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EMS162-2010-Homework1_Solutions

# EMS162-2010-Homework1_Solutions - UNIVERSITY OF CALIFORNIA...

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EMS 162 Prof. Y. Takamura, 2010 UNIVERSITY OF CALIFORNIA, DAVIS Department of Chemical Engineering and Materials Science EMS 162: Structure and Characterization of Engineering Materials Homework 1 - Solutions Due Date: Tuesday, Jan. 12 th , 2010, in class Note: You must show all the steps you used to obtain your answer in order to receive full credit. 1. Geometry of Crystals [13 pts] Consider the 2D lattice shown below. A number of different potential unit cells have been drawn. (i) Which of these schematics are valid unit cells? For each invalid unit cell, list the reason why it cannot be a valid unit cell. (ii) For each valid unit cell, identify the number of lattice points per unit cell. (iii) Which of the valid cells is a primitive unit cell? Solution: (i) The valid unit cells are (a), (c), (d), and (e). [1 pt each] (b) and (f) are invalid unit cells because they do not fill all of space upon translation since the opposite sides are not parallel to one another. [1 pt for reason] (ii) Unit cells (a) and (d) have 2 lattice points each [1 pt each] ; unit cells (c) and (e) have 1 lattice point each. [1 pt each] (iii) Unit cells (c) and (e) are primitive unit cells [1 pt each]. (iv) Consider the 2D lattice + basis shown on the left below where the unit cell is shown as the hatched parallelogram. For the 2D lattice + basis shown on the right, half of the smiling faces have been changed into frowning faces. This type of transition could represent the formation of a magnetically ordered structure, where the smiling and frowning faces represent opposite directions of magnetic moments. On the drawing, indicate the smallest possible unit cell.

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EMS162-2010-Homework1_Solutions - UNIVERSITY OF CALIFORNIA...

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