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EMS162-2010-Homework3_Solutions

# EMS162-2010-Homework3_Solutions - UNIVERSITY OF CALIFORNIA...

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EMS 162, page 1/3 Prof. Y. Takamura, 2010 UNIVERSITY OF CALIFORNIA, DAVIS Department of Chemical Engineering and Materials Science EMS 162: Structure and Characterization of Engineering Materials Homework 3 - Solutions Due Date: Tuesday, Jan. 26 th , 2010, in class Note: You must show all the steps you used to obtain your answer in order to receive full credit. 1. Topic for Term Paper (5% of term paper grade) 2. Finite Symmetry Element [15 pts] (a) Finite symmetry elements are symmetry elements which don’t involve a translation. The Pecharsky and Zavalij textbook uses a pyramid notation to represent the basis of a lattice, however, the International Tables for Crystallography use the notation of an open circle with and without a comma which was described in lecture. Starting with the generic position (x,y,z), complete the missing points for the finite symmetry elements shown below. The example for a mirror plane is shown in (i). It is not necessary to denote the (x,y,z) coordinates of the new positions, but do include a +/- sign to denote the z positions. The thin dotted lines are for reference. (ii) 4-fold roto-inversion with two perpendicular mirror planes (iii) 3-fold roto-inversion, and (iv) 6-fold roto-inversion with a mirror plane perpendicular to the rotation axis, and three in-plane mirror planes rotated by 60 degrees relative to one another. (b) Identify any additional symmetry elements which result from the original ones shown below. (c) Using the handout on Point Groups from the SmartSite, identify the point group associated with each set of finite symmetry elements shown below. Note that this handout uses the notation from a stereographic projection such that +/- z positions are shown as closed/open circles, and ignores the right/left handed and top/bottom notation from the other system. Solution:

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EMS162-2010-Homework3_Solutions - UNIVERSITY OF CALIFORNIA...

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