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

The correct option is 4,12.

Let's break down the explanation into two parts: finding the number of atoms per unit cell and finding the number of slip systems for a face-centered cubic (FCC) crystal structure.

1. Number of Atoms Per Unit Cell in FCC:
In a face-centered cubic (FCC) crystal lattice, the atoms are located at:
- The 8 corners of the cube.
- The centers of the 6 faces of the cube.

These atoms are shared with neighboring unit cells in the following way:
- Each corner atom is shared by 8 adjacent unit cells. Therefore, only 1/8th of each corner atom belongs to a single unit cell.
- Each face-centered atom is shared by 2 adjacent unit cells. Therefore, only 1/2 of each face atom belongs to a single unit cell.

Calculating the total number of atoms in a unit cell:

$$\text{Total atoms}=\left(8\times \frac{1}{8}\right)+\left(6\times \frac{1}{2}\right)$$

$$\text{Total atoms}=1+3=4$$

Thus, there are 4 atoms per unit cell in an FCC crystal.

2. Number of Slip Systems in FCC:
A slip system is a combination of a slip plane and a slip direction along which plastic deformation occurs. Slip typically occurs on the most closely packed planes and in the most closely packed directions.
- Slip Planes: In FCC, the close-packed planes are the family of {111} planes. There are 4 unique, non-parallel planes of this type: (111), (1-11), (11-1), and (-111).
- Slip Directions: Within each {111} plane, the close-packed directions belong to the ⟨110⟩ family. There are 3 close-packed directions in each {111} plane.

The total number of slip systems is calculated by multiplying the number of slip planes by the number of slip directions per plane:

$$\text{Total slip systems}=4\text{planes}\times 3\text{directions/plane}=12$$

Therefore, the number of slip systems for an FCC crystal is 12.

Combining these two results, the number of atoms per unit cell and the number of slip systems for an FCC crystal are 4 and 12, respectively.