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- HOMEWORK ASSIGNMENT#5 Return by Next WEDNESDAY MEMS1054 AY2101 NAME 1 A diffraction pattern has been obtained from a powder sample of a material

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HOMEWORK ASSIGNMENT #5 MEMS1054 AY2101 Return by: Next WEDNESDAY! NAME: 1) A diffraction pattern has been obtained from a powder sample of a material with a cubic crystal structure using an X-ray powder diffractometer instrument with Cu-K α X-radiation, wavelength λ =0.1542nm=1.542Å. The experimental X-ray diffraction pattern exhibits seven maxima for the following angles θ : 14.89˚ 21.31˚ 26.43˚ 30.93˚ 35.08˚ 39.01˚ 42.84˚ Determine the Bravais lattice and the corresponding lattice parameters of this material with a cubic crystal structure. You can assume that it has a mono-atomic (elemental) motif. The interplanar spacing of a plane (hkl), d hkl , for a cubic crystal relates to the lattice parameters as d hkl 2 = a 0 2 /(h 2 + k 2 + l 2 ). Suggested Solution on Next Page!
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Answer to Q1) (There are different ways to arrive at the correct answers to this question, namely cI lattice with a o =4.243Å=0.4243nm. Suggested solutions are outlined below) There are three possible Bravais lattices in the cubic crystal system: cP, cI and cF. With a mono-atomic motif this implies that atoms are distributed only on the lattice points of the respective Bravais lattice, namely, P-lattice unit cell with one atom at 000, I-lattice unit cell with two atoms at 000 and at ½ ½ ½ , and F-lattice unit cell with four atoms at 000 and at ½ ½ 0 and at 0 ½ ½ and at ½ 0 ½ , respectively. The corresponding structure factors for a P-, I- and F-lattice can then be calculated and are cP: F hkl = fa , where fa is the respective atomic scattering factor for X-rays; cI: F hkl = fa(1 + (-1) (h+k+l) )= 0, if h+k+l=2n+1, here n is an integer, and F hkl = fa(1 + (-1) (h+k+l) )= 2fa, if h+k+l=2n, here n is an integer; cF: F hkl = fa(1 + (-1) (h+k) + (-1) (k+l) + (-1) (h+l) )= 0, if h, k, l are a mix of even and odd integers, and F hkl = fa(1 + (-1) (h+k) + (-1) (k+l) + (-1) (h+l) )= 4fa, if h, k, l are either all even or all odd integers; Thus, the seven diffraction maxima for the three possible cubic Bravais lattices, cP, cI and cF, would have different indices, hkl, as summarized below in the table, providing different “fingerprints” in the ratio of the angles of the diffraction maxima. The table also includes the d- spacings determined from the experimental X-ray diffraction pattern,
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This note was uploaded on 01/07/2011 for the course MEMS Mems1054 taught by Professor Wiezorek during the Fall '10 term at Pittsburgh.

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- HOMEWORK ASSIGNMENT#5 Return by Next WEDNESDAY MEMS1054 AY2101 NAME 1 A diffraction pattern has been obtained from a powder sample of a material

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