International
Tables for
Crystallography
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. 118-149
https://doi.org/10.1107/97809553602060000940

Chapter 2.5. Two-dimensional powder diffraction

B. B. Hea*

aBruker AXS Inc., 5465 E. Cheryl Parkway, Madison, WI 53711, USA
Correspondence e-mail: bob.he@bruker.com

Two-dimensional X-ray diffraction, also referred to as 2D powder diffraction, covers X-ray diffraction applications with a 2D detector and corresponding data reduction and analysis. A two-dimensional diffraction pattern contains abundant information about the atomic arrangement, microstructure and defects of a solid or liquid material. In recent years, use of two-dimensional detectors has dramatically increased in academic research and various industries. When a 2D detector is used for X-ray powder diffraction, the diffraction cones are intercepted by the area detector and the X-ray intensity distribution on the sensing area is converted to a 2D diffraction pattern. A 2D diffraction pattern contains far more information than a conventional diffraction pattern, and therefore demands a special data-collection strategy and data-evaluation algorithms. This chapter covers the basic concepts and recent progress in two-dimensional X-ray diffraction theory and technologies, including geometry conventions, X-ray source and optics, two-dimensional detectors, and diffraction-data interpretation, and various applications, such as phase identification, texture, stress, crystallinity and crystallite-size analysis.

2.5.1. Introduction

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2.5.1.1. The diffraction pattern measured by an area detector

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The diffracted X-rays from a polycrystalline or powder sample form a series of cones in three-dimensional space, since large numbers of crystals oriented randomly in the space are covered by the incident X-ray beam. Each diffraction cone corresponds to the diffraction from the same family of crystal planes in all the participating grains. The apex angles of cones are given by Bragg's law for the corresponding crystal interplanar d-spacing. A conventional X-ray powder-diffraction pattern is collected by scanning a point or linear detector along the 2θ angle. The diffraction pattern is displayed as scattering intensity versus 2θ angle (Klug & Alexander, 1974[link]; Cullity, 1978[link]; Warren, 1990[link]; Jenkins & Snyder, 1996[link]; Pecharsky & Zavalij, 2003[link]). In recent years, use of two-dimensional (2D) detectors for powder diffraction has dramatically increased in academic and industrial research (Sulyanov et al., 1994[link]; Rudolf & Landes, 1994[link]; He, 2003[link], 2009[link]). When a 2D detector is used for X-ray powder diffraction, the diffraction cones are intercepted by the area detector and the X-ray intensity distribution on the sensing area is converted to an image-like diffraction pattern, also referred to as a frame. Since the diffraction pattern collected with a 2D detector is typically given as an intensity distribution over a two-dimensional region, so X-ray diffraction with a 2D detector is also referred to as two-dimensional X-ray diffraction (2D-XRD) or 2D powder diffraction. A 2D diffraction pattern contains far more information than a conventional diffraction pattern, and therefore demands a special data-collection strategy and data-evaluation algorithms. This chapter covers the basic concepts and recent progress in 2D-XRD theory and technologies, including geometry conventions, X-ray source and optics, 2D detectors, diffraction-data interpretation, and various applications, such as phase identification and texture, stress, crystallinity and crystallite-size analysis. The concepts and algorithms of this chapter apply to both laboratory and synchrotron diffractometers equipped with 2D detectors.

2.5.1.2. Comparison between 2D-XRD and conventional XRD

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Fig. 2.5.1[link] is a schematic of X-ray diffraction from a powder (polycrystalline) sample. For simplicity, it shows only two diffraction cones; one represents forward diffraction [(2\theta\leq90^\circ)] and one represents backward diffraction [(2\theta\,\gt\,90^\circ)]. The diffraction measurement in a conventional diffractometer is confined within a plane, here referred to as the diffractometer plane. A point (0D) detector makes a 2θ scan along a detection circle. If a line (1D) detector is used in the diffractometer, it will be mounted on the detection circle. Since the variations in the diffraction pattern in the direction (Z) perpendicular to the diffractometer plane are not considered in a conventional diffractometer, the X-ray beam is normally extended in the Z direction (line focus). Since the diffraction data out of the diffractometer plane are not detected, the structures in the material that are represented by the missing diffraction data will either be ignored, or extra sample rotation and time are needed to complete the measurement.

[Figure 2.5.1]

Figure 2.5.1 | top | pdf |

Diffraction patterns in 3D space from a powder sample and the diffractometer plane.

With a 2D detector, the diffraction measurement is no longer limited to the diffractometer plane. Depending on the detector size, the distance to the sample and the detector position, the whole or a large portion of the diffraction rings can be measured simultaneously. Diffraction patterns out of the diffractometer plane have for a long time been recorded using Debye–Scherrer cameras, so the diffraction rings are referred to as Debye rings. However, when a Debye–Scherrer camera is used, only the position of the arches in the 2θ direction and their relative intensities are measured for powder-diffraction analysis. The diffraction rings collected with a large 2D detector extend further in the `vertical' direction and the intensity variation in the vertical direction is also used for data evaluation. Therefore, the terms `diffraction cone' and `diffraction ring' will be often be used in this chapter as alternatives to `Debye cone' and `Debye ring'.

2.5.1.3. Advantages of two-dimensional X-ray diffraction

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A 2D diffraction frame contains far more information than a diffraction pattern measured using a conventional diffraction system with a point detector or a linear position-sensitive detector. In addition to the significantly higher data-collection speed, the intensity and 2θ variation along the diffraction rings can reveal abundant structural information typically not available from a conventional diffraction pattern. Fig. 2.5.2[link] shows a 2D pattern collected from a battery component containing multiple layers of different phases. Some diffraction rings have strong intensity variation due to preferred orientation, and the spotty diffraction rings are from a phase that contains large crystal grains. It is apparent that different diffraction-ring patterns are from different phases. 2D-XRD analyses commonly performed on polycrystalline materials include phase identification, quantitative phase analysis, preferred-orientation quantification and characterization of residual stresses.

[Figure 2.5.2]

Figure 2.5.2 | top | pdf |

Diffraction pattern from a battery component containing multiple layers.

Phase identification (phase ID) can be done by integration in a selected 2θ range along the diffraction rings (Hammersley et al., 1996[link]; Rodriguez-Navarro, 2006[link]). The integrated data give better intensity and statistics for phase ID and quantitative analysis, especially for those samples with texture or large grain sizes, or where the sample is small. Then the integrated diffraction profiles can be analysed with existing algorithms and methods: profile fitting with conventional peak shapes and fundamental parameters, quantification of phases, and lattice-parameter indexing and refinement. The results can be used to search and match to entries in a powder-diffraction database, typically the Powder Diffraction File.

Texture measurement with 2D-XRD is extremely fast compared to measurement using a point or linear detector. The area detector collects texture data and background values simultaneously for multiple poles and multiple directions. Owing to the high measurement speed, pole figures can be measured at very fine steps, allowing detection of very sharp textures (Smith & Ortega, 1993[link]; Bunge & Klein, 1996[link]; He, 2009[link]).

Stress measurement with 2D-XRD is based on a direct relationship between the stress tensor and distortion of the diffraction cones. Since the whole or a part of the diffraction ring is used for stress calculation, 2D-XRD can measure stress with high sensitivity, high speed and high accuracy (He & Smith, 1997[link]; He, 2000[link]). It is highly suitable for samples containing large crystals and textures. Simultaneous measurement of stress and texture is also possible, since 2D data contain both stress and texture information.

Concentrations of crystalline phases can be measured faster and more accurately with data analysis over 2D frames, especially for samples with an anisotropic distribution of crystallite orientations and/or amorphous content. The amorphous region can be defined by the user to consist of regions with no Bragg peaks, or the amorphous region can be defined with the crystalline region included when the crystalline region and the amorphous region overlap.

Microdiffraction data are collected with speed and accuracy. Collection of X-ray diffraction data from small sample amounts or small sample areas has always been a slow process because of limited beam intensity. The 2D detector captures whole or a large portion of the diffraction rings, so spotty, textured or weak diffraction data can be integrated over the selected diffraction rings (Winter & Squires, 1995[link]; Bergese et al., 2001[link]; Tissot, 2003[link]; Bhuvanesh & Reibenspies, 2003[link]; He, 2004[link]). The point beam used for microdiffraction allows diffraction mapping with fine space resolution, even on a curved surface (Allahkarami & Hanan, 2011[link]).

Data can be collected from thin-film samples containing a mixture of single-crystal and polycrystalline layers with random orientation distributions, and highly textured layers, with all the features appearing simultaneously in diffraction frames (Dickerson et al., 2002[link]; He, 2006[link]). The pole figures from different layers and the substrate can be overlapped to reveal the orientation relationships. The use of a 2D detector can dramatically speed up the data collection for reciprocal-space mapping on an in-plane reciprocal-lattice point (Schmidbauer et al., 2008[link]).

Because of the penetrating power of the X-ray beam, fast nondestructive data collection and the abundant information about atomic structure, two-dimensional X-ray diffraction can be used to screen a library of materials with high speed and high accuracy. Two-dimensional X-ray diffraction systems dedicated for combinatorial screening are widely used in the pharmaceutical industry for drug discovery and process analysis (Klein et al., 1998[link]; He et al., 2001[link]).

Forensic science and archaeology have benefited from using two-dimensional X-ray diffraction for identifying materials and structures from small specimens (Kugler, 2003[link]; Bontempi et al., 2008[link]). It is nondestructive and does not require special sample treatment, so the original evidence or sample can be preserved. Two-dimensional diffraction patterns contain abundant information and are easy to observe and explain in the courtroom.

2.5.2. Fundamentals

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A conventional powder-diffraction pattern is displayed as the scattering intensity versus scattering angle 2θ or d-spacing. A 2D-XRD pattern contains the scattering-intensity distribution as a function of two orthogonal dimensions. One dimension can be expressed in 2θ, which can be interpreted by Bragg's law. The distribution in the dimension orthogonal to 2θ contains additional information, such as the orientation distribution, strain states, and crystallite-size and -shape distribution. In order to understand and analyse 2D diffraction data, new geometry conventions and algorithms are introduced. The geometry conventions and algorithms used for 2D-XRD should also be consistent with conventional XRD, so that many existing concepts and algorithms are still valid when 2D diffraction data are used.

The geometry of a 2D-XRD system can be explained using three distinguishable and interrelated geometry spaces, each defined by a set of parameters (He, 2003[link]). The three geometry spaces are the diffraction space, detector space and sample space. The laboratory coordinate system XL, YL, ZL is the basis of all three spaces. Although the three spaces are interrelated, the definitions and corresponding parameters should not be confused. Except for a few parameters introduced specifically for 2D-XRD, many of these parameters are used in conventional X-ray diffraction systems. Therefore, the same definitions are maintained for consistency. The three-circle goniometer in Eulerian geometry is the most commonly used, and all the algorithms for data interpretation and analysis in this chapter are based on Eulerian geometry. The algorithms can be developed for the geometries of other types (such as kappa) by following the same strategies.

2.5.2.1. Diffraction space and laboratory coordinates

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2.5.2.1.1. Diffraction cones in laboratory coordinates

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Fig. 2.5.3[link](a) describes the geometric definition of diffraction cones in the laboratory coordinate system XL, YL, ZL. The laboratory coordinate system is a Cartesian coordinate system. The plane given by XL and YL is the diffractometer plane. The axis ZL is perpendicular to the diffractometer plane. The axes XL, YL and ZL form a right-handed rectangular coordinate system with the origin at the instrument centre. The incident X-ray beam propagates along the XL axis, which is also the rotation axis of all diffraction cones. The apex angles of the cones are determined by the 2θ values given by the Bragg equation. The apex angles are twice the 2θ values for forward reflection [(2\theta\leq90^\circ)] and twice the value of 180° − 2θ for backward reflection [(2\theta\,\gt\,90^\circ)]. For clarity, only one diffraction cone of forward reflection is displayed. The γ angle is the azimuthal angle from the origin at the six o'clock direction with a right-handed rotation axis along the opposite direction of incident beam (−XL direction). A given γ value defines a half plane with the XL axis as the edge; this will be referred to as the γ plane hereafter. The diffractometer plane consists of two γ planes at γ = 90° and γ = 270°. Therefore many equations developed for 2D-XRD should also apply to conventional XRD if the γ angle is given as a constant of 90° or 270°. A pair of γ and 2θ values represents the direction of a diffracted beam. The γ angle takes a value of 0 to 360° for a complete diffraction ring with a constant 2θ value. The γ and 2θ angles form a spherical coordinate system which covers all the directions from the origin of sample (instrument centre). The γ–2θ system is fixed in the laboratory system XL, YL, ZL, which is independent of the sample orientation and detector position in the goniometer. 2θ and γ are referred to as the diffraction-space parameters. In the laboratory coordinate system XL, YL, ZL, the surface of a diffraction cone can be mathematically expressed as[y_L^2 + z_L^2 = x_L^2\tan ^22\theta, \eqno(2.5.1)]with [{x_L} \ge 0] or [2\theta \le 90^\circ ] for forward-diffraction cones and [{x_L} \,\lt\, 0] or [2\theta\, \gt \,90^\circ ] for backward-diffraction cones.

[Figure 2.5.3]

Figure 2.5.3 | top | pdf |

The diffraction cone and the corresponding diffraction-vector cone.

2.5.2.1.2. Diffraction-vector cones in laboratory coordinates

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Fig. 2.5.3[link](b) shows the diffraction-vector cone corresponding to the diffraction cone in the laboratory coordinate system. C is the centre of the Ewald sphere. The diffraction condition can be given by the Laue equation as[{{{\bf{s}} - {{\bf{s}}_0}} \over \lambda } = {{\bf{H}}_{hkl}},\eqno(2.5.2)]where s0 is the unit vector representing the incident beam, s is the unit vector representing the diffracted beam and Hhkl is the reciprocal-lattice vector. Its magnitude is given as[\left| {{{{\bf{s}} - {{\bf{s}}_0}} \over \lambda }} \right| = {{2\sin \theta } \over \lambda } = \left| {{{\bf{H}}_{hkl}}} \right| = {1 \over {{d_{hkl}}}}, \eqno(2.5.3)]in which dhkl is the d-spacing of the crystal planes (hkl). It can be easily seen that it is the Bragg law in a different form. Therefore, equation (2.5.2)[link] is the Bragg law in vector form. In the Bragg condition, the vectors s0/λ and s/λ make angles θ with the diffracting planes (hkl) and Hhkl is normal to the (hkl) crystal plane. In order to analyse all the X-rays measured by a 2D detector, we extend the concept to all scattered X-rays from a sample regardless of the Bragg condition. Therefore, the index (hkl) can be removed from the above expression. H is then a vector which takes the direction bisecting the incident beam and the scattered beam, and has dimensions of inverse length given by [2\sin \theta /\lambda ]. Here 2θ is the scattering angle from the incident beam. The vector H is referred to as the scattering vector or, alternatively, the diffraction vector. When the Bragg condition is satisfied, the diffraction vector is normal to the diffracting lattice planes and its magnitude is reciprocal to the d-spacing of the lattice planes. In this case, the diffraction vector is equivalent to the reciprocal-lattice vector. Each pixel in a 2D detector measures scattered X-rays in a given direction with respect to the incident beam. We can calculate a diffraction vector for any pixel, even if the pixel is not measuring Bragg scattering. Use of the term `diffracted beam' hereafter in this chapter does not necessarily imply that it arises from Bragg scattering.

For two-dimensional diffraction, the incident beam can be expressed by the vector s0/λ, but the diffracted beam is no longer in a single direction, but follows the diffraction cone. Since the direction of a diffraction vector is a bisector of the angle between the incident and diffracted beams corresponding to each diffraction cone, the trace of the diffraction vectors forms a cone. This cone is referred to as the diffraction-vector cone. The angle between the diffraction vector and the incident X-ray beam is 90° + θ and the apex angle of a vector cone is 90° − θ. It is apparent that diffraction-vector cones can only exists on the −XL side of the diffraction space.

For two-dimensional diffraction, the diffraction vector is a function of both the γ and 2θ angles, and is given in laboratory coordinates as[{\bf{H}} = {{{\bf{s}} - {{\bf{s}}_0}} \over \lambda } = {1 \over \lambda }\left [{\matrix{ {\cos 2\theta - 1} \cr { - \sin 2\theta \sin \gamma } \cr { - \sin 2\theta \cos \gamma } \cr } } \right].\eqno(2.5.4)]The direction of the diffraction vector can be represented by its unit vector, given by[{{\bf{h}}_{{L}}} = {{\bf{H}} \over {\left| {\bf{H}} \right|}} = \left [{\matrix{ {{h_x}} \cr {{h_y}} \cr {{h_z}} \cr } } \right] = \left [{\matrix{ { - \sin \theta } \cr { - \cos \theta \sin \gamma } \cr { - \cos \theta \cos \gamma } \cr } } \right],\eqno(2.5.5)]where hL is a unit vector expressed in laboratory coordinates and the three components in the square brackets are the projections of the unit vector on the three axes of the laboratory coordinates, respectively. If γ takes all values from 0 to 360° at a given Bragg angle 2θ, the trace of the diffraction vector forms a diffraction-vector cone. Since the possible values of θ lie within the range 0 to 90°, hx takes only negative values.

2.5.2.2. Detector space and pixel position

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A typical 2D detector has a limited detection surface, and the detection surface can be spherical, cylindrical or flat. Spherical or cylindrical detectors are normally designed for a fixed sample-to-detector distance, while a flat detector has the flexibility to be used at different sample-to-detector distances so as to choose either high resolution at a large distance or large angular coverage at a short distance.

2.5.2.2.1. Detector position in the laboratory system

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The position of a flat detector is defined by the sample-to-detector distance D and the detector swing angle α. D and α are referred to as the detector-space parameters. D is the perpendicular distance from the goniometer centre to the detection plane and α is a right-handed rotation angle about the ZL axis. Detectors at different positions in the laboratory coordinates XL, YL, ZL are shown in Fig. 2.5.4[link]. The centre of detector 1 is right on the positive side of the XL axis (on-axis), α = 0. Both detectors 2 and 3 are rotated away from the XL axis with negative swing angles (α2 < 0 and α3 < 0). The detection surface of a flat 2D detector can be considered as a plane, which intersects the diffraction cone to form a conic section. Depending on the swing angle α and the 2θ angle, the conic section can appear as a circle, an ellipse, a parabola or a hyperbola.

[Figure 2.5.4]

Figure 2.5.4 | top | pdf |

Detector positions in the laboratory-system coordinates.

2.5.2.2.2. Pixel position in diffraction space for a flat detector

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The values of 2θ and γ can be calculated for each pixel in the frame. The calculation is based on the detector-space parameters and the pixel position in the detector. Fig. 2.5.5[link] shows the relationship of a pixel P(x, y) to the laboratory coordinates XL, YL, ZL. The position of a pixel in the detector is defined by the (x, y) coordinates, where the detector centre is defined as x = y = 0. The diffraction-space coordinates (2θ, γ) for a pixel at P(x, y) are given by[\eqalignno{2\theta &= \arccos {{x\sin \alpha + D\cos \alpha } \over {( {{D^2} + {x^2} + {y^2}})^{1/2}}}\quad(0 \,\lt\, 2\theta \,\lt\, \pi), &(2.5.6)\cr \gamma &= {{x\cos \alpha - D\sin \alpha } \over {\left| {x\cos \alpha - D\sin \alpha } \right|}}\arccos {{ - y} \over {[{{y^2} + {{(x\cos \alpha - D\sin \alpha)}^2}})]^{1/2} }}&\cr &\quad\quad(- \pi \,\lt\, \gamma \le \pi). &(2.5.7)}]

[Figure 2.5.5]

Figure 2.5.5 | top | pdf |

Relationship between a pixel P and detector position in the laboratory coordinates.

2.5.2.2.3. Pixel position in diffraction space for a curved detector

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The conic sections of the diffraction cones with a curved detector depend on the shape of the detector. The most common curved detectors are cylinder-shaped detectors. The diffraction frame measured by a cylindrical detector can be displayed as a flat frame, typically a rectangle. Fig. 2.5.6[link](a) shows a cylindrical detector in the vertical direction and the corresponding laboratory coordinates XL, YL, ZL. The sample is located at the origin of the laboratory coordinates inside the cylinder. The incident X-rays strike the detector at a point O if there is no sample or beam stop to block the direct beam. The radius of the cylinder is R. Fig. 2.5.6[link](b) illustrates the 2D diffraction image collected with the cylindrical detector. We take the point O as the origin of the pixel position (0, 0). The diffraction-space coordinates (2θ, γ) for a pixel at P(x, y) are given by[\eqalignno{2\theta &= \arccos \left [R\cos \left({x\over R}\right) \big/({R^2} + {y^2})^{1/2} \right],&(2.5.8)\cr \gamma &= {x\over |x|}\arccos \left \{ - y\big/\left[ y^2 + R^2 \sin ^2\left({x \over R}\right)\right]^{1/2} \right\}\quad(-\pi \,\lt\, \gamma\leq \pi).&\cr &&(2.5.9)}]The pixel-position-to-(2θ, γ) conversion for detectors of other shapes can also be derived. Once the diffraction-space coordinates (2θ, γ) of each pixel in the curved 2D detector are determined, most data-analysis algorithms developed for flat detectors are applicable to a curved detector as well.

[Figure 2.5.6]

Figure 2.5.6 | top | pdf |

Cylinder-shaped detector in vertical direction: (a) detector position in the laboratory coordinates; (b) pixel position in the flattened image.

2.5.2.3. Sample space and goniometer geometry

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2.5.2.3.1. Sample rotations and translations in Eulerian geometry

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In a 2D-XRD system, three rotation angles are necessary to define the orientation of a sample in the diffractometer. These three rotation angles can be achieved either by a Eulerian geometry, a kappa (κ) geometry or another kind of geometry. The three angles in Eulerian geometry are ω (omega), ψ (psi) and ϕ (phi). Fig. 2.5.7[link](a) shows the relationship between rotation axes (ω, ψ, ϕ) in the laboratory system XL, YL, ZL. The ω angle is defined as a right-handed rotation about the ZL axis. The ω axis is fixed in the laboratory coordinates. The ψ angle is a right-handed rotation about a horizontal axis. The angle between the ψ axis and the XL axis is given by ω. The ψ axis lies on XL when ω is set at zero. The ϕ angle defines a left-handed rotation about an axis on the sample, typically the normal of a flat sample. The ϕ axis lies on the YL axis when ω = ψ = 0. In an aligned diffraction system, all three rotation axes and the primary X-ray beam cross at the origin of the XL, YL, ZL coordinates. This cross point is also known as the goniometer centre or instrument centre.

[Figure 2.5.7]

Figure 2.5.7 | top | pdf |

Sample rotation and translation. (a) Three rotation axes in laboratory coordinates; (b) rotation axes (ω, ψ, ϕ) and sample coordinates.

Fig. 2.5.7[link](b) shows the relationship and stacking sequence among all rotation axes (ω, ψ, ϕ) and the sample coordinates S1, S2, S3. ω is the base rotation; all other rotations and translations are on top of this rotation. The next rotation above ω is the ψ rotation. The next rotation above ω and ψ is the ϕ rotation. The sample coordinates S1, S2, S3 are fixed to the sample regardless of the particular sample orientation given by the rotation angles (ω, ψ, ϕ). The ϕ rotation in the goniometer is intentionally chosen as a left-handed rotation so that the diffraction vectors will make a right-hand rotation observed in the sample coordinates S1, S2, S3.

2.5.2.4. Diffraction-vector transformation

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2.5.2.4.1. Diffraction unit vector in diffraction space and sample space

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In 2D-XRD data analysis, it is crucial to know the diffraction-vector distribution in terms of the sample coordinates S1, S2, S3. However, the diffraction-vector distribution corresponding to the measured 2D data is always given in terms of the laboratory coordinates XL, YL, ZL because the diffraction space is fixed to the laboratory coordinates. Fig. 2.5.8[link] shows the unit vector of a diffraction vector in both (a) the laboratory coordinates XL, YL, ZL and (b) the sample coordinates S1, S2, S3. In Fig. 2.5.8[link](a) the unit vector hL is projected to the XL, YL and ZL axes as hx, hy and hz, respectively. The three components are given by equation (2.5.5)[link]. In order to analyse the diffraction results relative to the sample orientation, it is necessary to transform the unit vector to the sample coordinates S1, S2, S3. Fig. 2.5.8[link](b) shows the same unit vector, denoted by hs projected to S1, S2 and S3 as h1, h2 and h3, respectively.

[Figure 2.5.8]

Figure 2.5.8 | top | pdf |

Unit diffraction vector in (a) the laboratory coordinates and (b) the sample coordinates.

2.5.2.4.2. Transformation from diffraction space to sample space

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The transformation of the unit diffraction vector from the laboratory coordinates XL, YL, ZL to the sample coordinates S1, S2, S3 is given by[{{\bf{h}}_{{s}}} = {\bi{A}}{{\bf{h}}_{{L}}}, \eqno(2.5.10)]where A is the transformation matrix. For Eulerian geometry in matrix form, we have[\eqalignno{&\left [{\matrix{ {{h_1}} \cr {{h_2}} \cr {{h_3}} \cr } } \right] = \left [{\matrix{ {{a_{11}}} & {{a_{12}}} & {{a_{13}}} \cr {{a_{21}}} & {{a_{22}}} & {{a_{23}}} \cr {{a_{31}}} & {{a_{32}}} & {{a_{33}}} \cr } } \right]\left [{\matrix{ {{h_x}} \cr {{h_y}} \cr {{h_z}} \cr } } \right] &\cr &= \left [\matrix{- \sin \omega \sin \psi \sin \varphi \hfill & \cos \omega \sin \psi \sin \varphi \hfill & -\cos \psi \sin \varphi \cr \quad - \cos \omega \cos \varphi \hfill &\quad - \sin \omega \cos \varphi \hfill\cr\cr \sin \omega \sin \psi \cos \varphi \hfill & - \cos \omega \sin \psi \cos \varphi \hfill & \cos \psi \cos \varphi \hfill\cr \quad - \cos \omega \sin \varphi\hfill & \quad - \sin \omega \sin \varphi \hfill\cr\cr{ - \sin \omega \cos \psi } \hfill & {\cos \omega \cos \psi } \hfill &{\sin \psi } \hfill }\right]&\cr&\times\left [\let\normalbaselines\relax\openup3pt\matrix{ - \sin \theta \cr\cr - \cos \theta \sin \gamma \cr\cr - \cos \theta \cos \gamma } \right]. &\cr &&(2.5.11)}]In expanded form:[\eqalignno{{h_1} &= \sin \theta (\sin \varphi \sin \psi \sin \omega + \cos \varphi \cos \omega) + \cos \theta \cos \gamma \sin \varphi \cos \psi\cr &\quad- \cos \theta \sin \gamma (\sin \varphi \sin \psi \cos \omega - \cos \varphi \sin \omega)\cr {h_2} &= - \sin \theta (\cos \varphi \sin \psi \sin \omega - \sin \varphi \cos \omega) &\cr&\quad- \cos \theta \cos \gamma \cos \varphi \cos \psi &\cr&\quad+ \cos \theta \sin \gamma (\cos \varphi \sin \psi \cos \omega + \sin \varphi \sin \omega)\cr {h_3} &= \sin \theta \cos \psi \sin \omega - \cos \theta \sin \gamma \cos \psi \cos \omega - \cos \theta \cos \gamma \sin \psi \cr &&(2.5.12)}]In addition to the diffraction intensity and Bragg angle corresponding to each data point on the diffraction ring, the unit vector hs{h1, h2, h3} provides orientation information in the sample space. The transformation matrix of any other goniometer geometry, such as kappa geometry (Paciorek et al., 1999[link]), can be introduced into equation (2.5.10)[link] so that the unit vector hs{h1, h2, h3} can be expressed in terms of the specified geometry. All equations using the unit vector hs{h1, h2, h3} in this chapter, such as in data treatment, texture analysis and stress measurement, are applicable to all goniometer geometries provided that the unit-vector components are generated from the corresponding transformation matrix from diffraction space to the sample space.

2.5.2.4.3. Transformation from detector space to reciprocal space

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Reciprocal-space mapping is commonly used to analyse the diffraction patterns from highly oriented structures, diffuse scattering from crystal defects, and thin films (Hanna & Windle, 1995[link]; Mudie et al., 2004[link]; Smilgies & Blasini, 2007[link]; Schmidbauer et al., 2008[link]). The equations of the unit-vector calculation given above can also be used to transform the diffraction intensity from the diffraction space to the reciprocal space with respect to the sample coordinates. The direction of the scattering vector is given by the unit vector hs{h1, h2, h3} and the magnitude of the scattering vector is given by [2\sin \theta /\lambda ], so that the scattering vector corresponding to a pixel is given by[{\bf{H}} = {{2\sin \theta } \over \lambda }{{\bf{h}}_{{s}}}.\eqno(2.5.13)]The three-dimensional reciprocal-space mapping can be obtained by applying the normalized pixel intensities to the corresponding reciprocal points. With various sample orientations, all pixels on the detector can be mapped into a 3D reciprocal space.

2.5.3. Instrumentation

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2.5.3.1. X-ray source and optics

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2.5.3.1.1. Beam path in a diffractometer equipped with a 2D detector

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The Bragg–Brentano (B-B) parafocusing geometry is most commonly used in conventional X-ray diffractometers with a point detector (Cullity, 1978[link]; Jenkins & Snyder, 1996[link]). In the Bragg–Brentano geometry, the sample surface normal is always a bisector between the incident beam and the diffracted beam. A divergent incident beam hits the sample surface with an incident angle θ. The area of the irradiated region depends on the incident angle θ and the size of the divergence slit. The diffracted rays leave the sample at an angle 2θ, pass through the anti-scatter slit and receiving slit, and reach the point detector. Soller slits are used on both the primary side and secondary side to minimize the effects of axial divergence due to the line-focus beam. The primary line-focus beam sliced by the Soller slits can also be considered as an array of point beams parallel to the diffractometer planes. Each of these point beams will produce a diffraction cone from the sample. The overlap of all the diffraction cones will create a smeared diffraction peak. The Soller slits on the receiving side allow only those diffracted beams nearly parallel to the diffractometer plane to pass through, so the smearing effect is minimized. In another words, the so-called `line-focus geometry' in conventional diffractometry is actually a superposition of many layers of `spot-focus geometry'.

The beam path in a diffractometer equipped with a 2D detector is different from that in a conventional diffractometer in many respects (He & Preckwinkel, 2002[link]). In a 2D-XRD system the whole or a large portion of the diffraction rings are measured simultaneously, and neither slits nor monochromator can be used between the sample and detector. Therefore, the X-ray source and optics for 2D-XRD systems have different requirements in terms of the beam spectral purity, divergence and beam cross-section profile. Fig. 2.5.9[link] shows the beam path in a 2D-XRD system with the θ–θ configuration. The geometry for the θ–2θ configuration is equivalent. The X-ray tube, monochromator and collimator assembly are all mounted on the primary side. The incident-beam assembly rotates about the instrument centre and makes an incident angle θ1 to the sample surface. The first main axis is also called the θ1 axis. The diffracted beams travel in all directions and some are intercepted by a 2D detector. The detector is mounted on the other main axis, θ2. The detector position is determined by the sample-to-detector distance D and the detector swing angle α (= θ1 + θ2).

[Figure 2.5.9]

Figure 2.5.9 | top | pdf |

X-ray beam path in a two-dimensional X-ray diffraction system.

All the components and space between the focal spot of the X-ray tube and sample are collectively referred to as the primary beam path. The primary beam path in a 2D-XRD system is typically sheltered by optical components except between the exit of the collimator and the sample. The X-rays travelling through this open incident-beam path are scattered by the air with two adverse effects. One is the attenuation of the primary beam intensity. The more harmful effect is that the scattered X-rays travel in all directions and some reach the detector, as is shown by the dashed lines with arrows in Fig. 2.5.9[link]. This air scatter introduces a background over the diffraction pattern. Weak diffraction patterns may be buried under the background. Obviously, the air scatter from the incident beam is significantly stronger than that from diffracted X-rays. The intensity of the air scatter from the incident beam is proportional to the length of the open incident-beam path. The effect of air scatter also depends on the wavelength of the X-rays. The longer the wavelength is, the more severe is the air scatter. The secondary beam path is the space between the sample and the 2D detector. The diffracted X-rays are also scattered by air and the diffraction pattern is both attenuated and blurred by the air scattering. In a conventional diffractometer, one can use an anti-scatter slit, diffracted-beam monochromator or detector Soller slits to remove most of the air scatter that is not travelling in the diffracted-beam direction. These measures cannot be used for a 2D-XRD system, which requires an open space between the sample and the 2D detector. Therefore, the open incident-beam path should be kept as small as possible. In order to reduce the air attenuation and air scatter of the incident beam, a helium-purged beam path or a vacuum beam path are sometimes used in a diffractometer. The air scatter from the diffracted X-rays is relatively weak and the effect depends on sample-to-detector distance. It is typically not necessary to take measures to remove air scatter from the diffracted X-rays between the sample and 2D detector if the sample-to-detector distance is 30 cm or less with Cu Kα radiation. However, if the sample-to-detector distance is larger than 30 cm or longer-wavelength radiation, such as Co Kα or Cr Kα, is used, it is then necessary to use an He beam path or vacuum beam path to reduce the air scatter.

2.5.3.1.2. Liouville's theorem

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Liouville's theorem can be used to describe the nature of the X-ray source, the X-ray optics and the coupling of the source and optics (Arndt, 1990[link]). Liouville's theorem can be stated in a variety of ways, but for X-ray optics the best known form is[{S_1}\alpha = {S_2}\beta, \eqno(2.5.14)]where S1 is the effective size of the X-ray source and α is the capture angle determined by the effective size of the X-ray optics and the distance between the source and optics. S2 is the size of the image focus. β is the convergence angle of the X-ray beam from the optics, which is also determined by the effective size of the X-ray optics and the distance between the optics and the image focus. The β angle is also called the crossfire of the X-ray beam. S2 and β are typically determined by experimental requirements such as beam size and divergence. Therefore, the product S1α is also determined by experimental conditions. In another expression of Liouville's theorem, the space volume containing the X-ray photons cannot be reduced with time along the trajectories of the system. Therefore, the brilliance of an X-ray source cannot be increased by optics, but may be reduced because of the loss of X-ray photons passing through the optics. In practice, no optics can have 100% reflectivity or transmission. Considering this, Liouville's theorem given in equation (2.5.14)[link] should be expressed as[{S_1}\alpha \le {S_2}\beta. \eqno(2.5.15)]This states that the product of the divergence and image size can be equal to or greater than the product of the capture angle and source size. If the X-ray source is a point with zero area, the focus image from focusing optics or the cross section of a parallel beam can be any chosen size. For focusing optics, the source size must be considerably smaller than the output beam size in order to achieve a gain in flux. In this case, the flux gain is from the increased capture angle. For parallel optics, the divergence angle β is infinitely small by definition, so it is necessary to use an X-ray source as small as possible to achieve a parallel beam. Focusing optics have an advantage over parallel optics in terms of beam flux. Using an X-ray beam with a divergence much smaller than the mosaicity of the specimen crystal does not improve the resolution, but does sacrifice diffraction intensity. For many X-ray diffraction applications with polycrystalline materials, a large crossfire is acceptable as long as the diffraction peaks concerned can be resolved. The improved peak profile and counting statistics can most often compensate for the peak broadening due to large crossfire.

2.5.3.1.3. X-ray source

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A variety of X-ray sources, from sealed X-ray tubes and rotating-anode generators to synchrotron radiation, can be used for 2D powder diffraction. The history and principles of X-ray generation can be found in many references (Klug & Alexander, 1974[link]; Cullity, 1978[link]). The X-ray beam intensity depends on the X-ray optics, the focal-spot brightness and the focal-spot profile. The focal-spot brightness is determined by the maximum target loading per unit area of the focal spot, also referred to as the specific loading. A microfocus sealed tube (Bloomer & Arndt, 1999[link]; Wiesmann et al., 2007[link]), which has a very small focal spot size (10–50 µm), can deliver a brilliance as much as one to two orders of magnitude higher than a conventional fine-focus sealed tube. The tube, which is also called a `microsource', is typically air cooled because the X-ray generator power is less than 50 W. The X-ray optics for a microsource, either a multilayer mirror or a polycapillary, are typically mounted very close to the focal spot so as to maximize the gain on the capture angle. A microsource is highly suitable for 2D-XRD because of its spot focus and high brilliance.

If the X-rays used for diffraction have a wavelength slightly shorter than the K absorption edge of the sample material, a significant amount of fluorescent radiation is produced, which spreads over the diffraction pattern as a high background. In a conventional diffractometer with a point detector, the fluorescent background can be mostly removed by either a receiving monochromator mounted in front of the detector or by using a point detector with sufficient energy resolution. However, it is impossible to add a monochromator in front of a 2D detector and most area detectors have insufficient energy resolution. In order to avoid intense fluorescence, the wavelength of the X-ray-tube Kα line should either be longer than the K absorption edge of the sample or far away from the K absorption edge. For example, Cu Kα should not be used for samples containing significant amounts of the elements iron or cobalt. Since the Kα line of an element cannot excite fluorescence of the same element, it is safe to use an anode of the same metallic element as the sample if the X-ray tube is available, for instance Co Kα for Co samples. In general, intense fluorescence is produced when the atomic number of the anode material is 2, 3, or 4 larger than that of an element in the sample. When the sample contains Co, Fe or Mn (or Ni or Cu), the use of Cu Kα radiation should be avoided; similarly, one should avoid using Co Kα radiation if the sample contains Mn, Cr or V, and avoid using Cr Kα radiation if the sample contains Ti, Sc or Ca. The effect is reduced when the atomic-number difference increases.

2.5.3.1.4. X-ray optics

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The function of the X-ray optics is to condition the primary X-ray beam into the required wavelength, beam focus size, beam profile and divergence. Since the secondary beam path in a 2D-XRD system is an open space, almost all X-ray optics components are on the primary side. The X-ray optics components commonly used for 2D-XRD systems include a β-filter, a crystal monochromator, a pinhole collimator, cross-coupled multilayer mirrors, a Montel mirror, a polycapillary and a monocapillary. Detailed descriptions of these optic devices can be found in Chapter 2.1[link] . In principle, the cross-sectional shape of the X-ray beam used in a 2D diffraction system should be small and round. In data-analysis algorithms, the beam size is typically considered to be a point. In practice, the beam cross section can be either round, square or another shape with a limited size. Such an X-ray beam is typically collimated or conditioned by the X-ray optics in two perpendicular directions, so that the X-ray optics used for the point beam are often called `two-dimensional X-ray optics'.

A pinhole collimator is normally used to control the beam size and divergence in addition to other optic devices. The choice of beam size is often a trade-off between intensity and the ability to illuminate small regions or resolve closely spaced sample features. Smaller beam sizes, such as 50 µm and 100 µm, are preferred for microdiffraction and large beam sizes, such as 0.5 mm or 1 mm, are typically used for quantitative analysis, or texture or crystallinity measurements. In the case of quantitative analysis and texture measurements, using too small a collimator can actually be a detriment, causing poor grain-sampling statistics. The smaller the collimator, the longer the data-collection time. The beam divergence is typically determined by both the collimator and the coupling optic device. Lower divergence is typically associated with a long beam path. At the same time, the beam flux is inversely proportional to the square of the distance between the source and the sample. There are two main factors determining the length of the primary beam path: the first is the required distance for collimating the beam into the required divergence, the second is the space required for the primary X-ray optics, the sample stage and the detector. On the condition that the above two factors are satisfied, the primary X-ray beam path should be kept as short as possible.

2.5.3.2. 2D detector

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Two-dimensional (2D) detectors, also referred to as area detectors, are the core of 2D-XRD. The advances in area-detector technologies have inspired applications both in X-ray imaging and X-ray diffraction. A 2D detector contains a two-dimensional array of detection elements which typically have identical shape, size and characteristics. A 2D detector can simultaneously measure both dimensions of the two-dimensional distribution of the diffracted X-rays. Therefore, a 2D detector may also be referred to as an X-ray camera or imager. There are many technologies for area detectors (Arndt, 1986[link]; Krause & Phillips, 1992[link]; Eatough et al., 1997[link]; Giomatartis, 1998[link]; Westbrook, 1999[link]; Durst et al., 2002[link]; Blanton, 2003[link]; Khazins et al., 2004[link]). X-ray photographic plates and films were the first generation of two-dimensional X-ray detectors. Now, multiwire proportional counters (MWPCs), image plates (IPs), charge-coupled devices (CCDs) and microgap detectors are the most commonly used large area detectors. Recent developments in area detectors include X-ray pixel array detectors (PADs), silicon drift diodes (SDDs) and complementary metal-oxide semiconductor (CMOS) detectors (Ercan et al., 2006[link]; Lutz, 2006[link]; Yagi & Inoue, 2007[link]; He et al., 2011[link]). Each detector type has its advantages over the other types. In order to make the right choice of area detector for a 2D-XRD system and applications, it is necessary to characterize area detectors with consistent and comparable parameters. Chapter 2.1[link] has more comprehensive coverage on X-ray detectors, including area detectors. This section will cover the characteristics specifically relevant to area detectors.

2.5.3.2.1. Active area and pixel size

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A 2D detector has a limited detection surface and the detection surface can be spherical, cylindrical or flat. The detection-surface shape is also determined by the detector technology. For example, a CCD detector is made from a large semiconductor wafer, so that only a flat CCD is available, while an image plate is flexible so that it is easily bent to a cylindrical shape. The area of the detection surface, also referred to as the active area, is one of the most important parameters of a 2D detector. The larger the active area of a detector, the larger the solid angle that can be covered at the same sample-to-detector distance. This is especially important when the instrumentation or sample size forbid a short sample-to-detector distance. The active area is also limited by the detector technology. For instance, the active area of a CCD detector is limited by the semiconductor wafer size and fabrication facility. A large active area can be achieved by using a large demagnification optical lens or fibre-optical lens. Stacking several CCD chips side-by-side to build a so-called mosaic CCD detector is another way to achieve a large active area.

In addition to the active area, the overall weight and dimensions are also very important factors in the performance of a 2D detector. The weight of the detector has to be supported by the goniometer, so a heavy detector means high demands on the size and power of the goniometer. In a vertical configuration, a heavy detector also requires a heavy counterweight to balance the driving gear. The overall dimensions of a 2D detector include the height, width and depth. These dimensions determine the manoeuvrability of the detector within a diffractometer, especially when a diffractometer is loaded with many accessories, such as a video microscope and sample-loading mechanism. Another important parameter of a 2D detector that tends to be ignored by most users is the blank margin surrounding the active area of the detector. Fig. 2.5.10[link] shows the relationship between the maximum measurable 2θ angle and the detector blank margin. For high 2θ angle measurements, the detector swing angle is set so that the incident X-ray optics are set as closely as possible to the detector. The unmeasurable blank angle is the sum of the detector margin m and the dimension from the incident X-ray beam to the outer surface of the optic device h. The maximum measurable angle is given by[2{\theta _{\max }} = \pi - {{m + h} \over D}.\eqno(2.5.16)]It can be seen that either reducing the detector blank margin or optics blank margin can increase the maximum measurable angle.

[Figure 2.5.10]

Figure 2.5.10 | top | pdf |

Detector dimensions and maximum measurable 2θ.

The solid angle covered by a pixel in a flat detector is dependent on the sample-to-detector distance and the location of the pixel in the detector. Fig. 2.5.11[link] illustrates the relationship between the solid angle covered by a pixel and its location in a flat area detector. The symbol S may represent a sample or a calibration source at the instrument centre. The distance between the sample S and the detector is D. The distance between any arbitrary pixel P(x, y) and the detector centre pixel P(0, 0) is r. The pixel size is Δx and Δy (assuming Δx = Δy). The distance between the sample S and the pixel is R. The angular ranges covered by this pixel are Δα and Δβ in the x and y directions, respectively. The solid angle covered by this pixel, ΔΩ, is then given as[\Delta \Omega = \Delta \alpha \Delta \beta = {D \over {{R^3}}}\Delta y \Delta x = {D \over {{R^3}}}\Delta A,\eqno(2.5.17)]where ΔA = ΔxΔy is the area of the pixel and R is given by[R = ({D^2} + {x^2} + {y^2})^{1/2} = ({D^2} + {r^2})^{1/2}. \eqno(2.5.18)]When a homogeneous calibration source is used, the flux to a pixel at P(x, y) is given as[F(x,y) = \Delta \Omega B = {{\Delta ADB} \over {{R^3}}} = {{\Delta ADB} \over {{{({D^2} + {x^2} + {y^2})}^{3/2}}}},\eqno(2.5.19)]where F(x, y) is the flux (in photons s−1) intercepted by the pixel and B is the brightness of the source (in photons s−1 mrad−2) or scattering from the sample. The ratio of the flux in pixel P(x, y) to that in the centre pixel P(0, 0) is then given as[{{F(x,y)} \over {F(0,0)}} = {{{D^3}} \over {{R^3}}} = {{{D^3}} \over {{{({D^2} + {x^2} + {y^2})}^{3/2}}}} = {\cos ^3}\phi, \eqno(2.5.20)]where [\phi] is the angle between the X-rays to the pixel P(x, y) and the line from S to the detector in perpendicular direction. It can be seen that the greater the sample-to-detector distance, the smaller the difference between the centre pixel and the edge pixel in terms of the flux from the homogeneous source. This is the main reason why a data frame collected at a short sample-to-detector distance has a higher contrast between the edge and centre than one collected at a long sample-to-detector distance.

[Figure 2.5.11]

Figure 2.5.11 | top | pdf |

Solid angle covered by each pixel and its location on the detector.

2.5.3.2.2. Spatial resolution of area detectors

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In a 2D diffraction frame, each pixel contains the X-ray intensity collected by the detector corresponding to the pixel element. The pixel size of a 2D detector can be determined by or related to the actual feature sizes of the detector structure, or artificially determined by the readout electronics or data-acquisition software. Many detector techniques allow multiple settings for variable pixel size, for instance a frame of 2048 × 2048 pixels or 512 × 512 pixels. Then the pixel size in 512 mode is 16 (4 × 4) times that of a pixel in 2048 mode. The pixel size of a 2D detector determines the space between two adjacent pixels and also the minimum angular steps in the diffraction data, therefore the pixel size is also referred to as pixel resolution.

The pixel size does not necessarily represent the true spatial resolution or the angular resolution of the detector. The resolving power of a 2D detector is also limited by its point-spread function (PSF) (Bourgeois et al., 1994[link]). The PSF is the two-dimensional response of a 2D detector to a parallel point beam smaller than one pixel. When the sharp parallel point beam strikes the detector, not only does the pixel directly hit by the beam record counts, but the surrounding pixels may also record some counts. The phenomenon is observed as if the point beam has spread over a certain region adjacent to the pixel. In other words, the PSF gives a mapping of the probability density that an X-ray photon is recorded by a pixel in the vicinity of the point where the X-ray beam hits the detector. Therefore, the PSF is also referred to as the spatial redistribution function. Fig. 2.5.12[link](a) shows the PSF produced from a parallel point beam. A plane at half the maximum intensity defines a cross-sectional region within the PSF. The FWHM can be measured at any direction crossing the centroid of the cross section. Generally, the PSF is isotropic, so the FWHMs measured in any direction should be the same.

[Figure 2.5.12]

Figure 2.5.12 | top | pdf |

(a) Point-spread function (PSF) from a parallel point beam; (b) line-spread function (LSF) from a sharp line beam.

Measuring the PSF directly by using a small parallel point beam is difficult because the small PSF spot covers a few pixels and it is hard to establish the distribution profile. Instead, the line-spread function (LSF) can be measured with a sharp line beam from a narrow slit (Ponchut, 2006[link]). Fig. 2.5.12[link](b) is the intensity profile of the image from a sharp line beam. The LSF can be obtained by integrating the image from the line beam along the direction of the line. The FWHM of the integrated profile can be used to describe the LSF. Theoretically, LSF and PSF profiles are not equivalent, but in practice they are not distinguished and may be referenced by the detector specification interchangeably. For accurate LSF measurement, the line beam is intentionally positioned with a tilt angle from the orthogonal direction of the pixel array so that the LSF can have smaller steps in the integrated profile (Fujita et al., 1992[link]).

The RMS (root-mean-square) of the distribution of counts is another parameter often used to describe the PSF. The normal distribution, also called the Gaussian distribution, is the most common shape of a PSF. The RMS of a Gaussian distribution is its standard deviation, σ. Therefore, the FWHM and RMS have the following relation, assuming that the PSF has a Gaussian distribution:[{\rm{FWHM}} = 2[- 2\ln (1/2)]^{1/2} {\rm{RMS}} = 2.3548\times {\rm{RMS}}.\eqno(2.5.21)]The values of the FWHM and RMS are significantly different, so it is important to be precise about which parameter is used when the value is given for a PSF.

For most area detectors, the pixel size is smaller than the FWHM of the PSF. The pixel size should be small enough that at least a 50% drop in counts from the centre of the PSF can be observed by the pixel adjacent to the centre pixel. In practice, an FWHM of 3 to 6 times the pixel size is a reasonable choice if use of a smaller pixel does not have other detrimental effects. A further reduction in pixel size does not necessarily improve the resolution. Some 2D detectors, such as pixel-array detectors, can achieve a single-pixel PSF. In this case, the spatial resolution is determined by the pixel size.

2.5.3.2.3. Detective quantum efficiency and energy range

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The detective quantum efficiency (DQE), also referred to as the detector quantum efficiency or quantum counting efficiency, is measured by the percentage of incident photons that are converted by the detector into electrons that constitute a measurable signal. For an ideal detector, in which every X-ray photon is converted to a detectable signal without additional noise added, the DQE is 100%. The DQE of a real detector is less than 100% because not every incident X-ray photon is detected, and because there is always some detector noise. The DQE is a parameter defined as the square of the ratio of the output and input signal-to-noise ratios (SNRs) (Stanton et al., 1992[link]):[{\rm DQE} = {\left({{{{{(S/N)}_{\rm out}}} \over {{{(S/N)}_{\rm in}}}}} \right)^2}.\eqno(2.5.22)]

The DQE of a detector is affected by many variables, for example the X-ray photon energy and the counting rate. The dependence of the DQE on the X-ray photon energy defines the energy range of a detector. The DQE drops significantly if a detector is used out of its energy range. For instance, the energy range of MWPC and microgap detectors is about 3 to 15 keV. The DQE with Cu Kα radiation (8.06 keV) is about 80%, but drops gradually when approaching the lower or higher energy limits. The energy range of imaging plates is much wider (4–48 keV). The energy range of a CCD, depending on the phosphor, covers from 5 keV up to the hard X-ray region.

2.5.3.2.4. Detection limit and dynamic range

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The detection limit is the lowest number of counts that can be distinguished from the absence of true counts within a specified confidence level. The detection limit is estimated from the mean of the noise, the standard deviation of the noise and some confidence factor. In order to have the incoming X-ray photons counted with a reasonable statistical certainty, the counts produced by the X-ray photons should be above the detector background-noise counts.

The dynamic range is defined as the range extending from the detection limit to the maximum count measured in the same length of counting time. The linear dynamic range is the dynamic range within which the maximum counts are collected within the specified linearity. For X-ray detectors, the dynamic range most often refers to linear dynamic range, since only a diffraction pattern collected within the linear dynamic range can be correctly interpreted and analysed. When the detection limit in count rate approaches the noise rate at extended counting time, the dynamic range can be approximated by the ratio of the maximum count rate to the noise rate.

Dynamic range is very often confused with the maximum count rate, but must be distinguished. With a low noise rate, a detector can achieve a dynamic range much higher than its count rate. For example, if a detector has a maximum linear count rate of 105 s−1 with a noise rate of 10−3 s−1, the dynamic range can approach 108 for an extended measurement time. The dynamic range for a 2D detector has the same definition as for a point detector, except that with a 2D detector the whole dynamic range extending from the detection limit to the maximum count can be observed from different pixels simultaneously. In order to record the entire two-dimensional diffraction pattern, it is necessary for the dynamic range of the detector to be at least the dynamic range of the diffraction pattern, which is typically in the range 102 to 106 for most applications. If the range of reflection intensities exceeds the dynamic range of the detector, then the detector will either saturate or have low-intensity patterns truncated. Therefore, it is desirable that the detector has as large a dynamic range as possible.

2.5.3.2.5. Types of 2D detectors

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2D detectors can be classified into two broad categories: photon-counting detectors and integrating detectors (Lewis, 1994[link]). Photon-counting area detectors can detect a single X-ray photon entering the active area. In a photon-counting detector, each X-ray photon is absorbed and converted to an electrical pulse. The number of pulses counted per unit time is proportional to the incident X-ray flux. Photon-counting detectors typically have high counting efficiency, approaching 100% at low count rate. The most commonly used photon-counting 2D detectors include MWPCs, Si-pixel arrays and microgap detectors. Integrating area detectors, also referred to as analogue X-ray imagers, record the X-ray intensity by measuring the analogue electrical signals converted from the incoming X-ray flux. The signal size of each pixel is proportional to the fluence of incident X-rays. The most commonly used integrating 2D detectors include image plates (IPs) and charge-coupled devices (CCDs).

The selection of an appropriate 2D detector depends on the X-ray diffraction application, the sample condition and the X-ray beam intensity. In addition to geometry features, such as the active area and pixel format, the most important performance characteristics of a detector are its sensitivity, dynamic range, spatial resolution and background noise. The detector type, either photon-counting or integrating, also leads to important differences in performance. Photon-counting 2D detectors typically have high counting efficiency at low count rate, while integrating 2D detectors are not so efficient at low count rate because of the relatively high noise background. An MWPC has a high DQE of about 0.8 when exposed to incoming local fluence from single photons up to about 103 photons s−1 mm−2. The diffracted X-ray intensities from a polycrystalline or powder sample with a typical laboratory X-ray source fall into this fluence range. This is especially true with microdiffraction, where high sensitivity and low noise are crucial to reveal the weak diffraction pattern. Owing to the counting losses at a high count rate, the DQE of an MWPC decreases with increasing count rate and quickly saturates above 103 photons s−1 mm−2. Therefore, an MWPC is not suitable for collecting strong diffraction patterns or for use with high intensity sources, such as synchrotron X-ray sources. An IP can be used in a large fluence range from 10 photons s−1 mm−2 and up with a DQE of 0.2 or lower. An IP is suitable for strong diffraction from single crystals with high-intensity X-ray sources, such as a rotating-anode generator or synchrotron X-ray source. With weak diffraction signals, the image plate cannot resolve the diffraction data near the noise floor. A CCD detector can also be used over a large X-ray fluence range from 10 photons s−1 mm−2 to very high fluence with a much higher DQE of 0.7 or higher. It is suitable for collecting diffraction of medium to strong intensity from single-crystal or polycrystalline samples. Owing to the relatively high sensitivity and high local count rate, CCDs can be used in systems with either sealed-tube X-ray sources, rotating-anode generators or synchrotron X-ray sources. With a low DQE at low fluence and the presence of dark-current noise, a CCD is not a good choice for applications with weak diffraction signals. A microgap detector has the best combination of high DQE, low noise and high count rate. It has a DQE of about 0.8 at an X-ray fluence from single photons up to about 105 photons s−1 mm2. It is suitable for microdiffraction when high sensitivity and low noise are crucial to reveal weak diffraction patterns. It can also handle high X-ray fluence from strong diffraction patterns or be used with high-intensity sources, such as rotating-anode generators or synchrotron X-ray sources.

2.5.3.3. Data corrections and integration

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2D diffraction patterns contain abundant information. In order to interpret and analyse 2D patterns accurately it is necessary to apply some data-treatment processes (Sulyanov et al., 1994[link]; Scheidegger et al., 2000[link]; Cervellino et al., 2006[link]; Boesecke, 2007[link]; Rowe, 2009[link]). Most data-treatment processes can be categorized as having one of the following four purposes: to eliminate or reduce errors caused by detector defects; to remove undesirable effects of instrument and sample geometry; to transfer a 2D frame into a format such that the data can be presented or further analysed by conventional means and software; and cosmetic treatment, such as smoothing a frame for reports and publications.

2.5.3.3.1. Nonuniform response correction

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A 2D detector can be considered as an array of point detectors. Each pixel may have its own response, and thus a 2D detector may exhibit some nonuniformity in intensity measurement when exposed to an isotropic source. The nonuniform response can be caused by manufacturing defects, inadequate design or limitations of the detector technology. For instance, a nonuniform phosphor screen or coupling fibre optic for a CCD detector may cause nonuniformity in quantum efficiency (Tate et al., 1995[link]). A gas-filled detector may have a different intensity response between the detector edge and centre due to the variation in the electric field from the centre to the edge. A thorough correction to the nonuniformity of the intensity response can be performed if the detector counting curves of all pixels are given. In practice, this is extremely difficult or impossible, because the behaviour of a pixel may be affected by the condition of the adjacent pixels and the whole detector. The practical way to correct the non­uniformity of the intensity response is to collect an X-ray image from an isotropic point source at the instrument centre and use the image data frame to generate a correction table for the future diffraction frames. The frame collected with the isotropic source is commonly referred to as a `flood-field' frame or a flat-field image, and the correction is also called a flood-field correction or flat-field correction (Stanton et al., 1992[link]). Another type of correction for a nonuniform response is background correction. Background correction is done by subtracting a background frame from the data frame. The background frame is collected without X-ray exposure. Integrating detectors, such as image plates or CCDs, have a strong background which must be considered in nonuniform response correction. Photon-counting detectors, such as MWPC and microgap detectors, have negligible background, so background correction is not necessary.

The X-ray source for calibration for flood-field correction should be a uniform, spherically radiating point source. Identical brightness should be observed at any pixel on the detector. The radiation strength of the source should match the intensity of the diffraction data to be collected. The photon energy of the source should be the same as or close to the X-ray beam used for diffraction-data collection so that the detector behaves the same way during calibration and data collection.

There are many choices of calibration sources, including X-ray tubes, radioactive sources, diffuse scattering or X-ray fluorescence. The radioactive source Fe-55 (55Fe) is the most commonly used calibration source for a diffraction system because of its major photon energy level of 5.9 keV. X-ray fluorescence is an alternative to a radioactive source. Fluorescence emission is generated by placing a fluorescent material into the X-ray beam. Fluorescence radiation is an isotopic point source if the irradiated area is a small point-like area. For example, Cu Kα can produce intense fluorescence from materials containing significant amounts of iron or cobalt and Mo Kα can produce intense fluorescence from materials containing yttrium. In order to avoid a high localized intensity contribution from X-ray diffraction, the fluorescent material should be amorphous, such as a glassy iron foil. An alternative to a glassy alloy foil is amorphous lithium borate glass doped with the selected fluorescent element up to a 10% concentration (Moy et al., 1996[link]).

There are many algorithms available for flood-field correction depending on the nature of the 2D detector. The correction is based on the flood-field frame collected from the calibration source. The simplest flood-field correction is to normalize the counts of all pixels to the same level assuming that all pixels have the same response curve. The corrected frame from an isotropic source is not flat, but maintains the cos3 [\phi] falloff effect, which will be considered in the frame integration. For gas-filled detectors, such as MWPC and microgap detectors, the pixel intensity response is not independent, but is affected by X-ray exposure to surrounding pixels and the whole detector. Flood-field correction is carried out by applying a normalization factor to each pixel in which a `rubber-sheet' kind of stretching and shrinking in regions along the x and y detector axes slightly alters the size of each pixel (He, 2009[link]). The total number of counts remains the same after the correction but is redistributed throughout the pixels so that the image from an isotropic source is uniformly distributed across the detector. The flood-field calibration must be done with the same sample-to-detector distance as for the diffraction-data collection.

2.5.3.3.2. Spatial correction

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In an ideal flat 2D detector, not only does each pixel have the same intensity response, but also an accurate position. The pixels are aligned in the x and y directions with equal spacing. In most cases we assume that the detective area is completely filled by pixels, so the distance between two neighbouring pixels is equivalent to the pixel size. The deviation from this perfect pixel array is called spatial distortion. The extent of spatial distortion is dependent on the nature and limitation of the detector technology. A CCD detector with 1:1 demagnification may have a negligible spatial distortion, but the barrel distortion in the coupling fibre-optic taper can introduce substantial spatial distortion. An image-plate system may have spatial distortion caused by imperfections in the scanning system (Campbell et al., 1995[link]). MWPC detectors typically exhibit more severe spatial distortion due to the window curvature and imperfections in the wire anode (Derewenda & Helliwell, 1989[link]).

The spatial distortion is measured from X-ray images collected with a uniformly radiating point source positioned at the instrument centre and a fiducial plate fastened to the front surface of the detector. The source for spatial correction should have a very accurate position, point-like shape and small size. The fiducial plate is a metal plate with accurately distributed pinholes in the x and y directions. The X-ray image collected with this setup contains sharp peaks corresponding to the pinhole pattern of the fiducial plate. Since accurate positions of the peaks are given by the fiducial plate, the spatially corrected image is a projection of the collected image to this plane. Therefore, the detector plane is defined as the contacting plane between the fiducial plate and detector front face.

Spatial correction restores the spatially distorted diffraction frame into a frame with correct pixel positions. Many algorithms have been suggested for spatial correction (Sulyanov et al., 1994[link]; Tate et al., 1995[link]; Stanton et al., 1992[link]; Campbell et al., 1995[link]). In the spatially corrected frame each pixel is generated by computing the pixel count from the corresponding pixels based on a spatial-correction look-up table. In a typical spatial-correction process, an image containing the spots from the calibration source passing through the fiducial plate is collected. The distortion of the image is revealed by the fiducial spots. Based on the known positions of the corresponding pinholes in the fiducial plate, the distortion of each fiducial spot can be determined. The spatial correction for all pixels can be calculated and stored as a look-up table. Assuming that the detector behaves the same way in the real diffraction-data collection, the look-up table generated from the fiducial image can then be applied to the real diffraction frames. The spatial calibration must be done at the same sample-to-detector distance as the diffraction-data collection.

2.5.3.3.3. Frame integration

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2D frame integration is a data-reduction process which converts a two-dimensional frame into a one-dimensional intensity profile. Two forms of integration are generally of interest in the analysis of a 2D diffraction frame from polycrystalline materials: γ integration and 2θ integration. γ integration sums the counts in 2θ steps (Δ2θ) along constant 2θ conic lines and between two constant γ values. γ integration produces a data set with intensity as a function of 2θ. 2θ integration sums the counts in γ steps (Δγ) along constant γ lines and between two constant 2θ conic lines. 2θ integration produces a data set with intensity as a function of γ. γ integration may also be carried out with the integration range in the vertical direction as a constant number of pixels. This type of γ integration may also be referred to as slice integration. A diffraction profile analogous to the conventional diffraction result can be obtained by either γ integration or slice integration over a selected 2θ range. Phase ID can then be done with conventional search/match methods. 2θ integration is of interest for evaluating the intensity variation along γ angles, such as for texture analysis, and is discussed in more depth in Chapter 5.3[link] .

The γ integration can be expressed as[I(2\theta) = \textstyle\int\limits_{{\gamma _1}}^{{\gamma _2}} {J(2\theta, \gamma)\,{\rm d}\gamma }, \quad 2{\theta _1} \le 2\theta \le 2{\theta _2},\eqno(2.5.23)]where J(2θ, γ) represents the two-dimensional intensity distribution in the 2D frame and I(2θ) is the integration result as a function of intensity versus 2θ. γ1 and γ2 are the lower limit and upper limit of integration, respectively, which are constants for γ integration. Fig. 2.5.13[link] shows a 2D diffraction frame collected from corundum (α-Al2O3) powder. The 2θ range is from 20 to 60° and the 2θ integration step size is 0.05°. The γ-integration range is from 60 to 120°. In order to reduce or eliminate the dependence of the integrated intensity on the integration interval, the integrated value at each 2θ step is normalized by the number of pixels, the arc length or the solid angle. γ integration with normalization by the solid angle can be expressed as[I(2\theta) = {{\textstyle\int_{{\gamma _1}}^{{\gamma _2}} {J(2\theta, \gamma)(\Delta 2\theta)\,{\rm d}\gamma } } \over {\textstyle\int_{{\gamma _1}}^{{\gamma _2}} {(\Delta 2\theta)\,{\rm d}\gamma } }},\quad 2{\theta _1} \le 2\theta \le 2{\theta _2}.\eqno(2.5.24)]Since the Δ2θ step is a constant, the above equation becomes[I(2\theta) = {{\textstyle\int_{{\gamma _1}}^{{\gamma _2}} {J(2\theta, \gamma)\,{\rm d}\gamma } } \over {{\gamma _2} - {\gamma _1}}},\quad 2{\theta _1} \le 2\theta \le 2{\theta _2}.\eqno(2.5.25)]

[Figure 2.5.13]

Figure 2.5.13 | top | pdf |

A 2D frame showing γ integration.

There are many integration software packages and algorithms available for reducing 2D frames into 1D diffraction patterns for polycrystalline materials (Cervellino et al., 2006[link]; Rodriguez-Navarro, 2006[link]; Boesecke, 2007[link]). With the availability of tremendous computer power today, a relatively new method is the bin method, which treats pixels as having a continuous distribution in the detector. It demands more computer power than older methods, but delivers much more accurate and smoother results even with Δ2θ integration steps significantly smaller than the pixel size. Depending on the relative size of Δ2θ to the pixel size, each contributing pixel is divided into several 2θ `bins'. The intensity counts of all pixels within the Δ2θ step are summarized. All the normalization methods in the above integration, either by pixel, arc or solid angle, result in an intensity level of one pixel or unit solid angle. Since a pixel is much smaller than the active area of a typical point detector, the normalized integration tends to result in a diffraction pattern with fictitiously low intensity counts, even though the true counts in the corresponding Δ2θ range are significantly higher. In order to avoid this misleading outcome, it is reasonable to introduce a scaling factor. However, there is no accurate formula for making the integrated profile from a 2D frame comparable to that from a conventional point-detector scan. The best practice is to be aware of the differences and to try not to make direct comparisons purely based on misleading intensity levels. Generally speaking, for the same exposure time, the total counting statistics from a 2D detector are significantly better than from a 0D or 1D detector.

2.5.3.3.4. Lorentz, polarization and absorption corrections

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Lorentz and polarization corrections may be applied to the diffraction frame to remove their effect on the relative intensities of Bragg peaks and background. The 2θ angular dependence of the relative intensity is commonly given as a Lorentz–polarization factor, which is a combination of Lorentz and polarization factors. In 2D diffraction, the polarization factor is a function of both 2θ and γ, therefore it should be treated in the 2D frames, while the Lorentz factor is a function of 2θ only. The Lorentz correction can be done either on the 2D frames or on the integrated profile. In order to obtain relative intensities equivalent to a conventional diffractometer with a point detector, reverse Lorentz and polarization corrections may be applied to the frame or integrated profile.

The Lorentz factor is the same as for a conventional diffractometer. For a sample with a completely random orientation distribution of crystallites, the Lorentz factor is given as[L = {{\cos \theta } \over {\sin ^22\theta }} = {1 \over {4\sin ^2\theta \cos \theta }}.\eqno(2.5.26)]

The Lorentz factor may be given by a different equation for a different diffraction geometry (Klug & Alexander, 1974[link]). The forward and reverse Lorentz corrections are exactly reciprocal and effectively cancel each other. Therefore, it is not necessary to perform the Lorentz correction to the frame before integration if relative intensities equivalent to a conventional Bragg–Brentano diffractometer are expected. The Lorentz correction can be done on the integrated diffraction profiles in the same way as on the diffraction profiles collected with conventional diffractometers.

When a non-polarized X-ray beam is scattered by matter, the scattered X-rays are polarized. The intensity of the diffracted beam is affected by the polarization; this effect is expressed by the polarization factor. In two-dimensional X-ray diffraction the diffraction vectors of the monochromator diffraction and sample crystal diffraction are not necessarily in the same plane or perpendicular planes. Therefore, the overall polarization factor is a function of both 2θ and γ. Fig. 2.5.14[link] illustrates the geometric relationship between the monochromator and detector in the laboratory coordinates, XL, YL, ZL. The monochromator is located at the coordinates [X_L, Y^\prime_L, Z^\prime_L], which is a translation of the laboratory coordinates along the XL axis in the negative direction. The monochromator crystal is rotated about the [Z^\prime_L] axis and 2θM is the Bragg angle of the monochromator crystal. The diffracted beam from the monochromator propagates along the XL direction. This is the incident beam to the sample located at the instrument centre O. The 2D detector location is given by the sample-to-detector distance D and swing angle α. The pixel P(x, y) represents an arbitrary pixel on the detector. 2θ and γ are the corresponding diffraction-space parameters for the pixel. Since a monochromator or other beam-conditioning optics can only be used on the incident beam, the polarization factor for 2D-XRD can then be given as a function of both θ and γ:[\eqalignno{&P(\theta, \gamma) =\cr&\quad {{(1 + \cos ^22\theta _M\cos ^22\theta)\sin ^2\gamma + (\cos ^22\theta _M + \cos ^22\theta)\cos ^2\gamma } \over {1 + \cos ^22\theta _M}}.&\cr&&(2.5.27)}]

[Figure 2.5.14]

Figure 2.5.14 | top | pdf |

Geometric relationship between the monochromator and detector in the laboratory coordinates.

If the crystal monochromator rotates about the [Y^\prime_L] axis, i.e. the incident plane is perpendicular to the diffractometer plane, the polarization factor for two-dimensional X-ray diffraction can be given as[\eqalignno{&P(\theta, \gamma) =\cr&\quad {{(1 + \cos ^22\theta _M\cos ^22\theta)\cos ^2\gamma + (\cos ^22\theta _M + \cos ^22\theta)\sin ^2\gamma } \over {1 + \cos ^22\theta _M}}.\cr&&(2.5.28)}]

In the above equations, the term cos2 2θM can be replaced by [\left| \cos ^n2\theta _M \right|] for different monochromator crystals. For a mosaic crystal, such as a graphite crystal, n = 2. For most real monochromator crystals, the exponent n takes a value between 1 and 2. For near perfect monochromator crystals, n approaches 1 (Kerr & Ashmore, 1974[link]). All the above equations for polarization factors may apply to multilayer optics. However, since multilayer optics have very low Bragg angles, [\left| \cos ^n2\theta _M \right|] approximates to unity. The γ dependence of the polarization factor diminishes in this case. The polarization factor approaches[P(\theta, \gamma) \simeq {{1 + \cos ^22\theta } \over 2}.\eqno(2.5.29)]

2.5.3.3.5. Air scatter

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X-rays are scattered by air molecules in the beam path between the X-ray source and detector. Air scatter results in two effects: one is the attenuation of the X-ray intensity, the other is added background in the diffraction pattern. Air scatter within the enclosed primary beam path – for instance, in the mirror, monochromator housing or collimator – results in attenuation of only the incident beam. The enclosed beam path can be purged by helium gas or kept in vacuum to reduce the attenuation so that no correction is necessary for this part of the air scatter. The open beam between the tip of the collimator and the sample generates an air-scatter background pattern, which is the major part of the air scatter. In the secondary beam path, the air scatter from the diffracted beam may generate background too, but the main effect of the air scatter is inhomogeneous attenuation of the diffraction pattern due to the different beam path lengths between the centre and the edge of the detector.

The background generated by air scattering from the open incident-beam path has a strong 2θ dependence. The specific scattering curve depends on the length of the open primary beam path, the beam size and the wavelength of the incident beam. There are two approaches to correct air scatter. One is to collect an air-scatter background frame under the same conditions as the diffraction frame except without a sample. The background frame is then subtracted from the diffraction frame. Another approach is to remove the background from the integrated profile, since the background is 2θ dependent.

The attenuation of the diffracted beam by air absorption depends on the distance between the sample and pixel. For a flat detector, air absorption can be corrected by[{p_c}(x,y) = {p_o}(x,y) \exp \left[{{\mu _{\rm air}}({{D^2} + {x^2} + {y^2}})^{1/2} } \right],\eqno(2.5.30)]where po(x, y) is the original pixel intensity of the pixel P(x, y) and pc(x, y) is the corrected intensity. The detector centre is given by (0, 0). μair is the linear absorption coefficient of air. The value of μair is determined by the radiation wavelength. By approximation, for air with 80% N2 and 20% O2 at sea level and at 293 K, μair = 0.01 cm−1 for Cu Kα radiation. Air scatter and absorption increases with increasing wavelength. For example, μair = 0.015 cm−1 for Co Kα radiation and 0.032 cm−1 for Cr Kα radiation. The absorption coefficient for Mo Kα radiation, μair = 0.001 cm−1, is only one-tenth of that for Cu Kα radiation, so an air-absorption correction is not necessary. Alternatively, the absorption correction may be normalized to the absorption level in the beam centre as[{p_c}(x,y) = {p_o}(x,y) \exp \left \{{{\mu _{\rm air}}\left[({D^2} + {x^2} + {y^2})^{1/2} - D\right]} \right\}.\eqno(2.5.31)]In this normalized correction the attenuation by air scatter is not fully corrected for each pixel, but rather corrected to the same attenuation level as the pixel in the detector centre. This means that the effect of path-length differences between the detector centre pixel and other pixels are eliminated.

2.5.3.3.6. Sample absorption

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The absorption of X-rays by the sample reduces the diffracted intensity. Many approaches are used to calculate and correct the absorption effect for various sample shapes and geometries [International Tables for Crystallography Volume C, Chapter 6.3[link] (Maslen, 1992[link]); Ross, 1992[link]; Pitschke et al., 1996[link]; Zuev, 2006[link]]. The sample absorption can be measured by the transmission coefficient (also referred to as the absorption factor):[A = (1 / V)\textstyle\int\limits_V {{\exp({ - \mu \tau })}} \,{\rm d}V,\eqno(2.5.32)]where A is the transmission coefficient, μ is the linear absorption coefficient and τ is the total beam path in the sample, which includes the incident-beam path and diffracted-beam path. Fig. 2.5.15[link](a) shows reflection-mode diffraction with a flat-plate sample. The thickness of the plate is t. z is the distance of the element dV from the sample surface. The normal to the reflection surface is n. The incident beam is represented by the unit vector so and the diffracted beam by the unit vector s. The transmission coefficient is given as (Maslen, 1992[link] [link])[A = {{1 - \exp \left \{{ - \mu t\left[{({1 /{\cos \eta })} + ({1 / {\cos \zeta }})} \right]} \right\}} \over {\mu \left[{({{\cos \zeta } /{\cos \eta }}) + 1} \right]}},\eqno(2.5.33)]where η is the angle between the incident beam and the normal to the sample surface, and ζ is the angle between the diffracted beam and the sample normal. For two-dimensional X-ray diffraction, there is a single incident-beam direction at a time, but various diffracted-beam directions simultaneously, so[\cos \eta = \sin \omega \cos \psi \eqno(2.5.34)]and[\eqalignno{\cos \zeta &= - \cos 2\theta \sin \omega \cos \psi - \sin 2\theta \sin \gamma \cos \omega \cos \psi &\cr&\quad - \sin 2\theta \cos \gamma \sin \psi. &(2.5.35)}]

[Figure 2.5.15]

Figure 2.5.15 | top | pdf |

Absorption correction for a flat slab: (a) reflection; (b) transmission.

The transmission coefficient from equation (2.5.33)[link] contains a length unit, which creates ambiguity if such transmission coefficients are used to correct the intensity pixel-by-pixel. In order to make the relative intensity comparable to the results from Bragg–Brentano geometry, we introduce a new transmission coefficient, which is normalized by the transmission coefficient of the Bragg–Brentano geometry, [A_{\rm BB} = 1/(2\mu)]. This normalized transmission coefficient is also a numerical factor without units. The transmission coefficient with normalization will be denoted by T hereafter in this chapter. The transmission coefficient for reflection-mode diffraction with a flat sample of thickness t is then given as[{{T}} = {{A/}}{{{A}}_{{\rm{BB}}}} = {{2\cos \eta \left( {1 - \exp \left \{{ - \mu {{t}}\left[{({1/ {\cos \eta })} + ({1 / {\cos \zeta })}} \right]} \right\}} \right)} \over {\cos \eta + \cos \zeta }}.\eqno(2.5.36)]For a thick plate or material with a very high linear absorption coefficient, the transmission through the sample thickness is negligible and the above equation becomes[T = {{2\cos \eta } \over {{\cos \eta + \cos \zeta }}}.\eqno(2.5.37)]Fig. 2.5.15[link](b) shows transmission-mode diffraction with a flat-plate sample. The thickness of the plate is t. The normal to the reflection surface is represented by the unit vector n. The incident beam is represented by the unit vector so and the diffracted beam by the unit vector s. η is the angle between the incident beam and the normal of the sample surface, and ζ is the angle between the diffracted beam and the sample normal.

The transmission coefficient normalized by [A_{\rm BB} = 1/(2\mu)] is given by (Maslen, 1992[link] [link]; Ross, 1992[link])[\eqalignno{T &= {{2\sec \eta \left [{\exp \left({ - \mu t\sec \eta } \right) - \exp \left({ - \mu t\sec \zeta } \right)} \right]} \over {\sec \zeta - \sec \eta }}&\cr&\quad{\rm for}\ \sec \zeta \ne \sec \eta. &(2.5.38)}]

For two-dimensional X-ray diffraction in transmission mode[\cos \eta = \sin \omega \sin \psi \sin \varphi + \cos \omega \cos \varphi \eqno(2.5.39)]and[\eqalignno{ \cos \zeta &= (\sin \omega \sin \psi \sin \varphi + \cos \omega \cos \varphi)\cos 2\theta \cr & \quad + (\cos \omega \sin \psi \sin \varphi - \sin \omega \cos \varphi)\sin 2\theta \sin \gamma \cr&\quad - \cos \psi \sin \varphi \sin 2\theta \cos \gamma. &(2.5.40)}]

It is very common practice to set the incident angle perpendicular to the sample surface, i.e. η = 0. For most transmission-mode data collection, equation (2.5.40)[link] becomes[T = {{2\left [{\exp \left({ - \mu t} \right) - \exp \left({ - \mu t\sec \zeta } \right)} \right]} \over {\sec \zeta - 1}}.\eqno(2.5.41)]When η = ζ, both the numerator and denominator approach zero, and the transmission coefficient should be given by[T = 2\mu t\sec \zeta \exp \left({ - \mu t\sec \zeta } \right).\eqno(2.5.42)]It is common practice to load the sample perpendicular to the incident X-ray beam at the goniometer angles [\omega = \psi = \varphi = 0]. Therefore, [\cos \eta = 1] and [\cos \zeta = \cos 2\theta ], and the transmission coefficient becomes[T = {{2\cos 2\theta \left [{\exp \left({ - \mu t} \right) - \exp \left({ - {{\mu t}/{\cos 2\theta }}} \right)} \right]} \over {1 - \cos 2\theta }}.\eqno(2.5.43)]The maximum scattered intensity occurs when[t = {{\cos 2\theta \ln \cos 2\theta } \over {\mu (\cos 2\theta - 1)}}.\eqno(2.5.44)]This equation can be used to select the optimum sample thickness for transmission-mode diffraction. For example, if the measurement 2θ range is between 3 and 50°, the preferred sample thickness should be given by μt = 0.8–1.0.

2.5.4. Applications

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2.5.4.1. Phase identification

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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 X-ray 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).

Two-dimensional X-ray diffraction has enhanced phase identification in many respects (Rudolf & Landes, 1994[link]; Sulyanov et al., 1994[link]; Hinrichsen, 2007[link]). Because of its ability to collect diffracted X-rays 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 point-beam 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 sample-to-detector 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 two-dimensional 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[link]; Rodriguez-Navarro, 2006[link]; Boesecke, 2007[link]; Fuentes-Montero et al., 2011[link]; Hammersley, 2016[link]). 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 lattice-parameter determination and refinement (Ning & Flemming, 2005[link]; Flemming, 2007[link]; Jabeen et al., 2011[link]). The results can be used to search a powder-diffraction database to find possible matches. Since there is a great deal of literature covering these topics (Cullity, 1978[link]; Jenkins & Snyder, 1996[link]; Pecharsky & Zavalij, 2003[link]), this section will focus on the special characteristics of two-dimensional X-ray diffraction as well as system geometry, data-collection 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 two-dimensional diffraction. In most applications, however, the γ-integrated profile can be used for phase identification without these corrections.

2.5.4.1.1. Relative intensity

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The integrated intensity diffracted from polycrystalline materials with a random orientation distribution is given by[{{{I}}_{hkl}} = {k_I}{{{p_{hkl}}} \over {{v^2}}}({\rm LPA}){\lambda ^3}F_{hkl}^2{g_{hkl}}(\alpha, \beta)\exp \left({ - 2{M_t} - 2{M_s}} \right),\eqno(2.5.45)]where kI is an instrument constant that is a scaling factor between the experimental observed intensities and the calculated intensity, phkl 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 X-ray wavelength, [F_{hkl}] is the structure factor for the crystal plane (hkl), [{g_{hkl}}(\alpha, \beta)] is the normalized pole-density distribution function and exp(−2Mt − 2Ms) is the attenuation factor due to lattice thermal vibrations and weak static displacements (Warren, 1990[link]; He et al., 1994[link]). 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.

Phase-identification 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 2D-XRD are largely due to texture effects. For B-B 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 B-B geometry. In 2D-XRD, 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 [{g_{hkl}}(\alpha, \beta)] should be replaced by the average normalized pole-density function within the γ integration range [\left\langle {{g_{hkl}}(\Delta \gamma)} \right\rangle ]. The relation between (α, β) and (2θ, γ) is given in Chapter 5.4[link] . The chance of having zero pole density over the entire γ-integration range is extremely small. Therefore, phase identification with 2D-XRD is much more reliable than with conventional diffraction.

2.5.4.1.2. Detector distance and resolution

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The 2θ resolution with B-B 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 two-dimensional X-ray diffraction system, the 2θ resolution is achieved with different approaches. A flat 2D detector has the flexibility to be used at different sample-to-detector distances. The detector resolution is determined by the pixel size and point-spread 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 sample-to-detector 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[link] 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.

[Figure 2.5.16]

Figure 2.5.16 | top | pdf |

Diffraction pattern merged from four 2D frames collected from a battery material.

2.5.4.1.3. Defocusing effect

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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 X-ray 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 reflection-mode diffraction can be expressed as[{{B} \over {b}} = {{\sin \theta _2} \over {\sin \theta _1}} = {{\sin (2\theta - \omega)} \over {\sin \omega }},\eqno(2.5.46)]where θ1 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 θ2 < θ1. The defocusing effect occurs when θ2 > θ1 and the defocusing factor increases with increasing θ2 or decreasing θ1. The maximum defocusing appears at θ2 = 90°. For the θ–2θ configuration, the incident angle ω (= θ1) is used in the equation.

For B-B 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[link]). However, the defocusing effect prevents it from being used for a large 2θ range for a flat sample. Fig. 2.5.17[link] compares the effect for a flat detector and a cylindrical detector. Fig. 2.5.17[link](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[link](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[link](c) compares the defocusing factors of the two configurations. The horizontal dot-dashed line with defocusing factor B/b = 1 represents the situation with B-B 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 B-B 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. Transmission-mode 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, transmission-mode 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 [t = 1/\mu ], where μ is the linear absorption coefficient. The scattering intensity drops dramatically when the thickness increases.

[Figure 2.5.17]

Figure 2.5.17 | top | pdf |

Defocusing effects: (a) cylindrical detector; (b) flat detector at various incident angles and detector swing angles; (c) comparison of defocusing factors.

2.5.4.1.4. Sampling statistics

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In powder X-ray diffraction, the number of crystallites contributing to each reflection must be sufficiently large to generate reproducible integrated peak intensities (see Chapter 2.10[link] ). 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 as[N_s = p_{hkl} {Vf_i \over v_i} {\Omega \over 4\pi },\eqno(2.5.47)]where phkl is the multiplicity of the diffracting planes, V is the effective sampling volume, fi is the volume fraction of the measuring crystallites (fi = 1 for single-phase materials), vi is the volume of individual crystallites and Ω is the angular window of the instrument (given as a solid angle). The multiplicity term, phkl, effectively increases the number of crystallites contributing to the integrated intensity from a particular set of (hkl) planes. The volume of individual crystallites, vi, 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 2D-XRD, the instrumental window is not only determined by the incident beam size and divergence, but also by the detective area and the sample-to-detector distance (γ angular coverage).

In B-B geometry, the effective irradiated volume is a constant,[V_{\rm BB} = A_oA_{\rm BB} = {A_o/2\mu },\eqno(2.5.48)]where Ao is the cross-section area of the incident beam measured on the sample surface, [A_{\rm BB} = 1/(2\mu)] is the transmission coefficient for B-B geometry, and μ is the linear absorption coefficient. For 2D-XRD, the effective volume is given as[V = A_oA = A_oT/2\mu,\eqno(2.5.49)]where A is the transmission coefficient and T is the transmission coefficient with B-B 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 β1 within the diffraction plane and β2 in the perpendicular direction. The angular window corresponding to the incident-beam divergence is given by[\Omega = {\beta _1}{\beta _2}/\sin \theta \ {\rm or}\ \Omega = {\beta ^2}/\sin \theta \ {\rm if}\ \beta = {\beta _1} = {\beta _2}.\eqno(2.5.50)]

For 2D-XRD, the angular window is not only determined by the incident-beam 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 sample-position 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 incident-beam divergence and the virtual oscillation is[\Omega = \beta \Delta \psi = 2\beta \arcsin [\cos \theta \sin (\Delta \gamma /2)],\eqno(2.5.51)]where 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 X-ray 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 powder-diffraction pattern can be generated from single-crystal sample if a sufficient angular window can be achieved by sample rotation in such as way as to simulate a Gandolfi camera (Guggenheim, 2005[link]). Sample oscillation is not always necessary if virtual oscillation can achieve sufficient sampling statistics.

2.5.4.2. 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.

2.5.4.2.1. 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]).

2.5.4.2.2. 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 2.5.4.2.4[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.

2.5.4.2.3. 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]).

2.5.4.2.4. 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 2.5.4.2.3[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.

2.5.4.2.5. 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.

2.5.4.2.6. 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.

2.5.4.3. Stress measurement

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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 d-spacing changes in the crystallites. These d-spacing changes can be measured by the changes in diffraction-peak positions based on Bragg's law. In this case, the d-spacing serves as a gauge of the deformation. Stress measurement by X-ray diffraction is typically done using a point detector or line detector (Walter, 1971[link]; James & Cohen, 1980[link]; Noyan & Cohen, 1987[link]; Lu, 1996[link]); this will be referred to as the conventional method. The stress or stress tensor is calculated from many strain measurements from diffraction-peak 2θ shifts of a specific lattice-plane 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[link]; Korhonen et al., 1989[link]; Yoshioka & Ohya, 1992[link]; Fujii & Kozaki, 1993[link]; He & Smith, 1997[link]; Kämpfe et al., 1999[link]; Hanan et al., 2004[link]). 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 stress-tensor measurement. The 2D method for stress measurement is based on the fundamental relationship between the stress tensor and the diffraction-cone distortion (He & Smith, 1997[link]; He, 2000[link]; European Standard, 2008[link]).

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 X-ray 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 X-ray 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 X-ray diffraction method for stress measurement at the macroscopic level.

2.5.4.3.1. Stress and strain relation

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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 S1, S2, S3 contains nine components, given by[{\sigma _{ij}} = \left [{\matrix{ {{\sigma _{11}}} & {{\sigma _{12}}} & {{\sigma _{13}}} \cr {{\sigma _{21}}} & {{\sigma _{22}}} & {{\sigma _{23}}} \cr {{\sigma _{31}}} & {{\sigma _{32}}} & {{\sigma _{33}}} \cr } } \right].\eqno(2.5.63)]

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:[{\sigma _{12}} = {\sigma _{21}},\ {\sigma _{23}} = {\sigma _{32}} \ {\rm and}\ {\sigma _{31}} = {\sigma _{13}}.\eqno(2.5.64)]Therefore, only six independent components define the stress state in a solid. The following stress states are typically measured:

  • Uniaxial: all stress components are zero except one normal stress component.

  • Biaxial: all nonzero components are within the S1S2 plane.

  • Biaxial with shear: [{\sigma _{33}} = 0], all other components are not necessarily zero.

  • Equibiaxial: a special case of biaxial stress where σ11 = σ22 = σ.

  • Triaxial: all components are not necessarily zero.

  • Equitriaxial: a special case of triaxial stress where σ11 = σ22 = σ33 = σ.

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:[{\varepsilon _{ij}} = \left [{\matrix{ {{\varepsilon _{11}}} & {{\varepsilon _{12}}} & {{\varepsilon _{13}}} \cr {{\varepsilon _{21}}} & {{\varepsilon _{22}}} & {{\varepsilon _{23}}} \cr {{\varepsilon _{31}}} & {{\varepsilon _{32}}} & {{\varepsilon _{33}}} \cr } } \right].\eqno(2.5.65)]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, X-ray 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:[{\sigma _{ij}} = {C_{ijkl}}{\varepsilon _{kl}},\eqno(2.5.66)]where [C_{ijkl}] are elastic stiffness coefficients. The stress–strain relations can also be expressed as[{\varepsilon _{ij}} = {S_{ijkl}}{\sigma _{kl}},\eqno(2.5.67)]where [S_{ijkl}] 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:[\eqalignno{ & {\varepsilon _{11}} = {1 \over E}[{\sigma _{11}} - \nu ({\sigma _{22}} + {\sigma _{33}})], \cr & {\varepsilon _{22}} = {1 \over E}[{\sigma _{22}} - \nu ({\sigma _{33}} + {\sigma _{11}})], \cr & {\varepsilon _{33}} = {1 \over E}[{\sigma _{33}} - \nu ({\sigma _{11}} + {\sigma _{22}})] ,\cr & {\varepsilon _{12}} = {{1 + \nu } \over E}{\sigma _{12}},\quad {\varepsilon _{23}} = {{1 + \nu } \over E}{\sigma _{23}},\quad {\varepsilon _{31}} = {{1 + \nu } \over E}{\sigma _{31}}.&(2.5.68)} ]It is customary in the field of stress measurement by X-ray diffraction to use another set of macroscopic elastic constants, S1 and ½S2, which are given by[{\textstyle{1 \over 2}}S_2 = (1 + \nu)/E \ \,{\rm and}\ \, S_1 = - \nu /E.\eqno(2.5.69)]

Although polycrystalline materials on a macroscopic level can be considered isotropic, residual stress measurement by X-ray 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 plane-specific elasticity constants, [S_1^{\{hkl\}}] and [{\textstyle{1 \over 2}}S_2^{\left\{hkl\right\}}], called X-ray elastic constants (XECs). XECs for many materials can be found in the literature, measured or calculated from microscopic elasticity constants (Lu, 1996[link]). In the case of materials with cubic crystal symmetry, the equations for calculating the XECs from the macroscopic elasticity constants ½S2 and S1 are[\eqalignno{{\textstyle{{{1}} \over {{2}}}}{{S}}_{{2}}^{{{\{ hkl\} }}} &= {\textstyle{{{1}} \over {{2}}}}{{{S}}_{{2}}}{{[1}} + 3(0.2 - {{\Gamma (hkl))\Delta]}} &\cr {{S}}_{{1}}^{{{\{ hkl\} }}} &= {{{S}}_{{1}}} - {\textstyle{{{1}} \over {{2}}}}{{{S}}_{{2}}}[0.2 - {{\Gamma (hkl)]\Delta }},&(2.5.70)}]where[{{\Gamma (hkl)}} = {{{{{h}}^{{2}}}{{{k}}^{{2}}} + {{{k}}^{{2}}}{{{l}}^{{2}}} + {{{l}}^{{2}}}{{{h}}^{{2}}}} \over {{{{{(}}{{{h}}^{{2}}} + {{{k}}^{{2}}} + {{{l}}^{{2}}}{{)}}}^{{2}}}}} \ {\rm and}\ {{\Delta }} = {{{{5(}}{{{A}}_{{\rm{RX}}}} - {{1)}}} \over {{{3}} + {{2}}{{{A}}_{{\rm{RX}}}}}}.]In the equations for stress measurement hereafter, either the macroscopic elasticity constants ½S2 and S1 or the XECs [{{S}}_{{1}}^{\left\{ {{{hkl}}} \right\}}] and [{\textstyle{{{1}} \over {{2}}}}{{S}}_{{2}}^{\left\{ {{{hkl}}} \right\}}] 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 (ARX) is a measure of the elastic anisotropy of a material (He, 2009[link]).

2.5.4.3.2. Fundamental equations

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Fig. 2.5.24[link] 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. [2\theta = 2\theta (\gamma, \omega, \psi, \varphi)]. This function is uniquely determined by the stress tensor. The strain measured by the 2θ shift at a point on the diffraction ring is [{{\varepsilon }}_{{{(\gamma, \omega, \psi, \varphi)}}}^{{{\{ hkl\} }}}], based on the true strain definition[\varepsilon _{(\gamma, \omega, \psi, \varphi)}^{\{ hkl\} } = \ln {d \over {{d_o}}} = \ln {{\sin {\theta _o}} \over {\sin \theta }} = \ln {\lambda \over {2{d_o}\sin \theta }},\eqno(2.5.71)]where do and θo are the stress-free values and d and θ are measured values from a point on the diffraction ring corresponding to [(\gamma, \omega, \psi, \varphi)]. The direction of [\varepsilon _{(\gamma, \omega, \psi, \varphi)}^{\{ {{hkl}}\} }] in the sample coordinates S1, S2, S3 can be given by the unit-vector components h1, h2 and h3. As a second-order tensor, the relationship between the measured strain and the strain-tensor components is then given by[\varepsilon _{(\gamma, \omega, \psi, \varphi)}^{\{ hkl\} } = {\varepsilon _{ij}} \cdot {h_i} \cdot {h_j}.\eqno(2.5.72)]The 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 in[\varepsilon _{(\gamma, \omega, \psi, \varphi)}^{\{ hkl\} } = h_1^2{\varepsilon _{11}} + 2{h_1}{h_2}{\varepsilon _{12}} + h_2^2{\varepsilon _{22}} + 2{h_1}{h_3}{\varepsilon _{13}} + 2{h_2}{h_3}{\varepsilon _{23}} + h_3^2{\varepsilon _{33}}.\eqno(2.5.73)]Or, taking the true strain definition,[\eqalignno{&h_1^2{\varepsilon _{11}} + 2{h_1}{h_2}{\varepsilon _{12}} + h_2^2{\varepsilon _{22}} + 2{h_1}{h_3}{\varepsilon _{13}} + 2{h_2}{h_3}{\varepsilon _{23}} + h_3^2{\varepsilon _{33}} &\cr&\quad= \ln \left({{{\sin {\theta _0}} \over {\sin \theta }}} \right),&(2.5.74)}]where θo corresponds to the stress-free d-spacing and θ are measured values from a point on the diffraction ring. Both θ and {h1, h2, h3} are functions of [(\gamma, \omega, \psi, \varphi)]. By taking γ values from 0 to 360°, equation (2.5.74)[link] establishes the relationship between the diffraction-cone distortion and the strain tensor. Therefore, equation (2.5.74)[link] is the fundamental equation for strain measurement with two-dimensional X-ray diffraction.

[Figure 2.5.24]

Figure 2.5.24 | top | pdf |

Diffraction-cone distortion due to stresses.

Introducing the elasticity of materials, one obtains[\eqalignno{& - {\nu \over E}({\sigma _{11}} + {\sigma _{22}} + {\sigma _{33}}) &\cr&\quad+ {{1 + \nu } \over E}({\sigma _{11}}h_1^2 + {\sigma _{22}}h_2^2 + {\sigma _{33}}h_3^2 + 2{\sigma _{12}}{h_1}{h_2} + 2{\sigma _{13}}{h_1}{h_3} + 2{\sigma _{23}}{h_2}{h_3}) &\cr&\quad\quad= \ln \left({{{\sin {\theta _0}} \over {\sin \theta }}} \right)&(2.5.75)}]or[\eqalignno{&{S_1}({\sigma _{11}} + {\sigma _{22}} + {\sigma _{33}}) &\cr&\quad+ {\textstyle{1 \over 2}}{S_2}({\sigma _{11}}h_1^2 + {\sigma _{22}}h_2^2 + {\sigma _{33}}h_3^2 + 2{\sigma _{12}}{h_1}{h_2} + 2{\sigma _{13}}{h_1}{h_3} + 2{\sigma _{23}}{h_2}{h_3})&\cr&\quad\quad = \ln \left({\sin \theta _0 \over \sin \theta } \right).&(2.5.76)}]

It is convenient to express the fundamental equation in a clear linear form:[{p_{11}}{\sigma _{11}} + {p_{12}}{\sigma _{12}} + {p_{22}}{\sigma _{22}} + {p_{13}}{\sigma _{13}} + {p_{23}}{\sigma _{23}} + {p_{33}}{\sigma _{33}} = \ln \left({{{\sin {\theta _0}} \over {\sin \theta }}} \right),\eqno(2.5.77)]where pij are stress coefficients given by[p_{ij} = \cases{(1/E)[(1 + \nu)h_i^2 - \nu] = {\textstyle{1 \over 2}}{S_2}h_i^2 + {S_1} & {\rm{if }}\ $i = j$,\hfill \cr 2(1/E)(1 + \nu){h_i}{h_j} = 2{\textstyle{1 \over 2}}{S_2}{h_i}{h_j}&{\rm{if }}\ $i \ne j$.\hfill} \eqno(2.5.78)]In the equations for the stress measurement above and hereafter, the macroscopic elastic constants ½S2 and S1 are used for simplicity, but they can always be replaced by the XECs for the specific lattice plane {hkl}, [S_1^{\{hkl\}}] and [{\textstyle{1 \over 2}}S_2^{\left\{hkl \right\}}], if the anisotropic nature of the crystallites should be considered. For instance, equation (2.5.76)[link] can be expressed with the XECs as[\eqalignno{&S_1^{\{ hkl\} }({\sigma _{11}} + {\sigma _{22}} + {\sigma _{33}}) &\cr&+ {\textstyle{1 \over 2}}S_2^{\{ hkl\} }({\sigma _{11}}h_1^2 + {\sigma _{22}}h_2^2 + {\sigma _{33}}h_3^2 + 2{\sigma _{12}}{h_1}{h_2} + 2{\sigma _{13}}{h_1}{h_3} + 2{\sigma _{23}}{h_2}{h_3}) &\cr&\quad\quad= \ln \left({{{\sin {\theta _0}} \over {\sin \theta }}} \right).&(2.5.79)}]

The fundamental equation (2.5.74)[link] 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 stress-tensor components from the measured strain (2θ-shift) values in various directions. The fundamental equation for stress measurement with 2D-XRD is a linear function of the stress-tensor components. The stress tensor can be obtained by solving the linear equations if six independent strains are measured or by linear least-squares regression if more than six independent measured strains are available. In order to get a reliable solution from the linear equations or least-squares analysis, the independent strain should be measured at significantly different orientations. Another function of the fundamental equation is to calculate the diffraction-ring distortion for a given stress tensor at a particular sample orientation [(\omega, \psi, \varphi)] (He & Smith, 1998[link]). The fundamental equation for stress measurement by the conventional X-ray diffraction method can also be derived from the 2D fundamental equation (He, 2009[link]).

2.5.4.3.3. Equations for various stress states

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The general triaxial stress state is not typically measured by X-ray diffraction because of low penetration. For most applications, the stresses in a very thin layer of material on the surface are measured by X-ray diffraction. It is reasonable to assume that the average normal stress in the surface-normal direction is zero within such a thin layer. Therefore, [{\sigma _{33}} = 0], 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:[{p_{11}}{\sigma _{11}} + {p_{12}}{\sigma _{12}} + {p_{22}}{\sigma _{22}} + {p_{13}}{\sigma _{13}} + {p_{23}}{\sigma _{23}} + {p_{\rm ph}}{\sigma _{\rm ph}} = \ln \left({{{\sin {\theta _0}} \over {\sin \theta }}} \right),\eqno(2.5.80)]where the coefficient [{{{p}}_{{{\rm ph}}}} = {\textstyle{{{1}} \over {{2}}}}{{{S}}_{{2}}} + {{3}}{{{S}}_{{1}}}] and σph is the pseudo-hydrostatic stress component introduced by the error in the stress-free d-spacing. In this case, the stresses can be measured without the accurate stress-free d-spacing, since this error is included in σph. The value of σph is considered as one of the unknowns to be determined by the linear system. With the measured stress-tensor components, the general normal stress (σϕ) and shear stress (τϕ) at any arbitrary angle ϕ can be given by[\eqalignno{\sigma_\varphi &=\sigma_{11}\cos^2\varphi +\sigma_{12}\sin 2\varphi +\sigma_{22}\sin^2\varphi,&(2.5.81)\cr \tau_\varphi &= \sigma_{13}\cos\varphi +\sigma_{23}\sin\varphi. &(2.5.82)}]

Equation (2.5.81)[link] 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 [{\sigma _{33}} = {\sigma _{13}} = {\sigma _{23}} = 0], so we have[{p_{11}}{\sigma _{11}} + {p_{12}}{\sigma _{12}} + {p_{22}}{\sigma _{22}} + {p_{\rm ph}}{\sigma _{\rm ph}} = \ln \left({{{\sin {\theta _0}} \over {\sin \theta }}} \right).\eqno(2.5.83)]

In the 2D stress equations for any stress state with σ33 = 0, we can calculate stress with an approximation of do (or 2θo). Any error in do (or 2θo) will contribute only to a pseudo-hydrostatic term σph. The measured stresses are independent of the input do (or 2θo) values (He, 2003[link]). If we use [d'_0] to represent the initial input, then the true do (or 2θo) can be calculated from σph with[\eqalignno{d_0&=d_0^\prime \exp\left({{1-2\nu}\over E}\sigma_{\rm ph}\right),&(2.5.84)\cr \theta_0&=\arcsin\left[\sin\theta_0^\prime\exp\left({{1-2\nu}\over E}\sigma_{\rm ph}\right)\right].&(2.5.85)}]

Care must be taken that the σph value also includes the measurement error. If the purpose of the experiment is to determine the stress-free d-spacing do, the instrument should be first calibrated with a stress-free standard of a similar material.

2.5.4.3.4. Data-collection strategy

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The practice of stress analysis with 2D-XRD involves the selection of the diffraction-system configuration and the data-collection 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 2D-XRD. 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 stress-measurement results can be considered as an extrapolation from the measured values. In the conventional sin2 ψ method, several ψ-tilt angles are required, typically at 15° steps from −45° to +45°. The same is true with a 2D-XRD system. The diffraction vectors corresponding to the data scan can be projected onto a 2D plot in the same way as the pole-density distribution in a pole figure. The 2D plot is called a data-collection strategy scheme.

By evaluating the scheme, one can generate a data-collection strategy suitable for the measurement of the intended stress components. Fig. 2.5.25[link] illustrates two schemes for data collection. In the bisecting condition ([\omega = \theta ] or [{\theta _1} = \theta ] and [\psi = 0^\circ ]), 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 biaxial-stress tensor. The data set with [\varphi = 0^\circ ], as shown within the box enclosed by the dashed lines, would be sufficient on its own to calculate [{\sigma _{11}}]. Since the diffraction-ring distortion at [\varphi = 0^\circ ] or [\varphi = 90^\circ ] is not sensitive to the stress component [{\sigma _{12}}], strategy (a) is suitable for the equibiaxial stress state, but is not able to determine [{\sigma _{12}}] 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 biaxial-stress tensor components and shear stress ([{\sigma _{11}},\,{\sigma _{12}},\,{\sigma _{22}},\,{\sigma _{13}},\,{\sigma _{23}}]). The scheme indicated by the boxes enclosed by the dashed lines is a time-saving alternative to scheme (b). The rings on two ϕ angles are aligned to S1 and S2 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 stress-gauge 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 data-collection time.

[Figure 2.5.25]

Figure 2.5.25 | top | pdf |

Data-collection strategy schemes: (a) ω + ϕ scan; (b) ψ + ϕ scan.

2.5.4.3.5. Data integration and peak evaluation

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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[link] illustrates data integration over a diffraction frame. The total integration region is defined by [2\theta _1], [2\theta _2], [{\gamma _1}] and [{\gamma _2}]. The integration region is divided into segments given by [\Delta \gamma ]. One data point on the distorted diffraction ring is generated from each segment. The γ value in the centre (denoted by the dot-dashed 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 ([\Delta \gamma ]) are selected based on the quality of the data frame. The larger the segment size [\Delta \gamma ] is, the better the integrated diffraction profile as more counts are being integrated. γ integration also produces a smearing effect on the diffraction-ring distortion because the counts collected within the segment size [\Delta \gamma ] are considered as a single γ value at the segment centre. The 2θ shift in the segment is averaged. The segment size [\Delta \gamma ] 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. [\Delta \gamma\, \le\, 2^\circ ], and still have a smooth profile for each segment. For data frames having low pixel counts, for example the frames collected from a micron-sized area, from a sample with large grains or with a short data-collection 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.

[Figure 2.5.26]

Figure 2.5.26 | top | pdf |

Data integration for stress measurement.

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 2D-XRD, 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[link].) The polarization and absorption correction is not always necessary if the error caused by absorption can be tolerated for the application, or if the data-collection 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 Kα1 and Kα2 radiation is used in the calculations. For samples with a broad peak width, diffraction of Kα1 and Kα2 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 d-spacing. For samples with a relatively narrow peak width, the diffraction profile shows strong asymmetry or may even reveal two peaks corresponding to the Kα1 and Kα2 lines, especially at high 2θ angles. In this case the profile fitting should include contributions from both the Kα1 and Kα2 lines. It is common practice to use the peak position from the Kα1 line and the Kα1 wavelength to calculate the d-spacing after Kα2 stripping.

Background correction is necessary if there is a strong background or the peak-evaluation algorithms are sensitive to the background, such as in Kα2 stripping, peak fitting, and peak-intensity and integrated-intensity 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, pseudo-Voigt or Pearson-VII functions (Lu, 1996[link]; Sprauel & Michaud, 2002[link]).

2.5.4.3.6. Stress calculation

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The final data set after integration and peak evaluation should contain many data points describing the diffraction-ring shape for all collected frames. Each measured data point contains three goniometer angles (ω, ψ, ϕ) and the diffraction-ring 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 least-squares method can be used to calculate the stresses. In a general least-squares regression, the residual for the ith data point is defined as[{r_i} = {y_i} - {\hat y_i},\eqno(2.5.86)]where [{y_i}] is the observed response value, [{\hat y_i}] is the fitted response value and [{r_i}] is the residual, which is defined as the difference between the observed value and the fitted value. The summed square of residuals is given by[S = \textstyle\sum\limits_{i = 1}^n {r_i^2 = } \textstyle\sum\limits_{i = 1}^n {{{({y_i} - {{\hat y}_i})}^2}}, \eqno(2.5.87)]where n is the number of data points and S is the sum-of-squares error to be minimized in the least-squares regression. For stress calculation, the observed response value is the measured strain at each data point,[{y_i} = \ln \left({{{\sin \theta _0} \over {\sin \theta _i}}} \right),\eqno(2.5.88)]and the fitted response value is given by the fundamental equation as[{\hat y_i} = {p_{11}}{\sigma _{11}} + {p_{12}}{\sigma _{12}} + {p_{22}}{\sigma _{22}} + {p_{13}}{\sigma _{13}} + {p_{23}}{\sigma _{23}} + {p_{33}}{\sigma _{33}} + {p_{\rm ph}}{\sigma _{\rm ph}},\eqno(2.5.89)]where all possible stress components and stress coefficients are listed as a generalized linear equation. Since the response-value function is a linear equation of unknown stress components, the least-squares problem can be solved by a linear least-squares 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 least-squares regression (He, 2009[link]).

2.5.4.3.7. Comparison between the 2D method and the con­ventional method

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Stress measurement on a polycrystalline material by X-ray diffraction is based on the strain measurements in a single or in several sample orientations. Each measured strain is calculated from the average d-spacing 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 X-ray beam. Lattice-plane 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 two-dimensional 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[link](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 Pearson-VII 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 [\sin ^2\psi ] 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 [\sin ^2\psi ] method and the 2D method are compared in Fig. 2.5.27[link](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.

[Figure 2.5.27]

Figure 2.5.27 | top | pdf |

Stress calculation with the 2D method and the [\sin ^2\psi ] method: (a) nine data points (abbreviated as pts) on the diffraction ring; (b) measured stress and standard deviation by different methods.

2.5.4.4. Quantitative analysis

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2.5.4.4.1. Crystallinity

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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 X-ray diffraction (Chung & Scott, 1973[link]; Alexander, 1985[link]; Murthy & Barton, 2000[link]; Kasai & Kakudo, 2005[link]). Assuming that the X-ray 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 by[{x_{\rm pc}} = 100\% {{{I_{\rm crystal}}} \over {{I_{\rm crystal}} + {I_{\rm amorphous}}}},\eqno(2.5.90)]where xpc is the per cent crystallinity, Icrystal is the integrated intensity of all crystalline peaks and Iamorphous 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[link] shows a 2D diffraction frame collected from an oriented polycrystalline sample. The diffraction is in transmission mode with the X-ray beam perpendicular to the plate sample surface. Fig. 2.5.28[link](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[link](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 2D-XRD system can measure per cent crystallinity more accurately and with more consistent results (Pople et al., 1997[link]; Bruker, 2000[link]) than a conventional system.

[Figure 2.5.28]

Figure 2.5.28 | top | pdf |

2D diffraction pattern from an oriented polycrystalline polymer sample. (a) Diffraction profile integrated from a horizontal region analogous to a profile collected with point detector. (b) Diffraction profile integrated from all parts of the 2D frame.

2.5.4.4.2. Crystallite size

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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. X-ray diffraction has been used for crystallite-size measurement for many years. Most methods are based on diffraction-line broadening and line-profile analysis (Wilson, 1971[link]; Klug & Alexander, 1974[link]; Ungár, 2000[link]). Another approach to crystallite-size measurement is based on the spotty diffraction rings collected with two-dimensional detectors when a small X-ray beam is used (Cullity, 1978[link]; He, 2009[link]). Line-profile analysis is based on the diffraction profile in the 2θ direction, while crystallite-size analysis with a spotty 2D diffraction pattern is based on the diffraction profile in the γ direction. The latter may be referred to as γ-profile analysis.

Fig. 2.5.29[link](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:[B = {{C\lambda } \over {t\cos \theta }},\eqno(2.5.91)]where λ is the X-ray 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[link]), and t is the crystallite size (also in Å). This equation shows an inverse relationship between crystallite size and peak-profile 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 sample-to-detector distance should be used to maximize the resolution. A small beam size and low convergence should also be used to reduce instrument broadening.

[Figure 2.5.29]

Figure 2.5.29 | top | pdf |

Crystallite-size analysis: (a) 2θ profile of a gold nanoparticle (grey) and regular gold metal (black); (b) γ profile of LaB6; (c) measurement range.

Fig. 2.5.29[link](b) shows a frame collected from an SRM660a (LaB6) sample with a 2D-XRD 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 incident-beam divergence and the calibration factor for the instrument, we obtain an equation for the crystallite size as measured in reflection mode:[d = k\left\{p_{hkl}b^2\arcsin [\cos \theta \sin (\Delta \gamma /2)] \over \mu N_s \right\}^{1/3},\eqno(2.5.92)]where d is the diameter of the crystallite particles, phkl 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 Ns is the number of spots within Δγ. For transmission mode, we have[d = k\left\{ {p_{hkl}b^2t\arcsin [\cos \theta \sin (\Delta \gamma /2)] \over N_s} \right\}^{1/3},\eqno(2.5.93)]where 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 [\exp (\mu t)] for crystallite-size analysis as long as the sample is thin enough for transmission-mode diffraction. The effective sampling volume reaches a maximum for transmission-mode diffraction when [t = 1/\mu ].

For both reflection and transmission,[k = \left({3\beta \over 8\pi } \right)^{1/3},\eqno(2.5.94)]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)[link] that a smaller X-ray beam size and low-multiplicity peak should be used if a smaller crystallite size is to be determined.

Fig. 2.5.29[link](c) shows the measurement ranges of 2θ-profile and γ-profile analysis. The 2θ-profile analysis is suitable for crystallite sizes below 100 nm (1000 Å), while γ-profile analysis is suitable for crystallite sizes from as large as tens of µm down to 100 nm with a small X-ray 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 as[k_n = n^{1/3}k,\eqno(2.5.95)]where 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 second-order 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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