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Chapter 11: Problem 34

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

Short Answer

Expert verified
The geometries of the cubic cells are as follows: Simple cubic cells just have atoms at the corners, body-centered cubic cells have atoms at the corners and in the center, face-centered cubic cells have atoms at the corners and on each face. With the same type of atoms, face-centered cubic structures would give the highest density, body-centered cubic structures would give the medium density, and simple cubic structures would give the lowest density.

Step by step solution

01

Describing the Simple Cubic Structure

In a simple cubic (sc) unit cell, each atom sits on a corner of the cube. Any atom in the cell has 6 nearest neighbors, up, down, left, right, in front and at the back. As a result, the packing efficiency or the density for this type of cell is lowest among the cubic cells at only about 52.4%.
02

Describing the Body-Centered Cubic Structure

A body-centered cubic (bcc) unit cell has an atom at each of the eight corners of a cube and a single atom at the center of the cube. This results in a coordination number of 8, meaning each atom has 8 nearest neighbors. The packing efficiency rises to about 68% in a body-centered cubic structure.
03

Describing the Face-Centered Cubic Structure

Lastly, a face-centered cubic(fcc) unit cell has an atom at each of its corners, and an atom in the center of each face of the cube. This gives a total of 14 nearest neighbors, as each atom is in contact with 6 others in the same layer, 3 in the layer above and 5 in the layer below. The coordination number for this type of structure is highest at 12, and the packing efficiency is the greatest among cubic cells, approximately 74%.
04

Concluding the Density Ranking

Given the packing efficiencies, the cubic arrangement with the highest density for the same atoms is face-centered cubic, then body-centered cubic and the lowest is simple cubic.

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