CHEMICAL APPLICATIONS OF GROUP THEORY

SPRING, 2002

GRADED EXERCISE 2

As noted in the syllabus, the due date for your work is 17:00 on Wednesday, 8 May. This is the last day of classes and no work will be accepted late. You may refer to the text and your notes but may not consult with others. Direct all questions to the instructor. Relevant entries from the International Tables will be provided in a handout that will be used in class as well as in the completion of these exercises.

1) Single crystals of 3 minerals are on display in the research laboratory of Professor Steinmetz. Please treat the samples with care. They come from his personal collection and some are irreplaceable. They should not be taken outside the room.

a) From a visual examination of the materials, determine the point group of each mineral. Document your evidence for the determination.

b) Draw a sketch of a crystal of sample #1 and determine the Miller indices of its faces.

c) What constraints on the lattice constants and the angles between the lattice vectors follow from the point group of each mineral?

d) Predict the symmetry (Laue symmetry) of the diffraction pattern for each mineral.

2) A group crystallized naphthalene, C₁₀H₈, in a form that was shown by X-ray diffraction to belong to the space group P2₁/c (#14). The dimensions of the unit cell were found to be a = 8.658 Å, b = 6.003Å, c = 8.235 Å with β = 122.92 º and α = γ = 90.00Ί. The density of the sample was 1.152 g/cm³.

a) Determine the point group and the crystal system.

b) Determine the number of molecules per unit cell and the number of molecules per asymmetric unit. What structural inferences can be drawn from this result and the identity of the space group?

3) The mineral aragonite is a form of calcium carbonate that forms in hot springs. It belongs to the space group Pnma (#62) and has the lattice dimensions a = 5.74 Å, b = 4.96 Å, and c = 7.97 Å.

a) Which symmetry operations are indicated by the systematic absences of certain reflections in the diffraction pattern?

b) Is the pattern of absent reflections sufficient to determine the space group? Explain.

c) An oxygen atom located at a General Position has the coordinates in units of the lattice dimensions (-0.08, 0.48, 0.67). Determine the coordinates of the oxygen atoms that lie at equivalent positions.

d) The four calcium ions in the unit cell lie at special positions. Provide possible coordinates in Å for these 4 cations.

e) Prepare a list of the symmetry operations for the crystal.

exer2_2002.htm

WES, 3 April 2002