Question

Imagine the primitive cubic lattice. Now imagine grabbing the top of it and stretching it straight up. All angles remain \(90^{\circ}\). What kind of primitive lattice have you made?

Short answer

We have created an orthorhombic primitive lattice (oP), as only the length of the z-axis is altered, while the angles between x, y, and z axes remain \(90^{\circ}\).

Step-by-step solution

Recall primitive lattice types and their properties

Simple cubic lattice (cP) has lattice points at all corners of the unit cell. The angles between the lattice vectors are \(90^{\circ}\). Body-centered cubic lattice (cI) has lattice points at all corners and one at the center of the unit cell. The angles between the lattice vectors are \(90^{\circ}\). Face-centered cubic lattice (cF) has lattice points at all corners and on the faces of the unit cell. The angles between the lattice vectors are \(90^{\circ}\). In addition to cubic lattices, there are three more types of primitive lattices: orthorhombic, tetragonal, and rhombohedral. For the orthorhombic lattice (oP), the angles between the lattice vectors are \(90^{\circ}\) and all sides have different lengths. Tetragonal lattice (tP) has one angle between lattice vectors equal to \(90^{\circ}\), and the other two angles are not equal to \(90^{\circ}\); all sides have different lengths. Rhombohedral lattice (rP) has all angles equal, but not equal to \(90^{\circ}\), and all sides have the same length.

Identify the resulting lattice after stretching

As we stretch the top of the primitive cubic lattice (cP) straight up, we are only changing the length of the z-axis, while keeping the x and y axes unchanged. Also, we are not altering any angles between the axes, so all angles remain at \(90^{\circ}\). As a result, we obtain a new lattice that has one side longer or shorter than the others, while maintaining all angles between the axes equal to \(90^{\circ}\). Considering the properties of different types of lattices, we find that the resulting lattice corresponds to an orthorhombic primitive lattice (oP). This lattice has different lengths for all sides, and all angles between the axes remain equal to \(90^{\circ}\). The final answer is: we have created an orthorhombic primitive lattice (oP).

Key concepts

primitive lattice

A primitive lattice is a fundamental concept in crystallography. It refers to a lattice that has only one lattice point per unit cell. Imagine it as a basic framework that forms the backbone of crystal structures. The lattice points are geometric points that define the repetitive pattern of a crystal's atomic or molecular arrangement. In a primitive lattice, each point represents an identical environment in the crystal.

Characteristics of primitive lattices include:

• Only one lattice point at each corner of the unit cell.
• The arrangement is repeated throughout the entire crystal structure.
• All primitive lattices have the smallest number of lattice points.

Primitive lattices serve as a foundational structure for understanding more complex lattice types. They are essential because all crystalline materials with lattices can be understood by considering the primitive variations.

orthorhombic lattice

The orthorhombic lattice is one among the seven crystal systems and is characterized by having lattice vectors of different lengths, but importantly, all the angles between these vectors are 90 degrees. This configuration can be visualized as a stretched version of a cubic lattice, where three axes have different lengths but remain perpendicular to each other.
Here are some key features:

• The unit cell shapes like a rectangular prism or stretched box.
• All axes intersect at right angles (90 degrees).
• A physical example could be the structures of crystals like sulfur and olivine.

By imagining a simple cubic structure and altering one or more of its side lengths while preserving perpendicularity, you create an orthorhombic lattice. It can be visualized as grabbing the top of a cubic lattice and stretching it upwards or sideways while keeping the angles unchanged, leading to a structure with unique symmetry and crystalline properties.

cubic lattice

Cubic lattices are perhaps the simplest and most symmetrical of all the lattice types. They serve as excellent starting points for those learning about crystal structures. A cubic lattice is where all edges are of equal length, and all angles between the edges are 90 degrees.
There are three main types of cubic lattices:

• Simple cubic (SC): Lattice points are only at the corners of the cube. It's the simplest form and less common in natural materials.
• Body-centered cubic (BCC): In addition to the corner points, there's an additional lattice point at the center of the cube. This structure is seen in metals like iron at certain temperatures.
• Face-centered cubic (FCC): Lattice points exist at each corner and the centers of each face of the cube. This dense arrangement is typical in metals like aluminum and copper.

Cubic lattices are crucial for understanding more complex crystal structures. They exhibit a high degree of symmetry due to equal edge lengths and angles, making them a foundational study for material properties.