How many "nearst" and "next nearst" neighbours, respectively, does potassium have in bcc lattice?

Options

A. 8,8
B. 8,6
C. 6,8
D. 6,6

Answer

To determine the number of nearest and next nearest neighbors for potassium in a body-centered cubic (BCC) lattice, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the BCC Structure**:
- In a BCC lattice, atoms are located at the eight corners of a cube and one atom is located at the center of the cube (body center).

2. **Identifying Nearest Neighbors**:
- For an atom located at a corner of the BCC unit cell, the nearest neighbors are the atoms that are closest to it.
- Each corner atom has one body-centered atom as its nearest neighbor.
- Additionally, there are 7 other corner atoms from adjacent unit cells that are also nearest neighbors.
- Therefore, the total number of nearest neighbors for a corner atom in a BCC lattice is:
\[ \text{Nearest neighbors} = 1 \text{ (body center)} + 7 \text{ (corner atoms from adjacent cells)} = 8 \]

3. **Identifying Next Nearest Neighbors**:
- Next nearest neighbors are the atoms that are one step further away than the nearest neighbors.
- For the corner atom, the next nearest neighbors are the corner atoms that are directly adjacent in the same unit cell and in the adjacent unit cells.
- There are 3 corner atoms in the same unit cell and 3 corner atoms in the adjacent unit cells that are next nearest neighbors.
- Therefore, the total number of next nearest neighbors is:
\[ \text{Next nearest neighbors} = 3 \text{ (same unit cell)} + 3 \text{ (adjacent unit cells)} = 6 \]

4. **Final Answer**:
- Thus, the number of nearest neighbors is 8 and the number of next nearest neighbors is 6.

### Conclusion:
The number of nearest neighbors for potassium in a BCC lattice is **8**, and the number of next nearest neighbors is **6**.

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