Introduction to XRD

TECHNOLOGYthink forward

Bruker AXS

Introductionto 2-dimensional X-ray Diffraction

Part Number: M86-E01055

Publication date: 31 January 2008

XRD

Introduction to 2-Dimensional X-ray Diffraction

Bruker AXS Inc.

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Phone +1 (800) 234-XRAY [9729] Fax +1 (608) 276-3006

E-mail: [email protected] www.bruker-axs.com

Bruker BioSciences

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Phone +1 (978) 663-3660 Fax: +1 (978) 667-5993

E-mail: [email protected] www.bruker-biosciences.com

This document introduces the concepts and applications of 2-dimensional X-ray diffraction.References to this document should be shown as M86-Exx055 Introduction to 2-dimensional X-ray Diffraction.

2008 Bruker AXS Inc., 5465 East Cheryl Parkway, Madison, WI 53711. All world rights reserved.

NoticeThe information in this publication is provided for reference only. All information contained in this publication is believed to be cor-rect and complete. Bruker AXS Inc. shall not be liable for errors contained herein, nor for incidental or consequential damages in conjunction with the furnishing, performance, or use of this material. All product specifications, as well as the information con-tained in this publication, are subject to change without notice.This publication may contain or reference information and products protected by copyrights or patents and does not convey any license under the patent rights of Bruker AXS Inc. nor the rights of others. Bruker AXS Inc. does not assume any liabilities arising out of any infringements of patents or other rights of third parties. Bruker AXS Inc. makes no warranty of any kind with regard to this material, including but not limited to the implied warranties of merchantability and fitness for a particular purpose.No part of this publication may be stored in a retrieval system, transmitted, or reproduced in any way, including but not limited to photocopy, photography, magnetic, or other record without prior written permission of Bruker AXS Inc.

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Revision History

Revision Date Changes

0 June 2002 Original release.

1 31 January 2008 Edited for style, form and content; addition of comprehensive index.

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Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract-1

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-1

2 X-ray Diffraction with Two-Dimensional Detectors. . . . . . . . . . . . . . . . . . . .2-1

2.1 X-ray Powder Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1

2.2 Comparison Between XRD and XRD2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3

3 Geometry Conventions in XRD2 Systems . . . . . . . . . . . . . . . . . . . . . . . . . . .3-1

3.1 Diffraction Cones in Laboratory Axes (Diffraction Space) . . . . . . . . . . . . . . . . . . . . . 3-13.1.1 Ideal Detector for Diffraction Pattern in 3D Space . . . . . . . . . . . . . . . . . . . . . . . . 3-23.1.2 Diffraction Cones and Conic Sections on 2D Detectors. . . . . . . . . . . . . . . . . . . . 3-3

3.2 Detector Position in the Laboratory System (Detector Space) . . . . . . . . . . . . . . . . . 3-43.3 Sample Orientation and Location in the Laboratory System (Sample Space) . . . . . 3-53.4 Summary of XRD2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-7

4 X-ray Optics for XRD2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-1

4.1 X-ray Beam Shape for XRD2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2

4.2 Beam Spread Over Flat Sample Surface (Geometry Broadening) . . . . . . . . . . . . . . . 4-34.3 Monochromator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4

4.4 Cross-Coupled Gbel Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5

4.5 Pinhole Collimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-7

4.6 Monocapillary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-10

4.7 Other Features of X-ray Optics for XRD2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-11

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5 Stress Measurement With XRD2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1

5.1 Fundamental Equations for Stress Measurement with XRD2 . . . . . . . . . . . . . . . . . . . 5-1

5.2 True Stress-Free Lattice D-Spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3

5.3 Anisotropy Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-4

5.4 Relationship between Conventional Theory and 2D Theory. . . . . . . . . . . . . . . . . . . . 5-5

5.5 Virtual Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-7

5.6 Data Collection Strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-9

5.7 2D Detectors for Stress Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-11

5.8 Summary of Stress Measurement with XRD2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-12

6 Texture Measurement With XRD2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1

6.1 Pole Density and Pole Figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-2

6.2 Fundamental Equations for Texture Measurement with XRD2 . . . . . . . . . . . . . . . . . . 6-5

6.3 Data Collection Strategy - Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-7

6.4 Texture Data Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-9

6.5 Texture Measurement in Transmission Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-11

6.6 Other Application Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-12

6.7 Summary of Texture Measurement with XRD2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-13

7 Combinatorial Screening with XRD2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1

7.2 XRD2 System for Combinatorial Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2

7.3 XYZ Sample Stage and Screening Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-3

7.4 Laser/Video Microscope Automatic Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-4

7.5 Diffraction Mapping and Results Display . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-5

8 Percent Crystallinity With XRD2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1

8.1 Principle of Percent Crystallinity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1

8.2 Comparison of Conventional XRD and XRD2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-3

8.3 Scatter Correction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-4

8.4 Percent Crystallinity: Internal and External Methods . . . . . . . . . . . . . . . . . . . . . . . . . 8-58.4.1 Internal Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-58.4.2 External Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-6

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8.5 Percent Crystallinity: Full Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-7

9 Summary of XRD2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9-1

10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10-1

Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index-1

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Abstract

Material in this Introduction was presented at the 50th Annual Denver X-ray Conference Steamboat Spring, Colorado, USA 30th July - 3rd August 2001.

Two-dimensional X-ray diffraction (XRD2) refers to X-ray diffraction applications with two-dimensional (2D) detectors and corresponding data reduction and analysis. The 2D diffraction pattern contains far more information than a one-dimensional profile collected with the conventional diffractometer.

In order to take advantage of the 2D detector, new approaches are necessary to configure the X-ray diffraction system and to analyze the 2D diffraction data. This Introduction discusses some fundamentals of XRD2, such as geometry conventions, diffraction data interpretation, point beam optics, and advantages of XRD2 in various applications.

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1 Introduction

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 (PSDs). Therefore, almost all X-ray powder diffraction applicationssuch 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. [1]

In recent years, usage of two-dimensional (2D) detectors has dramatically increased due to the advances in detector technology, point beam X-ray optics, and computing power [2,3]. Although a 2D image contains far more information than a 1D profile, the potential advantages of a 2D detector cannot be fully realized if the data interpretation and analysis methods are simply inherited from the conventional diffraction theory. Two-dimensional X-ray Diffraction (XRD2) is a new technique in the field of X-ray diffraction; it 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. Due to the unique nature of data collected with a 2D detector, a completely new concept and approach are necessary to configure a XRD2 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 [4].

An XRD2 system is a diffraction system with the capability of acquiring a diffraction pattern in 2D space and analyzing the 2D diffraction data accordingly.

XRD2 systems are available in a variety of configurations to fulfill the requirements of different applications and samples. A system normally consists of the following five major units (see typical system in Figure 1.1), each of which may have several options: X-ray Generator to produce X-rays with the required radiation energy, focal spot size, and intensity. X-ray Optics to condition the primary X-ray beam to the required wavelength, beam focus size,

beam profile, and divergence. Goniometer and Sample Stage to establish and manipulate the geometric relationship between

primary beam, sample, and detector. Sample Alignment and Monitor to assist users in positioning the sample to the instrument center

and in monitoring the sample state and position. Two-dimensional Detector to intercept and record the scattering X-rays from a sample, and to

save and display the diffraction pattern as a two-dimensional image frame.

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Figure 1.1 Five major components in an XRD2 system: an area detector, goniometer and sample stage, sample alignment and monitoring (laser/video) system, X-ray optics (monochromator and collimator), and an X-ray generator.

A typical XRD2 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. Figure 1.2 shows an XRD2 system: the Bruker D8 DISCOVER, configured for stress and texture [5]. Figure 1.3 shows a 2D diffraction pattern from corundum powder.

Figure 1.2 An XRD2 system for stress and texture

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Figure 1.3 A 2D diffraction pattern from corundum powder

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2 X-ray Diffraction with Two-Dimensional Detectors

2.1 X-ray Powder Diffraction

X-ray diffraction (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 X-ray scattering 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.

There are many theories and equations about the relationship between the diffraction pattern and the materials structure. Braggs law is a simple way to describe the diffraction of X-rays by a crystal. In Figure 2.1, the incident X-rays hit the crystal planes at an angle , and the reflection angle is also . The diffraction pattern is a delta function when the Bragg condition is satisfied:

where is the wavelength, d is the distance between each adjacent crystal planes (i.e., the d-spacing), and is the Bragg angle at which one observes a diffraction peak.

Figure 2.1 The incident X-rays and reflected X-rays make an angle of symmetric about the normal of crystal plane. The diffraction peak is observed at the Bragg angle .

2d sin=

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Figure 2.1 is an oversimplified model. For real materials, the diffraction patterns vary from theoretical delta functions, with discrete relationships between points, to continuous distributions with spherical symmetry. Figure 2.2 shows diffraction from a single crystal and from a polycrystalline sample. The diffracted rays from a single crystal point in discrete directions, each corresponding to a family of diffraction planes (Figure 2.2a). The diffraction pattern from a polycrystalline (i.e., powder) sample forms a series of diffraction cones if a large number of randomly-oriented crystals are covered by the incident X-ray beam (Figure 2.2b). Each diffraction cone corresponds to the diffraction from the same family of crystalline planes in all of the participating grains. Hereafter, the diffraction patterns from polycrystalline materials will be considered in this Introductions discussion of the theory and configuration of XRD2 systems. Polycrystalline materials can be single-phase or multi-phase in bulk solid, thin film, or fluid form. In practice, XRD2 applications are not limited to polycrystalline materials, and the sample can be a mixture of single-crystal, polycrystal, and amorphous materials.

Figure 2.2 Patterns of diffracted X-rays from (a) a single crystal and (b) a polycrystalline sample

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2.2 Comparison Between XRD and XRD2

First, we compare conventional X-ray diffraction (XRD) and two-dimensional X-ray diffraction (XRD2). Figure 2.3 is a schematic of X-ray diffraction from a polycrystalline (i.e., powder) sample. For simplicity, it shows only two diffraction cones: one for 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 a 2 scan along a detection circle. If a one-dimensional position-sensitive detector (PSD) is used in the diffractometer, it will be mounted along the detection circle. Since the diffraction patterns variation in the direction perpendicular to the diffractometer plane (i.e., the Z direction) is not considered in a conventional diffractometer, the X-ray beam is normally extended in the Z direction (called line focus). The actual diffraction pattern measured by a conventional diffractometer is an average over a range defined by beam size in the Z-direction. Since diffraction data outside the diffractometer plane is not detected, the materials structure represented by the missing diffraction data will be ignored, or extra sample rotation and time will be needed to complete the measurement.

Figure 2.3 The diffractometer planeand powder diffraction patterns in 3D spacein a conventional diffractometer

The conventional diffraction pattern, collected with either a scanning point detector or PSD, is a plot of X-ray scattering intensity at different 2 angles. Figure 2.4 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 (called Debye rings) can be measured simultaneously, depending on the detectors size and position. Figure 2.5 shows the diffraction pattern on a 2D detector compared with the diffraction measurement range of scintillation detector and PSD. Because 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 XRD2 is a point (called spot focus). In practice, the beam cross-section can be either circular or square in limited size.

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Figure 2.4 Conventional diffraction pattern of corundum, showing the intensity of X-ray scattering at different 2 angles

Figure 2.5 Comparison of diffraction pattern coverage between point (0D), linear PSD (1D), and area (2D) detectors

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3 Geometry Conventions in XRD2 Systems

The geometry of an XRD2 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 coordinate system XLYLZL is the basis of all three spaces. Although the three spaces are interrelated, the definitions and corresponding parameters should not be confused.

3.1 Diffraction Cones in Laboratory Axes (Diffraction Space)Figure 3.1 describes the geometric definition of diffraction cones in the laboratory coordinate system XLYLZL. Analogous to the conventional 3-circle and 4-circle goniometer, the direct X-ray beam propagates along the XL axis, ZL is up, and YL makes up a right-handed rectangular coordinate system. The axis XL 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 180-2 for backward reflection (2 > 90). The angle is the azimuthal angle from the origin at the 6 Oclock direction (-ZL direction) with a right-handed rotation axis along the opposite direction of the incident beam (-XL direction). Since has also been used to denote one of the goniometer angles in 4-circle convention, (gamma) 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 XL axis as the edge, hereafter referred to as the -plane. Intersections of any diffraction cones with a -plane have the same value. The conventional diffractometer plane consists of two -planes with one =90 plane on the negative YL side and =270 plane on the positive YL side.

The and 2 angles form a kind of spherical coordinate system that covers all the directions from the sample origin (i.e., the goniometer center). The -2 system is fixed in the laboratory system XLYLZL, which is independent of the sample orientation in the goniometer. This is a very important concept when we deal with the 2D diffraction data.

Figure 3.1 The geometric definition of diffraction rings in laboratory axes

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3.1.1 Ideal Detector for Diffraction Pattern in 3D Space

An ideal detector to measure the diffraction pattern in 3D space is a detector with a spherical detecting surface, covering all of the diffraction directions in 3D space as is shown in Figure 3.2. The sample is in the center of the sphere. The direction of a diffracted beam is defined by 2 (latitude) and (longitude). 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, many 2D detector technologies are available including photographic film, CCD, image plate (IP), and multi-wire proportional counter (MWPC). Each technique has its advantages over the others [2]. The detection surface can be spherical, cylindrical, or flat. Spherical or cylindrical detectors are normally designed for a fixed sample-to-detector distance, while the flat detector has the flexibility to be used at different sample-to-detector distances so that you may choose between higher resolution at long distance or higher angular coverage at short distance. This Introductions discussion of XRD2 geometry will focus on flat 2D detectors.

Figure 3.2 Schematic of an ideal detector covering 4 solid angle

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3.1.2 Diffraction Cones and Conic Sections on 2D Detectors

Figure 3.3 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 the Bragg relation. 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, also referred to as detector 2 angle. The conic section takes different shapes for different angles. When imaged on-axis ( = 0) the conic sections appear as circles, producing the Debye rings familiar to most diffractionists. When the detector is at an 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 Introduction. The 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 XRD2 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.

Figure 3.3 A diffraction cone, the 2D detector plane, and the conic section at their intersection


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3.2 Detector Position in the Laboratory System (Detector Space)The detector position is defined by the sample-to-detector 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 the ZL axis.

Figure 3.4 shows different detector positions in the laboratory coordinate system XLYLZL. At =0 (on-axis, position 1 in Figure 3.4), the positive side of the XL axis intersects with the center of the detector. Detector positions 2 and 3 are rotated away from XL axis with negative swing angles (2 < 0 and 3 < 0). The swing angle is sometimes called detector two-theta and denoted by 2D 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 3.4 Detector positions in the laboratory system XLYLZL: D is the sample-to-detector distance; is the detectors swing angle.


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3.3 Sample Orientation and Location in the Laboratory System (Sample Space)

In an XRD2 system, three rotation angles are necessary to define the orientation of a sample in the diffractometer. These three rotation angles can be achieved by an Eulerian cradle (4-circle) geometry, a kappa () geometry, or other kind of geometry. The 4-circle geometry will be discussed in this Introduction. The three angles in 4-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 previous publications (and some software), a subscript g indicates the angle is a goniometer angle. Figure 3.5(a) shows the relationship between rotation axes (, g, ) and the laboratory system XLYLZL. is defined as a right-handed rotation about the ZL 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 XL axis in the XLYL plane. The g axis lies on XL when is set at zero. is a left-handed rotation. The g angle is also the angle between the axis and the ZL axis.Figure 3.5(b) shows the relationship among all rotation axes (, g, , ) and translation axes XYZ. is 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 whenever possible in this Introduction. The next rotation above and g() is rotation about . The sample translation coordinates XYZ are so defined that, when = = = 0, the relationship to the laboratory axes becomes X= -XL, Y= ZL, and Z = YL. The rotation axis is always the same as the Z axis at any sample orientation.

Figure 3.5 Sample rotation and translation: (a) Three rotation axes in the XLYLZL coordinate system; (b) Rotation axes (, g, , ) and translation axes XYZ


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In an aligned diffraction system, all three rotation axes and the primary X-ray beam cross at the origin of XLYLZL. This cross point is also known as the goniometer center or instrument center. The XY plane is the sample surface and Z is the sample surface normal. In a preferred embodiment, the XYZ translations are above all the rotations, so the XYZ translations will not move any rotation axis away from the goniometer center. Instead, the XYZ translations bring different parts of the sample into the goniometer center. For example, in Figure 3.6 a sample translation of +x and +y will bring the sample spot P(-x, -y) into the instrument center. Therefore, the origin of XYZ is not fixed on the sample.Alternatively, the sample coordinates S1, S2, and S3 are also used in XRD2. The sample coordinates S1, S2, and S3 have the same directions as the sample translation coordinates X, Y, and Z respectively. The origin of the sample coordinates is considered fixed on the sample. The S1 S2 plane is the sample surface plane and S3 is the sample surface normal. In practice, there is no distinction between the two coordinate systems.

Figure 3.6 Sample translation axes XYZ and sample coordinates S1S2S3


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3.4 Summary of XRD2 Geometry

The definitions, parameters, and relationships between the three spaces are displayed in Figure 3.7. All three spaces are based on the laboratory coordinates. The diffraction space is mainly defined by the sample crystal structure and the X-ray beams 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 of the diffraction space to be measuredand the measurement resolutionbut not the diffraction space itself.

Changing the sample space will not change the diffraction space of an ideal polycrystalline (i.e., powder) sample that has no stress or texture.

Changing the sample orientation will change the diffraction space slightly when there is texture or stress in the sample.

Changing the sample location will change the diffraction space for an inhomogeneous sample.

Figure 3.7 The relationship between the three spaces and the laboratory coordinates


3 - 8 M86-E01055


M86-E01055 4 - 1

4 X-ray Optics for XRD2

The function of the X-ray optics is to condition the primary X-ray beam to the required wavelength, beam focus size, beam profile, and divergence. The X-ray optics components commonly used for an XRD2 system are a monochromator, a pinhole collimator, cross-coupled multilayer mirrors, and a monocapillary.

Figure 4.1 shows a typical X-ray optics assembly for an XRD2 system, which includes an X-ray tube, a monochromator, a collimator, and a beamstop [5]. It also shows the instrument center and the shadow of a fixed-chi stage. Using a point X-ray source with pinhole collimation enables you to examine small samples (microdiffraction) or small regions on larger samples (selected-area diffraction). This configuration enables you to measure crystallographic phase, texture, and residual stress from precise locations on irregularly-shaped parts, including curved surfaces.

Figure 4.1 Typical X-ray optics in a standard XRD2 system: X-ray tube, monochromator, collimator, and beamstop. Also shown are the instrument center and the shadow of a fixed-chi stage.

X-ray Optics for XRD2 Introduction to 2-Dimensional X-ray Diffraction

4 - 2 M86-E01055

4.1 X-ray Beam Shape for XRD2

In principle, the cross-section shape of the X-ray beam used in a two-dimensional diffraction system should be a zero-dimensional point. In practice, the beam cross-section can be either round or square in limited size. In a conventional diffraction system, a line focus beam is typically used because the variation vertical to the diffractometer plane is not considered. However, if a line focus beam is used in a two-dimensional diffraction system, the smearing effect will dramatically increase the peak width, especially at angles away from the diffractometer plane (=90 and =270). Figure 4.2a is a diffraction frame of corundum collected with a line focus beam. The diffraction rings are broadened at the parts of the frame away from the diffractometer plane. For comparison, Figure 4.2b shows the diffraction frame collected with a spot focus beam, where no smearing effect is observed. Therefore, every part of the diffraction ring can be used for data analysis.

Figure 4.2 A diffraction frame from corundum powder: (a) Smearing effect from line focus beam; (b) Diffraction rings from spot focus beam

(a) (b)

Introduction to 2-Dimensional X-ray Diffraction X-ray Optics for XRD2

M86-E01055 4 - 3

4.2 Beam Spread Over Flat Sample Surface (Geometry Broadening)Most conventional diffractometers use the Bragg-Brentano parafocusing geometry [1], in which the sample surface normal is always a bisector between the incident beam and the diffracted beam as shown in Figure 4.3. A divergent beam from the X-ray tube passes first through a divergent slit, then hits the sample surface with an incident angle . The incident X-rays spread over the sample surface with various incident angles in the vicinity of . The area of the irradiated region depends on the incident angle and beam divergence. The diffracted rays from the irradiated area leave the sample at an angle 2 from the corresponding incident rays, pass through the anti-scatter slit, and focus at the detector slit. A point X-ray detector can be mounted after the detector slit or after a crystal monochromator. You can see that the beam-spread over the sample varies with the incident angle , but the diffracted beams are focused back to the point detector as long as the sample surface normal bisects the incident and diffracted beams.

Figure 4.3 A conventional diffractometer in Bragg-Brentano geometry

In an XRD2 system, the diffracted X-rays are measured simultaneously in a two-dimensional range so that the Bragg-Brentano geometry cannot be achieved. The beam-spread over the sample surface cannot be focused back to the detector. Figure 4.4 shows the beam-spread at a low incident angle over a flat sample surface observed by a two-dimensional detector. Figure 4.4a is in reflection mode and shows that the diffracted beam at a low 2 angle is narrower than the diffracted beam at a high 2 angle. Figure 4.4b shows that the opposite is true in transmission mode. If the sample size is limited to a size comparable to the X-ray beam size and diffraction happens in both transmission and reflection modes, the peak broadening caused by beam-spread can be reduced or eliminated. Loading powder samples in capillaries is one way to achieve this effect. When collecting phase ID data with a flat sample, setting the sample angle as half of the median 2 can also reduce the beam-spread effect.

Figure 4.4 Beam spread on a flat sample surface with low incident angle: (a) Reflection mode, (b) Transmission mode.(a) (b)


4 - 4 M86-E01055

4.3 Monochromator

In a conventional diffraction system, the monochromator can be used on either the source side or the detector sideor on both sideswhile a monochromator can only be used on the source side for an XRD2 system.

A crystal monochromator allows only a selected characteristic wavelength (typically K radiation) to pass through. A crystal monochromator is illustrated in Figure 4.5. The single crystal has a d-spacing: d. The wavelength of the X-rays diffracted by the crystal is given by the Bragg law, =2dsinM. We can set the monochromator crystal to a diffraction condition such that only the wavelength of K satisfies the Bragg law. X-rays of other wavelengths are filtered out by the monochromator. As shown in Figure 4.5, the X-rays must also be in the correct direction to satisfy the diffraction condition. Therefore, the reflected beam from a monochromator with a perfect crystal will be a parallel X-ray beam.

Figure 4.5 Illustration of a crystal monochromator. Monochromatic X-rays are obtained by diffraction from a single crystal plate.

In practice, the reflected beam from a monochromator is not strictly monochromatic due to the mosaicity of the crystal (the mosaicity of a crystal is measured by rocking angle).The crystal type in a monochromator must be chosen based on the performance you require in terms of intensity and resolution. The monochromator crystal shape also varies from flat to bent to cut-to-curve. A flat crystal is used for parallel beams and a curved crystal for focus geometry. A typical XRD2 system uses a flat graphite monochromator, which gives the strongest beam intensity. The monochromator is designed to accept a limited angular range of X-rays about the takeoff angle. The monochromator can be used for takeoff angles from 3 to 6 (typically set to 6). The graphite crystal cannot resolve K1 and K2 lines, so it is aligned to the K line. The monochromator is designed to work with various anode materials; their 2M angles are listed in Table 4.1.

Table 4.1 Bragg angles of graphite crystal (002) plane for various anode materials

Target Materials K Wavelength Bragg angle 2MAg 0.560868 9.58

Mo 0.710730 12.14

Cu 1.541838 26.53

Co 1.790260 30.90

Fe 1.937355 33.51

Cr 2.29100 39.87


M86-E01055 4 - 5

4.4 Cross-Coupled Gbel Mirrors

Recent developments in X-ray optics include graded multilayer X-ray mirrors, known as Gbel mirrors [6]. A cross-coupled arrangement of these optics provides a highly parallel beam which is much more intense than can be obtained with standard pinhole collimation and a graphite monochromator. For applications such as microdiffraction (where a small spot size is desired), Gbel mirrors can offer greater intensity than conventional optics. The low divergence of the beam incident on the sample also decreases the width of crystalline peaks, improving resolution for XRD2 systems.

Figure 4.6 (a) A single parabolically-bent Gbel mirror transforms the sources divergent primary beam into a parallel beam. (b) In cross-coupled Gbel mirrors, the second Gbel mirror turned 90 collimates the beam in the direction perpendicular to the first mirror.

The Gbel mirror is a parabolic-shaped multilayer mirror. Multilayer mirrors reflect X-rays in the same way as Bragg diffraction does from crystals, so multilayer mirrors can be used as a monochromator. In contrast to a conventional crystal monochromator, Gbel mirrors are manufactured such that the d-spacing between the layers varies in a controlled manner. The appropriate variation in the d-spacing depends on factors such as wavelength, the location of the mirror with respect to the source, and the application for which the mirror is designed.

Figure 4.6(a) illustrates a single Gbel mirror. The Gbel mirror is parabolically bent, which causes a divergent beam striking the mirror at different locations and angles to yield an intense and highly parallel beam. With Bragg diffraction, the radiation is monochromatized to K while K and Bremsstrahlung are suppressed. The cross-coupled Gbel mirrors are used for X-ray sources with spot focus, in which a second Gbel mirror turned 90 collimates the beam in the direction perpendicular to the first mirror (Figure 4.6(b)).


4 - 6 M86-E01055

For all applications requiring strong collimation of the beam, Gbel mirrors provide considerable intensity gains. Experimental results show that the smaller the beam size, the stronger the intensity gains from cross-coupled Gbel mirrors compared with a monochromator (Figure 4.7). The intensity break-even point for Gbel mirrors versus standard monochromator with pinhole collimation is approximately 0.3 to 0.4 mm. In other words, for applications such as texture or phase identification from a bulk powdered specimen (which ordinarily employ collimators larger than 0.4 mm), you will get no benefit from using Gbel mirrors. In fact, the low divergence of the resulting beam can cause poor statistical grain sampling in such cases. Therefore, cross-coupled Gbel mirrors are especially suitable for microdiffraction and small angle X-ray scattering.

Figure 4.7 Comparison of X-ray intensity between cross-coupled Gbel mirrors and monochromator for various collimator size. The solid line represents experimental value, and the broken line is the computer-simulated values.

0

1

10

100

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Collimator Pinhole Size (mm)

Inte

nsity

Rat

io (m

irror

s/m

onoc

hrom

ator

)

Experiment

Simulation


M86-E01055 4 - 7

4.5 Pinhole Collimator

The pinhole collimator is normally used to control the beam size and divergence. In an XRD2 system, the pinhole collimator is normally used with a monochromator or a set of cross-coupled Gbel mirrors. Figure 4.8 shows the X-ray beam path in a pinhole collimator achieved with two pinhole apertures of the same diameter d separated by a distance h. F is the dimension of the projection of focal spot or beam focus projection from the monochromator or Gbel mirrors. The distance between the focus to the second pinhole is H. The distance from the second pinhole to the sample surface is g.

Figure 4.8 Schematic of the beam path in a pinhole collimator showing the parallel, divergent, and convergent X-rays and beam spot on sample surface

The beam consists of three components: parallel, divergent, and convergent X-rays. The parallel part of the beam has a size of d all the way from the focus to the sample. The anti-scattering pinhole is used to block the X-ray scattering from the second pinhole. The size of the anti-scattering pinhole must be such that it allows no exposure to direct X-rays from the focus.

The maximum divergence angle is given by

(4-1)

The maximum angle of convergence is given by

(4-2)

The maximum beam spot D on a flat sample facing the X-ray source is given by

(4-3)

As shown in Equation (4-3), the shorter the distance between the second pinhole and the sample (or the longer the distance between two pinholes), the smaller the beam spot on the sample. The effective beam focus size f is determined by the pinhole distance h and the distance between the X-ray source and the pinholes.

(4-4)

2dh

------=

dh g+------------=

D d 1 2gh

------+ =

f d 2Hh

------- 1 =


4 - 8 M86-E01055

If the actual X-ray source F is larger than the effective focus size f, the difference between F and f represents the wasted X-ray energy. Sometimes, a micro-focus tube is required when a small beam size is used. The actual beam divergence is also determined by the monochromator and mirrors before the collimator in the beam path. For example, when cross-coupled Gbel mirrors are used, the X-ray beam is almost a parallel beam, and the divergence of the beam is smaller than the value calculated from equation (4-1). When the actual beam focus on the source f is smaller than f, we have the following equations to calculate the maximum divergence (), convergence () and beam spot size on sample (D):

(4-5)

(4-6)

(4-7)

Table 4.2 lists the values of beam divergence, convergence, and beam spot on sample for a system with a 0.4 mm 0.8 mm fine focus tube. The graphite monochromator has a rocking curve of 0.4 and cross-coupled Gbel mirrors a rocking curve of 0.06. The beam divergence and convergence angles should not be above these values.

Table 4.2 X-ray beam divergence angle (), convergence angle (), and beam spot size on sample (D) for a 0.4 mm spot focus with graphite monochromator or cross-coupled Gbel mirrors

Table 4.2 also shows that the beam divergency decreases continuously with decreasing pinhole size for the combination of double-pinhole collimator and monochromator. In some cases, the application requires small beam size but not necessarily the small divergence. We recommend that you remove the pinhole 1 from the collimator to achieve higher beam intensity.

Collimator size Graphite Monochromator Gbel Mirrors

d (mm) () () D (mm) f (mm) () () D (mm)0.05 0.041 0.017 0.07 0.15 0.041 0.017 0.07

0.10 0.082 0.034 0.14 0.30 0.060 0.034 0.13

0.20 0.164 0.067 0.29 0.60 0.060 0.060 0.23

0.30 0.246 0.101 0.42 0.80 0.060 0.060 0.33

0.50 0.266 0.148 0.64 0.80 0.060 0.060 0.53

0.80 0.327 0.148 0.97 0.80 0.060 0.060 0.83

d f+H------------

=

fH g+-------------=

D H g+( ) f=


M86-E01055 4 - 9

Table 4.3 gives a comparison between double-pinhole collimators and single-pinhole collimators in terms of intensity gain (the approximate ratio of single-to-double pinhole), beam divergency, and beam spot size on sample.

Table 4.3 Comparison between single-pinhole collimator and double-pinhole collimator in terms of intensity gain, beam divergency angle (), and beam spot size on sample (D)

Microdiffraction collimators are 50 m and 100 m in diameter. For quantitative analysis, texture, or percent crystallinity measurements, 0.5 mm or 0.8 mm collimators are typically used. In the case of quantitative analysis and texture measurements, using too small a collimator can actually be a detriment, causing poor statistical grain sampling. In such cases, you can improve statistics by oscillating the sample. Crystallite size measurements are usually measured with a 0.2 mm collimator at a 30 cm sample-to-detector distance. The choice of collimator size is often a trade-off between intensity and the ability to illuminate small regions or to resolve closely-spaced lines. The smaller the collimator, the lower the photon flux that strikes the sample, and the longer the count time to acquire statistically-significant data.

Collimator size Intensity gain Single pinhole Double pinhole

d (mm) Single/double () D (mm) () D (mm)0.05 > 20 0.174 0.14 0.041 0.07

0.10 16 0.184 0.20 0.082 0.14

0.20 4 0.205 0.31 0.164 0.29

0.30 2.4 0.225 0.42 0.225 0.42

0.50 1.2 0.266 0.64 0.266 0.64

0.80 1.0 0.327 0.97 0.327 0.97


4 - 10 M86-E01055

4.6 Monocapillary

Capillary X-ray optics are based on the concept of total external reflection. X-rays can be reflected by a smooth surface when the angle of incidence is smaller than a critical angle c. The critical angle is a function of the wavelength and materials. The shorter the wavelength, the lower the critical angle. When X-rays are reflected by the inner surface of a capillary at a grazing angle smaller than the critical angle of the capillary materials, X-rays are reflected with little energy loss. The transmission efficiency depends upon the X-ray energy, incident beam divergence, the capillary materials, the reflection surface smoothness, and the capillarys inner diameter. K radiation, having higher energy than K, has less transmission efficiency. For typical capillary materials, the critical angle is about 0.2 for Cu-K radiation.

The monocapillary (trade name MonoCap) is a cylindrical tube with a smooth inner surface, mounted inside a steel tube. The tube is of the same design as the one used for the pinhole collimator, therefore it is easy to switch between the pinhole collimator and monocapillary. The monocapillary performs the following main functions: It collimates the beam spatially to a variety of beam sizes for different applications. You have a

choice of monocapillary sizes from 1.0 mm down to 0.01 mm. It collimates the beam divergency. The exit beam divergency is controlled by the capillary dimen-

sions (diameter and length) and the critical angle of total reflection. It can produce significant intensity gains on the sample relative to pinhole collimators.

Table 4.4 shows that 0.1 to 1.0 mm capillaries give practically the same spot sizes on the sample as the corresponding double-pinhole collimators. The capillaries produce large intensity gains relative to the corresponding double-pinhole collimators. In the case of small beam size, a special combination of capillary and pinhole may be desirable. A capillary of large diameter captures more radiation near the source and transmits it with less intensity loss. The smaller-diameter pinhole defines the final beam size. The combination can obtain more uniformly-distributed radiation energy on the sample.

Table 4.4 Intensity gain (calculated and experimental) and beam spot size including 90% energy on sample for monocapillaries compared with double-pinhole collimators

Capillary/Pinhole

size:d (mm)

Cu-K radiation (8.0 keV) Mo-K radiation (17.4 keV) Collimator

Gaincalc. Gainexp. Spot 90% Gaincalc. Gainexp. Spot 90% Spot 90%

0.10 110 66 0.18 39 40 0.14 0.10

0.30 15 10 0.34 5.6 5.9 0.31 0.31

0.50 7.4 6.0 0.50 2.6 3.0 0.49 0.50

1.00 3.4 4.2 0.89 1.2 1.5 0.97 0.98


M86-E01055 4 - 11

4.7 Other Features of X-ray Optics for XRD2

In addition to the features of X-ray optics for XRD2 described in the previous Sections, other phenomena exist, such as fluorescence and air scattering.

In a conventional diffractometer, fluorescence radiation from the X-ray-irradiated sample can be removed from the collected data using a diffracted-beam monochromator, or a detector with energy discrimination capability. While most two-dimensional detectors have a very limited energy resolution, you cannot add a diffracted-beam monochromator to remove fluorescence. Use of a thin-metal foil filter to cover the detector window can remove part of the fluorescence background. The best way to avoid fluorescence is to choose an X-ray tube anode material having K energy lower than the absorption edge of the sample materials.

Air scatter in an XRD2 system makes a significant contribution to the intensity background. In a conventional diffractometer, an anti-scatter slit (Figure 4.3), a diffracted-beam monochromator, or a Soller slit can be used to remove most air scatter not travelling in the diffracted beam direction. However, these measures cannot be used for an XRD2 system, which requires an open space between the sample and 2D detector. In order to reduce air scatter from the incident beam, the collimators beam exit port is located as near as possible to the sample. As shown in Figure 4.1, for a typical XRD2 optics design, the tip of the collimator is 6 mm from the sample. Typically, you would not need to (and it is not easy to) remove air scatter on the diffracted-beam side (i.e., between the sample and 2D detector). However, if the sample-to-detector distance is 30 cm or more, you may use a helium beampath or vacuum beampath to reduce air scatter. A helium beampath is available for SAXS (Small Angle X-ray Scattering) applications.


4 - 12 M86-E01055


M86-E01055 5 - 1

5 Stress Measurement With XRD2

5.1 Fundamental Equations for Stress Measurement with XRD2

A two-dimensional (2D) diffraction pattern contains far more information than a one-dimensional (1D) profile collected with a conventional diffractometer. When used for stress measurement, two-dimensional X-ray diffraction (XRD2) systems have many advantages over conventional 1D diffraction systems in dealing with highly-textured materials, large grain size, small sample area, weak diffraction, stress mapping, and stress tensor measurement [7-11]. 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, getting better measurement results 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 cones shapes so that they are no longer regular cones. Figure 5.1 shows a diffraction cone cross-section on a 2D detector plane. 2 becomes a function of : 2 = 2(). This function is uniquely determined by the stress tensor and the sample orientation.

Figure 5.1 Diffraction cone distortion due to stress

Stress Measurement With XRD2 Introduction to 2-Dimensional X-ray Diffraction

5 - 2 M86-E01055

The fundamental equation for strain measurement using a 2D detector is given [8] as(5-1)

with:

where strain coefficients fij are determined by the sample orientation and diffraction vector direction for each data point on the diffraction ring. ln(sin0 / sin) determines the diffraction cone distortion at the particular (, 2) position.

Equation (5-1) 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 statistical error. For isotropic materials, there are only two independent elastic constants: Youngs modulus E and Poissons ratio or the macroscopic elastic constants S2 = (1+ )/ E and S1 = -/ E. Then we have

(5-2)

where

(5-3)

Each diffraction frame corresponds to one set of sample orientations (, and ). The diffraction ring on each frame is integrated and peak-fitted over a selected number of sections along the ring to obtain a set of (, 2) data points representing the = 2() function. The stress tensor can be determined by fitting the data points to equation (5-2) with the least-squares method. For biaxial stress, the above equation becomes

(5-4)where the coefficient Pph=(1-2)/E and ph is a pseudo-hydrostatic stress component caused by the approximate d-spacing d0. For biaxial stress with shear, we have

(5-5)

The biaxial stress state corresponds to the straight line of the d-sin2 plot, and biaxial stress with shear is the case when a split occurs between the data points on the + side and - side. The general normal stress () and shear stress () at an arbitrary given angle are given by

(5-6)(5-7)

Strain Coefficients: f11 f12 f22 f13 f23 f33

= A2 2AB B2 2AC 2BC C2

A = acos - bcossin + csinsin a = sincos + sincossinB = asin + bcoscos - csincos b = -cos cos C = b sin + ccos c = sinsin - sincoscos

f1111 f1212 f2222 f1313 f2323 f3333+ + + + +0sinsin-------------

ln=

P1111 P1212 P1313 P2222 P2323 P3333+ + + + +0sinsin-------------

ln=

if

if

1E--- 1 v+( )fij v[ ]

12---S2fij S1+= i j=

1E--- 1 v+( )fij

12---S2fij= i j

pij =

P1111 P1212 P2222 Pphph+ + + 2d0 sin---------------------- ln=

P1111 P1212 P2222 P1313 P2323 Pphph+ + + + + 2d0 sin---------------------- ln=

11 cos2 12 2 22 sin2+sin+= 13 23+ sincos=

Introduction to 2-Dimensional X-ray Diffraction Stress Measurement With XRD2

M86-E01055 5 - 3

5.2 True Stress-Free Lattice D-Spacing

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 d0 (or 20). Any error in d0 (or 20) will contribute only to a pseudo-hydrostatic term ph.Figure 5.2 shows the biaxial stress tensor measured from a shot-peened Almen strip with different input d0 in the range of 1.165 to 1.175. The measured stress tensor is independent of the input d0 (11 = 623 MPa, 12 = 638 MPa, 22 = 80 MPa), where the pseudo-hydrostatic term ph changes with the input d0. The true d0 corresponds to the cross point of ph line and zero stress.

Figure 5.2 Measured biaxial stress tensor and pseudo-hydrostatic stress as a function of input d0

If we use d0 to represent the initial input, the true d0 (or 20) can be calculated from ph with the following equations:

(5-8)

or

(5-9)

-2500-2000-1500-1000-500

05001000150020002500

1.16

5

1.16

6

1.16

7

1.16

8

1.16

9

1.17

0

1.17

1

1.17

2

1.17

3

1.17

4

1.17

5

do input

Stre

ss (M

Pa)

sigma11

sigma12

sigma22

pseudo-hydro

d0 d0 1 2vE---------------ph exp=

0 arc 0 2v 1E---------------ph expsinsin=


5 - 4 M86-E01055

5.3 Anisotropy Factor

The anisotropy correction can also be included in the X-ray elastic constants S2(hkl) and S1(hkl) to replace the macroscopic elastic constants S2 and S1. The equations for calculating X-ray elastic constants are:

;

;

;

(5-10)

The factor of anisotropy (ARX) is a measure of the elastic anisotropy of a material. Values of ARX for the most important cubic materials are given in Table 5.1, and additional values may be found from literature.

Table 5.1 ARX values for cubic materials

Materials ARX

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

12---S2 hkl( ) 12---S2 1 3 0.2 hkl( )( )+[ ]=

S1 hkl( ) S1 12---S2 0.2 hkl( )[ ]=

hkl( ) h2k2 k2l2 l2h2+ +

h2 k2 l2+ +( )2-------------------------------------------=

5 ARX 1( )3 2ARX+

---------------------------=


M86-E01055 5 - 5

5.4 Relationship between Conventional Theory and 2D Theory

In order to find the relationship between the conventional theory and the new 2D theory, we first compare the configurations used for data collection in both cases. The conventional diffraction profile is collected with a point detector scanning in the diffractometer plane or a position-sensitive detector mounted in the diffractometer plane (see Figure 2.3). The 2D diffraction data consists of diffracted X-ray intensity distribution on the detector plane. The diffraction profiles at =90 and = -90 (i.e., 270) on the 2D detector are equivalent to the diffraction profiles collected in the conventional diffractometer plane. Therefore, you can use diffraction profiles at =90 and =-90 on a 2D detector to imitate a conventional diffractometer.

Figure 5.3 The stress values calculated by the conventional (1D) method and 2D method from the 10 samples, averaged over 9 measurements for each sample.

In theory, it has been proven that the conventional fundamental equation is a special case of the 2D fundamental equation [3]. In the same way, a conventional detector can be considered as a limited part of a 2D detector. Depending on the specific condition, you can choose either theory for stress measurement when a 2D detector is used. If the conventional theory is used, you have to get a diffraction profile at =90 or = -90, 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.

Experiment results also show a good correlation between the two methods. Ten Almen strips were used for the residual stress measurement. The Almen strips have a hardness of 55 HRC and had been shot peened in both faces for 30 minutes 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 5.3.

Conventional 1D vs. 2D Method -700

-650

-600

-550

-500

-450

-400

-350

-3001 2 3 4 5 6 7 8 9 10

Sample Number

Stre

ss (M

Pa)

1D2D


5 - 6 M86-E01055

Figure 5.4 shows the relationship between the conventional method and the 2D method.

Figure 5.4 The relationship between the conventional method and the 2D method. (a) The data points and the simulated diffraction ring corresponding to the measured stresses at one sample orientation. (b) The diffraction vectors (Hhkl) and sample coordinates S1S2S3 during the data collection scan for both the conventional method and 2D method. (c) The data points at =90 are displayed in d vs. sin2 plot.

Figure 5.4(b) shows the relationship between the diffraction vector (Hhkl) and sample coordinates S1S2S3 during the data collection scan for both the conventional method and the 2D method. In the conventional method, each sample orientation corresponds to one diffraction vector orientation. For instance, if the data is collected at = -45 to +45 with 15 steps, a total of 7 data points are collected. The head point of the diffraction vector scans along a curved line (=90). The data can be displayed in a d vs. sin2 plot as shown in Figure 5.4(c). The slope of the plot is determined by the stress. In the 2D method, however, at each sample orientation ( = 45, for instance), the diffraction ring represents a series of diffraction vectors for various angles. Figure 5.4(a) shows the data points collected at one sample orientation, along with a simulated diffraction ring. The simulated diffraction ring is calculated from the stress results based on a simulation equation, which is a reversion of the fundamental equation [5-1].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 to be in reflection mode [8], so that equations (5-1) and (5-2) can be used for transmission mode diffraction without any modification. More experiments are necessary to further explore the stress measurement with transmission mode diffraction.

sin2

d

(a) (b) (c)


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5.5 Virtual Oscillation

For materials with large grain size, or microdiffraction with a small X-ray beam size, determining the 2 position can be difficult due to poor counting statistics. To solve this problem with conventional detectors, some kind of sample oscillationseither by translation or by rotationare necessary to bring more crystallites into diffracting condition. In other words, the purpose of oscillations is to bring more crystallites into a condition where the normal of the diffracting crystal plane coincides with the instrument diffraction vector. 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 a so-called oscillation. Therefore, we may call this effect virtual oscillation. Figure 5.5 shows the relationship between the -integration range, , and the virtual oscillation angle, . The 2 value of the -integrated profile is an average over the Debye ring defined by the range. The average effect is over a region of orientation distribution, rather than volume distribution.

Figure 5.5 Relationship between the -range, , and the virtual oscillation angle


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The virtual oscillation angle can be calculated from the integration range and Bragg angle ,

(5-11)

For example, Figure 5.6 is a frame taken from stainless steel with a large grain size. If we integrate from = 80 to 100, = 20, 64, the virtual oscillation angle = 8.7. 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 has no such problem.

Figure 5.6 The virtual oscillation by -integration over =20 gives a smooth diffraction profile.

When the 2D method is used for stress measurement, the virtual oscillation effect is further enhanced due to the larger range. More importantly, the virtual oscillation effect for the 2D method is intrinsic, i.e., the data points along the diffraction ring are treated at almost their exact angle.In the conventional method, the virtual oscillation is extrinsic. Regardless of whether the profile is from -integration of a 2D frame or from a physical angular oscillation, the profile is treated as if the data were collected at one orientation. Thus, the measured 2 value is actually an average over the -integration range or angular oscillation range, called the smearing effect. For example, the 2 value of the profile in Figure 5.6 is an average over =20 (80-100), but is treated as if it were collected at the diffractometer plane (=90).However, in the 2D method, the virtual oscillation range is the total range of the selected diffraction ring, and the smearing effect is only within the step. For example, in the previous stress tensor measurement on the shot-peened Almen strip (recall Figure 5.4), the virtual oscillation range is 50, but the smearing effect is only over =2.5.

2arc 2------ sincossin=


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5.6 Data Collection Strategy

X-ray diffraction measures stress by measuring the d-spacing change caused by the stress. The diffraction vector is in the normal direction of the measured crystalline planes. Having the diffraction vector on the desired measurement direction is not always possible. In reflection-mode X-ray diffraction, having the diffraction vector normal to the sample surface is easy, but having the vector on the surface plane is impossible. The stress on the surface plane, or biaxial stress, is calculated by elasticity theory. The final stress result can be considered as an extrapolation from the measured values. Thus, in the conventional sin2 method, several -tilt angles are required, typically from -45 to +45. The same is true with an XRD2 system. The diffraction vectors corresponding to the data scan can be projected in a 2D plot in the same way as the pole density distribution in a pole figure.

Figure 5.7 shows the relationship between the conventional method and 2D method in terms of the diffraction vector (Hhkl) and sample coordinates S1S2S3 during the data collection scan.

Figure 5.7 Stress pole figure for the conventional method and the 2D method.

Bruker AXS GADDS (General Area Detector Diffraction System) software has a 2D Scheme function, which simulates the diffraction vector distribution relative to the sample orientation S1 and S2. The data scan strategy can be simulated to estimate the outcome from the stress calculation.

Figure 5.8 shows the input parameters for 2D scheme. Stress Peak is the approximate value of the stress-free 2, 2-theta is the detector position; Omega, Phi, and Chi are the goniometer angles; Distance is the sample-to-detector distance; #frames is the total number of frames collected in the data scan; Scan axis can be set to 2 Omega, 3 Phi, and 4 Chi; and Frame width is the scan step. The parameters in Figure 5.8 are for the (211) peak of a steel sample using Cr radiation.

Figure 5.8 Stress data collection strategy: 2D Scheme input panel

XRD

XRD2


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Figure 5.9 shows the 2D scheme plot from the above parameters. The diffraction vectors are clustered along the sample axis S1. Therefore, the data collected with the above setting will yield the best stress result for 11.

Figure 5.9 The 2D scheme plot simulated from the parameters in Figure 5.8. The diffraction vectors are clustered along the S1 direction.

If we collect the data with the same scan at =0, 45, and 90, we will see (as shown in the 2D scheme in Figure 5.10) that the data is good for a biaxial stress tensor including the components 11, 12, and 22. The scheme function can be used for a more complicated data collection strategy to reduce the data collection time, yet still achieve the best result.

Figure 5.10 The 2D scheme plot simulated from the same scan at =0, 45 and 90. The diffraction vectors are distributed in the S1, S2 and 45 directions. The data is good for a biaxial stress tensor.


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5.7 2D Detectors for Stress Measurement

Currently, three kinds of 2D detector techniques are available for XRD2 systems: image plate (IP), multi-wire proportional counter (MWPC), and charge-coupled device (CCD). Stress measurements using IP have been reported by several authors [6,8]. The advantages of using a multiwire area detector for stress measurements have also been discussed by the authors [7,8]. Experiments with the Bruker HI-STAR area detector show that the multiwire detector has high sensitivity with essentially no noise, which is very suitable for a stress measurement system using a laboratory source (e.g., sealed X-ray tubes or rotating anode generators). For stress measurements of ferrous metals, Cr or Co radiation is normally used to prevent fluorescence and, in this case, multiwire area detectors are currently the best choice.

CCD detectors have high flux capability, high spatial resolution, and sensitivity to high-energy X-rays. These features are suited to strong diffraction, especially from a synchrotron radiation source. Figure 5.11 is a diffraction frame collected in 30 seconds with 1 mm textured Cu film on a proprietary substrate using a 1.4 synchrotron beam. Two frames were collected for each stress measurement at =106.1 and 79.5. The total counts of the frame are 437 million. The profile by integration over a 2 (displayed as chi in the frame) (331) ring shows a peak intensity of 3408 counts/pixel. A total of 15 data points on each frame were used for stress calculation. The measured stress from the (331) ring is 415.6 MPa tensile with a standard error of 24.9 MPa. The stress calculated from (420) ring has an almost identical value of 414.6 MPa (30.9 MPa). In this measurement, the data collection time of 30 seconds is excessive, whereas a few seconds of data collection would be enough for counting statisticsthis particular stress measurement can be done within seconds when a CCD or a synchrotron is used.

Figure 5.11 The diffraction frame collected with 1 mm textured Cu film using a 1.4 synchrotron beam. The diffraction profile is an integration within 2 ring indicated by the integration box. The 15 data points used for stress calculation are displayed over the measured diffraction ring.


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5.8 Summary of Stress Measurement with XRD2

Two-dimensional X-ray diffraction (XRD2) 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 XRD2 system. For biaxial stress measurement, the approximation of the d-spacing or 2 input for the stress-free condition does not cause error in the stress calculation. The true stress-free d-spacing can be calculated from the pseudo-hydrostatic term. The conventional method and the 2D method are consistent both in theory and application. The same equation can be used for both reflection mode and transmission mode diffraction.

Stress measurements using CCD detectors and synchrotron radiation have very high speeds. The measurement of the residual stress of 1 mm Cu thin film takes one minute of data collection time. Stress measurements can be made in seconds. Considering the extremely high counts in the frame, it is expected that the measurement could have been done in seconds.

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6 Texture Measurement With XRD2

When used for texture measurement, two-dimensional (2D) X-ray diffraction systems have many advantages over conventional one-dimensional (1D) diffraction systems [12-13]. The texture of a sample is evaluated by measuring pole figures of one or more crystalline planes. The measurement is based on a fundamental relationship between the pole figure angles and the intensity distribution along a diffraction ring. The benefit of XRD2 is that several pole figures can be measured simultaneously, and all the data points on a diffraction ring are used to calculate a one-dimensional pole density mapping. This translates to better measurement results with less data collection time.

Texture Measurement With XRD2 Introduction to 2-Dimensional X-ray Diffraction

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6.1 Pole Density and Pole Figure

XRD results from a random powder normally serve as a basis for determining the relative intensity of each crystalline peak. In real life, however, polycrystalline materials usually do not have randomly-oriented grains. The deviation of the statistical 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 that satisfies the Bragg condition as shown in Figure 6.1a. The pole has the same orientation as the diffraction vector (Hhkl). Compared with the diffraction peak from a random powder (Figure 6.1b), the diffraction peak intensity change is from the texture, while the peak shift is due to stress.

Figure 6.1 (a) Definition of a pole; (b) diffraction peak intensity change due to texture and peak shift due to stress

The measured 2D diffraction pattern contains two very important parameters at each angle, the partially integrated intensity I and the Bragg angle 2. Figure 6.2 shows the diffraction cone distortion due to stress and diffraction intensity variation along due to texture: For a stressed sample, 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 of and the sample orientation (,,),

i.e., I = I(,,,). This function is uniquely determined by the orientation distribution function (ODF).

(a) (b)

Introduction to 2-Dimensional X-ray Diffraction Texture Measurement With XRD2

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Figure 6.2 Diffraction cone distortion due to stress and diffraction intensity variation along due to texture.

Texture is a measure of the orientation distribution of all grains in a sample with respect to a sample direction (e.g., the rolling direction in a sheet metal or the substrate normal in a thin film). Texture characterization by X-ray diffraction involves measurement of the peak intensity for a particular crystalline plane at all tilt angles with respect to a sample direction. Typically, one to four independent crystalline planes (different hkl values) are measured to quantify the major orientation distribution of a material.

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 shown in Figure 6.3, 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 point P would project to point P on the equatorial plane. The 2D map on the equatorial plane is the pole figure.


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Figure 6.3 Definition of the angles and and stereographic projection


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6.2 Fundamental Equations for Texture Measurement with XRD2

For a textured sample, the intensity is a function of and the sample orientation (,,), i.e., I = I(,,,). This function is uniquely determined by the orientation distribution function (ODF). Each pole direction is defined by the radial angle and azimuthal angle . The and angles are functions of , , , , and 2. The pole density at the pole figure angles (,) is proportional to the integrated intensity at the same angles:

(6-1)where Ihkl() is the integrated intensity corrected by absorption, polarization, background, and so forth. Nhkl is the normalization factor, and Phkl() is the pole density distribution function. The relationship between the pole figure angles (,), the sample orientation (,,) and diffraction cone (2,) is:

, (6-2)

where {h1,h2,h3} are components of the unit vector of the diffraction vector Hhkl. They are given by:

(6-3)

The diffraction intensity along the diffraction ring is then converted to the pole density at each and angle (from , , , ) and the 2 angles. The pole figures relative intensity can be normalized such that it represents a fraction of the total diffracted intensity integrated over the pole sphere.

Ihkl ( ) Nhkl Phkl ( )=

h3sin 1=

if

if

h1h2----- cot 1 h2 0 0 <


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Figure 6.4 is a comparison between the pole figure measurement with conventional X-ray diffraction and two-dimensional X-ray diffraction.

Figure 6.4 Comparison of pole figure measurement with the conventional X-ray diffraction and with two-dimensional X-ray diffraction

In conventional X-ray diffraction, one pole (marked by the diffraction vector Hhkl) is measured at each sample angle. In Figure 6.4, for example, with seven different positions, only seven poles are measured.

In two-dimensional X-ray diffraction, numerous poles are measured at each sample angle and a one-dimensional pole mapping is created at each exposure. As a result, 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 figures, and the data collection time can also be dramatically reduced.

S2

S1

S3Hhkl

Hhkl

XRD2

XRD


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6.3 Data Collection Strategy - Scheme

Because 1D pole density mapping is created with each exposure in XRD2, it is important to design a data-collection strategy that optimizes pole figure coverage and minimizes redundancy in data collection. A scheme function is available in Bruker AXS GADDS software to simulate pole figure coverage. A user can input the diffraction 2 angle, detector distance and swing angle, goniometer angles, and scanning step widths to simulate the pole figure coverage.

Figure 6.6 shows an example of a scheme generated at 2=40, =20, =35.26 (i.e., g=54.74), and D=7 cm with scan of 5 steps. The real data collection can use smaller scanning steps, such as 1 to 2. The data collected with a single exposure at =0 would generate a 1D pole figure as shown by the curve marked .

Figure 6.5 A scheme generated at 2=40, =20, =35.26(g=54.74) and D=7 cm with scan of 5 steps. The curve is pole figure mapping at =0.

The pole figure measured with the data collection strategy in the above example has a blank hole in the center. The pole density at the center represents the diffraction vector perpendicular to the sample surface. So, the pole figure angle at the center is =0. To achieve a filled center in a pole figure, you would collect the pole density for curve with:

(6-4)or

(6-5)To avoid redundancy in data collection, the best strategy is to put point A at the center of the pole figure. That is:

(6-6)

AB

)

A

B

AB

)AB

)

h3 0=sin 1=

h3 1=sincoscoscoscossincossincossin=

h3A A A 1=sincoscoscoscossincossincossin=


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If we modify the parameters in Figure 6.5 to =23, =30 (i.e., g=60) for the same 2=40 and D=7cm with scan of 5 steps, the angle measured on the same (HI-STAR) detector at point A is A = -122. Then we have . The scheme generated by the GADDS software is shown in Figure 6.6. The center of the pole figure is filled with pole density data. The data collection parameters may be optimized by trial-and-error or calculation with equation (6-6).

Figure 6.6 A scheme generated at 2=40, =23, =30 (g=60), and D=7 cm with scan of 5 steps.

h3A 1


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6.4 Texture Data Process

Figure 6.7 shows one of the diffraction frames collected on an aluminum sample for texture analysis. At a detector distance of 7 cm with Cu radiation, a total of five diffraction rings from the crystalline planes (111), (200), (220), (311), and (222) can be measured simultaneously. Therefore, five pole figures can be measured simultaneously. The low and high backgrounds and the (220) diffraction rings 2- range are defined by three boxes. The integrated diffraction intensities at various angles are mapped into a pole figure, as defined by equations (6-2) and (6-3). The background can be removed by the intensity values defined in the low and high background boxes, or ignored at the users discretion.

Figure 6.7 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 rings 2- range are defined for the (220) plane.

Figure 6.8 shows pole figures of the (111) and (222) planes processed from the same set of Al foil data frames. Both pole figures show the same trend because the (111) and (222) poles are identical.

Figure 6.8 Pole figures of (111) and (222) processed from the same set of Al foil data frames

(111) (200) (220) (311) (222)

Diffraction ring 2-

Low background High

background

(111) (222)


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Figure 6.9 is the same (111) pole figure in 3D surface plot.

Figure 6.9 Pole figure of (111) in 3D surface plot

Pole figure data can be further used for Orientation Distribution Function (ODF) analysis, which quantifies the directions of the crystallites and provides the (volume) percentage of crystallites oriented in specific directions. The ODF is similar to a single-crystal orientation matrix (see Sands, 1982), except that it describes a continuum of orientations based on the large number of crystallites illuminated (usually >106). While some samples require a three-dimensional representation, samples such as films and fibers can often be described with a compact description of the orientation since these samples are either one- or two-dimensional in nature. The texture of many films and fibers can be described by a representation known as a Fiber Texture Plot (FTP), while polymer orientation is often characterized with Hermans and White-Spruiell orientation indices.

rolling direction


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6.5 Texture Measurement in Transmission Mode

The fundamental equations (6-1), (6-2), and (6-3) for texture measurement with XRD2 were derived without requiring the diffraction to be in reflection mode. They apply to both reflection and transmission. But the condition defined by equations (6-4), (6-5), and (6-6) cannot be satisfied in transmission mode diffraction. Therefore, pole figures collected using only transmission mode will always have an unfilled blank region in the center.

Figure 6.10a shows one of the frames collected in transmission mode with Mo-K radiation on an aluminum plate. A SMART 1000 CCD detector is used due to the Mo X-ray source. Four complete diffraction rings from crystalline planes of (111), (200), (220), and (311) can be observed simultaneously. Figure 6.10b is a pole figure scheme of the (111) plane for an scan, in 5 steps, between -50 and +50.

Figure 6.10 (a) Diffraction frame from Al plate with Mo-K in transmission mode; (b) Scheme of (111) plane

The pole figures from (111), (200), and (220) planes are shown in Figure 6.11.

Figure 6.11 Pole figures from transmission measurement of Al plate

(a) (b)

(111) (200) (220)


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6.6 Other Application Examples

Because several crystalline planeseither from polycrystalline materials or single crystalscan be measured simultaneously with one set of data frames, XRD2 has many advantages over a conventional diffractometer for texture analysis. One advantage is a measurement of the epitaxial relationship of a Pd thin film with respect to its Si substrate. Figure 6.12a is a pole figure collected from a Pd thin film. Figure 6.12b is a pole figure collected from a Si wafers (220) peak. Figure 6.12c is an overl

Introduction to XRD

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