# Explanation: Counting Atoms in the Unit Cell Let's break down the problem step by step: --- ## 1. **Understanding the Image** - The image shows a 3D crystal lattice with a **unit cell** outlined in red. - There are **two types of atoms**: - **Purple atoms** (corners) - **Black atoms** (centers of the edges) --- ## 2. **Counting the Atoms** ### **Purple Atoms (Corners)** - Each **corner atom** is shared by **8 neighboring unit cells**. - There are **8 corners** in a cube. - **Contribution per unit cell**: \[ \text{Number of purple atoms in unit cell} = 8 \times \frac{1}{8} = 1 \] ### **Black Atoms (Edge Centers)** - Each **edge atom** is shared by **4 unit cells** (each edge belongs to 4 cubes). - There are **12 edges** in a cube. - **Contribution per unit cell**: \[ \text{Number of black atoms in unit cell} = 12 \times \frac{1}{4} = 3 \] --- ## 3. **Filling the Table** | Atom | Number in unit cell | |--------|---------------------| | Purple | 1 | | Black | 3 | --- ## 4. **Summary** - **Purple atoms:** 1 per unit cell - **Black atoms:** 3 per unit cell --- If you have any more questions about crystal structures or unit cells, let me know!