MSE 605, Crystallography & Crystal Chemistry
Name: _______________________________
Homework 5
Due 29 October 2015
The numbers following each question give the approximate percentage of marks
allocated to that question.
1. Define the terms:
a. lattice
b. motif
c. crystal
d. Bravais lattice
e. point group
f. plane group
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MSE 605, Crystallography & Crystal Chemistry
2. Identify the point symmetry of this snowflake. Use a diagram, either on the picture
itself or next to it, to illustrate the symmetry elements of this point group. This is a
photograph of an actual snowflake that fell from the sky.
SnowCrystals.com
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MSE 605, Crystallography & Crystal Chemistry
3. In the space below, draw objects in the 4/mmm (D4h) point group, placing all the
correct symmetry symbols in the appropriate places.
5
Calculate the coordinates for all the general positions and SHOW your work.
15
Is there a centre of symmetry? How can you tell mathematically?
5
Autumn 2015
MSE 605, Crystallography & Crystal Chemistry
4. Two unusual descriptions of point symmetry include 2m and 3/mm. What are the
conventional notations? Draw the point groups, indicating both atomic positions and
symmetry operations, and place the correct symmetry symbol in the centre.
10
5. In the image below:
Identify with the appropriate symbols all of the non-trivial proper rotation axes
and reflection planes present.
Sketch in the smallest possible primitive unit cell. What shape is it?
To which planar system does this pattern belong? What is the plane group?
5
5
5
Autumn 2015
MSE 605, Crystallography & Crystal Chemistry
6. Sketch an outline of the smallest primitive unit cell in the honeycomb below.
5
Show the positions of the proper rotation and mirror symmetry elements with the
appropriate symbols within one unit cell.
5
Identify the plane group of the honeycomb.
5
Autumn 2015
MSE 605, Crystallography & Crystal Chemistry
7. Draw in all of the non-trivial proper rotation axes and/or mirror planes present within
the pattern below.
5
Outline the smallest primitive unit cell in the pattern.
5
To what plane group does this pattern belong?
5
Autumn 2015
MSE 605, Crystallography & Crystal Chemistry
Autumn 2015
MSE 605, Crystallography & Crystal Chemistry
Rotational Symmetry
For proper rotations about a in Cartesian axes:
0
 x'   1
 y'  = 0 cosφ
  
 z'  0 - sinφ
0 x 
sinφ   y 
cosϕ   z 
For proper rotations about b in Cartesian axes:
 x'   cosφ 0 sinφ   x 
 y'  =  0
1
0   y 
  
 z'  - sinφ 0 cosφ   z 
For proper rotations about c in Cartesian axes:
 x'  cosφ - sinφ 0  x 
 y'  =  sinφ cosφ 0  y 
  
 
 z'   0
0
1  z 
 x'   1 - 1 0   x 
For proper 6-fold rotations about c in hexagonal axes:  y'  = 1 0 0   y 
 z'  0 0 1  z 
Mirror Symmetry
Mirror normal to x:
 x'  − 1 0 0  x 
 y'  =  0 1 0  y 
  
 
 z'   0 0 1  z 
Mirror normal to y:
 x'   1 0 0  x 
 y'  = 0 - 1 0  y 
  
 
 z'  0 0 1  z 
Mirror normal to z:
 x'   1 0 0   x 
 y'  = 0 1 0   y 
  
 
 z'  0 0 - 1  z 
Inversion Symmetry:
 x'  - 1 0 0   x 
 y'  =  0 - 1 0   y 
  
 
 z'   0 0 - 1  z 
Autumn 2015
MSE 605, Crystallography & Crystal Chemistry
Plane Groups
Autumn 2015
After G. Burns & A.M. Glazer