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FUNDAMENTALS OF TWO-DIMENSIONAL X-RAY DIFFRACTION

(XRD2)

Baoping Bob He, Uwe Preckwinkel and Kingsley L. Smith

Bruker Analytical X-ray Systems

Madison, Wisconsin, USA

ABSTRACT

Two-dimensional x-ray diffraction (XRD2) refers to x-ray diffraction applications with two-

dimensional (2D) 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 the advantages of the 2D detector, new approaches

are necessary to configure the two-dimensional x-ray diffraction system and to analyze the 2D

diffraction data. The present paper discusses some fundamentals about two-dimensional x-ray

diffraction, such as geometry convention, diffraction data interpretation, point beam optics, and

advantages of XRD2 in various applications.

1. INTRODUCTION

In the field of x-ray powder diffraction, data collection and analysis have been based mainly on

one-dimensional diffraction (1D) 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. [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 advantages of a 2D detector can not

be fully taken if the data interpretation and analysis methods are simply inherited from the

conventional diffraction theory. XRD2 is a new technique in the field of x-ray diffraction (XRD),

which is not simply a diffractometer with a two-dimensional (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 the data collected with a 2D detector, a completely

new concept and new approach are necessary to configure 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 also be used for conventional applications.

An XRD2 system is a diffraction system with the capability of acquiring diffraction pattern in 2D

space simultaneously and analyzing 2D diffraction data accordingly. A 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. Figure 1 is an example of an XRD2

system with five major components.

Copyright(c)JCPDS-International Centre for Diffraction Data 2000,Advances in X-ray Analysis,Vol.43 273

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Figure 1. Five major components in an XRD2 system, an area detector, an x-ray generator,

x-ray optics (monochromator and collimator), goniometer and sample stage, andsample alignment and monitoring (laser/video) system.

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 2a). The diffraction pattern from a polycrystalline (powder)

sample forms a series diffraction cones if large number of crystals oriented randomly in the space

are covered by the incident x-ray beam (Figure 2b). 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 thereafter in the further discussion of the

theory and configuration of XRD2 systems. Polycrystalline materials here refer to single-phase,

multi-phase, bulk and thin film materials.

Figure 2. The patterns of diffracted x-rays: (a) from a single crystal and (b) from a

polycrystalline sample.

First, we compare the conventional x-ray diffraction (XRD) and two-dimensional x-ray diffraction

(XRD2). Figure 3 is a schematic of x-ray diffraction from a powder (polycrystalline) sample. For

simplicity, it shows only two diffraction cones, one represents forward diffraction (2θ 90°) and one


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for backward diffraction (2θ>90°). The diffraction measurement in the conventional diffractometer

is confined within a plane, here referred to as diffractometer plane. A point detector makes 2θ scan

along a detection circle. If a one-dimensional position sensitive detector (PSD) is used in the

diffractometer, it will be mounted on the detection circle. Since the variation 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). Theactual 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 is not

detected, the corresponding structure information will be either ignored, or measured by various

additional sample rotations.

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

diffractometer plane in the conventional diffractometer.

Figure 4. Comparison of diffraction pattern coverage between point (0D), linear PSD (1D),

and area (2D) detectors.

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 can be measured simultaneously,

depending on the detector’s size and position. Figure 4 shows the diffraction pattern on 2D


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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.

2. GEOMETRY CONVENTIONS IN XRD2

SYSTEM

2.1. Diffraction Cones in Laboratory Axes

Figure 5 describes the geometric definition of diffraction cones in the laboratory coordinates

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 2θ values given by the Bragg equation. The apex angels are twice the 2θ

values for forward reflection (2θ≤90°) and twice the values of 180°-2θ for backward reflection

(2θ>90°). The γ angle is the azimuthal angle from the origin at the 6 o’clock direction (-ZL

direction) with right handed rotation axis along the opposite direction of incident beam ( -XL

direction). In the previous publications, χ has been used to denote this angle by the present

author(s). Since χ has also been used to denote one of the goniometer angles in 4-circle

convention, γ will be used thereafter to represent this angle. The γ angle here is used to define the

direction of the diffracted beam on the cone. The γ angle actually defines a half plane with the XL

axis as the edge, it will be referred to as γ -plane thereafter. Intersections of any diffraction cones

with a γ -plane have the same γ value. The conventional diffractometer plane consists of two γ -

planes with one γ =90° plane in the negative YL side and γ =270° plane in the positive YL side. γ and

2θ angles forms a kind of spherical coordinate system which covers all the directions from the

origin of sample (goniometer center). The γ -2θ system is fixed in the laboratory systems 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 5. The geometric definition of diffraction rings in laboratory axes.

2.2. Ideal Detector for Diffraction Pattern in 3D Space

An ideal detector to measure the diffraction pattern in 3D space is a detector with spherical

detecting surface covering all the diffraction directions in 3D space as is shown in Figure 6. The


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sample is in the center of the sphere. The direction of a diffracted beam is defined by γ (longitude)

and 2θ (latitude). The incident x-ray beam points to the center of the sphere through the detector

at 2θ = π. The detector surface covers the whole spherical surface, i.e. 4π 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 (IP) and multi-wire proportional counter (MWPC). Each technique has its advantages

over the others [2]. The detection surface can be a 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 XRD2 geometry will focus on flat 2D detectors.

Figure 6. Schematics of an ideal detector covering 4π solid angle.

2.1. Diffraction Cones and Conic Sections on 2D Detectors

Figure 7 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 2θ 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 α is the detector swing angle. The conic section takes the different shapes for

different α angles. When imaged on-axis (α = 0°) the conic sections appear as circles. When the

detector is at off-axis position (α ≠ 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 ringsalternatively hereafter in this paper. 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

γ -2θ coordinates. In an XRD2 system, γ and 2θ values at each pixel position are given according to

the detector position. The diffraction rings can be displayed in terms of γ and 2θ coordinates,

disregarding the actual shape of each diffraction ring.


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Figure 7. A diffraction cone and the conic section with a 2D detector plane.

2.4. Detector Position in the Laboratory System

The detector position is defined by the sample-to-detector distance D and the detector swing angle

α. D is the perpendicular distance from the goniometer center to the detection plane and α is a

right-handed rotation angle above ZL axis. In the laboratory coordinates XLYLZL, detectors at

different positions are shown in Figure 8. The center of the detector 1 is right on the positive side

of XL axis (on-axis), α=0. Both the detector 2 and 3 are rotated away from XL axis with negative

swing angles (α2<0 and α3<0). The swing angle is also called as detector two-theta and denoted by

2θD in the previous publications. It is very important to distinguish between the Bragg angle 2θand detector angle α. At a given detector angle α, a range of 2θ values can be measured.

Figure 8. Detector position in the laboratory system XLYLZL: D is the sample-to-

detector distance; α is the swing angle of the detector.


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2.3. Sample Orientation and Position in the Laboratory System

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 either by an Eulerian cradle (4-circle)

type geometry, a kappa (κ ) geometry or other kind of geometry. The 4-circle geometry will be

discussed in this paper. The three angles in 4-circle geometry are ω (omega), χg (goniometer chi)and φ (phi). Since the χ 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 9(a) shows the relationship between rotation axes (ω, χg, φ) and the laboratory

system XLYLZL. ω is defined as a right-handed rotation about ZL axis. The ω axis is fixed on the

laboratory coordinates. χg is a left-handed rotation about a horizontal axis. The χg axis makes an

angle of ω with XL axis in the XL-YL plane. The χg axis lies on XL when ω is set at zero. φ is a left-

handed rotation. The χg angle is also the angle between φ axis and the ZL.

Figure 9(b) shows the relationship among all rotation axes (ω, χg, ψ , φ) and translation axes XYZ.

ω is the base rotation, all other rotations and translations are on top of this rotation. The next

rotation above ω is the χg rotation. ψ is also a rotation above a horizontal axis. ψ and χg have the

same axis but different starting positions and rotation directions, and χg = 90°-ψ . The next rotation

above ω and χg(ψ ) is φ rotation. The sample translation coordinates XYZ are so defined that,

when ω = ψ = φ =0, the relationship to the laboratory axes becomes X= -XL, Y= ZL, and Z= YL.

The φ rotation axis is always the same as the Z-axis at any sample orientation.

Figure 9. Sample rotation and translation. (a) Three rotation axes in XLYLZL

coordinates; (b) Rotation axes (ω, χg, ψ , φ) and translation axes XYZ.

In an aligned diffraction system, all three rotation axes and the primary x-ray beam cross at the

origin of XLYLZL coordinates. This cross point is also known as goniometer center or instrument

center. X-Y plane is usually the sample surface and Z is the sample surface normal. In a preferred

embodiment, XYZ translations are above all the rotations so that the 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.


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3. SUMMARY OF XRD2 APPLICATIONS

Compared to a conventional one-dimensional diffraction system, an XRD2 system has many

advantages in various applications:

• Phase identification (Phase ID) can be done by integration over a selected range of 2θ and γ .

The integrated data gives better intensity and statistics for phase ID and quantitative analysis,especially for those samples with texture, large grain size, or small quantity [3].

• Texture measurement is extremely fast. An XRD2 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 [4,5].

• 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 [6,7].

• Percent crystallinity can be measured faster and more accurately with the data analysis over

the 2D pattern, especially for samples with anisotropic distribution of crystalline orientation.

• Small angle x-ray scattering (SAXS) data can be collected at high speed. Anisotropicfeatures from specimens, such as polymers, fibrous materials, single crystals and bio-materials,

can be analyzed and displayed in two-dimension. De-smearing correction is not necessary due

to the collimated point x-ray beam. Since one exposure takes all the SAXS information, it is

easy to scan over the sample to map the structure information [1,8].

• Microdiffraction data is collected with speed and accuracy. The 2D detector captures whole

or a large portion of the diffraction rings, so spotty, textured, or weak diffraction data can be

integrated over the selected diffraction rings.

• Thin film samples with a mixture of single crystal, random polycrystalline layers and highly

textured layers can be measured with all the features appearing simultaneously in 2D frames.

REFERENCES

[1] Ron Jenkins and Robert L. Snyder, Introduction to X-ray Powder Diffractometry, John Wiley

& Sons, New York, 1996.

[2] Philip R. Rudolf and Brian G. Landes, Two-dimensional X-ray Diffraction and Scattering of

Microcrystalline and Polymeric Materials, Spectroscopy, 9(6), pp 22-33, July/August 1994.

[3] S. N. Sulyanov, A. N. Popov and D. M. Kheiker, Using a Two-dimensional Detector for X-ray

Powder Diffractometry, J. Appl. Cryst. 27, pp 934-942, 1994.

[4] Hans J. Bunge and Helmut Klein, Determination of Quantitative, High-resolution Pole Figures

with the Aea Detector, Z. Metallkd. 87(6), pp 465-475, 1996.

[5] K. L. Smith and R. B. Ortega, Use of a Two-dimensional, Position Sensitive Detector for

Collecting Pole Figures, Advances in X-ray Analysis, Vol. 36 , pp 641-647, Plenum, 1993.[6] Baoping B. He and Kingsley L. Smith, Fundamental Equation of Strain and Stress

Measurement Using 2D Detectors, Proceedings of 1998 SEM Spring Conference on

Experimental and Applied Mechanics, Houston, Texas, USA, 1998.

[7] Baoping B. He, Uwe Preckwinkel and Kingsley L. Smith, Advantages of Using 2D Detectors

for Residual Stress Measurements, Advances in X-ray Analysis, Vol. 42, Proceedings of the

47th Annual Denver X-ray Conference , Colorado Springs, Colorado, USA, 1998.

[8] H. F. Jakob, S. E. Tschegg, P. Fratzl, J. Struct. Biology, Vol. 113. Pp. 13-22, 1994.


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