X-ray Diffraction
Introduction
When x-rays were discovered in 1895 by Wilhelm Conrad Röntgen (the first Nobel laureate in Physics 1901), their exact nature was not known. The question posed at the time was "are x-rays particles or are they waves like visible light?" Showing that x-rays could diffract would prove that x-rays have a wave-like nature. If x-rays were waves, then their wavelengths had already been estimated to be roughly 10-9 cm. Diffraction effects are observed only when the repeat distances in a material are of the order of magnitude of the wavelength of the radiation.
At that time it was strongly believed that crystals were made up of many repeating blocks and that each repeating block contained a constant number of the same type of atoms. From some simple calculations using density, atomic numbers, and Avogadro's number, researchers of the time were able to show that simple crystals should have repeating units of the size needed for their proof of the wave-like nature of x-rays. Friedrich and Knipping performed the first x-ray diffraction experiment using a crystal of copper sulfate.( W. Freidrich, P. Knipping, and M. Von Laue, Proc. Bavarian Acad. Sci., 1912, 303;
Reprinted in Naturewissenschaften, 1952, 39, 367.) They obtained a diffraction pattern and concluded that x-ray must be electromagnetic radiation. Bragg diffraction was first proposed by William Lawrence Bragg in 1912 as a means of analyzing the structure of crystals. Bragg and his father William Henry Bragg collimated x-rays to diffract off of different crystal planes. The x-rays were then collected in an ionization chamber and the level of ionization was measured as a function of the incident angle of the x-rays. Using this method, the Braggs were able to determine the crystalline spacing for a number of substances. In 1919 A. W. Hull gave a paper titled, “A New Method of Chemical Analysis.” Here he pointed out that “….every crystalline substance gives a pattern; the same substance always gives the same pattern; and in a mixture of substances each produces its pattern independently of the others.”
Why X-ray diffraction systems ?
X-ray diffraction (XRD) is an analytical technique looking at X-ray scattering from crystalline materials. Each material produces a unique X-ray "fingerprint" of X-ray intensity versus scattering angle that is characteristic of it's crystalline atomic structure. Qualitative analysis is possible by comparing the XRD pattern of an unknown material to a library of known patterns.
What X-ray diffraction systems ?
About 95% of all solid materials can be described as crystalline. When X-rays interact with a crystalline substance (phase), one gets a diffraction pattern. The X-ray diffraction pattern of a pure substance is, therefore, like a fingerprint of the substance. The powder diffraction method is thus ideally suited for characterization and identification of polycrystalline phases.
Today about 50,000 inorganic and 25,000 organic single component, crystalline phases, diffraction patterns have been collected and stored on magnetic or optical media as standards. The main use of
powder diffraction is to identify components in a sample by a search/match procedure. Furthermore, the areas under the peak are related to the amount of each phase present in the sample.
Theoretical Considerations
In order to better convey an understanding of the fundamental principles and buzz words of x-ray diffraction instruments, let us quickly look at the theory behind these systems. (The theoretical considerations are rather primitive, hopefully they are not too insulting.)
Solid matter can be described as:
Amorphous: The atoms are arranged in a random way similar to the disorder we find in a liquid.
Glasses are amorphous materials.
Crystalline: The atoms are arranged in a regular pattern, and there is as smallest volume element that by repetition in three dimensions describes the crystal. E.g. we can describe a brick wall by the shape and orientation of a single brick. This smallest volume element is called a unit cell. The dimensions of the unit cell are described by three axes: a, b, c and the angles between them alpha, beta, gamma.
About 95% of all solids can be described as crystalline.
An electron in an alternating electromagnetic field will oscillate with the same frequency as the field.
When an x-ray beam hits an atom, the electrons around the atom start to oscillate with the same frequency as the incoming beam. In almost all directions we will have destructive interference, that is, the combining waves are out of phase and there is no resultant energy leaving the solid sample. However the atoms in a crystal are arranged in a regular pattern, and in a very few directions we will have constructive interference. The waves will be in phase and there will be well defined x-ray beams leaving the sample at various directions. Hence, a diffract ed beam may be described as a beam composed of a large number of scattered rays mutually reinforcing one another. This model is complex to handle mathematically, and in day to day work we talk about x-ray reflections from a series of parallel planes inside the crystal. The orientation and interplanar spacing of these planes are defined by the three integers h, k, l called indices.
A given set of planes with indices h, k, l cut the a-axis of the unit cell in h sections, the b axis in k sections and the c axis in l sections. A zero indicates that the planes are parallel to the corresponding axis.
e.g. the 2, 2, 0 planes cut the a– and the b– axes in half, but are parallel to the c– axis.
Structural Characterization of Thin Film by X-ray Diffraction (XRD) Bragg’s Diffraction :
n = 2 d sin
a
b
c Miller indices of planes: Example (210)
Real space co-ordinate a b c
Intercepts 1 2 ∞
Reciprocals 1 1/2 0
Miller indices (clear fraction)
2(h) 1(k) 0(l)
For simple cubic structures:
2 2 2 2 2
1
a l k h d
(210) (111)
(001) (101) (110)
d
1
2
1’
2’
The two parallel incident rays 1 and 2 make an angle (THETA) with these planes. A reflected beam of maximum intensity will result if the waves represented by 1’ and 2’ are in phase. The difference in path length between 1 to 1’and 2 to 2’ must then be an integral number of wavelengths, (LAMBDA). We can express this relationship mathematically in Bragg’s law.
2d sin θ = n λ
The process of reflection is described here in terms of incident and reflected (or diffracted) rays, each making an angle THETA with a fixed crystal plane. Reflections occurs from planes set at angle THETA with respect to the incident beam and generates a reflected beam at an angle 2-THETA from the incident beam. The possible d-spacing defined by the indices h, k, l are determined by the shape of the unit cell.
Rewriting Bragg’s law we get : sin θ = λ/2d
Therefore the possible 2-THETA values where we can have reflections are determined by the unit cell dimensions. However, the intensities of the reflections are determined by the distribution of the electrons in the unit cell. The highest electron density are found around atoms. Therefore, the intensities depend on what kind of atoms we have and where in the unit cell they are located. Planes going through areas with high electron density will reflect strongly, planes with low electron density will give weak intensities.
SAMPLES
In x-ray diffraction work we normally distinguish between single crystal and polycrystalline or powder applications. The single crystal sample is a perfect (all unit cells aligned in a perfect extended pattern) crystal with a cross section of about 0.3 mm. The single crystal diffractometer and associated computer package is used mainly to elucidate the molecular structure of novel compounds, either natural products or man made molecules. Powder diffraction is mainly used for “finger print identification” of various solid materials, e.g. asbestos, quartz. In powder or polycrystalline diffraction it is important to have a sample with a smooth plane surface. If possible, we normally grind the sample down to particles of about 0.002 mm to 0.005 mm cross section. The ideal sample is homogeneous and the crystallites are randomly distributed (we will later point out problems which will occur if the specimen deviates from this ideal state). The sample is pressed into a sample holder so that we have a smooth flat surface.
Ideally we now have a random distribution of all possible h, k, l planes. Only crystallites having reflecting planes (h, k, l) parallel to the specimen surface will contribute to the reflected intensities. If we have a truly random sample, each possible reflection from a given set of h, k, l planes will have an equal number of crystallites contributing to it. We only have to rock the sample through the glancing angle THETA in order to produce all possible reflections.
GONIOMETER
The mechanical assembly that makes up the sample holder, detector arm and associated gearing is referred to as goniometer. The working principle of a Bragg-Brentano parafocusing (if the sample was curved on the focusing circle we would have a focusing system) reflection goniometer is shown below.
The distance from the x-ray focal spot to the sample is the same as from the sample to the detector. If we drive the sample holder and the detector in a 1:2 relationship, the reflected (diffracted) beam will stay focused on the circle of constant radius. The detector moves on this circle.
For the THETA:2-THETA goniometer, the x-ray tube is stationary, the sample moves by the angle THETA and the detector simultaneously moves by the angle 2-THETA. At high values of THETA small or loosely packed samples may have a tendency to fall off the sample holder.
For the THETA:THETA goniometer, the sample is stationary in the horizontal position, the x-ray tube and the detector both move simultaneously over the angular range THETA.
When a completely randomized powder sample is placed into an X-ray beam the result is a complete Debye cone. When analyzing this type of sample any linear scan through the Debye cones will give an accurate powder pattern. This "linear scan" is exactly how a conventional Bragg-Brentano powder diffraction system works.
However, when a system is not a completely randomized powder (texture measurement) or if there is only one or just a few crystal (microdiffraction measurement) or a combination of both, the result is an incomplete Debye cone. This situation can cause problems for a conventional diffractometer with point or line detector.
Mode of detection:
1) Linear scan – point or line detection (source)
High resolution (can be), sensitive and very accurate measurement of a particular structural property. It is slow and time consuming.
2) 2-dimensional (area) detection
Record virtually all-structural properties simultaneously, but inferior resolution and sensitivity.
Four circle mode
Texture measurement and stress analyses:
Pole figures
They provide a panoramic view of the structural properties of the films.
It is well known that materials have different properties in different directions. A major push has happened to engineer ways to process the orientation of materials in the direction of the desired property. This sample orientation is called texture and can be determined quite easily by pole figures.
Stress analyses
Residual stress is conventionally analyzed with X-ray diffraction by measuring the angular shifts of diffraction peaks with respect to the orientation of the sample relative to the primary X-ray beam. However, obtaining the complete stress state in three dimensions with point or line detectors requires very long measurement times. The use of area detector has made such analyses simple and fast.
Small Angle X-ray Scattering (SAXS)
We have considered that Bragg's Law, d = /(2 sin), supports a minimum size of measurement of /2 in a diffraction experiment (limiting sphere of inverse space) but does not predict a maximum size, i.e. the point (000) of inverse space reflects infinite size. In class we have discussed the use of diffraction to measure crystalline and amorphous structures on the atomic scale, but clearly, many morphologies are of importance that have characteristic sizes much larger than the atomic scale. In metals this was first noted by Guinier in his development of the theory of Guinier-Preston zones. Guinier was one of the fathers of an outgrowth of diffraction aimed at large-scale structures in the 1950's. Bragg's Law predicts that information pertaining to such nano- to colloidal-scale structures would be seen below 6° 2in the diffractometer trace. It is possible to design specialized instruments to measure down to less than 1/1000 of a degree for measurement of up to 1-micron scale structures using x-rays! The characteristics of materials at these larger size scales are fundamentally different than at atomic scales. Atomic scale structures are characterized by high degrees of order, i.e. crystals, and relatively simple and uniform building blocks, i.e. atoms. On the nano-scale, the building blocks of matter are rarely well organized and are composed of rather complex and non-uniform building blocks. The resulting features in x-ray
scattering or diffraction are sharp diffraction peaks in the XRD range and comparatively nondescript diffuse patterns in the SAXS range.
Diffraction Methods:
For a single crystal, a single wavelength and a single orientation of the crystal to the main beam it is unlikely that the Bragg condition will be met for a given plane. Rotation of the crystal will bring planes into the Bragg condition. This means that it is desirable to have one of these parameters have variability.
1) X-ray Spectrometer: Rotation of a perfect crystal in an x-ray beam is one method to determine the x-ray spectrum using Bragg's Law. By rotation of a single crystal with a fixed detector, or using a position sensitive detector with a fixed crystal we can perform x-ray spectroscopy experiments similar to how a diffraction grating can be used in an IR or UV spectrometer.
2) Laue Camera: With a polychromatic incident beam many planes will meet the Bragg condition and tracing the 2-d pattern on a photographic film will reveal the planes of a zone.
3) Rotating Crystal Method: For a mono chromatic beam and a single crystal. Rotate the crystal during diffraction experiment to bring Bragg planes into alignment.
4) Powder Method: Mono-chromatic beam, polycrystalline sample. Usually done with a flat film in pinhole arrangement.
5) Diffractometer Method: Similar to the powder method but uses a step-scanner and a line beam.
Usually involves a polycrystalline sample.
Description of Laue Camera:
The Laue method was the first diffraction method ever used. A beam of white radiation falls on a fixed single crystal. The Bragg angle theta is therefore fixed for every set of planes in the crystal, and each set picks out and diffracts that particular wavelength which satisfies the Bragg law for the particular values of the lattice distance d and the diffraction angle theta involved. There are two variants of the Laue method, depending on the relative positions of the source, crystal and film: transmission Laue method and back-reflection Laue method.
This Laue photo camera makes use of the back-reflection Laue method. In this case the film is placed between the crystal and the X-ray source, the incident beam passing through a hole in the film, and the beams diffracted in a backward direction are recorded.
The diffracted beams form an array of spots on the film as shown in the figure below. The positions of the spots on the film depend on the orientation of the crystal relative to the incident beam, and the spots become distorted and smeared out if the crystal has been bent or twisted in any way.
The two main uses of the Laue methods are the determination of crystal orientation and the assessment of crystal perfection.
Rotating Crystal Method:
Powder Method:
Diffractometer Method:
To understand more about how XRD functions, see Phase, Stress and Texture analysis below.
Phase analysis:
About 95% of all solid materials can be described as crystalline. When X-rays interact with a crystalline substance, one gets a diffraction pattern. The X-ray diffraction pattern of a substance is, therefore, like a fingerprint of the substance. It can be used or identification and quantitative determination of the various crystalline compounds, known as 'phases', present in solid materials and powders. The powder diffraction method is
thus ideally suited for characterization and identification of polycrystalline phases.
The main use of powder diffraction is to identify components in a sample by a search/match procedure.
Furthermore, the areas under the peak are related to the amount of each phase present in the sample.
The diagram (above) depicts a typical sample, comprising two crystalline phases (violet, grey), each with different average crystallite sizes, plus a proportion of amorphous material (beige).
How large are the crystallites?
The widths of the peaks in a particular pattern provide an indication of the average crystallite size. Large crystallites give rise to sharp peaks, while the peak width increases as crystallite size reduces. Peak broadening also occurs as a result of variations in d-spacing
caused by micro-strain. However, the relationship between broadening and diffraction angle 2-theta is different from that of crystallite size effects, making it possible to differentiate between the two phenomena.
The result of an XRD measurement is a diffractogram, showing phases present (peak positions), phase concentrations (peak heights), amorphous content (background hump) and crystallite size/strain (peak widths).
What is the structure of the crystals?
The Rietveld method refines a crystal structure by comparing the measured diffraction pattern with that calculated from a known crystal structure. A least-squares refinement is used to optimize the structure parameters.
Stress analysis
Stress is determined by recording the angular shift of a given Bragg reflection as a function of sample tilt (psi). This actually provides a measure of strain in the sample from which the stress can then be calculated by plotting the change in d-spacing against sin2psi.
Texture analysis
Measurement of texture (the non-random or preferred orientation of crystallites) involves measurement of the variations in intensity of a single Bragg reflection as the sample is both tilted (psi) and rotated (phi).
The result is plotted as a 'pole figure', in which the contours indicate intensity levels as a function of sample orientation.
X-ray reflectometry:
X-ray reflectivity is based on interference of X-ray reflected from two interfaces (i.e. vacuum-film or film-substrate). For angles above the critical one, the expression for the X-ray reflection coefficient from a film-substrate system is described by Fresnel theory. In semiconductor manufacturing, existing X-ray instruments for thin films are designed for ex-situ measurements.
The most common implementation of X-ray reflectivity rotates the sample being studied by some angle in the range from zero to several degrees while the detector is rotated at double angular velocity; this is
“-2” scanning.
The equation describing the variation of the reflection coefficient shows that interference can occur with changing angle of incidence, wavelength, or film thickness. In principle, variation of the wavelength is not different from variation of the angle of incidence. However, variation of film thickness (e.g. as occurs during thin-film deposition) with constant angle of incidence and wavelength provides a fundamental new approach to the application of X-ray reflectivity. By recording the time dependence of the reflection coefficient, one can find the same set of parameters as that determined from angular dependence.