EMS162-2010-Homework7_Solutions

# EMS162-2010-Homework7_Solutions - UNIVERSITY OF CALIFORNIA...

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EMS 162, page 1/5 Prof. Y. Takamura, 2010 UNIVERSITY OF CALIFORNIA, DAVIS Department of Chemical Engineering and Materials Science EMS 162: Structure and Characterization of Engineering Materials Homework 7 Due Date: Tuesday, March 2 nd , 2010, in class Note: You must show all the steps in order to receive full credit. Remember to include your units. 1. Powder Diffraction [34 pts] Consider the phase diagram shown below for the metal alloy formed between Au and Cu. In their pure forms, both metals have the face centered cubic (FCC) crystal structure. At high temperatures, the alloy retains the FCC crystal structure (denoted as the FCC_A1 phase) with the Au and Cu atoms randomly occupying the four atomic positions. When the alloy is cooled slowly, the ordered L1 2 or L1 0 phases are able to form, but if it is quenched rapidly from high temperatures then the random FCC alloy structure gets frozen in. In this problem, you will explore how x-ray diffraction can help you to determine the structure that has formed. (a) Vegard’s law states that the lattice parameter of an alloy varies linearly with the concentration of the constituent elements. For the end members, pure Au, a =4.08 Å, and pure Cu, a =3.61 Å. What is the lattice parameter for the 0.25 mol Cu/(Au+Cu) mixture (Au 3 Cu) when it is an alloy at temperatures above 550 K? (b) Calculate the structure factor for the Au 3 Cu alloy at temperatures above 550 K but below the melting temperature. Assume that the atomic scattering factor for each of the atomic positions can be approximated as the weighted average of pure Au and Cu based on the composition. (c) Calculate the structure factor for Au 3 Cu at temperatures below 550 K when it takes the L1 2 phase. The lattice parameter for this cubic phase is a =3.99 Å. (d) List the factors that affect the intensity of the reflections in a powder diffraction experiment. (e) Using the c , a 1 -a 4 and b 1 -b 4 parameters listed in the Atomic Scattering Factors file, create a plot of the atomic scattering factors for Au, Cu, and the Au 3 Cu alloy as a function of (sin θ )/ λ .

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