Tables for
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 2.5, pp. 136-140

Section Texture analysis

B. B. Hea*

aBruker AXS Inc., 5465 E. Cheryl Parkway, Madison, WI 53711, USA
Correspondence e-mail: Texture analysis

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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[link]).

When a conventional X-ray 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 two-dimensional X-ray 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 data-collection time (Smith & Ortega, 1993[link]; Blanton, 1994[link]; Bunge & Klein, 1996[link]; Helming et al., 2003[link]; Wenk & Grigull, 2003[link]; He, 2009[link]). 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. Pole density and pole figures

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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[link] 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 pole-density 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).

[Figure 2.5.18]

Figure 2.5.18 | top | pdf |

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[link](a), the sample orientation is defined by the sample coordinates S1, S2 and S3. For metals with rolling texture, the axes S1, S2 and S3 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[link](b). This two-dimensional 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 S1. 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[r = R\tan\left ({\pi \over 4} - {\alpha \over 2}\right) = R\tan {\chi \over 2}.\eqno(2.5.52)]

[Figure 2.5.19]

Figure 2.5.19 | top | pdf |

(a) Definition of pole direction angles α and β; (b) stereographic projection in a pole figure.

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 two-dimensional polar coordinate system. Other pole-figure mapping styles may be used, but must be properly noted to avoid confusion (Birkholz, 2006[link]). Fundamental equations

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The α and β angles are functions of γ, ω, ψ, ϕ and 2θ. As shown in Fig. 2.5.19[link](a), a pole has three components h1, h2 and h3, parallel to the three sample coordinates S1, S2 and S3, respectively. The pole-figure angles (α, β) can be calculated from the unit-vector components by the following pole-mapping equations:[\eqalignno{\alpha &= {\sin ^{ - 1}}\left| {{h_3}} \right| = {\cos ^{ - 1}}({h_1^2 + h_2^2})^{1/2},&(2.5.53)\cr \beta &= \pm {\cos ^{ - 1}}{{{h_1}} \over {({h_1^2 + h_2^2})^{1/2} }}\matrix{ {} & {} & {\left\{ {\matrix{ {\beta \ge 0^\circ } \ {\rm if} \ {{h_2} \ge 0} \cr {\beta\, \lt \,0^\circ } \ {\rm if} \ {{h_2} \,\lt\,0} \cr } } \right.}, \cr }&\cr&&(2.5.54)} ]where α takes a value between 0 and 90° ([0^\circ \le \alpha \le 90^\circ ]) and β takes values in two ranges ([0^\circ \le \beta \le 180^\circ ] when [{h}_2 \,\gt\, 0] and [ - 180^\circ \le \beta \,\lt \,0^\circ ] when [{h}_2\, \lt \,0]). The condition for reflection-mode diffraction is [h_3\, \gt \,0]. For transmission diffraction it is possible that [h_3\, \lt \,0]. In this case, the pole with mirror symmetry about the S1S2 plane to the diffraction vector is used for the pole-figure mapping. The absolute value of h3 is then used in the equation for the α angle. When [h_2 = 0] in the above equation, β takes one of two values depending on the value of h1 ([\beta = 0^\circ ] when [h_1 \ge 0] and [\beta = 180^\circ ] when [h_1\, \lt \,0]). For Eulerian geometry, the unit-vector components [\{h_1,h_2,h_3\}] are given by equation (2.5.11[link]).

The 2θ integrated intensity along the diffraction ring is then converted to the pole-density 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 2θ-integrated 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. [I_{hkl} = I_{hkl}(\omega, \psi, \varphi, \gamma, \theta)]. From the pole-figure angle-mapping equations, we can obtain the integrated intensity in terms of pole-figure angles as[{I_{hkl}}(\alpha, \beta) = {I_{hkl}}(\omega, \psi, \varphi, \gamma, \theta).\eqno(2.5.55)]The pole density at the pole-figure angles (α, β) is proportional to the integrated intensity at the same angles:[{P_{hkl}}(\alpha, \beta) = {K_{hkl}}(\alpha, \beta) {I_{hkl}}(\alpha, \beta),\eqno(2.5.56)]where [I_{hkl}(\alpha, \beta)] is the 2θ-integrated intensity of the (hkl) peak corresponding to the pole direction [(\alpha, \beta)], [K_{hkl}(\alpha, \beta)] is the scaling factor covering the absorption, polarization, background corrections and various instrument factors if these factors are included in the integrated intensities, and [P_{hkl}(\alpha, \beta)] is the pole-density distribution function. Background correction can be done during the 2θ integration and will be discussed in Section[link]. The pole figure is obtained by plotting the pole-density function based on the stereographic projection.

The pole-density function can be normalized such that it represents a fraction of the total diffracted intensity integrated over the pole sphere. The normalized pole-density distribution function is given by[g_{hkl}(\alpha, \beta) = {2\pi P_{hkl}(\alpha, \beta) \over \int_0^{2\pi } \int_0^{\pi /2} P_{hkl}(\alpha, \beta)\cos \alpha \, {\rm d}\alpha \, {\rm d}\beta }.\eqno(2.5.57)]The pole-density 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 pole-density distribution, we can obtain the normalized pole-density function by[{g_{hkl}}(\alpha, \beta) = {{{I_{hkl}}(\alpha, \beta)} \over {I_{hkl}^{\rm random}(\alpha, \beta)}}.\eqno(2.5.58)]

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 S3 direction, the pole-density distribution function becomes independent of the azimuthal angle β. For samples with fibre texture, or artificially formed fibre texture by rotating, the pole-density function is conveniently expressed as a function of a single variable, [{g_{hkl}}(\chi)]. Here, χ is the angle between the sample normal and pole direction.[\chi = 90^ \circ - \alpha \ {\rm or}\ \chi = {\cos ^{ - 1}}\left| {{h_3}} \right|.\eqno(2.5.59)]

The pole-density function for fibre texture can be expressed as a fibre plot. The fibre plot [{g_{hkl}}(\chi)] can be calculated from the relative intensity of several peaks (He, 1992[link]; He et al., 1994[link]) and artificial fibre texture can be achieved by sample spinning during data collection. Data-collection strategy

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Since a one-dimensional pole-density mapping is created from each 2D frame, it is important to lay out a data-collection strategy so as to have the optimum pole-figure coverage and minimum redundancy in data collection. The pole-figure 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[link](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 one-dimensional pole figure as shown in the curve marked by A and B. The pole figure can be generated by a full-circle rotation of 360°. The pole density at the centre represents the diffraction vector perpendicular to the sample surface. It is important to have the pole-density information in the centre region of the pole figure, especially for fibre texture. The pole-figure angle at the centre is α = 90°, and the best strategy is to put point A at the centre of pole figure. That is[\eqalignno{h_3^A &= \sin \theta \cos \psi \sin \omega - \cos \theta \sin \gamma _A\cos \psi \cos \omega &\cr&\quad- \cos \theta \cos \gamma _A\sin \psi = 1.&(2.5.60)}]

[Figure 2.5.20]

Figure 2.5.20 | top | pdf |

Data-collection 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 pole-figure 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[link](b) illustrates the data-collection 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 data-collection strategy should also be optimized for several crystallographic planes if all can be covered in a frame. The step size of the data-collection 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 two-dimensional data collection for a texture can be compared with that using a point detector with the data-collection strategy of the Cu thin film as an example. Fig. 2.5.20[link](c) shows the pole-figure data-collection strategy with a point detector. For the same pole-figure resolution, significantly more exposures are required with a point detector. Considering that several diffraction rings are measured simultaneously with a 2D detector, the pole-figure measurement is typically 10 to 100 times faster than with a point detector. Therefore, quantitative high-resolution pole-figure measurements are only practical with a 2D-XRD system (Bunge & Klein, 1996[link]). Texture-data processing

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For a specific diffraction ring, 2θ is a constant or at least assumed to be constant for texture analysis, and the sample-orientation angles (ω, ψ, ϕ) for a frame are also constants. Therefore, the pole-density information is given by the diffraction-intensity 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[link](a) shows a 2D diffraction ring for texture analysis. The low and high background and diffraction-ring 2θ–γ range are defined by three boxes, noted as BL, BH and I(γ), respectively. All three boxes have the same γ range from γ1 to γ2. 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[link](b) is the 2θ profile integrated over the section Δγ in Fig. 2.5.21[link](a). The background ranges on the low and high 2θ sides are given by 2θL1–2θL2 and 2θH1–2θH2, respectively. The 2θ-integrated diffraction intensities as a function of γ are plotted in Fig. 2.5.21[link](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.

[Figure 2.5.21]

Figure 2.5.21 | top | pdf |

Pole-figure 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 as[I(\gamma) = \textstyle\int_{2{\theta _1}}^{2{\theta _2}} {J(2\theta, \gamma)\, {\rm d}(2\theta)}, \quad {\gamma _1} \le \gamma \le {\gamma _2}.\eqno(2.5.61)]A similar equation can be used for 2θ integration of the low and high backgrounds BL(γ) and BH(γ). Assuming a linear background change in the vicinity of the 2θ peak, the background under the peak, B(γ), is then given by[\eqalignno{B(\gamma) &= {B_L}(\gamma){{(2{\theta _2} - 2{\theta _1})(2\theta_{H2}+2\theta_{H1}-2\theta_2 - 2\theta_1)} \over {(2{\theta _{L2}} - 2{\theta _{L1}})(2\theta_{H2}+2\theta_{H1}-2\theta_{L2}-2\theta_{L1})}} &\cr&\quad+ {B_H}(\gamma){{(2{\theta _2} - 2{\theta _1})(2{\theta _2} + 2{\theta _1} - 2{\theta _{L2}} - 2{\theta _{L1}})} \over {(2{\theta _{H2}} - 2{\theta _{H1}})(2{\theta _{H2}} + 2{\theta _{H1}} - 2{\theta _{L2}} - 2{\theta _{L1}})}}.&\cr&&(2.5.62)}]Then the background B(γ) can be subtracted from the integrated intensity distribution I(γ).

The algorithms of γ integration given in Section[link] can be easily modified for 2θ integration by exchanging γ and 2θ in the equations. Algorithms with solid-angle normalization should be used to get consistent integrated intensity over all areas of the detector. The 2θ-integrated intensity distribution can then be mapped onto a pole figure based on the fundamental equations (2.5.53)[link] and (2.5.54)[link]. When a pole-figure 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[link](b), the average value should be mapped to that pole-figure pixel. Fig. 2.5.22[link](a) shows pole-density mappings on the pole figure. There are big gaps between the measured pole-density data points due to the large ϕ-scan steps of 5°.

[Figure 2.5.22]

Figure 2.5.22 | top | pdf |

Pole-figure 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 Be-window 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 pole-density regions covered by two different incident angles may be observed if sample absorption is not properly corrected. Pole-figure interpolation and use of symmetry

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The pole figure is stored and displayed as a bitmap image. The pole-density data from the data set may not fill up all the pixels of the pole-figure 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 pole-density points are filled after this interpolation. For a sample with sharp texture, smaller ϕ-scan 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 symmetry-equivalent pole-figure pixels are filled by the average value of the measured pixels. For the unmeasured pole-figure 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[link](b) shows the results after both interpolation and use of symmetry. Orientation relationship

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A 2D-XRD 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 sputter-deposited Cu film on an Si wafer (He et al., 2005[link]). Fig. 2.5.23[link] 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 thin-film layers, the orientation relationships between the different layers of the films and substrate can be revealed by superimposing their pole figures.

[Figure 2.5.23]

Figure 2.5.23 | top | pdf |

Combined pole figure of a Cu (111) film on an Si (400) substrate: (a) regular 2D projection; (b) 3D surface plot.


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