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 performing the given transformations on the primitive cubic lattice, we have created a primitive orthorhombic lattice with lattice parameters \(a \neq b \neq c\) and angles \(\alpha = \beta = \gamma = 90^{\circ}\).

Step-by-step solution

Understand the initial lattice

We are given a primitive cubic lattice, which has a cubic unit cell with lattice points at the vertices. In a cubic lattice, all sides are equal in length, and all angles between the sides are \(90^{\circ}\). So, initially, we have a lattice with \(a = b = c\) and \(\alpha = \beta = \gamma = 90^{\circ}\).

Push down on top of the lattice

The first transformation applies pressure on the top face of the lattice, pushing it down. This will result in a change to the length of one of the sides. We can denote this new length as 'c' since 'c' usually refers to the vertical side in a lattice. Therefore, our lattice parameters change to \(a = b\) (as they remain equal) and \(c \neq a\). The angles remain unchanged at \(90^{\circ}\).

Stretch the lattice by pulling a face to the right

The second transformation pulls the face to the right, resulting in a change in the side length of another axis. We can denote this new length as 'b', and in this case, 'a' remains unchanged as we have not altered it. Now, our lattice parameters are \(a \neq b \neq c\) with \(\alpha = \beta = \gamma = 90^{\circ}\).

Identify the new lattice type

Now that we have transformed the lattice, we have new lattice parameters: \(a \neq b \neq c\), and the angles between the sides remain unchanged at \(90^{\circ}\). This describes a primitive orthorhombic lattice. In the orthorhombic lattice, all angles remain at \(90^{\circ}\), and all sides are unequal. So, after performing the given transformations, we have created a primitive orthorhombic lattice.