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

The correct answer is 4,12.

Let's break down the derivation of both the number of atoms per unit cell and the number of slip systems for a face-centered cubic (FCC) crystal step-by-step.

1. Number of Atoms per Unit Cell in FCC:
An FCC unit cell has atoms at two types of positions:
- Corner positions: There are 8 corners in a cube. Each corner atom is shared equally among 8 adjacent unit cells. Therefore, the contribution of corner atoms to a single unit cell is:
$8\times \frac{1}{8}=1\text{atom}$
- Face centers: There are 6 faces in a cube. Each face-centered atom is shared between 2 adjacent unit cells. Thus, the contribution of face-centered atoms to a single unit cell is:
$6\times \frac{1}{2}=3\text{atoms}$
Adding these contributions together, the total number of atoms per unit cell for an FCC crystal is:
$1+3=4\text{atoms}$

2. Number of Slip Systems in FCC:
A slip system is defined by a combination of a slip plane (usually the closest-packed plane) and a slip direction (the closest-packed direction within that plane).
- Slip Planes: In FCC crystals, the slip planes are the close-packed octahedron planes, which belong to the $\left\{111\right\}$ family. There are 4 unique non-parallel $\left\{111\right\}$ planes.
- Slip Directions: Within each $\left\{111\right\}$ plane, there are 3 close-packed directions, which belong to the $⟨110⟩$ family.
- Total Slip Systems: The total number of slip systems is the product of the number of unique slip planes and the number of slip directions per plane:
$4\text{planes}\times 3\text{directions/plane}=12\text{slip systems}$

Therefore, the number of atoms per unit cell is 4 and the number of slip systems is 12.