NANO 106 - Crystallography of Materials Spring 2015

Problem Set 2Due Fri, Apr 17

1 Graded Questions

Note: Questions marked with an asterisk (*) are harder than usual and meant to test yourunderstanding of the material.

1. Consider a 3-fold rotation (120◦) in a hexagonal crystal.

(a) Write down the matrix that represents a rotation of 120◦ about the c-axis in Carte-sian coordinates.

(b) Write down the matrix that represents a rotation of 120◦ about the c-axis in crystalcoordinates.

(c) Consider a point with fractional coordinates (0.5, 0.25, 0). What are its crystalcoordinates after undergoing a rotation of 120◦ about the c-axis?

(d) Attempt to work out the answer to part (c) in Cartesian space. Without loss ofgenerality, you may assume that your Cartesian x and z axes are parallel to your aand c lattice vectors. (The point of this exercise is just to show how much easier itis to work in crystal coordinates for this problem)

2. In Cartesian space, determine the matrix that represents a 2-fold rotation operation alongz-axis followed by a reflection operation in the plane formed by x and y axes. Whatother basic symmetry operation is this combination of symmetry operations equivalentto?

3. The 3m(C3v) point group can be generated from just two symmetry operations - a 3-foldrotation and the vertical mirror plane. As per usual convention, let the rotation axis bethe c axis and you may assume that the vertical mirror plane is in the b− c plane.

(a) Construct the group multiplication table for the 3m point group.

(b) Is this group Abelian?

(c) Are there any cyclic subgroups?

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NANO 106 - Crystallography of Materials Spring 2015

4. * Symmetry operations and their combination.

(a) Write down the matrix that represents a reflection in the plane formed by b andc axes. Without loss of generality, let this be the mirror plane indicated by thevertical line in the figure.

(b) Consider the mirror plane that is at an angle α from mirror plane in part (a). This isthe other (non-vertical) mirror plane indicated in the figure. Determine the matrixthat represent the reflection operation in the rotated mirror plane. Hint: This is aquestion similar to symmetry operations that do not pass through the origin.

(c) Show that the combination of two reflection operations in (a) and (b) results in a(counterclockwise) rotation operation of 2α along c-axis.

5. Consider the stereographic projection below:

(a) Write down the Hermann-Mauguin and Schonflies symbols for the correspondingpoint group.

(b) What is the 3D crystal system that is compatible with this point group?

(c) For the motif represented by the circle in the figure, draw the equivalent positionsin the figure using standard crystallographic notation.

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NANO 106 - Crystallography of Materials Spring 2015

6. Determine the point group for each of the molecule or shape pictured below, indicatingboth the International as well as Schonflies notation.

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

2 Supplemental Practice Questions from Structure of

Materials

We will sometimes include practice questions from the textbook. These questions are op-tional and meant to help you reinforce your understanding of the concepts with more practice.These questions will not be graded. Please do not hand in your solutions for these ques-tions. Solutions from the textbook itself will be posted for your reference. If you have anyquestions on the problems and solutions, you can bring them up during office hours.

• Chapter 8, Q (i) and (iii)

• Chapter 9, Q (i) and (ii)

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