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HW-CH3 - 92 0 Chapter-3 i REFERENCES Azaroff L E Elements...

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Unformatted text preview: 92 0 Chapter-3 i StructumofMotalsIndCeramics REFERENCES Azaroff, L. E, Elements of X Ray Crystallography, McGraw-I-Iill, New York. 1968. Reprinted by TechBooks, Marietta, OH, 1990. Buerger, M. 1., Elementary Crystallography, Wiley, New York, 1956. Chiang, Y. M., D. P. Birnie, 1]], and W. D. Kingery, Physical Ceramics: Principles for Ceramic Sci- ence and Engineering, Wiley, New York, 1997. Cullity, B. D., and S. R. Stock, Elements of X-Ray Dzflraction, 3rd edition, Prentice Hall, Upper Saddle River, NJ, 2001. Curl, R. F. and R. E. Smalley, “Fullerenes,” Scien- tific American. Vol. 265, No. 4, October 1991, pp. 54—63. QUESTIONS AND PROBLEMS Hauth; W. 5., “Crystal Chemistry in Ceramics,” American Ceramic Society Bulletin, Vol. 30, 1951:|No. 1, pp. 5—7; No. 2, pp. 47—49; No. 3, pp. ‘76—'17; No. 4, pp. 137-142; No. 5, pp. 165—167; No. 6, pp. 203—205. A good overview of silicate structures. Kingery, W. D., H. K. Bowen, and D. R. Uhlmann, Introduction to Ceramics; 2nd edition, Wiley, New York, 1976. Chapters 1—4. Richerson, D. W., The Magic ofCeramics, American Ceramic Society, Westerville, OH, 2000. Richerson, D. W., M odem Ceramic Engineering, 3rd edition, CRC Press, Boca Raton, FL, 2006. Additional problems and questions for this chapter may be found on both Student and Instructor Companion Sites at WW. M'leyxom/coflege/callirter. Unit Cells Metallic Crystal Structure: 3.1 If the atomic radius of lead is 0.175 nm, calcu- late the volume of its unit cell in cubic meters. 3.2 Show that the atomic packing factor for BCC is 0.68. Density Computations—Metals 3.3 Molybdenum has a BCC crystal structure, an atomic radius of 0.1363 nm, and an atomic weight of 95.94 g/mol. Compute its theoretical density and compare it with the experimental value found inside the front cover. 3.4 Calculate the radius of a palladium atom, given that Pd has an FCC crystal structure, a density of 12.0 glcma, and an atomic weight of 106.4 glmoi. 3.5 Some hypothetical metal has the simple cu- bic crystal structure shown in Figure 3.42. If its atomic weight is 14.5 glmol and the atomic radius is 0.145 nm, compute its density. 3.6 Using atomic weight, crystal structure, and atomic radius data tabulated inside the front cover, compute the theoretical densities of alu- minum, nickel, magnesium, and tungsten, and then compare these values with the measured densities listed in this same table. The cia ratio for magnesium is 1.624. 3.7 Below are listed the atomic weight, density, and atomic radius for three hypothetical al- loys. For each determine whether its crystal structure is FCC, BCC, or simple cubic and then justify your determination. A simple cu— bic unit cell is shown in Figure 3.42. Atomic Atomic Weight Density Radius Alloy _ _ (lg/moi) (5mm) fun) A 43.1 6.40 0.122 B 184.4 12.30 0.146 C 91.6 9.60 0.137 Figure 3.42 Hard-sphere unit cell representation of the simple cubic crystal structure. 3.8 Indium has a tetragonal unit cell for which the a and c lattice parameters are 0.459 and 0.495 urn. respectively. (a) If the atomic packing factor and atomic ra- dius are 0.693 and 0.1625 nm, respectively, determine the number of atoms in each unit cell. (Is) The atomic weight of indium is 114.82 g/mol; compute its theoretical density. 3.9 Magnesium has an HCP crystal structure a cfa ratio of 1. 624 and a density of 1.74 glcmll. Compute the atomic radius for Mg. Ceramic Crystal Structures 3.10 Show that the minimum cation-to—anion ra- dius ratio for a coordination number of 4 is 0.225. 3.11 Demonstrate that the minimum cation—to— anioni radius ratio for a coordination number of 8 is 0.732. 3.12 On the basis of ionic charge and ionic radii, predict crystal structures for the following ma- terials: (u) C30 (13) KBr. Justify your selections. Density Computations—Ceramics 3.13 Compute the atomic packing factor for the rock salt crystal structure in which refer: 0.414. 3.14 Compute the atomic packing factor for cesium chloride using the ionic radii in Table 3.4 and assuming that the ions touch along the cube diagonals. 3.15 Iron oxide (FeO) has the rock salt crystal struc- ture and a density of 5.?0 g/cm3. (a) Determine the unit cell edge length. (b) How does this result compare with the edge length as determined from the radii in Table 3 .4, assuming that the Fe2+ and 02 ions just touch each other along the edges? 3.16 One crystalline form of silica (SiOz) has a cubic unit cell, and from x-ray diffraction data it is known that the cell edge length is 0.700 nm. If the measured density is 2.32 glcrn3, Questions and Problems 0 93 how many Si“ and 02‘ ions are there per unit cell? 3.17 A hypothetical AX type of ceramic material is known to have a density of 2.10 glcm3 and a unit cell of cubic symmetry with a cell edge length of 0.57 nm. The atomic weights of the A and X' elements are 28.5 and 30.0 g/mol, re- spectively. On the basis of this information, which of the following crystal structures is (are) possible for this material: sodium chlo- ride, cesium chloride, or zinc blende? Justify your choice(s). Silicate Ceramics 3.18 Determine the angle between covalent bonds in an no? tetrahedron. Carbon 3.19 Compute the theoretical density of ZnS given that the Zn—S distance and bond angle are 0.234 nm and 1095", respectively. How does this value compare with the measured density? 3.20 Compute the atomic packing factor for the di- amond cubic crystal structure (Figure 3.16). Assume that bonding atoms touch one an- other. that the angle between adjacent bonds is 109.5“, and that each atom internal to the unit cell is positioned r114 of the distance away from the two nearest cell faces (a is the unit cell edge length). Crystal Systems 3.21 Sketch a unit cell for the face-centered or- thorhombic crystal structure. Point Coordinate: 3.22 List the point coordinates of both the sodium and chlorine ions for a unit cell of the sodium chloride crystal structure (Figure 3.5). 3.23 Sketch a tetragonal unit cell, and] within that cell indicate locations of the 1 1— Iand 1 1 % point coordinates. 3.24 Using the Molecule Definition Utility found in ._ lography“ and “Ceramic Crystal Structures” ' modules of VMSE, located on the book's web site [www.wileycomlcollegefcallister (Student @I both “Metallic Crystal Structures and Crystal- 4 ' Chnphr3 I mummcmnm Companion Site)], generate (and print out) a three-dimensional unit cell for lead oxide. PbO, given the following: (1) The unit cell is tetragonal with a = 0.397 nm and c = 0.502 nm, (2) oxygen atoms are located at the following point coordinates: 000 001 100 101 010 011 110 111 11 11 ii” 551 and (3) Pb atoms are located at the following point coordinates: ‘ 1 0 0 763 0 1 o 237 2 . 2 . 1 1 — . 3 — 0. 2 10 76 1 2 237 Crystallographic Directions 3.25 Sketch} monoclinic unit cell. and within that cell a [101] direction. 3.26 What are the indices for the direction in- dicated by the vector in the sketch below? (1:) [212], (a) [301]. 3.28 Determine the indices for the directions _.‘_ shown in the following cubic unit cell: +2 . T H .5! _.....+y top- ”In-- / +1 3.29 For tetragonal crystals, cite the inclines of di- rections that are equivalent to each of the fol- lowing directions: (a) [011] m)uam 3.30 Convert the [001] direction into the four- index Miller—Bravais scheme for hexagonal unit cells. 3.31 Determine the indiccs for the two directions shown in the following hexagonal unit cell: 3.32 Using Equations 3.7a, 3.7b. 3.?c, and 3.7d, de- rive expressions for each of the three primed indices set (14’ , v’, and w’) in terms of the four unprimed indices (u, v, t, and w). Crystallographic Planes 3.33 Draw an orthorhombic unit cell, and within that cell a (021) plane. Questions and Problems '- 95 3.34 What are the indices for the plane drawn in 3.38 For each of the following crystal structures, the sketch below? represent the indicated plane in the manner of Figures 3.26 and 3.27, showing both anions and cations: (in) (111) plane for the diamond cubic crystal structure, and (b) (110) plane for the fluorite crystal struc- ture. 3.39 Below are shown three different crystallo- graphic planes for a unit cell of some hypo- thetical metal. The circles represent atoms: 3.35 Sketch within a cubic unit cell the following T T T 5.: ' '2 planes: 5 g E 2:! (a) (012) . (c) (191), g a a 99 (b) (313) (a) (211). 1 j: 0‘ 3.36 Determine the Miller indices for the planes H0 30 MRI ' ' _-'shown tn the following unit cell. m ’ o. 25 m I (110) {101) {011} (a) To what crystal system does the unit cell belong? (b) What would this crystal structure be called? (c) If the density of this metal is 18.91 glen-13, determine its atomic weight. 3.40 Convert the (0T2) plane into the four-index Miller—Bravais scheme for hexagonal unit cells. / H: 3.41 Determine the indices for the planes shown in the hexagonal unit cells below: 3.37 Determine the Miller indices for the planes "I".3 shown in the following unit cell: I '1‘.- . a: 1"“ +3" 3.42 Sketch the (2110) plane in a hexagonal unit cell. 96 I Chephr3 I'Shuehrmanctalsand Ceramics Linearandflamrbemflies 3.43 (a) Derive linear density expressions for FCC [100] and [111] directions in terms of the atomic radius R. ' (b) Compute and compare linear density val- ues for these same two directions for cop- per. 3.44 (a) Derive planar density expressions for BCC (100) and (110) planes in terms of the atomic radius R. (1)) Compute and compare planar density val- ues for these same two planes for molyb— denum. I Close-Packed Crystal Structures 3.45 The comndum crystal structure, found for A1203, consists of an HCP arrangement of 02‘ ions; the Al“ ions Occupy octahedral po- sitions. (a) 'What fraction of the available octahedral positions are filled with A1“ ions? (Ir) Sketch two close-packed 02* planes stacked in an AB sequence, and note octa- hedral positions that will be filled with the Al3+ ions. 3.46 Iron titanate. FeTiOg, forms in the ilmenite crystal structure that consists of an l-lCP ar- rangement of Oz'ions. (In) Which type of interstitial site will the Fe“ ions- occupy? Why? (h) Which type of interstitial site will the 'Ii4+ ions occupy? Why? (c) What fraction of the total tetrahedral sites will be occupied? (d) What traction of the total octahedral sites will be occupied? g E 0.0 20.0 «0.0 50.0 50.0 Difiraction angle 29 100.0 X—Ray Dalmatian: Detemu'nalz'an of Cranial 5mm 3.47 Determine the expected diffraction angle for the first-order reflection from the (310) set of planes for BCC chromium when monochro- matic radiation of wavelength 0.0711 nm is used. 3.48 Using the data for ct-iron in Table 3.1, compute the interplanar spacings for the (11 1) and (21 1) sets of planes 3.49 The metal niobium has a BCC crystal struc- ture. If the angle of diffraction for the (211) set of planes occurs at 75.95” (first-order reflec- tion) when monochromatic x-radiation hav- ing a wavelength of 0.1659 nm is used, com- pute (a) the interplanar spacing for this set of planes, and (b) the atomic radius for the nio— bium atom. 3.50 Figure 3.39 shows an x—ray diffraction pat— tern for lead taken using a diffractometer and monochromatic x—radiation having a wave- length of 0.1542 run; each diffraction peak on the pattern has been indexed. Compute the interplanar spacing for each set of planes m- dexed; also determine the lattice parameter of Pb for each of the peaks. 3.51 Figure 3.43 shows the first five peaks of the x—ray diffraction pattern for tungsten, which has a BCC crystal structure; monochromatic x-radiation having a wavelength of 0.1542 nm was used. (a) Index (i.e.. give h, k, and l indices) for each of'these peaks. (b) Determine the interplanar spacing for each of the peaks. (c) For each peak, determine the atomic ra— dius for W and compare these with the value presented in Table 3.1. Figure 3.43 Diffraction pattern for powdered tungsten. (Courtesy of Wesley L. Holman.) ...
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