Chap 12 Solns-6E

Chap 12 Solns-6E - CHAPTER 12 STRUCTURES AND PROPERTIES OF...

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CHAPTER 12 STRUCTURES AND PROPERTIES OF CERAMICS PROBLEM SOLUTIONS 12.1 The two characteristics of component ions that determine the crystal structure are: 1) the magnitude of the electrical charge on each ion; and 2) the relative sizes of the cations and anions. 12.2 In this problem we are asked to show that the minimum cation-to-anion radius ratio for a coordination number of four is 0.225. If lines are drawn from the centers of the anions, then a tetrahedron is formed. The tetrahedron may be inscribed within a cube as shown below. The spheres at the apexes of the tetrahedron are drawn at the corners of the cube, and designated as positions A , B , C , and D . (These are reduced in size for the sake of clarity.) The cation resides at the center of the cube, which is designated as point E . Let us now express the cation and anion radii in terms of the cube edge length, designated as a . The spheres located at positions A and B touch each other along the bottom face diagonal. Thus, AB = 2r A But (AB ) 2 = α 2 + α 2 = 2α 2 65
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or AB = a 2 = 2r A And a = 2r A 2 There will also be an anion located at the corner, point F (not drawn), and the cube diagonal AEF will be related to the ionic radii as AEF = 2 r A + r C ( 29 (The line AEF has not been drawn to avoid confusion.) From the triangle ABF (AB ) 2 + (FB ) 2 = (AEF ) 2 But, FB = a = 2r A 2 and AB = 2r A from above. Thus, 2r A ( 29 2 + 2 ρ Α 2 2 = 2 ρ Α + ρ Χ ( 29 [ ] 2 Solving for the r C / r A ratio leads to r C r A = 6 - 2 2 = 0.225 12.3 This problem asks us to show, using the rock salt crystal structure, that the minimum cation-to-anion radius ratio is 0.414 for a coordination number of six. Below is shown one of the faces of the rock salt crystal structure in which anions and cations just touch along the edges, and also the face diagonals. 66
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From triangle FGH , GF = 2r A and FH = GH = r A + r C Since FGH is a right triangle (GH ) 2 + (FH ) 2 = (FG ) 2 or r A + r C ( 29 2 + ρ Α + ρ Χ ( 29 2 = Α ( 29 2 which leads to r A + r C = 2r A 2 Or, solving for r C / r A r C r A = 2 2 - 1 = 0.414 12.4 This problem asks us to show that the minimum cation-to-anion radius ratio for a coordination number of 8 is 0.732. From the cubic unit cell shown below 67
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the unit cell edge length is 2 r A , and from the base of the unit cell x 2 = 2r A ( 29 2 + Α ( 29 2 = 8 ρ Α 2 Or x = 2r A 2 Now from the triangle that involves x , y , and the unit cell edge x 2 + 2r A ( 29 2 = ψ 2 = 2 ρ Α + Χ ( 29 2 2r A 2 ( 29 2 + 4 ρ Α 2 = Α + 2 ρ Χ ( 29 2 Which reduces to 2r A 3 - 1 ( 29 = 2 ρ Χ Or r C r A = 3 - 1 = 0.732 12.5 This problem calls for us to predict crystal structures for several ceramic materials on the basis of ionic charge and ionic radii.
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Chap 12 Solns-6E - CHAPTER 12 STRUCTURES AND PROPERTIES OF...

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