Question

Of the seven three-dimensional primitive lattices, (a) which one has a unit cell where the \(a\) and \(b\) lattice vectors form a rectangular base, while the \(c\) lattice vector is perpendicular to the other two and of a different length? (b) What is the lattice if the \(a\) and \(b\) lattice vectors form a square base and the \(c\) lattice vector is perpendicular to the other two and of a different length?

Short answer

(a) The lattice with a rectangular base formed by the \(a\) and \(b\) lattice vectors and a perpendicular \(c\) lattice vector of different length is a Simple Orthorhombic lattice. (b) The lattice with a square base formed by the \(a\) and \(b\) lattice vectors and a perpendicular \(c\) lattice vector of different length is a Simple Tetragonal lattice.

Step-by-step solution

Primitive Lattice Types

There are seven three-dimensional primitive lattice types: 1. Simple Cubic (SC) 2. Body-Centered Cubic (BCC) 3. Face-Centered Cubic (FCC) 4. Simple Tetragonal (ST) 5. Body-Centered Tetragonal (BCT) 6. Simple Orthorhombic (SO) 7. Body-Centered Orthorhombic (BCO)

Problem (a) Solution

We are asked to find the lattice where \(a\) and \(b\) lattice vectors form a rectangular base, while the \(c\) lattice vector is perpendicular to the other two and of a different length. This description is characteristic of a simple orthorhombic lattice. With this lattice, the sides are perpendicular to each other, but \(a\), \(b\), and \(c\) have different lengths. Answer for Problem (a): Simple Orthorhombic lattice

Problem (b) Solution

Now, we are asked to find the lattice in which the \(a\) and \(b\) lattice vectors form a square base, and the \(c\) lattice vector is perpendicular to the other two and of a different length. This description is characteristic of a simple tetragonal lattice. In this lattice, the \(a\) and \(b\) lattice vectors have the same length, and they're perpendicular to the \(c\) lattice vector, which has a different length. Answer for Problem (b): Simple Tetragonal lattice

Key concepts

Primitive Lattices

In crystallography, a primitive lattice refers to the simplest repeating pattern of points in space that form a crystal structure. Primitive lattices are crucial because they represent the building blocks of more complex structures. Each point in a lattice is described by three lattice vectors, usually denoted as \(a\), \(b\), and \(c\), which define the periodicity and orientation of the lattice.
A primitive lattice has points solely at the corners of the unit cell. In three dimensions, there are seven crystal systems, each with its own set of primitive lattices, such as simple cubic and simple orthorhombic. Understanding primitive lattices helps in deciphering the geometric arrangement of atoms within a crystal, which is vital for predicting the material's properties.
- **Simple Cubic Lattice**: characterized by equal lattice parameters and angles.- **Simple Orthorhombic Lattice**: features three unequal axes that are all orthogonal.- **Simple Tetragonal Lattice**: has two axes of equal length and one of a different length, all meeting at right angles.

Tetragonal Lattice

A tetragonal lattice is one of the crystal systems where the lattice points are arranged in a three-dimensional structure that appears like an elongated cube. They are defined by their rectangular base, formed by equally spaced lattice vectors \(a\) and \(b\). The third lattice vector, \(c\), is perpendicular to the base and has a different length.
Tetragonal lattices are known for their simple symmetry and occur naturally in various minerals and synthetic materials. The defining feature of tetragonal lattices is the square base, where \(a = b eq c\). This creates a unit cell that is unique due to the different length along one axis, allowing materials with tetragonal lattices to have different physical properties compared to cubic structures.
- **Applications**: Widely present in ceramics and electronic materials, which often display anisotropic properties (varying with direction) due to the tetragonal arrangement.

Orthorhombic Lattice

The orthorhombic lattice is a type of primitive lattice characterized by having three unequal axes, all of which intersect at right angles. This means the \(a\), \(b\), and \(c\) lattice vectors all differ in length, compared to tetragonal or cubic lattices where some axes might be equal.
Orthorhombic lattices are less symmetrical than cubic or tetragonal lattices, providing a different set of attributes and physical properties. They can align differently in space to form extensive networks of atoms, affecting a material's crystallinity and functionality, particularly influencing mechanical and thermal properties.
- **Structural properties**: This diversity in axis lengths leads to unique mechanical behaviors in materials containing orthorhombic lattices. Examples can be found in various crystals and metals where stress distribution and thermal expansion vary along different axes.