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Answers to PS #1 - Classes at U. of L.

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Chemistry 4000 Problem set <strong>#1</strong> <strong>Answers</strong><br />

Fall2009<br />

1. Determine the interfacial angles <strong>of</strong> the faces <strong>of</strong> the cube, the regular octahedron, and the regular tetrahedron. (Hint:<br />

consider vec<strong>to</strong>rs normal <strong>to</strong> the surfaces, and how these intersect.)<br />

2. Wh<strong>at</strong> point groups result on adding a center <strong>of</strong> symmetry <strong>to</strong> the following point groups:<br />

1 2 3 4 222 mm2 4mm 6 6 6m 2<br />

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3. Wh<strong>at</strong> point groups result from the combin<strong>at</strong>ion <strong>of</strong> two intersecting mirror planes <strong>at</strong>:<br />

a) 90° b) 60° c) 45° d) 30° <strong>to</strong> each other? Hint: make all the planes vertical.<br />

4. Wh<strong>at</strong> point groups result from the combin<strong>at</strong>ion <strong>of</strong> two intersecting tw<strong>of</strong>old axes <strong>at</strong>:<br />

a) 90° b) 60° c) 45° d) 30° <strong>to</strong> each other? Hint: make both 2-folds perpendicular <strong>to</strong> z.<br />

5. An a<strong>to</strong>m in an orthorhombic unit cell has fractional xyz coordin<strong>at</strong>es (0.1, 0.15, 0.20). Give the coordin<strong>at</strong>es <strong>of</strong> a second<br />

a<strong>to</strong>m in the cell th<strong>at</strong> is rel<strong>at</strong>ed <strong>to</strong> the first by each <strong>of</strong> the following (do each separ<strong>at</strong>ely, i.e. 5 new positions in all):<br />

a) body centering b) a center <strong>of</strong> symmetry <strong>at</strong> the origin<br />

c) a 2 axis parallel <strong>to</strong> z and passing through the origin d) a 21 axis parallel <strong>to</strong> z and passing through the origin<br />

e) A-centering<br />

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6. Draw the following Bravais l<strong>at</strong>tices, and show th<strong>at</strong> they are equivalent (hence the former <strong>of</strong> each set is never used):<br />

a) C-tetragonal and P-tetragonal b) F-tetragonal and I-tetragonal<br />

c) B-monoclinic and P-monoclinic (b unique axis) d) C-monoclinic and I-monoclinic (b unique axis)<br />

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7. For each <strong>of</strong> the fourteen Bravais l<strong>at</strong>tices, calcul<strong>at</strong>e the primitive and unit cell volumes in terms <strong>of</strong> the unit vec<strong>to</strong>rs and<br />

the angles between the axes. I.e. in terms <strong>of</strong> a, α, etc.<br />

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8. Draw the primitive unit cell for the body-centered cubic l<strong>at</strong>tice.<br />

9. As the body-centered tetragonal Bravais l<strong>at</strong>tice is drawn in the notes (or Smart and Moore, Fig 1.24, p.23), it seems<br />

th<strong>at</strong> there are two sets <strong>of</strong> l<strong>at</strong>tice points, one set <strong>at</strong> the body centers and another set <strong>at</strong> the corners <strong>of</strong> the parallelepiped.<br />

Draw several adjacent unit cells and convince yourself th<strong>at</strong> the two sets are equivalent. We can call either set the bodycenter<br />

set and the other the corner set.<br />

10. The cube and the regular octahedron (m3m) have the full symmetry <strong>of</strong> the cubic group, 48 symmetry elements.<br />

Enumer<strong>at</strong>e all 48 elements.<br />

11. The tetrahedron ( 43m ) also is a member <strong>of</strong> the cubic class but contains only half the symmetry <strong>of</strong> the full cubic group.<br />

Find the 24 symmetry elements <strong>of</strong> the tetrahedron.<br />

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8. Draw the primitive unit cell for the body-centered cubic l<strong>at</strong>tice.<br />

9. As the body-centered tetragonal Bravais l<strong>at</strong>tice is drawn in the notes (or Smart and Moore, Fig 1.24, p.23), it seems<br />

th<strong>at</strong> there are two sets <strong>of</strong> l<strong>at</strong>tice points, one set <strong>at</strong> the body centers and another set <strong>at</strong> the corners <strong>of</strong> the parallelepiped.<br />

Draw several adjacent unit cells and convince yourself th<strong>at</strong> the two sets are equivalent. We can call either set the bodycenter<br />

set and the other the corner set.<br />

10. The cube and the regular octahedron (m3m) have the full symmetry <strong>of</strong> the cubic group, 48 symmetry elements.<br />

Enumer<strong>at</strong>e all 48 elements.<br />

11. The tetrahedron ( 43m ) also is a member <strong>of</strong> the cubic class but contains only half the symmetry <strong>of</strong> the full cubic group.<br />

Find the 24 symmetry elements <strong>of</strong> the tetrahedron.<br />

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12. a) Show the interrel<strong>at</strong>ionship between the cube and the tetrahedron (with clear sketches).<br />

b) Show the interrel<strong>at</strong>ionship between the cube and the octahedron.<br />

13. a) The contents <strong>of</strong> the unit cell <strong>of</strong> any compound must contain an integral number <strong>of</strong> formula units. Why?<br />

b) Note th<strong>at</strong> unit cell boundaries "slice" a<strong>to</strong>ms in<strong>to</strong> fragments: An a<strong>to</strong>m on a face will be split in half between two cells;<br />

one on an edge will be split in<strong>to</strong> quarters among four cells, etc. Identify the number <strong>of</strong> Na + and Cl - ions in the unit cell<br />

<strong>of</strong> sodium chloride and st<strong>at</strong>e how many formula units <strong>of</strong> NaCl the unit cell contains. The unit cell <strong>of</strong> NaCl is<br />

illustr<strong>at</strong>ed, e.g. in Smart and Moore, p. 31.<br />

c) Why is the unit cell <strong>of</strong> NaCl so large? Why can a smaller unit not be chosen? Demonstr<strong>at</strong>e this by trying it.<br />

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14. Using the d<strong>at</strong>a in Table 1.2 in Smart and Moore and the formulae in the Introduction <strong>to</strong> the Lecture Notes, calcul<strong>at</strong>e the<br />

volumes <strong>of</strong> the following (primitive) unit cells in the seven crystal systems:<br />

a) Cubic, a = 3.4Å b) Tetragonal, a = 4.0 Å, c = 2.5 Å<br />

c) Orthorhombic, a = 2.3, b = 2.7, c = 1.9 Å d) Hexagonal, a = 3.4, c = 1.8 Å<br />

e) Trigonal-b a = 2.1 Å, a = 87 f) Monoclinic, a = 1.7, b = 2.2, c = 5.3 Å; β = 109°<br />

g) Triclinic, a = 2.4, b = 2.6, c = 4.1 Å; a = 92, β = 98, γ = 103°<br />

15. Express the loc<strong>at</strong>ion <strong>of</strong> the ligands L <strong>of</strong> the following ELn molecules as x,y,z coordin<strong>at</strong>es, with E <strong>at</strong> 0,0,0. Use<br />

symmetry opera<strong>to</strong>rs <strong>to</strong> simplify your work, and st<strong>at</strong>e explicitly how you have used symmetry.<br />

a) CH4, d(C-H) = 1.10 Å tetrahedral b) PtCl4 2- , d(Pt-Cl) = 2.30 Å sq. planar<br />

c) NH3, d(N-H) = 1.08 Å, ∠(H-N-H) = 106° pyramidal d) H2O, d(O-H) = 1.06 Å, ∠(H-O-H) = 104° bent<br />

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16. As was done in class, assign the crystal system and loc<strong>at</strong>e symmetry elements for the macroscopic crystals depicted in<br />

the diagrams below. Do the same for the crystals depicted in Figure 2.11, "Vari<strong>at</strong>ion in habit <strong>of</strong> crystals", included in<br />

the notes.<br />

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