Question

If 'a' stands for the edge length of the cubic systems - The ratio of radii in simple cubic, body centered cubic and face centered cubic unit cells is

• A. \(1a:\sqrt3a:\sqrt2a\)
• B. \(\frac{1}{2}a:\frac{\sqrt3}{4}a:\frac{1}{2\sqrt2}a\)
• C. \(\frac{1}{2}a:\frac{\sqrt3}{2}a:\frac{\sqrt2}{2}a\)
• D. \(\frac{1}{2}a:\sqrt3a:\frac{1}{\sqrt2}a\)

Answer 1

The Correct Option is B

1. Simple cubic (SC): 

Atoms touch along the edges.

\[ 2r = a \]

Therefore, radius for SC:

\[ r = \frac{a}{2} \]

1. Body-centered cubic (BCC):

Atoms touch along the body diagonal of the cube.

\[ 4r = \sqrt{3}a \]

Therefore, radius for BCC:

\[ r = \frac{\sqrt{3}a}{4} \]

1. Face-centered cubic (FCC):

Atoms touch along the face diagonal.

\[ 4r = \sqrt{2}a \]

Therefore, radius for FCC:

\[ r = \frac{\sqrt{2}a}{4} = \frac{a}{2\sqrt{2}} \]

Combining these results:

Simple cubic : Body-centered cubic : Face-centered cubic

\[ \frac{a}{2} : \frac{\sqrt{3}a}{4} : \frac{a}{2\sqrt{2}} \]

Conclusion:

The correct ratio is:

\( \frac{1}{2}a : \frac{\sqrt{3}}{4}a : \frac{1}{2\sqrt{2}}a \)

Correct Answer: Option (B)

Answer 2

The edge length 'a' refers to the edge length of the unit cell. The relationship between the radii in different cubic structures can be derived as follows:
1. Simple cubic (SC):  In a simple cubic unit cell, the atoms are located at the corners. The radius of the atom \( r \) is half the edge length of the unit cell. Therefore, \( r_{\text{SC}} = \frac{a}{2} \).

2. Body-centered cubic (BCC): In a body-centered cubic unit cell, the atoms are at the corners and at the center of the cell. The diagonal of the cube passes through the center, and it is equal to \( 4r \). From geometry, the diagonal is \( \sqrt{3}a \), so: \[ \sqrt{3}a = 4r_{\text{BCC}} \quad \Rightarrow \quad r_{\text{BCC}} = \frac{\sqrt{3}}{4}a \]

3. Face-centered cubic (FCC): In a face-centered cubic unit cell, the atoms are at the corners and the centers of the faces. The diagonal across the face is equal to \( 4r \). The face diagonal is \( \sqrt{2}a \), so: \[ \sqrt{2}a = 4r_{\text{FCC}} \quad \Rightarrow \quad r_{\text{FCC}} = \frac{\sqrt{2}}{4}a \]

Thus, the ratio of the radii in simple cubic, body-centered cubic, and face-centered cubic unit cells is: \[ r_{\text{SC}} : r_{\text{BCC}} : r_{\text{FCC}} = \frac{a}{2} : \frac{\sqrt{3}}{4}a : \frac{\sqrt{2}}{4}a \]

The correct answer is option (B).