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XRD Basics

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Introduction

This document provides an introduction to the basics of x-ray diffraction (XRD), aimed primarily at scientists and engineers who are not experts in the field but who are interested in using XRD as a tool. After describing what can be learned from XRD, and how typical XRD instruments are constructed, we go into some depth in describing the analysis of data from x-ray "area" or "two-dimensional" detectors.

Experts in the field may identify exceptions to some statements that made in this document, but an effort has been made to strike the right balance between simplicity and accuracy.

For a more in-depth presentation of the same topics, Paul Heiney has produced a series of tutorial videos on X-ray scattering techniques.

Contents
What is x-ray diffraction?
What types of measurement are typically made?
What are the components of an x-ray diffraction instrument?
How are x-ray area detector data analyzed?
What is involved in calibrating an XRD instrument with an area detector?
References and further reading.

What is x-ray diffraction?

X-ray diffraction (XRD) is a non-destructive technique for analyzing the structure of materials, primarily at the atomic or molecular level. It works best for materials that are crystalline or partially crystalline (i.e., that have periodic structural order) but is also used to study non-crystalline materials.

XRD relies on the fact that X-rays are a form of light, with wavelengths on the order of nanometers. When X-rays scatter from a substance with structure at that length scale, interference can take place, resulting in a pattern of higher and lower intensities. This is qualitatively similar to the colorful patterns produced by soap bubbles, in which different colors are viewed in different directions.

XRD is quite different from X-ray radiography, or tomography. Tomography relies on the fact that the X-rays are absorbed more strongly by some materials than others--for example, bone or tumors absorb more than muscle or fat. Therefore, the transmitted image provides a direct image of the structure inside the body or object (typically a length scales of a millimeter or above), making it an invaluable tool for doctors. (X-ray tomography is also widely used in other fields such as materials science and metallurgy.) In contrast, the XRD produces a diffraction pattern, which does not superficially resemble the underlying structure, and provides information about the internal structure on length scales from 0.1 to 100 nm.

In its most simplified form, a generic X-ray scattering measurement is shown below.

Schematic of generic scattering experiment.
A beam of X-rays is directed towards a sample, and the scattered intensity is measured as a function of outgoing direction. By convention, the angle between the incoming and outgoing beam directions is called 2θ. For the simplest possible sample, consisting of sheets of charge separated by a distance d, constructive interference (greater scattered intensity) is observed when Bragg's Law is satisfied:
n λ = 2 d sin θ
Here n is an integer (1, 2, 3, ...), λ is the wavelength of the x-ray beam, and θ is half the scattering angle 2 θ shown above.

Real materials are more complicated, of course, but the general result holds that there is a relationship between interparticle distances within the sample and the angles at which the scattered intensity is the highest, with larger distances d corresponding to smaller scattering angles 2θ.

What types of measurement are typically made?

Books have been filled describing different specialized techniques! But here is a short glossary of the most important techniques.


What are the components of an x-ray diffraction instrument?

Although there are many possible permutations, essentially all XRD instruments incorporate the components shown in the following schematic: a means of producing the x-ray radiation, some kind of collimation, something to support the sample (and possibly orient it or maintain a desired environment), and a means for detecting the scattered radiation.

Schematic of generic x-ray diffraction instrument.
  • Production of X-rays: There are a variety of methods for producing a beam of x-rays.
    • X-ray Tube. This is the simplest and oldest approach, and is still occasionally used. A beam of electrons strikes a metallic target and X-rays are emitted. The intensity of the X-ray beam is limited by the heat released into the target by the electron beam.
    • Rotating anode X-ray Generator. This variant of the traditional X-ray tube, which became widely available in the 1970's, addresses the heat loading problem by replacing the fixed target with a rotating cylinder, water-cooled on the inside. Considerably more X-ray intensity is thereby made possible, but there are both literal and figurative costs: the engineering requirements are considerably more stringent, and rotating-anode generators are subject to breakdowns and require frequent maintenance.
    • Microfocus Tube. The most recent solution to the heat loading problem takes a different tack: the electron beam is focused down to a tiny spot (typically 50 μm or less in diameter), so that the total heat load on the anode is quite small. Microsource tubes started to become available around 2000, and are gradually replacing rotating anode generators.
    • Synchrotron. A synchrotron X-ray source uses a totally different mechanism from the tube sources described above: the radiation emitted from a relativistic beam of electrons (or positrons) accelerated by a magnetic field. The resulting beam is generally many orders of magnitude more intense than that produced by the tabletop sources described above. However, such a beam can be produced only at a large centralized facility, obliging most users to travel substantial distances and plan their usage well in advance. For this reason, tube/rotating anode/microfocus sources, which can be operated at the user's home institution, are best suited for relatively routine measurements, while synchrotron sources are required for experiments requiring extremely high intensity or other specialized conditions. Major synchrotron sources include the Advanced Photon Source and the National Synchrotron Light Source in the US, the European Synchrotron Radiation Facility in France, the Diamond Light Source in Britain, and the Photon Factory in Japan, among others.
  • Collimation: The radiation produced by any of the above mechanisms consists in general of rays traveling in a variety of directions and consisting of a spread of wavelengths. The purpose of the collimation portion of an XRD instrument is to produce a relatively thin beam of X-rays with a narrow spread of wavelengths, all traveling in essentially the same direction. Some commonly used components are described below.
    • Slits or pinholes. These form a part of almost every instrument, and act by geometrically restricting the beam. To be effective, they must be constructed from a heavy element such as tungsten. Care must be taken to minimize diffuse ("parasitic") scattering from the edges of the slits which can contribute to the background signal.
    • Crystal monochromator. The most common method for producing a "monochromatic" beam (containing only a narrow spread of wavelengths λ) is to insert a high quality single crystal of a material such as silicon or germanium into the beam and separate out only those components of the beam that satisfy Bragg's Law. Conversely, for a beam that is already largely monochromatic, this Bragg reflection from a crystal can be used as a means of collimation. The degree of collimation and spectral selection depend on the perfection of the crystal and also the characteristics of the incoming beam.
    • X-ray Mirror. X-ray mirrors rely on the same effect referred to in our discussion of X-ray reflectivity, namely that a beam which strikes a flat surface at a very low angle can be strongly reflected. X-ray mirrors are typically made of a metal such as gold and are gently curved so as to produce a beam that is focused along a vertical and/or horizontal axis. They also affect the spectral characteristics since shorter wavelengths are reflected much less effectively than long wavelengths.
    • Multilayer Optics. This approach, which is incorporated in many units currently on the market (especially those optimized for small-angle scattering) combines the benefits of a crystal monochromator and an X-ray mirror. A multilayer coating on a curved substrate results in a monochromatic, collimated beam, most often either parallel or slightly convergent focus. The optical unit must be closely coupled with the source, but when done properly this can result in a beam that is simultaneously more intense and better collimated than achievable with previous technologies.
Two representative XRD instruments.
Two representative instruments for X-ray scattering. In both cases (1) shows the source or generator, (2) shows the sample support, and (3) shows the detector.
Top: Bruker D8 Discover system with Cu tube and scintillation detector. Source: Materials Characterization Services of the University of Minho
Bottom: Nanostar Area Detector system for SAXS, with microfocus source and area detection.
Source: University of Connecticut.

How are X-ray area detector data analyzed?

The first step in a diffraction experiment using an area detector is to position the sample in the X-ray beam such that diffracted rays strike the detector and then to expose the sample for a fixed amount of time. The resulting area of intensities (usually photon counts) for each pixel on the detector is then read by the computer, and displayed as a false-color image.

For quantitative analysis, the (x,y) pixel coordinates must be converted to more useful units. The sketch to the right shows a common way of labeling the angles. A portion of the incident beam generally passes through the sample undeflected--this is called the "primary beam". It is usually necessary to have some kind of beamstop to block this beam from directly striking the detector, but the position where it would hit is well defined. Then, relative to the beam center position, other diffracted rays will be deflected by a scattering angle 2θ at an azimuthal angle χ as shown to the right.

Instead of the scattering angle 2θ, the amount by which the scattered beam has been deflected is often described by the momentum transfer Q:
Q = (4 π / λ ) sin θ

The next steps depend on the kind of measurement being performed. Some of the more frequently encountered measurements are single crystal scattering, powder diffraction, and solution SAXS:

  • Single Crystal: For single crystal measurements, the pattern on the detector will consist of a large number of sharp spots. The analysis software must determine the position (2θ, χ) of each spot and the total (integrated) intensity within that spot. This measurement is then repeated for many different sample orientations. Detailed analysis (beyond the scope of this article) can then invert this information to determine the atomic positions within the sample.
  • Powder diffraction For powder diffraction the pattern on the detector will consist of a set of concentric sharp rings. In this case the intensity is independent of χ. For further analysis, the 2D image is reduced to an x-y plot consisting of the intensity per pixel as a function of 2θ, or Q averaged over all values of χ. Producing a plot of this sort (with the option for export to other applications) is one of the central capabilities of Datasqueeze. The next task is to produce a list of scattering angles and intensities for each peak (overlapping peaks can be a problem). A commonly used approach is to perform a least-squares fit to the pattern, modeling each peak as a Gaussian or similar function together with a smooth background.
  • Solution SAXS For small angle scattering from particles embedded in a liquid or solid matrix, the scattered intensity is again independent of χ, and again the 2D image is reduced to an x-y plot consisting of the intensity per pixel as a function of 2θ, or Q averaged over all values of χ. In this case a smooth pattern without sharp peaks is generally observed, but least-squares fits can be used to compare the observed pattern to the functional forms predicted for the size and shape of the individual particles. For example, the intensity predicted for scattering from uniform solid spheres of radius R is the square of the "Rayleigh Function":
    I = (const) | P(Q) | 2
    P(Q) = ( sin ( q R) - Q R cos(Q R) ) / (Q R )3
    Datasqueeze incorporates a wide selection of functions used for fitting SAXS data.
Diffraction geometry for a 2D X-ray detector.
Diffraction geometry for a 2D X-ray detector.
(a) Undeflected (primary) beam.
(b) Beam scattered at angle 2θ.
(c) Spot on detector produced by beam at deflection
angle 2θ and azimuthal angle χ=0.
(d) Spot produced by beam at deflection angle 2θ and azimuthal angle χ .

What is involved in calibrating an XRD instrument with an area detector?
Our discussion of X-ray data analysis makes it clear that a number of parameters must be accurately determined to perform any meaningful analysis. The first, which is not really a detector characteristic, is the wavelength λ. This depends on the characteristics of the X-ray source and collimation.

The image to the top right shows the typical geometry for an area detector-based apparatus. We need to accurately map the (x,y) coordinates of a detector pixel to (2θ,χ).

The first parameter that must be established is the exact position on the detector where the primary beam hits (or, would hit if it were not blocked by the beamstop). You might think that visual examination of that region of the measured image would be good enough--for example, one could choose a pixel in the middle of the shadow provided by the beamstop. However, it turns out that this is not good enough; for accurate measurements one needs to know the beam center position to within a fraction of one pixel size.

To interpret a radial distance from the beam center to a particular pixel, we also need a scale factor: the relationship between the width of one pixel and the scattering angle 2θ. We can get a good approximate idea of this factor if we know the distance between the sample and the detector and the dimensions of each pixel (or, equivalently, the sample:detector position and the dimensions of the entire detector).

Another issue to consider is the issue that the detector face may not be exactly perpendicular to the primary beam, but be rotated away by some small angle β as shown on the figure to the top right. This will have the effect of converting circular Bragg rings into ellipses on the detector.

One of the best solutions to the accurate determination of these parameters, which is employed by the Datasqueeze calibration wizard, is to use the Bragg rings of a known calibration standard. By using the fact that these rings must be centered on the primary beam position, must be circular, and must appear at known values of 2θ, it is possible to establish all of the calibration parameters to high accuracy.

An obvious limitation of the configuration just discussed is that the detector has a limited angular range. The smallest scattering angle is determined by the beamstop size: if D is the diameter of the beamstop and L is the sample-to-detector distance then the minimum scattering angle is given by
2 θminimum = tan-1 (D / 2 L )
Similarly, the widest achievable angle, for a detector of width W (with the beam center in the center of the detector) is approximately given by
2 θmaximum = tan-1 (W / 2 L )
If this spread in minimum angles is insufficient, the most common solution is to make measurements at multiple sample-detector distances. However, another solution is shown on the bottom right: the detector is mounted on a moveable arm and rotated through some large angle 2 Θ. After stepping through many values of 2 Θ to obtain overlapping patterns, a complete profile of the scattered intensity over a wide scattering angle range can be obtained. However, the conversion between (x,y) pixel location and angular coordinates (2θ, χ) becomes more complicated. For example, Bragg "rings" now become more generally conic sections. Datasqueeze is unusual in providing software to accurately interpret such data.

Calibration geometry for a 2D X-ray detector in small and wide angle configurations.
Top: Typical configuration for an XRD apparatus
incorporating an area detector. The direct beam is
prevented by a beamstop from striking the detector.
The detector face is close to but in general not perfectly
perpendicular to the direct beam.
Bottom: Wide-angle configuration, in which the detector
is mounted on a moveable arm that rotates it well away from the direct beam.

References and further reading.

A variety of useful links have been provided in the text above. The following books and articles are recommended for those seeking a more in-depth understanding of the topics discussed in this article.

Last updated August 9, 2017

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