Page 130 - Materials Science and Engineering An Introduction

102 • Chapter 3 / The Structure of Crystalline Solids

3.55 Convert the (111) and (012) planes into the four- 3.61 (a) Derive planar density expressions for BCC
index Miller–Bravais scheme for hexagonal unit (100) and (110) planes in terms of the atomic
cells. radius R.
3.56 Determine the indices for the planes shown in (b) Compute and compare planar density values
the following hexagonal unit cells: for these same two planes for molybdenum (Mo).
3.62 (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).
Polycrystalline Materials
3.63 Explain why the properties of polycrystalline
materials are most often isotropic.

X-Ray Diffraction: Determination
of Crystal Structures
3.64 The interplanar spacing d hkl for planes in a unit
(a) (c) cell having orthorhombic geometry is given by
1 h 2 k 2 l 2
= + +
2 2 2 2
d hkl a b c
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?
3.65 Using the data for aluminum in Table 3.1, com-
pute the interplanar spacing for the (110) set of
planes.
(b) (d) 3.66 Using the data for a-iron in Table 3.1, compute
the interplanar spacings for the (111) and (211)
sets of planes.
3.57 Sketch the (0111) and (21 10) planes in a hexago- 3.67 Determine the expected diffraction angle for the
nal unit cell. first-order reflection from the (310) set of planes
for BCC chromium (Cr) when monochromatic
Linear and Planar Densities radiation of wavelength 0.0711 nm is used.
3.58 (a) Derive linear density expressions for FCC 3.68 Determine the expected diffraction angle for the
[100] and [111] directions in terms of the atomic first-order reflection from the (111) set of planes
radius R. for FCC nickel (Ni) when monochromatic radia-
(b) Compute and compare linear density values tion of wavelength 0.1937 nm is used.
for these same two directions for copper (Cu). 3.69 The metal rhodium (Rh) has an FCC crystal struc-
3.59 (a) Derive linear density expressions for BCC ture. If the angle of diffraction for the (311) set of
[110] and [111] directions in terms of the atomic planes occurs at 36.12 (first-order reflection) when
radius R. monochromatic x-radiation having a wavelength of
0.0711 nm is used, compute the following:
(b) Compute and compare linear density values
for these same two directions for iron (Fe). (a) The interplanar spacing for this set of planes
3.60 (a) Derive planar density expressions for FCC (b) The atomic radius for a Rh atom
(100) and (111) planes in terms of the atomic 3.70 The metal niobium (Nb) has a BCC crystal
radius R. structure. If the angle of diffraction for the (211)
(b) Compute and compare planar density values set of planes occurs at 75.99 (first-order reflec-
for these same two planes for aluminum (Al). tion) when monochromatic x-radiation having