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Chapter 12: Problem 24

What is the minimum number of atoms that could be contained in the unit cell of an element with a face-centered cubic lattice?

Short Answer

Expert verified
The minimum number of atoms in the unit cell of an element with a face-centered cubic lattice is 4. This includes 1 atom contributed from the 8 corners of the cube and 3 atoms contributed from the face centers of the cube.

Step by step solution

01

Understanding the face-centered cubic lattice structure

A face-centered cubic (FCC) lattice structure is a type of crystalline structure where atoms are located at each corner of the cube and also at the center of each face of the cubic unit cell.
02

Counting atoms at the corners

There are 8 corners in a cubic unit cell, and each corner is shared by 8 adjacent unit cells. Therefore, each corner contributes \(\frac{1}{8}\) of an atom to a unit cell. So, for all the 8 corners combined, the contribution of atoms to the unit cell from the corners is: \( 8 \times \frac{1}{8} = 1 \)
03

Counting atoms at the face centers

There are 6 faces in a cubic unit cell, and each face-center is shared by 2 adjacent unit cells. Therefore, each face-center contributes \(\frac{1}{2}\) of an atom to a unit cell. So, for all the 6 face-centers combined, the contribution of atoms to the unit cell from the face centers is: \( 6 \times \frac{1}{2} = 3 \)
04

Calculating the total number of atoms in the unit cell

Now we just add up the contributions from the corners and the face centers to find the total number of atoms in the unit cell: \( 1 (from \:corners) + 3(from\:face-centers)= 4 \) Thus, the minimum number of atoms contained in the unit cell of an element with a face-centered cubic lattice is 4.

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Most popular questions from this chapter

Briefly describe each of the following: (a) elastomer, (b) thermoplastic, (c) thermosetting plastic.

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