Question

The number of atoms per unit cell and the number of slip system, respectively, for a face-centered cubic (FCC) crystal are

• A. 4, 12
• B. 3, 12
• C. 3, 3
• D. 4, 48

Hint:

FCC metals (like Cu, Al, Au) have 12 slip systems, which makes them highly ductile.
BCC metals have 48 slip systems but lack close-packed planes, while HCP metals typically have 3 primary slip systems.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks for the effective number of atoms in a single unit cell and the total number of active slip systems in a Face-Centered Cubic (FCC) crystal lattice.

Step 2: Key Formula or Approach:
The effective number of atoms \(N\) in a cubic unit cell is:
\[ N = \frac{N_{\text{corner}}}{8} + \frac{N_{\text{face}}}{2} + N_{\text{body}} \] The number of slip systems is the product of the number of unique slip planes and the number of slip directions per plane:
\[ \text{Slip Systems} = (\text{No. of slip planes}) \times (\text{No. of slip directions}) \]

Step 3: Detailed Explanation:

• Atoms per unit cell:
In an FCC unit cell, there are 8 atoms at the corners and 6 atoms at the centers of the faces.
Each corner atom is shared among 8 adjacent unit cells, and each face atom is shared between 2 adjacent unit cells.
\[ N = \frac{8}{8} + \frac{6}{2} = 1 + 3 = 4\text{ atoms per unit cell} \]
• Slip systems:
Slip occurs on the closest-packed planes and in the closest-packed directions.
For FCC, the close-packed planes are the \(\{111\}\) family, which contains 4 unique planes.
The close-packed directions within each of these planes are of the \(\langle110\rangle\) family, containing 3 unique directions per plane.
\[ \text{Slip Systems} = 4\text{ planes} \times 3\text{ directions/plane} = 12\text{ slip systems} \]

Step 4: Final Answer:
The number of atoms per unit cell is 4 and the number of slip systems is 12.