(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).

(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).

(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).

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

(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 $(\mathrm{Cu})$.

Calculate and compare the planar densities for the $\{100\},\{110\}$, and $\{111\}$ planes in a BCC unit cell. Which plane is the most close-packed (dense)?

Find the planar density and packing fraction for FCC nickel in the $(100)$, $(110)$, and $(111)$ planes. Which, if any, of these planes are close packed?

Find the planar density and packing fraction for BCC lithium in the $(100)$, $(110)$, and $(111)$ planes. Which, if any, of these planes are close packed?

Find the indices for the planes in the hexagonal lattice shown in Figure $3-43$.

Find the indices for the planes in the hexagonal lattice shown in Figure $3-44$.

the first four peaks of the x-ray diffraction pattern for copper, which has an FCC crystal structure; monochromatic x-radiation having a wavelength of 0.1542 nm was used.

(a) Index (i.e., give h,k, and l indices for) each of these peaks.

(b) Determine the interplanar spacing for each of the peaks.

(c) For each peak, determine the atomic radius for Cu and compare these with the value presented.

For a simple cubic lattice, determine the Miller indices for the planes shown in the given figure.

What is the atomic packing factor for the bcc crystal structure?

A specimen of aluminum having a rectangular cross section $10 \mathrm{~mm} \times 12.7 \mathrm{~mm}(0.4$ in. $\times$ 0.5 in.) is pulled in tension with $35,500 \mathrm{~N}$ $\left(8000 \mathrm{lb}_{\mathrm{f}}\right.$ ) force, producing only elastic deformation. Calculate the resulting strain.