Introduction to two-dimensional X-ray diffraction

Transcription

1 Introduction to two-dimensional X-ray diffraction Bob Baoping He a) Bruker Advanced X-ray Solutions, Inc., 5465 East Cheryl Parkway, Madison, Wisconsin Received 7 February 2003; accepted 1 April 2003 Two-dimensional X-ray diffraction refers to X-ray diffraction applications with two-dimensional detector and corresponding data reduction and analysis. The two-dimensional diffraction pattern contains far more information than a one-dimensional profile collected with the conventional diffractometer. In order to take advantage of two-dimensional diffraction, new theories and approaches are necessary to configure the two-dimensional X-ray diffraction system and to analyze the two-dimensional diffraction data. This paper is an introduction to some fundamentals about two-dimensional X-ray diffraction, such as geometry convention, diffraction data interpretation, and advantages of two-dimensional X-ray diffraction in various applications, including phase identification, stress, and texture measurement International Centre for Diffraction Data. DOI: / I. INTRODUCTION a Electronic mail: bhe@bruker-axs.com In the field of X-ray powder diffraction, data collection and analysis have been based mainly on one-dimensional 1D diffraction profiles measured with scanning point detectors or linear position-sensitive detectors PSD. Therefore, almost all the X-ray powder diffraction applications, such as phase identification, texture orientation, residual stress, crystallite size, percent crystallinity, lattice dimensions, and structure refinement Rietveld, are developed in accord with the 1D profile collected by conventional diffractometers Jenkins and Snyder, In recent years, usage of two-dimensional 2D detectors has dramatically increased due to advances in detector technology, point beam X-ray optics, and computing power Rudolf and Landes, 1994; Sulyanov et al., Although a 2D image contains far more information than a 1D profile, the advantages of a 2D detector cannot be fully taken if the data interpretation and analysis methods are simply inherited from the conventional diffraction theory. Two-dimensional X-ray diffraction (XRD 2 ) is a new technique in the field of X-ray diffraction XRD, which is not simply a diffractometer with a 2D detector. In addition to the 2D detector technology, it involves 2D image processing and 2D diffraction pattern manipulation and interpretation. Because of the unique nature of data collected with a 2D detector, a completely new concept and new approach are necessary to configure the XRD 2 system and to understand and analyze the 2D diffraction data. In addition, the new theory should also be consistent with the conventional theory so that the 2D data can be used for conventional applications He et al., An XRD 2 system is a diffraction system with the capability of acquiring diffraction pattern in 2D space simultaneously and analyzing 2D diffraction data accordingly. XRD 2 systems are available in a variety of configurations to fulfill requirements of different applications and samples. As is shown in Figure 1, a typical two-dimensional X-ray diffraction system consists of at least one two-dimensional detector, X-ray source, X-ray optics, sample positioning stage, sample alignment, and monitoring device as well as corresponding computer control and data reduction and analysis software. II. X-RAY DIFFRACTION WITH TWO-DIMENSIONAL DETECTOR A. Two-dimensional diffraction pattern XRD is a technique used to measure the atomic arrangement of materials. When a monochromatic X-ray beam hits a sample, in addition to absorption and other phenomena, we observe X-ray scattering with the same wavelength as the incident beam, called coherent X-ray scattering. The coherent scattering of X-ray from a sample is not evenly distributed in space, but is a function of the electron distribution in the sample. The atomic arrangement in materials can be ordered like a single crystal or disordered like glass or liquid. As such, the intensity and spatial distributions of the scattered X-rays form a specific diffraction pattern which is the fingerprint of the sample. Figure 2 shows the pattern of diffracted X-rays from a single crystal and from a polycrystalline sample. The diffracted rays from a single crystal point to discrete directions each corresponding to a family of diffraction planes Figure 2 a. The diffraction pattern from a polycrystalline powder sample forms a series of diffraction cones if a large number of crystals oriented randomly in the space are covered by the incident X-ray beam Figure 2 b. Each diffraction cone corresponds to the diffraction from the same family of crystalline planes in all the participating grains. The diffraction patterns from polycrystalline materials will be considered hereafter in the further discussion of the theory and configuration of XRD 2 systems. Polycrystalline materials can be single-phase or multiphase in bulk solid, thin film, or fluid. In practice, the XRD 2 applications are not limited to polycrystalline materials, and the specimen can be a mixture of single crystal, polycrystal, and amorphous materials. B. Comparison between XRD and XRD 2 First, we compare the conventional XRD and twodimensional x-ray diffraction (XRD 2 ). Figure 3 is a schematic of X-ray diffraction from a powder polycrystalline 71 Powder Diffraction 18 (2), June /2003/18(2)/71/15/$ JCPDS-ICDD 71

2 Figure 1. Five major components in an XRD 2 system, an area detector, an X-ray generator, X-ray optics monochromator and collimator, goniometer and sample stage, and sample alignment and monitoring laser/video system. sample. For simplicity, it shows only two diffraction cones, one represents forward diffraction (2 90 ) and one for backward diffraction (2 90 ). The diffraction measurement in the conventional diffractometer is confined within a plane, here referred to as the diffractometer plane. A point detector makes 2 scan along a detection circle. If a onedimensional PSD is used in the diffractometer, it will be mounted on the detection circle. Since the variations of diffraction pattern in the direction (Z) perpendicular to the diffractometer plane are not considered in the conventional diffractometer, the X-ray beam is normally extended in Z direction line focus. The actual diffraction pattern measured by a conventional diffractometer is an average over a range defined by beam size in Z direction. Since the diffraction data out of the diffractometer plane are not detected, the corresponding structure information will be either ignored, or measured by various additional sample rotations. The conventional diffraction pattern, collected with either a scanning point detector or a PSD, is a plot of X-ray scattering intensity at different 2 angles. Figure 4 shows the conventional diffraction pattern of corundum powder. With a 2D detector, the measurable diffraction is no longer limited in the diffractometer plane. Instead, the whole or a large portion of the diffraction rings as called the Debye ring can be measured simultaneously, depending on the detector size and position. Figure 5 shows the diffraction pattern on 2D detector compared with the diffraction measurement range of scintillation detector and PSD. Since the diffraction rings are measured, the variations of diffraction intensity in all directions are equally important, the ideal shape of the X-ray beam cross-section for XRD 2 is a point point focus. In practice, the beam cross-section can be either circular or square in limited size. III. GEOMETRY CONVENTIONS IN XRD 2 SYSTEM The geometry of an XRD 2 system consists of three distinguishable geometry spaces, each defined by a set of parameters. The three geometry spaces are: diffraction space, detector space, and sample space. The laboratory coordinates system, X L Y L Z L is the basis of all three spaces. Although the three spaces are interrelated, the definitions and corresponding parameters should not be confused. A. Diffraction cones in laboratory axes diffraction space Figure 6 describes the geometric definition of diffraction cones in the laboratory coordinates system, X L Y L Z L. Analogous to the conventional three-circle and four-circle goniometer, the direct X-ray beam propagates along the X L axis, Z L is up, and Y L makes up a right-handed rectangular coordinate system. The axis X L is also the rotation axis of the cones. The apex angles of the cones are determined by the 2 values given by the Bragg equation. The apex angles are twice the 2 values for forward reflection (2 90 ) and twice the values of for backward reflection (2 90 ). The angle is the azimuthal angle from the origin at the 6 o clock direction ( Z L direction with the righthanded rotation axis along the opposite direction of incident beam ( X L direction. In the previous publications, has been used to denote this angle by the present author s. Since has also been used to denote one of the goniometer angles in four-circle convention, will be used hereafter to represent this angle. The angle here is used to define the direction of the diffracted beam on the cone. The angle actually defines a half plane with the X L axis as the edge, it will be referred to as the plane hereafter. Intersections of any dif- Figure 2. The patterns of diffracted X-rays: a from a single crystal and b from a polycrystalline sample. 72 Powder Diffr., Vol. 18, No. 2, June 2003 Bob Baoping He 72

3 Figure 3. Diffraction patterns in 3D space from a powder sample and the diffractometer plane in the conventional diffractometer. fraction cones with a plane have the same value. The conventional diffractometer plane consists of two planes with one 90 plane in the negative Y L side and 270 plane in the positive Y L side. and 2 angles forms a kind of spherical coordinate system which covers all the directions from the origin of sample goniometer center. The 2 system is fixed in the laboratory systems X L Y L Z L, which is independent of the sample orientation in the goniometer. This is a very important concept when we deal with the 2D diffraction data. The unit vector of a diffraction vector is given in the laboratory system as h L h x h y h z sin cos sin cos cos. The vector describes a diffraction cone when takes values from 0 to Figure 4. The conventional diffraction pattern of corundum shows the intensity of X-ray scattering at different 2 angles. B. Ideal detector for diffraction pattern in threedimensional space An ideal detector to measure the diffraction pattern in three-dimensional 3D space is a detector with a spherical detecting surface covering all the diffraction directions in 3D space as shown in Figure 7. The sample is in the center of the sphere. The direction of a diffracted beam is defined by longitude and 2 latitude. The incident X-ray beam points to the center of the sphere through the detector at 2. The detector surface covers the whole spherical surface, i.e., 4 in solid angle. The ideal detector should have large dynamic range, small pixel size, and narrow point spread function, as well as many properties for an ideal detector. In practice, such an ideal detector does not exist. However, there are many 2D detector technologies available, including photographic film, CCD, image plate, and multiwire proportional counter. Each technique has its advantages over the others Rudolf and Landes, The detection surface can be spherical, cylindrical, or flat. The spherical or cylindrical detectors are normally designed for a fixed sample to detector distance, while flat detector has the flexibility to be used at different sample-to-detector distance so as to choose between higher resolution at large distance or higher angular coverage at short distance. The following discussion on XRD 2 geometry will focus on flat 2D detectors. C. Diffraction cones and conic sections on 2D detectors Figure 8 shows the geometry of a diffraction cone. The incident X-ray beam always lies along the rotation axis of the diffraction cone. The whole apex angle of the cone is twice the 2 value given by Bragg law. For a flat 2D detector, the detection surface can be considered as a plane, which intersects the diffraction cone to form a conic section. D is the distance between the sample and the detector, and is the detector swing angle. The conic section takes different shapes for different angles. When imaged on-axis ( 0 ) the conic sections appear as circles. When the detector Figure 5. Comparison of diffraction pattern coverage between point 0D, linear PSD 1D, and area 2D detectors. 73 Powder Diffr., Vol. 18, No. 2, June 2003 Introduction to two-dimensional X-ray diffraction 73

4 Figure 6. The geometric definition of diffraction rings in laboratory axes. is at off-axis position ( 0 ), the conic section may be an ellipse, parabola, or hyperbola. For convenience, all kinds of conic sections will be referred to as diffraction rings or Debye rings alternatively hereafter in this paper. A 2D diffraction image collected in a single exposure will be referred to as a frame. The frame is normally stored as intensity values on 2D pixels. The determination of the diffracted beam direction involves the conversion of pixel information into the 2 coordinates. In an XRD 2 system, and 2 values at each pixel position are given according to the detector position. The diffraction rings can be displayed in terms of and 2 coordinates, disregarding the actual shape of each diffraction ring. D. Detector position in the laboratory system detector space The detector position is defined by the sample-todetector distance D and the detector swing angle. D is the perpendicular distance from the goniometer center to the detection plane and is a right-handed rotation angle above Z L axis. In the laboratory coordinates X L Y L Z L, detectors at different positions are shown in Figure 9. The center of detector 1 is right on the positive side of X L axis on-axis, 0. Both detectors 2 and 3 are rotated away from the X L axis with negative swing angles ( 2 0 and 3 0). The swing angle is also called as detector two-theta and denoted by 2 D in previous publications. It is very important to distinguish between the Bragg angle 2 and detector angle. At a given detector angle, a range of 2 values can be measured. Figure 8. A diffraction cone and the conic section with a 2D detector plane. E. Sample orientation and location in the laboratory system sample space In an XRD 2 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 an Eulerian cradle four-circle type geometry, a kappa geometry, or other kind of geometry. The four-circle geometry will be discussed in this paper. The three angles in four-circle geometry are omega, g goniometer chi, and phi. Since the symbol has also been used for the azimuthal angle on the diffraction cones in the previous publications and some software, a subscript g indicates the angle is a goniometer angle. Figure 10 a shows the relationship between rotation axes, g, and the laboratory system X L Y L Z L. is defined as a right-handed rotation about the Z L axis. The axis is fixed on the laboratory coordinates. g is a left-handed rotation about a horizontal axis. The g axis makes an angle of with the X L axis in the X L Y L plane. The g axis lies on X L when is set at zero. is a lefthanded rotation. The g angle is also the angle between axis and the Z L. Figure 10 b shows the relationship among all rotation axes, g,, and sample axes S 1, S 2, and S 3. The is Figure 7. Schematics of an ideal detector covering 4 solid angle. Figure 9. Detector position in the laboratory system X L Y L Z L : D is the sample-to-detector distance; is the swing angle of the detector. 74 Powder Diffr., Vol. 18, No. 2, June 2003 Bob Baoping He 74

5 Figure 10. Sample rotation and translation. a Three rotation axes in X L Y L Z L coordinates; b rotation axes, g,, and sample axes S 1 S 2 S 3. the base rotation, all other rotations and translations are on top of this rotation. The next rotation above is the g rotation. is also a rotation above a horizontal axis. and g have the same axis but different starting positions and rotation directions, and g 90. will be used hereafter whenever possible in this article. The next rotation above and g ( ) is rotation. When 0, the relationship between two coordinates S 1 X L, S 2 Z L, and S 3 Y L. The rotation axis is always the S 3 axis at any sample orientation. The S 1 S 2 plane is the sample surface plane and S 3 is the sample surface normal. In an aligned diffraction system, all three rotation axes and the primary X-ray beam cross at the origin of X L Y L Z L coordinates. This cross point is also known as goniometer center or instrument center. Alternatively, the sample translation coordinates X, Y, and Z are also used in the XRD 2. The sample coordinates S 1, S 2, and S 3 have the same directions as the sample translation coordinates X, Y, and Z, respectively. The sample coordinates S 1 S 2 S 3 are considered as fixed on the sample with the origin on the instrument center, while the XYZ translations bring different parts of the sample into the instrument center. In practice, the two coordinate systems may be used alternatively. However the differences between the two coordinates should not be ignored. In the two-dimensional diffraction data analysis, it is crucial to know the diffraction vector distribution in terms of the sample coordinates S 1 S 2 S 3. However, the diffraction vector distribution corresponding to the measured 2D data is always given in terms of the laboratory coordinates X L Y L Z L. The angular relationships between the laboratory coordinates X L Y L Z L and the sample coordinates S 1 S 2 S 3 are: X L Y L Z L S 1 a 11 a 12 a 13 S 2 a 21 a 22 a 23 2 S 3 a 31 a 32 a 33 Based on the above-defined Eulerian geometry, the transformation matrix from the laboratory coordinates X L Y L Z L to the sample coordinates S 1 S 2 S 3 is sin sin 11 a 12 a 13 cos cos A a a 21 a 22 a 23 sin cos a 31 a 32 a 33 sin cos sin cos sin sin sin cos cos sin cos sin sin cos sin sin cos cos cos sin The unit vector h S of the diffraction vector in the sample coordinates S 1 S 2 S 3 is given by. 3 cos cos h s Ah L. 4 In matrix form, we have h 1 h 2 sin sin cos sin sin cos sin cos cos sin cos h y sin cos a 31 a 32 a h 33 hx z sin cos sin cos cos cos cos sin sin sin sin cos cos cos sin a12 a13 a 21 a 22 a 23 h 3 a11 sin cos sin cos cos. 5 Or in expansion: h 1 sin sin sin sin cos cos cos cos sin cos cos sin sin sin cos cos sin, h 2 sin cos sin sin sin cos cos cos cos cos cos sin cos sin cos sin sin, 6 h 3 sin cos sin cos sin cos cos cos cos sin. 75 Powder Diffr., Vol. 18, No. 2, June 2003 Introduction to two-dimensional X-ray diffraction 75

6 Figure 11. 2D frames collected in transmission mode from NIST Standard Reference Material 660, LaB 6 : a no sample oscillation; b 10 oscillation. F. Summary of XRD 2 geometry All three spaces are based on the laboratory coordinates. The diffraction space is mainly defined by the sample crystal structure and X-ray beam direction and wavelength. The detector space is determined by the 2D detector size, distance to the sample, and swing angle. The sample space is defined by the sample location and orientation. Selection of detector space should be based on the diffraction space. Changing the detector space changes the part to be measured of the diffraction space and the measurement resolution, but not the diffraction space itself. Changing sample space will not change the diffraction space of an ideal powder polycrystalline sample which has no stress and texture. Changing the sample orientation will change the diffraction space when there is texture or stress in the sample. Changing the sample location will change the diffraction space for an inhomogeneous sample. IV. PHASE IDENTIFICATION WITH XRD 2 The diffraction profiles measured with an XRD 2 system have better statistics compared with the diffraction profiles collected by conventional diffractometer, especially for those samples with texture, large grain size, or of small quantity. A. 2D frame integration The diffraction pattern measured with a 2D detector consists of a series of diffraction rings as is shown in Figure 11. A diffraction profile similar to the conventional diffraction result can be obtained from the 2D frame by integration over a selected 2 range. Phase identification can then be done with the conventional search/match method. A flat 2D detector has the flexibility to be used at different sample-todetector distance. 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 large distance, and higher angular coverage at short distance. The sample-to-detector distance should be optimized depending on the 2 range and resolution. In case that 2 range is not enough at the desired detector distance, several frames at sequential 2 position can be collected. The integrated profiles then can be merged to achieve the large 2 range. For a flat sample in reflection mode, it is also desirable to collect several frames at different incident angles for different 2 range so as to reduce the defocusing effect. Figure 11 shows two 2D frames collected in transmission mode diffraction from NIST Standard Reference Material 660, LaB 6. The -integration box and the integrated diffraction profile are overlaid on the original frames. Comparing the frame a without oscillation and frame b with 10 oscillation, it shows that the oscillation generated smoother diffraction rings. B. Virtual oscillation In the case of materials with large grain size, preferred orientation or microdiffraction with a small X-ray beam size, it is difficult to determine the 2 position due to poor counting statistics. To solve this problem with conventional detectors, some kind of sample oscillations, either by translation or by rotation, is necessary to bring more crystallites into diffraction condition. In other words, the purpose of oscillations is to bring more crystallites in the condition that the normal of the diffracting crystal plane coincides with the instrument diffraction vector. Figure 11 b shows the effect of oscillation. A similar effect can also be achieved by oscillation. For 2D detectors, when the integration is used to generate the diffraction profile, it actually integrates the data collected in a range of various diffraction vectors. The angle between two extreme diffraction vectors is equivalent to the oscillation angle in oscillation. Therefore, we may call this effect virtual oscillation. Figure 12 shows the relation between the -integration range,, and the virtual oscillation angle,. The 2 value of the -integrated pro- 76 Powder Diffr., Vol. 18, No. 2, June 2003 Bob Baoping He 76

7 Figure 12. The relation between the range,, and the virtual oscillation angle. Figure 13. Geometry of XRD 2 : a reflection mode; b transmission mode. file is an average over the Debye ring defined by the range. The average effect is over a region of orientation distribution, rather than a volume distribution. The virtual oscillation angle can be calculated from the integration range and Bragg angle, 2 arcsin cos sin /2. 7 For example, in Figure 11, the center peak , the integration range , 30, so the virtual oscillation angle In the 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, the virtual oscillation can avoid this error. C. Transmission and reflection In an XRD 2 system, the diffracted X-rays are measured simultaneously in a two-dimensional range so no slit or scanning step can be used to control the instrument broadening. Figure 13 shows geometry of two-dimensional diffraction in reflection mode a and transmission mode b. Defocusing effect is observed with low incident angle over a flat sample surface in reflection mode diffraction. In reflection mode, the diffracted beam in low 2 angle is narrower than the diffracted beam in high 2 angle. In transmission mode with the incident beam perpendicular to the sample surface, no such defocusing effect is observed. If one looks at the cross section on the diffractometer plane and forward diffraction (2 90 ) only, the defocusing effect with reflection mode diffraction can be expressed as B sin 2, 8 b sin where is the incident angle, b is the incident beam size, and B is diffracted beam size. The defocusing with transmission mode with a perpendicular incident beam can be given as B b cos 2 t sin 2, b 9 where t is the sample thickness. If the sample thickness t is negligible compared to the incident beam size b, we have B cos 2 1. b 10 There should be no defocusing effect at all. Figure 14 is a comparison between reflection mode and transmission mode diffraction with data frames collected from corundum powder. With 5 incident angle a, the reflection pattern shows severe peak broadening compared with no defocusing in transmission mode pattern b. In many combinatorial screening applications, such as polymorphism study in pharmaceutical chemistry and catalysis development in the oil industry, a typical 2 measuring range is from 2 to 60. It is necessary to run the combinatorial XRD screening in transmission mode in order to avoid the defocusing effect. In the transmission mode X-ray diffraction measurement, the incident beam is typically perpendicular to the sample so the irradiated area on the specimen is limited to a size comparable to the X-ray beam size, allowing the X-ray beam to concentrate on the intended measuring area. In combinatorial screening applications, sample Figure 14. Diffraction pattern from corundum: a reflection mode diffraction 5 incident angle, b transmission mode diffraction with perpendicular incident beam. 77 Powder Diffr., Vol. 18, No. 2, June 2003 Introduction to two-dimensional X-ray diffraction 77

8 cells are located close to each other. Therefore, the transmission mode diffraction can also avoid cross contamination between adjacent samples. V. STRESS MEASUREMENT WITH XRD 2 When used for stress measurement, two-dimensional X-ray diffraction systems have many advantages over conventional one-dimensional diffraction systems in dealing with highly textured materials, large grain size, small sample area, weak diffraction, stress mapping, and stress tensor measurement He et al., 1998; He and Smith, 1998; Yoshioka and Ohya, 1994; Borgonovi, 1984; Sasaki et al., A. Fundamental equations for stress measurement with XRD 2 The stress measurement is based on the fundamental relationship between the stress tensor and the diffraction cone distortion. The benefit of the 2D method is that all the data points on diffraction rings are used to calculate stresses so as to get better measurement result with less data collection time. The diffraction cones from a stress-free polycrystalline sample are regular cones in which 2 is a constant. The stress in the sample distorts the diffraction cone shape so that they are no longer regular cones. Figure 15 shows a diffraction cone cross-section on a 2D detector plane. The 2 becomes a function of, 2 2 ( ), this function is uniquely determined by the stress tensor and the sample orientation. The fundamental equation for strain measurement using 2D detector is given He and Smith, 1998 as f f f f f f ln sin 0 sin with: Figure 15. Diffraction cone distortion due to stress. 11 f 11 h 2 1, f 12 2h 1 h 2, f 22 h 2 2, f 13 2h 1 h 3, f 23 2h 2 h 3, 2 f 33 h 3 a sin cos sin cos sin h 1 a cos b cos sin c sin sin b cos cos h 2 a sin b cos cos c sin cos c sin sin sin cos cos h 3 b sin c cos where f ij s are the strain coefficients determined by the sample orientation and diffraction vector direction for each data point on the diffraction ring, h 1,h 2,h 3 are components of the unit vector of the diffraction vector H hkl expressed in the sample coordinates, and ln(sin 0 /sin ) represents the diffraction cone distortion at a particular, 2 position. Equation 11 is the fundamental equation for strain and stress measurement by diffraction using 2D detectors, which gives a direct relation between the diffraction cone distortion and strain tensor. Since it is a linear equation, the least squares method can be used to solve the strain or stress tensor with very high accuracy and low statistics error. For isotropic materials, there are only two independent elastic constants, Young s modulus E and Poisson s ratio or the 1 macroscopic elastic constants 2 S 2 (1 )/E and S 1 /E. Then we have where p p p p p p ln sin 0 sin, 12 p ij 1/E 1 f ij 1 2 S 2 f ij S 1 1/E 1 f ij 1 2 S 2 f ij if i j if i j. 13 Each diffraction frame corresponds to one set of sample orientation,, and. The diffraction ring on each frame is integrated and peak-fitted over a selected number of sections along the ring, so as to obtain a set of, 2 data points representing the 2 ( ) function. The stress tensor can be determined by fitting the data points to Eq. 12 with the least-squares method. The fundamental equation for stress measurement with two-dimensional diffraction method can also be written as S S 2 11 h h h h 1 h h 1 h h 2 h 3 ln sin 0 sin. 14 B. Relationship between the 2D equation and conventional equation As shown in Figure 6, the conventional diffraction is limited in the diffractometer plane. The diffraction data col- 78 Powder Diffr., Vol. 18, No. 2, June 2003 Bob Baoping He 78

9 lected at 90 for negative detector swing and 90 ( 270) for positive detector swing are within the diffractometer plane. Considering the positive and negative detector swing angles and both the diffractometer and the diffractometer, we have a total of four configurations to imitate the conventional system. For example, at positive detector swing angle with a -diffractometer condition, we have the following conditions: 90 using positive detector swing angle ; no omega rotation with a diffractometer ; the -tilt angles are achieved by rotation ; 90 the tilt direction is 90 away from S 1 direction at 0 when using a diffractometer. With the above-mentioned conditions, the unit vector in sample coordinates is given as sin sin h cos cos 1 h 2 sin cos h 3 sin cos sin cos sin sin sin cos cos sin cos sin cos cos cos cos sin sin sin 0 sin cos cos cos sin sin sin sin cos sin cos sin sin. cos cos 15 Considering at any and angles, the measured 2 shift on diffraction cone is corresponding to the rational strain in that direction, we replace ln(sin 0 /sin ) by, then we have exactly the same equation as the formula for strain measurement using in X-ray diffraction:, S S 2 33 cos S 2 11 cos 2 22 sin 2 12y sin 2 sin S 2 13 cos 23 sin sin We can see that the fundamental equation for stress measurement with two-dimensional diffraction is a general equation which defines the relationship between stress tensor and diffraction cone distortion. When dealing with a single diffraction profile measured with a conventional diffractometer, the general equation reduces to the conventional equation. It can be easily proved that the same is true for all the other three configurations. All the above-mentioned results show that the 2D fundamental equation covers the conventional fundamental equation, or in other words, the conventional fundamental equation is a special case of the more general 2D fundamental equation. The conditions for imitating all four configurations are summarized in the following: Detector swing diffractometer diffractometer Positive: 0 0 /2 /2 Negative: 0 0 /2 /2 C. Biaxial stress state The biaxial stress state, in which one assumes that there is no force acting on the free surface in the direction perpendicular to the sample surface, is the most common stress state in practice. Since the strain components , 33 /(1 ) ( ), Eq. 5-2 becomes p p p p ph ph ln sin 0 sin, 17 where the coefficient p ph (1 2 )/E, ph is a pseudohydrostatic stress component caused by the approximate d-spacing d 0. For biaxial stress with shear, we have p p p p p p ph ph ln sin 0 sin. 18 The biaxial stress state corresponds to the straight line of the d-sin 2 plot. And the biaxial stress with shear is the case when there is a split between the data points in side and side. The general normal stress ( ) and shear stress ( ) at any arbitrary given angle are given by 11 cos 2 12 sin 2 22 sin 2, cos 23 sin. D. True stress-free lattice D spacing 20 In the biaxial 2D and biaxial shear 2D calculation, we have assumed that 33 is zero so that we can calculate stress with an approximation of d 0 or 2 0 ). Any error in d 0 or 2 0 ) will contribute only to a pseudohydrostatic term ph. Figure 16 shows the biaxial stress tensor measured from a shot peened Almen strip with different input d 0 in the range of Å. The measured stress tensor is independent of the input d 0 ( MPa, MPa, MPa), where the pseudohydrostatic term ph changes with the input d 0. The true d 0 corresponds to the cross point of ph line and zero stress. If we use d 0 to represent the initial input, then the true d 0 or 2 0 ) can be calculated from ph with the following: d 0 d 0 exp 1 2 E ph 21 or 79 Powder Diffr., Vol. 18, No. 2, June 2003 Introduction to two-dimensional X-ray diffraction 79

10 Figure 16. The measured biaxial stress tensor and pseudohydrostatic stress as a function of input d 0. E. Anisotropy factor 0 arcsin sin 0 exp 2 1 E ph. 22 The anisotropy correction can also be included in the X-ray elastic constants 1 2 S 2 (hkl) and S 1 (hkl) to replace the macroscopic elastic constants 1 2 S 2 and S 1. The equations for calculating X-ray elastic constants are: 1 2 S 2 hkl 1 2 S hkl, S 1 hkl S S hkl, 23 hkl h2 k 2 k 2 l 2 l 2 h 2 h 2 k 2 l 2 2, 5 A RX A RX The factor of anisotropy (A RX ) is a measure for the elastic anisotropy of a material. Values of A RX for the most important cubic materials are given in the following, additional values may be found from literature. Materials A RX Body-centered cubic bcc Fe-base materials 1.49 Face-centered cubic fcc Fe-base materials 1.72 Face-centered cubic fcc Cu-base materials 1.09 Ni-base materials fcc 1.52 Al-base materials fcc 1.65 F. Comparison between conventional method and 2D method In theory, it has been previously proved that the conventional fundamental equation is a special case of the 2D fundamental equation. In the same way, a conventional detector can be considered as a limited part of a 2D detector. Depending on the specific condition, one can choose either theory for stress measurement when a 2D detector is used. If the conventional theory is used, one has to get a diffraction profile at 90 or 90, this is normally done by integrating the data in a limited range. The disadvantage is that only part of the diffraction ring is used for stress calculation. When the new 2D theory is used, all parts of the diffraction ring can be used for stress calculation. Experimental results also show a good correlation between the two methods. Ten almen strips were used for the residual stress measurement. The almen strips have hardness of 55 HRC and had been shot peened in both faces for 30 min with S170 cast steel shot. The samples were loaded by three operators, and each sample was measured three times by each operator. A total of 90 stress measurements were taken. The discrepancy between the conventional method and the 2D method is very small. The correlation between the conventional 1D method and 2D method for the 10 samples is shown in Figure 17. In the conventional -tilt method for stress measurement, the tilt is achieved by either rotation on an diffractometer iso-inclination or rotation on a diffractometer side-inclination. Both methods collect diffraction data in reflection mode. The fundamental equation for 2D detectors was derived without requiring the diffraction in reflection mode, so that Eqs. 11 and 12 can be used for transmission mode diffraction without any modification. More experiments are necessary to further explore the stress measurement with transmission mode diffraction. G. Summary of stress measurement with XRD 2 Two-dimensional X-ray diffraction (XRD 2 ) systems, when used for residual stress measurement, have many advantages over the conventional one-dimensional diffraction systems in dealing with highly textured materials, large grain size, small sample area, weak diffraction, stress mapping, and stress tensor measurement. The 2D fundamental equation is the basis of stress measurement in an XRD 2 system. For biaxial stress measurement, the approximation of the d spacing or 2 input for stress-free condition does not cause error in the stress calculation. The true stress-free d-spacing can be calculated from the pseudohydrostatic term. The conventional method and the 2D method are consistent both in theory and applications. The same equation can be used for both reflection mode and transmission mode diffraction. VI. TEXTURE MEASUREMENT WITH XRD 2 When used for texture measurement, two-dimensional X-ray diffraction systems have many advantages over the conventional one-dimensional diffraction systems 80 Powder Diffr., Vol. 18, No. 2, June 2003 Bob Baoping He 80

11 Figure 17. The stress values calculated by the conventional 1D and 2D methods from the ten samples, average over nine measurements for each sample. Bunge and Klein, 1996; Smith and Ortega, The texture measurement is based on a fundamental relationship between the pole-figure angles and the intensity distribution along diffraction ring. The benefit of XRD 2 is that several pole figures can be measured simultaneously and that all the data points on a diffraction ring are used to calculate a onedimensional pole density mapping so as to get better measurement result with less data collection time. A. Pole density and pole figure XRD results from ideally random powder normally serve as a basis for determining the relative intensity of each crystalline peak. In real life, polycrystalline materials usually do not have randomly oriented grains. The deviation of the statistic grain orientation distribution of a polycrystalline material from the ideal random powder is measured as texture or preferred orientation. The pole-figure for a particular crystalline plane is normally used to represent the texture of a sample. If all grains or crystallites have the same volume, each pole represents a grain which satisfies the Bragg condition as shown in Figure 18 a. The pole has the same orientation as the diffraction vector (H hkl ). If we take the diffraction peak from a random powder as reference Figure 18 b, the diffraction peak intensity change is from the texture while the peak shift is due to stress. The measured 2D-diffraction pattern contains two most important parameters at each angle, the partially integrated intensity I and the Bragg angle 2. Figure 19 shows the diffraction cone distortion due to stress and diffraction intensity variation along due to texture. For a stressed sample, the 2 becomes a function of and the sample orientation,,, i.e., 2 2 (,,, ), this function is uniquely determined by the stress tensor. For a textured sample, the intensity is a function and the sample orientation,,, i.e., I I(,,, ), which is uniquely determined by the orientation distribution function. 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 Figure 20, each pole direction is defined by the radial angle and azimuthal angle. The pole densities at all directions are mapped on the equatorial plane by stereographic projection. The pole density at the point P would project to the point P on the equatorial plane. The two-dimensional map on the equatorial plane is called pole figure. B. Fundamental equations for texture measurement with XRD 2 For a textured sample, the intensity is a function of and the sample orientation,,, i.e., I I(,,, ), which is uniquely determined by the orientation distribution function. Each pole direction is defined by the radial angle and Figure 18. a The definition of pole and b diffraction peak intensity change due to texture and peak shift due to stress. Figure 19. Diffraction cone distortion due to stress and diffraction intensity variation along due to texture. 81 Powder Diffr., Vol. 18, No. 2, June 2003 Introduction to two-dimensional X-ray diffraction 81

12 Figure 21. Comparison between the pole-figure measurement with the conventional X-ray diffraction and two-dimensional X-ray diffraction. Figure 20. Definition of the angles and and stereographic projection. created at each exposure. For the same seven positions, the poles measured can map a large area in the pole figure. Therefore, when a two-dimensional diffraction system is used for texture measurement, much smaller scan steps can be used to achieve high resolution pole figure and the data collection time can also be dramatically reduced. azimuthal angle. The and angles are functions of,,,, and 2. The pole density at the pole figure angles, is proportional to the integrated intensity in the same angles: I hkl N hkl P hkl, 24 where I hkl ( ) is the integrated intensity corrected by absorption, polarization, background, etc. N hkl is the normalization factor, and P hkl ( ) is the pole density distribution function. The relationship between the pole figure angles,, the sample orientation,,, and diffraction cone 2, is given by sin 1 h 3, h cot 1 1 if h h cot h, 1 1 if h h where h 1,h 2,h 3 are components of the unit vector of the diffraction vector H hkl. They are given by Eq. 5 or Eq. 6. The diffraction intensity along the diffraction ring is then converted to the pole density at each and angles from,,,, and 2 angles. The pole-figure s relative intensity can be normalized such that it represents a fraction of the total diffracted intensity integrated over the pole sphere. Figure 21 is a comparison between the pole-figure measurement with conventional X-ray diffraction and two-dimensional X-ray diffraction. With the conventional X-ray diffraction, one pole marked by the diffraction vector H hkl ) is measured at each sample angle. As an example, with seven different positions, only seven poles are measured. With twodimensional X-ray diffraction, numerous poles are measured at each sample angle. A one-dimensional pole mapping is C. Data collection strategy scheme Since one-dimensional pole density mapping is created with each exposure with XRD 2, it is important to layout a data collection strategy so as to have the optimum polefigure coverage and less redundancy in data collection. Figure 22 shows a scheme generated at 2 40, 20, 35.26, and D 7 cm with scan of 5 steps for Bruker GADDS General Area Detector Diffraction System. The real data collection can use smaller scanning steps, such as 1 2. The data collected with a single exposure at 0 would generate a one-dimensional pole figure as shown as the curve marked by A and B. The pole figure measured with the data collection strategy in the above-given example has a blank hole in the center. The pole density at the center represents the diffraction vector perpendicular to the sample surface. The pole-figure angle at the center is 90, so the condition to achieve the filled center in a pole figure is that one pole density within A B curve is collected at or sin 1 h 3 90 h 3 sin cos sin cos sin cos A cos cos cos sin In order to avoid redundancy in the data collection, the best strategy is to put point A at the center of pole figure. That is h 3 A sin cos sin cos sin A cos cos cos cos A sin Powder Diffr., Vol. 18, No. 2, June 2003 Bob Baoping He 82

13 Figure 22. A scheme generated at 2 40, 20, ( g ), and D 7 cm with scan of 5 steps. The curve A B is pole-figure mapping at 0. If we modify the parameters in the example of Figure 22 to 23, 30 for the same 2 40 and D 7 cm with scan of 5 steps. The angle measured on the same detector at point A is A 122. Then we have h 3 A 1. The scheme generated by GADDS software is shown in Figure 23. The center of the pole figure is filled with pole density data. The data collection parameters may be optimized by trial-anderror or calculation with Eq. 28. Figure 23. A scheme generated at 2 40, 23, 30 ( g 60 ), and D 7 cm with scan of 5 steps. 83 Powder Diffr., Vol. 18, No. 2, June 2003 Introduction to two-dimensional X-ray diffraction 83

14 Figure 25. Pole figures of 111 and 222 processed from the same set of data frames collected on Al foil. Figure 24. Diffraction frame collected from Al sample at D 6 cm. Diffraction rings from five crystalline planes are collected simultaneously. The low and high background and diffraction ring 2 range are defined for 220 plane. D. Texture data process Figure 24 shows one of the diffraction frames collected on an aluminum sample for texture analysis. At detector distance of 6 cm with Cu radiation, a total of five diffraction rings from the crystalline planes of 111, 200, 220, 311, and 222 can be measured simultaneously. Therefore, five pole figures can be measured simultaneously. The low and high background and diffraction ring 2 range are defined by three boxes. The integrated diffraction intensities at various angles are mapped into pole figure as defined by Eqs. 24 and 25. The background can be removed by the intensity values defined in the low and high background boxes or ignored at user s discretion. Figure 25 shows the pole figures of 111 and 222 planes processed from the same set of data frames collected on an Al foil. Both pole figures show the same trend because the 111 and 222 poles are identical. The two-dimensional diffraction frames can also directly reveal texture and grain size information qualitatively even before the data processing. For example, Figure 26 shows two frames collected from two -TiAl alloy samples, one with large grain and weak texture, the other with small grain and strong texture. One can immediately tell that a is from the sample with larger grain and weak texture, while b is from the sample with fine grain and strong texture. E. Summary of texture measurement with XRD 2 Two-dimensional X-ray diffraction (XRD 2 ) systems have many advantages over the conventional one- Figure 26. Frames from -TiAl alloys: a Large grain and weak texture, b small grain and strong texture. 84 Powder Diffr., Vol. 18, No. 2, June 2003 Bob Baoping He 84

15 dimensional diffraction systems when used for texture measurement. A 2D detector can measure several diffraction rings simultaneously, and each diffraction ring represents a continuous pole density distribution compared to one pole per exposure with the conventional system. Therefore an XRD 2 can measure texture from samples with polycrystal, single crystal, and mixture of both with high resolution and high speed. The orientation relationship between different phases, or thin film and substrate can be revealed because the measurements from all phases of the sample are done simultaneously. Qualitative texture and grain size information can be observed directly from the 2D frame without processing the data. VII. SUMMARY OF XRD 2 Compared to a conventional one-dimensional diffraction system, an XRD 2 system has many advantages in various applications. The following three applications are discussed in this paper. Some applications are not covered here. Please read the reference or contact the author for details. 1 Phase identification (phase ID) can be done by integration over a selected range of 2 and. The integrated data give better intensity and statistics for phase ID and quantitative analysis, especially for those samples with texture, large grain size, or small quantity Sulyanov et al., Texture measurement is extremely fast. An XRD 2 system collects texture data and background values simultaneously for multiple poles and multiple directions. Due to the high measurement speed, pole figure can be measured at very fine steps for sharp textures Bunge and Klein, 1996; Smith and Ortega, Stress can be measured using the 2D fundamental equation, which gives the direct relationship between the stress tensor and the diffraction cone distortion. Since the whole or a part of the Debye ring is used for stress calculation, it can measure stress with high sensitivity, high speed, and high accuracy. It is very suitable for large grain and textured samples He et al., 1998; He and Smith, Borgonovi, G. M Determination of Residual Stress from Two- Dimensional Diffraction Pattern, Nondestructive Methods for Material Property Determination Plenum, New York, p.47. Bunge, H. J., and Klein, H Determination of quantitative, highresolution pole figures with the Aea detector, Z. Metallkd. 87, He, B. B., Preckwinkel, U., and Smith, K. L Advantages of using 2D detectors for residual stress measurements, Advances in X-ray Analysis Vol. 42, the 47th Annual Denver X-ray Conference, Colorado Springs, CO. He, B. B., Preckwinkel, U., and Smith, K. L Fundamentals of two-dimensional x-ray diffraction (XRD 2 ), Advances in X-ray Analysis Vol. 43, the 48th Annual Denver X-ray Conference, Steamboat Springs, CO. He, B. B., and Smith, K. L Fundamental equation of strain and stress measurement using 2D detectors, Proceedings of 1998 SEM Spring Conference on Experimental and Applied Mechanics, Houston, TX. Jenkins, R., and Snyder, R. L Introduction to X-ray Powder Diffractometry Wiley, New York. Rudolf, P. R., and Landes, B. G Two-dimensional X-ray diffraction and scattering of microcrystalline and polymeric materials, Spectroscopy 9, Sasaki, T. et al Influence of image processing conditions of Debye Scherrer ring images in x-ray stress measurement using imaging plate, Adv. X-Ray Anal. 40, the 45th Annual Denver X-ray Conference, Denver, CO. Smith, K. L., and Ortega, R. B Use of a two-dimensional, position sensitive detector for collecting pole figures, Adv. X-ray Anal. 36, Sulyanov, S. N., Popov, A. N., and Kheiker, D. M Using a twodimensional detector for X-ray powder diffractometry, J. Appl. Crystallogr. 27, Yoshioka, Y., and Ohya, S X-ray analysis of stress in a localized area by use of image plate, Proceedings of ICRS-4, Baltimore, MD. 85 Powder Diffr., Vol. 18, No. 2, June 2003 Introduction to two-dimensional X-ray diffraction 85