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Chapter 4: Problem 40

For a BCC single crystal, would you expect the surface energy for a (100) plane to be greater or less than that for a (110) plane? Why? (Note: You may want to consult the solution to Problem \(3.61\) at the end of Chapter 3.)

Short Answer

Expert verified
Answer: The (110) plane has higher surface energy than the (100) plane in a BCC single crystal due to the higher number of broken bonds per unit area on the (110) plane.

Step by step solution

01

Identify the crystal geometry

The given crystal is a Body-Centered Cubic (BCC) single crystal. In a BCC structure, there is one atom at the center and one at each corner of the cube (a total of 2 atoms per unit cell).
02

Describe the (100) and (110) planes

The (100) plane represents a plane perpendicular to the x-axis that cuts through the cubic cell. The atoms in this plane are only those in the corners, and 1 entire row of (100) plane contains 4 corner atoms. On the other hand, the (110) plane represents a plane perpendicular to the x+y diagonal (along the x=y line) that cuts through the cubic cell. The atoms in this plan are 2 corner atoms and 1 center atom, making a total of 3 atoms in the plane. For every 2 rows of (110) plane, this plane has 4 corner atoms and 2 central atoms.
03

Calculate broken bonds per unit area in each plane

In the (100) plane, there are 4 broken bonds (e.g. bonds between 2 corner atoms) per unit area, and in the (110) plane, 6 bonds are broken per unit area. Two bonds originate from the 4 corner atoms (equivalent to the (100) plane) and additional 4 bonds arise from the 2 central atoms.
04

Compare surface energies

As surface energy depends on the number of broken bonds, which leads to increased energy contribution from atomic interactions, we can conclude by comparing bonds broken per unit area between the two planes. Since the (110) plane has more broken bonds per unit area (6) than the (100) plane (4), we can expect the surface energy of the (110) plane to be greater than that of the (100) plane. So, for a BCC single crystal, the surface energy for a (100) plane is less than that for a (110) plane due to the higher number of broken bonds per unit area on the (110) plane.

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Most popular questions from this chapter

(a) For a given material, would you expect the surface energy to be greater than, the same as, or less than the grain boundary energy? Why? (b) The grain boundary energy of a small-angle grain boundary is less than for a high-angle one. Why is this so?

Calculate the number of vacancies per cubic meter in gold (Au) at \(900^{\circ} \mathrm{C}\). The energy for vacancy formation is \(0.98 \mathrm{eV} /\) atom. Furthermore, the density and atomic weight for Au are \(18.63 \mathrm{~g} / \mathrm{cm}^{3}\) (at \(900^{\circ} \mathrm{C}\) ) and \(196.9 \mathrm{~g} /\) mol, respectively.

What is the composition, in atom percent, of an alloy that contains \(33 \mathrm{~g}\) of \(\mathrm{Cu}\) and \(47 \mathrm{~g}\) of \(\mathrm{Zn}\) ?

Calculate the unit cell edge length for an \(80 \mathrm{wt} \% \mathrm{Ag}-\) \(20 \mathrm{wt} \%\) Pd alloy. All of the palladium is in solid solution, the crystal structure for this alloy is \(\mathrm{FCC}\), and the room-temperature density of \(\mathrm{Pd}\) is \(12.02 \mathrm{~g} / \mathrm{cm}^{3} .\)

For a BCC iron-carbon alloy that contains \(0.1\) wt \(\%\) C, calculate the fraction of unit cells that contain carbon atoms.
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