If an atom crystallizes in bcc lattice with `r=4 Å` then the edge length will be

Options

A. `2 Å`
B. `8 Å`
C. `2.39 Å`
D. `9.23 Å`

Answer

To find the edge length of an atom crystallizing in a body-centered cubic (BCC) lattice with a radius of 4 Å, we can follow these steps:

### Step 1: Understand the BCC Structure
In a body-centered cubic (BCC) lattice, atoms are located at each corner of the cube and one atom is located at the center of the cube. The atoms touch along the body diagonal of the cube.

### Step 2: Use the Relationship Between Edge Length and Atomic Radius
In a BCC lattice, the relationship between the edge length (a) and the atomic radius (r) is given by the equation:
\[ \sqrt{3}a = 4r \]
This equation arises because the body diagonal of the cube can be expressed in terms of the edge length and the atomic radius.

### Step 3: Rearrange the Equation to Solve for Edge Length
We can rearrange the equation to find the edge length (a):
\[ a = \frac{4r}{\sqrt{3}} \]

### Step 4: Substitute the Given Radius
We are given that the radius \( r = 4 \, \text{Å} \). Now, substitute this value into the equation:
\[ a = \frac{4 \times 4 \, \text{Å}}{\sqrt{3}} \]
\[ a = \frac{16 \, \text{Å}}{\sqrt{3}} \]

### Step 5: Calculate the Edge Length
Now, we calculate the numerical value:
\[ a \approx \frac{16 \, \text{Å}}{1.732} \]
\[ a \approx 9.24 \, \text{Å} \]

### Conclusion
The edge length of the BCC lattice is approximately \( 9.24 \, \text{Å} \).

### Final Answer
The edge length will be \( 9.23 \, \text{Å} \) (rounded to two decimal places).

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