles with 45° intervals. This scheme produces comprehensive coverage on the scheme chart in a symmetric distribution. The data set collected with this strategy can be used to calculate the complete biaxialstress tensor components and shear stress (). The scheme indicated by the boxes enclosed by the dashed lines is a timesaving alternative to scheme (b). The rings on two ϕ angles are aligned to S_{1} and S_{2} and the rings on the third ϕ angle make 135° angles to the other two arrays of rings. This is analogous to the configuration of a stressgauge rosette. The three ϕ angles can also be separated equally by 120° steps. Suitable schemes for a particular experiment should be determined by considering the stress components of interest, the goniometer, the sample size, the detector size and resolution, the desired measurement accuracy and the datacollection time.
The purpose of data integration and peak evaluation is to generate a set of data points along distorted diffraction rings. Data integration for stress analysis is γ integration over several defined segments so as to generate diffraction profiles representing the corresponding segments. The peak position can be determined by fitting the diffraction profile to a given analytic function. Fig. 2.5.26 illustrates data integration over a diffraction frame. The total integration region is defined by , , and . The integration region is divided into segments given by . One data point on the distorted diffraction ring is generated from each segment. The γ value in the centre (denoted by the dotdashed line) of each segment is taken as the γ value of the data point. γ integration of the segment produces a diffraction profile and the 2θ value is determined from the profile. The number of segments and the segment size () are selected based on the quality of the data frame. The larger the segment size is, the better the integrated diffraction profile as more counts are being integrated. γ integration also produces a smearing effect on the diffractionring distortion because the counts collected within the segment size are considered as a single γ value at the segment centre. The 2θ shift in the segment is averaged. The segment size should be sufficient to produce a smooth diffraction profile, but not so large as to introduce too much smearing. For data frames containing high pixel counts, the integration segment can be small, e.g. , and still have a smooth profile for each segment. For data frames having low pixel counts, for example the frames collected from a micronsized area, from a sample with large grains or with a short datacollection time, it is critical to choose a sufficiently large segment size. The segment size can be determined by observing the smoothness of the integrated profile.
Peak evaluation in each segment can be done using the same algorithm used in the conventional method. The corrections to the integrated profiles are performed before or during the peak evaluation. Absorption correction eliminates the influence of the irradiated area and the diffraction geometry on the measured intensity distribution. The absorption for a given material and radiation level depends on the incident angle to the sample and the reflected angle from the sample. For 2DXRD, the reflected angle is a function of γ for each frame. The polarization effect is also a function of γ. Therefore, the correction for polarization and absorption should be applied to the frame before integration. (Details of these corrections were discussed in Section 2.5.4.3.4.) The polarization and absorption correction is not always necessary if the error caused by absorption can be tolerated for the application, or if the datacollection strategy involves only ψ and ϕ scans.
In most cases, Kα radiation is used for stress measurement, in which case the weighted average wavelength of Kα_{1} and Kα_{2} radiation is used in the calculations. For samples with a broad peak width, diffraction of Kα_{1} and Kα_{2} radiation is merged together as a single peak profile, and the profile can be evaluated as if there is a single Kα line without introducing much error to the measured dspacing. For samples with a relatively narrow peak width, the diffraction profile shows strong asymmetry or may even reveal two peaks corresponding to the Kα_{1} and Kα_{2} lines, especially at high 2θ angles. In this case the profile fitting should include contributions from both the Kα_{1} and Kα_{2} lines. It is common practice to use the peak position from the Kα_{1} line and the Kα_{1} wavelength to calculate the dspacing after Kα_{2} stripping.
Background correction is necessary if there is a strong background or the peakevaluation algorithms are sensitive to the background, such as in Kα_{2} stripping, peak fitting, and peakintensity and integratedintensity evaluations. Background correction is performed by subtracting a linear intensity distribution based on the background intensities at the lower 2θ side and the higher 2θ side of the diffraction peak. The background region should be sufficiently far from the 2θ peak so that the correction will not truncate the diffraction profile. The 2θ ranges of the low background and high background should be determined based on the width of the 2θ peak and available background in the profile. Based on a normal distribution, a 2θ range of 2 times the FWHM covers 98% of the peak intensity, and 3 times the FWHM covers more than 99.9%, so the background intensity should be determined at more than 1 to 1.5 times the FWHM away from the peak position. The background correction can be neglected for a profile with a low background or if the error caused by the background is tolerable for the application. The peak position can be evaluated by various methods, such as gravity, sliding gravity, and profile fitting by parabolic, pseudoVoigt or PearsonVII functions (Lu, 1996; Sprauel & Michaud, 2002).
The final data set after integration and peak evaluation should contain many data points describing the diffractionring shape for all collected frames. Each measured data point contains three goniometer angles (ω, ψ, ϕ) and the diffractionring position (γ, 2θ). The peak intensity or integrated intensity of the diffraction profile is another value to be determined and may be used in the stress calculation. In most cases the number of data points is more than the number of unknown stress components, so a linear leastsquares method can be used to calculate the stresses. In a general leastsquares regression, the residual for the ith data point is defined aswhere is the observed response value, is the fitted response value and is the residual, which is defined as the difference between the observed value and the fitted value. The summed square of residuals is given bywhere n is the number of data points and S is the sumofsquares error to be minimized in the leastsquares regression. For stress calculation, the observed response value is the measured strain at each data point,and the fitted response value is given by the fundamental equation aswhere all possible stress components and stress coefficients are listed as a generalized linear equation. Since the responsevalue function is a linear equation of unknown stress components, the leastsquares problem can be solved by a linear leastsquares regression. In order to reduce the impact of texture, large grains or weak diffraction on the results of the stress determination, the standard error of profile fitting and the integrated intensity of each profile may be introduced as a weight factor for the leastsquares regression (He, 2009).
Stress measurement on a polycrystalline material by Xray diffraction is based on the strain measurements in a single or in several sample orientations. Each measured strain is calculated from the average dspacing of specific lattice planes {hkl} over many crystallites (grains). A larger number of contributing crystallites gives better accuracy and sampling statistics (also referred to as particle statistics). The sampling statistics are determined by both the crystal structure and the instrumentation. The instrument window is mainly determined by the divergence of the incident Xray beam. Latticeplane families with high multiplicity will also effectively improve the sampling statistics. The number of contributing crystallites measured by a conventional diffractometer is limited by the sizes and divergences of the incident and diffracted beams to the point detector. In a twodimensional diffraction system, more crystallites can contribute to the diffraction because of the larger γ range.
An example of a stress calculation is provided by the measurement of the residual stress on the end surface of a carbon steel roller. One of the seven frames taken with an ω scan is shown in Fig. 2.5.27(a). The (211) ring covering the γ range 67.5 to 112.5° was used for stress analysis. First, the frame data were integrated along γ with an interval of Δγ = 5°. A total of nine diffraction profiles were obtained from γ integration. The peak position 2θ for each γ angle was then obtained by fitting the profile with a PearsonVII function. A total of 63 data points can be obtained from the seven frames. The data points at γ = 90° from seven frames, a typical data set for an ω diffractometer, were used to calculate the stress with the conventional method. In order to compare the gain from having increased data points with the 2D method, the stress was calculated from 3, 5, 7 and 9 data points on each frame. The results from the conventional method and the 2D method are compared in Fig. 2.5.27(b). The measured residual stress is compressive and the stress values from different methods agree very well. With the data taken from the same measurement (seven frames), the 2D method gives a lower standard error and the error decreases with increasing number of data points from the diffraction ring.
The crystallinity of a material influences many of its characteristics, including mechanical strength, opacity and thermal properties. Crystallinity measurement provides valuable information for both materials research and quality control in materials processing. The diffraction pattern from a material containing both amorphous and crystalline solids has a broad feature from the amorphous phase and sharp peaks from the crystalline phase. The weight percentage of the crystalline phases in a material containing both crystalline and amorphous phases can be determined by Xray diffraction (Chung & Scott, 1973; Alexander, 1985; Murthy & Barton, 2000; Kasai & Kakudo, 2005). Assuming that the Xray scattering intensity from each phase in such a material is proportional to its weight percentage, and that the scattering intensities from all phases can be measured within a given 2θ range, the per cent crystallinity is given bywhere x_{pc} is the per cent crystallinity, I_{crystal} is the integrated intensity of all crystalline peaks and I_{amorphous} is the integrated intensity of the amorphous scattering. The accuracy of the measured per cent crystallinity depends on the integrated diffraction profile. Since most crystalline samples have a preferred orientation, it is very difficult to obtain a consistent measurement of crystallinity with a conventional diffractometer. Fig. 2.5.28 shows a 2D diffraction frame collected from an oriented polycrystalline sample. The diffraction is in transmission mode with the Xray beam perpendicular to the plate sample surface. Fig. 2.5.28(a) shows a diffraction profile integrated from a horizontal region analogous to a profile collected with a conventional diffractometer. Only one peak from the crystalline phase can be observed in the profile. It is also possible that a different peak or no peak is measured if the sample is loaded in other orientations. Fig. 2.5.28(b) is the diffraction profile integrated from the region covering all peaks from the crystalline phase over almost all azimuthal angles. A total of four peaks from the crystalline phase are observed. This shows that a 2DXRD system can measure per cent crystallinity more accurately and with more consistent results (Pople et al., 1997; Bruker, 2000) than a conventional system.
The size of the crystallites in a polycrystalline material has a significant effect on many of its properties, such as its thermal, mechanical, electrical, magnetic and chemical properties. Xray diffraction has been used for crystallitesize measurement for many years. Most methods are based on diffractionline broadening and lineprofile analysis (Wilson, 1971; Klug & Alexander, 1974; Ungár, 2000). Another approach to crystallitesize measurement is based on the spotty diffraction rings collected with twodimensional detectors when a small Xray beam is used (Cullity, 1978; He, 2009). Lineprofile analysis is based on the diffraction profile in the 2θ direction, while crystallitesize analysis with a spotty 2D diffraction pattern is based on the diffraction profile in the γ direction. The latter may be referred to as γprofile analysis.
Fig. 2.5.29(a) shows a diffraction profile collected from gold nanoparticles and regular gold metal. The 2θ profile from the gold nanoparticles is significantly broader than the profile from regular gold metal. The crystallite size can be calculated by measuring the broadening and using the Scherrer equation:where λ is the Xray wavelength (in Å), B is the full width at half maximum (FWHM) of the peak (in radians) corrected for instrumental broadening and strain broadening, θ is the Bragg angle, C is a factor, typically from 0.9 to 1.0, depending on the crystallite shape (Klug & Alexander, 1974), and t is the crystallite size (also in Å). This equation shows an inverse relationship between crystallite size and peakprofile width. The wider the peak is, the smaller the crystallites. The 2θ diffraction profiles can be obtained either by using a conventional diffractometer with a point or line detector, or by γ integration from a diffraction pattern collected with 2D detector. When a 2D detector is used, a long sampletodetector distance should be used to maximize the resolution. A small beam size and low convergence should also be used to reduce instrument broadening.

Crystallitesize analysis: (a) 2θ profile of a gold nanoparticle (grey) and regular gold metal (black); (b) γ profile of LaB_{6}; (c) measurement range. 
Fig. 2.5.29(b) shows a frame collected from an SRM660a (LaB_{6}) sample with a 2DXRD system. The spotty diffraction rings are observed with average crystallite size of 3.5 µm. The number of spots in each diffraction ring is determined by the crystallite size and diffraction volume. Introducing a scaling factor covering all the numeric constants, the incidentbeam divergence and the calibration factor for the instrument, we obtain an equation for the crystallite size as measured in reflection mode:where d is the diameter of the crystallite particles, p_{hkl} is the multiplicity of the diffracting planes, b is the size of the incident beam (i.e. its diameter), Δγ is the γ range of the diffraction ring, μ is the linear absorption coefficient and N_{s} is the number of spots within Δγ. For transmission mode, we havewhere t is the sample thickness. In transmission mode with the incident beam perpendicular to the sample surface, the linear absorption coefficient affects the relative scattering intensity, but not the actual sampling volume. In other words, all the sample volume irradiated by the incident beam contributes to the diffraction. Therefore, it is reasonable to ignore the absorption effect for crystallitesize analysis as long as the sample is thin enough for transmissionmode diffraction. The effective sampling volume reaches a maximum for transmissionmode diffraction when .
For both reflection and transmission,where β is the divergence of the incident beam. Without knowing the precise instrumental broadening, k can be treated as a calibration factor determined from the 2D diffraction pattern of a known standard. Since only a limited number of spots along the diffraction ring can be resolved, it can be seen from equation (2.5.94) that a smaller Xray beam size and lowmultiplicity peak should be used if a smaller crystallite size is to be determined.
Fig. 2.5.29(c) shows the measurement ranges of 2θprofile and γprofile analysis. The 2θprofile analysis is suitable for crystallite sizes below 100 nm (1000 Å), while γprofile analysis is suitable for crystallite sizes from as large as tens of µm down to 100 nm with a small Xray beam size. By increasing the effective diffraction volume by translating the sample during data collection or multiple sample integration (or integrating data from multiple samples), the measurement range can be increased up to millimetres. Multiple sample integration can deal with large crystallite sizes without recalibration. The new calibration factor is given aswhere n is the number of targets that are integrated. The number of crystallites can be counted by the number of intersections of the γ profile with a threshold line. Every two intersections of the γ profile with this horizontal line represents a crystallite. In order to cancel out the effects of preferred orientation and other material and instrumental factors on the overall intensity fluctuation along the γ profile, one can use a trend line fitted to the γ profile by a secondorder polynomial. It is always necessary to calibrate the system with a known standard, preferably with a comparable sample geometry and crystallite size. For reflection mode, it is critical to have a standard with a comparable linear absorption coefficient so as to have similar penetration.
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