Question

In which of the following crystal systems the endcentred unit cell exists? (a) monoclinic (b) triclinic (c) cubic (d) all of these

Short answer

The end-centred unit cell exists in the monoclinic crystal system (option a).

Step-by-step solution

Understanding Unit Cells

A unit cell is the smallest repeating unit in the crystal lattice that shows the entire structure's symmetry. In crystallography, there are seven crystal systems, each with unique attributes.

Types of Unit Cells

There are different types of unit cells based on the lattice points present: (1) Primitive (P) - lattice points at corners; (2) Body-Centred (I) - additional point at the center; (3) Face-Centred (F) - points at faces; (4) Base-Centred (C) or End-Centred - additional point on two opposite faces, which is the focus here.

Identify Crystal Systems with End-Centred Cells

Not all crystal systems can support all types of unit cells. End-centred unit cells typically exist in the orthorhombic, monoclinic, and sometimes tetragonal systems, but not in cubic or triclinic systems.

Check Against Given Options

Review each option based on end-centred unit cell potential: (a) Monoclinic - can have end-centred (C-type) cells. (b) Triclinic - does not support end-centred unit cells due to lack of symmetry. (c) Cubic - cannot have end-centred cells because all its sides are equal, and faces are symmetrical. (d) The "all of these" option is incorrect as not all listed can have end-centred cells.

Key concepts

Crystal Systems

Crystals, the beautiful solid substances we often see adorning various items, are more than just aesthetics. They are a perfect arrangement of atoms and ions, organized into specific _crystal systems_. Understanding these crystal systems is essential in crystallography, the study of crystals and their structure. There are seven recognized crystal systems, each defined by unique characteristics of symmetry and the lengths and angles of their axes. These include:

• Cubic - where all axes are equal in length and intersect at right angles.
• Tetragonal - similar to cubic but one axis is longer than the other two.
• Orthorhombic - has three unequal axes, all intersecting at right angles.
• Monoclinic - has axes of unequal length, with two axes at right angles and the third at an oblique angle.
• Triclinic - all axes are of different lengths and none are perpendicular.
• Hexagonal - comprises four axes, three of which are of equal length at 120-degree angles, with the fourth perpendicular.
• Trigonal - similar to hexagonal, but with a threefold rotational symmetry.

Each of these systems can further be broken down into specific unit cell types, which are the building blocks for understanding crystal structures.

Unit Cells

A unit cell is like a tiny three-dimensional blueprint that reveals how atoms pack together in a crystal. Think of it as the smallest portion of a crystal lattice that, when repeated, forms the whole crystal. Every crystal system is constructed from these unit cells. There are three basic types of unit cells:

• Primitive (P) - where lattice points are only at the corners of the cell.
• Body-Centred (I) - featuring a lattice point at the center of the cell, in addition to those at the corners.
• Face-Centred (F) - having additional lattice points at the center of each face of the cell.

Some variations include _Base-Centred_ or _End-Centred (C)_, where additional lattice points exist on two opposite faces. Each type provides a different arrangement and symmetry, playing a crucial role in determining the crystal's properties.

End-Centred Unit Cells

End-centred or base-centred unit cells are special configurations where there is a lattice point on two opposite faces of the cell. They offer a unique symmetry that is only possible in some crystal systems. Typically, _end-centred unit cells_ can exist in:

• Orthorhombic System - due to its three unequal axes and ability to bear this type of symmetry.
• Monoclinic System - with axes at oblique angles, which can accommodate end-centred cells.
• The _tetragonal system_ may occasionally have end-centred cells, though this is less common.

However, the cubic and triclinic systems do not support end-centred cells. Cubic is too symmetric with equal axes, and triclinic lacks the necessary symmetry altogether. Understanding where end-centred cells exist helps scientists and researchers tailor materials for specific applications based on their structural characteristics.