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Chemistry 4000 Problem set #1 Answers Spring 2004 1. Determine the interfacial angles of the faces of the cube, the regular octahedron, and the regular tetrahedron. (Hint:

consider vectors normal to the surfaces, and how these intersect.)

2. What point groups result on adding a center of symmetry to the following point groups:

1 2 3 4 222 mm2 4mm 6 6 6 2m

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3. What point groups result from the combination of two intersecting mirror planes at: a) 90° b) 60° c) 45° d) 30° to each other? Hint: make all the planes vertical.

4. What point groups result from the combination of two intersecting twofold axes at: a) 90° b) 60° c) 45° d) 30° to each other? Hint: make both 2-folds perpendicular to z.

5. An atom in an orthorhombic unit cell has fractional xyz coordinates (0.1, 0.15, 0.20). Give the coordinates of a second

atom in the cell that is related to the first by each of the following (do each separately, i.e. 5 new positions in all): a) body centering b) a center of symmetry at the origin c) a 2 axis parallel to z and passing through the origin d) a 21 axis parallel to z and passing through the origin e) A-centering

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6. Draw the following Bravais lattices, and show that they are equivalent (hence the former of each set is never used): a) C-tetragonal and P-tetragonal b) F-tetragonal and I-tetragonal c) B-monoclinic and P-monoclinic (b unique axis) d) C-monoclinic and I-monoclinic (b unique axis)

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7. For each of the fourteen Bravais lattices, calculate the primitive and unit cell volumes in terms of the unit vectors and the angles between the axes. I.e. in terms of a, α, etc.

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8. Draw the primitive unit cell for the body-centered cubic lattice.

9. As the body-centered tetragonal Bravais lattice is drawn in the notes (or Smart and Moore, Fig 1.24, p.23), it seems

that there are two sets of lattice points, one set at the body centers and another set at the corners of the parallelepiped. Draw several adjacent unit cells and convince yourself that the two sets are equivalent. We can call either set the body-center set and the other the corner set.

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

Enumerate all 48 elements.

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

Find the 24 symmetry elements of the tetrahedron.

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8. Draw the primitive unit cell for the body-centered cubic lattice.

9. As the body-centered tetragonal Bravais lattice is drawn in the notes (or Smart and Moore, Fig 1.24, p.23), it seems

that there are two sets of lattice points, one set at the body centers and another set at the corners of the parallelepiped. Draw several adjacent unit cells and convince yourself that the two sets are equivalent. We can call either set the body-center set and the other the corner set.

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

Enumerate all 48 elements.

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

Find the 24 symmetry elements of the tetrahedron.

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12. a) Show the interrelationship between the cube and the tetrahedron (with clear sketches). b) Show the interrelationship between the cube and the octahedron.

13. a) The contents of the unit cell of any compound must contain an integral number of formula units. Why? b) Note that unit cell boundaries "slice" atoms into fragments: An atom on a face will be split in half between two cells;

one on an edge will be split into quarters among four cells, etc. Identify the number of Na+ and Cl- ions in the unit cell of sodium chloride and state how many formula units of NaCl the unit cell contains. The unit cell of NaCl is illustrated, e.g. in Smart and Moore, p. 31.

c) Why is the unit cell of NaCl so large? Why can a smaller unit not be chosen? Demonstrate this by trying it.

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14. Using the data in Table 1.2 in Smart and Moore and the formulae in the Introduction to the Lecture Notes, calculate the volumes of the following (primitive) unit cells in the seven crystal systems:

a) Cubic, a = 3.4Å b) Tetragonal, a = 4.0 Å, c = 2.5 Å c) Orthorhombic, a = 2.3, b = 2.7, c = 1.9 Å d) Hexagonal, a = 3.4, c = 1.8 Å e) Trigonal-b a = 2.1 Å, a = 87 f) Monoclinic, a = 1.7, b = 2.2, c = 5.3 Å; β = 109° g) Triclinic, a = 2.4, b = 2.6, c = 4.1 Å; a = 92, β = 98, γ = 103°

15. Express the location of the ligands L of the following ELn molecules as x,y,z coordinates, with E at 0,0,0. Use

symmetry operators to simplify your work, and state explicitly how you have used symmetry. a) CH4, d(C-H) = 1.10 Å tetrahedral b) PtCl4

2-, d(Pt-Cl) = 2.30 Å sq. planar 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

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16. As was done in class, assign the crystal system and locate symmetry elements for the macroscopic crystals depicted in

the diagrams below. Do the same for the crystals depicted in Figure 2.11, "Variation in habit of crystals", included in the notes.

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17. Practice with space groups: for the following problems, determine and sketch the unit cell geometry from the information given (unit cell dimensions, number of formulae per cell, measured positional parameters, etc.). Use the space group information reproduced with this problem set. In each case, discuss which equivalent positions you have chosen, what type they are, and what symmetry constraints, if any, are imposed on the ions at those sites. If the choice of positions seem ambiguous, work out all the possibilities and try to guess which is the right one. a) Sodium oxide, Na2O, crystallizes in space-group Fm m3 , with a = 5.56 Å and four formula units per unit cell.

After sketching the unit cell and providing plausible positions for the ions, calculate (i) the sodium-oxygen bond length, (ii) the oxygen-oxygen bond length and (iii) the density of Na2O.

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b) Calcium iodide, CaI2, crystallizes in space-group P m3 1, with a = 4.49; c = 6.97 Å, and one formula unit per unit cell. After sketching the unit cell and providing plausible positions for the ions, calculate (i) the calcium-iodide bond length, (ii) the iodide-iodide bond length and (iii) the density of CaI2.

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c) Calcium carbide, CaC2 crystallizes in space-group I4/mmm, with a = 3.89 and b = 6.38 Å, and 2 formula units per

cell. The C22- ion is a linear diatomic ion, with a carbon-carbon bond length of 1.20 Å. Treat this ion as a unit, and

locate the midpoint of the C-C bond in the lattice. Suggest how the linear unit is oriented in the lattice. After sketching the unit cell and providing plausible positions for the ions, calculate (i) the calcium-carbon bond length(s), (ii) the distance between midpoints of neighboring carbide ions (iii) the density of CaC2.

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18. SrTiO3 has the Perovskite structure, a = 3.905 Å. Calculate (i) the Sr-O bond length, (ii) the Ti -O bond length and (iii) the density of SrTiO3. What is the lattice type (Bravais lattice?)

19. Metallic gold and platinum both have face centered cubic unit cells with dimensions a = 4.08 and 3.91 Å, respectively.

Calculate the metallic radii of the gold and platinum atoms.

20. Identify the following cubic structure types from the information on unit cells and atomic coordinates (sketch the

cells!): (i) MX: M ½00, 0½0, 00½, ½½½ X 000, ½½0, ½0½, 0½½ (ii) MX: M 000, ½½0, ½0½, 0½½ X ¼¼¼, ¾¼¾, ¾¾¼, ¼¾¾ (iii) MX: M 000 X ½½½ (iv) MX2: M 000, ½½0, ½0½, 0½½ X ¼¼¼, ¾¼¼, ¼¾¼, ¼¼¾, ¾¾¼, ¾¼¾, ¼¾¾, ¾¾¾

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