Clays
Problems
3.1. The set of {110} planes in an FCC unit cell passes through all corners of the cubic lattice and some face-centered atoms. The {110} set, however, misses the remaining face atoms. Write the Miller indices for a set of planes which is parallel to (110) but passes through all atoms of the FCC lattice.
3.2. List the Miller–Bravais indices of the six prism planes in a hexagonal lat-tice (show how you arrived at your result). What is the relationship to mutually opposite planes?
3.3. Show that the rhombohedral unit cell becomes an FCC or a BCC structure for specific angles . Calculate the values of these angles. Hint: Draw the lattice vectors of the primitive (non-cubic) unit cells of the FCC and BCC structures. The conventional (FCC, BCC) unit cells are not the smallest possible (primitive) unit cells.
3.4. Calculate the number of atoms for an HCP unit cell.
3.5. State the coordinates of the center atom in an HCP crystal structure (a1, a2, c) and show how one arrives at these values.
3.6. From the information given in Problem 3.5 above, show that the c/a ratio for the HCP structure is generally 兹8苶/3苶if the atoms are assumed to be spherical.
Compare this result with the experi-mental c/a ratios for Zn, Mg, and Zr.
3.7. State the slip plane and the slip di-rection in a hypothetical simple cu-bic lattice.
3.8. Show that a 具111典 direction lies in the (110) plane by sketching the appro-priate direction and plane. Which one of the two (direction or plane) are close-packed assuming a BCC crystal structure? Discuss the impli-cations for slip.
and covalent atomic bonds, both of which are very strong. Fur-ther, dislocation movement is almost impossible because of the directionality of the covalent bonds or the tendency toward main-taining charge neutrality in the case of ionic bonds. All of this contributes to stone being hard and brittle.
Finally, the elastic properties which some materials may ex-hibit are caused by the stretching of metallic, ionic, or covalent bonds. In other words, during elastic deformation the atoms are slightly and temporarily displaced from their equilibrium positions.
Since the metallic bond is relatively weak, metals and alloys have a higher elasticity compared to covalently or ionically bond mate-rials, such as glass, ceramics, etc. When an applied stress that has induced elastic elongation is removed, the deformation is reversed and the recovery to the original shape occurs almost instantly (ex-cept in viscoelastic polymers, as we shall learn in Section 16.4).
Historically, copper was eventually replaced to a certain extent by bronze. We shall focus our attention on this material and on al-loys in general in the chapters to come.
Suggestions for Further Study
M.F. Ashby and D.R.H. Jones, Engineering Materials I and II, Pergamon, Oxford (1986).
G.E. Dieter, Mechanical Metallurgy, 3rd Edition, McGraw-Hill, New York (1986).
R.W. Hertzberg, Deformation and Fracture Mechanics of Engi-neering Materials, Wiley, New York (1976).
J.P. Hirth and J. Lothe, Theory of Dislocations, Wiley-Inter-science, New York (1982).
E. Hornbogen and H. Warlimont, Metallkunde, 2nd Edition, Springer-Verlag, Berlin (1991).
D. Hull, Introduction to Dislocations, 3rd Edition, Pergamon, Elmsford, NY (1984).
Metals Handbook, American Society for Metals (1979).
D. Peckner (Editor), The Strengthening of Metals, Reinhold, New York (1964)
W.T. Read, Jr., Dislocations in Crystals, McGraw-Hill, New York (1953).
R.E. Reed-Hill and R. Abbaschian, Physical Metallurgy Principles, 3rd Edition, PWS-Kent, Boston (1992).
3.9. Determine (a) the coordination num-ber, (b) the number of each ion per unit cell, and (c) the lattice constant (expressed in ionic radii) for the CsCl and the NaCl crystal structures.
3.10. Determine (a) the coordination num-ber, (b) the number of each ion per unit cell, and (c) the lattice constant (expressed in ionic radii) for the zinc blende structure.
3.11. Calculate the packing factor for dia-mond cubic silicon.
3.12. Calculate the linear packing fraction of the [100] direction in (a) an FCC and (b) a BCC material assuming one atom per lattice point.
3.13. Calculate the planar packing fraction in the (111) plane in (a) the BCC and (b) the FCC structure.
3.14. Calculate the density of copper from its atomic mass. Compare your value
with the density given in the Appen-dix. (The lattice parameter for Cu is a 3.6151 Å.)
3.15. Sketch the [11苶0] direction in the hexagonal crystal system. What is the four-digit index for this direction?
Hint: The transformation equations
3.16. Calculate the distance between two nearest face atoms of the conven-tional FCC unit cell.
B. Se˘sták and A. Seeger, Glide and Work-Hardening in BCC Met-als and Alloys, I–III (in German), Zeitschrift für Metallkunde, 69 (1978), pp. 195ff, 355ff, and 425ff.
B. Se˘sták, “Plasticity and Crystal Structure,” in Strength of Met-als and Alloys,” Proc. 5th. Int. Conf. Aachen (Germany) August 1979, P. Haasen, V. Gerold, and G. Kostorz, Editors, Perga-mon, Toronto (1980).
4
Chalcolithic man was clearly aware of the many useful features of copper that made it preferable to stone or organic materials for some specialized applications. Among these properties were its elasticity and particularly plasticity, which allowed sheets or chunks of copper to be given useful shapes. Chalcolithic man also exploited the fact that copper hardens during hammering, that is, as a result of plastic deformation. Last but not least, molten copper can be cast into molds to obtain more intricate shapes.
On the negative side, surface oxidation and gases trapped dur-ing meltdur-ing and castdur-ing which may form porosity were probably of some concern to Chalcolithic man. More importantly, how-ever, cast copper is quite soft and thus could hardly be used for strong weapons or tools. Eventually, the time had come for a change through innovation. A new material had to be found. This material was bronze; see Fig. 4.1.
It is not known whether Chalcolithic man discovered by ex-perimentation or by chance that certain metallic additions to cop-per considerably improved the hardness of the cast alloy. (An al-loy is a combination of several metals.) In other words, cast bronze has a higher hardness than pure copper without necessi-tating subsequent hammering. Further, it had probably not es-caped the attention of Chalcolithic man that the melting tem-perature of certain copper alloys is remarkably reduced compared to pure copper (by about 100°C as we know today) and that molten alloys flow more easily during casting.
Naturally, some impurities that were already present in the copper ore transferred into the solidified copper. Among them were arsenic, antimony, silver, lead, iron, bismuth, and occa-sionally even tin. These impurities, however, were not present in sufficient quantities that one could refer to the resulting product as an alloy. Small quantities of these impurities rarely change