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Chapter 3: Problem 66

Using the data for \(\alpha\)-iron in Table \(3.1\), compute the interplanar spacings for the (111) and (211) sets of planes.

Short Answer

Expert verified
Answer: The interplanar spacing for the (111) plane is approximately 1.657 Å, and the interplanar spacing for the (211) plane is approximately 1.172 Å.

Step by step solution

01

Identify the lattice constant of α-iron.

From Table 3.1, the lattice constant for α-iron (a) is given as \(\approx 2.87 \, \text{Å}\).
02

Compute the interplanar spacing for the (111) plane.

To compute the interplanar spacing for the (111) plane, we can substitute \(h = 1\), \(k = 1\), \(l = 1\), and \(a = 2.87\, \text{Å}\) in the formula for \(d_{hkl}\): \(d_{111} = \frac{2.87 \, \text{Å}}{\sqrt{1^2 + 1^2 + 1^2}} = \frac{2.87 \, \text{Å}}{\sqrt{3}} \approx 1.657 \, \text{Å}\) So, the interplanar spacing for the (111) plane is around \(1.657\, \text{Å}\).
03

Compute the interplanar spacing for the (211) plane.

To compute the interplanar spacing for the (211) plane, we can substitute \(h = 2\), \(k = 1\), \(l = 1\), and \(a = 2.87\, \text{Å}\) in the formula for \(d_{hkl}\): \(d_{211} = \frac{2.87 \, \text{Å}}{\sqrt{2^2 + 1^2 + 1^2}} = \frac{2.87 \, \text{Å}}{\sqrt{6}} \approx 1.172\, \text{Å}\) So, the interplanar spacing for the (211) plane is around \(1.172\, \text{Å}\).

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Most popular questions from this chapter

(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).

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 \(3.12 b\) and \(3.13 b\) ).

(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).

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 \(0.0711 \mathrm{~nm}\) is used.

Molybdenum (Mo) has a BCC crystal structure, an atomic radius of \(0.1363 \mathrm{~nm}\), and an atomic weight of \(95.94 \mathrm{~g} / \mathrm{mol}\). Compute and compare its theoretical density with the experimental value found inside the front cover of the book.
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