1 5 Part 1. X-RAY DIFFRACTION MICROSTRUCTURE ANALYSIS 1.1. INTERACTION OF ELECTROMAGNETIC WAVES WITH MATTER The different parts of the electromagnetic spectrum have very different effects upon interaction with matter. Starting with low frequency radio waves, the human body is quite transparent. (You can listen to your portable radio inside your home since the waves pass freely through the walls of your house and even through the person beside you!). As you move upward through microwaves and infrared to visible light, you absorb more and more strongly. In the lower ultraviolet range, all the UV from the sun is absorbed in a thin outer of your skin. As you move further up into the X-ray region of the spectrum, you become transparent again, because most of the mechanisms for absorption are gone. You then absorb only a small fraction of the radiation, but that absorption involves the more violent ionization events. Each portion of the electromagnetic spectrum has quantum energies appropriate for the excitation of certain types of physical processes. The energy levels for all physical processes at the atomic and molecular levels are quantized, and if there are no available quantized energy levels with spacing which match the quantum energy of the incident radiation, then the material will be transparent to that radiation, and it will pass through Microwave interactions The quantum energy of microwave photons is in the range to ev which is in the range of energies separating the quantum states of molecular rotation and torsion. The interaction of microwaves with matter other than metallic conductors will be to rotate molecules and produce heat as result of that molecular motion. Conductors will strongly adsorb microwaves and any lower frequencies because they will cause electric currents which will heat the material. Most matter, including the human body, is largely transparent to microwaves. High intensity microwaves, as in a microwave oven where they pass back and forth through the food millions of times, will heat the material by producing molecular rotations and torsions. Since the quantum energies are a million times lower than those of X- rays, they cannot produce ionization and the characteristic types of radiation damage associated with ionizing radiation Infrared interactions Frequencies: ( ) Hz. Wavelengths: 1 mm-750 nm. The quantum energy of infrared photons is in the range to 1.7 ev which is in the range of energies separating the quantum states of molecular vibrations. Infrared is absorbed more strongly than microwaves, but less strongly than visible light. The result of infrared absorption is heating of the tissue since it increases molecular vibrational activity. Infrared radiation does penetrate the skin further than visible light and can thus be used for photographic imaging of subcutaneous blood vessels.
2 Visible light interactions Frequencies: (4-7.5) Hz. Wavelengths: nm. Quantum energies: ev. The primary mechanism for the absorption of visible light photons is the elevation of electrons to higher energy levels. There are many available states, so visible light is adsorbed strongly. With a strong light source, red light can be transmitted through the hand or a fold of skin, showing that the red end of the spectrum is not adsorbed as strongly as the violet end. While exposure to visible light causes heating, it does not cause ionization with its risks. You may be heated by the sun through a car windshield, but you will not be sunburned that is an effect of the higher frequency UV part of sunlight which is blocked by the glass of the windshield Ultraviolet interactions Frequencies: Hz. Wavelengths: nm. Quantum energies: ev. The near ultraviolet is adsorbed very strongly in the surface layer of the skin by electron transitions. As you go to higher energies, the ionization energies for many molecules are reached and the more dangerous photoionization processes take place. Sunburn is primarily an effect of UV, and ionization produces the risk of skin cancer. The ozone layer in the upper atmosphere is important for human health because it adsorbs most of the harmful ultraviolet radiation from the sun before it reaches the surface. The higher frequencies in the ultraviolet are ionizing radiation and can produce harmful physiological effects ranging from sunburn to skin cancer. Health concerns for UV exposure are mostly for the range nm in wavelength, the range called UVB. According to Scotto, et al, the most effective biological wavelength for producing skin burns is 297 nm. Their research indicates that the biological effects increase logarithmically within the UVB range, with 330 nm being only 0.1% as effective as 297 nm for biological effects. So it is clearly important to control exposure to UVB X-ray interactions Frequencies: Hz upward. Wavelengths: 10 nm downward. Quantum energies: 124 ev upward. Since the quantum energies of X-ray photons are much too high to be absorbed in electron transitions between states for most atoms, they can interact with an electron only by knocking it completely out of the atom. Thus is, all X-rays are classified as ionizing radiation. This can occur by giving all of the energy to an electron (photoionization) or by giving part of the energy to the photon and the remainder to a lower energy photon (Compton scattering). At sufficiently high energies, the X-ray photon can create an electron positron pair.
3 BASICS OF X-RAY DIFFRACTION X-ray tube X-rays for diagnostic procedures or for research purposes are produced in a standard way: by accelerating electrons with a high voltage and allowing them to collide with a metal target Fig. 1. X-rays are produced when the electrons are suddenly decelerated upon collision with the metal target; these X-rays are commonly called bremsstrahlung or braking radiation. If the bombarding electrons have sufficient energy, they can knock an electron out of an inner shell of the target metal atoms. Then electrons from higher states drop down to fill the vacancy, emitting X-ray photons with precise energies determined by the electron energy levels. These X-rays are called characteristic X-rays. Heat filament emits electrons by thermionic emission Electrons are accelerated by a high voltage Cooper rod for heat dissipation Glass envelope Fig. 1. X-ray tube X-rays produced when high speed electrons hit the metal target Characteristic X-rays Characteristic X-rays are emitted from heavy elements when their electrons make transitions between the lower atomic energy levels. The characteristic X-rays emission which shown as two sharp peaks in the illustration (Fig. 2) occur when vacancies are produced in the n = 1 or K-shell of the atom and electrons drop down from above to fill the gap. The X-rays produced by transitions from the n = 2 to n = 1 levels are called K-alpha X-rays, and those for the n = 3 > 1 transition are called K-beta X-rays. Transitions to the n = 2 or L-shell are designated as L X-rays (n = 3 > 2 is L-alpha, n = 4 > 2 is L-beta, etc.). The continuous distribution of X-rays which forms the base for the two sharp peaks at left is called bremsstrahlung radiation. X-ray production typically involves bombarding a metal target in an X-ray tube with high speed electrons which have been accelerated by tens to hundreds of kilovolts of potential. The bombarding electrons can eject electrons from the inner shells of the atoms of the metal target. Those vacancies will be quickly filled by electrons dropping down from higher levels, emitting X-rays with sharply defined frequencies associated with the difference between the atomic energy levels of the target atoms.
4 8 K α 3 Relative intensity 2 1 Bremsstrahlung continuum K β Characteristic X-rays Wavelength, nm Fig. 2. X-rays from a molybdenum target at 35 kv The frequencies of the characteristic X-rays can be predicted from the Bohr model. Moseley measured the frequencies of the characteristic X-rays from a large fraction of the periodic table and produces a plot of them which is now called a Moseley plot. Characteristic X-rays are used for the investigation of crystal structure by X-ray diffraction. Crystal lattice dimensions may be determined with the use of Bragg s law in a Bragg spectrometer Bremsstrahlung X-rays Bremsstrahlung means breaking radiation and is retained from the original German to describe the radiation which is emitted when electrons are decelerated or braked when they are fired at a metal target. Accelerated charges give off electromagnetic radiation, and when the energy of the bombarding electrons is high enough, that radiation is in the X-ray region of the electromagnetic spectrum. It is characterized by a continuous distribution of radiation which becomes more intense and shifts toward higher frequencies when the energy of the bombarding electrons is increased. The curves in Fig. 3 are from the 1918 data of Ulrey, who bombarded tungsten targets with electrons of four different energies. The bombarding electrons can also eject electrons from the inner shells of the atoms of the metal target, and the quick filling of those vacancies by electrons dropping don from higher levels gives rise to sharply defined characteristic X-rays.
5 9 50 kv 40 kv 30 kv 20 kv Fig. 3. X-ray continuous radiation (Bremsstrahlung) The Bragg s law When X-rays interact with a crystalline substance (phase), one gets a diffraction pattern. Solid matter can be described as: (i) amorphous: the atoms are arranged in a random way similar to the disorder we find in a liquid, glass are amorphous materials; (ii) crystalline: the atoms in a regular pattern, and there is as smallest volume element that by repetition in three dimensions describes the crystal. 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 α, β, γ. 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 combing 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 be well defined X-ray beams leaving the sample at various directions. Hence, a diffracted 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, consider X-ray reflections from a series of parallel planes inside the crystal. The orientation and interplanar spacings 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 (Fig. 4). E.g., the 2, 2, 0 planes cut the a- and the b- axes in half, but are parallel to the c- axis.
6 10 c c c b 100 a b 110 a b 200 a Fig. 4. The notation of planes The three dimensional diffraction grating as a mathematical model is used. The three indices h, k, l become the order of diffraction along the unit cell axes a, b and c, respectively. Depending on what mathematical model we have in mind, the terms X-ray reflection and X- ray diffraction are used as synonyms. Let us consider an X-ray beam incident on a pair of parallel planes P1 and P2, separated by an interplanar spacing d d d sin 2 P 1 P 2 Fig. 5. Schematic presentation of incident (1 and 2) and reflected (1 and 2 ) beams The two parallel incident rays 1 and 2 make an angle (theta) with these planes (Fig. 5). 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 λ (lambda). We can express this relationship mathematically in Bragg s law. 2 d sin = nλ (1) The process of reflection is described here in terms of incident and reflected (or diffracted) rays, each making an angle with a fixed crystal plane. Reflections occur from planes set at angle with respect to the incident beam and generates a reflected beam at an angle 2 from the incident beam. The possible d spacing defined by the indices h, k, l are determined by he shape of the unit cell. Rewriting Bragg s we get:
7 11 sin = nλ 2d (2) Therefore the possible 2 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 is 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. Bragg s law refers to the simple equation (Eq. (1)) derived by the English physicists W.H. Bragg and his son W.L. Bragg in 1913 to explain why the cleavage faces of crystals appear to reflect X-ray beams at certain angles of incidence (). The variable d is the distance between atomic layers in a crystal, and the variable λ is the wavelength of the incident X-ray beam; n is an integer. This observation is an example of X-ray wave interference (Roentgenstrahlinterferenzen), commonly known as X-ray diffraction (XRD), and was direct evidence for the periodic atomic structure of crystals postulated for several centuries. The Braggs were awarded the Nobel Prize in physics in 1915 for their work in determining crystal structures beginning with NaCl, ZnS and diamond. Although Bragg s law was used to explain the interference pattern of X-rays scattered by crystals, diffraction has been developed to study the structure of all states of matter with any beam, e.g., ions, electrons, neutrons, and protons, with a wavelength similar to the distance between the atomic or molecular structures of interest. Bragg s law can easily be derived by considering the conditions necessary to make the phases of the beams coincide when the incident angle equals and reflecting angle. The rays of the incident beam are always in phase and parallel up to the point at which the top beam strikes the top layer at atom z (Fig. 6). z A C d B z d A d sin B Fig. 6. Bragg s law derivation using the reflection geometry and applying trigonometry. The lower beam must travel the extra distance (AB+BC) to continue travelling parallel and adjacent to the top beam
8 12 The second beam continues to the next layer where it is scattered by atom B. The second beam must travel the extra distance AB+BC if two beams are to continue travelling adjacent and parallel. This extra distance must be an integral (n) multiple of the wavelength (λ) for the phase of the two beams to be the same n λ = AB + BC (3) Recognizing d as the hypotenuse of the right triangle ABz, we can trigonometry to relate d and to the distance (AB+BC). The distance AB is opposite so, AB = d sin (4) Because AB = BC, Eq. (3) becomes n λ = 2 AB (5) Substituting Eq. (4) in Eq. (5) we have Eq. (6) nλ = 2d sin (6) The magnitude of the distance between two adjacent and parallel planes of atoms d (i.e. the interplanar spacing d hkl ) is a function of the Miller indices (h, k and l) as well as the lattice parameter(s). For example, for crystal structures having cubic symmetry a h l = (7) h + k + l d k in which a is the lattice parameter (unit cell edge length). Relationships similar to Eq. (7) but more complex exist for the other six crystal systems. Bragg s law Eq. (6), is a necessary but not sufficient condition for diffraction by real crystals. It specifies when diffraction will occur for unit cells having atoms positioned only at cell corners. However, atoms situated at other sides (e.g., face and interior unit cell positions as with FCC and BCC) act as extra scattering centers, which can produce out-of-phase scattering at certain Bragg s angles. The net results are the absence of some diffracted beams that, according to Eq. (6), should be present. For example, for the BCC crystal structure, h + k + l must be even if diffraction is to occur, whereas for FCC, h, k and l must all be either odd or even. Example. Interplanar spacing and diffraction angle computations. For BCC iron, compute: (i) the interplanar spacing and (ii) the diffraction angle for the (220) set of planes. The lattice parameter for Fe is nm. Also, assume that monochromatic radiation having a wavelength of nm is used, and the order of reflection is 1.
9 13 Solution: (i) the value of interplanar spacing d hkl is determined using Eq. (7), with a = nm, and h = 2, k = 2, and l = 0 since we are considering the (220) planes. Therefore d h k l = = = h a + k + l nm (ii) the value of may now be computed using Eq. (6) with n = 1, since this is a first order reflection nλ sin = = = d h k l = sin = o The diffraction angle is 2, or 2 = EXPERIMENT :2 goniometer The mechanical assembly that makes up the sample holder, detector arm and associated gearing is referred to as goniometer (Fig. 7). 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 :2 goniometer, the X-ray tube is stationary, the sample moves by the angle and the detector simultaneously moves by the angle 2. At high values of small or loosely samples may have a tendency to fall off the sample holder.
10 14 X-ray tube Sample θ Monochromator Photomultiplier Fig. 7. Schematic diagram of an X-ray diffractometer Sample preparation 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 cell aligned in a perfect extend 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 mm to mm cross section. The ideal sample is homogeneous and the crystallites are randomly distributed. The sample is pressed into a sample holder so that we have of 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 intensities. If we have a truly random sample, each possible reflection from a given set of h, k, l planes will have a equal number of crystallites contributing to it. We only have to rock the sample through the glancing angle in order to produce all possible reflections.
11 Diffraction spectra A typical diffraction spectrum consists of a plot of reflected intensities versus the detector angle 2 or depending on the goniometer configuration. The typical XRD diffractograms are included in the paragraph 1.4 XRD standards. The 2 values for the peak depend on the wavelength of the anode material of the X-ray tube. It is therefore customary to reduce a peak position to the interplanar spacing d that corresponds to the h, k, l planes that caused the reflection. The value of the d spacing depends on the shape of the unit cell. We get the d spacing as a function of 2 from Bragg s law. nλ d = (8) 2 sin Each reflection is fully defined we know the d spacing, the intensity (area under the peak) and h, k, l. If we know the d spacing and the corresponding indices h, k, l, we can calculate the dimension of the unit cell using Eq. (7) Practice: do we have diamonds? If we use X-rays with a wavelength (λ) of 1.54 Ǻ and we have diamonds in the material we are testing, we will find peaks on our X-ray pattern at values that correspond to each of the d spacings that characterize diamond. These d spacings are Ǻ Ǻ and 2.06 Ǻ. To discover where to expect peaks if diamond is present, you vary it until you find a Bragg s condition. Do the same with each of the remaining d spacings. Do the same with each of the remaining d spacings. Remember that you are varying, while on the X-ray pattern printout, the angles are given as 2. Consequently, when the Bragg s condition is fulfilled at a particular angle, you must multiply that angle by 2 to locate the angle on the X-ray pattern where you would expect a peak Diffraction data base International Center Diffraction Data (ICDD) or formerly known as (JCPDS) Joint Committee on Powder Diffraction Standards is the organization that maintains the data base of inorganic and organic spectrum. The data base is available from the Diffraction equipment manufacturers or from ICDD direct. The PDF I data base contains information on d spacing, chemical formula, relative intensity, RIR quality information and routing digit. The information is stored in an ASCII format in a file called PDF1.dat The PDF II data base contains full information on a particular phase cell parameters. Scintag s newest search/match and look-up software package is using the PDF II format. Today about inorganic and 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.
12 Applications Phase identification: Crystallinity: Residual stress: Texture analysis: The most common use of powder (polycrystalline) diffraction is chemical analysis. This can include phase identification (search/match), investigation of high /low temperature phases, solid solutions and determinations of unit cell parameters of new materials. A polymer can be considered partly crystalline and partly amorphous. The crystalline domains act as a reinforcing grid, like the iron framework in concrete, and improve the performance over a wide range of temperature. However, too much crystallinity causes brittleness. The crystallinity parts give sharp narrow diffraction peaks and the amorphous component gives very broad peak (halo). The ratio between these intensities can be used to calculate the amount of crystallinity in the material. Residual stress is the stress that remains in the material after the external force that caused the stress has been removed. Stress is defined as force per unit area. Positive values indicate tensile (expansion) stress, and negative values indicate a compressive state. The deformation per unit length is called strain. The residual stress can be introduced by any mechanical, chemical or thermal process. E.g., machining, plating and welding. The principals of stress analysis by the X-ray diffraction is based on measuring angular lattice strain distribution. That is, we choose a reflection at high 2 and measure the change in the d spacing with different orientations of the sample. Using Hooke s law the stress can be calculated from the strain distribution. The determination of the preferred orientation of the crystallites in polycrystalline aggregates is referred to as texture analysis, and the term texture is used as a broad synonym for preferred crystallographic orientation in the polycrystalline material, normally a single phase. The preferred orientation is usually described in terms of polefigures.