X RAY DIFFRACTION- XRD

1. X RAY DIFFRACTION- XRD SOLID MATTER- AMORPHOUS: Atoms arranged in a random manner , like in liquids- eg: Glass CRYSTALLINE: Atoms arranged in a regular pattern. Smallest volume element repeats in three dimensions describing the crystal. The smallest volume element is UNIT CELL. Dimensions of the unit cell described by the edges a,b, and c and the angles between them alpha, beta and gamma.

2. X - RAYS • German scientist Rontgen discovered X-rays in 1895 accidentally when working with discharge tube. • Barium platinocyanide screen placed near the tube began to glow, Glow continued even when a wooden screen was placed between them. • As cause was not known, called as X-rays. • It could pass through opaque bodies. Wave length shorter than that of ultraviolet light.

4. Essential elements of a coolidge X- ray vacuum tube: • Cathode- tungsten filament heated to incandescence by a low voltage AC from a step down transformer/ storage battery. • Emits large number of electrons known as thermions focused on a target using cylindrical shields (molybdenum) • Shield maintained at a negative potential surrounding the cathode. • Electrons accelerated to very high speeds by DC potential difference about 50kV to 100kV applied between cathode and anode (anticathode). The high DC from a step up transformer. Electrons Tungsten filament Shield Cooling water X rays

6. The Coolidge tube (1913) is also called hot cathode tube • It works with a very good quality vacuum (about 10-4 Pa, or 10-6 Torr). • The filament is the cathode of the tube. The high voltage potential is between the cathode and the anode, the electrons are accelerated and then hit the anode. • There are two designs: end-window tubes and side-window tubes. • In the end-window tubes, the filament is around the anode, the electrons have a curved path. • Special about side-window tubes is: • An Electrostatic lens focuses the beam onto a very small spot on the anode • The anode is specially designed to dissipate the heat and wear resulting from this intense focused barrage of electrons: • Mechanically spun to increase the area heated by the beam. • Cooled by circulating coolant. • The anode is precisely angled at 1-20 degrees off perpendicular to the electron current so as to allow escape of some of the X-ray photons which are emitted essentially perpendicular to the direction of the electron current. • The anode is usually made out of tungsten or molybdenum. • The tube has a window designed for escape of the generated X-ray photons. • The power of a Coolidge tube usually ranges from 1 to 4 kW

7. Introduction to X-ray Diffraction • References: • Elements of Modern X-ray Physics, Jens Als-Nielsen and Des McMorrow, John Wiley & Sons, Ltd., 2001 (Modern x-ray physics & new developments) • X-ray Diffraction, by B.E. Warren, General Publishing Company, 1969, 1990 (Classic x-ray physics book) • Elements of X-ray Diffraction,2nd Ed., by B.D. Cullity, Addison-Wesley, 1978 (Covers most techniques used in traditional material characterization) • High Resolution X-ray Diffractometry and Topography, by D. Keith Bowen and Brian K. Tanner, Taylor & Francis, Ltd., 1998 (Semiconductors and thin film analysis) • Modern Aspects of Small-Angle Scattering, by H. Brumberger, Editor, Kluwer Academic Publishers, 1993 (SAXS techniques) • Principles of Protein X-ray Crystallography, by Jan Drenth, Springer, 1994 (Crystallography)

9. The incoming beam (coming from upper left) causes each scatterer to re-radiate a small portion of its energy as a spherical wave. If scatterers are arranged symmetrically with a separation d, these spherical waves will be in synch only in directions where their path-length difference 2 d sin θ equals an integer multiple of the wavelength λ. In that case, part of the incoming beam is deflected by an angle 2θ, producing a reflection spot in the diffraction pattern

10. An intuitive understanding of XRD can be obtained from the Bragg Model of Diffraction. • In this model, a given reflection is associated with a set of evenly spaced sheets running through the crystal, usually passing through the centers of the atoms of the crystal lattice. • The orientation of a particular set of sheets is identified by its three MILLER INDICES (h, k, l), and let their spacing be noted by d. • WILLIAM LAWARENCE BRAGG proposed a model in which the incoming X-rays are scattered specularly (mirror-like) from each plane; from that assumption, X-rays scattered from adjacent planes will combine constructively when the angle θ between the plane and the X-ray results in a path-length difference that is an integer multiple n of the X-ray wave length λ.

11. A reflection is said to be indexed when its Miller indices have been identified from the known wavelength and the scattering angle 2θ. Such indexing gives the unit cell parameters, the lengths and angles of the unit-cell, as well as its space group. Since BRAGG’S LAW does not interpret the relative intensities of the reflections, however, it is generally inadequate to solve for the arrangement of atoms within the unit-cell; for that, a Fourier transform method must be carried out.

12. BRAGG’S LAW

13. Theoretical Considerations

14. An X-ray diffraction pattern formed when X-rays are focused on a crystalline material, (a protein). Each dot, called a reflection, forms from the coherent interference of scattered X-rays passing through the crystal. X-ray scattering techniques are a family of non-destructive analytical techniques which reveal information about the crystallographic structure, chemical composition, and physical properties of materials and thin films. These techniques are based on observing the scattered intensity of an X-RAY beam hitting a sample as a function of incident and scattered angle, polarization, and wavelength or energy.

15. X-ray diffraction techniques • X-ray diffraction finds the geometry or shape of a molecule using x-rays. X-ray diffraction techniques are based on the elastic scattering of x-rays from structures that have long range order. • Single-crystal X-ray diffraction is a technique used to solve the complete structure of crystalline materials, ranging from simple inorganic solids to complex macromolecules, such as proteins. • Powder diffraction (XRD) is a technique used to characterize the crystallographic structure, crystallite size (grain size), and preferred orientation in polycrystalline or powdered solid samples. Powder diffraction is commonly used to identify unknown substances, by comparing diffraction data against a database maintained by the International Centre for Diffraction Data. It may also be used to characterize heterogeneous solid mixtures to determine relative abundance of crystalline compounds and, when coupled with lattice refinement techniques, such as Rietveld refinement, can provide structural information on unknown materials. Powder diffraction is also a common method for determining strains in crystalline materials.

16. Thin film diffraction and grazing incidence x-ray diffraction may be used to characterize the crystallographic structure and preferred orientation of substrate-anchored thin films. • High-resolution x-ray diffraction is used to characterize thickness, crystallographic structure, and strain in thin epitaxial films. It employs parallel-beam optics. • X-ray pole figure analysis enables one to analyze and determine the distribution of crystalline orientations within a crystalline thin-film sample. • X-ray rocking curve analysis is used to quantify grain

17. Scattering techniques Elastic scattering • Materials that do not have long range order may also be studied by scattering methods that rely on elastic scattering of monochromatic x-rays. • Small angle X-ray scattering (SAXS) probes structure in the nanometer to micrometer range by measuring scattering intensity at scattering angles 2θ close to 0°. • X-ray reflectivity is an analytical technique for determining thickness, roughness, and density of single layer and multilayer thin films. • Wide angle X-ray scattering (WAXS), a technique concentrating on scattering angles 2θ larger than 5°. Inelastic scattering • When the energy and angle of the inelastically scattered x-rays are monitored scattering techniques can be used to probe the electronic band structure of materials. • Compton scattering • Resonant inelastic x-ray scattering (RIXS) • X-ray Raman scattering

18. X-ray Generation & Properties • Lattice Planes and Bragg's Law • Powder Diffraction • Thin Film Diffraction • Texture Measurement (Pole Figures) • Residual Stress Measurements • Small Angle X-ray Scattering (SAXS) • X-ray Crystallography

19. 1. X-ray Generation & Properties • X-rays are electromagnetic radiation with typical photon energies in the range of 100 eV - 100 keV. For diffraction applications, only short wavelength x-rays (hard x-rays) in the range of a few angstroms to 0.1 angstrom (1 keV - 120 keV) are used. • Because the wavelength of x-rays is comparable to the size of atoms, they are ideally suited for probing the structural arrangement of atoms and molecules in a wide range of materials. The energetic x-rays can penetrate deep into the materials and provide information about the bulk structure. • X-rays are produced generally by either x-ray tubes or synchrotron radiation. In a x-ray tube, which is the primary x-ray source used in laboratory x-ray instruments, x-rays are generated when a focused electron beam accelerated across a high voltage field bombards a stationary or rotating solid target. As electrons collide with atoms in the target and slow down, a continuous spectrum of x-rays are emitted, which are termed Bremsstrahlung radiation. The high energy electrons also eject inner shell electrons in atoms through the ionization process. When a free electron fills the shell, a x-ray photon with energy characteristic of the target material is emitted.

20. Common targets used in x-ray tubes include Cu and Mo, which emit 8 keV and 14 keV x-rays with corresponding wavelengths of 1.54 Å and 0.8 Å, respectively. (The energy E of a x-ray photon and it's wavelength is related by the equation E = hc/l, where h is Planck's constant and c the speed of light) • In recent years synchrotron facilities have become widely used as preferred sources for x-ray diffraction measurements. Synchrotron radiation is emitted by electrons or positrons travelling at near light speed in a circular storage ring. These powerful sources, which are thousands to millions of times more intense than laboratory x-ray tubes, have become indispensable tools for a wide range of structural investigations and brought advances in numerous fields of science and technology.

21. 2. Lattice Planes and Bragg's Law • X-rays primarily interact with electrons in atoms. When x-ray photons collide with electrons, some photons from the incident beam will be deflected away from the direction where they original travel, much like billiard balls bouncing off one anther. If the wavelength of these scattered x-rays did not change (meaning that x-ray photons did not lose any energy), the process is called elastic scattering (Thompson Scattering) in that only momentum has been transferred in the scattering process. These are the x-rays that we measure in diffraction experiments, as the scattered x-rays carry information about the electron distribution in materials. On the other hand, In the inelastic scattering process (Compton Scattering), x-rays transfer some of their energy to the electrons and the scattered x-rays will have different wavelength than the incident x-rays.

22. Diffracted waves from different atoms can interfere with each other and the resultant intensity distribution is strongly modulated by this interaction. If the atoms are arranged in a periodic fashion, as in crystals, the diffracted waves will consist of sharp interference maxima (peaks) with the same symmetry as in the distribution of atoms. Measuring the diffraction pattern therefore allows us to deduce the distribution of atoms in a material.

23. The peaks in a x-ray diffraction pattern are directly related to the atomic distances. Consider an incident x-ray beam interacting with the atoms arranged in a periodic manner as shown in 2 dimensions • The atoms, represented as green spheres in the graph, can be viewed as forming different sets of planes in the crystal (colored lines). For a given set of lattice plane with an inter-plane distance of d, the condition for a diffraction (peak) to occur can be written as known as the Bragg's law, after W.L. Bragg, who first proposed it. n is an integer representing the order of the diffraction peak. The Bragg's Law is one of most important laws used for interpreting x-ray diffraction data. • Here, atoms are used as scattering points in this example, Bragg's Law applies to scattering centers consisting of any periodic distribution of electron density. Ie., the law holds true if the atoms are replaced by molecules or collections of molecules, such as colloids, polymers, proteins and virus particles

24. 3. Powder Diffraction • Powder XRD (X-ray Diffraction) is perhaps the most widely used x-ray diffraction technique for characterizing materials. As the name suggests, the sample is usually in a powdery form, consisting of fine grains of single crystalline material to be studied. The technique is used also widely for studying particles in liquid suspensions or polycrystalline solids (bulk or thin film materials).

25. The term 'powder' really means that the crytalline domains are randomly oriented in the sample. Therefore when the 2-D diffraction pattern is recorded, it shows concentric rings of scattering peaks corresponding to the various d spacings in the crystal lattice. The positions and the intensities of the peaks are used for identifying the underlying structure (or phase) of the material. For example, the diffraction lines of graphite would be different from diamond even though they both are made of carbon atoms. This phase identification is important because the material properties are highly dependent on structure (just think of graphite and diamond). • .

26. Powder diffraction data can be collected using either transmission or reflection geometry, as shown below. Because the particles in the powder sample are randomly oriented, these two methods will yield the same data. In the MRL x-ray facility, powder diffraction data are measured using the Philips XPERT MPD diffractometer, which measures data in reflection mode and is used mostly with solid samples, or the custom built 4-circle diffractometer, which operates in transmission mode and is more suitable for liquid phase samples

27. MOUNTING THE CRYSTAL DIFFRACTOMETER

28. A powder XRD scan from a K2Ta2O6 sample is as shown -as a plot of scattering intensity v/s. the scattering angle 2theta or the corresponding d-spacing. The peak positions, intensities, widths and shapes all provide important information about the structure of the material.

29. 4. Thin Film Diffraction • Thin film diffraction refers not to a specific technique but rather a collection of XRD techniques used to characterize thin film samples grown on substrates. These materials have important technological applications in microelectronic and optoelectronic devices, where high quality epitaxial films are critical for device performance. Thin film diffraction methods are used as important process development and control tools, as hard x-rays can penetrate through the epitaxial layers and measure the properties of both the film and the substrate. • There are several special considerations for using XRD to characterize thin film samples. (i) reflection geometry is used for these measurements as the substrates are generally too thick for transmission. (ii) high angular resolution is required because the peaks from semiconductor materials are sharp due to very low defect densities in the material. Multiple bounce crystal monochromators are used to provide a highly collimated x-ray beam for these measurements. Eg: in the Philips MRD used in the x-ray facility, a 4-crystal monochromator made from Ge is used to produce an incident beam with less than 5 arc seconds of angular divergence.

30. Basic XRD measurements made on thin film samples include: • Precise lattice constants measurements derived from 2q - q scans, which provide information about lattice mismatch between the film and the substrate and therefore is indicative of strain & stress • Rocking curve measurements made by doing a q scan at a fixed 2q angle, the width of which is inversely proportionally to the dislocation density in the film and is therefore used as a gauge of the quality of the film. • Superlattice measurements in multilayered heteroepitaxial structures, which manifest as satellite peaks surrounding the main diffraction peak from the film. Film thickness and quality can be deduced from the data. • Glancing incidence x-ray reflectivity measurements, which can determine the thickness, roughness, and density of the film. This technique does not require crystalline film and works even with amorphous materials. • Texture measurements-(discussed separately)

31. The graph shows the high resolution XRD data of the superlattice peaks on the GaN (002) reflections. Red line denotes results of computer simulation of the structure.

32. 5. Texture Measurement (Pole Figure) • Texture measurements are used to determine the orientation distribution of crystalline grains in a polycrystalline sample. A material is termed textured if the grains are aligned in a preferred orientation along certain lattice planes. One can view the textured state of a material (typically in the form of thin films) as an intermediate state in between a completely randomly oriented polycrystalline powder and a completely oriented single crystal. The texture is usually introduced in the fabrication process (e.g. rolling of thin sheet metal, deposition,etc.) and affect the material properties by introducing structural anisotropy.

33. A texture measurement is also referred to as a pole figure as it is often plotted in polar coordinates consisting of the tilt and rotation angles with respect to a given crytallographic orientation. A pole figure is measured at a fixed scattering angle (constant d spacing) and consists of a series of f -scans (in- plane rotation around the center of the sample) at different tilt or Y -(azimuth) angles, as illustrated below.

34. The pole figure data are displayed as contour plots or elevation graphs with zero angle in the center. Below we show two pole figure plots using the same data set. An orientation distribution function (ODF) can be calculated using the pole figure data.

35. 6. Residual Stress Measurement • Structural and residual stress in materials can be determined from precision lattice constants measurements. For polycrystalline samples high resolution powder diffraction measurements generally will provide adequate accuracy for stress evaluation. For textured (oriented) and single crystalline materials, 4-circle diffractometry is needed in which the sample is rotated so that measurements on multiple diffraction peaks can be carried out. The interpretation of stress measurement data is complicated and model dependent. Consult the reference literature for more details

36. 7. Small Angle X-ray Scattering (SAXS) • SAXS measurements typically are concerned with scattering angles < 1o. As dictated by Bragg's Law, the diffraction information about structures with large d-spacings lies in the region. Therefore the SAXS technique is commonly used for probing large length scale structures such as high molecular weight polymers, biological macromolecules (proteins, nucleic acids, etc.), and self-assembled superstructures (e.g. surfactant templated mesoporous materials). • SAXS measurements are technically challenging because of the small angular separation of the direct beam (which is very intense) and the scattered beam. Large specimen-to-detector distances (0.5 m - 10 m) and high quality collimating optics are used to achieve good signal-to-noise ratio in the SAXS measurement.

37. The MRL x-ray facility has cutting edge capabilities for SAXS measurements with three custom-built SAXS instruments including one 3.5-meter long ultra-small angle SAXS instrument with state-of-the-art optics and area detector for low scattering density samples (see instrumentation section for more details)

38. 8. X-ray Crystallography • X-ray crystallography is a standard technique for solving crystal structures. Its basic theory was developed soon after x-rays were first discovered more than a century ago. However, over the years it has gone through continual development in data collection instrumentation and data reduction methods. In recent years, the advent of synchrotron radiation sources, area detector based data collection instruments, and high speed computers has dramatically enhanced the efficiency of crystallographic structural determination. Today x-ray crystallography is widely used in materials and biological research. Structures of very large biological machinery (e.g. protein and DNA complexes, virus particles) have been solved using this method.

39. In x-ray crystallography, integrated intensities of the diffraction peaks are used to reconstruct the electron density map within the unit cell in the crystal. To achieve high accuracy in the reconstruction, which is done by Fourier transforming the diffraction intensities with appropriate phase assignment, a high degree of completeness as well as redundancy in diffraction data is necessary, meaning that all possible reflections are measured multiple times to reduce systematic and statistical error. The most efficient way to do this is by using an area detector which can collect diffraction data in a large solid angle. The use of high intensity x-ray sources, such as synchrotron radiation, is an effective way to reduce data collection time.

40. One of the central difficulties in structural determination using x-ray crystallography is referred to as the "phase problem", which arises from the fact that the diffraction data contains information only on the amplitude but not the phase of the structure factor. Over the years many methods have been developed to deduce the phases for reflections, including computationally based direct methods, isomorphous replacement, and multi-wavelength anormalous diffraction (MAD) methods.

41. X-ray crystallography

42. Procedure • The technique of single-crystal X-ray crystallography has three basic steps. The first — and often most difficult — step is to obtain an adequate crystal of the material under study. The crystal should be sufficiently large (typically larger than 100 micrometres in all dimensions), pure in composition and regular in structure, with no significant internal imperfections such as cracks or twinning. A small or irregular crystal will give fewer and less reliable data, from which it may be impossible to determine the atomic arrangement.

43. In the second step, the crystal is placed in an intense beam of X-rays, usually of a single wavelength (monochromatic X-rays), producing the regular pattern of reflections. As the crystal is gradually rotated, previous reflections disappear and new ones appear; the intensity of every spot is recorded at every orientation of the crystal. Multiple data sets may have to be collected, with each set covering slightly more than half a full rotation of the crystal and typically containing tens of thousands of reflection intensities.

44. In the third step, these data are combined computationally with complementary chemical information to produce and refine a model of the arrangement of atoms within the crystal. The final, refined model of the atomic arrangement — now called a crystal structure — is usually stored in a public database.

45. A real 3-dimensional crystal contains many sets of planes. For diffraction, crystal must have the correct orientation with respect to the incoming beam. • Perfect, infinite crystal and perfectly collimated beam: diffraction condition must be satisfied ``exactly.'' • Strains, defects, finite size effects, instrumental resolution: diffraction peaks are broadened. More formally, the scattered intensity is proportional to the square of the Fourier transform of the charge density:

46. where is the charge density. For perfect crystals, I(q) consists of delta functions (perfectly sharp scattering). For imperfect crystals, the peaks are broadened. For liquids and glasses, it is a continuous, slowly varying function

47. Features of Electron, X-ray, or Neutron Diffraction • For a known structure, pattern can be calculated exactly. • Symmetry of the diffraction pattern given by symmetry of the lattice. • Intensities of spots determined by basis of atoms at each lattice point. • Sharpness and shape of spots determined by perfection of crystal. • Liquids, glasses, and other disordered materials produce broad fuzzy rings instead of sharp spots. • Defects and disorder in crystals also result in diffuse scattering.

48. The ``Ultimate'' (Technically Challenging) Experiment • Sample is tiny (micron-sized). • The effect is weak (light elements, small modulations, subtle modifications of the long-range order). • Instrumental resolution (angle and energy) is ``perfect'' allowing detailed measure- ments of structural disorder. • Measurement is time-resolved (nanosecond time scale). • To achieve all of the above, will need lots of intensity in the primary beam together with sensitive detection systems.

49. Powder vs. Single Crystal X-ray Diffraction SINGLE CRYSTAL • Put a crystal in the beam, observe what reflections come out at what angles for what orientations of the crystal with what intensities. • Advantages • In principle you can learn everything there is to know about the structure. • Disadvantages • You may not have a single crystal. It is time-consuming and difficult to orient the crystal. If more than one phase is present, you will not necessarily realize that there is more than one set of reflections.

50. POWDER • Samples consists of a collection of many small crystallites with random orientations. Average over crystal orientations and measure the scattered intensity as a function of outgoing angle. • Disadvantage • Inversion of the measured intensities to find the structure is more difficult and less reliable. • Advantages • It is usually much easier to prepare a powder sample. You are guaranteed to see all reflections. The best way to follow phase changes as a function of temperature, pressure, or some other variable.