
Concept explainers
Interpretation:
The packing efficiencies of a simple cubic, body-centered, and face-centered cell are to be calculated
Concept introduction:
A crystal lattice is made of small repeating unit cells.
A unit cell is of primitive or centered type. The spheres in a unit cell can represent an atom, a molecule, or an ion.
Each sphere is shared by adjacent cells. A sphere at corners is shared by eight unit cells. A sphere in the center of each face is shared by two unit cells.
The volume of the unit cell is calculated as:
Here,
is the edge length.
The formula to calculate the volume of a sphere is as:
Here,
is the radius of the sphere.
Packing efficiency is the ratio of the volume of atoms in a unit cell to the volume of the unit cell. Its expression is:

Answer to Problem 136AP
Solution:
Explanation of Solution
Given information: For SCC,
In a simple cubic cell, spheres are present only at the eight corners of the unit cell.
The contribution of 8 spheres at the corners to one unit cell is:
Thus, one sphere is present in the unit cell.
The formula to calculate the volume of a unit cell is as follows:
Substitute
for
in the above expression as:
The volume of a sphere is
The formula to calculate packing efficiency is as:
Substitute
for the volume of atoms in the unit cell and
for the volume of the unit cell in the above expression as:
In BCC, there are spheres at the eight corners of the unit cell and one sphere is present in the center of the unit cell.
The contribution of 8 spheres at the corners to one unit cell is:
One sphere is in the center of the unit cell.
Thus, two spheres are present in a body centered unit cell.
The volume of a sphere is
Thus, the volume of two spheres will be as:
Substitute
in equation (1) as:
Substitute
for the volume of atoms in the unit cell and
for the volume of the unit cell in equation (2) as follows:
In FCC cubic cell, there are spheres at the eight corners of the unit cell and one sphere in the center of six faces.
The contribution of
The contribution of
spheres present in the center of each face to one-unit cell is:
Thus, four spheres are present in a face-centered unit cell.
The volume of a sphere is
Thus, the volume of four spheres is:
Substitute
for
in equation (1):
Substitute
for the volume of atoms in the unit cell and
for the volume of the unit cell in equation (2) as follows:
The packing efficiencies of simple cubic, body-centered, and face-centered cell are
and
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