Show that the reciprocal lattice for a body centred cubic is a face centred cubic.

A body-centred cubic (BCC) lattice is a common crystal structure. Imagine a cube. In this cube, there's a lattice point (or atom) at every corner of the cube. Besides these corner points, there's also an atom right in the center of the cube.

BCC can be visualized as:

• Eight atoms at the corners of the cube
• One atom at the center of the cube

The structure is important in materials science and physics because it helps understand certain properties of materials, like metals that often crystallize in the BCC structure. Some metals with BCC structures are iron at room temperature and vanadium. In crystallography, the concept of BCC helps explain how atoms arrange themselves in a solid to minimize energy and maximize efficiency.

A face-centred cubic (FCC) lattice is another critical arrangement in crystallography. In this lattice, there are atoms at each corner of the cube and in the center of each face of the cube.

You can think of an FCC lattice as having:

• Eight atoms at the corners of the cube
• Six atoms at the center of each face of the cube

This arrangement is more densely packed than the BCC lattice. Metals like aluminum, copper, and gold crystallize in the FCC structure. Knowing about FCC structures helps scientists understand the behavior of these metals, like how they deform and their mechanical properties.

Reciprocal lattices are essential for understanding phenomena like X-ray diffraction. The reciprocal lattice of a crystal lattice simplifies understanding of wave propagation in the lattice. Reciprocal lattice vectors are calculated using the primitive vectors of the direct lattice.

• For a given lattice, we can find primitive vectors (a1, a2, a3).
• The reciprocal lattice vectors (b1, b2, b3) are calculated using these primitive vectors.

For example, for a BCC lattice, the primitive vectors a1, a2, and a3 can be used to compute reciprocal vectors using the formula:

b1 = 2π (a2 × a3) / (a1 · (a2 × a3))
b2 = 2π (a3 × a1) / (a1 · (a2 × a3))
b3 = 2π (a1 × a2) / (a1 · (a2 × a3))

When the calculations are done, you find that the reciprocal lattice of a BCC is an FCC. This shows a direct relationship between the two types of lattices and helps us understand crystal structures much better.