To determine the number of atoms per unit cell in a simple cubic, face-centered cubic (FCC), and body-centered cubic (BCC) structure, we will analyze each type of unit cell step by step.
### Step 1: Simple Cubic Unit Cell
1. In a simple cubic unit cell, atoms are located at the corners of the cube.
2. There are a total of 8 corners in a cube.
3. Each corner atom is shared among 8 adjacent unit cells, contributing \( \frac{1}{8} \) of an atom to the unit cell.
4. Therefore, the total contribution from the corner atoms is:
\[
\text{Total atoms from corners} = 8 \times \frac{1}{8} = 1
\]
5. Thus, the number of atoms per unit cell in a simple cubic structure is **1**.
### Step 2: Face-Centered Cubic Unit Cell
1. In a face-centered cubic unit cell, atoms are located at the corners and at the centers of each face of the cube.
2. Again, there are 8 corner atoms, contributing \( \frac{1}{8} \) of an atom each.
3. Additionally, there are 6 faces, and each face-centered atom is shared between 2 unit cells, contributing \( \frac{1}{2} \) of an atom to the unit cell.
4. Therefore, the total contribution is:
\[
\text{Total atoms from corners} = 8 \times \frac{1}{8} = 1
\]
\[
\text{Total atoms from face centers} = 6 \times \frac{1}{2} = 3
\]
5. Adding these contributions gives:
\[
\text{Total atoms in FCC} = 1 + 3 = 4
\]
6. Thus, the number of atoms per unit cell in a face-centered cubic structure is **4**.
### Step 3: Body-Centered Cubic Unit Cell
1. In a body-centered cubic unit cell, atoms are located at the corners and one atom is located at the center of the cube.
2. As before, there are 8 corner atoms, each contributing \( \frac{1}{8} \) of an atom.
3. The body-centered atom is entirely within the unit cell, contributing 1 full atom.
4. Therefore, the total contribution is:
\[
\text{Total atoms from corners} = 8 \times \frac{1}{8} = 1
\]
\[
\text{Total atoms from body center} = 1
\]
5. Adding these contributions gives:
\[
\text{Total atoms in BCC} = 1 + 1 = 2
\]
6. Thus, the number of atoms per unit cell in a body-centered cubic structure is **2**.
### Final Answer
The number of atoms per unit cell in a simple cubic, face-centered cubic, and body-centered cubic are respectively: **1, 4, and 2**.
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