Problem 99 - Chemistry: The Central Science

Question

A face-centered tetragonal lattice is not one of the 14 threedimensional lattices. Show that a face-centered tetragonal unit cell can be redefined as a body-centered tetragonal lattice with a smaller unit cell.

Step-by-Step Solution

Answer

A face-centered tetragonal lattice can be redefined as a body-centered tetragonal lattice with a smaller unit cell by adding a lattice point at the center of the unit cell and removing the face-centered lattice points. The new lattice vectors a' and b' are then defined along the diagonal of the faces of the original unit cell, resulting in a smaller unit cell volume. Specifically, \(V_{new} = \frac{a^2c}{2}\), which is less than the original volume, \(V_{original} = a^2c\).

Step 1: Understand the structure of face-centered tetragonal lattice

A face-centered tetragonal lattice has lattice points at the corners and at the center of each face of the unit cell. This lattice can be characterized by three lattice vectors a, b, and c, with a = b ≠ c. The structure can be visualized as follows: - There are 8 lattice points at the corners of the unit cell - There are 6 lattice points at the center of each face

Step 2: Redefining a face-centered tetragonal lattice as a body-centered tetragonal lattice

To redefine the face-centered tetragonal lattice as a body-centered tetragonal lattice with a smaller unit cell, let's consider the following steps: 1. Add new lattice points at the center of the unit cell and remove the face-centered lattice points. This will produce a body-centered lattice structure. 2. Adjust the lattice vectors to create the new tetragonal unit cell, as follows: - The new lattice vector a' runs from the original corner lattice point to the previous face-centered lattice point along the x-axis. - The new lattice vector b' runs from the original corner lattice point to the previous face-centered lattice point along the y-axis. - The new lattice vector c' is the same as the original lattice vector c since the height remains unchanged.

Step 3: Demonstrating that the redefined unit cell is smaller

Since the redefined unit cell has its lattice vectors running along the diagonal of the faces of the original unit cell, the length of the new lattice vectors a' and b' is less than the lengths of the original lattice vectors a and b. Therefore, the volume of the redefined unit cell will be smaller than the original face-centered tetragonal unit cell. Mathematically, The length of a' (and b') can be calculated as follows: \(a' = \sqrt{(\frac{a}{2})^2 + (\frac{b}{2})^2}\) Since a=b, we can rewrite the equation as: \(a' = \frac{a}{\sqrt{2}}\) The volume of the redefined tetragonal unit cell can be calculated using the new lattice vectors: \(V_{new} = a'b'c' = \frac{a^2c}{2}\) Comparing this to the volume of the original face-centered tetragonal unit cell, \(V_{original} = abc = a^2c\) Clearly, \(V_{new} < V_{original}\), as \(\frac{a^2c}{2} < a^2c\) Thus, we have successfully shown that a face-centered tetragonal unit cell can be redefined as a body-centered tetragonal lattice with a smaller unit cell.

Key Concepts

Face-Centered Tetragonal Lattice

A face-centered tetragonal lattice features atoms located at each of the eight corners of a cell and also at the center of each face of the cell, giving it a distinct symmetry and shape.

Consider a room where in each corner sits an atom, and precisely in the center of each wall, there's another atom. Now imagine this setup being repeated in a pattern extending across three dimensions. This configuration is significant in solid-state physics and materials science because it affects the physical properties of the material such as density, melting point, and electrical conductivity.

Body-Centered Tetragonal Lattice

In contrast, the body-centered tetragonal lattice is slightly simpler: you still have the eight corner atoms, but now, there's just a lone atom smack in the middle of the cell. No wall-center atoms in this case. It's like the core or heart of the room is occupied while the walls are left bare compared to the face-centered version.

This difference in atom location leads to a distinct crystal geometry and can result in different physical properties for the material. Adapting one structure to the other, as in our exercise, involves a creative rethinking of how atoms are arranged in space.

Unit Cell Redefinition

Now, if you're keen on transforming the spacious face-centered unit into its cozier body-centered counterpart, you're essentially playing a game of atom hide-and-seek. You're hiding the face-centered atoms and seeking a new center for the body atoms. The trick here is to spot that by positioning a new atom in the heart of the cell and ignoring those at the halfway point of the walls, you can draw new borders around a smaller, yet still tetragonal cell.

This redefinition retains the crucial aspects of lattice symmetry, but on a smaller scale, which would imply a change in the material's properties. Yes, it's like you've downsized a mansion (face-centered) into a cottage (body-centered) without changing the architectural style (tetragonal).

Lattice Vectors

Understanding Lattice Vectors

Picture the lattice vectors as the invisible lines that define the edges of your cell – or the arms and legs of the atoms' dance space. They outline the length, width, and height of the cell. In our case, during the transformation from face-centered to body-centered, we kind of folded these vectors inwards.

These vectors are not just for looks though, they dictate how the entire crystal is built up from the unit cells, like bricks in a wall. By adjusting these vectors, materials scientists can conceive new crystal structures, which might seem like an easy task on paper, but in reality, it's nature that usually calls the shots on how these vectors align in most materials.