Crystallography Due 15 Sept 2014

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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 identied on the drawing. Another type of symmetry operation is also present. It is a combination of reection 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 denitions of the angles..

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MITOCW | ocw-3.60-27sep2005-part2-220k_512kb.mp4 PROFESSOR: Examine every possible means for combining the symmetry at once, but there is a seemingly paradoxical trick that we can yet pull. And let me indicate what is true here for 2mm. OK, so this is a twofold axis. Has mirror planes perpendicular to it. If one of these is a mirror plane, the other one has to be a mirror plane as well. So there's no way we could make one a mirror plane and one a glide plane. OK, that requires a net that is exactly rectangular. So let's put in the twofold axis. And I add one to the corner of the cell. As we well know we have to have twofold axes at all of these other locations. We want to put a mirror plane in the cell. We could pass it through the twofold axis, and that would be the same as P getting to P2mn back again. But why do we have to put the mirror plane through the twofold axis? We have to have the twofold axis left unchanged when we add the mirror plane, because if we created a new twofold axis we create a new lattice and we'd wreck the lattice that we've constructed. But why, why, oh why couldn't we put the mirror plane in like this? That's going to leave the twofold axis alone. It's going to leave the translations invariant. Why don't we do that? Why not? So here, trick number five, or wherever we are now.
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