Question

Which of the following statements are correct:
A. In SC structure a = 2r
B. In SC structure a = r/2
C. In BCC structure a = \( 4r/\sqrt{3} \)
D. In FCC structure a = \( 2\sqrt{2} r \)
Where a is a lattice parameter and r is the atomic radius.

• A. A, C and D
• B. A, B and C
• C. A, B, C and D
• D. B, C and D

Hint:

Visualize the unit cell for each structure and identify the direction along which the atoms are in contact. For SC it's the edge, for BCC it's the body diagonal, and for FCC it's the face diagonal. Use basic geometry (Pythagorean theorem) to relate this direction's length to the lattice parameter 'a'.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question asks for the correct relationship between the lattice parameter (a) and the atomic radius (r) for different cubic crystal structures. This relationship is determined by how the atoms are packed and where they touch each other within the unit cell.
Step 2: Detailed Explanation:
A. Simple Cubic (SC) Structure:
In an SC structure, atoms are located at the corners and touch along the edge of the cube.
The length of the edge 'a' is equal to the sum of two atomic radii.
\[ a = 2r \]
So, statement A is correct, and statement B is incorrect.
C. Body-Centered Cubic (BCC) Structure:
In a BCC structure, atoms touch along the body diagonal of the cube.
The length of the body diagonal is \( \sqrt{3}a \).
This diagonal accommodates one full atom diameter (2r) and two radii (r) from the corner atoms. So, the total length is 4r.
\[ \sqrt{3}a = 4r \implies a = \frac{4r}{\sqrt{3}} \]
So, statement C is correct.
D. Face-Centered Cubic (FCC) Structure:
In an FCC structure, atoms touch along the face diagonal of the cube.
The length of the face diagonal is \( \sqrt{2}a \).
This diagonal accommodates one full atom diameter (2r) and two radii (r) from the corner atoms. So, the total length is 4r.
\[ \sqrt{2}a = 4r \implies a = \frac{4r}{\sqrt{2}} = \frac{4\sqrt{2}r}{2} = 2\sqrt{2}r \]
So, statement D is correct.
Step 3: Final Answer:
The correct statements are A, C, and D.