WHY STUDY Structures of Metals and Ceramics?
X- Ray Diffraction and Bragg’s Law
3.52 Using the Molecule Definition Utility found in both
“Metallic Crystal Structures and Crystallography”
and “Ceramic Crystal Structures” modules of VMSE, located in WileyPLUS, 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
1 2 1
20 12 121
and (3) Pb atoms are located at the following point coordinates:
1
2 0 0.763 0 12 0.237
1
2 1 0.763 1 12 0.237 Crystallographic Directions
3.53 Draw an orthorhombic unit cell, and within that cell, a [211] direction.
3.54 Sketch a monoclinic unit cell, and within that cell, a [101] direction.
3.55 What are the indices for the directions indicated by the two vectors in the following sketch?
0.5 nm
+y 0.4 nm
+z
0.3 nm
+x Direction 1
Direction 2
Questions and Problems • 109 3.56 Within a cubic unit cell, sketch the following
directions:
(a) [101] (e) [111]
(b) [211] (f) [212]
(c) [102] (g) [312]
(d) [313] (h) [301]
3.57 Determine the indices for the directions shown in the following cubic unit cell:
A
1 2
1
2 B
C
D
1 2
1 2 ,
+x
+y +z
3.58 Determine the indices for the directions shown in the following cubic unit cell:
3.59 (a) What are the direction indices for a vector that passes from point 14012 to point 341212 in a cubic unit cell?
(b) Repeat part (a) for a monoclinic unit cell.
3.60 (a) What are the direction indices for a vec- tor that passes from point 13120 to point 233412 in a tetragonal unit cell?
(b) Repeat part (a) for a rhombohedral unit cell.
3.61 For tetragonal crystals, cite the indices of direc- tions that are equivalent to each of the following directions:
(a) [011]
(b) [100]
3.62 Convert the [110] and [001] directions into the four-index Miller–Bravais scheme for hexagonal unit cells.
3.63 Determine the indices for the directions shown in the following hexagonal unit cells:
a1 a2
a3
z
(a)
a1 a2
a3
z
(b)
a1 a2
a3
z
(c)
a1 a2
a3
z
(d)
3.64 Using Equations 3.12a—3.12d, derive expres- sions for each of the three U, V, and W indices in terms of the four u, v, t, and w indices.
Crystallographic Planes
3.65 (a) Draw an orthorhombic unit cell, and within that cell, a (021) plane.
(b) Draw a monoclinic unit cell, and within that cell, a (200) plane.
3.66 What are the indices for the two planes drawn in the following sketch?
3.67 Sketch within a cubic unit cell the following planes:
(a) (101) (e) (111) (b) (211) (f) (212) (c) (012) (g) (312) (d) (313) (h) (301)
3.68 Determine the Miller indices for the planes shown in the following unit cell:
3.69 Determine the Miller indices for the planes shown in the following unit cell:
3.70 Determine the Miller indices for the planes shown in the following unit cell:
3.71 Cite the indices of the direction that results from the intersection of each of the following pairs of planes within a cubic crystal:
(a) The (110) and (111) planes (b) The (110) and (110) planes (c) The (111) and (001) planes.
3.72 Sketch the atomic packing of the following:
(a) The (100) plane for the FCC crystal structure (b) The (111) plane for the BCC crystal structure
(similar to Figures 3.25b and 3.26b).
3.73 For each of the following crystal structures, represent the indicated plane in the manner of Figures 3.25b and 3.26b, showing both anions and cations:
(a) (100) plane for the cesium chloride crystal structure
(b) (200) plane for the cesium chloride crystal structure
(c) (111) plane for the diamond cubic crystal structure
(d) (110) plane for the fluorite crystal structure 3.74 Consider the reduced-sphere unit cell shown in
Problem 3.44, having an origin of the coordinate system positioned at the atom labeled O. For the following sets of planes, determine which are equivalent:
(a) (100), (010), and (001) (b) (110), (101), (011), and (101) (c) (111), (111), (111), and (111)
Questions and Problems • 111 3.75 The accompanying figure shows three different
crystallographic planes for a unit cell of a hypo- thetical metal. The circles represent atoms:
(a) To what crystal system does the unit cell belong?
(b) What would this crystal structure be called?
3.76 The accompanying figure shows three different crystallographic planes for a unit cell of some hypothetical metal. The circles represent atoms:
(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 g/cm3, de- termine its atomic weight.
3.77 Convert the (111) and (012) planes into the four- index Miller–Bravais scheme for hexagonal unit cells.
3.78 Determine the indices for the planes shown in the following hexagonal unit cells:
(a) (b)
(c) (d)
3.79 Sketch the (0111) and (2110) planes in a hex- agonal unit cell.
Linear and Planar Densities
3.80 (a) Derive linear density expressions for FCC [100] and [111] directions in terms of the atomic radius R.
(b) Compute and compare linear density values for these same two directions for copper (Cu).
3.81 (a) Derive linear density expressions for BCC [110] and [111] directions in terms of the atomic radius R.
(b) Compute and compare linear density values for these same two directions for iron (Fe).
3.82 (a) Derive planar density expressions for FCC (100) and (111) planes in terms of the atomic radius R.
(b) Compute and compare planar density values for these same two planes for aluminum (Al).
3.83 (a) Derive planar density expressions for BCC (100) and (110) planes in terms of the atomic radius R.
(b) Compute and compare planar density val- ues for these same two planes for molybdenum (Mo).
3.84 (a) Derive the planar density expression for the HCP (0001) plane in terms of the atomic radius R.
(b) Compute the planar density value for this same plane for titanium (Ti).
Close-Packed Structures
3.85 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?
3.86 The corundum crystal structure, found for Al2O3, consists of an HCP arrangement of O2− ions; the Al3+ ions occupy octahedral positions.
(a) What fraction of the available octahedral positions are filled with Al3+ ions?
(b) Sketch two close-packed O2− planes stacked in an AB sequence, and note octahedral positions that will be filled with the Al3+ ions.
3.87 Beryllium oxide (BeO) may form a crystal struc- ture that consists of an HCP arrangement of O2−
ions. If the ionic radius of Be2+ is 0.035 nm, then (a) Which type of interstitial site will the Be2+
ions occupy?
(b) What fraction of these available interstitial sites will be occupied by Be2+ ions?
3.88 Iron titanate, FeTiO3, forms in the ilmenite crys- tal structure that consists of an HCP arrangement of O2− ions.
(a) Which type of interstitial site will the Fe2+
ions occupy? Why?
(b) Which type of interstitial site will the Ti4+
ions occupy? Why?
(c) What fraction of the total tetrahedral sites will be occupied?
(d) What fraction of the total octahedral sites will be occupied?
Polycrystalline Materials
3.89 Explain why the properties of polycrystalline materials are most often isotropic.
X-Ray Diffraction: Determination of Crystal Structures
3.90 The interplanar spacing dhkl for planes in a unit cell having orthorhombic geometry is given by
1 d2hkl=h2
a2+k2 b2 +l2
c2
where a, b, and c are the lattice parameters.
(a) To what equation does this expression reduce for crystals having cubic symmetry?
(b) For crystals having tetragonal symmetry?
3.91 Using the data for aluminum in Table 3.1, com- pute the interplanar spacing for the (110) set of planes.
3.92 Using the data for 𝛼-iron in Table 3.1, compute the interplanar spacings for the (111) and (211) sets of planes.
3.93 Determine the expected diffraction angle for the first-order reflection from the (310) set of planes for BCC chromium (Cr) when monochromatic radiation of wavelength 0.0711 nm is used.
3.94 Determine the expected diffraction angle for the first-order reflection from the (111) set of planes for FCC nickel (Ni) when monochromatic radia- tion of wavelength 0.1937 nm is used.
3.95 The metal rhodium (Rh) has an FCC crystal structure. If the angle of diffraction for the (311) set of planes occurs at 36.12° (first-order reflec- tion) when monochromatic x-radiation having a wavelength of 0.0711 nm is used, compute the following:
(a) The interplanar spacing for this set of planes (b) The atomic radius for a Rh atom
3.96 The metal niobium (Nb) has a BCC crystal structure. If the angle of diffraction for the (211) set of planes occurs at 75.99° (first-order reflec- tion) when monochromatic x-radiation having a wavelength of 0.1659 nm is used, compute the following:
(a) The interplanar spacing for this set of planes (b) The atomic radius for the Nb atom.
3.97 For which set of crystallographic planes will a first-order diffraction peak occur at a diffrac- tion angle of 44.53° for FCC nickel (Ni) when monochromatic radiation having a wavelength of 0.1542 nm is used?
3.98 For which set of crystallographic planes will a first-order diffraction peak occur at a diffraction angle of 136.15° for BCC tantalum (Ta) when monochromatic radiation having a wavelength of 0.1937 nm is used?
3.99 Figure 3.42 shows the first five peaks of the x-ray diffraction pattern for tungsten (W), 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) each of these peaks.
(b) Determine the interplanar spacing for each of the peaks.
(c) For each peak, determine the atomic radius for W, and compare these with the value pre- sented in Table 3.1.
Questions and Problems • 113
3.100 The following table lists diffraction angles for the first four peaks (first-order) of the x-ray dif- fraction pattern for platinum (Pt), which has an FCC crystal structure; monochromatic x-radiation having a wavelength of 0.0711 nm was used.
Plane Indices Diffraction Angle (2𝜽)
(111) 18.06°
(200) 20.88°
(220) 26.66°
(311) 31.37°
(a) Determine the interplanar spacing for each of the peaks.
(b) For each peak, determine the atomic radius for Pt, and compare these with the value pre- sented in Table 3.1.
3.101 The following table lists diffraction angles for the first three peaks (first-order) of the x-ray dif- fraction pattern for some metal. Monochromatic x-radiation having a wavelength of 0.1397 nm was used.
(a) Determine whether this metal’s crystal struc- ture is FCC, BCC, or neither FCC or BCC, and explain the reason for your choice.
(b) If the crystal structure is either BCC or FCC, identify which of the metals in Table 3.1 gives this diffraction pattern. Justify your decision.
Peak Number Diffraction Angle (2θ)
1 34.51°
2 40.06°
3 57.95°
3.102 The following table lists diffraction angles for the first three peaks (first-order) of the x-ray dif- fraction pattern for some metal. Monochromatic x-radiation having a wavelength of 0.0711 nm was used.
(a) Determine whether this metal’s crystal struc- ture is FCC, BCC, or neither FCC or BCC, and explain the reason for your choice.
(b) If the crystal structure is either BCC or FCC, identify which of the metals in Table 3.1 gives this diffraction pattern. Justify your decision.
Peak Number Diffraction Angle (2θ)
1 18.27°
2 25.96°
3 31.92°
Noncrystalline Solids
3.103 Would you expect a material in which the atomic bonding is predominantly ionic to be more likely or less likely to form a noncrystalline solid upon solidification than a covalent material?
Why? (See Section 2.6.)
Spreadsheet Problem
3.1SS For an x-ray diffraction pattern (having all