Problem 33 - Chemistry in 30 days

Question

For a unit cell, it is given that \(a=10.8 \mathrm{~A}, b=9.46 \mathrm{~A}, c=5.3 \mathrm{~A}\) and also \(\alpha=41^{\circ}, \beta=82.9^{\circ}\), \(y=93.1^{\circ}\). What type of crystal system it belongs to? (a) Monoclinic (b) Triclinio (c) Hexagonal (d) Tetragonal

Step-by-Step Solution

Answer

The crystal system is Triclinic.

Step 1: Identify Given Parameters

The unit cell dimensions are provided as: \(a = 10.8 \, \text{Å}\), \(b = 9.46 \, \text{Å}\), \(c = 5.3 \, \text{Å}\). The angles are given as \(\alpha = 41^{\circ}, \beta = 82.9^{\circ}, \gamma = 93.1^{\circ}\). We must identify the crystal system these parameters correspond to.

Step 2: Review Crystal System Characteristics

Different crystal systems have specific constraints on side lengths and angles. The question involves determining which of the given crystal systems (Monoclinic, Triclinic, Hexagonal, Tetragonal) fits these constraints.

Step 3: Check Against Triclinic System

A triclinic system has no constraints on angles or side equality: \(a eq b eq c\) and \(\alpha eq \beta eq \gamma eq 90^{\circ}\). The given parameters fit since \(\alpha = 41^{\circ}, \beta = 82.9^{\circ}, \gamma = 93.1^{\circ}\), all of which are not equal to 90°.

Step 4: Verify and Conclude

The given parameters satisfy the characteristics of the triclinic system, in which there are no 90-degree angles, and all sides are of different lengths.

Key Concepts

Triclinic

The triclinic crystal system is unique in its simplicity and lack of symmetry. It is the most general type of crystal system among the seven systems. What makes it special is that it has no constraints when it comes to the lengths of its sides or the measures of its angles. This means:

• The edges of the unit cell—represented by the variables \(a\), \(b\), and \(c\)—do not need to be equal. They can vary individually.
• The angles \(\alpha\), \(\beta\), and \(\gamma\) between these edges are also unrestricted and do not equal 90°.

These features make the triclinic system the most flexible and less symmetric than other crystal systems like monoclinic or hexagonal. In the exercise, the angles are given as \(41^{\circ}\), \(82.9^{\circ}\), and \(93.1^{\circ}\), none of which are 90°, perfectly matching the triclinic description.

Unit Cell Dimensions

The unit cell is a fundamental concept in understanding crystal structures. It acts as the building block of a crystal. To identify the type of crystal system a unit cell belongs to, we examine its dimensions and angles.

The dimensions of a unit cell are defined by three parameters: \(a\), \(b\), and \(c\). These are the lengths of the edges that form the cell. For instance, in our problem:

• \(a = 10.8 \, \text{Å}\)
• \(b = 9.46 \, \text{Å}\)
• \(c = 5.3 \, \text{Å}\)

Besides the edge lengths, we also need to consider the angles between these edges, represented as \(\alpha\), \(\beta\), and \(\gamma\). These angles provide further insight into the geometry and symmetry of the unit cell.

Crystal System Identification

Identifying a crystal system is like solving a puzzle using the information of unit cell dimensions and angles. Each crystal system—triclinic, monoclinic, hexagonal, etc.—has distinct characteristics. Our task is to match the given cell parameters against these unique traits.

For instance, if all angles are right angles \((90°)\) and the three edge lengths are not equal, one might think of a tetragonal or orthorhombic system. However, when none of the angles are 90° and all edges have different lengths, like in this exercise, it fits the profile of a triclinic crystal system:

• No angle equals 90°: \(\alpha = 41^{\circ}, \beta = 82.9^{\circ}, \gamma = 93.1^{\circ}\)
• The edge lengths \(a\), \(b\), and \(c\) are all different.

Using these characteristics as a guide, the process of identifying the crystal system becomes straightforward and intuitive.