Matching Crystal Systems with Crystallographic Parameters

Analysis of Crystal Systems and Parameters:

1. (A) Simple cubic and face-centred cubic:

• Both are Bravais lattice types belonging to the same crystal system, namely the cubic system ($s$).
• Both have axial lengths $a = b = c$ and interaxial angles $\alpha = \beta = \gamma = 90^\circ$ ($p$).
• Matching: $(\text{A}) \rightarrow (\text{p}, \text{s})$.

1. (B) Cubic and rhombohedral (trigonal):

• Cubic: $a = b = c$ and $\alpha = \beta = \gamma = 90^\circ$.
• Rhombohedral: $a = b = c$ and $\alpha = \beta = \gamma \neq 90^\circ$.
• Both satisfy the condition of equal edges and identical angles $a = b = c$ and $\alpha = \beta = \gamma$ ($p$).
• Furthermore, cubic and rhombohedral are classified as two distinct crystal systems ($q$).
• Matching: $(\text{B}) \rightarrow (\text{p}, \text{q})$.

1. (C) Cubic and tetragonal:

• Cubic ($a = b = c, \alpha = \beta = \gamma = 90^\circ$) and tetragonal ($a = b \neq c, \alpha = \beta = \gamma = 90^\circ$) are two distinct crystal systems ($q$).
• Neither has $a=b=c$ shared, nor only two $90^\circ$ angles (both have all three angles $= 90^\circ$).
• Matching: $(\text{C}) \rightarrow (\text{q})$.

1. (D) Hexagonal and monoclinic:

• Hexagonal and monoclinic are two distinct crystal systems ($q$).
• In hexagonal: $\alpha = \beta = 90^\circ, \gamma = 120^\circ$ (exactly two $90^\circ$ angles).
• In monoclinic: $\alpha = \gamma = 90^\circ, \beta \neq 90^\circ$ (exactly two $90^\circ$ angles).
• Hence, both have only two crystallographic angles equal to $90^\circ$ ($r$).
• Matching: $(\text{D}) \rightarrow (\text{q}, \text{r})$.

Matching:
$(\text{A}) \rightarrow (\text{p}, \text{s}),\; (\text{B}) \rightarrow (\text{p}, \text{q}),\; (\text{C}) \rightarrow (\text{q}),\; (\text{D}) \rightarrow (\text{q}, \text{r})$

Correct Answer: B