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

After stretching the primitive cubic lattice vertically while maintaining all angles at \(90^{\circ}\), we form a Primitive Orthorhombic lattice. It has two edges of equal length 'a' (width and depth), and a vertical edge of different length 'c', with angles remaining unchanged at \(90^{\circ}\).

Step-by-step solution

Understand the initial unit cell

Initially, we have a primitive cubic lattice. This means that the unit cell is a cube with all angles equal to \(90^{\circ}\) and all edges have the same length.

Stretching the lattice vertically

As per the exercise, we stretch the top of the lattice along the vertical axis. All angles remain the same, so they continue to be at \(90^{\circ}\). However, stretching changes the edge length along this axis. Let's assume that the initial edge length was 'a' and after stretching the vertical edge length is now 'c'.

Identify the new lattice

Now we have a unit cell with two edges (width and depth) having the same length 'a' and the vertical edge (height) with a different length 'c'. The angles remain unchanged at \(90^{\circ}\). Thus, we have a lattice system with right angles and different edge lengths along different axes. This system is known as an Orthorhombic lattice system.

Identifying the type of orthorhombic lattice

As we started with a primitive cubic lattice and only stretched the vertical edge, there is only one lattice point per unit cell. Therefore, the new lattice we have formed is a Primitive Orthorhombic lattice. So, after stretching the primitive cubic lattice, a Primitive Orthorhombic lattice is formed.