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Chapter 7: Problem 6

(a) Compare planar densities (Section \(3.11\) and Problem 3.54) for the (100), (110), and (111) planes for FCC. (b) Compare planar densities (Problem 3.55) for the (100), (110), and (111) planes for BCC.

Short Answer

Expert verified
Answer: In FCC crystal structure, the (111) plane has the highest planar density. In BCC crystal structure, the (110) plane has the highest planar density.

Step by step solution

01

Determine the number of atoms per unit cell

For FCC crystal structures, there are 4 atoms per unit cell.
02

Calculate the planar density for the (100) plane

In the (100) plane, 1 atom is lying in the unit cell section. The area of this unit cell section can be calculated as follows: Area = \(a^2\), where a is the lattice parameter. Therefore, the planar density for the (100) plane is 1 / \(a^2\).
03

Calculate the planar density for the (110) plane

In the (110) plane, 2 atoms are lying in the unit cell section. The area of this unit cell section can be calculated as follows: Area = \(a \times \frac{a\sqrt{2}}{2}\). Therefore, the planar density for the (110) plane is 2 / (\(a \times \frac{a\sqrt{2}}{2}\)).
04

Calculate the planar density for the (111) plane

In the (111) plane, 3 atoms are lying in the unit cell section. The area of this unit cell section can be calculated as follows: Area = \(\frac{a\sqrt{2}}{2} \times \frac{a\sqrt{2}}{2}\). Therefore, the planar density for the (111) plane is 3 / (\(\frac{a\sqrt{2}}{2} \times \frac{a\sqrt{2}}{2}\)). #BCC Crystal Structure#
05

Determine the number of atoms per unit cell

For BCC crystal structures, there are 2 atoms per unit cell.
06

Calculate the planar density for the (100) plane

In the (100) plane, 1 atom is lying in the unit cell section. The area of this unit cell section can be calculated as follows: Area = \(a^2\), where a is the lattice parameter. Therefore, the planar density for the (100) plane is 1 / \(a^2\).
07

Calculate the planar density for the (110) plane

In the (110) plane, 2 atoms are lying in the unit cell section. The area of this unit cell section can be calculated as follows: Area = \(a \times \frac{a\sqrt{2}}{2}\). Therefore, the planar density for the (110) plane is 2 / (\(a \times \frac{a\sqrt{2}}{2}\)).
08

Calculate the planar density for the (111) plane

In the (111) plane, 1 atom is lying in the unit cell section. The area of this unit cell section can be calculated as follows: Area = \(\frac{a\sqrt{3}}{2} \times \frac{a\sqrt{3}}{2}\). Therefore, the planar density for the (111) plane is 1/(\(\frac{a\sqrt{3}}{2} \times \frac{a\sqrt{3}}{2}\)). The solution provides the clear comparison of planar densities for the specified planes in FCC and BCC crystal structures. To summarize, for FCC structure, the (111) plane has the highest planar density, followed by the (110) and (100) planes. For BCC structure, the (110) plane has the highest planar density, followed by the (100) planes. The (111) plane has the lowest planar density in the BCC structure.

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Most popular questions from this chapter

Consider a single crystal of silver oriented such that a tensile stress is applied along a \([001]\) direction. If slip occurs on a (111) plane and in a [ \(\overline{101}\) ] direction, and is initiated at an applied tensile stress of \(1.1 \mathrm{MPa}\) (160 psi), compute the critical resolved shear stress.

Consider a metal single crystal oriented such that the normal to the slip plane and the slip direction are at angles of \(43.1^{\circ}\) and \(47.9^{\circ}\), respectively, with the tensile axis. If the critical resolved shear stress is \(20.7\) MPa (3000 psi), will an applied stress of 45 MPa (6500 psi) cause the single crystal to yield? If not, what stress will be necessary?

One slip system for the BCC crystal structure is \(\\{110\\}(111\rangle\). In a manner similar to Figure \(7.6 b\), sketch a \\{110\\}-type plane for the BCC structure, representing atom positions with circles. Now, using arrows, indicate two different \(\langle 111\rangle\) slip directions within this plane.

Briefly cite the differences between recovery and recrystallization processes.

List four major differences between deformation by twinning and deformation by slip relative to mechanism, conditions of occurrence, and final result.
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