TNM 5823 – Análise de Materiais Inorgânicos por Difração de Pó 2ª Lista de exercícios

TNM 5823 – Análise de Materiais Inorgânicos por Difração de Pó 2ª Lista de exercícios Prazo para entrega: 20/10/16

1) Starting from the general equation that correlates cell parameters (a, b and c), cell angles (α, β and γ) and interplanar spacings (d) for the triclinic crystal symmetry and the axial lengths and angles for other crystal systems, simplify the expression of triclinic cell for the other six crystal systems.

where:

2) Considering the general equation that correlates cell parameters (a, b and c), cell angles (α, β and γ) and unit cell volume (V) for the triclinic crystal symmetry:

show that the expressions for cell volume for the other six crystal systems can be expressed by:

3) Determine the interplanar spacings and Bragg angles for the reflexions 110, 200, 211, 220, 310 and 222 of α-Iron ( BCC structure, a0 = 2-8664 Å) measured with Mo-Kα radiation (λ = 0.7093 Å). 4) Define unit cell of a crystal lattice. 5) Determine and list in order of increasing angle, the values of Bragg angle ( 2θ ) and Miller indexes (hkl) for the first three lines (those of lowest 2θ values) on the powder patterns of substances with the following structures (incident radiation Cu Kα (λ = 1.54 Å ): a) simple cubic (a = 3.00 Å), b) simple tetragonal (a = 2.00 Å, c = 3.00 Å), c) simple tetragonal (a = 3.00 Å, c = 2.00 Å), d) simple rhombohedral (a = 3.00 Å, α = 80°). 6) The figure below represents the x-ray diffraction pattern of copper (FCC structure) measured with Co-Kα radiation (λ = 1.542 Å). Determine: a) The interplanar distance dhkl, for each peak. b) The cell parameter, a0.

7) The structure factor for an x-ray diffraction reflexion hkl) can be calculated by the expression:

Where: N is the number of atoms of the motif; fj is the atomic scattering factor for the atom j; (uj , vj , wj) are the fractional coordinates for atom j; h, k e l are the Miller index for the reflexion. Considering a FCC structure with one atom in each net site, at positions (0, 0, 0), (½, ½, 0), (½, 0, ½) e (0, ½, ½). Prove that: a) Fhkl = 4f when the values of h, k and l are all odd or even (unmixed); b) Fhkl = 0 for values of h, k and l mixed (odd and even). 8) Simulate the X-ray diffraction pattern for a sample of polycrystalline aluminum (facecentered cubic symmetry, space group Fm3m with 1 atom / lattice point), whose cell parameter is 4.0494 Å, determining: a) The possible XRD reflexions measured with Cu-kα1 radiation (λ = 1.54056 Å) and 2θ ranging from 20° to 100°, their angular positions (2θhkl) and interplanar spacings (dhkl). b) Calculate the structure factors and the relative intensities of these reflections by assigning a value of 100% to the more intense reflection. 9) Calculate the relative intensities for the 10 first XRD reflexions of NaCl (Space Group: Fm-3m; a0 = 5.6402 Å) in a diffraction pattern measured with Cu-Ka radiation (λ = 1.54056 Å). Detail the calculations used and values. 10) Optional question (but highly important for the final concept). If you have ability to use spreadsheets, as EXCEL and Origin, you can construct a spreadsheet to calculate the structure factors and the relative intensities for any structure (including exercises 8 and 9). Using your spreadsheet, determine the reflexions and its relative intensities for powder patterns measured with Co-Kα1 (λ = 1.78897 Å), in the Bragg angle range (2θ) from 20º to 120º, for the crystal phases: a) LiF (Lithium Fluoride; System: cubic; Space Group: Fm-3m; a0 = 4.0270 Å) b) V (Vanadium; System: cubic; Space Group: Im-3m; a0 = 3.0274 Å). For do this read carefully the Chapter 12 of the book “Structure of Materials – An Introduction to Crystallography Diffraction and Symmetry – M. De Graef and M. E. McHenry” (included in the books you received) and use the Equation 12.3 and Table 12.1 for determine the atomic scattering factors. You can find the crystal structures and cif files at the Crystallography Open Database (http://www.crystallography.net/cod/).

SOME USEFUL RELATIONS In calculating structure factors by complex exponential functions, many particular relations occur often enough to be worthwhile stating here.They may be verified by means of Eq. (7). a) 𝑒𝑥𝑝𝜋𝑖 = 𝑒𝑥𝑝3𝜋𝑖 = 𝑒𝑥𝑝5𝜋𝑖 ... = -1 b) 𝑒𝑥𝑝2𝜋𝑖 = 𝑒𝑥𝑝4𝜋𝑖 = 𝑒𝑥𝑝6𝜋𝑖 ... = +1 c) In general, 𝑒𝑥𝑝𝑛𝜋𝑖 = (-1)n , (where n is any integer) d) 𝑒𝑥𝑝𝑛𝜋𝑖 = 𝑒𝑥𝑝−𝑛𝜋𝑖 , (where n is any integer) e) 𝑒𝑥𝑝𝜋𝑖 + 𝑒𝑥𝑝−𝜋𝑖 = 2 cos x