What is the condition number of an atom for an element crystallizing with a cubic lattice? Calculate the corresponding coordination number for the simple, fcc and bcc lattics:

Options

A. 12sc, 12fcc, 8bc c
B. 6sc, 14fcc, 8bc c
C. 8sc, 12fcc, 6bc c
D. sc, 12fcc, 8bc c

Answer

To solve the question regarding the coordination number of an atom for an element crystallizing with a cubic lattice, we will analyze three types of cubic lattices: simple cubic (SC), face-centered cubic (FCC), and body-centered cubic (BCC).

### Step-by-Step Solution:

1. **Understanding Coordination Number**:
- The coordination number (CN) is defined as the number of nearest neighboring particles surrounding a central particle in a crystal lattice.

2. **Simple Cubic (SC) Lattice**:
- In a simple cubic lattice, each atom is located at the corners of a cube.
- Each corner atom is shared by eight adjacent cubes, but it has 6 nearest neighbors (one on each face of the cube).
- Therefore, the coordination number for a simple cubic lattice is **6**.

3. **Face-Centered Cubic (FCC) Lattice**:
- In a face-centered cubic lattice, atoms are located at each corner and the centers of each face of the cube.
- Each corner atom has 4 face-centered neighbors and 4 corner neighbors from adjacent cubes, totaling 12 nearest neighbors.
- Thus, the coordination number for a face-centered cubic lattice is **12**.

4. **Body-Centered Cubic (BCC) Lattice**:
- In a body-centered cubic lattice, there is one atom at each corner and one atom at the center of the cube.
- Each corner atom is surrounded by 8 nearest neighbors (the center atom and the 7 corner atoms of adjacent cubes).
- Therefore, the coordination number for a body-centered cubic lattice is **8**.

### Summary of Coordination Numbers:
- Simple Cubic (SC): **6**
- Face-Centered Cubic (FCC): **12**
- Body-Centered Cubic (BCC): **8**

### Final Answer:
- The coordination numbers for the different cubic lattices are:
- Simple Cubic (SC): 6
- Face-Centered Cubic (FCC): 12
- Body-Centered Cubic (BCC): 8