Geology 310 Mineralogy October 30, 2014 Homework Problem Set 2
Crystallography due 15 Sept 2014
Examine the two-dimensional crystal structure, below. This is the same structure that you are examining in the lab exercise for this week. The primitive unit cell is shown by the lines. The unit cell contains one small and one large circle. Convince yourself that this is true. The combination of one large and one small circle is called the motif. It is the primitive object that is repeated by all the symmetry operations to make the crystal structure. The number of motifs per unit cell, called Z, is one. There are numerous symmetry elements present, including 4-fold rotation axes, 2-fold rotation axes, and mirror planes. These are identi ed on the drawing. Another type of symmetry operation is also present. It is a combination of re ection across a mirror plane and translation one-half unit cell distance parallel to the mirror plane. It is called a glide plane and is labeled with the g and the dashed lines. This crystal structure in two dimensions has one of seventeen possible sets of symmetry elements. These are called the two-dimensional space groups. See pages 67 in the text and the supplemental handout on plane space groups. This crystal belongs to the space group p4mm; its symmetry elements and general positions are shown in the handout.
m
g g m g g
m
m
m
Unit Cell
Primitive Square Lattice Z = 1 Geology 310 Mineralogy October 30, 2014 Homework Problem Set 2
1. Find the unit cell, motif, and symmetry elements for the following pattern, just as above.
2. Cerussite is a lead carbonate, PbCO3, which forms orthorhombic crystals. Cerussite is an impor- tant ore of lead. It typically forms in zones of alteration of galena. Crystals of cerussite are tabular and elongated, usually on [001], and have prominent 111 dipyramids (see example crystal). The face (111) has a normal normal with an angle φ with the c axis of 54.2° and its projection on the ab plane makes an angle θ with the b axis of 58.6°. A sketch of the (111) face is shown below with the angles indicated. Determine the axial ratio a/b : 1 : c/b for cerussite. Show all your work! Geology 310 Mineralogy October 30, 2014 Homework Problem Set 2
c ϕ = 54.2°
b
θ = 58.6°
a
3. Zircon is a common accessory mineral in granite and similar igneous rock. It is tetragonal with and axial ratio of c/a = 0.905. Zircon commonly has the tetragonal dipyramid 111 as shown in the sample below from Brasil. Find the angles φ and θ for the plane (111). Refer to the above drawing for the de nitions of the angles.