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Chap 8 Analytical Instruments

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XRD

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Measure X-Rays “Diffracted” by the specimen and obtain a diffraction pattern Interaction of X-rays with sample creates secondary “diffracted” beams (actually generated in the form of cones) of X-rays related to interplanar spacings in the crystalline powder according to a mathematical relation called “Bragg’s Law”: nλ = 2d sinθ where n is an integer λ is the wavelength of the X-rays d is the interplanar spacing generating the diffraction and θ is the diffraction angle λ and d are measured in the same units, usually angstroms. We will derive the Bragg law a bit more rigorously later but for a powder specimen in a diffractometer having a statistically infinite amount of randomly oriented crystallites, diffraction maxima (or peaks) are measured along the 2θ diffractometer circle.

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The aspects of crystallography most important to the effective interpretation of XRD data are: conventions of lattice description, unit cells, lattice planes, d-spacing and Miller indices, crystal structure and symmetry elements, the reciprocal lattice (covered in a separate document)

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Lattice Planes Lattice planes are defined in terms of the Miller indices, which are the reciprocals of the intercepts of the planes on the coordinate axes. In Fig. 1-5, the plane shown intercepts a at 100, b at 010 and c at 002. The Miller index of the plane is thus calculated as 1/1(a), 1/1(b), 1/2(c), and reduced to integers as 2a,2b,1c. Miller indices are by convention given in parentheses, i.e., (221). If the calculations result in indices with a common factor (i.e., (442)) the index is reduced to the simplest set of integers (221). This means that a Miller index refers to a family of parallel lattice planes defined by a fixed translation distance (defined as d) in a direction perpendicular to the plane. If directions are negative along the lattice, a bar is placed over the negative direction, i.e. (2 2 1) Families of planes related by the symmetry of the crystal system are enclosed in braces { }. Thus, in the tetragonal system {110} refers to the four planes (110), ( 1 10), ( 1 1 0) and (1 1 0). Because of the high symmetry in the cubic system, {110} refers to twelve planes. As an exercise, write the Miller indices of all of these planes.

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Spacing of Lattice Planes

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In a and c are in the plane of the paper, and b is perpendicular to the plane of the page. The notation shown for the d spacing and the relationship to the particular lattice plane (i.e., d 001, d 101, d 103 ) with the Miller index for the particular plane shown in the subscript (but usually without parentheses) are standard notation used in crystallography and x-ray diffraction.

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Scherrer’s Formula for Estimation of Crystallite Size If there is no inhomogeneous strain, the crystallite size D can be estimated from the peak width with the Scherrer’s formula: D = kλ/Bcosθ B Where λis the X-ray wavelength, B is the full width of height maximum of a diffraction peak, θ B is the diffraction angle, and k is the Scherrer’s constant of the order unity for usual crystal.

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Disadventages Compared with electron diffraction, XRD is the low intensity of diffracted X-rays, particularly for low Z materials. XRD is more sensitive to high Z materials. For low Z materials, neutron or electron diffraction is more suitable. Because of the small diffraction intensity, XRD requires large amount of specimens for measurements.

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Electron Spectroscopy EDS: Energy Dispersive X-ray Spectroscopy AES: Auger Electron Spectroscopy XPS: X-ray Photoelectron Spectroscopy, similar to EDS but has a lower energy X-ray is used to eject the electrons from an atom via photoelectric effect. RBS: Rutherford Backscattering Spectrometry, use of high energy beams of low mass ions to penetrate into the sample and cause back scattering of the ions. SIMS: secondary ion mass spectrometry,

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EDSAES

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Each atom in the Periodic Table has a unique electronic structure with a unique set of energy levels, both X-ray and Auger spectral lines are characteristic of the element in question.

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Basic Crystallography for X-ray Diffraction Earle Ryba.

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