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: Concept Overview:
The problem requires identifying the accurate relationships between the lattice parameter (a) and the atomic radius (r) for different cubic crystal structures. These relationships are dictated by atomic packing and contact points within the unit cell.
Step 2: Detailed Analysis:
A. Simple Cubic (SC):
In SC structures, atoms reside at the corners and make contact along the cube's edge.
The edge length 'a' equals twice the atomic radius.
\[ a = 2r \]
Therefore, statement A is correct, and B is incorrect.
C. Body-Centered Cubic (BCC):
In BCC structures, atoms touch along the body diagonal.
The body diagonal's length is \( \sqrt{3}a \).
This diagonal spans one full atom diameter (2r) plus two radii (r) from corner atoms, totaling 4r.
\[ \sqrt{3}a = 4r \implies a = \frac{4r}{\sqrt{3}} \]
Hence, statement C is correct.
D. Face-Centered Cubic (FCC):
In FCC structures, atoms touch along the face diagonal.
The face diagonal's length is \( \sqrt{2}a \).
This diagonal accommodates one full atom diameter (2r) and two radii (r) from the corner atoms. Thus, the total length is 4r.
\[ \sqrt{2}a = 4r \implies a = \frac{4r}{\sqrt{2}} = \frac{4\sqrt{2}r}{2} = 2\sqrt{2}r \]
Thus, statement D is correct.
Step 3: Conclusion:
The correct statements are A, C, and D.