For a body centred lattice, edge length is equal to

Options

A. `2sqrt(2)r `
B. `2r`
C. `(sqrt3)/(4) r`
D. `4/(sqrt3) r`

Answer

To find the edge length (A) of a body-centered cubic (BCC) lattice, we can follow these steps:

### Step 1: Understand the Structure of BCC
In a body-centered cubic lattice, there are atoms located at the corners of the cube and one atom at the center of the cube.

### Step 2: Determine the Number of Atoms
- There are 8 corner atoms in a cube, and each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell.
- Therefore, the contribution from the corner atoms is:
\[ 8 \times \frac{1}{8} = 1 \text{ atom} \]
- There is 1 atom at the center of the cube, contributing fully:
\[ 1 \text{ atom} \]
- Thus, the total number of effective atoms in a BCC unit cell is:
\[ 1 + 1 = 2 \text{ atoms} \]

### Step 3: Identify the Body Diagonal
In a BCC structure, the atoms touch along the body diagonal of the cube. The body diagonal can be calculated using the Pythagorean theorem in three dimensions.

### Step 4: Calculate the Length of the Body Diagonal
The length of the body diagonal (d) of a cube with edge length A is given by:
\[ d = \sqrt{A^2 + A^2 + A^2} = \sqrt{3A^2} = A\sqrt{3} \]

### Step 5: Relate the Body Diagonal to Atomic Radii
In a BCC lattice, the body diagonal is equal to four times the radius (r) of the atoms because there are two atomic radii from the corner atom to the center atom and two from the center atom to the opposite corner atom:
\[ d = 4r \]

### Step 6: Set the Equations Equal
Now, we can set the two expressions for the body diagonal equal to each other:
\[ A\sqrt{3} = 4r \]

### Step 7: Solve for Edge Length (A)
To find the edge length A, rearrange the equation:
\[ A = \frac{4r}{\sqrt{3}} \]

### Final Result
Thus, the edge length (A) of a body-centered cubic lattice is given by:
\[ A = \frac{4r}{\sqrt{3}} \]