Explanation:

Packing fraction:
The packing fraction is a measure of how efficiently particles are packed in a lattice. It is defined as the ratio of the volume occupied by the particles to the total volume of the unit cell.

FCC lattice:
In a face-centered cubic (FCC) lattice, there are particles at the corners and at the center of each face of the unit cell. The coordination number (number of nearest neighbors) in FCC is 12. The packing fraction for FCC is given by:

Packing fraction (FCC) = (number of particles * volume of particle) / (volume of unit cell)

BCC lattice:
In a body-centered cubic (BCC) lattice, there is a particle at each corner of the unit cell and one particle at the center of the unit cell. The coordination number in BCC is 8. The packing fraction for BCC is given by:

Packing fraction (BCC) = (number of particles * volume of particle) / (volume of unit cell)

Simple cubic lattice:
In a simple cubic lattice, there is a particle at each corner of the unit cell. The coordination number in simple cubic is 6. The packing fraction for simple cubic is given by:

Packing fraction (simple cubic) = (number of particles * volume of particle) / (volume of unit cell)

Diamond lattice:
In a diamond lattice, there are particles at the corners and at the center of each face of the unit cell, similar to FCC lattice. However, the particles in diamond lattice are not in contact with each other along the body diagonal of the unit cell. The coordination number in diamond lattice is 4. The packing fraction for diamond lattice is given by:

Packing fraction (diamond) = (number of particles * volume of particle) / (volume of unit cell)

Comparison:
To determine which lattice has the least packing fraction, we compare the packing fractions of each lattice.

FCC: The FCC lattice has a higher coordination number (12) and the particles are in close contact with each other. Therefore, it has a higher packing fraction compared to the other lattices.

BCC: The BCC lattice has a lower coordination number (8) compared to FCC, so it is expected to have a lower packing fraction.

Simple cubic: The simple cubic lattice has the lowest coordination number (6) among the given lattices. Therefore, it is expected to have the lowest packing fraction.

Diamond: The diamond lattice has a coordination number of 4, which is lower than that of FCC and BCC. Therefore, it is expected to have a lower packing fraction compared to the other lattices.

Conclusion:
From the comparison, it can be concluded that the diamond lattice has the least packing fraction among the given lattices (FCC, BCC, simple cubic, diamond).