91Ó°ÊÓ

For a ceramic compound, what are the two characteristics of the component ions that determine the crystal structure?

Show that the minimum cation-to-anion radius ratio for a coordination number of 4 is 0.225.

Show that the minimum cation-to-anion radius ratio for a coordination number of 6 is \(0.414 .[\) Hint: Use the NaCl crystal structure (Figure 12.2 ), and assume that anions and cations are just touching along cube edges and across face diagonals.

Demonstrate that the minimum cation-toanion radius ratio for a coordination number of 8 is 0.732

Compute the atomic packing factor for the rock salt crystal structure in which \(r_{\mathrm{C}} / r_{\mathrm{A}}=0.414\).

The zinc blende crystal structure is one that may be generated from close- packed planes of anions. (a) Will the stacking sequence for this structure be FCC or HCP? Why? (b) Will cations fill tetrahedral or octahedral positions? Why? (c) What fraction of the positions will be occupied?

The corundum crystal structure, found for \(\mathrm{Al}_{2} \mathrm{O}_{3},\) consists of an \(\mathrm{HCP}\) arrangement of \(\mathrm{O}^{2-}\) ions; the \(\mathrm{Al}^{3+}\) ions occupy octahedral positions. (a) What fraction of the available octahedral positions are filled with \(\mathrm{Al}^{3+}\) ions? (b) Sketch two close-packed \(\mathrm{O}^{2-}\) planes stacked in an \(A B\) sequence, and note octahedral positions that will be filled with the \(\mathrm{Al}^{3+}\) ions.

Iron oxide (FeO) has the rock salt crystal structure and a density of \(5.70 \mathrm{g} / \mathrm{cm}^{3}\) (a) Determine the unit cell edge length. (b) How does this result compare with the edge length as determined from the radii in Table \(12.3,\) assuming that the \(\mathrm{Fe}^{2+}\) and \(\mathrm{O}^{2-}\) ions just touch each other along the edges?

(a) Using the ionic radii in Table \(12.3, \mathrm{com}-\) pute the theoretical density of CsCl. (Hint: Use a modification of the result of Prob\(\operatorname{lem} 3.3 .)\) (b) The measured density is \(3.99 \mathrm{g} / \mathrm{cm}^{3} .\) How do you explain the slight discrepancy between your calculated value and the measured one?

Compute the atomic packing factor for the diamond cubic crystal structure (Figure 12.15). Assume that bonding atoms touch one another, that the angle between adjacent bonds is \(109.5^{\circ},\) and that each atom internal to the unit cell is positioned \(a / 4\) of the distance away from the two nearest cell faces \((a\) is the unit cell edge length).