Question

The crystal system of a compound with cell dimensions \( a = 0.387 \), \( b = 0.387 \), \( c = 0.504 \text{ nm} \); \( \alpha = \beta = 90^{\circ} \) and \( \gamma = 120^{\circ} \) is

• A. cubic
• B. rhombohedral
• C. orthorhombic
• D. hexagonal

Hint:

To identify the crystal system, memorize the seven crystal systems and their corresponding axial lengths and angles. A quick recall table can be very helpful: \begin{itemize} \item Cubic: $a=b=c, \alpha=\beta=\gamma=90^\circ$ \item Tetragonal: $a=b \ne c, \alpha=\beta=\gamma=90^\circ$ \item Orthorhombic: $a \ne b \ne c, \alpha=\beta=\gamma=90^\circ$ \item Hexagonal: $a=b \ne c, \alpha=\beta=90^\circ, \gamma=120^\circ$ \item Trigonal (Rhombohedral): $a=b=c, \alpha=\beta=\gamma \ne 90^\circ$ \item Monoclinic: $a \ne b \ne c, \alpha=\gamma=90^\circ, \beta \ne 90^\circ$ \item Triclinic: $a \ne b \ne c, \alpha \ne \beta \ne \gamma \ne 90^\circ$ \end{itemize}

Answer 1

The Correct Option is D

Step 1: Understand Crystal Systems and Their Parameters

Crystal systems are classified based on the lengths of the unit cell edges \((a, b, c)\) and the angles between them \((\alpha, \beta, \gamma)\). There are seven basic crystal systems, each with unique axial lengths and interfacial angles.

Step 2: List the Given Cell Dimensions and Angles

• Edge lengths: \(a = 0.387 \, \text{nm}, \quad b = 0.387 \, \text{nm}, \quad c = 0.504 \, \text{nm}\)
• Interfacial angles: \(\alpha = 90^\circ, \quad \beta = 90^\circ, \quad \gamma = 120^\circ\)

Step 3: Compare Given Parameters with Characteristics of Different Crystal Systems

• Cubic: \(a = b = c\), \(\alpha = \beta = \gamma = 90^\circ\)
  Does not match — here, \(a = b \ne c\), and \(\gamma = 120^\circ\).
• Rhombohedral (Trigonal): \(a = b = c\), \(\alpha = \beta = \gamma \ne 90^\circ\)
  Does not match — here, edge lengths are unequal.
• Orthorhombic: \(a \ne b \ne c\), \(\alpha = \beta = \gamma = 90^\circ\)
  Does not match — angles do not all equal 90° and \(a = b\).
• Hexagonal: \(a = b \ne c\), \(\alpha = \beta = 90^\circ, \gamma = 120^\circ\)
  This matches perfectly with the given unit cell dimensions and angles.

Step 4: Conclude the Crystal System

The given parameters clearly correspond to the hexagonal crystal system.

Step 5: Analyze Options

• Option (1): Cubic — Incorrect
• Option (2): Rhombohedral — Incorrect
• Option (3): Orthorhombic — Incorrect
• Option (4): Hexagonal — Correct

✅ Final Answer: Hexagonal