X-ray Diffraction and Its Applications in Science & Engineering CONVENTIONS OF LATTICE DESCRIPTION Unit cell is the smallest unit of a crystal, which, if repeated, could generate the whole crystal. A crystal’s unit cell dimensions are defined by six numbers, the lengths of the 3 axes, a, b, and c, and the three interaxial angles, , and . CRYSTAL STRUCTURE A crystal lattice is a 3-D arrangement of unit cells. Space lattice is an imaginative grid system in three dimensions in which every point (or node) has an environment that is identical to that of any other point or node. Space Lattice + Basis = Crystal structure CRSYSTALLOGRAPHIC DIRECTIONS A crystallographic direction is defined as the line between two vectors. The following steps are utilized in the determination of the three dimensional indices: 1. A vector of conventional length is positioned such that it passes through the origin of the coordinate system. 2. The length of the vector projection on each of the three axes is determined; these are measured in terms of the unit cell dimensions a,b and c. 3. These three numbers are multiplied or divided by a common factor to reduce them to the smallest integer values. 4. The three indices are enclosed in square brackets : [uvw]. The u, v and w integers correspond to the reduced projections along the x, y and z axes, respectively CRYSTALLOGRAPHIC PLANES The crystallographic planes are specified by the three miller indices (hkl). Any two planes parallel to each other are equivalent and have identical indices. The following procedure is used to determine h, k and l index numbers : 1. The length of the intercept for each axis is determined in terms of the lattice parameters a, b and c. 2. The reciprocals of these numbers are taken. 3. These numbers are changed to the set of smallest integers by multiplying or division by a common factor. 4. Finally, the integer indices, not separated by commas, are enclosed within parentheses, thus: (hkl). INDEXING OF PLANES AND DIRECTIONS IN CUBIC SYSTEMS z (100) c (110) y a x z c (111) b b a X y CRYSTAL SYSTEMS Seven Types of crystal systems are : Crystal system Cell Length Cell Angles Cubic a=b=c α = β = γ =90º Tetragonal a=b#c α = β = γ =90º Orthorhombic a#b#c α = β = γ =90º Hexagonal a=b#c α = β =90º, γ = 120º Rhombohedral a=b=c α = β = γ # 90º Monoclinic a#b#c β = γ =90º # α Triclinic a#b#c α#β#γ BASICS OF X-RAY DIFFRACTION WHAT ARE X-RAYS? X-rays are electromagnetic waves having wavelength in the range of 0.1-100 Aº and energies in the range of 120 eV to 120 keV. X-rays up to about 10 keV (1-100 Aº wavelength) are classified as "soft" X-rays, and from about 10 to 120 keV ( 0.1-1 Aº ) as "hard" X-rays, due to their penetrating abilities. ELECTROMAGNETIC SPECTRUM PRODUCTION OF X-RAYS A beam of electrons is generated from the hot ungsten filament and these electrons are accelerated towards the anode with a high potential difference between the cathode and anode (Target). Anode is mainly Cu, Mo, Al and Mg. After striking the anode the electrons generate the X-rays. While monochromatic source is preffered, the X-ray beam actually consists of several characteristic X-ray lines. ORIGIN OF X-RAY ee- e- Continuous X-ray e- Characteristics X-ray SPECTRAL CONTAMINATION IN DIFFRACTION PATTERNS K1 K2 K1 K2 Kβ will give extra peak in the XRD pattern which can be eliminated by adding filters. BRAGG’s LAW Bragg’s Law is used to expalin the intereference pattern of the X-rays scattered by the crystals n 2d hkl sin Where, n an integer λ wavelength of the incident X-ray dhkl interplanar spacing WHAT IS X-RAY DIFFRACTION ? The periodic lattice found in crystalline structure may act as diffraction grating for wave particles of electromagnetic radiation with wavelength of a similar order of magnitude (1Aº). The atomic planes of a crystal causes an incident beam of X-rays to interfere with one another as they come out from the crystal. This phenomenon is called X-ray diffraction. ESSENTIAL PARTS OF THE DIFFRACTOMETER X-ray Tube: the source of X rays Incident-beam optics: condition the X-ray beam before it hits the sample The goniometer: the platform that holds and moves the sample, and detector. The sample & sample holder Receiving-side optics: condition the X-ray beam after it has encountered the sample Detector: count the number of X rays scattered by the sample APPLICATIONS OF XRD XRD is a nondestructive technique To identify crystalline phases and orientation To determine structural properties: strain, grain size, epitaxy, phase composition, preferred orientation, order-disorder transformation, thermal expansion To measure thickness of thin films and multilayers To determine atomic arrangement Detection limits: ~ 3% in a two phase mixture; can be ~ 0.1 % with synchrotron radiation SAMPLE PREPARATION FOR XRD An ideal powder sample should have many crystallites in random orientations If the crystallites in a sample are very large, there will not be a smooth distribution of crystal orientations. You will not get a powder average diffraction pattern. Crystallites should be <10 mm in size to get good powder statistics Large crystallite sizes and non-random crystallite orientations both lead to peak intensity variation. Intensity X-RAY DIFFRACTION PATTERN OF AIR 20 40 2θ (degrees) 60 X-RAY DIFFRACTION PATTERN OF AMORPHOUS SOLIDS XRD PATTERNS OF NANO-PARTICLES Ceria Dried ZrO2 CERIA ZrO2 INTENSITY DIFFRACTION PATTERN OF A SINGLE CRYSTAL A single crystal will produce only one family of peaks in the diffraction pattern INTENSITY DIFFRACTION PATTERN OF A POLYCRYSTALLINE SAMPLE A polycrystalline samples contain thousands of crystallites, therefore all possible diffraction peaks should be observed. EXTINCTION RULES FOR CUBIC CRYSTALS Bravais Lattice Allowed Reflections SC All BCC (h + k + l) even FCC h, k and l unmixed DC h, k and l are all odd Or all are even & (h + k + l) divisible by 4 h2 + k2 + l2 SC 1 100 2 110 3 111 4 200 5 210 6 211 211 8 220 220 9 300, 221 10 310 11 311 12 222 13 320 14 321 BCC FCC DC 111 111 110 200 200 7 220 220 311 311 310 222 321 222 INFORMATION PROCURRED FROM X-RAY DATA 1. Phase identification 2. Volume fraction of the phases 3. Crystallite size 4. Strain PHASE IDENTIFICATON The diffraction pattern for every phase is as unique as your fingerprint Phases with the same chemical composition can have drastically different diffraction patterns. Obtain XRD pattern Measure d-spacings Obtain integrated intensities Compare data with known standards in the JCPDS file, which are for random orientations (there are more than 50,000 JCPDS cards of inorganic materials). JCPDS CARD 1.file number 2.three strongest lines 3.lowest-angle line 4.chemical formula and name 5.data on diffraction method used 6.crystallographic data 7.optical and other data 8.data on specimen 9.data on diffraction pattern. Joint Committee on Powder Diffraction Standards, JCPDS (1969) Replaced by International Centre for Diffraction Data, ICDF (1978) QUANTITATIVE PHASE ANALYSIS The four main methods of quantitative phase analysis: (1) External standard method (2) direct comparison method (3) internal standard method (4) Reference intensity ratio method (RIR) With high quality data, you can determine how much of each phase is present. The ratio of peak intensities varies linearly as a function of weight fractions for any two phases in a mixture. RIR method is fast and gives semi-quantitative results. Whole pattern fitting/Rietveld refinement is a more accurate but more complicated analysis. CRYSTALLITE SIZE Crystallites smaller than ~120nm create broadening of diffraction peaks. This peak broadening can be used to quantify the average crystallite size of nano particles using the Scherrer ‘s equation 00-043-1002> Cerianite- - CeO 2 Intensity (a.u.) B 2 K cos wavelength of X - ray Full width at half maxima 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 2 (deg.) EFFECT OF LATTICE STRAIN IN DIFFRACTION PEAK AND POSITION NO STRAIN Uniform Strain (d1-d0)/d0 Peak moves, no shape changes Non-Uniform Strain d1# constant Peak broadens Continued…………….. Uniform strain causes the unit cell to expand/contract in an isotropic way. This simply leads to a change in the unit cell parameters and shift of the peaks. There is no broadening associated with this type of strain. Non-uniform strain leads to systematic shifts of atoms from their ideal positions and to peak broadening. This type of strain arises from the following sources: . Point defects (vacancies, site-disorder) . Plastic deformation (cold worked metals, thin films) . Poor crystallinity STRUCTURAL DETERMINATION To determine the structure of monoatomic cubic crystals, the following equation is used: sin 2 2 4a 2 (h 2 k 2 l 2 ) n is assumed to be 1 Θ values are determined from the diffraction pattern Λ is wavelength of X-ray UNIT CELL LATTICE PARAMETER REFINEMENT By accurately measuring peak positions over a long range of 2theta and d spacings, we can determine the unit cell lattice parameters of the phases in our sample by using the following formulas for different the crystal system. INSTRUMENTAL SOURCES OF ERROR Specimen displacement Instrument misalignment Error in zero 2θ position Peak distortion due to Kα2 and Kβ wavelengths CONCLUSIONS Non-destructive, fast, easy sample prep High-accuracy for d-spacing calculations Can be done in-situ Single crystal, poly, and amorphous materials Standards are available for thousands of material systems