Question

Imagine the primitive cubic lattice. Now imagine pushing on top of it, straight down. Next, stretch another face by pulling it to the right. All angles remain \(90^{\circ} .\) What kind of primitive lattice have you made?

Short answer

After applying the transformations of pushing from the top and stretching a face horizontally to the initial cubic lattice, we obtain a primitive lattice with three different side lengths (\(a\), \(b\), and \(c\)) and all angles between them equal to \(90^{\circ}\). This results in an orthorhombic lattice.

Step-by-step solution

Identify the initial lattice type

Initially, we have a primitive cubic lattice. Its side lengths are equal (let's denote them as \(a\)), and the angles between these sides are all \(90^{\circ}\).

Transformation 1: Pushing from the top

We push the lattice from the top, straight down. This causes a compression of the lattice along the vertical axis, resulting in a shorter side length along this axis. Let's assume that the new length of this side is \(b\). Since we pushed straight down without any angular deviation, the angles between sides remain \(90^{\circ}\).

Transformation 2: Stretching a face to the right

Next, we stretch one of the faces horizontally by pulling it to the right. This transforms one of the remaining side lengths of the lattice. Let's assume that this new side length is \(c\). As mentioned, all angles between sides still remain \(90^{\circ}\).

Determine the transformed lattice type

Now, we have a primitive lattice with three different side lengths (\(a\), \(b\), and \(c\)) and all the angles between them equal to \(90^{\circ}\). This defines an orthorhombic lattice. Therefore, after applying the two transformations, the resulting primitive lattice is an orthorhombic lattice.

Key concepts

Orthorhombic Lattice

An orthorhombic lattice is a type of crystal structure where the unit cell is defined by three axes of different lengths, yet all angles between these axes measure exactly 90°. This forms a distinct rectangular prism shape that is essential for numerous crystallography studies. The orthorhombic structure can be visualized as a stretched and compressed version of a cube.

In simple terms, imagine a shoe box; its length, width, and height are different, but all angles at the corners remain right angles.

• Three different side lengths: The axes lengths are typically labelled as a, b, and c, where each can have a unique measure.
• Right angles: Each internal angle formed where two axes meet is an unbending right angle.
• Types of orthorhombic cells: Within orthorhombic crystals, cells can be primitive, base-centered, body-centered, or face-centered, depending on how atoms are arranged within the cell.

The orthorhombic lattice reflects not just a shape, but a host of properties governing how materials behave. Such structures are foundational in understanding more complex lattice forms.

Primitive Cubic Lattice

A primitive cubic lattice is the simplest of the crystal lattice types that outlines a repeating pattern in three dimensions by aligning its corners with atom positions. This basic unit cell is characterized by having all sides of equal length and meeting at 90° angles.

Imagine a stack of identical boxes piled up symmetrically, where every corner of each box has an identical atom. It's a bit like stacking dice, each with their corners touching.

• Equal side lengths: All sides are given the same measurement, often represented by 'a'.
• Right angles: The corners form perfect cubes as all angles remain fixed at 90°.
• Single lattice point per cell: As a primitive lattice, it has atoms only at the vertices of the cube.

The simplicity of the primitive cubic lattice makes it a fundamental concept. It serves as a stepping stone to understanding more diverse and intricate lattice structures. However, due to limited packing efficiency, it is not as common in nature compared to other lattice types.

Lattice Transformations

Lattice transformations involve changing the dimensions or angles of a crystal lattice while maintaining its periodic nature. They are often used to explore new material properties and understand structural behaviors.

For instance, starting with a primitive cubic lattice, simple transformations can lead to the formation of vastly different lattice structures like the orthorhombic.

• Compression: Applying pressure along a particular axis can shorten specific lattice dimensions. However, the angles, as in this exercise, remain unchanged at 90°.
• Stretching: Pulling a lattice face along a horizontal plane results in longer side dimensions. This does not affect right angles, preserving the lattice's structural integrity.
• Preservation of angles: Despite changes in side lengths, maintaining the perfect 90° angles is crucial in these transformations.

Lattice transformations are a fundamental part of material science, providing insights into how materials can be manipulated to achieve desired physical properties. This understanding is vital for technological applications ranging from semiconductors to synthetic crystals.