Describe the difference between a face-centered cubic structure (FCC) and a body-centered cubic structure \((\mathrm{BCC})\)

Face-Centered Cubic (FCC) and Body-Centered Cubic (BCC) are two types of cubic crystal structures, which means they have equal sides with angles of 90 degrees between them on their unit cells. The core difference between FCC and BCC lies in how their atoms are arranged within the unit cell.

In the \(FCC\) structure, each corner of the unit cell is occupied by an atom, and also an additional atom is located at the center of each face of the unit cell, hence the name face-centered cubic. In the FCC structure, there are a total of 14 atoms in the unit cell, with eight corner atoms contributing 1/8 each, and six face-centered atoms contributing 1/2 each, which sums up to 4 atoms in the unit cell. In contrast, in the \(BCC\) structure, there are atoms at each corner of the unit cell and an additional atom located at the center of the unit cell, justifying the name body-centered cubic. In the BCC structure, there are a total of 9 atoms in the unit cell, with eight-corner atoms contributing 1/8 each and one body-centered atom contributing 1, which sums up to 2 atoms in the unit cell.

The coordination number represents the number of nearest neighboring atoms that an atom is directly bonded with or is in contact with. In the \(FCC\) structure, the coordination number is 12, which means that each atom is in direct contact with 12 other atoms. The 12 surrounding atoms consist of 6 atoms in the same layer, 3 atoms in the layer above, and 3 atoms in the layer below. In the \(BCC\) structure, the coordination number is 8, which means that each atom is in direct contact with 8 other atoms. The 8 surrounding atoms form a cube around the central atom, with atoms located at each corner of the cube.

Packing efficiency refers to the fraction of the total volume of the unit cell occupied by the atoms. In a \(FCC\) lattice, the packing efficiency is about \(74\% \), which means that \(74\%\) of the total volume is filled with atoms, leaving \(26\%\) empty space. This results in a higher atomic packing factor (APF) compared to the BCC lattice. The FCC lattice is more tightly packed, leading to less empty space between the atoms. In a \(BCC\) lattice, the packing efficiency is approximately \(68\% \), which means that \(68\%\) of the total volume is filled with atoms, leaving \(32\%\) empty space. The FCC structure is more efficient in terms of atomic packing compared to BCC, which leads to a lower density of materials with a BCC structure as compared to FCC. In summary, the main differences between face-centered cubic (FCC) and body-centered cubic (BCC) structures are their lattice geometry, atomic positions, coordination numbers, and packing efficiency. FCC structures have higher packing efficiency and tighter atomic arrangements than BCC structures, resulting in higher density materials when in FCC structures compared to BCC structures.