he number of atoms per unit cell, coordination number, and atomic packing factor, for a FCC crystal are……………

To find the number of atoms per unit cell, coordination number, and atomic packing factor for a Face-Centered Cubic (FCC) crystal structure, we can analyze each aspect systematically.

1. Number of Atoms per Unit Cell

• In a FCC structure, atoms are located at each corner of the cube and at the center of each face.
• Each corner atom is shared by 8 adjacent unit cells, contributing $\frac{1}{8}$ of an atom to each unit cell.
• Each face-centered atom is shared by 2 unit cells, contributing $\frac{1}{2}$ of an atom to the unit cell.

Calculating the total number of atoms:

• 8 corner atoms contribute: $8 \times \frac{1}{8} = 1$ atom.
• 6 face-centered atoms contribute: $6 \times \frac{1}{2} = 3$ atoms.

Thus, the total number of atoms per unit cell in a FCC structure is: $\text{Total atoms} = 1 + 3 = 4$

2. Coordination Number

• The coordination number refers to the number of nearest neighbors surrounding an atom.
• In the FCC structure, each atom touches 12 other atoms.

Thus, the coordination number for FCC is: $\text{Coordination Number} = 12$

3. Atomic Packing Factor (APF)

• The atomic packing factor is given by the ratio of the volume occupied by atoms in the unit cell to the volume of the unit cell itself.
• The volume of one atom is given by $V_{\text{atom}} = \frac{4}{3} \pi r^3$, where $r$ is the atomic radius.
• In a FCC lattice, the relation between the atomic radius $r$ and the lattice parameter $a$ (edge length) is $a = 2\sqrt{2}r$.

Calculating the volume:

• The volume of the unit cell is $V_{\text{cell}} = a^3 = (2\sqrt{2}r)^3 = 16\sqrt{2}r^3$.
• The volume occupied by the atoms in the unit cell is $4 \times V_{\text{atom}} = 4 \times \frac{4}{3} \pi r^3 = \frac{16}{3} \pi r^3$.

Thus, the atomic packing factor (APF) can be calculated as: $\text{APF} = \frac{\text{Volume of atoms in unit cell}}{\text{Volume of unit cell}} = \frac{\frac{16}{3} \pi r^3}{16\sqrt{2}r^3} = \frac{\pi}{3\sqrt{2}} \approx 0.7405$

Final Answer

1. Number of atoms per unit cell: 4
2. Coordination number: 12
3. Atomic Packing Factor (APF): Approximately 0.7405