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)
4fold rotoinversion with two perpendicular mirror planes
(iii) 3fold rotoinversion, and
(iv) 6fold rotoinversion with a mirror plane perpendicular to the rotation axis, and three inplane
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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 Winter '08
 Howitt
 Crystallography, Cubic crystal system, Diamond cubic, Atomic packing factor, close packed directions

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