Match the crystal systems in Column I with the corresponding axial lengths $a, b, c$ and interaxial angles $\alpha, \beta, \gamma$ provided in Column II.
Column I Column II
(P) Tetragonal (1) $a \neq b \neq c,\ \alpha = \beta = \gamma = 90^{\circ}$
(Q) Rhombohedral (2) $a = b \neq c,\ \alpha = \beta = \gamma = 90^{\circ}$
(R) Orthorhombic (3) $a \neq b \neq c,\ \alpha = \gamma = 90^{\circ} \neq \beta$
(S) Monoclinic (4) $a = b = c,\ \alpha = \beta = \gamma \neq 90^{\circ}$

Question

The length of Burgers vector b in copper (Cu), which has an FCC crystal structure, is

Answer 1

Each crystal system is fixed by how many of its three axial lengths are equal and how many of its angles sit at 90 degrees. Instead of deriving the match name by name, check each of the four given combinations against what we know of tetragonal, rhombohedral, orthorhombic and monoclinic cells.

(A) P-3, Q-1, R-2, S-4: puts tetragonal at (3), which reads $a \neq b \neq c$ with one angle off 90 degrees. That description belongs to monoclinic, not tetragonal. Wrong.
(B) P-2, Q-3, R-4, S-1: tetragonal at (2) is right, but rhombohedral is placed at (3), which is the monoclinic-type description, not the all-axes-equal description. Wrong.
(C) P-4, Q-3, R-2, S-1: tetragonal at (4) means all three axes equal, which is rhombohedral's property, not tetragonal's. Wrong.
(D) P-2, Q-4, R-1, S-3: tetragonal at (2) ($a=b\neq c$, all angles 90), rhombohedral at (4) ($a=b=c$, angles equal but not 90), orthorhombic at (1) ($a\neq b\neq c$, all angles 90) and monoclinic at (3) ($a\neq b\neq c$, one angle off 90). Every pairing fits the standard definition. Correct.

The self-consistent set is P-2, Q-4, R-1, S-3, matching option (D).

Let's summarize:

Tetragonal: two equal axes, all angles 90 degrees.
Rhombohedral: three equal axes, angles equal but not 90 degrees.
Orthorhombic: three unequal axes, all angles 90 degrees.
Monoclinic: three unequal axes, one angle away from 90 degrees.

So the correct match is P-2, Q-4, R-1, S-3, option (D).

Answer 2

Each crystal system is fixed by how many of its three axial lengths are equal and how many of its angles sit at 90 degrees. Instead of deriving the match name by name, check each of the four given combinations against what we know of tetragonal, rhombohedral, orthorhombic and monoclinic cells.

(A) P-3, Q-1, R-2, S-4: puts tetragonal at (3), which reads $a \neq b \neq c$ with one angle off 90 degrees. That description belongs to monoclinic, not tetragonal. Wrong.
(B) P-2, Q-3, R-4, S-1: tetragonal at (2) is right, but rhombohedral is placed at (3), which is the monoclinic-type description, not the all-axes-equal description. Wrong.
(C) P-4, Q-3, R-2, S-1: tetragonal at (4) means all three axes equal, which is rhombohedral's property, not tetragonal's. Wrong.
(D) P-2, Q-4, R-1, S-3: tetragonal at (2) ($a=b\neq c$, all angles 90), rhombohedral at (4) ($a=b=c$, angles equal but not 90), orthorhombic at (1) ($a\neq b\neq c$, all angles 90) and monoclinic at (3) ($a\neq b\neq c$, one angle off 90). Every pairing fits the standard definition. Correct.

The self-consistent set is P-2, Q-4, R-1, S-3, matching option (D).

Let's summarize:

Tetragonal: two equal axes, all angles 90 degrees.
Rhombohedral: three equal axes, angles equal but not 90 degrees.
Orthorhombic: three unequal axes, all angles 90 degrees.
Monoclinic: three unequal axes, one angle away from 90 degrees.

So the correct match is P-2, Q-4, R-1, S-3, option (D).

Show Hint

The Correct Option is D

Solution and Explanation

Top Questions on Crystal Structures

Questions Asked in GATE MT exam

Column I	Column II
(P) Tetragonal	(1) \(a \neq b \neq c,\ \alpha = \beta = \gamma = 90^{\circ}\)
(Q) Rhombohedral	(2) \(a = b \neq c,\ \alpha = \beta = \gamma = 90^{\circ}\)
(R) Orthorhombic	(3) \(a \neq b \neq c,\ \alpha = \gamma = 90^{\circ} \neq \beta\)
(S) Monoclinic	(4) \(a = b = c,\ \alpha = \beta = \gamma \neq 90^{\circ}\)