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Chapter 12: Problem 34

Describe the geometries of these cubic cells: simple cubic cell, body-centered cubic cell, face-centered cubic cell. Which of these cells would give the highest density for the same type of atoms?

Short Answer

Expert verified
The geometry for the simple cubic, body-centered cubic, and face-centered cubic cells are as described. The face-centered cubic cell gives the highest density for the same type of atoms as it contains the most atoms.

Step by step solution

01

Simple Cubic Cell

A simple cubic cell has atoms at each of its 8 corners. Each corner is shared by 8 cells so a single cell contains \( \frac{1}{8}*8 = 1 \) atom.
02

Body-Centered Cubic Cell

A body-centered cubic cell has atoms at the 8 corners like in a simple cubic cell as well as an additional atom in the center of the cell, so a single cell contains \( \frac{1}{8}*8 + 1 = 2 \) atoms.
03

Face-Centered Cubic Cell

A face-centered cubic cell has atoms at each of its 8 corners and one additional atom at the center of each of the 6 faces. So a single cell contains \( \frac{1}{8}*8 + \frac{1}{2}*6 = 1+3 = 4 \) atoms. However, note that each face atom is shared by two cells.
04

Comparing Densities

Density depends on the number of atoms per unit volume. Since the type of atom is the same in this case, the cell with the highest number of atoms will have the highest density. Thus, the face-centered cubic cell has the highest density as it contains the most atoms.

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